Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T10:11:28.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Deterministic Modelling of Three-Dimensional Rarefied Gas Flows

Published online by Cambridge University Press:  20 August 2015

V. A. Titarev
V. A. Titarev*
Affiliation:
Dorodnicyn Computing Centre of Russian Academy of Sciences, Vavilov st. 40, Moscow, Russia, 119333 Cranfield University, Cranfield, UK, MK43 0AL
*
*Corresponding author.Email:titarev@ccas.ru, titarev@mail.ru
Get access

Abstract

The paper is devoted to the development of an efficient deterministic framework for modelling of three-dimensional rarefied gas flows on the basis of the numerical solution of the Boltzmann kinetic equation with the model collision integrals. The framework consists of a high-order accurate implicit advection scheme on arbitrary unstructured meshes, the conservative procedure for the calculation of the model collision integral and efficient implementation on parallel machines. The main application area of the suggested methods is micro-scale flows. Performance of the proposed approach is demonstrated on a rarefied gas flow through the finite-length circular pipe. The results show good accuracy of the proposed algorithm across all flow regimes and its high efficiency and excellent parallel scalability for up to 512 cores.

Keywords

82D0576P0582B4082C4076M1265N0865B99KineticS-modelunstructuredmixed-elementmicro channelrarefiedimplicit
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 162 - 192
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andries, P., Aoki, K., and Perthame, B.. A consistent BGK-type model for gas mixtures. Journal of Statistical Physics, 106:993–1018, 2002.CrossRefGoogle Scholar
[2] Yu.Anikin, A., Kloss, Yu.Yu., Martynov, D.V., and Tcheremissine, F.G.. Computer simulation and analysis of the Knudsen experiment of the 1910 year. Journal of nano and microsystem technique, (8), 2010.Google Scholar
[3]Aoki, K., Yoshida, H., and Nakanishi, T.. Inverted velocity profile in the cylindrical Couette flow of a rarefied gas. Physical Review E, 68(016302), 2003.Google Scholar
[4]Aristov, V.V. and Zabelok, S.A.. A deterministic method for solving the Boltzmann equation with parallel computations. Comp. Math. Math. Phys., 42(3):406–418, 2002.Google Scholar
[5]Arkhipov, A. S. and Bishaev., A. M.Three-dimensional numerical simulation of the plasma plume from a stationary plasma thruster. Computational Mathematics and Mathematical Physics, 47(3):472–486, 2007.Google Scholar
[6]Arslanbekov, R., Kolobov, V., Frolova, A., and Zabelok, S.. Evaluation of unified kinetic/continuum solver for computing heat flux in hypersonic blunt body flows. AIAA-2007-4544.Google Scholar
[7]Bhatnagar, P.L., Gross, E.P., and Krook, M.. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(511):1144–1161, 1954.Google Scholar
[8]Bird, G.A.. Molecular Gas Dynamics and Direct Simulation of Gas Flows. Clarendon Press: Oxford, 1994.Google Scholar
[9]Dumbser, M. and Käser, M.. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics, 221(2):693–723, 2007.Google Scholar
[10]Dumbser, M., Käser, M., Titarev, V.A., and Toro, E.F.. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics, 226:204–243, 2007.CrossRefGoogle Scholar
[11]Frezzotti, A.. Numerical investigation of the strong evaporation of a polyatomic gas. In Proc. 17th Symp. Rarefied Gas Dynamics, pages 1243–1250, 1991.Google Scholar
[12]Garzo, V., Santos, A., and Brey, J.J.. A kinetic model for a multicomponent gas. Phys. Fluids, 1(2):380–383, 1988.Google Scholar
[13]Godunov, S.K.. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sbornik, 47:357–393, 1959.Google Scholar
