Why do anguilliform swimmers perform undulation with wavelengths shorter than their bodylengths?

Natural aquatic species have distinct physical mechanisms and swimming habits that have inspired engineers to explore them in developing robots capable of efficient and sustainable swimming. Recently, the research community have focused on answering the question of whether bio-mimicry or bio-inspiration should be preferred in developing next generation of underwater autonomous vehicles and energy harvesting technologies. Answering this question comprehensively requires a deep understanding of the swimming mechanisms of various species in nature. Some recent studies (van Rees et al., 2013) have explored the prospects of engineered swimmers outperforming natural species. For instance, van Rees et al. (2013) argued that optimal shapes of anguilliform swimmers could be different from their natural morphologies. This view has motivated the researchers to strive for better designs of such engineering systems. But on the other hand, it challenges the process of natural evolution in swimming species that resulted in their current physiological forms and propulsion and maneuvering capabilities. Because swimming is an essential and primary activity in marine life, we need to uncover relations between morphology and kinematics of different swimmers. Just recently, Fish (2020) presented a thorough review to illustrate where the research community stands with regard to bio-inspired and biomimicking robots in comparison with what we observe in nature. He concluded that the current technology would still need significant advancements and improvements to match the hydrodynamic performance of with marine animals in terms of speed, efficiency, and maneuverability. It also demonstrates existing knowledge gaps in our current understanding of underwater swimming, addressing which requires more efforts in terms of both kinematics of swimmers and associated flow physics and wake dynamics.

Numerous research investigators have directed their efforts to explore hydrodynamic mechanisms and performance of anguilliform (Tytell, 2004a,b; Tytell and Lauder, 2004; Kern and Koumoutsakos, 2006; Borazjani and Sotiropoulos, 2009), carangiform (Borazjani and Sotiropoulos, 2008; Liu et al., 2017; Wang et al., 2020; Lucas et al., 2020; Han et al., 2020; Zhang et al., 2020), and thunniform (Xia et al., 2015) swimming modes. Anguilliform swimmers with slender and skinnier bodies actuate their muscles to produce traveling waves opposite to their swimming directions, which make their whole bodies contribute to perform different propulsion maneuvers. Several remarkable features of anguilliform swimming mode have been discussed earlier by biologists and engineers. Generally, wavelengths (λ) of anguilliform swimmers’ undulating kinematics are shorter than their bodylengths (L) (Sfakiotakis, Lane, and Davies, 1999). Gillis (1997) found that undulating wave-speeds remained constant throughout the bodylengths for eels and salamanders with $λ=0.64L$⁠. Müller et al. (2001) found that the wavelength of undulating motion of eels, Anguilla anguilla LeSueur, vary in the range of $0.69L$–$0.87L$ for different swimming speeds. Moreover, Liao (2002) used experiments to show that needlefish and eels exhibited undulation of their bodies with $λ=0.49L−0.73L$ and $0.44L−0.54L$⁠, respectively. Similarly, Tytell and Lauder (2004) and Tytell (2004a) reported $λ=0.604L$ and $0.597L$⁠, respectively, for steadily swimming American eels. Hultmark et al. (2007) observed that Lampreys, another anguilliform species, exhibited $λ=0.642L$⁠. Their kinematic profile was further utilized by Borazjani and Sotiropoulos (2009, 2010) and Ogunka et al. (2020) for their three-dimensional (⁠$3D$⁠) numerical simulations.

Another important characteristic parameter in unsteady hydrodynamics is Strouhal number (St), which is a function of a swimmer’s tail-beat frequency (f), its swimming speed (U), and tail-beat amplitude (⁠$A°$⁠). It is defined as $St=2A°f/U$⁠. Tytell and Lauder (2004) found that the preferred St of American eels for their steady swimming was 0.314. Next, Tytell (2004a) reported that eels swam in the range of $St=0.314−0.41$⁠, while employing different tail-beat amplitudes. A living silver lamprey was also found to swim steadily with $St=0.30−0.35$ (Hultmark et al., 2007). These observations give us important leads to investigate the natural swimming characteristics of marine species in more details.

