Advection-diffusion-reaction benchmark: Semi-infinite 1D domain

The movement of a tracer in a semi-infinite column is a classical benchmark problem in transport phenomena and porous media hydrology.

References: Jacob Bear. Dynamics of Fluids in Porous Media. New York: Elsevier, 1972. isbn: 978-0-486-65675-5.

It serves to validate the accuracy of the LBM framework for solving advection-diffusion-reaction (ADR) equations. The setup considers a tracer initially injected at the inlet of a semi-infinite column, which then advects, diffuses, and reacts along the flow direction.

The governing ADR equation is

$$\partial_t c + U \,\partial_x c = D \,\partial_{xx} c - \lambda c, \label{adr}$$

where $c(x,t)$ is the concentration, $U$ the advective velocity, $D$ the molecular diffusion coefficient, and $\lambda$ the reaction rate.

The boundary and initial conditions are

$$c(x,0) = 0, \qquad c(0,t) = C_0, \qquad c(\infty,t) = 0, \label{adr_bc}$$

corresponding to a constant concentration injection at the inlet.

The analytical solution for this problem is given as

$$\begin{split} C(x,t) = \frac{C_0}{2} \exp\!\left(\frac{Ux}{2D}\right) \Bigg[ & \exp(-\beta x)\, \mathrm{erfc}\!\left(\frac{x - \sqrt{U^2 + 4 \lambda D}\, t}{2\sqrt{Dt}}\right) \\ +\, & \exp(\beta x)\, \mathrm{erfc}\!\left(\frac{x + \sqrt{U^2 + 4 \lambda D}\, t}{2\sqrt{Dt}}\right) \Bigg], \end{split} \label{adr_solution}$$

with $\beta^2 = \tfrac{U^2 + 4\lambda}{4D}$ and $\mathrm{erfc}(x) = 1 - \mathrm{erf}(x)$ the complementary error function.

In this benchmark, the simulation parameters are set in params.lua, and the analytical solution is implemented in func.lua. To view the profile of the concentration along the column at a specific time, run the example and plot with GNUplot:

gnuplot -p -e "
    plot $(printf "'%s' " tracking/*.res) using 1:4 title 'calculated' with points,
         $(printf "'%s' " tracking/*.res) using 1:5 title 'analytical' with lines
"

This will generate a plot comparing the calculated concentration profile from the LBM simulation against the analytical solution at the specified time.

If your parameters are set as in the provided params.lua, you should observe the same results as shown in media/reference.png.