Assessment of two vortex formulations for computing forces of a flapping foil at high Reynolds numbers

Several vortex formulations for computing the lift and drag/thrust forces on a flapping foil are compared in terms of simplicity, accuracy, and suitability for using with experimental data of measured velocity and vorticity fields. These methods are very useful to physically understand the aerodynamic performance of the flapping foil, and therefore the mechanisms of animal swimming and flight. In particular, we consider the case of a two-dimensional heaving foil at high Reynolds numbers and, using the results of accurate numerical simulations, we compare the performance of three force formulations based on the vortical impulse and Lamb's vector, projection of Lamb's vector, and wake integrals. We find that the formulation based on the vortical impulse is appropriate when all vortices generated by the flapping foil remain close to the foil, such as in a starting motion or in hovering, but may become quite inaccurate for analyzing forward flight or swimming. In these latter cases, the projection method is much more appropriate because it only takes into account vorticity close to the moving solid body at each instant. In fact, we find that, for high Reynolds numbers, the performance of the projection method is always better, even in hovering, due to its simplicity and because it neatly separates the contribution of the added-mass force from the circulatory or vortical force component. Finally, it is shown that the standard integral momentum method, which is quite simple to implement in a forward motion using experimental data of the velocity field in the wake behind the solid body, is much less accurate than the methods based on vortex force formulas.

Figure 1

Schematic of the heaving foil bounded by $S_s$, the integration domain delimited by the surfaces $S_{\infty }$ and $S_s$, and the control volume $V_c$ delimited by $S_e$ and $S_s$.

Figure 2

$C_L\left(t\right)$ during a cycle computed for $\text{Re}=10000$, $h_0=0.025$, and $k=7.86$ compared with the experimental results of McGowan et al. [26] and the theoretical ones by Theodorsen [9].

Figure 3

Averaged $C_T$ vs $k{h}_0$ computed for $h_0=0.175$ and $\text{Re}=20000$ compared with experimental results of Heathcote and Gursul [27] for the same values of $h_0$ and $k$ and three Reynolds numbers, and the numerical results by Young and Lai [25] for the same conditions. Also included are the theoretical results by Garrick [28] and Fernandez-Feria [10], both corrected with $C_{T0}\simeq -0.028$. The shaded area corresponds to the values of $k{h}_0$ for which our numerical results are no longer periodic with period $T$, and error bars are included for the present numerical results.

Figure 4

$\eta$ vs $k{h}_0$ for the same cases plotted in Fig. 3, except the numerical results by Young and Lai (same legend).

Figure 5

Functions ${\phi }_x(x,z)$ (left) and ${\phi }_z(x,z)$ (right) for a NACA 0012 foil.

Figure 6

Comparison of $C_T\left(t\right)$ (a) and $C_L\left(t\right)$ (b) during three periods, well after a periodic state has been reached, computed with Eq. (5) (Num.), with Eq. (11) (Imp.), and with Eq. (21) (Proj.) for $h_0=0.05$, $k=6$ and $\text{Re}=20000$. The size of the control volume $V_c$ used in Eqs. (11) and (21) corresponds to $r=l=10$.

Figure 7

Averaged values of $C_T$ (a) and $C_L$ (b) vs $k$ for $\text{Re}=20000$ and $h_0=0.05$, computed with Eq. (5) (Num.), with Eq. (11) (Imp.), and with Eq. (21) (Proj.). The error bars are for the values computed with Eq. (5) when the flow is not longer periodic (shaded area).

Figure 8

Vorticity fields at $t/T=2.75$ (a) and $t/T=50.75$ (b), and mean velocity $\left|\mathbf{v}\right|$ field at $t/T=50.75$ (c), of the flow starting at $t=0$ with a periodic developed wake from the steady case, for $h_0=0.05$, $\text{Re}=20000$ and $k=6$. $S_E$ is located ten chord lengths downstream from the trailing edge, but the control volume is not shown in its entirety.

Figure 9

Comparison of $C_T\left(t\right)$ (a) and $C_L\left(t\right)$ (b) along two periods around $t/T=2$ [corresponding to the flow depicted in Fig. 8] computed numerically using Eq. (5) (Num.) with the results from the Impulse formulation Eqs. (13, 14, 15, 16) (Imp.) and their four components from ${\mathbf{F}}_v$, ${\mathbf{F}}_I$, ${\mathbf{F}}_{oE}$, and ${\mathbf{F}}_V$ (the last one vanishes for $C_T$). $h_0=0.05$, $\text{Re}=20000$, and $k=6$. $S_E$ is located ten chord lengths downstream from the trailing edge.

Figure 10

As in Fig. 9 but for two periods around $t/T=50$, corresponding to the flow depicted in Fig. 8.

Figure 11

As in Fig. 9 but for several cycles around the transition time $t/T=17$, when the wake reaches $S_E$. Note that the curves Num. and Imp. practically coincide for all time.

Figure 12

Density fields of thrust (left) and lift (right) corresponding to the different terms of ${\mathbf{F}}_{\text{vf}}$ in Eq. (26): $\omega v_z(\partial {\phi }_x/\partial x)$ (a), $\omega v_z(\partial {\phi }_z/\partial x)$ (b), $-\omega v_x(\partial {\phi }_x/\partial z)$ (c), $-\omega v_x(\partial {\phi }_z/\partial z)$ (d). $t/T=50.75$ (midupstroke) for $h_0=0.05$, $\text{Re}=20000,$ and $k=6$.

Figure 13

Comparison of $C_T\left(t\right)$ (a) and $C_L\left(t\right)$ (b) along several periods around $t/T=17$ (corresponding to the instant when the airfoil's wake reaches $S_E$) computed numerically using Eq. (5) (Num.) with the results from the projection formulation Eqs. (24, 25, 26) (Proj.), and their two components from ${\mathbf{F}}_{\text{am}}$ and ${\mathbf{F}}_{\text{vf}}$. $h_0=0.05$, $\text{Re}=20000,$ and $k=6$. $S_E$ is located ten chord lengths downstream from the trailing edge.

Figure 14

Comparison of $C_T\left(t\right)$ (a) and $C_L\left(t\right)$ (b) along two periods around $t/T=50$ computed numerically using Eq. (5) (Num.) with the results from the integral momentum approximation in the control volume (CV) and their two components from ${\mathbf{F}}_{tv}$ and ${\mathbf{F}}_{vE}$. $h_0=0.05$, $\text{Re}=20000,$ and $k=6$. $S_E$ is located ten chord lengths downstream from the trailing edge.

Figure 15

As in Fig. 14, but for $S_E$ located three chord lengths downstream from the trailing edge.

Figure 16

Averaged values of $C_T$ (a) and $C_L$ (b) vs $k$ for $\text{Re}=20000$ and $h_0=0.05$, computed with Eq. (5) (Num.), with the impulse/Lamb approximation Eq. (13) (Imp.), with the projection approximation Eq. (24) (Proj.), and with the integral momentum approximation Eq. (27) when $S_E$ is located three chord lengths downstream from the leading edge (CV 3). The error bars are for the values computed with Eq. (5) when the flow is not longer periodic (shaded area).