Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-18T21:11:02.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards the development of a multiscale, multiphysics method for the simulation of rarefied gas flows

Published online by Cambridge University Press:  02 August 2010

DAVID A. KESSLER ,
ELAINE S. ORAN  and
CAROLYN R. KAPLAN
DAVID A. KESSLER*
Affiliation:
Laboratory for Computational Physics and Fluid Dynamics, US Naval Research Laboratory, Washington, DC 20375, USA
ELAINE S. ORAN
Affiliation:
Laboratory for Computational Physics and Fluid Dynamics, US Naval Research Laboratory, Washington, DC 20375, USA
CAROLYN R. KAPLAN
Affiliation:
Laboratory for Computational Physics and Fluid Dynamics, US Naval Research Laboratory, Washington, DC 20375, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continuum field, and the cycle is repeated. The target applications for this type of method are transition-regime gas flows for which standard continuum models (e.g. Navier–Stokes equations) cannot be used, but solution of Boltzmann's equation is prohibitively expensive. The use of molecular-level data to close the conservation laws significantly extends the range of applicability of the continuum conservation laws. In this study, the CM3 is used to perform two proof-of-principle calculations: a low-speed Rayleigh flow and a thermal Fourier flow. Velocity, temperature, shear-stress and heat-flux profiles compare well with direct-simulation Monte Carlo solutions for various Knudsen numbers ranging from the near-continuum regime to the transition regime. We discuss algorithmic problems and the solutions necessary to implement the CM3, building upon the conceptual framework of the heterogeneous multiscale methods.

JFM classification

Mathematical Foundations: Computational methods Compressible Flows: Gas dynamics Rarefied Gas Flow: Kinetic theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010 , pp. 262 - 293
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Al-Mohssen, H. A., Hadjiconstantinou, N. G. & Kevrekidis, I. G. 2007 Acceleration methods for coarse-grained numerical solution of the Boltzmann equation. ASME J. Fluid Engng 129, 908–912.CrossRefGoogle Scholar
van Albada, G. D., van Leer, B. & Roberts, W. W. 1982 A comparative study of computational methods in cosmic gas dynamics. Astron. Astrophys. 108, 76–84.Google Scholar
Andullah, L. S. & Babovsky, H. 2003 A discrete Boltzmann equation based on hexagons. Math. Models Methods Appl. Sci. 13, 1537–1563.CrossRefGoogle Scholar
Ansumali, S. & Karlin, I. V. 2002 Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev. E 66, 026311.CrossRefGoogle ScholarPubMed
Ansumali, S. & Karlin, I. V. 2005 Consistent lattice Boltzmann method. Phys. Rev. Lett. 95, 260605.CrossRefGoogle ScholarPubMed
Ansumali, S., Karlin, I. V., Arcidiacono, S., Abbas, A. & Pasianakis, N. I. 2007 Hydrodynamics beyond Navier–Stokes: exact solution to the lattice Boltzmann hierarchy. Phys. Rev. Lett. 98, 124502.CrossRefGoogle Scholar
Babovsky, H. 1998 Discretization and numerical schemes for steady kinetic model equations. Comput. Math. Appl. 35, 29–40.CrossRefGoogle Scholar
Baker, L. L. & Hadjiconstantinou, N. G. 2005 Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17, 051703.Google Scholar
