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In this paper we apply a finite volume method, together with a cost-effective segregated solution algorithm, to solve for the primitive velocities and pressure in a set of incompressible Navier -Stokes equations. The well-categorized workshop problem of lid-driven cavity flow is chosen for this exercise, and results focus on the Reynolds number. Solutions are given for a depth-to-width aspect ratio of 1:1 and a span-to width aspect ratio of 3:1. Upon increasing the Reynolds number, the flows in the cavity of interest were found to comprise a transition from a strongly two-dimensional character to a truly three-dimensional flow and, subsequently, a bifurcation from a stationary flow pattern to a periodically oscillatory state. Finally, viscous (Tollmien-Schlichting) travelling wave instability further induced longitudinal vortices, which are essentially identical to Taylor -Go ¨rtler vortices. The objective of this study was to extend our understanding of the time evolution of a recirculatory flow pattern against the Reynolds number. The main goal was to distinguish the critical Reynolds number at which the presence of a spanwise velocity makes the flow pattern become three-dimensional. Secondly, we intended to learn how and at what Reynolds number the onset of instability is generated.

Numerical simulations are performed to investigate three-dimensional unsteady vortex-vortex and vortex-surface interactions in the near field of a wall-mounted rectangular cylinder placed inside a channel. The generation mechanism of the upstream and the trailing vortices from the topologically important critical points and their near-wall evolution pattern have been examined in detail. In the upstream region, a laminar necklace vortex system formed around the junction between the rectangular block (cylinder) and the flat channel floor. A sequence of streamwise vortical rollers dominated the downstream interaction region, and they exhibited strong unsteady vortex-surface interaction. Streamwise vortices which formed upstream of the obstacle exhibited quadrupole structure with the dominant pair being central downwash, whereas those lifting the flow behind the obstacle were of predominantly central upwash. Notably, at some downstream location, the near-wall wake structure was observed to locally disappear due to mutual interaction and annihilation by opposite strength vortices on either side of the wake centerline. During the entire course of unsteady flow evolution, such a disappearance of the wake remained closely associated with local contraction of the limiting streamlines on the channel floor, the development of a pair of topologically important floor critical points (saddles), and the presence of a near-wall node on the vertical symmetry plane. The dominance of inward transverse flow toward these saddles together with flow evolution from the downstream node on the vertical symmetry plane were found to be particularly responsible for facilitating the local interaction of various vortices of opposite strength, leading to significant vorticity cancellation in the region. Moreover, the basic source of the wake vortices and their nature of evolution behind the cylinder were also investigated here, and they were found to be fundamentally different from what one usually observes in the near-wake of a transverse jet. However, the growth of a pair of vertically lifting vortices from the spiraling shear layer nodes just behind the downstream edge of the cylinder base was detected in this flow configuration also.

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Present computational investigation reports a steady bifurcation phenomenon for three-dimensional flows through a plane-symmetric sudden expansion. When the channel aspect ratio exceeds a critical value, the well-known step height (pitchfork) bifurcation evolves with different symmetry breaking orientations on the left and right sides of the channel and bifurcates in the spanwise direction. For the channel aspect ratio less than the critical value, the originally occurring spanwise bifurcation cannot be stably retained and evolves eventually to a step height bifurcation. Compared to step height bifurcation, the spanwise bifurcation is found to be more difficult to obtain, because the symmetric flow present on the spanwise symmetry plane is unstable in two dimensions. For completeness, an extensive analysis of the observed spanwise bifurcation, covering its transient behavior, dependence on flow Reynolds number, channel aspect ratio, and expansion ratio, is included.

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