Effect of Geometry on Direct and Adjoint Linear Global Modes of Low Reynolds Number Laminar Flow over Body

We study a two-dimensional flow over a cylinder and three ellipseswith the aspect ratio a/b= 2.04 ,3.65 ,and 5 at a Reynolds number Re= 100 based on the inflow velocity and hydraulic diameter using a directnumerical simulation (DNS) and linear stability analysis. The DNS showsthat increase in a/bleads to stabilization of the flow due to decrease in the recirculationzone. The linear stability analysis based on the time-averaged velocityfield shows that the modes of interest describe a Karman vortex street.The oscillation frequency reconstructed from the linear stabilityanalysis is in excellent agreement with the DNS. Solving the linearizedadjoint equations made it possible to identify the flow area wheredirect and adjoint modes overlap. These areas of the fieldS_w,usually called “wavemaker,” change their shape with increase ina/b.Non-zero values of S_wtend to approach the bottom part of the “tail” of the ellipse due to theasymmetry of the recirculation region, attached to the top side of theellipse.