# Consistent forcing scheme in the cascaded lattice Boltzmann method

## TL;DR

This paper presents a clear derivation of the cascaded lattice Boltzmann method (CLBM) using a general MRT framework, introduces a consistent forcing scheme, and validates its effectiveness through canonical simulations.

## Contribution

It provides a new, unified derivation of CLBM, develops a consistent forcing scheme, and clarifies the relation between MRT LBM and CLBM under a general framework.

## Key findings

- The forcing scheme achieves second-order spatial convergence.
- The scheme maintains isotropy and satisfies the non-slip rule.
- Validation through canonical problems demonstrates the scheme's effectiveness.

## Abstract

In this paper, we give a more pellucid derivation for the cascaded lattice Boltzmann method (CLBM) based on a general multiple-relaxation-time (MRT) frame through defining a shift matrix. When the shift matrix is a unit matrix, the CLBM degrades into an MRT LBM. Based on this, a consistent forcing scheme is developed for the CLBM. The applicability of the non-slip rule, the second-order convergence rate in space and the property of isotropy for the consistent forcing scheme is demonstrated through the simulation of several canonical problems. Several other existing force schemes previously used in the CLBM are also examined. The study clarifies the relation between MRT LBM and CLBM under a general framework.