Brinkman flow in a channel is a classical benchmark problem for validating the Navier--Stokes--Brinkman (NSB) solver, as it combines viscous diffusion and Darcy drag effects characteristic of porous media. The problem considers a steady laminar flow through a channel filled with a porous matrix, where the additional Darcy drag term significantly alters the velocity distribution. The governing one-dimensional Brinkman equation reads
where $u_x$ is the velocity in the streamwise ($x$) direction, and $F_0 = \tfrac{\mu_0}{k\mu}$ is the normalized Darcy coefficient, with $\mu$ the fluid viscosity, $\mu_0$ the effective viscosity, and $k$ the permeability.
The boundary conditions are imposed as a no-slip condition at the lower wall and a symmetric velocity condition at the upper wall,
The analytical solution of this boundary-value problem is
with $k_1 = \tfrac{u_m}{e^{\sqrt{F_0} h} - e^{-\sqrt{F_0} h}}$ and $k_2 = -k_1$.
The analytical solution is written in func.lua. The values of $F_0$ is and
other parameters are set in params.lua.
In this example, BGK collision model is adopted. Meanwhile, the source term is expanded up to the second Hermite order. Separating it according to lattice parity gives
where the first-order, antisymmetric contribution is
and the second-order, symmetric contribution is