Some research efforts were devoted to explain the formation of coherent structures and how they produce hydrodynamic forces to help swimmers propel themselves. Müller et al. (2001) reported that eels shed two vortices in the wakes per half tail-beat. They noticed that those vortices tend to travel more on lateral sides instead of going downstream, which would not be helpful for improving their thrust or efficiency. For American eels, Tytell and Lauder (2004) and Tytell (2004a) found similar wake patterns, where the two vortices shed per tail-stroke were aligned in such a way that the jet would be lateral, making $90°$ angle with the direction of steady swimming. Through their experiments using an undulating robotic lamprey, Hultmark et al. (2007) showed the presence of stream-wise components of lateral jets in between vortices, due to which thrust production exceeded drag. Larval zebrafish (Müller et al., 2008) also exhibited similar wake dynamics with two rows of vortex rings diverging from the mean path of its motion. Later, Borazjani and Sotiropoulos (2009) studied wake structures behind anguilliform swimmers. They determined that the dynamics of these structures was closely related to the swimming motion, and particularly their St. In their three-dimensional simulations, single-vortex rows were formed behind a lamprey swimming at lower Strouhal numbers. The wake at higher Strouhal numbers, however, consisted of double-vortex rows.

These research investigations have provided us with critical information about natural swimming mechanisms of various marine animals belonging to the class of anguilliform swimmers. However, our present knowledge lacks in many aspects and it is of utmost importance to know more about their underlying flow characteristics. A few questions in this context include (1) why anguilliform swimmers choose to swim steadily using undulatory wavelengths smaller than those employed by carangiform and other swimmers, (2) how does undulatory wavelengths affect the hydrodynamic performance of anguilliform swimmers, and (3) how does the wavelength qualitatively impact the formation of vortices and wake dynamics? This study addresses these aspects of anguilliform swimming. Previously, Thekkethil et al. (2018) and Khalid et al. (2020b) performed numerical simulations to investigate the role of λ in hydrodynamics of two-dimensional undulating foils. It is evident from their conclusions that the thrust production capacity of those oscillating bodies depends strongly on their wave-forms and amplitude envelops. Those studies were limited in their scope due to the absence of real fish morphology, which consequently did not enable the modeling of important three-dimensional flow features in the wake of undulating bodies.

We organize the manuscript in the following manner. Section II explains our computational methodology for constructing a physiological model of American eel and prescribing particular kinematics for its locomotion in our simulations. It also presents details for our computational solver. In Sec. III, we quantitatively analyze hydrodynamic performance of the eel undulating with different wavelengths and Strouhal numbers. Section IV presents our analysis for the formation of different vortex structures and their dynamics in its wake. We discuss how these coherent structures affect the hydrodynamic performance of the anguilliform swimmer in Sec. V. Finally, we summarize our findings and conclusion in Sec. VI.

In this section, we present our computational methodology to reconstruct and simulate the flow around a physiological model of an American eel. We also explain the numerical techniques based on a sharp interface immersed-boundary method.

We use a physiological model similar to anatomy of American eels. As shown in Fig. 1(b), we include its membrane-like caudal fin extended from the anterior regions to its posterior body. Figures 1(c) and 1(d) show that this model is discretized into small elements. We have 33112 triangular elements with 16558 nodes on the body of the American eel.

