Fish larvae exploit edge vortices along their dorsal and ventral fin folds to propel themselves

Abstract

Larvae of bony fish swim in the intermediate Reynolds number (Re) regime, using body- and caudal-fin undulation to propel themselves. They share a median fin fold that transforms into separate median fins as they grow into juveniles. The fin fold was suggested to be an adaption for locomotion in the intermediate Reynolds regime, but its fluid-dynamic role is still enigmatic. Using three-dimensional fluid-dynamic computations, we quantified the swimming trajectory from body-shape changes during cyclic swimming of larval fish. We predicted unsteady vortices around the upper and lower edges of the fin fold, and identified similar vortices around real larvae with particle image velocimetry. We show that thrust contributions on the body peak adjacent to the upper and lower edges of the fin fold where large left–right pressure differences occur in concert with the periodical generation and shedding of edge vortices. The fin fold enhances effective flow separation and drag-based thrust. Along the body, net thrust is generated in multiple zones posterior to the centre of mass. Counterfactual simulations exploring the effect of having a fin fold across a range of Reynolds numbers show that the fin fold helps larvae achieve high swimming speeds, yet requires high power. We conclude that propulsion in larval fish partly relies on unsteady high-intensity vortices along the upper and lower edges of the fin fold, providing a functional explanation for the omnipresence of the fin fold in bony-fish larvae.

1. Introduction

Actively flying and swimming animals propel themselves by accelerating and decelerating their appendages and body. The oscillating and undulating surfaces generate unsteady forces, causing unsteady flows and varying shear stresses and pressures on the body. Under such circumstances, time-dependent steep pressure gradients often occur in concert with separation vortices formed near the moving wings or fins [1–3], especially when they operate at high angles of attack. The generated force on the wing or fin may substantially exceed quasi-steady force estimates, which ignore the flow history [4].

A swimmer undulating in the frontal plane generates a varying three-dimensional vorticity field around its body. Vortical patterns in the frontal plane received most attention [5–7], with few studies on axial vorticity [7,8] and vorticity near the leading edge of the tail fin [9]. Computational studies predict that different vortical flows blend in the wake [8] and may interact to enhance performance in flyers [10] and swimmers [11], yet we lack studies that quantify the propulsive contributions of these flows, such as edge vortices, in undulatory swimmers.

Larvae from oviparous bony fish swim in the intermediate flow regime, where both inertial and viscous forces are important [7,12,13]. Larvae of virtually all bony-fish species possess a median fin fold that extends along the dorsal, caudal and ventral sides of the body. The fin fold transforms into separate median fins as the larvae grow into juveniles (with a few exceptions, such as eels). The functional significance of the omnipresent fin fold has been attributed to its role in respiration and propulsion [14]. In this study, we aim to quantify the still enigmatic propulsive role of the fin fold.

Vortices are generated when foils impulsively start moving at a high angle of attack or rapidly increase angle of attack, and may persist when the foil experiences significant rotations [15]. We hypothesize that fish larvae can generate vortices along the dorsal and ventral edge of their fin fold with high-pressure differences across the fin fold. Computational studies found indications for edge vortices for adult anguilliform [8] and larval swimmers [7], but their propulsive role was not addressed.

To advance the understanding of swimming in larval fish, the varying distribution of viscous stresses and pressures on the undulating larval body needs to be quantified and converted into drag and thrust contributions over the body. Yet, these stresses cannot be measured. Fortunately, in the intermediate flow regime, unsteady computational fluid dynamics (CFD) can provide an accurate representation of the fluid dynamics around larval fish [7].

Using high-resolution CFD in combination with an experimental test of the simulated flow, we explore to what extent larval fish exploit edge vortices and use unsteady drag-based mechanisms for propulsion. First, we show that larval fish generate strong edge vortices along the upper and lower edge of their fin fold, which are formed in tune with the travelling body wave and shed at the local maxima of this wave as the lateral body motion reverses direction. Second, we demonstrate that the presence of vortices relates to substantial unsteady pressure differences across the fin fold, providing significant drag-based thrust, implying an important propulsive role of the fin fold. Third, we demonstrate that these pressure differences across the fin fold contribute less to total thrust as the flow regime becomes less viscous. Finally, we compare the swimming performance of a morph with fin fold with two counterfactual morphs with reduced fin fold areas across a range of Reynolds numbers, and demonstrate that the presence of the fin fold provides higher swimming speed at the expense of an elevated power input in the intermediate flow regime.

