Global stability analysis and direct numerical simulations of boundary-layer flows past roughness elements

Investigations of roughness-induced transition are conducted for a laminar boundary layer past an isolated cubic roughness element. The ratio between the roughness height and the displacement boundary layer thickness is h/{\delta^*}=2.86. Global linear stability analysis is performed using a time-stepper method in conjunction with the implicitly restarted Arnoldi iteration method. The base flow is computed using the selective frequency damping method. The global stability results using the base flows are compared to those using the mean flows obtained using time-averaging. The critical Reynolds number of global instability based on the roughness height is found to be Re_h=475. The leading unstable global mode exhibits varicose instability. The production terms in the disturbance kinetic energy equation show that the varicose mode extracts most energy from the central low-speed streak and the lateral streaks also make a contribution. The corresponding adjoint mode is located in the upstream vicinity of the cube as well as on top of the element, highlighting the most receptive regions to momentum forcing. The nonlinear evolution downstream of the cube is examined using direct numerical simulations. The effect of the roughness elements are examined in the presence of upstream forcing.