Now I try to follow the model to express vapor-liquid phase transition by LBM in reference to Gong&Cheng “A lattice Boltzamann method for simulation of liquid-vapor phase-change heat transfer”(International Journal of Heat and Mass Transfer 55(2012) 4923-2927). To sum up this paper, the domain is filled with liquid water and heated on the bottom boundary under constant temperature(Isothermal). Then, both of side boundaries are set on periodic boundary, and top is kept constant temperature and pressure. On these condition, I cannot implement the boundary condition on the top which is Isobar and Isothermal, because local velocity and density distribution function are unknown by this condition in my opinion. So, if anyone know the method to make clear the unknown parameter(velocity and desity distribution function), please tell me.

There are several methods of applying boundary conditions of constant properties (mostly density or velocity) by finding values for ‘missing’ distribution functions. The schemes by Inamuro et al. [ Phys. Fluids, 7, 2928-2930 (1995)] and Zou and He [ Phys. Fluids, 9, 1591-1598 (1997)] are among the most straightforward and reasonably accurate, and involve using known distribution functions for the boundary point and the desired density to obtain the fluid velocity and values for the missing distribution functions. (The former is calculated by assuming flow only exists tangentially to the boundary.)

These schemes can be used to apply constant pressure and temperature boundaries, since (1) the pressure is directly related to density by the equation of state, and (2) the temperature effectively substitutes the density for the thermal field distribution. If you apply the boundary for the density distribution functions first, you will be able to get the velocity needed for the thermal distribution functions.

Thank you for reply quickly. I have just read them(Inamuro 1995, Zou&He 1997), but I cannot imagine and constract the detailed boundary condition because in the reference paper(Gong&Cheng 2012) vapor generates from the bottom surface. And it separetes, goes up and reaches at the top. So, the density and velocity on the top must be changed. It makes me confusing. Also the relation of both distribution function(density, energy) are unclear.

In the description of (Gong&Cheng 2012), constant pressure is realized to apply (Zou&He) scheme, and constant tempareture is (Inamuro) scheme.

(Inamuro 1995) reports the counter-slip sheme, which can derive all of parameters to assume the fictitious density and velocity. But how can I apply to isothermal condition? In my perception this sheme is not for the thermal condition but for the density and velocity condition. Perhaps density distribution function is replaced with energy distribution function?

(Zou&He 1997) also showes the scheme to obtain the unknowns from known combination [specified both velocity(ux,uy)] or [specified density and one velocity]. In case multiphase flow is not always satisfied constant density and velosity as I mention above.

I am sorry for many questions, if someone would cooperate, really helpful.

In terms of constant pressure systems with phase change, the equation of state should allow you to work out what the densities of vapour and liquid phases should be at a particular temperature. If you are following the Gong and Cheng paper, this should be the cubic Peng-Robinson equation: below the critical point you should get three densities for a given temperature and pressure, of which the lowest and highest values are for the vapour and liquid respectively. (The middle value does not really have any physical significance.)

The two distribution functions for the fluid density and energy are coupled together by the fluid velocity, i.e. the velocity used in the local equilibrium distribution functions during the collision step must be the same for both. In practice, only the fluid density distribution functions are used to calculate the velocity, which is used in both equilibrium distribution functions.

Inamuro’s scheme can be modified to use it as a constant temperature scheme, and you correctly guessed that you substitute the density distribution functions with energy distribution functions. (The following is based on a later paper by Inamuro et al, J Comput Phys,179, 201-215 (2002).) The velocity of the fluid at that boundary point will already be known - you would apply either a constant density or a constant velocity scheme first - so no counter-slip velocity is needed. However, you will need an adjusting temperature (similar to the unknown density in the standard Inamuro scheme), which you can find by expressing the unknown energy distribution functions as equilibrium values with the adjusting temperature. The sum of all energy distribution functions at that boundary point should give the required temperature: rearranging this sum should give you the adjusting temperature as a function of the known energy distribution functions and the fluid velocity, which you can then plug into the expression for the equilibrium distribution function for the unknown values.

If there is only one phase in contact with a particular boundary, the Zou/He scheme should be OK if you can find out what the density of that phase should be for the given temperature: for the system you want to model, it seems likely the liquid phase will dominate at the bottom boundary. A complication might arise if both liquid and vapour phases are in contact with the boundary for constant pressure/density, but you could use the grid point next to a boundary grid point to check which phase is needed (e.g. if the neighbouring point is vapour, you can assume the vapour density for the given temperature and pressure).