Recent Progress on Supercomputer Modelling of High-Speed Rarefied Gas Flows Using Kinetic Equations

Anna A. Frolova, Vladimir A. Titarev


Numerical solution of the Boltzmann equation for stationary high-speed flows around complex three-dimensional bodies is an extremely difficult computational problem. This is because of high dimension of the equation and lack of efficient implicit methods for the calculation of the collision integral on arbitrary non-uniform velocity grids. Therefore, the use of the so-called model (approximate) kinetic equations appears to be more appropriate and attractive. This article uses the numerical methodology recently developed by the second author which includes an implicit method for solving the approximating kinetic equation of E.M. Shakhov (S-model) on arbitrary unstructured grids in both velocity and physical spaces. Since most of model equations have a well-known drawback associated with the velocityindependent collision frequency it is important to determine the deviations of solutions of these equations from the solution of the complete Boltzmann equation or DSMC for high-speed gas flows. Our recent comparison of the DSMC and S-model solutions for monatomic gases with a soft interaction potential shows good agreement of surface coefficients of the pressure, heat transfer and friction, which are most important for industrial applications. In this paper, we compare the solution of model equations and the Boltzmann equation for the problem of supersonic gas flow around a cylinder when molecules interact according to the law of hard spheres. Since this law of molecular interaction is the most rigid, the difference in solutions can show the maximum error that can be obtained by using model equations instead of the exact Boltzmann equation in such problems. Our high-fidelity computations show that the use of model kinetic equations with adaptation in phase space is very promising for industrial applications.

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Kashkovsky, A., Bondar, Y., Zhukova, G., Ivanov, M., Gimelshein, S.: Object-oriented software design of rReal gas effects for the DSMC method. In: 24th International Symposium on Rarefied Gas Dynamics. AIP Conference Proceedings. vol. 762, pp. 583–588 (2004). DOI: 10.1063/1.1941599

Kolobov, V., Arslanbekov, R., Aristov, V., Frolova, A., Zabelok, S.: Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J. Comput. Phys. 223, 589–608 (2007). DOI: 10.1016/

Lofthouse, A.: Nonequilibrium hypersonic aerothermodynamics using the direct simulation Monte Carlo and Navier-Stokes models. In: Ph.D. thesis. The University of Michigan (2008)

Sadovnichy, V., Tikhonravov, A., Voevodin, Vl., Opanasenko, V.: ”Lomonosov”: Supercomputing at Moscow State University. In: Contemporary High Performance Computing: From Petascale toward Exascale. pp. 283–307. Chapman & Hall/CRC Computational Science, CRC Press, Boca Raton, USA (2013)

Shakhov, E.: Generalization of the krook kinetic relaxation equation. Fluid Dynamics 3(5), 95–96 (1968), DOI: 10.1007/bf01029546

Tcheremissine, F.G.: Solution to the Boltzmann kinetic equation for high-speed flows. Computational Mathematics and Mathematical Physics 46(2), 315–329 (2006). DOI: 10.1134/s0965542506020138

Titarev, V.A.: Efficient deterministic modelling of three-dimensional rarefied gas flows. Commun. Comput. Phys. 12(1), 161–192 (2012). DOI: 10.4208/cicp.220111.140711a

Titarev, V.A.: Application of model kinetic equations to hypersonic rarefied gas flows. Computers & Fluids, Special issue “Nonlinear flow and transport” 169, 62–70 (2018).

DOI: 10.1016/j.compfluid.2017.06.019

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