Convective Heat Transfer Solved Problems
Michel FavreMarinet Sedat Tardu
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Convective Heat Transfer
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Convective Heat Transfer Solved Problems
Michel FavreMarinet Sedat Tardu
First published in France in 2008 by Hermes Science/Lavoisier entitled: Écoulements avec échanges de chaleur volumes 1 et 2 © LAVOISIER, 2008 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 2737 St George’s Road London SW19 4EU UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2009 The rights of Michel FavreMarinet and Sedat Tardu to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress CataloginginPublication Data FavreMarinet, Michel, 1947[Ecoulements avec échanges de chaleur. English] Convective heat transfer : solved problems / Michel FavreMarinet, Sedat Tardu. p. cm. Includes bibliographical references and index. ISBN 9781848211193 1. HeatConvection. 2. HeatTransmission. I. Tardu, Sedat, 1959 II. Title. TJ260.F3413 2009 621.402'25dc22 2009016463 British Library CataloguinginPublication Data A CIP record for this book is available from the British Library ISBN: 9781848211193 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.
Table of Contents
Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Chapter 1. Fundamental Equations, Dimensionless Numbers . . . . . . . .
1
1.1. Fundamental equations . . . . . . . . . . . . . . . . . . . 1.1.1. Local equations . . . . . . . . . . . . . . . . . . . . . 1.1.2. Integral conservation equations. . . . . . . . . . . . 1.1.3. Boundary conditions . . . . . . . . . . . . . . . . . . 1.1.4. Heattransfer coefficient . . . . . . . . . . . . . . . . 1.2. Dimensionless numbers . . . . . . . . . . . . . . . . . . 1.3. Flows with variable physical properties: heat transfer in a laminar Couette flow . . . . . . . . . . . . . . . . . . . . 1.3.1. Description of the problem . . . . . . . . . . . . . . 1.3.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Flows with dissipation . . . . . . . . . . . . . . . . . . . 1.4.1. Description of the problem . . . . . . . . . . . . . . 1.4.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Cooling of a sphere by a gas flow. . . . . . . . . . . . . 1.5.1. Description of the problem . . . . . . . . . . . . . . 1.5.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . .
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1 1 4 7 7 8
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9 9 10 10 14 14 14 15 20 20 21 21
vi
Convective heat Transfer
Chapter 2. Laminar Fully Developed Forced Convection in Ducts . . . . . 2.1. Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Characteristic parameters . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Flow regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Thermal boundary conditions. . . . . . . . . . . . . . . . . . . . . . 2.2.2. Bulk temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Heattransfer coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Fully developed thermal region . . . . . . . . . . . . . . . . . . . . 2.3. Heat transfer in a parallelplate channel with uniform wall heat flux . 2.3.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Flow in a plane channel insulated on one side and heated at uniform temperature on the opposite side . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
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31 31 32 33 33 34 34 34 35 35 36 37
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46 46 47 47
Chapter 3. Forced Convection in Boundary Layer Flows . . . . . . . . . . .
53
3.1. Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Prandtl equations . . . . . . . . . . . . . . . . . . . 3.1.2. Classic results . . . . . . . . . . . . . . . . . . . . . 3.2. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Equations of the thermal boundary layer . . . . . 3.2.2. Scale analysis . . . . . . . . . . . . . . . . . . . . . 3.2.3. Similarity temperature profiles . . . . . . . . . . . 3.3. Integral method . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Integral equations . . . . . . . . . . . . . . . . . . . 3.3.2. Principle of resolution using the integral method 3.4. Heated jet nozzle. . . . . . . . . . . . . . . . . . . . . . 3.4.1. Description of the problem . . . . . . . . . . . . . 3.4.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Asymptotic behavior of thermal boundary layers . . 3.5.1. Description of the problem . . . . . . . . . . . . . 3.5.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Protection of a wall by a film of insulating material . 3.6.1. Description of the problem . . . . . . . . . . . . . 3.6.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . 3.6.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . .
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53 53 55 58 58 58 59 62 62 64 65 65 67 68 68 69 69 74 74 76 77
Table of Contents
3.7. Cooling of a moving sheet . . . . . . . . . . . . . . . . 3.7.1. Description of the problem . . . . . . . . . . . . . 3.7.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . 3.7.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Heat transfer near a rotating disk . . . . . . . . . . . . 3.8.1. Description of the problem . . . . . . . . . . . . . 3.8.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . 3.8.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Thermal loss in a duct. . . . . . . . . . . . . . . . . . . 3.9.1. Description of the problem . . . . . . . . . . . . . 3.9.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . 3.9.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . 3.10. Temperature profile for heat transfer with blowing 3.10.1. Description of the problem . . . . . . . . . . . . 3.10.2. Solution . . . . . . . . . . . . . . . . . . . . . . . .
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83 83 84 86 93 93 94 96 106 106 107 108 117 117 118
Chapter 4. Forced Convection Around Obstacles . . . . . . . . . . . . . . . .
119
4.1. Description of the flow . . . . . . . . . . . . . . . . . . . . 4.2. Local heattransfer coefficient for a circular cylinder . . 4.3. Average heattransfer coefficient for a circular cylinder 4.4. Other obstacles . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Heat transfer for a rectangular plate in crossflow . . . . 4.5.1. Description of the problem . . . . . . . . . . . . . . . 4.5.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Heat transfer in a stagnation plane flow. Uniform temperature heating . . . . . . . . . . . . . . . . . . . 4.6.1. Description of the problem . . . . . . . . . . . . . . . 4.6.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Heat transfer in a stagnation plane flow. Stepwise heating at uniform flux . . . . . . . . . . . . . . . . 4.7.1. Description of the problem . . . . . . . . . . . . . . . 4.7.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Temperature measurements by coldwire . . . . . . . . . 4.8.1. Description of the problem . . . . . . . . . . . . . . . 4.8.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . .
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vii
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119 121 123 125 126 126 126
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128 128 129 130
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131 131 132 133 135 135 136 137
viii
Convective heat Transfer
Chapter 5. External Natural Convection . . . . . . . . . . . . . . . . . . . . . . 5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Boussinesq model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Dimensionless numbers. Scale analysis . . . . . . . . . . . . . . 5.4. Natural convection near a vertical wall . . . . . . . . . . . . . . 5.4.1. Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Similarity solutions . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Integral method for natural convection. . . . . . . . . . . . . . . 5.5.1. Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Correlations for external natural convection . . . . . . . . . . . 5.7. Mixed convection . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8. Natural convection around a sphere . . . . . . . . . . . . . . . . 5.8.1. Description of the problem . . . . . . . . . . . . . . . . . . . 5.8.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Heated jet nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1. Description of the problem . . . . . . . . . . . . . . . . . . . 5.9.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10. Shear stress on a vertical wall heated at uniform temperature 5.10.1. Description of the problem . . . . . . . . . . . . . . . . . . 5.10.2. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11. Unsteady natural convection . . . . . . . . . . . . . . . . . . . . 5.11.1. Description of the problem . . . . . . . . . . . . . . . . . . 5.11.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12. Axisymmetric laminar plume . . . . . . . . . . . . . . . . . . . 5.12.1. Description of the problem . . . . . . . . . . . . . . . . . . 5.12.2. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13. Heat transfer through a glass pane. . . . . . . . . . . . . . . . . 5.13.1. Description of the problem . . . . . . . . . . . . . . . . . . 5.13.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14. Mixed convection near a vertical wall with suction . . . . . . 5.14.1. Description of the problem . . . . . . . . . . . . . . . . . . 5.14.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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141
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141 142 142 145 145 146 149 149 150 152 152 155 155 155 157 157 158 161 161 162 164 164 166 167 176 176 177 183 183 183 184 189 189 190 190
Chapter 6. Internal Natural Convection . . . . . . . . . . . . . . . . . . . . . .
195
6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Scale analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195 195
Table of Contents
6.3. Fully developed regime in a vertical duct heated at constant temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Enclosure with vertical walls heated at constant temperature . . . . . 6.4.1. Fully developed laminar regime . . . . . . . . . . . . . . . . . . . . 6.4.2. Regime of boundary layers . . . . . . . . . . . . . . . . . . . . . . . 6.5. Thermal insulation by a doublepane window . . . . . . . . . . . . . . 6.5.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Natural convection in an enclosure filled with a heat generating fluid 6.6.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 6.6.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Onedimensional mixed convection in a cavity. . . . . . . . . . . . . . 6.7.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 6.7.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
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197 198 198 199 199 199 200 201 201 203 206 206 207 208
Chapter 7. Turbulent Convection in Internal Wall Flows . . . . . . . . . . .
211
7.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Hydrodynamic stability and origin of the turbulence . . . . . . 7.3. Reynolds averaged NavierStokes equations . . . . . . . . . . . 7.4. Wall turbulence scaling. . . . . . . . . . . . . . . . . . . . . . . . 7.5. Eddy viscositybased one point closures. . . . . . . . . . . . . . 7.6. Some illustrations through direct numerical simulations . . . . 7.7. Empirical correlations. . . . . . . . . . . . . . . . . . . . . . . . . 7.8. Exact relations for a fully developed turbulent channel flow. . 7.8.1. Reynolds shear stress. . . . . . . . . . . . . . . . . . . . . . . 7.8.2. Heat transfer in a fully developed turbulent channel flow with constant wall temperature. . . . . . . . . . . . . . . . . . . . . 7.8.3. Heat transfer in a fully developed turbulent channel flow with uniform wall heat flux . . . . . . . . . . . . . . . . . . . . . . . 7.9. Mixing length closures and the temperature distribution in the inner and outer layers . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1. Description of the problem . . . . . . . . . . . . . . . . . . . 7.9.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10. Temperature distribution in the outer layer . . . . . . . . . . . 7.10.1. Description of the problem . . . . . . . . . . . . . . . . . . 7.10.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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211 211 213 215 216 227 231 233 233
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243 245 245 246 252 252 252 252
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x
Convective heat Transfer
7.11. Transport equations and reformulation of the logarithmic layer . . . . 7.11.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 7.11.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12. Nearwall asymptotic behavior of the temperature and turbulent fluxes 7.12.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 7.12.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13. Asymmetric heating of a turbulent channel flow . . . . . . . . . . . . . 7.13.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 7.13.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14. Natural convection in a vertical channel in turbulent regime . . . . . . 7.14.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 7.14.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 8. Turbulent Convection in External Wall Flows . . . . . . . . . . . 8.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Transition to turbulence in a flat plate boundary layer . . . . . . . . . 8.3. Equations governing turbulent boundary layers . . . . . . . . . . . . . 8.4. Scales in a turbulent boundary layer . . . . . . . . . . . . . . . . . . . . 8.5. Velocity and temperature distributions. . . . . . . . . . . . . . . . . . . 8.6. Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8. Temperature measurements in a turbulent boundary layer . . . . . . . 8.8.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 8.8.2. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9. Integral formulation of boundary layers over an isothermal flat plate with zero pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . . 8.9.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10. PrandtlTaylor analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . 8.10.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11. Turbulent boundary layer with uniform suction at the wall . . . . . . 8.11.1. Description of the problem . . . . . . . . . . . . . . . . . . . . . . 8.11.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
255 257 257 258 261 261 261 261 264 264 265 266 270 270 271 274 281
. . . . . . . . . .
281 281 282 284 284 285 286 289 289 290
. . . . . . . . . . . .
292 292 293 294 297 297 297 298 301 301 301 302
Table of Contents
8.12. Turbulent boundary layers with pressure gradient. Turbulent FalknerSkan flows . . . . . . . . . . . . . . . . . . 8.12.1. Description of the problem . . . . . . . . . . . . . . 8.12.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 8.12.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 8.13. Internal sublayer in turbulent boundary layers subject to adverse pressure gradient . . . . . . . . . . . . . . . . . . . . 8.13.1. Description of the problem . . . . . . . . . . . . . . 8.13.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 8.13.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 8.14. Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.14.1. Description of the problem . . . . . . . . . . . . . . 8.14.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 8.14.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
. . . .
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. . . .
. . . .
. . . .
. . . .
. . . .
306 306 306 307
. . . . . . . .
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. . . . . . . .
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. . . . . . . .
312 312 312 313 319 319 320 320
Chapter 9. Turbulent Convection in Free Shear Flows . . . . . . . . . . . . .
323
9.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. General approach of free turbulent shear layers . . . . . . . . . . 9.3. Plumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Twodimensional turbulent jet. . . . . . . . . . . . . . . . . . . . . 9.4.1. Description of the problem . . . . . . . . . . . . . . . . . . . . 9.4.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Mixing layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Description of the problem . . . . . . . . . . . . . . . . . . . . 9.5.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. Determination of the turbulent Prandtl number in a plane wake . 9.6.1. Description of the problem . . . . . . . . . . . . . . . . . . . . 9.6.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7. Regulation of temperature . . . . . . . . . . . . . . . . . . . . . . . 9.7.1. Description of the problem . . . . . . . . . . . . . . . . . . . . 9.7.2. Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3. Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
323 323 326 328 328 329 330 335 335 335 336 340 340 341 342 348 348 350 351
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
363
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371
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Foreword
It is a real surprise and pleasure to read this “brainy” book about convective heat transfer. It is a surprise because there are several books already on this subject, and because the book title is deceiving: here “solved problems” means the structure of the field and the method of teaching the discipline, not a random collection of homework problems. It is a pleasure because it is nononsense and clear, with the ideas placed naked on the table, as in elementary geometry. The field of convection has evolved as a sequence of solved problems. The first were the most fundamental and the simplest, and they bear the names of Prandtl, Nusselt, Reynolds and their contemporaries. As the field grew, the problems became more applied (i.e. good for this, but not for that), more complicated, and much more numerous and forgettable. Hidden in this stream, there are still a few fundamental problems that emerge, yet they are obscured by the large volume. It is here that this book makes its greatest contribution: the principles and the most fundamental problems come first. They are identified, stated and solved. The book teaches not only structure but also technique. The structure of the field is drawn with very sharp lines: external versus internal convection, forced versus natural convection, rotation, combined convection and conduction, etc. The best technique is to start with the simplest problem solving method (scale analysis) and to teach progressively more laborious and exact methods (integral method, selfsimilarity, asymptotic behavior). Scale analysis is offered the front seat in the discussion with the student. This is a powerful feature of the book because it teaches the student how to determine (usually on the back of an envelope) the proper orders of magnitude of all the physical features (temperature, fluid velocity, boundary layer thickness, heat flux) and the correct dimensionless groups, which are the fewest such numbers. With
xiv
Convective Heat Transfer
them, the book teaches how to correlate in the most compact form the results obtained analytically, numerically and experimentally. In summary, this book is a real gem (it even looks good!). I recommend it to everybody who wants to learn convection. Although the authors wrote it for courses at the MS level, I recommend it to all levels, including my colleagues who teach convection. Adrian BEJAN J. A. Jones Distinguished Professor Duke University Durham, North Carolina April 2009
Preface
Heat transfer is associated with flows in a wide spectrum of industrial and geophysical domains. These flows play an important role in the problems of energy and environment which represent major challenges for our society in the 21st century. Many examples may be found in energyproducing plants (nuclear power plants, thermal power stations, solar energy, etc.), in energy distribution systems (heat networks in towns, environmental buildings, etc.) and in environmental problems, such as wasteheat release into rivers or into the atmosphere. Additionally, many industrial processes use fluids for heating or cooling components of the system (heat exchangers, electric components cooling, for example). In sum, there are a wide variety of situations where fluid mechanics is associated with heat transfer in the physical phenomena or in the processes involved in industrial or environmental problems. It is also worth noting that the devices implied in the field of heat transfer have dimensions bounded by several meters, as in heat exchangers up to tenths of microns in micro heattransfer applications which currently offer very promising perspectives. Controlling fluid flows with heat transfer is essential for designing and optimizing efficient systems and requires a good understanding of the phenomena and their modeling. The purpose of this book is to introduce the problems of convective heat transfer to readers who are not familiar with this topic. A good knowledge of fluid mechanics is clearly essential for the study of convective heat transfer. In fact, determining the flow field is most often the first step before solving the associated heat transfer problem. From this perspective, we first recommend consulting some fluid mechanics textbooks in order to get a deeper insight into this subject. Therefore, we recommend the following references (the list of which is not exhaustive): – general knowledge of fluid mechanics [GUY 91], [WHI 91] [CHA 00] and, in particular, of boundary layer flows [SCH 79]; – turbulent flows [TEN 72], [REY 74], [HIN 75].
xvi
Convective Heat Transfer
The knowledge of conductive heat transfer is, obviously, the second necessary ingredient for studying convective transfer. Concerning this topic, we refer the reader to the following textbooks: [ECK 72], [TAI 95], [INC 96], [BIA 04]. The intention of this book is to briefly introduce the general principles of theory at the beginning of each chapter and then to propose a series of exercises or problems relating to the topics of the chapter. The summary presented at the beginning of each chapter will usefully be supplemented by reading textbooks on convective heat transfer, such as: [BUR 83], [CEB 88], [BEJ 95]. Each problem includes a presentation of the studied case and suggests an approach to solving it. We also present a solution to the problem. Some exercises in this book are purely applications of classical correlations to simple problems. Some other cases require further thought and consist of modeling a physical situation, simplifying the original problem and reaching a solution. Guidelines are given in order to help the reader to solve the presented problem. It is worth noting that, in most cases, there is no unique solution to a given problem. In fact, a solution results from a series of simple assumptions, which enable rather simple calculations. The object of the book is to facilitate studying flows with heat transfer and to propose some methods to calculate them. It is obvious that numerical modeling and the use of commercial software now enable the treatment of problems much more complex than those presented here. Nevertheless, it seems to us that solving simple problems is vital in order to acquire a solid background in the domain. This is a necessary step in order to consistently design systems or to correctly interpret results of the physical or numerical experiments from a critical point of view. Industrial projects and geophysical situations involve relatively complex phenomena and raise problems with a degree of difficulty depending on the specificity of the case under consideration. We restrict the study of the convective heat transfer phenomena in this book to the following set of assumptions: – singlephase flows with one constituent; – Newtonian fluid; – incompressible flows; – negligible radiation; – constant fluid physical properties; – negligible dissipation. However, in Chapter 1 only the last two points will be discussed. The first chapter presents the fundamental equations that apply with the above list of assumptions, to convective heat transfer and reviews the main dimensionless numbers in this topic.
Preface
xvii
Most flows present in industrial applications or in the environment are turbulent so that a large section at the end of the book is devoted to turbulent transfer. The study of laminar flows with heat transfer is, however, a necessary first step to understanding the physical mechanisms governing turbulent transfer. Moreover, several applications are concerned with laminar flows. This is the reason why we present convective heat transfer in fully developed laminar flows in Chapter 2. A good knowledge of boundary layers is extremely important to understanding convective heat transfer, which most usually concerns flows in the vicinity of heated or cooled walls. Consequently, Chapter 3 is devoted to these flows and several problems are devoted to related issues. This chapter is complemented by the next one, which is concerned with heat transfer in flows around obstacles. Chapters 5 and 6 deal with natural convection in external and internal flows. The coupling between the flow field and heat transfer makes the corresponding problems difficult and we present some important examples to clarify the key points relative to this problem. Turbulent transfer is presented in Chapters 7 to 9, for flows in channels and ducts, in boundary layers and finally in free shear flows. Scale analysis [BEJ 95] is widely used in this textbook. It is quite an efficient tool to use to get insight into the role played by the group of parameters of a given physical situation. Scale analysis leads to the relevant dimensionless numbers and enables a quick determination of the expected trends. The information given by this analysis may be used as a guideline for simplifying the equations when a theoretical model is implemented and for interpreting the results of numerical simulations or physical experiments. This approach has the notable advantage of enabling substantial economy in the number of studied cases since it is sufficient to vary few dimensionless numbers instead of all the parameters to specify their influence on, for example, a heat transfer law. Other classical methods of solving are presented in the review of the theoretical principles and are used in the presented problems (autosimilarity solutions, integral method). This book is addressed to MSc students in universities or engineering schools. We hope that it will also be useful to engineers and developers confronted with convective heat transfer problems.
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Chapter 1
Fundamental Equations, Dimensionless Numbers
1.1. Fundamental equations The equations applying to incompressible flows and associated heat transfer are recalled hereafter. The meaning of symbols used in the fundamental equations is given in the following sections, otherwise the symbols are listed at the end of the book. 1.1.1. Local equations The local equations express the conservation principles for a fluid particle in motion. The operator d dt represents the Lagrangian derivative or material derivative of any physical quantity. It corresponds to the derivative of this quantity as measured by an observer following the fluid particle: d dt
w w uj wt wx j
[1.1]
1.1.1.1. Mass conservation The continuity equation expresses the mass conservation for a moving fluid particle as: dU dt
G U div u 0
[1.2]
2
Convective Heat Transfer
For the applications presented in this book, the density U may be considered as constant so that the continuity equation reduces to: G div u 0
[1.3]
1.1.1.2. NavierStokes equations The NavierStokes equations express the budget of momentum for a fluid particle. Without loss of generality, we can write:
U
G du
G
UF divV
[1.4] dt G where F represents the body force vector per unit mass (the most usual example is G G that of gravity, with F g ). V is the stress tensor, expressed with index notations for a Newtonian fluid by:
V ij
pG ij
2P 3
G
divu G ij 2P d ij
[1.5]
In equation [1.5], Gij is the Kronecker symbol (Gij = 1 if i = j, Gij = 0 if i j) and 1 §wu i wu j · ¸). ¨ dij is the pure strain tensor ( d ij ¸ 2 ¨ ©wx j wx i ¹ The NavierStokes equations are then obtained for an incompressible flow of a fluid with constant dynamic viscosity μ. They are expressed in vector notations as:
U
G du
dt
G
G
UF grad p P'u
[1.6]
In some flows influenced by gravitational forces it is usual to introduce the modified pressure p* p Ugz , where z represents the altitude with respect to a fixed origin. 1.1.1.3. Energy equation Inside a flow, a fluid particle exchanges heat by conduction with the neighbouring particles during its motion. It also exchanges heat by radiation with the environment, but this mode of transfer is not covered in this book.
Fundamental Equations, Dimensionless Numbers
3
The conductive heat transfer is governed by Fourier’s1 law: [1.7]
qcc k gradT
where qcc is the heat flux vector at a current position. Its components are expressed G in W/m2. The heat transfer rate flowing through a surface element dS of normal n is G q cc.ndS . Combining the first principle of thermodynamics, the kinetic energy equation, Fourier’s law, and introducing some fluid physical properties, the energy equation is obtained without loss of generality as:
UC p
dT dt
div k grad T qccc ET
dp dt
DQ
[1.8]
This equation shows that the temperature2 variations of a moving fluid particle are due to: – conductive heat exchange with the neighbouring particles (first term of righthand side); – internal heat generation ( qccc: Joule effect, radioactivity, etc.); – mechanical power of the pressure forces during the particle fluid compression or dilatation (third term of the righthand side); – specific viscous dissipation (DQ, power of the friction forces inside the fluid). It is worth noting that the lefthand side represents the transport (or advection) of enthalpy by the fluid motion. All the terms of equation [1.8] are expressed in W m3. The specific viscous dissipation is calculated for a Newtonian fluid by: DQ
G 2 2P d ij d ij 2 P 3 div u
[1.9]
For flows with negligible dissipation or for gas flows at moderate velocity (DQ and dp dt are assumed to be negligible), with constant thermal conductivity k and without internal heat generation ( qccc 0 ), the energy equation reduces to: dT dt
D'T
1. JeanBaptiste Joseph Fourier, French mathematician and physicist, 1768–1830. 2. In fact, the equation is first derived for fluid enthalpy.
[1.10]
4
Convective Heat Transfer
Using Cartesian coordinates, the terms of equation [1.8] are expressed in the following form:
dT
u
dt
wT wT wT v w wx wy wz
div kgrad T
DQ
w § wT · w § wT · w § wT · ¸ ¨k ¸ ¨k ¸ ¨k wx © wx ¹ wy © wy ¹ wz © wz ¹
2 ª º § wu ·2 §wv · §ww ·2 » « 2P ¨ ¸ ¨ ¸ ¨ ¸ DQ 2 «© wx ¹ ©wy ¹ ©wz ¹ » ¬ ¼
2 2 2 ª§ 2 º wv wu · §ww wv · §wu ww · » 2P §wu wv ww · DQ 2 = μ «¨ ¸ ¨ ¸ ¨ ¨ ¸ ¸ «© wx wy ¹ ©wy wz ¹ ©wz wx ¹ » 3 ©wx wy wz ¹ ¬ ¼
Equation [1.10] reads: u
§w 2T w 2T w 2T wT wT wT v w = D ¨¨ 2 2 2 wx wy wz wy wz ©wx
· ¸ ¸ ¹
[1.11]
1.1.2. Integral conservation equations Integral equations result from the application of the conservation principles to a G finite volume of fluid V, delimited by a surface S of outer normal n (Figure 1.1).
n u
V
Figure 1.1. Definition of a control volume
Fundamental Equations, Dimensionless Numbers
5
1.1.2.1. Mass conservation In the case of constant fluid density, the mass conservation equation may be simplified by U and, in absence of sinks or sources inside the volume V, then reads: G G
³ u.n dS 0
[1.12]
S
1.1.2.2. Momentum equation The Lagrangian derivative of the fluid momentum contained in the volume V is G in equilibrium with the external forces resultant R :
d § G · G ¨³ Uu dv ¸ R ¹ dt ©V
[1.13]
We recall that, for any scalar physical quantity: d
³ f dv =
dt v
w G G ³ f dv + ³ f u.n dS wt v S
[1.14]
The momentum budget may also be written as:
w § G · G G G ¨³ Uu dv ¸ ³ Uu u.n dS ¹ S wt ©V
G
G G
³ UF dv ³ T n dS V
[1.15]
S
G G G whereF is the body force vector per unit mass inside the fluid and T n is the stress vector at a current point of the surface S. The first term of the lefthand side is zero in the case of steady flow.
1.1.2.3. Kinetic energy equation The kinetic energy K of the fluid contained in the volume V satisfies: dK dt
pe pi
with: –K= ³ V
1 2
Uu 2 dv ;
– Pe = power of external forces (volume and surface forces);
[1.16]
6
Convective Heat Transfer
– Pi = power of internal forces. It can be shown that Pi may be decomposed into two parts: Pi = Pc  DQ
[1.17]
– Pc = mechanical power of the pressure forces during the fluid volume compression or dilatation (Pc may be positive or negative): Pc = ³ p v
d §1 · ¨ ¸dm dt ©U ¹
[1.18]
– DQ = viscous dissipation inside the volume V, DQ =
³
DQdv.
V
The viscous dissipation DQ is always positive and corresponds to the fluid motion irreversibilities. 1.1.2.4. Energy equation As for a fluid particle (section 1.1.1.3), the first principle of thermodynamics may be combined to the kinetic energy equation and the Fourier law applied to a finite fluid volume V. The variation of enthalpy H contained in the volume V is then expressed as: dH dt
G
³ qcccdv ³ qcc.ndS ³ V
S
V
1 dp
U dt
dm DQ
In usual applications, H = ³ U CpT dv and v
dp dt
[1.19]
Tw1 x T x, K @
uK Um
dS
Tw1 x Tm x
1 2
§
1¨q1cc q2cc ³1 ¨
©
k
ª § K 4 · «3¨K 2 ¸ ¸ ¨ 6 8 « ¹ © ¬ e
· º cc cc q q1 3 » 2 eK 1 ¸ 1 K 2 dK ¸ 2 » 2k ¼ ¹2 5
Laminar Fully Developed Forced Convection in Ducts
41
Finally, we find that: Tw1 x Tm x
º e ª17 1 « q1cc q2cc q1cc q2cc » ¼ k ¬70 2
[2.24]
2.3.3.4. Nusselt number The Nusselt number may be based on q1ccor q2cc, while the characteristic dimension is the hydraulic diameter (Dh = 4e):
k
q1cc Tw1 Tm
k
q2cc Tw2 Tm
Nu1
Nu2
4e
4e
Using [2.24], we find that: Nu1
140 26 9
[2.25]
q2cc q1cc
2.3.3.5. Study of the different cases 2.3.3.5.1. Symmetric heating ( q1cc= q2cc) The two channel walls are obviously at same temperature. The temperature difference between the wall and the fluid ([2.23]) is: Tw1 (x) Tm x
17eq1cc
[2.26]
35k
The fluid temperature profile is symmetric with respect to the symmetryaxis of the channel: T x, K Tw1 x
q1cc e ª § 2 K 4 · «3¨K ¸ ¨ k 4 « 6 ¸ ¹ ¬ ©
5 º » 2 » ¼
42
Convective Heat Transfer
The temperature may be normalized with
T K
Tm
Nu1
K
Tw1 x T x, K q1cc 2e k
Tw1 x Tm x q1cc 2e k 140 17
q1cc 2e as: k
1 ª § 2 K 4 · ¸ «3¨ ¨K 6 ¸ 8 « ¹ ¬ ©
5 º » 2 » ¼
[2.27]
17 70
8.235
[2.28]
1 heating wall
0.8 0.6 0.4 0.2
TK
0 0.2 0.4 0.6
bulk temperature
0.8 heating wall 1 0.00
0.05
0.10
0.15
0.20
0.25
0.30
T
0.35
Figure 2.3. Temperature profile in a parallelplate channel. Symmetric heating
Figure 2.3 shows the fluid temperature profile in a channel crosssection. This distribution is symmetric with respect to Oxaxis for the thermal conditions studied in this section. It is worth recalling that the dimensionless temperature varies in the opposite direction to the actual temperature (according to definition [2.27]) when the walls are heating the fluid. The plane of symmetry of the duct corresponds to the maximum value of T, i.e. to the minimum value of the fluid temperature T.
Laminar Fully Developed Forced Convection in Ducts
43
2.3.3.5.2. Wall 2 insulated ( q2cc = 0) The fluid temperature is at its minimum on the adiabatic wall. With the same normalization as in the previous case:
T K
Tw1 x T x, K q1cc2e
1 ª § 2 K 4 · «3¨K ¸ ¨ 6 ¸ 16 « ¹ ¬ ©
5 º K 1 » 2 » 4 ¼
k
Tm
Nu1 K
1
Tw1 x Tm x q1cc 2e k
70 13
13 35
5.384
adiabatic wall
0.8 0.6 0.4
TK
0.2 0 0.2 0.4
bulk temperature
0.6 0.8 heating wall
1 0.0
0.1
0.2
0.3
0.4
0.5
T
0.6
Figure 2.4. Temperature profile in a parallelplate channel. Unsymmetric heating
In this case, the temperature profile presents a vertical tangent at the adiabatic wall due to the Fourier law (Figure 2.4).
44
Convective Heat Transfer
2.3.3.5.3. Simultaneous heating on one wall and cooling on the opposite wall ( q1cc q2cc) Heat is transferred by pure conduction. The crosssection temperature profile is linear:
T K
K 1
Tw1 T q1cc2e
2
k
4.
According to [2.25], Nu1 2.3.3.5.4. Comment
Equation [2.25] leads to an infinite value of Nu1 when q2cc q1cc 26 9 . This simply means that Tm Tw1 in this type of heating and that the denominator of Nu1 is zero in the definition given in 2.3.3.4. Recently, [NIE 04] has shown that this unusual behavior may be avoided when the Nusselt number is defined with the mean heat flux q1cc q2cc 2 and the mean wall temperature Tw1 Tw2 2 . Moreover, he has shown that, using this definition, Nu does not depend on the wall heat flux ratio q2cc q1cc when the velocity profile is symmetric with respect to the symmetryaxis of the channel. 2.3.3.6. Temperature distribution along the channel for unsymmetric heating ( q2cc 0 ) The bulk temperature varies linearly along the channel from the start of heating when axial conduction effects are negligible. This assumption is generally well verified, except in the case of liquid metal flows. According to [2.16]: Tm ( x )
Tm0
q1cc
x
UC pU m 2 e
qcce 2 x Tm0 1 k Pe e
[2.29]
Otherwise, according to [2.23] and [2.24], we may write in the fully developed region: Tw1 (x) Tm x
Tw2 ( x ) Tw1 ( x )
26 q1cce 35 k
q1cce k
[2.30]
Laminar Fully Developed Forced Convection in Ducts
45
or accounting for: qcc2e § 1 x 26 1 · Tm0 1 ¨ ¸ k ©Pe e 70 2 ¹
Tw2 (x)
[2.31]
Since the adiabatic wall temperature cannot be lower than the fluid inlet temperature, it follows that the fully developed region cannot be reached before the abscissa x1, so that the term in the parentheses of equation [2.31] is positive, that is: x1 1 e Pe
1
!
2
26 70
0.13
The fluid and wall temperature distribution along the channel is sketched in Figure 2.5.
T 26 q1cce
Tw1(x)
70 k
q1cce §1 26 · ¸ ¨ k ©2 70 ¹
Tm(x)
Tw2(x)
Figure 2.5. Fluid and wall temperature distribution along the channel
2.3.3.7. Heattransfer coefficient in an annular duct As the gap spacing is very small compared to the radius R, curvature effects can be ignored so that the calculation may be performed by using the previous results obtained for unsymmetric heating in a plane channel. The fluid is heated by the inner wall. The outer wall is adiabatic. Let us first calculate the bulk velocity, the Reynolds and the Péclet numbers: Um
Re
Q
0.1 x 10 3
2 SR 2 e
2S x 0.1 x 0.5 x 10 2
U m 4e
Q
0.032 x 10 2 5 x 10 7
= 0.032 m/s
640 Pe
640 x 3.5
2230
46
Convective Heat Transfer
The heat flux q1cc, supplied by the heating wall, is obtained by dividing the total heat transfer rate by the inner wall surface: q1cc
q
10 4
2SRL
2S x 0.1 x 1
15.9 kW/m 2
q1cc2e k
132.5qC
The following temperatures are deduced from [2.29], [2.30] and [2.31] for the distance x = 1 m: Tm = 33.7°C, Tw1 = 82.9°C, Tw2 = 16.5°C.
2.4. Flow in a plane channel insulated on one side and heated at uniform temperature on the opposite side 2.4.1. Description of the problem
In an industrial facility, fluid flows through a rectangular channel at the inlet temperature T0 (Figure 2.6). The channel spacing 2e is very small compared to the length Lz in the spanwise direction (e T0 for x ! x0 ). The section S0 is located far downstream from the channel inlet so that the flow is hydrodynamically fully developed in this region. As a consequence of heating, a thermal boundary layer develops on P1 and reaches the opposite wall P2 at abscissa x1 . A thermally fully developed region is observed at some distance downstream from x1 . Further downstream, the temperature homogenizes in the channel. The aim of the task is to estimate the distance necessary for the fluid to become nearly isothermal. y
P2 2e
T0
Tw(x)
u(y) x0
T1
P1
x1
Figure 2.6. Flow in a channel insulated on one side and heated at uniform temperature on the opposite side. Notations
x
Laminar Fully Developed Forced Convection in Ducts
47
Determine the temperature distribution Tw(x) on wall P2 in the thermally fully developed region. Axial conduction effects are assumed to be negligible. Calculate the distance X x x1 corresponding to the wall temperature difference equal to 1% of its initial value. The fluid is helium gas of thermal diffusivity D 1.6 104 m2 s1; Pr = 0.67. The flow conditions are 2e = 3 mm, Um = 5 m s1. 2.4.2. Guidelines
Calculate the velocity distribution u(y) in the channel. Simplify the energy equation when the assumptions indicated previously are taken into account. Integrate the energy equation over the channel height. It is recommended that dimensionless variables are used. Propose a dimensionless temperature profile that accounts for the thermal boundary conditions on the two walls (it is suggested a polynomial form is used) and determine the temperature distribution Tw(x)along the channel by using the integral energy equation 2.4.3. Solution
2.4.3.1. Velocity profile The velocity profile is given by [2.13] in the fully developed laminar regime: u(K) Um
1 K2 with K 2 3
y e
2.4.3.2. Energy equation PRELIMINARY NOTE.– The temperature of wall P2 starts to increase for x ! x1 . This situation seems surprising at first view since this wall is insulated and is therefore assumed not to exchange heat with the fluid. However, it must be considered as the limiting case of a very weak heat transfer from the fluid to wall P2 . The steady temperature distribution on P2 corresponds to the thermal equilibrium reached after a very long time. In a practical application, the wall P2 is built of an insulating material so that the heat flux through it is very weak, but not strictly zero. P2 is
48
Convective Heat Transfer
slowly heated up to its equilibrium temperature. The current problem considers the situation of thermal equilibrium where the temperature field is steady. The energy equation simplifies, as in section 2.3.3.2, as: u(y)
wT wx
D
w T wy
[2.32]
Normalizing the fluid temperature with that of the wall P1 does not modify this equation, which is integrated from one wall to the other. Since the velocity is independent of x, the integral equation becomes: d dx
³
ªwT ºe u(y)>T x, y T1 @dy D« » e ¬wy ¼e e
The temperature gradient wT wy y
e
is zero since the wall P2 is adiabatic. The
bulk temperature [2.8] is introduced into the integral equation as in section 2.3.3.3: 2U m e
d wT Tm x T1 @ D > wy dx
[2.33] y e
This equation is then expressed with dimensionless variables. The distance y to P1 was already normalized with the channel halfwidth as K y e . The Péclet number is based on the hydraulic diameter ( Dh 4 e ): Pe
U m 4e
D
The abscissa x along the channel is normalized as [
x x1 1
. The 4 e Pe temperature is normalized by the difference Tw x T1 in a channel crosssection:
T [ , K
T ( x , y ) T1 Tw x T1
[2.34]
The dimensionless bulk temperature is calculated by the integral:
T m [
1 1 uK T [ , K dK ³ 2 1 U m
[2.35]
Laminar Fully Developed Forced Convection in Ducts
49
The temperature of wall P2 may also be written in the dimensionless form: T1 Tw x
4x
with 4 0
40
T1 T 0
[2.36]
The temperature scale 4 0 simplifies in the integral equation, which after the transformation of variables becomes:
1 d
wT wK K
>T m [ 4[ @
8 d[
4[
[2.37] 1
In the fully developed regime, the dimensionless temperature profile is independent of x and the same statement is true for T m versus [ . Finally, the differential equation satisfied by the dimensionless wall temperature 4[ yields: d4 d[
O4[
[2.38]
The solution is 4[
exp O[
[2.39]
where the condition 40 number is defined by
Nu k
q1cc T1 Tm 4e
0 for x
x1 has been accounted for. The Nusselt
kwT wy y e T Tm k 1 4e
[2.40]
where q1cc is the heat flux at wall P1 . The Nusselt number is constant and, after substituting the dimensionless variables, is given by: Nu
O
4 wT
2
T m wK K
[2.41] 1
2.4.3.3. Approximate solution An approximate solution is obtained by estimating a temperature profile for calculating the two unknowns T m , wT wK K 1 in equation [2.42] and then obtaining
50
Convective Heat Transfer
the Nusselt number. Some conditions are prescribed to the selected temperature profile in order to obtain a shape close to the physical solution. With definition [2.34], the temperature continuity at the walls implies T 1 0 , T 1 1. The adiabaticity condition on wall P2 implies Tc1
0.
Among all the possible functions that satisfy these three conditions, it is 1 convenient to choose a seconddegree polynomial T K K 2 2K 3 . 4
The bulk temperature is then calculated by using equation [2.35] T m = 7/10. Moreover, wT wK K 1 = 1. Finally, the Nusselt number is obtained by using equation [2.42], Nu = 40/7 = 5.71. This approach may be refined by noting that the lefthand side of equation [2.32] is zero on the wall (u = 0) for the exact solution, so that an additional condition may be considered for the approximate temperature profile, w 2T wK 2 r1 0 . It is possible to account for one or two of these additional conditions by selecting a polynomial of higher degree than the previous approximate solution. The results depend on the degree of the chosen polynomial and are shown in Table 2.2. The thirddegree polynomial was determined by taking the condition w 2T wK 2 1 0 into account. [SHA 78] indicates Nu = 4.86. T
Tm
Nu
Deviation relative to [SHA 78]
1 4 K 2 2K 3
0.7
5.71
+17%
0.712
4.21
13%
0.743
5.38
+10%
1 16 K 3 3K 9K 11 1 16 K 4 6K 8K 13 2
2
Table 2.2. Flow in a channel insulated on one side and heated at uniform temperature on the opposite side. Nusselt number in fully developed region obtained with the integral method
Figure 2.7 compares the temperature profiles approximated by second or thirddegree polynomials to the exact profile obtained by a seriessolution. The seconddegree polynomial underestimates the temperature in the half of the channel near the adiabatic wall whereas the thirdorder polynomial overestimates the temperature in the whole channel.
Laminar Fully Developed Forced Convection in Ducts
2.4.3.4. Numerical application Using the data of the problem, the Péclet number is Pe
5 x 6 x 10 3 1.6 x 10 4
51
= 187.5.
Let us calculate the distance X corresponding to 4X 0.01 . According to Ln100 Ln100 [2.40], the corresponding value of [ verifies [ . The distance X 2Nu O is calculated by using Nu = 5.38 deduced from Table 2.2: X
4ePe
Ln100 2Nu
= 0.48 m
The accuracy of the calculation is the same as for the determination of Nu, i.e. about 10%. The exact value of Nu (= 4.86) gives X = 0.53 m. It is worth noting that this distance is proportional to the Péclet number since Nu is a constant. K
1
0.8 seriessolution seconddegree polynomial thirddegree polynomial
0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0
0.2
0.4
0.6
0.8
TK
Figure 2.7. Flow in a channel insulated on one side and heated at uniform temperature on the opposite side. Temperature profile
1
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Chapter 3
Forced Convection in Boundary Layer Flows
3.1. Hydrodynamics Flows at a high Reynolds number are generally characterized by large irrotational regions adjacent to thin boundary layers, which develop on walls and concentrate vorticity and viscosity effects. These velocity boundary layers connect the nearwall region, where the velocity tends to zero at the wall due to the noslip condition, to the free stream characterized by the external velocity uf . The velocity uf is given by the theory of potential flows and is the slip velocity at the wall in this model. Boundary layers are very important because they play a major role in heat transfer phenomena, which take place when the walls and the free stream have different temperatures. In fact, the nearwall flow mainly controls the heat exchange at a high Reynolds number. As for velocity, the fluid temperature variations are concentrated in a very thin nearwall region, called the thermal boundary layer. This region constitutes the main contribution to the thermal resistance between the wall and the fluid. 3.1.1. Prandtl equations The NavierStokes equations simplify in boundary layers because the transverse length scale (boundary layer thickness G) is much shorter than the longitudinal length scale (abscissa x or wall length L). For a twodimensional constantproperty
54
Convective Heat Transfer
flow without dissipation or buoyancy, the following Prandtl equations are used for calculating the flow:
wu wv wx wy u
wu wu v wx wy
0
[3.1]
0
1 dp
U dx
Q
w 2u
[3.2]
wy 2
1 wp
[3.3]
U wy
It is worth recalling the important statement that pressure1 is considered as constant across the boundary layer (in ydirection). The pressure is therefore calculated with the Bernoulli equation applied to the irrotational external flow: p(x)
1 2
Uuf2 (x)
[3.4]
Constant
Irrotational external flow
Boundary layer
y
u (x) f
u(x,y)
(x) x
Figure 3.1. Twodimensional boundary layer: notations
The velocity field must satisfy both the noslip condition ( u(y 0) 0 ) and the zero transverse velocity condition at the wall when this wall is impermeable. The boundary layer must also connect the wall to the external flow. The potential flow theory determines the external flow and gives the law of external velocity uf x . The external flow and, consequently, the velocity uf x depend on the geometry of the solid body on which the boundary layer develops. The velocity in the boundary layer must tend to uf x far from the wall relative to the boundary layer thickness G.
1. p is the modified pressure in these equations (the superscript * has been omitted).
Forced Laminar Boundary Layer Flows
55
3.1.2. Classic results 3.1.2.1. Scale analysis Considering the Prandtl equations, scale analysis shows that the velocity boundary layer thickness verifies the relationship:
G (x) x
1

[3.5]
Re(x)
uf (x)x
. The symbol  means that the two sides of equation [3.5] have Q the same order of magnitude (apart from a numerical coefficient of the order of 1). with Re(x)
In the same way, the shear stress exerted by the fluid on the wall is estimated by
W 0 (x)
P
wu(x, y) wy y
u (x) P f G (x) 0
[3.6]
where G(x) is given by [3.5]. 3.1.2.2. Blasius boundary layer The simplest type of boundary layer corresponds to uniform velocity external flow along a plane plate. In these conditions, equation [3.5] shows that the boundary layer thickness grows as a parabolic function against x. The G0.99 thickness is defined as the distance to the wall corresponding to u( x , y ) 0.99uf . The numerical solution of the Prandtl equations gives:
G0.99 x
4.92
x
Re x
[3.7]
The skinfriction coefficient is given by: C f x
W 0 (x)
0.332
Uuf
Re x
2
[3.8]
3.1.2.3. Influence of pressure gradient The spatial development of a boundary layer is controlled by the law of external velocity uf (x) or, equivalently, by the longitudinal pressure gradient dp(x) dx . For accelerated external flows (flow in a diverging duct, in the downstream region of an obstacle), the boundary layer has a tendency to thin under the influence of the
56
Convective Heat Transfer
pressure gradient ( dp(x) dx 0 , “favorable” pressure gradient) so that the wall shear stress increases along the wall. Conversely, for retarded flows (flow in a diverging duct, downstream region of an obstacle), the boundary layer has a tendency to thicken under the influence of the pressure gradient ( dp(x) dx ! 0 , “unfavorable” pressure gradient) so that the wall shear stress decreases along the wall. When the unfavorable pressure gradient becomes too strong, the flow separates from the wall. The boundary layer model is then no longer valid. Prandtl equations have exact solutions (similarity velocity profiles) when the external velocity has the form of a powerlaw function as: uf (x)
Kx m
[3.9]
This law characterizes irrotational flows around wedges with an apex angle of ES (radians), as illustrated in Figure 3.2. The potential flow theory gives the relation 2m . between the coefficient E and the exponent m, E m 1
x
uf (x)
Stagnation point Figure 3.2. Irrotational flow around a wedge
The similarity velocity profiles in this type of boundary layer are given by u(x, y) uf (x)
F c(K)
[3.10]
with
K
y
G (x)
and G (x)
Qx uf (x)
[3.11]
Forced Laminar Boundary Layer Flows
57
The function F(K), which is proportional to the stream function and which defines the shape of the velocity profile, is the solution of the FalknerSkan equation
m 1
F cc
FF c m F c2 1
2
[3.12]
0
with the following boundary conditions written for an impermeable wall: F c(0)
F (0)
1.0
F c(f)
0,
[3.13]
1
E 1.0 E 0.3
FcK1
0.8
Accelerated Flows
E
0
E
0.6
E
0.18 0.1988
0.4 Retarded Flows
0.2
K1
y
m 1 uf x 2 Qx
0.0 0
1
2
3
4
5
Figure 3.3. Influence of pressure gradient on boundary layer velocity profiles, adapted from [WHI 91]
The FalknerSkan equation has solutions for E ! 0.1988 or, equivalently, m ! 0.0904 . The solutions of [3.12] are plotted in Figure 3.3, with the abscissa K1, which is slightly different from definition [3.11]. The wall shear stress is an important physical quantity; it is presented in a dimensionless form as the skinfriction coefficient C f x
where F0c
W 0 (x)
F0cc
Uuf
Rex
d 2F dK 2 K
2
is given in Table 3.1. 0
[3.14]
58
Convective Heat Transfer
E
0.1988
0.18
0
0.3
1
F0c
0
0.087
0.332
0.594
1.232
Table 3.1. Influence of pressure gradient on boundary layer skinfriction (from [WHI 91])
3.2. Heat transfer 3.2.1. Equations of the thermal boundary layer For very high Péclet numbers ( Pe uf L D , where L is the longitudinal length scale), heat transfer is concentrated in a thermal boundary layer whose thickness GT is much shorter than L. The temperature field verifies the simplified energy equation: u
wT wT v wx wy
D
w 2T
[3.15]
wy 2
Boundary conditions depend on the mode of heating. When the wall and external temperatures are prescribed, the boundary conditions are: T x, y
0 Tw ( x ) , T x , y
f Tf
Equation [3.15] complements the Prandtl equations. The thermal field can be calculated when the velocity field has been determined in a first step. 3.2.2. Scale analysis The thermal problem is characterized by a typical temperature difference 4 Tw Tf . The local Nusselt number is defined using the heat flux q0cc( x ) , supplied by the wall to the fluid and by the longitudinal length scale x. The heattransfer coefficient h (in W m2 K1) is related to the heat flux by
q0cc(x) Nu x
[3.16]
h(x)4 q0cc( x )
h( x ) x
k4 x
k
[3.17]
The analysis of equations [3.1], [3.2] and [3.15] gives the order of magnitude of the main variables shown in Table 3.2. It is worth noting that the scale 4 does not play any role in these results, since the thermal problem is linear in temperature.
Forced Laminar Boundary Layer Flows
G GT
Nu x
1/2
Pr 1 or Pr ~ 1
59
Re x
1/2
Pr
1/2
Pr
1/2 1/3
Table 3.2. Results of scale analysis in laminar thermal boundary layers
3.2.3. Similarity temperature profiles 3.2.3.1. Energy equation For a flow characterized by the external velocity law of the form uf (x) Kx m , energy equation [3.15] has similarity solutions if the wall temperature also has the form of a powerlaw function Tw (x) Tf Hx n . In this case, the temperature profile is: T(x, y) Tf Tw (x) Tf
T (K)
[3.18]
The dimensionless distance to the wall K is defined by equation [3.11]. The function T (K) , which defines the shape of the temperature profile, satisfies
Tcc
m 1 2
PrFTc nPrF cT
[3.19]
0
with the boundary conditions
T (0) 1,
T (f)
0
The solution of [3.19] depends on m, n and Pr. The wall heat flux is deduced from the temperature field. Setting Tc0 Tc(0) , it is expressed as
q0cc(x)
k
wT wy
k Tw (x) Tf y 0
uf (x)
Qx
T0cm, n, Pr
[3.20]
and the Nusselt number is inferred from the above equation using definition [3.17] ( Tc0 0 ).
