NR$xx$ Simulation of Microflows with Shakhov Model

Cai, Zhenning; Li, Ruo; Qiao, Zhonghua

doi:10.1137/110828551

Cited by 33 publications

(55 citation statements)

References 31 publications

(79 reference statements)

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“…The G26 results show several improvements over the R13 system at the expense of extended equations. It is also possible to numerically generate large systems of regularized moment equations, as demonstrated by Cai & Li (2010) and Cai et al (2012). In this way, it is possible to approximate the actual solution of the Boltzmann equation quite accurately.…”

Section: Further Systems Of Moment Equationsmentioning

confidence: 99%

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Torrilhon

2016

Annu. Rev. Fluid Mech.

192

158

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This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number-the ratio between the mean free path of the particles and a macroscopic length scale-becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann's equation in kinetic gas theory. With moment approximations, extended fluid dynamic equations can be derived, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low-Mach number setting for which both evolution equations and boundary conditions are well established. Conversely, nonlinear, high-speed processes require special closures that are still under development. Current approaches are examined, along with the challenge of computing shock wave profiles based on continuum equations. 429

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Section: Further Systems Of Moment Equationsmentioning

confidence: 99%

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Torrilhon

2016

Annu. Rev. Fluid Mech.

192

158

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“…A list of relevant publications can be found in the references of [13]. Recently we became interested in the large moment system together with its numerical methods [2,3,4,5], and it was found that the lack of the well-posedness due to the loss of global hyperbolicity is a major obstacle in our simulations, especially for large Mach number gas flows [5]. Torrilhon [14] provided a 13-moment moment system based on the multivariate Pearson IV distributions, which is hyperbolic when reduced to one dimension, but it seems unlikely that the same technique can be extended to systems with a large number of moments.…”

Section: Introductionmentioning

confidence: 99%

Globally Hyperbolic Regularization of Grad's Moment System

Cai

Fan

2013

Comm Pure Appl Math

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118

162

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In this paper, we propose a globally hyperbolic regularization to the general Grad's moment system in multidimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment system can be analytically given and depend only on the macroscopic velocity and the temperature. The structure of the eigenvalues and eigenvectors of the coefficient matrix is fully clarified. The regularization together with the properties of the resulting moment systems is consistent with the simple onedimensional case discussed in [1]. In addition, all characteristic waves are proven to be genuinely nonlinear or linearly degenerate, and the studies on the properties of rarefaction waves, contact discontinuities, and shock waves are included. GLOBALLY HYPERBOLIC MOMENT SYSTEM 465 Ã C PROOF. Let i D N D .˛/, with j˛j Ä M . Then we need only to verify that (3.20) z A M .i; 1WN / r D r w i is always valid. Since A M is determined by (3.2), (3.3), and (3.4), and y A M and z A M are defined as in (3.11) and (3.13), respectively, we can write all entries of z A M . Now let us verify equation (3.20) case by case: (1) For˛D 0, z A M .1; N D .e 1 // D 1 is the only nonzero entry of z A M .1; 1WN /; hence, z A M .i; 1WN / r D 1 r u 1 D r D r w i :(2) For˛D e 1 , z A M .i; 1WN / r D 2 r p 2e 1 =2 D 2 r D r u 1 D r w i :(3) For˛D e k , k D 2; : : : ; D, z A M .i; 1WN / r D 1 r p e 1 Ce k =2 D r u k D r w i : ÃHe M 1 . / ZHENNING CAI

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“…In this direction, people are also starting to concentrate on collision models other than the simple Maxwell molecules [31]. It has been shown in [5,6] that BGK and Shakhov collision terms can be easily modeled by the moment method with almost arbitrary number of moments. However, the construction of moment methods with the original collision operator is obviously not so trivial as the BGK and Shakhov operators [33].…”

Section: Introductionmentioning

confidence: 99%