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Updated: Jul 2, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Generalized local equilibrium in the cascaded lattice Boltzmann method.

Pietro Asinari¹

¹Department of Energetics, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 4, 2008

Summary

The cascaded lattice Boltzmann method (LBM) uses a generalized equilibrium, maintaining LBM consistency. This approach simplifies to a two-relaxation-time operator with standard moment relaxation.

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Area of Science:

Computational fluid dynamics
Numerical methods for fluid flow
Lattice Boltzmann methods

Background:

Traditional multiple-relaxation-time collision operators exhibit insufficient Galilean invariance.
Geier's cascaded lattice Boltzmann method (LBM) addresses this by differentially relaxing velocity-shifted moments.

Purpose of the Study:

To analyze the theoretical underpinnings of the cascaded LBM.
To clarify the relationship between cascaded LBM and traditional LBM formulations.
To investigate the implications for collision operators and equilibrium distributions.

Main Methods:

Theoretical analysis of moment-based collision operators in LBM.
Examination of equilibrium distribution functions in different reference frames.
Derivation of collision operators based on moment relaxation properties.

Main Results:

The cascaded LBM employs a generalized local equilibrium in the frame at rest.
This generalized equilibrium does not compromise the overall consistency of the LBM.
Reducing relaxation frequencies and relaxing raw moments in the rest frame yields a two-relaxation-time operator with a proper polynomial equilibrium.

Conclusions:

The cascaded LBM offers a consistent framework by utilizing a generalized equilibrium.
The method provides a theoretical link to established two-relaxation-time collision operators.
This analysis deepens the understanding of LBM collision dynamics and Galilean invariance.