Lattice Boltzmann equation: what Do We Know and What Can We Do With It?

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Description:	Luo, LS (Old Dominion) Tuesday 21 September 2010, 14:00-14:45

Created:

2010-09-30 17:06

Collection:

Partial Differential Equations in Kinetic Theories

Publisher:

Isaac Newton Institute

Luo, LS

Language:

eng (English)

Credits:

Author:	Luo, LS
Producer:	Steve Greenham


Abstract:	The primary purpose of this talk is to satisfy the curiosity of those who may be interested in the lattice Boltzmann equation (LBE): What is LBE? And what can it do for us? Thus, I will first show how to derive the LBE from the continuous linearized Boltzmann equation. Through the derivation, some basic mathematical features of the LBE become apparent. First, it is a discrete moment system suitable for solving near incompressible (or low speed) flows and it can be related to the method of artificial compressibility. Second, it cannot be used to solve the continuous Boltzmann equation, because it cannot accurately model the higher order moments of the distribution function due to the constraint imposed by the symmetry of the underlying lattice. And third, the LBE can be related to finite-difference method; it is a second-order accurate solver for incompressible Navier-Stokes equation. To demonstrate the capability of the LBE, I will show a comparative study of the LBE and the pseudo-spectral (PS) method for direct numerical simulations (DNS) of the decaying turbulence in a three dimensional periodic cube. Not only some statistical quantities, but also the instantaneous flow fields (velocity and vorticity) are compared in details. Our study demonstrates that the results obtained by the LBE and PS methods agree very well when the flow is properly resolved in both methods. Our results indicate that the LBE requires twice as many grid points in each dimension as in the PS method. LBE simulations of complex flows (such as particulate suspensions, free-surface flows, and multi-component flow through porous media, etc.) will also be shown if time permits.

Abstract:

The primary purpose of this talk is to satisfy the curiosity of those who may be interested in the lattice Boltzmann equation (LBE): What is LBE? And what can it do for us?
Thus, I will first show how to derive the LBE from the continuous linearized Boltzmann equation. Through the derivation, some basic mathematical features of the LBE become apparent. First, it is a discrete moment system suitable for solving near incompressible (or low speed) flows and it can be related to the method of artificial compressibility. Second, it cannot be used to solve the continuous Boltzmann equation, because it cannot accurately model the higher order moments of the distribution function due to the constraint imposed by the symmetry of the underlying lattice. And third, the LBE can be related to finite-difference method; it is a second-order accurate solver for incompressible Navier-Stokes equation. To demonstrate the capability of the LBE, I will show a comparative study of the LBE and the pseudo-spectral (PS) method for direct numerical simulations (DNS) of the decaying turbulence in a three dimensional periodic cube. Not only some statistical quantities, but also the instantaneous flow fields (velocity and vorticity) are compared in details. Our study demonstrates that the results obtained by the LBE and PS methods agree very well when the flow is properly resolved in both methods. Our results indicate that the LBE requires twice as many grid points in each dimension as in the PS method. LBE simulations of complex flows (such as particulate suspensions, free-surface flows, and multi-component flow through porous media, etc.) will also be shown if time permits.

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