Gas kinetic algorithm using Boltzmann model equation. (English) Zbl 1100.82520
Summary: Based on the Boltzmann-BGK model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented by the aid of the basic characteristic on molecular movement and collision approaching to equilibrium. The optimum golden section principle is extended and applied to the discrete velocity ordinate method in order to discretize the corresponding velocity components, and then the molecular velocity distribution function equation will be cast into hyperbolic conservation laws form with nonlinear source terms. In view of the unsteady characteristic of molecular convective movement and colliding relaxation, the time-splitting method is applied to decompose the velocity distribution function equations into the colliding relaxation equations with non-linear source terms and the convective motion equations. Based on the second-order Runge-Kutta method and the non-oscillatory, containing no free parameters, and the dissipative (NND) finite difference method, a gas kinetic finite difference second-order scheme is constructed to directly solve the discrete velocity distribution functions. Four types of discrete velocity quadrature rules, such as the modified Gauss-Hermite formula and the Golden Section number-theoretic integral method, are developed and applied to evaluate the macroscopic flow moments of the distribution functions over the velocity space. As a result, a unified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime. To test the reliability of the present method, the one-dimensional shock wave structures, the flows past two-dimensional circular cylinder and the three-dimensional flows over sphere with various Knudsen numbers are simulated. The computational results are found in high resolution of the flow fields and good agreement with the theoretical, DSMC, and experimental results.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
References:
| [1] | Chapmann, S.; Cowling, T. G., The mathematical theory of non-uniform gases (1990), Cambridge Press: Cambridge Press Cambridge |
| [2] | Cercignani C. The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer-Verlag, New York Inc., 1988; Cercignani C. The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer-Verlag, New York Inc., 1988 · Zbl 0646.76001 |
| [3] | Bird, G. A., Molecular gas dynamics (1976), Claredon Press: Claredon Press Oxford, England |
| [4] | Ivanov MS, Kotov VM, Reshetin AG, Shelkonogov AM. Aerodynamics of complex shape bodies within a wide range of supersonic flows of rarefied gases. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, 1990. p. 703-708; Ivanov MS, Kotov VM, Reshetin AG, Shelkonogov AM. Aerodynamics of complex shape bodies within a wide range of supersonic flows of rarefied gases. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, 1990. p. 703-708 |
| [5] | Bird, G. A., Molecular gas dynamics and the direct simulation of gas flows (1994), Oxford University Press: Oxford University Press London · Zbl 0709.76511 |
| [6] | Moss JN, Price JM, Dogra VK, Hash DB. Comparison of DSMC and experimental results for hypersonic external flow. AIAA Paper 95-2028, 1995; Moss JN, Price JM, Dogra VK, Hash DB. Comparison of DSMC and experimental results for hypersonic external flow. AIAA Paper 95-2028, 1995 |
| [7] | Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model collision processes in gases. I. Small amplitude processes is charged and neutral one-component system, Phys. Rev., 94, 511-525 (1954) · Zbl 0055.23609 |
| [8] | Holway, L. H., New statistical models for kinetic theory, methods of construction, Phys. Fluids, 9, 1658-1673 (1966) |