[14]Gradoboev, M.I. and Rykov, V.A.. Convervative method for numerical solution of the kinetic equations for small Knudsen numbers. Comp. Math. Math. Phys., 34(2):246–266, 1994.Google Scholar
[15]Gropp, W., Lusk, E., and Skjellum, A.. Using MPI. Portable parallel programming with the message-passing interface. The MIT Press, second edition, 1999.CrossRefGoogle Scholar
[16]Gusarov, A.V. and Smurov, I.. Gas-dynamic boundary conditions of evaporation and condensation: numerical analysis of the Knudsen layer. Phys. Fluids, 14(12):4242–4255, 2002.Google Scholar
[17]Ivanov, M.Ya. and Nigmatullin, R.Z.. Implicit scheme of S.K. Godunov with increased order of accuracy for Euler equaitons. USSR Comp. Math. Math. Phys., 27(11):1725–1735, 1987.Google Scholar
[18]Kloss, Yu.Yu., Cheremisin, F.G., Khokhlov, N.I., and Shurygin, B.A.. Programming and modelling environment for studies of gas flows in micro- and nanostructures based on solving the Boltzmann equation. Atomic Physics, 105(4), 2008.Google Scholar
[19]Kloss, Yu.Yu., Cheremisin, F.G., and Shuvalov, P.V.. Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels. Computational Mathematics and Mathematical Physics, 50(6), 2010.Google Scholar
[20]Kolgan, V.P.. Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics. Transactions of the Central Aerohydrodynamics Institute, 3(6):68–77, 1972. in Russian.Google Scholar
[21]Kolgan, V.P.. Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics. Journal of Computational Physics, 230(7):2384–2390, 2011.Google Scholar
[22]Kolobov, V.I., Arslanbekov, R.R., Aristov, V.V., Frolova, A.A., and Zabelok, S.A.. Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J. Comp. Phys., 223:589–608, 2007.Google Scholar
[23]Kolobov, V.I., Bayyuk, S.A., Arslanbekov, R.R., Aristov, V.V., Frolova, A.A., and Zabelok, S.A.. Construction of a unified continuum/kinetic solver for aerodynamic problems. AIAA Journal of Spacecraft and Rockets, 42(4):598, 2005.Google Scholar
[24]Kulikovskii, A.G., Pogorelov, N.V., and Semenov, A.Yu.. Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman and Hall, 2002. Monographs and Surveys in Pure and Applied Mathematics, Vol. 118.Google Scholar
[25]Larina, I.N. and Rykov, V.A.. A numerical method for calculating axisymmetric rarefied gas flows. Comp. Math. Math. Phys., 38(8):1335, 1998.Google Scholar
[26]Larina, I.N. and Rykov, V.A.. Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom. Computational Mathematics and Mathematical Physics, 50(12):2118–2130, 2010.Google Scholar
[27]Li, Z.-H. and Zhang, H.-X.. Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation. International Journal for Numerical Methods in Fluids, 42(4):361–382, 2003.Google Scholar
[28]Li, Z.-H. and Zhang, H.-X.. Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. J. Comput. Phys., 193(2):708–738, 2004.Google Scholar
[29]Men’shov, I.S. and Nakamura, Y.. An implicit advection upwind splitting scheme for hypersonic air flows in thermochemical nonequilibrium. In A Collection of Technical Papers of 6th Int. Symp. on CFD, volume 2, page 815. Lake Tahoe, Nevada, 1995.Google Scholar
[30]Men’shov, I.S. and Nakamura, Y.. On implicit Godunov’s method with exactly linearized numerical flux. Computers and Fluids, 29(6):595–616, 2000.Google Scholar
[31]Mieussens, L.. Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models and Meth. Appl. Sci., 8(10):1121–1149, 2000.Google Scholar
[32]Mieussens, L.. Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys., 162(2):429–466, 2000.Google Scholar
[33]Rykov, V.A.. A model kinetic equation for a gas with rotational degrees of freedom. Fluid Dynamics, 10(6):959–966, 1975.Google Scholar
[34]Shakhov, E.M.. Approximate kinetic equations in rarefied gas theory. Fluid Dynamics, 3(1):156–161, 1968.Google Scholar
[35]Shakhov, E.M.. Generalization of the Krook kinetic relaxation equation. Fluid Dynamics, 3(5):142–145, 1968.Google Scholar
