Vortex dynamics and fin-fin interactions resulting in performance enhancement in fish-like propulsion

The leading-edge vortex (LEV) formation on the caudal fin (CF) has been identified as playing a key role in efficient lift-based thrust production of fish-like propulsion. The enhancement of the CF LEV through its interaction with vortices formed upstream due to a median fin with a distinct shape is the focus of this paper. High-speed, high-fidelity videos and particle imaging velocimetry (PIV) were obtained from rainbow trout during steady forward swimming to visualize the undulatory kinematics and two-dimensional flow behavior. Body kinematics are quantified using a traveling-wave formulation that is used to prescribe the motion of a high-fidelity three-dimensional surface model of the fish body for a computational fluid dynamics (CFD) study. The pressure field of the CFD result is compared and validated with the PIV result from the experiment. Using CFD, the vortex forming and shedding behaviors of the anal fin (AF) and their capturing and interaction with the trunk (TK) and the CF are visualized and examined. Coherent AF-bound LEVs are found to form periodically, leading to thrust production of the AF. The vortices subsequently shed from the AF are found to help stabilize and reinforce the LEV formation on the CF by aiding LEV initiation at stroke reversal and enhancing LEV during a tail stroke, which leads to enhancement of lift-based thrust production. The CF is found to shed vortex tubes (VTs) that create backward-facing jets, and the ventral-side VT and the associated backward jets are both strengthened by vortices shed by the AF. An additional benefit of the AF is found to be reduction of body drag by reducing the lateral crossflow that leads to loss of beneficial pressure gradient across the body. Through varying AF-CF spacing and AF height, we find that CF thrust enhancement and TK drag reduction due to the AF are both affected by the position and size of the AF. The position and area of the AF that led to the most hydrodynamic benefit are found to be the original, anatomically accurate position and size. In this paper, we demonstrate the important effect of vortex interaction among propulsive surfaces in fish-like propulsion.

Figure 1

(a1) and (b1) Original frames from the video recording taken during particle imaging velocimetry (PIV) data collection. (a2) and (b2) Surface mesh (coarsened and untriangulated for visualization) is highlighted in cyan, while the virtual skeleton is highlighted in yellow. (a3) and (b3) Fully refined and triangulated surface mesh of the fish trunk (blue) and two-dimensional membrane bodies representing the caudal fin (CF, red), anal fin (AF, green), and dorsal fin (DF, green).

Figure 2

(a) Lateral of the computational model with abbreviations of body parts labeled. (b) Parametric study setup with (b1) defining key geometric parameters on a coarse mesh model overlaid on a picture of the real fish, (b2) demonstrating cases of varying the anal fin (AF)-caudal fin (CF) spacing $d$ to 1.5 (cyan), 1.25 (blue), and 0.75 (red) times the original position $d_0$. (b3) Parametric study cases of varying the AF aspect ratio through variations of $b_{\mathrm{AF}}$ to 0.75 (purple) and 1.25 (red) times the original. (c) Midline kinematics generated by the traveling-wave (TW) model. (d) Validation of the TW model by tracking two points [labeled in (a)] in an entire undulation cycle, solid lines representing the TW kinematics and dots representing measurements based on the video recording (experimental).

Figure 3

Computation grid setup: (a) Nominal Cartesian grid with base mesh and nested adaptive mesh refinement (AMR) blocks, with a total grid count of 12 million. Every second grid point is shown for visibility. (b) Grid-dependence study with instantaneous $C_T$ for reconstructed kinematics for a fine grid (20 million grid points), a coarse grid (7 million grid points), and the nominal grid (12 million grid points) used for subsequent analysis.

Figure 4

(a) Pressure contour and (b) net velocity vector fields on a coronal slice cut at $t/T=0.17$ showing (a1) and (b1) experimental [particle imaging velocimetry (PIV)] results and (a2) and (b2) computational fluid dynamics (CFD) results.

Figure 5

Time history of (a) coefficient of thrust and (b) coefficient of power for M1 and M2 during one typical undulating cycle.

Figure 6

Wake structure and vortex shedding with $Q$-criterion isosurfaces at $t/T=0.18$ shown in (a) ventral view and (b) perspective view for $Q=50$ (blue) and $Q=5$ (white, transparent), and $z$ vorticity ${\omega }_z$ of (c) M1 and (d) M2 at the plane denoted by the red line.