60
Convective Heat Transfer
3.2.3.2. Results 3.2.3.2.1. Uniform wall temperature For Pr > 1, it is found that: ªF cc(0)Pr º1/ 3 1.12« (m 1)» Pr1 3 Re x1 2 ¬ 12 ¼
Nu x
[3.22]
In the case of flow at uniform external velocity (m = 0, Blasius layer) and Pr >> 1, heat transfer law [3.22] is:
0.339Pr1 3 Re x1 2
Nu x
[3.23]
For Pr ~ 1 or Pr > 1, it is found that:
0.332Pr1 3 Re x1 2
Nu x
[3.24]
3.2.3.2.2. Influence of pressure gradient The Nusselt number increases for an accelerated flow (m > 0). For a fluid at low Prandtl number heated at uniform wall temperature, equation [3.21] gives: Nu x (m) Nu x (m 3 2.6
m 1
0)
for Pr 1
[3.25]
Nu x (m) Nu x (m
Pr
0) x
10 5
2.2 1.8
f
stagnation point
0
1.4 1 0.6 0.2 0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
E 1.6
Figure 3.4. Influence of pressure gradient on Nusselt number, after [ECK 72]
Forced Laminar Boundary Layer Flows
E
0.199
0.18
0
0.3
1
m
0.090
0.083
0
0.176
1
0.01
0.882
0.915
1
1.108
1.473
0.1
0.787
0.857
1
1.145
1.568
0.72
0.680
0.789
1
1.192
1.696
2
0.625
0.755
1
1.215
1.761
6
0.571
0.724
1
1.235
1.818
10
0.547
0.712
1
1.242
1.839
100
0.451
0.670
1
1.263
1.900
Pr
61
0) as a function of Pr and Efrom [WHI 91]
Table 3.3. Nu x (m) Nu x (m
Figure 3.4 shows the Nusselt number for a fixed value of the exponent m, normalized by the result obtained for a plate with zero pressure gradient (m = 0). The velocity of the reference flow is chosen as equal to the local velocity of the actual flow (Kxm). Heating is performed at uniform wall temperature. The ratio Nu x (m) Nu x (m 0) is also given by Table 3.3. It is worth noting that the heattransfer coefficient is significantly increased in the case of a strong favorable pressure gradient. The computation of Nu is no longer possible for E 0.1988 (condition for flow separation). 3.2.3.2.3. Influence of heating mode Equation [3.20] shows that similarity solutions to [3.19] exist for uniform flow (m = 0) when q0cc(x) varies as x n 1 2 . Hence, heating with uniform wall heat flux ( n 1 2 ) is a particular case of such heating modes. The wall temperature varies as x 1 2 in this mode of heating. For a flow with zero pressure gradient (m = 0) and uniform wall flux heating, the increase in Nu is ([ECK 72]): Nu x (n
1 2)
Nu x (n Nu x
0)
[3.26]
1.31
0.435Pr1 3 Re x1 2
for n
12
[3.27]
62
Convective Heat Transfer
3.2.3.2.4. Overall heat transfer rate The overall heat transfer rate between the wall and a boundary layer flow is easily calculated when similarity temperature profiles are known. If Lz is the length of the wall in the spanwise direction, the overall heat transfer rate is calculated by:
qcL
q(L) Lz
³0Lq0cc(x)dx
The overall Nusselt number is defined by:
NuL
qL / L z L
[3.28]
k >Tw (L) Tf @ L
For a similarity solution, it is found that: Nu L
Tc0 m, n, Pr
ReL m /2 n 1/2
[3.29]
For a boundary layer with zero pressure gradient and uniform wall temperature (m = n = 0), the overall Nusselt number is:
NuL
0.664 ReL
2Nu x x
L)
[3.30]
3.3. Integral method 3.3.1. Integral equations The local conservation equations are integrated in a slice of fluid of length dx and height H (much larger than the velocity and temperature boundary layer thicknesses, Figure 3.5).
Figure 3.5. Integral method. Control domain
Forced Laminar Boundary Layer Flows
63
For incompressible flow, the momentum equation yields in the control domain:
w H w H ³ 0 udy ³ 0 uu uf (x) dy wt wx duf (x) dx
>³
H 0
§wu · ¸ ©wy ¹0
uf (x) u
[email protected] Q ¨
In steady regime ( w wt dT u (x) dx
[3.31]
0 ), the Karman equation is obtained
uc (x) f 2T u (x) G * (x) uf (x)
[3.32]
C f (x)
with the following definitions: §
u(x, y) ·
– displacement thickness
G * (x)
f ¸¸dy ³ 0 ¨¨1 © uf (x) ¹
– momentum thickness
T u (x)
³ 0f
– skinfriction coefficient
C f (x)
u(x, y) § u(x, y) · ¨¨1 ¸¸dy uf (x) © uf (x) ¹
W 0 (x) Uuf (x) 2
The integral energy equation reads for the same control domain
w H w ³ UC pTdy ³0H UC p uT Tf dy wt 0 wx
§wT · q0ccx k ¨ ¸ ©wy ¹y
[3.33] 0
where q0ccx is the local heat flux. In a steady regime, this equation may be written (Pohlhausen equation) uf (x) dG h2
Q
dx
2
G h2 duf (x) dx Q
2 Gh
[3.34]
Pr G c
with the following definitions: – convection thickness
Gh (x)
– conduction thickness
Gc (x)
f T(x, y) Tf
³0
u(x, y)
Tw (x) Tf uf (x)
Tw (x) Tf
wT wy 0
dy ;
64
Convective Heat Transfer
3.3.2. Principle of resolution using the integral method The integral method is an approximate method, which makes it possible to calculate global quantities (skinfriction coefficient, heattransfer coefficient) with a reasonable accuracy. Concerning hydrodynamics, the problem consists of one equation ([3.32]) in three unknowns: G * (x) , T u (x) and C f x . An approximate velocity profile is selected to approach at best the shape of the exact one. Some conditions are prescribed for the velocity profile in this purpose. A polynomial function is conveniently selected for the shape of the velocity profile. For example, the following thirddegree polynomial is often used in the velocity boundary layer u(x, y) uf (x)
T (K )
f (K)
3 2
K
K3 2
1 for K t 1
[3.35]
with K y G(x) , where G (x) is proportional to the velocity boundary layer thickness. The polynomial selected above satisfies the conditions f (0)
f c(1)
f cc(0)
0, f (1)
1
which express the noslip condition for an impermeable wall and matching conditions with the free stream. The condition f cc(0) 0 is a consequence of the local Prandtl equation in xdirection, written in y = 0, for a flow with zero pressure gradient. The three unknowns G * (x) , T u (x) and C f x are expressed as functions of G (x) by using their definitions. Replacing the result in equation [3.32], an equation in one unknown G (x) is obtained. The initial value of the boundary layer thickness is given by G (x 0 ) 0 . Solving the new equation gives G (x) and consequently the quantities to be calculated, G * (x) , T u (x) and C f x . The principle of resolution is the same for the thermal problem. In this case, a temperature profile is selected, for example of the polynomial form T(x, y) Tf Tw (x) Tf
T (] )
§3 ] 3 · ¸ for ] d 1 T (] ) 1 ¨ ] ¸ ¨2 2 ¹ ©
0 for ] t 1
[3.36]
Forced Laminar Boundary Layer Flows
65
with ] y GT (x) , where GT (x) is proportional to the thermal boundary layer thickness. The polynomial selected above satisfies the conditions:
T (0) 1, Tc(1) Tcc(0) T (1)
0
The method is then the same as for hydrodynamics. The unknowns G h (x) and G c (x) are expressed as functions of GT (x) and the results are reported in equation [3.34]. Solving the new equation gives GT (x) and, consequently, the quantities to be determined. The heattransfer coefficient is deduced from the conduction thickness, according to its definition: h(x)
k
[3.37]
G c (x)
The boundary condition that GT (x) must satisfy also concerns the beginning of the thermal boundary layer and is of the form GT (x1 ) 0 . The integral method is very convenient for solving the Eckert problem, where the start of heating is shifted downstream relative to the beginning of the velocity boundary layer ( x1 ! x 0 ). As an example, the integral method may be checked when applied to the Blasius boundary layer. Using profile [3.35], the skinfriction coefficient is found to be Cf
W 0 (x)
0.323
Uuf
Re x
2
which is 3% lower than the exact value [3.8]. For heat transfer, when the velocity and temperature boundary layers develop simultaneously ( x1 x 0 ) and, for Pr > 1, the Nusselt number is found to be
Nu x
0.332Pr1 3 Re x1 2
in perfect agreement with the exact result [3.24]; this coincidence is, however, accidental. 3.4. Heated jet nozzle 3.4.1. Description of the problem A facility consists of a large tank with a nozzle attached; the nozzle is composed, first, of a converging duct, followed by a cylindrical tube of length L and diameter D (Figure 3.6). The tank is filled with air at pressure p1 and temperature T1, which
66
Convective Heat Transfer
exits through the nozzle into the ambient atmosphere at pressure p0 and temperature T0 : T1 = 40°C T0 = 20°C p1 p0 = 30 N m2 L = 12 cm
D = 8 cm
The physical properties of air are given in Table 3.4. Density
U = 1.165 kg m3
Kinematic viscosity
Q = 16 106 m2 s1
Specific heat at constant pressure
Cp = 103 J kg1 K1
Prandtl number
Pr = 0.7
Thermal conductivity
k = 0.025 W m1 K1 Table 3.4. Physical properties of air
Figure 3.6. Heated jet. Sketch of the facility
The flow is laminar in the nozzle and it is assumed that the boundary layer theory applied to a plate with zero pressure gradient gives a reasonable approximation of the flow and the associated heat transfer between the fluid and the tube. The starting up of the flow from rest is considered in this problem, so that the tube temperature is assumed to be constant and equal to T0 . Calculate the velocity in the central part of the jet and the thermal boundary layer thickness in the exit plane of the nozzle. Calculate the overall heat transfer rate q0 lost by the fluid (on length L) when it passes across the nozzle by assuming that the wall is at constant temperature T0 .
Forced Laminar Boundary Layer Flows
67
3.4.2. Solution 3.4.2.1. Jet exit velocity A velocity boundary layer develops on the tube wall. As a first approximation, it may be assumed that the boundary layer starts at the tube inlet. A more precise calculation should account for the development of the boundary layer in the converging part of the nozzle. It is also assumed that the boundary layer thickness is much narrower than the tube radius. This issue will be checked at the end of the calculation. These assumptions make it possible to use Bernoulli’s relation between the tank, where the velocities are negligible, and a current point located in the central part of the nozzle exit crosssection, where the velocity is U. It is important to remember that the pressure is constant in a jet crosssection and is therefore equal to the atmospheric pressure p0 at the nozzle exit. The Bernoulli’s relation reads: p1
p0
1 2
UU 2
The exit velocity is deduced from the above equation, U = 7.18 m s1. 3.4.2.2. Thermal boundary layer thickness The characteristic Reynolds number is based on the longitudinal length L for a boundary layer flow and not on the tube diameter D, as in the fully developed regime in a duct (Chapter 2). Re L
UL
5.38 x 10 4
Q
The flow is laminar in the tube for this value of the Reynolds number (Chapter 8). Assuming that the results relative to a boundary layer with zero pressure gradient may be applied to the actual flow, the velocity boundary layer thickness is calculated by [3.7]
G  4.9
L Re L
so that G  2.5 mm . This value is actually much lower that the tube radius (R = 40 mm).
68
Convective Heat Transfer
The thermal boundary layer thickness is obtained by using the order of magnitude given in Table 3.2. For air (Pr = 0.7), the ratio of the velocity and temperature boundary layer thicknesses follows the law
G GT  Pr1 3 so that GT  2.9 mm. 3.4.2.3. Overall heat transfer rate lost by the fluid across the tube The overall Nusselt number is given by equation [3.30]: NuL
0.664 Re L Pr1 3
136.8
The heat transfer rate lost by the fluid towards the tube wall on length L is calculated by using equation [3.28] with the equivalent transverse length Lz 2SR . q0
2SRk T1 T0 Nu L = 17.2 W
This heat transfer rate may be compared to the transfer of gas enthalpy q1 by fluid flow to the ambient air: q1
UC pUS
D2 4
T1 T0 = 839 W
We find that q0 represents a loss of 2.1% relative to q1 . 3.5. Asymptotic behavior of thermal boundary layers 3.5.1. Description of the problem We will consider the behavior of thermal boundary layers for very low or very high Prandtl numbers. The free stream velocity may vary with the longitudinal coordinate x . This external flow is, however, not necessarily of the FalknerSkan type (section 3.1.2.3). Dissipation is negligible. The fluid is Newtonian and the flow is incompressible. The wall and free stream temperatures are uniform. The two limiting cases are to be considered separately, when Pr o 0 and Pr o f respectively. The main issue is to determine the similarity variable, which results in similarity solutions. Determine the Nusselt number for these two asymptotic limits.
Forced Laminar Boundary Layer Flows
69
3.5.2. Guidelines The velocity boundary layer thickness is much smaller than the thermal boundary layer thickness when Pr o 0 . It is then possible to approximate the actual velocity by u uf x in the thermal boundary layer. Write the energy equation by using this approximation. Introduce a function gx and a similarity variable K y gx . Transform the energy equation by using the new variable K . Deduce a differential equation for gx , so that the problem has similarity solutions. For Pr o f , on the other hand, the situation is inverted and the thermal boundary layer thickness is much smaller than G . It is then possible to consider that the velocity varies approximately linearly in the thermal boundary layer. Repeat the same procedure as for Pr o 0 . 3.5.3. Solution 3.5.3.1. Very small Prandtl numbers For Pr o 0 , the velocity boundary layer thickness is much smaller than the thermal boundary layer thickness. It is then possible to approximate the actual velocity by u uf x in the thermal boundary layer (Figure 3.7). The distribution of the velocity component normal to the wall is deduced from the continuity equation
wu wv wx wy
duf
wx
wv wy
0
so that v
duf
wx
[3.38]
y
The energy equation becomes: u
wT wT v wx wy
uf x
wT duf wT y wx wx wy
The dimensionless temperature is T
D
w 2T wy 2
T Tf . Tw Tf
[3.39]
70
Convective Heat Transfer
Introducing the arbitrary length scales x 0 and y 0 , and setting x * y y y 0 , equation [3.39] becomes *
uf y02 wT
Dx0 wx *
yy0 duf wT
w 2T
dx wy *
wy *2
D
x x0 ,
which shows that
T
§ x
T ¨¨
©x 0
,
u y 2 yy du · , f 0 , 0 f ¸ y0 Dx 0 D dx ¸¹ y
[3.40]
T u(x)
Pr o 0
GT
G Figure 3.7. Thermal boundary layers. Asymptotic behavior for Pr o 0
The length scales x 0 and y 0 have been introduced to nondimensionalize the energy equation; they have, however, no physical meaning and are eliminated in [3.40] by successive multiplications/divisions to finally obtain:
T
§u y 2 y 2 du · f f ¸ , ¸ D x D dx © ¹
T ¨¨
[3.41]
This analysis shows that there is no unique similarity variable. Let us choose the following dimensionless variable
K
y gx
Forced Laminar Boundary Layer Flows
and assume that T
wT wx wT wy
71
f K . The transformation of variables gives
wK dg f cyg 2 dx wx wK f cK f cg 1 wy f cK
w 2T wy 2
f ccK g 2
Substituting these expressions in the energy equation, we find: · 1 § dg 2 2 duf ¸§ K · ¨uf 2 g ¨ f c¸ f cc 0 D ¨© dx dx ¸¹© 2 ¹
[3.42]
Similarity solutions are clearly possible if: uf
dg 2 dx
2g2
duf dx
Constant
The constant is arbitrary; the resolution of the problem is, however, made easier if the constant is chosen as equal to the fluid thermal diffusivity: uf
dg 2 dx
2g2
duf dx
D
[3.43]
With this choice, the dimensionless temperature is the solution to the differential equation: §K · ¨ f c¸ f c 0 ©2 ¹
[3.44]
Multiplying the two sides of equation [3.43] by uf , and integrating the resulting equation, the function g( x ) is determined by
g( x )
ª x º1/ 2 D u x dx « ³ f 1 » uf x ¬ 0 ¼ 1
[3.45]
72
Convective Heat Transfer
which guarantees the existence of a similarity solution. Integration of [3.44] gives the temperature field: § z 2 · ¸¸dz © 4 ¹
1 K
1 T K
S
³ exp¨¨
0
[3.46]
The similarity variable is:
K
ª x º1/ 2 yuf x «D ³ uf x1 dx» ¬ 0 ¼
The Nusselt number is easily calculated by Nu
hx k
§dT · wK ¨ ¸ x ©dK ¹0 wy
and may be presented in the form
Nu
1
S 1/ 2
ª x º1 /2 uf x «D x ³ uf x1 dx» ¬ 0 ¼
[3.47]
The Nusselt number on a flat plate in the limit Pr o 0 is deduced from the above equation ( uf constant ) Nu
§Pr Rex ·1/ 2 ¨ ¸ © S ¹
where Rex Uf x Q . This approach is a good approximation of heat transfer in liquid metal boundary layers, with or without pressure gradient. 3.5.3.2. Very high Prandtl numbers Let us now consider the case Pr o f . For this condition, the following inequality is verified: GT G , as illustrated by Figure 3.8. The velocity may then be expressed in the thermal boundary layer, by a linear law, as u
§wu · ¨ ¸ y ©wy ¹0
W w x P
y
Forced Laminar Boundary Layer Flows
73
and the velocity component normal to the wall is deduced from the continuity equation.
v
y 2 dW w 2P dx
u(x) Pr o f
T
G
Tw
GT
Figure 3.8. Thermal boundary layers. Asymptotic behavior for Pr o f
Replacing these relations in the energy equation, we find:
u
wT wT v wx wy
W w x wT y 2 dW w wT y P wx 2P dx wy
D
w 2T wy 2
[3.48]
The same procedure as for Pr o 0 is used. The similarity variable K y gx is introduced and the function T f K is considered. The necessary condition for the existence of similarity solutions is reduced to:
W w dg 3 3 3 dW w g P dx 2P dx
D
[3.49]
The differential equation in f becomes: 3 f c K 2 f c 0
[3.50]
74
Convective Heat Transfer
The solution of [3.50] is 1 T K
where K
§ z 3 · K 0.5384 ³ exp¨¨ ¸¸dz 0 © 9 ¹
[3.51]
§ x ·1/ 3 2 yW 1w/ 2 ¨DP ³ W 1/ dx . ¸ w © 0 ¹
Consequently, the Nusselt number for Pr o f is:
Nu
§ x ·1/ 3 1/2 1/ 2 0.5384W w ¨DP ³ W w dx ¸ © 0 ¹
x
[3.52]
According to Blasius solution
Ww Uuf2
1/ 2
0.332Re x
on a flat plate. Replacing this expression in [3.52], we find:
Nu
2 0.3387Pr1/ 3 Re1/ x
[3.53]
This relation is valid for a flat plate in the limit Pr o f . 3.6. Protection of a wall by a film of insulating material 3.6.1. Description of the problem A flat plate is surrounded by a flow of hot gas parallel to the wall. This arrangement simulates the problem of turbine blades, which must be maintained at a sufficiently low temperature to avoid the destructive effects of very hot fluid in gas turbines. In order to maintain the plate at an acceptable temperature, several channels have been machined in the solid wall. Water passes through these channels to evacuate the heat released by the hot gas to the plate (Figure 3.9). It is proposed that we calculate the total heat transfer rate between the wall and the external gas in the two following situations: – when the wall is in direct contact with gas; – when a film of insulating material is placed on the wall surface.
Forced Laminar Boundary Layer Flows
75
Hypotheses: – the physical properties of the gas are considered as constant; – the velocity and temperature fields are considered as twodimensional and steady; – dissipation and compressibility effects are ignored; – the plate is made of a highly heatconductive metal so that the solid temperature may be considered as uniform and equal to T0 ; – the gas flow is laminar near the wall. Notations and data: – solid temperature: – gas temperature in the free stream: – uniform gas velocity of the free stream:
T0 = 50°C Tf = 150°C uf = 30 m s1
The plate dimensions are: – length: L = 0.1 m – width: Lz = 0.1 m
Figure 3.9. Insulation of a wall. Direct contact between gas and wall
The physical properties of the fluids are given in Table 3.5.
76
Convective Heat Transfer
3.6.2. Guidelines 3.6.2.1. Direct contact between gas and wall Recall the heat transfer law to be used under the problem conditions. Calculate the heat flux in ydirection as measured by a fluxmeter that is 5 cm from the leading edge of the plate. Calculate the total heat transfer rate q0 released by the gas to the solid. Determine the water flow rate necessary to maintain a temperature variation of 0.2°C between the channel’s inlet/outlet. The cooling of the plate is provided by 5 cylindrical channels of diameter D = 5 mm. Show that the flow in the channels is turbulent. The corresponding Nusselt number is 53. It is assumed that the heat flux is uniform on the channel’s walls. Calculate the difference between the wall and water bulk temperature at the channel’s outlet. Gas
Water
Density
Uf = 0.946 kg m3
Ul = 988 kg m3
Kinematic viscosity
Qf = 0.230 104 m2 s1 Ql = 5.5 107 m2 s1
Specific heat at constant pressure Cpf = 103 J kg1 K1
Cpl = 4.05 x 103 J kg1 K1
Prf = 0.7
Prandtl number Thermal conductivity
Prl = 3.57 1
1
kf = 0.032 W m K
kl = 0.64 W m1 K1
Table 3.5. Physical properties of the fluids
3.6.2.2. Insulating film on the plate A film of insulating material (thermal conductivity kins = 0.1 W m1 K1) is placed on the wall surface (Figure 3.10). The film thickness e is uniform and sufficiently small to make it possible to ignore the heat conduction in the xdirection relative to heat conduction in the ydirection. The insulating film temperature on the gas side is denoted Tw x . The solid temperature is assumed to be uniform and maintained at the same value T0 as in section 3.6.2.1. It is assumed that the heat flux qccx at the film surface may be calculated by using the heat transfer law valid for laminar forced convection at uniform wall temperature. Express qccx as a function of the problem variables.
Forced Laminar Boundary Layer Flows
77
Express the conductive heat flux through the insulating film and write the continuity condition for the heat flux at the gas/solid interface. Calculate the temperature distribution Tw x . Show that the normalized temperature 4x
Tf Tw x
depends on a Tf T0 dimensionless number (Br, Brun number) in order to be interpreted. Study the variation of 4 (x L) or 4 ( x *) as a function of x/L or of a new dimensionless variable x * built with x/L and the Brun number. Calculate the temperature of the wall/gas interface at the downstream end of the plate for the numerical values given above and for e = 5 mm. Calculate the overall rate of heat transfer q supplied by the gas to the solid. Study the variation of q q0 as a function of the Brun number. What are the asymptotic limits for q q0 when Br > 1? Calculate q q0 for the numerical values given above.
Figure 3.10. Insulating film on the plate
3.6.3. Solution 3.6.3.1. Direct contact between gas and wall The Reynolds number is first calculated at the downstream end of the plate, using the data of the problem, as: Re x
UL
30 x 0.1
Q
0.23 x 10 4
= 1.3 x 10 5
78
Convective Heat Transfer
The laminarturbulent transition is taking place for Re x ~ 3 x 10 5 in a boundary layer with zero pressure gradient; the gas flow is laminar along the plate. 3.6.3.1.1. Heat flux Heat transfer between the wall and the gas flow is governed by the law relative to laminar boundary layers with zero pressure gradient at uniform wall temperature [3.24]: 0.332 Pr f 1 3 Re x1 2
Nu x
[3.54]
In the section of the fluxmeter (x = 0.05 m), we find uf x
Re x
Qf
30 x 0.05 0.23 x 10 4
= 6.52 x 10 4
so that Nu x = 75.3. The local heattransfer coefficient is deduced from [3.17] h
Nu x
kf x
= 75.3 x
0.032 0.05
and, according to [3.16], qccx
= 48.2 W m2 K1.
0.05m = 48.2 x (15050) = 4818 W m2
3.6.3.1.2. Total heat transfer rate on the plate The total heat transfer rate q0 received by the plate is obtained from equations [3.28] and [3.30] as: 0.664 Re L = 0.664 1.3 x 10 5 = 213
NuL
The average heattransfer coefficient is calculated by: h
q0
Nu
kf L
= 213 x
0.032 0.1
= 68.1 W m2 K1
hLLz Tf T0 = 68.1 x 0.1 x 0.1 x (15050) = 68.1 W
Forced Laminar Boundary Layer Flows
79
3.6.3.1.3. Increase in the cooling water temperature The heat budget is written between the channel’s inlet/outlet. We assume that the total heat transfer rate received by the plate surface is evacuated by the cooling water (there are no heat losses)
q0
Ul C pl QToutlet Tinlet
[3.55]
where Q, Tinlet and Toutlet denote the water flow rate and the temperature at the channel’s inlet/outlet respectively. The water flow rate necessary to maintain a temperature variation of 0.2°C between the channel’s inlet/outlet is deduced from the above equation as: Q
q0
Ul C pl Toutlet Tinlet
68,1
=
3
= 8.5 x 105 m3 s1
988 x 4.05 x 10 x 0.2
where Q = 5.1 l min1. 3.6.3.1.4. Temperature variations in a crosssection of the cooling channels The flow rate in a cooling channel is Q 5 = 1.7 x 105 m3 s1 and the bulk Q 5 velocity is U m = 0.86 m s1. The Reynolds number characteristic of the SD 2 4 flow in a channel is: Re
Um D
Ql
=
0.86 x 0.005 5.5 x 10 7
= 7820.
The flow is therefore turbulent in the channels (Chapter 7). With Nuchannels = 53, k 0.64 the heattransfer coefficient is h Nuchannels l = 53 x = 6784 Wm2 K1. D 0.005 Assuming that the heat flux is uniform on the channels surface, it is given by: cc qchannels
q0
5 SDL
= 8658 W m2
80
Convective Heat Transfer
At the outlet, where the thermal regime is supposed to be fully developed ( Nuchannels = 53), the difference between the wall and water bulk temperatures is given by: Tw Tm
cc qchannels h
=
8658
= 1.28°C
6784
Since this temperature difference is weak, it is justifiable to ignore the coupling phenomenon between convection in the channels and conduction in the solid plate. 3.6.3.2. Film of insulating material on the wall surface 3.6.3.2.1. Temperature of the wall/fluid interface An approximate solution consists of assuming that the convective heat transfer on the wall surface is governed by the law relative to a laminar boundary layer on a plate heated at uniform temperature. The convective heat flux is then deduced from [3.24] as: cc ( x ) qconv
0.332 Pr f 1 3 Re x1 2
kf x
>Tf Tw ( x )@
[3.56]
It is convenient to introduce the reference heat flux qLcc
0.332 Pr f 1 3 Re L1 2
kf L
Tf T0
hL Tf T0
where hL is the heattransfer coefficient for x = L ( hL
h 2 , for an isothermal wall, Tf Tw x according to [3.30]). Introducing the dimensionless temperature 4x , Tf T0 equation [3.56] becomes cc ( x ) qconv
qLcc x L
4( x )
h( x )4( x )Tf T0
[3.57]
where h(x) is the heattransfer coefficient at abscissa x. The conductive heat flux across the insulating film is calculated with the Fourier law. Assuming that the heat flux is mainly in the perpendicular direction to the film, it may simply be written as cc ( x ) qcond
kins
>Tw ( x ) T0 @ e
[3.58]
Forced Laminar Boundary Layer Flows
cc or, denoting qins cc ( x ) qcond
kins
81
Tf T0 , e
cc >1 4( x )@ qins
[3.59]
The heat flux must be continuous at the gas/wall interface: cc ( x ) qconv
cc ( x ) qcond
qcc( x )
Using equations [3.57] and [3.59], the interface temperature distribution is obtained as 4 (x)
1
[3.60]
1 Br(x)
where the dimensionless Brun2 number has been introduced Br ( x )
qcLc
[3.61]
cc x L qins
When the Brun number Br(x) is written in the form 1 2
Br(x)
0.332Pr1 3 Re L1 2 k f Lx L kins e
h(x) kins e
it is clearly seen that it represents the ratio of the convective to the conductive thermal resistance for heat transfer between the fluid and the solid. The wall temperature variations are obtained as a function of x, or in a condensed qLcc hL x L form, by denoting BrL and x * . cc qins kins e BrL 2 4 ( x *)
x*
1 x*
[3.62]
The above calculation is, however, a rather crude approximation since it uses the heat transfer law valid for uniform wall temperature whereas the resulting wall temperature is a function of x. [LUI 74] and [LIM 92] have performed more accurate 2. This is also a Biot number.
82
Convective Heat Transfer
computations by taking the coupling between convection and conduction into account. Lim and coauthors [LIM 92] have investigated the problem of optimizing insulation with a film of variable thickness along the plate. Equation [3.62] shows that the gas/wall interface temperature Tw is equal to Tf ( 4 0 ) for x = 0. In this initial region, the convective thermal resistance is negligible relative to the conductive thermal resistance of the insulating film. The solid is only protected by the film, which prevents excessive heat flux. While the boundary layer develops along the plate, the corresponding thermal resistance increases and the surface temperature Tw decreases. If the value reached by x * at the end of the plate is sufficiently high, the surface temperature becomes close to T 0 . This means that the heat flux between the gas and the plate becomes very weak and that the temperature drop is concentrated in the boundary layer whose thermal resistance is predominant. With the data of the problem, BrL
hL e
34 x 5 x 10 3
kins
0.1
= 1.7. At the
downstream end of the plate, x * 1 BrL 2 1 1.7 2 = 0.346. According to [3.62], 4 ( x L) = 0.37. The surface temperature is 113 °C. 3.6.3.2.2. Total heat transfer rate The total heat transfer rate q released by the gas to the solid is obtained by integrating one of the two equations [3.57] or [3.59]. For example, [3.57] and [
[email protected] give the following law for the heat flux qcc( x )
qc Lc
1
[3.63]
BrL 1 x *
and, after transformation of variables, the total rate of heat transfer is calculated by q
1 qLccL z LBrL ³0
BrL2
It is easily shown that: ³
1 1 x
*
dx *
1 1 x*
dx *
[3.64]
ª § ·º 2« x * Ln¨1 x * ¸» © ¹¼ ¬
Forced Laminar Boundary Layer Flows
83
The total rate of heat transfer is: q
ª § 1 ·º 2qc ¸¸» LcL z L«1 BrL Ln¨¨1 «¬ © BrL ¹»¼
[3.65]
The heat transfer rate exchanged in the case of direct contact is obtained for e = 0, namely for BrL o 0 , q0 2qcLcLz L . Finally, the total heat transfer rate may be written in the form: q q0
ª § 1 ·º «1 BrL Ln¨¨1 ¸¸» «¬ © BrL ¹»¼
[3.66]
This model provides the asymptotic trends: BrL 1
BrL !! 1
q q0 q q0
 >1 BrL LnBrL @ 
1 2BrL
[3.67]
[3.68]
with BrL = 1.7, q q0 = 0.21. The total heat transfer rate received by the plate is reduced by about 80% due to the insulating film. 3.7. Cooling of a moving sheet 3.7.1. Description of the problem The problem of cooling a solid strip of metal moving in surrounding fluid is present in several industrial processes. We assume that the fluid is at rest far from the moving strip, so that cooling is due to convective heat transfer in the boundary layer generated on the strip surface. Some other cooling processes are clearly more efficient (impacting jets, for example), but are not investigated in this problem. In order to determine the heattransfer coefficient for the fluid/solid strip exchange, let us consider the twodimensional problem of a strip (Figure 3.11) at uniform temperature Tw , moving at constant velocity U in a constantproperty fluid (kinematic viscosity Q, density U, thermal conductivity k). Buoyancy and dissipation
84
Convective Heat Transfer
effects are assumed to be negligible. The flow is in the laminar regime. This problem is adapted from a paper of A. M. Jacobi >JAC
[email protected] The surrounding fluid temperature is denoted Tf . The coordinates in the moving sheet direction and in the perpendicular direction are denoted x, y, respectively. The corresponding fluid velocity components are u and v. The local Reynolds number is defined by Rex Ux Q . The velocity U is assumed to be sufficiently high so that the flow entrained by the strip is of the boundary layer type ( Rex !! 1 ). The velocity and temperature boundary layer thicknesses are denoted Gx and GT x , respectively. We propose to determine the heat transfer law for the cooling of the sheet and to specify the influence of the Prandtl number on this law.
Figure 3.11. Sketch of a moving strip
3.7.2. Guidelines Show that the strip motion gives rise to a stream perpendicular to the wall, characterized by the velocity vf x far from the strip. Use scale analysis to link this velocity to the abscissa x, the strip velocity U and the fluid kinematic viscosity. For a fluid at Pr > 1. Determine the distribution of vf x by using the integral method (modeling the longitudinal velocity profile by a seconddegree polynomial is suggested). For a fluid at Pr > 1, consider the energy equation written with the similarity variables [3.19]. Propose an approximation of the function F used in this latter equation. Solve the energy equation and determine the Nusselt number (it is recalled 2 K m 2 that erf (K) ³ 0 e dm ; erf (f) 1 ).
S
Compare the results to Figure 3.12. The Churchill and Usagi >CHU
[email protected] method is used to obtain a correlation at intermediate Prandtl numbers. Let Nu =
b RePr 1n n ½ § ° b RePr · ° ¸¸ ¾ ®1 + ¨¨ °¯ ©aPr Re ¹ °¿
. Calculate the coefficients a and b by using the
asymptotic trends previously found for the heat transfer law. The exponent n will be determined by adjusting the value of Nu as given by the above correlation to an experimental point on Figure 3.12. It is suggested that a point whose abscissa is close to that of the intersection point of the asymptotes is chosen. 10.00 Nu Re Pr >> 1 1.00 Experimental results reported by Jacobi (1993) 0.10
Pr JAC
[email protected] Pr
100.00
86
Convective Heat Transfer
3.7.3. Solution 3.7.3.1. Description of the flow The strip motion gives rise to a driving force, which entrains the surrounding fluid in the longitudinal direction in its vicinity. This entrainment of fluid near the strip induces, in turn, a fluid motion in the transverse direction towards the strip (Figure 3.13). This transverse motion is opposite to that of a classical boundary layer. In the classic boundary layer situation, the fluid moves away from the wall, i.e. the streamlines are slightly diverging, contrary to the present case.
Figure 3.13. Moving sheet. Sketch of the flow
3.7.3.2. Estimation of the velocity vf x Scale analysis is used to estimate the velocity field. The analysis of equation [3.2] shows that inertia and viscous terms have the order of magnitude U 2 x and
QU G 2 respectively. The classical order of magnitude of the boundary layer thickness is recovered:
G
Qx
[3.69]
U
The continuity equation gives the order of magnitude of the transverse velocity (velocity scale V) in the velocity boundary layer V
G

U x
V U
G x

QU
[3.70]
x
or in dimensionless form V U

1 Re x
with Re x
Ux
Q
[3.71]
Forced Laminar Boundary Layer Flows
87
This is also the order of magnitude of the transverse velocity at infinity vf x . Results [3.70] and [3.71] must be used with the absolute value of the transverse velocity. The classical order of magnitude of the transverse velocity is recovered; the velocity v is, however, negative in the present problem. 3.7.3.3. Scale analysis of the thermal boundary layer 3.7.3.3.1. Pr 1, conversely to the previous case, it is expected that the velocity boundary layer is much thicker than the thermal boundary layer and, consequently, that velocities are much smaller than vf in the present case. In fact, the continuity equation reads:
wu wv wx wy
0
The longitudinal velocity is constant on the strip wall so that wu wx y therefore, wv wy y
0
0 and,
0
0
A development in series of the velocity v near the wall gives: v y
v 0
wv w 2v y 2 wy 0 wy
The first two terms are zero;
y2 0
2
w 2v wy
.....
is estimated by
2 0
vf
G2
. The transverse velocity
is therefore estimated in the thermal boundary layer by:3 vGT 
Let V1
UG GT 2 x G2
[3.74]
vGT .
Equation [3.74] shows that the order of magnitude of the transverse velocity V1 is much smaller than vf ([3.70]) when GT G , as expected for Pr >> 1. 3. For sake of consistency, this estimation is written without numerical coefficients.
Forced Laminar Boundary Layer Flows
89
Taking this result into account, the different terms of the energy equation are estimated by: u

wT wx
U4

x
v
wT wy
D
U4 GT
G
x
D
w 2T wy 2 4
GT 2
The dominant term of the lefthand side is the first one, which represents advection in the longitudinal direction. The thermal boundary layer thickness is given by
GT 2 
Dx
[3.75]
U
and the ratio of the boundary layer thicknesses is r
G  Pr1 2 GT
[3.76]
The Nusselt number is estimated as in the previous case and is found as:
Nu  Re1 2 Pr 1 2
[3.77]
3.7.3.4. Calculation of the transverse velocity at infinity with the integral method The conservation of flow rate is written for a slice dV of length dx, height H and unit depth in the spanwise direction (Figure 3.13) d ³0 u( x , y)dy vf ( x )dx H
0
from which, we infer the transverse velocity at infinity vf (x)
d dx
³ u(x,y)dy H
[3.78]
The velocity profile is modeled by using a seconddegree polynomial, u (x, y) U
f (K)
a 2K 2 a1K a 0 with K
y
G (x)
90
Convective Heat Transfer
The coefficients of this polynomial are determined with the following conditions: f (0)
1, f (1)
f c(1)
0
The result f (K) K 1 transverse velocity at infinity:
2
vf ( x )
is reported in equation [3.78], which gives the
1 dG ( x) U 3 dx
[3.79]
The integral method is also used to calculate G (x) . The budget of momentum is written for the control volume dV. In this flow with zero pressure gradient, it reduces to: §wu · P ¨ ¸ dx ©wy ¹0
d ³ 0H Uu 2 ( x, y ) dy
This budget results from the variation of momentum between the crosssections of abscissas x and x + dx and from the shear stress exerted by the wall on the flow. Introducing the dimensionless velocity profile and simplifying by U, the momentum equation becomes U2
d dx
G ( x ) ³ 01 f (K) 2 dK Q
U
G ( x)
f c(0)
since the velocity U is independent of x. Replacing f by its expression and integrating in x, we find that:
G ( x)
20
x
Rex
Reporting in [3.79], the transverse velocity at infinity is finally obtained as: vf ( x ) U
5
1
3
Re x
0.745 Re x
[3.80]
The result of scale analysis ([3.71]) is recovered, as expected; in addition, the integral method has given the numerical coefficient in the relation between vf ( x ) and 1
Rex .
Forced Laminar Boundary Layer Flows
91
3.7.3.5. Determination of the heat transfer law for a fluid at Pr > 1 Considering the hydraulic and thermal conditions of the problem, it is possible to find a similarity solution of form [3.18] to energy equation [3.19], written with m = n = 0:
Tc
1 2
PrFTc 0
92
Convective Heat Transfer
For fluids at Pr >> 1, the longitudinal component of the fluid velocity may be approximated by a constant in the thermal boundary layer ( u  U ) or, according to [3.10], F c 1. Integrating and using the boundary condition F (0) 0 (the wall is a particular streamline), we find F (K) K . The equation to be solved becomes:
Tc
1 2
PrKTc 0
The boundary conditions are T (0) 1, T (f) 0 . The equation is integrated by means of two successive quadratures. Taking the boundary conditions into account, the solution reads
T K 1 Tc0
S Pr
erf K
and, considering the boundary condition, we find Tc0
Pr
S
.
The Nusselt number is deduced from the above relation by using equation [3.20]: Nu x
1
S
Re x1 2 Pr1 2
0.564 Re x1 2 Pr1 2
[3.83]
3.7.3.7. Comparison with Jacobi’s results and heat transfer law for intermediate Pr Jacobi’s results [JAC 93] show that: – for Pr > 1, Nu x
0.545 Re x1 2 Pr1 2
[3.85]
The results as given by correlations [3.82] and [3.83] are respectively slightly lower, within 7.7%, and higher, within 3.7%, than those of Jacobi. The coefficients a and b of the law Nu =
b RePr 1n n ½ § ° b RePr · ° ¸¸ ¾ ®1 + ¨¨ °¯ ©aPr Re ¹ °¿
[3.86]
are obtained by using the asymptotic trends Pr > 1. Using Jacobi’s results, we find a = 0.807, b = 0.545.
Forced Laminar Boundary Layer Flows
93
The Churchill and Usagi >CHU
[email protected] method consists of determining the exponent n by adjusting a value of correlation [3.86] to an experimental result. The exponent n is easily calculated by choosing the Prandtl number corresponding to the abscissa of the intersection point of the two asymptotes. Figure 3.12 shows that no experimental point exactly corresponds to this abscissa. The method is, however, applied by using the closest point of coordinates Pr = 0.674, Nu Re 0.327 . The result is n = 1.72. The law [3.86] is plotted in Figure 3.15 for a = 0.807, b = 0.545, n = 1.72. 10.00 Nu Re Pr >> 1 1.00 Experimental results reported by Jacobi (1993) Eq. 3.86 0.10
Pr T r, z Tf @ >Tw r Tf @ if the wall temperature follows a law of the form Tw r Tf GT is independent of r in this solution.
Hr n . Show that the thickness
The case of an isothermal disk is considered in the following discussion. Show that advection in the radial direction then does not play any role in heat transfer. As a consequence, simplify further the local energy equation. Interpret the two sides of the resulting equation. How is the heat flux distributed on the disk surface? Apply scale analysis to this problem. Consider the extreme cases Pr > 1 . In the latter case, take the continuity equation into account to show that wu z wz z 0 0 and estimate the order of magnitude of u z in the thermal boundary layer. Compare the resulting Nusselt number to that given by [KRE 68], as shown in Figure 3.17. Conduct calculations by using the integral method. Integrate the local energy equation across the boundary layer from the disk wall up to a distance H sufficiently large to recover the farfield conditions (temperature Tf and velocity u zf ). Consider successively the two cases Pr > 1 to model the velocity component u z in the thermal boundary layer. Compare the results to those shown in Figure 3.17. Determine the energy budget for a cylinder of axis Oz, radius R and height H and check the results obtained previously.
96
Convective Heat Transfer
For experimental results on this problem, [HAR 98] and quoted references are suggested readings. 10.00
1.00
0.10
0.01 0.01
0.10
1.00
10.00
100.00 Pr
Figure 3.17. Nusselt number on an isothermal rotating disk [KRE 68]
3.8.3. Solution 3.8.3.1. Velocity field Differentiating the stream function with respect to z and r successively, the velocity components are obtained as: ur
1 w\ r wz
=
1 r
\ 0 r
ur
ZrF cK
uz
uz
2 QZ F K
F cK
G [3.89]
1 w\
1 \c0 r F K r wr = r
[3.90]
The radial velocity component is proportional to r, whereas the vertical velocity component is independent of r. As a matter of interest, we note that the tangential or
Forced Laminar Boundary Layer Flows
97
azimuthal velocity component is obviously also proportional to r. The three velocity components depend on z through the function F K . 3.8.3.2. Similarity solution for the thermal field Assuming that the temperature field is axisymmetric, advection does not play any role in the azimuthal direction. The energy equation is written with cylindrical coordinates and simplifies into: ur
wT wT uz wr wz
ª1 w § wT ¨r « ¬r wr © wr
D«
· w 2T º ¸ 2 » ¹ wz ¼ »
The first term in the bracket represents radial diffusion and is of the order of 4 R 2 with 4 Tw Tf . The second term represents diffusion in the normal direction and is of the order of 4 GT 2 . Since it was assumed that GT R , the energy equation is governed by the second term and further simplifies into: ur
wT wT uz wr wz
D
w 2T
[3.91]
wz 2
We try a solution of the form:
T K
T r, z Tf Tw r Tf
with Tw r Tf
Hr n
Introducing expressions [3.89] and [3.90] and the transformation of variables defined above for the temperature, equation [3.91] becomes:
ZrF cK nHr n1T K 2 QZ F K Hr n
Z Z TcK DHr n TccK Q Q
Simplifying by Hr n , it is found that T satisfies the secondorder ordinary differential equation:
TcK Pr >2F K TcK nF cK T K @ 0 The boundary conditions are:
T 0 1 , T f 0
[3.92]
98
Convective Heat Transfer
It can be concluded that equation [3.92] has, in effect, a similarity solution since the variable r has been eliminated in this equation. According to the form of the dimensionless temperature T K , the thermal boundary layer thickness is independent of r, like G. 3.8.3.3. Uniform disk temperature When the disk is isothermal, n = 0 in [3.92]. The energy equation further simplifies into:
TcK 2PrF K TcK 0
[3.93]
The thermal field is independent of r and advection in the radial direction does not play any role in heat transfer. Equivalently, equation [3.91] simplifies into: uz
wT wz
D
w 2T
[3.94]
wz 2
Thermal equilibrium is maintained near the disk by the fluid motion in the zdirection, which is opposite to thermal diffusion in the same direction. The temperature gradient
wT wz
T p Tf z 0
G
Tc0 is independent of r, as well as
the heat flux. Note that Tc0 is a function of the Prandtl number, since Pr is present in [3.93]. 3.8.3.3.1. Scale analysis The thermal boundary layer is characterized by its thickness GT . The temperature gradient wT wz is estimated by 4 GT , as it is in the general case. The Nusselt qcc wT number, defined by Nu with qcc k , is estimated by: wz z 0 k Tw Tf G Nu 
G GT
[3.95]
Pr 1 It is expected that GT G in this case. It is necessary to examine the behavior of the normal velocity component in the thermal boundary layer, i.e. very near the wall. This is the same situation as in section 3.7.3.3.2, so the same estimation of the velocity component perpendicular to the wall may be made.
100
Convective Heat Transfer
The role of the corresponding fluid motion in heat transfer is, however, different. In the moving sheet problem, heat transfer was governed by longitudinal advection because the velocity and thermal boundary layer thicknesses varied with x. In the present situation, these thicknesses are independent of x, so radial advection does not play any role in heat transfer. Only forced convection in direction z contributes to heat transfer. In the present axisymmetric flow, the continuity equation reads: 1 w r wr
ru r
wu z wz
0
On the disk, the radial velocity component is zero, whatever the distance r, 0 and therefore owing to the noslip condition. As a result, 1 r w wr ru r
wu z wz z
z 0
0
0.
A development in series of the velocity component u z near the disk wall gives: 2
wuz w u z2 uz z uz 0 z 2z ... wz 0 w z 0 2 The first two terms are zero; w 2 u z wz 2 is estimated by u zf G 2 . The velocity 0
is estimated in the thermal boundary layer by setting z GT in the above development. The numerical coefficient 1/2 is ignored in this order of magnitude calculation. uz GT 
uzf
G2
GT 2  QZ
GT 2 G2
The ratio of the velocities uz GT and u zf is proportional to GT 2 G 2 (> GT ): H
³ 0H
ªw 2T º wT uz dz D« 2 » wz «wz » ¬ ¼0
D
wT wz
[3.100] 0
Pr > 1 The calculation of the normal velocity component near the wall performed above by scale analysis suggests the model for u z z to be u z z
w 2uz wz 2
z2 0
2
.
Taking [3.90] into account, we obtain u zf F ccK
w 2uz wz so that
with b
G 2 F f
2
uz
b
u zf
F c0 F f
K2
[3.103]
2
=
0.51
= 1.15.
0.443
The temperature profile is modeled, by using [3.36], as: T(z) Tf Tw Tf
§3 ] 3 · ¸ with ] T (] ) 1 ¨ ] ¨2 2 ¸ © ¹
z
GT
.
Equation [3.100] becomes: § K 2 ·Tw Tf 3 2 T T H ¸ u b ] 1 dz D w f T0c ³0 ¨ z ¨ f ¸ ©
2 ¹
GT
2
] is kept as integration variable and we set Y is then: u zf b
Y3 2
G ³ 01 ] 2 ] 2 1 d]
D
GT
GT G . The equation to be solved
Forced Laminar Boundary Layer Flows
103
The value of the integral is  2/15. The boundary layer thickness ratio is obtained by: § ·1 3 1 15 2.45 GT G = ¨ = ¸ ¨ ¸ 13 Pr1 3 ©2 x 1.15 x 0.443 ¹ Pr
Y
The heat flux exchanged between the fluid and the wall is given by: qcc k
wT wz
z 0
3 T T T Tf k w T0c = k w f GT GT 2
Using its definition, the Nusselt number is given by: Nu
qcc
3
k Tw Tf G
2Y
Nu
0.612 Pr1 3
[3.104]
These results are logically in perfect agreement with those of scale analysis since the physical ingredients are the same in both approaches. Figure 3.19 also shows an excellent agreement with the results of [KRE 68] for the asymptotic trends Pr > 1. 10.00
1.00
0.10
0.01 0.01
0.10
1.00
10.00
Pr
100.00
Figure 3.19. Nusselt number on a rotating disk. Results of integral method. Solid line: results of [KRE 68]. Dashed line: equation [3.102], dotted line: equation [3.104]
104
Convective Heat Transfer
3.8.3.3.3. Heat budget for a cylinder of Ozaxis, radius R and height H Let us consider a control domain consisting of a cylinder of Ozaxis, radius R and height H ( ! MaxG, GT ) with its bottom on the rotating disk. The fluid flows into the cylinder through its top with uniform normal velocity u zf and uniform temperature Tf . It flows radially outwards through the cylinder lateral surface Slateral with the velocity distribution ur R, z , which is independent of the azimuthal direction. The heat flux supplied by the disk to the fluid is uniform since the temperature profiles are independent of r. The heat budget of the control domain yields: qccSR 2
GG
³ UC pTu.ndS
Scylinder
qccSR 2
UC p ¨uzf TfSR 2 ³ u.nTdS ¸ ¨ ¸
§ ©
GG
Slateral
· ¹
[3.105]
Pr > 1
The thermal boundary layer is then much thinner than the velocity boundary layer ( GT G ). Heat budget [3.105] is written by taking the conservation of flow rate into account, or equivalently, by taking the temperature Tf as reference:6 qccSR 2
GG
UC p ³ u.nT Tf dS Slateral
UC p ³0H 2SRur ( R, z )T z Tf dz
In order to calculate the integral, it is possible to replace the radial velocity profile by its tangent at the origin in the thermal boundary layer (beyond GT , the 5. The positive sign for u z corresponds to Ozdirection. f 6. The same calculation may be carried out with a cylinder of radius r.
106
Convective Heat Transfer
contribution to the energy flow rate is zero and GT G ), so that, according to [3.89], the approximation is ur R, z ZRF c0 z G . The temperature profile is modeled, as it is in section 3.8.3.3.2, by: T ( z ) Tf Tw Tf
§3 ] 3 · T (] ) 1 ¨¨ ] ¸¸ with ] 2 ¹ ©2
z
GT
Substituting into the energy budget equation, we find
UC p 2SZR 2 F0cTw Tf ³0GT or, setting Y
T Tf z ª §3 ] 3 ·º «1 ¨ ] ¸»dz k w T0cSR 2 ¨ ¸ » 2 G « 2 G ¹¼ T ¬ ©
GT G and simplifying the equation
ª § 3 ] 3 ·º» ¸ d] 2ZF0cY 3G 2 ³ 01] «1 ¨¨ ] «¬ ©2 2 ¸¹»¼
DTc0
The value of the integral is 1/10. F0c 0.510
T0c 3 2 . We find Y
2.45 Pr1 3
and result [3.104] is obtained. 3.9. Thermal loss in a duct 3.9.1. Description of the problem Oil is flowing with bulk velocity U and temperature T0 in fully developed laminar regime in a channel of spacing e delimited by two parallel plates. The length L z of the plates in the spanwise direction is sufficiently long so that the flow is considered as twodimensional. The walls are perfectly insulated, except on a small surface Sl of dimensions l x L z on one of the walls (Figure 3.22). It is assumed that the local wall temperature is uniform and equal to Tw0 on Sl . It is proposed that the total heat transfer rate lost by oil on Sl and the wall temperature distribution Tw x in the region downstream from the heat sink are calculated. Data: e = 10 cm, l = 10 cm, L z = 1 m, U cm s1 T0 = 80°C, Tw0 = 20°C. The physical properties of oil are given in Table 3.6.
