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The influence of surface roughness on transition to turbulence in a Mach 4.5boundary layer is studied using direct numerical simulations. Transition isinitiated by the nonlinearly most dangerous inflow disturbance, which causesthe earliest possible breakdown on a flat plate for the prescribed inflowenergy and Mach number. This disturbance is primarily comprised of two normalsecond-mode instability waves and an oblique first mode. When localizedroughness is introduced, its shape and location relative to the synchronizationpoints of the inflow waves are confirmed to have a clear impact on theamplification of the second-mode instabilities. The change in modalamplification coincides with the change in the height of the near-wall regionwhere the instability wave-speed is supersonic relative to the mean flow; thenet effect of a protruding roughness is destabilizing when placed upstream ofthe synchronization point and stabilizing when placed downstream. Assessment ofthe effect of the roughness location is followed by an optimization of theroughness height, abruptness and width with the objective of achieving maximumtransition delay. The optimization is performed using an ensemble-variational(EnVar) approach, while the location of the roughness is fixed upstream of thesynchronization points of the two second-mode waves. The optimal roughnessdisrupts the phase of the near-wall pressure waves, suppresses theamplification of the primary instability waves, and mitigates the nonlinearinteractions that lead to breakdown to turbulence. The outcome is a sustainednon-turbulent flow throughout the computational domain.

Optimal two-dimensional roughness for transition delay in high-speed boundary layer