2. Material and methods

We developed an in-house three-dimensional numerical approach programmed in Fortran 90 to simulate cyclic swimming of larval fish [7,16]. The model fish swims in the horizontal plane (3 degrees of freedom): centre-of-mass (CoM) movements and body orientation are determined by the hydrodynamic forces on the body, obtained by coupling the hydrodynamic and body-dynamic solutions (electronic supplementary material, §§B1 and 2). Thus, the hydrodynamic force and moment on the body determine the fish's motion. The approach comprises a surface model of the changing fish shape, and a local fine-scale body-fitted grid plus a stationary global grid to calculate the flow patterns around the fish with sufficient resolution (electronic supplementary material, figure E2b,c). The body was modelled on the silhouette of a larval zebrafish (Danio rerio, Hamilton, 1822), made dorsoventrally symmetric with a ventral and dorsal fin fold of constant height (figure 1a, intermediate-flow-regime morph). The instantaneous body shape (figure 1b) is described by

2.1

where ℓ is the dimensionless distance from the snout along the longitudinal axis of the fish based on the length of the fish model L; H(ℓ, t) is the lateral excursion at dimensionless time t; A(ℓ) = 0.272 ℓ² is the dimensionless amplitude envelope function at ℓ; λ is the length of the body wave; ω is the angular frequency defined as ω = 2πfL/U_ref, where f is the absolute tail beat frequency. Equation (2.1) may cause ‘total body length along the midline’ to vary over the tail beat; this variation is corrected by a function that keeps body length constant at 1 L. Based on measurements of swimming zebrafish larvae [7,17], we set

2.2

Figure 1.

Fish model body shape and deformation. (a) Simplified dorsoventrally and bilaterally symmetric surface model with ventral and dorsal fin-fold edges of constant height (top), based on larval zebrafish (mid and bottom). (b) The lateral bending of the longitudinal axis of the fish model was prescribed by a sinusoidal deformation function. Black curves: amplitudes along the longitudinal axis of the fish at different instances in a fish frame of reference (indicated by grey arrows; origin at snout). Note that the black curves do not show the actual shape of the longitudinal axis of the fish. Red curves: amplitude envelope. The function prescribes only the deformation of the model fish, while the changes in CoM and body angle follow from computed external fluid forces on the body (electronic supplementary material, §B). (c) Two counterfactual models. Inertial-flow-regime morph: reduced fin fold with only a tail fin. Viscous-flow-regime morph: no fin fold, except for a thin slice ensuring a constant body length. (Online version in colour.)

The Reynolds number is defined as

2.3

with density of fluid ρ, viscosity of fluid μ, reference speed U_ref and reference length L_ref. In the experimental observation [17], ρ = ρ_w = 10³ kg m⁻³ (ρ_w: density of water), L_ref = L = 3.8 × 10⁻³ m, U_ref = U_exp = 18 L s⁻¹ and μ = μ_w = 8.3 × 10⁻⁴ kg m⁻¹ s⁻¹ (μ_w: viscosity of water). Our simulations yield swimming speed as an output, so we do not know swimming speed a priori. Hence, we use U_exp = 18 L s⁻¹ as the initial reference speed for all simulations, then recalculate Re once we know resultant speeds.

Our simulations were validated against experimental kinematic and flow data [17], confirming that our simulations yield (2.1) realistic swimming speeds for a given body wave kinematics, morphology and Reynolds number [17], and (2.2) realistic flow fields (see electronic supplementary material for details). To validate the vortex structures predicted in our simulations, we visualized the flow around the fin fold using particle image velocimetry (PIV; approach: [17]). To validate our predicted pressure- and shear-stress distributions as well as the predicted vortex structures, we compared our computed hydrodynamic forces and flow field against measurements [18] of the cross section of an oscillating cylinder at a similar Re (electronic supplementary material, §B3).

To explore the propulsive function of the fin fold, we quantify the pressure differences across the fin fold. To determine how the fin fold affects propulsive performance, we calculated instantaneous and tail-beat-cycle-averaged vertical and horizontal force distributions. To explore the effect of Re and whether the fin fold is a morphological adaptation for the intermediate flow regime, we created a range of Re by altering viscosity (to 10, 3.2, 1, 0.32 and 0.1 times μ_w) while keeping all other input parameters constant (virtual viscosity test; [19,20]). By changing viscosity (rather than body length) and by using real units, we can compare results directly without the need to convert. To characterize physical performance, we follow [21] and use a dimensionless parameter system based on individual resultant speed(s) indicated with a superscript ‘asterisk’.

Owing to the nonlinear nature of fluid phenomena, we cannot isolate the propulsive function of the fin fold. However, we can determine how swimming performance is affected by change in the fin fold area by using two counterfactual morphologies (figure 1c) typical for undulatory swimmers in the viscous and inertial flow regime. We compare the performance of the intermediate-flow-regime morph (figure 1a) with two other counterfactual body morphs (figure 1c): the ‘inertial-flow-regime morph’ has only a tail fin; the ‘viscous-flow-regime’ morph retains a narrow strip of fin fold at the tail to keep body length constant across morphs.

3. Results

We computed the flow fields around swimming fish larvae for a range of Reynolds numbers and body shapes to explore the larvae's potential for generating edge vortices along their fin fold. We verified that our simulations accurately model the swimming performance of the actual fish larva at age 5 days post-fertilization [17]. The non-dimensional average swimming speed of the simulated larvae is 1.081 U_exp (Re = 340).

