Improvement of the meshless method for numerical simulation of supersonic viscous gas flows

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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 $2.5 \cdot 10^7$.

About the authors

Dmitry L. Reviznikov

Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: reviznikov@inbox.ru
ORCID iD: 0000-0003-0998-7975
SPIN-code: 9763-9898
Scopus Author ID: 6602701797
ResearcherId: T-4571-2018
https://www.mathnet.ru/rus/person26170

Dr. Phys. & Math. Sci., Professor; Professor; Dept. of Computational Mathematics and Programming

Russian Federation, 125993, Moscow, Volokolamskoe Shosse, 4

Andrey V. Sposobin

Moscow Aviation Institute (National Research University)

Email: spise@inbox.ru
ORCID iD: 0009-0004-7720-2556
SPIN-code: 3028-0859
Scopus Author ID: 42462268200
ResearcherId: AAV-1002-2020
https://www.mathnet.ru/rus/person34344

Dr. Phys. & Math. Sci.; Senior Researcher; R&D Dept. 806

Russian Federation, 125993, Moscow, Volokolamskoe Shosse, 4

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Supplementary files

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2. Figure 1. Distribution of computational nodes in the central cross-section of the computational domain

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3. Figure 2. Distribution of computational nodes on the sphere surface

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4. Figure 3. Schematic diagram of a computational node and its neighborhood (cloud of neighboring nodes)

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5. Figure 4. Distribution of neighbor node clouds in the computational domain

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6. Figure 5. Schlieren image of supersonic flow past a sphere

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7. Figure 6. Gas density distribution in the central cross-section of the computational domain for supersonic flow past a sphere

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8. Figure 7. Gas pressure distribution in the central cross-section of the computational domain for supersonic flow past a sphere

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9. Figure 8. Mach number distribution in the central cross-section of the computational domain for supersonic flow past a sphere

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10. Figure 9. Comparison of the calculated surface pressure distribution on the sphere with benchmark data

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11. Figure 10. Comparison of the calculated convective heat flux to the sphere surface with an approximate analytical solution

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