Chapman–Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media

Abstract

The gray lattice Boltzmann equation (GLBE) method has recently been used to simulate fluid flow in porous media. It employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) in the evolution equation to account for the linear drag of the medium. Several particular GLBE schemes have been proposed in the literature and these schemes are very easy to implement; but there exists uncertainty about the need for redefining the macroscopic velocity as there has been no systematic analysis to recover the Brinkman equation from the various GLBE schemes. Rigorous Chapman–Enskog analyses are carried out to show that the momentum equation recovered from these schemes can satisfy Brinkman equation to second order in $\epsilon$ only if $\theta = {\rm O}\left( \epsilon \right)$ in which $\epsilon$ is the ratio of the lattice spacing to the characteristic length of physical dimension. The need for redefining macroscopic velocity is shown to be scheme-dependent. When a body force is encountered such as the gravitational force or that caused by a pressure gradient, different forms of forcing redefinitions are required for each GLBE scheme.

Appendices

Appendix A: Chapman–Enskog Analysis of GLBE Schemes to Recover Darcy-Extended Brinkman Equation

By introducing the expansion Eq. (15) and

$$ f_{\alpha } \left( {{\mathbf{x}} + {\mathbf{e}}_{\alpha } \Delta t,t + \Delta t} \right) = \sum\limits_{n = 0}^{\infty } {\frac{{ \epsilon^{n} }}{n!}{\mathbf{D}}^{n} f_{\alpha }^{\left( n \right)} } \left( {{\mathbf{x}},t} \right), $$

(A-1)

where D = e _α ·∇ (omitting the time dependency for steady-state condition), we can rewrite the WBS scheme proposed by Walsh et al. [16] including a body force term

$$ \begin{aligned} f_{\alpha } ({\mathbf{x}} + {\mathbf{e}}_{\alpha } \Delta t,t + \Delta t) - f_{\alpha } ({\mathbf{x}},t) = - \left( {1 - \theta } \right)\left( {{\mathbf{M}}^{ - 1} {\mathbf{SM}}} \right)_{\alpha \beta } \left[ {f_{\beta } \left( {{\mathbf{x}},t} \right) - f_{\beta }^{eq} \left( {{\mathbf{x}},t} \right)} \right] + \left( {1 - \theta } \right)\Delta tF_{\alpha } - \theta \left[ {f_{\alpha } \left( {{\mathbf{x}},t} \right) - f_{{\bar{\alpha }}} \left( {{\mathbf{x}},t} \right)} \right] \hfill \\ \end{aligned} $$

(A-2)

in the consecutive orders of the parameter $\epsilon$ as follows:

$$ {\rm O}\left( { \epsilon^{0} } \right):f_{\alpha }^{\left( 0 \right)} {=} f_{\alpha }^{eq} , $$

(A-3a)

$$ {\rm O}\left( { \epsilon^{1} } \right):D_{1\alpha } f_{\alpha }^{\left( 0 \right)} {=} - \frac{1}{\Delta t}\left( {{\mathbf{M}}^{ - 1} {\mathbf{SM}}} \right)_{\alpha \beta } f_{\beta }^{\left( 1 \right)} - \frac{{\theta_{1} }}{\Delta t}\left( {f_{\alpha }^{\left( 0 \right)} - f_{{\bar{\alpha }}}^{\left( 0 \right)} } \right) + F_{1\alpha } , $$

(A-3b)

$$ {\rm O}\left( { \epsilon^{2} } \right):\left( {{\mathbf{M}}^{ - 1} \left( {{\mathbf{I}} - \frac{{\mathbf{S}}}{2}} \right){\mathbf{M}}} \right)_{\alpha \beta } D_{1\beta } f_{\beta }^{\left( 1 \right)} + \frac{\Delta t}{2}D_{1\alpha } F_{1\alpha } = - \frac{1}{\Delta t}\left( {{\mathbf{M}}^{ - 1} {\mathbf{SM}}} \right)_{\alpha \beta } \left( {f_{\beta }^{\left( 2 \right)} } \right) + \frac{{\theta_{1} }}{\Delta t}\left( {{\mathbf{M}}^{ - 1} {\mathbf{SM}}} \right)_{\alpha \beta } \left( {f_{\beta }^{\left( 1 \right)} } \right) - \frac{{\theta_{1} }}{\Delta t}\left( {f_{\alpha }^{\left( 1 \right)} - f_{{\bar{\alpha }}}^{\left( 1 \right)} } \right) - \theta_{1} F_{1\alpha } + \frac{1}{2}\frac{\partial }{{\partial x_{1i} }}D_{1\alpha } \left[ {\theta_{1} \left( {f_{\alpha }^{\left( 0 \right)} - f_{{\bar{\alpha }}}^{\left( 0 \right)} } \right)} \right]. $$

(A-3c)

where D_1α = e _α ·∇₁ and $F_{1\alpha } = \omega_{\alpha } \left[ {A + B\frac{{{\mathbf{e}}_{\alpha } \cdot {\mathbf{F}}_{1} }}{{c_{s}^{2} }} + BC\frac{{\left( {{\mathbf{e}}_{\alpha } {\mathbf{e}}_{\alpha } - c_{s}^{2} {\mathbf{I}}} \right):\left( {{\mathbf{uF}}_{1} + {\mathbf{F}}_{1} {\mathbf{u}}} \right)}}{{2c_{s}^{4} }}} \right]$ with F₁ = (F_x1, F_y1) in 2-D space.

