The flow behind rings: bluff body wakes without end effects

Abstract

Recent studies have demonstrated the strong influence of end effects on low-Reynoldsnumber bluff body wakes, and a number of questions remain concerning the intrinsic nature of three-dimensional phenomena in two-dimensional configurations. Some of them are answered by the present study which investigates the wake of bluff rings (i.e. bodies without ends) both experimentally and by application of the phenomenological Ginzburg–Landau model. The model turns out to be very accurate in describing qualitative and quantitative observations in a large Reynolds number interval. The experimental study of the periodic vortex shedding regime shows the existence of discrete shedding modes, in which the wake takes the form of parallel vortex rings or ‘oblique’ helical vortices, depending on initial conditions. The Strouhal number is found to decrease with growing body curvature, and a global expression for the Strouhal–Reynolds number relation, including curvature and shedding angle, is proposed, which is consistent with previous straight cylinder results. A secondary instability of the helical modes at low Reynolds numbers is discovered, and a detailed comparison with the Ginzburg–Landau model identifies it as the Eckhaus modulational instability of the spanwise structure of the near-wake formation region. It is independent of curvature and its clear observation in straight cylinder wakes is inhibited by end effects.The dynamical model is extended to higher Reynolds numbers by introducing variable parameters. In this way the instability of periodic vortex shedding which marks the beginning of the transition range is characterized as the Benjamin–Feir instability of the coupled oscillation of the near wake. It is independent of the shear layer transition to turbulence, which is known to occur at higher Reynolds numbers. The unusual shape of the Strouhal curve in this flow regime, including the discontinuity at the transition point, is qualitatively reproduced by the Ginzburg–Landau model. End effects in finite cylinder wakes are found to cause important changes in the transition behaviour also: they create a second Strouhal discontinuity, which is not observed in the present ring wake experiments.

Three-dimensional Floquet stability analysis of the wake of a circular cylinder

“…The development of each mode is associated with a distinct change in measurable quantities such as the shedding frequency, discussed above, and the base pressure coefficient (see figure 4). Leweke & Provansal (1995) note additional changes in measured spanwise correlations at R e = 180 and Re = 260.…”

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

“…CGL model equation describe correctly most non-linear wave systems such as low-Prandtl-number oscillatory instability [43,7,8], binary fluid convection [4,5], convection with rotation [9], cylinder wake [10] and so on. A single equation is enough for systems with broken x → −x symmetry, i.e., when a single wave is present.…”

“…This recalls the slow chaotic state [4] for Eckhaus transition of subcritical TW. The non-saturated growth of waves-modulations have been described as the basic mechanism for the development of the Eckhaus instability in several systems [4,6,7,9,10] leading to spatio-temporal defects and subsequent variation of the mean wavenumber. The only difference in our system is the basic state which is already slightly modulated.…”

“…On the other hand, non-linear traveling-waves have exhibited a fascinating variety of behaviors and patterns. Wave systems have been studied in binary-fluid convection (subcritical traveling waves bifurcation) [4][5][6], oscillatory instability in low Prandtl number convection [7,8], oscillatory rotating convection [9], cylinder wake [10], Taylor dean vortices. [11,12] Among these different waves systems, thermocapillary flows and in particular hydrothermal waves [13][14][15][16][17][18] or hot wire waves [19][20][21][22] appear as a very simple tool, owing to their supercritical bifurcation.…”

Mode competition in cylindrical flows driven by sidewall oscillations

“…Therefore, it would be very convenient to have a flow with the mentioned symmetries and the SO(2) symmetry fulfilled exactly. Axisymmetric bluff body wakes are an option, like the flow past a ring [16,17]. Unluckily, in this setting the spatiotemporal symmetry is only satisfied in the limit of zero curvature, and the effects of the broken symmetry result in the presence of subharmonic modes that are absent in the presence of the exact symmetry.…”