Hydrodynamically Beneficial School Configurations in Carangiform Swimmers: Insights from a Flow-Physics Informed Model

We start by examining the thrust enhancement map for a two-fish configuration where both fish have exactly the same swimming kinematics in terms of frequency, amplitude, and phase. We summarize the process used to predict these hydrodynamic interactions in fish (see Fig. 4). Once the wake velocity field (i.e., $\left(u_{L}(x,y,t),v_{L}(x,y,t)\right)$ ) for the leading fish is obtained via a direct-numerical simulation, a trailing fish is virtually placed in this wake field at a location with its caudal fin located at $\left(X_{T},Y_{T}\right)$ relative to the caudal fin of the leading fish on the center plane of the leading fish. The wake velocity at $\left(X_{T},Y_{T}\right)$ is then combined with the kinematics of the caudal fin of the virtual trailing fish to estimate the interaction effect on the thrust enhancement factor $\Delta\Lambda_{T}$ via the expression in Eqs. 7$-$12. Based on the linear relationship between $\Lambda_{T}$ and $C_{T}$ (see Eq. 7), the $\Delta\Lambda_{T}$ is used as a surrogate for the change in thrust of the trailing fish. A thrust enhancement map is then generated by placing the virtual trailing fish at various locations within the domain with minimal computational expense.

Fig. 4(C) shows the thrust enhancement map generated based on the above procedure. We highlight a small rectangular region behind the leading fish where the trailing fish cannot be placed since this would lead to collisions between two fish. We also exclude positions of the trailing fish that would place it ahead of the leading fish. Fig. 4(D) shows two notional positions of a trailing fish on the thrust enhancement map that would correspond to either a beneficial or a detrimental interaction. In Fig. 4(D), the blue trailing fish is positioned with its caudal fin within the positive $\Delta\Lambda_{T}$ region and would benefit from a constructive interaction with the leading fish’s wake, generating more thrust than a solitary swimmer. In contrast, the pink trailing fish, with its caudal fin located in a negative $\Delta\Lambda_{T}$ region, would experience a reduction in thrust.

As pointed out earlier, several assumptions are inherent in the prediction of thrust for the trailing fish based on the LEVBM, and we have carried out comprehensive verifications of the model predictions to assess these assumptions.

As noted above, a known limitation of the LEVBM is that it does not account for the effect of the body of the trailing fish on the wake perturbations encountered by the caudal fin of the trailing fish. Other key assumptions are neglecting the effect of the trailing fish’s caudal fin on the encountered wake perturbation and the inability to account for the effect of the 3D shape of the caudal fin of the trailing fish on thrust enhancement. Thus, in the first set of DNS simulations, we exclude the body of the trailing fish (see Fig. 5(A) for the configuration) to test the effect of the latter two assumptions on the LEVBM predictions.

For 14 selected locations of the trailing fish, which cover a range of placements with beneficial and detrimental interactions (as noted in Fig. 6(A)), we compare the thrust enhancement predictions from the LEVBM directly to the thrust enhancement of the pressure component on the fin calculated from the DNS. Fig. 6(C) shows the correlation between the predicted $\Delta\Lambda_{T}$ values and the thrust change ($\Delta T$) obtained from the DNS results corresponding to the case where the body of the trailing fish is excluded. A best-fit line between the two data sets suggests a linear relationship with a high degree of correlation ($R^{2}=0.9$). The equation of the best-fit line has a slope of 0.91 with zero intercept corresponding to 0.05%, which signifies an excellent one-to-one relationship between the LEVBM prediction and the DNS. All in all, the above comparison suggests that the latter two assumptions discussed in the previous paragraph do not significantly deteriorate the predictions of the LEVBM, and it performs quite well despite the complex shape and effect of the caudal fin.

