Hopf bifurcation in wakes behind a rotating and translating circular cylinder

Abstract

A low-dimensional Galerkin method, initiated by Noack and Eckelmann ͓Physica D 56, 151 ͑1992͔͒, for the prediction of the flow field around a stationary two-dimensional circular cylinder in a uniform stream at low Reynolds number is generalized to the case of a rotating and translating cylinder. The Hopf bifurcation describing the transition from steady to time-periodic solution is investigated. A curve indicating the transitional boundary is given in the two-dimensional parameter plane of Reynolds number Re and rotating parameter ␣. Our results show that rotation may delay the onset of vortex street and decrease the vortex-shedding frequency. © 1996 American Institute of Physics. ͓S1070-6631͑96͒00107-9͔The problem of the flow around a uniformly rotating and translating circular cylinder has been investigated by several researchers due to its engineering importance and academic interest. Badr et al. 1 numerically simulated the steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers Re with rotating parameter ␣, in which Re is based on the cylinder radius R and the incoming velocity U ϱ and ␣ϭR /U ϱ , where represents the angular velocity of the rotating cylinder. Ingham, 2 Ingham and Tang, 3 D'Alessio and Dennis 4 considered numerical solutions of the steadystate N-S equation at subcritical Re. The investigations of the unsteady flow for supercritical Reynolds numbers are relatively fewer than the case of the steady-state flow. Badr et al. 5 numerically studied the time-dependent flow past an impulsively rotating and translating circular cylinder started from rest for ReϾ200, while Coutanceau and Menard 6 gave corresponding experimental results. Chang and Chen 7 investigated the same problem at some higher Re for 0р␣Ͻ2, and suggested there are three modes of vortex shedding existing in wakes depending on Re and ␣.Meanwhile, the research on the bifurcation structure in an open-flow at low Reynolds numbers is of great interest. Provansal et al., 8 Sreenivasan et al., 9 and Schumm et al. 10 experimentally studied the onset of 2-D vortex shedding in the wakes behind a stationary circular cylinder and showed that the transition from the steady to the periodic flow is characterized by a Hopf bifurcation and can be described by the Stuart-Landau equation. Jackson, 11 Zebib 12 and Noack et al. 13,14 numerically investigated the onset of vortex shedding in flow past a stationary circular cylinder by applying the linear stability analysis to an autonomous dynamical system.Following Noack's work, 13 a low-dimensional Galerkin method ͑LDGM͒ is generalized to the case of a 2-D uniformly rotating and translating circular cylinder. Although the LDGM cannot compete with grid-based computational techniques for high accurate simulations of the velocity fields or the resolution of far-wake properties, it is confirmed to be an ideal tool for investigations on global stability and chaos-theoretical analysis. 13,14 In the present Galerkin method, the streamfunction is approximated by a finite ex- BRIEF CO...

Laminar flow of power-law fluids past a rotating cylinder

“…While many studies are available at moderate to high Reynolds numbers at which the flow field is three-dimensional (e.g., see [9][10][11][12]), since the present work is concerned with the steady flow regime, the ensuing discussion is limited to the steady flow regime. At Re = 46-47, the flow becomes unsteady even for a [13]. However, subsequent more detailed studies [6,8] also point to the occurrence of two more transitions.…”

Effect of Rotation Rates on the Laminar Flow and Heat Transfer Past a Circular Cylinder

“…Their results show that, at higher rotational velocity, the Nusselt number is almost independent of Reynolds number and the thermal boundary conditions. The suppression of the von Karman vortex street was also reported numerically by Hu et al (1996) and experimentally by Dol et al (2008) and Lam (2009) for isothermal flow.…”

Effect of cross-stream buoyancy and rotation on the free-stream flow and heat transfer across a cylinder

“…For aiding buoyancy at a super-critical Re, the VS suppression was found experimentally [2] for Re ¼ 800 and numerically [3e6] at Ri ¼ 0. 15 for Re ¼ 100. VS suppression under the effect of aiding buoyancy at Ri ¼ 0.12 for Re ¼ 100 and at Ri ¼ 0.2 for Re ¼ 200 was reported by Patnaik et al [5].…”

“…For a super-critical Re ¼ 50, suppression of vortex shedding at a critical rotational velocity a > a I was first predicted by Hu et al [15] using a lowdimensional Galerkin method. Later, Kang et al [16] observed no onset of vortex shedding regardless of a at a sub-critical Re ¼ 40.…”