Forced Laminar Boundary Layer Flows Density
U = 850 kg m3
Kinematic viscosity
Q = 0.4 104 m2 s1
Specific heat at constant pressure
Cp = 2 103 J kg1 K1
Thermal conductivity
k = 0.14 W m1 K1
Thermal diffusivity
D=0.8 107 m2 s1
107
Table 3.6. Physical properties of oil
Figure 3.22. Heat sink in a parallelplate channel
3.9.2. Guidelines Determine the velocity profile. It is suggested that the velocity profile be replaced by its tangent at the origin in the thermal boundary layer that develops along the region of heat loss (Lévêque7 solution). Solve the energy equation by trying a solution of the form T ( x , y ) Tw0 T0 Tw0
with ]
y
GT ( x )
T (] )
.
Calculate the thermal boundary layer thickness and the wall heat flux at x = l. Calculate the overall heat transfer rate lost by the fluid through Sl . Calculate the fluid bulk temperature for x = l. 7. André Marcel Lévêque, French engineer, 1896–1930.
108
Convective Heat Transfer
Propose a method to calculate Tw x for x > l. It is suggested that two functions be superposed in order to satisfy the condition of zero wall heat flux for x > l. 3.9.3. Solution 3.9.3.1. Velocity distribution in the channel In a hydraulically fully developed regime, the solution to NavierStokes equations gives the following velocity profile (see Chapter 2, equation [2.13]) u (K) U
6K1 K
[3.106]
with K y e . Note that the origin of ordinates y has been chosen in the plane containing Sl and that y is normalized by the overall channel spacing. The velocity profile is replaced by its tangent at the origin: u (K) U
6K
6
y e
3.9.3.2. Lévêque solution to the energy equation A thermal boundary layer is developing on the part of the wall lacking insulating material. The energy equation is written in this region with the usual boundary layer approximations 6U
y wT
D
e wx
w 2T wy 2
or using the transformation of variables ]
y
GT ( x )
,
T ( x , y ) Tw0 T0 Tw0
T (] )
G Gc D 6U T ] 2 t Tc Tcc e GT GT 2 Introducing the Péclet number Pe
x*
x e , GT* x *
Ue D and the dimensionless variables
GT x e , the energy equation becomes:
2 c 6PeGT* GT* ] 2Tc Tcc
[3.107]
Forced Laminar Boundary Layer Flows
109
Separating the functions of x * and the functions of ], we find that: 2
c
6PeGT* GT*
[3.108]
Constant
The constant is arbitrary and for sake of convenience, we choose Constant = 3. Equation [3.108] is easily integrated as § 3 * ·1 3 x ¸ ¨ ©2Pe ¹
[3.109]
where the condition GT* 0 becomes
0 has been accounted for. Equation [3.107] then
GT* x *
3] 2Tc Tcc
which is integrated by two successive quadratures
with A
[3.110]
[3.111]
T c ]
A exp ] 3
T ]
A³0 exp z 3 dz
]
1
f ³0 exp
z 3 dz
1,12 .
Equations [3.109] and [3.111] constitute the Lévêque solution of the problem of a thermal boundary layer developing in a plane channel in a hydraulically fully developed flow. The temperature profile is plotted in Figure 3.23. 3.9.3.3. Thermal boundary layer thickness Equation [3.109] gives the characteristic thermal boundary layer thickness:
GT* l *
Pe
§ 3 * ·1 3 l ¸ with l * ¨ ©2Pe ¹
0.01 x 0,1 0.8 x 10 7
12500 l *
l e 1
110
Convective Heat Transfer
GT* l *
0.0493 GT l
0.0049 m = 4.9 mm
Figure 3.23 shows that the oil temperature T0 is reached for ]  1.5 ( T  1). The thermal boundary layer thickness may be evaluated graphically as 1.5GT 7.4 mm. More precisely, the thickness GT 0.99 is defined as the distance to the wall corresponding to T 0.99 . Using the numerical solution to equation [3.111], we find ] 1.4 , which leads to GT 0.99 6.9 mm. The relative difference between the actual velocity profile and its tangent at the origin is evaluated by using equation [3.106]. For this distance to the wall, it is found that GT 0.99 e 0.069 , which is a moderate value and justifies the approximation used in the calculations. 3.9.3.4. Wall heat flux ccx at abscissa x is obtained by using equations [3.109] and The wall heat flux qw [3.110]. Denoting 4 T0 Tw0 , the heat flux is given by: 1.2
1.0
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Figure 3.23. Temperature profile in a twodimensional boundary layer. Hydraulically fully developed regime in a duct. Lévêque solution
cc(x) qw
§wT x, y · k¨ ¸ © wy ¹y
= k 0
Equation [3.110] shows that T 0c
4
GT (x)
Tc0
Tc0
A
[3.112]
1.12 .
Forced Laminar Boundary Layer Flows
111
§ 3 * ·1 3 e¨ x ¸ ©2Pe ¹
According to [3.109], GT x
Introducing the reference heat flux q0cc k 4 e , we obtain: cc(x * ) qw
q0cc q0cc 0.14
§2 ·1 3 §Pe ·1 3 ¨ ¸ A¨ * ¸ ©3 ¹ ©x ¹
60
§Pe · 0.978¨ * ¸ ©x ¹
13
84 W m2 with x *
0.1
l*
[3.113]
cc(l) = 1906 W m2 1, qw
The overall heat transfer rate q(x) L z lost by the fluid on the length x per cc(x) over x transverse unit is obtained by integrating the local heat flux qw *
³0 qcc(x1 )dx1
q(x) L z
§x · 0.978k4Pe1 3 ¨ ¸ ©e ¹ 2
x
x q0cce ³ 0 0.978 Pe x1*
q(x) L z
23
3
1/ 3
dx1*
§x · Bk4Pe1 3¨ ¸ ©e ¹
23
[3.114]
with B = 1.467. For x = l, we obtain
q(l) L z
3 2
and finally, q(l) L z
ccl l qw
286 W m1.
3.9.3.5. Fluid bulk temperature in the region downstream from the heat sink The energy budget is written for a control domain delimited by an upstream crosssection, a crosssection of abscissa x (0 < x < l), and the channel walls. Using the fluid bulk temperature Tm(x) in the crosssection of abscissa x, the energy equation yields: q(x) L z
UC pUeT0 Tm x
[3.115]
In the crosssection corresponding to the downstream end of Sl (x = l), we obtain: q(l) L z
UC pUeT0 Tm l
112
Convective Heat Transfer
The energy budget gives T0 Tm l 0.17 °C. The fluid bulk temperature Tm0 Tm l does not vary any more in the downstream region of the channel since the walls are perfectly insulated for x ! l . 3.9.3.6. Distribution of wall temperature downstream from the heat sink Since the walls are perfectly insulated for x ! l , the temperature field must satisfy the condition of zero wallheat flux in this region. As a consequence, the temperature profile is modified and the wall temperature increases for x ! l . It is convenient to represent these changes as resulting from a second boundary layer embedded in the first one and developing from the abscissa x = l (Figure 3.24). The temperature profile for x ! l may then be considered as resulting from the superposition of two profiles corresponding to the two boundary layers CL1 and CL2 that are developing on the wall from the abscissas x = 0 and x = l respectively. 3.9.3.6.1. Superposition of two Lévêque solutions An approximate solution consists of superposing Lévêquetype solutions, as presented in section 3.9.3.2. This is not an exact solution since the wall temperature is not uniform in the region downstream from the heat sink, contrary to the condition prescribed for 0 < x < l in CL1. Solution for CL1: T (x, y) Tw0 1
4
T (]1 ) with ]1
y
[3.116]
GT1 (x)
Solution for CL2: 2 T (x, y) Tw2 (x)
Tw2 (x)
T (] 2 ) with ] 2
y
GT 2 (x)
[3.117]
where the function T is the Lévêque solution, as given by [3.111]. According to [3.109], the Lévêque solution also gives GT1 x changing the origin of x for CL2, GT 2 x
§ 3 x l ·1 3 e¨ ¸ . ©2Pe e ¹
§ 3 x ·1 3 e¨ ¸ or, ©2Pe e ¹
Forced Laminar Boundary Layer Flows
113
The superposition gives:
T (x, y)
1 2 T (x, y) T (x, y)
T(x, y) Tw0
4 T (]1 ) Tw2 (x)1 T (] 2 )
Figure 3.24. Development of temperature profile in the region downstream from the thermal sink
The matching condition for the temperature to T0 is verified since T (f) 1 (Figure 3.23). The wall temperature Tw x is obtained by writing the condition of zero heat flux at the wall for x > l. cc1 (x) qw cc2 (x) qw
0
Using the expression for the heat flux ([3.112]), the condition yields: k
4
GT1 (x)
T 0c k
T p2 (x)
GT 2 (x)
T 0c
0
Thus: Tw2 (x) 4
GT 2 (x) GT1 (x)
§x l ·1 3 ¨ ¸ © x ¹
The dimensionless temperature is therefore T(x, y) Tw0 4
§x l · T (]1 ) ¨ ¸ © x ¹
13
1 T (] 2 )
114
Convective Heat Transfer
and, since T (0)
0 , the corresponding wall temperature is
Tw (x) Tw0 4
§ l ·1 3 ¨1 ¸ © x ¹
[3.118]
The wall temperature grows rapidly and tends to the fluid temperature of the channel central region (Figure 3.25). The initial temperature difference 4 is recovered within about 10% for x/l = 4. 3.9.3.6.2. Integral method It is also possible to apply the principle of superposition in freeing the function
GT 2 x , whereas it had a prescribed form in the previous approach. We then
superpose a Lévêque solution [3.116], corresponding to CL1 and a function T 2 (x, y) corresponding to CL2. For this latter function, it is possible to use the formalism of equation [3.117]; however, the function corresponding to the shape of the temperature profile, denoted T (] ) , has to be modeled. Moreover, the function GT 2 x has to be determined simultaneously with Tw2 (x) . We therefore
must write two equations in two unknown functions. For T (] ) , it is suggested that a polynomial be used, matching to 0 for ] 2 1 .
The conservation of enthalpy flow rate through a crosssection downstream from the heat sink yields: e ³0 UC p u(x, y)>T (x, y) T0 @dy UC pU Tm0 T0 e
[3.119]
Introducing the decomposition T (x, y) T 1 (x, y) T 2 (x, y) into [3.119], the calculation of the integral of the lefthand side may be performed in two parts. According to [3.114] and [3.115], the Lévêque solution gives: e 13§ · ª 1 º ³0 U C p u ( x, y ) «¬T ( x, y ) T0 »¼ dy U C pUe T0 Tm x Bk 4Pe ¨ e ¸
x
23
© ¹
Using a linear approximation of the velocity profile together with equation 2 [3.117] (the shape of the function T (2) will be specified later), the profile T (x, y) gives the following calculation ª º 2 2 e e y ³0 UC p u(x, y)T (x, y)dy UC pU ³0 6 Tw2 (x)«¬1 T ] 2 »¼G T 2 x d] 2
e
6 e
aUC pUTw2 (x)>GT 2 x @
2
Forced Laminar Boundary Layer Flows 1
ª
³0 ] 2 «¬1 T
where a
1
115
2 ] º d] . 2 »¼ 2
Tw x Tw0
0.9
4
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10 x/l
Figure 3.25. Distribution of wall temperature in the region downstream from the heat sink. Solid line: superposition of two Lévêque solutions. Dashed line: combination of a Lévêque solution and integral method
The last term of equation [3.119] is calculated like the first one: 23 1 3§l ·
UC pUeT m 0 T 0 Bk4Pe
¨ ¸ ©e ¹
Finally, equation [3.119] becomes: §GT 2 x ·2 Tw2 x ¨ ¸ 4 © e ¹
ª§ ·2 3 § ·2 x l «¨ ¸ ¨ ¸ ©e ¹ 6a Pe 2 3 «¬© e ¹
B
1
3 º
» »¼
[3.120]
The second equation is obtained by writing that the heat flux is zero at the wall for x > l, which gives, after simplifying by the thermal conductivity k § 1 · § 2 · ¨wT ¸ ¨wT ¸ ¨ w y ¸ ¨ ¸ © ¹y 0 © wy ¹y 0
0
116
Convective Heat Transfer
4
GT1 (x)
or denoting T 0c
Tw2 (x) 4
T 0c
2
Tw2 (x) dT (2)
GT 2 (x) d] 2
0 ]2 0
dT (2) d] 2
]2 0
T 0c GT 2 (x) 2 T 0c GT1 (x)
[3.121]
Substituting [3.121] in [3.120] and using law [3.109] for the thickness GT1 x of the boundary layer CL1 as a function of x, we obtain the thickness of the boundary layer CL2
GT 2 x e
with C
13
§l e ·1 3ª§ x · §x ·1 3 º C¨ ¸ «¨ ¸ ¨ ¸ » ©Pe ¹ «¬© l ¹ ©l ¹ »¼
§ ·1 3 2 B ¸ ¨3 T 0c ¨ ¨2 T 0c 6a ¸ ¸ © ¹
[3.122]
§ 2 ·1 3 ¨T 0c ¸ ¨ ¨ 4a ¸ ¸ © ¹
and the law giving Tw2 (x) Tw2 (x)
ª § ·2 l D«1 ¨ ¸ « ¬ ©x ¹
13 3 º
» » ¼
[3.123]
It is worth recalling that T 0c
1.12 in the Lévêque solution. It is now possible to
4
with D
T 0c C 2 §3 ·1 3 T 0c ¨ ¸ ©2 ¹
specify the shape of the temperature profile T (] ) . Choosing a form similar to
Forced Laminar Boundary Layer Flows
[3.35] ( T 2 (] 2 )
3 2
laws:
GT 2 x e
]2
] 23 2
), we obtain T 0c
2
3 2
, a
1 10
117
, and the following
13
§l e ·1 3ª§x · §x ·1 3 º 1.554¨ ¸ «¨ ¸ ¨ ¸ » ©Pe ¹ « ¼ ¬©l ¹ ©l ¹ »
Tw (x) Tw 0
Tw2 (x)
4
4
ª § ·2 l 1.013«1 ¨ ¸ « ¬ ©x ¹
[3.124]
13 3 º
» » ¼
[3.125]
Equation [3.125] is plotted in Figure 3.25. The curve exhibits a moderate deviation relative to the solution in section 3.9.3.6.1. The relative difference decreases from 12% for the smallest values of x/l to 3% for the highest values. It is consistent to find the highest relative deviation in the region of the fastest variations in wall temperature since the actual thermal condition in this region is far from that used in the Lévêque theory, namely uniform wall temperature. Note that the approximation that consists of replacing the velocity profile by its tangent at the origin becomes less and less appropriate in the downstream direction. It is possible to improve the calculation by using the actual velocity profile and the integral method. 3.10. Temperature profile for heat transfer with blowing 3.10.1. Description of the problem Let us consider convective heat transfer near a porous wall at temperature Tw higher than that of the fluid in the far region Tf . A transverse flow is generated through the wall with the local velocity v0. How does the energy equation simplify on the wall (y = 0+)? Study the sign of w T wy 2 and wT wy across the boundary layer and show that the temperature profile presents a point of inflexion at some distance to the wall. 2
118
Convective Heat Transfer
3.10.2. Solution The energy equation in a boundary layer [3.15] simplifies on the wall into v0
wT wy
D y 0
w 2T wy 2
[3.126] y 0
where the noslip condition at the wall (u = 0) and the blowing condition have been taken into account. In the classical situation where the wall is impermeable w 2T 0 , which leads to Tcc(0) 0 in the conditions prescribed for the ( v0 0 ), wy 2 y 0 temperature profile in the integral method (section 3.3.2). This is not the case in the present situation. Accounting for heating conditions, it is expected that the heat flux is directed wT 0. from the wall towards the fluid, say wy y 0 Since v0 ! 0 for blowing, we deduce from [3.126] that
w 2T wy 2
0 . The y 0
conductive heat flux supplied by the wall to the fluid ( qcc! 0 ) is proportional to wT wy and therefore increases with y near the wall. Far from the wall, the conductive heat flux tends to 0 (the fluid temperature tends to Tf ); it then exhibits a
w 2T wy 2
maximum at a point M, where
0 . Hence, the temperature profile y yM
presents a point of inflexion in M. The situation is shown in Figure 3.26.
T y M
u T(x,y) v0
Tp
x
Figure 3.26. Boundary layer with blowing through a porous wall. Temperature profile
Chapter 4
Forced Convection Around Obstacles
4.1. Description of the flow This chapter is devoted to heat transfer on bodies immersed in a stream. We consider a solid characterized by the length scale L placed in a stream characterized by the reference velocity U, which is generally the velocity far upstream from the obstacle (several times the length L in practice). If the solid is heated to a temperature Tf different from that of the fluid, heat transfer occurs in the vicinity of the solid surface and in the downstream wake. The thermal field depends strongly on the flow field, which is characterized by the Reynolds number Re = UL Q . We first distinguish the flows at Re > 1. The first type of flow is present in limited domains of application (sedimentation, thermoanemometry). These creeping flows are governed by viscous effects. Flows of the second type (Re >> 1) are more often encountered in practical applications. These flows are characterized by thin velocity and thermal boundary layers in the upstream part of the obstacle. A particular streamline ends at a stagnation point A located in front of the obstacle. Boundary layers originate in the stagnation region and then grow in the downstream direction up to the trailing edge for streamlined bodies (Figure 4.1). A velocity wake (velocity defect) and a thermal wake (temperature excess when the solid is heated) develop behind the obstacle. The wake region follows the development of the boundary layers when the obstacle is well profiled. When the obstacle is not well profiled (bluff body), the flow separates at some distance downstream from the stagnation point, where the pressure would have
120
Convective Heat Transfer
tended to increase in the absence of separation. The circular cylinder is a typical example of such obstacles (Figure 4.2). For bluff bodies, the wake is much broader than for streamlined airfoils.
Boundary layers U A
Wake L
Figure 4.1. Flow near a streamlined body. Re >> 1
Figure 4.2. Flow near a circular cylinder. Re >> 1
For a circular cylinder of diameter D, the Reynolds number is defined by Re = UD Q . When Re < 3 x 105, the separation point is located at an angle I ~ 80°, counted from the stagnation point A. When Re > 3 x 105, the boundary layer becomes turbulent in the upstream part of the cylinder and the separation point moves downstream, say at an angle I ~ 120°. The case of a circular cylinder has been investigated in many studies. Several flow regimes may be distinguished in the range Re >1 . Alternate vortices are generated at the rear side of the cylinder for Re higher than about 50 (Karman vortex street). The complexity of the flow field topology increases in the wake along with the Reynolds number.
Forced Convection Around Obstacles
121
4.2. Local heattransfer coefficient for a circular cylinder We consider a circular cylinder of radius R (diameter D) at uniform temperature Tw immersed in a stream of uniform velocity at temperature T (Figure 4.3). A current point M of the cylinder surface is determined by its curvilinear abscissa x ( RI ), counted from the stagnation point A. The local heattransfer coefficient hx is defined with the local heat flux qccx exchanged between the cylinder and the fluid as qccx
hx Tw Tf
[4.1]
The local heattransfer coefficient depends on the position of M, the Reynolds and the Prandtl numbers. The local Nusselt number is proportional to hx : Nu
h(x)D k
[4.2]
u(x) U
M x A I
B
Figure 4.3. Circular cylinder in a uniform stream. Definitions
In the upstream region close to the cylinder, the flow is characterized by laminar boundary layers so that Nu  Re , or equivalently Nu Re , is independent of Re. The distribution of Nu on the cylinder surface is shown in Figure 4.4 for air flow (Pr = 0.7). Experimental results (symbols in Figure 4.4) show that Nu Re  1 at the front stagnation point A ( I 0 ). The heattransfer coefficient decreases in the downstream direction, which corresponds to boundary layer thickening. Contrary to the upstream part of the cylinder, the ratio Nu Re depends on Re in the downstream part of the flow, where the wake configuration depends on Re. Figure 4.5 shows experimental results for the heattransfer coefficient at the downstream stagnation point B on the rear side of the cylinder ( I S ). It is worth noting that Nu Re is smaller at point B than at point A for Re 10 4 .
122
Convective Heat Transfer
Figure 4.4. Nusselt number on the upstream part of a cylinder in air; adapted from [KHA 05]
For high Reynolds number flows it is possible to calculate the heattransfer coefficient hx in the laminar boundary layer that develops on the upstream side of the cylinder. The result depends on the external velocity law uf ( x ) that is chosen for the calculation (Figure 4.3). Potential flow theory gives the velocity distribution outside the boundary layer: uf ( x ) U
2sinx R
[4.3]
However, separation has a significant effect on the flow on the upstream side of the cylinder and the following law is closer than [4.3] to the actual velocity distribution near the cylinder surface: uf ( x ) U
2
5
3.631x d 3.275x d 0.168x d
[4.4]
Several approaches have been proposed for calculation of hx . A review may be found in >SPA
[email protected] The different approaches may be classified according to the principles that they used: – local similarity; – integral method using two equations; – method using one equation, which combines the integral energy equation and similarity solutions to boundary layer equations [SMI 58].
Forced Convection Around Obstacles
123
Figure 4.4 shows the results obtained by [KHA 05] with the integral method and the external velocity law [4.3]. This calculation slightly overpredicts (  15% ) the Nusselt number when compared to experimental results. This discrepancy may be due to the method accuracy or to the velocity law chosen by the authors.
Figure 4.5. Nusselt number at the rear stagnation point of a cylinder in air [NAK 04]. CHF: constant heat flux, CWT: constant wall temperature. Reprinted from Int. J. of Heat and Mass Transfer, Vol. 47, NAKAMURA H., IGARASHI T., Variation of Nusselt number with flow regimes behind a circular cylinder for Reynolds numbers from 70 to 30,000, pages 51695173, copyright 2004, with permission from Elsevier
4.3. Average heattransfer coefficient for a circular cylinder The overall rate of heat exchanged by transverse unit, qc q Lz , between the cylinder and the flow is represented by the average heattransfer coefficient h 1 q SDL z
h Tw Tf
1. The average heattransfer coefficient
[4.5]
h is defined here by a spatial average and should not be confused with the time average used in Chapter 7 for a timevarying quantity in turbulent flows.
124
Convective Heat Transfer
or in dimensionless form
Nu
q SDL z
hD k
[4.6]
k Tw Tf D
Nu depends on Re and Pr. Many correlations are available in the literature. For example, Churchill and Bernstein [CHU 77a] recommend:
Nu
0.3
0.62Re1 2 Pr1 3
>
23
1 + 0.4 Pr
@
14
ª § Re ·5 8 º4 «1 ¨ ¸ » © ¹ » « 282 000 ¼ ¬
5
[4.7]
for 0.2 < Pe = Re Pr. Nu , Re, Pr are calculated with the fluid physical properties at the film temperature T f Tw Tf 2 . For uniform wall heat flux, equation [4.7] may be used with the temperature averaged on the cylinder perimeter. [BEJ 95] indicates that equation [4.7] underestimates the Nusselt number by up to 20% when compared to experimental results in the range 7 x 104 < Re < 4 x 105.
The law of Collis and Williams [COL 59] may be applied to low Reynolds number air flows and consequently to thermoanemometry applications Nu
A BRe T n
f
Tf
a
[4.8]
where the coefficients n, A, B and a are given in Table 4.1. 0.02 < Re < 44
44 < Re < 140
n
0.45
0.51
A
0.24
0
B
0.56
0.48
a
0.17
0.17
Table 4.1. Coefficients of the Collis and Williams law [COL 59]
The fluid physical properties are evaluated at Tf = Tw Tf 2 . Nakamura and Igarashi [NAK 04] give the results in Figure 4.6.
Forced Convection Around Obstacles
125
Figure 4.6. Overall Nusselt number for a circular cylinder [NAK 04]. Reprinted from Int. J. of Heat and Mass Transfer, Vol 47, NAKAMURA H., IGARASHI T., Variation of Nusselt number with flow regimes behind a circular cylinder for Reynolds numbers from 70 to 30,000, pages 5169–5173, copyright 2004, with permission from Elsevier
4.4. Other obstacles Heat transfer between a uniform stream and a sphere is represented by [WHI 91] Nu
2 0.4 Re1 2 0.06 Re 2 3 Pr 0.4 P f P w
14
[4.9]
for 0.71 < Pr < 380, 3.5 < Re < 7.6 x 104, 1 P f P w 3.2 . A review of several papers on this topic and the following correlation may be found in [MEL 05] Nu
2 0.47 Re1 2 Pr 0.36
for 3 x 103 < Pr < 10, 102 < Re < 5 x 104.
[4.10]
126
Convective Heat Transfer
4.5. Heat transfer for a rectangular plate in crossflow 4.5.1. Description of the problem A heated rectangular plate (height d, span length Lz, negligible thickness) is placed in a uniform crossflow (Figure 4.7). We consider the rate of heat exchanged between the rear side of the plate and the fluid. The lateral sides of the plate are assumed to be perfectly insulated. In order to determine the law governing heat transfer between the plate and the fluid, a series of experimental tests is performed in a flow of water at different velocities. The plate is maintained at the constant temperature Tw = 70°C by electrical heating. A film that is electrically heated covers the rear side of the plate. Measurements give the electric power q necessary to keep the film at the constant temperature Tw = 70°C when the temperature of water is 20°C. The dimensions of the plate are d = 2 cm, Lz = 20 cm. The physical properties of water are: – kinematic viscosity – thermal conductivity – specific heat at constant pressure
Q = 6 x 107 m2 s1 k = 0.63 W m1 K1 Cp = 4.18 103 J kg1 K1
Using the test results shown in Table 4.2, propose a correlation for heat transfer between the plate and the flow. 4.5.2. Solution Since the plate is insulated at its lateral sides and uniformly heated in the spanwise direction, we assume that the temperature field is twodimensional, both in the plate and in the fluid domains. The span length Lz therefore neither plays any role in the flow nor in the global heattransfer coefficient between the plate and the flow. This statement would be incorrect if heat losses take place at the lateral sides of the plate. In this case, transverse conduction would occur along the plate so that the temperature field would be threedimensional.
Forced Convection Around Obstacles
127
U d Lz
Figure 4.7. Rectangular plate placed in a uniform crossflow
U (m/s)
0.2
0.4
0.8
1.2
1.6
q(watts)
485
805
1150
1650
2250
Table 4.2. Electrical power supplied to the rear film against velocity
With these assumptions, the flow is only characterized by the Reynolds number hd Re = Ud Q and heat transfer by the global Nusselt number Nu = with k qconv dL z h Tw Tf . If heat losses are ignored (transverse conductive flux, radiation), the convective heat transfer rate at the rear side of the plate balances the electrical power due to the Joule effect in the film, qconv q . As the working fluid is unchanged during the experiments, the Prandtl number is kept constant. The heat transfer law is then of the form Nu = f(Re) . The two dimensionless numbers are given in Table 4.3. The results are plotted using logarithmic scales because a heat transfer correlation of the powerlaw type, like [4.7], is expected. U (m s1) Re
0.2 3.3 x 10
q (watts)
0.4 3
6.67 x 10
0.8 3
1.33 x 10
2 x 10 1650
1.6 4
2.67 x 104
485
805
h (W m )
4850
8050
11500
16500
22500
Nu
77
127.8
182.5
261.9
357.1
2
1150
1.2 3
2250
Table 4.3. Interpretation of the results by using dimensionless variables
128
Convective Heat Transfer
1000
Nu 0.707
y = 0.244x 2
R = 0.987
100
10 1000
10000
Re
100000
Figure 4.8. Representation of the results with dimensionless variables (logarithmic scales)
The experimental points are roughly aligned in this logarithmicscale plot in Figure 4.8. We deduce the empirical heat transfer correlation for air: Nu
0.24Re 0.7
[4.11]
Using their experimental results obtained with uniform flux heating in air, [RAM 02] propose the following correlation for the rear side of a plate of aspect ratio Lz d 6 : For 5.6 x 103 < Re < 3.85 x 104
Nu
0.16Re 0.72
4.6. Heat transfer in a stagnation plane flow. Uniform temperature heating 4.6.1. Description of the problem A twodimensional body is placed in a stream of air (temperature far from the obstacle T). We consider heat transfer near the upstream stagnation line, where the flow is assumed to be laminar. The problem is restricted to the case of a plane plate of length 2Lx perpendicular to the stream (Figure 4.9). The similarity solutions to the FalknerSkan equations are used to determine the flow (Figure 3.2 for the general case). Velocity and shear stress profiles are shown in dimensionless variables in Figure 3.3 and in Table 3.1 as a function of the parameter m.
Forced Convection Around Obstacles
129
The physical properties of air are: – density U = 1.29 kg m3 – kinematic viscosity Q =13 x 106 m2 s1 – thermal conductivity k = 0.024 W m1 K1 – Prandtl number Pr = 0.72
x T y
Stagnation point
Figure 4.9. Flow perpendicular to a plane plate
4.6.2. Guidelines Give the external velocity law for the boundary layer that develops on the plate. It is assumed that the velocity scale is fixed at some point of the plate, for example at its edges in xdirection. In other words, the coefficient K of the external velocity law is assumed to be known. Give the order of magnitude G of the boundary layer thickness. Calculate this thickness by using Figure 3.3. Calculate the shear stress W exerted by the flow on the wall as a function of x using Table 3.1. The wall is heated at uniform temperature Tw . Using the similarity solutions to the energy equation in the forced laminar regime, express the local Nusselt number Nux as a function of the local Reynolds number and the Prandtl number. Express the heattransfer coefficient as a function of the parameters that define the problem conditions. Numerically calculate the thickness G for K = 1000 s1 and the shear stress for x = 5 cm. Calculate the local heat flux exchanged between the fluid and the plate for these values of x, K and Tw = 5 K, Tf =  5 K (the plate is heated to avoid icing on the wall). Calculate the overall heat transfer rate exchanged by the plate of length 2Lx = 10 cm and span length Lz = 50 cm.
130
Convective Heat Transfer
4.6.3. Solution 4.6.3.1. Flow field For a twodimensional stagnationpoint flow on a flat plate, the angle of the reference wedge is ES S (Figure 3.2), from which it follows that E 1 and m 1. The external velocity law is uf (x) Kx for the boundary layer that develops on the plate. For a laminar boundary layer in forced convection, scale analysis gives the order of magnitude G x  1 Rex with Re x uf ( x ) x Q (equation [3.5]). Replacing uf ( x ) with the above expression in Re x , we find that the boundary layer thickness is constant with x. Let us denote:
Q
G1
[4.12]
K
Figure 3.3 shows that the thickness may be estimated more precisely by K  2.5 , which gives the thickness G2  2.5G1 . The wall shear stress is related to the similarity function F by:
W 0 (x) P
wu(x, y) wy y
P
uf (x)
0
G1
F cc0
[4.13]
Table 3.1 gives F cc0 1.232 . Thus, the wall shear stress varies linearly with x as:
W0
1.232P
K3
Q
[4.14]
x
4.6.3.2. Heat transfer A favorable pressure gradient leads to an increase in the Nusselt number when compared to a flow with zero pressure gradient. In the case of uniform temperature heating, Table 3.3 gives the ratio: Nu( m
1, Pr
0, 7)
Nu( m
0, Pr
0, 7)
 1.7
Forced Convection Around Obstacles
131
Using the heat transfer law for a plane plate (equation [3.24]), the local Nusselt number is obtained as:
Nu x  1.7 x 0.332Re x1 2 Pr1 3  0.564Re x1 2 Pr1 3
[4.15]
The heattransfer coefficient is therefore constant with x: h
0.564k
K
Q
Pr1 3
[4.16]
With the data given above, we calculate for x = 5 cm:
G2
0.28 mm , W 0
9.1 Nm 2 , h
106.5 Wm2 K 1
The local heat flux is qcc hTf Tw = 1065 Wm2 and the total heat transfer rate exchanged between the fluid and the plate is q qcc2L x L z = 53 W. 4.7. Heat transfer in a stagnation plane flow. Stepwise heating at uniform flux 4.7.1. Description of the problem We again consider the problem in section 4.6 with, however, two important differences. The plate heating is now started at a location downstream from the stagnation point (Figure 4.10). Moreover, the plate is heated at uniform flux instead of uniform temperature. The plate is unheated on the length x 0 so that a thermal boundary layer develops beyond this starting length and is therefore embedded in the velocity boundary layer that starts at the leading edge of the plate. The plate is heated at uniform flux q0cc for x x0 . The wall temperature is then Tf for x < x0 and increases further downstream. The unknown wall temperature is denoted Tw x in the heated region. Let us denote 4x Tw x Tf . We plan to determine the heat transfer law between the wall and the fluid for x t x0 .
132
Convective Heat Transfer
x
x0
T
Stagnation point
y
Figure 4.10. Stepwise heating at uniform flux
4.7.2. Guidelines The problem is restricted to the initial region of the thermal boundary layer. Propose a velocity profile u(x,y) inside the thermal boundary layer by using Figure 3.3 and replace the actual velocity profile near the wall by its tangent at the origin. Use the integral energy equation applied to a boundary layer flow. Assume that the heat flux q0cc is independent of x and determine the rate of enthalpy convected through a section perpendicular to the wall as a function of x. It is suggested that the following polynomial be used to represent the temperature profile in the thermal boundary layer of thickness GT x T(x, y) Tf Tw (x) Tf
§3
1
·
©2
2
¹
T (] ) 1 ¨ ] ] 3 ¸
[4.17]
with ] = y GT x , 0 ] 1. Determine a relation between 4(x) and GT x (take the boundary condition in x = x0 into account). Use the profile given by equation [4.17] in the relation between the heat flux and the temperature gradient at the wall and find a second equation between 4(x) and GT x . Combine these equations to calculate 4(x) and GT x . Determine the heat transfer law giving the local Nusselt number Nux as a function of Rex and the ratio x 0 x . Verify that when x 0 = 0, the heat transfer law is close to that issuing from the similarity solution.
Forced Convection Around Obstacles
133
4.7.3. Solution 4.7.3.1. Modeling the velocity profile Replacing the actual velocity profile near the wall by its tangent at the origin and using [4.13], we obtain u(x, y) uf (x)
F0cc
y
[4.18]
G1
with F0cc 1.232 or u(x, y) Oxy
with O
1.232
K
G1
[4.19]
1.232
K3
Q
.
4.7.3.2. Integral energy equation The integral energy equation reads (see Chapter 3, equation [3.33]): §wT ·
d
f ³0 UC p ux, y >T x, y Tf @ qccx k ¨ ¸
[4.20]
©wy ¹0
dx
In the conditions of the present problem, the heat flux is independent of x ( qccx q0cc), so that the above equation may be integrated as:
³0f UC p ux, y >T x, y Tf @ q0ccx C1
[4.21]
The expressions chosen for the velocity and temperature profiles are then reported in [4.21]:
³0f UC p Oxy4x T ] dy q0ccx C1 We eliminate y in favor of ] by using y GT x ] and we obtain a first equation satisfied by the unknown functions 4(x) and GT x , 2
AOUC p xGT x 4x
q0ccx C1
[4.22]
1
with A
³ T (] )] d] . The modeled temperature profile [4.17] gives A 0
1 10 .
134
Convective Heat Transfer
4.7.3.3. Heat flux The modeled temperature profile [4.17] is also used in the expression of the heat flux as: §wT · q0cc k ¨ ¸ ©wy ¹0
k
4x
GT x
Tc0
3 The chosen profile gives Tc0 . 2
The second equation satisfied by the unknowns of the problem is therefore:
3
k
4x
q0cc
2 GT x
[4.23]
4.7.3.4. Heat transfer law The solution of the system of equations [4.22] and [4.23] gives for x t x0 the distribution of: – the boundary layer thickness:
GT x
13 ª Q § x0 ·º «12.2 D ¨1 ¸» «¬ x ¹»¼ K 3 ©
[4.24]
– the wall temperature: 4x
13 2 q0cc ª Q § x 0 ·º «12.2 D ¨1 ¸» 3 k «¬ x ¹»¼ K 3 ©
[4.25]
– and the Nusselt number: Nux k
q0cc 4x x
ª Q § x ·º1 3 0.65x«D 1 0 ¸» 3 ¨ © « x ¹» K ¬ ¼
[4.26]
Forced Convection Around Obstacles
– or, as a function of Re x
Nux k
For x 0
q0cc 4x x
0.65
uf x x
Kx 2
Q
Q
135
:
Re x1 2 Pr1 3
[4.27]
§ x 0 ·1 3 ¨1 ¸ © x ¹
0 , the heat transfer law becomes:
Nux
0.65Re x1 2 Pr1 3
A similarity solution to the energy equation is available in this particular case. Table 3.3 gives the ratio Nu x (m) Nu x (m 0) for uniform temperature heating. When the fluid is air (Pr = 0.72), this ratio has the value 1.696. Nu x (m 0) is given by equation [3.23]. The result must be further corrected for uniform flux heating (equation [3.26]). Thus, the Nusselt number issuing from the similarity solution is: Nux
0.332 x 1.696 x 1.31Re x1 2 Pr1 3
0.74Re x1 2 Pr1 3
The integral method underestimates the heattransfer coefficient by 12%. 4.8. Temperature measurements by coldwire 4.8.1. Description of the problem Temperature measurements in a turbulent flow are often performed with the coldwire method (see, for example, [BRU 95]). The measurements are performed with an electric wire with a very small diameter d, length l, and electric resistance Rw. The probe consists of a cylindrical body equipped with two prongs onto which the wire is soldered. The probe is connected to an electrical source, which supplies a current of intensity I to the wire (Figure 4.11). The sensing element is placed in a stream in order to measure its temperature T g (t) varying with time.
136
Convective Heat Transfer
I
small wire, resistance
to electrical supply
prong Figure 4.11. Coldwire measurements. Sketch of the probe
The temperature Tw of the electric wire is measured via its electric resistance Rw , which is related to the temperature by Rw
R0 >1 E w T w T 0 @
[4.28]
where R0 is the wire resistance at the reference temperature T0 and Ew , the temperature coefficient of the wire material. Measurements are correct if the wire temperature is very close to that of the fluid ( T g ) to be measured. Hence, the current intensity I (I = Constant) used to measure the resistance Rw must be very small. Typical values for a platinum wire are: d = 1.5 Pml = 1 mm, R0 = 100 :, I = 0.3 mA, Ew = 3.8 103 K1 solid density Uw = 21.5 103 kg m3, solid heat capacity cw = 133 J kg1 K1 4.8.2. Guidelines Estimate the difference T w T g in steady conditions, i.e. when the wire is placed in a stream of velocity U and temperature T g , constant with time. We assume that heat losses by radiation and axial conduction to the prongs along the wire are negligible. Numerical application: air at 20°C, kinematic viscosity Q = 15 x 106 m2 s1, U = 15 m s1. When the fluid temperature T g fluctuates with time, the probe temperature also varies, but due to the thermal inertia of the wire, the measured signal is damped and shifted with respect to T g . Evaluation of the response time of the probe is proposed. Write the equation giving the variations of T w with time. Calculate the time constant of the probe as a function of velocity U and diameter d. Calculate the damping and phase lag of T w relative to T g for a sinusoidal temperature signal T g of frequency f. Numerical application: f = 1 kHz; 10 kHz.
Forced Convection Around Obstacles
137
4.8.3. Solution 4.8.3.1. Steady conditions If heat losses are ignored, the electric power dissipated by Joule effect in the wire is balanced at thermal equilibrium by the rate of heat exchanged by the wire to the ambient fluid. This heat transfer rate corresponds to forced convection around the cylindrical wire. The heatgeneration rate due to Joule effect is Rw I 2 . The heat transfer rate between the cylindrical wire and the fluid is given by [4.6] qconv
Slk Tw0 Tg0 Nu
where T w 0 , T g0 denote the equilibrium temperature of the wire and the constant fluid temperature respectively. The Reynolds number is Re The Collis and Williams law [4.8] gives Nu
15 x10 6 15 x10 6
= 1.
0.24 0.56Re 0.45 = 0.8.
The equilibrium temperature is therefore T w 0 T g0
or Tw0 Tg0
Rw I 2
SlkNu
0.14K with the conditions as given in section 4.8.1.
4.8.3.2. Unsteady conditions When the fluid/wire temperatures fluctuate with time, the thermal budget of the wire yields mc w
dT w (t ) dt
Rw (t ) I 2 Slk>T w (t ) T g (t )@Nu
where m is the mass of the cylindrical wire m
Uw S
d2 4
[4.29]
l.
At any instant there is a difference between the heat rate dissipated by Joule effect and the convective heat rate removed by the fluid. This difference is stored in or lost by the wire (lefthand side of equation [4.29]). We assume that the heat
138
Convective Heat Transfer
transfer law of the steady regime is still available in the present situation. This assumption then gives Nu 0.8 . Introducing relation [4.28] between the resistance Rw t and the fluid temperature T w (t) , equation [4.29] yields: mcw
dTw (t ) dt
R0 I 2 R0 E w I 2 S lk Nu Tw (t )
[4.30]
2
R0 E w I T0 S lk NuTg (t )
This firstorder equation shows that the wire temperature T w (t ) follows the fluid temperature variations T g (t ) with the time constant M
mc w
SlkNu R0 E w I
2

mc w
[4.31]
SlkNu
or, replacing the cylinder mass in favor of d M 
Uwcw d 2
[4.32]
4kNu
The Nusselt number varies roughly as Re1 2 . Hence, the time constant varies as 1 U and as d 3 2 . It is therefore recommended to use a wire with a diameter as small as possible in order to reduce M. For a velocity of 15 m s1 and the data given above, equation [4.32] gives M = 0.03 ms. In order to characterize the probe response, we consider sinusoidal variations of the fluid temperature T g (t) T g0 a cos Zt . The probe temperature also varies sinusoidally with a phase lag M such that T w (t) T w 0
b cosZt M .
Substituting into equation [4.30], we identify the terms in cos Zt and sin Zt to obtain the signal damping and phase lag: b
1
a
1 M 2Z 2
tgM
MZ
[4.33] [4.34]
Forced Convection Around Obstacles
139
With the data of the problem, we find for: f = 1 kHz,
b a = 0.98, M = 10.7°
f = 10 kHz,
b a = 0.47, M = 62°
With a coldwire of 1 Pm, it is then possible to measure temperature fluctuations of the order of 1 kHz. In order to perform measurements at higher frequency, a wire of smaller diameter must be used (0.6 Pm).
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Chapter 5
External Natural Convection
5.1. Introduction Natural convection corresponds to situations where the fluid is set in motion by buoyancy forces due to density variations. These density variations may result from heat transfer inside the fluid or between the fluid and heated or cooled solid walls. It may also result from mass transfer when the fluid is a multicomponent mixture. Contrary to the situations described in the previous chapters, natural convection flows are not due to external mechanical power, but solely to heat or mass transfer, which leads to density variations inside the fluid. It is worth underlining two main properties of natural convection flows: – there is a strong coupling between the flow and heat transfer. In natural convection problems, it is not possible to calculate successively the velocity field, then the temperature field, as it is in forced convection when the fluid properties are considered as constant. The two fields must be calculated simultaneously, which makes the problems of natural convection rather difficult; – buoyancy forces are generally weak so that the characteristic velocities are small compared to those which are found in forced convection. As a result, natural convection flows are mostly laminar. However, laminarturbulent transition may occur like in forced convection. Natural convection is very often encountered in many practical situations (cooling of electrical or electronic components, solar energy, domestic heating, etc.).
142
Convective Heat Transfer
5.2. Boussinesq model1 The relevant fundamental equations are those of fluid mechanics [1.2] and [1.6] and heat transfer [1.8] of the general type. Simplifying these equations is, however, possible in most applications and the resulting equations constitute the Boussinesq model. This model is available for moderate density variations (typically for relative variations within about 10%). The assumptions are the following: – the variations of density U are considered as linear against temperature T. Using an arbitrary reference state defined by p, U, T (these variables are linked by the fluid state equation), density variations are given by:
U Uf UfE T Tf
[5.1]
where E is the coefficient of thermal expansion at constant pressure; – density variations are ignored in the continuity equation, which keeps the usual expression: G div u 0
[5.2]
– density variations are also negligible at first order in the inertia term of momentum equation [1.6] so that U is replaced by U in this term; – in this chapter, the longitudinal axis Ox is chosen along the ascending vertical axis and we introduce the modified pressure: p* p Ugx . After decomposing pressure and accounting for the above assumptions, the momentum equation reads: G G G 1 du grad p* E T Tf g Q'u Uf dt
[5.3]
– velocities being weak, the term dp dt and the dissipation D are ignored with respect to the convection term in the energy equation. Without any heat source or sink, we then consider the following equation [1.10] instead of [1.8]: dT dt
D'T
[5.4]
5.3. Dimensionless numbers. Scale analysis Let us consider a situation involving only one length scale L along the vertical axis and a temperature scale 4as in forced convection, this is a characteristic 1. Joseph Valentin Boussinesq, French mathematician and physicist, 1842–1929.
External Natural Convection
143
temperature difference between a solid and the surrounding fluid for a given problem). Normalizing the equations by using reference scales shows that two dimensionless numbers are relevant to natural convection flows: – the Prandtl number Pr = QD, like in forced convection; – the Grashof number Gr
gE4L3
Q2
.
[5.5]
The Rayleigh number is obtained by combining the two previous ones and is often more relevant than the Grashof number for characterizing natural convection: Ra
gE4L3
QD
[5.6]
Gr Pr
Heat transfer is characterized in natural convection by correlations of the form: Nu
f Gr , Pr or, preferably, Nu
f Ra, Pr .
In a very large reservoir, heat transfer is generally concentrated in boundary layers that develop near vertical walls. For a given temperature difference 4 between a wall and the ambient fluid, scale analysis [BEJ 95] states the order of magnitude shown in Table 5.1 for: – the velocity and thermal boundary layer thicknesses, G and GT respectively; – the characteristic velocity U in the direction parallel to the wall; – the Nusselt number. The Rayleigh number is defined by using the distance x counted along the vertical axis from the wall leading edge (beginning of the boundary layers): Ra x
by:
gE4x 3
QD
[5.7]
The local heat flux at abscissa x being q0cc( x ) , the Nusselt number is defined
Nu x
q0cc(x) k4 x
[5.8]
144
Convective Heat Transfer
G GT Pr 1 or Pr ~ 1
Pr1/2
Ux D
GT x Ra x
1/4
Ra x
Pr
1/4
1/4
1/2
Pr
Ra x
1/2
Ra x
Nu x 1/2
1/4
Pr
Ra x
1/4
Ra x
1/4
Table 5.1. Results of scale analysis for natural convection
Figure 5.1. Sketch of the boundary layers in natural convection. Pr > 1
The flow is represented schematically in Figure 5.1 and Figure 5.2 for the two cases where Pr > 1 and Pr T0cn, Pr @RaL1 4
[5.31]
where Tw represents the average temperature over the plate in the case of nonuniform wall temperature heating.
External Natural Convection
149
For uniform wall flux heating, the governing system of equations enables a similarity solution with n = 1/5 (Table 5.2). In this case, it is preferable to characterize the flow and heat transfer by a modified Rayleigh number Ra x *
gEq0ccx 4
[5.32]
QDk
where q0cc is the uniform heat flux. Using the modified Rayleigh number changes the formalism in the scales of Table 5.1. [BEJ 95] gives the asymptotic trends obtained by an integral method: 15
Pr >> 1, Nu(x)
0.616Ra*x
Pr T x, y Tf @dy Q ¨ ¸ ©wy ¹0
[5.36]
150
Convective Heat Transfer
For pure convection, where the longitudinal velocity component is zero far from the wall ( uf 0 ), this equation simplifies in: d dx
2 H ³ 0 ux, y dy
§wu ·
H ³ 0 gE >T x, y Tf @dy Q ¨ ¸
©wy ¹0
[5.37]
The energy equation is the same as in forced convection: d dx
§wT ·
H ³0 UC p ux, y >T x, y Tf @ qccx k ¨ ¸
©wy ¹0
[5.38]
Figure 5.3. Integral method. Control domain
5.5.2. Solution Satisfying results are obtained by choosing velocity and temperature profiles characterized by the same length scale:
ux, y U 0 x
T Tf T p Tf
f >y G (x)@
[5.39]
T >y G (x)@
[5.40]
External Natural Convection
151
Contrary to the case of forced convection, the velocity scale U 0 x is now unknown. Modeling the flow and thermal fields with only one length scale leads to a system of two equations in two unknowns G x and U 0 x . Accounting for the conditions that the functions f and T must satisfy and choosing a polynomial form for these functions, we find the velocity and temperature shapes
1 K 2
f K
[5.41] 2
T K K1 K with K
[5.42]
y G ( x) .
For a vertical plate heated at uniform temperature, [BUR 83] gives the following results § C ·1/ 4 3.94¨ ¸ ¨ ¸ ©Ra x Pr ¹
Gx x
U max x x
D
§Ra Pr ·1/ 2 0.765¨ x ¸ © C ¹
[5.43]
[5.44]
1/ 2
Nu x
§Ra Pr · 0.508¨ x ¸ © C ¹
[5.45]
Nu
§Ra Pr ·1/ 2 0.677¨ L ¸ © C ¹
[5.46]
with C
20 21
Pr
These results are in good agreement with the similarity solutions.
[5.47]
152
Convective Heat Transfer
5.6. Correlations for external natural convection Heat transfer correlations for external natural convection are shown in Table 5.4 in addition to the previous results. They are taken from [CHU 75], [FIS 50] and have been collected by [BEJ 95]. They take turbulence effects into account when the Grashof number based on the solid height becomes larger than a critical value. Typically, GrL  10 9 . crit
The fluid physical properties are taken at the film temperature 1 2 Tw Tf . The global Nusselt number Nu is based on the total heat transfer rate exchanged by the solid of surface S: Nu
q S
k Tw Tf
[5.48]
L
For a vertical plate heated at nonuniform temperature, Tw is the average temperature over the plate height. The reference length scale L is specified in the index of the Nusselt, Rayleigh and Grashof numbers and in the different cases is: – the vertical plate height L; – the diameter d of a cylinder or a sphere; – a dimension calculated as the ratio of the surface to the perimeter in the case of a horizontal plate A S p . 5.7. Mixed convection The influence of buoyancy forces on a flow characterized by the velocity scale U is estimated by the Richardson number: Ri
gE4L U
2
=
buoyancy force per mass unit inertia force per mass unit
=
Gr Re 2
When Ri 1 , buoyancy forces are negligible (forced convection). When Ri !! 1 , the flow and heat transfer are governed by buoyancy forces (pure natural convection). This is the case presented in the first sections of this chapter, where there is no velocity imposed to the fluid far from a heated solid (U = 0).
