Lattice Boltzmann Methods for Navier-Stokes/Korteweg model

Target macroscopic NS/K model

The target macroscopic PDEs are fully described in Model of Navier-Stokes/Korteweg (NSK).

Since the historical method of “Shan-Chen” [1] & [2] many methods have been published to simulate those equations. An important family of methods is called Pseudo-potential methods. See a review e.g. in [3]. We present here, one of them implemented in LBM_Saclay, called the Kupershtokh’s forcing term [4]. Next, we present an alternative method, more intuitive which discretizes the potential form of the tensor pressure. Finally we present a scheme, based on a modification of the equilibrium distribution function and called “Well-Balanced LBM” [5].

Standard LBM with a forcing term

(229)\[f_{i}(\boldsymbol{x}+\boldsymbol{c}_{i}\delta t,t+\delta t)=f_{i}(\boldsymbol{x},\,t)-\frac{1}{\tau}\left[f_{i}(\boldsymbol{x},\,t)-f_{i}^{eq}(\boldsymbol{x},\,t)\right]+\delta t\mathcal{F}_{i}(\boldsymbol{x},t)\]
(230)\[f_{i}^{eq}(\boldsymbol{x},t)=w_{i}\rho\left[1+\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}+\frac{(\boldsymbol{c}_{i}\cdot\boldsymbol{u})^{2}}{2c_{s}^{4}}-\frac{\boldsymbol{u}\cdot\boldsymbol{u}}{2c_{s}^{2}}\right]\]
(231)\[\rho=\sum_{i}f_{i}\qquad\]
(232)\[\boldsymbol{u}=\frac{1}{\rho}\sum_{i}f_{i}\boldsymbol{c}_{i}+\frac{\delta t}{2}\boldsymbol{F}\]

Pseudo-potential methods

Kupershtokh’s forcing term

In Eq. (229), the Kupershtokh’s forcing term writes

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

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

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

The force term \(\boldsymbol{F}_{int}\) is defined in Eq. (235). The subscript \(int\) for stands for interaction

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

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

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

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

The discrete version of (235) writes

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

Discretization of Potential form of pressure tensor

Standard scheme with Guo’s forcing term (STD)

In Eq. (229), the forcing term is defined by

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

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

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

The chemical potential \(\mu_{\rho}\) depends on the Equation of State which is used. They are defined for van der Waals and Carnahan-Starling in Equations of state (EoS). In Eq. (239) \(\boldsymbol{\nabla}\rho c_{s}^{2}\) is added to cancel the perfect gas eos which arises from Chapman-Enskog when the standard EqDF Eq. (230) is used. Both gradients \(\boldsymbol{\nabla}\rho\) and \(\boldsymbol{\nabla}\mu_{\rho}\) are discretized by directional finite difference (see Gradients and Laplacian).

Well-balanced LBM

Well-balanced LBM

The Lattice Boltzmann Equation is unchanged:

(240)\[f_{i}(\boldsymbol{x}+\boldsymbol{c}_{i}\delta t,t+\delta t)=f_{i}(\boldsymbol{x},t)-\frac{1}{\tau}\left[f_{i}(\boldsymbol{x},t)-f_{i}^{eq}(\boldsymbol{x},t)\right]+\left(1-\frac{1}{2\tau}\right)\delta t{\color{blue}\mathcal{F}_{i}(\boldsymbol{x},t)}\]

But, in order to avoid parasitic currents arising from the finite difference discretization of \(\boldsymbol{\nabla}\rho c_{s}^{2}\) in the previous scheme (STD), a new equilibrium distribution function is defined [5]:

(241)\[\begin{split}f_{i}^{eq}&=\begin{cases} \rho-(1-w_{0})\rho_{0}+w_{0}\rho\Gamma_{0}(\boldsymbol{u}) & i=0\\ w_{i}\left[\rho_{0}+\rho\Gamma_{i}(\boldsymbol{u})\right] & i\neq0 \end{cases}\end{split}\]

where \(\rho_0\) is a constant often set as \(\rho_0=0\) and the function \(\Gamma_i(\boldsymbol{u})\) is:

(242)\[\Gamma_{i}(\boldsymbol{u})=\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}+\frac{1}{2}\left(\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}\right)^{2}-\frac{\boldsymbol{u}\cdot\boldsymbol{u}}{2c_{s}^{2}}\]

The forcing term must also be modified:

(243)\[\mathcal{F}_{i}=w_{i}\left[\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{F}}{c_{s}^{2}}+\frac{\boldsymbol{u}(\boldsymbol{F}+c_{s}^{2}\boldsymbol{\nabla}\rho):(\boldsymbol{c}_{i}\boldsymbol{c}_{i}-c_{s}^{2}\overline{\overline{\boldsymbol{I}}})}{c_{s}^{4}}+\frac{1}{2}\left(\frac{\boldsymbol{c}_{i}^{2}}{c_{s}^{2}}-D\right)(\boldsymbol{u}\cdot\boldsymbol{\nabla}\rho)\right]\]

where \(D\) is the spatial dimension (\(D=2\) for D2Q9).

References

Section author: Alain Cartalade