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