Hydrodynamics of tandem flapping pectoral fins with varying stroke phase offsets

We show how phasing between tandem bioinspired fins flapping at high-stroke amplitudes modulates rear fin thrust production and wake characteristics. Load cell thrust measurements show that the rear fin generates 25% more thrust than the front fin when it lags the latter by a quarter cycle, and performs 8% worse when it leads the front fin by the same amount. The flow interactions between the fins responsible for these observations are analyzed using two-dimensional particle image velocimetry measurements and three-dimensional computational fluid dynamics simulations. Distributions of velocity elucidate variations in the effective flow induced on the rear fin for different phase offsets. Vortex structure interactions and particle rakes reveal the contributions of the leading- and trailing-edge vortices shed by the front fin in modulating the suction at the rear fin leading edge. Furthermore, the wake structure far downstream of the fins changes in its coherence, axial and radial extents for the different phase offsets. These findings are relevant for the design and performance optimization of various unmanned underwater vehicles that utilize such tandem systems.

Figure 1

(a) Schematic of experimental setup showing side/axial-vertical ($x-z$) view of the rectangular glass tank. Origin in this view is defined at the intersection of the leading edge of the front fin at a zero stroke angle (vertical center of the PIV domain). (b) Top view of the setup highlighting the geometry and arrangement of the front and rear fins. Leading (LE) and trailing edges (TE), as well as the tip of the trail fin are labeled for clarity. Origin is defined as the intersection of the lead fin LE with the midspan (laser sheet). Flow arrows indicate direction of front-fin induced flow.

Figure 2

3D CAD view of the experimental setup including the tank, laser sheet, and fin orientations.

Figure 3

Cyclic variation of stroke ($\varphi$) and pitch angle ($\alpha$). $\varphi$ is positive when the fin is stroking downward in $z$ and $\alpha$ is positive when the fin leading edges are pitching upward in $z$. Grid lines in the $x$ axis represent the ten time-steps for which PIV data is acquired per cycle.

Figure 4

Views in the $x-z$ and $y-z$ planes of the front fin for ten time-steps in a single cycle when PIV data was recorded. For convenience, times corresponding to up- and down-stroke are shown in the left and right columns, respectively. Flow arrows indicate direction of front-fin induced flow.

Figure 5

Measured thrust profiles at the (a) front and (b) rear fins at stroke phase offsets $\delta$ = $-{90}^{\circ }, 0^{\circ }$, and $+{90}^{\circ }$. Average thrust value, $\langle T\rangle$ is provided at the top for each $\delta$ case. (c) Rear fin $\varphi$ for stroke phase offsets $\delta$ = $-{90}^{\circ }, 0^{\circ }$, and $+{90}^{\circ }$. Note that the time axis is with respect to the front fin, and the staggering of profiles in (b) corresponds to the staggering of $\varphi$ in (c). A direct comparison of the rear fin thrust values for different upstream conditions is provided in Fig. 6.

Figure 6

(a) Measured rear fin thrust profiles time-shifted to match the front fin stroke cycle for $\delta$ = $-{90}^{\circ }, 0^{\circ }$, and $+{90}^{\circ }$. (b) Measured rear fin thrust gains for $\delta$ = $-{90}^{\circ }$ and $+{90}^{\circ }$ with respect to the $\delta =0^{\circ }$ case.

Figure 7

Contour distributions of $W/V_{\text{tip}}$ from PIV data overlaid with velocity vector field (every fifth vector is shown in each direction for clarity) for five time-steps spanning the rear fin upstroke at $\delta$ = $-{90}^{\circ }$ (a–e), $0^{\circ }$ (f–j), and $+{90}^{\circ }$(k–o). Horizontal and vertical axes represent $x/c$ and $z/c,$ respectively. Time-steps correspond to the front fin timing as shown in Fig. 5. Arrows at the front and rear fins represent the direction of their vertical motion.

Figure 8

Contour distributions of $c\mathrm{\Omega }/V_{\text{tip}}$ from PIV data for five time-steps spanning the rear fin upstroke at $\delta$ = $-{90}^{\circ }$ (a–e), $0^{\circ }$ (f–j), and $+{90}^{\circ }$(k–o). Horizontal and vertical axes represent $x/c$ and $z/c,$ respectively. Time-steps correspond to the front fin timing as shown in Fig. 5. Arrows at the front and rear fins represent the direction of their vertical motion.

