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Abstract
In this thesis, we focus on a linear and weakly non-linear stability analysis of the parallel, fully developed convection flow in a rectangular cavity whose boundaries are subjected to crossed uniform heat fluxes. For numerical modeling of this system, the Navier-Stokes equation with the assumption of the Boussinesq approximation, and the energy equation are coupled to simulate the natural convective motion. Parametric investigations based on the Rayleigh number, the Prandtl number, and the aspect ratio are performed to characterize the Stokes problem. The effect of initial conditions and boundary conditions on the convection process are also investigated.
First, we examined the parallel flown pattern within a cavity with large values of the aspect ratios (a >> 1 or a << 1). The results showed that dependently of Rayleigh number and heat flux ratio (q), multiple solutions can exist, some of which are unstable (anti-natural). Numerical calculations were carried out to determine the minimum aspect ratio above which the flow can be assumed to be quasi-parallel. The effects of the governing parameters on the intensity of the flow were examined. The multiples states of natural convection and different type of bifurcations were also discussed.
Second, we studied the stability of the parallel flow pattern in order to predict the thresholds for Hopf bifurcation. The critical Rayleigh numbers for the onset of convection in the layer are predicted for a wide range of Prandtl numbers and heat flux ratios. Depending on the value of Pr and the heat flux ratio, two instability modes are predicted, thermal, for Pr > 1 and hydrodynamic, for Pr < 1. For small Prandtl numbers (Pr ∈ [0,1]), both modes can occur at the codimension-2 intersection points of the critical branches. The transition between the shear-driven and buoyancy-driven instabilities occurs with a jump in the critical wavelength number and the critical oscillation frequency.
For the horizontal fluid layer and for the range of parameters considered, stable, super-critical bifurcation occur solely for longitudinal disturbances with three velocity components. According to the value of Pr and the heat flux ratio, the instability is hydrodynamic and oscillating at small Pr, and thermal and steady for larger Pr. For small heat flux ratios (q < 1), there exists an unstable, subcritical branches of the hydrodynamic modes, for oblique disturbances. At the codimension-2 points, computations of the four Landau coefficients revealed that the hydrodynamic mode is the only stable mode.
In addition, for the vertical slot, the critical disturbances are the transverse across the base flow (βc). For small and mild values of Prandtl (Pr < 53.325), instabilities are always oscillating. When Prandtl increases (Pr > 53.325), stationary supercritical branches are detected. Computations of the Landau coefficients for the two interacting modes revealed that the thermal mode is the only stable mode at the codimension-2 points. (Abstract shortened by UMI.)
