Self-sustained oscillations in the wake of a sphere

Abstract

The periodic wake of a sphere maintained in a uniform flow is experimentally characterized. Visualizations of the vortical structures periodically shed in this regime are presented and measurements of amplitude and frequency of the streamwise velocity fluctuations are performed. The experiments show that the resulting self-sustained oscillations of the wake flow appear via a supercritical Hopf bifurcation. Then, the relevance of a Landau-type equation to describe the temporal dynamics of the shedding modes near the instability threshold is discussed. For this purpose transient experiments of growth of the shedding modes are performed under supercritical conditions. From the evolutions during these transients of the instantaneous growth rate and frequency, the parameters of the Landau model are estimated. Dependence on the Reynolds number of these parameters is also deduced from the measurements.

Sedimenting sphere in a variable-gap Hele-Shaw cell

“…For our largest plate separation, Γ = 1.40, we observed timeindependent motion at Re = 83 and oscillatory motion at Re = 125 (table 5); this result can be compared to Re = 135 found for the onset of oscillations in a threedimensional numerical simulation of a sphere sedimenting without sidewalls (Pan 1999). Further, the flow field and vorticity we observed for a sedimenting sphere with Γ = 1.40 (figure 5) were similar to the results obtained for the wake behind a sphere for Re = 345 (Schouveiler & Provansal 2002, figure 6). Our visualizations using Kaliroscope particles for Γ = 1.20 and Re = 310 (not shown here) reveal that the structure of the wake is composed of sequences of hairpin vortices akin to those observed in the wake of spheres (Schouveiler & Provansal 2002) and bubbles (Brucker 1999).…”

“…Further, the flow field and vorticity we observed for a sedimenting sphere with Γ = 1.40 (figure 5) were similar to the results obtained for the wake behind a sphere for Re = 345 (Schouveiler & Provansal 2002, figure 6). Our visualizations using Kaliroscope particles for Γ = 1.20 and Re = 310 (not shown here) reveal that the structure of the wake is composed of sequences of hairpin vortices akin to those observed in the wake of spheres (Schouveiler & Provansal 2002) and bubbles (Brucker 1999).…”

“…At Re ≈ 270, the parallel vortex tubes begin to form hairpin vortices that are periodically swept away. Schouveiler & Provansal (2002) interpreted this phenomenon in terms of the Crow vortex pair instability (Crow 1970;Widnall, Bliss & Zalay 1971) and established that this transition occurs with no hysteresis and with a well-defined frequency, St = 0.127. Ormieres & Provansal (1999) observed the vortical structure and stability in the proximity of this transition.…”

“…1.1 6 Γ 6 1.4 The transition from non-periodic to periodic behaviour observed for this case is indicated in figure 2 by the dotted line. We analysed the data using the method of Schouveiler & Provansal (2002), who measured for flow past a fixed sphere (without confining sidewalls) the fluctuations of the fluid velocity in the vertical direction directly behind the sphere. Schouveiler & Provansal (2002) determined that the energy of these fluctuations at the primary shedding frequency increased linearly with increasing Re, indicating that the transition was a supercritical Hopf bifurcation.…”

High-speed imaging in fluids

“…De Vries [78] was one of the first to report on high-speed imaging of wake visualizations behind rising spheres using a Schlieren setup. One important aspect is the non-intrusive nature of the Schlieren technique, as opposed to PIV [79] and dye injection [80]. The latter two are widely used and described in literature for the visualization of wake structures behind bodies [refs], both for solid spheres and bubbles, however the introduction of particles and fluorophores interacts with the slip boundary layers and interfaces of the rising objects, changing the wake vorticity and bubble shape.…”