Mathematical statement of boundary conditions on a burning surface in the Godunov method

Аuthors

Koroleva M. R.^*, Tenenev V. A.^**

Udmurt Federal Research Center Ural Branch of the Russian Academy of Sciences,

*e-mail: koroleva@udman.ru
**e-mail: v.tenenev@gmail.ru

Abstract

The work is devoted to the formulation of boundary conditions on the surface of a burning material when solving a problem using the Godunov method. The purpose of the work is to obtain the basic mathematical relationships that allow estimating the parameters on the combustion surface taking into account gas-dynamic processes in the flow. Most often, for determining the combustion rate of a material, the pressure in the boundary cell of the mesh is used. This is done in order to reduce the numerical complexity of the applied numerical algorithms. In the proposed approach, the numerical solution is based on the exact solution of the Riemann problem. When setting the boundary conditions, this problem is implemented in the opposite direction. Partially knowing the parameters on the contact discontinuity, the values in the fictitious cell of the mesh are restored and the missing values on the decay are determined. In this case, the equations of state of real gases are used. They more accurately describe the physicochemical properties of media at high pressures compared to the equation of state of a perfect gas, but lead to a complication of the mathematical model of the problem. As a result of the conducted research, the basic mathematical equations were obtained, which allow solving the problem of setting boundary conditions using two equations of state – Dupre-Abel and van der Waals. Algorithms for finding all the parameters in the Riemann problem are presented step by step. The described approach was used to solve a model problem of the motion of an inert body, to the end of which a burning element is attached. The problem was solved in a one-dimensional statement on a moving calculation mesh. The combustion rate obtained within the framework of the proposed approach is lower than the combustion rate obtained using a simplified boundary condition by 2.5%. This difference tends to increase in the case of a longer time interval. The obtained relations can be used in the numerical solution of practical problems with burning surfaces and make the use of equations of state of real gases more accessible for solving such problems.

Keywords:

boundary conditions, burning surface, Godunov method, Riemann problem, Dupre’s equation of state, van-der-Waals’s equation of state

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