×

zbMATH — the first resource for mathematics

Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows. (English) Zbl 1349.76752
Summary: This paper presents a new gas-kinetic scheme for simulation of compressible inviscid flows. It starts to simplify the integral domain of Maxwellian distribution function over the phase velocity \(\xi\) and phase energy \(\zeta\) to the integral domain of modified Maxwellian function over the phase velocity \(\xi\) only. The influence of integral over phase energy \(\zeta\) is embodied as the particle internal energy \(e_p\). The modified Maxwellian function is further simplified to a circular function with the assumption that all the particles are concentrated on a circle. Then two circular function-based gas-kinetic schemes are presented for simulation of compressible inviscid flows. In the new schemes, no error and exponential functions, which are often appeared in the Maxwellian function-based gas-kinetic schemes, are involved. As a result, the new schemes can be implemented in a more efficient way. To validate the proposed new gas-kinetic schemes, test examples in the transonic flow, supersonic flow and hypersonic flow regimes are solved. Numerical results showed that the solution accuracy of the circular function-based gas-kinetic schemes is comparable to that of corresponding Maxwellian function-based gas-kinetic schemes. However, the circular function-based gas-kinetic schemes need less computational effort.

MSC:
76M28 Particle methods and lattice-gas methods
76N15 Gas dynamics (general theory)
76H05 Transonic flows
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gou, Z. L.; Liu, H. W.; Luo, L. S.; Xu, K., A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows, J. Comput. Phys., 227, 4955-4976, (2008) · Zbl 1388.76291
[2] Tang, L., Progress in gas-kinetic upwind schemes for the solution of Euler/Navier-Stokes equations, I: overview, Comput. Fluids, 56, 39-48, (2012) · Zbl 1365.76279
[3] Chen, S. Z.; Xu, K.; Lee, C. B.; Cai, Q. D., A unified gas kinetic scheme with moving mesh and velocity space adaptation, J. Comput. Phys., 231, 6643-6664, (2012)
[4] Kumar, G.; Girimaji, S. S.; Kerimo, J., WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence, J. Comput. Phys., 234, 499-523, (2013)
[5] Pan, L.; Zhao, G. P.; Tian, B. L.; Wang, S. H., A gas kinetic scheme for the Baer-Nunziato two-phase flow model, J. Comput. Phys., 231, 7518-7536, (2012) · Zbl 1284.76271
[6] Pullin, D. I., Direct simulation methods for compressible inviscid ideal-gas flow, J. Comput. Phys., 34, 231-244, (1980) · Zbl 0419.76049
[7] Mandal, J. C.; Deshpande, S. M., Kinetic flux vector splitting for Euler equations, Comput. Fluids, 23, 447-478, (1994) · Zbl 0811.76047
[8] Chou, S. Y.; Baganoff, D., Kinetic flux-vector splitting for the Navier-Stokes equations, J. Comput. Phys., 130, 217-230, (1997) · Zbl 0873.76057
[9] Tao, T.; Xu, K., Gas-kinetic schemes for the compressible Euler equations: positivity-preserving analysis, Z. Angew. Math. Phys., 50, 258-281, (1999) · Zbl 0958.76080
[10] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[11] Osher, S.; Chakravarthy, S., Upwind schemes and boundary conditions with applications to Euler equations in general geometries, J. Comput. Phys., 50, 447-481, (1983) · Zbl 0518.76060
[12] Xu, K., Gas-kinetic schemes for unsteady compressible flow simulations, (1998), VKI for Fluid Dynamics Lecture Series 1998-03
[13] Prendergast, K. H.; Xu, K., Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys., 109, 53-66, (1993) · Zbl 0791.76059
[14] 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
[15] 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, 289-335, (2001) · Zbl 1058.76056
[16] Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases. I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, 511-525, (1954) · Zbl 0055.23609
[17] Jiang, S.; Ni, G. X., A second-order γ-model BGK scheme for multimaterial compressible flows, Appl. Numer. Math., 57, 597-608, (2007) · Zbl 1190.76159
[18] Jiang, J.; Qian, Y. H., Implicit gas-kinetic BGK scheme with multigrid for 3D stationary transonic high-Reynolds number flows, Comput. Fluids, 66, 21-28, (2012) · Zbl 1365.76253
[19] Tang, H. Z.; Xu, K., A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics, J. Comput. Phys., 165, 69-88, (2000) · Zbl 0995.76066
[20] Lian, Y. S.; Xu, K., A gas-kinetic scheme for multimaterial flows and its application in chemical reactions, J. Comput. Phys., 163, 349-375, (2000) · Zbl 0994.76082
[21] Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys. Rev. E, 69, 056702, (2004)
[22] Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E, 69, R035701, (2004)
[23] Dellar, P. J., Two routes from the Boltzmann equation to compressible flow of polyatomic gases, Prog. Comput. Fluid Dyn., 8, 84-96, (2008) · Zbl 1187.76725
[24] Qu, K.; Shu, C.; 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, 036706, (2007)
[25] Qu, K.; Shu, C.; Chew, Y. T., Simulation of shock-wave propagation with finite volume lattice Boltzmann method, Int. J. Mod. Phys. C, 18, 447-454, (2007) · Zbl 1137.76463
[26] Yang, L. M.; Shu, C.; Wu, J., Development and comparative studies of three non-free parameter lattice Boltzmann models for simulation of compressible flows, Adv. Appl. Math. Mech., 4, 454-472, (2012)
[27] Venkatakrishnan, V., On the accuracy of limiters and convergence to steady state solutions, (1993), AIAA paper 93-0880
[28] Venkatakrishnan, V., Convergence to steady-state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118, 120-130, (1995) · Zbl 0858.76058
[29] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173, (1984) · Zbl 0573.76057
[30] Wu, H.; Shen, L. J.; Shen, Z. J., A hybrid numerical method to cure numerical shock instability, Commun. Comput. Phys., 8, 1264-1271, (2010) · Zbl 1364.76163
[31] Meister, A., Euler equations: transonic flow past a RAE2822, profile (2D)
[32] Xu, K.; Sun, Q.; Yu, P., Valid physical processes from numerical discontinuities in computational fluid dynamics, Int. J. Hypersonics, 1, 157-172, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.