Meshless algorithm for calculating supersonic viscous gas flows

DOI: 10.34759/trd-2021-121-09

Аuthors

Sposobin A. V.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: spise@inbox.ru

Abstract

Тhe article presents in detail the algorithm for the numerical solution of the Navier-Stokes equations [12] by the meshless method [8, 10]. The described method is used for numerical simulation of blunt bodies flow-around by supersonic viscous gas flow [1, 2]. Cartesian-grid-based immersed boundary method presented in previously published works was successfully applied to simulate such flows in the two-dimensional planar and an axisymmetric formulation [3]. Numerical studies of the gas thermal impact change on the streamlined surface while the highly inertial particle motion against the incoming flow along the symmetry axis were performed by dint of it [4, 5, 6]. Particles motion along various trajectories required gas-dynamics problems solving in the 3D formulation, for which the Cartesian-grid based methods application required too much computer memory.

When solving gas-dynamics problems by the meshless method, a finite sets of nodes is being selected in the space. Approximation by the least square method is applied for spatial derivatives computing in computational nodes. The said approach is being used for convective and viscous fluxes computing. The convection fluxes are being computed by the AUSMPW+ method in conjunction with the MUSCL scheme and van Abada limiter. The system of equations time integration is being performed by the explicit Runge-Kutta method of the third order of accuracy. The flow-around surface is being represented by the model of isothermal wall with the specified temperature. Conditions of gas adhesion, as well as pressure gradient equality to zero, which modeling also employs the least square method, is realized on the said model.

The meshless method gives the opportunity to compute the gas flow in the areas with complex geometry, it is simpler in realization herewith compared to the finite volumes method, since it does not require generation of the high-quality computational grid. It allows setting anisotropic distribution of nodes in space, which is of vital importance for qualitative resolution of the boundary layer near the flow-around surface.

Along with the 3D realization, adaptation of meshless method for computing flat and axisymmetric viscous gas flows was performed.

The software implementation of the described method is realized in the C++ programming language and employs the OpenMP technology for computations parallelization.

A computational experiment was performed on modeling the sphere flow-around by the supersonic airflow at the Mach number of M = 3 and Reynolds number of Re = 10⁵ was conducted for the said method verification. The spatial shadow patterns of the flow, pressure field and Mach number are presented. Comparison of gas parameters in the boundary layer, obtained by the meshless method with the computational results of continuous flow combined with the boundary layer equations is presented. Correspondence of the gas computational parameters on the flow-around surface to the reference data and approximate-analytical expressions is demonstrated. The heat flow value in the critical point coincides with the one calculated with the Fay and Riddell formula, and the heat flow distribution curve along the surface is close to the approximate-analytical one.

The next stage in the development of the meshless method is planned to solve numerically the unsteady multiphase flows problem, in particular, to simulate the solid particles motion in the shock layer.

Keywords:

numerical simulation, meshless method, Navier-Stokes equation, supersonic flows around bodies, heat flux

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