3.1. Edge vortices form along the undulating fin fold

Our simulations show that attached vortices are periodically formed and shed along the dorsal and ventral edges of the fin fold (figure 2a). This vorticity is of similar magnitude to the vorticity generated close to the body in the mid-frontal plane, but considerably stronger than the vorticity shed in the wake (figure 2b). Our simulated streamlines show that the flow separates at the ventral and dorsal edge of the fin fold and forms a swirl (figure 3d) with a longitudinal flow component along the fin edge (electronic supplementary material, §C1, figure E10), consistent with an attached vortex.

Figure 2.

Edge–vortices topology. (a) Vertical cross sections through the edge vortices at four instances during one cycle. Numbers indicate vortex pairs. (b) Vorticity in mid-horizontal plane. Values above the lower and upper extremes of the scale bar were clipped. (c) Isosurface of Q = 10, coloured by the direction of circulation. (d) Isosurface of Q = 0.1, at this Q value, three vortex rings in wake are visible.

Figure 3.

(a) Surface pressure stress on the left side of the fish at four instances, in relation to major edge vortex structures. (b) Surface pressure stress on the left side of the fish at the four instances of panel (a). Surface stress on the right side at each instant t equals the surface stress on the left side at t plus a half-cycle interval. The grey-dashed lines highlight the motion of main pressure zones following the body wave. (c) Surface shear stress on the left side of the fish at the four instances of panel (a). Surface stress on the right side at each instant t equals the surface stress on the left side at t plus a half-cycle interval. (d) In-plane streamline and pressure field on a typical cross section (position marked in figure 2a and figure 3a), and the vertices occupying the fin edges coexist with the previously shed vortices. Green dots: seeds for streamlines. The low-pressure zones on the edges of the fin fold are related to the flow separation and reattachment. The actual three-dimensional streamlines are shown in the electronic supplementary material, §C1, figure E10.

To demonstrate that the flow forms true vortices along the fish, we visualize the vorticity topology based on the iso-Q-criterion (figure 2c,d). The Q-criterion is defined as

3.1

where Ω and Φ denote the asymmetric and symmetric parts of the dimensionless velocity gradient, respectively, ‖‖ is the matrix norm [22]. Positive Q isosurfaces isolate areas where rotation dominates strain [23]. These surfaces form an envelope around vortices but not dipoles [24], whose vorticity fields might be mistaken for vortices. The Q isosurfaces reveal turgid edge vortices along the fin fold (at Q = 10) and in the wake (Q = 0.1; [16]). The vortices identified by the Q-criterion agree with the vortices shown by the x-axis vorticity value.

The size, strength and rotational sense of the vortices varies along the body and in tune with the body wave (figure 2a): in a fish frame of reference, the vortices start at the lateral extrema of the body wave as a particular body segment begins to move from one side to the other (figure 2, vortices to ); the vortices are shed when the segment reaches the opposite lateral extreme of the body wave (figure 2, vortices to ) or when they reach the tail tip (figure 2, vortices , , ); vortices of the same sense travel backward along with the body wave until they are shed. As old edge vortices are shed, these vortices interact with the newly forming vortices in a complex manner (figure 3d). The overall vortex pattern along the body comprises a series of edge vortices: given that fish larvae typically swim with body wavelengths between 0.8 and 1.2 L [12], there are up to three distinct edge vortices present with alternating sense along each fin-fold edge. Elevated vorticity around the head (figure 2, vortex at 0/4 T) merges into the edge vortices. At the tail, the shed vorticity forms vortex rings in the wake (figure 2).

Our simulations are consistent with flow visualizations in a transverse plane of a zebrafish larva, supporting the existence of edge vortices along the fin fold (figure 4). In the PIV recording, the larva swam approximately vertically through a horizontal light sheet (slight turn to left, slight roll). The resulting flow field contains a stable vortex pair (figure 4 left) in a position and orientation similar to a section through the simulated flow field (figure 4 bottom right) of a fish in a similar position and orientation as in the PIV recording (figure 4 top right). Both flow fields show two edge vortices (formed by vectors) and a jet flow between them.

Figure 4.

Comparison of experimental and computational flow fields showing edge vortices. (left) PIV data of a larval fish that swam in an approximately vertical direction through a horizontal light sheet, in combination with a slight change of swimming direction towards the left side of the fish. From frame 19 to 22, a stable vortex pair was recorded in the laser sheet. (right) Comparison of the flow pattern between a PIV observation and a computational result in a corresponding cross section of the flow field, at a similar position and orientation. Both flow fields showed the dorsal and ventral edge vortices and jet flow between them.

3.2. Larvae generate thrust along the undulating fin fold

Areas of high vorticity along the fin fold correspond to areas of strong pressure differentials (figure 3a,d). The transverse view through the flow shows highly negative pressure areas at the fin fold edges on the ‘leeward’ side of the body (side facing away from the flow caused by the lateral movement of the body segment) and areas of high positive pressure along the body near the midline on the ‘windward’ side of the body (side facing into the flow; figure 3d). Overall, the instantaneous distribution of pressure across the body shows a ribbon of positive (negative) pressure along the fin fold on the side facing into (away from) the flow caused by the undulating movement of the body (figure 3b). Such a pressure distribution is consistent with the edge vortices producing thrust.

