Three-dimensional biological hydrodynamics study on various types of batoid fishlike locomotion

The present study is on periodic three-dimensional biological flow during batlike flapping as well as fishlike undulation based spanwise symmetric three-dimensional (3D) kinematics and propulsion of batoid fish. For a single parameter based unified study on various types of real and hypothetical batoid fish, a 3D unified-kinematics model is proposed here. Using our immersed interface method based in-house code, the present numerical study is on the effect of various wavelengths (of the wavy undulation of a hydrofoil-shaped lateral cross section) and aspect ratios (of the elliptical span of the body)—corresponding to various types of batoid fishlike locomotion—at a constant nondimensional frequency of 0.5. Furthermore, for Dasyatis and a hypothetical batoid fish, the effect of various nondimensional frequencies is studied at an aspect ratio of 1. The study is done at a maximum nondimensional amplitude of 0.15 of the pitching/undulation and a Reynolds number of 10 000. A 3D vortex structure demonstrates a spanwise symmetric double row of vortex structure at the various wavelengths while a larger wavelength also results in a horseshoe type of vortices with multiple vortex rings. For propulsive performance, the maximum thrust force (propulsive efficiency) is obtained at an intermediate (smaller) wavelength. Further, with increasing aspect ratio, an increase in the thrust force and the propulsive efficiency are found for various wavelengths. A single-row (double-row) vortex structure leading to smaller (larger) thrust force is obtained at a smaller (larger) nondimensional frequency. Using the more realistic 3D kinematics, as compared to the 2D kinematics used earlier, more realistic flow structures and the associated propulsive performance parameters are presented that can be used for the efficient design of underwater vehicles.

Figure 1

Three different types of batoid fish, with vertical ($z$-direction) undulation and/or pitching motion of the body along the streamwise ($x$) direction and vertical ($z$) pitching motion of the body along the spanwise ($y$) direction (modified from the figure presented by Rosenberger [4]). The red lines show the center of the body ($y=0$).

Figure 2

Batoid fishlike body: (a) 3D view, and (b)–(d) various 2D views: (b) top view, (c) front cross-sectional view (at $y=0.25b$), and (c) side cross-sectional view (at $x=0.5c$) of the batoidlike body.

Figure 3

Nondimensional computational setup for the tethered propulsion of batoid fishlike body and kinematics. The hydrofoil-shaped frontal cross sections are undulating with a chordwise flexibility of ${\lambda }^{*}$ and the aspect ratio of the ellipse-shaped top cross section is $A_R$. The nondimensional frequency of the undulation of the frontal cross section and two-sided symmetric-pitching motion of the streamwise cross section is St while the nondimensional amplitude of the undulation/pitching is $A_{\text{max}}$.

Figure 4

Vortex topology from (a) present study and (b) Dong et al. [24], for a pitching as well as heaving ellipsoid at $\text{Re}=200$, pitching amplitude ${\theta }_{\text{max}}={30}^{\circ }$, heaving amplitude $h_{\text{max}}=0.5$, and Strouhal number $\text{St}=0.6$.

Figure 5

For a 3D anguilliform fishlike body and kinematics, comparison of present and published results [25] for temporal variation in the drag coefficient $C_D$, at ${\lambda }^{*}=1.0$, $\text{St}=0.75$, $\text{Re}=2857$, and $A_{\text{max}}=0.16$.

Figure 6

Temporal variation in instantaneous thrust force coefficient $C_T$ for three different grid sizes, at ${\lambda }^{*}=4.0$, $A_R=1.0$, $\text{St}=0.5$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$.

Figure 7

Schematic representation of the formation of vortex ring, vortex contrail, and horseshoe vortex from a 3D body oscillating in the vertical direction ($z$ direction). Formation of spanwise (streamwise) vortices on the trailing edge (sides) of the body is shown in (a). The spanwise and streamwise vortices together form a vortex ring, shown in (c); and a vortex contrail is formed between the two vortex rings [shown in (d)]. When the streamwise vortices are farther apart at the trailing edge, a horseshoe vortex is formed instead of the vortex ring as shown in (e).

Figure 8

Effect of ${\lambda }^{*}$ and $A_R$ on the instantaneous $Q$-criterion based 3D vortex structure, for the batoid fishlike body, kinematics, and tethered propulsion, at $\text{Re}=10000$. The hydrofoil-shaped frontal cross sections are undulating with various chordwise flexibilities ${\lambda }^{*}$ and aspect ratio of the spanwise elliptical cross section $A_R$. The nondimensional frequency of the undulation of the frontal cross section and the two-sided symmetric-pitching motion of the streamwise cross section is $\text{St}=0.5$ while the maximum nondimensional amplitude of the undulation/pitching is $A_{\text{max}}=0.15$. Note that the time instant shown in the figures correspond to the starting of the undulating/pitching cycle, i.e., $\tau =0$; and the tail position varies with ${\lambda }^{*}$ at $\tau =0$.

