Three-dimensional transition in the wake of bluff elongated cylinders

Abstract

Despite little supporting evidence, there appears to be an implicit assumption that the wakes of two-dimensional bluff bodies undergo transition to three-dimensional flow and eventually turbulence, through the same sequence of transitions as observed for a circular cylinder wake. Previous studies of a square cylinder wake support this assumption. In this paper, the transition to three-dimensional wake flow is examined for an elongated cylinder with an aerodynamic leading edge and square trailing edge. The three-dimensional instability modes are determined as a function of aspect ratio ($\hbox{\it AR}\,{=}\,$length to width). Floquet analysis reveals that three distinct instabilities occur. These are referred to as Modes A, B$^\prime$ and S$^\prime$ through analogy with the modes for circular and square cylinders. For aspect ratios less than approximately 7.5, Mode A is the most unstable mode. For aspect ratios greater than this, the most unstable mode switches to Mode B$^\prime$. This has the same spatio-temporal symmetry as Mode B for a circular cylinder, but a spanwise wavelength and near-wake features more in common with Mode S for a square cylinder. The dominant wavelength for this mode is approximately two cylinder thicknesses, much longer than for Mode B for a circular cylinder. It is found that the critical Reynolds number for the onset of the Mode A instability varies approximately with the square root of the aspect ratio. On the other hand, the critical Reynolds number for Mode B$^\prime$ is almost independent of aspect ratio. For large aspect ratios, the separation in Reynolds number between the critical Reynolds numbers is substantial; for instance, for $\hbox{\it AR}\,{=}\,17.5$, these values are approximately 450 and 700. In fact, for this aspect ratio, the third instability mode, Mode S$^\prime$, is more unstable than Mode A. These results suggest that the transition scenario for elongated bluff bodies may be distinctly different to short bodies such as circular or square cylinders. At the very least, the dominant spanwise wavelength in the turbulent wake is likely to be much longer than that for a circular cylinder wake. In addition, the reversal of the ordering of occurrence of the two modes with the different spatial symmetries is likely to affect the development of spatio-temporal chaos as a precursor to fully turbulent flow.

In conjunction with prior work, the current results indicate that nearly all three-dimensional instabilities of the vortex street can be identified as one of only a handful of transition modes.