| [9] | Abe T, Oguchi H. A hierarchy kinetic model and its applications, Rarefied Gas Dynamics. In: Potter JL, editor. Progress in Astronautics and Aeronautics, vol. 51, Pt. II, 1977. New York: AIAA. p. 781-93; Abe T, Oguchi H. A hierarchy kinetic model and its applications, Rarefied Gas Dynamics. In: Potter JL, editor. Progress in Astronautics and Aeronautics, vol. 51, Pt. II, 1977. New York: AIAA. p. 781-93 |
| [10] | Shakov, E. M., Generalization of the krook kinetic relaxation equation, Fluid Dyn., 3, 1, 158-161 (1968) |
| [11] | Vincenti, W. G.; Kruger, C. H., Introduction to physical gas dynamics (1965), Wiley: Wiley New York |
| [12] | Cercignani, C., Mathematical methods in kinetic theory (1990), Plenum Press: Plenum Press New York, revised edition · Zbl 0726.76083 |
| [13] | Park, D. C., The kinetic theory of Gases with applications in rarefied gas dynamics (1981), University of Strathclyde: University of Strathclyde Glasgow, Scotland |
| [14] | Cercignani, C., Rarefied gas dynamics: from basic concepts to actual calculations (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0961.76002 |
| [15] | Pullin, D. I., Direct simulation methods for compressible inviscid ideal-gas flow, J. Comput. Phys., 34, 231-244 (1980) · Zbl 0419.76049 |
| [16] | Reitz, R. D., One-dimensional compressible gas dynamics calculations using the Boltzmann equation, J. Comput. Phys., 42, 108-123 (1981) · Zbl 0466.76069 |
| [17] | Prendergast, K. H.; Xu, K., Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys., 109, 53-66 (1993) · Zbl 0791.76059 |
| [18] | Kim, C.; Jameson, A., A robust and accurate LED-BGK solver on unstructured adaptive meshes, J. Comput. Phys., 143, 2, 598-627 (1998) · Zbl 0936.76042 |
| [19] | Xu K. Gas-kinetic schemes for unsteady compressible flow simulations, 29th CFd Lecture Series 1998-03, Von Karman Institute for Fluid Dynamics, Lecture Series, Belgium, February 23-27, 1998; Xu K. Gas-kinetic schemes for unsteady compressible flow simulations, 29th CFd Lecture Series 1998-03, Von Karman Institute for Fluid Dynamics, Lecture Series, Belgium, February 23-27, 1998 |
| [20] | Chae, D.; Kim, C.; Rho, O. H., Development of an improved gas-kinetic BGK scheme for inviscid and viscous flows, J. Comput. Phys., 158, 1-27 (2000) · Zbl 0974.76056 |
| [21] | Xu, K., A gas-kinetic BGK schemes for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171, 289-335 (2001) · Zbl 1058.76056 |
| [22] | Morinishi K, Oguchi H. A computational method and its application to analyses of rarefied gas flows. In: Oguchi H, editor. Proceedings of 14th International Symposium on Rarefied Gas Dynamics, vol. 1. Tokyo: University of Tokyo Press; 1984. p. 149-158; Morinishi K, Oguchi H. A computational method and its application to analyses of rarefied gas flows. In: Oguchi H, editor. Proceedings of 14th International Symposium on Rarefied Gas Dynamics, vol. 1. Tokyo: University of Tokyo Press; 1984. p. 149-158 |
| [23] | Bergers, D., Kinetic model solution for axisymmetric flow by the method of discrete ordinates, J. Comput. Phys., 57, 285-302 (1985) · Zbl 0586.76136 |
| [24] | Yang, J. Y.; Huang, J. C., Rarefied flow computations using nonlinear model Boltzmann equtions, J. Comput. Phys., 120, 323-339 (1995) · Zbl 0845.76064 |
| [25] | Deng ZT, Liaw GS, Chou LC. Numerical Investigation of Low-Density Nozzle Flow by Solving the Boltzmann Model Equation, NASA-TM-110492. In: Proceedings of 33rd Aerospace Science Meeting and Exhibit, January 9-12, 1995; Deng ZT, Liaw GS, Chou LC. Numerical Investigation of Low-Density Nozzle Flow by Solving the Boltzmann Model Equation, NASA-TM-110492. In: Proceedings of 33rd Aerospace Science Meeting and Exhibit, January 9-12, 1995 |
