Gregory J. Sheard

Cylinders with square cross-section: wake instabilities with incidence angle variation

The wakes behind square cylinders with variation in incidence angle are computed over a range of Reynolds numbers to elucidate the three-dimensional stability and dynamics up to a Reynolds number of Re = 300, based on the projected height of the inclined square cylinder. Three-dimensional instability modes are predicted and computed using a linear stability analysis technique and three-dimensional simulations, respectively. Depending on the incidence angle, the flow is found to transition to three-dimensional flow through either a mode A instability, or a subharmonic mode C instability. The mode A instability is predicted as the first-occurring instability at incidence angles smaller than 12° and greater than 26°, with the mode C instability predicted between these incidence angles. At a zero-degree angle of incidence, the wake instabilities closely match modes A, B and a quasi-periodic mode predicted in earlier studies behind square and circular cylinders. With increasing angle of incidence, the three-dimensional wake transition Reynolds number first increases from Re = 164 as the mode A instability weakens, before decreasing again beyond an incidence angle of 12° as the wake becomes increasingly unstable to the mode C instability, and then again to the mode A instability as the incidence angle approaches 45°. A spanwise autocorrelation analysis from computations over a cylinder span 20 times the square cross-section side length reveals that beyond the onset of three-dimensional instabilities, the vortex street breaks down with patterns consistent with spatio-temporal chaos. This effect was more pronounced at higher incidence angles.

From spheres to circular cylinders: the stability and flow structures of bluff ring wakes

The low-Reynolds-number wake dynamics and stability of the flow past toroids placed normal to the flow direction are studied numerically. This bluff body has the attractive feature of behaving like the sphere at small aspect ratios, and locally like the straight circular cylinder at large aspect ratios. Importantly, the geometry of the ring is described by a single parameter, the aspect ratio (Ar), defined as a ratio of the torus diameter to the cross-sectional diameter of the ring. A rich diversity of wake topologies and flow transitions can therefore be investigated by varying the aspect ratio. Studying this geometry allows our understanding to be developed as to why the wake transitions leading to turbulence for the sphere and circular cylinder differ so greatly. Strouhal–Reynolds-number profiles are determined for a range of ring aspect ratios, as are critical Reynolds numbers for the onset of flow separation, unsteady flow and asymmetry. Results are compared with experimental findings from the literature. Calculated Strouhal–Reynolds-number profiles show that ring wakes shed at frequencies progressively closer to that of the straight circular cylinder wake as aspect ratio is increased from Ar =3 . For Ar > 8, the initial asymmetric transition is structurally analogous to the mode A transition for the circular cylinder, with a discontinuity present in the Strouhal–Reynolds-number profile. The present numerical study reveals a shedding-frequency decrease with decreasing aspect ratio for ring wakes, and an increase in the critical Reynolds numbers for flow separation and the unsteady flow transition. A Floquet stability analysis has revealed the existence of three modes of asymmetric vortex shedding in the wake of larger rings. Two of these modes are analogous to mode A and mode B of the circular cylinder wake, and the third mode, mode C, is analogous to the intermediate wavelength mode found in the wake of square section cylinders and circular cylinder wakes perturbed by a tripwire. Furthermore, three distinct asymmetric transition modes have been identified in the wake of small aspect ratio bluff rings. Fully developed asymmetric simulations have verified the unsteady transition for rings that exhibit a steady asymmetric wake.

From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings

Non-axisymmetric simulations verify and extend the results from a previous linear Floquet stability analysis of the wakes of rings. The wakes corresponding to the saturated state of each predicted non-axisymmetric instability mode over the entire aspect ratio range are successfully computed, and isosurface plots are presented elucidating the vortical wake structures. The existence of three non-axisymmetric flow regimes (Modes I, II and III) for the flow past rings with aspect ratios

Computations of the drag coefficients for low-Reynolds-number flow past rings

\textit{Ar\/} \lesssim 3.9

On quasiperiodic and subharmonic Floquet wake instabilities

is verified, as is the existence of non-axisymmetric instabilities of vortex streets in the flow past rings with