External Natural Convection
Configuration
Correlation
Vertical plate
L
Nu L
Uniform temperature heating
§ ·2 16 ¨ ¸ 0.387RaL ¨0.825 ¸ 9 16 8 27 ¸ ¨ ¨ ¸ 1 0.492 Pr © ¹
>
@
[5.49]
101 Ra L 1012
Vertical plate
Nu L Uniform temperature heating
0.68
0.67 RaL
14
>1 0.492 Pr @ 9 16
49
[5.50]
GrL 10 9
Laminar regime Vertical plate
Nu L
Uniform flux heating
§ ·2 16 ¨ ¸ 0.387 RaL ¨0.825 ¸ 9 16 8 27 ¸ ¨¨ ¸ 1 0.437 Pr © ¹
>
@
[5.51] Vertical cylinder Uniform temperature heating
d L
Nu d
14 § d · 0.525¨Ra d ¸ © L ¹
[5.52]
Air, Pr = 0.7 Vertical cylinder Uniform temperature heating
Equations [5.49]–[5.51] are valid for a d vertical cylinder if ! Ra L 1 4 L
Table 5.4. Correlations for external natural convection
153
154
Convective Heat Transfer
Configuration Horizontal cylinder
Correlation
d
Nu d
0.52Ra d1 4 [5.53]
L
Horizontal cylinder
Nu d
§ ·2 16 0.387Rad ¨ ¸ ¨0.6 ¸ 8 27 9 16 ¨ ¸ ¨ ¸ 1 0.437 Pr © ¹
>
@
[5.54]
d
Sphere
Nud
2 0.45Grd1 4 Pr1 3 [5.55]
1 Grd 10 6
Nu d Sphere
2
0.589Rad
14
>1 0.469 Pr @ 9 16
49
[5.56] 11
Pr 0.7, Ra d 10 Horizontal plate Hot side upward Laminar regime
Horizontal plate Hot side upward Turbulent regime
Horizontal plate Heated side downward
Nu A
0.54Ra A1 4 [5.57]
4
10 Ra A 10 Nu A
7
0.15Ra A1 3 [5.58]
10 7 Ra A 10 9 Nu A
0, 27Ra A 1 4 [5.59]
10 5 Ra A 1010
Table 5.4 (continued) Correlations for external natural convection
External Natural Convection
155
When Ri  1 , buoyancy and inertia forces have the same order of magnitude. This is the case of mixed convection. [CHU 77b] suggests the empirical estimation of the heattransfer coefficient: 3 Numixed
3 Nu 3forced conv. Nunatural conv.
[5.60]
5.8. Natural convection around a sphere 5.8.1. Description of the problem In order to determine the natural convection heat transfer law between a sphere and the surrounding fluid, experiments are performed with several spheres of different diameters d. The spheres are electrically heated and kept at uniform temperature Tw = 40°C while the surrounding air is at 20°C. Measurements of the electrical energy rate q are performed and the results are displayed below. d, cm
4
12
20
40
q, watts
0.65
4
9.5
31
Table 5.5. Experimental results for natural convection around a sphere
The physical properties of air are given in Table 5.6. Density
U = 1.2 kg m3
Kinematic viscosity
Q = 1.5 105 m2 s1
Specific heat at constant pressure
Cp = 103 J kg1 K1
Thermal conductivity
kf = 0.032 W m1 K1
Coefficient of thermal expansion
E= 1/300 K1
Table 5.6. Physical properties of air
Propose a heat transfer law by using the results of Table 5.5. 5.8.2. Solution The Rayleigh number based on the sphere diameter is first calculated: Ra
gE4d 3
QD
156
Convective Heat Transfer
The thermal diffusivity is calculated by D
k
UC p
=
0.025 1.2 x 10 3
m2 s1.
D= 2.08 x 105 m2 s1 The temperature scale is prescribed, 4
Tw Tf = 20°C.
Denoting d 0 the diameter of the smallest sphere, the corresponding Rayleigh number is: Ra 0
gE4d 0 3
QD
=
9.81 x 1 300 x 20 x 0.04 3 1.5 x 10
5
x 2.08 x 10
5
= 1.34 x 105
Since the temperature scale is kept constant, the Rayleigh number relative to the § d ·3 other spheres is obtained by Ra Ra 0 ¨¨ ¸¸ . ©d 0 ¹ The Nusselt number is calculated with the electrical energy rate dissipated by Joule effect. Assuming that heat losses are negligible, this energy rate is balanced by the heat transfer rate removed by the natural convection air flow. The sphere surface is Sd 2 .
Nu
q Sd 2
k Tw Tf d
=
q
Sk Tw Tf d
Denoting q0 the heat transfer rate associated with the sphere of diameter d 0 , the corresponding Nusselt number is Nu 0
q0
Sk Tw Tf d 0
and, for the other spheres, Nu
=
0.65
S x 0.025 x 40 20 x 0.04
= 10.34
d q , hence Table 5.7. Nu 0 0 d q0
d in cm
4
12
20
40
Ra
1.34 x 105
3.6 x 106
1.68 x 107
1.34 x 108
Nu
10.34
21.1
30.1
49.1
Table 5.7. Natural convection around a sphere. Calculation of the dimensionless numbers
External Natural Convection
157
100
Nu
10 1.0E+05
1.0E+06
1.0E+07
1.0E+08
Rad
1.0E+09
Figure 5.4. Display of experimental results. Dashed line: equation [5.56]
The results are plotted in Figure 5.4. For a prescribed law of the form Nu A BRa d 1 4 , a linear regression gives A = 1.96, B = 0.44. Law [5.56] is displayed in the same figure for the sake of comparison. 5.9. Heated jet nozzle 5.9.1. Description of the problem This is the continuation of problem in section 3.4. A facility consists of a large tank with a nozzle attached; the nozzle is composed, first, of a converging duct, followed by a cylindrical tube of length L and diameter D (L = 12 cm, D = 8 cm, Figure 5.5). The tank is filled with air at pressure p1 and temperature T1 (= 40°C), which exits through the nozzle into the ambient atmosphere at pressure p0 and temperature T0 (= 20°C). An electrical resistance is wrapped around the tube in order to compensate for heat losses toward the surrounding air and to approach a uniform temperature profile in the nozzle exit crosssection. The heating system is modeled as a tube with zero thickness around the nozzle. It is considered that the external surface of this tube is at temperature T1 when the heating system is correctly adjusted. Calculate the heat transfer rate q1 supplied by the electrical resistance in these operating conditions when heat transfer to the surrounding air is governed by natural convection.
158
Convective Heat Transfer
Heating by electrical resistance
p0, T0
D p1, T1
U
Reservoir L Nozzle exit Figure 5.5. Heated jet. Sketch of the facility
The electrical resistance now supplies the energy rate q2 (= 5 W). Calculate the new wall temperature and the heat transfer rate removed from the tube by the natural convection flow. The physical properties of air are given in Table 5.8. Density
U = 1.165 kg m3
Kinematic viscosity
Q = 16 106 m2 s1
Specific heat at constant pressure
Cp = 103 J kg1 K1
Prandtl number
Pr = 0.7
Thermal conductivity
k = 0.025 W m1 K1
Table 5.8. Physical properties of air
5.9.2. Solution 5.9.2.1. Energy rate necessary to maintain the temperature T1 at the external side of the heating system In the present case there is no heat transfer between the fluid inside the nozzle and the heating system because they have the same temperature. The energy rate supplied by the electrical system is balanced by the heat transfer rate removed by the natural convection flow of ambient air. The configuration is that of a horizontal cylinder, for which the heat transfer law is, according to [5.54]
Nu D1
§ ¨ ¨ 0, 6 ¨¨ ©
0.387 RaD11 6 ª1 0.437 Pr 9 16 º «¬ »¼
8 27
· ¸ ¸ ¸¸ ¹
2
External Natural Convection
gE T1 T 0 D 3
with Ra D1
QD
=
9.81 x 1 303 x 40 20 x 0.08 3 16 x 10
6
x 16 x 10
6
0.7
159
= 9.06 x 105.
The result is Nu D1 = 14.5.
According to [5.48], the heat transfer rate exchanged with the ambient fluid is:
q1
SLk T1 T0 Nu D1 = S x 0.12 x 0.025 x 40 20 x 14.5 = 2.7 W.
It is also the electrical energy rate necessary to maintain the external side of the heating system at temperature T1 . It is worth noting that the problem in section 3.4 considered the start of heating and that the wall was at temperature T0 in these conditions. The heat transfer rate lost by the inner fluid was used to heat the nozzle tube and was higher than the heat transfer rate exchanged with the ambient fluid in the present problem. In the present case, the wall is at temperature T1 and the electrical energy rate is entirely lost toward the ambient fluid. 5.9.2.2. Overheating of the nozzle When a higher electrical energy rate q2 ! q1 is supplied to the nozzle, the heating system becomes hotter than both the ambient fluid and the fluid inside the nozzle. It follows that the electrical energy rate q2 , dissipated by Joule effect, is used to heat both the ambient fluid (heat transfer rate qnat. conv. ) and the fluid inside the nozzle (heat transfer rate q forced . conv. ). q2
qnat. conv. q forced . conv.
[5.61]
We assume that the heating system temperature is uniform. It is denoted T2 or in dimensionless form:
T
T2 T0 T1 T 0
[5.62]
The new Rayleigh number may be expressed as a function of Ra D1 , which is already calculated in section 5.9.2.1
Ra D2
Ra D1 T
160
Convective Heat Transfer
and the corresponding Nusselt number is
Nu D2
where B
0.6 BT
16 2
9 16 0.387 RaD11 6 ª«1 0.437 Pr º» ¬ ¼
8 27
.
The heat transfer rate exchanged by natural convection is finally:
SLk T1 T0 Nu D2 T T
qnat. conv.
The heat transfer rate exchanged by forced convection is given (problem in section 3.4) by q forced.conv.
SDk T2 T1 Nu L
0.664 Re L Pr1 3
with NuL
136.8 .
Accounting for the definition of T, we have T 2 T1
T1 T 0 T 1 .
Thus, according to [5.61], the dimensionless temperature satisfies
q2
SLk T1 T0 Nu D2 T T SDk T1 T0 NuL T 1
A
0.6 BT T Nu T 1 D
or
where A
16 2
L
L
q2
SDk T1 T0
[5.63]
.
With the numerical data of the problem, we find A = 39.8, B = 3.22 so that equation [5.63] becomes: 39.8
1.5 0.6 3.22T
T 136.8T 1
16 2
External Natural Convection
161
As a first approximation, we consider T 1 6  1 in this equation, which then becomes linear and gives T = 1.113. Taking T 1 6 into account, it is found that:
T = 1.109. The wall temperature is then T2 = 42.2°C. The heat transfer rate supplied to the inner fluid is given by q forced . conv.
SDk T1 T0 Nu L T 1 = 2.17 W
and the heat transfer rate removed by the ambient fluid is qnat. conv. = 2.83 W.
This result is close to q1 , because the new wall temperature is not very different from T1 . 5.10. Shear stress on a vertical wall heated at uniform temperature 5.10.1. Description of the problem A vertical wall is heated at uniform temperature Tw in a very large reservoir filled with a fluid of Prandtl number larger than 1. The fluid temperature far from the wall is Tf ( Tw ! Tf ). The fluid is characterized by the density U, the kinematic viscosity Q and the coefficient of thermal expansion Eassumed to be constant. Coordinates are x along the vertical axis and y in the direction normal to the wall. For Pr > 1, the velocity and temperature profiles have the shape shown in Figure 5.1. How is the Boussinesq equation in x direction at the wall (y = 0) simplified? Express the viscous term in this equation by using the wall shear stress W 0 and the thermal boundary layer thickness GT . Estimate W 0 as a function of the Rayleigh number and the parameters of the problem. Also estimate W 0 with Figure 5.6 that represents the similarity velocity profiles for two different values of the Prandtl number. Show that the two estimations of W 0 are equivalent.
162
Convective Heat Transfer
ux Rax1/2 D
y Rax1/4 x Figure 5.6. Similarity velocity profiles near a vertical plate. Uniform wall temperature heating [BEJ 95]. A Bejan, Convection Heat Transfer, 2nd edn, John Wiley & Sons, 1995 (New York) copyright, reprinted with permission of John Wiley & Sons, Inc
5.10.2. Solution Boussinesq equation [5.11] simplifies on the wall, where u 0
UgE Tw Tf
0 and becomes
w 2u
gE Tw Tf Q
wy 2
or multiplying by U, since W 0
v
wW wy
y 0
P wu wy with P
constant [5.64]
y 0
Figure 5.7 shows that the local shear stress W decreases from W 0 on the wall to 0 for y  GT , which is the approximate position where the velocity reaches its maximum value in a cross section. The shear stress gradient is therefore estimated by:
wW wy
 y 0
W0 GT
External Natural Convection
163
Reporting in [5.64], the order of magnitude of the wall shear stress is given by
W 0  UgE Tw Tf GT
[5.65]
or taking the order of magnitude of Table 5.1 into account, 4 W 0  UgE Tw Tf xRa 1 x
[5.66]
Another calculation may be achieved by using similarity solutions. Figure 5.6 shows that the different velocity profiles plotted for Pr > 1 have the same slope of the tangent at the origin, close to 1. This is expressed by accounting for the definition of the abscissa and ordinate in Figure 5.6 by §ux 1 · ¸ d ¨ ¨D 1 / 2 ¸ © Ra x ¹ 1 §y 1 / 4 · d ¨ Ra x ¸ ©x ¹
or
x2
du
DRa x3 / 4 dy
= y 0
W 0x2
PDRa x3 / 4
The resulting shear stress is:
W0 
PD x2
Ra x3 / 4
[5.67]
Expressions [5.66] and [5.67] are equivalent, because
UgE Tw Tf xRa x 1 4 and [5.67] is recovered.
Ra x Ra x
UgE Tw Tf x
gE Tw Tf x 3 QD
Ra x3 4
164
Convective Heat Transfer
Figure 5.7. Sketch of the velocity boundary layer (natural convection)
5.11. Unsteady natural convection 5.11.1. Description of the problem A vertical plane plate of thickness e, length L and span length Lz is initially at temperature T0 in a very large reservoir filled with a fluid at the same temperature. The fluid physical properties (kinematic viscosity Q, thermal diffusivity D are assumed to be constant. The plate is suddenly electrically heated with a condition of uniform volumetric heat source. The plate is assumed to be thin so that its thermal inertia is negligible. As a result, heat transfer between the plate and the surrounding fluid is supposed to be at heat flux q0cc, both constant with time and uniform along the plate. The starting time of heating is chosen as the time origin. The problem is focused on the flow and heat transfer during the very first instants of heating. The axes are Ox along the ascending vertical direction, Oy in the direction normal to the plate and Oz perpendicular to the plane of Figure 5.8. The plate temperature is denoted Tw t at time t. The fluid velocity component parallel to the plate and the fluid temperature are denoted u and T respectively. The flow is assumed to be independent of z.
External Natural Convection
165
We propose the determination of the evolution of the plate temperature and heattransfer coefficient as a function of time and the length of time of the unsteady regime at given position x. The results may be compared to the experimental data of Goldstein and Eckert [GOL 60] obtained with a plate placed in water and shown in Figure 5.9.
x
Tw(t)
T0
qcct
g qcc
q0cc
t y Figure 5.8. Unsteady natural convection. Notations
The physical properties of water are given in Table 5.9. Density
U = 103 kg m3
Prandtl number
Pr
Thermal diffusivity
D = 1.4 107 m2 s1
Specific heat at constant pressure
Cp = 4.18 x 103 J kg1 K1
Thermal conductivity
kf = 0.6 W m1 K1
Coefficient of thermal expansion
E= 1.8 104 K1
Table 5.9. Physical properties of water
166
Convective Heat Transfer 1000 h(t) W m2 K1
x=9.67 cm x=5.34 cm x=2.2 cm x=0.14 cm
qcc 49.3Wm2
100 0.01
0.10
1.00
10.00 t (min)
Figure 5.9. Unsteady natural convection. Heattransfer coefficient as a function of time, drawn from the experimental results of [GOL 60]
5.11.2. Guidelines Analyze the phenomena governing the transfer of heat and momentum by using the results of Figure 5.9. It is assumed that advection is negligible during the very first instants. What are the variables on which the velocity u and temperature T depend? Simplify the local equations governing the variations of u and T. By using scale analysis, determine the order of magnitude of the velocity and thermal boundary layer thicknesses, respectively G(t)and GT t , the temperature difference 4(t ) Tw (t ) T0 , the heattransfer coefficient h(t) and the velocity scale U(t) at time t. Implement the integral method by using only one length scale Gt for the boundary layer thicknesses. Write the integral conservation equations of momentum and heat for a slice of thickness dx, extending from the wall to the reservoir. Model the temperature profile T T 0 4(t ) T (K) with K y G (t ) Write two equations between 4and G and calculate these two quantities as a function of the parameters of the problem. Model the velocity profile. It is suggested that a velocity scale U 0 t to be determined be introduced. Solve the integral momentum equation. Show that U 0 t varies as t3/2.
External Natural Convection
167
Calculate the heattransfer coefficient h(t). The exact solution to the thermal S k problem gives h (t ) . Compare this expression to the result of the integral 2 Dt method and to the experimental results in Figure 5.9. Compare the experimental results that correspond to very long time in Figure 5.9, to those obtained from a heat transfer law in steady conditions. Propose a method to determine the length of time t d of the unsteady regime. Compare to the experimental results of Goldstein and Eckert [GOL 60] 5.11.3. Solution 5.11.3.1. Analysis of physical phenomena, equations and scale analysis During the first instants of heating, heat is transferred by pure diffusion into water through a thin layer whose thickness only depends on time. The fluid is then set into motion by buoyancy forces and a boundary layer develops from the bottom edge of the plate. However, a fluid particle located at height x is initially unaware of the extremity effect coming from the plate bottom. This information arrives at the xlevel with a time delay which depends on x. At given height, the transfer phenomena are purely diffusive and unsteady during the very first instants. Later on, advection comes into play and, finally, the steady regime is reached after a time delay which is increasing with x. Figure 5.9 shows that the heattransfer coefficient h actually does not depend on x during the initial stage of heating, and then does not depend on t beyond a time length which is increasing with x. In other words, the temperature and velocity fields do not depend on x during the initial stage of transfer. The velocity component u and the temperature T initially satisfy:
wu wx
0 , wT wx
0
The normal velocity component v is zero, according to the continuity equation. It follows that the advection terms are zero in the momentum and energy equations, which simplify in:
wu wt
gE T T 0 Q
w 2u wy 2
[5.68]
168
Convective Heat Transfer
wT wt
D
w 2T
[5.69]
wy 2
It is worth noting that during this initial stage of transfer, the thermal and dynamic problems are uncoupled, contrary to the general case of natural convection. It is therefore possible to first solve the thermal problem, and, in a second step, the temperature field being calculated, to solve the dynamical problem. It is also worth noting that the heat transfer problem is that of a stepwise variation of heat flux on a semiinfinite solid, whose solution is well known [ECK 72]. The above reasoning suggests representing the flow as shown in Figure 5.10. Scale analysis gives the following orders of magnitude
wT 4  , wt t
w 2T wy
2

4
GT 2
so that equation [5.69] gives the variation of GT t with time
GT t  Dt
[5.70]
GT t is the depth of penetration by diffusion into a material of thermal diffusivity D in unsteady conduction regime [ECK 72]. The heat flux is given by q0cc k wT wy y thus:
0
, and estimated by q0cc  k 4 GT ,
qcc 4t  0 Dt k
[5.71]
The wall temperature increases as t1 2 . The heattransfer coefficient is defined by q0cc h4 , so that: ht 
k
Dt
[5.72]
External Natural Convection
169
Figure 5.10. Unsteady natural convection. Sketch of the flow at very short time. Pr > 1
The analysis of equation [5.68] for 0 d y d GT gives the following orders of magnitude:
wu U  wt t
gE T T 0  gE4
Q
w 2u wy
2
Q
U
GT
2
 Pr
U t
For Pr > 1, the fluid motion is governed by the viscous term and the velocity scale is deduced from [5.68] U
gE4 Pr
or U t 
t
gE q0cc Pr k
Dt 3 2
[5.73]
A last estimation, which is not necessary to solve the problem, is made for T 0 ), and the fluid is
GT d y d G . In this region, the buoyancy force is zero ( T entrained by viscosity forces. It follows that G t  Qt .
170
Convective Heat Transfer
5.11.3.2. Integral method We already know an exact solution to the thermal problem. For the sake of consistency with the calculation of the velocity field, an approximate solution is, however, calculated by the integral method. The basic equations are written, as in the general case, for a control domain of height dx, length H, and span length Lz, in the directions Ox, Oy and Oz respectively (Figure 5.3). Let us consider integral equations [5.37] and [5.38], in which the unsteady term replaces the advection term (see lefthand side of [3.31] and [3.33]). Simplifying by Lz, we obtain:
w H ³ uy , t dy wt 0
§wu ·
H ³ 0 gE >T y, t T 0 @dy Q ¨ ¸
©wy ¹0
§wT · q0cc k ¨ ¸ ©wy ¹0
w H ³ UC p >T y, t T0 @dy wt 0
[5.74]
[5.75]
The temperature and velocity profiles are modeled by using one length scale only, like in the steady case (section 5.5.2) uy, t U 0 t
T y, t T 0 4t
with K
2
[5.76]
1 K 2
[5.77]
f >y G (t)@ K1 K
T >y G (t)@
y G (t) .
Heat transfer is first considered. Introducing [5.77] into equation [5.75] and dividing by UC p , a first equation is obtained as
w 4 4G ³ 01T dK D Tc0 wt G or, denoting a1
³ 01T dK
External Natural Convection
4t G t c
D
Tc0 4t
171
[5.78]
a1 G t
A second equation is obtained from the definition of the wall heat flux ( q0cc k wT wy y 0 ) as:
q0cc k
4t
G t
Tc0
[5.79]
4t is therefore proportional to G t . Reporting this result in [5.78], the boundary layer thickness is obtained as
>G t @c 2
aD , or G t
aDt , which accounts for G t
0
0
with a
Tc0
[5.80]
a1
Temperature profile [5.77] gives the constants:
Tc0 2 , a1 = 1/3, a = 6 Finally:
G t
6Dt
[5.81]
3
4t
q0cc
ht
2
k
3
Dt
2
Dt
[5.82]
= 0.816
k
Dt
[5.83]
The exact solution to the thermal problem [ECK 72] is h(t)
S
k
2
Dt
= 0.886
k
Dt
[5.84]
172
Convective Heat Transfer
or Nu(t)
h(t)x
S
x
k
2
Dt
= 0.886
x
[5.85]
Dt
The integral method provides a satisfying approximation of the heattransfer coefficient since this latter is underestimated by up to 8% relative to the exact solution. It is worth noting the good agreement with the experimental results of [GOL 60] in Figure 5.11. 1000
h(t) x=9.67 cm x=5.34 cm x=2.62 cm x=0.814 cm
W m2 K1
qcc 49.3Wm2
100 0.01
0.10
1.00
10.00 t (min)
Figure 5.11. Heattransfer coefficient vs time. Solid line: equation [5.84]
The velocity scale is obtained by solving equation [5.74]. After introducing the velocity profile [5.76], the momentum equation becomes
w U 0G ³ 01 f dK wt
gE4G ³ 01T dK Q
U0
G
f c0
or
b1U 0 t G t c
a1gE4G Q
U 0 t
G t
f c0
[5.86]
External Natural Convection
173
1
³ 0 f dK .
where b1
G t and 4t are replaced by their expressions [5.81] and [5.82], respectively.
The velocity U 0 t verifies
U t §1 f c0 · gEM 0 U 0 t c 0 ¨¨ Pr Dt ¸¸ C t ©2 ab1 ¹ k
with C
a1
[5.87]
a
>Tc0 @
Scale analysis has shown that U t  At 3 2 ([5.73]). We verify that equation [5.87] is satisfied by this power function and we obtain: C
A
2 Pr
f c0
gEq0cc k
D
ab1
With the modeled profiles, the velocity scale finally reads: U 0 t
gEq0cc
6 1 Pr
k
Dt 3 2
[5.88]
5.11.3.3. Steady regime The results of [GOL 60] may be compared to correlation [5.89] obtained by [BEJ 95] by using the integral method for uniform heat flux heating:
Nu(x)
with Ra x *
§ Pr ·1 5 15 ¨ ¸¸ Ra *x 1 5 ¨ 360 ©4 5 Pr ¹
2
gEq0ccx 4
QDk
[5.89]
.
Figure 5.12 demonstrates a good agreement between the two types of results.
174
Convective Heat Transfer 100
Nu
0.20
y = 0.60x 10
Theoretical calculation, 1 Theoretical calculation, 2 Exp. [GOL 1960], 1 Exp. [GOL 1960], 2
1 1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09 Ra*
Figure 5.12. Natural convection. Steady regime. Theoretical computation: equation [5.89]. 1: q0cc 49.3 W m2, 2: q0cc 132 W m2
5.11.3.4. Length of time of the unsteady regime It is possible to exploit the reasoning presented in section 5.11.3.1 for calculating the time taken by a fluid particle to reach the height x, counted from the plate bottom. We call X(t) the abscissa at instant t on the vertical axis of a fluid particle that left the bottom plate at t = 0. X(t) is related to the velocity field in a complex way. An order of magnitude of X(t) is obtained by using the velocity scale U 0 t calculated above dX t dt
with U 0 t
U 0 t 6
[5.90]
gEq0cc
1 Pr
k
D t 3 2 as given by equation [5.88].
Integrating with time gives
X t
2
6
5 1 Pr
gEq0cc k
Dt 5 2
External Natural Convection
175
from which it follows that the time taken by a fluid particle to reach the height X = x is ª º2 5 x B1« » ¬gEq0cc D k ¼
t d1 x
[5.91]
ª 5 º2 5 1 Pr » « ¬2 6 ¼
with B1
Another method consists of determining the time after which the Nusselt number calculated for the unsteady regime [5.85] is equal to the corresponding value in the steady regime [5.89].
0.886
x
Dt
=
§ Pr ·1 5 15 ¨ ¸¸ Ra *x 1 5 ¨ 360 ©4 5 Pr ¹
2
We obtain a second determination of the length of time of the unsteady regime as t d 2 x
with B2
§ ·2 5 x B2 ¨¨ ¸¸ ©gEq0cc D k ¹
[5.92]
25 §0.886 ·2 ª §4 ·º ¨ ¸ «360¨ Pr ¸» . © 2 ¹ ¬ ©5 ¹¼
For Pr = 7, we find B1 = 2.3 and B2 = 4.7. The two expressions are identical in the sense of scale analysis. It is logical to find t d2 x ! t d1 x since the first calculation was achieved with the velocity
U 0 t , which is close to the maximum value in a crosssection. The time taken by particles located elsewhere in the crosssection is obviously longer.
Figure 5.13 has been plotted with the time t d2 x and experimental results of [GOL 60]. A very good agreement is found for the two types of results.
176
Convective Heat Transfer 100 90 t (s)
80 70 60 50
Exp. [GOL 60]
40
Theoretical calculation, Eq. [5.92]
30 20 10 0 0
0.02
0.04
0.06
0.08
0.1
0.12 x (m)
Figure 5.13. Length of time of the unsteady regime
5.12. Axisymmetric laminar plume 5.12.1. Description of the problem An axisymmetric laminar plume develops upon a sphere placed in a very large reservoir where the fluid is at rest far from the sphere (Figure 5.14). x
u v
fluid at rest T r
Figure 5.14. Laminar plume upon a sphere
External Natural Convection
177
The plume is caused by the buoyancy forces generated by the temperature difference between the sphere and the surrounding fluid. The sphere is maintained at uniform temperature Tw whereas the ambient fluid is at temperature Tf ( Tw > Tf ). We denote x and r, the coordinates, ux, r and vx, r , the velocity components in the vertical and radial directions respectively, T x, r the fluid temperature at a current point. Write the local equations governing the fluid motion and heat transfer. For a fluid at Pr >> 1, determine the scaling laws for a typical velocity U, temperature difference 4 between the plume and the ambient fluid, and the velocity and thermal radii of the plume denoted G and GT , respectively, as a function of height x. 5.12.2. Solution 5.12.2.1. Local equations Supposing axisymmetry about Oxaxis, the Boussinesq equations yield in cylindrical coordinates: – continuity equation
wu 1 w rv wx r wr
0
[5.93]
– momentum equation in xdirection u
wu wu v wx wr
1 dp*
Uf dx
gE T Tf
Q w § wu · ¨r ¸ r wr © wr ¹
[5.94]
– momentum equation in rdirection 0
wp* wr
[5.95]
– energy equation u
wT wT v wx wr
D w § w T · ¨r ¸ r wr © wr ¹
[5.96]
Equation [5.95], combined with the boundary condition p o pf when r o f,
shows that the pressure is uniform in the whole plume (thus dp* dx
0 in [5.94]).
178
Convective Heat Transfer
5.12.2.2. Scaling laws In the case where Pr >> 1, the velocity and temperature profiles have the shape as shown in Figure 5.15 because the velocity plume radius (denoted G) is expected to be larger than the thermal plume radius (denoted GT ). x U, Taxis
G fluid at rest
u
T
r
GT
T
r
Figure 5.15. Laminar plume. Velocity and temperature profiles. Pr >> 1
We denote 4 the temperature scale in a crosssection at height x. It is chosen as the difference between the fluid temperature on the plumeaxis and the ambient fluid temperature, 4 Taxis Tf . In the same way, the longitudinal velocity scale is chosen as the velocity U on the plumeaxis. The plume radii (G and GT ) are defined as typical length scales in a plume crosssection. All these scales depend on the vertical length scale L ( x ). In the thermal plume ( 0 r GT ), the different terms of equation [5.96] have the following order of magnitude: – advection term, u wT v wT  U 4
wx wr L § · D w w T 4 – diffusion term, ¨r ¸ D 2 r wr © wr ¹ GT
A first relation between the scales is deduced from the balance between these two terms:
GT 2 
DL U
[5.97]
External Natural Convection
179
In the velocity plume ( GT r G ) there are no buoyancy forces and the flow is due to entrainment by viscous forces. Equation [5.94] simplifies in wu w u Q w § w u · u v ¨r ¸. The viscous term is estimated by developing the wx wr r wr © wr ¹ derivative as:
Q w § wu · ª1 wu w 2 u º U 2 » Q 2 ¨r ¸ Q « r wr © wr ¹ « G ¬r wr wr » ¼ The balance between inertia and viscous forces therefore yields a result, the velocity plume radius satisfies:
G2 
QL
U2 L
U  Q 2 . As G
[5.98]
U
Equations [5.97] and [5.98] show that G GT  Pr1 2 . In the thermal plume ( 0 r GT ), velocity variations are of the order of U over the transverse distance GT . The viscous term is therefore estimated by:
Q w § wu · ª1 wu w 2 u º U 2 » Q 2 ¨r ¸ Q « r wr © wr ¹ ¬ «r wr wr » GT ¼ The order of magnitude of the different terms of equation [5.94] is obtained by accounting for [5.98] as: u
wu wu v wx wr
U2
gE4
L
gE T Tf
Q
Q w § wu · ¨r ¸ r wr © wr ¹
U G2 U2  Q  Pr L GT 2 G 2 GT 2 U
The viscous term is dominant with respect to the inertia term, from which it follows that: U2 L

gE4 Pr
[5.99]
180
Convective Heat Transfer
A last relation between the scales comes from the conservation of enthalpy rate convected through a crosssection at height x. Considering a control domain delimited by a crosssection below the sphere, a crosssection at height x and a cylinder of radius R (>> G) as shown in Figure 5.16, the energy budget yields
³0f UC p uT Tf 2 Srdr q
[5.100]
where the temperature is referred to Tf and q is the heat transfer rate supplied by the sphere to the fluid. It is worth noting that the stream entering through the lateral surface of the control domain does not contribute to the energy budget ( T Tf ).
Figure 5.16. Axisymmetric plume. Control domain for the energy budget
In equation [5.100], the different variables are estimated by: u  U , T Tf  4, r  GT .
These scales depend on the height x only and can be taken out of the integrand so that they satisfy: U4GT 2
Constant independent of L
[5.101]
External Natural Convection
181
More precisely, there is a similarity solution for the velocity and thermal fields to the plume problem [GEB 71] so that U4GT 2
a
q
UC p
[5.102]
where a is a numerical constant, which depends on the shape of the velocity and temperature profiles. Combining [5.97] and [5.102], we determine the temperature scale on the plume axis: 4
c
q kL
[5.103]
Combining [5.97], [5.99] and [5.102], the velocity scale is found to be: U
§gEq ·1 2 1 b¨ ¸ © k ¹ Pr1 2
[5.104]
The velocity scale U is independent of the vertical length scale, i.e. of the height of the crosssection considered for the computation. Finally, the transverse length scale is:
GT
§kQDL2 ·1 4 ¸ d ¨ ¨ gEq ¸ ¹ ©
[5.105]
Relations [5.103] and [5.105] show that the temperature scale 4 decreases as 1/x, whereas the plume radius increases as x . Introducing the Rayleigh number Ra L
gEqL2 kQD
[5.106]
equations [5.104] and [5.105] are expressed in the form UL
D
 Ra L 1 2
[5.107]
182
Convective Heat Transfer
GT L
 Ra L 1 4
[5.108]
[GEB 71] gives the following result, obtained from the similarity solution 4
cPr
q
[5.109]
2Skx
where c(Pr) is given in Table 5.10. Pr c(Pr)
102
0.7
1
2
10
0.759
0.687
0.667
0.625
0.561
0.5
Table 5.10. Axisymmetric laminar plume. Coefficient in the temperature law
Table 5.10 shows that the coefficient c slightly depends on the Prandtl number, in agreement with scale analysis (equation [5.103] with L = x). [PAN 03] conducted a numerical modeling of axisymmetric and twodimensional plumes with or without fluid physical properties variations. When the plume axisvelocity is presented in the form U axis x x D b( Pr )Ra1x1 2 , the author gives the results set out in Table 5.11 in the axisymmetric case and the constant fluid gx 3 Uf U axis is physical properties. The Rayleigh number as given by Ra1x
QD
identical to Ra x ([5.106] with L = x) apart from a multiplying coefficient. Pr
1263
2310
4696
10785
b(Pr)
3.45
3.61
3.84
4.09
Uf
Table 5.11. Laminar plume. Coefficient in the temperature law, from [PAN 03]
We note that the variation of b with Pr is weak (variation of 18% for a variation of Pr by a factor 8.5), confirming again the scale analysis results (equation [5.107]).
External Natural Convection
183
5.13. Heat transfer through a glass pane 5.13.1. Description of the problem A vertical glass pane of thickness e, height L and span length Lz separates a room, where the ambient air is at temperature Ti , from the external air at temperature Te . It is assumed that heat transfer is due to laminar natural convection on both sides of the pane. The problem is considered as twodimensional. Numerical data: L = 1 m, e = 2 mm, Ti = 20 °C, Te = 0 °C, glass thermal conductivity kv = 0.70 W m1 K1. The physical properties of air are supposed to be constant and are given in Table 5.12. Density Kinematic viscosity Specific heat at constant pressure Prandtl number Thermal conductivity Coefficient of thermal expansion
Uf = 1.247 kg m3 Qf = 1.41 105 m2 s1 Cpf = 103 J kg1 K1 Prf = 0.7 kf = 0.025 W m1 K1 E = 1/283 K1
Table 5.12. Physical properties of air
It is asked that the heat transfer rate through the pane is calculated. 5.13.2. Guidelines Describe qualitatively the flow on each side of the pane. What is the heat flux distribution expected along the pane? What are the dimensionless numbers that characterize heat transfer? Show that the conductive thermal resistance of the pane is negligible compared to the thermal resistance due to natural convection. It is proposed that a first simplified calculation is conducted by assuming uniform heat flux along the pane. Choose a height to determine the heat flux and use a known heat transfer law of natural convection in the calculation. A more precise calculation is possible by using a principle of local similarity. Calculate the heat flux distribution by using locally the heat transfer law of natural convection along a vertical flat plate at uniform heat flux. Calculate the total heat transfer rate through the pane in both approaches.
184
Convective Heat Transfer
5.13.3. Solution 5.13.3.1. Description of the flow and scale analysis The pane temperature is obviously intermediate between the two extreme temperatures Ti and Te . Inner air is hotter than the pane and is therefore cooled at its contact. It then flows downwards under the influence of buoyancy forces. Conversely, external air is colder than the pane and is heated at its contact. It then flows upwards. We therefore expect the configuration of two conjugate boundary layers, as shown in Figure 5.17. In the pane, the transverse conduction is dominant with respect to the longitudinal conduction because e L . At given height, the thermal resistance opposing the transfer of heat from the room to outside consists of three resistances placed in series. Two of them correspond to the boundary layers generated by natural convection. The third one is the conductive thermal resistance of the pane. Denoting G i and G e the boundary layer thicknesses on the two sides of the pane, the convective thermal resistances are of the order of G i k f and G e k f respectively. Figure 5.17 shows that the convective thermal resistances, which depend on x, partially compensate along the pane. At the pane bottom, G e k f 0 , but G i k f is maximum. The opposite occurs at the pane top. On the other hand, it is known that the boundary layers grow slowly except very near the origin.
Figure 5.17. Boundary layers generated by natural convection near a pane. Longitudinal section
As a result, on the hot side for example, G i k f slowly varies between the middle and the bottom of the pane. A maximum in the convective thermal resistance and a subsequent minimum in the heat flux are expected in the central part of the pane since the sum of the two thermal resistances associated with the boundary layers is likely to be maximum in this region. It is also expected that the total convective thermal resistance and the heat flux vary slightly in the central region of the pane.
External Natural Convection
185
Scale analysis of section 5.3 (Table 5.1) shows that for Pr ~ 1, GT L  Ra L 1/4 . Otherwise, the conductive thermal resistance of the pane is e k v . The ratio of the thermal resistances is therefore: Thermal resistance by convection Thermal resistance by conduction
Denoting 4 0
kv L 1 k f e RaL1 4
T i T e , we obtain with the data of the problem:
gE4 0 L3
Ra L

QD
= 2.44 109,
Thermal resistance by convection Thermal resistance by conduction
 63
This estimation demonstrates that the conductive thermal resistance may be ignored with respect to the convective thermal resistance. This is equivalent to ignoring temperature variations in a crosssection of the pane, in other words to consider a uniform pane temperature Tw x at given height x. 5.13.3.2. Global approach In a first approach, we ignore the variations of heat flux along the pane and we use the heat transfer law [5.35] for natural convection along a vertical plane heated at uniform wall heat flux. Thus
Nu(x)
Nu(x)
15 Ra *x
bPr
q0cc
with b(Pr)
k Tw (x) Tf x
§ ·1 5 Pr ¨ ¸ , ¨4 9Pr 1 2 10Pr ¸ © ¹
and Ra x *
gEq0ccx 4
QDk
.
where q0cc denotes the heat flux. The reasoning of the previous paragraph suggests that we apply this relation with x = L/2, where the pane temperature is T i T e 2 , for symmetry reasons. Nu(x
L 2) rk
q0cc with Tf Tw (x L 2) Tf
T i or Tf
T e , according to
L2 whether we consider the heat exchange with inner or external air. In both cases, Tw (x L 2) Tf 4 0 2 .
186
40
Convective Heat Transfer
Heat transfer is characterized by a global Nusselt number, built with T i T e and the total height of the pane L: q0cc
NuL
[5.110]
k f 40 L
With this definition, we find Nu L Nu(x L 2) . The reason is that Nu L has been defined with 4 0 and not with the temperature difference between the wall and the far fluid. Denoting Ra L4 0
gE4 0 L3
QD
equations is given by Ra *x
L 2
, the Rayleigh number used in the previous Nu L 24
Ra L4 0 .
Heat transfer law [5.35] becomes §Nu ·1 5 bPr ¨ 4L Ra L4 0 ¸ © 2 ¹
Nu L
or 54
Nu L Ra L4 0 1 4
bPr
[5.111]
2
For Pr = 0.72, equation [5.35] gives b(Pr )
NuL RaL4 0 1 4
0.519, from which it follows:
0.220
With the numerical data, Ra L4 0 = 2.44 109, Nu L = 48.9. q0cc q Lz
k f 40 L NuL = 24.4 W m
2
q0ccL = 24.4 W m1
External Natural Convection
187
5.13.3.3. Local similarity approach The heat transfer law [5.35] is used with the local heat flux qccx , which is now a function of x. We write a relation between qccx and the temperature difference Ti Tw x on the inner side on the one hand, and between qccx and the temperature difference Tw x Te on the external side on the other hand. As a result, we obtain a system of two equations in the two unknowns qccx , Tw x . x
Denoting q1cc k f 4 0 L , x *
L
, T x*
Tw x * Te 40
and qcc* x *
,
qcc x * q1cc
the Nusselt and Rayleigh numbers are
T x
M * x * x*
Nu x
*
, Rax*
RaL40 M * x* x*4
so that the heat transfer law, written on the external side becomes
*
qcc x
*
T x* x
54 54
bPr
*1 4
Ra L4 0 1 4
[5.112]
The heat transfer law on the inner side is obtained by substituting x * by 1 x * and T by 1 T in the above relation (note that it is the same heat flux on the internal and external sides of the pane, owing to continuity of heat flux through the pane):
q " * x*
1 x
ª1 T x* º ¬ ¼ *
54
14
b Pr
54
RaL40 1 4
Combining the two equations, we obtain
T x* x
54
*1 4
1 x
54 ª * º 1 T x « » ¬ ¼ *
14
[5.113]
188
Convective Heat Transfer
from which we deduce the temperature and heat flux distribution along the pane 15
x*
T x
*
15
x*
1 x*
qcc* x * *
qcc max
1 ª *1 5 1 x* «x ¬
54
bPr
with qcc*max
[5.114]
15
[5.115]
54 1 5 º
» ¼
RaL4 0 1 4 .
Figure 5.18 confirms that the heat flux is minimum at the middle of the pane and slightly varies in its central part ( 0,1 x * 0, 9 ). Conversely, qcc* strongly increases near the ends of the pane ( qcc*max 2qcc*min ) and the pane temperature varies most in these regions. Owing to its definition, the dimensionless temperature is zero at the pane bottom. The pane is not insulated by the external boundary layer (Figure 5.17) and is at the external ambient temperature at this end. Conversely, T 1 at the pane top where the temperature is that of internal ambient air. 1.0
qcc* qcc* max
0.9
T
0.8 0.7
qcc* qcc* max
0.6 0.5 0.4
T
0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 x*
Figure 5.18. Natural convection near a pane. Temperature and heat flux distribution along the pane
1.0
External Natural Convection
189
The total heat transfer rate per transverse unit length is:
³
q Lz
L
0
q" x dx q1" L
³ q x dx 1
"
*
*
0
The integral is easily calculated by a numerical method,
³ q x dx 1
0
"
*
*
= 0.533.
Using the definition of Nu L , the result is: Nu L RaL4 0
14
54
0.533bPr
[5.116]
It is worth noting that the first approach gave a very similar result ([5.111]) (coefficient 0.5 instead of 0.533). For Pr = 0,72, the heat transfer law is
Nu L Ra L4 0 1 4
= 0.235.
When the same calculation as in the first approach is conducted, the previous result gives the heat transfer rate par transverse length:
q Lz = 26 W m1 5.14. Mixed convection near a vertical wall with suction 5.14.1. Description of the problem We consider mixed convection near a vertical plate, as shown in Figure 5.19. The flow results from a combination of natural and forced convection. The wall is porous and is crossed by a uniform suction velocity flow (velocity v0 , counted positively for a suction). Suction prevents the thermal and velocity boundary layers from growing and, as a consequence, they are spatially uniform dG dx dGT dx 0 . Determine: – the temperature and velocity distribution; – the friction coefficient C f W w Uv02 . Separate the contributions of natural and forced convection to C f . Discuss. This problem is inspired by [ARP 84].
190
Convective Heat Transfer
5.14.2. Guidelines Introduce the dimensionless variable K v 0 y D . Determine the temperature distribution. Look for solutions of the type § K · uK C1 expK C 2 exp¨ ¸ Uf © Pr ¹
for the velocity. Determine the constants C1 and C 2 .
dG dx
x T0
0
u(y) Uf
u
Tf
v
v0
g
y
0
Figure 5.19. Mixed convection on a porous vertical wall
5.14.3. Solution Considering that the flow is fully developed with dG dx we may write: u
uy
T Tf
T y Tf
T y
Moreover, the continuity equation
wu wv wx wy
0
0 and dGT dx
0,
External Natural Convection
191
implies
dv dy
0 and consequently v
Constant
v0
The momentum and energy equations become: u
wu wu v wx wy
v 0
wT wT u v wx wy
du dy
dT v 0 dy
Q
d 2u dy 2
D
gE T Tf
d 2T
[5.117]
dy 2
As a result, the temperature and velocity fields are uncoupled. System [5.117] is subjected to the following boundary conditions:
u0 0, uf Uf
[5.118]
T 0 T 0 , T f Tf Denoting T 0
T0 Tf , the energy equation is easily integrated as:
T y T y Tf
§ v 0 y · ¸ © D ¹
T 0 exp¨
[5.119]
This equation shows that the dimensionless variable K v 0 y D is relevant for the temperature distribution T K T 0 expK , which is completely governed by suction at the wall. The momentum equation takes the form of a nonlinear inhomogeneous differential equation:
Q
d 2u dy
2
v0
du dy
§ v y · gE Tf T 0 exp¨ 0 ¸ © D ¹
[5.120]
The velocity distribution uy is the sum of the solution to the homogeneous part of this equation and a particular solution. Following the guidelines given in section 5.14.2, uy is of the form: § K · uK C1 expK C 2 exp¨ ¸Uf © Pr ¹
192
Convective Heat Transfer
The coefficients C1 and C 2 are determined from the boundary conditions [5.118]. We obtain:
uK Uf
v 02
DgET 0
1 Pr Uf
e K e K 1 e K
Pr
The limit of expression [5.121] for Pr 1 ( Q
uK
QgET 0 v 02Uf
Uf
KeK 1 eK
Pr
[5.121]
D ) is:
[5.122]
This is a problem of mixed convection, i.e. a combination of natural and forced convection. We expect therefore to recover the classical dimensionless numbers which govern the mechanisms involved in these phenomena, in other words the Grashof number and the Reynolds number respectively. Let us recall that the first one is given by: Gr
gET 0
Q
2
L3
The length scale L is clearly imposed in the present problem by the characteristic thermal scale Lt D v 0 , which appears in K . The dimensionless group that appears in equation [5.121] may be written in the form
v 02
DgET 0
1 Pr Uf
§ 3 º¨ ª § · «gET 0 ¨D ¸ »¨ Q 2 « Q ©v 0 ¹ »¨U D ¼¨ f ¬ © v 0
· ¸ Pr ¸ ¸1 Pr ¸ ¹
which obviously may be transformed as v 02
DgET 0
1 Pr Uf
GrLt Pr ReLt 1 Pr
(Pr z 1)
For Pr 1 , the characteristic thermal scale also becomes the viscous length LQ Q v 0 Lt . The dimensionless group in equation [5.122] may in turn be written in the form:
QgET 0 v 02Uf
§ ª § Q ·3º¨ Q ET g 0 « ¨ ¸ »¨ Q « Q 2 ©v 0 ¹ »¨ ¬ ¼¨Uf © v 0
· ¸ ¸ ¸ ¸ ¹
§ ª § Q ·3º¨ Q g ET 0 « ¨ ¸ »¨ D « Q 2 ©v 0 ¹ »¨ ¬ ¼¨Uf © v 0
· ¸ ¸ GrLt ¸ ReLt ¸ ¹
External Natural Convection
193
In sum, the velocity distribution is expressed as: uK
e K e K 1 e K
Uf
GrLt Pr ReLt 1 Pr
Uf
GrLt K Ke 1 eK ReLt
uK
Pr
Pr
Pr z 1 [5.123]
Pr
1
These relations are easily interpreted. The first groups of terms in the righthand side represent the contribution of natural convection to the velocity field. This contribution is canceled when GrLt o 0 and/or ReLt o f . It is worth noting that the dimensionless number governing this problem of mixed convection is Gr Re and not Gr Re 2 , as in the general case (section 5.7). This comes from the fact that the inertia term is proportional to U and not to U2 (equation [5.120]). The friction coefficient is Cf
Ww Uv02
P wu wy 0 Uv02
QgET 0 v03 Pr
Uf v0
and may be rearranged as Cf
GrLt Pr 2 ReLt Pr
[5.124]
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Chapter 6
Internal Natural Convection
6.1. Introduction The previous chapter was devoted to external natural convection flows. In unconfined configurations, flows can develop freely under the influence of the various forces brought into play. In many situations, however, confinement effects are present because the space offered to flows and heat transfer has limited dimensions, so that distances between walls are of the order of the length scale of the flow or are even smaller. These confined situations correspond to natural convection in open ducts (chimneys) or in closed cavities. Main applications are in the domain of housing, in particular solar energy, and also in the domain of nuclear safety (in the case of pump failure) or concern the cooling of electrical or electronic components. In these configurations, which we restrict to twodimensional problems, a new length scale (2e) in the transverse direction comes in addition to the scales relevant to external natural convection. The flow and associated heat transfer then depend on the dimensionless numbers RaL (or Ra2e ), Pr and 2e/L, where L is the vertical length scale. 6.2. Scale analysis Internal natural convection may be present in two very different types of situations (Figure 6.1): – flows in open ducts (chimneytype); – flows in closed cavities.
196
Convective Heat Transfer
x
x T1 Tw
L
G
2e
T2 y
L
g
2e T0
y
Figure 6.1. Natural convection in ducts or enclosures
In the first case, a duct is delimited by two plane walls (in twodimensional geometry) or a cylindrical wall, heated (or cooled) at temperature Tw higher (or lower) than that of ambient fluid, T 0 . Buoyancy forces are generated near the walls and, in the case of heating, give rise to an upward flow inside the duct. The external fluid enters the duct at temperature T 0 and then is heated by the walls. Boundary layers grow on the walls from the bottom of the duct. The fluid is rejected as a plume into the environment at the top end of the duct. In the second case, corresponding to closed spaces, the fluid recirculates inside the enclosure. Several wall heating modes are possible. In the example of Figure 6.1, a vertical wall is at temperature T1 and the opposite wall is at temperature T2 ( T1 ! T2 ). Horizontal walls that close the box may be insulated. In these conditions, the fluid rises along the hot wall and goes down along the cooled wall. Temperature profiles rearrange themselves along the adiabatic walls. These two types of situations show common characteristics, which are presented below. The choice of reference temperature is not obvious in the case of internal flows. Contrary to the case of external natural convection, for which the ambient fluid temperature was the natural choice of reference, an arbitrary temperature has to be chosen for confined flows. In the example of the closed enclosure shown in Figure 6.1, it is possible to choose the average T1 T 2 2 , which presents the advantage of establishing a symmetric temperature field. For a duct heated at constant temperature, the wall temperature may be chosen as a reference. It is worth underlining the fact that the reference temperature T ref must be linked to the reference density U ref in the definition of the modified pressure p* p U ref gz ( U ref is the fluid density at temperature T ref ).