We define the amplitude envelop of anguilliform kinematics using the following relation (Maertens et al.,2017; Khalid et al., 2020c):

Then, the undulatory motion is modeled by the following mathematical form:

Here, x denotes the stream-wise coordinate of each node used to discretize the eel’s model with A being the maximum oscillation amplitude in the lateral direction, f the oscillation frequency, and t the time. We define nondimensional wavelength as $λ*=λ/L$⁠. Figure 2 presents motion profiles of the mid-line passing through the body when the fish undulates with different wavelengths. Here, solid and dashed lines represent orientations of the body during rightward and leftward strokes of the tail, respectively. It is important to note that body of the real fish is inextensible, whereas the nature of the prescribed wavy motion requires the presently employed physiological model to be extensible. However, a more recent study (Khalid et al., 2020a) on carangiform swimmers has shown that this extensibility does not affect vortex structures significantly. This observation was based on detailed analysis of wake features formed via the prescribed motion and recorded kinematics through high-speed photogrammetry. Furthermore, this method of prescribing motion is an established technique to investigate fish hydrodynamics and the consequent flow characteristics and force measurements resemble those observed through experiments (Zhu et al., 2002; Borazjani and Sotiropoulos, 2008, 2009, 2010; Liu et al., 2017; Han et al., 2020).

We perform three-dimensional (3D) numerical simulations for flows over an undulating eel with anguilliform motion. The nondimensional forms of the continuity and incompressible Navier–Stokes equations constitute the mathematical model for fluid flows,

where ${i,j}={1,2,3}$⁠, x_i shows the Cartesian directions, u_i denotes the Cartesian components of the fluid velocity, p is the pressure, and Re represents the Reynolds number. We define $Re=LU∞/ν$⁠, where ν indicates kinematic viscosity of the fluid, $U∞$ stands for free-stream velocity, and L is the entire bodylength of the eel. For the current numerical formulation, the term f_b shows a discrete forcing term.

We solve this governing mathematical model for fluid flows using a Cartesian grid-based sharp interface immersed boundary method (Mittal et al., 2008), where the spatial terms are discretized using a second-order central difference scheme and a fractional-step method is employed for time marching. Thus, the solutions are second-order accurate in both time and space. We utilize Adams–Bashforth and implicit Crank–Nicolson schemes for numerical approximations of convective and diffusive terms, respectively. The prescribed wavy kinematics is enforced as a boundary condition for the swimmer’s body. We impose such conditions on immersed bodies through a ghost-cell procedure (Mittal et al., 2008) that is suitable for both rigid and membranous body-structures (Liu et al., 2017; Han et al., 2020; Khalid et al., 2020b). More details of this solver and its employment to solve numerous bio-inspired fluid flow problems are available in studies of Liu et al. (2017), Wang et al. (2019), Wang et al. (2020), Han et al. (2020), and Pan and Dong (2020).

Next, Dirichlet boundary conditions are used for flow velocities at all sides except for the boundary located on the left-hand side of the domain, where Neumann conditions are used for the outflow boundary [see Fig. 1(a)]. The slices on the back and left boundaries show regions with high mesh density in order to adequately resolve flow features around the body and its wake. The size of the fine region is $2.0L×0.16L×0.6L$⁠, with the finest core being in the vicinity of the model with dimensions of $1.2L×0.12L×0.3L$ within which the minimum spacing along three Cartesian directions (x, y, z) are $0.0036L, 0.0019L$⁠, and $0.0027L$⁠, respectively. We use a mesh size $(Nx,Ny,Nz)=(385,129,161)$ for the whole flow domain, translating to a total of 7.99 million nodes in the entire flow domain. For the mesh-independence study, the readers are referred to Liu et al. (2017).

The real advantage of using our advanced computational solver lies in its ability to handle large-amplitude oscillations of complex-shaped bodies and their interaction with surrounding fluids. Because numerous parameters cannot be controlled with real animals in laboratory conditions, our numerical solver gives us more freedom to prescribe different kinematic profiles over real fish-like anatomical models and examine their hydrodynamic performance. Such techniques are also very helpful in revealing how dynamics of coherent structures around these bodies alter temporal profiles of hydrodynamic forces (Xiong and Liu, 2019; Bi and Zhu, 2019; Luo et al., 2020; Wang et al., 2020; Khosronejad et al., 2020).