| [26] | Aoki, K.; Kanba, K.; Takata, S., Numerical analysis of a supersonic rarefied gas flow past a flat plate, Phys. Fluids, 9, 4, 1144-1161 (1997) |
| [27] | 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) · Zbl 0984.76070 |
| [28] | Huang AB, Giddens DP. The discrete ordinate method for the linearized boundary value problems in kinetic theory of gases. In: Brundin CL et al., editors. Proceedings of 5th International Symposium on Rarefied Gas Dynamics. New York: Academic Press; 1967. p. 481-6; Huang AB, Giddens DP. The discrete ordinate method for the linearized boundary value problems in kinetic theory of gases. In: Brundin CL et al., editors. Proceedings of 5th International Symposium on Rarefied Gas Dynamics. New York: Academic Press; 1967. p. 481-6 |
| [29] | Zhang, H. X., Non-oscillatory and non-free-parameter dissipation difference scheme, Acta Aerodyn. Sin., 6, 2, 143-165 (1988) |
| [30] | Huang, A. B.; Giddens, D. P., A new table for a Modified (Half-Range) Gauss-Hermite quadrature with an evaluation of the integral, J. Math. Phys., 47, 213-218 (1968) · Zbl 0179.22001 |
| [31] | Hua, L. K.; Wang, Y., Some topics on mathematical modeling (1998), Hunan’s Education Press: Hunan’s Education Press Hunan |
| [32] | Shakov EM. Kinetic model equations and numerical results. In: Oguchi H, editor. Proceedings of 14th International Symposium on Rarefied Gas Dynamics, vol. 1. Tokyo: University of Tokyo Press; 1984. p. 137-48; Shakov EM. Kinetic model equations and numerical results. In: Oguchi H, editor. Proceedings of 14th International Symposium on Rarefied Gas Dynamics, vol. 1. Tokyo: University of Tokyo Press; 1984. p. 137-48 |
| [33] | Li ZH. Study on the Unified Algorithm from Rarefied Flow to Continuum. PhD Dissertation. China Aerodynamics Researchment and Development Center, Mianyang, China, 2001; Li ZH. Study on the Unified Algorithm from Rarefied Flow to Continuum. PhD Dissertation. China Aerodynamics Researchment and Development Center, Mianyang, China, 2001 |
| [34] | Riedi, P. C., Thermal physics: an introduction to thermodynamics, statistical mechanics and kinetic theory (1976), The Macmillan Press Ltd: The Macmillan Press Ltd London |
| [35] | Brittin WE. Lectures in theoretical physics: Kinetic theory, Volume IX C, Gordon and Breach, Science publishers, New York/London/Paris, 1967; Brittin WE. Lectures in theoretical physics: Kinetic theory, Volume IX C, Gordon and Breach, Science publishers, New York/London/Paris, 1967 |
| [36] | Shizgal, B., A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems, J. Comput. Phys., 41, 309-327 (1981) · Zbl 0459.65096 |
| [37] | Kopal, Z., Numerical analysis (1955), Chapman & Hall Ltd: Chapman & Hall Ltd London · JFM 67.0792.02 |
| [38] | Wang, Y., On approximation evaluation and application to integration (Applications of the number theoretic method), Adv. Math., 5, 1, 1-40 (1962) |
| [39] | Hua, L. K.; Wang, Y., On uniform distribution and numerical analysis (number-theoretic method), Sci. Sin., 16, 4, 483-505 (1973), 1974;17(3):331-48, 1975;18(2):184-98 · Zbl 0343.10034 |
| [40] | Li ZH, Zhang HX. Numerical algorithm from rarefied flow to continuum. In: Zhang HX, editor. Proceedings of the 10th National Symposium on Computational Fluid Dynamics, Mianyang, China, 2000. p. 165-270 [in Chinese]; Li ZH, Zhang HX. Numerical algorithm from rarefied flow to continuum. In: Zhang HX, editor. Proceedings of the 10th National Symposium on Computational Fluid Dynamics, Mianyang, China, 2000. p. 165-270 [in Chinese] |
