圆柱尾迹的时空结构英文文献和中文翻译(4)

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As already mentioned, very near the cylinder (x/d between 1 and 4), the energy of the global mode presents a double hump structure (see Figure 3). In fact, they correspond to measurements which were realised at longitudinal positions inside the recirculation eddies formed behind the cylinder. Therefore, the double hump structure corresponds to a wave propagation on each side of the cylinder. We show such a space-time measurement for R=120 and x/d=2, in Figure 7-a). As in Figure 2, we use the Fourier Transform and a band-pass filter to enhance the wake structure (Figure 7-b). It can be observed that the oscillations on each side of the cylinder, are in phase, but for some Reynolds numbers, it has also been possible to observe for some periods of time, oscillations in phase opposition. In this case, the shedding is a mixture of a symmetric and anti-symmetric oscillations. Figure 8 shows for a Reynolds number of R=150 and a longitudinal abscissa x/d=3, such a case of mixing between the varicose (anti-symmetric transversal velocity fluctuations) and the sinuous (symmetric transversal velocity fluctuations) mode. We will confirm further the presence of the varicose mode of vortex shedding in section 5, where a visual study of very short aspect ratio cylinders is presented.

4.2

Longitudinal velocity profiles

The ultrasound probe is now placed inside the water channel, aligned with the x axis in order to measure the longitudinal velocity t1uctuations from the longitudinal location x/d=20 up to the cylinder. Because this mode is odd, its amplitude is null along the x axis, and we placed the probe at the transversal position y/d= 0.5. Figure 9-a and 9-b give space-time diagrams obtained by the ultrasound profiler. As it can be clearly observed on these figures, two systems of waves which are generated from a position around x/d=3.5 downstream of the cylinder. The waves propagating in the downstream direction between x/d=3.5 and 20 correspond to the alternate shedding of the travelling vortices. The other waves propagate to the upstream direction between x/d=3.5 and the cylinder on the edge of the stationnary recirculating eddies. These experimental observations of the cylinder wake, may confirm the theoretical predictions of the existence of a region of the flow where the instability is of "absolute type" as it is defined in the review of Huerre and Monkewitz (1990).

Visualization of very short aspect ratio cylinder wakes

To verify the presence of the varicose mode of vortex shedding, we placed in the water channel, cylinders having aspect ratios r as small as 0.5. The wake is visualized by injecting (at an appropriate very small rate) white dye by small holes placed at the middle of the transversal plane, at the rear face of the cylinder. Figure 10 presents a sequence of snapshots obtained at different Reynolds numbers. As predicted by Lee and Budwig (1990), the threshold of the vortex shedding is pushed toward high Reynolds numbers. Moreover, as it can be seen on figure 10, the first instability of the wake in such conditions is of varicose type. For this particular cylinder, this instability occurs for a Reynolds number around 440. Although the main features of the now are two-dimensional, we believe that it is the three-dimensional structure (and not shedding) of the near wake which is at the origin of this varicose mode. Under this threshold, the wake remains stationary with the formation of large recirculating eddies behind the cylinder (see Figure 10-a). Above the threshold, a symmetric oscillation develops on each side of the recirculating eddies (Figure l0-b-c). We note also on these snapshots the existence next to the cylinder of a dead water zone which traps the dye. The extension of this region is approximatively one half of a diameter. When increasing the Reynolds number, the varicose mode generates a symmetric vortex street. Finally for R=800 (Figure l0-e), the classical alternate shedding appears. First both modes exist together, but for R=900 (see Figure 10-f) the alternate shedding and its associated Benard-Von Kilrmim vortex street dominate the wake. A physical interpretation of this phenomenon can be proposed if we consider the oscillating wake as the result of the coupling between two shear layers which have a thickness given by viscous effects and which are located on each side of the cylinder. Following the work we made on two coupled wakes (see Peschard and Le Gal (1996), we propose to model these two unstable shear layers by two linearly coupled Hopf bifurcations of the type described by equation (1). As it was demonstrated for two wakes, the in-phase mode appears for strongly coupled oscillators and the phase-opposition mode appears for weakly coupled wakes. We can therefore extrapolate these observations to the case of a single wake. The two shear layers have a thickness approximately given by the viscous scale of the flow: ð =γ/U where v is the kinematic viscosity of the fluid and U its velocity. It is then easily shown that the ratio between the distance separating the shear layers and their thickness is d/ ð = R. Therefore, near the threshold of very small aspect ratio cylinders (Rc≈500), this ratio which gives the coupling strengh between the oscillators, is large and our model predicts a phase-opposition locking which gives a varicose mode of vortex shedding. On the contrary, large aspect ratio cylinders have relatively close shear layers: d/ð≈50 and then are strongly coupled. The sinuous mode appears in this case. Note that a similar oscillators model based on coupled Van der Pol oscillators has been recently proposed by Lopez-Ruiz and Pomeau (1996). They obtained similar results to ours ( Peschard and Le Gal (1996) to interpret the transition between a varicose and a sinuous mode of vortex shedding phenomena. Note also that a similar mode selection (see for instance Thomas and Prakash (1990) works generally for two-dimensional jets: undeveloped "flat" profiles jets are unstable to the varicose mode (weakly coupled shear layers) contrary to well developed parabolic profile jets (strongly coupled shear layers) which are unstable to the sinuous mode.