Equilibrium distribution functions for Transport equations

For standard Advection-Diffusion Equation (ADE), the EqDF are presented in Continuous Boltzmann equation. We present here the EqDF for several ADE-type equations such as the Conservative Allen-Cahn equation, the Cahn-Hilliard equation and phase-field equation for crystal growth. For those phase-field equations, a new distribution function \(g_i\) is introduced. The lattice Boltzmann equation for that \(g_i\) with the BGK collision operator writes:

(211)\[g_{i}(\boldsymbol{x}+\boldsymbol{c}_{i}\delta t,t+\delta t)=g_{i}(\boldsymbol{x},t)-\frac{1}{\tau_{g}+0.5}\left[g_{i}(\boldsymbol{x},t)-g_{i}^{eq,\iota}(\boldsymbol{x},t)\right]+\delta t\mathcal{G}_i\]

where \(\mathcal{G}_i\) is the microscopic source term. In \(g_i^{eq,\iota}\), the superscript \(\iota\) can be \(ADE,CH,CAC\). Those options are described below. In the rest of this section, the coefficient \(M_{\phi}\) involved in the transport equation is related to the collision rate \(\tau_g\) by:

(212)\[M_{\phi}=\tau_{g}c_{s}^{2}\delta t\]

where \(c_s=c/\sqrt{3}\) and \(c=\delta x/\delta t\).

\(g_i^{eq}\) for ADE with source term

Target macroscopic equation

The ADE with a source term writes:

(213)\[\partial_{t}\underbrace{\phi}_{\mathcal{M}_{0}}+\boldsymbol{\nabla}\cdot(\underbrace{\boldsymbol{u}\phi}_{\boldsymbol{\mathcal{M}}_{1}})=\boldsymbol{\nabla}\cdot\Bigl[M_{\phi}\boldsymbol{\nabla}\cdot(\underbrace{\phi\boldsymbol{I}}_{\boldsymbol{\mathcal{M}}_{2}})\Bigr]+\mathscr{S}_{\phi}\]

where the notation \(\phi\) is used here as the conserved quantity (e.g. concentration), \(\boldsymbol{u}\) is the velocity, the notation \(M_{\phi}\) represents the diffusion coefficient and \(\mathscr{S}_{\phi}\) is the macroscopic source term to be defined. Inside the bracket, \(\boldsymbol{I}\) is the identity tensor of second order. In that equation, the moments are noted:

• \(\mathcal{M}_{0}\): is the moment of order 0 (scalar)

• \(\boldsymbol{\mathcal{M}}_{1}\): is the moment of order 1 (vector)

• \(\boldsymbol{\mathcal{M}}_{2}\): is the moment of order 2 (tensor of rank 2)

Source term

The microscopic source term is simply defined by

(214)\[\mathcal{G}_i=w_i\mathscr{S}_{\phi}\]

Its moment of order zero is \(\mathscr{S}_{\phi}\) because \(\sum_iw_i=1\).

Equilibrium with source term

The EqDF \(g_i^{eq,ADE}\) must be defined such as \(\mathcal{M}_{0}=\phi\), \(\boldsymbol{\mathcal{M}}_{1}=\boldsymbol{u}\phi\) and \(\boldsymbol{\mathcal{M}}_{2}=\phi\boldsymbol{I}\). One can prove that one possible solution is

(215)\[g_{i}^{eq,ADE}(\boldsymbol{x},t)=w_{i}\phi\left[1+\frac{\boldsymbol{u}\cdot\boldsymbol{c}_{i}}{c_{s}^{2}}\right]=w_{i}\underbrace{\phi}_{\mathcal{M}_{0}\text{ and }\boldsymbol{\mathcal{M}}_{2}}+w_{i}\frac{(\overbrace{\boldsymbol{u}\phi}^{\boldsymbol{\mathcal{M}}_{1}})\cdot\boldsymbol{c}_{i}}{c_{s}^{2}}\]

When using a source term in LBE (211), a second order of accuracy is obtained provided that the following variable change \(\overline{g}_{i}^{eq,ADE}\) is used:

(216)\[\overline{g}_{i}^{eq,ADE}(\boldsymbol{x},t)=g_{i}^{eq,ADE}-\frac{\delta t}{2}\mathcal{G}_{i}\]

Moment

After collision and streaming, the new \(\phi\) function is obtained by the moment of order zero corrected by the source term \(\mathcal{G}_{i}\delta t/2\):

(217)\[\phi=\sum_{i}g_{i}+\frac{\delta t}{2}\mathscr{S}_{\phi}\]

\(g_i^{eq}\) for Cahn-Hilliard equation

Target macroscopic equation

The Cahn-Hilliard equation writes:

(218)\[\partial_{t}\underbrace{\phi}_{\mathcal{M}_{0}}+\boldsymbol{\nabla}\cdot(\underbrace{\boldsymbol{u}\phi}_{\boldsymbol{\mathcal{M}}_{1}})=\boldsymbol{\nabla}\cdot\Bigl[M_{\phi}\boldsymbol{\nabla}\cdot(\underbrace{\mu_{\phi}\boldsymbol{I}}_{\boldsymbol{\mathcal{M}}_{2}})\Bigr]\]

where The chemical potential is a function of \(\phi\) defined by

(219)\[\mu_{\phi}=4H\phi(\phi-1)(\phi-1/2)-\zeta\underbrace{\boldsymbol{\nabla}^{2}\phi}_{\text{Laplacian}}\]

The first term of the right-hand side is the derivative of double-well and the second one involves the Laplacian of \(\phi\). Compared to the ADE (213), in Cahn-Hilliard equation (218), the source term is null \(\mathscr{S}_{\phi}=0\), and the chemical potential \(\mu_{\phi}\) replaces \(\phi\) for the moment of second order.