Internal Natural Convection
197
Depending on the geometry, the fluid physical properties and heating conditions, two asymptotic cases may be distinguished: – a regime of distinct boundary layers; – a fully developed regime in the main part of the duct or the enclosure. The distinction between the two cases may be simply made by considering the thermal boundary layer thickness for a wall of length L, as given in the laminar regime by the scale analysis conducted in Chapter 5 (for practical cases, we consider Pr > 1)
G L
 Ra L 1 4
with Ra L
gE'TL3
QD
, 'T
T1 T 2 for an enclosure, 'T
Tw T0 for a duct.
The limiting case corresponds to the conditions for which the boundary layers on the two walls merge at the end of the duct or at the middle of the enclosure. In both cases, this criterion may be expressed by the dimensionless number e L Ra L 1 4 . When e L Ra L 1 4 !! 1, the boundary layers thickness is much smaller than e at the end of the duct or at the middle of the enclosure. The flow has the configuration of two distinct boundary layers (Figure 6.1) so that the results presented in Chapter 5 may be used to calculate heat transfer between the fluid and the walls. 4
When e L Ra L 1, the boundary layers merge very close to the duct inlet so that the flow is fully developed in the main part of the duct. In a closed enclosure, the boundary layers interfere close to the corners where they originate. It is worth noting the different exponents for the dimensionless number e L Ra L 1 4 in the two situations. These come from the different criterion used to define the limiting cases. 6.3. Fully developed regime in a vertical duct heated at constant temperature Assuming that the velocity field is independent of x and that the fluid temperature is equal to that of the walls ( Tw ) on most parts of the duct, the velocity profile is found to be parabolic uy
2 gE Tw T0 e 2 ª §y · º «1 ¨ ¸ » « 2Q ¬ ©e ¹ » ¼
[6.1]
198
Convective Heat Transfer
where T0 is the ambient fluid temperature. The flow rate per unit transverse length is given by: Q
2 gE Tw T0 e 3
Lz
3Q
[6.2]
Denoting by q the total heat transfer rate, the heat transfer law may be written in the form Ra 2e
Nu L
[6.3]
24
with Nu L k
q 2Le and Ra2e Tw T0
3
gE Tw T0 2e
QD
.
L
6.4. Enclosure with vertical walls heated at constant temperature 6.4.1. Fully developed laminar regime We consider an enclosure (Figure 6.1) of very long span length Lz (!! L, e ) where the flow and temperature fields are twodimensional. When the aspect ratio e L is sufficiently high and the Rayleigh number sufficiently weak so that 4
e L
RaL 1 , the flow and temperature fields are fully developed in most parts of
the enclosure ( w wx 0 ). The velocity profile is symmetric with respect to the vertical axis originating at the center of the enclosure: uy
gE T1 T 2 e 2 ª§y · §y ·3º «¨ ¸ ¨ ¸ » «¬© e ¹ ©e ¹ »¼ 12Q
[6.4]
The temperature profile is linear and, consequently, the heat transfer rate q between the walls is the same as in pure conduction: T y
Nu
T1 T 2 2
q Lz L
T1 T 2 §y · ¨ ¸ 2 ©e ¹
[6.5]
1
[6.6]
k T1 T2 2e
Internal Natural Convection
199
6.4.2. Regime of boundary layers When the enclosure is sufficiently large and the Rayleigh number sufficiently high that e L Ra L 1 4 !! 1, the flow is characterized by boundary layers along the walls (Figure 6.1). The analysis of [BEJ 95] shows that the heat transfer law is: Nu
0.364
2e L
RaL1 4
[6.7]
[BEJ 95] indicates that equation [6.7] is valid for:
L
4 7
2e
Ra L 1 7 ! 100
6.5. Thermal insulation by a doublepane window 6.5.1. Description of the problem We consider the flow and heat transfer in a doublepane window. The two panes are separated by an air layer where the fluid is set in motion by buoyancy forces, as illustrated in Figure 6.2. The air layer thickness 2e is very small compared to the height of the panes L. It is assumed that the temperatures of the two panes are uniform, and they are denoted T1 and T2 respectively. The purpose of the problem is to determine the optimal spacing, which gives the best thermal insulation. The physical properties of air are given in Table 6.1.
Figure 6.2. Flow in a doublepane window. T1 > T2 (T1 = 20°C, T2 = 10°C, L = 1 m)
200
Convective Heat Transfer
Kinematic viscosity
Pr Q = 1.5 105 m2 s1
Thermal conductivity Coefficient of thermal expansion
kf = 0.024 W m1 K1 E= 1/283 K1
Prandtl number
Table 6.1. Physical properties of air
6.5.2. Solution 6.5.2.1. Qualitative analysis When the air layer thickness is increased in the doublepane window, the dimensionless parameter e L Ra L 1 4 also increases and there is a transition from the conductive regime to the regime of boundary layers for the flow and associated heat 4 transfer. In the conductive regime corresponding to e L Ra L 1, air circulates in the layer between the two panes. The streamlines are parallel to the vertical walls, except at the top and bottom ends of the panes (Figure 6.2). If end effects are not taken into account, the flow does not play any role in heat transfer, which is governed by equation [6.6]. The thermal resistance is proportional to the air layer thickness in this regime. Conversely, when e L Ra L 1 4 !! 1, the thermal resistance corresponds to the boundary layers that develop on the vertical walls. These distinct boundary layers and associated thermal resistance do not change when e is further increased. It is therefore inefficient to increase the doublepane spacing too much. The thermal resistance of the doublepane window R is defined with the reference surface S Lz L as 'T T1 T2 Rq RS qcc and is plotted in Figure 6.3 for the two limiting cases described above. The previous reasoning indicates that RS is represented in this graph by a straight line passing through the origin for the weak values of e and by a horizontal line for the high values of e. S Regime of boudary layers
Conductive regime e eoptimum Figure 6.3. Doublepane window. Variation of the thermal resistance as a function of spacing
Internal Natural Convection
201
In a first approximation, the optimal spacing is obtained as the abscissa of the intersection point of the two lines corresponding to the limiting cases. 6.5.2.2. Calculation of the optimal spacing Writing the identity of the thermal resistances as given by [6.6] and [6.7] leads to the estimation of eoptimum 2eoptimum
L
k
0.364kRaL1 4
or 2eoptimum L
RaL1 4
1
with the data of the problem, RaL 2eoptimum L
[6.8]
0.364 9.81 x 10 x 0.7
283 x 1.5 x 10
5 2
= 1.08 109, and we find
= 1.5 102.
For a window 1 m in height, the optimum thickness is 15 mm. The heat transfer rate per transverse unit is calculated by using [6.6] or, equivalently, [6.7]. Using [6.6]: q Lz
0.024 x
10 1.5 x 10 2
= 16 W m1
6.6. Natural convection in an enclosure filled with a heat generating fluid 6.6.1. Description of the problem Heat generation inside a fluid plays an important role in nuclear power plants and in chemical reactors. In the first case, natural convection is present in the problems of nuclear safety. We consider the very simplified situation of a parallelepipedic enclosure of height L, width D and span length Lz (Lz >> D, L) filled with a radioactive fluid. Heat is generated at uniform volumetric rate qccc(W m3) inside the fluid. The fluid is cooled by the vertical walls at constant temperature T0 . The horizontal walls are adiabatic.
202
Convective Heat Transfer
For the high Prandtl number fluid (Pr >> 1) considered, the physical properties are the densityU, the specific heat at constant pressure Cp, the kinematic viscosityQ, the thermal diffusivity D and the thermal conductivity k. The xaxis is chosen along the ascending vertical direction. The vertical and horizontal velocity components are denoted u and v respectively. It is observed (in Figure 6.4) that: – the flow consists of a central part with upward velocities surrounded by boundary layers developing on the cooled walls with downward velocities; – the temperature field is stratified in the central part (horizontal isotherms) from T0 at the bottom of the reservoir to T1 at the top. This last temperature depends on the parameters of the problem. For further details, see for example [BER 80].
x T1 T0 U
L
G qccc
U1 y T0 D Figure 6.4. Cavity filled with a heat generating fluid
It is proposed that a very simplified model of the flow is used in order to obtain the relevant dimensionless numbers of the problem. We assume that: – the flow and heat transfer are twodimensional, steady and symmetric with respect to the vertical axis of symmetry of the cavity; – the fluid motion is governed by the perfect fluid equations in the central part of the cavity and is characterized by the uniform velocity U (independent of coordinates x and y);
Internal Natural Convection
203
– the axial diffusion is negligible in the whole cavity; – the heat volumetric source is negligible in the boundary layers; – the boundary layer thickness is small compared to D. Scale analysis is conducted by considering a horizontal plane at the middle of the cavity. We denote: – U the vertical velocity scale in the central region; – U1 the velocity scale in the boundary layer; –G the boundary layer thickness (the velocity and thermal boundary layer thicknesses are not distinguished); – 'T = T1 T0 the temperature scale. Find four relations between the different scales and the parameters of the problem by considering: – the energy equation in the central region; – the energy equation in the boundary layer; – the Boussinesq equation along x in the boundary layer; – the conservation equation for the flow rate. Solve the system. Deduce from the results an order of magnitude for the Nusselt number that characterizes heat transfer between the fluid and the walls and for the total heat transfer rate per transverse unit q L z . 6.6.2. Solution 6.6.2.1. Energy equation in the central region Energy equation [1.8] simplifies into U
wT wx
qccc
UC p
[6.9]
from which it follows that U
'T L

qccc
UC p
[6.10]
204
Convective Heat Transfer
6.6.2.2. Energy equation in the boundary layer u
wT wT v wx wy
w 2T
D
wy
2
qccc
[6.11]
UC p
The vertical velocity scale is U1 in the boundary layer. We suppose that the two advection terms (lefthand side) have the same order of magnitude, as in the general theory of boundary layer flows. Then, the order of magnitude of the various terms of equation [6.11] is: U1
'T
D
L
'T
qccc
2
UC p
G
According to [6.10], the term of internal heat generation is estimated by U 'T L . It is expected that the velocity U1 in the boundary layers is much greater than that of the core region owing to the conservation of flow rate. The term of internal heat generation is then negligible in the boundary layers so that thermal equilibrium is given by: U1
'T
'T D 2 L G
L U1  D 2 G
[6.12]
6.6.2.3. Boussinesq equation along the vertical direction in the boundary layer The Boussinesq equation along the vertical direction (equation [5.3]) reads u
wu wu v wx wy
1 dp*
U dx
gE T T 0 Q
w 2u wy 2
[6.13]
with p* p U 0 gx , U 0 being the fluid density at temperature T 0 . The pressure gradient may be eliminated by noting that the Boussinesq equation simplifies in the core region when the velocity is uniform
0
1 dp*
U dx
gE >T c ( x) T 0 @
[6.14]
Internal Natural Convection
205
where T c ( x ) denotes the core temperature at height x. Combining [6.13] and [6.14], we obtain u
wu wu v wx wy
gE >T T c ( x )@ Q
w 2u wy 2
[6.15]
with the following order of magnitude U 12
gE'T
L
Q
U1
G2
Taking [6.12] into account, the last term is also estimated by Pr U12 L . This is therefore the dominant term when Pr >> 1. As a result, we obtain a third equation between the characteristic scales: U12 
gE'TL Pr
[6.16]
6.6.2.4. Conservation of flow rate The total flow rate is zero through a horizontal crosssection:
³ 0G udy ³GD 2 Udy 0 Supposing that G D , it follows that:
U1G  UD
[6.17]
6.6.2.5. Calculation of the characteristic scales We have the system of four equations [6.10], [6.12], [6.16] and [6.17] to calculate the four characteristic scales for velocities (U, U1), length ( GT ) and temperature ( 'T ). The system is easily solved by elimination to find
G5 
QDk L gEqccc D
or denoting RaL 
gEqcccL5
QDk
[6.18]
206
Convective Heat Transfer
§L ·1 5  Ra L 1 5 ¨ ¸ ©D ¹ L
G
[6.19]
§L · 2 5  Ra L 2 5 ¨ ¸ ©D ¹ D
U 1L
[6.20]
4 5
§L ·  Ra L 1 5 ¨ ¸ ©D ¹ D
UL
'T 
4 4 5 §L · Ra ¨ ¸ L ©D ¹ gEL3
QD
[6.21] 5
The Nusselt number, defined by Nu case by Nu 
L
G
[6.22] qcc k 'T x
, is estimated as in the general
, which leads to:
§L · 1 5 Nu  Ra L 1 5 ¨ ¸ ©D ¹
[6.23]
The heat flux received by a wall is estimated by qcc k 'T G . Replacing 'T and G by the above results, we find qcc qcccD . The total heat transfer rate per transverse unit q L z is therefore of the order of qccL  qcccDL . This means that the total heat generated inside the cavity ( qcccx DLL z ) is transferred to the walls ( qccx LLz ). This is a verification of the global thermal equilibrium of the cavity. 6.7. Onedimensional mixed convection in a cavity 6.7.1. Description of the problem A fluid is confined in a vertical cavity, as illustrated in Figure 6.5. The walls are separated by a distance l and are maintained at uniform temperatures T1 and T 2 . We assume that the velocity and temperature fields only depend on the horizontal distance to the wall y . This hypothesis leads to a great simplification of the problem. It is, however, valid only if the cavity is very slender in the x direction. One of the walls moves with constant velocity U (Figure 6.5). Determine the flow in the cavity. This problem is inspired by [ARP 84].
Internal Natural Convection
207
6.7.2. Guidelines Write the conservation equation for the flow rate. Note that the energy equation reduces to a simplified conduction equation. Express the momentum equation in the form: 0
1 d gE d 2u p pr T T r 2 Q P dx dy
Explain why the reference quantities pr and T r are included in this equation. It is proposed that the velocity field be decomposed into two parts, namely: uy u1y u 2 y
T1
Moving wall
g x
U
y l
T2
Figure 6.5. Onedimensional flow with buoyancy forces in a slender cavity. One of the walls moves with the velocity U
The component u1 corresponds to the flow forced by the cavity wall motion. The component u 2 is generated by buoyancy forces. Write the equations and the boundary conditions for u1 and u 2 . Determine the unknowns pr and T r by using the integral equation for mass conservation.
208
Convective Heat Transfer
6.7.3. Solution We suppose that the velocity u uy only depends on y because the cavity is very slender. The continuity equation then reads:
wu wv wx wy
dv dy
0
[6.24]
As a result, v 0 . Otherwise, since the flow is confined in an enclosure, the integral continuity equation shows that: l
³ udy Constant 0
[6.25]
0
The constant is obviously zero since there is no flow across the impermeable cavity walls. We also suppose that T conduction equation:
u
wT wT v wx wy
0 D
T y . The energy equation then reduces to the
d 2T dy 2
[6.26]
The solution is very simple: T
T1
y y T 2 T1 T1 'T l l
[6.27]
The momentum equation along x yields:
0
1 d gE d 2u p pr T T r 2 Q P dx dy
[6.28]
It is worth noting that we have kept two parameters, namely the reference pressure pr and the reference temperature T r , in equation [6.28]. We recall that, in the classical formulation of natural convection along a vertical plate, the pressure gradient at infinity is linked to the buoyancy term at infinity, which results in the net term gE T Tf in the momentum equation. This procedure is not valid for the confined flow of the present problem. The term d p pr dx is kept in the equation for the same reason. There is effectively a pressure gradient induced by the moving wall because this is not a simple channel Couette flow, but a confined flow in a cavity, subject to constraint [6.25].
Internal Natural Convection
209
Equation [6.28] is linear with respect to velocity. Consequently, it is possible to use a superposition method. The velocity is decomposed as: uy u1y u 2 y
The first component u1 corresponds to the isothermal flow induced by the cavity wall motion. The component u 2 is generated by buoyancy forces. The system of equations with the associated boundary conditions is:
0
1 d
P dx
u1 0 gE
0
Q
u2 0
p pr
d 2 u1 dy 2
l
0, u1 l U, ³ u1 y dy
[6.29] 0
0
T Tr 0, u2 0
d 2 u2 dy 2
l
0, ³ u2 y dy
[6.30] 0
0
Integrating equation [6.29] twice and combining it with the associated boundary conditions, we obtain a velocity CouettePoiseuille profile: § y 1 §l 2 · d u1y U ¨¨ ¸¸ p pr ¨1 © l 2 ©P ¹dx
y · y ¸ l ¹ l
[6.31]
l
The unknown d p pr dx is determined by the condition ³ u1y dy
0.
0
We obtain:
d U p pr 6 P 2 dx l
Thus, the component u1y is: ª y §y ·2 º u1y U«2 3¨ ¸ » ©l ¹ » « ¬ l ¼
[6.32]
210
Convective Heat Transfer
We proceed in the same way with equation [6.30] to obtain:
u 2 y
3 2 ½ gE'Tl 2 °1 ªy §y · º 1 §T 0 T1 ·ªy §y · º° ® « ¨ ¸ » ¨ ¸« ¨ ¸ »¾ Q ° ¼ 2 © 'T ¹« ¼° ¬l ©l ¹ » ¬l ©l ¹ » ¿ ¯6 «
[6.33]
l
The unknown T 0 is deduced from the mass conservation equation ³ u 2 y dy
0:
0
T 0 T1
1 'T 2
[6.34]
2 §y ·3º gE'Tl 2 ªy §y · « 3¨ ¸ 2¨ ¸ » ©l ¹ ©l ¹ » Q « ¬l ¼
[6.35]
Finally: u 2 y
The velocity distribution is the sum of u1y and u 2 y : uy ª y §y ·2 º gE'Tl 2 ªy §y ·2 §y ·3º «2 3¨ ¸ » « 3¨ ¸ 2¨ ¸ » ©l ¹ » ©l ¹ ©l ¹ » U QU « « ¬ l ¼ ¬l ¼
[6.36]
The first component comes from forced convection and the second from natural convection. The coefficient of the second term may be rearranged in order to bring out the physical parameters as gE'Tl 2 QU
gE'Tl 3 Q Q 2 Ul
Gr Re1
where the Grashof and Reynolds numbers are respectively Gr
gE'Tl 3
Q
2
, Re
Ul
Q
.
It is worth noting that the dimensionless number that governs this mixed convection problem is Gr Re and not Gr Re2 , as in the general case (Chapter 5). This comes from the fact that the inertia terms are zero in the momentum equation so that the velocity contributes linearly to the viscosity term and not by its square to the inertia term.
Chapter 7
Turbulent Convection in Internal Wall Flows
7.1. Introduction This chapter deals with internal turbulent flows in channels and ducts. The turbulence is significantly affected by the presence of the wall. The wall turbulence can be classified into two main categories. The first category deals with flows confined in space, such as turbulent flows in channels or ducts. Turbulent boundary layers constitute the second class. Both flows are similar in many aspects, especially near the wall, but there are also some significant differences between them, especially in the outer layer. Turbulent flows are of considerable industrial importance. Despite important progress made in the domain, in particular in the last four decades, turbulence still remains enigmatic in many aspects, both in the physical comprehension of the phenomena and its modeling. 7.2. Hydrodynamic stability and origin of the turbulence One of the classical tools used to investigate the transition from laminar to turbulent flows is the theory of linear stability. The theory deals with the amplification or decay of small perturbations imposed to the basic laminar flow. If the perturbations are amplified in time and/or in space, the flow is linearly unstable and there is transition to turbulence. The hydrodynamic stability theory is an entire branch of fluid dynamics and is beyond the scope of this book. The interested reader may consult the vast literature on the subject and, for example, [DRA 81] to begin with. We restrict ourselves to a short discussion on some basic aspects of the linear stability of Poiseuille flow here.
212
Convective Heat Transfer
Consider the laminar Poiseuille flow in a twodimensional channel. Suppose that the flow is slightly perturbed by twodimensional unsteady protuberances. The instantaneous velocity and pressure field will be ux, y,t U y ucx, y,t v x, y, t vcx, y,t wx, y,t 0
px, y,t P(x, y) pcx, y,t
where U y is the base (Poiseuille) flow and the perturbations c v H are small. Replacing these quantities into the NavierStokes equation and ignoring the terms of the order of H 2 (i.e. linearizing) we obtain:
w uc w uc dU 1 w p c U vc Q 2 u c wt wx dy U w x w vc w vc 1 w pc U Q 2 v c wt wx U wx
[7.1]
The stream function \ x, y, t ] y exp ª¬i J x E t º¼ is introduced to solve
1 , the perturbation the resulting system of equations. In the \ expression, i wavelength J is real, while E E r iE i and ] y ] r i] i are complex numbers. According to the form of the stream function, it is obvious that perturbations with E i ! 0 will be amplified, leading to the linear instability of the system. Using uc w\ wy , vc w\ wx and eliminating the pressure in [7.1], we obtain the OrrSommerfeld equation:
U c ] cc J 2] U cc]
i ] cccc 2J 2] cc J 4] J Re
[7.2]
The derivatives with respect to the wall distance y are denoted by c in equation [7.2]. The Reynolds number Re is based on the centerline velocity and the half channel height and c E J . In order to resolve the initial value problem associated with [7.2], we fix * 2S J and Re , and calculate c c r ic i by using [7.2] with the appropriate boundary conditions. The flow is unstable if c i ! 0 . Figure 7.1 shows the neutral stability curve of a Poiseuille flow. Critical Reynolds numbers and wave numbers emerge from this figure and are respectively Rec = 5,772 and *c 1.02 . The flow is linearly unstable when Re t Rec . The transitional Reynolds number Retr is, however, significantly smaller because the Poiseuille flow is subcritical due to the nonlinearity. It is often agreed that
Turbulent Convection in Internal Wall Flows
213
Retr = 2,000, but the lower critical number beyond which perturbations can be sustained to lead to a transitional spot is in fact Retr = 1,000.
UNSTABLE
Figure 7.1. Neutral (linear) stability curve of a Poiseuille flow
7.3. Reynolds averaged NavierStokes equations
Consider the instantaneous NavierStokes equation for an incompressible fluid, for example, in the streamwise x direction
wu wu 2 wuv wuw wt wx wy wz
1 wp Q 2 u U wx
[7.3]
where u , v and w are the instantaneous components of velocity in the directions x, y and z , respectively, p is the instantaneous pressure and Q 2 u regroups the molecular viscous diffusion terms. This equation is valid in time and space anywhere in the flow, and must be associated, of course, with the appropriate G boundary and initial conditions. The instantaneous components u i u i x, t of the turbulent velocity field are not only a function of time, but also of space, and the flow is strongly unsteady, random (although some coherence may persist) and threedimensional. The instantaneous equations have to be averaged in time, G if we deal with the long time behavior of the flow field u . Thus, each G G G quantity qx , t is decomposed into a time mean q x and fluctuating qcx, t component with, obviously, qc 0 . This decomposition yields, for example,
214
Convective Heat Transfer
ui u j
ui uic u j u cj
ui u j uicu cj and the crosscorrelation uci ucj is generally
different from zero and plays a fundamental role. Proceeding in the same way for each term of equation [7.3], averaging and rearranging gives: u
wu wu wu v w wx wy wz
1 wp
U wx
Q 2 u
w ucuc w ucvc w ucwc wx wy wz
[7.4]
Generalizing to the other components of the velocity field enables us to write: uj
w ui wx j
1 wp
U wx i
Q 2 u i
w uci ucj wx j
[7.5]
This averaging procedure was been introduced by Reynolds more than a century ago, and equation [7.5] is called the Reynolds averaged NavierStokes equation. The continuity equation, on the other hand, holds true both for the time mean flow field with wu i wx i 0 and the fluctuating velocity field through wuci wx i 0 . The most important feature of equation [7.5] is the presence of the correlation terms uci ucj that distinguish turbulent and laminar flows. The terms U uci ucj are called Reynolds shear stresses and introduce six unknown quantities, for which there are no supplementary equations. We can, of course, continue the averaging procedure and obtain a transport equation for the Reynolds shear stress tensor, but then triple correlations will appear and so on. We therefore need to close the system at a given step, and the only way to proceed is to use physical or phenomenological arguments. This is the closure problem. One of the more or less direct methods is to relate U uci ucj to the shear wu i wx j in some way. The basic problem of turbulence is in finding adequate relationships to model the Reynolds shear stress tensor. The wall turbulence closure is the key problem addressed in this chapter. G Decomposing the temperature in a time mean T x and fluctuating component G T cx , t , and using the same procedure as for the velocity field raises into the scalar transport equation:
Ucu j
wT wx j
k 2 T Uc
w ucjT c wx j
[7.6]
The three Uc ucj T c terms are connected to the heat flux regenerated by turbulence. All these aspects will be clarified later on in this chapter.
Turbulent Convection in Internal Wall Flows
215
7.4. Wall turbulence scaling
We will begin by introducing relevant velocity and length scales to correctly describe the physical process of the wall turbulence and wall turbulent transfer. What we mean by scaling will become progressively clearer when we introduce the notions of eddy viscosity and mixing length hypothesis. There are basically two distinct zones in a turbulent wall flow: a region next to the wall wherein the turbulence is governed by the inner scales and a zone relatively far away from the wall in which the outer scales are relevant. The wall shear stress Ww
P wu wy w and the kinematic viscosity Q characterize
the zone immediately adjacent to the wall. We introduce a fictitious velocity scale uW Ww U based on the shear at the wall, and which is called the shear velocity. This velocity scale well represents the wall turbulence physics since without the shear (vorticity), the turbulence cannot be sustained and Ww PZzw , where Zzw wu wy w , is the spanwise component of the mean vorticity field, which is the quantity adequate near the wall. Thus, the length scale near the wall is based on the shear velocity and the viscosity and it reduces to lQ Q uW . The couple uW , lQ constitutes the inner scales (also called wall variables). A quantity q , nondimensionalized by the inner variables, will be denoted by q , like, for example, the mean velocity u
t
u uW , the crosscorrelations uci ucj
t uW2 Q , or the frequency f
uci ucj uW2 , the time
f Q uW2 given in wall units.
The equivalent of the wall shear stress in the context of wall heat transfer is the wall heat flux qccw k wT wy . The temperature scale in inner variables can w
qccw
qccw
, with l d D uW lQ Pr being the UcuW diffusive length scale. We assume here that qccw ! 0 . We will call Tqccw the wall flux therefore be expressed as Tqccw
k
ld
temperature. Note that there is a similarity between the wall shear (friction) velocity and Tqccw . In duct or channel flows, the velocity scale in the outer layer is either the centerline U c or the bulk velocity U m . The length scale in this zone is obviously the half channel height e . The outer scales are the large global scales of the flow, while the inner scales are related to the local near wall phenomenology of the turbulence.
216
Convective Heat Transfer
7.5. Eddy viscositybased one point closures Consider the terms
Q
w 2 u w u cvc wy w y2
· 1 w § wu U u cvc ¸ ¨P U wy© wy ¹
of the Reynolds averaged NavierStokes equation (called the Reynolds equation hereafter). The term U ucvc plays the physical role of a shear stress, as indicated before, and as it is underlined in the form presented in the preceding equation. We already indicated that the Reynolds shear stress is positive in a flow where wu wy ! 0 and that it increases the shear. We will give some qualitative arguments to show that U ucvc! 0 . Imagine that the fluctuating wall normal velocity is vct 0 , i.e. directed towards the wall at a given time and distance y . Since vct 0 at the wall by wvc 0 . We have impermeability, it can be easily seen that locally wy wuc wvc wwc wwc by continuity, and if we ignore the effect of the term , we wx wy wz wz wuc conclude that the zones wherein vct 0 coincide with ! 0 , so that globally wx uct ! 0 and vice versa. This is, of course, a rough argument. In reality, events with uct ! 0 associated with vct ! 0 , called quadrant 1 ( Q1 ) events (quadrant refers here to the uc vc distribution), together with Q2 ( uct 0 , vct ! 0 ), Q3 ( uct 0 , vct 0 ) and finally Q4 ( uct 0 vct 0 ) events, exist altogether. However, it turns out that the contribution Q2 Q4 to the turbulent shear stress dominates, resulting in U ucvc! 0 . The turbulent shear stress U ucvc is added to the viscous shear stress P wu wy to give the total shear Wtot . A usual and easy way to model U ucvc is to connect it to the local wu wy , since the turbulence production is inconceivable without mean shear. Thus, a fictitious viscosity Q t y , called the eddy viscosity, is introduced. The Reynolds shear stress is consequently expressed as: ucvc Q t y
wu wy
[7.7]
Turbulent Convection in Internal Wall Flows
217
The eddy viscosity is certainly not a physical characteristic of the fluid. It is not a constant, but varies in space. A universal eddy viscosity formulation does not (cannot) exist since it is strongly dependent on the turbulence phenomenology. The total shear stress can now be written as:
W tot
U ª¬Q Q t y º¼
wu wy
We can at the same time introduce an eddy diffusivity to model the turbulent transfer process. The related terms of the Reynolds averaged convection equation read:
D
w 2T w vcT c wy w y2
º 1 w ª wT U cvcT c» «k Uc w y ¬ w y ¼
The term Uc vcT c plays the physical role of a heat flux and, for the reasons given before, we obtain Uc vcT c! 0 when wT wy ! 0 . Let us introduce a thermal eddy diffusivity in such a way that: vcT c D t y wT wy
The total heat flux qcctot is therefore qcctot
k
wT wT Uc v cT c Uc>D Dt y @ wy wy
in a way that is very similar to the total shear stress. Incidentally, note the sign appearing in front of qcctot . This conveys the fact that the flux is conventionally opposite to the temperature gradient. We basically distinguish two regions in the near wall turbulence, depending on whether the eddy viscosity is an order of magnitude smaller or larger than the molecular viscosity. The molecular viscosity is dominant in a thin layer adjacent to the wall wherein Q t y Q . We can expand the velocity u y into a Taylor series in this region u y
§wu · ¨ ¸ © w y ¹y
0
1 § w 2u y ¨ 2 2 ¨© w y
· ¸¸ ¹y
0
1 § w 3u y2 ¨ 3 6 ¨© w y
· ¸¸ ¹y
y 3 .... 0
218
Convective Heat Transfer
and keep only the first term because of the small thickness y  H of this layer §wu · u y  ¨ ¸ ©wy ¹y
y 0
Ww y P
We use the wall variables (Q , uW ) introduced in [7.4] to write
u y
y
[7.8]
for this region, commonly called the viscous sublayer. The linearity of the velocity distribution with respect to the wall normal distance is not an extraordinary result, and such a zone exists in a wall bounded laminar flow as well. The fact that the molecular viscosity is dominant in this layer does not of course mean that the flow is laminar in the viscoussublayer: the turbulent fluctuations are very significant in this region. The nondimensionalized turbulent fluctuation wall shear stress intensity WcWc W is for instance as large as 0.40. Now consider the opposite case, i.e. the layer in which the eddy viscosity is dominant with Q t !! Q . This layer in which the direct molecular viscosity effects are negligible is relatively far away from the wall yet not too far, in such a way that the total shear stress Wtot remains approximately constant and equal to the wall shear stress Ww at the wall (Figure 7.2). This is why this region is also called the constant stress sublayer. These hypotheses lead to:
Wtot
Ww
UQ t y wu wy
The eddy viscosity Q t y remains to be modeled. Dimensional analysis enables us to write Q t ( y)  Av , where A and v are respectively the length and velocity scales characterizing the turbulent mixing. The natural choice for v in the inner layer is the wall shear stress velocity uW . The length scale of the eddy viscosity, in return is the wall normal distance y to the wall. The turbulent mixing becomes progressively important as we move off the wall and A Ny where N is a constant. The turbulence closures are generally based on phenomenological arguments. The flow quantities modeled this way have to be subsequently confronted to the experiments or the direct numerical simulations when available. There is no universal and general approach to the problem. The closure A Ny , where N 0.40 is the universal von Karman constant, and is one of the historical propositions in the domain.
Turbulent Convection in Internal Wall Flows
219
W +tot
1 Viscous sublayer
Logarithmic layer Buffer layer
Q t y > Q
30
y+
Figure 7.2. Division of the internal zone of a turbulent wall flow into multiple sublayers
Q t
Expressing these hypotheses by using the wall variables and using, in particular, Q t Q results in W tot
du dy
1 Q t y
N y
du dy
whose integration gives a logarithmic distribution of the mean velocity
u y
A ln y B
1
N
ln y B
[7.9]
The constant A 1 N 2.5 is universal.1 The constant B slightly depends on the Reynolds number and varies between B 4.5 and B 5.5 . Equation [7.9] refers to the logarithmic sublayer. The thickness of the viscous sublayer is typically GQ 5 . The logarithmic sublayer starts at approximately y 30 and extends up to the outer layer. The matching zone 5 y 30 is called the buffer layer. The latter plays a fundamental role in the wall turbulence. The vortical structures named as coherent structures are concentrated in this region. The coherent structures are responsible for the turbulent shear stress regeneration and mixing. A semiempirical
1. The dependence of the von Karman constant upon the Reynolds number and its universality
are matters of current academic research.
220
Convective Heat Transfer
relationship for the mean velocity distribution that significantly agrees with the experiments in the buffer layer is:
u y
§ y · 14.5 tanh ¨ ¨ 14.5 ¸¸ © ¹
[7.10]
Figure 7.3. Mean velocity profile in a fully developed turbulent channel flow inferred from direct numerical simulations at ReW uW h Q 180 [DOC 06]
Figure 7.3 shows the distribution of the mean velocity profile versus the wall normal distance in semilogarithmic coordinates obtained by direct numerical simulations (DNS) in a fully developed turbulent channel flow. It is clearly seen that the velocity is linear until y 5 and logarithmic beyond y 30 until practically the centerline with: u
2.5 ln y 5.5
A similar methodology can be adopted to determine the mean temperature profile T y . The physical role played by the wall shear stress is now taken by the wall qcctot
heat
flux
qccw
Uc>D Dt y @
k wT wy . w
Recall
that
the
total
heat flux
is
wT . We will nondimensionalize this relation by using the wy
Turbulent Convection in Internal Wall Flows
wall variables Q and uW together with the wall temperature scale Tqccw
221
qccw
UcuW
introduced in section 7.4. Let us define the dimensionless temperature by Tw T T . The total heat flux is Tqccw
qcctot
qcctot qccw
wT D Dt y Q wy
wT § 1
1 · Q t y ¸ ¨ ¸ ¨ wy ©Pr Prt ¹
[7.11]
where Pr Q D is the molecular Prandtl number and Prt Q t y D t y is the turbulent Prandtl number. We can express the latter in a different way, namely §wT ¸ · ¨ ucvc © wy ¹ Prt , by making use of the definitions of the eddy viscosity and vcT c § ¨wu · ¸ © wy ¹ diffusivity. There is no a priori reason indicating that Prt is y independent. The turbulent Prandtl number varies differently in different sublayers and its exact distribution is not known. Indeed it depends upon the wall normal distance, the molecular Prandtl number and the Reynolds number, i.e. Prt
Prt y , Pr , Re . It
can strongly vary in the inner layer and reach large values when Pr 1 . However, it is relatively uniform and independent of the Reynolds number when Pr !! 1 . The simplified procedure is to assume that Prt is constant and to directly connect the problem to the eddy viscosity closure. The consensual values found in the literature are Prt 0.9 for turbulent wall flows and Prt 0.7 in free shear layers. Figure 7.4 shows the results obtained by [KAW 04] through DNS for different Reynolds numbers and for a large and small molecular Prandtl number that are respectively Pr 0.7 and Pr 0.025 . It is seen that there are large variations of Prt in the whole layer in the small molecular Prandtl and Reynolds numbers case with values as large as 3. For Pr 0.7 , however, Prt  1 . Let us be clear that there are some significant differences between DNS data and some empirical correlations, such as those given by [CEB 73] and [CEB 88], that overestimate Prt for small molecular Prandtl numbers and do not take into account the Reynolds number dependence. These aspects are still not entirely clarified and are subjects of current academic research.
222
Convective Heat Transfer Prt
4
ReW 1020 180
3
Pr=0.025 2
1
0
Pr=0.71 .1
1
10
y + 100
(a)
4
Prt 3
Pr=0.025
2
1
Pr=0.71 0 0.
0.1
0.2
y h 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.
(b)
Figure 7.4. Turbulent Prandtl number distribution for different Reynolds and molecular Prandtl numbers. Data adapted from DNS conducted by [KAW 04]: (a) distribution in the inner layer versus the normal distance to the wall in wall variables; (b) distribution in the whole channel versus y scaled by the half height
The wall transfer mechanism is analogous to the momentum transfer except that the temperature distribution depends not only upon y , but also on Pr and Prt . Consider a first sublayer next to the wall wherein the eddy diffusivity is negligible
compared with D , i.e. Q t y Prt Pr . The total flux in this zone is approximately
equal to the wall flux. Consequently qcctot  1 and dT dy
T
Pr y
Pr , implying
[7.12]
for y Gc , where Gc stands for the thickness of conductive sublayer. The turbulent flux is negligible in the conductive sublayer that plays a role in the wall
Turbulent Convection in Internal Wall Flows
223
transfer similar to the viscous sublayer in the momentum transport. The thickness of the conductive sublayer depends on the molecular Prandtl number. The conditions
resulting in the linearity of the mean velocity and temperature are Q t y 1 and
Q t y Prt Pr respectively. In the case Prt  1 , it is logical that Gc d Gv for Pr ! 1 and vice versa. We can therefore assume that Gc Gv { Pr p with p ! 0 . We will clarify this point in more details in the problem proposed in section 7.9.
The molecular diffusivity is negligible in the fully developed turbulent mixing region starting at y ! Gmt . We also assume that the total flux is constant and equal
to the wall flux through qcctot 1 in this region in a way quite similar to the assumption we made in the logarithmic sublayer with constant shear. These hypotheses reduce equation [7.11] to:
qcctot
1 dT Qt y Prt dy
1
The eddy viscosity is Q t y
N y in this region provided that, typically,
Gmt ! 30 . Thus, a logarithmic distribution of the dimensionless temperature is
obtained under these circumstances
T
Prt
ln y C
N
1
NT
ln y C
[7.13]
which is entirely similar to the velocity distribution given by equation [7.9]. The constant C
C Pr , G c
depends on the molecular Prandtl number and the
thickness of the conductive sublayer. In other respects, since Gc { Pr pGv , that dependence reduces to C C Pr . The thermal von Karman constant N T N Prt is also a function of the Prandtl number. Indeed, although we assumed that the turbulent Prandtl number is constant in the constant flux zone and obtained equation [7.13], we have to take into account that Prt strongly depends on Pr itself. Recent results indicate that N T only slightly
depends upon Pr once Pr ! 0.2 and that NT  0.34 in the layer y ! 50 , leading to T 2.94 ln y CPr . There are strong similarities between T y and the
mean velocity distribution, but important differences have to be pointed out. The
224
Convective Heat Transfer
molecular Prandtl number plays a key role in the T y profile, and the thicknesses that demarcate the thermal sublayers are schematically shown in Figure 7.5. +
q"tot
1 Conductive sublayer
Logarithmic thermal sublayer
Thermal buffer layer
Q +t > Prt
Pr
Pr
G c+= Pr p G v+
+ G mt
y+
Figure 7.5. Thermal sublayers in the internal region of a wall bounded turbulent flow
These points will be further discussed in section 7.9. The Spalding modeling of the eddy viscosity [SPA 61], combined with Prt gives the following equation for the constant C: C
12.8Pr 0.68 7.3
1,
[7.14]
The total shear and flux Wtot , qcctot do not stay constant and decrease towards the centerline. Therefore, the logarithmic velocity and temperature distributions are no more valid in this fourth zone, which is called the outer layer. The outer layer is far away from the wall and the molecular and eddy viscosities are no longer relevant in this region. Thus, the mean velocity depends on the shear velocity uW , the centerline velocity U c , the distance to the wall y and the channel half width e . A similarity analysis leads to Uc u y uW
U c u y
§ y· gu ¨ ¸ ©h¹
[7.15]
in the outer layer. As the time mean temperature does not depend upon the diffusivity (therefore the Prandtl number) in the outer layer, this results in: T y Tc Tqccw
Tc T
§y · gT ¨ ¸ ©h ¹
[7.16]
Turbulent Convection in Internal Wall Flows
225
Figure 7.6 shows the mean temperature distribution expressed in outer variables for Pr 0.025 and Pr 0.7 and different Reynolds numbers. These results have been obtained by [KAW 04] through direct numerical simulations. It is seen that the Tc T profiles collapse reasonably well in the investigated range of ReW when the molecular Prandtl number is not significantly smaller than 1. However, Tc T increases with ReW in the small Prandtl number Pr 0.025 case. The conductive
sublayer becomes thick according to Gc v Pr 1 (see section 7.9). The arguments leading to temperature distribution [7.16] in the outer layer are obviously questionable when the molecular Prandtl number is small. The temperature profiles given in Figure 7.6 are compared with the [KAD 81] empirical correlation.
Tc+T +
ReW
20
180 395 1020 [KAD 81]
Pr=0.025 10
0 0.
y h 0.2
0.4
0.6
0.8
1.
Tc+T +
20
Pr=0.71 10
0 0.
y h 0.2
0.4
0.6
0.8
1.
Figure 7.6. Mean temperature distribution in outer variables for Pr 0.025 (a) and Pr 0.71 (b). The data is due to the direct numerical simulations of [KAW 04]
226
Convective Heat Transfer
T y
Pr y e*
ª 1.5 2 y h « ° « 2.12 ln ® 1 y ° « 1 2 1 y h ¯ ¬
c
3.85Pr 1.3 10 Pr y 1/3
2
*
2
2
½º °» c ¾» e 1/ * °» ¿¼
[7.17]
2.12 ln Pr
4
1 5 Pr 3 y
This correlation is widely used in the literature. It is based on considerable experimental data that cover 0.7 d Pr d 60 and Re d 4 10 4 . The Kader correlation can be applied to both internal and external layers. The turbulence structure in the outer layer is analogous in many respects to the wake turbulence. [COL 56] proposed a relationship for the mean velocity distribution that is presumably valid both in the logarithmic sublayer and outer region. He added a functional wake term to distribution [7.9]
u y
1
N
ln y B
§y · w ¨ ¸ N ©e ¹
3
[7.18]
where the argument of the wake function w is the wall normal distance scaled by the outer length scale e , and where 3 is constant. Coles further proposed an empirical function for the wake component §y · 2 y e 1 S w ¨ ¸ 1 sin ©e ¹ 2
The constant 3 inferred from the measurements conducted for Re U c e Q ! 4 10 4 is 3  0.55 . The equation equivalent to [7.18] for the mean temperature distribution is simply:
T
1
NT
ln y C
§y · w¨ ¸ N T ©e ¹
3T
The coefficient 3T  0.3 is independent of Pr when Re ! 4 10 4 .
[7.19]
Turbulent Convection in Internal Wall Flows
227
7.6. Some illustrations through direct numerical simulations
We will have a short discussion about the wall transfer mechanism and give some illustrations using some DNS results obtained in a fully developed turbulent channel flow at ReW 180 and Pr 1 . The temperature being a scalar passive not affecting the flow, the instantaneous threedimensional velocity field u i is first calculated at each time step, and the local temperature is subsequently determined wT wu iT w 2T by solving D 2 in time and space. The T x i , t calculation by wt wx i wx i direct numerical simulations at large molecular Prandtl numbers is difficult. The smallest passive scalar scale, called the Batchelor scale, is A T A K Pr 1/ 2 , where A K is the Kolmogorov dissipative scale related to the dynamic flow field. Thus, the
number of calculation modes has to be increased by a factor Pr 3 / 2 at a given Re as the molecular Prandtl number increases. This is the reason why existing direct numerical simulations dealing with turbulent heat or momentum transfer are generally limited to Pr < 10. The initial temperature profile used in the DNS under discussion is given in Figure 7.7. The upper and lower walls are maintained at constant temperatures T1 and T1 respectively.
'T
T1
e
y x
T1
Figure 7.7. Boundary conditions related to the direct numerical simulations discussed in section 7.6
We show in Figure 7.8 the mean temperature distribution T y
T T q ccw 2 in
wall units versus y . The effect of the turbulence on the mean temperature field is clearly seen here by comparing T y and the laminar conduction profile. The 2. The flux q ccw ! 0 from the fluid to the lower wall in this section, see sections 7.8.2 and 7.8.3.
228
Convective Heat Transfer
turbulent mixing makes the temperature rapidly vary near the wall, while the variations are weaker near the centerline. We note that, due to the symmetry, the temperature gradient is not zero at the centerline contrary to the mean velocity. The vorticity is zero at the centerline, while wT wy z 0 in the present configuration.
Figure 7.8. Mean temperature in wall units corresponding to the case shown in Figure 7.7. ReW = 180, Pr = 1 [DOC 06]
We compare the mean temperature distribution T
T Tw Tqccw
to the velocity
profile in Figure 7.9 in semilogarithmic coordinates. It is seen that the T profile collapses well with the u distribution, both in the conductive and buffer layers. This can be explained by the fact that Pr 1 in these calculations and, consequently, Gc Gv { Pr p 1 . Indeed, we will see in section 7.9 that the thicknesses of the conductive and buffer layers strongly depend on Pr . A logarithmic distribution T 3 ln y 5 starting at y 50 is clearly seen in Figure 7.9. The temperature slope ( N T 1 3) is slightly different than that of the mean velocity profile (N1=2.5). The mean temperature distribution over a wall with uniform flux is similar in the inner layer. It is, however, different in the outer layer because T is then symmetric and the mean temperature gradient becomes zero at the centerline, opposite to what happens in the case of uniform wall temperature. There is a large literature on these effects and the reader can, for example, consult [KAW 98] for further details.
Turbulent Convection in Internal Wall Flows
229
We will now briefly discuss some results related to the fluctuating velocity and temperature fields. Figure 7.10 shows the distribution of the root mean squares of the streamwise
wc
uc
ucuc uW , wall normal
vc
vcvc uW
and spanwise
wcwc uW velocity fluctuations obtained through DNS in a fully developed 180 . The centerline is at y
turbulent channel flow at ReW
180 . The streamwise
12 in the buffer layer. We will see turbulent intensity reaches its maximum at y in the solution of section 7.11 that the turbulent kinetic energy has its maximum at the same position. The Reynolds shear stress distribution is also shown at the bottom
of Figure 7.10, of course with ucvc t 0 in the whole channel.
Figure 7.9. Modified time mean temperature corresponding to the case in Figure 7.7 [DOC 06]
Figure 7.11 shows the ucTc and vcTc profiles ( Tc T c Tqccw ) versus y for
the example given in Figure 7.7. The turbulent heat flux is clearly vcTc t 0 in the whole channel, like the Reynolds shear stress distribution. The asymptotic behaviors
of ucTc and vcTc near the wall are also indicated in Figure 7.11. Globally, the
turbulent flux terms and Tc TcTc vary considerably in the channel crosssection, both in turbulent flows subject to uniform wall temperature or heat flux. We will discuss these aspects in section 7.12.
230
Convective Heat Transfer
Figure 7.10. Turbulent intensities (top) and Reynolds shear stress (bottom) distributions in wall units in a fully developed turbulent channel flow at ReW 180 [DOC 06]
Figure 7.11. Turbulent heat fluxes and their asymptotic behaviors near the wall for the case corresponding to Figure 7.7 [DOC 06]
Turbulent Convection in Internal Wall Flows
231
7.7. Empirical correlations
We will give some empirical correlations relating the Nusselt number to the Reynolds and Prandtl numbers in this section. The Nusselt number is defined by Nu
qccw
k Tw Tm Dh
where it is recalled that Dh is the hydraulic diameter and T m is the bulk temperature. The Reynolds number is based on the bulk velocity U m and the hydraulic diameter, i.e. ReDh U m Dh Q . Remember that the DarcyWeisbach drag 1 coefficient is / Ww UU m2 and that the pressure loss for a duct of length L in the 8 fully developed regime is: 'p*
/
1 2
UU m 2
L Dh
The Blasius empiric correlation for / is: 0.3164
/
0.25 Re Dh
Re Dh d 10 5
[7.20]
This correlation is valid for a developed turbulent flow over smooth walls. For rough walls, in particular in the presence of sand roughness of typical height3 k s , we often use the Nikuradse law: 1 /
§ · 2.51 ¸ k D 2 log10 ¨ s h ¨ 3.7 Re Dh / ¸ ¹ ©
[7.21]
The existing empirical correlations for the heat transfer often refer to fully developed turbulent flows over smooth walls. For instance, the DittusBoelter correlation is quite popular: Nu
0.0243Pr 0.4 Re Dh 0.8
3. Do not confuse ks with the conductivity.
[7.22]
232
Convective Heat Transfer
It has the advantage of being simple and quite precise in the range: 0.7 Pr 160
Re ! 10 4
The KarmanBoelterMartinelli correlation Nu
Re Dh Pr / 8
>
@
0.833 5Pr 5 ln5Pr 1 2.5 ln Re Dh / 8 60
[7.23]
was originally obtained in a turbulent pipe flow under uniform wall heat flux. It corresponds well to the eddy diffusivity predictions for Pr ! 0.7 [CEB 88]. The Sleicher and Rouse correlation is recommended for liquid metals ( Pr 1 ) Nu
0.85 0.93 6.3 0.0167Re Dh Pr
[7.24]
for fully developed turbulent pipe flows under constant wall flux. We recommend the Kader and Yaglom correlation be used in internal turbulent wall flows subject to constant temperature at the wall Nu
Re Dh Pr / 2
4.24 ln Re Dh / /16 25 Pr 2 / 3 4.24 ln Pr 20.2
[7.25]
in the range 1 d Pr d 10 6 and 10 4 d ReDh d 10 6 . A useful correlation of acceptable precision between the Stanton number defined as St
qccw
h
UcU m Tw Tm
UcU m
and the drag coefficient,4 C f St
Ww UU m2 , is the Colburn relationship
C f Pr 2 / 3
Note that Nu parameters.