To assess net thrust generation, we need to consider also the instantaneous shear stresses distributed across the body (figure 3c). Shear stresses oscillate with the body wave cycle and are mainly produced at the head and along the posterior body. At the head, shear is high, because the developing boundary layer causes high velocity gradients. At the tail, shear is even higher, implying that the boundary layer around a swimmer with a large-amplitude tail beat is different from a Blasius boundary layer. Shear is largely contributing drag except in a small region on the upper, lower and posterior edges of the fin fold.

To tally pressure, shear and net thrust over an entire tail beat cycle, we calculate time-averaged distributions along the body. Averaged over a tail beat cycle t_p, the forward component of pressure stress (), shear stress () and net thrust () are

3.2

where n_pres, n_shear and X, respectively, denote the unit direction vectors of pressure, shear and forward direction.

At the Re of an experimentally observed larval fish (at Re = 340), is positive and peaks at the fin fold edges at all four calculated cross sections (figure 5a, green lines). The distribution of cycle-averaged positive pressure (thrust) along the body shows that the larva generates pressure-based thrust along the fin-fold edge, the body and the tail, whereas the head generates pressure drag (figure 5c, top). When pressure is tallied against shear, the cycle-averaged net thrust distribution shows a similar result, with the fin-fold edge contributing net thrust and the head contributing net drag (figure 5c, bottom). The main body and the tail have regions that generate low net thrust or even net drag. Overall, the fin fold is the most consistent generator of net thrust (figure 5c, bottom), with high thrust values along its dorsal and ventral edge (figure 5c, top).

Figure 5.

(a) Cycle-averaged forward component of pressure distribution () on four vertical cross sections (positive = thrust). The vertical axis shows normalized height along the cross section. At the lower Re values, 14 and 70, two strong thrust peaks form near the dorsal and ventral edges, corresponding to the edge–vortex locations. (b) The distributions on the body of the cycle-averaged forward component of pressure () and total forward stress component () for Re = 14. Less thrust was generated at the fin fold at (b) Re = 340 and (c) Re = 5500 compared with (a).

3.3. Flow regime affects thrust contribution by edge vortices

Having shown that the fin fold generates strong edge vortices and consistently contributes net thrust at Re = 340, we next explore how the fin fold performs across Re. To this end, we computed flow fields and force distributions for two lower and two higher Re (14, 70 < 340 < 1450, 5500). The lower and upper extremes are beyond the actual values of larval fish. This Re range allows us to detect trends without moving beyond the validity of our CFD simulations. Parameters in real values and non-dimensionalized based on individual resultant speeds are presented in table 1. Our simulations agree with experimental findings [12]—instantaneous swimming speed decreases and fluctuates more strongly as Re decreases (table 1 and electronic supplementary material §C1, figure E9).

Table 1.

Computational swimming parameters at different viscosity and Re. U, resultant speed; fluctuation, amplitude of the speed curve; slip ratio, ratio of U to body wave speed; A, amplitude of tail tip; St = Af/U, Strouhal number; T, thrust; P, total power; η = TU/P, Froude efficiency; C = P/U, cost of transport. Dimensionless parameters based on resultant speed have superscript ‘asterisk’: T* = T/ρU²L²; P* = P/ρU³L². P_m and C_m are respectively power and transport cost per unit muscle mass, based on 40% relative weight of muscle in total mass [25].

viscosity	10×	3.2×	1×	0.32×	0.1×
Re	14	70	340	1450	5500
U (L s−1)	8.0	12.4	19.5	26.4	31.3
(fluctuation (L s−1))	5.8	3.2	2.8	2.4	2.1
slip ratio	0.13	0.21	0.32	0.44	0.52
A (L)	0.34	0.33	0.35	0.36	0.36
St	2.56	1.60	1.07	0.81	0.69
T (µN)	9.3	4.9	3.2	2.2	1.6
T*	0.693	0.154	0.040	0.015	0.008
P (µW)	2.77	1.24	0.74	0.49	0.36
P*	6.723	0.820	0.127	0.034	0.015
η	0.10	0.19	0.32	0.45	0.55
C (µJ m−1)	90.6	26.4	10.0	4.9	3.0
Pm (W kg−1)	29.1	13.1	7.8	5.1	3.7
Cm (µJ m−1 kg−1)	952	277	105	51	31

To explore how flow regime affects the role of the fin fold in thrust production, we calculated the distribution of pressure stress, shear stress and net thrust. To this end, we calculated the tail-beat-cycle-averaged forward component of pressure stress () along four vertical sections through the fish larva; in this context, positive approximately equals thrust per unit area (figure 5a). Reynolds number affects both the magnitude and the distribution of thrust along these cross sections. Across Re, the highest thrust occurs at the fin fold near its ventral and dorsal edge. At Re lower (higher) than those of the actual fish larva, the pressure difference across the fin fold contributes more (less) to the cycle-averaged thrust (figure 5a)—which directly relates to the intensity of the edge vortices. Hence, the relative thrust contribution by the pressure difference across the fin surface at the four transverse sections increases as the flow regime becomes more viscous. While these sections provide clear snapshots of the vertical distribution, they do not provide a complete picture of the total thrust and drag generated along the entire fish.

