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A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

  • C. T. Hsu (a1), S. W. Chiang (a1) and K. F. Sin (a1)


The Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.


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[1]Broadwell, J. E., Shock structure in a simple discrete velocity gas, Phys. Fluids, 7(8) (1964), 12431247.
[2]Broadwell, J. E., Study of rarefied shear flow by the discrete velocity method, J. Fluid Mech., 19(3) (1964), 401414.
[3]Krook, M., On the solution of equation of transfer I, Astrophys. J., 122 (1955), 488497.
[4]Luo, L.-S., Some recent results on discrete velocity models and ramifications for lattice Boltz-mann equation, Comput. Phys. Commun., 129 (2000), 6374.
[5]Yang, J. Y. and Huang, J. C., Rarefied flow computations using nonlinear model Boltzmann equations, J. Comput. Phys., 120 (1995), 323339.
[6]Tjon, J. and Wu, T. T., Numerical aspects of the approach to a Maxwellian distribution, Phys. Rev. A, 19(2) (1978), 883888.
[7]Benzi, R., Succi, S. and Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys. Rep., 222(3) (1992), 145197.
[8]Chen, H., Chen, S. and Matthaeus, W. H., Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method, Phys. Rev. A, 45(8) (1992), R5339.
[9]Succi, S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford, Clarendon Press, 2001.
[10] G. R. McNamara and Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61(20) (1988), 23322335.
[11]Succi, S., Mohammad, A. A. and Horbach, J., Lattice-Boltzmann simulation of dense nanoflows: a comparison with molecular dynamics and Navier-Stokes solutions, Int. J. Mod. Phys. C, 18(4) (2007), 667675.
[12]Lallemand, P. and Luo, L. S., Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions, Phys. Rev. E, 68(3) (2003), 36706013670625.
[13]Karniadakis, G., Beskok, A. and Aluru, N., Microflows and Nanoflows: Fundamentals and Simulation, New York, Springer, 2007.
[14]Albright, B. J., Lemons, D. S., Jones, M. E. and Winske, D., Quiet direct simulation of Eulerian fluid, Phys. Rev. E, 65 (2002), 055302.
[15]Smith, M. R., Cave, H. M., Wu, J. S., Jermy, M. C. and Chen, Y. S., An improved quiet direct simulation method for Eulerian fluids using a second-order scheme, J. Comput. Phys., 228 (2009), 22132224.
[16]Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, 3rd ed, Cambridge University Press, Cambridge, 1970.
[17]Sod, G. A., Survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27 (1978), 131.
[18]Hsu, C. T., Chiang, S. W. and Sin, K. F., Dynamic lattice Boltzmann method for the simulation of cavity flows with high performance computing, 22nd International Conference on Parallel Computational Fluid Dynamics, Kaohsiung, Taiwan, 2010.
[19]Sin, K. F., Chiang, S. W. and Hsu, C. T., Evaluation of dynamic discrete ordinate method for Boltzmann equation, The 8th Asian Computational Fluid Dynamics Conference, Hong Kong, 2010.
[20]Chiang, S. W., Sin, K. F. and Hsu, C. T., Numerical simulation of Rayleigh-Benard instability using dynamic lattice Boltzmann method, The 8th Asian Computational Fluid Dynamics Conference, 1014 January 2010, Hong Kong, 2010.
[21]Drazin, P. G., Introduction to Hydrodynamic Stability, Cambridge, Cambridge University Press, 2002.
[22]Hsu, C. T., Sin, K. F. and Chiang, S. W., Parallel computation for Boltzmann equation simulation with Dynamic Discrete Ordinate Method (DDOM), Computers and Physics, (2011), (in press).
[23]Hsu, C.T., K. F. SinandChiang, S. W., Dynamic discrete ordinate method for Boltzmann equation in viscous flow, The 21st International Symposium on Transport Phenomena, November 2010, Taiwan, 2010.


A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

  • C. T. Hsu (a1), S. W. Chiang (a1) and K. F. Sin (a1)


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