The response of a laminar boundary layer in supersonic flow to small-amplitude progressive waves

Abstract

In this paper the effect of a small-amplitude progressive wave on the laminar boundary layer on a semi-infinite flat plate, due to a uniform supersonic free-stream flow is considered. The perturbation to the flow caused by the wave divides into two streamwise zones. In the first, relatively close to the leading edge of the plate, on a transverse scale comparable with the boundary-layer thickness, the perturbation flow is described by a form of the unsteady linearized compressible boundary-layer equations. In the free stream, this component of flow is governed by the wave equation, the solution of which provides the outer velocity conditions for the boundary layer. This boundary-layer system is solved numerically, and also the asymptotic structure in the far downstream limit is studied. This reveals a breakdown and a subsequent second streamwise zone, where the flow disturbance is predominantly inviscid. The two zones are shown to match in a proper asymptotic sense.

Aeroacoustics of Supersonic Boundary Layers

“…It was shown by Gaponov [13] that its value can vary within wide limits, and it can either exceed unity in modulus or vanish. A possibility of the reflection with a factor being less then unity is in contradiction with the data of Duck [18]. It should be noted that the results of Duck [18] were obtained without regard for the viscosity effect which in the stability theory is related to the quantity αR.…”

Acoustic Receptivity of Compressible Tollmien-Schlichting Waves with an Efficient Time-Harmonic Linearized Navier-Stokes Method

“…This approach will henceforth be denoted local LNS. The Stokes layer has a double-layer asymptotic structure in the high Strouhal number limit [15,26]. This means that, for a fixed frequency, the Stokes layer has a constant thickness, as well as constant velocity and temperature disturbance profiles in the downstream limit.…”

“…While our profiles agree qualitatively, we have not attempted to establish a quantitative comparison because it proved hard to extract reliable data from the figures of TR. Another source of comparison comes from the work of Duck [15]. Duck established the linearized compressible boundary layer equations through a direct linearization of the unsteady form of Eq.…”

“…Federov [14] used an adjoint-based approach to determine acoustic receptivity coefficients for Mach numbers up to 0.8. Duck [15] focused on the forced acoustic wave signature in the boundary layer. He first considered the boundary layer approximation governing an acoustic wave traveling parallel to the flat plate.…”

“…Consequently, the LUBLE are suitable to model the forced acoustic response. But when one considers an incident wave at an angle, this approximation ceases to be valid [15,16,18].…”