On the instability of Ekman boundary-layer flow over a rough disk with a wavy surface profile

We apply the method of lines and the backward difference method to solve highly nonlinear governing mean flow equations of the Ekman boundary-layer flow over a rough disk with a wavy surface profile. These equations are derived using the YHP roughness model that imposes a wavy surface profile in terms of the radial position r, assuming rotational symmetry. Our main interest is to determine the effect of roughness on the local linear convective instability of the Ekman flow, and we solve the local linear stability equations of the disturbances stationary relative to the disk using the Chebyshev polynomials. The crossflow and viscous types of instability modes, the Type I and Type II modes respectively, are analyzed. Our theoretical findings prompts that an adequately designed rough disk can be a useful tool for a passive flow-control method for the Ekman flow. The underlying physical mechanisms are investigated applying a fundamental energy analysis.