Abdelkader Mojtabi

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/GrT=−1, where GrS and GrT are, respectively, the solutal and thermal Grashof numbers). We demonstrate that the equilibrium solution is linearly stable until the parameter RaT|Le−1| reaches a critical value, which depends on the aspect ratio of the cell, A. For the square cavity we find a critical value of Rac|Le−1|=17u2009174 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 kc=2.53, 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 pr...

Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer

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 monocellular 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, characterizing the Soret effect and the normalized porosity, 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 normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio >0(Le,), while for 0, while a stationary or an oscillatory bifurcation occurs if mono the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and <0, the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters.

Onset of oscillatory flows in double-diffusive convection

A numerical study is presented of unsteady double-diffusive convection in a square cavity with equal but opposing horizontal temperature and concentration gradients. The boundary conditions along the vertical side-walls are imposed in such a way that the buoyancy ratio N = GrS⧹GrT is equal to −1, where GrS and GrT are the solutal and thermal Grashof numbers, respectively. In this situation, steady-state convective flow is stable up to a threshold value Grc1 of the thermal Grashof number which depends on the Lewis number Le. Beyond Grc1, oscillatory convective flows occur. Here we study the transition, steady-state flow–oscillatory flow, as a function of the Lewis number. The Lewis number varies between 2 and 45. Depending on the values of the Lewis number, the oscillatory flow occurring for GrT slightly larger than Grc1 is either centro-symmetric ( for Le ⩾ 17) or asymmetric single frequency flow ( for Le ⩽ 17) . For larger values of the thermal Grashof number, the two regimes occur for fixed values of GrT and Le. Furthermore, computations show that Grc1 reaches a minimum equal to 4.75×104 for Le ≈ 7

Double diffusive instability in an inclined cavity

Double diffusive convection in a rectangular two-dimensional cavity with imposed temperatures and concentrations along two opposite sidewalls is considered. The study is performed for two-dimensional cavities in which the thermal and solutal buoyancy forces have the same magnitude, but are of opposite sign. The influence on the convective instability of the aspect ratio A (height/length) of the cavity and the cavity inclination α with respect to gravity is discussed. The onset of convection is computed for an infinite layer and compared to that for bounded boxes. The study is completed by the continuation of bifurcating solutions. It is found that, due to centrosymmetry, steady bifurcations are either pitchfork or transcritical depending on A and α. However, a primary pitchfork bifurcation is found to create unstable steady solutions, even if it is the first bifurcation. For the aspect ratios we studied, and close to the onset of convection, the stable solutions are mainly one-roll structures that can be ...

Analyse du transfert de chaleur en convection mixte laminaire entre deux cylindres coaxiaux horizontaux

The mixed convection flow is analysed after it has been dynamically and thermally established. The solution is obtained using both the perturbation method and finite differences. The results obtained permit us to characterize the streamlines and their helicoidal motion is tied to the Rayleigh, Prandtl and Reynolds numbers.

Some properties of convective oscillations in porous medium

Convective oscillations in porous media are studied numerically. A two-dimensional square, differentially heated cavity, filled with a saturated porous medium, is considered subject to linear harmonic oscillations in the vertical direction. The formulation is based on the Darcy-Boussinesq model. The problem includes three nondimensional parameters: the Rayleigh number for porous media Ra, its vibrational analog Ra, and the nondimensional frequency f. The time-dependent Darcy-Boussinesq equations have been solved using a pseudo-spectral Chebyshev collocation method. The instantaneous fields of the established oscillatory regimes are presented. Also, some instantaneous and mean characteristics are studied and discussed. The distinctions from the case of viscous fluid atone are emphasized.

Energy stability of a natural convective flow in a horizontal annular space

The conditions for global asymptotic stability are determined by energy theory. The basic flow is obtained using the perturbation method. The results are compared to those obtained by the linear theory and to those derived from various experimental investigations.

Convection entre deux cylindres coaxiaux en regime laminaire permanent

Resume La resolution de lequation de lenergie a ete effectuee dans le cas dun ecoulement laminaire entre deux cylindres coaxiaux isothermes maintenant un gradient constant dans la couche annulaire. Levolution du champ de temperature le long des tubes a ete obtenue a laide de la methode de Galerkin ainsi quavec un modele numerique aux differences finies pour plusieurs rapports de rayons. Le transfert de chaleur entre les deux tubes, caracterise par les nombres de Nusselt exterieur et interieur, ainsi que la temperature de melange ont ete determines en fonction de la longueur reduite des deux tubes. La comparaison des solutions obtenues par differences finies et par la methode de Galerkin, pour plusieurs fonctions dessais, fait apparaitre une bonne concordance.

Onset of Free Convection in Solutions with Variable Soret Coefficients

Abstract It is not unusual for a Soret coefficient to change sign with temperature. We develop the theory for the onset of convection in such systems, heated from below or from above, provided that the mean temperature is precisely that at which the change of sign occurs. We also consider the realistic case of rigid, conducting, impervious boundaries for later comparison with laboratory experiments.

Thermosolutal convection in a horizontal porous layer heated from below in the presence of a horizontal through flow

In this paper, we study the effect of a homogeneous longitudinal through flow on the onset of convection in a horizontal porous layer saturated by a binary fluid and heated from below or above. The layer boundaries are subjected to a constant heat flux. The investigation is made by taking the Soret effect into account. It is found that in the case of positive separation ratio when the denser component moves toward the cooler wall, through flow has no effect on the stability threshold but exerts an orientating effect on the convective patterns. For negative separation ratio, a strong destabilization occurs of the spatially homogeneous state with respect to long-wave disturbances. The stability range for long-wavelength convective rolls is defined.