Numerical Hydrodynamics from Gas-Kinetic Theory

Prendergast, K. H.; Xu, Kun

doi:10.1006/jcph.1993.1198

Cited by 195 publications

(130 citation statements)

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“…(8) is only one of the finite-difference scheme of the kinetic equation (6).Therefore, we use Eq. (6) The specific models that satisfy the above constraints (9a)-(9e) are presented in the following section (Sec.IIIB).…”

Section: Lattice Boltzmann Modelmentioning

confidence: 99%

“…Huang et al [5] used flow-adapted discrete velocities, a non-unique equilibrium distribution constrained by a set of linear moments and used interpolated nodes. Prendergast and Xu [6], Kim et al [7] and Koltelnikov and Montgomery used Bhatnagar-Gross-Krook type model [8] to establish new type flux and employed TVD flux limitation with the neighborhood cells, Rena et al [9], Vahala et al [10] , Qian [11,12],Sun [13], De Cicco et al [14], Mason [15,16], Yan [17,18], and Kataoka and Tsutahara [19,20] proposed many models by using additional techniques to achieve higher Mach number for the compressible flows [21]. Among the existing models for the compressible flows, the one by Kataoka and Tsutahara (KT) has a simple and rigorous theoretical background [22].…”

Section: Introductionmentioning

confidence: 99%

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Entropy and Weak Solutions in the Thermal Model for the Compressible Euler Equations

Zheng

2009

Int. J. Mod. Phys. C

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Among the existing models for compressible fluids, the one by Kataoka and Tsutahara (KT model, Phys. Rev. E 69, 056702, 2004) has a simple and rigorous theoretical background. The drawback of this KT model is that it can cause numerical instability if the local Mach number exceeds 1. The precise mechanism of this instability has not yet been clarified. In this paper, we derive entropy functions whose local equilibria are suitable to recover the Euler-like equations in the framework of the lattice Boltzmann method for the KT model. Numerical examples are also given, which are consistent with the above theoretical arguments, and show that the entropy condition is not fully guaranteed in KT model. The negative entropy may be the inherent cause for the non-physical oscillations in the vicinity of the shock. In contrast to these Karlin's microscopic entropy approach, the corresponding subsidiary entropy condition in the LBM calculation could also be deduced explicitly from the macroscopic version, which provides some insights on the numerical instability of the lattice Boltzmann model for shock calculation.

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Section: Lattice Boltzmann Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Entropy and Weak Solutions in the Thermal Model for the Compressible Euler Equations

Zheng

2009

Int. J. Mod. Phys. C

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“…Physically, upwinding can be viewed in two di erent ways: Firstly, as resulting from a Gudonov methodology of resolving discontinuities in data at cell interfaces using an approximate Riemann solver 11,12]. And secondly, as resulting from the local solution of the collisionless Boltzmann equations 20,21,22]. The present scheme falls into the second category but di ers from other such schemes in having a discretized velocity space.…”

Section: The Recovery Of Characteristic-based Ux-splittingmentioning

confidence: 99%

An Euler solver based on locally adaptive discrete velocities

Nadiga

1995

J Stat Phys

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A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature|both the origin of the discrete-velocity space and the magnitudes of the discrete-velocities are dependent on the local ow| and are used in a nite volume context. The numerical implementation of the model follows the near-equilibrium ow method of Nadiga and Pullin 1] and results in a scheme which is second order in space (in the smooth regions and between rst and second order at discontinuities) and second order in time.(The three-dimensional code is included.) For one choice of the scaling between the magnitude of the discrete-velocities and the local internal energy of the ow, the method reduces to a uxsplitting scheme based on characteristics. As a preliminary exercise, the result of the Sod shock-tube simulation is compared to the exact solution.

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“…Gas-kinetic schemes developed from the BGK model have been successfully applied to 1-D and 2-D flows [1,2]. One of the advantages of the gas-kinetic approach over more conventional methods is realized when one considers the multidimensional Euler and Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

“…For an equilibrium gas flow with f = g, the Euler equations in three dimensional space can be obtained by taking the moments of Ca in Eq. (1). On the other hand, to the first order of r, the Navier-Stokes equations, with a dynamic viscosity coefficient of v = rp (where p is the pressure), can be derived from the Chapman-Enskog expansion.…”

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confidence: 99%

Gas-kinetic finite volume methods

Martinelli

Jameson

Lecture Notes in Physics

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I n t r o d u c t i o nGas-kinetic schemes developed from the BGK model have been successfully applied to 1-D and 2-D flows [1,2]. One of the advantages of the gas-kinetic approach over more conventional methods is realized when one considers the multidimensional Euler and Navier-Stokes equations. In fact, by requiring only a single scalar gas distribution function f, the gas-kinetic approach greatly simplifies the calculation of mass, momentum, energy densities and their fluxes.In this paper we construct a novel 3-D method for the system of conservation laws of fluid flow. A Lax-Wendroff type scheme is developed from gas-kinetics and applied to solve time dependent problems. In the limit of infinite collision time (v = ~) our formulation reduces precisely to the kinetic representation of flux vector splitting for the Euler equations. Moreover, we introduce a new time-independent flux formulation which, when applied in conjunction with Jameson's multigrid time stepping scheme, yields efficient and accurate solutions of steady compressible flows.

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Numerical Hydrodynamics from Gas-Kinetic Theory

Cited by 195 publications

References 0 publications

Entropy and Weak Solutions in the Thermal Model for the Compressible Euler Equations

Entropy and Weak Solutions in the Thermal Model for the Compressible Euler Equations

An Euler solver based on locally adaptive discrete velocities

Gas-kinetic finite volume methods

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