I am working with the double-population thermal lattice Boltzmann method for simulating heat transfer in a channel whose geometry repeats periodically (a series of identical modules). For this case, it is possible to assume that in a fully developed region, the flow structure repeats from module to module, allowing to impose periodic boundary conditions. However, since both top and bottom walls are kept isothermal with a constant temperature that is higher than the temperature of the fluid, the fluid temperature does not repeat itself from module to module, but the normalized temperature profile (T-Tw)/(Tb-Tw) repeats identically. Tw is wall temperature, and Tb is the cross-sectional local bulk temperature. For the inlet and outlet boundaries there is the following relationship:
T(x+L,y,t) - Tw T(x,y,t) - Tw
----------------------- = --------------------------
Tb(x+L) - Tw Tb(x) - Tw
where L is the length of a periodic module.
Now my question, does anyone know how to impose this kind of periodic boundary condition using the LBM. For the flow BC’s there is no problem, but with the thermal I am getting so many troubles. Is there any good paper about this topic?
I will be very grateful for your help.