Delaying leading edge vortex detachment by plasma flow control at topologically critical locations

Flapping wing propulsion offers unrivaled maneuverability and efficiency at low flight speeds and in hover. These advantages are attributed to the leading edge vortex developing on an unsteady wing, which induces additional lift. We propose and validate a manipulation hypothesis that allows prolongation of the leading edge vortex growth phase, by delaying its detachment with the aid of flow control. This approach targets an overall lift increase on unsteady airfoils. A dielectric barrier discharge plasma actuator is successfully used to compress secondary structures upstream of the main vortex on a pitching and plunging flat plate. To determine flow control timing and location, the tangential velocity on the airfoil surface is used, which is also used to quantify topological effects of flow control. This flow control is then tested for different motion kinematics on a NACA 0012 airfoil. An increase of the peak circulation of the leading edge vortex of about 20% for all cases with flow control indicates that this approach is applicable for various kinematics, dynamics, and airfoil types.

Figure 1

Topological sketch of the flow field around a pitching and plunging flat plate with a leading edge vortex (LEV), denoted as node ${\mathrm{N}}_1$, growing on the airfoil. Nodes are indicated by circles, a full saddle by a square, and half saddles by diamonds. (a) LEV detachment caused by growing secondary vortices (also termed secondary structures and shown as nodes ${\mathrm{N}}_2$ and ${\mathrm{N}}_3$), in accordance with the boundary-layer eruption mechanism. (b) Manipulation hypothesis to delay LEV detachment by compression of secondary vortices with the plasma-induced body force $f_{\mathrm{PA}}$.

Figure 2

Tangential velocity $u$ on the airfoil surface over dimensionless time $t/T$, shown in a plate fixed frame of reference over the dimensionless chord position $x/c$ with the origin at the leading edge ($x/c=0$). (a) Evolution without flow control. The leading edge vortex center position ${(x/c)}_{\mathrm{LEV}}$ and LEV confining half saddles (${\mathrm{H}}_1$ and ${\mathrm{H}}_2$) are indicated. The location of the plasma actuator ${(x/c)}_{\mathrm{PA}}$ and the actuation period $\mathrm{\Delta }{(t/T)}_{\mathrm{PA}\mathrm{active}}$ are also indicated. (b) Evolution with flow control applied. The secondary structure confining half saddle in the uncontrolled (${\mathrm{H}}_{2\mathrm{Base}}$) and the controlled case (${\mathrm{H}}_{2\mathrm{FC}}$) are compared, to evaluate flow control effects.

Figure 3

Schematic design of a DBD plasma actuator and the electric circuit to operate it. Electrodes are depicted in blue and the dielectric layer between them in yellow.

Figure 4

Effect of different domain sizes $S$ used to compute the vortex boundary from ${\mathrm{\Gamma }}_2$ scalar field. (a) $|{\mathrm{\Gamma }}_2|=2/\pi$ vortex contours for different domain sizes $S$ superimposed on normalized vorticity $\omega c/U_{\infty }$ field around the pitching and plunging flat plate at the dimensionless time instant $t/T=0.26$. (b) Evolution of the LEV area $A_{\mathrm{LEV}}$ for different domain sizes $S$ over dimensionless time $t/T$ of the airfoil motion. Occurrences of gaps in area evolution caused by vortex center detection outside of its identified boundary are marked with blue arrows.

Figure 5

Velocity field and normalized vorticity $\omega c/U_{\infty }$ for different dimensionless time instants $t/T$. (a)–(d) Without flow control (Base); (e)–(h) controlled case (FC). An inactive plasma actuator is indicate by a magenta line and an active one by a magenta arrow. While (a)–(c) and (e)–(g) depict the entire field with only every sixth velocity vector for clarity, (d) and (h) show the leading edge region with every second vector as indicated by a red square in (b) and (f). The inflow is from the left, the airfoil is masked out in gray, and the laser light shadow caused by the airfoil in black.

Figure 6

Leading edge vortex characteristics evolution over dimensionless time $t/T$. The case without flow manipulation (Base) is shown in black and the controlled case (FC) in blue. The standard deviation between single runs is indicated by shaded areas of the respective color. (a) Normalized leading edge vortex circulation ${\mathrm{\Gamma }}_{\text{LEV}}/U_{\infty }c$. The peak circulation ${\mathrm{\Gamma }}_{\max }$ of the uncontrolled and controlled case is highlighted by dashed lines of the respective color in addition to the derived peak circulation increase $\mathrm{\Delta }{\mathrm{\Gamma }}_{\max }$ and temporal delay of peak circulation $\mathrm{\Delta }{(t/T)}_{{\mathrm{\Gamma }}_{\max }}$. (b) Normalized LEV center position $x_{\mathrm{LEV}}/c$.

Figure 7

(a)–(d) Unactuated (Base in the top row) and actuated (FC in the bottom row) flow field in terms of normalized vorticity $\omega c/U_{\infty }$ for ID 2 [(a) and (c)] and ID 4 [(b) and (d)] cases at different dimensionless time instants $t/T$. (e) Relative peak circulation increase $\mathrm{\Delta }{\mathrm{\Gamma }}_{\max }$ between controlled and uncontrolled cases and the temporal delay of peak circulation $\mathrm{\Delta }{(t/T)}_{{\mathrm{\Gamma }}_{\max }}$. Error bars indicate the standard deviation of circulation increase.