Figure 7

Three-dimensional wake structures of (a1), (b1), and (c1) M1 and (a2), (b2), and (c2) M2 during the left-right stroke of the caudal fin at timesteps (a) $t/T=0.063$, (b) $t/T=0.25$, and (c) $t/T=0.44$. The wake structures are visualized by the isosurface $Q$ criterion, with $Q$ = 50 (colored by streamwise vorticity contour) and $Q$ = 5 (white).

Figure 8

Three-dimensional wake structures of (a1), (b1), and (c1) M1 and (a2), (b2), and (c2) M2 during the right-left stroke of the caudal fin at timesteps (a) $t/T=0.63$, (b) $t/T=0.87$, and (c) $t/T=1.00$. The wake structures are visualized by the isosurface $Q$ criterion, with $Q$ = 50 (colored by streamwise vorticity contour) and $Q$ = 5 (white).

Figure 9

Slice cut parallel to the $yz$ plane made at the location denoted by the red line in M1 and M2 shown in lateral view above (a1) and (a2), showing the $x$ vorticity ${\omega }_x$ for (a1)–(c1) M1 and (a2)–(c2) M2, at time steps (a1) and (a2) $t/T$ = 0.44, (b1) and (b2) $t/T$ = 0.50, and (c1) and (c2) $t/T$ = 0.56. Green arrow indicating the direction of flow in a channel created by two closely set counterrotating vortices.

Figure 10

At time step $t/T=0.50$, six equally spaced $x$ slices made on the caudal fin (CF) from near the root ($x=0.86L_{\mathrm{B}}$ from the snout of the trout) to near the tip ($x=0.98L_{\mathrm{B}}$), showing dominant vortex formations on (a1) M1 and (a2) M2, with the cores traced by directional dotted lines also indicating the direction of rotation, the solid black lines denoting the boundaries of leading-edge vortices (LEVs). The strengths of the LEVs on the slices are plotted in (b).

Figure 11

Pressure plots at (a) and (b) $t/T=0.50$ and (c) and (d) $t/T=0.88$, with (a) a slice cut through the lower body, the precise position denoted in (a1) and (a2) top-left corner with a red line, and dashed contour lines separating regions with negative pressure, (b) surface pressure on the suction side of the peduncle-caudal fin (CF) section, (c) pressure isosurface with $C_P=-0.20$ (blue) and $C_P=-0.05$ (white), and (d) surface pressure contour of the whole right side of the fish body on (a1), (b1), and (c1) M1 and (a2), (b2), and (c2) M2.

Figure 12

(a) Time history of instantaneous $C_T$ over a typical cycle for varying anal fin (AF)-caudal fin (CF) spacing. (b) Performance trend in terms of cycle-averaged coefficient of thrust (${\overline{C}}_T)$ for the sum of body parts and deviation from baseline case ($\mathrm{\Delta }{\overline{C}}_T={\overline{C}}_T-{\overline{C}}_{T_{M1}}$) for individual body parts with respect to AF-CF spacing $d/d_0$.

Figure 13

(a) Pressure isosurface at $t/T$ = 0.88, with $C_P=-0.05$ (white, transparent) and $C_P=-0.20$ (blue), (b) pressure contour on fish body at the same time step, for (a1) and (b1) $d/d_0=0.75$ and (a2) and (b2) $d/d_0=1.50$.

Figure 14

Three-dimensional wake structure visualized by the isosurface $Q$ criterion, with $Q$ = 50, colored by streamwise vorticity (${\omega }_x$) contour, and $Q$ = 5 (white) for (a) $d/d_0=0.75$. (b) $d/d_0=1.50$, at $t/T$ = 0.13.

Figure 15

(a) and (b) ${\omega }_x$ and ${\omega }_z$ showing on the slicecut made at the same location as in Figs. 9, 6, and 6 at $t/T=0.13$. (a1) and (b1) $d/d_0=0.75$. (a2) and (b2) $d/d_0=1.50$.

Figure 16

Relative flapping motion of the anal fin (AF) and caudal fin (CF) at different locations. (a) Lateral excursion of the CF and AF tips was tracked for an entire cycle, showing cases of the maximum and minimum AF-CF spacings for AF. (b) Global phase difference between AF and CF for different AF-CF spacing cases.