Bifurcation for flow past a cylinder between parallel planes

Abstract

Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented.A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem.A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion.Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.

The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

“…Figure ll For a fixed twodimensional cylinder embedded in a Poiseuille flow, St was observed to increase even more strongly with S, becoming three times larger ｦ ｯ ｲ ｓ ｾ ＠ 0.5 than in the unconfined case (Chen et al 1995;Sahin & Owens 2004).…”

“…Below the threshold of wake instability, Tavener (1994) and Maheshwari, Chhabra & Biswas (2006) have shown that the length of the recirculating wake of a fixed sphere decreases when the confinement ratio is increased. Numerical investigations for fixed two-dimensional cylinders (Chen, Pritchard & Tavener 1995 ;Sahin & Owens 2004) and fixed spheres (Tavener 1994;Cliffe, Spence & Tavener 2000) placed in an incoming confined flow indicate that for S < 0.5, wake instability is delayed. For 0.5 < S < 0.7, the threshold for wake instability decreases but remains larger than its value in the unconfined case.…”

Unsteady Mixed Convection from Two Isothermal Semicircular Cylinders in Tandem Arrangement

“…(10), u is the vertical component of the velocity field specified on the upstream boundary and u D is the average longitudinal velocity based on the diameter of the semicylinder. The accuracy of the numerical algorithm was tested by comparing results of the mean Nusselt number against available analytical [2] and numerical results [35] for the standard case of a symmetrically confined isothermal circular cylinder in a plane channel. Details about the numerical solution, validation of the algorithm and the grid employed can be found elsewhere [36,37].…”

The motion of a 2D pendulum in a channel subjected to an incoming flow

“…The effect of confinement on the vortex-shedding instability The effect of confinement on the vortex shedding in the wake of a two-dimensional cylinder placed inside a channel has been described in the literature, (see e.g. Chen, Pritchard & Tavener 1995;Sahin & Owens 2004;Camarri & Giannetti 2007;Biancofiore, Gallaire & Pasquetti 2011). Constraint of the confining walls has a stabilizing effect, due to the interaction between the cylinder wake and the wall boundary-layer vorticity.…”