Benzi, R., Succi, S. & Vergassola, M. 1992 The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145–197.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford Science Publications.CrossRefGoogle Scholar
Boris, J., Landsberg, A., Oran, E. S. & Gardner, J. 1993 LCPFCT: a flux-corrected transport algorithm for solving generalized continuity equations. Memorandum Rep. 6410-93-7192, US Naval Research Laboratory, Washington, DC.CrossRefGoogle Scholar
Bourgat, J. F., Le Tallec, P. & Tidriri, M. D. 1996 Coupling Boltzmann and Navier–Stokes equations by friction. J. Comput. Phys. 127, 227–245.CrossRefGoogle Scholar
Broadwell, J. E. 1964 Study of rarefied flow by the discrete velocity method. J. Fluid Mech. 19, 401–414.CrossRefGoogle Scholar
Cercignani, C. 1975 Theory and Applications of the Boltzmann Equation. Elsevier.Google Scholar
Chikatamarla, S. S. & Karlin, I. V. 2006 Entropy and Galilean invariance of lattice Boltzmann theories. Phys. Rev. Lett. 97, 190601.CrossRefGoogle ScholarPubMed
Chun, J. & Koch, D. 2005 A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number. Phys. Fluids 17, 87–132.CrossRefGoogle Scholar
Degond, P. & Jin, S. 2005 A smooth transition model between kinetic and diffusion equations. SIAM J. Numer. Anal. 42, 2671–2687.Google Scholar
Degond, P., Jin, S. & Mieussens, L. 2005 A smooth transition model between kinetic and hydrodynamic equations. J. Comput. Phys. 209, 665–694.Google Scholar
Degond, P., Liu, J.-G. & Mieussens, L. 2006 Macroscopic fluid models with localized kinetic upscaling effects. Multiscale Model. Simul. 5, 940–979.Google Scholar
Deshpande, S. M. 1986 Kinetic-theory based method for inviscid compressible flows. Tech. Paper 2613. NASA Langley.Google Scholar
Dierckx, P. 1975 An algorithm for smoothing, differentiation and integration of experimental data using spline functions. J. Comput. Appl. Math. 1, 165–184.CrossRefGoogle Scholar
Dierckx, P. 1982 A fast algorithm for smoothing data on a rectangular grid while using spline functions. SIAM J. Numer. Anal. 19, 1286–1304.CrossRefGoogle Scholar
Dierckx, P. 1993 Curve and Surface Fitting with Splines. Monographs on Numerical Analysis, Oxford University Press.Google Scholar
Doolen, G. D. (Ed.) 1990 Lattice Gas Methods for Partial Differential Equations. Addison-Wesley.Google Scholar
E, W. & Engquist, B. 2003 The heterogeneous multiscale methods. Commun. Math. Sci. 1, 87–132.CrossRefGoogle Scholar
Fan, J. & Shen, C. 2001 A direct simulation method for subsonic, microscale gas flows. J. Comput. Phys. 167, 393–412.CrossRefGoogle Scholar
Fiscko, K. A. & Chapman, D. R. 1989 Comparison of Burnett, super-Burnett and Monte Carlo solutions for hypersonic shock structure. Prog. Aeronaut. Astronaut. 118, 374–395.Google Scholar
Garcia, A. L., Bell, J. B., Crutchfield, W. Y. & Alder, B. J. 1999 Adaptive mesh and algorithm refinement using direct simulation Monte Carlo. J. Comput. Phys. 154, 134–155.CrossRefGoogle Scholar
Gatignol, R. 1970 Théorie cinétique d'un gaz à répartition discrète de vitesses. Z. Flugwiss. 18, 93–97.Google Scholar
Goldstein, D., Sturtevant, B. & Broadwell, J. E. 1989 Investigation of the motion of discrete-velocity gases. Prog. Astronaut. Aeronaut. 118, 100–117.Google Scholar
Gorban, A. N., Karlin, I. V. & Zinovyev, A. Y. 2004 Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396, 197–403.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407.Google Scholar
Harten, A., Lax, P. D. & van Leer, B. 1983 On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61.Google Scholar
