Engineering

Two-dimensional simulations of flow past both an elastically-mounted cylinder and an externally-driven oscillating cylinder were performed at a Reynolds number of Re = 200. The results were compared to determine if the oscillations of the driven-oscillation model were consistent with the oscillations observed in the elastically-mounted system. It was found that while this is the case, there is considerable sensitivity to input forcing. This sensitivity could explain observed discrepancies between experimental results for the two systems.

To date, the majority of studies of vortex-induced vibration (VIV) of a cylinder constrained to oscillate transverse to the freestream have been experimental, and at a Reynolds number, Re, where the flow is inherently three-dimensional. At these higher Re, an upper and lower branch of response are observed. Mostly, these branches have not previously been investigated at lower Re where the flow is two-dimensional. Therefore, two-dimensional simulations have been performed at Re ¼ 200 to thoroughly investigate the response of a cylinder to VIV. It was found that two regimes of response were present, similar in nature to the upper and lower branch at higher Re. While some differences were present, it was found that the genesis of the higher-Re flow behaviour was present in the low-Re two-dimensional flow, with evidence for this found in the amplitude, frequency and phase response of the cylinder. r

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.

A study of the flow past an oscillatory rotating cylinder has been conducted, where the frequency of oscillation has been matched to the natural frequency of the vortex street generated in the wake of a stationary cylinder, at Reynolds number 300. The focus is on the wake transition to three-dimensional flow and, in particular, the changes induced in this transition by the addition of the oscillatory rotation. Using Floquet stability analysis, it is found that the fine-scale three-dimensional mode that typically dominates the wake at a Reynolds number beyond that at the second transition to three-dimensional flow (referred to as mode B) is suppressed for amplitudes of rotation beyond a critical amplitude, in agreement with past studies. However, the rotation does not suppress the development of three-dimensionality completely, as other modes are discovered that would lead to three-dimensional flow. In particular, the longer-wavelength mode that leads the three-dimensional transition in the wake of a stationary cylinder (referred to as mode A) is left essentially unaffected at low amplitudes of rotation. At higher amplitudes of oscillation, mode A is also suppressed as the two-dimensional near wake changes in character from a single-to a doublerow wake; however, another mode is predicted to render the flow three-dimensional, dubbed mode D (for double row). This mode has the same spatio-temporal symmetries as mode A.

The full non-linear evolution of the tidal instability is studied numerically in an ellipsoidal fluid domain relevant for planetary cores applications. Our numerical model, based on a finite element method, is first validated by reproducing some known analytical results. This model is then used to address open questions that were up to now inaccessible using theoretical and experimental approaches. Growth rates and mode selection of the instability are systematically studied as a function of the aspect ratio of the ellipsoid and as a function of the inclination of the rotation axis compared to the deformation plane. We also quantify the saturation amplitude of the flow driven by the instability and calculate the viscous dissipation that it causes. This tidal dissipation can be of major importance for some geophysical situations and we thus derive general scaling laws which are applied to typical planetary cores.

A series of direct numerical simulations, both in two-and three-dimensions, of the flow past a circular cylinder for Reynolds numbers Re 6 600 has been conducted. From these simulations, the time-mean (and, for the three-dimensional simulations, the spanwise spatial-mean) flow has been calculated. A global linear stability analysis has been conducted on these mean flows, showing that the mean cylinder wake for Re 6 600 is marginally stable and the eigenfrequency of the leading global mode closely predicts the eventual saturated vortex shedding frequency. A local stability analysis has also been conducted. For this, a series of streamwise velocity profiles has been extracted from the mean wake and the stability of these profiles has been analysed using the Rayleigh stability equation. The real and imaginary instability frequencies gained from these profiles have then been used to find the global frequency selected by the flow using a saddle-point criterion. The results confirm the success of the saddle-point criterion when the mean flow is quasi-parallel in the vicinity of the saddle point; however, the limitations of the method when the mean flow exhibits higher curvature are also elucidated. †

Simulations of a cylinder undergoing externally controlled sinusoidal oscillations in the free stream direction have been performed. The frequency of oscillation was kept equal to the vortex shedding frequency from a fixed cylinder, while the amplitude of oscillation was varied, and the response of the flow measured. With varying amplitude, a rich series of dynamic responses was recorded. With increasing amplitude, these states included wakes similar to the Kármán vortex street, quasiperiodic oscillations interleaved with regions of synchronized periodicity (periodic on multiple oscillation cycles), a period-doubled state and chaotic oscillations. It is hypothesized that, for low to moderate amplitudes, the wake dynamics are controlled by vortex shedding at a global frequency, modified by the oscillation. This vortex shedding is frequency modulated by the driven oscillation and amplitude modulated by vortex interaction. Data are presented to support this hypothesis.

