buoyancy terms in bubble rising simulation

Recently, I am simulating a single bubble rising in water with large density radio, according to the article of “A lattice Boltzmann model for multiphase flows with large density ratio”.
But I have a big question about this article(about the bouyancy): obviously, in the article the body force has been considered in the equation[34] , but why the author also said to consider the bouyancy, we should add the δF and δG terms according another article “Simulation of Bubble Motion under Gravity by Lattice Boltzmann Method” ? And, in some other articles about bubble rising, I also see they only considered the body force while did not add the δF and δG terms.
Could you help me to know why? Is there some wrong in the article “A lattice Boltzmann model for multiphase flows with large density ratio” ? Or should we consider both the body force and δF, δG ?
Thank you !

It is the same doubt about the forcing term, and I want to know that how you solve this problem.
In this paper, the forcing term is considered in the equation of f,and not in the equation of g. And the author also said to consider the bouyancy, we should add the δF and δG terms. Should I add both the forcing term in the equations of f and g? My understanding about the effection of the forcing term is that, g has an effect on the f, but f has not that on g. Is it right?
Looking forward to help from everyone, Thanks!

I don’t know these models but I assume you have two distribution functions, one for the flow (f_i) and one for the order parameter (g_i). The governing equation for the order parameter will probably be some kind of convection-diffusion equation, possibly the Cahn-Hilliard equation. The velocity in this equation is the flow velocity, which is found from the flow equation.

This flow equation is some kind of multiphase Navier-Stokes equation. It will include a term for the phase effects/density variations. The LBE for this equation (f_i) will incorporate such effects into a surface forcing term, or perhaps into the pressure tensor.

Gravity is an additional, separate, force - a body force. It is an acceleration and effects the velocity. The equation which you use to find the velocity, ie the momentum (Navier-Stokes) equation, needs to include this force. Thus, f_i should include a body to mimic gravity.

The order parameter equation advects the scalar parameter with the flow velocity, u. This velocity will now already include the effects of gravity, because u has been found by f_i. Thus, the body force for gravity only appears in the momentum equation. I haven’t read that paper, but this is what I would assume.

Thanks very much for your reply.
I agree with your idea. But I still have some questions. The order parameter equation also correspond to a velocity, and how the velocity to be understand?

The order parameter gets advected by the flow, or so I assume. So the velocity in this equation is the flow velocity. It is an equation for the order parameter, not for u. So one would guess that f_i solves for u and pressure, and g_i solves for the order parameter. This, if true, would mean that you find u from f_i, and then put this u in g_i, right?

You are right, and I am working in this way. Thank you very much.