Vorticity transfer in a leading-edge vortex due to controlled spanwise bending

Many natural flyers and swimmers routinely flex their lifting or propulsive surfaces to control the leading-edge vortex (LEV) that forms on the suction side during maneuvering at a high angle of attack. In this paper, we studied the effect of a similar bending on the vortex dynamics of a flat-plate airfoil of aspect ratio 3 (chord $5\text{cm}$) and held at an angle of attack of ${30}^{\circ }$. This flat plate is accelerated from rest to a Reynolds number of 2400, while being dynamically bent along the span in a controlled manner with a bending ratio of 0.65 and a maximum bending angle of ${30}^{\circ }$. We investigated the effect of such spanwise bending on the resultant vorticity transfer via both experiments and numerical simulation. It shows that a dynamic spanwise bending induces a change in the effective shear layer velocity along the span's bent part and creates spanwise vorticity convection. As a result, the growth of circulation in the LEV gets delayed along the bent part, and the final circulation is smaller than the no-bending case.

Figure 1

The construction of the morphing plate: (a) grooves inside a 3D printed plate for holding prebent rods; (b) the plate-rod assembly is connected to servos, which bend the plate along the span; (c) the definition of bending angle and bending ratio. The tips are marked by AB and GH; the 50% span and the 80% span are, respectively, marked by CD and EF.

Figure 2

The kinematics of the plate motion. The plate is accelerated for $t^{*}=1$ and the bending is completed at $t^{*}=2$. $u^{*}$ is obtained by normalizing $u$ (velocity) by $U_{\infty }$ (0.05 m/s).

Figure 3

Experimental setup: a 1-m-long towing tank facility equipped with Velmex traverse. The camera travels along with the plate. A skim plate is used to prevent the formation of surface waves during towing. PIV measurements were conducted at 50% and 80% span of the plate (CD and EF in Fig. 1).

Figure 4

Schematics of (a) the computational domain and Cartesian mesh employed in the current simulation. The mesh along the $x$ direction is stretched and the length of the uniform part is $5.8c$; (b) the dimensions of the plate used in the current study. The solid arrow stands for the moving direction of the plate, which is along the $x$-axis direction.

Figure 5

Details of the analytical model: (a) The LEV is modeled as a semicylindrical blockage in Wong et al. [29], where vorticity is being supplied by a shear layer; (b) the modification implemented in the analytical model in the present work, where the component of the bending velocity was subtracted from the effective velocity on the leading edge.

Figure 6

The development of the LEV on the midspan CD (50% span) of the plate from PIV measurements: when no bending was applied (left column) and when bending was applied (right column). The contour plots are created from normalized spanwise vorticity. The cutoffs were selected at each $t^{*}$ to exclude the boundary layer.

Figure 7

The development of the LEV on EF (80% span) of the plate from PIV measurements: when no bending was applied (left column) and when bending was applied (right column). The contour plots are created from normalized spanwise vorticity. When the plate was dynamically bent, the growth of the LEV at 80% span was delayed.

Figure 8

The LEV development on CD (50% span) of the plate from DNS: when no bending was applied (left column) and when bending was applied (right column).

Figure 9

The LEV development on EF (80% span) of the plate from DNS: when no bending was applied (left column) and when bending was applied (right column).

Figure 10

Comparison of the temporal growth of LEV circulation computed from the analytical model, PIV experiment, and DNS study.

Figure 11

The comparison of the spanwise vorticity convection flux through the LEV cores of the flat and bending cases in (a) a global view and (b) a zoomed view when $t^{*}=1.9$.

Figure 12

The comparison of the spanwise vorticity convection flux of the flat (left) and bending (right) cases at different moments in a zoomed view from DNS.

Figure 13

The vorticity contours and the vorticity convection flux at EF (80% span) when $t^{*}=1.5$ (top) and $t^{*}=1.9$ (bottom). The solid lines and dashed lines represent the LEV and the secondary vortex, respectively. The vorticity convection flux is colored.