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Title: Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

Abstract

The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, andmore » has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier-Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier-Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.« less

Authors:
 [1];  [2];  [3]
  1. School of Mathematical Sciences, Beijing Normal University, 100875 (China)
  2. Department of Engineering Mechanics, Tsinghua University, 100084 (China)
  3. Department of Mathematics, Hong Kong University of Science and Technology, Kowloon (Hong Kong)
Publication Date:
OSTI Identifier:
21499747
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 230; Journal Issue: 12; Other Information: DOI: 10.1016/j.jcp.2011.03.028; PII: S0021-9991(11)00179-3; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0021-9991
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; BOLTZMANN EQUATION; CALCULATION METHODS; COMPARATIVE EVALUATIONS; COMPRESSIBLE FLOW; COMPUTERIZED SIMULATION; CONTROL THEORY; FUNCTIONS; MACH NUMBER; MATHEMATICAL EVOLUTION; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; NAVIER-STOKES EQUATIONS; STEADY-STATE CONDITIONS; TIME DEPENDENCE; TRANSPORT THEORY; DIFFERENTIAL EQUATIONS; DIMENSIONLESS NUMBERS; EQUATIONS; EVALUATION; EVOLUTION; FLUID FLOW; INTEGRO-DIFFERENTIAL EQUATIONS; KINETIC EQUATIONS; MATHEMATICAL LOGIC; PARTIAL DIFFERENTIAL EQUATIONS; SIMULATION; SPACE; VELOCITY

Citation Formats

Li Jiequan, E-mail: jiequan@bnu.edu.c, Li Qibing, E-mail: lqb@tsinghua.edu.c, and Xu Kun, E-mail: makxu@ust.h. Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations. United States: N. p., 2011. Web.
Li Jiequan, E-mail: jiequan@bnu.edu.c, Li Qibing, E-mail: lqb@tsinghua.edu.c, & Xu Kun, E-mail: makxu@ust.h. Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations. United States.
Li Jiequan, E-mail: jiequan@bnu.edu.c, Li Qibing, E-mail: lqb@tsinghua.edu.c, and Xu Kun, E-mail: makxu@ust.h. 2011. "Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations". United States.
@article{osti_21499747,
title = {Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations},
author = {Li Jiequan, E-mail: jiequan@bnu.edu.c and Li Qibing, E-mail: lqb@tsinghua.edu.c and Xu Kun, E-mail: makxu@ust.h},
abstractNote = {The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier-Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier-Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.},
doi = {},
url = {https://www.osti.gov/biblio/21499747}, journal = {Journal of Computational Physics},
issn = {0021-9991},
number = 12,
volume = 230,
place = {United States},
year = {2011},
month = {6}
}