Boussinesq approximation And variable density model
 

I have a question about the Boussinesq approximation.
I know that the approximation is rho = rho0* ( 1+beta*(T-T0))
where rho0 is the reference density at T0 (reference temperature)
And beta is the thermal expansion coefficient.

My question is do T0 change with the system or is it constant?
Imagine you have a natural circulation system single-phase water system where the pressure is atmospheric pressure and you have a heat source and a sink(cooling).
If you give initial condition T0 = 300 ,rho0 = 1000, T0=T , rho0 = rho, beta =-0.0007
Beta will obviously be updated based on beta= - (1/rho0 )* delta rho/delta T

So the rho will change based on the equation but do I change T0 based on the previous time step value or keep it constant.

Also , can any body provide a variable density model to calculate the density based on temperature and pressure change other than the Boussinesq approximation? With reference literature please.

I read in Relap7 manual some equation where they used the equation of state to get the density change. There was a linear equation of state that seemed to be for water but they did not provide any reference to the equation source to get the correct parameters.

Any help will be much appreciated

The very idea of the Boussinesq model is to keep T0, beta and rho0 constant, as it is a linearization around the 0 state. If you change stuff during iterations it is not Boussinesq anymore, you are actually using the true model in a certain sense.

Other than the classical Bousinnesq model, have also a look to the low Mach formulation, classically used in cases of combustion.

I just want to echo this statement. Yes, the point of Boussinesq is to keep all those those parameters constant. If you do any more (i.e. allow any of those to vary), you very quickly approach solving a variable density model. At that point you are no longer following the Boussinesq philosophy, you're just picking and choosing what dependencies & complexities you want to include in your model.




I'm confused by the question because these are really common. People use all the time the full ideal gas law, incompressible ideal gas law, polynomial functions of temperature and so on. What kind of novelties are you looking for that isn't readily available?

Yes, two papers I read a couple of months ago dealing with variable density flow, actually metal alloy. The densities of both solid and liquid change with temperature and composition. The density dependence on composition is very strong. Check out the follwoing 2 coupled papers: Numerical method for solution of strongly coupled binary alloy solidification problems, Daming Xu & Qingchun Li, 1991; Gravity-and solidification-shrinkage-induced liquid flow in a horizontally solidified alloy ingot, Daming Xu & Qingchun Li, 1991. Hope it helps.

Mingming