[7.26]
St Re Pr according to the definitions of these dimensionless
2
4. Cf is often given in the literature in the form scaled by 1 2 UU m .
Turbulent Convection in Internal Wall Flows
233
These relationships can be applied as a first approximation to fully developed turbulent flows in ducts of arbitrary sections by using the hydraulic diameter definition. The effect of the wall roughness has to be analyzed separately. Some indications are given in Chapter 8. The reader may consult [WEB 05] for correlations in turbulent flows over rough walls. 7.8. Exact relations for a fully developed turbulent channel flow 7.8.1. Reynolds shear stress
We will obtain three precise relations that govern the heat transfer process in the three following problems. We begin, first, by establishing a precise equation for the Reynolds shear stress distribution that is also often used to check the quality of the experimental data or the convergence of direct numerical simulations. 7.8.1.1. Reynolds shear stress in a fully developed turbulent channel flow. Description of the problem The aim is to show that the total shear stress (i.e. the sum of turbulent and viscous stresses) varies linearly with the distance to the wall in a fully developed steady turbulent channel flow that is homogenous in the streamwise and spanwise directions. 7.8.1.2. Guidelines The flow is homogenous in the streamwise (longitudinal) and spanwise directions. The first step is to determine the exact relationship that governs the mean velocity that has to be subsequently analyzed to find out the timemean pressure gradient and the required equation for the Reynolds shear stress. 7.8.1.3. Solution The homogenity in the streamwise x and spanwise z directions implies 0 . The timemean spanwise velocity is consequently w 0 . The continuity therefore leads to v 0 and u u y . The precise Reynolds equations in the streamwise and wall normal directions read:
w wx w wz
1wp w 2 u w u cvc Q U wx wy w y2
0
0
1 w p w vcvc U wy wy
[7.27]
234
Convective Heat Transfer
It is seen that the pressure p depends not only upon x , but that there is also a wall normal distance dependence induced by the vcvc gradient. The integration of the second equation given above from the wall y 0 to a point y in the flow leads to y
wp dy U wy 1
³ 0
y
³ 0
w vcvc dy wy
vcvc y
and, consequently, px, y U vcvcy p 0 x where p0 x is the pressure at the wall. Therefore, and since vcvc depends only on y , we have
wp wx
dp0 dx
by
homogenity. Thus, at this step it is straightforward to integrate the u y equation from the wall to the channel centerline y e . Noticing that the shear wu wy y e 0 , we obtain 0
1 dp0
U dx
e
1
U
Ww
because, by symmetry, the Reynolds shear stress U ucvc is zero both at the wall and at the centerline. The last equation connects the pressure gradient dp0 dx to the wall shear stress Ww Pwu wy y 0 . The use of the last relationship in the u equation and the integration of the latter from the wall to y lead to
P
§ y · wu y U ucvcy Ww ¨1 ¸ © e ¹ wy
wherein the total shear stress clearly appears at the right hand side, i.e.
Wtot
P
wu y U ucvcy wy
which is the sum of the viscous and Reynolds shear stresses. Scaling this equation by the inner variables (based on the length and velocity scales that are respectively uW and lQ Q uW ) results in the simple equation: Wtot
wu y ucvc 1 wy e
[7.28]
Turbulent Convection in Internal Wall Flows
235
Figure 7.12. Reynolds shear distribution in inner variables versus the normaltothewall distance in wall units. The solid line represents the total shear stress. These results have been obtained by direct numerical simulations of a homogenous turbulent channel flow at ReW
huW
Q
180 by [DOC 06]
The total shear stress is therefore a linear function of the normaltothewall distance. Equation [7.28] is precise for a turbulent channel flow homogenous in the streamwise and spanwise directions. Figure 7.12 shows the distributions of the
obtained by DNS in a fully Reynolds shear stress ucvc and the total shear Wtot
that developed turbulent channel flow. The results show a perfect linearity of Wtot provides a quick check of the experimental or numerical results (such as the statistical convergence of the computations, for instance). The direct numerical simulations resolve the entire spectrum of turbulent scales imposed by the discretization of the computational domain, and do not, by definition, require any closure. We mainly use the DNS results obtained at low Reynolds numbers to illustrate some basic results in this book, and the calculation details are omitted. There are a number of excellent works on this particular subject, and the interested reader may consult, for example, [ORL 00] for further details.
The direct numerical simulations resolve the three components of the velocity field both in time and space. They are undeniably limited by the memory and calculation time requirements. First, the size of the calculation domain has to be larger than the largest turbulent scale present in the flow. Second, each calculation mode has to be able to resolve the smallest dissipative scales that decrease as the Reynolds number increases. The ratio of the largest to the smallest scale in turbulence is proportional to Re 3 / 4 . Since the required mesh size has to be reduced
236
Convective Heat Transfer
when the Reynolds number is increased, the total number of modes increases resulting in significantly higher computational cost. The total number of calculation modes is related to Re 3 . This is why existing DNS of wallbounded turbulent shear flows are limited to relatively low Reynolds numbers (typically ReW huW Q 10 3 , at the moment). The results shown in Figure 7.12 have been obtained for ReW 180 . Let us state, however, that the wall turbulence structure remains mainly unmodified for 150 ReW 2000 . At larger Reynolds numbers, largescale structures become somewhat important and the contribution of the near wall eddies to the turbulent wall and Reynolds shear stresses is subsequently modified. These details that are important at large Reynolds number turbulent wall flows constitute an intensive research topic. We can go a step further and determine the drag coefficient. The integral of equation [7.28] from the wall to some y position in the flow gives the timemean velocity distribution:
u y
y
y 2 2e
y
³ ucvc K dK
[7.29]
0
Integrating again this relationship from y e
³ u y dy
e 2 3
0
e
e leads to:
ª y º « u cvc K dK »dy « » ¬0 ¼
³ ³ 0
0 to the centerline y
The last integral appearing in the righthand side is calculated by integration by parts: e
ª y º « u cvc K dK »dy « » ¬0 ¼
³ ³ 0
e
³ e
e 3
e
§
y ·
Um
³ ¨©1 e ¸¹§¨© ucvc ·¸¹dy 0
y u cvc
0
Using the bulk velocity defined by
Um
1 e
y dy
e
³ u y dy
0
, we have:
Turbulent Convection in Internal Wall Flows
237
It is useful to rewrite the last equation in outer variables ( e and U m ) in order to obtain an equivalent relationship for the drag coefficient, defined here as 1 C f Ww UU m2 2 U m2 . We obtain, by introducing Re eU m Q , noting that 2 * e ReW euW Q Re U m and using the notation for the quantities nondimensionalized by e and U m : 1
Cf
³
* 6 6 1 y* u cvc dy* Re 0
[7.30]
The quantity 6 Re is just the drag coefficient for a laminar Poiseuille flow.5 It can be interpreted as the laminar contribution C fl to the total drag in the present case. The second term on the righthand side of [7.30] is a weighted integral of the Reynolds shear stress. It represents the direct contribution of the turbulence to the drag and will be denoted by C ft . The turbulence significantly increases the drag because of C ft . We will see hereafter that this is also the case for the turbulent heat transfer at the wall, although the corresponding relationships are not a priori analogue. Equation [7.30] has already been obtained by [FUK 02], albeit in a different way. We can go further in the analysis and see whether or not it is possible to decompose the velocity distribution into its laminar and turbulent counterparts. Equation [7.29] nondimensionalized by the outer variables e and U m is:
u
*
y *
Re Cf 2
§ * y*2 ¨¨ y 2 ©
· ¸¸ ¹
y*
³ ucvc K dK *
*
*
0
Let us assume that the velocity distribution can be decomposed as u * where for a given Re (or U m ), ul*
3 2 y * 2 y*
u l* u t*,
is the laminar Poiseuille flow
velocity distribution and u t* is the turbulent velocity contribution to u * to be determined. Rearranging equation [7.30] by splitting the drag coefficient into C f C fl C ft results in:
5. The Re number is not based on the hydraulic diameter of the channel. The coefficient is
therefore 6 instead of 24 in equation [7.30].
238
Convective Heat Transfer
ut*
y* 1 ½ * * 3Re * ° ° y 2 y* ® 1 y* u cv c dy* u cvc K * dK * ¾ 2 °¯ 0 °¿ 0
³
³
[7.31]
The turbulent u t* component obviously becomes zero in the absence of Reynolds shear stress ucvc. 7.8.2. Heat transfer in a fully developed turbulent channel flow with constant wall temperature
7.8.2.1. Description of the problem We consider a fully developed turbulent channel flow that is homogenous in the streamwise and spanwise directions. The lower and upper walls are at y 0 and y 2e , respectively. The lower wall is cooled at the constant temperature Tw T2 T1 , while the upper wall is maintained at T1 (Figure 7.7). We wish to know the Nusselt number and identify the contribution of the turbulence to the wall heat flux. This problem has been analyzed in a slightly different manner by [FUK 05]. 7.8.2.2. Guidelines Begin by nondimensionalizing the relevant quantities by using the temperature scale 'T T1 T 2 2T1 , the bulk velocity and the channel halfwidth. Note that the temperature distribution is antisymmetric. Therefore, the bulk temperature is zero, and the temperature is homogenous in the streamwise direction. We subsequently need to manipulate the heat transfer equation in an attempt to obtain a relationship for the Nusselt number. 7.8.2.3. Solution We define the dimensionless temperature T e
T Tw 'T
T T1 . The velocity is 2T1
1 ³ u y dy , with u * u U m , and the length scale e 0 is the halfchannel height e . We have, according to [7.6]:
scaled by the bulk velocity U m
ww xT
u * y*
*
1 w 2 T w v c*T c RePr w y*2 w y*
Turbulent Convection in Internal Wall Flows
The corresponding boundary conditions are T y*
0
0 and T y*
239
2
1.
The integration of this relationship from the lower to the upper wall yields d 2 * ³ u Tdy* dx * 0
2
ª§ · º § wT · «¨ wT ¸ » 0 ¨ ¸ ¨ * ¸ * ¸ » RePr «¨ ¬©wy ¹y* 2 ©wy ¹y* 0 ¼
dT m
1
dx *
due to the antisymmetry. Thus, not only the bulk temperature T m is constant, but also Tm 0 , because the velocity distribution is symmetric, while the temperature profile is antisymmetric with respect to the channel centerline. The energy budget clearly indicates that wT wx * 0 . Hence, the transport equation takes the simple form: 1
0
w 2T
RePr wy
*2
w vc*Tc
[7.32]
wy *
The temperature profile in the related laminar flow ( v c*Tc 0 ) reduces to the simple linear conduction distribution. The integration of the preceding equation from the upper wall to y * somewhere in the flow results in: §wT · ¨ ¨ ¸ ¸ RePr ©wy * ¹y * 0 1
1
Nu
wT
RePr wy *
RePr
vc*Tc
The Nusselt number is defined by Nu
wT
[7.33]
w y*
y* 0
wT
w y*
y* 2
. The
global transfer process is governed by the following equation obtained by the integral of the local equation [7.33] from the upper wall to the centerline y * 1 1
Nu
1 RePr ³ vc*T ' dy *
[7.34]
0
wherein we made use of the boundary condition Ty
at the centerline through T y
*
1
0
0 and the antisymmetry
1 . Equation [7.34] is exact; it does not
necessitate any closure and it is instructive in several aspects. It is to be noted that the Nusselt number is simply Nu l 1 in the absence of turbulent flux terms, which is the Nu of the related laminar flow in the same configuration. It is clearly seen that the Nusselt number can easily be decomposed into two parts, namely Nu t
Nu l Nu mt , where Nu mt
contribution.
1
RePr ³ vc*T ' dy * stays for the turbulent mixing 0
240
Convective Heat Transfer
The turbulence considerably increases the wall transfer since vcT ' ! 0 . It is important to note that equations [7.34] and [7.30], which correspond respectively to heat and momentum transfer processes, are different. The contribution of the *
turbulence to mixing is a simple integral of vc*T ' while the ucvc contribution to the drag coefficient is a weighted integral. 7.8.3. Heat transfer in a fully developed turbulent channel flow with uniform wall heat flux
7.8.3.1. Description of the problem The problem is identical to that in section 7.8.2, but with different boundary conditions. The channel is now subject to constant uniform heat flux qccw 6 of the same intensity on the upper and lower walls. 7.8.3.2. Guidelines Both the upper and lower walls are maintained at the same heat flux; the wall temperature Tw (x) varies linearly with x in the same way as the bulk temperature, dTm wT x, y dT as well as T x, y , in the fully developed thermal regime ( w ). dx dx wx The energy budget in the channel is obtained by integrating the convection equation from the lower to the upper wall y 2e d 2e
dTw 2e
dx 0
dx 0
³ uT dy
³ udy
ªwT º2e 2e D« » v cT c 0 ¬wy ¼0
2qccw
Uc
because u does not depend upon x . That results in dTw dx
qccw UceU m ,
suggesting the use of the temperature scale 'T e dTw dx qccw UcU m . We therefore propose to formulate the problem by using the dimensionless temperature Tw x T x, y T . The transport equation has to be integrated in a second step in 'T a way similar to that in section 7.8.2.3 with the aim of determining the turbulence contribution to the Nusselt number. Note that the boundary conditions here are
T y*
0
0 and T y*
2
0.
6. qccw is positive from the wall to the fluid in the channel.
Turbulent Convection in Internal Wall Flows
241
7.8.3.3. Solution The temperature scale suggested above gives rise to the nondimensionalized transport equation
u * dTw 'T dx *
d 2T
d vc*Tc RePr dy *2 dy * 1
u *
[7.35]
where we used the outer scales e and U m to obtain dimensionless parameters noted * by , in a way very similar to that in section 7.8.2.3. A first integration of the last equation from the lower wall to the channel centerline yields
0
§dT · ¨¨ ¸¸ RePr ©dy * ¹y * 1
1 0
since the temperature profile is symmetric; therefore,
dT
dy*
y* 1
0 and the
integral of the lefthand side of [7.35] is 1, due directly to the definition of the bulk velocity. The Nusselt number based on the bulk temperature and the half height of 1 §dT · the channel is Nu , with the dimensionless bulk temperature ¨ ¸ Tm ¨©dy * ¸¹y * 0
Tm
Tw x Tm x
'T
. Using [7.35] results in Nu
RePr
Tm
.
The rest of the solution is very similar to the procedure used in section 7.8.1.3. A first integration of equation [7.35] enables us to obtain a relationship establishing equilibrium between different flux terms: y*
1 dT * y 1 v c*T c y* RePr dy*
³
u *dK * 0
y*
Let us denote
³u 0
[7.36]
y*
*
*
³ u dK . The temperature field is calculated
dK by F y
*
*
*
0
by integrating [7.36] from the wall to y * y*
³ dK
F K 0
*
*
1 T y* y* RePr
y*
³ vc T c K dK *
0
*
*
242
Convective Heat Transfer
and using the boundary condition T0
0 . We proceed by weighting this equation
by the velocity u y* y*
³ F K dK
u y
*
*
*
0
³ vc T c K dK
1 u y* T y* u y* y* u y* RePr
and by integrating it again from y *
[7.37]
y*
0 to y *
*
*
*
0
1 (centerline).
Our aim is to attain a relationship for the Nusselt number that we have in mind. The calculation is long, but not difficult. We will systematically use integration by parts as we did in section 7.8.1.3. Let us first go through the last term of [7.37] ª «u y* 0« ¬
y*
ª « F y* « ¬
y*
³ ³ vc T c K 1
*
*
0
º dK dy* » ¼
ª y* º « vc*T c K * dK * » d F » 0« ¬0 ¼
1
³ ³
*»
1
º
³ vc T c K dK » ³ F y vc T c y dy *
³ ¬ª1 F y ¼º vc T c y dy *
*
*
*
*
*
*
0
¼0
0
1
1
*»
*
*
0
since F 1 1. Proceeding in the same way, we obtain: 1
³
1
u * y* dy*
0
1
³
1 F y* dy* and 0
³
1 u * y* T y* dy* RePr 0
Tm RePr
1 Nu
The integration by parts of the lefthand side of equation [7.37] implies that ª y* º « F K * dK * » u * y* dy* « » 0¬0 ¼
1
³³
1
³
1
³
F y* dy* F 2 y* dy*
0
0
Turbulent Convection in Internal Wall Flows
243
which can be rearranged as
1
1
Nu
§
1
·
vc*Tc¸dy * ³ F 2 F dy* ³01 1 F ¨ © ¹
[7.38]
0
The resulting form of equation [7.38] does not allow us to express the Nusselt number as a function of laminar and turbulent contributions. The reason is that the function
F y*
contains terms coming from both laminar and turbulent
contributions. By decomposing
F
Fl Ft y*
with Fl
³
y*
³
y*2 3 y* 2
3 K * 2 K * dK * 2
ul*dK *
0
0
(see section 7.8.1.3), it is
possible to rewrite the last term of [7.38] as 1
³ F 2 F dy *
0
1 ª º * ³ « y *3 3y *2 2 F t F t2 »dy ¬ ¼ 70
36
0
and ultimately we have 1 Nu
34 70
³
1
1
³
1 F vc*T c dy* ª¬ y*3 3 y*2 2 Ft Ft2 º¼dy* 0 0
[7.39]
Now, Nul 70 34  2.06 is simply the Nusselt number of the related laminar flow subject to a uniform wall heat flux constant.7 Therefore, the integrals at the righthand side of equation [7.39] thoroughly represent the turbulent contribution 1 Nu t . 7.9. Mixing length closures and the temperature distribution in the inner and outer layers
Prandtl was the first person who proposed a closure based on the phenomenology of mixing length, a notion that was originally used in kinetic theory. Consider a fluid particle in a shear flow as shown in Figure 7.13. The particle is displaced with a velocity vc to the point B after a traveling distanceA . It will 7. Nu = 8.235 by using the hydraulic diameter.
244
Convective Heat Transfer
maintain its velocity from the origin, provided that the distance A is short compared to the local relaxation length scale of the turbulence. A local fluctuation uc u A u B  Awu wy at the point B is subsequently generated. The characteristic length scales of the turbulent eddy are of the same order of magnitude in all directions x i , with l tx  l ty  l tz . The continuity wuci wx i 0 therefore implies that the turbulent intensities uc  vc  wc are also of the same order of magnitude. Incidentally, note that a particle coming from a zone of low speed with vc! 0 induces a local fluctuation that is uc 0 as shown in Figure 7.13. Consequently, ucvc 0 and combining gives:
§wu ·2 ucvc A ¨ ¸ ©wy ¹ 2
[7.40]
Van Driest’s mixing length formulation is quite popular and is successfully used in turbulent wall flows. It reads A
ª
§ y ·º ¸» ¸ © A ¹»¼
Ny «1 exp¨¨ «¬
[7.41]
where A 26 is an empirical constant. The term in the square brackets models the viscous damping near the wall. The eddy viscosity deduced from the van Driest mixing length hypothesis is:
Q t
2 ª · § y ·º §wu · 2 2 « ¸ ¨ ¸» ¨ A ¨ ¸ ¸ N y «1 exp¨ A ¸» ¨wy ¸ ¹ ©wy ¹ © ¹¼ © ¬
§
2 ¨wu
B
v' >0
uA
[7.42]
u'>1
Consider now the case Pr 1 . The equation governing the conductive sublayer is obviously unaltered. In the thermal buffer layer, in return, the term 1 Pr is still predominant
in
the
equation
wT wy
1
Pr ·¸¹
§¨ 1 Pr Q t y ©
t
provided
that Q t y Prt Pr . Therefore, the conductive sublayer is extended into the buffer layer. The distribution in the logarithmic layer is unmodified. To conclude, we have
T y T y
Pry
0 d y d Gc
Prt
y ! Gc
N
ln y C
when Pr 1 . The constant C is now, C
continuity of T y implies Gc
Prt
NPr
at y
PrGc
Prt
ln Gc , obtained by the
N Gc . Finally, the wT wy continuity at y Gc
 2.25Pr 1 . The thickness of the conductive sublayer increases
with Pr 1. There are important differences in the dimensionless temperature distribution depending on the molecular Prandtl number. This problem is also analyzed in [ARP 84] in a slightly different way.
252
Convective Heat Transfer
7.10. Temperature distribution in the outer layer 7.10.1. Description of the problem
We have already introduced the temperature distribution in the outer layer in equation [7.16]. We will now show that more precise relationships can be obtained in the outer layer for the two classical boundary conditions that are constant wall temperature or uniform heat flux at the wall. We seek to determine the temperature distribution in the outer layer for a fully developed turbulent channel flow with constant temperature at the wall by assuming that the turbulent flux terms are preponderant. Section 7.8.2 can be useful to this end. Then solve the same problem when the wall flux is uniform. The departure point can be section 7.8.3. 7.10.2. Guidelines
The problem can be solved by using relationships [7.33] and [7.36]. It can further be shown that the temperature distribution in the outer layer depends on the boundary conditions. The eddy viscosity first increases linearly and subsequently reaches a constant that is typically Q t e 15 once y e t 0.5 . The molecular viscosity can be ignored in this region. Similarly, the eddy diffusivity D t Q t Prt is also constant in this zone providing that the turbulent Prandtl number does not vary with the wall normal distance y and that is the case for Pr t 1 (Figure 7.4). 7.10.3. Solution
We will first consider the case with constant wall temperature. Equation [7.33], indicating that the total flux is constant, is written *
qcctot
wT wy *
RePr vc*Tc Nu
[7.51]
It is recalled that the terms appearing in this equation are scaled by the outer variables introduced in section 7.8.2.3. The turbulent flux term RePr vc*Tc can be expressed as RePr vc*Tc RePrDt*
wT wy *
Turbulent Convection in Internal Wall Flows
253
in the outer layer. Note that the eddy diffusivity D t is nondimensionalized by the
outer scales, i.e. D *t D t U m e . The dimensionless turbulent flux is larger than
wT wy* which can consequently be ignored. The integration of [7.51], from y * being in this zone to the centerline ( y * 1), leads to:
Tc T y
*
Tc T y*
Nu
2Tw
RePrDt*
1 y*
Nu RePrDt*
§ y · ¨1 ¸ © e ¹
The term ReD *t becomes: Um e D t D t Q Um e
ReD *t
Scaling with the flux temperature Tqccw results in an equation similar in form to [7.16]:
Tc T y*
Tc T y* Tqccw
2Tw Nu § y · ¨1 ¸ ¸ PrDt ¨ © e ¹
[7.52]
It is clearly seen that the temperature varies linearly with y in the outer layer when the wall temperature is kept constant. The temperature distribution is different in the outer layer in fully developed turbulent channel flow subject to uniform heat flux at the wall. Let us consider again the exact equation [7.36] y*
³ u dK
F y
*
*
*
0
Tw x T x, y
with T
1 dT * y 1 vc*T c y* RePr dy*
and 'T
cc qw
(see section 7.8.3.2). The second term 'T UcU m on the righthand side (molecular diffusion) is negligible in the outer layer and the eddy diffusivity is constant in the outer layer. Consequently dT dy
*
y *
1 F y *
D *t
[7.53]
254
Convective Heat Transfer
resulting in 1 ª º 1 « y * *» 1 F K K d » D t* « e y* ¬ ¼
Tc T y
³
*
after integration. We use the heat flux temperature T q ccw result in wall variables with, for instance, Tc
T c T y*
qcwc UcuW to rewrite this
Tc 'T Tqccw
1 ª º «1 y F K * dK * » » D t*U m « e y* ¬ ¼
³
1
1 ª º e « y * *» 1 d F K K » D t « e y* ¬ ¼
[7.54]
³
since
Dt U m
Dt*U m
U m e uW
Dt e
It is clearly seen that the last relationship is totally different from equation [7.52] which corresponds to the constant wall temperature case. To simplify equation [7.54], let us assume that the velocity is constant and equal to the bulk velocity (which is indeed a very crude approximation). Then we have K*
³ u dK '
F K
*
*
*
 K * , rising to:
0
Tc T
y *
e § y ·  1 ¨ ¸ 2D t ¨© e ¸¹
2
[7.55]
Note that there is a close similarity between equations [7.55] and [7.45] that correspond to the temperature and velocity distributions in the outer layer, respectively.
Turbulent Convection in Internal Wall Flows
255
7.11. Transport equations and reformulation of the logarithmic layer
The instantaneous local kinetic energy of the turbulence is defined § · as K 1 2 uci uci , with the time mean K 1 2 uci uci 1 2 ¨uc2 vc2 wc2 ¸. The K © ¹ and T cT c dynamics play a key role both in the physical understanding of and modeling of the wall transfer mechanisms. The relations that govern these quantities are called turbulent transport equations. We propose to determine these equations. We will first determine the transport equation of the turbulent kinetic energy K and leave the reader to establish the relation governing T cT c. Some characteristics of the transport equations will subsequently allow us to obtain the logarithmic distributions in the fully turbulent mixing sublayers. We consider a fully developed turbulent internal flow in a twodimensional channel. The streamwise component of the NavierStokes equation reads for
wuc wuc §wu wuc· wuc 1 w p 1 wpc u uc vc¨ Q 2 u uc ¸ wc wt wx wz U wx U w x ©wy wy ¹ with the timemean average 0
1 wp w 2u w Q 2 ucvc wy U wx wy
The elimination of
1 wp
U wx
between these equations leads to the relation that
governs ucx, y, z, t in time and space:
wuc wuc §wu wuc· wuc w 1 wpc u uc vc¨ Q 2 uc ucvc ¸ wc wt wx w y w y w z U w x y w © ¹
[7.56]
Concurrently, vcx, y, z, t and wcx, y, z, t are governed by: 1 wpc wvc wvc wvc wvc u uc vc wc Q 2 vc wt wx wy wz U wy 1 wpc wwc wwc wwc wwc u uc vc wc Q 2 wc wt wx wy wz U wz
[7.57]
256
Convective Heat Transfer
We multiply equations [7.56] and [7.57] by uc, vc and wc, and proceed to timemean averaging. Adding the resulting equations leads to: PK TK DK N K H K wu PK ucvc wy
0
w uci K wx i
TK
DK
Q
NK
HK
w2 K wx kwx k 1 w uck pc U wx k
Q
[7.58]
wuci wuci wx k wx k
Each term of [7.58] has a clear physical meaning. The term PK represents the production. It regenerates the turbulence. The terms TK and N K correspond, respectively, to the turbulent transport and the pressurevelocity gradients redistribution. The molecular diffusion is denoted by DK . The turbulent dissipation H K regroups the correlations between the fluctuating velocity gradients. It plays a role as important as production does in the transport mechanisms. Figure 7.16 shows the distributions of the turbulent kinetic energy transport budget terms in wall units. These results have been reported by [MAN 88] and obtained in a turbulent channel flow at ReW 180 through direct numerical simulations. It is seen that production and dissipation dominate the kinetic energy transport. The production reaches its maximum at y
12 where the streamwise turbulent
intensity ucuc is also at its maximum (Figure 7.10). We note, in particular, that dissipation is in equilibrium with production at y ! 30 . This observation constitutes the basis of the present problem.
Turbulent Convection in Internal Wall Flows
257
DK+
0.25
PK+
0.15 T+K
0.05 0.05 0.15
HK+ y+
0.25 0
20
40
60
80
100
120
140
160
180
Figure 7.16. Turbulent kinetic energy budget in a fully developed turbulent channel flow at ReW 180 . DNS data from [MAN 88]
7.11.1. Description of the problem
We want to first show that the equilibrium between dissipation and production Pu cv c H u cv c at y ! 30 (Figure 7.16) results in the logarithmic velocity distribution. The second part of the problem deals with the transport equation of the temperature fluctuations. We seek to clearly identify the terms appearing in the temperature fluctuations transport equation, once it is determined. The production PT c and dissipation HT c terms are particularly important because, as we want to demonstrate, the equilibrium PT c HT c leads to the logarithmic mean temperature profile in a way quite similar to the velocity distribution. 7.11.2. Guidelines
We recommend expressing the dissipation as H u cv c v uH 3 lH , where uH and lH represent the characteristic velocity and length scales. The arguments to be developed should then be based on these scales in the fully turbulent mixing zone. The transport equation of the temperature fluctuations T cT c can be obtained by following a procedure similar to that which leads to the kinetic energy budget. We
258
Convective Heat Transfer
qcc 2 1 then propose to model the turbulent thermal dissipation term as HT ' v H2T ' 2 , U c uHT 'lHT ' where qHccT ' represents a typical flux scale related to the dissipation, together with the associated velocity uHT' and length lHT' scales. Once more, we have to be careful when we determine these scales in the fully turbulent region. Both H u cv c and HT ' , for instance, should be independent of molecular viscosity and diffusivity in this zone.
7.11.3. Solution
Consider the production term PK
ucvcwu wy of the turbulent kinetic energy
2
budget [7.58]. One has ucvc  uW in the constant shear zone. This region is fully turbulent; it is sufficiently far away from the wall in order that the molecular viscosity is negligible compared to the eddy viscosity, yet close enough to the wall so that the total shear does not vary appreciably. Thus any quantity should depend on Q in this region. Given that the dissipation can be put in the form H u cv c v uH 3 lH (by simple dimensional analysis), we only have to correctly determine the velocity and length scales, uH and lH . The choice for uH is, naturally, uW . The dissipation length scale has to be based on a local scale, such that lH v y Ny . The equality wu uW PK H K then implies , or in other words the logarithmic u distribution. wy Ny Let us now determine the budget for the temperature fluctuations. The temperature field is governed in time and space by
w T T c w T T c w T T c wTc u uc vc wc wt wx wy wz
D 2 T T c
with the Reynolds average u
wT wx
D 2T
w vcT c wy
Taking the difference between these equations yields the following relationship governing the instantaneous T c field:
wTc wt
u
w T T c w T T c wTc wT w vcT c uc vc wc D 2T c wx wx wy wz wy
Turbulent Convection in Internal Wall Flows
259
Multiplying the last equation by T c, and averaging the result in the intensity of temperature fluctuations budget 0
AT c TT c PT c DT c HT c
AT c TT c
1 2
uk
wT c2 wx k
1 w § · ¨uckT c2 ¸ ¹ 2 wx k ©
wT wx k
[7.59]
PT c uckT c
D w 2 T c2 2 wx kwx k
DT c
HT c D
wT c wT c wx k wx k
wherein we distinguish, in order, between the advection, turbulent transport, production, molecular diffusion and the dissipation. Equation [7.59] is not difficult to obtain, yet we have to decompose the term
w 2T c w xi w xi
DT c
ª 1 w 2 T c2 wTc wTc º » «¬ 2 w xk w xk w xk w xk »¼
D«
to correctly sort out the expressions of the diffusion and dissipation. The terms appearing in equation [7.59] reduce to AT c 0 , TT c
wT D w 2T c2 , DT c wy 2 wy 2
PT c vcT c
and HT c D
1 w 2 wy
vcT c2 ,
wT c wT c in a fully developed wx k wx k
turbulent channel flow homogenous in the streamwise and spanwise directions. That is, for instance, the case for the turbulent channel flow with uniform wall temperature presented in Figure 7.7. Figure 7.17 shows the energetic budget obtained in the configuration of Figure 7.7 with Pr 1 . The budget terms are scaled by inner variables. It is clearly seen that PT c  HT c, at y ! 30 , in the transport of turbulent kinetic energy in the logarithmic layer.
260
Convective Heat Transfer
Figure 7.17. Temperature intensity T cT c budget in a fully developed turbulent channel flow with uniform wall temperature at ReW 180 and Pr 1 [DOC 06]
qcc wT cc Uc and PT c  w In the fully turbulent mixing region v cT c qw . The Uc wy dissipation HT c should be independent of the molecular diffusivity and related to the local flux in this layer. A dimensional analysis shows that:
§qcc ·2 1 HT c v ¨ HT c¸ © Uc ¹ lHT cuHT c cc , The characteristic heat flux, length and velocity scales are qHccT c qw lHT c v y N T y and uHT c uW respectively. The equilibrium PT c  HT c therefore wT § qwcc · 1 . It is interesting to note that the wall flux temperature implies ¨ ¸ wy ¨©UcuW ¸¹NT y cc qw Tqccw is recovered independently in this analysis. The preceding equation UcuW
reduces to
wT wy
1
NT y
temperature distribution.
in wall variables and rises into the logarithmic timemean
Turbulent Convection in Internal Wall Flows
261
7.12. Nearwall asymptotic behavior of the temperature and turbulent fluxes 7.12.1. Description of the problem
We first want to establish the asymptotic behavior of the turbulent fluxes
ucTc v y 2 and vcTc v y 3 as y o 0 in a fully developed turbulent channel flow with uniform wall temperature, as is shown in Figure 7.11. The second part of the problem deals with the near wall form of the timemean temperature distribution
T Tw Tqccw
T y
in two separate cases, namely uniform heat flux and
temperature at the wall. 7.12.2. Guidelines
We propose using Taylor series expansions of the velocity and temperature fluctuations near the wall. To this end, it is necessary to use the instantaneous local continuity equation at the wall. The timemean temperature equation will constitute the basic equation to determine the asymptotic form of T as y o 0 . The key point is that the successive derivatives of T at the wall may be different, depending on the boundary conditions. 7.12.3. Solution
We will begin by analyzing the asymptotic behavior of the ucTc and vcTc correlations over a wall maintained at uniform temperature. Remember that the non
T Tw Tqccw . The expansion
dimensionalized temperature is defined as T y
in a Taylor series of the streamwise velocity fluctuations uc near the wall is u c x, y , z , t
§ w uc · 2 u cp ¨ ¸ yO y y w © ¹p
a1 y O y 2
[7.60]
§wuc· 0 , but ¨ ¸ z 0 . The continuity equation written at the wall (over ©wy ¹w which uc 0 and wc 0 ) gives
since ucw
§wuc wv c ww c· ¨ ¸ wz ¹w ©wx wy
§wv c· ¨ ¸ ©wy ¹w
0
262
Convective Heat Transfer
Implying v c x, y , z , t
§ w vc · 1 § w 2 vc · 2 3 vcp ¨ ¸ y O y ¸ y ¨¨ 2 © w y 2 ¸¹ © w y ¹p p
2
a2 y O y
[7.61]
3
The near wall behavior of the temperature fluctuations is given by: § wT c · 2 ¸ yO y y w © ¹p
T c x, y, z , t T pc ¨
a3 y O y 2
[7.62]
Indeed, we have Twc 0 by definition if the temperature is maintained constant at the wall. Combining these equations shows that
u' Tc v y 2
[7.63]
v' Tc v y 3
which is in agreement with the results presented in Figure 7.11.
The near wall behavior of T y to the first order is
T y
Pry
whatever the boundary conditions, according to equation [7.12]. The higher order terms depend, however, on the boundary conditions. Let us first consider the case of the uniform wall temperature. Equation [7.32] in wall variables is: 0
1 w 2T
w vcTc 2 Pr wy wy
[7.64]
Integration in y results in
wT wy
Pr vcTc Pr
3
c1 y Pr
Turbulent Convection in Internal Wall Flows
263
where we used the first order behavior of T and the result [7.63] concerning v cTc . Integrating the preceding equation again gives the asymptotic temperature behavior over a wall maintained at uniform temperature:
T y
Pry C1 y 4 O y 5
[7.65]
Coupling between the convective heat transfer to the fluid and conduction to the solid is plausible in the uniform wall flux case. The temperature at the fluidsolid interface may vary in time resulting in: §wTc· Tcx, y , z , t Tcw ¨ ¸ y O y 2 ©wy ¹w
a4 O y
Combining with the near wall behavior of the wall normal velocity fluctuations gives:
v cT c v y 2
c2 y 2
On the other hand, the timemean temperature is now governed by: u
wT wx
1 w 2T w vcTc Pr wy 2 wy
[7.66]
Therefore, at the wall we have § 2 · ¨w T ¸ ¨ 2 ¸ ©wy ¹w
§ · w vcTc wT Pr¨ u ¸ ¨ wy wx ¸ © ¹w
0
because vcTc v y 2 . The third derivative of the temperature at the wall is however different from zero, in contrast to the uniform wall temperature case. Indeed, deriving equation [7.66] with respect to y gives u
w §wT ·
¨ wy ¨©wx
¸ ¸ ¹w
u
w §wT · ¨
¸
wx ¨©wy ¸¹w
0,
264
Convective Heat Transfer
since the heat flux at the wall is uniform, but the remaining terms are different from zero § 3 · ¨w T ¸ ¨ 3 ¸ ©wy ¹w
§
Pr
·
w ¨w vcTc wT ¸ u wy ¨ wx ¸ © wy ¹
w
§ wu wT · ¸ Pr¨ 2c ¨ 2 wy wx ¸ z 0 © ¹w
Thus, the mean temperature varies as in
T y
Pr y C2 y 3 O y 4
[7.67]
when y o 0 over a uniform flux wall. 7.13. Asymmetric heating of a turbulent channel flow 7.13.1. Description of the problem We consider a fully developed flow of water in a smooth turbulent regime through a channel with a rectangular crosssection (spacing 2e, span length Lz, e y1), the turbulent viscosity is assumed to be constant and equal to Qtc. Experimental results show that uW 2e/Qtc = 26. Calculate y1 2e by matching the turbulent viscosity at plane P1 between the regions 1 and 2. Calculate the mean velocity U1. cc by applying the Colburn Relate the temperature difference Tw1 T1 to qw analogy in region 1, between the plane P1 and the heated wall.
Calculate the mean temperature distribution in region 2 by using the mean heat flux distribution. Calculate the wall temperature difference Tw1 Tw2 as a function of the parameters of the problem. 7.13.3. Solution 7.13.3.1. Wall shear stress
/ 2e 1 m s1 where Q Q / L . The The bulk velocity is calculated by U m Q z hydraulic diameter of the channel is Dh = 4e = 10 cm. The Reynolds number is calculated by Re U m Dh Q 1 x 0.1 8 x 10 7 = 1.25 105. The flow is therefore in turbulent regime. Assuming that roughness is sufficiently small for the flow to be in the smooth turbulent regime, the head loss coefficient is given by the Blasius law /
0.316 Re 1 4
from which the skinfriction coefficient is deduced as Cf
/ 8 , with C f
Ww UU m2 .
The wall shear stress is obtained by:
Ww
2 0.039 UU m Re 1 4 = 2.07 Pa.
The friction velocity is given by Ww
UuW 2 . Hence, uW = 0.045 m s1.
Turbulent Convection in Internal Wall Flows
267
7.13.3.2. Thermal budget for a slice of fluid We consider a slice of fluid of height 2e, length dx and span length Lz. Replacing T x by T1x T 2 y , the thermal budget yields: Lz
d dx
2e ³0 UC p u y >T1 x T2 y @dy qwccLz
Owing to effects of turbulence, the mean velocity profile is nearly flat in the central region of the channel and the high velocity gradient is concentrated in the very thin nearwall regions. Ignoring velocity variations in the above integral, the thermal balance becomes:
UC pU m 2e
dT1
cc qw
dx
[7.68]
We find Thus, the longitudinal temperature gradient is dT1 dx = qW" / U C p Q. dT1 dx = 0.24 °C m1. 7.13.3.3. Heat flux distribution across the channel For the present flow where the mean velocity field is onedimensional, the local energy equation reduces to
UC p u
wT wx
wqccy wy
[7.69]
where qccy is the total heat flux in the direction normal to the wall, namely the sum of the molecular ( k wT wy ) and the turbulent heat flux components ( UC p vcT c). Integrating [7.69] across the channel, we obtain: qccy ³ UC p u
wT dy + Constant wx
In the central region, where the velocity profile is nearly flat, the mean velocity may be approximated by u  U m and may be taken out the integral. The same holds wT wT1 . As a result, the heat flux distribution is nearly linear across the for wx wx central region of the channel.
268
Convective Heat Transfer
The heat flux must satisfy the boundary conditions: qccy
0
cc qw
qccy
2e
0
We adopt the approximation that the linear distribution is also valid in the nearwall regions. The distribution of heat flux across the channel is then: cc 1 y 2e qccy qw
[7.70]
It is worth noting that the unsymmetric heating induces different laws of distribution for the heat flux and the shear stress. In fact, the shear stress distribution is symmetric with respect to the plane of symmetry of the channel (y = e), Wy Ww 1 y e (see section 7.8.1). This property is not satisfied by the heat flux. cc 2 , according to [7.70]. It is In particular, Wy e 0 whereas qccy e  qw therefore not possible to use calculation methods based on the statement that these cc Ww . two quantities have similar distributions across the channel as qccy Wy  qw 7.13.3.4. Determination of the boundary between the nearwall and the central regions The logarithmic law is based on the turbulent viscosity model Q t NuW y ( N = 0.4). Matching the laws for Q t at ordinate y1 in the nearwall and the central regions leads to:
Q t1
NuW y1 = Q t c with uW 2e Q tc = 26.
y1
1
2e
26N
= 0.096
[7.71]
which is rounded off to 0.1 because of the approximations of the model. Thus, the boundary between the two regions is located at y1  5 mm, or in inner y1uW variables, y1  281.
Q
The mean velocity U1 at the boundary between the two regions may be calculated by using the logarithmic law: U1
1 0.4
Lny1 5.5 = 19.6
Turbulent Convection in Internal Wall Flows
269
U1 = 0.88 m s1. As expected, this velocity is slightly smaller than U m . 7.13.3.5. Colburn analogy in the nearwall region We apply the Colburn analogy in the nearwall region 1. This approach accounts for the molecular transfer, which plays an important role near the heated wall. The reference velocity is then U1 and the reference temperature scale is Tw1 T1 .
St1
Pr 2
C f1 with C f1
Ww UU12
3
[7.72]
and St1
cc qw
UC pU1 Tw1 T1
We find: Tw1 T1
U qcc Pr 2 3 1 w C p Ww
[7.73]
With the data of the problem, this relation gives Tw1 T1 = 15.8°C. 7.13.3.6. Temperature distribution in the central region Heat transfer is governed by turbulence in the central region so that the total heat flux is reduced to its turbulent component qccy
UC p vcT c= UC pD t
wT wy
where D t is the thermal turbulent diffusivity. Assuming Prt = 1, D t is constant in the central region as is Q t . Accounting for [7.70], the mean temperature satisfies
wT wy
cc qw
U C pD t
1 y
2 e
270
Convective Heat Transfer
which is integrated in T
qW" y y 2 / 4e C U C pD 1
This temperature profile is matched to that of region 1 at y1: T T1
§ y 2 y12 · ¨y y1 ¸ UC pDt ¨© 4 e ¸¹ cc qw
Assuming that this temperature distribution is valid up to the adiabatic wall, we obtain the temperature difference T1 Tw2 as:
T1 Tw2
2 cce § y1 · qw ¨1 ¸ UC pDt © 2e ¹
[7.74]
The numerical data lead to T1 Tw2 = 2.8°C. Combining [7.73] and [7.74], the wall temperature difference is finally obtained as: Tw1 Tw2
2 º ª U1 e § y1 · cc«Pr 2 3 qw ¨1 ¸ » «¬ C pWw UC pDt © 2e ¹ »¼
[7.75]
The numerical value is Tw1 Tw2 = 18.6°C. We note the important temperature drop in region 1 that constitutes the main resistance to heat transfer and the weaker temperature drop in region 2 owing to the efficient turbulent mixing in the central region. 7.14. Natural convection in a vertical channel in turbulent regime 7.14.1. Description of the problem
We consider the cooling system of the fast neutron reactor core in a nuclear power plant in the situation when the coolant flow is accidentally lost. The system is simulated by a loop with mercury at moderate temperature as the coolant. In a lossofcoolant accident, the fluid is maintained in motion by natural convection. The problem concerns the flow and heat transfer in a vertical channel (height L, spacing
Turbulent Convection in Internal Wall Flows
271
2e, span length Lz >> e), heated at uniform heat flux qcc along the lateral walls (Figure 7.19). The ratio L/e being much larger than 1, entrance effects are ignored so that the flow is assumed to be thermally and hydrodynamically fully developed through the whole channel height. We assume that the flow is in turbulent regime (this assumption will be checked at the end of the calculation). We denote T 0 the temperature and U 0 the density at the channel inlet. We propose calculating the flow rate through the channel and the difference between the wall and the fluid bulk temperature. Geometrical data: e = 2 cm, L = 1 m. The surface roughness is assumed to be negligible. Physical properties of mercury: – Density – Kinematic viscosity – Thermal conductivity – Prandtl number – Specific heat at constant pressure – Coefficient of thermal expansion
U 0 = 13.6 103 kg m3 Q= 107 m2 s1 k = 8.7 W m1K1 Pr = 0.0249 C p = 140 J kg1 K1 E = 104 K1
The channel is heated at uniform heat flux qcc = 103 W m2. 7.14.2. Guidelines
The velocity and thermal fields are assumed to be twodimensional. It is proposed that we first consider the momentum budget for the flow. The modified pressure is defined by
p*
p U 0 gx
where x denotes the vertical abscissa, with the channel inlet being taken as the origin. We denote Tw x the wall temperature and W y the total shear stress at any point in a channel crosssection. Write the local Reynolds equation in a channel crosssection. Integrate this equation over the crosssection to obtain the global momentum budget for a slice of fluid of height dx.
272
Convective Heat Transfer
We introduce the spatial average of the mean fluid temperature in a channel crosssection as:
Tmean x T0
1 e ªT x, y T0 º¼ dy 2e ³ e ¬
[7.76]
x
Tw
T u
L qcc
qcc
y e e T0 Figure 7.19. Natural convection in a vertical channel in turbulent regime
Relate the wall shear stress to the fluid bulk velocity U m using a classical law of forced convection in smooth turbulent regime. Integrate the momentum equation written for a slice of fluid over the total channel height. The fluid exits from the channel into a very large reservoir where the ambient fluid is at rest far from the channel. Show that the modified pressure p* is the same at the channel ends and simplify the momentum equation accordingly.
Turbulent Convection in Internal Wall Flows
273
The mean velocity and temperature profiles are assumed to obey the classical power law with the exponent 1/7 as
u y1 Ua
Tw x T x, y1 Tw x Ta x
K1 7 with K
y1 e
[7.77]
where U a and T a are, respectively, the velocity and temperature on the axis of symmetry of the channel, and y1 is the distance to a wall. Show that the difference between the fluid bulk temperature T m x and the spatially averaged temperature Tmean x is very weak in fully developed regime. These two quantities are not distinguished in the following discussion. Starting from the energy budget written for a slice of fluid of height dx and width 2e, determine the temperature distribution T m x along the channel when axial conduction is ignored. Calculate the expression of the buoyancy term in the momentum equation as a function of the parameters of the problem. Represent the
momentum equation in the form Re = f Gr* , Pr , where Re = UDh Q ( Dh denotes the hydraulic diameter of the channel). Express Gr * as a function of the parameters of the problem. Use a correlation of forced convection for metal liquid flows to calculate the heattransfer coefficient h and the difference between the wall and the fluid bulk temperature. Numerical application Calculate: – the bulk velocity; check that the flow regime is turbulent; – the heattransfer coefficient h and the difference between the wall and the fluid bulk temperature; check that the Colburn analogy has poor accuracy in the present case; – the temperature difference between the channel inlet and outlet; – the friction velocity; – the viscous sublayer thickness; – the temperature fluctuation scale; – the temperature drop across the viscous sublayer. Discuss the result.
274
Convective Heat Transfer
7.14.3. Solution
7.14.3.1. Momentum budget In a fully developed regime, the mean longitudinal velocity is independent of x and the mean normal velocity is zero everywhere in the channel. As a result, the inertia term is zero in the Reynolds equation along x, which becomes 0
w p* wW U 0 gE T T0 wx wy
[7.78]
where T x , y is the mean temperature and Wy the total shear stress at a current point in the channel. Note that Wy does not depend on x since the flow is fully
developed. The Reynolds equation in normal direction gives w p* wy mean pressure only depends on x.
0 , so that the
We integrate equation [7.78] between the two channel walls: 0
w p* 2e U0 g E wx
e
³ T T dy W e W e e
0
[7.79]
Equation [7.79] results from equilibrium between the resultant of pressure forces, buoyancy forces and friction forces on the walls for a slice of fluid delimited by the channel walls (Figure 7.20). We introduce the spatial average of T x , y over a channel crosssection ([7.76]). Assuming that the mean velocity profile is symmetric with respect to the axis of symmetry of the channel ( We W e ), we obtain: 0
w p* e U 0 gEeTmean T0 We wx
[7.80]
Let us remember that the wall shear stress is related to the mean velocity profile by We P wu wy y e . This quantity is negative (Figure 7.19). Equation [7.80] is finally integrated between the channel inlet and outlet. End effects are ignored; in other words, the regime is assumed to be fully developed over the whole channel length: 0
ª p* 0 p* L º e U g E e 0 ¬« ¼»
³
L
0
ª º ¬Tmoy x T0 ¼dx W e L
[7.81]
Turbulent Convection in Internal Wall Flows
275
Resultant of buoyancy forces
* § · * wp ¸ ¨ ¨p wx dx ¸ ¹ ©
dx W e
W e
p*
2e
Figure 7.20. Control domain for the momentum budget
Ignoring inertia effect at channel inlet, the modified pressure p* 0 is equal to the reservoir pressure p0* . Moreover, p* L is equal to the reservoir pressure p0* at the channel outlet (parallel streamlines at channel exit). The reservoir is in hydrostatic equilibrium so that the modified pressure is the same at the channel inlet and outlet. As a result, the pressure term vanishes in equation [7.81]. The momentum budget results from equilibrium between the resultant of buoyancy forces (positive, driving forces) and friction forces on the walls (negative, resisting forces) integrated over the whole channel length: 0
U0 g E e
³
L
0
ªTmoy x T0 ºdx W e L ¬ ¼
[7.82]
We assume that the laws governing friction in turbulent forced flow in a duct apply with satisfactory accuracy to the present natural convection flow. The Blasius law [7.19] gives: /
0.3164
Re
0.25
with /
We
1 8
UU m2
Consequently, We 0.0395 UU m2 Re 1 4 . 7.14.3.2. Spatiallyaveraged and bulk temperature in a crosssection The previous calculations involve the spatial average of the mean temperature in a crosssection whereas the energy budget is expressed with the bulk temperature, as seen in Chapter 2. We demonstrate below that these two quantities are very close together in turbulent regime.
276
Convective Heat Transfer
The velocity and temperature profiles being symmetric, equation [7.76] may be restricted to a halfpart of the channel crosssection. Moreover, the temperature may be conveniently referred to the wall temperature Tw , which is independent of y, instead of T0 , to calculate the integral. We assume that the halfpart of the velocity and temperature profiles corresponding to 0 d y1 d e ( y1 e y ) is well described by [7.77]. The spatial average of the mean temperature in a crosssection is then obtained by: 1
Tw Tmean x
e
³0e >Tw T x, y1 @dy1
Using [7.77], we find:
Tw Ta ³01 K1 7 dK
Tw Tmean
7 8
Tw Ta
The bulk velocity is calculated with the same procedure and the result is 7 Um U a , where U a denotes the mean velocity on the channel axis. 8 According to its definition, the bulk temperature is calculated by: 1
Tw Tm
³0 u y1 >Tw T x, y1 @dy1 e
Um e
Using [7.77] again, integration gives: Tw Tm
Tw Ta
Ua Um
1
³0 K1 7K1 7 dK
8 7 79
Tw Ta
8 9
Tw Ta
Finally: Tw Tmean Tw Tm
63 64
0.984
[7.83]
We conclude that the difference between the fluid bulk temperature T m x and the spatially averaged temperature Tmean x is very weak and we replace Tmean x with T m x in the following calculations.
Turbulent Convection in Internal Wall Flows
277
7.14.3.3. Energy budget The energy budget for a slice of fluid of height dx and width 2e (Figure 7.20) yields
UC pU m 2e dTm
2qccdx
from which it follows that
dTm
qcc
dx
UC pU m e
Thus, since the heat flux is uniform: qcc
Tm x
UC pU m e
x T0
[7.84]
Hence, for these conditions of hydrodynamically fully developed regime and uniform wall heat flux heating, the energy budget gives a simple relation between the bulk temperature and the unknown bulk velocity. 7.14.3.4. Bulk velocity The linear temperature distribution [7.84] is introduced into the integral of equation [7.82].
