Fluid dynamics of moving fish in a two-dimensional multiparticle collision dynamics model

The fluid dynamics of animal locomotion, such as that of an undulating fish, are of great interest to both biologists and engineers. However, experimentally studying these fluid dynamics is difficult and time consuming. Model studies can be of great help because of their simpler and more detailed analysis. Their insights may guide empirical work. Particularly the recently introduced multiparticle collision dynamics method may be suitable for the study of moving organisms because it is computationally fast, simple to implement, and has a continuous representation of space. As regards the study of hydrodynamics of moving organisms, the method has only been applied at low Reynolds numbers (below 120) for soft, permeable bodies, and static fishlike shapes. In the present paper we use it to study the hydrodynamics of an undulating fish at Reynolds numbers 1100–1500, after confirming its performance for a moving insect wing at Reynolds number 75. We measure (1) drag, thrust, and lift forces, (2) swimming efficiency and spatial structure of the wake, and (3) distribution of forces along the fish body. We confirm the resemblance between the simulated undulating fish and empirical data. In contrast to theoretical predictions, our model shows that for steadily undulating fish, thrust is produced by the rear 2/3 of the body and that the slip ratio $U/V$ (with $U$ the forward swimming speed and $V$ the rearward speed of the body wave) correlates negatively (instead of positively) with the actual Froude efficiency of swimming. Besides, we show that the common practice of modeling individuals while constraining their sideways acceleration causes them to resemble unconstrained fish with a higher tailbeat frequency.

Figure 1

Overview of the system for (a) the flapping insect wing and (b) the swimming fish. The wing moves along the open arrow, over the distance $A_0$. The height $H$ and width $W$ of the box (insect wing) and the width $W$ and length $L_{\mathrm{box}}$ of the box (fish) are not to scale.

Figure 2

Scrambling boundary condition. Within the boundary zone of thickness $B$ (not to scale), a particle $p$'s probability to be randomly displaced depends on its distance Dist to the outer system edge. Displacement is parallel to the nearest system edge. Dotted lines indicate the boundaries between horizontal and vertical shuffling zones. The new position is along the interval $R$ which lies between these boundaries at distance Dist from the outer system edge.

Figure 3

Elimination of vorticity by the boundary zone for (a) wake of the flapping insect wing and (b) wake of the swimming fish (Fig. 2). The arrow shows overall flow direction. Black lines are truncated streamlines. Localized flow phenomena are eliminated by the boundary zone (BZ) as fluid travels through it from one part of the simulation (1) to another (2).

Figure 4

Effects of size of simulation box on (a) drag coefficient of the cross-section of an insect wing (box size in chord length $c$ of the wing) and (b) equilibrium swimming speed (in simulation units) of the undulating fish (box size in fish body length $L$).

Figure 5

(a) “Fake particle” boundary condition. Fake particles (indicated in gray) are included in cells which partially overlap the object (indicated in light gray), and where the number of particles is below the mean fluid density. Mean fluid density $\rho$ is 8 (Table ). (b) Schematic overview of the intersection between the path of a moving particle $\mathbf{r}$ and a moving edge $\mathbf{E}$. The precise intersection point depends on the movement speed of both the particle ($d\mathbf{r}$) and of the edge ($d\mathbf{p}$ and $d\mathbf{q}$ for the end points of the edge). The area through which the edge $E$ moves during the time step $\Delta t$ is indicated in gray.

Figure 6

Schematic overview of a hovering fruit fly. (a) Side view: A series of snapshots of the moving cross-section of the wing as it moves back and forth over distance $A_0$. The leading edge of the wing is represented by a dot. Movement and rotation of the wing are in phase. The elliptical path of motion is a visual aid: In the simulations the wing does not displace vertically. (b) Top view of the insect: The dashed line shows which cross-section (of chord length $c$) of the wing we simulate. (c) Forces on the moving wing. The forces act on the center of gravity.

Figure 7

Schematic overview of the deviation from the central axis of the spine of an undulating mullet.

Figure 8

Sideways speed (in simulation units) over time of the center of gravity of the fish and its tail tip, for fish that either are or are not constrained from sideways acceleration.

Figure 9

Wake of the swimming fish in our model, with truncated streamlines. Distance between vortices in the swimming direction and perpendicular to it are indicated as $dx$ and $dy$, respectively. Vorticity is shown by the color, with blue indicating clockwise and red indicating counterclockwise vorticity.

Figure 10

Decomposition of the force $F_i$ on the skin of the fish into pressure ($F_n$) and viscous ($F_t$) components (see Methods). The surface normal is indicated as $n$ and the unit vectors pointing forward and sideways are labeled $e_f$ and $e_s$.

Figure 11

Drag and lift coefficients of a flapping insect wing over time, at $A_0/c=2.8$. Light gray areas indicate that the wing moves to the right. Real wing data taken from Wang et al. [49]. (a) $\phi =-\frac{\pi }{4}$ (delayed rotation). (b) $\phi =0$. Force peaks associated with wake capture and rotational forces are labeled “w” and “r”, respectively. (c) $\phi =\frac{\pi }{4}$ (advanced rotation).

Figure 12

Force in the swimming direction of the freely swimming fish, decomposed into pressure and viscous component. Positive values indicate thrust, negative drag.

Figure 13

Force in the swimming direction as distributed over several segments of the skin of the fish, without constraints (a) and with constraint of lateral and forward acceleration (b). Positive values indicate thrust, negative drag. Error bars indicate one standard deviation.

Figure 14

Drag and thrust forces on the skin of the fish over time (in seconds, s), for tailbeat frequency of 3.8 Hz, unconstrained acceleration. Black indicates drag, gray thrust. Force areas are composed of lines perpendicular to the skin, with the length of the line indicating the relative size of the force on that segment of the skin.

Figure 15

Results of swimming fish that are either constrained or free in their forward and sideways acceleration, for several tailbeat frequencies. (a) Equilibrium forward swimming speed. (b) Average lateral power exerted (in simulation units) by the fish. (c) Average forward thrust component (in simulation units). (d) Slip ratio $U/V$. (e) Froude efficiency of the swimming fish Eq. (19).

Figure 16

Results of swimming fish that are either constrained or free in their forward and sideways acceleration, for several tailbeat frequencies. (a) Radius of the vortex rings in the wake of the fish. (b) Angle $\phi$ of the vortex rings with the forward swimming direction (Fig. 17). (c) Circulation $\Gamma$ in the vortex rings of the wake. (d) Equilibrium Strouhal number of the swimming fish as a function of the Reynolds number.

Figure 17

Schematic overview of the wake structure of a mullet, after [21]. $A1$ and $A2$ are the centers of two adjacent vortices in the two-dimensional plane that comprise a vortex ring in a three-dimensional wake. $R$ indicates the radius of the vortex ring, $\phi$ its angle relative to the swimming direction and $\alpha$ the angle relative to the swimming direction of the jet through the vortex ring.