Receptivity Of Supersonic Boundary Layers To Acoustic Waves: A Combined Theoretical And Numerical Investigation

This paper investigates,through high-Reynolds-number asymptotic analysis and direct numerical simulations,the local receptivity of viscous and inviscid modes in supersonic boundary layers due to the interaction between a weak free stream acoustic wave and a small streamwise isolated wall roughness element.In a supersonic boundary layer with the freestream Mach number M>1,there exists a multitude of modes of different physical and mathematical characteristics.Among those,the lower-branch first modes with obliqueness angles(?)>tan^-1M²-1 are of viscous nature and governed by the triple-deck structure,while the first mode with(?)>tan^-1M²-1and the second mode are essentially inviscid and governed by the Rayleigh equation.In comparison with subsonic boundary layers,the local receptivity in supersonic regime exhibits a number of new interesting phenomena.For the local receptivity of the viscous first modes,we are interested in the slow acoustic waves whose characteristic frequencies and wavelengths are on the triple-deck scales,and their phase speeds are thus asymptotically small.When their amplitude in the freestream are of O(?_u)1,the induced perturbation velocities in the near-wall region are amplified toO(?_uR^1/8),where R is the Reynolds number based on the distance of the roughness site to the leading edge.Thus,their interaction with streamwise localized roughness elements leads to stronger receptivity.The resulting strong receptivity is described mathematically using triple-deck theory,and direct numerical simulations(DNS)are also carried out.The theoretical predictions are found to be in quantitative agreement with the DNS results at moderate values of R when the roughness elements are located near the lower branch of the instability.The amplitude of the instability mode excited is proportional to the streamwise velocity amplitude of the acoustic signature inside the boundary layer.For the receptivity of inviscid modes,our large-Reynolds-number asymptotic analysis show that the distortion of the acoustic signature by the curved wall contributes to the leading-order receptivity,while the interactions between the roughness-induced mean-flow distortion and the acoustic signature contribute to the second-order receptivity.Interestingly,the leading-order receptivity is equivalent to a canonic receptivity problem:the excitation by time-periodic blowing/suction through a local slot on the wall with the equivalent blowing/suction being expressed explicitly in terms of the roughness shape function.This equivalence holds even when the roughness height is large enough to cause a nonlinear mean-flow distortion.The receptivity of inviscid modes is investigated as well by using finite-Reynolds-number theory and DNS.The finite-Reynolds-number calculations agree with the DNS results for moderate Reynolds numbers,and approach the asymptotic solutions when the Reynolds number is sufficiently large.