## Peer review process

**Not revised:** This Reviewed Preprint includes the authors’ original preprint (without revision), an eLife assessment, and public reviews.

## Editors

- Reviewing EditorMarcus SeldinUniversity of California, Irvine, Irvine, United States of America
- Senior EditorChristian RutzUniversity of St Andrews, St Andrews, United Kingdom

**Reviewer #1 (Public Review):**

Summary:

The study seeks to establish accurate computational models to explore the role of hydrodynamic interactions on energy savings and spatial patterns in fish schools. Specifically, the authors consider a system of (one degree-of-freedom) flapping airfoils that passively position themselves with respect to the streamwise direction, while oscillating at the same frequency and amplitude, with a given phase lag and at a constant cross-stream distance. By parametrically varying the phase lag and the cross-stream distance, they systematically explore the stability and energy costs of emergent configurations. Computational findings are leveraged to distill insights into universal relationships and clarify the role of the wake of the leading foil.

Strengths:

(1) The use of multiple computational models (computational fluid dynamics, CFD, for full Navier-Stokes equations and computationally efficient inviscid vortex sheet, VS, model) offers an extra degree of reliability of the observed findings and backing to the use of simplified models for future research in more complex settings.

(2) The systematic assessment of the stability and energy savings in multiple configurations of pairs and larger ensembles of flapping foils is an important addition to the literature.

(3) The discovery of a linear phase-distance relationship in the formation attained by pairs of flapping foils is a significant contribution, which helps compare different experimental observations in the literature.

(4) The observation of a critical size effect for in-line formations of larger, above which cohesion and energetic benefits are lost at once, is a new discovery in the field.

Weaknesses:

(1) The extent to which observations on one-degree-of-freedom flapping foils could translate to real fish schools is presently unclear so some of the conclusions on live fish schools are likely to be overstated and would benefit from some more biological framing.

(2) The analysis of non-reciprocal coupling is not as novel as the rest of the study and potentially not as convincing due to the chosen linear metric of interaction (that is, the flow agreement).

Overall, this is a rigorous effort on a critical topic: findings of the research can offer important insight into the hydrodynamics of fish schooling, stimulating interdisciplinary research at the interface of computational fluid mechanics and biology.

**Reviewer #2 (Public Review):**

The document "Mapping spatial patterns to energetic benefits in groups of flow-coupled swimmers" by Heydari et al. uses several types of simulations and models to address aspects of stability of position and power consumption in few-body groups of pitching foils. I think the work has the potential to be a valuable and timely contribution to an important subject area. The supporting evidence is largely quite convincing, though some details could raise questions, and there is room for improvement in the presentation. My recommendations are focused on clarifying the presentation and perhaps spurring the authors to assess additional aspects:

(1) Why do the authors choose to set the swimmers free only in the propulsion direction? I can understand constraining all the positions/orientations for investigating the resulting forces and power, and I can also understand the value of allowing the bodies to be fully free in x, y, and their orientation angle to see if possible configurations spontaneously emerge from the flow interactions. But why constrain some degrees of freedom and not others? What's the motivation, and what's the relevance to animals, which are fully free?

(2) The model description in Eq. (1) and the surrounding text is confusing. Aren't the authors computing forces via CFD or the VS method and then simply driving the propulsive dynamics according to the net horizontal force? It seems then irrelevant to decompose things into thrust and drag, and it seems irrelevant to claim that the thrust comes from pressure and the drag from viscous effects. The latter claim may in fact be incorrect since the body has a shape and the normal and tangential components of the surface stress along the body may be complex.

(3) The parameter taudiss in the VS simulations takes on unusual values such as 2.45T, making it seem like this value is somehow very special, and perhaps 2.44 or 2.46 would lead to significantly different results. If the value is special, the authors should discuss and assess it. Otherwise, I recommend picking a round value, like 2 or 3, which would avoid distraction.

(4) Some of the COT plots/information were difficult to interpret because the correspondence of beneficial with the mathematical sign was changing. For example, DeltaCOT as introduced on p. 5 is such that negative indicates bad energetics as compared to a solo swimmer. But elsewhere, lower or more negative COT is good in terms of savings. Given the many plots, large amounts of data, and many quantities being assessed, the paper needs a highly uniform presentation to aid the reader.

(5) I didn't understand the value of the "flow agreement parameter," and I didn't understand the authors' interpretation of its significance. Firstly, it would help if this and all other quantities were given explicit definitions as complete equations (including normalization). As I understand it, the quantity indicates the match of the flow velocity at some location with the flapping velocity of a "ghost swimmer" at that location. This does not seem to be exactly relevant to the equilibrium locations. In particular, if the match were perfect, then the swimmer would generate no relative flow and thus no thrust, meaning such a location could not be an equilibrium. So, some degree of mismatch seems necessary. I believe such a mismatch is indeed present, but the plots such as those in Figure 4 may disguise the effect. The color bar is saturated to the point of essentially being three tones (blue, white, red), so we cannot see that the observed equilibria are likely between the max and min values of this parameter.

(6) More generally, and related to the above, I am favorable towards the authors' attempts to find approximate flow metrics that could be used to predict the equilibrium positions and their stability, but I think the reasoning needs to be more solid. It seems the authors are seeking a parameter that can indicate equilibrium and another that can indicate stability. Can they clearly lay out the motivation behind any proposed metrics, and clearly present complete equations for their definitions? Further, is there a related power metric that can be appropriately defined and which proves to be useful?

(7) Why do the authors not carry out CFD simulations on the larger groups? Some explanations should be given, or some corresponding CFD simulations should be carried out. It would be interesting if CFD simulations were done and included, especially for the in-line case of many swimmers. This is because the results seem to be quite nuanced and dependent on many-body effects beyond nearest-neighbor interactions. It would certainly be comforting to see something similar happen in CFD.

(8) Related to the above, the authors should discuss seemingly significant differences in their results for long in-line formations as compared to the CFD work of Peng et al. [48]. That work showed apparently stable groups for numbers of swimmers quite larger than that studied here. Why such a qualitatively different result, and how should we interpret these differences regarding the more general issue of the stability of tandem groups?

(9) The authors seem to have all the tools needed to address the general question about how dynamically stable configurations relate to those that are energetically optimal. Are stable solutions optimal, or not? This would seem to have very important implications for animal groups, and the work addresses closely related topics but seems to miss the opportunity to give a definitive answer to this big question.

(10) Time-delay particle model: This model seems to construct a simplified wake flow. But does the constructed flow satisfy basic properties that we demand of any flow, such as being divergence-free? If not, then the formulation may be troublesome.