The transformation matrix M is used to map the distribution function f _α and its equilibrium f ^eq _α in the velocity space onto the moment space (see Ref. [35] for details):

$$ {\mathbf{m}}:\, = \,{\mathbf{M}}f_{\alpha } \, = \,\left( {\rho , e, \varepsilon , \rho u_{x} \, - \,nF_{x} \Delta t, q_{x} , \rho u_{y} \, - \,nF_{y} \Delta t, q_{y} , p_{xx} , p_{xy} } \right)^{T} , $$

(A-4a)

$$ {\mathbf{m}}^{eq} : = {\mathbf{M}}f_{\alpha }^{eq} = \left( {\rho , - 2\rho ,\rho ,\rho u_{x} , - \rho u_{x} ,\rho u_{y} , - \rho u_{y} ,0,0} \right)^{T} . $$

(A-4b)

Similarly, the projection of its opposite component $f_{{\bar{\alpha }}}$ in the moment space can be found as

$$ {\bar{\mathbf{m}}}: = {\mathbf{M}}f_{{\bar{\alpha }}} = \left( {\rho ,e,\varepsilon , - \rho u_{x} + nF_{x} \Delta t, - q_{x} , - \rho u_{y} + nF_{y} \Delta t, - q_{y} ,p_{xx} ,p_{xy} } \right)^{T} . $$

(A-4c)

The corresponding equations of consecutive orders in the moment space can be written as

$$ {\rm O}\left( { \epsilon^{0} } \right):{\mathbf{m}}^{\left( 0 \right)} {=} {\mathbf{m}}^{eq} , $$

(A-5a)

$$ {\rm O}\left( { \epsilon^{1} } \right):{\tilde{\mathbf{D}}}_{1} {\mathbf{m}}^{\left( 0 \right)} {=} - \frac{1}{\Delta t}{\mathbf{Sm}}^{\left( 1 \right)} - \frac{{\theta_{1} }}{\Delta t}\left( {{\mathbf{m}}^{\left( 0 \right)} - {\bar{\mathbf{m}}}^{\left( 0 \right)} } \right) + {\mathbf{MF}}_{1} , $$

(A-5b)

$$ {\rm O}\left( { \epsilon^{2} } \right):{\tilde{\mathbf{D}}}_{1} {\tilde{\mathbf{S}}\mathbf{m}}^{\left( 1 \right)} + \frac{\Delta t}{2}{\tilde{\mathbf{D}}}_{1} {\mathbf{MF}}_{1} = - \frac{1}{\Delta t}{\mathbf{Sm}}^{\left( 2 \right)} + \frac{{\theta_{1} }}{\Delta t}{\mathbf{Sm}}^{\left( 1 \right)} - \frac{{\theta_{1} }}{\Delta t}\left( {{\mathbf{m}}^{\left( 1 \right)} - {\bar{\mathbf{m}}}^{\left( 1 \right)} } \right) - \theta_{1} {\mathbf{MF}}_{1} + \frac{1}{2}{\tilde{\mathbf{D}}}_{1} \left[ {\theta_{1} \left( {{\mathbf{m}}^{\left( 0 \right)} - {\bar{\mathbf{m}}}^{\left( 0 \right)} } \right)} \right], $$

(A-5c)

where ${\tilde{\mathbf{D}}}_{1} = {\mathbf{MD}}_{1} {\mathbf{M}}^{ - 1}$, D₁ = ∇_1idiag(e_0i, e_1i, …, e_8i) and ${\tilde{\mathbf{S}}} = {\mathbf{I}} - {\mathbf{S}}/2$.

Based on Eqs. (A-4) and (A-5a), we find

$$ {\mathbf{m}}^{\left( 1 \right)} = \left( {0,e^{\left( 1 \right)} ,\varepsilon^{\left( 1 \right)} , - n\Delta tF_{x1} ,q_{x}^{\left( 1 \right)} , - n\Delta tF_{y1} ,q_{y}^{\left( 1 \right)} ,p_{xx}^{\left( 1 \right)} ,p_{xy}^{\left( 1 \right)} } \right). $$

(A-6)

