Propulsion performance of a two-dimensional flapping airfoil with wake map and dynamic mode decomposition analysis.

In the present study, we use the dynamic mesh method based on the radial basis function interpolation for the two-dimensional simulation of harmonically oscillating NACA0015 airfoil. Under various flapping frequencies, heaving and pitching amplitudes, the observed wake flows can be divided into seven types, including the Bénard-von Kármán (BvK) vortex street, the reversed BvK (RBvK) vortex street, the 1P wake, the mP wake, the 2P+mS wake, the 2S+mS wake, and the mS wake, where m is around 4 and xS+yP signifies x single vortices and y vortex pairs shedding per oscillation period. Then we have constructed two phase diagrams of the wake types in terms of the flapping frequency, heaving and pitching amplitudes. Importantly, we have combined the propulsion performance of the flapping airfoil with the wake map and found that α(T/4), the angle of attack at t=T/4, can determine the wake type: negative value corresponding to drag-dominated wakes, while positive value corresponding to thrust dominated flow wakes. With the increase of α(T/4), the wake transforms from the mP to 2S+mS then to RBvK and eventually to 1P wake. Furthermore, the coherent structure analysis and spectral analysis are conducted for all the types of wakes by using dynamic mode decomposition. And there is a positive correlation between the strengths of vortices shedding at i times flapping frequency and the modulus of the ith dynamic mode decomposition mode, which can further reveal the differences among different types of wakes.