Equilibrium for CH equation

To design the EqDF, its moment of order zero ant its moment of first order must remain unchanged i.e. \(\mathcal{M}_{0}=\phi\) and \(\boldsymbol{\mathcal{M}}_{1}=\boldsymbol{u}\phi\). But now its moment of second order must be equal to \(\boldsymbol{\mathcal{M}}_{2}=\mu_{\phi}\boldsymbol{I}\). One solution writes:

(220)\[g_{i}^{eq,\,CH}=\mathcal{A}_{i}(\phi,\,\mu_{\phi})+w_{i}\frac{\boldsymbol{c}_{i}\cdot(\overbrace{\boldsymbol{u}\phi}^{\boldsymbol{\mathcal{M}}_{1}})}{c_{s}^{2}}\]

where

(221)\[\begin{split}\mathcal{A}_{i}(\phi,\,\mu_{\phi})=\begin{cases} \phi-3\mu_{\phi}(1-w_{0}) & \text{if }i=0\quad\mathcal{M}_{0}\\ 3w_{i}\mu_{\phi} & \text{if }i\neq0\quad\boldsymbol{\mathcal{M}}_{2} \end{cases}\end{split}\]

The chemical potential \(\mu_{\phi}\) appears in (221). For the computation of the Laplacian which is involved in its definition (219), see Section Gradients and Laplacian.

Moment

After collision and streaming, the new \(\phi\) function is obtained by the moment of order zero

(222)\[\phi=\sum_{i}g_{i}\]

\(g_i^{eq}\) for Conservative Allen-Cahn equation

Target macroscopic equation

The Conservative Allen-Cahn equation writes:

(223)\[\partial_{t}\underbrace{\phi}_{\mathcal{M}_{0}}+\boldsymbol{\nabla}\cdot(\underbrace{\boldsymbol{u}\phi}_{\boldsymbol{\mathcal{M}}_{1}})+\boldsymbol{\nabla}\cdot\biggl[\underbrace{M_{\phi}\frac{4}{W}\phi(1-\phi)\boldsymbol{n}}_{\boldsymbol{j}_{ct}\equiv\boldsymbol{\mathcal{M}}_{1}}\biggr]=\boldsymbol{\nabla}\cdot\Bigl[M_{\phi}\boldsymbol{\nabla}\cdot(\underbrace{\phi\boldsymbol{I}}_{\boldsymbol{\mathcal{M}}_{2}})\Bigr]\]

Compared to ADE with source term, a new term appears: the last one in the left-hand side. That term involves a counter term noted \(\boldsymbol{j}_ct\) which cancels the interface movement due to the curvature. The EqDF is designed such as its moments of order 0 and 2 are unchanged compared to ADE. Its moment of order 1 must be shlightly modified to account for the new counter term. Two methods are possible for that purpose.

Method 1: equilibrium for CAC without source term (\(\mathcal{G}_i=0\))

The first method consists in modifying the EqDF by adding a new equilibrium for that counter term. That \(g_{i}^{eq,CAC}\) is the sum of the classical ADE equiblibrium plus an equilibrium designed for \(\boldsymbol{j}_{ct}\):

(224)\[\begin{split}g_{i}^{eq,CAC}=g_{i}^{eq,ADE}+g_{i}^{eq,ct}\qquad\text{with }\begin{cases} g_{i}^{eq,ADE} & =w_{i}\phi+(\boldsymbol{u}\phi)\cdot\left(\frac{w_{i}\boldsymbol{c}_{i}}{c_{s}^{2}}\right)\\ g_{i}^{eq,ct} & =\boldsymbol{j}_{ct}\cdot\left(\frac{w_{i}\boldsymbol{c}_{i}}{c_{s}^{2}}\right) \end{cases}\end{split}\]

where \(g_{i}^{eq,ct}\) is simply obtained by the scalar product of \(\boldsymbol{j}_{ct}\) with \(w_i\boldsymbol{c}_{i}/c_{s}^{2}\) like the advective term \(\boldsymbol{u}\phi\).

(225)\[g_{i}^{eq,CAC}(\boldsymbol{x},\,t)=\underbrace{w_{i}\phi\left[1+\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}\right]}_{g_{i}^{eq,ADE}}+\underbrace{M_{\phi}\left[\frac{4}{W}\phi(1-\phi)\right]\boldsymbol{n}}_{\boldsymbol{j}_{ct}}\cdot\left(\frac{w_{i}\boldsymbol{c}_{i}}{c_{s}^{2}}\right)\]

The evolution of \(g_i\) follows (211) with \(\mathcal{G}_i=0\).

Method 2: equilibrium for ADE with source term (\(\mathcal{G}_i\neq0\))

Because the moments of order 0, 1 and 2 are the same than ADE, the second method uses the \(g_i^{eq,ADE}\) and considers the divergence of the counter term as a source term \(\mathcal{G}_i\):

(226)\[\overline{g}_{i}^{eq,ADE}(\boldsymbol{x},t)=w_{i}\phi\left[1+\frac{\boldsymbol{c}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}\right]-\frac{\delta t}{2}\mathcal{G}_{i}\]

where the source term is defined by

(227)\[\mathcal{G}_{i}=\frac{4}{W}\phi(1-\phi)w_{i}\boldsymbol{c}_{i}\cdot\boldsymbol{n}\]

Let us notice that in the right-hand side, the mobility coefficient does not appear compared to the expression which appears in the equilibrium (225).

Moment

Finally, after collision and streaming, the new \(\phi\) is updated by

(228)\[\phi=\sum_{i}g_{i}+\frac{\delta t}{2}\mathcal{G}_{i}\]

For the first method \(\mathcal{G}_i=0\).

Section author: Alain Cartalade