Supersonic Boundary Layer in Regions with Small Temperature Perturbations on the Wall
The problem of disturbed flow in the laminar boundary layer when heating elements are located on the surface is investigated. The flow is described by the theory of viscous-inviscid interaction with full Navier–Stokes equations. We apply the triple-deck model of the flow and describe asymptotic multiscale structure for the supersonic case. Using this structure as a base, we construct the solution to the flat stationary problem by means of Fourier transform and analysis in the complex plane. It is also shown that localization of a heating element may arrive at a singularity of the pressure downstream. Results of numerical and analytic analysis are presented. Using these elements gives us the opportunity to develop a novel method for the separation control of the boundary layer.
Local temperature perturbations of the boundary layer in the regime of free viscous–inviscid interaction
Abstract We analyse the disturbed flow in the subsonic laminar boundary layer, disturbances being generated by local heating elements, which are placed on the surface. It is exhibited that these flows are described in terms of free interaction theory for specific sizes of thermal sources. We construct the numerical solution for the case of a flat subsonic stream in the viscous asymptotic layer, in which the flow is described by nonlinear equations for vorticity, temperature and an interaction condition which provides the influence of perturbations to the pressure in the main order. The obtained solutions are compared with those for corresponding linear problems with small perturbations. It is demonstrated that strong temperature perturbations in some situations allow us to obtain the flow close to the separated flow.
Self-induced separation
A rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock. In this theory, which is firmly based on the linearized theory of Lighthill (1953), the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary layer in which the pressure variation is small; and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is large but is controlled by the incompressible boundary-layer equations, together with novel boundary conditions. We find that the first two parts can be handled in a straightforward way and the problem of self-induced separation reduces, in its essentials, to the solution of a single problem in the theory of incompressible boundary layers. It is found that this problem has three solutions, one of which corresponds to undisturbed flow and another describes a boundary layer which, spontaneously, generates an adverse pressure gradient and a decreasing skin friction which eventually vanishes and then downstream a reversed flow is set up. The third solution generates a favourable pressure gradient and is not relevant to the present study. Although there has hitherto been no valid numerical method of integrating a boundary layer with reversed flow, we find that an ad hoc method seems to lead to a stable solution which has a number of the properties to be expected of a separated boundary layer. Comparison with experiment gives qualitatively good agreement, but quantitatively errors of the order of 20% are found. It is believed that these errors arise because the Reynolds numbers at which the experiments were carried out are too small.