HEAT- AND MASS-TRANSFER EFFECTS ON THREE-DIMENSIONAL HYPERSONIC VISCOUS SHOCK-LAYER FLOWS

The fully viscous shock-layer equations have been solved for hyperboloids at zero lift and spherically blunted cones at large angles of attack. The calculations include the effects of slip and/or mass transfer. Comparisons were made with previous shock-layer calculations, experimental data, and solutions of the parabolized Navier-Stokes equations. Mach numbers ranging from 10 to 23, Reynolds numbers in the range 5.2 x 103 to 3.8 x 1Q4, and angles of attack up to 40 deg were considered in the calculations. A case with an asymmetric mass-transfer distribution also is included to demonstrate the capabilities of the method. Nomenclature A-J = coefficients of a parabolic differential equation Aref = reference area based on local r 2 2 o cS = specific heat at constant pressure, ft /sec R CA = 2 axial force/pooUlAref 2 CD = drag coefficient, 2 drag/A^fpJJoo Cf = streamwise friction coefficient, 2T*/pooU£ Cf = transverse friction coefficient, 2T,/pooU£ CM = 2 rolling moment/pooU£ArefZ CMY = 2 yawing CMZ = 2 pitching moment/pooU£ArefZ Presented as Paper 78-844 at the 2nd AIAA/ASME Thermophysics and Heat Transfer Conference, Palo Alto, Calif., May 2426, 1978. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1978. All rights reserved. ^Research Associate, Aerospace and Ocean Engineering Department; currently at Acurex Corporation, Mountain View, Calif. Professor, Aerospace and Ocean Engineering Department. 165 166 A.L MURRAY AND C. H. LEWIS CN = 2 normal force/p~u|Aref Cz = 2 side force/p00U^Av:ef h = static enthalpy, h /U£ hi = shape factor in the streamwise direction, h-j = 1 + shape factor in the cross-flow direction, coses)/cose ( j ) H = stagnation enthalpy, H*/u£ i = step counter in the streamwise direction j = step counter in the normal direction k = ratio of step sizes in the normal direction m = mass of one gas molecule M = Mach number n = normal distance from body surface, n*/Rn p = pressure, p*/PTOU2 Pr = Prandtl number q = heat-transfer rate, q*/pjj3 ^ ^ r = distance from and normal to body axis, r /Rn R = gas constant, ft/sec2-QR Rn = nose radius, ft St = Stanton number, -qVpJ T = temperature, T /T*ef T*ef = reference temperature, U / C p or u = tangential velocity in the streamwise direction, u /U v = normal velocity, v /Uoo w = tangential velocity in the cross-flow direction, wVl W = general dependent variable X3 = h3/r Z = axial length, Z*/Rn a = angle of attack, deg y = ratio of specific heats e = Reynolds number parameter, e = y*(T*ef )/vJJjtf\ C,cj) = transverse coordinate in the coordinate system (£,n»5 n = normalized coordinate in the direction normal to the body surface, n/nsh es = body angle in the streamwise direction (see Fig. 1) e^ = body angle between the radius vector and the line tangent to the body surface in the plane 5 = const (see Fig. 1 ) KS = body curvature in the £ direction (Fig. 1) K^ = body curvature ir\ the c direction (Fig. 1) y = viscosity, y*/y*(T*ef) £,s = streamwise coordinate in the system (5 ,n»c ) p = density, P*/POO a = angle between the freestream velocity vector and the vector tangent to the shock surface in the plane formed by the shock-normal vector T = shear-stress component 3-D VISCOUS SHOCK-LAYER FLOWS 167 Subscripts aw = adiabatic wall conditions I = conditions in the injection chamber s = streamwise direction sh = conditions behind the bow shock wave w = wall condition 0 = stagnation-point conditions 5 = derivative with respect to £ $ = transverse direction °° = dimensional freestream conditions Superscripts (") = variable that has been normalized by the value behind the bow shock wave * = dimensional quantity Introduction The viscous shock-layer approach has received considerable attention in recent years for studies in re-entry aerodynamics. 1-6 Since the shock-layer equations treat the entire flowfield from the body to the shock in a uniform manner, the displacement effects of the viscous region and the boundary conditions are simple to treat. This is not the case with a classical two-layer model, where the flowfield is divided into an inviscid outer flow and a viscous boundary-layer region. At the low Reynolds numbers encountered during re-entry, the effect of the displacement thickness interaction must be considered in the inviscid calculation because the viscous region is a major portion of the flowfield. The viscous shock-layer equations are parabolic in both streamwise and cross-flow directions and may be solved by efficient methods. The parabolic characteristic gives the shock-layer approach the ability to march downstream within reasonable computing times. The parabolized Navier-Stokes (PNS) equations? are parabolic in the streamwise direction but are elliptic in the cross-flow direction and require matrix inversion to obtain solutions. The PNS procedure also requires a small step in the streamwise direction for stability and convergence and large amounts of computing time. The viscous shock-layer approach has been applied to axisymmetric re-entry vehicles at altitudes up to 300 kft, where low Reynolds number effects such as shock and wall slip become important. In addition, the severe heating encountered during re-entry causes the body material to ablate, adding mass to the flowfield. Both slip and mass transfer can be modeled with 168 A.L MURRAY AND C. H. LEWIS modifications to the boundary conditions of the shock-layer equations. In the present work, a three-dimensional viscous shocklayer (VSL3D) procedure has been used to calculate flowfields about sphere-cones at angles of attack with slip and mass transfer. Comparisons were made with an axisymmetric shocklayer calculation to determine differences in the solution procedures. Predictions including the effects of slip were compared with experimental heat-transfer and wall pressure distributions and with experimental drag data. Finally a calculation was made for a sphere-cone at angle of attack with an asymmetric mass-transfer distribution to demonstrate the capabilities of the code. Analysis The viscous shock-layer equations are derived from the steady Navier-Stokes equations in a surface-oriented coordinate system (s,n,(j)) (see Fig. 1). The system is an orthogonal coordinate system such that n is the coordinate normal to the surface, s is tangent to the body in the streamwise direction, and $ is the angle around the b h., an h.,h0 a I 0 1 10 2 pw aho 1 ap p 3 h,h~ as has an 3-D VISCOUS SHOCK-LAYER FLOWS 169 _L • + e Fig. 1 Body geometry. ' _L ° 1 U — —— + — —— I ruy — I — h, 8n 8n 3n '1 The normal momentum equation requires some special treatment, since retaining terms of 0(e) in the right-hand side include cross derivatives that keep the equation from being parabolic. For this reason, the right-hand side of the normal momentum equation is only first order. n Momentum 2 2 pU 8V 8V pW 8V pU 8h-, pW 8h~ ____ ____ + y ____ _|_ ____ ____ _ ______ ____L ______ ____—— 8s 8n ., 8n 8n 8P 8n The ^-momentum and energy equations are completely 0(e) and are derived in the same manner as the s momentum: cj) Momentum pU 8W 8W pW 8W pUW 8h, —— —— + p V —— + — — — — + ————^ h-t 8s 8n hQ 9(f) hnhQ 8s pVW pU