Higuera, F. J. & Jimenez, J. 1989 Boltzmann approach to lattice gas simulations. Europhys. Lett. 9, 663–668.CrossRefGoogle Scholar
Homolle, T. M. M. & Hadjiconstantinou, N. G. 2007 A low-variance deviational simulation Monte Carlo for the Boltzmann equation. J. Comput. Phys. 226, 2341–2358.CrossRefGoogle Scholar
Kaplan, C. R. & Oran, E. S. 2002 Nonlinear filtering for low-velocity gaseous microflows. AIAA J. 40, 82–90.CrossRefGoogle Scholar
Karlin, I. V., Gorban, A. N., Dukek, G. & Nonnenmacher, T. F. 1998 Dynamic correction to moment approximations. Phys. Rev. E 57, 1668–1672.CrossRefGoogle Scholar
Kevrekidis, I. G., Gear, C. W. & Hummer, G. 2004 Equation-free: the computer-assisted analysis of complex, multiscale systems. AICHE J. 50, 1346–1355.Google Scholar
Kevrekidis, I. G., Gear, C. W., Hyman, J. M., Kevrekidis, P. G., Runborg, O. & Theodoro-poulos, K. 2003 Equation-free coarse-grained multiscale computation: enabling macroscopic simulators to perform system-level tasks. Commun. Math. Sci. 1, 715–762.Google Scholar
Kim, S. H., Pitsch, H. & Boyd, I. D. 2008 Accuracy of higher order lattice Boltzmann methods for microscale flows with finite Knudsen numbers. J. Comput. Phys. 227, 8655–8671.CrossRefGoogle Scholar
Laso, M. & Ottinger, H. C. 1993 Calculation of viscoelastic flow using molecular models: the CONNFESSIT approach. J. Fluid Mech. 47, 1–20.Google Scholar
Le Tallec, P. & Mallinger, F. 1997 Coupling Boltzmann and Navier–Stokes equations by half fluxes. J. Comput. Phys. 136, 51–67.CrossRefGoogle Scholar
Lian, Y.-Y., Wu, J.-S., Cheng, G. & Koomullil, R. 2005 Development of a parallel hybrid method for the DSMC and Navier–Stokes solver. AIAA Paper 2005-0435.CrossRefGoogle Scholar
Macrossan, M. N. 1989 The equilibrium flux method for the calculation of flows with non-equilibrium chemical reactions. J. Comput. Phys. 80, 204–231.CrossRefGoogle Scholar
McNamara, G. R. & Zanetti, G. 1988 Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61, 2332–2335.CrossRefGoogle ScholarPubMed
Mieussens, L. 2000 Discrete-velocity models and numerical schemes for the Boltzmann–BGK equation in plane and axisymmetric geometries. J. Comput. Phys. 162, 429–466.Google Scholar
Nordsieck, A. & Hicks, B. L. 1967 Monte Carlo evaluation of the Boltzmann collision integral. In Rarefied Gas Dynamics (ed. Brundin, L.), pp. 675–710. Academic.Google Scholar
Oran, E. S. & Boris, J. P. 2001 Numerical Simulation of Reactive Flows. Elsevier.Google Scholar
Oran, E. S., Oh, C. K. & Cybyk, B. 1998 Direct simulation Monte Carlo: recent advances and applications. Annu. Rev. Fluid Mech. 30, 403–441.CrossRefGoogle Scholar
Pan, L., Ng, T. & Lam, K. 2001 Molecular block model direct simulation Monte Carlo method for low velocity microgas flows. J. Micromech. Mircroengng 11, 181–188.CrossRefGoogle Scholar
Panton, R. L. 1996 Incompressible Flow. Wiley–Interscience.Google Scholar
Perthame, B. 1992 Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29, 1–19.CrossRefGoogle Scholar
Pham-Van-Diep, G. C., Erwin, D. A. & Muntz, E. P. 1991 Testing continuum descriptions of low-Mach-number shock structures. J. Fluid Mech. 232, 403–413.Google Scholar
Plapp, M. & Karma, A. 2000 Multiscale finite-difference-diffusion-Monte-Carlo method for simulating dendritic solidification. J. Comput. Phys. 165, 592–619.Google Scholar