Thus Eq. (A-5b) can be rewritten as

$$ \frac{\partial }{{\partial x_{1} }}\left( {\begin{array}{*{20}c} {\rho u_{x} } \\ 0 \\ { - \rho u_{x} } \\ p \\ { - \rho /3} \\ 0 \\ 0 \\ {2\rho u_{x} /3} \\ {\rho u_{y} /3} \\ \end{array} } \right) + \frac{\partial }{{\partial y_{1} }}\left( {\begin{array}{*{20}c} {\rho u_{y} } \\ 0 \\ { - \rho u_{y} } \\ 0 \\ 0 \\ p \\ { - \rho /3} \\ { - 2\rho u_{y} /3} \\ {\rho u_{x} /3} \\ \end{array} } \right) = - \frac{1}{\Delta t}\left( {\begin{array}{*{20}c} 0 \\ {s_{e} e^{\left( 1 \right)} } \\ {s_{\varepsilon } \varepsilon^{\left( 1 \right)} } \\ { - s_{j} n\Delta tF_{x1} } \\ {s_{q} q_{x}^{\left( 1 \right)} } \\ { - s_{j} n\Delta tF_{y1} } \\ {s_{q} q_{y}^{\left( 1 \right)} } \\ {s_{v} p_{xx}^{\left( 1 \right)} } \\ {s_{v} p_{xy}^{\left( 1 \right)} } \\ \end{array} } \right) - \frac{{2\theta_{1} }}{\Delta t}\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {\rho u_{x} } \\ { - \rho u_{x} } \\ {\rho u_{y} } \\ { - \rho u_{y} } \\ 0 \\ 0 \\ \end{array} } \right) + B\left( {\begin{array}{*{20}c} 0 \\ {6C{\mathbf{u}} \cdot {\mathbf{F}}_{1} } \\ { - 6C{\mathbf{u}} \cdot {\mathbf{F}}_{1} } \\ {F_{x1} } \\ { - F_{x1} } \\ {F_{y1} } \\ { - F_{y1} } \\ {2C\left( {u_{x} F_{x1} - u_{y} F_{y1} } \right)} \\ {C\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \\ \end{array} } \right) $$

(A-7)

in which the first-order equations corresponding to the mass and momentum conservation are

$$ \nabla_{1} \cdot \left( {\rho {\mathbf{u}}} \right) = 0, $$

(A-8a)

$$ \nabla_{1} p = \left( {s_{j} n + B} \right){\mathbf{F}}_{1} - \frac{{2\theta_{1} }}{\Delta t}\rho {\mathbf{u}}, $$

(A-8b)

where p = ρc ² _s  = ρ/3 is applied.

