A hyperbolic model for viscous mixed flows

Abstract

1. INTRODUCTIONThere is a wide class of steady-state viscous gasflows that are important to industrial application inwhich the disturbance propagation upstream is insig-nificant. This class includes limited mixed flows, themost part of which are overlapped by a sonic surface.Since small (acoustic) disturbances do not propagatecounter to a supersonic flow, the physical conditions atthe right-hand boundary situated downstream withrespect to the sonic surface weakly affect the main-stream region [1, 2]. For internal flows, such a regimeis realized, for example, in a Laval nozzle that repre-sents a component of chemical or gas-dynamic lasersor of rocket or other-type engines [1, 3]. In the case ofexternal flows, this effect occurs in the shock layerbeing formed by a supersonic flow around blunted bod-ies [2, 4]. It is unreasonable to use the completeNavier–Stokes equations for numerical simulation ofsuch flows. This is especially true while calculatingchemically and thermally nonequilibrium gas-mixtureflows at moderate or large values of the Reynolds num-ber [2, 3].The most efficient models describing these flows arebased on systems of parabolic or hyperbolic equations.These equations are evolutionary with respect to thelongitudinal coordinate along the dominating flowdirection. Therefore, they can be solved by fast space-marching methods for one downstream run [2, 3].For internal viscous flows in Laval nozzles, suchmodels were proposed in [5–7]. However, models [5, 6]are inadequate for flows with considerable transversepressure gradients, while the applicability of model [7]is limited by moderate values of the longitudinal duct-wall curvature.For the problem of a supersonic viscous gas flowaround a blunted body, nonelliptic models were pro-posed in [8, 9]. However, their efficiency depends onthe azimuth angle counted off from the frontal stagna-tion point.The hyperbolic model [10] proposed recently forinternal mixed viscous flows is free of disadvantagesintrinsic to models [5–7] and allows pressure fieldswith considerable transverse pressure gradients to beadequately reproduced. Here, we propose a new gas-dynamic model for external mixed viscous flows. It isbased on hyperbolic-type equations and is intended todescribe the shock layer being formed by a supersonicflow near a blunted body at large or moderate values ofthe Reynolds number. In contrast to [8, 9], this modelwell reproduces distributions of pressure and heat fluxalong the surface of a body placed into a flow andmakes it possible to calculate flows around thin bodieswith lengths hundreds of times their diameters. Wedemonstrate, as an example, the calculation data for ashock layer formed near a sphere and a very longblunted (in the hemispherical-shape) cylinder in con-tact with a moving viscous fluid. These calculationsagree sufficiently well with the corresponding calcula-tions according to the equations of the full viscousshock layer (FVSL) [11] and the Navier–Stokes equa-tions.2. A FLOW MODELWe consider a steady-state flow of a viscous heat-conducting perfect gas in a shock layer, which formsnear an unyawed either smooth axisymmetric or planeblunted body. According to [2], at moderate or largevalues of the Reynolds number, the descriptions offlows by the FVSL model [11] and by the Navier–Stokes equations are close to each other. Therefore, asinput equations, we use equations which describe theFVSL and are written out in the curvilinear coordinatesystem ( ξ , η ). These equations are (1)(2)g∂∂ξ-----()ρu∂∂η++ 0,------()fρ hρu =g∂∂ξ-----()ρu