A Gas-Kinetic BGK Scheme for the Navier–Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method

Xu, Kun

doi:10.1006/jcph.2001.6790

Cited by 644 publications

(749 citation statements)

References 41 publications

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“…This is again an extension of g in [15]. It will be shown in Section 3 that the inclusion of both the external source term and the flow gradients in the tangential direction in the expressions of f 0 (9) and g (10) is very important to obtain a well-balanced scheme and to improve the accuracy and stability of the scheme.…”

Section: Gas Evolutionmentioning

confidence: 96%

“…The above model is an extension of the Chapman-Enskog expansion in [15], where the gravitational force term has been added in the modification of the non-equilibrium state. The term proportional to τ corresponds to the dissipative terms in the Navier-Stokes equations.…”

Section: Gas Evolutionmentioning

confidence: 99%

“…To make the BGK scheme simulate any heat-conducting flow, the energy flux related to the heat conduction part in (12) can be modified [15] to…”

Section: Gas Evolutionmentioning

confidence: 99%

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A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

Tian

Chan

et al. 2007

Journal of Computational Physics

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This paper extends the gas-kinetic scheme for one-dimensional inviscid shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to multidimensional gas dynamic equations under gravitational fields. Four important issues in the construction of a well-balanced scheme for gas dynamic equations are addressed. First, the inclusion of the gravitational source term into the flux function is necessary. Second, to achieve second-order accuracy of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann equation with the inclusion of the external force term is used. Third, to avoid artificial heating in an isolated system under a gravitational field, the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface. Fourth, the multidimensional approach with the inclusion of tangential gradients in two-dimensional and three-dimensional cases becomes important in order to maintain the accuracy of the scheme. Many numerical examples are used to validate the above issues, which include the comparison between the solutions from the current scheme and the Strang splitting method. The methodology developed in this paper can also be applied to other systems, such as semi-conductor device simulations under electric fields.

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Section: Gas Evolutionmentioning

confidence: 96%

Section: Gas Evolutionmentioning

confidence: 99%

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A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

Tian

Chan

et al. 2007

Journal of Computational Physics

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“…Since in kinetic regions the numerical fluxes in 'b' and 'B' cells are computed using an upwind formula on the distribution function, we choose to compute the numerical fluxes as in gas kinetic schemes ( [31]). Thus for cell interfaces inside the Euler region, the numerical flux will be given by:…”

Section: Transport Step In Euler Cellsmentioning

confidence: 99%

“…These fluxes can be computed analytically in terms of macroscopic quantities, see [31]. The computations are tedious and can be found in [1].…”

Section: Transport Step In Euler Cellsmentioning

confidence: 99%

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

Alaia

Puppo

2012

Journal of Computational Physics

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In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows.In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms.Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen numbers.

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Unified Treatment of Rarefied and Continuum Fluid Flows

2012

Encyclopedia of Aerospace Engineering

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The unified scheme is a newly developed multiscale gas‐kinetic scheme for both continuum and rarefied flow simulations. Different from many kinetic solvers that discretize the Boltzmann‐type equations directly, the unified scheme uses the local evolution solution of a gas distribution function for the flux modeling across a cell interface. The integral solution of the kinetic model covers both hydrodynamical and kinetic scale particle evolution physics. The weights between these two scale solutions used in the unified scheme depend on the ratio of local numerical cell size to the particle mean free path. Since the integral solution is valid in all flow regimes, the unified scheme can present accurate solution from free molecule transport to the Navier–Stokes gas evolution. As a result, the unified method has advantages to simulate flow with both continuum and rarefied regimes in a single computation. This advantage of using the unified scheme in microflow simulations is more obvious because of the absence of statistical noise. The unified scheme has been validated through extensive numerical tests. New microflow phenomena have been discovered through its calculation. The unified scheme provides a valuable tool in the study of nonequilibrium flows.

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A Gas-Kinetic BGK Scheme for the Navier–Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method

Cited by 644 publications

References 41 publications

A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

Unified Treatment of Rarefied and Continuum Fluid Flows

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