In the second and final stage of verification, we reintroduce the body of the trailing fish into the DNS to examine the effect of the body on the LEVBM thrust enhancement map predictions, and Fig. 6(D) shows the correlation between the DNS estimation of the pressure thrust on the fin and the LEVBM predictions. With the body of the trailing fish included, the linear correlation reduces to a $R^{2}$ of 0.7, which suggests that while the presence of the fish body does diminish the predictive power of the LEVBM, the linear correlation remains acceptable, and the model is still useful for predicting the thrust changes for fish schools. We also note that the slope and intercept of the best-fit line is 0.29 and 3.7%, respectively. This significant reduction in slope from the expected value of unity indicates that the body acts to diminish the effect of the hydrodynamic interactions on the thrust of the caudal fin. This is likely due to the fact that the most significant effect on the effective angle-of-attack of the trailing caudal fin is via the perturbation in the lateral velocity, and the body acts as a “wall” and diminishes these lateral perturbations of vortex structure that convects past it to the fin. This effect of the body would be an interesting issue to explore in a future study.

The periodic distribution of positive and negative regions of the $\Delta\Lambda_{T}$ in the streamwise direction is a result of the alternating shedding of vortices from the leading fish’s tail, which convects downstream with the flow and drives the velocity perturbation pattern in the wake. As shown by Seo & Mittal (2022), the thrust enhancement is associated with the effective phase difference between the tail beats of the leading and trailing fish $\Delta\phi_{\mathrm{eff}}$, which is estimated by the following expression:

Here, $\phi_{T}$ and $\phi_{L}$ are the tailbeat phases of the trailing and leading fish, respectively, and $\lambda$ represents the wavelength of the velocity perturbation in the wake, which may be estimated as $\lambda\approx U/f_{L}$, where $U$ is the swimming speed and $f_{L}$ is the tailbeat frequency of the leading fish. The effective phase difference is directly related to the phase difference between $-\dot{h}_{T}(t)$ and $v_{L}^{\prime}(X_{T},t)$ in Eq. 9, and it determines the changes in $\alpha_{\text{eff}}$. This suggests that the phase difference between the tail beats of the two fish, as well as the trailing fish location $X_{T}$ and the tail beat frequency $f_{L}$ of the leading fish are all factors that affect the thrust enhancement map.

The above expression also suggests that the effects of differences in tail beat phase $\Delta\phi=\left(\phi_{T}-\phi_{L}\right)$ and separation $X_{T}$ between the two fish are essentially interchangeable in the near wake. In Fig. 7(A), thrust enhancement maps for two additional $\Delta\phi$ are shown alongside the $\Delta\Lambda_{T}$ plots along the locus of the extreme values of these quantities for four different choices of $\Delta\phi$. We note that the topology of the thrust enhancement maps for the different phases is very similar except for a shift in the streamwise direction. This is confirmed by the peak locus plot in Fig. 7(B), which shows that all the different cases of $\Delta\phi$ are essentially the same except for this shift. This suggests that a trailing swimmer can improve thrust at any given streamwise location by suitably adjusting its flapping phase. Similarly, for a given phase difference, the trailing swimmer can gain thrust benefit by moving to an appropriate location in the wake.

Several other features of the thrust enhancement map are worth noting for their implication for schooling. First, there are multiple regions where the thrust is enhanced in the wake, but these regions are interspersed with regions where the thrust is reduced due to the hydrodynamic interactions. The extreme values of thrust change are located along two oblique angles from the center of the wake that correspond well to the double-vortex loop wake that is generated behind the flapping fin. The region near the wake center has relatively low values of $\Delta\Lambda_{T}$.

Fig. 7(B) shows the value of $\Delta\Lambda_{T}$ along the local peaks in this parameter and this pattern also decays relatively slowly in the wake. Indeed, while the peak in $\Delta\Lambda_{T}$ around $X_{T}\sim 0.5$ corresponds to a 25% increase in $\Delta\Lambda_{T}$, the peak at a distance of 2.5 body-lengths is still 20%. Thus, even in a minimal two-fish school, the trailing fish could achieve comparable propulsion benefits in many different locations within the wake of a leading fish. Second, even small changes in the movement of the leading fish would perturb the entire wake pattern and require the trailing fish to make larger adjustments in its location and/or flapping kinematics to recover to a beneficial state.