To explore how Re affects net thrust generation along the body, we compare the cycle-averaged pressure and net thrust distribution for Re = 340 (Re of actual larva), 14 (low Re) and 5500 (high Re; figure 5b–d). Both counterfactual cases differ mainly quantitatively from the factual case: in all cases, tail and fin fold contribute pressure-based thrust, whereas the snout generates pressure drag. The head generates net drag, whereas the body and tail generate mostly net thrust; net thrust occurs at 0.3–0.6 L and along the tail, except at its trailing edge, where high shear stresses outweigh a relatively low pressure differential. While at Re = 5500, the contributions of the trunk and fin fold are about equal, the fish larva at Re = 14 generates its net thrust primarily along the fin fold, suggesting that the fin fold becomes more important for thrust generation as the Re decreases.

Given the merely quantitative differences in spatial distribution among the cases at different Re, we reverted to generating one-dimensional distributions (distribution along the body axis) for all five Re. To this end, we calculated the tail-beat-period averaged force per unit length owing to (i) the pressure differences across the body and (ii) shear stresses (colour-coded curves in panels a and b), as well as the sum of these components (figure 6) according to

3.3

where means the integral along the closed curve around the cross section at position l (0 < ℓ < 1L) along the body.

Figure 6.

Computed force distribution along fish body. (a) Distribution of cycle-averaged forward component of pressure (δ_pres). Negative values at the snout indicate pressure drag, whereas the positive values from CoM to tail tip indicate thrust. (b) Distribution of cycle-averaged forward component of shear stress (δ_shear). The continuously negative values from snout to tail tip suggest that the shear stresses resulted merely in viscous drag. Viscous drag increased sharply in the tail region. (c) Distribution of cycle-averaged forward component of total stress (δ_total). Anterior to 0.35 L and just in front of the tail tip, the body generated net drag. The summed force contribution between CoM and 0.8 L was near zero, whereas thrust mainly originates between 0.8 L and 0.95 L.

Concerning the distribution of thrust and drag along the body, these tail-beat-period averages show that the head region generates pressure drag, whereas pressure-related thrust is generated near and posterior to the CoM. Shear friction occurs along the entire body, indicating that shear forces contribute largely drag. Overall, the body anterior to the CoM generates net drag, the body posterior to the CoM generates net thrust. The main source of the drag is friction (shear) rather than pressure drag. Absolute dimensionless thrust and drag increase with decreasing Re, consistent with a more viscous flow regime—the fish balances the increase in mean drag by increasing mean thrust.

Our simulations reveal both new and familiar trends. A surprising finding was that pressure and net thrust distribution have a dip around 0.7 L. Posterior to the CoM, net thrust is generated along 0.35–0.6 L and at the tail (0.8–0.95 L); between these two net thrust zones, low net thrust or even net drag occurs between 0.6 and 0.8 L (figure 6c). This dip is most likely due to recoil caused by fluctuating torques on the body [13]: although the body wave amplitude envelope increases at a steady rate in the fish frame of reference (figure 1b), in the earth frame of reference recoil causes the rate at which the envelope widens to be much lower around 0.6–0.8 L [12], reducing local body speed, peak instantaneous thrust and hence cycle-average thrust between 0.6 and 0.8 L.

Well-known trends that are also present in our simulations are: with decreasing Re, hydrodynamic efficiency decreases as total dimensionless drag increases (table 1). Hence, total dimensionless thrust increases to match total drag, consistent with a constant cycle-averaged swimming speed. At lower Re, edge-vortices-related pressure differences across the fin fold are the main source of thrust, not those across the trunk, compensating for the increasing drag in a more viscous flow regime. This increased dimensionless thrust production to overcome higher dimensionless drag requires higher dimensionless power and causes a higher cost of transport.

3.4. Larval fin fold generates superior performance in the intermediate flow regime

In §3.3 (figure 5), we saw that net thrust generation is distributed unevenly across the body and that the net thrust generated near the fin-fold edges increases monotonically as Re decreases. However, owing to the nonlinear nature of fluid phenomena, we cannot isolate the force provided by the fin fold. So in order to explore why the fin fold is prevalent in the intermediate but not the viscous or inertial flow regime, we compare the swimming performance of the factual intermediate-flow-regime morph (fin fold, FM) with two counterfactual morphs (an inertial-flow-regime morph (IM) and a viscous-flow-regime morph (VM)) across a range of Re just beyond the actual values of larval fish (figure 7). We create a range of Re by varying viscosity while keeping the input motion of the longitudinal axis deformation constant. This approach should enable us to detect the effects of morphology, which might be otherwise overwhelmed by body kinematics effects—previous studies suggest that body kinematics has a stronger effect on performance than morphology, in particular when maximizing swimming speed [26]. The effect of Re on swimming performance of a particular morph can be assessed by following the coloured lines; the effect of morph at a fixed water viscosity can be assessed by comparing along the black lines; Re is an output parameter of the simulations and therefore, Re differs between morphs, because the three morphs have different swimming speeds (at the same body length, body-wave kinematics and viscosity).

