Flow transitions and mapping for undulating swimmers

Natural swimmers usually perform undulations to propel themselves and perform a range of maneuvers. These include various biological species ranging from microsized organisms to large-sized fish that undulate at typical kinematic patterns. In this paper, we consider anguilliform and carangiform swimming modes to perform numerical simulations using an immersed-boundary method based computational solver at various Reynolds number (Re) regimes. We carry out thorough studies using wavelength and Strouhal frequency as the governing parameters for the hydrodynamic performance of undulating swimmers. Our analysis shows that the anguilliform kinematics achieves better hydrodynamic efficiency for viscous flow regime, whereas for flows with higher Re, the wavelength of a swimmer's wavy motion dictates which kinematics will outperform the other. We find that the constructive interference between vortices produced at anterior parts of the bodies and corotating vortices present at the posterior parts plays an important role in reversing the direction of Benard–von Karman vortex street. Since most of the thrust producing conditions appear to cause wake deflection; a critical factor responsible for degrading the hydrodynamic efficiency of a swimmer, we discuss the underlying mechanics that would trigger this phenomenon. Moreover, we also find that resistive thrust force due to the frictional drag for anguilliform swimmers plays a substantial role in their propulsion at a low Reynolds number. As we approach more inertial flow conditions, its role is minimum and our carangiform swimmers primarily takes advantage of the thrust force due to the added mass effect under all the flow and kinematic conditions. Our findings are expected to provide a guideline on their selection for bioinspired underwater vehicles.

Figure 1

(a) Amplitude envelops for carangiform and anguilliform undulations, (b) backbone motion of an anguilliform swimmer, and (c) backbone motion of a carangiform swimmer.

Figure 2

Flow domain and the boundary conditions.

Figure 3

These Phase maps show flow regimes and their transitions for both the undulating swimmers. Here, the left (a), (c), and (e) and right columns (b), (d), and (f) correspond to the anguilliform and carangiform swimmers, respectively. The top row (a) and (b), middle (c) and (d), and bottom (e) and (f) rows present the characterization of flow regimes for $\mathrm{Re}={10}^2$ (viscous regime), $\mathrm{Re}={10}^3$ (transitional regime), and $\mathrm{Re}=5\times {10}^3$ (inertial regime). The symbols $\times$ (blue), $\mathrm{\Delta }$ (black), and $\nabla$ (filled with red) represent the markers for parameters producing drag, thrust, and thrust with jet deflection, respectively. Moreover, dashed (green) line is to show points for the transformation of BvKVS to its RBvKVS. Solid (violet) line demarks the thrust-generating parameters from those producing drag and the dotted (red) line separates the region for thrust production along with the jet deflection.

Figure 4

Time-averaged hydrodynamic parameters at $\mathrm{Re}={10}^2$ for undulating swimmers where the left (a), (c), and (e) and right (b), (d), and (e) columns show the data for anguilliform and carangiform swimmers, respectively.

Figure 5

Time-averaged hydrodynamic parameters at $\mathrm{Re}={10}^3$ for undulating swimmers where the left (a), (c), and (e) and right (b), (d), and (e) columns show the data for anguilliform and carangiform swimmers, respectively.

Figure 6

Time-averaged hydrodynamic parameters at $\mathrm{Re}=5\times {10}^3$ for undulating swimmers where the left (a), (c), and (e) and right (b), (d), and (e) columns show the data for anguilliform and carangiform swimmers, respectively.

Figure 7

Optimum efficiency parameters in $f^{*}-{\lambda }^{*}$ phase plane.

Figure 8

Vorticity dynamics (Vorticity contours ranging from −5; blue, to +5; red) in the close vicinity of the swimmers where the first (a)–(d) and second rows (e)–(f) shows flow fields around the anguilliform and carangiform swimmers, respectively, at $\frac{t}{\tau }=0.50$, whereas the third (i)–(l) and fourth rows (m)–(p) shows flow fields around the anguilliform and carangiform swimmers, respectively, at $\frac{t}{\tau }=1.00$. Here, first, second, third, and fourth columns belong to the cases 1, 2, 3, and 4, respectively, as described in Table 4.

Figure 9

(a)–(d) shows the circulation of positive and negative vortices for our undulating swimmers. Here, red symbols show data for the positive vortices, whereas blue symbols are for negative vortices.

Figure 10

These contours show time-averaged the horizontal velocity component $(\overline{U}/U_{\infty })$ in the wake of the two swimmers for $\mathrm{Re}=5\times {10}^3$ and ${\lambda }^{*}=1.00$. Left and right columns correspond to anguilliform and carangiform swimmers, respectively, whereas the top, middle, and bottom rows show the results for $f^{*}=0.40$, $0.50$, and $0.60$.

Figure 11

These plots present time-averaged lateral force coefficients for both the swimmers where left and right columns are for anguilliform and carangiform swimmers, respectively. The upper and lower rows show $\overline{C_Y}$ computed for flows at $\mathrm{Re}={10}^3$ and $5\times {10}^3$.

Figure 12

These are the vorticity contours in the wake of our undulating swimmers at the end of the tenth oscillation cycle where the left and right columns show the wake dynamics of anguilliform and carangiform swimmers, respectively. These top, middle, and bottom rows are taken for $f^{*}=0.30$, $0.40$, and $0.50$ with ${\lambda }^{*}=1.0$ at $\mathrm{Re}=5\times {10}^3$.

Figure 13

A schematic for vortex pairing mechanisms.

Figure 14

These figures provide temporal histories for coordinates of the centers of the vortices and their respective circulation in the wake of our anguilliform swimmer at $\mathrm{Re}=5\times {10}^3$ and ${\lambda }^{*}=1.0$ where the left (a), (d), and (g), middle (b), (e), and (h), and right (c), (f), and (i) columns present data for ${\mathrm{f}}^{*}=0.40$, $0.50$, and $0.60$, respectively.

Figure 15

These plots (a), (c), and (e) show the distance between the centers of vortices forming dipoles 1 and 2, whereas the plots in the right column present symmetry-breaking and symmetry-holding velocities. Top, middle, and bottom rows are for $f^{*}=0.40$, $0.50$, and $0.60$.

Figure 16

Contours of time-averaged frictional drag coefficient as a function of $\mathrm{St}$ and ${\lambda }^{*}$ where the left, middle, and right columns show data for $\mathrm{Re}=100$, 1000, and 5000. The top and bottom rows belong to anguilliform and carangiform swimmers, respectively.

Figure 17

Contours of normalized mean reactive force component $\left(\overline{T_{am}}\right)$ as a function of $\mathrm{St}$ and ${\lambda }^{*}$ where the left, middle, and right columns show data for $\mathrm{Re}=100$, 1000, and 5000. The top and bottom rows belong to anguilliform and carangiform swimmers, respectively.