Approximate Solutions for Shock Waves in a Steady, One-Dimensional, Viscous and Compressible Gas

Abstract

In this paper the equations of motion for a steady, one-dimensional, viscous and compressible gas are simplified according to two procedures and are then integrated. In one procedure the viscosity term is retained in the momentum equation and omitted in the energy equation. In the second procedure the viscosity term is omitted in the momentum equation and retained in the energy equation. The equations used in the first case are the momentum equation, into which is introduced the frictional force [= (4/3)judfc-rdx]; the continuity equation; and a polytropic relationship between pressure and density, the last expression replacing the energy equation in its usual form. The value of the exponent n, in the polytropic relationship for which the Rankine-Hugoniot points are satisfied, is shown to be a function of the initial Mach Number, Mo, only. I t is strongly indicated that retaining the viscosity terms in the momentum equation and neglecting them in the energy equation would offer a significant and considerably simplified mathematical representation of a steady compressible viscous flow in two and three dimensions. In the second case, the Eulerian equation, together with the continuity and energy equations for a one-dimensional motion, describes approximately the flow of a viscous, heat-conducting gas. This is analogous to the treatment of Boley and Lieber for two-dimensional flow. An exact solution is obtained for the shock-wave structure in both cases, and, although the velocity distribution in the latter case is continuous everywhere, the singularity in the velocity gradient is not removable. However, while the present report shows that retaining the viscous term in the energy equation alone is not sufficient for the removal of this singularity, it is shown that the frictional force, when considered in the momentum equation, does remove the singularity. Since the shock-wave thickness, according to the Prandtl definition, is zero in the second treatment (a physical impossibility), it was not found necessary to calculate the variation of the remaining flow variables (p, p, S, T) with Mach Number. Good agreement with the exact one-dimensional solutions of Morduchow and Libby and H. Reissner and MeyerhofF with regard to the structure of the shock wave and its thickness is obtained when the effect of viscosity is considered only in the momentum equation. Moreover, indications are such as to justify the conclusion that the presence of the viscous stress term in the momentum equation is a necessary condition for the removal of the singular velocity gradient in one-dimensional flow.