Transient growth of secondary instabilities in parallel wakes: Anti lift-up mechanism and hyperbolic instability Phys. Fluids 23, 114106 (2011) Steady interaction of a vortex street with a shear flow Phys. Fluids 22, 096601 (2010) On quasiperiodic and subharmonic Floquet wake instabilities Phys. Fluids 22, 031701 (2010) Using hyperbolic Lagrangian coherent structures to investigate vortices in bioinspired fluid flows Chaos 20, 017510 (2010) Effects of background noise on generating coherent packets of hairpin vortices

This paper studies the fluid-structure interaction of an elastically mounted square crosssection cylinder immersed in a free stream. The cross-section is mounted such that its sides are at 451 to the free stream direction, in a ''diamond'' configuration, and its motion is constrained to the transverse direction relative to the flow direction. Apart from the cross-section, this setup is the same as the majority of single-degree-of-freedom vortexinduced vibration studies of cylinders. Two-dimensional direct numerical simulations of this system have been performed. The Reynolds number based on the point-to-point distance of the cross-section has been fixed at Re ¼200). Simulations at this Reynolds number allow a direct comparison with previous results from circular cylinders, and therefore focus directly on the impact of the geometry.

The wake of a rotating circular cylinder in a free stream is investigated for Reynolds numbers Re 400 and non-dimensional rotation rates of α 2.5. Two aspects are considered. The first is the transition from a steady flow to unsteady flow characterized by periodic vortex shedding. The two-dimensional computations show that the onset of unsteady flow is delayed to higher Reynolds numbers as the rotation rate is increased, and vortex shedding is suppressed for α 2.1 for all Reynolds numbers in the parameter space investigated. The second aspect investigated is the transition from two-dimensional to three-dimensional flow using linear stability analysis. It is shown that at low rotation rates of α 1, the three-dimensional transition scenario is similar to that of the non-rotating cylinder. However, at higher rotation rates, the threedimensional scenario becomes increasingly complex, with three new modes identified that bifurcate from the unsteady flow, and two modes that bifurcate from the steady flow. Curves of marginal stability for all of the modes are presented in a parameter space map, the defining characteristics for each mode presented, and the physical mechanisms of instability are discussed.

A study of the flow past an oscillatory-rotating cylinder has been conducted, where the frequency of oscillation has been matched to the natural frequency of the vortex street generated in the wake of a stationary cylinder, for Reynolds number (Re= 300). Using Floquet stability analysis, it was found that the fine-scale three-dimensional mode that typically dominates the wake as Re is increased beyond that at transition to three-dimensional flow (referred to as mode B) was suppressed for amplitudes of rotation beyond a critical amplitude, confirming past studies. However, the rotation did not suppress the three-dimensional transition completely, as other modes were discovered that would lead to three-dimensional flow. In particular, the longer-wavelength mode that leads the three-dimensional transition in the wake of a stationary cylinder (referred to as mode A) was left essentially unaffected at low amplitudes of rotation. At higher amplitudes of oscillation, mode A was also suppressed, however other modes were predicted to render the flow three-dimensional, one of these modes appearing to be a spatial harmonic of mode A.

A series of direct numerical simulations, both in two-and three-dimensions, of the flow past a circular cylinder for Reynolds numbers Re 6 600 has been conducted. From these simulations, the time-mean (and, for the three-dimensional simulations, the spanwise spatial-mean) flow has been calculated. A global linear stability analysis has been conducted on these mean flows, showing that the mean cylinder wake for Re 6 600 is marginally stable and the eigenfrequency of the leading global mode closely predicts the eventual saturated vortex shedding frequency. A local stability analysis has also been conducted. For this, a series of streamwise velocity profiles has been extracted from the mean wake and the stability of these profiles has been analysed using the Rayleigh stability equation. The real and imaginary instability frequencies gained from these profiles have then been used to find the global frequency selected by the flow using a saddle-point criterion. The results confirm the success of the saddle-point criterion when the mean flow is quasi-parallel in the vicinity of the saddle point; however, the limitations of the method when the mean flow exhibits higher curvature are also elucidated. †

The richness of the wake structures and transitions of bluff bodies is enhanced when geometries and relative flow motions different to the generic fixed circular cylinder are involved. In this paper, an overview of the results of bluff body studies at moderate Reynolds numbers in recent years obtained by the FLAIR and IRPHE groups is presented. These include the effect on wake structure and transition due to bluff body geometry changes and of rocking and rolling of circular cylinders. In particular, the two dimensional wake structures and the order of wake transitions to three-dimensionality are found to vary enormously.