[36]Shakhov, E.M.. Couette problem for the generalized Krook equation. Stress-peak effect. Fluid Dynamics, 4(5):9–13, 1969.Google Scholar
[37]Shakhov, E.M.. Transverse flow of a rarefield gas around a plate. Fluid Dynamics, 7(6):961–966, 1972.CrossRefGoogle Scholar
[38]Shakhov, E.M.. Solution of axisymmetric problems of the theory of rarefied gases by the finite-difference method. USSR Comp. Math. Math. Phys., 14(4):970–981, 1974.Google Scholar
[39]Shakhov, E.M.. Linearized two-dimensional problem of rarefied gas flow in a long channel. Computational Mathematics and Mathematical Physics, 39(7):1192–1200, 1999.Google Scholar
[40]Shakhov, E.M.. Rarefied gas flow in a pipe of finite length. Computational Mathematics and Mathematical Physics, 40(4):618, 2000.Google Scholar
[41]Sharipov, F. and Seleznev, V.. Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data, 27(3):657–706, 1998.Google Scholar
[42]Sharipov, F. and Seleznev, V.. Flows of rarefied gases in channels and microchannels. Russian Academy of Science, Ural Branch, Institute of Thermal Physics, 2008. in Russian.Google Scholar
[43]Takata, S., Sone, Y., and Aoki, K.. Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equaion for hard-sphere molecules. Physics of Fluids, 5(3):716–737, 1992.Google Scholar
[44]Tillaeva, N.I.. A generalization of the modified Godunov scheme to arbitrary unstructured meshes. Transactions of the Central Aerohydrodynamics Institute, 17(2):18–26, 1986. in Russian.Google Scholar
[45]Titarev, V.A.. Towards fully conservative numerical methods for the nonlinear model Boltz-mann equation. In Preprint NI03031-NPA, page 13. Isaac Newton Institute for Mathematical Sciences, University of Cambridge, Cambridge, UK, 2003.Google Scholar
[46]Titarev, V.A.. Conservative numerical methods for advanced model kinetic equations. In Onate, E.Wessling, P. and Periaux, J., editors, Proceedings of the ECCOMAS 2006. TU Delft, The Netherlands, 2006. ISBN: 90-9020970-0.Google Scholar
[47]Titarev, V.A.. Conservative numerical methods for model kinetic equations. Computers and Fluids, 36(9):1446–1459, 2007.Google Scholar
[48]Titarev, V.A.. Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped domains. Computational Mathematics and Mathematical Physics, 49(7):1197–1211, 2009.Google Scholar
[49]Titarev, V.A.. Implicit numerical method for computing three-dimensional rarefied gas flows using unstructured meshes. Computational Mathematics and Mathematical Physics, 50(10):1719–1733, 2010.Google Scholar
[50]Titarev, V.A.. Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas. Communications in Computational Physics, 8(2):427–444, 2010.Google Scholar
[51]Titarev, V.A. and Shakhov, E.M.. Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic S-model. Computational Mathematics and Mathematical Physics, 50(12):2131–2144, 2010.Google Scholar
[52]Varoutis, S., Valougeorgis, D., and Sharipov, F.. Simulation of gas flow through tubes of finite length over the whole range of rarefaction for various pressure drop ratios. J. Vac. Sci. Technol. A, 27(6):1377–1391, 2009.Google Scholar
[53]Venkatakrishnan, V.. On the accuracy of limiters and convergence to steady-state solutions. In AIAA paper 93-0880, 31st Aerospace Science Meeting &Exhibit, January 11-14, 1993, Reno, NV, 1993.Google Scholar
[54]Yang, J.Y. and Huang, J.C.. Rarefied flow computations using nonlinear model Boltzmann equations. J. Comput. Phys., 120(2):323–339, 1995.Google Scholar
[55]Zhang, Y.-T. and Shu., C.-W.Third order WENO scheme on three dimensional tetrahedral meshes. Communications in Computational Physics, 5(2-4):836–848, 2009.Google Scholar
[56]Zhuk, V.I.. Spherical expansion of vapor during evaporation of a droplet. Fluid Dynamics, 11(2):251–255, 1976.Google Scholar