Accuracy of asymptotic solutions for nonuniform supersonic flow past a blunt body

Abstract

532.526.2The study of flow field and heat exchange about blunt bodies when the oncoming supersonic stream is substantially nonuniform has been of much practical interest lately. In [1] we considered the results of experimental and theoretical study of the resistance, heat exchange, and the gasdynamic picture of the flow past a pair of bodies, one of which is behind the other in a supersonic stream. The experimental results in [1] were obtained for a relatively small separation between the bodies (no more than 20 calibers). The study is carried out by theoretical methods when the bodies are separated by a large distance (several hundred calibers) [2][3][4][5][6]. In [2, 3] we obtained asymptotic solutions for the problem of the flow of a nonuniform wake-type stream past a blunt body for moderate (Re** < 103) and high (Re** > 105) Reynolds numbers. In [1,[4][5][6] as a result of numerical solution of the equations of a thin viscous shock layer (on the assumption that the shock wave is equidistant from the surface of the body) we obtained relations for the heat exchange, friction, and the criterion of flow without separation as functions of the parameters of the problem.For a uniform mainstream the method of a thin viscous shock layer (TVSL) gives results that are in satisfactory agreement with calculations with more exact methods [7, 8]. The applicability of the TVSL method [4-6] and asymptotic formulas [2, 3] to the case of a nonuniform supersonic stream of the far wake type flowing past a blunt body has not been examined sufficiently.It is a particularly complicated matter to prove that the asymptotic expansions converge to an exact solution of the problem when there are several small parameters in which the expansions are made (t = P**/Ps, Re** -lcz, M** -2, etc., where p** and Ps are the density in the mainstream and behind the step, Re** and M** are the Reynolds and Mach numbers). The answer to these questions can be obtained either from systematic comparisons of the calculated gasdynamic parameters over a wide range with the results of specially designed aerodynamic experiments (which are often complex or impossible) or from a comparison with numerical solutions of more exact (unsimplified) equations of gasdynamics.This study of a nonuniform supersonic flow past blunt bodies is based on the equations of a complete viscous shock layer (CVSL), which are solved numerically by using the effective method of global iterations [7][8][9][10][11]. The high accuracy and speed of this method as applied to CSVL equations have been confirmed by a comparison with experiment and the results of numerical solution of the Navier-Stokes equations by the method of fixing [7, 9, 10].As shown here, in the case of low Reynolds numbers (Re** = 50-100) the asymptotic formulas for a heat flux from [2] give practically the same results as does the numerical solution of TVSL equations. A comparison of the numerical solution of TVSL and CVSL equations showed that for a wake-type nonuniform mainstream the TVSL method leads to substantia...