Similarly, the second-order hydrodynamic equation in $\epsilon$ can be derived

$$ \begin{aligned} \frac{\partial }{{\partial x_{1} }}\left( {\begin{array}{*{20}c} { - \tilde{s}_{j} n\Delta tF_{x1} } \\ { - \tilde{s}_{j} n\Delta tF_{x1} + \tilde{s}_{q} q_{x}^{\left( 1 \right)} } \\ {\tilde{s}_{q} q_{x}^{\left( 1 \right)} } \\ {\tilde{s}_{e} e^{\left( 1 \right)} /6 + \tilde{s}_{v} p_{xx}^{\left( 1 \right)} /2} \\ {\tilde{s}_{e} e^{\left( 1 \right)} /3 + \tilde{s}_{\varepsilon } \varepsilon^{\left( 1 \right)} /3 - \tilde{s}_{v} p_{xx}^{\left( 1 \right)} } \\ {\tilde{s}_{v} p_{xy}^{\left( 1 \right)} } \\ {\tilde{s}_{v} p_{xy}^{\left( 1 \right)} } \\ { - \tilde{s}_{j} n\Delta tF_{x1} /3 - \tilde{s}_{q} q_{x}^{\left( 1 \right)} /3} \\ { - 2\tilde{s}_{j} n\Delta tF_{y1} /3 + \tilde{s}_{q} q_{y}^{\left( 1 \right)} /3} \\ \end{array} } \right) + \frac{\partial }{{\partial y_{1} }}\left( {\begin{array}{*{20}c} { - \tilde{s}_{j} n\Delta tF_{y1} } \\ { - \tilde{s}_{j} n\Delta tF_{y1} + \tilde{s}_{q} q_{y}^{\left( 1 \right)} } \\ {\tilde{s}_{q} q_{y}^{\left( 1 \right)} } \\ {\tilde{s}_{v} p_{xy}^{\left( 1 \right)} } \\ {\tilde{s}_{v} p_{xy}^{\left( 1 \right)} } \\ {\tilde{s}_{e} e^{\left( 1 \right)} /6 - \tilde{s}_{v} p_{xx}^{\left( 1 \right)} /2} \\ {\tilde{s}_{e} e^{\left( 1 \right)} /3 + \tilde{s}_{\varepsilon } \varepsilon^{\left( 1 \right)} /3 + \tilde{s}_{v} p_{xx}^{\left( 1 \right)} } \\ {\tilde{s}_{j} n\Delta tF_{y1} /3 + \tilde{s}_{q} q_{y}^{\left( 1 \right)} /3} \\ { - 2\tilde{s}_{j} n\Delta tF_{x1} /3 + \tilde{s}_{q} q_{x}^{\left( 1 \right)} /3} \\ \end{array} } \right) + \frac{B\Delta t}{2}\frac{\partial }{{\partial x_{1} }}\left( {\begin{array}{*{20}c} {F_{x1} } \\ 0 \\ { - F_{x1} } \\ {2Cu_{x} F_{x1} } \\ {2C\left( {u_{y} F_{y1} - u_{x} F_{x1} } \right)} \\ {C\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \\ {C\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \\ {2F_{x1} /3} \\ {F_{y1} /3} \\ \end{array} } \right) + \frac{B\Delta t}{2}\frac{\partial }{{\partial y_{1} }}\left( {\begin{array}{*{20}c} {F_{y1} } \\ 0 \\ { - F_{y1} } \\ {C\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \\ {C\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \\ {2cu_{y} F_{y1} } \\ {2C\left( {u_{x} F_{x1} - u_{y} F_{y1} } \right)} \\ { - 2F_{y1} /3} \\ {F_{x1} /3} \\ \end{array} } \right) \hfill \\ = - \frac{1}{\Delta t}\left( {\begin{array}{*{20}c} 0 \\ {s_{e} e^{\left( 2 \right)} } \\ {s_{\varepsilon } \varepsilon^{\left( 2 \right)} } \\ 0 \\ {s_{q} q_{x}^{\left( 2 \right)} } \\ 0 \\ {s_{q} q_{y}^{\left( 2 \right)} } \\ {s_{v} p_{xx}^{\left( 2 \right)} } \\ {s_{v} p_{xy}^{\left( 2 \right)} } \\ \end{array} } \right) + \frac{{2\theta_{1} }}{\Delta t}\left( {\begin{array}{*{20}c} 0 \\ {s_{e} e^{\left( 1 \right)} /2} \\ {s_{\varepsilon } \varepsilon^{\left( 1 \right)} /2} \\ {\tilde{s}_{j} n\Delta tF_{1x} } \\ { - \tilde{s}_{q} q_{x}^{\left( 1 \right)} } \\ {\tilde{s}_{j} n\Delta tF_{1y} } \\ { - \tilde{s}_{q} q_{y}^{\left( 1 \right)} } \\ {s_{v} p_{xx}^{\left( 1 \right)} /2} \\ {s_{v} p_{xy}^{\left( 1 \right)} /2} \\ \end{array} } \right) - \theta_{1} B\left( {\begin{array}{*{20}c} 0 \\ {6C{\mathbf{u}} \cdot {\mathbf{F}}_{1} } \\ { - 6C{\mathbf{u}} \cdot {\mathbf{F}}_{1} } \\ {F_{x1} } \\ { - F_{x1} } \\ {F_{y1} } \\ { - F_{y1} } \\ {2C\left( {u_{x} F_{x1} - u_{y} F_{y1} } \right)} \\ {C\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \\ \end{array} } \right) + \frac{\partial }{{\partial x_{1} }}\left( {\begin{array}{*{20}c} {\theta_{1} \rho u_{x} } \\ 0 \\ { - \theta_{1} \rho u_{x} } \\ 0 \\ 0 \\ 0 \\ 0 \\ {2\theta_{1} \rho u_{x} /3} \\ {\theta_{1} \rho u_{y} /3} \\ \end{array} } \right) + \frac{\partial }{{\partial y_{1} }}\left( {\begin{array}{*{20}c} {\theta_{1} \rho u_{y} } \\ 0 \\ { - \theta_{1} \rho u_{y} } \\ 0 \\ 0 \\ 0 \\ 0 \\ { - 2\theta_{1} \rho u_{y} /3} \\ {\theta_{1} \rho u_{x} /3} \\ \end{array} } \right). \hfill \\ \end{aligned} $$

(A-9)

Neglecting the terms of the order O(Ma³), we can obtain the following equations for e⁽¹⁾, p ⁽¹⁾ _xx and p ⁽¹⁾ _xy with the aid of Eq. (A-7):

$$ - \frac{1}{\Delta t}s_{e} e^{\left( 1 \right)} = - 6BC{\mathbf{u}} \cdot {\mathbf{F}}_{1} , $$

(A-10a)

$$ - \frac{1}{\Delta t}s_{v} p_{xx}^{\left( 1 \right)} = \frac{2}{3}\left( {\frac{{\partial \rho u_{x} }}{{\partial x_{1} }} - \frac{{\partial \rho u_{y} }}{{\partial y_{1} }}} \right) - 2BC\left( {u_{x} F_{x1} - u_{y} F_{y1} } \right), $$

(A-10b)

$$ - \frac{1}{\Delta t}s_{v} p_{xy}^{\left( 1 \right)} = \frac{1}{3}\left( {\frac{{\partial \rho u_{y} }}{{\partial x_{1} }} + \frac{{\partial \rho u_{x} }}{{\partial y_{1} }}} \right) - BC\left( {u_{x} F_{y1} + u_{y} F_{x1} } \right). $$

(A-10c)

Then the second-order equations corresponding to the mass and momentum conservation are

$$ - \nabla_{1} \cdot \left( {\theta_{1} \rho {\mathbf{u}}} \right) = \frac{{\left[ {\left( {2 - s_{j} } \right)n - B} \right]\Delta t}}{2}\Delta t\left( {\nabla_{1} \cdot {\mathbf{F}}_{1} } \right), $$