Previous studies (Borazjani and Sotiropoulos, 2009; Liu et al., 2017; Zhong et al., 2019) have revealed that wake dynamics for different marine animals remain inertia-dominated for Reynolds numbers of order 10³ and higher. They have also shown that primary flow features substantially resemble those observed in experiments with real swimmers. Recently, Zhong et al. (2019) have simulated a model fish swimming at $Re=2100$⁠, and the flow features obtained from their simulations have shown strong similarities to their experimental observations using the same model at 20–50 times higher Re values. Although Reynolds number of real anguilliform swimmers is 5500–120 000 (Müller et al., 2001; Tytell, 2004a,b; Tytell and Lauder, 2004; Hultmark et al., 2007), the selected Re of 3000 for a detailed direct numerical simulation of the flow provides sufficient insights into the dominating coherent wake structures without the additional complexities of high Reynolds number turbulent flow. It is apparent that there will be some variations imposed at higher Reynolds numbers, but previous studies suggest that such variations do not substantially alter the dominant wake dynamics. To quantify hydrodynamic performance parameters, we project relevant fluid flow variables around body of the fish to obtain surface pressure and shear stress. These parameters are further integrated to compute hydrodynamic forces and power consumed by the anguilliform swimmer. This expended power is mathematically defined as

where $∮$ is the surface integral operator, $σ¯¯$ shows the stress tensor, n denotes the vector normal to the eel’s body surface, and V shows the fluid velocity vector adjacent to the swimmer’s body. We obtain the nondimensionalized coefficients of axial force, drag(F_D) or thrust (F_T), and power using

where A_s denotes the surface area of the eel’s body.

The next important parameter in this study is Strouhal number. The parameter space in the current study is $St=0.30$ and 0.40 following the experimental observations of anguilliform swimming (Tytell, 2004a,b; Hultmark et al., 2007). Furthermore, in real conditions, steadily swimming fish experiences no net force in the axial direction because its drag and thrust forces are balanced. We present data for thrust coefficient (⁠$CT=−CD$⁠) for a two-dimensional undulating foil having anguilliform gait (Khalid et al., 2020c) as a function of St at $Re=100$⁠, 1000, and 5000 in Fig. 3. Lines at C_T = 0 in these plots present information about St for free-swimming or self-propulsion. As we increase Re from 100 (viscous flow regime) to 1000 (intermediate flow condition), we observe a drop in St for free-swimming conditions. Strouhal number greater than 0.50 for steady swimming of larval zebrafish under viscous flow regimes have been reported previously (Müller et al., 2008). Considering the objectives of our current study, we focus more on details at $Re∼103$⁠. It is apparent that most kinematic conditions with different undulatory wavelengths satisfy self-propulsion conditions for St ranging from 0.30 to 0.40. Besides, it is also important to mention that fish experience small variations in their swimming speeds due to their oscillatory movements. In the work of Borazjani and Sotiropoulos (Borazjani and Sotiropoulos, 2010), it is evident using three-dimensional numerical simulations at various Reynolds numbers that these oscillations in velocity profiles have very small amplitudes in comparison with tether velocities or averaged steady-state speed. Moreover, these oscillations are reduced as the Reynolds number is increased. Most importantly, the oscillatory character of this swimming speed is more prominent for the mackarel compared to the lamprey. This may happen due to the larger aspect ratio of the carangiform swimmer. It seems that anguilliform swimmers do not experience significant oscillations in their swimming speed due to their slender bodies. Hence, only this forward velocity is used for all practical purposes, including computations of relevant Strouhal number and Reynolds number. However, this topic requires future investigations.

Figure 4 presents the temporal variations of axial force coefficients, C_D, and its components from pressure (C_DP) and friction (C_DF) for the undulating eel at different $λ*$⁠. For $St=0.30$⁠, C_DP shows greater values for $λ*=0.50$ compared to other wavelengths. It is interesting to see that C_DF under this kinematic profile is only higher than those for $λ*=0.65$ and 0.80. Another interesting feature of the axial force is its higher root-mean-squared (rms) values for larger $λ*$⁠. Such higher fluctuations can make swimmers uncomfortable during their propulsion. We also notice that the primary change in C_DP for $λ*>0.65$ seems to be only in terms of higher rms values. C_DF, however, decreases when $λ*$ varies from 0.50 to 0.65 and 0.80. Analyzing the axial force production for the eel at $St=0.40$⁠, C_DP shows a similar pattern to the case of $St=0.30$⁠. Moreover, C_DF gradually increases with longer undulatory wavelengths of the anguilliform swimmer.

