A universal three-dimensional instability of the wakes of two-dimensional bluff bodies

Abstract

Linear stability analysis of a wide range of two-dimensional and axisymmetric bluff-body wakes shows that the first three-dimensional mode to became unstable is always mode E. From the studies presented in this paper, it is speculated to be the universal primary 3D instability, irrespective of the flow configuration. However, since it is a transition from a steady two-dimensional flow, whether this mode can be observed in practice does depend on the nature of the flow set-up. For example, the mode E transition of a circular cylinder wake occurs at a Reynolds number of Re 96, which is considerably higher than the steady to unsteady Hopf bifurcation at Re 46 leading to Bénard-von-Kármán shedding. On the other hand, if the absolute instability responsible for the latter transition is suppressed, by rotating the cylinder or moving it towards a wall, then mode E may become the first transition of the steady flow. A well-known example is flow over a backward-facing step, where this instability is the first global instability to be manifested on the otherwise two-dimensional steady flow. Many other examples are considered in this paper. Exploring this further, a structural stability analysis (Pralits et al. J. Fluid Mech., vol. 730, 2013, pp. 5-18) was conducted for the subset of flows past a rotating cylinder as the rotation rate was varied. For the non-rotating or slowly rotating case, this indicated that the growth rate of the instability mode was sensitive to forcing between the recirculation lobes, while for the rapidly rotating case, it confirmed sensitivity near the cylinder and towards the hyperbolic point. For the non-rotating case, the perturbation, adjoint and structural stability fields, together with the wavelength selection, show some similarities with those of a Crow instability of a counter-rotating vortex pair, at least within the recirculation zones. On the other hand, at much higher rotation rates, Pralits et al. (J. Fluid Mech., vol. 730, 2013, pp. 5-18) have suggested that hyperbolic instability may play a role. However, both instabilities lie on the same continuous solution branch in Reynolds number/rotation-rate parameter space. The results suggest that this particular flow transition at least, and probably others, may have a number of different physical mechanisms supporting their development.

Linear global and asymptotic stability analysis of the flow past rectangular cylinders moving along a wall

“…The largest values are placed along the symmetry plane dividing the top and bottom wake recirculating regions and two further local maxima of lower intensity are placed just downstream of the TE corners. Similar results are reported by Rao et al (2016) for the circular cylinder case. When the gap height is reduced, the maximum of the sensitivity moves progressively inside the top recirculating region delimited by the ψ = 0 streamline.…”

“…Figure 15 plots the real part of the streamwise vorticity Re( ωx ) of the direct mode to show the three-dimensional nature of the mode. Figure 16 plots the real part of the vertical In free stream, the three-dimensional mode corresponds to mode E described by Rao et al (2016) for the circular cylinder. As shown in figure 15, the direct mode propagates downstream and, unlike the two-dimensional unsteady mode associated with the Hopf bifurcation, is symmetric, i.e.…”

“…Experimental investigations at larger Reynolds numbers showed that the scenario does not change in the turbulent regime, and that the critical gap height is almost independent on Re (Nishino, Roberts & Zhang 2007). Rao, Thompson & Hourigan (2016) referred to the three-dimensional regular bifurcation as mode E. They speculated that this mode is the universal primary three-dimensional instability in the flow around two-dimensional bluff bodies, irrespective of the flow configuration. They noticed that mode E is actually observed only when the two-dimensional Hopf bifurcation is delayed or suppressed.…”

Flow past a circular cylinder and a downstream sphere for Re < 300

“…1995; Jiang et al. 2016; Rao, Thompson & Hourigan 2016; Jiang, Cheng & An 2018). Vortex shedding occurs when the Reynolds number ( Re ) exceeds 46, and the transition of the wake from two-dimensional to three-dimensional occurs as the Reynolds number exceeds a critical value between 140 and 190 in experimental studies.…”

Experimental and numerical investigation of flames stabilised behind rotating cylinders: interaction of flames with a moving wall

“…Kelvin-Helmholtz (KH) instabilities develop in shear layers where velocity gradients induce coherent structures whereas Bénard-von Kármán (BVK) instabilities lead to an asymmetric vortex shedding in the wake of the obstacle (Cantwell & Coles 1983;Monkewitz 1988;Plaschko, Berger & Peralta-Fabi 1993;Kelso, Lim & Perry 1996). The transition to observe such instabilities does not depend only on the Reynolds number Re (Rao, Thompson & Hourigan 2016), but also on the blockage ratio determined by the ratio of the cylinder diameter to the width of the flow passage (Sahin & Owens 2004), or the position of the cylinder with respect to walls (Rao et al 2013). One method to cancel the KH or BVK instabilities is to rotate the cylinder along its axis.…”