(A-11a)

$$ \begin{aligned} BC\Delta t\left\{ {\frac{\partial }{{\partial x_{1} }}\left[ {\left( {s_{e}^{ - 1} + s_{v}^{ - 1} } \right)\left( {u_{x} F_{x1} } \right) + \left( {s_{e}^{ - 1} - s_{v}^{ - 1} } \right)\left( {u_{y} F_{y1} } \right)} \right] + \frac{\partial }{{\partial y_{1} }}\left[ {s_{v}^{ - 1} \left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \right]} \right\} \hfill \\ = \frac{\partial }{{\partial x_{1} }}\left[ {\frac{\Delta t}{3}\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\left( {\frac{{\partial \rho u_{x} }}{{\partial x_{1} }} - \frac{{\partial \rho u_{y} }}{{\partial y_{1} }}} \right)} \right] + \frac{\partial }{{\partial y_{1} }}\left[ {\frac{\Delta t}{3}\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\left( {\frac{{\partial \rho u_{y} }}{{\partial x_{1} }} + \frac{{\partial \rho u_{x} }}{{\partial y_{1} }}} \right)} \right] + \theta_{1} \left[ {\left( {2 - s_{j} } \right)n - B} \right]F_{x1} , \hfill \\ \end{aligned} $$

(A-11b)

$$ \begin{aligned} BC\Delta t\left\{ {\frac{\partial }{{\partial x_{1} }}\left[ {s_{v}^{ - 1} \left( {u_{x} F_{y1} + u_{y} F_{x1} } \right)} \right] + \frac{\partial }{{\partial y_{1} }}\left[ {\left( {s_{e}^{ - 1} - s_{v}^{ - 1} } \right)u_{x} F_{x1} + \left( {s_{e}^{ - 1} + s_{v}^{ - 1} } \right)u_{y} F_{y1} } \right]} \right\} \hfill \\ = \frac{\partial }{{\partial x_{1} }}\left[ {\frac{\Delta t}{3}\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\left( {\frac{{\partial \rho u_{y} }}{{\partial x_{1} }} + \frac{{\partial \rho u_{x} }}{{\partial y_{1} }}} \right)} \right] + \frac{\partial }{{\partial y_{1} }}\left[ {\frac{\Delta t}{3}\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\left( {\frac{{\partial \rho u_{y} }}{{\partial y_{1} }} - \frac{{\partial \rho u_{x} }}{{\partial x_{1} }}} \right)} \right] + \theta_{1} \left[ {\left( {2 - s_{j} } \right)n - B} \right]F_{y1} . \hfill \\ \end{aligned} $$

(A-11c)

Combining the terms in Eq. (A-9) and Eq. (A-11) of orders ${\rm O}\left( \epsilon \right)$ and ${\rm O}\left( { \epsilon^{2} } \right)$, we have the macroscopic mass and momentum equations

$$ \nabla \cdot \left[ {\left( {1 - \theta } \right)\rho {\mathbf{u}}} \right] = \frac{{\left[ {\left( {2 - s_{j} } \right)n - B} \right]\Delta t}}{2}\nabla \cdot {\mathbf{F}}, $$

(A-12a)

$$ \nabla p + BC\Delta t\left( {\nabla {\varvec{\uppsi}} + \nabla \cdot {\varvec{\varphi }}} \right) = \nabla \cdot {\varvec{\uptau}} + \left[ {2n\theta + \left( {s_{j} n + B} \right)\left( {1 - \theta } \right)} \right]{\mathbf{F}} - \frac{2\theta }{\Delta t}\rho {\mathbf{u}}, $$

(A-12b)

where τ is the shear stress defined as

$$ {\varvec{\uptau}} = \rho c_{s}^{2} \Delta t\left( {s_{v}^{ - 1} - \tfrac{1}{2}} \right)\left[ {2{\dot{\mathbf{e}}} - \left( {\nabla \cdot {\mathbf{u}}} \right){\mathbf{I}}} \right], $$

(A-13)

where ${\dot{\mathbf{e}}} = \frac{1}{2}\left[ {\nabla {\mathbf{u}} + \left( {\nabla {\mathbf{u}}} \right)^{T} } \right]$ is the strain rate tensor. For incompressible flow, Eq. (A-13) can be further simplified as ${\varvec{\uptau}} = 2\rho c_{s}^{2} \Delta t\left( {s_{v}^{ - 1} - \tfrac{1}{2}} \right){\dot{\mathbf{e}}}$.

