Instability at the onset of natural convection
 

I undertake a numerical study using VCEF method in the goal to determinate entropy generation (Entropy generation is proportional to (Grad(T)**2)) at the onset of naturel convection for a Newtonien incompressible fluid enclosed in a differentiel heated square cavity( submitted to an horizontal thermal gradient, the horizontal walls are adiabatic). It is found that for small Rayleigh number total entropy generation increases to reach a maximum , then deceases asymptotically toward a constant value in the steady state. For higher Rayleigh number total entropy generation increases and reach its maximum, but it deacreases with an oscillatory regime before reaching the steady state. This case( oscillatory regime) is necessary due to the fact that for higher Ray the system is far from equilibrium ,then the system is in a spiral approach toward the steady state.These oscillations indicate that the system has instability at the onset of natural convection (instability aroud the steady state).I would like to know the physical causes of these oscillations (instability) at the onset of natural convection for important Rayleigh numbers by using two approaches, the one hydrodynamics and the other thermodynamics

Best regards

Mag