More meaningful information can be extracted by looking at the time-averaged force coefficients in Fig. 5. The plot of $CDP¯$ vs Strouhal number [see Fig. 5(c)] clearly indicates that an increase in St reduces pressure drag or increases thrust. We also find that swimming with a larger $λ*$ will enable eels to enhance their performance in terms of thrust. It also demonstrates that there is a great improvement in $CDP¯$ when $λ*$ is varied from 0.50 to 0.65. However, further increase in $λ*$ does not affect $CDP¯$⁠. On the other hand, $CDF¯$ rises almost linearly (with the slope $∼1$⁠) by increasing $λ*$ for both Strouhal numbers. Most importantly, Fig. 5(a) illustrates why eels and other anguilliform swimmers exhibit swimming conditions, in which $λ*$ is in the range of 0.60–0.80 (Tytell, 2004a,b). It is evident that the overall hydrodynamic performance in terms of axial force production improves for $λ*=0.65$ and then, 0.80 [see Fig. 5(a)]. Afterward, it starts deteriorating again. The reason behind this phenomenon is that larger wavelengths increase pressure components of the axial force for thrust enhancement, but it is countered by substantially increased frictional drag over the body of anguilliform swimmers. It implies that these swimmers may prefer a compromise between these two physical quantities to improve their overall thrust production.

Now, we proceed with showing temporal variations in power consumed by the undulating eel for different $λ*$ at $St=0.30$ and 0.40 in Fig. 6. These pressure profiles are similar to those for C_D in Fig. 4. For $λ*=0.50$⁠, eel exhibits deteriorated performance [see Fig.4(a)] despite consuming more power. However, further increase in $λ*$ does not seem to demand more power from the swimmer in a time-averaged sense. This pattern also remains similar for C_P at $St=0.40$⁠. Similarly, C_D and C_P also show more fluctuations in temporal profiles of power consumption for greater undulatory wavelengths.

We turn our attention now to the formation of vortices, their topology, and dynamics in the wake of an undulating eel. Borazjani and Sotiropoulos (2009) previously observed the wake of a lamprey consisting of a single-row street behind lampreys when they swam at $St=0.20$ at different viscous (⁠$Re=200$ and 4000) and inertial (⁠$Re=∞$⁠) flow conditions. Here, we also find similar configuration of coherent structures in the wake of an eel swimming with $St=0.30$ and 0.40 at $Re=3000$⁠. Additionally, we discover distinct characteristics of flow dynamics around this anguilliform swimmer, which can be critical in understanding their hydrodynamic performance.

