Numerical solution of the system of one-dimensional unsteady Navier-Stokes equations for a compressible gas

Abstract

In many problems encountered in modern gasdynamics, the boundary layer approximations are inadequate to account for the dissipative factors-viscosity and thermal conductivity of the gas-and the solution of the complete system of Navier-Stokes equations is required. This includes, for example, flows with large longitudinal pressure gradients, which in order of magnitude are comparable with or exceed the transverse gradients (temperature jumps, sharp flow rotations, compression shocks, etc.). In many cases, for example in flows with low density, the scale of action of the longitudinal gradients becomes significant, which leads to the need for considering the flow structure in the vicinity of the large gradients. The formulation of certain problems of this type leads to a system of one-dimensional Navier-Stokes equations. We present a difference scheme for the solution of the system of one-dimensional stationary and nonstationary Navier-Stokes equations and give examples of the calculation of the structure of the stationary shock wave front, unsteady gas flow under the influence of sudden heating of one of the boundaries, and unsteady gas flow in the vicinity of the decay of an initial discontinuity. The solution of the stationary problems is accomplished as a result of stabilization as t → ∞.