Instabilities in compressible fluids

Linear stability analysis in combination with the Boussinesq approximation is usually used in order to determine the critical conditions for the onset of convection for the Rayleigh-Benard problem. In this way a non-dimensional number, the Rayleigh number, is obtained, which reflects whether the fluid is at rest (stability) or in motion (instability). From a thermodynamic point of view the Boussinesq approximation has a shortcoming that contradicts thermodynamic stability. Therefore a linear stability analysis for the compressible Navier-Stokes-Fourier equations has been carried out numerically for rigid-rigid, rigid-free and free-free boundaries. The results show that the critical value of the Rayleigh number is not constant anymore as in the Boussinesq case, but that it depends on the thickness of the fluid layer.