As noted earlier, for each region in the wake where the thrust is enhanced due to the hydrodynamic interactions, there is an adjoining region where the thrust is reduced due to these interactions. The presence of regions in the wake where the interactions are detrimental to the trailing fish is not surprising and was shown in our earlier work as well Seo & Mittal (2022). However, the LEVBM thrust enhancement map shows that the detrimental effects exceed the beneficial effects, and they also extend over larger regions of the wake than the beneficial regions. This feature is unexpected and we therefore examine this in more detail. Fig. 6(C), which shows the correlation between the LEVBM and the DNS for the cases where the body of the trailing fish is excluded, confirms this bias towards detrimental interactions since the negative peaks in relative thrust reach -43.3% whereas the positive peaks are limited to +26.1%. Thus, the negative bias is not an erroneous prediction from the LEVBM but is confirmed by the DNS. We now examine the flow physics that results in this negative bias.

For a fish swimming in the wake of another fish, the flow perturbations due to the wake vortices, experienced by the caudal fin of the trailing fish will modify $\sin({\alpha_{\text{eff}}(t)})$ and through it, the thrust force. To understand this effect, we consider the velocity field in the wake of the leading fish in terms of a mean (denoted by “bar”) and a fluctuation (denoted by a double-prime) as follows

We can now define an effective angle-of-attack due to the effect of the mean flow ($\bar{\alpha}_{\textrm{eff}}(t)$) as follows

Similarly, the change in the thrust factor can be decomposed into

where $\Delta\bar{\Lambda}_{T}=\langle\sin{(\bar{\alpha}_{\textrm{eff}})}\sin(\theta)\rangle$ is the change in thrust factor due to the mean wake and $\Delta\Lambda_{T}^{\prime\prime}=\Delta\Lambda_{T}-\Delta\bar{\Lambda}_{T}$ is the remaining component that is primarily associated with the velocity fluctuation in the wake. This simple decomposition now allows us to dissect the effect of the wake on the performance of the trailing fish.

Fig. 8(A) shows contour plots of $\bar{u}_{L}(X_{T},Y_{T})$ and $\bar{v}_{L}(X_{T},Y_{T})$ and we note that the mean streamwise velocity in the wake symmetric about the centerline and the values are within $\pm 10\%$ of the swimming speed. In contrast, the mean lateral velocity is anti-symmetric about the wake centerline with values up to $\pm 25.6\%$ of the swimming speed. The lower part of the wake has a mean lateral component that has a negative (downwards) induced velocity, and vice-versa for the upper part of the wake.

Fig. 9(A) shows contours of $\Delta\bar{\Lambda}_{T}$ for the case with $\Delta\phi=0$ and this figure shows that the mean wake has a predominantly detrimental effect on the thrust factor and would therefore result in a reduction in the thrust of the trailing fish. In fact, the reduction in the thrust factor due to the mean flow reaches magnitudes of 10.7%. Fig. 9(B) shows the corresponding contour plot for $\Delta\Lambda_{T}^{\prime\prime}$ and it shows that with the effect of the mean flow removed, the fluctuations generate nearly similar magnitudes of beneficial and detrimental interactions. Thus, the negative bias in the thrust factor for the trailing fish is clearly due to the effect of the mean wake, which has a strong negative bias on thrust generation.

What remains now is to explain why the mean wake leads to a negative bias in the thrust factor. We begin by noting that $\Lambda_{T}$ is an inner product of $\sin({\theta(t))}$ with $\sin{(\alpha_{\text{eff}}(t))}$ (see Eq. 8) and large positive values of $\Lambda_{T}$ are generated when these two periodic functions are similar to each other in shape and phase. We plot the $\alpha_{\textrm{eff}}(t)$ for a solitary fish (SF) and the $\bar{\alpha}_{\textrm{eff}}(t)$ for a trailing fish (TF) in two-fish configurations at location $(X_{T}=0.63,Y_{T}=-0.2)$, which is location d on Fig. 6(A) and which corresponds to a location close to the peak of thrust enhancement for the trailing fish. Also plotted is the pitch angle $\theta(t)$, which is the same for the two fish. As Fig. 10(A) shows, for the solitary fish, we find that $\alpha_{\textrm{eff}}(t)$ and $\theta(t)$ are very much in phase as evidenced by the fact that they cross the abscissa at the exact locations. This is not a coincidence since both functions result from the same BCF motion of the fish. In fact, as evident from Eqs. 2 and 5, for BCF motion, $\alpha_{\textrm{eff}}(t)$ and $\theta(t)$ are both related to the arctangent of $\dot{h}$ and are therefore expected to be in phase. Thus, the simple sinusoidal flapping motion of the caudal fin generates temporal profiles of $\sin({\theta(t)})$ and $\sin({\alpha_{\text{eff}}(t)})$ that are intrinsically well-suited for thrust generation. We note that BCF swimmers in Nature have arrived at this swimming motion through hundreds of millions of years of evolution, and it would, in fact, be puzzling if this swimming motion was not well suited for thrust generation. Indeed, the thrust factor parameter that emerges from the LEVBM provides a strong theoretical underpinning and understanding of why such a swimming mode is ubiquitous in Nature.

