# Shock wave interaction with viscous wake in supersonic flow

Battam, N. W. and Gorounov, D. G. and Korolev, G. L. and Ruban, A. I. (2004) Shock wave interaction with viscous wake in supersonic flow. Journal of Fluid Mechanics, 504. pp. 301-341. ISSN 0022-1120

A theoretical investigation of the breakdown of the viscous wake downstream of a flat plate in supersonic flow is performed in this paper based on the large Reynolds number ($Re \to \infty$) asymptotic analysis of the Navier–Stokes equations. The breakdown is provoked by an oblique shock wave impinging on the wake a small distance $l_s$ downstream of the plate trailing edge. Two flow regimes are considered. In the first $l_s$ is assumed to be an $O(Re^{-3/8})$ quantity in which case the shock impinges on the wake within the region of viscous–inviscid interaction that is known to occupy a vicinity of the trailing edge with longitudinal extent of $O(Re^{-3/8})$. Under these conditions the interaction process may be described by the equations of the triple-deck theory. To obtain a numerical solution of these equations we used a rapid matrix Thomas technique in conjunction with Newton iterations. The results of the calculations not only predict the wake breakdown near the shock location but also reveal a hysteresis behaviour of the flow as the shock is moved downstream, giving rise to three solution branches. The second part of the paper is concerned with the flow regime when the shock interacts with the wake further downstream of the trailing edge triple-deck region: $l_s \gg Re^{-3/8}$. In this case the fluid motion proves to be inviscid to leading order not only in the upper deck of the interaction region but also everywhere inside the wake. Due to this simplification the interaction problem can be reduced to a single integro-differential equation governing the pressure distribution along the interaction region. With known pressure the Bernoulli equation may be used to find the velocity field. The Bernoulli equation also allows us to formulate a simple criterion which may be used to predict the onset of wake breakdown. We found that viscosity becomes important again in a smaller vicinity of the breakdown point where the flow reversal takes place. It is remarkable that the viscous–inviscid interaction problem governing the flow in this vicinity admits an analytical solution.