Add temperature field variable to vertical component of force vector in Navier Stokes

Question

Hi!

I'm currently constructing a Chorin/IPCS Navier Stokes solver using Fenics. The temperature/energy equation is also incorporated.

So, my question is, I'm going to use the Boussinesq bouyancy approximation for incompressible fluids, but how do I apply this to my force vector?

The temperature field T1 in the last time step is known, and I want to add it to my force vector so it becomes f = 0i + T1[i,j]j in the spatial point i, j for the next time step. Any tips on the easiest way of doing this?

This is just for testing purposes, the full model is going to be implemented at a later stage.

Thanks!

Comments

Hi, so you have f, T1 possibly in different function spaces and you want to achieve something along
the lines of f += T1?

Answer 1

Hi,

If your known temperature is T1 (a Function), then a directional Boussinesq force in your momentum equation can be implemented as

V = VectorFunctionSpace(mesh, ...)
v = TestFunction(V)
alpha = Constant(1.)
T0 = Constant(1.)
f = as_vector((0, alpha*(T1-T0)))
L = inner(f, v)*dx

where T0 is a reference temperature and v is TestFunction for the velocity vector. L must be reassembled as part of your rhs linear form each timestep. How you do it depends on what your solver actually looks like, but as long as f is defined before your momentum form you should be ok. I don't remember the exact Boussinesq model, but there should definately be some gravity, g, there.