The plot of $\bar{\alpha}_{\textrm{eff}}(t)$ for a trailing fish shows some interesting differences from that of a solitary fish. The entire curve for this quantity is essentially shifted downwards by values ranging from $2^{\circ}$ to $10^{\circ}$. We note that at this location $(\bar{u}_{L},\bar{v}_{L})=(1.045U,-0.168U)$ and this induces a flow angle of $-9^{\circ}$ which is consistent with the shift in $\bar{\alpha}_{\textrm{eff}}(t)$. This downwards lateral velocity at this location increases the effective angle-of-attack during the upstroke, but reduces the effective angle-of-attack during the downstroke which is akin to the fin flapping with a biased pitch angle.

Fig. 10(B) shows the variation of $\sin({\theta(t)})\cdot\sin{(\alpha_{\text{eff}}(t))}$ for these two cases and we note that compared to the solitary fish, the trailing fish sees a significant reduction in this quantity during the downstroke and a smaller increase during the upstroke. This asymmetry is due to the fact that the downward shift of $\bar{\alpha}_{\textrm{eff}}(t)$ results in a phase mismatch with $\theta(t)$, thereby diminishing the product of these two functions. Thus, the net result of the mean wake is to reduce the thrust of the trailing fish. The fluctuation in the flow velocity has nearly equal potential to either increase or decrease the thrust depending on the location. However, the negative bias effect due to the mean flow results in detrimental interactions being more significant.

In the previous section, we examined the case where the leading and trailing fish swim with identical kinematics (i.e., the same amplitude and frequency). For these cases, the hydrodynamic interaction effects are determined almost exclusively by $\Delta\phi_{\mathrm{eff}}$, which depends on a combination of the difference in tail beat phases and the distance between the two fish. However, depending on its position in the wake, a swimmer in the wake of a leading swimmer will experience changes in thrust (and therefore the total surge force), which would lead to acceleration or deceleration of the swimmer. One way to maintain the position at a beneficial location or to move from a detrimental position to a beneficial position is via modification in tailbeat amplitude. Indeed, a swimmer in a beneficial location could reduce their power expenditure while maintaining position by reducing their tailbeat amplitude. It is, therefore, of interest to examine the thrust enhancement map for a fish that is swimming in the wake of another fish using a tailbeat amplitude that is different from the leading fish. The current LEVBM provides the opportunity to easily examine this question since the kinematics of the trailing fish can be modified without requiring any additional high-fidelity simulations.

Fig. 11 shows the effect of the tail beat amplitude of the trailing fish on thrust enhancement for the trailing fish. The $\Delta\Lambda_{T}$ maps for $A_{T}/A_{L}=0.5$ and $A_{T}/A_{L}=1.5$ reveal that the topology of the thrust enhancement map is similar to that for the same amplitude but a smaller flapping amplitude in the trailing fish ($A_{T}/A_{L}=0.5$) leads to larger relative percentage increase in the $\Delta\Lambda_{T}$ values. This is expected since for a trailing fish with a reduced tail beat amplitude, the wake perturbations $(u^{\prime}_{L},v^{\prime}_{L})$ from the leading fish’s wake are more significant relative to the trailing fish’s fin velocity, which is itself proportional to $\dot{h}_{T}$. Consequently, the effective angle-of-attack is modified more significantly by the wake interaction, leading to greater changes in $\Delta\Lambda_{T}$. Conversely, with higher tail beat amplitudes, the trailing fish’s fin motion dominates, thereby reducing the influence of the perturbation from the wake vortices. Thus, the current analysis shows that a difference in amplitude between the two fish does not fundamentally change the flow physics of thrust enhancement, and this can serve as a viable strategy for maintaining or reaching a beneficial location in the wake.

