Comparing macroscopic continuum models for rarefied gas dynamics

a new test method

Yingsong Zheng, Jason Reese, Henning Struchtrup

Research output: Contribution to journalArticle

14 Citations (Scopus)
66 Downloads (Pure)

Abstract

We propose a new test method for investigating which macroscopic continuum models, among the many existing models, give the best description of rarefied gas flows over a range of Knudsen numbers. The merits of our method are: no boundary conditions for the continuum models are needed, no coupled governing equations are solved, while the Knudsen layer is still considered. This distinguishes our proposed test method from other existing techniques (such as stability analysis in time and space, computations of sound speed and dispersion, and the shock wave structure problem). Our method relies on accurate, essentially noise-free, solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation; in this paper, the BGK model and the ES-BGK model equations are considered. Our method is applied to test whether one-dimensional stationary Couette flow is accurately described by the following macroscopic transport models: the Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and the regularized 13 moment equations (two types: the original, and that based on an order of magnitude approach). The gas molecular model is Maxwellian. For Knudsen numbers in the transition-continuum regime (Kn less-than-or-equals, slant 0.1), we find that the two types of regularized 13 moment equations give similar results to each other, which are better than Grad's original 13 moment equations, which, in turn, give better results than the Burnett equations. The Navier-Stokes-Fourier equations give the worst results. This is as expected, considering the presumed accuracy of these models. For cases of higher Knudsen numbers, i.e. Kn > 0.1, all macroscopic continuum equations tested fail to describe the flows accurately. We also show that the above conclusions from our tests are general, and independent of the kinetic model used.
Original languageEnglish
Pages (from-to)748-769
Number of pages21
JournalJournal of Computational Physics
Volume218
Issue number2
DOIs
Publication statusPublished - 1 Nov 2006

Fingerprint

rarefied gas dynamics
Rarefied Gas Dynamics
Gas dynamics
Continuum Model
Moment Equations
continuums
Knudsen number
BGK Model
Knudsen flow
Kinetic Model
Burnett equations
Navier-Stokes
BGK model
moments
Continuum
Rarefied Gas Flow
Kinetics
Couette Flow
Kinetic Equation
Boltzmann Equation

Keywords

  • non-continuum effects
  • rarefied gas flows
  • microfluidics
  • burnett equations
  • moment equations

Cite this

Zheng, Yingsong ; Reese, Jason ; Struchtrup, Henning. / Comparing macroscopic continuum models for rarefied gas dynamics : a new test method. In: Journal of Computational Physics. 2006 ; Vol. 218, No. 2. pp. 748-769.
@article{350e782db435423d98e837aab6b30308,
title = "Comparing macroscopic continuum models for rarefied gas dynamics: a new test method",
abstract = "We propose a new test method for investigating which macroscopic continuum models, among the many existing models, give the best description of rarefied gas flows over a range of Knudsen numbers. The merits of our method are: no boundary conditions for the continuum models are needed, no coupled governing equations are solved, while the Knudsen layer is still considered. This distinguishes our proposed test method from other existing techniques (such as stability analysis in time and space, computations of sound speed and dispersion, and the shock wave structure problem). Our method relies on accurate, essentially noise-free, solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation; in this paper, the BGK model and the ES-BGK model equations are considered. Our method is applied to test whether one-dimensional stationary Couette flow is accurately described by the following macroscopic transport models: the Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and the regularized 13 moment equations (two types: the original, and that based on an order of magnitude approach). The gas molecular model is Maxwellian. For Knudsen numbers in the transition-continuum regime (Kn less-than-or-equals, slant 0.1), we find that the two types of regularized 13 moment equations give similar results to each other, which are better than Grad's original 13 moment equations, which, in turn, give better results than the Burnett equations. The Navier-Stokes-Fourier equations give the worst results. This is as expected, considering the presumed accuracy of these models. For cases of higher Knudsen numbers, i.e. Kn > 0.1, all macroscopic continuum equations tested fail to describe the flows accurately. We also show that the above conclusions from our tests are general, and independent of the kinetic model used.",
keywords = "non-continuum effects, rarefied gas flows, microfluidics, burnett equations, moment equations",
author = "Yingsong Zheng and Jason Reese and Henning Struchtrup",
year = "2006",
month = "11",
day = "1",
doi = "10.1016/j.jcp.2006.03.005",
language = "English",
volume = "218",
pages = "748--769",
journal = "Journal of Computational Physics",
issn = "0021-9991",
number = "2",