Figure 7.

Comparison of swimming performance between the factual intermediate-regime morph (FM, green) and two counterfactual morphs (inertial-regime: IM, blue; viscous-regime: VM, orange). Each dot in the logarithmic plots (a(i)), (b(i)), (c(i)), (d(i)) represents the simulation of a swimming event. The deformations of the longitudinal axis of the fish models are the same for all events, but five different viscosities were applied per morph. Black curves in (a(i)), (b(i)), (c(i)), (d(i)) connect points with equal viscosity but changed morph. (a(i)) Resultant swimming speed U against resultant Reynolds number Re. (a(ii)) Swimming speed as percentage of that of FM against the applied viscosities. (b(i)) Power per unit muscle mass P_m against Re. (b(ii)) Power per unit muscle mass P_m of the different morphs as a percentage of that of FM against the applied viscosities. (c(i)) Cost of transport per unit muscle mass C_m = P_m/U against resultant Re. (c(ii)) C_m of the different morphs as a percentage of that of FM against the viscosities used in the simulations. (d(i)) Froude efficiency η = TU/P against Re. (d(ii)) Froude efficiency as a percentage of that of FM against the applied viscosities. Note that comparing morphs at a given Re implies a comparison at different water viscosities—the morphs have the same length but generate different swimming speeds, so viscosity must be different to keep Re the same across morphs.

To assess the effect of the fin fold on swimming performance, we evaluated swimming speed and swimming power per unit mass (defined as the sum of fluid-dynamic power and body kinematic power per unit mass). At the five examined viscosity values used to traverse the considered Re range, the ‘intermediate-flow-regime’ morph swims faster than the other two morphs (figure 7a(i)), yet both counterfactual morphs catch up with increasing Re (decreasing viscosity) and the inertial morph reaches 87% of the intermediate morph's speed (defined as 100%; figure 7a(ii)) at the lowest viscosity. Owing to being the fastest swimmer and having the largest surface area, the intermediate morph also requires the most power (figure 7b(i)), with the inertial morph closest in power requirement (around 80%; figure 7b(ii)) to the intermediate morph (defined as 100%). Given the identical deformation of the body axis between morphs, the intermediate morph exerts higher forces at a particular viscosity owing to its larger surface area, resulting in higher power expenditure, higher swimming speeds and a higher resulting Re. The computed powers are below 30 W kg⁻¹, within the physiological range for fish muscle [27]. Hence, we predict from our model that power production is not limiting for the different morphs. The presence of the fin fold enables the intermediate morph to operate at a higher power level and swim faster than the other morphs.

Furthermore, we evaluated two performance measures—cost of transport (energy spent to travel unit distance per unit mass) and Froude efficiency (the ratio of the produced thrust times swimming speed and power). The cost of transport (figure 7c(i),(ii)) can be computed directly from the swimming speed (figure 7a) and the power per unit mass (figure 7b). The three morphs differ relatively little in their cost of transport at the five selected viscosities. However, this small difference is caused by a confluence of differences. The intermediate morph has the highest swimming speed with the highest power requirement, resulting overall in a cost-of-transport value that is similar to that of the slow, low-power, viscous morph because it covers the same distance in a shorter time. The corresponding values of the inertial morph are intermediate. Cost of transport decreases with increasing Re (decreasing viscosity) for all three morphs. All three morphs have similar costs of transport near the Re (and viscosity of real water) typical for larval fish. Yet at the lowest Re (highest viscosity), the viscous morph becomes most economical and the inertial morph less economical than the intermediate morph. At high Re (low viscosity), outside the range of larval swimmers, the intermediate morph becomes the least economical. Interestingly, at high Re, the viscous morph has low transport costs owing to its low swimming speed.

For all three morphs, Froude efficiency increases with a lower viscosity and a higher Re (figure 7d(i)). The intermediate morph is superior to the counterfactual morphs at Re typical for larval fish. However, the inertial morph is catching up to the intermediate morph as viscosity drops and Re increases and at the highest counterfactual Re, its performance is close to the intermediate morph, with the viscous morph falling well behind.

Cost of transport may be a better performance measure than Froude efficiency, our findings are consistent with previous findings in eel [28], showing that low speed and low power lead to low cost of transport, but not high Froude efficiency. This finding is consistent with previous studies indicating that Froude efficiency might not be the best indicator for evaluating swimming performance [7].

