Double diffusive convection in a vertical rectangular cavity

In the present work, we study the onset of double diffusive convection in vertical enclosures with equal and opposing buoyancy forces due to horizontal thermal and concentration gradients ~in the case GrS /GrT521, where GrS and GrT are, respectively, the solutal and thermal Grashof numbers!. We demonstrate that the equilibrium solution is linearly stable until the parameter RaTuLe21u reaches a critical value, which depends on the aspect ratio of the cell, A. For the square cavity we find a critical value of RacuLe21u517 174 while previous numerical results give a value close to 6000. When A increases, the stability parameter decreases regularly to reach the value 6509, and the wave number reaches a value kc52.53, for A!`. These theoretical results are in good agreement with our direct simulation. We numerically verify that the onset of double diffusive convection corresponds to a transcritical bifurcation point. The subcritical solutions are strong attractors, which explains that authors who have worked previously on this problem were not able to preserve the equilibrium solution beyond a particular value of the thermal Rayleigh number, Rao1. This value has been confused with the critical Rayleigh number, while it corresponds in fact to the location of the turning point