A Reynolds number of 5000 corresponds to a 2-5 cm caudal fin swimmer (such as a Giant danio) swimming at O(1) BL/s (body length per second), and as we have seen that the wake at these low Reynolds number is well organized and highly periodic. It is of interest to see if the thrust enhancement maps are affected by an increase in Reynolds numbers, which would result in a complex, non-periodic wake with transitional/turbulent flow characteristics. To examine this, we employ our solver to compute the flow past the swimmer at a Reynolds number of 50,000. These simulations are carried out on an extensive 233 million point grid, which is chosen after a grid convergence study ((Mittal et al., 2024)). To simulate a condition corresponding to steady swimming at a terminal speed, we have conducted a series of simulations at different freestream velocities and selected a velocity for which the mean thrust and drag are nearly balanced out, and the net mean hydrodynamic force on the swimmer is almost zero. This condition for Re=50,000 corresponds to a Strouhal number of 0.28, and it nominally represents a  15-20 cm carangiform swimmer such as a medium-sized trout.

Fig. 12 shows two views of the vortex structures in the wake, and we note that while the wake still exhibits the characteristic oblique dual-vortex street structure, the flow is significantly more complicated with a wide range of smaller vortex structures that are a result of various instabilities in the wake.

This lack of strict periodicity in the wake could impact the ability of a trailing swimmer to harness the induced velocities from the wake to enhance its LEV and thrust. To examine this issue, the wake velocity field from this simulation is extracted and used to generate a thrust enhancement map of a swimmer in the wake. Fig. 13(A) shows the thrust enhancement map for a swimmer that is swimming with kinematics and phases that are identical to the leading swimmer. Remarkably, we find that despite the one order-of-magnitude increase in the Reynolds number and the significantly increased complexity of the wake, the thrust enhancement map looks very similar in form to that at the lower Reynolds number. Fig. 13(B) shows $\alpha_{\textrm{eff}}$ for a trailing fish at a beneficial location (marked in Fig. 13(A)) along with the $\alpha_{\textrm{eff}}$ for a solitary fish and $\theta(t)$. We observe from this figure that while $\alpha_{\textrm{eff}}$ for the trailing fish reflects the oscillatory and stochastic nature of the wake flow at this higher Reynolds number, there is an increase in the amplitude of this quantity, which ultimately results in a larger value of $\Delta\Lambda_{T}$ at this location. All of these observations indicate that the flow physics associated with wake-induced enhancement of the LEV is quite robust and effective for caudal fin swimmers over a large range of scales.

Based on the successful demonstration of the use of $\Delta\Lambda_{T}$ map for two-fish schools, we have applied the same methodology to predict beneficial configurations for three-fish schools, where the three are swimming with identical swimming strokes. For this, we first performed direct numerical simulations of two different optimal two-fish configurations, specifically $a$ and $d$ in Fig. 6(B)) and obtained their wake velocity fields. Using these velocity fields, beneficial locations for the third fish are explored by computing the respective $\Delta\Lambda_{T}$ maps (Fig. 14(A)) for these cases. We note that there are many other two-fish configurations that could be examined, but we cannot explore all of them using DNS. The two configurations chosen here are however distinct enough that analysis of these two should provide general insights into the application of the LEVBM to larger fish schools.

Fig. 4(C) shows the thrust enhancement maps for the two two-fish configurations, and we note that while the overall pattern of benefit and detriment is observed, the thrust enhancement maps have a significantly more complex topology. As shown in Fig. 14(A), the third fish can be positioned in many beneficial locations. We select five locations: 3-Fish-a to 3-Fish-e for a detailed analysis. Fig. 14(B) illustrates the schematic of each configuration. Placing the third fish in position $a-c$ results in the well-documented diamond formation, which has been the subject of many previous studies (Liao, 2007; Pavlov & Kasumyan, 2000; Timm et al., 2024). Alternatively, positioning the third fish at $e$ leads to a diagonal configuration, theoretically allowing for further extension into larger schooling formations.

To verify the configurations suggested by the $\Delta\Lambda_{T}$ maps for the third fish, we have conducted DNS of those configurations illustrated in Fig. 14(B), and the metric related to the hydrodynamic performance of each swimmer is shown in Fig. 14(C). The availability of the DNS for these configurations allows us to perform a more comprehensive assessment of the swimming performance of the fish, including a quantification of the effect of schooling on drag, power, and efficiency for all the fish in these configurations. We note that these additional quantities are unavailable from the LEVBM-based thrust enhancement maps.

