Abstract
This investigation addresses a systematic numerical method based on the finite volume method and a full multigrid technique to study two-dimensional and three-dimensional flow of an incompressible fluid inside a cavity driven by the motion of the upper lid. Quantitative aspects of two and three dimensional flows in lid-driven cavities are analyzed by encompassing descriptive Reynolds numbers Re bounded by 1000 and 8000 for 2D case, 100 and 1000 for 3D case. Furthermore, the effects of inner spherical shape on fluid flow characteristics have been also investigated. An analysis of the flow evolution demonstrates that, by further increasing Reynolds beyond a certain critical value, the steady flow becomes unstable and bifurcates to the unsteady flow. Results show that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation, giving rise to a perfectly periodic state. Flow periodicity has been verified through time history plots for the velocity component and phase-space trajectories as a function of Reynolds number for both cases 2D and 3D. It is also concluded that for 2D and 3D configurations, whether the cavity is obstructed or non-obstructed, the refinement of the grid accelerates the appearance of the unsteady regime and delays it in the opposite case. A one-to-one comparison results between a cavity induced by a centrally located inner shape (a sphere for the 3D case and a circle for the 2D case) and a cavity without an inner shape confirms that by using a grid size 64(2) for the 2D case or 80(3) for the 3D case, critical Reynolds number for a cavity with an inner shape are less than that without an inner shape. Whilst, for the grid sizes 128(2) and 256(2) (2D case) the opposite phenomena occurs and it can be inferred that a bidimensional cavity with an inner circle at the center has a critical Reynolds number higher than that without an inner circle. Hence, the unsteadiness is delayed in this case.