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Liu ZJ, Shu C, Chen SY, Liu W, Yuan ZY, Yang LM. Development of explicit formulations of G45-based gas kinetic scheme for simulation of continuum and rarefied flows. Phys Rev E 2022; 105:045302. [PMID: 35590639 DOI: 10.1103/physreve.105.045302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/14/2021] [Accepted: 03/04/2022] [Indexed: 06/15/2023]
Abstract
In this work, the explicit formulations of the Grad's distribution function for the 45 moments (G45)-based gas kinetic scheme (GKS) are presented. Similar to the G13 function-based gas kinetic scheme (G13-GKS), G45-GKS simulates flows from the continuum regime to the rarefied regime by solving the macroscopic governing equations based on the conservation laws, which are widely used in conventional Navier-Stokes solver. These macroscopic governing equations are discretized by the finite volume method, where the numerical fluxes are evaluated by the local solution to the Boltzmann equation. The initial distribution function is reconstructed by the G45 distribution function, which is a higher order truncation of the Hermite expansion of distribution function compared with the G13 distribution function. Such high order truncation of Hermite expansion helps the present solver to achieve a better accuracy than G13-GKS. Moreover, the reconstruction of distribution function makes the development of explicit formulations of numerical fluxes feasible, and the evolution of the distribution function, which is the main reason why the discrete velocity method is expensive, is avoided. Several numerical experiments are performed to examine the accuracy of G45-GKS. Results show that the accuracy of the present solver for almost all flow problems is much better than G13-GKS. Moreover, some typical rarefied effects, such as the direction of heat flux without temperature gradients and thermal creep flow, can be well captured by the present solver.
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Affiliation(s)
- Z J Liu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
- Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - C Shu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
| | - S Y Chen
- Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - W Liu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
| | - Z Y Yuan
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
| | - L M Yang
- Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, Jiangsu, China
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Yuan ZY, Shu C, Liu ZJ, Yang LM, Liu W. Variant of gas kinetic flux solver for flows beyond Navier-Stokes level. Phys Rev E 2021; 104:055305. [PMID: 34942831 DOI: 10.1103/physreve.104.055305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2021] [Accepted: 10/18/2021] [Indexed: 11/07/2022]
Abstract
In this paper, a variant of gas kinetic flux solver (GKFS) is presented for simulation of flows beyond the Navier-Stokes (NS) level. The method retains the framework of GKFS and reconstructs the numerical fluxes by the moments of distribution function at the cell interface, which is given from the local solution of the Boltzmann equation. In the conventional GKFS, the first-order Chapman-Enskog (CE) expansion is utilized to approximate the initial distribution function. By using the differential chain rule, it was found that the CE expansion form could be linked to the stress tensor and the heat flux. For flows in the NS level, the stress tensor and heat flux can be simply calculated from the linearized constitutive relationship and Fourier's law, respectively. However, for flows beyond the NS level, due to the strong nonequilibrium effect, the linearized constitutive relationship and Fourier's law are insufficient to predict the stress tensor and the heat flux. To overcome this difficulty, this paper introduces correction terms to the stress tensor and heat flux in the initial distribution function. These correction terms will take effect in the strong nonequilibrium region for flows beyond the NS level. To avoid finding complex expressions or solving complicated partial differential equations for the correction terms, a simple and iterative procedure is proposed to update the correction terms based on the framework of GKFS. The proposed method is validated by three benchmark cases which cover the flow from the continuum regime to the transition regime. Numerical results show that the present solver can provide accurate solution in the continuum regime. It is indeed the correction terms that take effect in the strong nonequilibrium region for flows beyond the NS level, which enables the present solver to capture the nonequilibrium phenomenon with reasonable accuracy for rarefied flows at moderate Knudsen number.
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Affiliation(s)
- Z Y Yuan
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore
| | - C Shu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore
| | - Z J Liu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore
| | - L M Yang
- Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
| | - W Liu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore
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Lin C, Luo KH, Xu A, Gan Y, Lai H. Multiple-relaxation-time discrete Boltzmann modeling of multicomponent mixture with nonequilibrium effects. Phys Rev E 2021; 103:013305. [PMID: 33601619 DOI: 10.1103/physreve.103.013305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2020] [Accepted: 12/16/2020] [Indexed: 06/12/2023]
Abstract
A multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for multicomponent mixtures, where compressible, hydrodynamic, and thermodynamic nonequilibrium effects are taken into account. It allows the specific heat ratio and the Prandtl number to be adjustable, and is suitable for both low and high speed fluid flows. From the physical side, besides being consistent with the multicomponent Navier-Stokes equations, Fick's law, and Stefan-Maxwell diffusion equation in the hydrodynamic limit, the DBM provides more kinetic information about the nonequilibrium effects. The physical capability of DBM to describe the nonequilibrium flows, beyond the Navier-Stokes representation, enables the study of the entropy production mechanism in complex flows, especially in multicomponent mixtures. Moreover, the current kinetic model is employed to investigate nonequilibrium behaviors of the compressible Kelvin-Helmholtz instability (KHI). The entropy of mixing, the mixing area, the mixing width, the kinetic and internal energies, and the maximum and minimum temperatures are investigated during the dynamic KHI process. It is found that the mixing degree and fluid flow are similar in the KHI process for cases with various thermal conductivity and initial temperature configurations, while the maximum and minimum temperatures show different trends in cases with or without initial temperature gradients. Physically, both heat conduction and temperature exert slight influences on the formation and evolution of the KHI morphological structure.
