LBM for Navier-Stokes/Korteweg model

(233)\[f_{i}(\boldsymbol{x}+\boldsymbol{c}_{i}\delta t,t+\delta t)=f_{i}(\boldsymbol{x},\,t)-\frac{1}{\tau}\left[f_{i}(\boldsymbol{x},\,t)-f_{i}^{eq}(\boldsymbol{x},\,t)\right]+\delta t\mathcal{F}_{i}(\boldsymbol{x},t)\]

(234)\[f_{i}^{eq}(\boldsymbol{x},t)=w_{i}\rho\left[1+\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}+\frac{(\boldsymbol{c}_{i}\cdot\boldsymbol{u})^{2}}{2c_{s}^{4}}-\frac{\boldsymbol{u}\cdot\boldsymbol{u}}{2c_{s}^{2}}\right]\]

(235)\[\rho=\sum_{i}f_{i}\qquad\]

(236)\[\boldsymbol{u}=\frac{1}{\rho}\sum_{i}f_{i}\boldsymbol{c}_{i}+\frac{\delta t}{2}\boldsymbol{F}\]

Pseudo-potential methods

Since the historical method of “Shan-Chen” many methods have been published. They are called Pseudo-potential methods. We present here, one of them implemented in LBM_Saclay.

Kupershtokh’s forcing term

In Eq. (233), the Kupershtokh’s forcing term writes

\[\mathcal{F}_{i}(\boldsymbol{x},t)=f_{i}^{eq}(\boldsymbol{u}^{\star}+\Delta\boldsymbol{u})-f_{i}^{eq}(\boldsymbol{u}^{\star})\]

where \(\boldsymbol{u}^{\star}\) and \(\Delta\boldsymbol{u}\) are defined by

\[\begin{split}\boldsymbol{u}^{\star}&=\frac{1}{\rho}\sum_{i}f_{i}\boldsymbol{c}_{i}\\ \Delta\boldsymbol{u}&=\frac{1}{\rho}\boldsymbol{F}_{int}\delta t\end{split}\]

The force term \(\boldsymbol{F}_{int}\) is defined in Eq. (237).

The subscript \(int\) for stands for interaction

\[\hspace{4mm}\]

Definition of \(\boldsymbol{F}_{int}\)

(237)\[\boldsymbol{F}_{int}=\psi(\rho)\boldsymbol{\nabla}\psi(\rho)\]

where \(\psi(\rho)\equiv \psi^{eos}(\rho)\) is called the pseudo-potential which is defined by

(238)\[\psi(\rho)=\sqrt{\frac{2(\rho c_{s}^{2}-p^{eos}(\rho))}{c_{s}^{2}}}\]

where \(p^{eos}(\rho)\) is the Other equations of state (EoS).

The discrete version of (237) writes

(239)\[\boldsymbol{F}_{int}(\boldsymbol{x},t)=\psi(\boldsymbol{x},t)\frac{1}{\delta x}\sum_{i}w_{i}\psi(\boldsymbol{x}+\boldsymbol{c}_{i}\delta t,t)\boldsymbol{c}_{i}\]

Potential form of pressure tensor

Guo’s forcing term

In Eq. (233), the forcing term is defined by

(240)\[\mathcal{F}_{i}(\boldsymbol{x},t)=w_{i}\left[\frac{\boldsymbol{c}_{i}-\boldsymbol{u}}{c_{s}^{2}}+\frac{(\boldsymbol{c}_{i}\cdot\boldsymbol{u})\boldsymbol{c}_{i}}{c_{s}^{4}}\right]\cdot\boldsymbol{F}_{tot}\]

where the total force \(\boldsymbol{F}_{tot}\) is defined by Eq. (241)

\[\hspace{4mm}\]

Definition of \(\boldsymbol{F}_{tot}\)

(241)\[\boldsymbol{F}_{tot}=\boldsymbol{\nabla}\rho c_{s}^{2}-\rho\boldsymbol{\nabla}\mu_{\rho}\]

where \(\mu_{\rho}\) are defined for van der Waals and Carnahan-Starling In

Other equations of state (EoS)

Section author: Alain Cartalade