}

Comparing macroscopic continuum models for rarefied gas dynamics : a new test method. / Zheng, Yingsong; Reese, Jason; Struchtrup, Henning.

In: Journal of Computational Physics, Vol. 218, No. 2, 01.11.2006, p. 748-769.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Comparing macroscopic continuum models for rarefied gas dynamics

T2 - a new test method

AU - Zheng, Yingsong

AU - Reese, Jason

AU - Struchtrup, Henning

PY - 2006/11/1

Y1 - 2006/11/1

N2 - We propose a new test method for investigating which macroscopic continuum models, among the many existing models, give the best description of rarefied gas flows over a range of Knudsen numbers. The merits of our method are: no boundary conditions for the continuum models are needed, no coupled governing equations are solved, while the Knudsen layer is still considered. This distinguishes our proposed test method from other existing techniques (such as stability analysis in time and space, computations of sound speed and dispersion, and the shock wave structure problem). Our method relies on accurate, essentially noise-free, solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation; in this paper, the BGK model and the ES-BGK model equations are considered. Our method is applied to test whether one-dimensional stationary Couette flow is accurately described by the following macroscopic transport models: the Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and the regularized 13 moment equations (two types: the original, and that based on an order of magnitude approach). The gas molecular model is Maxwellian. For Knudsen numbers in the transition-continuum regime (Kn less-than-or-equals, slant 0.1), we find that the two types of regularized 13 moment equations give similar results to each other, which are better than Grad's original 13 moment equations, which, in turn, give better results than the Burnett equations. The Navier-Stokes-Fourier equations give the worst results. This is as expected, considering the presumed accuracy of these models. For cases of higher Knudsen numbers, i.e. Kn > 0.1, all macroscopic continuum equations tested fail to describe the flows accurately. We also show that the above conclusions from our tests are general, and independent of the kinetic model used.

AB - We propose a new test method for investigating which macroscopic continuum models, among the many existing models, give the best description of rarefied gas flows over a range of Knudsen numbers. The merits of our method are: no boundary conditions for the continuum models are needed, no coupled governing equations are solved, while the Knudsen layer is still considered. This distinguishes our proposed test method from other existing techniques (such as stability analysis in time and space, computations of sound speed and dispersion, and the shock wave structure problem). Our method relies on accurate, essentially noise-free, solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation; in this paper, the BGK model and the ES-BGK model equations are considered. Our method is applied to test whether one-dimensional stationary Couette flow is accurately described by the following macroscopic transport models: the Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and the regularized 13 moment equations (two types: the original, and that based on an order of magnitude approach). The gas molecular model is Maxwellian. For Knudsen numbers in the transition-continuum regime (Kn less-than-or-equals, slant 0.1), we find that the two types of regularized 13 moment equations give similar results to each other, which are better than Grad's original 13 moment equations, which, in turn, give better results than the Burnett equations. The Navier-Stokes-Fourier equations give the worst results. This is as expected, considering the presumed accuracy of these models. For cases of higher Knudsen numbers, i.e. Kn > 0.1, all macroscopic continuum equations tested fail to describe the flows accurately. We also show that the above conclusions from our tests are general, and independent of the kinetic model used.

KW - non-continuum effects

KW - rarefied gas flows

KW - microfluidics

KW - burnett equations

KW - moment equations

UR - http://www.sciencedirect.com/science/article/pii/S0021999106001318

U2 - 10.1016/j.jcp.2006.03.005

DO - 10.1016/j.jcp.2006.03.005

M3 - Article

VL - 218

SP - 748

EP - 769

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -