Computational framework for the regularized 20‐moment equations for non‐equilibrium gas flows

Mizzi, Simon; Gu, Xiaojun; Emerson, David R.; Barber, Robert W.; Reese, Jason M.

doi:10.1002/fld.1747

Cited by 7 publications

(6 citation statements)

References 11 publications

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“…Without confusion, the superscript "n" will be omitted below, and Π u β ,θ β is simplified asΠ β . Mimicing the method in [32], suppose 20) and then we can reconstruct the spatial partial derivative by a central difference in the smooth case: 21) or the van Leer reconstruction in the discontinuous case:…”

Section: The Numerical Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

Cai¹,

Li²

2010

SIAM J. Sci. Comput.

139

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We introduce a numerical method for solving Grad's moment equations or regularized moment equations for arbitrary order of moments. In our algorithm, we do not explicitly need the moment equations. Instead, we directly start from the Boltzmann equation and perform Grad's moment method [12] and the regularization technique [27] numerically. We define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions. For the collision term, the BGK model is adopted so that the production step can be done trivially based on the Hermite expansion. Extensive numerical examples for one-and two-dimensional problems are presented. Convergence in moments can be validated by the numerical results for different number of moments.

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Section: The Numerical Methodsmentioning

confidence: 99%

“…The regularized moment equations are also considered, and to our knowledge, no results about the generation of regularized equations for arbitrary number of moments have been reported. Numerical methods for R13 equations are discussed in [32,15,37], and the computational framework for the R20 equations can be found in [21].…”

Section: Introductionmentioning

confidence: 99%

Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

Cai¹,

Li²

2010

SIAM J. Sci. Comput.

139

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“…As the Knudsen layer is a linear superposition of many exponential layers (Struchtrup 2008), the R13 equations only resolve one such contribution. Mizzi et al (2008) introduced m ij k into the governing equations to alleviate this problem. In the present approach, the GTMs for m ij k and R ij near the wall are provided by extending the GMM from 13 to 26 moments, which includes the moments m ij k , R ij and as extended hydrodynamic variables.…”

Section: Extended Thermodynamic Governing Equationsmentioning

confidence: 99%

A high-order moment approach for capturing non-equilibrium phenomena in the transition regime

2009

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The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

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“…The indices in the angular brackets denote the symmetric trace-free parts of tensors. The R13 equations closure contains, therefore, additional quantities , and ∆, corresponding to the higher order moments, given by (Struchtrup 2005, Mizzi et al 2008, Rana et al 2013:…”

Section: Continuum Model Based Descriptionmentioning

confidence: 99%

Continuum Analysis of Rarefaction Effects on a Thermally Induced Gas Flow

Hssikou

Baliti

Alaoui

2019

Mathematical Problems in Engineering

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A Maxwell gas confined within a micro cavity with non-isothermal walls is investigated in the slip and early transition regimes using the classical and extended continuum theories. The vertical sides of the cavity are kept at the uniform and environmental temperature 0 , while the upper and bottom ones are linearly heated in opposite directions from the cold value 0 to the hot one . The gas flow is, therefore, induced only by the temperature gradient created along the longitudinal walls. The problem is treated from a macroscopic point of view by solving numerically the so-called regularized 13-moment equations (R13) recently developed as an extension of Grad 13-moments theory to the third order of the Knudsen number powers in the Chapman-Enskog expansion. The gas macroscopic properties obtained by this method are compared with the classical continuum theory results (NSF) using the first and second order of velocity slip and temperature jump boundary conditions. The gas flow behavior is studied as a function of the Knudsen number ( ), nonlinear effects, for different heating rates 0 ⁄ . The micro cavity aspect ratio effect is also evaluated on the flow fields in this study.

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Computational framework for the regularized 20‐moment equations for non‐equilibrium gas flows

Cited by 7 publications

References 11 publications

Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

A high-order moment approach for capturing non-equilibrium phenomena in the transition regime

Continuum Analysis of Rarefaction Effects on a Thermally Induced Gas Flow

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