High Resolution Entropy Consistent Scheme For Two-dimensional Euler Equations Based On Unstructured Meshes

Two-dimensional Euler equation is a basic equation of gas dynamics and a class of classical hyperbolic conservation law equations.Its characteristic is that even if the initial condition is continuous,the solution may be discontinuous,such as shock wave,rarefaction wave and contact discontinuity.In general,it is difficult to find its analytical solution,which requires in-depth study on numerical solution.In recent years,for two-dimensional Euler equations,the entropy consistent scheme of structured grids has been developed greatly.In practical problems,the computational boundaries are complex,and the structured grids are difficult to meet the computational needs,so it is necessary to divide the computational area into unstructured grids.In this paper,the numerical scheme based on entropy theory is studied in detail,and the entropy consistent scheme is constructed on the basis of the entropy stability scheme of unstructured grids.By linear reconstruction of variables,the high resolution entropy consistent scheme is constructed.Numerical experiments show that the method has less smear phenomenon and non-oscillation.The main work of this paper is as follows:1)Distmesh is used for triangular mesh partitioning of the calculation area.And the cellvertex type finite volume method is used to the two-dimensional Euler equation which select the area bounded by the lines of the triangle center as the control body.2)Based on the entropy theory,this paper analyzes the entropy changes across the discontinuity.Based on the entropy stability scheme of unstructured grids,a new dissipative term modified entropy stability scheme is proposed,and a more accurate entropy consistent scheme is proposed.3)The entropy consistent scheme constructed is first-order.Although the appearance of non-oscillations is suppressed,there is excessive smear phenomenon.Therefore,in this paper,the least square method is used to approximate the gradient.Then the limiter is used to linearly reconstruct the variables at the interface of the lines of the triangle center.And the reconstructed variables are applied to the entropy consistent scheme.Therefore,a high resolution entropy consistent scheme for solving two-dimensional Euler equations is obtained.4)The entropy consistent scheme of the constructed unstructured mesh and the high resolution entropy consistent scheme are numerically tested.The good properties of the two schemes in solving two-dimensional Euler equations are proved.The comparison of the numerical results of the two schemes shows that the high resolution entropy consistent schemes have higher resolution and accuracy for discontinuities.