Competing roughness effects on the non-stationary crossflow instability of the boundary-layer over a rotating broad cone

Abstract

There are two primary aims of this paper: the first aim is to investigate the effects of the roughness types of the Miklavčič and Wang model on stationary disturbances of the boundary-layer flow over a broad rotating cone in still fluid. The second aim is to examine similar effects of surface roughness, but on non-stationary modes of crossflow instability. This study begins with the formulations of the mean-flow system based on the cone geometry. These equations are solved using a spectral numerical method based on Chebyshev polynomials and then used to formulate the linear stability system, which are computed for obtaining neutral curves of the unsteady flows. For the stationary modes, our results indicate that the inviscid instability (type I mode) is more stable, while the viscous instability (type II mode) entirely eliminates, as concentric grooves or isotropic roughness and the cone half-angle increase. In contrast, streamwise grooves have a slight stabilizing effect on the type I mode and a significant destabilizing effect on the viscous instability. Another finding indicates that decreasing the half-angle leads to a greater stabilizing effect of isotropic roughness on the type I modes. Our outcomes are also confirmed by the growth rate and the energy analysis, which shows a large reduction of the total energy balance as a result of increasing concentric grooves or isotropic roughness for the crossflow mode. For non-stationary modes, similar effects are observed in that increasing all levels of roughness stabilizes the type I branch (with concentric grooves and isotropic roughness having a much stronger effect than streamwise grooves), and although increasing concentric grooves and isotropic roughness stabilizes the type II branch, increasing streamwise grooves destabilizes this branch. For modes traveling slower than the rotation of the cone, concentric grooves generally have a stronger stabilization effect for the type I branch, whereas for modes traveling faster than the cone, isotropic grooves have a stronger stabilization effect on the type I branch for all broad half angles. Importantly, increasing concentric grooves and isotropic roughness increases the frequency at which the most dangerous modes occur, whereas increasing the streamwise grooves reduces the frequency at which the most dangerous modes occur.