We begin our discussion by presenting vortex topology around an eel swimming with $λ*=0.50$ at $St=0.30$ in Fig. 7 (Multimedia view) taken from video 1. For this purpose, we employ Q-criterion for visualization of three-dimensional vortices (Hunt et al., 1988). This quantity is defined as $Q=0.5(||Ω||2−||S||2)$ with Ω being the anti-symmetric and S as the symmetric part of the velocity gradient. Regions with Q > 0, where the rotation rate dominates the strain rate, represent distinct coherent vortices. Using this tool, we can see different vortical structures around the body and in the wake. It is evident that the anterior part of the body is immersed in shear layers. Also, pairs of coherent structures emerge over first two crests on the left and right sides. Here, we focus on their development at the left side because the same dynamics hold on both sides due to symmetric undulation. We name these two vortex cores as dorsal-body vortex (DBV) and ventral-body vortex (VBV). The subscript L denotes their production on the fish’s left side. The formation of a trailing-edge vortex (TEV) is also visible over the posterior body region. This TEV is formed by the shear layer wrapping around the posterior body. At this time instant (⁠$t/τ=0.042$ in this case), when eel’s tail completes its half rightward stroke, two vortex structures are apparent near the tail ready to be shed in the wake. In fact, the stream-wise directed arms of these two L-shaped coherent structures are formed by $DBVL1$ and $VBVL1$ (not shown in Fig. 7 because of their merger with the TEV to form $VR1$⁠). These two vortex cores form a hairpin-like coherent structure, which we have combined as one flow feature and represent it as $VR1$⁠. The subscript R1 shows that this complete vortex will be shed in the wake by the end of this rightward tail-beat. There are two other hairpin-like vortices with each having two additional extended arms in the stream-wise direction (along x-axis). The vortices $VL0$ and $VR0$ have been shed during previous leftward and rightward tail-beats, respectively. It is interesting to observe that lateral legs of these vortices are formed by TEVs produced by the wrapping of the shear layer on the opposite sides. The orientation of these legs is governed by the angle at which they separate from the tail and the rate of tail’s reversal. Furthermore, stream-wise oriented upper and lower arms of $VR0$ are primarily constituted by $DBVL1$ and $VBVL1$⁠, respectively, as explained earlier. Such hairpin-like flow structures were also observed by Borazjani and Sotiropoulos (2009) in the wake of lampreys.

A similar process prevails on the right side of the body, which leads to the development of $VL1$ in Fig. 7. The curved-shaped heads of these hairpin-like vortices (⁠$VR0$ and $VL1$⁠) are developed as their lateral and stream-wise features traverse in the wake. In Fig. 7 taken from video 1, blue solid arrows inside different vortices directed horizontally, vertically, and laterally show the directional sense of respective components of vorticity (ω), i.e., ω_x, ω_y, and ω_z. Clockwise and anti-clockwise orientation of these components of vorticity can be found accordingly by the right-hand rule. We now turn our attention toward vortex topology and wake dynamics for the anguilliform swimming with larger λ. Figure 8 (Multimedia view) taken from video 2 exhibits flow features around eel undulating with $λ*=0.65$ and $St=0.30$⁠. In this case, its tail arrives in the middle of its rightward stroke at a different time-instant (⁠$t/τ=0.583$⁠), because variations in λ alter both amplitudes and phase of the oscillatory motion of the body. Here, we observe only the presence of $VBVL2$⁠, while the formation of $DBVL2$ on the dorsal side is delayed. Quite interestingly, there are no stream-wise directed extended arms of $VR0$ and $VL1$⁠, which is contrary to the case with $λ*=0.50$⁠. Similarly, $VBV$ and $DBV$ traverse to the posterior part of the body to become parts of the main structure of the TEV, and thus, lateral parts for the hairpin-like vortex structures for $λ*=0.65$⁠. This is attributed to the timing of their arrival at the trailing edge, at which point they interact with TEVs and transform them in lateral parts of the hairpin-like structure, which is different from their dynamics at a lower λ. For this undulation pattern, it is evident that these parts are not much laterally aligned as they were observed in the case of $λ*=0.50$⁠. Moreover, hairpin-like vortices $VR0$ and $VL1$ have elongated stream-wise parts.

By further increasing $λ*$ to 0.80 in Fig. 9 (Multimedia view) taken from video 3, these dorsal and ventral body vortices appear to be weaker despite their growth in size. This is due to the fact that they do not have inner vortex cores as we observe for the cases with smaller λ. When they arrive at the trailing edge of the body, the TEV has already separated from the tail. Vortices DBV and VBV will only be shed in the wake as two distinct but weaker coherent structures. In this scenario, hairpin-like TEVs are shed in the wake with long stream-wise legs. As they travel downstream, their stream-wise elongated parts tend to dissipate earlier than their curved-shaped heads. These curvy structures become more laterally oriented as they go farther in the wake. As $λ*$ is further increased, the only change we observe is the elongation of stream-wise legs of hairpin-like vortices (see supplementary movies 1–3). The higher the wavelength of the undulatory anguilliform kinematics, the quicker is the diffusion of these vortices.

