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Book ChapterDOI

Macroscopic Transport Equations for Rarefied Gas Flows

01 Jan 2005-pp 145-160
About: The article was published on 2005-01-01. It has received 473 citations till now.

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Journal ArticleDOI
TL;DR: A strategy capable of simulating polydisperse flows in complex geometries is employed where the fluid transport equations are solved in an Eulerian framework and the dispersed phase is represented as Lagrangian particles.

414 citations

Journal ArticleDOI
TL;DR: This survey considers the development and mathematical analysis of numerical methods for kinetic partial differential equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods, and an overview of the current state of the art.
Abstract: In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

339 citations


Cites background or methods from "Macroscopic Transport Equations for..."

  • ...We refer to Struchtrup (2005) for alternative approaches that avoid some of the shortcomings of the classical high-order closures....

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  • ...The problem of finding high-order closures to the moment system for small and moderate Knudsen numbers has been tackled by several authors with the goal of avoiding the expensive solution of the kinetic equation (Müller and Ruggeri 1993, Struchtrup 2005)....

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  • ...H. Struchtrup (2005), Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics, Springer....

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  • ...Higher-order fluid models, such as the compressible Navier– Stokes model, can be derived using the expansions due to Chapman–Enskog and to Grad (Müller and Ruggeri 1993, Struchtrup 2005)....

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Journal ArticleDOI
TL;DR: A novel moment-inversion algorithm, based on 1-D adaptive quadratures of conditional velocity moments, is introduced and shown to always yield realizable distribution functions (i.e. non-negative quadrature weights).

220 citations


Cites background or methods from "Macroscopic Transport Equations for..."

  • ...In most applications we are not interested in knowing the exact form of the velocity distribution function, rather knowledge of its lower-order moments is sufficient [70]....

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  • ...The BGK model does not provide the correct value of Pr, which should be 2/3 for a mono-atomic gas [70]....

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  • ...Note that most moment methods are designed to work for systems near equilibrium where the collisions are dominant [41,52,70,71] (i....

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  • ...For inelastic hard-sphere collisions, a kinetic model can be used to approximate the Boltzmann collision integral in terms of a closed set of lower-order moments [10,70]....

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  • ...The advantage of using the BGK approximation is that the moment equations derived from the collision term are closed [70]....

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Journal ArticleDOI
TL;DR: In this article, a systematic approach for developing large-eddy-simulation (LES) tools for dispersed multiphase flows starting from the microscale model is presented, where a key intermediate step is the mesoscopic model for the dispersed phase, formulated in terms of kinetic equations, that contains the physical models for the flow.
Abstract: Multiphase flows occurring in nature and in technological applications are often turbulent. The large range of length scales and timescales in turbulent multiphase flows makes direct numerical simulation of the microscale governing equations intractable for many applications. In this article we review a systematic approach for developing large-eddy-simulation (LES) tools for dispersed multiphase flows starting from the microscale model. A key intermediate step is the mesoscopic model for the dispersed phase, formulated in terms of kinetic equations, that contains the physical models for the flow. Owing to the phase-space variables, direct solution of the mesoscopic model is usually intractable, and additional mathematical approximations are introduced to arrive at a macroscopic model. We show that self-conditioned LES models can be derived for both the mesoscopic and macroscopic models, but the former is preferred to ensure consistency and physical accuracy. The principal difficulties and open challenges ...

197 citations

Journal ArticleDOI
TL;DR: In this article, the basic tenets and procedures in implementing phonon hydrodynamics in nanoscale heat transport are presented through a review of its recent wide applications in modeling thermal transport properties of nanostructures.

195 citations


Cites background or methods from "Macroscopic Transport Equations for..."

  • ...The small parameter ε usually denotes Kn number (the ratio of mean free path to the characteristic size of fluid system) in classical hydrodynamics [154,155]....

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  • ...Velocity slip boundary in microscale gas flow Inmicroscale gas flow, theKnudsennumber (Kn) is defined as the ratio ofmolecularmean free pathΛ to the characteristic dimension L of gas flow (such as the height of a channel, or the diameter of a pipe) [154]: Kn = Λ/L....

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  • ...In addition, the firstorder Chapman–Enskog expansion to Boltzmann transport equation can also yield the Navier–Stokes equation [154]; the entropy balance equation and expressions of entropy flux in CIT can be derived from the kinetic theory of gases [40,219]....

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  • ...Finally, an interesting aspect is noted that nine-moment phonon hydrodynamic model corresponds to the classical thirteen-moment model in hydrodynamics [153,154]....

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