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A Hybrid Lattice Boltzmann Flux Solver for Simulation of Viscous Compressible Flows

  • L. M. Yang (a1) (a2), C. Shu (a3) and J. Wu (a2)

Abstract

In this paper, a hybrid lattice Boltzmann flux solver (LBFS) is proposed for simulation of viscous compressible flows. In the solver, the finite volume method is applied to solve the Navier-Stokes equations. Different from conventional Navier-Stokes solvers, in this work, the inviscid flux across the cell interface is evaluated by local reconstruction of solution using one-dimensional lattice Boltzmann model, while the viscous flux is still approximated by conventional smooth function approximation. The present work overcomes the two major drawbacks of existing LBFS [28–31], which is used for simulation of inviscid flows. The first one is its ability to simulate viscous flows by including evaluation of viscous flux. The second one is its ability to effectively capture both strong shock waves and thin boundary layers through introduction of a switch function for evaluation of inviscid flux, which takes a value close to zero in the boundary layer and one around the strong shock wave. Numerical experiments demonstrate that the present solver can accurately and effectively simulate hypersonic viscous flows.

Copyright

Corresponding author

*Corresponding author. Email: mpeshuc@nus.edu.sg (C. Shu), yangliming_2011@126.com (L. M. Yang)

References

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[1] Li, X. L., Fu, D. X. and Ma, Y. W., Optimized group velocity control scheme and DNS of decaying compressible turbulence of relative high turbulent Mach number, Int. J. Numer. Meth. Fluids, 48 (2005), pp. 835852.
[2] Qiu, J. X., Khoo, B. C. and Shu, C. W., A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes, J. Comput. Phys., 212 (2006), pp. 540565.
[3] Zheng, H. W., Shu, C. and Chew, Y. T., An object-oriented and quadrilateral-mesh based solution adaptive algorithm for compressible multi-fluid flows, J. Comput. Phys., 227 (2008), pp. 68956921.
[4] Yang, L. M., Shu, C., Wu, J., Zhao, N. and Lu, Z. L., Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows, J. Comput. Phys., 255 (2013), pp. 540557.
[5] Chen, S. Z., Xu, K., Lee, C. B. and Cai, Q. D., A unified gas kinetic scheme with moving mesh and velocity space adaptation, J. Comput. Phys., 231 (2012), pp. 66436664.
[6] Main, A. and Farhat, C., A second-order time-accurate implicit finite volume method with exact two-phase Riemann problems for compressible multi-phase fluid and fluid-structure problems, J. Comput. Phys., 258 (2014), pp. 613633.
[7] McDonald, P. W., The computation of transonic flow through tow-dimensional gas turbine cascades, ASME Paper 71-GT-89, 1971.
[8] Patankar, S. V. and Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transfer, 15 (1972), pp. 17871806.
[9] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), pp. 1226.
[10] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357372.
[11] Van Leer, B., Flux vector splitting for the Euler equations, Lecture Notes in Physics, 170 (1982), pp. 507512.
[12] Liou, M. S. and Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107 (1993), pp. 2339.
[13] Kitamura, K., Shima, E. and Roe, P. L., Evaluation of Euler fluxes for hypersonic heating computations, AIAA J., 48 (2010), pp. 763776.
[14] Van Leer, B., Thomas, J. L., Roe, P. L. and Newsome, R. W., A comparison of numerical flux formulas for the Euler and Navier-Stokes equations, AIAA Paper, 87-1104, 1987.
[15] Chou, S. Y. and Baganoff, D., Kinetic flux-vector splitting for the Navier-Stokes equations, J. Comput. Phys., 130 (1997), pp. 217230.
[16] Chae, D., Kim, C. and Rho, O. H., Development of an improved gas-kinetic BGK scheme for inviscid and viscous flows, J. Comput. Phys., 158 (2000), pp. 127.
[17] Xu, K., A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), pp. 289335.
[18] Xu, K., Gas-kinetic schemes for unsteady compressible flow simulations, VKI for Fluid Dynamics Lecture Series, 1998-03 (1998).
[19] Benzi, R., Succi, S. and Vergassola, M., The lattice Boltzmann equation: theory and application, Physics Report, 1992.
[20] Guo, Z. L. and Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, World Scientific Publishing, 2013.
[21] Kataoka, T. and Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys. Rev. E., 69 (2004), 056702.
[22] 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.
[23] Li, Q., He, Y. L., Wang, Y. and Tang, G. H., Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows, Phys. Lett. A, 373 (2009), pp. 21012108.
[24] Zhong, C. W., Li, K., Sun, J. H., Zhou, C. S. and Xie, J. F., Compressible flow simulation around airfoil based on lattice Boltzmann method, Transactions of Nanjing University of Aeronautics and Astronautics, 26 (2009), pp. 206211.
[25] Xi, H. W., Peng, G. W. and Chou, S. H., Finite-volume lattice Boltzmann method, Phys. Rev. E., 59 (1999), pp. 62026205.
[26] Ubertini, S., Bella, G. and Succi, S., Lattice Boltzmann method on unstructured grids: further developments, Phys. Rev. E., 68 (2003), 016701.
[27] Stiebler, M., Tölke, J. and Krafczyk, M., An upwind discretization scheme for the finite volume lattice Boltzmann method, Comput. Fluids, 35 (2006), pp. 814819.
[28] 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. 313316.
[29] Yang, L. M., Shu, C. and Wu, J., Development and comparative studies of three non-free parameter lattice Boltzmann models for simulation of compressible flows, Adv. Appl. Math. Mech., 4 (2012), pp. 454472.
[30] Yang, L. M., Shu, C. and Wu, J., A moment conservation-based non-free parameter compressible lattice Boltzmann model and its application for flux evaluation at cell interface, Comput. Fluids, 79 (2013), pp. 190199.
[31] Shu, C., Wang, Y., Yang, L. M. and Wu, J., Lattice Boltzmann flux solver: an efficient approach for numerical simulation of fluid flows, Transactions of Nanjing University of Aeronautics and Astronautics, 31 (2014), pp. 115.
[32] 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. 511525.
[33] Xu, K. and He, X. Y., Lattice Boltzmann method and gas-kinetic BGK scheme in the low-Mach number viscous flow simulations, J. Comput. Phys., 190 (2003), pp. 100117.
[34] Barth, T. J. and Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, AIAA Paper, 89-0366, 1989.
[35] Blazek, J., Computation Fluid Dynamics: Principle and Application, Elsevier, 2001.
[36] Swanson, R. C. and Radespiel, R., Cell centered and cell vertex multigrid schemes for the Navier-Stokes equations, AIAA J., 29 (1991), pp. 697703.
[37] Venkatakrishnan, V., Convergence to steady-state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118 (1995), pp. 120130.
[38] Bristeau, M. O., Glowinski, R., Periaux, J. and Viviand, H., Numerical simulation of compressible Navier-Stokes flows, Vieweg and Sonh Braunschweig, Wiesbaden, (1987).
[39] Jawahar, P. and Kamath, H., A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids, J. Comput. Phys., 164 (2000), pp. 165203.
[40] Van Leer, B., Toward the ultimate conservative difference scheme iv, a new approach to numerical convection, J. Comput. Phys., 23 (1977), pp. 276299.
[41] Yoon, S. and Jameson, A., Lower-upper Symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations, AIAA J., 26 (1988), pp. 10251026.
[42] Wieting, A. R., Experimental study of shock wave interface heating on a cylindrical leading edge, NASA TM-100484, 1987.
[43] Xu, K., Mao, M. L. and Tang, L., A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow, J. Comput. Phys., 203 (2005), pp. 405421.

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