Mark C. Thompson

Three-dimensional instabilities in the wake of a circular cylinder

Abstract The two- and three-dimensional wake structure behind a circular cylinder has been computed using a high-order spectral element technique. For the two-dimensional computations the predictions are compared with accurate experimental results and agree to within experimental uncertainty for the Strouhal number and base pressure coefficient. For the three-dimensional simulations the two modes of three-dimensional instability designated as modes A and B both found experimentally but not previously computationally have been captured. Mode A appears first at a Reynolds number slightly less than 200. As the Reynolds number is increased there is a transfer of energy to mode B which has a wavelength approximately one-fourth that of mode A.

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.

Flow past a cylinder close to a free surface

Two-dimensional flow past a cylinder close to a free surface at a Reynolds number of 180 is numerically investigated. The wake behaviour for Froude numbers between 0.0 and 0.7 and for gap ratios between 0.1 and 5.0 is examined. For low Froude numbers, where the surface deformation is minimal, the simulations reveal that this problem shares many features in common with flow past a cylinder close to a no-slip wall. This suggests that the flow is largely governed by geometrical constraints in the low-Froude-number limit. At Froude numbers in excess of 0.3–0.4, surface deformation becomes substantial. This can be traced to increases in the local Froude number to unity or higher in the gap between the cylinder and the surface. In turn, this is associated with supercritical to subcritical transitions in the near wake resulting in localized free-surface sharpening and wave breaking. Since surface vorticity is directly related to surface curvature, such high surface deformation results in significant surface vorticity, which can diffuse and then convect into the main flow, altering the development of Strouhal vortices from the top shear layer, affecting wake skewness and suppressing the absolute instability. The variations of parameters such as Strouhal number and formation length are provided for Froude numbers spanning the critical range. At larger Froude numbers, good agreement is obtained with recently published experimental investigations. The previously seen metastable wake states are observed to occur for similar system parameters to the experiments despite the difference in Reynolds numbers by a factor of about 40. The wake state switching appears to be controlled by a feedback loop. Important elements of the feedback loop include the cyclic generation and suppression of the absolute instability of the wake, and the role of surface vorticity and vortices formed from the bottom shear layer in controlling vortex formation from the top shear layer. The proposed mechanism is presented. Shedding ceases at very small gap ratios (∼ 0.1–0.2). This behaviour can be explained in terms of the fluid flux through the gap, vorticity diffusion into the surface and opposite-signed surface vorticity from the strong surface deformation.

Three-dimensional transition in the wake of a transversely oscillating cylinder

A Floquet stability analysis of the transition to three-dimensionality in the wake of a cylinder forced to oscillate transversely to the free stream has been undertaken. The effect of varying the oscillation amplitude is determined for a frequency of oscillation close to the natural shedding frequency. The three-dimensional modes that arise are identified, and the effect of the oscillation amplitude on their structure and growth rate quantified. It is shown that when the two-dimensional wake is in the 2S configuration (which is similar to the Karman vortex street), the three-dimensional modes that arise are similar in nature and symmetry structure to the modes in the wake of a fixed cylinder. These modes are known as modes A, B and QP and occur in this order with increasing Re. However, increasing the amplitude of oscillation causes the critical Reynolds number for mode A to increase significantly, to the point where mode B becomes critical before mode A. The critical wavelength for mode A is also affected by the oscillation, becoming smaller with increasing amplitude. Elliptic instability theory is shown also to predict this trend, providing further support that mode A primarily arises as a result of an elliptic instability. At higher oscillation amplitudes, the spatio-temporal symmetry of the two-dimensional wake changes and it takes on the P + S configuration, with a pair of vortices on one side of the wake and a single vortex on the other side, for each oscillation cycle. With the onset of this configuration, modes A, B and QP cease to exist. It is shown that two new three-dimensional modes arise from this base flow, which we call modes SL and SS. Both of these modes are subharmonic, repeating over two base-flow periods. Also, either mode can be the first to become critical, depending on the amplitude of oscillation of the cylinder. The emergence of these two new modes, as well as the reversal of the order of inception of the three-dimensional modes A and B, leads to the observation that for an oscillating cylinder wake there are four different modes that can lead the transition to three-dimensionality, depending on the amplitude of oscillation

Wake state and energy transitions of an oscillating cylinder at low Reynolds number

This paper reports on an extensive parameter space study of two-dimensional simulations of a circular cylinder forced to oscillate transverse to the free-stream. In particular, the extent of the primary synchronization region, and the wake modes and energy transfer between the body and the fluid are analyzed in some detail. The frequency range of the primary synchronization region is observed to be dependent on Reynolds number, as are the wake modes obtained. Energy transfer is primarily dependent on frequency at low amplitudes of oscillation, but primarily dependent on amplitude at high amplitudes of oscillation. However, the oscillation amplitude corresponding to zero energy transfer is found to be relatively insensitive to Reynolds number. It is also found that there is no discernible change to the wake structure when the energy transfer changes from positive to negative.

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

Cardiogenesis of Embryonic Stem Cells with Liquid Marble Micro-Bioreactor

\textit{Ar\/} \lesssim 3.9

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

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

Sphere-wall collisions: vortex dynamics and stability

\textit{Ar\/} \gtrsim 3.9

The sensitivity of steady vortex breakdown bubbles in confined cylinder flows to rotating lid misalignment

. 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.