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Multiple-relaxation-time discrete Boltzmann modeling of multicomponent mixture with nonequilibrium effects

Chuandong Lin, Kai H. Luo, Aiguo Xu, Yanbiao Gan, and Huilin Lai
Phys. Rev. E 103, 013305 – Published 11 January 2021
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  • INTRODUCTION
  • DISCRETE BOLTZMANN MODEL
  • VERIFICATION AND VALIDATION
  • KELVIN-HELMHOLTZ INSTABILITY
  • CONCLUSIONS AND DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    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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    • Received 24 January 2020
    • Revised 9 June 2020
    • Accepted 16 December 2020

    DOI:https://doi.org/10.1103/PhysRevE.103.013305

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Complex fluidsKelvin-Helmholtz instabilityTurbulent mixing
    Fluid Dynamics

    Authors & Affiliations

    Chuandong Lin1, Kai H. Luo2,*, Aiguo Xu3,4,5, Yanbiao Gan6, and Huilin Lai7,†

    • 1Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China
    • 2Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom
    • 3Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009-26, Beijing 100088, China
    • 4State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China
    • 5Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of Engineering, Peking University, Beijing 100871, China
    • 6North China Institute of Aerospace Engineering, Langfang 065000, China
    • 7College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350007, China

    • *K.Luo@ucl.ac.uk
    • †hllai@fjnu.edu.cn

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    Vol. 103, Iss. 1 — January 2021

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    Images

    • Figure 1
      Figure 1

      Sketch of the discrete velocities and mesh grids.

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    • Figure 2
      Figure 2

      Initial configuration of the three-component diffusion.

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    • Figure 3
      Figure 3

      Grid convergence analysis: (a) the horizontal distribution of mole fractions XA at the time t=0.05, (b) relative errors under various spatial steps.

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    • Figure 4
      Figure 4

      Molar fractions in the diffusion process: XA (top), XB (middle), and XC (bottom). Squares, circles, triangles, and diamonds denote DBM results at time instants t=0.005, 0.02, 0.06, and 0.2, respectively. Solid lines stand for the corresponding analytical solutions.

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    • Figure 5
      Figure 5

      Nonequilibrium quantities at time t=0.02 in the diffusion process. Squares, circles, and triangles denote DBM results of f̂5Asneq, f̂5Bsneq, and f̂5Csneq, respectively. Solid lines stand for the corresponding analytical solutions.

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    • Figure 6
      Figure 6

      Initial configuration of the thermal Couette flow.

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    • Figure 7
      Figure 7

      Vertical distribution of the horizontal speed ux (a) and nonequilibrium quantity f̂6Asneq (b) in the thermal Couette flow. Squares, circles, triangles, and diamonds represent DBM results at time instants t=0.1, 0.4, 2.0, and 30, respectively. Solid lines stand for the corresponding analytical solutions.

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    • Figure 8
      Figure 8

      Vertical distribution of the temperature in the steady Couette flow. (a) Cases with Pr=1.0 and γ=1.3, 1.4, and 1.5, respectively. (b) Cases with γ=1.4 and Pr=0.5, 1.0, and 2.0, respectively.

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    • Figure 9
      Figure 9

      Initial configuration of the Sod shock tube.

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    • Figure 10
      Figure 10

      Profiles of density at a time instant t=0.2 in the Sod shock tube with various spatial steps (a) and temporal steps (b).

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    • Figure 11
      Figure 11

      Profiles of density (a), horizontal speed (b), internal energy (c), and pressure (d) at a time instant t=0.2 in the Sod shock tube. Symbols represent DBM results, and solid lines stand for Riemann solutions.

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    • Figure 12
      Figure 12

      Initial configuration of the KHI.

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    • Figure 13
      Figure 13

      Contours of the entropy of mixing at time instants t=0.0, 0.5, 1.0, 1.5, and 3.0 in the evolution of KHI.

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    • Figure 14
      Figure 14

      Physical quantities in the evolution of KHI with TL=TR: (a) the entropy of mixing and its growth rate, (b) the mixing area and its growth rate, (c) the mixing width and its growth rate, (d) the kinetic energy, (e) the internal energy, (f) the total energy, (g) the maximum temperature, (h) the minimum temperature, and (i) the temperature difference. The inserts in (d) and (e) correspond to the rectangles, respectively. The lines with squares, circles, upper triangles, lower triangles, and diamonds indicate Pr=0.25, 0.5, 1.0, 2.0, and 4.0, respectively.

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    • Figure 15
      Figure 15

      Physical quantities in the evolution of KHI with TL≠TR: (a) the entropy of mixing and its growth rate, (b) the mixing area and its growth rate, (c) the mixing width and its growth rate, (d) the kinetic energy, (e) the internal energy, (f) the total energy, (g) the maximum temperature, (h) the minimum temperature, and (i) the temperature difference. The inserts in (d) and (e) correspond to the rectangles, res- pectively. The lines with squares, circles, upper triangles, lower triangles, and diamonds indicate Pr=0.25, 0.5, 1.0, 2.0, and 4.0, respectively.

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    • Figure 16
      Figure 16

      Comparison of physical fields at the time t=1.5 in the KHI process. From top to bottom are the mass fraction of species A, vorticity, and temperature in the three rows, respectively. From left to right are the cases (TL=TR and Pr=0.25), (TL=TR and Pr=4.0), (TL≠TR and Pr=0.25), and (TL≠TR and Pr=4.0) in the four columns, respectively. Only a part of the horizontal range 0.5≤x≤1.0 is shown in each subfigure.

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