Substitution of Eq. (A-13) into Eq. (A-12b) gives

$$ \nabla p + BC\Delta t\left( {\nabla {\varvec{\uppsi}} + \nabla \cdot {\varvec{\varphi }}} \right) = \rho c_{s}^{2} \Delta t\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\nabla^{2} {\mathbf{u}} + \left[ {2n\theta + \left( {s_{j} n + B} \right)\left( {1 - \theta } \right)} \right]{\mathbf{F}} - \frac{2\theta }{\Delta t}\rho {\mathbf{u}}. $$

(A-12b*)

On LHS of Eq. (A-12b*), ψ and φ are given as

$$ {\varvec{\uppsi}} = \left( {s_{e}^{ - 1} - s_{v}^{ - 1} } \right){\mathbf{u}} \cdot {\mathbf{F}}, $$

(A-14a)

$$ {\varvec{\varphi }} = s_{v}^{ - 1} \left[ {{\mathbf{uF}} + \left( {{\mathbf{uF}}} \right)^{T} } \right]. $$

(A-14b)

Now, let us compare Eq. (A-12a) with the steady-state mass conservation equation

$$ \nabla \cdot \left( {\rho {\mathbf{v}}} \right) = 0. $$

(A-15)

It is clear that a redefinition of macroscopic velocity v = (1 − θ)u is necessary such that Eq. (A-12a) becomes consistent with Eq. (A-15) and only force Treatments 1 and 2 guarantee the mass transport to be conserved with a non-divergence-free force field (See Table 1). By taking into account the macroscopic velocity redefinition and comparing Eq. (A-12b*) with the Brinkman equation Eq. (A-16) term by term,

$$ \nabla p = \tilde{\mu }\nabla^{2} {\mathbf{v}} - \frac{\mu }{K}{\mathbf{v}} + {\mathbf{G}}, $$

(A-16)

one can obtain the relations between of GLBE parameters with the macroscopic variables (see Eqs. (18, 19)) as well as the force redefinition (see Table 2).

Following the same procedure, the recovered macroscopic mass and momentum equations and the corresponding redefinitions for the macroscopic velocity and the forcing term can be derived for the other two GLBE schemes. For YH scheme in particular, since its θ is defined on half-link between two neighboring nodes instead of being on the nodes, it essentially becomes θ(x + e _α Δt/2) and thusly needs to be further expanded. Here we give their recovered macroscopic mass and momentum equations:

ZM scheme [17]:

$$ \nabla \cdot \left[ {\left( {1 - \theta } \right)\rho {\mathbf{u}}} \right] = \frac{{\left[ {\left( {2 - s_{j} } \right)n - B} \right]\Delta t}}{2}\nabla \cdot {\mathbf{F}}, $$

(A-17a)

$$ \nabla p + BC\Delta t\left( {\nabla {\varvec{\uppsi}} + \nabla \cdot {\varvec{\varphi }}} \right) = \rho c_{s}^{2} \Delta t\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\nabla^{2} {\mathbf{u}} + \left[ {2n\theta + \left( {s_{j} n + B} \right)\left( {1 - 2\theta } \right)} \right]{\mathbf{F}} - \frac{2\theta }{\Delta t}\rho {\mathbf{u}}. $$

(A-17b)

YH scheme [18]:

$$ \nabla \cdot \left( {\rho {\mathbf{u}}} \right) = \frac{{\left[ {\left( {2 - s_{j} } \right)n - B} \right]\Delta t}}{2}\nabla \cdot {\mathbf{F}}, $$

(A-18a)

$$ \left( {1 - \theta } \right)\nabla p + BC\Delta t\left( {\nabla {\varvec{\uppsi}} + \nabla \cdot {\varvec{\varphi }}} \right) = \rho c_{s}^{2} \Delta t\left( {s_{v}^{ - 1} - \frac{1}{2}} \right)\nabla^{2} {\mathbf{u}} + \left[ {2n\theta + \left( {s_{j} n + B} \right)\left( {1 - 2\theta } \right)} \right]{\mathbf{F}} - \frac{2\theta }{\Delta t}\rho {\mathbf{u}}. $$

(A-18b)

Comparing Eqs. (A-17a, A-18a) and (A-17b, A-18b) with the respective steady-state mass conservation equation and the Brinkman equation, one can obtain the respective macroscopic velocity redefinition described in Sect. 2.2.2 and force redefinition summarized in Table 2.

Appendix B: Analytical Solutions to the Recovered Momentum Eqs. (28, 29) for Flow Through Porous Blocks in “Parallel”

The analytical solutions to Eqs. (28) and (29) with the velocity and shear stress continuity conditions discussed in Sect. 3.3 are provided here.

Recall that Eq. (28) is in the forms of

$$ \begin{aligned} \frac{dp}{dy} = \frac{{\nu^{*} }}{{1 - \theta_{1} }}\frac{{d^{2} V_{y1} }}{{dx^{2} }} - \frac{{2\theta_{1} }}{{1 - \theta_{1} }}V_{y1} \quad - L \le x \le 0 \hfill \\ \frac{dp}{dy} = \frac{{\nu^{*} }}{{1 - \theta_{2} }}\frac{{d^{2} V_{y2} }}{{dx^{2} }} - \frac{{2\theta_{2} }}{{1 - \theta_{2} }}V_{y2} \quad 0 \le x \le L \hfill \\ \end{aligned} $$

(B-1)

with the following boundary conditions

• (i) Velocity continuity: V_y1|_x=0 = V_y2|_x=0 and V_y1|_x=-L = V_y2|_x=L;

• (ii) Shear stress continuity $\left. {\frac{{\nu^{*} }}{{1 - \theta_{1} }}\frac{{dV_{y1} }}{dx}} \right|_{x = 0} = \left. {\frac{{\nu^{*} }}{{1 - \theta_{2} }}\frac{{dV_{y2} }}{dx}} \right|_{x = 0}$ and $\left. {\frac{{\nu^{*} }}{{1 - \theta_{1} }}\frac{{dV_{y1} }}{dx}} \right|_{x = - L} = \left. {\frac{{\nu^{*} }}{{1 - \theta_{2} }}\frac{{dV_{y2} }}{dx}} \right|_{x = L}$.

