Buoyancy: Boussinesq approximation or temperature dependence?
 

Hi folks,

I am dealing with a simulation of an incompressible fluid, where I have natural convection.
Most of the people use the Boussinesq approximation, where they use as the expansion coefficient.
But, in the case where the density (or temperature) changes a lot, this model should not be applied, and temperature dependence of density must be taken into account.

When I check the validity of the Boussinesq approximation, some people says a small density difference (ie, temperature), but never said the exact value, or a reference, or a difference that allows .

Mi question is if you know any criteria to choose between one of those ways?

Thanks for your answer

as the approximation is based on a linear expansion, the analysis generally says no more than some degree of variation in temperature

Hi

in my case, in the range of temperatures I am working, the density varies linearly. So, in the case I use Boussinesq, it will be only valid, if there is a linear variation?

no, it depends on the range of variation, you cannot consider Bousinnesq if you get more than some degree

I agree with you in that, but every journal, report... they only apply it, without justifying if under their conditions it's ok to use Boussinesq or not.

Since there are not more indications about when it should be used, the ideal situation should be consider always temperature dependence on density.

The Boussinesq approximation is limited to small temperature differences because it only accounts for the changes in forces caused by temperature/density. The Boussinesq model does not actually take into account density changes (it's nearly an incompressible fluid). If your density change was purely linear, the Bounssinesq approximation would predict the correct change in buoyancy forces and in that sense it is valid to always use the Boussinesq approximation. However, if there are density changes then the constant density assumption is not valid in the sense of the remaining terms in the continuity, momentum, and energy equations (the terms involving the convective derivative). In this sense, it is always incorrect to use the Boussinesq approximation and in general, any changes in density should always be taken into account. In other words, even if your density change was purely linear, the Bounssinesq aproximation is still invalid as it does not allow for density changes (only buoyancy force changes).

The criteria to determine the validity of the Bounssinesq approximation is therefore, to limit the temperature range to be small enough so that the density change is not significant. I don't know of a general criteria that is not subjective but it is analogous to: for what Mach number can a flow considered incompressible flow? It depends on the criteria that you are interested in studying (M=0.1 for some, 0.3 for others).