³0L>Tmean x T0 @dx
³0L>Tm x T0 @dx
qcc
L2
UC pU m e 2
Equation [7.82], completed with Blasius law, becomes: 0
gE
qcc Cp
L 0.079 U 0U m3 Re
1 4
The second term of the above equation may be expressed as a function of the Reynolds number only. The equation becomes: 11 4
0.079 Re
gE
qcc LDh 3
U 0C p Q 3
278
Convective Heat Transfer
Rearrangement of the righthand side reveals the Grashof number associated with uniform heat flux heating: Gr *
gEqcc k LDh 3
[7.85]
Q2
The dimensionless bulk velocity is expressed by:
Re
§12.66Gr * ·4 11 ¨ ¸ ¨ Pr ¸ © ¹
[7.86]
7.14.3.5. Heattransfer coefficient We use a correlation of forced convection for metal liquid flows [7.23] with the Reynolds number calculated above: Nu
6.3 0.0167 Re 0.85 Pr 0.93
[7.87]
We deduce the heattransfer coefficient h Nu k Dh and the temperature difference Tw Tm qcc h from the above equation. Note that Tw Tm is constant along the channel in a fully developed regime. 7.14.3.6. Numerical application Hydraulic diameter: for a channel of high aspect ratio ( e Lz ), the hydraulic diameter is given by: Dh
4 x 2eL z
2e L z
 4 e = 0.08 m
Grashof number: Gr *
9.81 x 1.82 x 10 4 x 12 x 10 3 x 0.08 3
8.7 x 1.14 x 10
7 2
Reynolds number [7.86]: § 9.7 x 1010 ·4 11 ¸ Re = ¨¨ ¸ ©0.079 x 0.0249 ¹
9.5 x 10 4
9.7 x 1010
Turbulent Convection in Internal Wall Flows
279
This result shows that the flow is effectively in a turbulent regime. Bulk velocity: U m = 9.5 x 10 4
x
1.14 x 10 7
= 0.134 m s1
0.08
Flow rate per transverse length unit: Q L z = U m 2e = 0.135 x 0.04 = 5.4 x 103 m3 s1 m1
Nusselt number [7.87]: Nu
6.3 0.0167 9.5 x 10 4
0.85
0.0249 0.93
15.5
It is worth noting the low value of Nu which is due to the very small Prandtl number of a liquid metal. This result is, however, compensated for by the high value of thermal conductivity:
h 15.5 x 8.7 0.08 = 1680 W m2K1 Difference between the wall and bulk temperature in a channel crosssection: Tw Tm
12 x 10 3 1680 = 7.1 K
This difference is constant along the channel in the fully developed regime. Between the channel inlet and outlet, the fluid bulk temperature increase is calculated by [7.84]: Tm x T0
12 x 10 3 13.6 x 10 3 x 140 x 0.135 x 0.02
= 2.3 K
For better accuracy, axial conduction effects should be taken into account for this fluid at Pr Q Q t y @
wu wy
Uc>D Dt y @
wT wy
The ratio of these quantities gives:
Wtot U qcctot Uc
>Q Q t y @ wu >D Dt y @ wT
The Reynolds analogy assumes that the ratio
>Q Q t y @ >D D t y @
C
[8.5]
Turbulent Convection in External Wall Flows
287
is constant in the entire turbulent layer. This assumption is coupled with the hypothesis cc , which means that a particular accent is put on the fully that Wtot Ww and qcctot qw turbulent mixing zone. Simple velocity and temperature profiles, such as u v y and T T w v y ,
are considered in a layer of thickness Gw , that includes the viscous and conductive sublayers. The profiles are assumed to be uniform in the turbulent core immediately adjacent ( y t Gw ) and in which the velocity and the temperature are equal to their respective bulk values: 1 G
³ u dy
u Um T
G 0
1 G ³ u T dy GU m 0
Tm
The last equation is valid for Pr ! 1 ; the thermal boundary layer thickness GT should be substituted in G in the opposite case. These hypotheses simplify the integration of [8.5], leading to
>Q Q t y @ Um >D Dt y @ Tw Tm
Ww U cc Uc qw
[8.6]
that can be rearranged as St
>D Dt y @ C f >Q Q t y @
[8.7]
where the Stanton number and the drag coefficient are respectively defined as St
h
UcU m
cc qw
Tw Tm UcU m
,
Cf
Ww UU m2
The Reynolds analogy goes through further simplifications by ignoring the molecular viscosity and diffusivity and assuming that the turbulent Prandtl number is approximately one
>Q Q t y @  Q t y >D D t y @ D t y
Prt y 1
288
Convective Heat Transfer
leading to the simple equation St
[8.8]
Cf
This relation ignores the role played by the viscous and conductive sublayers. Consequently, the Reynolds analogy has limited area of application. We know, for instance (from, for example, the Colburn correlation, see below), that the relationship between the Stanton number and the drag coefficient has to contain a dependence on the Prandtl number. That is found in the PrandtlTaylor analogy that considers equation [8.5] in each sublayer separately. It can be expressed as St
Cf
1 Prt · § 2 Pr 112C 1/ 1¸ f ¨ ¹ ©Prt
[8.9]
and is valid for Pr t 1 . We establish this relationship in section 8.10. We also have the following useful correlations: – Colburn: St
C f Pr 2 / 3
[8.10]
– Kader and Yaglom:
St
Cf 2.12 ln2Rex C f 12.5Pr 2 / 3 2.12 ln Pr 7.2
[8.11]
for 5 x 10 5 Rex 5 x 10 6
One of these correlations gives the heat exchange coefficient h once the drag coefficient is known. The drag coefficient can be found from either the classical Blasius relationship Cf
0.0296Re x 1 5
[8.12]
or the von Karman law 1
Cf
a blog10 C f Re x
with a = 2.4, b = 5.87.
[8.13]
Turbulent Convection in External Wall Flows
289
8.8. Temperature measurements in a turbulent boundary layer 8.8.1. Description of the problem A flat plate is maintained at constant temperature ( Tw = 50°C) in an air stream of uniform velocity ( uf = 10 m/s) and temperature ( Tf = 20°C) at a distance from the plate. Roughness triggers the flow near the leading edge of the plate so that the flow is assumed to be turbulent from x = 0 (Figure 8.3). At distance x (= 1 m) from the plate leading edge, measurements give the wall shear stress W w equal to 0.23 N m2. In the same crosssection, a cold wire of diameter 1 Pm is placed at 0.19 mm from the wall and measures the fluid mean temperature equal to 45°C. Using these experimental results, calculate the wall heat flux in the measurements crosssection. Compare with the Colburn correlation. Mean temperature measurements are performed with a thermocouple. The sensor has a spherical shaped diameter d § 50 Pm. What is the expected temperature when the thermocouple is placed at 4.5 mm from the wall? What is the uncertainty in the mean temperature measurement induced by the sensor size at this distance from the wall?
uf ,
Tf
thermocouple roughness
cold wire
Tw
Ww Figure 8.3. Turbulent boundary layer on a flat plate
Physical properties of air: – density
U = 1.13 kg m3
– kinematic viscosity Q= 106 m2 s1 – thermal conductivity k = 0.026 W m1 K1 – Prandtl number Pr = 0.7 – specific heat at constant pressure C p = 103 J kg1 K1
290
Convective Heat Transfer
8.8.2. Solution The friction velocity is calculated with the measured wall shear stress W w :
Ww
UuW 2 0.23 1.13 = 0.45 m s1.
Hence, uW
8.8.2.1. Measurements at 0.19 mm from the wall The distance of the measurement point to the wall is expressed in wall units by: y
0.19 x 10 3 x 0.45
yuW
16.7 x 10 6
Q
= 5.1
This result shows that the measurement point is located in the viscous sublayer. It is therefore possible to calculate the wall heat flux by using the temperature gradient calculated over the distance of 0.19 mm between the measurement point and the wall: cc qw
§wT · k ¨ ¸ ©wy ¹0
0.026 x
50 45 0.19 x 10 3
= 684 W m2
The corresponding Stanton number is given by: St
cc qw
UC p uf Tw Tf
=
684 1.13 x 10 3 x 10 x 30
= 2.0 x 103
The skinfriction coefficient is calculated from the experimental wall shearstress: Cf
Ww Uuf
2
=
0.23 1.13 x 10 2
= 2 x 103
The Colburn analogy gives St
C f Pr 2 / 3 = 2.53 x 103,
that is to say a difference of 26% relative to the direct measurement.
Turbulent Convection in External Wall Flows
291
8.8.2.2. Measurements at 4.5 mm from the wall The distance of the measurement point from the wall is expressed in wall units by: 4.5 x 10 3 x 0.45
yuW
y
16.7 x 10 6
Q
= 121
The measurement point is therefore located in the logarithmic layer. In wall units, the temperature is given by 1
T
NT
with 1 NT
ln y C
[8.14]
2.5 , C = 12.8 Pr0.68 – 7.3 for Prt
1 (equations [7.13] and [7.14]).
The logarithmic law gives: 1
T C
NT
ln y C = 2.5 ln121 C
12.8 x 0.7 0.68 7.3 = 2.74
T
14.7
It is worth noting that combining equations [7.13] and [7.50] gives a result in close agreement with this value. In fact, with 1 NT 0.9 x 2.5 2.25 , C
14.5 Pr 2
T
3
7.86 for Prt
0.9 , we find:
14.4
We use the definition of T
Tqccw Tw T
684 1.13 x 10 3 x 0.45
Tw T Tqccw
= 1.35°C
14.7 x 1.35 = 20°C
, with Tqccw
cc qw
UC p uW
292
Convective Heat Transfer
For a point located at 4.5 mm from the wall, the temperature is then expected to be: 30 °C
T
The thermocouple diameter is expressed in wall units as: d
duW
50 x 10 6 x 0.45 16.7 x 10 6
Q
= 1.35.
The temperature variation over this distance is obtained by differentiating relation [8.14]: 'T
1 'y
1 d
NT y
NT y
= 2.5
1.35 121
= 2.8 x 102
Therefore, the temperature difference over a distance equal to the thermocouple diameter is: 'T
Tqccw 'T = 1.35 x 2.8 x 10 2 = 3.8 x 10 2 °C
This variation is negligible relative to the measured temperature differences. 8.9. Integral formulation of boundary layers over an isothermal flat plate with zero pressure gradient 8.9.1. Description of the problem We consider a thermal boundary layer over an isothermal flat plate. The free stream velocity is therefore uniform. We want to proceed through an integral analysis to determine the thermal and dynamic boundary layer thicknesses. The wall is maintained at uniform temperature Tw . We will assume that local similarity solutions exist, with: u x , y uf
§ y · f u ¨¨ ¸¸ ©Gx ¹
Turbulent Convection in External Wall Flows
T x, y Tf Tw Tf
293
§ y · ¸ fT ¨ ¨G x ¸ © T ¹
Integral equations connected to turbulent boundary layers are similar in form to those governing laminar boundary layers. We recall that polynomial f u and fT profiles are generally chosen in the integral laminar boundary layer formulation (see Chapter 3). Polynomial distributions are, however, not convenient in turbulent boundary layers unless we take polynomials of very high degrees that are practically untreatable. The exact f u and fT profiles have to take into account the characteristics of different sublayers. We will therefore approach the problem in an approximate way by choosing power law distributions: u x, y uf
§ y ·1/ n ¨ ¨G x ¸ ¸ © ¹
T x, y Tf Tw Tf
§ y ·1 / n ¸ 1 ¨ ¨G x ¸ © T ¹
Experiments show that these distributions are partially acceptable in the log and external sublayers (constituting of 80% of the total flow field) if we take n  7 , but they are obviously not valid in the internal layer. In fact, the shear and flux deduced from these relations become indefinite at the wall. We will address this drawback by assuming that the drag coefficient is given by the Blasius law: §u ·2 C f x ¨ W ¸ ©uf ¹
C1ReGm
§u G · m C1¨ f ¸ © Q ¹
The empirical coefficients appearing in this relationship are C1 0.0225 and m 0.25 . The aim is to determine Gx and GT x . We also assume that the thermal and dynamic boundary layers start at the same point x 0 . This problem is somewhat classical. It is treated in more or less different ways in several textbooks such as [SCH 79] or [ARP 84]. 8.9.2. Guidelines We have to use the integral equations dealing with velocity and temperature fields developing over a flat plate. These equations have to be modified to sort out the drag coefficient and the Stanton number. It will be necessary at this point to use
294
Convective Heat Transfer
an analogy between the wall heat flux and shear stress. We recommend the Colburn analogy, St x C f x Pr 2 / 3 . It is also assumed that the ratio between the thermal
and dynamic boundary layer thicknesses GT x Gx is a constant depending on the molecular Prandtl number. 8.9.3. Solution The integral streamwise momentum and energy equations in a turbulent boundary layer are similar in form to those governing a laminar boundary layer. The integral momentum equation is: d dx
G x
³ Uu uf u dy
Ww
[8.15]
0
The timemean temperature distribution is governed by: d GT x
³ UC p u T Tf dy qwcc
dx
[8.16]
0
Dividing the two sides of equation [8.15] by uf2 (which does not depend on x ), using the definition of the skinfriction velocity Ww coefficient C f x
uW
2
uf leads to
ª1 u § u · ºdG «³ ¨1 ¸d K» ¬0 uf © uf ¹ ¼ dx
UuW2 and of the drag
C f x
where we introduced the variable K
[8.17] y Gx and assumed a similarity form
u uf fu K . We will use the algebraic mean velocity profile f u K K1/ n . Thus the integral appearing in the preceding equation is calculated as: 1
u § u · ¨1 ¸dK uf ¹ f ©
³u 0
1
³ K1/ n 1 K1/ n dK n 1 n 2 0 n
Turbulent Convection in External Wall Flows
295
Using the drag coefficient given by the Blasius law, and rearranging equation [8.17] gives:
n 1 n 2 C
dG dx
n
§u G · m f ¸ 1 ¨ © Q ¹
This equation now has to be integrated from the leading edge of the plate to some streamwise x position. The result is:
G ªm 1 n 1 n 2 x
« ¬
n
1
m
ºm 1 §u x · 1 m C1» ¨ f ¸ ¼ © Q ¹ 1 ºm 1
ªm 1 n 1 n 2 C1» « n ¬ ¼
[8.18]
m Rex 1 m
The first conclusion that can be drawn from this relationship is that the turbulent
boundary layer thickness varies like G v x x 4 / 5 ( m 0.25 ), significantly more rapidly than the laminar boundary layer for which G v x1/ 2 (Chapter 3). Using the empirical coefficients m 0.25 , n 7 and C1 0.0225 in [8.18], it is further concluded that: 1/ m 1
G x
5  0.37 Re 1/ x
[8.19]
Integral equation [8.16] can be rearranged to give
d GT x u T Tf dx
³
0
uf Tw Tf
dy
cc qw
UC p uf Tw Tf
[8.20]
because the wall temperature is uniform, the physical fluid characteristics are uniform and uf does not depend upon x . We note the presence of the Stanton number at the righthand side of this equation: St x
h x
Uc p uf
ccx qw
Uc p uf Tw Tf
296
Convective Heat Transfer
We further opt for the following selfsimilar distributions:
u uf
§ y · ¸ fu¨ ¨G x ¸ © ¹
and
T Tf Tw Tf
§ y · ¸ fT ¨ ¨G x ¸ © T ¹
[8.21]
We will now assume that the ratio of the thermal to dynamic boundary layer thicknesses is independent of the streamwise position x and is a function of the molecular Prandtl number only with GT x Gx F Pr . Defining the new integration variable by Kc y GT x , using the empirical Colburn relation and combining with the Blasius law, we obtain: St
C f x Pr 2 / 3
C1ReGm Pr 2 / 3
Thus, equation [8.20] now takes the form ª1 ºdG c fT Kc dKc» T «³ f u KF ¬0 ¼ dx
§u · m C1ReGm Pr 2 / 3 C1¨ f ¸ F mGT m Pr 2 / 3 © Q ¹
where the definition of F has been used to eliminate G . We will use simple similarity distributions f u K K1/ n and fT Kc 1 Kc1/ n
resulting in ª1 ºdG c fT Kc dKc» T «³ f u KF ¬0 ¼ dx
F 1/ n
n
n 1 n 2
dGT dx
Combining and integrating leads to: 1
GT x
§ 1 ·ºm1 ª m ¨m ¸ «m 1 n 1 n 2 2 / 3 © n ¹» 1m C1Pr F Rex « » n « » ¬ ¼
[8.22]
Turbulent Convection in External Wall Flows
297
m 1m GT x v Rex
5 The conclusion is that Re1/ , exactly as for the boundary x layer thickness Gx . Dividing equation [8.22] by [8.18] and rearranging enables us to reach a relationship for the ratio of the boundary layer thicknesses, namely: 2n
F
GT x Pr 3(n1) Gx
[8.23]
An equation that can be used in practical situations dealing with dimensionless thermal boundary layer thickness is finally obtained by using the empirical coefficients m 0.25 , n 7 and C1 0.0225 :
GT x
5  0.37 Pr 0.58 Re 1/ x
[8.24]
8.10. PrandtlTaylor analogy 8.10.1. Description of the problem We discussed the Reynolds analogy in section 8.7. This is a crude approximation since it does not take into account the effect of the wall layer in which the turbulent viscosity and diffusivity are negligible compared to their molecular analogs. The PrandtlTaylor analogy divides the whole layer into two parts. The molecular viscosity and diffusivity are predominant in the wall layer of thickness Gw . This layer extends to typically G w 13 in inner variables. The turbulent bulk zone constitutes the adjacent sublayer in which Q t and D t govern the flow and heat transfer. To simplify the formulation of the problem, it will be assumed that both the mean velocity and the temperature are constant and equal to their consecutive bulk values in the fully developed turbulent sublayer as shown in Figure 8.4. The solution requires a procedure similar to that conducted in section 8.7, but here we have to take into account the supplementary effect of the wall layer. 8.10.2. Guidelines It is recommended that equation [8.5] is used as the departure point. This equation has to be integrated in the wall layer and subsequently the turbulent bulk
298
Convective Heat Transfer
zone (Figure 8.4). The resulting relationships should then be conveniently combined to obtain an equation relating the Stanton number to the drag coefficient. This approach is valid only for Pr t 1 . Discuss why.
Tm Fully developed turbulent mixing zone
Um
uG w
TG w
Gw
Near wall sublayer
Tw
Figure 8.4. Simplified velocity and temperature profiles to be used to obtain PrandtlTaylor analogy
8.10.3. Solution Reconsider equation [8.5]:
Wtot U qcctot Uc
>Q Q t y @ wu >D Dt y @ wT
The reader should again consult section 8.7 to be familiar with the concept of analogy. The difference between the PrandtlTaylor and Reynolds analogies is that the former takes into account the effect of the wall layer that is ignored in the latter. The whole turbulent layer is divided into two main zones, as shown in Figure 8.4, in order to sufficiently simplify the problem. We further assume that the total shear cc . The wall sublayer, stress and heat flux are constant with Wtot Ww , and qcctot qw wherein the eddy viscosity and diffusivity are ignored, is extended slightly above the viscous sublayer until y Gw 13 and thus includes the buffer layer. The preceding equation takes the following form in this region
Ww U cc Uc qw
Q wu D wT
Pr
wu wT
Turbulent Convection in External Wall Flows
299
whose integral is cWw
Pr
cc qw
uG p
[8.25]
Tw TG w
where Gw stands for the wall sublayer thickness. The molecular viscosity and diffusivity are negligible in the fully turbulent core. We further assume that the turbulent Prandtl number is constant
Q t y Dt y
Prt
Ct
with
Q y wu t Dt y wT
Ww U cc Uc qw
Prt
wu wT
Integrating this equation from y G p to y G leads to: cWw cc qw
Prt
U m uG w
[8.26]
TG w Tm
Eliminating TG w between equations [8.25] and [8.26] results in TG w
where ]
] Tw Tm 1 ]
Prt U m uG w
uG w Pr
Substituting this expression into equation [8.26] gives Tw Tm
cc qw
Prt Um uG uG Pr cWw
that may be rearranged to give
w
w
cc ª uG w U m Prt qw «1 cWw «¬ U m
§Pr ·º ¨¨ 1¸¸» ©Prt ¹»¼
300
Convective Heat Transfer
ª º « » 1 Prt « »C St « uG w » f 1 Pr Pr 1 t « U » ¬ ¼ m
[8.27]
This relationship is known as the PrandtlTaylor analogy. Remember that the Stanton number and the drag coefficient are respectively defined by: St
cc qw
Tw Tm UcU m
,
Cf
Ww UU m2
The term under bracket at the righthand side of equation [8.27] is the weighting factor of the drag coefficient. It represents the PrandtlTaylor correction of the Reynolds analogy (equation [8.8]). The present approach is valid only for Pr t 1 . Indeed, in the opposite case of Pr 1 , the conductive sublayer may be considerably thicker than the viscous sublayer, which is contradictory to ignoring the molecular diffusivity at y t Gw . For Pr t 1 the conductive sublayer thickness
decreases with Pr as G c v Pr 1/ 3 (see section 7.9). Ignoring the eddy diffusivity in the wall layer y d Gw is therefore incorrect, in particular when Pr !! 1 , but that is the price to pay for having a simple approach. We further have: uG w Um
uGw
G
 w Um Um
2 Gw C1/ f
2 12C1/ f
We finally recover the form of the PrandtlTaylor analogy as given by equation [8.9], namely: St
Cf
1 Prt § · 2 Pr 1 12C 1/ 1¸ f ¨ ¹ ©Prt
Note that St C f Reynolds number.
f Re, Pr , Prt , because the drag coefficient depends upon the
Turbulent Convection in External Wall Flows
301
8.11. Turbulent boundary layer with uniform suction at the wall 8.11.1. Description of the problem Control of turbulent boundary layers is of major importance from both fundamental and applicative points of view. One category of the classical openloop active control strategies consists of applying uniform blowing or suction at the wall. Let us consider a turbulent boundary layer with zero pressure gradient subject to uniform wall suction with constant wall normal velocity v w , as shown in Figure 8.5. Experiments show that wall suction annihilates the streamwise growth of the boundary layer in such a way that the thermal and dynamic boundary layer thicknesses stay constant when the critical suction parameter is c v v w uf 5 u 103 . We will be limited to this critical case here. The aim is to determine the velocity and temperature distributions in the fully turbulent mixing sublayer. The wall is subject to either a constant temperature or uniform heat flux. uf, Tf c v =5u103 ==> Boundary layer thicknesses become asymptotically constant
y x Uniform vw Figure 8.5. Turbulent boundary layer over a flat plate subject to uniform wall suction
8.11.2. Guidelines Our departure point will be the turbulent boundary layer equations. The thicknesses of the thermal and dynamic boundary layers are asymptotically uniform in the present circumstances. We first have to determine the wall’s normal velocity distribution. We also propose to apply two different onepoint closure schemes, namely the mixing length hypothesis and the eddy viscosity formulation (see Chapter 7), and to compare the emerging results.
302
Convective Heat Transfer
8.11.3. Solution Recall the turbulent boundary layer equation: u
· w § w u ucvc¸ ¨Q wy © wy ¹
wu wu v wx wy
The boundary layer thickness G no longer varies with x and therefore wu wx 0 in the socalled asymptotically fully developed zone. The continuity equation leads to:
wu wv wx wy
wv wy
0
Thus, we conclude that the wall’s normal velocity distribution is v vw d 0 , since v does not depend upon x and is consequently constant in the entire layer. The boundary layer equation takes the form
vw
du dy
· d § du ucv c¸ ¨Q dy © dy ¹ 0 ) to a point y in the boundary layer
that can be integrated from the wall ( y y
vw ³
0
du dy
y
dy
³
0
· w § du ucvc¸dy ¨Q wy © dy ¹
giving ª du ºy vw u y «Q ucv c» ¬ dy ¼0
Q
du dy
y ucvcy
Ww U
[8.28]
because ucvc 0 at the wall and, by definition, Q wu wy y 0 Ww U . We seek out the velocity and temperature distributions in the fully turbulent region. We can consequently ignore the molecular viscous stress with respect to the Reynolds shear stress and rewrite equation [8.28] in wall units Q and uW :
u ucvc 1 vw
[8.29]
Turbulent Convection in External Wall Flows
303
The final result will obviously depend upon the closure scheme that is used for
ucvc . We have
§du ·2 ¸ ©dy ¹
N 2 y 2 ¨
ucvc
if we opt for the Prandtl mixing length hypothesis (see section 7.9). Therefore
1 vw u
§du · ¸ ¸ ©dy ¹
1/ 2
Ny ¨¨
[8.30]
which clearly shows that u is a function of y and vw . The preceding equation
takes the following form by the transformation of the integration variable F
dF
1 F 1 2
vw
vw u
1 dy
N y
whose integral is 1/ 2
21 F
>
@
C c vw ln y C v p
Now, F 0 when v w 0 and therefore the integration constant is C c 2 . We consequently have the solution º 1 2 ª 1/ 2 1 vw u 1» ln y C vw «¬ ¼ N vw
[8.31]
taking the explicit form u
2 1 ª1 º ln y C vw vw ln y C v « w » ¼ N 4 ¬N
1
>
[8.32]
@ decreases the mean velocity with respect to
The term 1 4 vw 1 N ln y C vw
2
the logarithmic distribution in the standard turbulent boundary layer because vw 0 . The classical logarithmic distribution should be recovered from equation
[8.31] when vw o 0 . In fact, a Taylor series expansion to the first order gives
304
Convective Heat Transfer
1 vw u
1/ 2
 1 1 2 vw u for small v w u , and we find u
1 N ln y C when
vw o 0.
A similar procedure enables us to determine the temperature field. The equation governing the temperature in the asymptotically developed thermal boundary layer wherein dGT dx 0 is: vw
dT dy
· d § dT v cT c¸ ¨D dy © dy ¹
The integration of this equation from the wall to some y position in the thermal boundary layer gives rise to: cc qw
vw >T y Tw @
Uc
D
dT dy
v cT c
ccx and Tw x This relationship is local. We have to consider local values of qw in the boundary conditions dealing with uniform wall temperature and heat flux respectively. Scaling the temperature by the wall variables T Tw T Tqccw and ignoring the molecular heat flux D dT dy compared to vcT c in the fully turbulent mixing zone, we obtain a relationship analog to equation [8.29], namely:
1 vw T
The
vcT c
v cT c
integration
[8.33] of
this
Q t Prt wT wy and Q 1 vw
ln 1 vw T
Prt
N
t
equation
combined
with
the
closure2
Ny , gives rise to:
ln y CT vw
1
NT
ln y CT vw
[8.34]
o 0 is the classical logarithmic The asymptotic form of this distribution for vw profile since ln 1 vwT  vwT for small vwT . The assumptions we made to
obtain equation [8.34] are: – the turbulent Prandtl number is constant;
2. T
varies in the opposite direction to T .
Turbulent Convection in External Wall Flows
305
– the logarithmic part of the thermal boundary layer merges into the logarithmic velocity sublayer justifying the use of Q t Ny in the eddy diffusivity formulation. Equation [8.34] can be modified to: 1
vw N y T
1 vw
[8.35]
The coefficient CT depends on CT
CT Pr, Prt , vw . The difference between
T
vw
exp
vw CT
distributions [8.31] and [8.34] comes from the fact that we used a Prandtl mixing length closure for the mean velocity and an eddy diffusivity approach for the temperature. The reader can easily verify that using
Q t
ucvc
du dy
Ny
du dy
results in
u
1 vw
exp
vw N y
vw C
1 vw
[8.36]
which is qualitatively equivalent to [8.35]. The constants C and CT appearing in the velocity and temperature profiles expressions are functions of the wall suction velocity, and they can consequently be adjusted to collapse the results inferred from the models and the experiments. We show in Figure 8.6 the distributions given by equations [8.32] and [8.35] with Prt 1, N NT 0.4 and C CT 5. We took v w 0,1 ; in wall units that
roughly corresponds to cv vw uf 5 x 10 3 rising into asymptotically constant boundary layer thicknesses sufficiently far away from the leading edge. The profiles are compared to the classical logarithmic distribution of the canonical turbulent boundary layer. We notice that suction flattens the velocity and temperature distributions and considerably reduces them compared to the standard law. The eddy viscosity model underestimates u and/or T with respect to the Prandtl mixing length closure.
306
Convective Heat Transfer
8.12. Turbulent boundary layers with pressure gradient. Turbulent FalknerSkan flows 8.12.1. Description of the problem We are considering a turbulent FalknerSkan boundary layer with a potential velocity variation uf x Kx m . We wish to show that similarity solutions exist for both the velocity and temperature distributions. 20
u + or T +
Eddy viscosity
15
Without aspiration Mixing length
10
5
y+
0 10
100
1000
Figure 8.6. Velocity and (or) temperature distributions in the fully turbulent mixing zone of a turbulent boundary layer over a flat plate subject to uniform wall suction. Comparison of the profiles obtained using equations [8.32] and [8.35] and the logarithmic distribution of the canonical turbulent boundary layer. See section 8.11.3 for details
8.12.2. Guidelines We have to begin with the Reynolds averaged twodimensional turbulent boundary layer equations. The closure will be performed through an eddy viscositydiffusivity formulation. We therefore recommend using the similarity variables introduced in Chapter 3 to check whether they are suitable for solving the problem. We will also discuss the possibility of using mixing length closures by providing some key elements in the solution.
Turbulent Convection in External Wall Flows
307
8.12.3. Solution Let us reconsider the Reynolds averaged equations [8.4] and combine them with the eddy viscositydiffusivity closure: ucvc Q t y vcT c
wu wy
Q t y wT Prt y wy
The boundary layer equations of the problem are:
wu wv wx wy
0
u
wu wu v wx wy
u
wT wT v wx wy
w ª duf w u º Q «1 Q t » wy ¬ dx wy ¼ w ª§ 1 Q ·wT º Q «¨¨ t ¸¸ » wy «¬© Pr Prt ¹wy »¼
uf
[8.37]
The potential velocity is uf x Kx m in the FalknerSkan flows. They have already been analyzed in Chapter 3 in laminar boundary layers. Equations [8.37] differ in form from their laminar counterparts due to the presence of eddy viscositydiffusivity terms and the turbulent Prandtl number. The supplementary terms with respect to the laminar boundary layer are:
w ª wu º » «Q wy ¬ t wy ¼
w ªQ t wT º » « wy « ¬Prt wy » ¼
[8.38]
It is therefore sufficient to duplicate the similarity analysis conducted in Chapter 3 by adding supplementary terms [8.38] to the laminar dynamic and thermal boundary layer equations. The similarity variable is still: §u ·1/ 2
K y ¨ f ¸ ©Qx ¹
308
Convective Heat Transfer
The velocity distribution is of the form:
u uf x
F cK
We remember that a wall temperature variation of the form Tw Tf Hx n is necessary to ensure similarity solutions of the temperature field. The dimensionless temperature is defined as T x, y Tf
T K
Tw x Tf
in the same way as in Chapter 3. This representation is not appropriate in the case of ccx as in that case Tw x is unknown. The specified heat flux at the wall qw ccx is deduced from dimensional analysis as temperature scale related to qw
qcc x x k Re where Rex w
uf x Q . Therefore, the dimensionless temperature for
1/ 2 x
a specified wall heat flux is:
T x, y Tf
2 ccx x k Re1/ qw x
T K
2 ccx x k Re1/ The condition for similarity is reduced to qw x in laminar FalknerSkan flows.
H cx n in this case as
Equation [8.38] is transformed into:
w ª wu º «Q t » wy ¬ wy ¼ w ªQ t wT º « » wy « ¬Prt wy » ¼
2 c uf Qx
Q t F cc
§Q
>Tw x Tf @¨¨
·c u f
Tc¸ ¸
Tw x specified
w ªQ t wT º §Q t · qwccx 1/ 2 « » ¨ Tc¸ Re x ¨ ¸ wy « ¬Prt wy » ¼ ©Prt ¹ kx
qwccx specified
t
©Prt
c
¹ Qx
[8.39]
Turbulent Convection in External Wall Flows
309
The remaining terms are determined in exactly the same way as for laminar boundary layers. Performing the transformation of variables and substituting the resulting expressions in equations [8.37] give:
ª ºc m 1 FF cc m F c2 1 0 «1 Q t F cc» ¬ ¼ 2 ºc ª§ m 1 1 Q · «¨ t ¸Tc» FTc nF cT 0 ¨ ¸ 2 « ¼ ¬©Pr Prt ¹ »
[8.40]
The associated boundary conditions are: F (0)
F c(0)
0, F c(f) 1
T (0) 1, T (f) 0 Tc(0) 1, T (f) 0
Tw x specified
[8.41]
qccx specified w
We, of course, recover equations [3.12] and [3.19] when Q t 0 . The advantage of the present procedure lies in the fact that an existing computational code for the laminar case can easily be adapted to the turbulent FalknerSkan flows. It is, however, obviously necessary to introduce closure schemes for eddy viscosity and Prt . We know the general relationship Prt
Prt y ,Re,Pr , but in order to
simplify the problem we will assume that the turbulent Prandtl number is constant with Prt 0.85 . The most popular closure is the Cebeci and Smith’s mixing length hypothesis. The eddy viscosity is expressed as
Q t
§wu · A2 ¨¨ ¸¸ ©wy ¹
(see section 7.9); it is combined with a van Driest relationship for the mixing length A
ª
§ y ·º ¸» © A ¹» ¼
Ny «1 exp¨ « ¬
The damping coefficient A depends upon the pressure gradient: A
26
1 c3
1/ 2
3
1 §Q dp · ¨ ¸ uW3 ©U dx ¹
[8.42]
310
Convective Heat Transfer
The empirical value of the coefficient c is c 12 . We will now briefly discuss some characteristics of turbulent boundary layers subject to mean pressure gradient. We will distinguish between two cases, namely flows with adverse pressure gradients ( 3 ! 0 ) and flows with favorable dp dx ( 3 0 ). We will focus on the case 3 ! 0 which can lead to separation. A favorable pressure gradient accelerates the nearwall flow and stabilizes the turbulence. The flow can be entirely relaminarized when 3 0 is sufficiently large. In the turbulent FalknerSkan flows with uf v x m , the separation can take place when m d 0.27 [SCH 79]. A general relationship for the flow detachment is hard to obtain because the adverse pressure gradient has to act for a sufficiently long time near the wall to lead to separation that is significantly sensitive to the upstream history of the turbulent boundary layer. The shear basically becomes zero in a thin layer near the wall when separation occurs. A subsequent layer, wherein the total shear is nonzero and constant, still exists in separated turbulent boundary layers. The turbulence production takes place in the external layer and affects the wall transfer by diffusion. The detachment has to be avoided whenever possible because it causes serious problems of energy loss and induces structural instabilities (for example in problems connected to fluidstructure interactions). The effect of adverse pressure gradient on heat transfer in turbulent wall flows is an area of current research. We summarize hereafter some characteristics on which a current consensus is established: – the drag coefficient decreases in turbulent flows subject to adverse pressure gradients. The Stanton number, in return is not significantly affected; – the temperature distribution is profoundly different from the velocity field, which points to a strong deviation from the Reynolds analogy; – the Reynolds shear stress expressed in wall variables increases drastically in the wall layer, with its maximum pushed away from the wall. There is no constant total shear zone, and therefore turbulent adverse pressure gradient boundary layers do not provide universal distributions.
Turbulent Convection in External Wall Flows
311
u+ 30
3+ 0.
20
9.12u103 1.93u102 2.56u102
10
0
0
u + =2.44 lny++5 0
u +=y + y+
0 1
10
100
1000
Figure 8.7. Velocity profiles scaled by inner variables in a turbulent boundary layer subject to adverse pressure gradient for different values of 3 , according to the experiments of [HOU 06]
Figure 8.7 shows experimental velocity profiles obtained by [HOU 06] in a turbulent boundary layer subject to adverse pressure gradient. The experiments were conducted in the range 0 d 3 d 2.56 u 10 2 and continued until the separation limit. The viscous sublayer is still persistent under dp dx as expected. We notice, however, important deviations in u with respect to the canonical logarithmic profile ( 3 0 ). The Coles wake zone becomes noticeably significant at 3 2.56 u 10 2 , and the velocity is consequently much higher than the classical u 2.44 ln y 5 distribution in the far logColes region. Figure 8.8 shows the temperature distribution under the same conditions. The conductive sublayer with T Pry ( Pr 0.7 here) is present whatever 3 is. However, and contrary to the velocity profiles, T is systematically lower than the corresponding distribution in the canonical turbulent boundary layer without adverse pressure gradient. It is impossible under these conditions to look for an analogy between u and T . In section 8.13 we will analyze the reasons for the lack of analogy between the velocity and temperature fields in noncanonical adverse pressure gradient turbulent boundary layers.
312
Convective Heat Transfer 20
T+ 10
0
0
T +=2.075 ln y + + 3.8
0
T +=Pr y + y+
0 1
10
100
1000
Figure 8.8. Temperature profiles in inner variables for different pressure gradient parameters 3 , according to the measurements of [HOU 06]. For legend, see Figure 8.7
8.13. Internal sublayer in turbulent boundary layers subject to adverse pressure gradient 8.13.1. Description of the problem We wish to determine more precisely the velocity and temperature distributions in the wall zone of a turbulent boundary layer subject to adverse pressure gradient. We will assume, as usual, that the advective terms are locally negligible. The adverse pressure gradient modifies the characteristic velocity scale in the wall region, and the usual shear velocity uW x is no longer appropriate to physically describe the mean velocity characteristic. 8.13.2. Guidelines We will begin by writing the Reynolds averaged equations and ignore the inertial terms. We will see after the integration of the velocity equation that the total shear stress is no longer equal to 1 as in canonical boundary layers, but that it varies with y . This shows that the shear velocity is not an appropriate scale in the presence of adverse pressure gradient. We must therefore define another velocity scale u* for the total shear stress to be 1. Similarity solutions can then be found in new variables
Turbulent Convection in External Wall Flows
313
nondimensionalized by this velocity scale (that will not be a constant). It is therefore proposed to use u* to define an eddy viscosity that will enable us to calculate the velocity distribution in the fully turbulent layer. The same procedure has to be followed to investigate the temperature distribution. It must be noted that the passive scalar is not directly affected by the adverse pressure gradient in the conductive sublayer. We recommend the use of an eddydiffusivity closure by using the eddy viscosity that has already been determined and the turbulent Prandtl number. This problem has been studied by [TAR 09]. 8.13.3. Solution The equation that governs the velocity in the inner layer is 0
1 dp w 2u w Q 2 ucvc U dx wy wy
where we locally ignored the inertial terms at the lefthand side. Integrating this equation from the wall to some y in the flow gives:
Wtot
§ w u · U ucvc¸ y ¨P © wy ¹
dp dx
y Ww
[8.43]
The total shear nondimensionalized by the local friction velocity uW x reads Wtot 1
dp y dx
1 3 y
[8.44]
where
3
3 1 §Q dp · 1 § duf · §u3 · ¨Q uf ¸ ¨ ¸ ¨ ¸ dx ¹ ©uW ¹ uW3 ©U dx ¹ uW3 ©
[8.45]
A new velocity scale, that will be called the pressure gradient velocity u3 , emerges from [8.45]. Since the Reynolds shear stress is negligible in the viscous sublayer, we have Wtot
wu wy
1 3 y
314
Convective Heat Transfer
and u
y
1 2 3 y 2
[8.46]
which reduces to the classical solution u y in the absence of pressure gradient. We can see that the contribution from dp dx in the viscous sublayer is only slightly important since it is roughly 6% in the case 3 2.56 u 10 2 at y 5. The situation is different in the fully turbulent mixing zone wherein the adverse pressure gradient has significant effects, as will be shown below. The nondimensionalization of u by the shear velocity leads to similarity solutions in zero adverse pressure gradient turbulent boundary layers through Wtot 1 . Equation [8.44] clearly indicates that uW is no longer a similarity velocity scale under dp dx . Let us consider, instead of uW , a new velocity scale defined as u*2
u3 uW2 3 y uW
[8.47]
that directly takes into account the adverse pressure gradient effects. The total shear given by [8.43] and nondimensionalized by u* , is: * Wtot
Wtot Uu*2
· 1 § u33 2 ¨¨U ¸¸ 1 y U u W Uu*2 © uW ¹
[8.48]
Using u* therefore enables us to obtain similarity solutions, as does uW in zero pressure gradient turbulent boundary layers. This velocity scale should also intervene in the formulation of eddy viscosity. The turbulent velocity scale is Q t v A t u t , where A t and u t are, respectively, local turbulent length and velocity scales. Taking as usual A t Ny , ignoring molecular viscosity terms in the fully turbulent mixing zone, and finally introducing u t u* (instead of u t uW ), gives rise to
Q t Nyu*
[8.49]
together with
Wtot
Uu*2 U ucvc UQ t
wu wy
U Nyu*
wu wy
Turbulent Convection in External Wall Flows
u* Ny . The shear scaled by inner variables Q
resulting in the simple form du dy and uW is consequently: du
dy
u*
1 3 y
1/ 2
[8.50]
Ny
Ny
315
The integral of this equation gives the velocity distribution in the fully turbulent mixing zone:
u
ª º § · 1 3 y 1 ¸ § ·» 1 « ¨ ln y 2 ln 2¨ 1 3 y 1¸ B ¨ ¸ © ¹» N « 2 © ¹ ¬ ¼
[8.51]
B is the constant appearing in the classical logarithmic law ([7.9]). The pressure gradient does not directly intervene in the temperature distribution, contrary to the velocity field; dp dx indirectly affects the thermal field in the logarithmic region through the eddy viscosity. We have the convection equation that simplifies to 0 D
w 2T w vcT c wy wy 2
following the same reasoning as in velocity field, thus ignoring the advective terms. The integration of this equation leads to:
qcctot
wT § 1
·
¸¸¹
1 ¨ Qt y ¨ wy ©Pr Prt
1
We remember that the nondimensionalization used in the preceding equation is based on the inner variables Q and uW . More precisely:
T
Tw T Tqccw
, Tqccw
cc qw
UcuW
and qcctot
qcctot cc qw
.
In the conductive sublayer, we have
T
Pry
as, incidentally, in the zero pressure gradient turbulent boundary layer.
[8.52]
316
Convective Heat Transfer
However, in the fully turbulent mixing zone, the use of the classical approach of eddy diffusivity results in:
dT dy
Prt
Prt
Prt
Ny u*
Q t
[8.53]
Ny 1 3 y
It would be better at this point to reconsider the discussions leading to [8.53] by directly using the dimensional form of the original expressions in order to convince the reader about the accuracy of the solution obtained here. The transfer equation in the fully turbulent mixing zone is:
UcQ t wT Prt wy
cc qw
The wall temperature is defined by
Tqccw
cc qw
k
ld
where l d stands for the diffusive length scale (see Chapter 7). Substituting gives:
Q t dT Prt dy
l D Q
ld
Q
ud uW
[8.54]
The formulation now depends on the characteristic length scale l d D u d , where u d is the diffusive velocity scale in the wall region. We have the choice between u d uW and u d u* . Since the viscosity is Q t Nyu* (equation [8.49]), using u d u* would result in:
Ny dT Prt dy
1 and T
Prt
N
ln y C
The effect of the pressure gradient would disappear under these circumstances,
except for an eventual dependence through the constant C 3 . The diffusive velocity scale is therefore the shear velocity uW , and not u* , for the thermal boundary layer. Furthermore, the pressure gradient does not intervene in the
Turbulent Convection in External Wall Flows
317
conductive sublayer, but the use of u* instead of uW would give contradictory
results and destroy the similarity form T
Pry .
We opt for a twolayer model to simplify the problem. Noting by G c the thickness of the conductive sublayer, we arrange equations [8.52] and [8.53] into: y d Gc
T
Pry
y t Gc
T
PrGc
[8.55]
ª§ ·§ ·º Prt «¨ 1 3 y 1 ¸¨ 1 3 Gc 1 ¸» ln N «¨ 1 3 G 1 ¸¨ 1 3 y 1 ¸» ¹© ¹¼ c ¬©
We clarify that G c  13 in the twolayer model corresponding to the canonical turbulent boundary layer with 3 0 . The temperature distribution is fundamentally different from the velocity profile [8.51]. We return to the classical form:
T
PrG c
y ln N Gc
Prt
for 3 o 0 . u 40
30
Experiments Model
20
u +=2.44ln y + + 5
10
y+
0 1
10
100
1000
Figure 8.9. Velocity distribution at 3 2.6 u 102 , obtained using equations [8.46] and [8.51] according to [TAR 09]. Comparison with experiments of [HOU 06]
318
Convective Heat Transfer
Figure 8.9 shows the velocity distribution deduced from equations [8.46] and [8.51] with 1 N 2.5 and B 2.5 for a severe imposed pressure gradient 3
2.6 u 10 2 . The best value of the constant appearing in [8.51] to fit the
experimental data is B 3
2.5 smaller than B
5 of the zeropressure gradient
turbulent boundary layer. The reader should refer to [TAR 09] for further details. There is a reasonable correspondence between the model and the recent experimental results of [HOU 06], in particular at y t 100 . The shift of the velocity profile beyond the standard loglaw u 2.44 ln y 5 in the high loglayer is a fundamental characteristic of turbulent boundary layers subject to adverse pressure gradient. This particularity is well predicted by the model. Figure 8.10 shows the temperature distribution for the same value of the pressure parameter 3 . The Prandtl number is Pr 0.7 (air), N 0.41; the turbulent Prandtl number is constant with Prt 0.85 and G c 13 in equation [8.55]. There is a satisfactory agreement with the experiments of [HOU 06] from the conductive sublayer until the wake zone. The comparison of T with respect to the standard boundary layer is made by comparing the profiles to T 2.075 ln y 3.8 that fit the measurements of [HOU 06] at 3 0 (see Figure 8.8). There is a strong disparity between the behaviors of velocity and temperature distributions. The temperature profile lies systematically under the canonical distribution in the fully turbulent mixing zone, while that is clearly not the case for u . The good correspondence between the predictions and the measurements of both the velocity and temperature profiles shows that the closure scheme presented here may constitute a convenient first approach to model turbulent boundary layers in adverse pressure gradient. 20
T
Experiments Model T +=2.075 ln y + + 3.8
15
10
5
y+
0 1
10
100
1000
Figure 8.10. Temperature distribution for 3 2.6 u 102 obtained from [8.55]. Comparison with experimental data of [HOU 06]. See [TAR 09] for further details
Turbulent Convection in External Wall Flows
319
8.14. Roughness 8.14.1. Description of the problem The relations we established and analyzed in the frame of turbulent wall flows are all valid over a smooth surface. Roughness plays a fundamental role in wall transfer mechanisms. Its effect depends upon its mean height expressed in inner variables as k r k r uW Q . For sandgrain type of roughness, three regimes can be distinguished, namely: – hydraulically smooth, k r d G v 5; – transitional, 5 d k r d 70 ; – fully rough, k r t 70 . Roughness does not affect the wall layer when it is embedded in the viscous sublayer. In the transitional and full rough regimes, roughness simultaneously increases the drag coefficient3 and the Stanton number, but not always in the same proportions. A simple approach to the rough wall layers, which has the advantage of being easily comprehensible, is to introduce the idea of the logarithmic distribution virtual origin. The flow is relatively inert in a layer of thickness proportional to k r and the velocity profile is therefore shifted towards the outer layer as schematically shown in Figure 8.11. The virtual origin of the logarithmic velocity profile over a smooth surface is: u
1
N
ln y C s
0 y s0
eN C s
We obtain y s0 0.13 by using the conventional parameters C s 5 and N 0.41. The virtual origin over a rough wall is y r0 v kr ! y s0 . We wish to determine the velocity and temperature profiles in the fully turbulent mixing zone over a rough surface in the light of arguments provided here.
3. We are not dealing with regular “randomly” distributed roughness of the Nikuradse type. Regular roughness such as the riblets may under some conditions decrease the drag and increase the Nusselt number. See, for example, [JIM 04] for further details.
320
Convective Heat Transfer
Roughness
Virtual origin of the logarithmic velocity profile over a smooth wall + =e N C s ys0
kr+
Virtual origin of the logarithmic velocity distribution over a rough surface + =f k + yr0 r
Figure 8.11. Displacement of the virtual origin of logarithmic distribution over a rough wall
8.14.2. Guidelines Proceed by using an eddy viscosity type closure. The wall layer at y !! k r is unaffected by the roughness. We have to think about the consequences of this fact. We want to determine the velocity and temperature profiles in the logarithmic sublayer. One of the main questions is the dependence of the constant, which appears in the T distribution, upon the roughness parameter and the molecular Prandtl number. 8.14.3. Solution The eddydiffusivity closure rises into
Q t
du dy
1
in the fully turbulent mixing zone. The eddy viscosity is unaffected by the roughness at y !! k r , as confirmed by the experiments. Consequently, Q t Ny and u
1
N
ln y C r
[8.56]
Turbulent Convection in External Wall Flows
321
where the integration coefficient C r is unknown. The virtual origin of distribution [8.56] is a function of k r according to the arguments and discussion in section 8.14.1. With the aid of Figure 8.11 we find y r0
eN C r v kr
f kr
Cr
C r kr
[8.57]
showing that the coefficient C r depends upon the mean roughness height k r . The procedure to obtain the temperature distribution is similar. A virtual origin is determined from the logarithmic distribution
T
Prt
N
ln y CT Pr
where we insist on the fact that CT is a function of molecular Prandtl number. Following the same reasoning, we find the temperature distribution to be
T
Prt
N
ln y C rT Pr,k r
[8.58]
over a rough wall. The velocity and temperature profiles over a rough wall are shifted with respect to the standard logarithmic law. The former are often expressed as u
T
1
N
ln y C 'u k r
Prt
N
ln y CT 'T
[8.59]
Prk
r
in the literature. The coefficients C and CT refer to the distributions over a smooth
wall and 'u ks
together with
C C r kr
'T Pr k r
CT C rT Pr, kr .
Both 'u and 'T are positive because the virtual origin is shifted towards the flow from the wall. Thus, the profiles over a rough wall are simply shifted compared to the standard distributions, as schematically shown in Figure 8.12. The experiments indicate that the functions 'u and 'T Prkr are proportional to
when
ln k r
kr
is sufficiently large [CEB 88]. These functions subsequently
become zero when k r d 5 . The drag coefficient and the Stanton number both vary
322
Convective Heat Transfer
as C f  St v k r
0.175
in a turbulent boundary layer over a flat plate in the fully
rough regime [CEB 88]. kr+d5 u + or T +
kr+>5
'u + or 'T +
ln y +
Figure 8.12. Profile displacements over a rough wall
Chapter 9
Turbulent Convection in Free Shear Flows
9.1. Introduction Free turbulent shear flows play an important role in many industrial and environmental problems. Typical examples can be found in the domains of combustion (turbulent mixing layers), dispersion of pollutants (jets, release of pollutants into rivers, plumes), etc. Turbulent shear layers that are far from walls have very different properties from wall boundary layers and must be treated differently. However, these various types of free shear flows, although physically different, may be analyzed with the same methods and have closure equations in common. After a general approach that is presented in the next section, thi chapter deals with plumes, turbulent jets, turbulent shear layers and turbulent wakes, and uses the form of various problems. 9.2. General approach of free turbulent shear layers Let us consider the turbulent interface between a fluid at rest and a plane jet near the nozzle exit, as shown in Figure 9.1. Two shear layers originate at the nozzle sides. They are laminar up to a critical distance and become turbulent farther downstream. Turbulent mixing layers are composed of a row of large structures whose characteristic size grows with the distance x .