The plots show that the thrust for every fish in all cases is enhanced, with the highest enhancement reaching 28% ($d$, Fish-2, in Fig. 14(C)). Noticeably, as we selected specific beneficial configurations from optimal two-fish schools, Fish-2, and even the leading fish, Fish-1, still attain a distinct thrust enhancement in configurations $a-e$ in Fig. 14(C). The interactive effects on drag reveal distinct trends depending on the swimmer’s position within the school. For the leading fish, Fish-1, the drag increases significantly in compact configurations, such as $a$. As the school becomes less compact ($d$ and $e$, for instance), the drag on Fish-1 returns close to the baseline, highlighting the influence of the trailing fish on upstream flow dynamics. This observation underscores the fact that the trailing fish can induce measurable hydrodynamic effects on the leading swimmer for compact configurations. In contrast, the drag changes for Fish-2 and Fish-3 are less pronounced, with variations remaining around 10% across the various configurations. This suggests that the intermediate and trailing positions experience more stable drag conditions regardless of the compactness of the school.

The trend for power expenditure follows thrust, with increases observed for all fish in every configuration. The consistent alignment between thrust and power, combined with relatively stable drag changes, results in efficiency trends that follow similar patterns, except for Fish-1. For Fish-1, the significantly increased drag in configurations $a$ and $b$ negates the thrust gains, leading to an efficiency that is not much different from that of a solitary fish. A similar effect is also observed for Fish-3 in configuration $d$. Beyond these isolated effects, every configuration offers benefits in terms of improved thrust and efficiency for at least two of the three swimmers. Some configurations are particularly beneficial for one of the fish (for instance, $c$ for Fish-1, $d$ for Fish-2, and $b$ for Fish-3.

In summary, these results demonstrate that the $\Delta\Lambda_{T}$ map not only reliably predicts thrust improvements across different configurations but also shows that these thrust improvements, which are associated with the enhancement of LEV on the caudal fin, are also often accompanied by concurrent improvements in efficiency.

In the previous sections, we have shown how LEVBM-based thrust enhancement maps provide an understanding of the effect of relative location and phase for fish in small schools comprising up to 3 swimmers. We have also shown that these maps apply over a wide range of scales. The final question that we address is - Can these thrust enhancement maps apply to larger schools, or is the flow in larger schools so complicated that it would not lend itself to the relatively simple phenomenology that forms the basis of the LEVBM thrust enhancement maps?

To address this question, we consider two 9-fish schools that have been the subject of a previous study by us (Zhou et al., 2024, 2023). Each individual fish in these schools corresponds exactly to the configuration examined in Sec. 3.1. In both schools, the fish are arranged exactly in the same tight diamond configuration, but in the first school, all nine fish are swimming with the same phase, and in the second school, fish in each row are swimming with a $0.5\pi$ phase difference from the fish in the previous row. Thus, these two schools provide two significantly distinct configurations for the application of the LEVBM. Indeed, the plots of the vortex structures for these two schools (Fig. 15(A) and (B)) confirm the fact that the wake flows for these two schools are not only very complex but also quite distinct.

Thrust enhancement maps are computed for both these schools, and these are done for a trailing fish swimming with identical kinematics and a fin flapping phase that matches that of the leading fish in the schools. Fig. 15(C) and (D) show the thrust enhancement maps for these two schools, and we observe that while the two maps are not identical, they exhibit a surprising similarity with each other and with the thrust enhancement maps for a single leading fish. In particular, both maps are dominated by laterally oriented alternating bands of thrust enhancement and decrement. Thus, despite the tremendous complexity of the vortex wake of the 9-fish schools, the effect on the thrust of a trailing fish is quite similar to that for a single fish. The emergence of this simplicity from complexity is quite remarkable and connected with two facts: first, vortices from the various fish in the school tend to self-organize in banded structures due to mutual induction. This effect is visible in the vortex structure plots for the two schools. Second, the vorticity tends to highlight small-scale structures. However, the thrust enhancement is associated primarily with transverse velocity, which is still primarily driven by the sinusoidal movement of the caudal fin.