Figure 1: a) Variation of the wake frequency with the Reynolds number. b) Variation of the energy of the oscillations versus the Reynolds number. Note the saturation around R=160.

4

Ultrasound measurements of the spatial shape of the global mode

Modelling the Benard-Von Kilrmim instability in terms of global modes, needs to represent the velocity wake fluctuations by the product of two independent functions linked to the temporal and to the spatial evolutions of the velocity field. As we saw in the previous section, the main temporal features of the wake can be described by local measurements of the velocity flow. On the contrary, the spatial behavior of the periodic disturbances is much more difficult to study because traditionnal anemometry techniques (hot wire or laser anemometry) need heavy mappings of the tlow. The ultrasound profile monitor presents in this context a new and interesting alternative.

4.1

Transversal velocity profiles

The early experimental studies of Mathis (1983) showed already that the envelope of the fluctuations of the wake possesses a maximum whose position varies with the Reynolds number. More recently, the experimental work of Goujon-Durand et al (1994) and Wesfreid et al. (1996) and the numerical investigations of Zielinska and Wesfreid (1995) have shown that the wakes of triangular bodies present a critical behavior at Reynolds numbers close to the instability threshold. In particular, it is shown that the amplitude and the position of the maxima of the transversal velocity oscillations obey power laws of the Reynolds number.

Using the ultrasound profiler, we measured the complete profiles of the transversal velocity at 8 different longitudinal positions downstream the cylinder and for several Reynolds numbers. Figure 2-a presents such a space-time diagram recorded at a Reynolds number of 132 at the position x/d = 6. The 2-D Fourier Transform of this space-time diagram is then calculated and band-pass filtered, in order to improve the visualisation of the vortex shedding (see Figure 2-b). The gray level coding permits to observe the periodic oscillation of the transversal velocity due to the vortex shedding. As shown by Dusek (1996), the transversal velocity mode is symmetric versus the flow axis as it can be seen by our measurements. This corresponds to the classical alternate vortex shedding or to the sinuous mode of the wake. Then, taking the temporal Fourier Transform for each of the 128 space points, it is possible to compute the spatial distribution of the squared amplitude of the fundamental mode at those longitudinal positions. These profiles are then displayed on the same diagram giving a three-dimensionnal view of the global mode of the circular cylinder wake. Figure 3 shows these spatial shapes of the wake as the Reynolds number increases. It can be observed that the energy profiles of the first x-location profiles has a double hump near the cylinder. Further downstream, these profiles are transformed into a single peak. We will come back later on this double hump structure, and analyse for now the evolution of the main peak. Plotting the amplitudes taken by these profiles at y=O, we show on Figure 4 the deformation of the global mode with the Reynolds number. The deformation which can be easily observed, is at the origin of the amplitude saturation displayed on Figure I-b. We observe a rapid increase of the amplitude of oscillation from zero near the cylinder, up to a maximum Amax whose position xmax varies between 9 and 5, when increasing the Reynolds number. Then, the decrease of these envelopes far away from the cylinder can be interpreted as a viscous relaxation of the far wake.

It is then easy to compute the evolution of the position and level of the maximum of the mode as a function of the Reynolds number. The logarithmic representations of figure 5-a and 5-b show the critical behavior of the envelops as already observed by Goujon-Durand et al (1994), Zielinska and Wesfreid (1995) and Wesfreid et aL(1996). Moreover, we confirm the exponent of the power laws (1/2 for the amplitude Amax and -1/2 for the position xmax) observed by these authors. Note that these exponents are in complete agreement with a second-order phase transition model of the wake. An improvement of the Landau model can be acchieved when adding to equation (1) a second-derivative-term in the x direction. In order to enhance the scale-invariance structure of the wake, we renormalize the amplitude and the space by using Amax and xmax. Figure 6 shows that the different envelopes now collapse onto a single curve.