# The computation of strain rate tensor in multiple-relaxation-time lattice Boltzmann model

## TL;DR

This paper compares two formulas for computing the strain rate tensor in MRT-LB models, analyzing their relationship, advantages, and applicability to different lattice patterns, to guide better choice in complex flow simulations.

## Contribution

It clarifies the relationship between Yu and Chai formulas and discusses their respective advantages, aiding in selecting the appropriate method for MRT-LB simulations.

## Key findings

- Yu formula is more computationally efficient.
- Chai formula applies to more lattice patterns.
- Yu formula can be derived from Chai formula in specific cases.

## Abstract

Multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is an important class of LB model with lots of advantages over traditional single-relaxation-time (SRT) LB model. In addition, the computation of strain rate tensor is crucial in MRT-LB simulations of some complex flows. Up to now, there are only two formulas to compute the strain rate tensor in the MRT LB model. One is to compute the strain rate tensor by using non-equilibrium parts of macroscopic moments (Yu formula). The other is to compute the strain rate tensor by using non-equilibrium parts of density distribution functions (Chai formula). The mathematical expressions of these two formulas are so different that we do not know which formula to choose for computing the strain rate tensor in MRT LB model. In this paper, we study the relationship of these two formulas. It is found that Yu formula can be deduced from Chai formula in a particular procedure. However, these two formulas have their own advantages and disadvantages. Yu formula is more efficient in the computation aspect while Chai formula can be applied to more lattice patterns of MRT LB models. It is also found that deducing Yu formula for a particular lattice pattern from Chai formula is more convenient than the way proposed by Yu et al.