ABSTRACT The wake of a short aspect ratio cylinder placed in a uniform flow is experimentally investigated. After having characterized the temporal behavior of the Benard-Von Karmim vortex shedding by the use of a classical hot-wire anemometer, an ultrasound anemometry technique is applied to study the spatial critical behavior of the envelope of the transversal velocity of the wake. It is shown that this envelope which represents the spatial form of the global mode of the wake, follows universal scaling lawswhich are in agreement with a second order phase transition. In a second set of experiments, the behavior of the longitudinal velocity fluctuations is also investigated. It has also been discovered that there is a special point several diameters behind the cylinder, which plays a role of a wave maker. Finally, for very small aspect ratio cylinders, symmetric vortex shedding is reported and modeled using a system of coupled oscillators equations.49889

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Introduction

The goal of this study is to analyse the temporal and the spatial features of the classical Benard-Von Karman wake of a cylinder. Although this wake is certainly the most popular flow in fluid mechanics, many problems about its spatio-temporal structure remain open. Recently, it has been proved by several authors (Sreenivasan et al (1986), Provansal et al (1987), Dusek et al (1994) and Schumm et al (1994 that the Benard-Von Karman vortex street shed by a cylinder, appears via a Hopf bifurcation at a critical Reynolds number Re. When neglecting the three-dimensional features of the wakes, the normal form for this bifurcation is called the Landau equation and reads simply:

dA/dt = (Ɛ+ i*Wo ) A - (1+iC2)∣A∣² A       (1)

where A is a complex order parameter representing the lateral velocity measured at one location in the wake, Ɛthe reduced Reynolds number (R - Rc)/Rc), Wo the natural frequency (without the non linear correction) of the shedding at Reynolds number Rand c2 a parameter corresponding to this frequency variation 'With the finite amplitude oscillations of the velocity field. The universal value for c2 is equal to -2.7 (see for instance Dusek et al (1994)) for infinitely long circular cylinder. In fact, Albarede and Monkewitz (1992) have shown that this model of two-dimensionnal wake can be extended to the case of the three-dimensionnal wake, and in particular in order to interpret the "chevron shedding" observed by Williamson (1989). Moreover, it has also been shown by Albarecte and Provansal (1995), that when the aspect ratio of the cylinder is small enough, only the first mode (along the axis of the cylinder) of vortex shedding can be observed. In these conditions, the temporal behavior of the wake is accurately described by the Landau equation. Of course, the coefficients of the model have to be estimated again as they depend on the aspect ratio of the cylinder. In particular, it is known from Lee and Budwig (1990) that the critical Reynolds number increases when the aspect ratio of the cylinder is decreased. Although these substantial progress achieved by these theoretical and experimental investigations concerning the temporal behavior of the wake, Goujon-Durand et al (1994) and later, Zielinska and Wesfreid (1995) and Wesfreid et al. (1996) stressed on the signification of the amplitude A of the global mode. They showed experimentally and numerically that A has to be a norm of the spatial amplitude distribution of the wake which is different from a local amplitude as it was usually considered in experiments. Analysing the deformation of the spatial envelope of the wake oscillations with the Reynolds number, they were able to prove by hot wire mappings of the flow or direct numerical simulation of the wake, that the position xmax (on the downstream direction) of the maximum Amax of the velocity oscillation, approaches the cylinder as the Reynolds number increases. Moreover, they showed that Amax obeys the Landau equation and that xmax varies with a power law of the Reynolds number. These results are in complete agreement with the prediction from a second order phase transition.