Figure 9

Effect of nondimensional chordwise flexibility ${\lambda }^{*}$ and aspect ratio $A_R$ on the instantaneous spanwise vorticity (${\omega }_y$) contours (at $Y=0.25A_R$), for the batoid fishlike locomotion, at $\text{St}=0.5$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$.

Figure 10

Effect of nondimensional chordwise flexibility ${\lambda }^{*}$ and aspect ratio $A_R$ on the instantaneous streamwise velocity ($U$) contours (at $Y=0.25A_R$), for the batoid fishlike locomotion, at $\text{St}=0.5$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$.

Figure 11

Temporal variation of instantaneous (a),(c),(e) thrust coefficient $C_T$ and (b),(d),(f) lift coefficient $C_L$, with increasing ${\lambda }^{*}$ during one cycle of undulation/pitching, at (a),(b) $A_R=0.5$, (c),(d) $A_R=0.75$, and (e),(f) $A_R=1.0$ for $\text{Re}=10000$, $\text{St}=0.5$, and $A_{\text{max}}=0.15$.

Figure 12

Variation of the (a),(b) mean thrust coefficient $C_{Tm}$, (c),(d) rms value of lift force coefficient $C_{Lrms}$, and (e),(f) propulsive efficiency ${\eta }_P$ with increasing (a),(c),(e) ${\lambda }^{*}$ and (b),(d),(f) $A_R$, for the batoid fishlike locomotion at $\text{St}=0.5$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$. Unfilled symbols in (a), (c), and (e) represent the values corresponding to the pitching motion of the hydrofoil cross section (${\lambda }^{*}\to \infty$).

Figure 13

Effect of nondimensional frequency of undulation/pitching St on the instantaneous $Q$-criterion based 3D vortex structure of the batoid fishlike locomotion for ${\lambda }^{*}=0.8$ and ${\lambda }^{*}=\infty$, at $A_R=1.0$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$.

Figure 14

Effect of nondimensional frequency of undulation/pitching St on the instantaneous spanwise vorticity (${\omega }_y$) contours (at $Y=0.25A_R$) of the batoid fishlike locomotion for ${\lambda }^{*}=0.8$ and ${\lambda }^{*}=\infty$, at $A_R=1.0$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$.

Figure 15

Effect of nondimensional frequency of undulation/pitching St on the instantaneous streamwise velocity ($U$) contours (at $Y=0.25A_R$) of the batoid fishlike locomotion for ${\lambda }^{*}=0.8$ and ${\lambda }^{*}=\infty$, at $A_R=1.0$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$.

Figure 16

Variation of the (a) mean thrust coefficient $C_{Tm}$ and (b) rms value of lift force coefficient $C_{Lrms}$ with increasing St for ${\lambda }^{*}=0.8$ and ${\lambda }^{*}=\infty$, at $A_R=1.0$, $\text{Re}=10000$, and $A_{\text{max}}=0.15$.

Figure 17

(a)–(d) Temporal variation of $Q$-criterion based 3D vortex structure within one time period of the periodic 3D kinematics for the batoid fishlike locomotion at ${\lambda }^{*}=0.8$, $A_R=1.0$, $\text{St}=0.5$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$. Zoomed views of (b) are shown in (e)–(g): (e) 3D, (f) side, and (g) top views. Note that the zoomed view shows the spanwise symmetric 3D vortex structure in one of the sides, from the midplane to the front side. Here, the red arrow represents the trailing edge spanwise vortex (TESPV) ${\omega }_y$, the blue arrow represents side streamwise vortex (SSTV) ${\omega }_x$, and the yellow arrow represents the $y$ midplane streamwise vortex (MPSTV) ${\omega }_x$. The time instant corresponding to each subfigure is marked in the top left corner.

Figure 18

Temporal variation of $Q$-criterion based 3D vortex structure within one time period of the periodic 3D kinematics for the batoid fishlike locomotion at ${\lambda }^{*}=\infty$, $A_R=1.0$, $\text{St}=0.5$, $A_{\text{max}}=0.15$, and $\text{Re}=10000$. The time instant corresponding to each subfigure is marked in the top left corner.

Figure 19

Schematic representation of the spanwise symmetric 3D vortex structures and the induced jet flow, obtained in the present work, at (a) smaller ${\lambda }^{*}$ and (b) intermediate as well as larger ${\lambda }^{*}$. The various vortices marked as TESPV, SSTV, and MPSTV correspond to trailing edge spanwise vortex, side streamwise vortex, and midplane streamwise vortex, respectively. The block arrows represent the direction of a jet flow induced by a vortex ring (formed by the combination of TESPV, SSTV, and MPSTV) and the horseshoe vortices.