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 Gr S /Gr T ϭϪ1, where Gr S and Gr T are, respectively, the solutal and thermal Grashof numbers͒. We demonstrate that the equilibrium solution is linearly stable until the parameter Ra T ͉LeϪ1͉ reaches a critical value, which depends on the aspect ratio of the cell, A. For the square cavity we find a critical value of Ra c ͉LeϪ1͉ϭ17 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 k c ϭ2.53, for A→ϱ. 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, Ra o1. This value has been confused with the critical Rayleigh number, while it corresponds in fact to the location of the turning point.

This is an author's version published in: http://oatao.univ-toulouse.fr/20645To cite this version: A numerical and experimental study of buoyancy-driven flow in the annulus between two horizontal coaxial cylinders at Rayleigh numbers approaching and exceeding the critical values is presented. The stability of the flow is investigated using linear theory and the energy method. Theoretical predictions of the critical Rayleigh number for onset of secondary flows are obtained for a wide range of radius ratio R and are verified by comparison with results of previous experimental studies. A subcritical Rayleigh number which provides a necessary condition for global flow stability is also determined. The three-dimensional transient equations of fluid flow and heat transfer are solved to study the manifestation of instabilities within annuli having impermeable endwalls, which are encountered in various applications. For the first time, a thorough examination of the development of spiral vortex secondary flow within a moderate gap annulus and its interaction with the primary flow is performed for air. Simulations are conducted to investigate factors influencing the size and number of post-transitional vortex cells. The evolution of stable three-dimensional flow and temperature fields with increasing Rayleigh number in a large gap annulus is also studied. The distinct flow structures which coexist in the large gap annulus at high Rayleigh numbers preceding transition to oscillatory flow, including transverse vortices at the end walls which have not been previously identified, are established numerically and experimentally. The solutions for the large-gap annulus are compared to those for the moderate-gap case to clarify fundamental differences in behaviour. Heat transfer results in the form of local Nusselt number distributions are presented for both the moderate-and large-gap cases. Results from a series of experiments performed with air to obtain data for validation of the numerical scheme and further information on the flow stability are presented. Additionally, the change from a crescent-shaped flow pattern to a unicellular pattern with centre of rotation at the top of the annulus is investigated numerically and experimentally for a Prandtl number of 100. Excellent agreement between the numerical and experimental results is shown for both Prandtl numbers studied. The present work provides, for the first time, quantitative three-dimensional descriptions of spiral convection within a moderate-gap annulus containing air, flow structures preceding oscillation in a large-gap annulus for air, and unicellular flow development in a large-gap annulus for large Prandtl number fluids.

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the mono-cellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, ψ , characterizing the Soret effect and the normalized porosity, ε , are investigated theoretically and numerically.From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the 2 normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced.