Analysis of the instabilities induced by an isolated roughness element in a laminar high-speed boundary layer

The disturbances evolving in the wake induced by an isolated roughness element are investigated on a flat plate inside a cold Mach 6 flow. Different instability modes are characterized by means of two-dimensional local linear stability computations for a cuboid and a ramp-shaped roughness element. A single pair of sinuous and varicose disturbances dominates the wake instability in the vicinity of each roughness geometry. A temporal growth-rate decomposition, extended to base flows depending on two spatial inhomogeneous directions, reveals that the roughness-induced wake modes extract most of their potential energy from the transport of disturbance entropy across the base-flow temperature gradients and most of their kinetic energy from the work of the disturbance Reynolds stresses against the base-flow velocity gradients. The growth rate of such instabilities is found to be influenced by the presence of Mack-mode disturbances developing on the flat plate. Evidence is observed of a continuous synchronization between the wake instabilities and the Mack-mode perturbations which resembles the second mechanism hypothesized by De Tullio & Sandham (J. Fluid Mech., vol. 763, 2015, pp. 136–145) for the excitation of wake disturbances. The evolution of the relevant production and dissipation terms of the temporal growth-rate decomposition shows that under this continuous synchronization process, the energy signature of the wake instabilities progressively shifts towards that of Mack-mode instabilities. This leads to an enhancement of the amplification rate of the wake instabilities far downstream of the roughness element, ultimately increasing the associated N-factors for some of the investigated conditions.