Its general solution can be expressed as

$$ \begin{aligned} V_{y1} (x) = C_{1} \exp \left( {\sqrt {\frac{{2\theta_{1} }}{{\nu^{*} }}} x} \right) + C_{2} \exp \left( { - \sqrt {\frac{{2\theta_{1} }}{{\nu^{*} }}} x} \right) - \frac{{1 - \theta_{1} }}{{2\theta_{1} }}\frac{dp}{dy} \quad - L \le x \le 0 \hfill \\ V_{y2} (y) = C_{3} \exp \left( {\sqrt {\frac{{2\theta_{2} }}{{\nu^{*} }}} x} \right) + C_{4} \exp \left( { - \sqrt {\frac{{2\theta_{2} }}{{\nu^{*} }}} x} \right) - \frac{{1 - \theta_{2} }}{{2\theta_{2} }}\frac{dp}{dy} \quad 0 \le x \le L \hfill \\ \end{aligned} $$

(B-2)

where C _i , i $\in$ {1, 2, 3, 4} are the coefficients to be determined.

By applying the velocity continuity boundary conditions, the following two constraints are obtained

$$ C_{1} + C_{2} - C_{3} - C_{4} = \Delta $$

(B-3a)

$$ \alpha_{1} C_{1} + {{C_{2} } \mathord{\left/ {\vphantom {{C_{2} } {\alpha_{1} }}} \right. \kern-0pt} {\alpha_{1} }} - \alpha_{2} C_{3} - {{C_{4} } \mathord{\left/ {\vphantom {{C_{4} } {\alpha_{2} }}} \right. \kern-0pt} {\alpha_{2} }} = \Delta $$

(B-3b)

where $\Delta = \left( {\frac{{1 - \theta_{1} }}{{2\theta_{1} }} - \frac{{1 - \theta_{2} }}{{2\theta_{2} }}} \right)\frac{dp}{dy}$ and $\alpha_{i} = \exp \left( { - \sqrt {\frac{{2\theta_{i} }}{{\nu^{*} }}} L} \right),\;i \in \left\{{1,2} \right\}$ for simplicity. Similarly, two other constraints can be derived with the shear stress boundary conditions.

$$ \beta_{1} C_{1} - \beta_{1} C_{2} - \beta_{2} C_{3} + \beta_{2} C_{4} = 0 $$

(B-3c)

$$ \alpha_{1} \beta_{1} C_{1} - \left( {{{\beta_{1} } \mathord{\left/ {\vphantom {{\beta_{1} } {\alpha_{1} }}} \right. \kern-0pt} {\alpha_{1} }}} \right)C_{2} - \alpha_{2} \beta_{2} C_{3} + \left( {{{\beta_{2} } \mathord{\left/ {\vphantom {{\beta_{2} } {\alpha_{2} }}} \right. \kern-0pt} {\alpha_{2} }}} \right)C_{4} = 0 $$

(B-3d)

where $\beta_{i} = \frac{{\sqrt {2\theta_{i} \nu^{*} } }}{{1 - \theta_{i} }},\;i \in \left\{ {1,2} \right\}$. The linear system with four unknowns C _i , $i \in \left\{ {1,2,3,4} \right\}$ given by Eq. (B-3) has the following solution

$$ \begin{array}{*{20}c} {C_{1} = - \frac{{\Delta \beta_{2} \left( {\alpha_{2} - 1} \right)}}{\varLambda }} \\ {C_{2} = - \frac{{\Delta \alpha_{1} \beta_{2} \left( {\alpha_{2} - 1} \right)}}{\varLambda }} \\ {C_{3} = - \frac{{\Delta \beta_{1} \left( {\alpha_{1} - 1} \right)}}{\varLambda }} \\ {C_{4} = - \frac{{\Delta \alpha_{2} \beta_{1} \left( {\alpha_{1} - 1} \right)}}{\varLambda }} \\ \end{array} , $$

(B-4)

where Λ = β₁(α₁ − 1)(α₂ + 1) − β₂(α₁ + 1)(α₂ − 1).

