Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T10:38:52.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Development and Comparative Studies of Three Non-free Parameter Lattice Boltzmann Models for Simulation of Compressible Flows

Published online by Cambridge University Press:  03 June 2015

L. M. Yang ,
C. Shu  and
J. Wu
L. M. Yang
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, Jiangsu, China
C. Shu*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
J. Wu
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, Jiangsu, China
*
Corresponding author. URL:http://serve.me.nus.edu.sg/shuchang/, Email: [email protected]
Get access

Abstract

This paper at first shows the details of finite volume-based lattice Boltzmann method (FV-LBM) for simulation of compressible flows with shock waves. In the FV-LBM, the normal convective flux at the interface of a cell is evaluated by using one-dimensional compressible lattice Boltzmann model, while the tangential flux is calculated using the same way as used in the conventional Euler solvers. The paper then presents a platform to construct one-dimensional compressible lattice Boltzmann model for its use in FV-LBM. The platform is formed from the conservation forms of moments. Under the platform, both the equilibrium distribution functions and lattice velocities can be determined, and therefore, non-free parameter model can be developed. The paper particularly presents three typical non-free parameter models, D1Q3, D1Q4 and D1Q5. The performances of these three models for simulation of compressible flows are investigated by a brief analysis and their application to solve some one-dimensional and two-dimensional test problems. Numerical results showed that D1Q3 model costs the least computation time and D1Q4 and D1Q5 models have the wider application range of Mach number. From the results, it seems that D1Q4 model could be the best choice for the FV-LBM simulation of hypersonic flows.

Keywords

76T10FV-LBMnon-free parameter modelscompressible inviscid flows
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 4 , August 2012 , pp. 454 - 472
Copyright
Copyright © Global-Science Press 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]Nabovati, A., Sellan, D. P. and Amon, C. H., On the lattice Boltzmann method for phonon transport, J. Comput. Phys., 230 (2011), pp. 5864–5876.Google Scholar
[2]Malaspinas, O., Fietier, N. and Deville, , Lattice Boltzmann method for the simulation of viscoelastic fluid flows, Non-Newtonian. Fluid. Mech., 165 (2010), pp. 1637–1653.Google Scholar
[3]Wu, J. and Shu, C., A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows, J. Comput. Phys., 230 (2011), pp. 2246–2269.Google Scholar
[4]Ricot, D., Marie, S., Sagaut, P. and Bailly, C., Lattice Boltzmann method with selective viscosity filter, J. Comput. Phys., 228 (2009), pp. 4478–4490.Google Scholar
[5]He, X. and Luo, L. S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88 (1997), pp. 927–944.Google Scholar
[6]Guo, Z. L., Shi, B. C. and Wang, N. C., Lattice BGK model for the incompressible Navier-Stokes equation, J. Comput. Phys., 165 (2000), pp. 288–306.Google Scholar
[7]Xu, H., Tao, W. Q. and Zhang, Y., Lattice Boltzmann model for three-dimensional decaying homogeneous isotropic turbulence, Phys. Lett. A., 373 (2009), pp. 1368–1373.Google Scholar
[8]Chen, S., A large-eddy-based lattice Boltzmann model for turbulent flow simulation, Appl. Math. Comput., 215 (2009), pp. 591–598.Google Scholar
[9]Inamuro, T., Ogata, T., Tajima, S. and Konishi, N., A lattice Boltzmann method for incompressible two-phase with large density differences, J. Comput. Phys., 198 (2004), pp. 628–644.CrossRefGoogle Scholar
[10]Premnath, K. N. and Abraham, J., Three-dimentional multi-relaxation time (MRT) lattice Boltzmann models for multiphase flow, J. Comput. Phys., 224 (2007), pp. 539–559.CrossRefGoogle Scholar
[11]Mehravaran, M. and Hannani, S. K., Simulation of incompressible two-phrase flows with large density differences employing lattice Boltzmann and level set methods, Comput. Methods. Appl. Mech. Eng., 198 (2008), pp. 223–233.Google Scholar
[12]Kataoka, T. and Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys. Rev. E., 69 (2004), 056702.Google Scholar
[13]Kataoka, T. and Tsutahara, M., Lattice Boltzmann method for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E., 69 (2004), R035701.Google Scholar
[14]Qu, K., Shu, C. and Chew, Y. T., Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number, Phys. Rev. E., 75 (2007), 036706.Google Scholar
[15]Qu, K., Shu, C. and Chew, Y. T., Simulation of shock-wave propagation with finite volume lattice Boltzmann method, Int. J. Mod. Phys. C., 18 (2007), pp. 447–454.Google Scholar
[16]Li, Q., He, Y. L., Wang, Y. and Tao, W. Q., Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations, Phys. Rev. E., 76 (2007), 056705.Google Scholar
[17]Ji, C. Z., Shu, C. and Zhao, N., A lattice Boltzmann method-based flux solver and its application to solve shock tube problem, Mod. Phys. Lett. B., 23 (2009), pp. 313–316.Google Scholar
[18]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), pp. 357–372.CrossRefGoogle Scholar
[19]van Leer, B., Flux vector splitting for the Euler equations, Lect. Notes. Phys., 170 (1982), pp. 507–512.CrossRefGoogle Scholar
[20]Liou, M. S. and Steffen, C. J., A new flux vector splitting scheme, J. Comput. Phys., 107 (1993), pp. 23–39.Google Scholar
[21]Liou, M. S., A sequel to AUSM: AUSM+, J. Comput. Phys., 129 (1996), pp. 364–382.Google Scholar
[22]Liou, M. S., A sequel to AUSM, part II: AUSM+-up for all speeds, J. Comput. Phys., 214 (2006), pp. 137–170.Google Scholar
[23]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 (1954), pp. 511–525.CrossRefGoogle Scholar
[24]Xu, K., A gas-kinetic BGK scheme for the compressible Navier-Stokes equations, NASA/CR, 38 (2000), 210544.Google Scholar
[25]Xu, K., A gas-kinetic BGK scheme for the Navier-Stocks equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), pp. 289–335.Google Scholar
[26]Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115–173.Google Scholar
[27]Venkatakrishnan, V., On the accuracy of limiters and convergence to steady state solutions, AIAA Paper, (1993), 93–0880.Google Scholar
[28]Venkatakrishnan, V., Convergence to steady-State solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118 (1995), pp. 120–130.Google Scholar