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JOURNALS // Computer Research and Modeling // Archive

Computer Research and Modeling, 2022 Volume 14, Issue 1, Pages 23–43 (Mi crm953)

This article is cited in 1 paper

NUMERICAL METHODS AND THE BASIS FOR THEIR APPLICATION

Relaxation model of viscous heat-conducting gas

V. S. Surov

South Ural State University (National Research University), 76, Lenin prospect, Chelyabinsk, 454080, Russia

Abstract: A hyperbolic model of a viscous heat-conducting gas is presented, in which the Maxwell–Cattaneo approach is used to hyperbolize the equations, which provides finite wave propagation velocities. In the modified model, instead of the original Stokes and Fourier laws, their relaxation analogues were used and it is shown that when the relaxation times $\tau_{\sigma}$ and $\tau_w$ tend to the hyperbolized equations are reduced to zero to the classical Navier–Stokes system of non-hyperbolic type with infinite velocities of viscous and heat waves. It is noted that the hyperbolized system of equations of motion of a viscous heat-conducting gas considered in this paper is invariant not only with respect to the Galilean transformations, but also with respect to rotation, since the Yaumann derivative is used when differentiating the components of the viscous stress tensor in time. To integrate the equations of the model, the hybrid Godunov method (HGM) and the multidimensional nodal method of characteristics were used. The HGM is intended for the integration of hyperbolic systems in which there are equations written both in divergent form and not resulting in such (the original Godunov method is used only for systems of equations presented in divergent form). A linearized solver's Riemann is used to calculate flow variables on the faces of adjacent cells. For divergent equations, a finite-volume approximation is applied, and for non-divergent equations, a finite-difference approximation is applied. To calculate a number of problems, we also used a non-conservative multidimensional nodal method of characteristics, which is based on splitting the original system of equations into a number of one-dimensional subsystems, for solving which a one-dimensional nodal method of characteristics was used. Using the described numerical methods, a number of one-dimensional problems on the decay of an arbitrary rupture are solved, and a two-dimensional flow of a viscous gas is calculated when a shock jump interacts with a rectangular step that is impermeable to gas.

Keywords: viscous heat conducting gas, hyperbolic model, Stokes, Fourier relaxation laws, hybrid Godunov's method, multidimensional nodal method of characteristics.

UDC: 532.529.5

Received: 18.09.2021
Revised: 28.12.2021
Accepted: 03.01.2022

DOI: 10.20537/2076-7633-2022-14-1-23-43



© Steklov Math. Inst. of RAS, 2025