Following the same process, Eq. (29) expressed as

$$ \begin{array}{*{20}c} {\frac{dp}{dy} = \nu^{*} \frac{{d^{2} u_{y1} }}{{dx^{2} }} - 2\theta_{1} u_{y1} } & { - L \le x \le 0} \\ {\frac{dp}{dy} = \nu^{*} \frac{{d^{2} u_{y2} }}{{dx^{2} }} - 2\theta_{2} u_{y2} } & {0 \le x \le L} \\ \end{array} , $$

(B-5)

has the general solution in the forms of

$$ \begin{array}{*{20}c} {u_{y1} (x) = C_{1}^{\prime } \exp \left( {\sqrt {\frac{{2\theta_{1} }}{{\nu^{*} }}} x} \right) + C_{2}^{\prime } \exp \left( { - \sqrt {\frac{{2\theta_{1} }}{{\nu^{*} }}} x} \right) - \frac{1}{{2\theta_{1} }}\frac{dp}{dy}} &\quad { - L \le x \le 0} \\ {u_{y2} (y) = C_{3}^{\prime } \exp \left( {\sqrt {\frac{{2\theta_{2} }}{{\nu^{*} }}} x} \right) + C_{4}^{\prime } \exp \left( { - \sqrt {\frac{{2\theta_{2} }}{{\nu^{*} }}} x} \right) - \frac{1}{{2\theta_{2} }}\frac{dp}{dy}} &\quad {0 \le x \le L} \\ \end{array} , $$

(B-6)

where C ^′ _i , $i \in \left\{ {1,2,3,4} \right\}$ are the coefficients to be determined. By applying the velocity and shear stress continuity boundary conditions (1 − θ₁)u_y1|_x=0 = (1 − θ₂)u_y2|_x=0, (1 − θ₁)u_y1|_x=−L = (1 − θ₂)u_y2|_x=L and $\left. {\nu^{*} \frac{{du_{y1} }}{dx}} \right|_{x = 0} = \left. {\nu^{*} \frac{{du_{y2} }}{dx}} \right|_{x = 0}$, $\left. {\nu^{*} \frac{{du_{y1} }}{dx}} \right|_{x = - L} = \left. {\nu^{*} \frac{{du_{y2} }}{dx}} \right|_{x = L}$, the following system of equations can be obtained.

$$ \left[ {\begin{array}{*{20}c} {1 - \theta_{1} } & {1 - \theta_{1} } & { - \left( {1 - \theta_{2} } \right)} & { - \left( {1 - \theta_{2} } \right)} \\ {\left( {1 - \theta_{1} } \right)\alpha_{1} } & {{{\left( {1 - \theta_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \theta_{1} } \right)} {\alpha_{1} }}} \right. \kern-0pt} {\alpha_{1} }}} & { - \left( {1 - \theta_{2} } \right)\alpha_{2} } & { - {{\left( {1 - \theta_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \theta_{2} } \right)} {\alpha_{2} }}} \right. \kern-0pt} {\alpha_{2} }}} \\ {\beta_{1}^{\prime } } & { - \beta_{1}^{\prime } } & { - \beta_{2}^{\prime } } & {\beta_{2}^{\prime } } \\ {\alpha_{1} \beta_{1}^{\prime } } & {{{ - \beta_{1}^{\prime } } \mathord{\left/ {\vphantom {{ - \beta_{1}^{\prime } } {\alpha_{1} }}} \right. \kern-0pt} {\alpha_{1} }}} & { - \alpha_{2} \beta_{2}^{\prime } } & {{{\beta_{2}^{\prime } } \mathord{\left/ {\vphantom {{\beta_{2}^{\prime } } {\alpha_{2} }}} \right. \kern-0pt} {\alpha_{2} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {C^{\prime}_{1} } \\ {C^{\prime}_{2} } \\ {C^{\prime}_{3} } \\ {C^{\prime}_{4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\Delta^{\prime}} \\ {\Delta^{\prime}} \\ 0 \\ 0 \\ \end{array} } \right], $$

(B-7)

where $\beta_{i} = \sqrt {2\theta_{i} \nu^{*} } ,\;i \in \left\{ {1,2} \right\}$ and $\Delta^{\prime} = \left( {\frac{1}{{2\theta_{1} }} - \frac{1}{{2\theta_{2} }}} \right)\frac{dp}{dy}$. Its solution is

$$ \begin{array}{*{20}c} {C_{1}^{\prime } = - \frac{{\Delta^{\prime } \beta_{2}^{\prime } \left( {\alpha_{2} - 1} \right)}}{{\varLambda^{\prime } }}} \\ {C_{2}^{\prime } = - \frac{{\Delta^{\prime } \alpha_{1} \beta_{2}^{\prime } \left( {\alpha_{2} - 1} \right)}}{{\varLambda^{\prime } }}} \\ {C_{3}^{\prime } = - \frac{{\Delta^{\prime } \beta^{\prime }_{1} \left( {\alpha_{1} - 1} \right)}}{{\varLambda^{\prime } }}} \\ {C_{4}^{\prime } = - \frac{{\Delta^{\prime } \alpha_{2} \beta_{1}^{\prime } \left( {\alpha_{1} - 1} \right)}}{{\varLambda^{\prime } }}} \\ \end{array} , $$

(B-8)

where Λ^′ = β₁(α₁ − 1)(α₂ + 1)(θ₂ − 1) − β₂(α₁ + 1)(α₂ − 1)(θ₁ − 1). To solve the macroscopic velocity, the relation V _yi (x) = (1 − θ _i )u _yi (x), $i \in \{1, 2\}$ needs to be applied.