How to add or delete fluid in LBM(lattice boltzmann method)

Question

LBM focuses on fluid clusters, and uses the macro fluid density and velocity to calculate the equilibrium distribution function, and then uses the evolution equation to achieve system iteration. But if we add the same fluid to the lattice grid points in the LBM or reduce the existing fluid continuously, how should we recalculate the macro fluid density and velocity? Or how should the distribution function at the lattice grid point be recalculated? Can LBM simulate a scenario where fluid is continuously added or reduced to the system? For example, water keeps flowing from the tap.

Answer 1

The traditional lattice-Boltzmann method (e.g. the D2Q9 lattice in 2D) can only be applied to incompressible flows. Put in simple terms this means that there can't be more mass entering the domain than exiting it: The mass inside the domain is roughly the same throughout the simulation. This simplification of the generally compressible Navier-Stokes equations can not only be applied to incompressible fluids (such as water) but also to low-Mach number flows like the flow around a car (for more details see here). Yet the traditional lattice-Boltzmann method can't describe multi-phase and free-surface flows as well as flows with sinks and sources (which all result in a change of density of the system).

Any inlet or outlet conditions in the incompressible lattice-Boltzmann method falls in one of the following categories:

• Periodic boundaries (the populations that exit the domain on one side enter it again on the other side)
• Pressure-drop-periodic boundaries (such as Zhang/Kwok) for periodic flow but with an additional term for compensating for a pressure drop inside the domain due to friction
• Velocity and pressure boundaries (generally a velocity inlet and a pressure outlet): There exist various formulations of these to make sure that the moments of the distribution are actually conserved and they have different characteristics regarding numeric stability. Most of them enforce some sort of symmetry and extrapolation of higher moments. The simplest ones are the ones by Zou/He but others like Guo's extrapolation method are significantly more stable for under-resolved and turbulent (high Reynolds number flows). This review discusses different ones in more detail.

You can have a look at this small code I have written in C++ for 2D and 3D simulations if you are interested in more details on how this actually works.

That being said there exist though several variations of lattice-Boltzmann methods in research that allow for multi-component or multi-phase flows (e.g. by introducing additional distributions) or compressible flows (with lattices with more discrete velocities and potentially a second lattice) but they are still exotics and you won't find many implementations around.