| [41] | Hua, L. K.; Wang, Y., Applications of number theory to approximation analysis (1978), Science Press: Science Press Beijing |
| [42] | Li ZH, Xie YR. Technique of molecular indexing applied to the direct simulation Monte Carlo method. In: Shen C, editor. Proceedings of 20th International Symposium on Rarefied Gas Dynamics, China, Beijing: Peking University Press; 1997. p. 205-9; Li ZH, Xie YR. Technique of molecular indexing applied to the direct simulation Monte Carlo method. In: Shen C, editor. Proceedings of 20th International Symposium on Rarefied Gas Dynamics, China, Beijing: Peking University Press; 1997. p. 205-9 |
| [43] | Li ZH, Zhang HX. Study on gas kinetic algorithm for flows from rarefied transition to continuum. In: Bartel TJ, Gallis MA, editors. Proceedings of 22nd International Symposium on Rarefied Gas Dynamics, Sydney, Australia, (AIP Conference Proceedings, Melville, New York, 2001), vol. 585, 2000. p. 628-36; Li ZH, Zhang HX. Study on gas kinetic algorithm for flows from rarefied transition to continuum. In: Bartel TJ, Gallis MA, editors. Proceedings of 22nd International Symposium on Rarefied Gas Dynamics, Sydney, Australia, (AIP Conference Proceedings, Melville, New York, 2001), vol. 585, 2000. p. 628-36 |
| [44] | Zhang, H. X.; Zhuang, F. G., NND schemes and their application to numerical simulation of two- and three-dimensional flows, Adv. Appl. Mech., 29, 193-256 (1992) · Zbl 0749.76050 |
| [45] | Deng, X. G.; Zhang, H. X., The extentions of NND schemes and application to viscous flow calculations, Acta Aerodyn. Sin., 12, 2, 121-129 (1994) |
| [46] | Li ZH. DSMC simulation of hypersonic flow past spacecraft. In: Zhang FG, editor. Proceedings of ASIA Workshop on Computational Fluid Dynamics, Sichuan, China, 1994. p. 97-100; Li ZH. DSMC simulation of hypersonic flow past spacecraft. In: Zhang FG, editor. Proceedings of ASIA Workshop on Computational Fluid Dynamics, Sichuan, China, 1994. p. 97-100 |
| [47] | Danny PH. Numerical solution of a three-dimensional cubic cavity flow using the Boltzmann equation. In: Proceedings of 18th International Symposium on Rarefied Gas Dynamics, 1992. p. 469-81; Danny PH. Numerical solution of a three-dimensional cubic cavity flow using the Boltzmann equation. In: Proceedings of 18th International Symposium on Rarefied Gas Dynamics, 1992. p. 469-81 |
| [48] | Li ZH, Zhang HX. Numerical Investigation from Rarefied Transition Flow to Continuum by Solving the Boltzmann Model Equation. In: Zhang HX, editor. Proceeding of the Third Asia Workshop on Computational Fluid Dynamics, Third China-Japan Joint Workshop on CFD, Mianyang, China, 2000. p. 71-76; Li ZH, Zhang HX. Numerical Investigation from Rarefied Transition Flow to Continuum by Solving the Boltzmann Model Equation. In: Zhang HX, editor. Proceeding of the Third Asia Workshop on Computational Fluid Dynamics, Third China-Japan Joint Workshop on CFD, Mianyang, China, 2000. p. 71-76 |
| [49] | Alsmeyer, H., Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid. Mech., 74, 3, 497-513 (1976) |
| [50] | ; |
| [51] | Vogenitz, F. W.; Bird, G. A.; Broadwell, J. E., Theoretical and experimental study of rarefied supersonic flows about several simple shapes, AIAA J., 6, 12, 2388-2394 (1968) · Zbl 0177.55402 |
| [52] | Kinslow M, Potter JL. The Drag of Sphere in Rarefied Hypervelocity Flow. AEDC-TDR-62-205, 1962. p. 1-46; Kinslow M, Potter JL. The Drag of Sphere in Rarefied Hypervelocity Flow. AEDC-TDR-62-205, 1962. p. 1-46 |
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