324
Convective Heat Transfer
Let us consider the averaged continuity and momentum equations in the twodimensional turbulent region by using the boundary layer approximation in the region of shear flow,
wu wv wx wy
0
u
wu wu v wx wy
u
wT wT v wx wy
w 2 u w ucvc wu º 1 dp 1 dp w ª Q 2 «Q Q t » U dx wy U dx wy ¬ wy ¼ wy
[9.1]
w ª wT º «D D t » wy ¬ wy ¼
Laminar shear layer
Turbulent shear layer
u0 ,T0 y Transition x
Ambient fluid
uf
0
Tf Figure 9.1. Development of free shear layers in a plane jet
The boundary layer approximations and closures based on the Boussinesq eddy viscosity model are used in the system of equations [9.1]. Later, an order of magnitude analysis will lead to a formulation of the turbulent viscosity Q t and the turbulent thermal diffusivity D t . We note that as a first approximation, the pressure gradient is zero since dp dx dpf dx and dpf dx Uuf duf dx 0 .1 In the
1. Normal fluctuations may induce a pressure gradient in the ydirection; see section 9.5.3 for more details.
Turbulent Convection in Free Shear Flows
325
fully turbulent region, the molecular viscosity and thermal diffusivity are negligible relative to Q t and D t . In this region, [9.1] becomes:
wu wv wx wy
0
u
wu wu v wx wy
u
wT wT v wx wy
w ª wu º «Q » wy ¬ t wy ¼
[9.2]
w ª wu º «D » wy ¬ t wy ¼
Experiments show that the shear layer thickness G increases with x and that, as a first approximation, G v x . Denoting u0 the initial jet velocity, inertia terms in the momentum equation are of the order of: u
wu u 02  wx x
In the same way, the Reynolds stress is estimated by:
w ucv c wy
w ª wu º u «Q t » Q t 02 wy ¬ wy ¼ G
Since it must be in equilibrium with the inertia terms in the fully turbulent region, we find:
Qt 
u0 2 G  u0 x x
We arrive at the same expression by using a mixing length model. In fact, considering:
wu wu u0 , with l  G and  wy wy G Q t  u0G  u0 x Qt
l2
The characteristic length scale relative to the turbulent viscosity in a free shear layer is proportional to the distance x , in the main flow direction. We use the classical relation D t Q t Prt to determine the temperature field. In our first approach, we assume that the turbulent Prandtl number is constant with Prt 0.9 .
326
Convective Heat Transfer
Using this formulation, the problem generally enables similarity solutions so that the approach is not fundamentally different from that of laminar free shear flows. In sections 9.4 and 9.5 we propose to achieve the same formulation for turbulent jets and for turbulent mixing layers, using exercises to do so. 9.3. Plumes Let us consider a vertical axisymmetric flow, generated by buoyancy forces above a point source of power q , as shown in Figure 9.2. The plume radius becomes proportional to x sufficiently far from the origin, as in most free turbulent shear flows. We suggest solving the problem by using cylindrical coordinates and an integral formulation [BEJ 95]. Coordinates are defined in Figure 9.2. The axial and radial velocity components are u and v , respectively.
x u v fluid at rest T r q
Figure 9.2. Turbulent plume
The continuity, momentum and energy equations are given by:
wu 1 wrv 0 wx r wr wu 2 1 w ru v wx r wr wu T 1 w rv T wx r wr
1 w ª wu º «r Q Q t » gE T Tf r w r ¬ w r ¼ 1 w ª w T º «r D Dt » r wr ¬ wr ¼
[9.3]
Turbulent Convection in Free Shear Flows
r
327
The continuity equation is multiplied by r and then integrated at given x , from 0 to a sufficiently large distance from the plumeaxis r R , as
d
R
dx
0
³ u rdr rv R
[9.4]
since rv 0 0 for sake of symmetry. Moreover, the momentum equation is also multiplied by r and integrated between the same limits, which gives R
R
wu º ª « r Q Q t w r » g E T Tf rdr ¬ ¼R 0
d u 2 rdr ruv R dx
³
³ 0
where the conditions of symmetry again have been used. For R o f , u o 0 , which leads to the approximation: R
d u 2 rdr dx
³
R
gE
0
³ T T rdr
[9.5]
f
0
The integration of energy equation, also multiplied by r, yields for the same reasons: d
R
ª
dx
0
¬
³ u T rdr rv T R «r D Dt
wT º »  0 wr ¼R
[9.6]
Combining [9.6] with [9.4] and letting R o f , we obtain the conservation of the flow of enthalpy through any crosssection of height x f
d u T Tf rdr dx
³
0
0
which is a general result for most free turbulent shear flows (see turbulent jets, section 9.4). The energy budget applied to a control domain, including the heat source, is expressed as: f
³ u T T rdr f
0
) 2SU c
[9.7]
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Convective Heat Transfer
The next step of the procedure depends on the shape of the chosen profiles, which is a general feature of integral analysis. [BEJ 95] considers the Gaussian profiles: u
ª § ·2 º r uc exp«¨ ¸ » « ¬ ©b ¹ » ¼
T Tf
ª § ·2 º r Tc Tf exp««¨¨b ¸¸ »» ¬ © t ¹ ¼
[9.8]
Equations [9.4], [9.5] and [9.7] enable us to determine the unknowns uc , T c and b , if we are able to model rv R , which remains finite for R o f . In particular, equation [9.4] gives: d dx
ucb2
2rv R
The plume radius satisfies b v x , according to observations. We deduce that in a order of magnitude sense rv R v uc b . It is then straightforward to show [BEJ 95] that: uc v x 1/ 3 Tc Tf v x 5 / 3
9.4. Twodimensional turbulent jet 9.4.1. Description of the problem
We consider a twodimensional turbulent jet, as shown in Figure 9.1. The problem consists of determining the necessary conditions for the basic equations to enable similarity solutions for the velocity and temperature fields. We introduce the similarity variable K y G x , where y is the distance to the jetaxis and G x the jet thickness: – By integrating the momentum and energy equations, show that the flow rates of momentum and enthalpy are constant through any jet crosssection. Calculate the relations relating G x to the jet axisvelocity u c x and temperature T c x . Show that G v x and compare this result to the corresponding law for a laminar jet. Show also that T c Tf v x 1/ 2 .
Turbulent Convection in Free Shear Flows
329
– Propose similarity profiles for u x, y and T x, y . Calculate the necessary conditions for the existence of such solutions. – Introduce a closure of the Boussinesq eddy diffusivity type for the flow in the fully turbulent region. Analyze the resulting equations and determine u x, y and T x, y . This is a classical problem and a procedure similar to that in [CEB 88] is given here. 9.4.2. Guidelines
Use the functions
f c K g K
u x, y uc x
T x, y Tf Tc x Tf
< x , y uc x G x f K where G x represents the jet thickness, u c x and T c x are the jetaxis velocity and temperature, respectively, and < is the stream function. Introduce two functions to be determined in the closures relative to Reynolds stress and to heat flux, namely: u cvc uc2 F K
vcT c uc Tc Tf G K
Combine with a closure of the Boussinesq eddy diffusivity type. Calculate the necessary conditions for the existence of similarity solutions.
330
Convective Heat Transfer
9.4.3. Solution
Let us again consider the system of equations [9.1] in the form
wu wv 0 wx wy wu wu u v wx wy u
wT wT v wx wy
1 w
U wy
Wtot
1 w qcctot Uc wy
[9.9]
where, as usual, the total shear stress and heat flux are defined by
Wtot
P
qcctot
wu U ucv c wy k
wT Uc v cT c wy
The molecular shear stress and heat flux are ignored in the fully turbulent region. The similarity variable is
K
y G x
where G x represents the jet thickness and the origin of y axis is on the jetcenterline. We define: f cK gK
u x, y u c x
T x, y Tf T c x Tf
[9.10]
<x, y u c x G x f K
In equation [9.10], u c and T c are, respectively, the jetaxis velocity and temperature and <x, y is the stream function.
Turbulent Convection in Free Shear Flows
331
First, we consider the integral momentum equation f
f wu v wu 2 dy ³ dy f wx f wy
d f 2 ³ u dy dx f
³
f
wWtot dy 0 f wy
³
so that the integral f
³ u 2 dy
J
[9.11]
f
is constant. In fact, the fluid being at rest at infinity, we can write: f
f
wu v f dy u vf f wy
wWtot dy Wtotf f f wy
³
0, ³
0
Combining [9.10] with [9.11] requires, for symmetry reasons, that: f
f
f
f
f
0
0
0
³ u 2 dy 2 ³ u 2 dy 2 ³ u c2 f c2 K GdK 2u c2 x G x ³ f c2 K dK
J
As a result, the product u c2 x G x C
[9.12]
is constant so that its derivative with respect to x is zero 2
dG du c G uc dx dx
[9.13]
0
We proceed in the same way for the energy equation. Integrating this equation gives: f wu T
³
f
wx
f
wv T dy f wy
dy ³
d
f
dx
f
f
³ u T dy >v T @f
According to the continuity equation, we may write
d f
f
³ u dy >v @ f dx f
0
f
w qcctot
f
wy
³
dy
0
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Convective Heat Transfer
from which it follows, after multiplying by Tf , that f
ª¬vT º¼ f
d dx
f
³ uT
f
dy
f
and finally
d f
³ u T Tf dy dx f
0
which confirms that the flow rate of enthalpy through a jet crosssection remains constant when x varies. We calculate by using the transformation of variables [9.10] that f
H
2 ³ u T Tf dy 0
f
2 ³ uc x f cK >Tc x Tf @gK G x dK 0
or f
H
2u c x G x >T c x Tf @³ f cK gK dK
[9.14]
0
is also constant. Since H is constant, the product u c x G x >T c x Tf @ is also constant. Its derivative with respect to x is therefore zero, which gives a second equation, after combining with [9.13] and dividing by uc x : 1 dG dT T c Tf G c dx 2 dx
0
Using the definition of the stream function leads to: u
w< wy
v
w< wx
u c x f cK
ª §du º dG · dG y «f ¨ c G u c ¸ u c f c» dx ¹ dx G ¬ © dx ¼
Reporting these expressions in the Reynolds equation, we find:
[9.15]
Turbulent Convection in Free Shear Flows
uc
333
du c 2 1 2 dG f cc 1 wWtot f c u c f dx dx G UG wK 2
Using the conservation of momentum [9.13], replacing du c dx
u dG c 2G dx
in the above expression and rearranging, we obtain
u c2 dG f c2 ff cc 2 dx
1 wWtot
[9.16]
U wK
This expression requires a closure equation for Wtot and additional assumptions in order to obtain a similarity solution. We proceed in the same fashion for the energy equation. Combining [9.9] and [9.10], we obtain:
G uc f cg
dTc dx
dG uc fgc Tc Tf dx 2
1
1 w qcctot
UcG
wK
Combining with equation [9.15], which expresses the conservation of energy, leads to:
G x uc x
dTc dx
f g c
1 w qcctot Uc wK
[9.17]
Let us now introduce closure equations for Wtot and qcctot . Using the guidelines given in section 9.4.2, we write
Wtot U qcctot
Uc
ucv c uc2 F K vcT c uc Tc Tf GK
which enables us to eliminate u c x in equations [9.16] and [9.17]. Finally, we find:
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Convective Heat Transfer
1 dG
f c2 ff cc F c 2 dx
G x
0
>Tc x Tf @ f g c Gc dx d
Tc x Tf
[9.18] 0
These equations enable similarity solutions if: – dG dx is independent of x , which requires G v x . Moreover, since u c2 x G x C according to [9.12], we obtain the law of distribution for the jetaxis
velocity as u c x v x 1/ 2 . We indicate that, for a laminar plane jet, G v x 2 / 3 and u c x v x 1/ 3 . Turbulence accelerates the jet expansion.
– In these conditions, the energy equation also enables a similarity solution if the jetaxis temperature varies as T c x Tf v x s , where s is a constant. In fact,
u cG v x 1/ 2 and the jetaxis temperature must satisfy T c x Tf v x 1/ 2 so that the integral flow of enthalpy remains constant, according to [9.14]. We choose a closure of the Boussinesq eddy diffusivity type, namely:
Wtot U
uc2 F K Q t
qcctot
Uc
wu wy
Qt
uc Tc Tf GK
uc
G
f cc
Q t wT Prt wy
Qt Prt
gc
Tc Tf G
Replacing these expressions in [9.18] and rearranging leads to: uc
G x dG 2 f c ff cc f cc 0 2Q t dx
2 Prt u cG x d >T x Tf @ f g c gcc 0 Q t T c x Tf dx c
[9.19]
The necessary conditions for the existence of similarity solutions now clearly appear. The turbulent viscosity must fulfill the condition
Q t v u cG
dG dx
Turbulent Convection in Free Shear Flows
335
and since G v x , the necessary condition is Q t v u cG v u c x . We recover the behavior of Q t as obtained by the order of magnitude analysis in section 9.2. The associated boundary conditions are:
Kof K
u
f c g f
0
0 (u
0 (v
0, T
Tf )
0), f cc 0 (
wu wy
0), gc 0 (
wT wy
0)
[9.20]
In practice, the jet thickness is defined as the distance to the axis for which 0.5uc . Experiments show that:
Q t x 0.037ucG A semiempirical solution is given by [CEB 88]: u
sech 2 0.881K
uc x
T x , y Tf Tc x Tf
For Prt
>sech0.881K @
[9.21]
2Prt
1, the velocity and temperature profiles become identical.
9.5. Mixing layer 9.5.1. Description of the problem
Two irrotational streams with uniform velocities and temperatures U1, T1 and U 2 , T 2 join and mix in the downstream direction, as shown in Figure 9.3. For Re
'u x
Q
! 7 u 10 4
where 'u U1 U 2 2 , the flow becomes turbulent. Determine the velocity and temperature distributions in the fully turbulent mixing layer and demonstrate the existence of similarity solutions.
336
Convective Heat Transfer
9.5.2. Guidelines
Choose the velocity and length scales involved in the turbulent viscosity by taking the physical phenomena governing the flow into consideration. Introduce the stream function <x, y 'u x f K u m y , where K similarity variable, 'u U1 U 2 2 and um U1 U 2 2 .
y x is the
Express the temperature in a similar form T x, y T m 'TgK , with the mean temperature T m T1 T 2 2 and 'T T1 T 2 2 . 9.5.3. Solution
The complete Reynolds equations for a twodimensionnal mixing layer yield: u 0 u
wu wu v wx wy
1wp w 2 u w u cu c w u cvc Q U wx wx wy w y2
1 w p w vcvc U wy wy
wT wT v wx wy
D
[9.22]
w 2T w vcT c wy w y2
U1, T1
y
x
G
G
U2, T2 Figure 9.3. Mixing layer
Turbulent Convection in Free Shear Flows
337
It is worth noting that we used the boundarylayertype approximations, in particular for the second equation concerning v , in which the pressure gradient wp wy , generated by the Reynolds stress Uw vcvc wy , appears. This pressure term also exists in the formulation of plane jets. It was ignored in section 9.2 and we will discuss its effect in the present problem. Integrating the second equation gives
1 rf wp ³ dy U y wy
rf w vcvc
³
y
wy
dy
which results in 1
U
>px,rf px, y @
vcvcx,rf vcvcx, y vcvcx, y
[9.23]
since vcvcx,rf 0 (in the irrotational flow u U1 and u U 2 ). Deriving [9.23] with respect to x leads to: 1 dp x, rf 1 w p x, y dx U U wx
urf
durf 1 w p x, y dx U wx
w vcvc wx
Eliminating the pressure gradient in the momentum equation in the x direction, we obtain: u
wu wu v wx wy
urf
durf w 2u w w u cvc u cu c vcvc Q 2 dx wy wy wx
[9.24]
This relation is valid for free turbulent shear layers. In the present problem: urf
durf dx
0
Moreover, we assume that the turbulent diffusion terms w wx ucuc vcvc are negligible in the momentum equation. This hypothesis is verified by experiments. It is also necessary in order to obtain simple solutions. Furthermore, turbulent diffusion must be taken into account in strongly anisotropic turbulent flows at high Reynolds numbers. If we focus on the fully turbulent region, we obtain the following system of equations: u
wu wu v wx wy
wT wT u v wx wy
w ucvc w § wu · ¨Q ¸ wy wy © t wy ¹
w vcT c w §Q t wT · ¨ ¸ wy wy ©Prt wy ¹
[9.25]
338
Convective Heat Transfer
Let us note that we have used a closure of the turbulent viscosity type. Modeling the turbulent viscocity requires physical considerations. We recall that the turbulent viscosity is of the form Q t v uQt A Qt . The velocity scale clearly is uQt v 'u U1 U 2 2 because the turbulent mixing is directly related to 'u (for 'u 0 , the potential flow is continuously maintained without possible transition). The characteristic length A Qt is of the form A Qt x according to the arguments stated in section 9.2. A simplified approach consists of setting:
Qt
cx'u
[9.26]
These hypotheses lead to similarity solutions. Let us introduce the stream function <x, y 'u x f K u m y
where um U1 U 2 2 is the mean velocity and K components u and v are expressed by:
y x . The velocity
w< u m 'u f cK wy w< v 'u Kf c f wx u
We adopt a similar form for the temperature field T x , y T m 'Tg K
with the mean temperature T m T1 T 2 2 and 'T T1 T 2 2 . Following the same procedure as before and assuming that the turbulent Prandtl number is constant, equations [9.25] reduce to um Kf c ff c 0 'u u c gc fgc m Kgc 0 Prt 'u
cf cc
[9.27]
Turbulent Convection in Free Shear Flows
339
which obviously enable similarity solutions. The associated boundary conditions are, for the velocity field f 0
v f cf 1 u f c0 0 u
0 for symmetry reason
0
u1
[9.28]
um
and for the temperature field g0 0 gf
T Tm 1 T T1
[9.29]
The differential equation governing the function f K is nonlinear and must be solved by numerical calculation. It is, however, possible to find approximate solutions valid for sufficiently large distances x and/or for sufficiently weak velocity difference 'u u m . In fact, far downstream u o u m and the streamlines become nearly parallel. It is possible to linearize the equation that governs u as u
wu wu v wx wy
wK wK 'uKf c f 'uf cc wx wx
u m 'uf c 'uf cc
wK 'u O 'u 2 wx
u m f c
groups the terms of order 'u
where O 'u 2
2
. Ignoring these terms and processing
as we did for the temperature equation, we obtain the system of equations: u f ccc m Kf cc 0 c'u Pr u gc t m Kgc 0 c'u
This system may be analytically integrated. For example, let us consider: Pr u gcc t mK gc c'u
[9.30]
340
Convective Heat Transfer
Denoting C the integration constant, a first integration gives: § Pr u · gc C exp¨ t m K 2 ¸ © 2c'u ¹
Accounting for the associated boundary conditions, a second integration results in: g K
ª§ Pr u ·1/2 º erf «¨ t m ¸ K » «¬© 2c'u ¹ »¼
The determination of f cK is identical. The longitudinal velocity component and temperature take the similar form:
u
T
ª§ u ·1/ 2 º u m 'u erf «¨ m ¸ K» « » ¬©2c'u ¹ ¼ 1/ 2 º ª§ Pr u · T m 'T erf «¨ t m ¸ K» « » ¼ ¬© 2c'u ¹
[9.31]
Experiments show that the constant c present in the expression of turbulent viscosity [9.26] is well represented by the following relationship [ARP 84]: c
0.055
'u um
This expression is physically consistent, since it implies that for 'u consequently, the turbulent viscosity are zero.
[9.32] 0 , c and
9.6. Determination of the turbulent Prandtl number in a plane wake 9.6.1. Description of the problem
The problem is devoted to an experimental method aimed at determining the turbulent Prandtl number Prt in the wake of a high aspect ratio obstacle placed in a stream. The flow is uniform in the region upstream from the obstacle (velocity U 0 , temperature T 0 ). The solid is heated and, consequently, the flow in the region downstream from the obstacle is a turbulent wake at a temperature higher than T 0 .
Turbulent Convection in Free Shear Flows
341
Figure 9.4 shows the shape of the velocity and temperature profiles in crosssections behind the obstacle. Respectively, the velocity and temperature match U 0 and T 0 when y o rf . The wake slowly grows in the downstream direction due to turbulent diffusion. Measurements of velocity and temperature profiles are performed in the farwake region. The object of the problem is to find a method for determining Prt from experimental results. 9.6.2. Guidelines
For a highaspect ratio obstacle, the mean velocity and thermal fields may be assumed to be twodimensional (plane wake). Let us consider the mean velocity defect u1x, y U 0 u x, y (> 0) and the mean temperature excess T1 x, y T x, y T0 in the wake. In the farwake region that is of interest where u1x, y U 0 and T1x, y T 0 , the basic equations governing the velocity defect and the temperature excess enable a similarity solution.
Figure 9.4. Turbulent wake. Sketch of the flow
Simplify the Reynolds and energy equations by taking the above assumptions into account. Express the relations representing the similarity profiles for the velocity defect, the temperature excess, the Reynolds stress and the turbulent heat flux by introducing velocity, temperature and length scales characteristic of these profiles at given distance x from the obstacle. Write the momentum and energy budget for a slice of fluid perpendicular to the wakeaxis and find a relation between the scales. Determine the relation satisfied by the wake thickness and the velocity defect scale so that the basic equations enable a similarity solution. Show that the functions that characterize the profiles of velocity defect, temperature excess, Reynolds stress and turbulent heat flux satisfy a straightforward relation.
342
Convective Heat Transfer
Relate the turbulent Prandtl number to the above functions and their derivatives by using the turbulent diffusivity model for the Reynolds stress and the turbulent heat flux. Propose a method for determining the turbulent Prandtl number from the measurements of mean velocity and temperature crosssection profiles. 9.6.3. Solution
9.6.3.1. Reynolds and energy equations Experiments show that viscous stresses are negligible relative to turbulent stresses in the Reynolds equations for a turbulent flow far from walls. Moreover, turbulent wakes, like jets, belong to the category of thin shear flows, which means that the transverse length scale G x (wake halfwidth) is much smaller than the longitudinal length scale x. In other words, the wake grows slowly in x direction. In this respect, the situation is the same as for laminar boundary layers. In these conditions, theory shows that the pressure varies slightly in the transverse direction. Since the pressure is constant outside the wake, it may be considered as constant everywhere in the flow. The influence of the mean pressure gradient in thin shear flows is discussed in section 9.5.3. The Reynolds equation in the longitudinal direction is simplified for the present twodimensional mean flow as: u
wu wu v wx wy
w ucuc w ucvc w ucwc wx wy wz
Turbulent stresses are twodimensional, like the mean flow, so that w ucwc wz 0 . The Reynolds stresses ucuc and ucvc have the same order of magnitude and, taking into account the inequality of length scales ( G x x ), we may write
w ucuc wx w ucvcwy . Finally, the Reynolds equation along x reduces to: u
wu wu v wx wy
w ucvc wy
[9.33]
It is convenient to introduce the velocity defect u1x, y U 0 u x, y . Since U 0 does not depend on x, we obtain: u
wu wx
u1
wu1 wu U 0 1 wx wx
Turbulent Convection in Free Shear Flows
343
Moreover, since u1x, y U 0 , only the second term is retained in the above equation. After introducing the velocity defect, the continuity equation yields:
wu1 wv wx wy
0
Denoting U1 the velocity defect scale in a given crosssection, the order of magnitude of the mean transverse velocity is V  U1G x . The two terms of the lefthand side of equation [9.33] thus have the respective order of magnitude U12 x
U 0 U1 x
so that the inertia term corresponding to v is negligible relative to the first term corresponding to u . Finally, the longitudinal Reynolds equation further simplifies as: U 0
wu1 wx
w ucvc wy
[9.34]
The same calculations applied to the energy equation lead to: U0
wT1 wx
w vcT c wy
[9.35]
9.6.3.2. Momentum and energy budget for a slice of fluid We consider a slice of fluid delimited by two crosssections of abscissa x and x dx and by the plane of symmetry of the wake (Figure 9.5). The conservation of flow rate shows that the flow necessarily has a velocity component vf at infinity to compensate for the difference in the flow rate between the two crosssections. vf U0
y x
T0
x +dx x
O
Figure 9.5. Turbulent wake. Control domain
344
Convective Heat Transfer
In fact, for the control domain the flow rate budget reads
d ª dx «¬
³
f
0
º u x, y dy » dx vf dx ¼
so that vf x
d dx
0
f ³ 0 u x, y dy .
This velocity is positive since the region of velocity defect extends more and more in the downstream direction. For the same control domain, since the pressure is constant everywhere in the flow, the momentum budget yields d dx
or
d dx
³ 0f Uu 2 x, y dy UU 0 vf
0
f ³ 0 u x, y >u x, y U 0 @dy 0
when the conservation of flow rate is accounted for. Introducing the velocity defect and noting that u1 U 0 , we obtain U0
d dx
f ³ 0 u1x, y dy 0
f ³ 0 u1x , y dy C1
[9.36]
The same procedure is used for the energy budget applied to the same control domain. Ignoring longitudinal heat flux, the flow rate of enthalpy is invariant through every crosssection. Normalizing the fluid temperature by the external temperature, it follows that d dx
³ 0f UC p u x, y >T x, y T 0 @dy 0
and consequently f ³ 0 T1x, y dy C 2
[9.37]
Turbulent Convection in Free Shear Flows
345
9.6.3.3. Similarity solution Let us assume that equations [9.34] and [9.35] enable a solution of the form u1x, y U a x
T1x, y T a x
ucvcx, y U a 2 x
f1K
[9.38]
f 2 K
[9.39]
g1 K
[9.40]
v cT cx, y g2 K U a x Ta x
[9.41]
where U a x and Ta x represent, respectively, the velocity defect and the temperature excess on the wakeaxis. The distance to the symmetryaxis is normalized by the wake halfwidth as K y G x . Substituting into the system [9.36] and [9.37], we obtain
³ 0f U a x G x f1K dK C1 from which it follows that U a x G x c1
[9.42]
and taking equation [9.37] into account f ³ 0 T a x G x f 2 K dK C 2
Taking the functions of x only out of the integral, we find: T a x G x c 2
[9.43]
346
Convective Heat Transfer
The above relations lead to: U ac x
T ac x
U a x
T a x
Gcx
[9.44]
G x
Substituting the assumed similarity solution [9.38][9.41] into the system [9.34][9.35] gives:
ª º U 0 «U a c x G x f1 K Gcx Kf1c K » g1c K U a x « » ¬U a x ¼
ª º U 0 «Ta c x G x f 2 K Gcx Kf 2c K » g2 c K U a x « » ¬Ta x ¼
[9.45]
[9.46]
Relations [9.44] are introduced into the two equations immediately above to obtain: U 0Gcx U a x
>f K Kf c K @ 1
g1c K
1
U Gcx 0 f 2 K Kf 2c K U a x
>
@
g 2c K
[9.47]
[9.48]
Clearly, a similarity solution is possible only if: U 0Gcx U a x
c3
[9.49]
The following calculation is not necessary to determine the turbulent Prandtl number; it gives, however, the farwake laws for U a x , Ta x and Gx . Combining [9.42] and [9.49], we obtain U 0G x Gcx c1c 3
Gx
2c1c 3 x U0
where x refers to a virtual origin.
[9.50]
Turbulent Convection in Free Shear Flows
347
The distribution of the velocity defect and temperature excess on the symmetryaxis of the wake are deduced from the previous equations: U a x
c1U 0 2c3 x
[9.51]
Ta x
c2 2U 0 2c1c 3 x
[9.52]
Let us again consider equations [9.47] and [9.48]. Substituting [9.49] into these equations, we obtain
>
@
c 3 f1 K Kf1c K
>
g1c K
@
c 3 f 2 K Kf 2c K
g 2c K
[9.53] [9.54]
which are integrated as c 3Kf1K g1K d1
c 3Kf 2 K g2 K d 2
On the symmetryaxis of the wake ( K 0 ), the Reynolds stress ucvc and the turbulent heat flux vcT c are zero, simultaneously with the mean velocity and temperature gradients (turbulent diffusivity model, as will be seen later). Furthermore, according to expressions [9.38] and [9.39], f10 f 2 0 1 . As a result, we find that the two constants d1 and d 2 are zero. It follows that: f1K
f 2 K
g1K
g 2 K
The definition of turbulent diffusivities is: ucvc Q t wu wy vcT c D t wT wy
[9.55]
348
Convective Heat Transfer
These relations may be transformed by using the similarity solution: U a 2 x g1K Q t
U a x
G x
f1c K
T x U a x T a x g 2 K D t a f 2c K G x
Dividing these two relations side by side, we find: g1K
g 2 K
Q t f1c K D t f 2c K
[9.56]
We recall that the turbulent Prandtl number is the ratio of the momentum to the heat diffusivity Prt Q t D t . Combining [9.55] and [9.56] the final result is: Prt
f1K f 2c K
f 2 K f1c K
[9.57]
This relation shows that the turbulent Prandtl number may be determined experimentally by using measured mean velocity and temperature profiles. In the first step, the velocity defect U a x and the temperature excess Ta x are determined on the symmetryaxis of the wake in a given crosssection. The profiles are then normalized as in equations [9.38] and [9.39] for numerically calculating the functions f1 K , f 2 K and their derivatives, which are reported in [9.57] in order to obtain finally Prt . 9.7. Regulation of temperature 9.7.1. Description of the problem
In many facilities, it is necessary to regulate the temperature of the working fluid for stable use conditions. We propose to examine the case of a closedloop circuit including a system for heating and regulating the fluid temperature. In order to damp temperature fluctuations generated by the heating system, the circuit also includes a cylindrical tank of diameter D and length L, which plays the role of a hydraulic capacity. This tank is assumed to be perfectly insulated. The working fluid is water. The fluid flow rate Q0 in the circuit is assumed to be constant with time. Water flows into the tank at temperature Ti t through a circular opening and then inside the tank as a turbulent axisymmetric jet. Finally, the liquid exits at temperature Te t
Turbulent Convection in Free Shear Flows
349
through an outlet opening at the opposite side of the tank (Figure 9.6). The two openings have the same diameter d. The jet expands in the downstream direction from the inlet opening by entrainment and mixing with the fluid of the tank. Near the tank exit, only a portion of the jet flow rate exits to the circuit while the remainder is recycled inside the tank. We propose to implement a simplified model in order to predict the exit fluid temperature as a function of the inlet temperature, the fluid physical properties and the geometric characteristics of the tank. r D
Q0 Ti(t)
O d
Q0
x
Te(t)
L Figure 9.6. Regulation of temperature. Sketch of the capacity
The inlet temperature variations are assumed to be periodic (period TZ ) with the mean value T0. We consider the two following cases: Case 1: the variations of Ti t are given by Figure 9.7. Case 2: the variations of Ti t are sinusoidal. TZ = 10 min. Numerical data: – diameter of the tank inlet and outlet openings d = 2 cm; – diameter of the tank D = 30 cm (radius R); – length of the tank L = 50 cm; – jet exit mean velocity U0 = 1 m/s.
350
Convective Heat Transfer
T0+ 'T t
T0 T0 'T 5s
0.5 s
Figure 9.7. Variations of inlet temperature with time. Case 1
9.7.2. Guidelines
9.7.2.1. Model assumptions The model uses the following assumptions: – the Reynolds number of the flow ( Re = U 0 d Q , where Q is the fluid kinematic viscosity) is high enough so that the jet is turbulent; – up to a distance X from the tank inlet, the jet development is approximately the same as that of the free jet; – for x > X, the jet no longer entrains fluid from the tank, but the total flow rate is redistributed between the portion that exits from the tank and the remainder that is recycled in the tank; – mean velocity and temperature profiles are assumed to be similar; – the distance l = L  X is assumed to be proportional to the jet radius r1 2 L which would be observed for x = L, in absence of the opposite wall of the tank. The radius is defined by:
u x, r1 2 ua x
1 2
[9.58]
where x and r are the axial and radial coordinates respectively, ua x is the mean centerline velocity. Experiments suggest using l 1.6r1 2 x L . The mean velocity field of the free turbulent round jet is given by a similarity solution [BEJ 95], which is obtained with a model of constant turbulent viscosity u x, r u a x
1
1 K 4 2
with K V r x , V = 15.2.
2
[9.59]
Turbulent Convection in Free Shear Flows
351
9.7.2.2. Suggested approach Develop the hydraulic model to represent the flow in the tank. Complete the theory of the turbulent free jet. Using the momentum budget, determine the laws of mean centerline velocity and jet flow rate as a function of the distance to the tank inlet. Estimate the characterictic time constants for the thermal regime of the jet and the tank. Propose a model for calculating the fluid exit temperature Te t in the two cases of the problem. 9.7.3. Solution
9.7.3.1. Hydraulic model 9.7.3.1.1. Turbulent jet development Up to the distance X from the tank inlet, it is assumed that the jet development is not influenced by confinment. First, we calculate the free jet radius for x = L. According to relation [9.59], the dimensionless radius K1 2 is given by 1
1 K 4 2
2
1 2
or K 1/2 = 1.287. The resulting radius is r1 2 x
K1 2 x = 0.085 x. V
The jet therefore expands linearly. For x = L, we obtain r1 2 L
4.25 cm.
The distribution of mean centerline velocity is deduced from the momentum budget in a jet crosssection. With the assumption that the flow is axisymmetric with respect to Oxaxis, the flow of momentum J through a tank crosssection is given by: J
2S ³ 0R Uu 2 x , r rdr
Ignoring the presence of the walls and the recycled back flow, the mean axial velocity distribution is given by [9.59] in a tank crosssection so that the previous equation becomes: J
2SUu a 2 x
x2
V
2
³
f
0
K
1 K 4 2
4
dK
352
Convective Heat Transfer
The calculation of the integral is straightforward by setting u 1 K 2 4 and gives the resulting value 2/3. Finally: J
4 x2 SUu a 2 x 2 3 V
In the turbulent jet theory, the mean pressure is uniform in the reservoir, which is assumed to have very large dimensions relative to the jet diameter. As a result, the flow of momentum is constant in any jet crosssection. We assume that the same result holds in the tank up to the abscissa X. J is also calculated at the tank inlet. J
SUU 0 2
d2 4
where the initial jet velocity U0 is assumed to be uniform in the tank inlet. Equaling the two expressions of J, we find: U0
ua x
4
x
3V d
0.152
x
[9.60]
d
The mean centerline velocity is inversely proportional to the distance x. It is worth noting that this law only holds in the region of similar crosssection velocity profiles, in other words where the velocity is given by equation [9.59]. In fact, there is a fictitious origin x0 for the jet, which depends on the boundary conditions at the tank inlet. In the present very simplified model, the shift in the jet origin x0 is ignored. It could be accounted for in law [9.60]. The jet flow rate in a crosssection limited to the radius r is calculated by again using velocity distribution [9.59]. Qx , r 2S
x2
³ u x, r rdr = 2Su x V ³ r
a
0
2
Vr x 0
K
1 K 4 2
2
dK
The calculation of the integral is again straightforward by setting u 1 K 2 4 and we obtain:
Vr x Qx,r 4 Sua x 2 V 4 Vr x 2 x2
2
Turbulent Convection in Free Shear Flows
353
Replacing ua x with expression [9.60] and normalizing Qx,r by its initial d2 (with the data of the problem, Q0 4 dimensionless flow rate is given by
value Q0 U 0 S
Qx, r Q0
with Kr
3.14 10 4 m3 s1), the
4 3 x Kr 2 V d 4 Kr 2
[9.61]
Vr x .
Using the same assumptions, the total flow rate through the jet crosssection of abscissa x is calculated by letting Kr tend to infinity in [9.61]: Qx Q0
4 3 x V d
[9.62]
The flow rate of the turbulent round free jet therefore increases linearly as a function of x. Obviously, this similarity regime is possible only up to a distance l to the tank outlet opening. Beyond this distance, the flow is influenced by the tank outlet sidewall. Experiments suggest that the order of magnitude of l is r1 2 L and, more precisely, l  1.6r1 2 L , which leads to X L l  43 cm. For this distance, equation [9.62] gives QL Q0  10 . An important part (90%) of the flow rate QL is recycled inside the tank since the initial flow rate Q0 only exits in the circuit. It is now possible to represent the flow by the sketch in Figure 9.8, where a longitudinal crosssection of the half tank has been drawn.
Figure 9.8. Sketch of the flow. Longitudinal crosssection of the half tank
A mean streamline2 (bold line in Figure 9.8) starts from the inlet opening wall, deviates from the jetaxis under turbulent diffusion effects (the jet expands and velocities decrease in the downstream direction) and ends on the tank outlet wall. 2. This streamline is obviously a fictitious line in a turbulent flow.
354
Convective Heat Transfer
The area open for the recycled flow is a cylindrical annulus, delimited by the tank lateral wall and the jet boundary. Its smallest value is in the downstream region of the jet. In the most unfavorable conditions for the model, the recycled flow rate 9 Q0
passes through an area of the order of S 0.15 2 0.04 2
, where the jet radius has
been estimated as r1 2 L . The corresponding backward bulk velocity is then
9Q0 S 0.15 2 0.04 2  0.04 m s1. In the same region ( x  L ), the jet centerline
velocity is of the order of 0.3 m s1, according to [9.60]. In the upstream region of the tank (x < X), the area open to the backward flow is larger than in the previous calculation and the backward flow rate is smaller than in the corresponding crosssection leading to smaller backward velocities. These estimations suggest that the backward velocities are small enough relative to the jet velocities so that the jet development is well described by equations [9.59] to [9.62]. 9.7.3.2. Thermal model 9.7.3.2.1. Time constants When the jet initial temperature varies with time, heat transfer associated with the flow in the tank involves two time constants related to two different phenomena. – On the one hand, the jet convects the information that the inlet conditions vary with time. The time response of the jet associated with this physical phenomenon is the travel time of a fluid particle in the tank. On length L, this time is of the order of: t tr  L U 0 = 0.5 s
In fact, this order of magnitude overestimates the time taken by a particle to flow across the tank since the fluid velocity decreases in the downstream direction. Restricting the calculation to a fluid particle on the jetaxis, the travel time is easily calculated as t tr
³
L 0
dx ua x =
0.152 U0
x
L ³0 dx =
d
0.152L L 2U 0
d
where equation [9.60] has been taken into account. With the numerical data of the problem: t tr  1 s
[9.63]
– On the other hand, the tank fluid temperature varies under the influence of the backward flow. The associated velocities being much smaller than those in the jet, the time response of the water tank is much larger than t tr .
Turbulent Convection in Free Shear Flows
355
The order of magnitude of this response time is estimated by the global energy equation applied to the tank. Denoting qi and qe the inlet flow and exit flow of enthalpy, respectively, and Tr , the mean fluid temperature in the tank, the energy budget reads:
UcSR 2 L
dTr dt
qi qe
Assuming that the temperature is uniform in the outlet opening, the righthand side is equal to UcQ0 Ti Te . Hence:
SR 2 L
dTr dt
Q0 >Ti t Te t @
[9.64]
As a first approximation, the tank time response is calculated from [9.64] and estimated by: tr 
SR 2 L Q0
 112 s
[9.65]
It is worth noting that tr is the ratio of the tank volume to the jet flow rate. As a first approximation, the mean fluid temperature Tr in the tank is not distinguished from the temperature of the back flow. The model of heat transfer to be implemented depends on the order of magnitude of the period TZ of the inlet fluid temperature variations compared to the two time constants found above. If TZ < ttr or TZ  ttr, the time variations of the jet temperature field must be taken into account. In other words, the unsteady term w T ! wt must be retained in the energy equation in turbulent regime. It is then necessary to consider phase averages < g > instead of the classical time averages in turbulence theory. If ttr Ta x, t Tr t @2S
SUc>Ti t Tr t @ 3d ³0L We set [ H2
V2
VR x 2 2 4 VR x
dx
VR x 2 dx V 4 VR x 2 x
2x VR :
SUc>Ti t Tr t @ 3d
S 3 8
R2 4
V ³02L VR
[ 1 [2
d[
>
2
>
2
Uc>Ti t Tr t @dR 2VLn 1 2L VR
Thus, finally, denoting a1 H
x2
3 Vd 8
L
@ @
Ln 1 2L VR :
UcSR 2 L^Tr t a1 >Ti t Tr t @`
[9.75]
Convective Heat Transfer
360
This expression of enthalpy is reported in [9.74] to obtain a first equation satisfied by T r t and Te t a1
dTi dt
1 a1
dTr
bTi Te
dt
[9.76]
Q0 SR 2 L = 1 t r .
with b
A second equation is obtained by writing the enthalpy budget on the same control domain as in model 1 (Figure 9.9)
Uc2 S ³0r u x, r >T X, r, t Tr t @rdr = UcQ0 >Te t Tr t @
qr X, r, t
where T r t has been chosen as reference temperature instead of T 0 to calculate the integral like in the first model. The calculation is identical to that of model 1 so that we find, as in [9.71], Te t Tr t Ti t Tr t
[9.77]
c1
with c1 = 0.27 for X d = 21.6. Eliminating T r t between [9.76] and [9.77], we obtain the equation satisfied by
Te t
a
with a
dTe dt
bTe
1 a1 1 c1
,c
c
dTi dt
bTi
[9.78]
ac1 a1 .
The case of sinusoidal variations of inlet temperature is easily calculated: Ti t T0
Ai cos Zt , with Z
2S TZ
We find Te t T0
Ae cosZt M
[9.79]
with the damping factor Ae Ai cos M m sin M , and the phase shift M given 1 mn by tan M , where we have set m aZ b , n b cZ . nm
Turbulent Convection in Free Shear Flows
361
With the data of the problem, we find:
a1 = 0.023, b = 0.0089 s1. Figure 9.10 shows the damping factor and the phase shift of the exit temperature relative to the inlet temperature, as given by the present model. For low frequencies, the tank does not play the role of damping any more because the backflow temperature T r t follows the inlet temperature variations. Beyond a value of the pulsation of the order of 0.06 rad s1, the damping factor no longer varies and we find a value slightly smaller than that of case 1, due to the jet fluid inertia, which is taken into account in model 2. For a period TZ of 10 min ( Z = 0.0105 rad s1), the damping factor is 57.6% and the phase shift is 10% of the period (0.626 rad). 1.0 0.9
Ae Ai
0.8 0.7 0.6 0.5
M rad
0.4 0.3 0.2 0.1 0.0 0.001
0.010
0.100
Z rad s1
1.000
Figure 9.10. Regulation of temperature. Damping factor and phase shift of the exit temperature relative to the inlet temperature as a function of the pulsation
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List of Symbols
Cf
skinfriction coefficient
dimensionless
Cp
specific heat at constant pressure
J kg1 K1
DQ
specific viscous dissipation
W m3
DQ
viscous dissipation in volume V
W
D
duct diameter
m
Dh G F
hydraulic diameter
m
body force vector per unit mass
m s2
g
gravitational acceleration
m s2
Gr
Grashof number
dimensionless
h
heattransfer coefficient
W m2 K1
k
thermal conductivity
W m1 K1
ks
roughness height
m
Nu
Nusselt number
dimensionless
p
pressure
N m2
p*
modified pressure = p + Ugz
N m2
Pm
mechanical power supplied by a machine (> 0 for a pump, < 0 for a turbine)
W
Pf
power developed by viscous forces
W
Pe
Peclet number
dimensionless
Pr
Prandtl number
dimensionless
364
Convective Heat Transfer
Prt
turbulent Prandtl number
dimensionless
q
heat transfer rate
W
qc
heat transfer rate per unit length
W m1
qcc
heat flux
W m2
qccc
rate of internal heat generation
W m3
Q
flow rate
m3 s1
Ra
Rayleigh number
dimensionless
Re
Reynolds number
dimensionless
S
crosssectional area
m2
T
temperature
K
Tm
bulk temperature
K
T
freestream temperature
K
Tqccw or TW friction temperature G velocity vector u
K
Um
bulk velocity
m s1
u
freestream velocity
m s1
uW
friction velocity
m s1
x
longitudinal coordinate
m
y
transversal coordinate
m
Greek letters
D
thermal diffusivity
m2 s1
E
coefficient of thermal expansion
K1
G
velocity boundary layer thickness
m
GT
thermal boundary layer thickness
m
/
DarcyWeisbach coefficient (/ = 4 Cf)
dimensionless
K
dimensionless distance to the wall
dimensionless
T
dimensionless temperature
dimensionless
List of Symbols
P
dynamic viscosity
kg m1 s1
Q
kinematic viscosity
m2 s1
U
density
kg m3
Vij
stress tensor
N m2
W
shear stress
N m2
Subscripts ( )f
fluid
( )i
velocity component
( )w
wall
( )t
turbulent
Superscripts ( )'
turbulent fluctuation
average of a fluctuating quantity
365
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References
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[COL 56] D.E. COLES, “The law of the wake in the turbulent boundary layer”, J. Fluid Mech., vol. 1, 191226, 1956. [COL 59] D.C. COLLIS and M.J. WILLIAMS, “Twodimensional convection from heated wires at low Reynolds numbers”, J. Fluid Mech., vol. 6, 357384, 1959. [DOC 06] O. DOCHE, Contrôle actif dual des écoulements turbulents pariétaux ; expériences et simulations numériques directes, Phd Thesis, Joseph Fourier University, Grenoble, 2006. [DRA 81] P.G. DRAZIN and W. REID, Hydrodynamic Stability, Cambridge University Press, 1981. >ECK
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Index
AB Blasius boundary layer, 55, 281 law, 266, 275, 277, 293296 Boundary conditions, 33, 35, 47, 5759, 9197, 146, 147, 191, 192, 207, 209, 212, 239, 240, 252268, 304, 309, 335340, 352 Boundary layer, 58, 199, 204 temperature, 68, 84 thermal, 34, 35, 46, 53, 58110, 119, 131, 132, 282307, 316 thickness, 129, 130149, 16171, 184, 197, 203, 305 velocity, 57 Boussinesq eddy viscosity model, 324 equation, 162, 177, 203, 204 model, 142 Buffer layer, 219, 229, 245251, 298 Bulk temperature, 34, 40, 76, 107112, 231, 238, 239, 240, 241, 271279 velocity, 3246, 79, 106, 215, 231241, 254, 264278, 354 CD Colburn analogy, 266273, 290, 294 Conductive sublayer, 222225, 245251, 300, 311318 Couette flow, 915
Cylinder, 95105, 120124, 154, 158, 180 DarcyWeisbach drag coefficient, 231 Direct numerical simulations, 218235 DittusBoelter correlation, 231 Drag coefficient, 232240, 287300, 310, 319, 321 Duct flow, 3132 EF Eddy diffusivity, 217, 222, 232, 252, 253, 300, 305, 316, 329, 334 closure, 216, 221 spalding model, 224 viscosity, 215223, 244258, 284, 298320 Energy budget, 1928, 38, 95, 106112, 180, 239258, 273277, 327, 341358 Energy equation, 2, 6, 19, 3648, 5873, 84118, 142, 150, 177, 191, 203, 204, 265, 267, 294, 326343, 355 integral, 122133 Enthalpy flow rate, 114 FalknerSkan flows, 306 310 Forced convection, 53, 119, 141160, 189, 192, 210, 272278 Friction coefficient, 189, 193 Fully developed flow, 264 hydrodynamically, 36 thermally, 36
372
Convective Heat Transfer
GH
OP
Grashof number, 8, 143, 152, 192, 278 Head loss coefficient, 33, 266 Heat flux, 323, 3449, 58118, 121134, 143149, 164188, 206, 267, 283308, 329347 Heattransfer coefficient, 7, 34, 45, 5883, 148172, 273, 278 average, 123 local, 34, 78, 121 Hydraulic diameter, 3148, 231243, 265278
Outer layer, 211228, 245254, 281, 285, 319 Overall heat transfer rate, 36, 62, 68, 129 Plume, 176182, 196, 326 Poiseuille flow, 211213, 237 Prandtl number, 8, 35, 36, 121, 127, 129, 143, 158165, 182, 183, 200, 202, 221227, 249279 PrandtlTaylor analogy, 288300 Pressure gradient, 11, 5578, 90, 130, 204, 208, 233, 234, 284318, 324342 Principle of superposition, 114
IJ QR Integral method, 50, 62, 114, 122135, 149, 166173 Integration of energy equation, 327
Kolmogorov dissipative scale, 227 Laminarturbulent transition, 78, 141 Lévêque solution, 107 Logarithmic layer, 245259, 291
Rayleigh number, 9, 143161, 181186, 198, 199 Reynolds analogy, 286310 equations, 216, 271, 274, 233, 332, 342, 343 number, 8, 32, 33, 53, 6784, 119129, 137, 192, 212221, 231236, 266278 shear stress, 214, 216, 229238 Rough walls, 231, 233
MN
ST
Mixed convection, 152, 189, 190, 206, 210 Mixing length, 215, 243247, 301309, 325 Modified pressure, 2, 33, 142, 196, 271275 Natural convection, 141156, 174, 188, 195, 270, 272 in an enclosure, 196, 201 unsteady, 164169 NavierStokes equation, 2, 37, 53, 94, 108, 212216, 255, 282 Newtonian fluid, 2, 3, 12 Nusselt number, 9, 3450, 58103, 121138, 143160, 175, 186, 203, 206, 231, 238, 239, 240, 241, 242, 243, 279
Scale analysis, 55, 58, 87, 98, 142, 167, 184, 195 temperature boundary layer, 62 velocity boundary layer, 32, 64, 62 Shear stress, 11, 12, 18, 5557, 90, 128130, 161, 214220, 233274, 289313, 330 velocity, 312316, 215, 224 Similarity solutions, 5973, 122129, 146, 292314, 326339 Skinfriction, 914, 5565, 266, 290, 294 Stagnation plane flow, 128, 131 point, 119, 120, 121, 123, 131 Thermal insulation, 199 resistance, 53, 81, 82, 183185, 200
KL
Index Turbulent channel flow, 220, 227240, 252265 heat flux, 229, 267, 283, 341347 jet, 351 kinetic energy, 229, 255259 kinetic energy budget, 258 mixing layers, 323 Prandtl number, 221, 223, 245, 252, 265, 287309, 313, 318, 325, 338348 viscosity, 265268, 324340, 350, 297 wall flux temperature, 215, 260 wall variables, 215222, 254262, 284, 304, 310
373
UZ Van Driest relationship, 309 Viscous sublayer, 218223, 246250, 273, 290, 298319 Wake velocity, 119 function, 226 thermal, 119