Problem when imposing pressure related boundary condition

[Vincent]

Hi,

I am trying to impose the boundary condition in doi.org/10.1017/jfm.2020.867 for Rayleigh-Benard convection. The boundary conditions are as follows:

in which, \bar{p} denotes the pressure of base state in which u=w=0. And I've implemented these boundary conditions using the following code

problem.add_equation("ez@u(z=0) + beta_u*p(z=0) = 0")
problem.add_equation("ez@u(z=Lz) - beta_u*p(z=Lz) = - beta_u*1/2")

Not sure if I am getting the pressure gauge condition wrong or setting the pressure gauge condition wrong. For now, I've just got a result (Ra=1e4, which should be large enough for onset) with a strange flow field in which no convection occurs and the hot fluid seems to "leak" through the bottom plate, 

Any guidance would be greatly appreciated.

Thanks in advance,

Vincent

[Keaton Burns]

Hi Vincent,

I haven’t looked too closely at the literature cited in that paper, but I think the pressure fluctuation is meant be calculated around the instantaneous horizontal mean, rather than the hydrostatic value which I think is what you have on the RHS. That is, maybe \bar{rho} should be something like integ(ρ,’x’)/Lx, and that way there is zero mean mass flux at each boundary at each point in time.

Best,

-Keaton

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