Im pretty new to the LBM method and i’ve got a (most probable very silly) question about the basic algorithm:

What is being done to assure the conservation of mass ?

Assume u’ve got a squarish grid consisting of 9 cells.

Further assume the distribution-functions of the middle cell to be zero except those ones representing the fraction of paricles that are not moving.

If now there is even just one single distribution-function of an adjacent cell with direction towards the middle cell that is different from zero, after the stream-step the Density of the middle Fluid Cell is >1.

And this cannot be regulated by the following collision-step.

The only chance, i saw was to normalize the distribution-functions at the end of a LBM step by the sum of all distribution functions of one cell so that the density will become 1 again.

But this is either problematic if a cell totally empties and together with common “Free Surface Boundary Conditions”, such a treatment could cause a loss of mass not just in very artificial situations.

*Imagine a flat surface fluid in a pot-shaped vessel and all interface-cells to have distribution-functions equal to zero except the ones heading straight to the bottom. This would cause them to empty. The underneath fluid-cells would absorb the additional mass by the above suggestet normalisation and thereafter they would be changed into interface-cells themselved (which again would have distributionfunctions different from zero only with the bottom-direction). Then the whole precedure would start again at * until the vessel would only contain one layer that would be reflected at the bottom (obstacle boundary) and the vessel would fill up infinite (unless, i dont know, the equilibrium-functions would damp the distribution-functions in such a way, that the surface would be convergent to some height-level

If sb would have a suggestion/answer on this id be very glad.