We now focus on discussing the flow physics for eel swimming at $St=0.40$⁠. For the case of $λ*=0.50$⁠, the dorsal and ventral body vortices (DBV and VBV) become the lateral legs of the TEV to form a hairpin-like structure. It is also evident from Fig. 10 (Multimedia view) taken from video 4 that the coherent structures in the wake (⁠$VR1, VL1$⁠, and $VR0$⁠) have longer stream-wise parts (compared to those shown in Fig. 7) combined with extensions in the lateral directions. This dependence of wake dynamics of an anguilliform swimmer on Strouhal number is in complete agreement with what Borazjani and Sotiropoulos (2009) have reported previously. However, their numerical simulations involved flows over lampreys with exponential waveforms proposed by Tytell and Lauder (2004). The agreement of hydrodynamic characteristics justifies the generality of our results for various anguilliform species swimming with the flow and kinematic parametric space considered here. Another significant observation is the thickening of the curve-shaped head of hairpin vortices when the eel swims with a larger λ. We do not show this trend here for brevity, but readers are encouraged to notice this phenomenon in supplementary movies 4–8. Because we do not consider higher Strouhal numbers in this study, we do not observe a double-row wake as reported by Tytell and Lauder (2004), Tytell (2004a), Hultmark et al. (2007), and Borazjani and Sotiropoulos (2009).

The vortex topology and dynamics certainly play a crucial role in determining the hydrodynamic performance of this anguilliform swimmer. In this context, legs of vortices $VR0$ and $VL1$ are oriented more in the lateral direction (an indication of strong lateral flows), which influence how eels experience large drag when swimming at $λ*=0.50$ and $St=0.30$ [see Figs. 4(a1) and 5(a)]. As we increase $λ*$⁠, stream-wise structures shed in the wake grow, which is more likely to enhance the axial force and improve the hydrodynamic performance of eel. This characteristic can be adequately associated with the reduction in pressure drag as reflected in Fig. 5(c).

In order to explore this aspect of anguilliform swimming, we present contours of the stream-wise component of vorticity (ω_x) on different sections over the body in Fig. 11. All these flow features are developed when the tail is in the middle of its rightward stroke at different λs and Strouhal number. It is evident that stronger vortices are formed on the posterior regions of the body for all kinematic and flow conditions. Focusing more on the section at 90% of the body, we observe that negative and positive coherent structures gain strength when eel employs higher undulation wavelength for swimming at both Strouhal numbers. This means that an increasing λ motivates vortices to grow in size and in strength based on vorticity magnitude.

Besides this qualitative visualization, we compute circulation (Γ) of the vortices formed on 90% of the body for one complete oscillation cycle in each case shown in Fig. 11. For this purpose, we employ our methodology previously reported by Khalid et al. (2020c). Circulation is a measure of the strength of a vortex and is mathematically defined as the line integral of the velocity field over its boundary (⁠$Γ=∮V·dl$⁠) or the surface integral of the vorticitiy field over the area of this vortex (⁠$Γ=∫∫Sω·ds$⁠). We devise that technique to avoid an overlap with a vortex boundary with another one in its vicinity (Godoy-Diana et al., 2009). This method allows us to prescribe coordinates of four points around it through visual inspection. Then, our computer program performs searching for all the data points having positive or negative vorticity greater or smaller, respectively, than a threshold value. In this study, we define this threshold as 0.1% of the maximum value of ω_x in a flow field. It enables us to extract a single vortex without intermingling with its surrounding vortices. Next, we draw out a boundary encompassing that vortex without predefining its geometric outline. Finally, we perform numerical integration of the resultant dot product between the velocity components with the respective displacement entities. Hence, we obtain more accurate and reliable information about the strength of the vortex.

