Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-Variate Pearson-IV-Distributions

Torrilhon, Manuel

doi:10.4208/cicp.2009.09.049

Cited by 54 publications

(45 citation statements)

References 31 publications

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“…A particular interesting case is the Pearson-IV distribution, which allows for anisotropic variances and skewness in multiple dimensions; hence, it can model stress and heat flux. Torrilhon (2010a) demonstrated how a Pearson-IV distribution can be used to formulate moment equations for rarefied gases. Whereas the equations showed global hyperbolicity for processes in one space dimension, complex eigenvalues still occur in higher-dimensional situations.…”

Section: Nonlinear Moment Closuresmentioning

confidence: 99%

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Torrilhon

2016

Annu. Rev. Fluid Mech.

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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: Nonlinear Moment Closuresmentioning

confidence: 99%

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Torrilhon

2016

Annu. Rev. Fluid Mech.

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192

158

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“…A list of relevant publications can be found in the references of [13]. 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. 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

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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“…whereû i = ξ if is a given vector of moments [25]. The closing flux moment is then given byŜ =Ŝ(Q,R) = ξ 5f (Model) .…”

Section: Closure Problemmentioning

confidence: 99%

“…accounting for the effects of macroscopic force fields. An important feature of moment equations is the hyperbolicity of the transport operator as loss of hyperbolicity renders the system ill-posed and numerically unstable, see [25].…”

Section: Introductionmentioning

confidence: 99%

On Singular Closures for the 5-Moment System in Kinetic Gas Theory

Schaerer

Torrilhon

2015

Commun. Comput. Phys.

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Moment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. A more recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

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Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-Variate Pearson-IV-Distributions

Cited by 54 publications

References 31 publications

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Globally Hyperbolic Regularization of Grad's Moment System

On Singular Closures for the 5-Moment System in Kinetic Gas Theory

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