mcyrus10/LBM_1D_diffusion_semi_infinite

1D Diffusion (sort of)

1D semi-infinite diffusion (Lab 3 From Alexiades PDE course). Sort of a little hack: its really a D2Q5 2d lattice (not a D1Q3), but both sides are periodic so it only varies along one axis. This is just a little sanity check/tutorial for using Palabos and getting used to the units.

parameter	description	expression [unit]
$\tau$	Relaxation time	variable $\rightarrow$ controls D [-]
$D$ (or $\kappa$)	Diffusivity (or Conductivity)	$\frac{1}{3} (\tau - \frac{1}{2})$ [-]
$resolution$	mesh resolution (lattice units per physical unit)	[-]
$\Delta t$	time step (seconds per iteration)	$(\frac{1}{resolution})^2$
$max_{iter}$	iterations to yield 1 second simulation	$\frac{1}{\Delta t}$ (integer)

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Example Calculations

For the base case we will use the regular D2Q5 BGK lattice to fix tau, ch and ct, then we will modify the diffusion coefficient without having to change tau. We want to make Diffusivity / Conductivity = 0.1 $\frac{cm^2}{s}$ and have a lattice spacing of 64 lattice units per cm.

For this case $\tau$ is dependent on D: $tau$

parameter	expression [unit]
$\tau$	$1.0$
$D$	$\frac{1}{3} (\tau-\frac{1}{2}) = \frac{1}{6}$
$Resolution$	64
$\Delta x$	$\frac{1}{64}$
$\Delta t$	$\frac{1}{64^2} = \frac{1}{4096}$
$max_{iter}$	4096

Testing out the program: this means that the LBM program must undergo 4096 iterations to equal 1 second of physical time.

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Initial Condition and Boundary Conditions

To be honest I am not exactly sure why you use the pointer syntax to make the boundary condition, but this is how they do it in Palabos. You can look at the documentation for the OnLatticeAdvectionDiffusionBoundaryCondition to see more.

void domain_setup( MultiBlockLattice2D<T,ADESCRIPTOR>& adLattice, IncomprFlowParam<T>& parameters )
{
  ...
  ...
  ...
  // INITIAL CONDITION
  initializeAtEquilibrium(  adLattice,....)

  // BOUNDARY CONDITION 
  //  bc is a pointer to the boundary condition
  OnLatticeAdvectionDiffusionBoundaryCondition2D<T, ADESCRIPTOR>*
    bc = createLocalAdvectionDiffusionBoundaryCondition2D<T,ADESCRIPTOR>();

  // the '->' syntax in c++ references the virtual method of a pointed to object
  bc->addTemperatureBoundary1N(south_boundary,adLattice);

  setBoundaryDensity(adLattice,south_boundary,(T)1.0);

  ...

}