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Affiliation(s)
- Chuandong Lin
- Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China
| | - Kai H Luo
- Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom
| | - Aiguo Xu
- Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009-26, Beijing 100088, China
- State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China
- Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of Engineering, Peking University, Beijing 100871, China
| | - Yanbiao Gan
- North China Institute of Aerospace Engineering, Langfang 065000, China
| | - Huilin Lai
- College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350007, China
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Abstract
We present a general framework for constructing trans-scale discrete Boltzmann models (DBMs) for high-speed compressible flows ranging from continuum to transition regime. This is achieved by designing a higher-order discrete equilibrium distribution function that satisfies additional nonhydrodynamic kinetic moments. To characterize the thermodynamic nonequilibrium (TNE) effects and estimate the condition under which the DBMs at various levels should be used, two measures are presented: (i) the relative TNE strength, describing the relative strength of the (N+1)th order TNE effects to the Nth order one; (ii) the TNE discrepancy between DBM simulation and relevant theoretical analysis. Whether or not the higher-order TNE effects should be taken into account in the modeling and which level of DBM should be adopted is best described by the relative TNE intensity and/or the discrepancy rather than by the value of the Knudsen number. As a model example, a two-dimensional DBM with 26 discrete velocities at Burnett level is formulated, verified, and validated.
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Affiliation(s)
- Yanbiao Gan
- North China Institute of Aerospace Engineering, Langfang 065000, China
- College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350007, China
| | - Aiguo Xu
- National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China
- Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of Engineering, Peking University, Beijing 100871, China
| | - Guangcai Zhang
- National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China
| | - Yudong Zhang
- National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China
- Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China
| | - Sauro Succi
- Center for Life Nano Science at La Sapienza, Fondazione Istituto Italiano di Tecnologia, Viale Regina Margherita 295, 00161 Roma, Italy
- Physics Department and Institute for Applied Computational Science, John A. Paulson School of Applied Science and Engineering, Harvard University, Oxford Street 29, Cambridge, Massachusetts 02138, USA
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Yang LM, Shu C, Yang WM, Wu J. Development of an efficient gas kinetic scheme for simulation of two-dimensional incompressible thermal flows. Phys Rev E 2018; 97:013305. [PMID: 29448389 DOI: 10.1103/physreve.97.013305] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2017] [Indexed: 06/08/2023]
Abstract
In this work, an efficient gas kinetic scheme is presented for simulation of two-dimensional incompressible thermal flows. In the scheme, the macroscopic governing equations for mass, momentum, and energy conservation are discretized by the finite volume method and the numerical fluxes at the cell interface are reconstructed by the local solution of the Boltzmann equation. To compute these fluxes, two distribution functions are involved. One is the circular function, which is used to calculate the numerical fluxes of mass and momentum equations. Due to the incompressible limit, the circle at the cell interface can be approximately considered to be symmetric so that the expressions for the conservative variables and numerical fluxes at the cell interface can be given explicitly and concisely. Another one is the D2Q4 model, which is utilized to compute the numerical flux of the energy equation. By following the process for derivation of numerical fluxes of mass and momentum equations, the numerical flux of the energy equation can also be given explicitly. The accuracy, efficiency, and stability of the present scheme are validated by simulating several thermal flow problems. Numerical results showed that the present scheme can provide accurate numerical results for incompressible thermal flows at a wide range of Rayleigh numbers with less computational cost than that needed by the thermal lattice Boltzmann flux solver (TLBFS), which has been proven to be more efficient than the thermal lattice Boltzmann method (TLBM).
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Affiliation(s)
- L M Yang
- Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, Jiangsu, China
- Sembcorp-NUS Corporate Laboratory, 1 Engineering Drive 2, Singapore 117576
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
| | - C Shu
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
| | - W M Yang
- Sembcorp-NUS Corporate Laboratory, 1 Engineering Drive 2, Singapore 117576
- Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
| | - J Wu
- Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, Jiangsu, China
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