Abstract
The planar shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations. As in the case of a planar shock wave we suggest such boundary conditions that the linear initial-boundary value problem on the stability of a curvilinear shock wave (subject to these boundary conditions) is well-posed. We also propose a new effective computational algorithm for investigation the stability. This algorithm uses the nonstationary regularization, the method of lines, the stabilization method, the spline function technique and the sweep method. Applying it we succeed to obtain the stationary solution of the considered boundary-value problem justifying the stability of shock wave.
Highlights
The motion of continuous media is often acco mpanied by the formation of transitional zones of strong gradients, where flow parameters vary rapidly
If d issipative mechanis ms are neglected, such thin zones are usually treated as surfaces of strong discontinuity
Note that motions of ideal continuous media are usually described by hyperbolic conservation laws for which the mathematical theory of shock waves has been well d iscovered for one-dimensional[1,2,3,4,5,6,7,8,9,10] and for mult i-d imensional flows[11,12,13,14,15,16,17,18,19,20,21]
Summary
The motion of continuous media is often acco mpanied by the formation of transitional zones of strong gradients, where flow parameters (velocity, density, pressure, temperature, etc.) vary rapidly. The Navier-Stokes equations are applied for solving the problem on shock structure in a v iscous and heat conducting gas (see, e.g., the classical approach in[22]) In this problem, instead of a surface of strong discontinuity one considers a thin transitional zone (viscous profile) where flo w parameters vary continuously. For this purpose one studies the initial boundary value problem (IBVP) obtained by the linearization of the Navier-Stokes equations and the jump conditions with respect to their piecewise constant solution This piecewise constant solution describes the following flow reg ime for a v iscous gas: a supersonic steady viscous flow (for x > 0 ) is separated fro m a subsonic one (for x < 0 ) by a planar shock discontinuity (with the equation x = 0 ). In the present paper we consider a curvilinear shock wave
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