Prendergast, K. H. & Xu, K. 1993 Numerical hydrodynamics from gas-kinetic theory. J. Comput. Phys. 109, 53–66.CrossRefGoogle Scholar
Pullin, D. I. 1980 Direct simulation methods for compressible inviscid ideal-gas flow. J. Comput. Phys. 34, 231–244.Google Scholar
Qian, Y. H., d'Humières, D. & Lallemand, P. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479–484.CrossRefGoogle Scholar
Reitz, R. D. 1981 One-dimensional compressible gas dynamics calculations using the Boltzmann equation. J. Comput. Phys. 42, 108–123.CrossRefGoogle Scholar
Roveda, R., Goldstein, D. B. & Varghese, P. L. 2000 Hybrid Euler/particle approach for continuum/rarefied flows. J. Spacecr. Rockets 35, 258–265.Google Scholar
Sanders, R. H. & Prendergast, K. H. 1974 The possible relation of the 3-kiloparsec arm to explosions in the galactic nucleus. Astrophys. J. 188, 489–500.CrossRefGoogle Scholar
Schwartzentruber, T. E. & Boyd, I. D. 2006 A hybrid particle-continuum method applied to shock waves. J. Comput. Phys. 215, 402–416.CrossRefGoogle Scholar
Shen, C., Tian, D. B., Xie, C. & Fan, J. 2004 Examination of the LBM in simulations of microchannel flow in transitional regime. Microscale Thermophys. Engng 8, 423–432.Google Scholar
Steger, J. L. & Warming, R. F. 1981 Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods. J. Comput. Phys. 40, 263–293.Google Scholar
Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grad's 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668.Google Scholar
Sun, Q. & Boyd, I. 2002 Statistical simulation of low-speed rarefied gas flows. J. Comput. Phys. 179, 400–425.Google Scholar
Theodoropoulos, K., Qian, Y. H. & Kevrekidis, I. G. 2000 ‘Coarse’ stability and bifurcation analysis using time-steppers: a reaction–diffusion example. Proc. Natl. Acad. Sci. 97, 9840–9843.Google Scholar
Torrilhon, M. & Struchtrup, H. 2004 Regularized 13-moment equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171.CrossRefGoogle Scholar
Verhaeghe, F., Luo, L.-S. & Blanpain, B. 2009 Lattice Boltzmann modeling of microchannel flow in slip flow regime. J. Comput. Phys. 228, 147–157.CrossRefGoogle Scholar
Wadsworth, D. C. & Erwin, D. A. 1992 Two-dimensional hybrid continuum/particle approach for rarefied flows. AIAA Paper 92-2975.Google Scholar
Wagner, W. 1992 A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation. J. Statist. Phys. 66, 1011–1044.CrossRefGoogle Scholar
Watari, M. 2009 Velocity slip and temperature jump simulations by the three-dimensional thermal finite-difference lattice Boltzmann method. Phys. Rev. E 79, 066706.Google Scholar
Wijesinghe, H. S., Hornung, R. D., Garcia, A. L. & Hadjiconstantinou, N. G. 2004 Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics. ASME J. Fluids Engng 126, 768–777.Google Scholar
Xu, K. & Prendergast, K. 1994 Numerical Navier–Stokes solutions from gas kinetic theory. J. Comput. Phys. 114, 9–17.Google Scholar
Yasuda, S. & Yamamoto, R. 2010 Multiscale modeling and simulation for polymer melt flows between parallel plates. Phys. Rev. E 81, 036308.Google Scholar
Yen, S.-M. 1971 Monte Carlo solutions of nonlinear Boltzmann equation for problems of heat transfer in rarefied gases. Intl J. Heat Mass Transfer 14, 1865–1869.Google Scholar
Yen, S. M. 1984 Numerical solution of the nonlinear Boltzmann equation for nonequilibrium gas flow problems. Annu. Rev. Fluid Mech. 16, 67–97.CrossRefGoogle Scholar