To date, the majority of studies of vortex-induced vibration (VIV) of a cylinder constrained to oscillate transverse to the freestream have been experimental, and at a Reynolds number, Re, where the flow is inherently three-dimensional. At these higher Re, an upper and lower branch of response are observed. Mostly, these branches have not previously been investigated at lower Re where the flow is two-dimensional. Therefore, two-dimensional simulations have been performed at Re ¼ 200 to thoroughly investigate the response of a cylinder to VIV. It was found that two regimes of response were present, similar in nature to the upper and lower branch at higher Re. While some differences were present, it was found that the genesis of the higher-Re flow behaviour was present in the low-Re two-dimensional flow, with evidence for this found in the amplitude, frequency and phase response of the cylinder. r

The circular cylinder has been the generic geometry employed to understand many aspects of flows around bluff bodies, including the transition to a turbulent wake flow. The FLAIR group has been interested in the types and orders of instabilities and transitions for a range of bluff body flows. In this paper, variations of the flow from the generic case of a fixed circular cylinder are considered: contraction to a thin plate whilst preserving elliptical profile; elongation of cylinders with elliptical leading edge and square trailing edge: contraction of tori from large major radius to a sphere; and transverse oscillation of a circular cylinder. For elongated cylinders with streamlined leading edges, the analogs of the instability modes for a circular cylinder become unstable in the reverse order. As well, the analogue of mode B has a significantly increased relative spanwise wavelength and appears to have a different near-wake structure. At the other extreme, for a normal flat plate, the wake first becomes unstable to a non-periodic mode that appears distinct from either of the dominant circular cylinder wake modes. For tori, which have a local geometry approaching a two-dimensional circular cylinder for large aspect ratios, the sequence of transitions with increasing Reynolds number is a strong function of aspect ratio. For intermediate aspect ratios, the first occurring wake instability mode is a subharmonic mode. In the case of transversely oscillating cylinders, it is shown that when the two-dimensional wake is in the 2S configuration, modes A, B and QP are present, similar in nature and symmetry structure to the modes of the same names for a fixed cylinder. However, increasing the amplitude of oscillation sees the critical Reynolds number for mode A increase significantly, to the point where mode B becomes critical before mode A. For higher Reynolds numbers, the two-dimensional wake loses its spatio-temporal symmetry and takes on the P + S configuration. With the onset of this configuration, modes A, B and QP cease to exist. It is shown that two new three-dimensional modes (SL and SS) arise from this base flow. The variety of transition modes and order of transition for these different cases may have implications for the route to wake turbulence for such bodies, distinct from that found for a fixed circular cylinder.

Two-dimensional simulations of flow past both an elastically-mounted cylinder and an externally-driven oscillating cylinder were performed at a Reynolds number of Re = 200. The results were compared to determine if the oscillations of the driven-oscillation model were consistent with the oscillations observed in the elastically-mounted system. It was found that while this is the case, there is considerable sensitivity to input forcing. This sensitivity could explain observed discrepancies between experimental results for the two systems.

A lack of information regarding the connections between energy transfer and wake mode of oscillating bodies in low Reynolds number flow prompted a two-dimensional numerical investigation at Re = 200. The region in which vortex shedding synchronized with the cylinder oscillation frequency was defined and the energy transfers calculated. The mode of Kármán vortex shedding, in which two vortices of opposite sign were shed each cycle (2S), displayed a gradual change from positive to negative energy transfer with increasing amplitude of motion. This change in energy transfer was directly related to the relative velocity of the fluid, which altered the position of the stagnation point and upper and lower shear layers on the body. As amplitude was increased the 2S shedding mode evolved into an asymmetric mode of shedding in which a pair and a single vortex (P+S) were shed each motion cycle. Energy transfers were calculated in the region of primary lock-in. The P+S mode occurred only in the region of negative energy transfer.

The flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode E) appears consistent with a hyperbolic instability in the wake, while the second (named mode F) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode F ) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased. flow, two three-dimensional instabilities were discovered using linear stability analysis, named mode E and mode F. Mode E was shown to have an aspect of hyperbolic instability occurring at the cusp of a single recirculation region in the wake. Mode E is stationary, representing the transition from a steady two-dimensional flow to a steady three-dimensional flow. Mode F was shown to be due to a centrifugal instability of the closed region encircling the cylinder. Unlike mode E, mode F varies with time, introducing a new frequency to the flow. This frequency appears to be associated with a travelling or modulated wave solution along the span of the cylinder. A number of studies exist at higher rotation rates, particularly in the range 2.5 < α < 6. covered this range for Re = 200 in detail, explicitly showing the existence of two steady solutions, with a hysteretic transition between the two in the range 4.8 α 5.0. These two solutions were differentiated by a change in length of the trailing shear layers that form the wake, and a movement of the hyperbolic point that forms in the flow where the fluid that orbits the cylinder rejoins the fluid that flows past. Using this movement of the hyperbolic point as a distinguishing feature, the results of at Re = 100 also show evidence of a change in the steady solution in the range 4.5 α 5.5.