Abstract:
A meshless method is developed and implemented for the three-dimensional numerical solution of the unsteady Navier–Stokes equations. The method is based on the discretization of the computational domain using a finite set of distributed computational nodes. To enhance accuracy, a combined approximation of spatial derivatives is employed: for convective fluxes, the Polynomial Least Squares (PLS) method is used, while for viscous fluxes, the Taylor Least Squares (TLS) approximation is applied. A key feature that eliminates asymmetry in the calculation of flow around axisymmetric bodies is the transformation of the orthonormal coordinate system for each pair of nodes during convective flux computation. Reconstruction of state vectors using the MUSCL scheme and gradient vectors ensures second-order spatial accuracy for convective fluxes. Time integration is performed by an explicit Runge–Kutta method. The software implementation in C++ utilizing OpenCL enables computations on graphics processing units (GPUs). The method is validated for the problem of supersonic flow around a sphere; the results demonstrate good agreement with benchmark data, and the deviation of the convective heat flux does not exceed 2 % as the number of nodes increases to $2.5 \cdot 10^7$.
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