Numerical dissipation control of a hybrid large-particle method in vortex instability problems

Abstract

Current trends in the development of numerical schemes are associated with a decrease in dissipative and dispersion errors, as well as an improvement in the grid convergence of the solution. Achieving the computational properties is not an easy problem since a decrease in scheme viscosity is often associated with an increase in the oscillations of gas dynamic parameters. The paper presents a study of the issues of numerical dissipation control in gas dynamics problems in order to increase the resolution in numerical reproduction of vortex instability at contact boundaries. To solve this problem, a hybrid large-particle method of the second order of approximation in space and time on smooth solutions is used. The method is constructed with splitting by physical processes in two stages: gradient acceleration and deformation of the finite volume of the medium; convective transfer of the medium through its facets. An increase in the order of approximation in time is achieved by a time correction step. The regularization of the numerical solution of problems at the first stage of the method consists in the nonlinear correction of artificial viscosity which, regardless of the grid resolution, tends to zero in the areas of smoothness of the solution. At the stage of convective transport, the reconstruction of fluxes was carried out by an additive combination of central and upwind approximations. A mechanism for regulating the numerical dissipation of the method based on a new parametric limiter of artificial viscosity is proposed. The optimal adjustment of the method by the ratio of dissipative and dispersive properties of the numerical solution is achieved by setting the parameter of the limiting function. The efficiency of the method was tested on two-dimensional demonstrative problems. In one of them, the contact surfaces are twisted into a spiral on which the Kelvin-Helmholtz vortex instability develops. Another task is the classic problem with a double Mach reflection of a strong shock wave. Comparison with modern numerical schemes has shown that the proposed variant of the hybrid large-particle method has a high competitiveness. For example, in the problem with double Mach reflection, the considered version of the method surpasses in terms of vortex resolution the popular WENO (Weighted Essentially Non-Oscillatory) scheme of the fifth order and is comparable to the numerical solution of WENO of the ninth order of approximation. The proposed method can be the basis of a convective block of a numerical scheme when constructing a computational technology for modeling turbulence.