Derivation of the Lattice Boltzmann Method by Means of the Discrete Ordinate Method for the Boltzmann Equation

1997 ◽  
Vol 131 (1) ◽  
pp. 241-246 ◽  
Author(s):  
Takashi Abe
Keyword(s):  
Boltzmann Equation ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
The Boltzmann Equation ◽  
Boltzmann Method

A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

2012 ◽  
Vol 11 (4) ◽  
pp. 1397-1414 ◽  
Author(s):  
C. T. Hsu ◽  
S. W. Chiang ◽  
K. F. Sin
Keyword(s):  
Boltzmann Equation ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Physical Space ◽  
Velocity Space ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
Mean Velocity ◽  
Quadrature Scheme ◽  
Boltzmann Method

AbstractThe Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

DEVELOPING LBM-BASED FLUX SOLVER AND ITS APPLICATIONS IN MULTI-DIMENSION SIMULATIONS

2012 ◽  
Vol 19 ◽  
pp. 90-99
Author(s):  
KUN QU ◽  
CHANG SHU ◽  
JINSHENG CAI
Keyword(s):  
Boltzmann Equation ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Lattice Boltzmann Model ◽  
Navier Stokes ◽  
Lattice Boltzmann Equation ◽  
One Dimensional ◽  
Volume Method ◽  
Boltzmann Method ◽  
Boltzmann Model

In this paper, a new flux solver was developed based on a lattice Boltzmann model. Different from solving discrete velocity Boltzmann equation and lattice Boltzmann equation, Euler/Navier-Stokes (NS) equations were solved in this approach, and the flux at the interface was evaluated with a compressible lattice Boltzmann model. This method combined lattice Boltzmann method with finite volume method to solve Euler/NS equations. The proposed approach was validated by some simulations of one-dimensional and multi-dimensional problems.

A Central Difference Finite Volume Lattice Boltzmann Method for Simulation of 2D Inviscid Compressible Flows on Triangular Meshes

2018 ◽  
Author(s):  
Alireza Karbalaei ◽  
Kazem Hejranfar
Keyword(s):  
Boltzmann Equation ◽  
Lattice Boltzmann Method ◽  
Finite Volume ◽  
Lattice Boltzmann ◽  
Compressible Flows ◽  
Lattice Boltzmann Equation ◽  
Triangular Meshes ◽  
Central Difference ◽  
Inviscid Compressible Flows ◽  
Boltzmann Method

In this work, a central difference finite volume lattice Boltzmann method (CDFV-LBM) is developed to compute 2D inviscid compressible flows on triangular meshes. The numerical solution procedure adopted here for solving the lattice Boltzmann equation is nearly the same as the procedure used by Jameson et al. for the solution of the Euler equations. The integral form of the lattice Boltzmann equation using the Gauss divergence theorem is applied on a triangular cell and the numerical fluxes on each edge of the cell are set to the average of their values at the two adjacent cells. Appropriate numerical dissipation terms are added to the discretized lattice Boltzmann equation to have a stable solution. The Boltzmann equation is discretized in time using the fourth-order Runge-Kutta scheme. The computations are performed for three problems, namely, the isentropic vortex and the supersonic flow around a NACA0012 airfoil and over a circular-arc bump. The effect of changing the grid resolution and the dissipation coefficients on the accuracy of the results is also studied. Results obtained by applying the CDFV-LBM are compared with the available numerical results which show good agreement.

A Novel Dynamics Lattice Boltzmann Method for Gas Flows

2010 ◽  
Author(s):  
C. T. Hsu ◽  
S. W. Chiang ◽  
K. F. Sin
Keyword(s):  
Boltzmann Equation ◽  
Coordinate System ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Physical Space ◽  
Efficient Implementation ◽  
Gas Flows ◽  
Lattice Node ◽  
Quadrature Scheme ◽  
Boltzmann Method

The lattice Boltzmann method (LBM), where discrete velocities are specifically assigned to ensure that a particle leaves one lattice node always resides on another lattice node, has been developed for decades as a powerful numerical tool to solve the Boltzmann equation for gas flows. The efficient implementation of LBM requires that the discrete velocities be isotropic and that the lattice nodes be homogeneous. These requirements restrict the applications of the currently-used LBM schemes to incompressible and isothermal flows. Such restrictions defy the original physics of Boltzmann equation. Much effort has been devoted in the past decades to remove these restrictions, but of less success. In this paper, a novel dynamic lattice Boltzmann method (DLBM) that is free of the incompressible and isothermal restrictions is proposed and developed to simulate gas flows. This is achieved through a coordinate transformation featured with Galilean translation and thermal normalization. The transformation renders the normalized Maxwell equilibrium distribution with directional isotropy and spatial homogeneity for the accurate and efficient implementation of the Gaussian-Hermite quadrature. The transformed Boltzmann equation contains additional terms due to local convection and acceleration. The velocity quadrature points in the new coordinate system are fixed while the correspondent points in the physical space change from time to time and from position to position. By this dynamic quadrature nature in the physical space, we term this new scheme as the dynamic quadrature scheme. The lattice Boltzmann method (LBM) with the dynamic quadrature scheme is named as the dynamic lattice Boltzmann method (DLBM). The transformed Boltzmann equation is then solved in the new coordinate system based on the fixed quadrature points. Validations of the DLBM have been carried with several benchmark problems. Cavity flows problem are used. Excellent agreements are obtained as compared with those obtained from the conventional schemes. Up to date, the DLBM algorithm can run up to Mach number at 0.3 without suffering from numerical instability. The application of the DLBM to the Rayleigh-Bernard thermal instability problem is illustrated, where the onset of 2D vortex rolls and 3D hexagonal cells are well-predicted and are in excellent agreement with the theory. In summary, a novel dynamic lattice Boltzmann method (DLBM) has been proposed with algorithm developed for numerical simulation of gas flows. This new DLBM has been demonstrated to have removed the incompressible and isothermal restrictions encountered by the traditional LBM.

Symmetry Breaking of Flow in 2D Symmetric Channels: Simulations by Lattice-Boltzmann Method

1997 ◽  
Vol 08 (04) ◽  
pp. 859-867 ◽  
Author(s):  
Li-Shi Luo
Keyword(s):  
Boltzmann Equation ◽  
Symmetry Breaking ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Numerical Study ◽  
Nonlinear Flow ◽  
Lattice Boltzmann Equation ◽  
Two Dimensional ◽  
Flow Phenomena ◽  
Boltzmann Method

In this paper, a numerical study of nonlinear flow phenomena in two-dimensional symmetric channels using the lattice-Boltzmann equation method is presented. The results are compared with both experimental results and other numerical results using some traditional methods. Comparisons are found to be quantitatively accurate.

A REVIEW OF THE LATTICE BOLTZMANN METHOD

1993 ◽  
Vol 04 (02) ◽  
pp. 409-415 ◽  
Author(s):  
S. SUCCI ◽  
R. BENZI ◽  
F. MASSAIOLI
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Porous Media ◽  
Boltzmann Equation ◽  
Lattice Boltzmann Method ◽  
Turbulent Flows ◽  
Lattice Boltzmann ◽  
Lattice Boltzmann Equation ◽  
Boltzmann Method ◽  
Qualitative Discussion

In this paper, the basic elements of the theory of the Lattice Boltzmann equation are reviewed. Representative applications, such as turbulent flows and low-Reynolds flows in porous media are presented, along with a qualitative discussion on the most recent advances of this recent tool for computational fluid dynamics.

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