To sum up, at intermediate Re, the intermediate morph has the widest functional scope, assuming that its axial muscles can deliver the required power. Our simulations support this viewpoint as they predict power output to be within the physiological range of fish muscle. The intermediate morph is able to swim fast by pushing off against a large mass of water and using the edge vortex to generate high thrust to overcome high drag. This high speed comes at the cost of a high power, but not at a higher cost of transport. The ability of the fin-fold morph to swim fast provides a survival advantage as it enhances the ability to escape from predators and to capture fast prey. As Re increases, the edge vortex effect decreases. As Re decreases, the edge vortex effect increases, yet the concomitant increase in drag requires too much power and the fin fold may lose its advantage. The inertial morph performs poorly at low Re, but provides a good balance of speed, power and transport cost at high Re. Across the considered Re range, the viscous morph is consistently the slowest swimmer with the lowest Froude efficiency, but it consistently requires the least power.

As Re decreases, power output rises sharply (table 1) and may rise beyond the capability of animal swimmers, making the viscous morph superior to the intermediate and inertial morphs in the viscous flow regime—at low Re low swimming speed is the price for low power requirements and low cost of transport. As Re increases, the inertial morph catches up to the intermediate morph in swimming speed and Froude efficiency, and surpasses it in having lower costs of transport.

4. Discussion

4.1. An unsteady mechanism at the undulating fin fold enhances thrust

Theoretical approaches based on quasi-steady estimation on thrust and drag were developed either for the inertial or viscous flow regime (e.g. inertial: [29,30]; viscous: [31]). Neither approach effectively captures the intermediate flow regime nor provides detailed distribution of stress. The limitations of these analytical approaches in the intermediate flow regime have been largely overcome by CFD approaches (e.g. [7,21], particularly when turbulence is absent, as is the case for larval fish). Our CFD model shows that reliable pressure and shear stress distributions on the body cannot be obtained in the intermediate flow regime with classical analytical models owing to the nonlinear nature of the fluid-dynamic problem and the inability to provide surface stress distributions.

Our simulations show that fish larvae generate edge vortices in combination with low-pressure zones on the fin fold that enhance thrust. While these vortices are attached, high fluid-dynamic forces operate along the fin, exceeding steady values at intermediate Re (i.e. at equivalent swimming speeds). Body movements and flow interact, resulting in a complex relationship between body wave and thrust distribution along the body. However, owing to the nonlinear nature of such flow phenomena creating this thrust enhancement and owing to this effect taking place in the intermediate flow regime, this thrust cannot be decomposed into trunk versus fin fold contributions or into steady versus unsteady contribution. Nevertheless, what our CFD model is able to show is that edge vortices correlate with strong pressure differentials resulting in net thrust.

The thrust-enhancement by the fin fold found in our study is confined to the intermediate flow regime. A previous study of the edge vortices along an undulating foil in the inertial flow regime (Re 5000–18 000) found no such thrust enhancement [32]. Instead, a major part of the total drag came from the edge vortex, and swimming performance improved with increasing height of the foil, moving it away from the low-bodied anguilliform shape of a larval fish. This detrimental effect does not dominate at intermediate Re, where edge vortices increase net thrust. Hence, this unsteady mechanism differs from those described previously for undulatory swimming (for reviews: [33,34]), and is not caused just by an enlargement of the area of the undulating body.

By exploring the performance of the fin fold, plus an inertial- and a viscous-flow-regime morph across a range of Re, we were able to show that the fin fold is most effective in the intermediate-flow-regime; eliminating the fin fold (viscous-flow-regime morph) sharply reduces forward speed and efficiency (figure 7). The high efficiency of the intermediate morph might be explained by the relatively large body of water that it interacts with for propulsion. At higher Re, the edge vortex effect becomes weak, and a fin fold becomes unnecessary; at lower Re, the edge vortex is stronger but power-consuming, forcing biological swimmers to reduce the fin fold and accept low swimming speeds to avoid excessive power requirements during cyclic swimming.

4.2. A comparison with unsteady edge vortices on oscillating appendages

Undulating fin folds exploit powerful edge vortices to enhance their thrust at intermediate Re. These edge vortices not only share interesting similarities, but also differences with the edge vortices generated by oscillating structures, such as insect wings [35]. These differences are due to differences in mechanism as well as in motion.

Our simulations show that fish larvae can generate powerful edge vortices during undulatory swimming that enhance thrust using an unsteady mechanism that is similar to delayed stall [35] on oscillating appendages (wings, paired fins and tail fin). Delayed stall occurs when a sharp edge impulsively starts moving at a high angle of attack, causing the flow to separate and form a vortex. While this vortex remains attached, it generates high fluid-dynamic forces, more than twice the steady values at the intermediate Re typical for swimming fish larvae and flying insects [36]. This attached vortex increases both the shear and pressure components, which are both strongly time-dependent, but only the thrust component is strongly Re-dependent [37]. The motion of a given fin-fold segment shares characteristics with this simple case of delayed stall—the fin fold has a high angle of attack, and body undulation causes lateral movements.

