Global stability analysis and direct numerical simulation (DNS) are performed
to study boundary layer flows with an isolated roughness element. Wall-attached
cuboids with aspect ratios η=1 and η=0.5 are investigated for fixed
ratio of roughness height to displacement boundary layer thickness
h/δ∗=2.86. Global stability analysis is able to capture the frequency
of the primary vortical structures. For η=1, only varicose instability is
seen. For the thinner roughness element (η=0.5), the varicose instability
dominates the sinuous instability, and the sinuous instability becomes more
pronounced as Reh​ increases, due to increased spanwise shear in the
near-wake region. The unstable modes mainly extract energy from the central
streak, although the lateral streaks also contribute. The DNS results show that
different instability features lead to different behavior and development of
vortical structures in the nonlinear transition process. For η=1, the
varicose mode is associated with the shedding of hairpin vortices. As Reh​
increases, the breakdown of hairpin vortices occurs closer to the roughness and
sinuous breakdown behavior promoting transition to turbulence is seen in the
farther wake. A fully-developed turbulent flow is established in both the inner
and outer layers farther downstream when Reh​ is sufficiently high. For
η=0.5, the sinuous wiggling of hairpin vortices is prominent at higher
Reh​, leading to stronger interactions in the near wake, as a result of
combined varicose and sinuous instabilities. A sinuous mode captured by dynamic
mode decomposition (DMD) analysis, and associated with the `wiggling' of
streaks persists far downstream