Journal of Statistical Physics

, Volume 155, Issue 2, pp 277–299

# Transmission–Reflection Coefficient in the Lattice Boltzmann Method

• Hiroaki Yoshida
• Hidemitsu Hayashi
Article

## Abstract

We consider the permeable bounce-back scheme in the lattice Boltzmann (LB) method for incompressible flows, in which a fraction of the distribution function is bounced back and the remainder travels to the neighboring lattice points. An asymptotic analysis of the scheme is carried out in order to show that the fractional coefficient, referred to as the transmission–reflection coefficient, relates the pressure drop to the flow velocity. The derived relation, which clarifies the role played by the transmission–reflection coefficient in the macroscopic description, is helpful in using the scheme to simulate flows involving a pressure drop or gradient. The scheme is compared with the existing methods in which the transmission–reflection coefficient is employed, and the difference is clarified. As an application of the permeable bounce-back scheme, we perform an LB simulation for flows through porous media described by the Brinkman model.

## Keywords

Lattice Boltzmann method Permeable bounce-back rule  Asymptotic analysis Brinkman model

## Notes

### Acknowledgments

The authors would like to thank Professor Li-Shi Luo and Professor Pierre Lallemand for their valuable comments and useful discussions. The present work was partially supported by MEXT program “Elements Strategy Initiative to Form Core Research Center” (since 2012). (MEXT stands for Ministry of Education Culture, Sports, Science and Technology, Japan.)

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