Our simulations show that the fin fold most effectively enhances force at intermediate Re (prominent thrust peaks in figure 5a). In addition, delayed stall is most effective at intermediate Re—transient forces peak between Re 10¹ and 10⁴ [36,38]. The transient nature of delayed stall implies that it works best when the timescale of the motion matches the timescale of the unsteady effect. In delayed stall, peak force occurs at approximately one chord length of travel [38]; the time-to-peak-force increases with Re in the intermediate flow regime [37] and appears to plateau at high Re [36].

Despite similarities in time course and Re effects, the unsteady phenomenon on the undulatory fin fold differs in important ways from delayed stall. ‘Delayed stall’ is a concept of lift enhancement that treats drag as an undesired ‘side-effect’; by contrast, in the unsteady mechanism of the undulatory fin fold, the enhanced drag on cross sections is the desired main effect.

Undulation and oscillation of a propulsive structure differ in their kinematics, which has consequences for the generated flow. In a purely oscillating structure, all its cross sections move synchronously. In an undulating structure, the motions of cross sections are phase shifted along the structure. Hence, the undulatory fin fold forms more complicated vortex structures than those found near oscillating appendices: multiple pairs of vortices form in a staggered pattern and they move relative to the body owing to the body wave.

Another difference in motion is that undulating fins do not have a clear leading and trailing edge. Instead, each vertical cross section of the fish body is oriented at a constant 90° to the horizontal plane, so the dorsal and ventral edge of the fin fold are equivalent. In contrast, an oscillating appendix not only has a clear leading and trailing edge, but also its angle with the horizontal plane is time-dependent. A constant angle of incidence in the horizontal plane, however, does not mean that body undulations are equivalent to a simple heave. Instead, body undulations also cause rotation as the body-wave amplitude waxes and wanes.

The edge vortices of an undulatory fin fold are to some extent similar to the tip vortex of a flapping plate. Shyy et al. [10] reported that tip vortices may conditionally enhance lift of a low aspect-ratio flapping plate. The computed lift distribution was concentrated near the tip edge (figure 5 in [10]), similar to the force distribution along the edge of the fin fold. However, for the simulated plate, the tip vortices interact with leading and trailing edge vortices, which does not occur in a similar fashion along the waving larval body.

4.3. Fin folds are nearly universal among fish larvae

Many aspects of larval morphology and kinematics come together to exploit edge vortices for propulsion in the intermediate flow regime. First, fish larvae are small enough for edge vortices to remain attached on cross sections over a half-cycle. This stability of attachment is primarily owing to the flow regime. At these Re, plates and cylinders in a stable cross flow generate stable attached vortices [39], but such flow separation is effective at generating a considerable low-pressure zone only if the object's cross section has a high height–width ratio and a sharp edge (see electronic supplementary material, §C2). The fin fold must be tall enough to accommodate a sufficiently large low-pressure zone to generate net thrust effectively.

Second, fish larvae's wide body wave amplitude is essential for exploiting edge vortices for propulsion. The total thrust of the swimming larva is largely the result of the pressure drag at each cross section. The thrust per unit length at each cross section is equal to the cosine of the angle between the local drag vector and the direction of body motion. A large-amplitude body wave increases this cosine and hence the conversion of drag into thrust.

Third, the fin fold's variable height in actual fish larvae might be functional. Our model with a constant fin-fold height shows that net thrust along the body is positive at approximately 0.35–0.6 and 0.8–0.95 L (figure 6). Intriguingly, actual larval fish have a relatively high fin fold in these two propulsive zones, whereas fin-fold height is reduced between them (figure 1a). Fin fold height along the body in real larvae might be optimized to enhance swimming performance.

The fin fold enhances the fluid-dynamic efficacy of locomotion (high thrust per unit length). Yet, it requires little material investment because it is a thin, fibre-reinforced structure [14]. For the survival of the larval zebrafish, this seems to be particularly beneficial given its limited resources in the yolk that must be redistributed to develop and grow the rest of the body (including the muscles that power locomotion). In zebrafish, yolk is the only source of nutrition until 5 dpf. The prominent propulsive role of the fin fold in combination with the low material investment provides a functional explanation for the omnipresence of the fin fold in bony-fish larvae.

Authors' contributions

G.L. made all the simulations. J.L.v.L. and H.L. provided critical advice on the design of simulations. H.L. programmed the solution module for the Navier–Stokes equations, whereas G.L. programmed the other CFD modules. G.L. analysed the CFD results, identified the edge vortex, and produced the figures. U.K.M. performed the PIV experiment. All authors participated in the discussion and explanation of the results. J.L.v.L., U.K.M. and G.L. wrote most of the paper; all authors participated in the revisions.

Competing interests

We declare we have no competing interests.

Funding

This work is partly supported by the grant-in-aid for Scientific Research on Innovative Areas of no. 24120007, JSPS, Japan. G.L. and U.K.M. were supported in part by NSF-IOS 1440576. G.L. was supported by a visitors grant by the Wageningen Institute of Animal Sciences.