I'm new to the LBE, but have been finding them very helpful for doing some fluid dynamics simulations.

I was wondering how pressure would be calculated. I'm guessing this is quite easy, but it's not obvious to me. We know the momentum in each direction at each lattice point, and the density. Would I need to just work out the change in momentum and then divide by a fixed area size (i.e. the distance between latices squared?)



From a multiscale analysis it is found that the lattice Boltzmann equation solves the continuity and Navier-Stokes equations in the incompressible limit:


$$\partial_t\rho\boldsymbol{u}+\boldsymbol{\nabla}\cdot\rho\boldsymbol{u}\boldsymbol{u}=-\boldsymbol{\nabla}p + \boldsymbol{\nabla}\cdot\left[\rho\nu\left(\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^T\right)\right]$$

where the pressure and viscosity is given by:

$$p=\rho c_s^2 \quad \nu=c_s^2\Delta t\left(\frac{\tau}{\Delta t}-\frac{1}{2}\right)$$

Here $c_s$ is the so-called 'speed of sound' which is defined by which lattice you use. D2Q9 has $c_s=1/\sqrt{3}$:

$$\sum_i w_i \boldsymbol{e}_i \boldsymbol{e}_i=c_s^2\boldsymbol{I}$$

The pressure is therefore directly proportional with the density and can be directly calculated from the density field.

In case you are dealing with multiphase systems using Shan-Chen method the pressure has an 'ideal' contribution as above and a 'non-ideal' contribution due to interaction with surrounding lattice nodes:

$$p=\rho c_s^2 + \frac{1}{2}Gc_s^2\psi^2$$

here $\psi$ is the chosen interaction potential.

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