Three-dimensional wake transition
Authors: Williamson C. H. K. 
Published: April 2006

It is now well-known that the wake transition regime for a circular cylinder involves two modes of small-scale three-dimensional instability (modes A and B), depending on the regime of Reynolds number (Re), although almost no understanding of the physical origins of these instabilities, or indeed their effects on near-wake formation, have hitherto been made clear. We address these questions in this paper. In particular, it is found that the two different modes A and B scale on different physical features of the flow. Mode A has a larger spanwise wavelength of around 3–4 diameters, and scales on the larger physical structure in the flow, namely the primary vortex core. The wavelength for mode A is shown to be the result of an ‘elliptic instability’ in the nearwake vortex cores. The subsequent nonlinear growth of vortex loops is due to a feedback from one vortex to the next, involving spanwise-periodic deformation of core vorticity, which is then subject to streamwise stretching in the braid regios. This mode gives an out-of-phase streamwise vortex pattern.

In contrast, mode-B instability has a distinctly smaller wavelength (1 diameter) which scales on the smaller physical structure in the flow, the braid shear layer. It is a manifestation of an instability in a region of hyperbolic flow. It is quite distinct from other shear flows, in that it depends on the reverse flow of the bluff-body wake; the presence of a fully formed streamwise vortex system, brought upstream from a previous half-cycle, in proximity to the newly evolving braid shear layer, leads to an in-phase stream-wise vortex array, in strong analogy with the ‘Mode 1’ of Meiburg & Lasheras (1988) for a forced unseparated wake. In mode B, we also observe amalgamation of streamwise vortices from a previous braid with like-sign vortices in the subsequent braid.

It is deduced that the large scatter in previous measurements concerning mode A is due to the presence of vortex dislocations. Dislocations are triggered at the sites of some vortex loops of mode A, and represent a natural breakdown of the periodicity of mode A instability. By minimizing or avoiding the dislocations which occur from end contamination or which occur during wake transition, we find an excellent agreement of both critical Re and spanwise wavelength of mode A with the recent secondary stability analysis of Barkley & Henderson (1996).

Wake transition is further characterized by velocity and pressure measurements. It is consistent that, when mode-A instability and large-scale dislocations appear, one finds a reduction of base suction, a reduction of (two-dimensional) Reynolds stress level, a growth in size of the formation region, and a corresponding drop in Strouhal frequency. Finally, the present work leads us to a new clarification of the possible flow states through transition. Right through this regime of Re, there exist two distinct and continuous Strouhal frequency curves: the upper one corresponds with purley small- scale instabilities (e.g. denoted as mode A), while the lower curve corresponds with a combination of small-scale plus dislocation structures (e.g. mode A*). However, some of the flow states are transient or ‘unstable’, and the natural transitioning wake appears to follow the scenario: (2D→A*→B).