Flow around an impulsively arrested circular cylinder

\textit{Ar\/} \gtrsim 3.9

The evolution of a subharmonic mode in a vortex street

. Wakes are computed which correspond to the Mode A and B instabilities found in the flow past a circular cylinder, and a wake is computed which develops from a subharmonic Mode C instability. This wake features an azimuthal wavelength of approximately 1.7 ring cross-section diameters, which is between the azimuthal wavelengths of the Mode A and B instabilities. This mode cannot occur, at least in a pure state, in the flow past a circular cylinder. Nonlinear transition characteristics are predicted by evaluating coefficients of the truncated Landau equation, and transition hysteresis is verified by studying the mode amplitude variation with Reynolds number in the vicinity of the transitions. The regular Mode I and Mode III transitions are found to occur through supercritical and subcritical bifurcations, respectively, and the secondary Hopf bifurcations to these transitions, as well as the Mode II Hopf transition, are found to be supercritical. We verify that the Mode A and Mode B transitions are subcritical and supercritical, respectively, and we determine that the nature of the Mode C transition is dependent on aspect ratio. Landau constants are evaluated for the Hopf transitions throughout the aspect ratio range studied.

Pressure-driven flow past spheres moving in a circular tube

The variation in the drag coefficient for low-Reynolds-number flow past rings orientated normal to the direction of flow is investigated numerically. An aspect ratio parameter is used for a ring, which describes at its limits a sphere and a circular cylinder. This enables a continuous range of bodies between a sphere and a circular cylinder to be studied. The computed drag coefficients for the flow past rings at the minimum and maximum aspect ratio limits are compared with the measured and computed drag coefficients reported for the sphere and the circular cylinder. Some interesting features of the behaviour of the drag coefficients with variation of Reynolds number and aspect ratio emerge from the study. These include the decrease in the aspect ratio at which the minimum drag coefficient occurs as the Reynolds number is increased, from A R 5 at Re = 1 to A R 1 at Re = 200. In addition, a substantial decrease in the pressure component of the drag coefficient is observed after the onset of three-dimensional flow while the viscous contribution is similar to that for flow with imposed axisymmetry

Subharmonic mechanism of the mode C instability

The physical characteristics of bifurcated states in systems with inherent symmetry are constrained in ways that those in systems with broken symmetry are not. Here we examine the issue of quasiperiodic versus subharmonic instability modes of time-periodic laminar wakes, and how the relationship between them is influenced by weak symmetry breaking. The examples used are the vortex street wake of a circular cylinder, where symmetry is broken by distorting the cylinder into a ring, and the wake of a square cylinder, where symmetry is broken by a small fixed rotation of the cylinder about its axis. In both cases the symmetric wakes exhibit a quasiperiodic instability mode, with a pair of complex-conjugate Floquet multipliers and which manifests as a traveling wave. As symmetry is broken these multipliers migrate continuously to the real axis, coalesce, and split into a pair of subharmonic multipliers that move apart along the negative real axis. This behavior resolves an apparent dichotomy between the previo...

A coupled Landau model describing the Strouhal–Reynolds number profile of a three-dimensional circular cylinder wake

The vortex dynamics of the flow around a suddenly arrested translating circular cylinder is investigated by direct numerical simulation and water tank experiments. In the numerical study, a method of visualization of streaklines in simulated-particle tracking computations is proposed, which is based on a variable-variance two-dimensional Gaussian-weighted summation of particles in the vicinity of each interpolation point, and for which a close similarity with physical dye visualizations is found. This technique is used to identify the trajectory of both the wake vortices, as well as the secondary vortices induced as the original wake convects over the arrested cylinder. Observations show that, in a fashion similar to the flow past an arresting sphere, each wake vortex induces a counter-rotating vortex pair, which subsequently self-propels over a range of sometimes surprising trajectories as the Reynolds number and cylinder translation distance are varied. At low Reynolds numbers and short translation dist...