In Fig. 12, we plot nondimensional circulation (⁠$Γ/LU∞$⁠) for the positive vortices (red-colored coherent structures in Fig. 11). Because the wavelength affects both phase and amplitude of the undulatory motion of the swimmer, the tail undergoes rightward and leftward strokes at different time-instants for different λ values. These plots exhibit that a higher λ helps the swimmer produce vortices with stronger stream-wise components. This characteristic illustrates the production of more axial force by an anguilliform swimmer using a larger wavelength. This observation also provides an explanation for what Du Clos et al. (2019) has reported very recently about sea lampreys, explaining that they used larger wavelengths for escaping phenomenon during acceleration phase. More importantly, those anguilliform swimmers adopted a lower λ for steadily swimming.

Another interesting characteristic revealed by the plots in Fig. 12 is that TEVs are shed in the wake at a later stage when the swimmer undulates its body with a higher wavelength. We call these vortices in this manner because they are formed near the trailing part of the body. When these vortices attain their maximum circulation, they are shed in the wake and the circulation drops to increase in the next tail-beat. The peak values of Γ in each plot give us information about the timing for detachment of TEVs. We notice that a larger wavelength and St help the TEV grow by delaying its detachment from the body. This quantification of the strength of vortices can also elucidate the formation of longer stream-wise legs of hairpin-like vortices for larger λ and higher Strouhal numbers (see Figs. 7–10). Moreover, this anguilliform swimmer experiences more fluctuations in the unsteady axial force for greater λ. This may be due to the attachment of growing TEVs to the body’s posterior region for longer time periods. Because these fluctuations can make the swimmer uncomfortable (Borazjani and Sotiropoulos, 2009), they only undulate their slender bodies with larger wavelengths during escape maneuvers.

This work addresses questions about hydrodynamic advantages for anguilliform swimmers by using undulatory wavelengths shorter than their bodylengths. While performing numerical simulations of flows over an anatomically accurate geometric model of an American eel, we vary the wavelength of its prescribed fish-like wavy kinematics. Our quantitative analysis reveals that larger wavelengths improve the generation of the pressure component of the axial force. But, it comes at the cost of larger frictional drag over the swimmer’s body. Due to this, the advantage of employing higher wavelength is lost. It implies that natural anguilliform swimmers, such as eels and lampreys, may find it hard to swim with higher undulatory wavelengths. They are observed to employ such techniques only for their escape maneuvers. Moreover, larger wavelengths also cause the production of more fluctuations in axial forces. It can make swimmers uncomfortable by putting more stress on their muscles. These kinematic profiles also make such swimmers expend more power for steady swimming. We also elucidate the formation of hairpin-like vortices for different kinematic and flow conditions. For very low wavelengths, vortices produced over the body form the lateral parts of these coherent structures. However, an increasing wavelength does not allow them to affect the formation of strong trailing-edge vortices and help the swimmer produce elongated stream-wise legs, which can be more assistive in enhancing its hydrodynamic performance. Comparing our results from those presented by Borazjani and Sotiropoulos (2009), the wavelength of the wavy kinematics does not seem to transform wakes with single-vortex rows into ones with double-vortex rows. Based on the circulation of trailing-edge vortices, it appears that larger wavelengths or Strouhal number helps trailing-edge vortices delay their separation from the body. These findings explain various important features about underlying mechanics of anguilliform swimming to tackle many challenges associated with modern fish-like underwater vehicles.

See the supplementary material for animations of the formation of three-dimensional vortices and wake dynamics for American eel undulating with $λ*=0.95$⁠, 1.10, and 1.25 at $St=0.30$ and $λ*=0.65$⁠, 0.80, 0.95, 1.10, and 1.25 at $St=0.40$⁠.

M. S. U. Khalid is International Exchange Postdoctoral Research Fellow sponsored by China National Science Postdoc Foundation and Peking University. H. Dong acknowledges the support from NSF CNS Grant No. CPS-1931929 and SEAS Research Innovation Awards of the University of Virginia.

The data that support the findings of this study are available from the corresponding author upon reasonable request.