Effects of aspect ratio on rolling and twisting foils

Flapping flight and swimming are increasingly studied due to both their intrinsic scientific richness and their applicability to novel robotic systems. Strip theory is often applied to flapping wings, but such modeling is only rigorously applicable in the limit of infinite aspect ratio ($AR$) where the geometry and kinematics are effectively uniform. This work compares the flow features and forces of strip theory and three-dimensional flapping foils, maintaining similitude in the rolling and twisting kinematics while varying the foil $AR$. We find the key influence of finite $AR$ and spanwise varying kinematics is the generation of a time-periodic spanwise flow which stabilizes the vortex structures and enhances the dynamics at the foil root. An aspect-ratio correction for flapping foils is developed analogous to Prandtl finite wing theory, enabling future use of strip theory in analysis and design of finite aspect ratio flapping foils.

Figure 1

(a) Kinematic illustration for a 3D foil undergoing rolling and twisting motions whose cross sections are comparable to 2D foil experiencing linearly increased heave and pitch amplitude along the span. (b) Illustration of 4 phases in one cycle period ($T$) for pure-rolling kinematic and twisting-rolling combination.

Figure 2

The gradient of maximum angle of attack ${\alpha }_{max}$ ($\pi$ rad) toward the tip is positive with linear rate ${\dot{\alpha }}_{max}$ ($\pi$ rad/$t$).

Figure 3

Instantaneous vortex wake visualization at $t/T=0.5$ for both kinematics and $S=3C$. Vortices are visualized with isocontours of ${\lambda }_2$ colored by dimensionless spanwise vorticity ${\omega }_{z}C/U_{\infty }$. Free-stream flow $U_{\infty }$ is flowing right to left and spanwise scale given in $z/S$.

Figure 4

Phase-averaged velocities for pure-rolling and twisting-rolling motions for $S=3C$. Visualization as in Fig. 3. See video 1 of the Supplemental Material [37] for comparison of pure-rolling and twisting-rolling motions for $S=3C$.

Figure 5

Instantaneous flow at $t/T=0.5$ for finite foils undergoing the same pure-roll kinematics. Labels contrast the relatively weaker wake vortices on the center line ($z=0$) and less turbulent tip flow on the $S=6C$ foil compared to $S=3C$. Visualization as in Fig. 3.

Figure 6

Phase-averaged flow for pure-roll and twist-roll motions for different aspect ratios at $t/T=0.5$. Flow visualization as in Fig. 3. Inset figure $I_1$ shows the spanwise vorticity on a center-line slice and $I_2$ shows the streamwise vorticity on a slice through the tip. The inset tables give the viscosity-scaled circulation of the shed vortices. See videos 2 and 3 of the Supplemental Material [37] for the comparison of $S/C=1...6$ for pure-rolling and twisting-rolling motions.

Figure 7

Phase-averaged pressure coefficient of twisting-rolling foil for cross sections $z/S=$ (a) 0.02, (b) 0.27, and (c) 1.0, at $t/T=0.5$ for upper and lower side of the foil. Section close to tip is kinematic dominated, whereas the one close to root is gradient dominated.

Figure 8

Phase-averaged lift coefficient $C_L$ evolution for various $S/C$ of foil undergoing (a) roll and (b) twist-roll motions over one cycle.

Figure 9

Sectional spanwise velocity normalized by the heave velocity at each $z/S$ section for (a) roll, and (b) twist-roll at $t/T=0.5$. Solid/dashed lines indicate flow on the upper/lower side of the foil. Positive values indicate flow toward the tip and vice versa.

Figure 10

Peak sectional lift coefficients against $U_k^2={\dot{\mathcal{H}}}_{max}^2/U_{\infty }^2$ for (a) roll and (b) twist-roll kinematics for 3D foils of different spans. Strip theory results from 2D foils with the same kinematics are also shown.

Figure 11

Peak sectional $C_l$ at the root $z/S=0$.

Figure 12

Grid convergence statistics for 3D and 2D simulations of infinite foils in a combination of heaving-pitching motion at $\mathrm{Re}=5300, A_D=1.0, {\mathrm{St}}_D=0.3$, and ${\theta }_{\mathrm{bias}}={10}^{\circ }$. The statistics are presented in zero-mean standard deviation ${\sigma }_0$ for (a) 2D $C_L$, (b) 2D $C_T$, (c) 3D $C_L$, and (d) 3D $C_T$. $\epsilon$ is relative error to $C/\mathrm{\Delta }x=256$.