Aerodynamic Shape Optimization of a Wavy Airfoil for Ultra-Low Reynolds Number Regime in Gliding Flight

The effect of the number of waves and the width of the ridge and valley in chord direction for a wavy airfoil was investigated at the angle of attack of $0^{\circ }$ and Reynolds number of ${10}^3$ through using the two-dimensional direct numerical simulation for four kinds of wavy airfoil shapes. A new method for parameterizing a wavy airfoil was proposed. In comparison with the original corrugated airfoil profile, the wavy airfoils that have more distinct waves show a lower aerodynamic efficiency and the wavy airfoils that have less distinct waves show higher aerodynamic performance. For the breakdown of the lift and drag concerning the pressure stress and friction stress contributions, the pressure stress component is significantly dominant for all wavy airfoil shapes concerning the lift. Concerning the drag, the pressure stress component is about $75\%$ for the wavy airfoils that have more distinct waves, while the frictional stress component is about $70\%$ for the wavy airfoils that have less distinct waves. From the distribution of pressure isoline and streamlines around wavy airfoils, it is confirmed that the pressure contributions of the drag are dominant due to high pressure on the upstream side and low pressure on the downside; the frictional contribution of the drag is dominant due to large surface areas of the airfoil facing the external flow. The effect of the angle of attack on the aerodynamic efficiency for various wavy airfoil geometries was studied as well. Aerodynamic shape optimization based on the continuous adjoint approach was applied to obtain as much as possible the highest global aerodynamic efficiency wavy airfoil shape. The optimal airfoil shape corresponds to an increase of $60\%$ and $62\%$ over the aerodynamic efficiency and the lift from the initial geometry, respectively, when optimal airfoil has an approximate drag coefficient compared to the initial geometry. Concerning an fixed angle of attack, the optimal airfoil is statically unstable in the range of the angle of attack from $-1^{\circ }$ to $6^{\circ }$ , statically quasi-stable from $-6^{\circ }$ to $-2^{\circ }$ , where the vortex is shedding at the optimal airfoil leading edge. Concerning an angle of attack passively varied due to the fluid force, the optimal airfoil keeps the initial angle of attack value with an initial disturbance, then quickly increases the angle of attack and diverges in the positive direction.