Figure 9

Contour distributions of $c\mathrm{\Omega }/V_{\text{tip}}$ from PIV data overlaid with velocity vector field (every third vector shown in each direction for clarity) focusing near the rear fin during its midupstroke for (a) $\delta$ = $-{90}^{\circ }$ and (b) $+{90}^{\circ }$.

Figure 10

Snapshot of evolving particle rakes released from the front fin LE and TE acquired from the CFD data when the rear fin is at midupstroke for $\delta$ values (a, d) $-{90}^{\circ }$, (b, e) $0^{\circ }$, and (c, f) $+{90}^{\circ }$, respectively. Particles are color-coded with their velocity magnitude (red: high, blue: low).

Figure 11

Profiles of $x/c$ versus $C_p$ along the rear fin acquired from the CFD data when the rear fin is at midupstroke for the three $\delta$ values. Area enclosed inside each curve is provided in the legend.

Figure 12

Contour distributions of $\langle U\rangle /V_{\text{tip}}$ covering an extended upper half of the tandem fins extracted from the CFD data for $\delta$ values (a) $-{90}^{\circ }$ (b) $0^{\circ }$, and (c) $+{90}^{\circ }$.

Figure 13

Profiles of $\langle U\rangle /V_{\text{tip}}$ along axial cuts taken at $x/c$ = 2 and 5 are shown in (a) and (b), respectively.

Figure 14

Steps involved in the image enhancement procedure. (a) Raw image, (b) after stroke-averaged background subtraction, (c) after modified histogram equalization (MHE) enhancement and range-filter based masking. Note that images shown in (a) and (b) are saturated at $40\%$ for viewing.

Figure 15

Mean subtracted axial ($x^{\prime }$) and vertical ($z^{\prime }$) front fin leading edge tip positions for the $\delta$ = $-{90}^{\circ }, t=60$ ms case.

Figure 16

Snapshots of the CFD mesh for $\delta$ = $-{90}^{\circ }$.

Figure 17

Average rear fin thrust profiles, $\langle T\rangle$ [N] for different $\delta$ values for experimentally measured thrust and CFD results at $U_{\infty }$ = 0.

Figure 18

Rear fin power over different $\delta$ values, measured in water ($P$=$P_m+P_h$), in air ($P$=$P_m$) and using CFD ($P$=$P_h$, right axis). Please note the different in scales between the left and right axes.

Figure 19

Average thrust profiles, $\langle T\rangle$ [N] for different $\delta$ values from CFD simulations performed at $U_{\infty }$ = 0.5 m/s for the front and rear fins.

Figure 20

Stroke-cycle averaged profiles of experimentally measured and frequency filtered axial thrust (N) compared with CFD results for the front (a–c) and rear (d–e) fins for (a, d) $\delta$ = $-{90}^{\circ }$, (b, e) $\delta =0^{\circ }$, and (c, f) $\delta$ = $+{90}^{\circ }$.

Figure 21

Evolution of ${U|}_x/V_{\text{tip}}$ averaged over all $z$ at $x/c$ = $-1.7$, 0, 3, and 5, for $\delta$ = $-{90}^{\circ }, 0^{\circ }$, and $+{90}^{\circ }$ cases over ten equally spaced time-steps ($t$ = 60, 160,...960 ms) spanning one stroke cycle for the PIV ($\square$) and CFD ($\circ$) data. Error bars denote uncertainty in instantaneous PIV data.

Figure 22

Evolution of ${W|}_x/V_{\text{tip}}$ averaged over all $z$ at $x/c$ = $-1.7$, 0, 3, and 5, for $\delta$ = $-{90}^{\circ }, 0^{\circ }$, and $+{90}^{\circ }$ cases over ten equally spaced time-steps ($t=60,160,...960$ ms) spanning one stroke cycle for the PIV ($\square$) and CFD ($\circ$) data. Error bars denote uncertainty in instantaneous PIV data.