# Local Vorticity Computation in Double Distribution Functions based Lattice Boltzmann Methods for Flow and Scalar Transport

## TL;DR

This paper introduces a novel local vorticity computation method within double distribution function lattice Boltzmann frameworks, enabling accurate, fully local flow analysis suitable for parallel processing, demonstrated through benchmark flow simulations.

## Contribution

It presents a new local vorticity computation approach using off-diagonal moments in DDF-LBMs, avoiding finite difference derivatives and supporting parallel computation.

## Key findings

- Accurate vorticity fields with second order convergence.
- Method is fully local and suitable for parallel implementation.
- Demonstrated effectiveness on benchmark flow problems.

## Abstract

Computation of vorticity in conjunction with the strain rate tensor, plays an important role in fluid mechanics in vortical structure identification and in the modeling of various complex fluids. For the simulation of flows accompanied by the advection-diffusion transport of a scalar field, double distribution functions (DDF) based lattice Boltzmann methods (LBMs) are commonly used. We present a new local vorticity computation approach by introducing an intensional anisotropy of the scalar flux in the third order, off-diagonal moment equilibria of the LB scheme for the scalar field, and then combining the second order non-equilibrium components of both the LBMs. As such, any pair of lattice sets in the DDF formulation that can independently support the third order off-diagonal moments would enable local determination of the complete flow kinematics, with the LBMs for the fluid motion and the transport of the passive scalar respectively providing the necessary moment relationships to determine the symmetric and skew-symmetric components of the velocity gradient tensor. Since the resulting formulation is completely local and do not rely on finite difference approximations for velocity derivatives, it is by design naturally suitable for parallel computation. As an illustration of our approach, we formulate a DDF-LB scheme for local vorticity computation using a pair of multiple relaxation times (MRT) based collision approaches on two-dimensional, nine velocity (D2Q9) lattices, where the necessary moment relationships to determine the velocity gradient tensor and the vorticity are established via a Chapman-Enskog analysis. Simulations of various benchmark flows demonstrate good accuracy of the predicted vorticity fields, with a second order convergence. Furthermore, extensions of our formulation for a variety of collision models to enable local vorticity computation are presented.