Multiphase periodic pressure difference boundary condition enhanced by a proportional-integral-derivative controller for the lattice Boltzmann method

A new pressure difference boundary condition between a pair of inlet and outlet boundaries in an immiscible multiphase periodic flow is introduced. This boundary condition is globally mass conservative and makes use of a simple proportional-integral-derivative controller to accurately control the pressure difference. For a droplet/bubble that crosses the outlet to reenter at the inlet of a periodic slit flow, the stability and accuracy of the boundary condition are numerically studied for a wide range of flow conditions. A visualization of a droplet crossing the outlet boundary provides qualitative evidence that the boundary condition works effectively. More importantly, quantitative results also confirm that the targeted average pressure difference is adequately set and that the total momentum in the pressure drop direction is nearly constant (which is what is expected from a perfect multiphase periodic pressure difference boundary condition). This new type of multiphase boundary condition is expected to open up new avenues to simulate complex interfacial multiphase systems using the lattice Boltzmann method.