Shear rate calculation in LBM

nipun.arora · October 8, 2020, 1:56pm

I want to calculate the shear rate magnitude in LBM. The individual components of the shear rate tensor can be calculated in LBM as given below in the first equation. Now I am confused about how to calculate the magnitude? I have seen 3 different formulae (as shown in Equations 2, 3, and 4).

$\tiny S_{\alpha\beta}=\frac{-3}{2\rho_0\tau}\sum_{i}^{}\left ( f_i-f_i^{eq} \right )\\ \dot{\gamma}=\sqrt{2D_{II}} \ or \ \dot{\gamma}=2\sqrt{D_{II}}\ or\ \dot{\gamma}=\sqrt{D_{II}}\ \ where\ D_{II}=\sum_{\alpha,\beta}^{}S_{\alpha\beta}S_{\alpha\beta}$

is the second invariant of the tensor. Please suggest which one is the correct formula with proper explanation.

orestis · October 28, 2020, 8:34am

To my knowledge it’s the first variant. I also think that’s the way it’s computed in Palabos.