Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element

We study the full three-dimensional instability mechanism past a hemispherical roughness element immersed in a laminar Blasius boundary layer. The inherent three-dimensional flow pattern beyond the Hopf bifurcation is characterized by coherent vortical structures usually called hairpin vortices. Direct numerical simulation results are used to analyze the formation and the shedding of hairpin vortices inside the shear layer. The first bifurcation is investigated by global-stability tools. We show the spatial structure of the linear direct and adjoint global eigenmodes of the linearized Navier-Stokes equations and use the structural-sensitivity field to locate the region where the instability mechanism acts. The core of this instability is found to be symmetric and spatially localized in the region immediately downstream of the roughness element. The effect of the variation of the ratio between the obstacle height k and the boundary layer thickness δk∗ is also considered. The resulting bifurcation scenario is found to agree well with previous experimental investigations. A limit regime for k/δk∗<1.5 is attained where the critical Reynolds number is almost constant, Rek ≈ 580. This result indicates that, in these conditions, the only important parameter identifying the bifurcation is the unperturbed (i.e., without the roughness element) velocity slope at the wall.