PMM U.S.S.R.,Vo1.53,No.6,pp.762-767,1989 0021-8928/89 $lO.OO+O.OO Printed in Great Britain 01991 Pergamon Press plc A COMPARISON OF THE MODELS OF A THIN AND A COMPLETE VISCOUS SHOCK LAYER IN THE PROBLEM OF THE SUPERSONIC FLOW OF A VISCOUS GAS PAST BLUNT CONES* G.A. TIRSKII and S.V. UTYUZHNIKOV The flow of a viscous heat conducting supersonic gas past spherically blunted cones is used to compare the solutions of the equations of a thin (hypersonic) viscous shock layer (TVSL) with a given form of the shock wave (SW), with the solutions of the complete equations of a viscous shock layer (CVSL) in which the assumption that the shock layer is thin is not made and, which is important, the form of the SW is determined in the course of solving the problem. It is shown that a "successful" description of the form of the SW in solving the problem of hypersonic flow past a blunt cone within the framework of the equations of a TVSL provides, firstly, the possibility of obtaining the solution at considerable distances downstream and secondly, of sharpening the solution considerably, assuming that it can be obtained at all within the framework of the equations of the TVSL, compared with the commonly used asymptotic approach in which the form of the SW is assumed, in solving these equations, to be an equidistant form of the body. The description of the supersonic flow of a viscous, heat conducting gas past a body using the simplified (parabolized) Navier-Stokes equations employs, as a rule, the Cheng two-layer model /l, 2/. According to Cheng, the whole perturbed region of the gas in front of the body can be divided into a region of viscous shock layer, and a transitional region corresponding to a density jump. The transition region is described by a system of ordinary differential equations which transforms, after appropriate simplifications, into the generalized Rankine-Hugoniot conditions. The region of the viscous shock layer is described using various systems of equations which are obtained either by an asymptotic method involving the separation of one or several small parameters of the problem (see e.g., /3/l, or by a heuristic method involving an estimation of the contribution of each term of the system of equations /4/. In both cases the domain of applicability of the model is not clear in advance, and can be determined only be comparison with the complete Navier-Stokes equations. It should be noted that the domain of applicability of the simplified Navier-Stokes equations used to obtain a number of basic aerodynamic and thermal characteristics is found, as a rule, to be much wider than that of the formal asymptotic estimates. 1. Modelof a thin viscous shock layer U'VSL). Historically, the first system of simpli- fied Navier-Stokes equations, which is also the one most often used by virtue of its mathemat- ical simplicity, is the system of equations of the TVSL (see e.g., /l-6/), obtained under the assumption that Y+ 1, M, --f 00, Re,- x (y is the adiabatic ratio, M is the Mach number and Re is the Reynolds number). The equations of the TVSL contain all terms appearing in the equations of a non-viscous hypersonic shock layer /7/. The system of two-dimensional equations of the TVSL in a curvilinear system of coordinates attached to the body, has the form (1.1) Here x is the arc length of the contour of the body, y is the distance between the normal and the surface of the body, u and v are the physical velocity components in the x and y directions, H is the total specific enthalpy, o is the Prandtl number, R (5) is the radius of curvature of the contour of the body, H, and 21are the metric Lam& coefficients, HI=1 + Y/Q. r is the distance between the given point of space and the axis of the body, v .:0 "PrikZ.Matem.Mekhan.,53,6,963-969,1989 762