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A systematic review of the role of penicillin versus penicillin plus metronidazole in the management of peritonsillar abscess
- C M Moen, K Paramjothy, A Williamson, H Coleman, X Lou, A Smith, C M Douglas
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- Journal:
- The Journal of Laryngology & Otology / Volume 137 / Issue 9 / September 2023
- Published online by Cambridge University Press:
- 17 May 2023, pp. 992-996
- Print publication:
- September 2023
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Background
Peritonsillar abscess is a localised infection in the peritonsillar space. Pus from the abscess can contain anaerobes. Many clinicians prescribe metronidazole in addition to penicillin, but evidence to support this is limited. This review assessed the evidence of benefit of metronidazole for the treatment of peritonsillar abscess.
MethodsA systematic review was conducted of the literature and databases including Ovid Medline, Ovid Embase, PubMed and Cochrane library. Search terms included all variations of peritonsillar abscess, penicillin and metronidazole.
ResultsThree randomised, control trials were included. All studies assessed the clinical outcomes after treatment for peritonsillar abscess, including recurrence rate, length of hospital stay and symptom improvement. There was no evidence to suggest additional benefit with metronidazole, with studies suggesting increased side effects.
ConclusionEvidence does not support the addition of metronidazole in first-line management of peritonsillar abscess. Further trials to establish optimum dose and duration schedules of oral phenoxymethylpenicillin would benefit clinical practice.
Impingement of a counter-rotating vortex pair on a wavy wall
- Sarah E. Morris, C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 895 / 25 July 2020
- Published online by Cambridge University Press:
- 21 May 2020, A25
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In this paper, we investigate the impingement of a two-dimensional (2-D) vortex pair translating downwards onto a horizontal wall with a wavy surface. A principal purpose is to compare the vortex dynamics with the complementary case of a wavy vortex pair (deformed by the long-wavelength Crow instability) impinging onto a flat surface. The simpler case of a 2-D vortex pair descending onto a flat horizontal ground plane leads to the well known ‘rebound’ effect, wherein the primary vortex pair approaches the wall but subsequently advects vertically upwards, due to the induced velocity of secondary vorticity. In contrast, a wavy vortex pair descending onto a flat plane leads to ‘rebounding’ vorticity in the form of vortex rings. A descending 2-D vortex pair, impinging on a wavy wall, also generates ‘rebounding’ vortex rings. In this case, we observe that the vortex pair interacts first with the ‘hills’ of the wavy wall before the ‘valleys’. The resulting secondary vorticity rolls up into a concentrated vortex tube, ultimately forming a vortex loop along each valley. Each vortex loop pinches off to form a vortex ring, which advects upwards. Surprisingly, these rebounding vortex rings evolve without the strong axial flows fundamental to the wavy vortex case. The present research is relevant to wing tip trailing vortices interacting with a non-uniform ground plane. A non-flat wall is shown to accelerate the decay of the primary vortex pair. Such a passive, ground-based method to diminish the wake vortex hazard close to the ground is consistent with Stephan et al. (J. Aircraft, vol. 50 (4), 2013a, pp. 1250–1260; CEAS Aeronaut. J., vol. 5 (2), 2013b, pp. 109–125).
The relative-velocity version of the Morison equation for obstacle arrays in combined steady, low and high frequency motion
- H. Santo, P. H. Taylor, C. H. K. Williamson, Y. S. Choo
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- Journal:
- Journal of Fluid Mechanics / Volume 842 / 10 May 2018
- Published online by Cambridge University Press:
- 07 March 2018, pp. 188-214
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This paper revisits the problem of forces on obstacle arrays in combined waves, an in-line steady current and structural dynamic motions. The intended application is the design and re-assessment of dynamically responding offshore platforms. Planar grids of perforated plates are moved in forced motion on three scales through otherwise stationary water. A new analytical wave–current–structure blockage model is developed by building on the existing wave–current blockage model presented by Santo et al. (J. Fluid Mech., vol. 739, 2014b, pp. 143–178) using a similar set of experiments but with forced motion on two scales. The new model, which is an improved Morison relative-velocity formulation, is tested against the experimental data for a range of structural to wave oscillation frequency ratios, $f_{s}/f_{w}=2$, 2.5 and 3. For relatively small current speed ($u_{c}$) and oscillatory structural velocity ($u_{s}$) compared with the oscillatory wave velocity ($u_{w}$), the drag force time history on grids is well approximated by a summation of the wave drag and the current drag components independently, without a $u_{w}\times u_{c}$ cross-term, consistent with the previous model. The wave drag component contains an additional $u_{s}$ contribution, while the current drag component may or may not contain an additional $u_{s}$ contribution depending on $f_{s}/f_{w}$. The measured drag force is observed to be asymmetric in time due to biasing from the mean flow. This is supported by numerical simulation using a porous block as a numerical model of the grids, although the simulated force asymmetry is weaker. All these effects can be sufficiently accounted for in the analytical model. The new model is shown to fit the variation of the experimental forces and force harmonics in time well for a wide range of cases, requiring only calibration of the Morison type drag and inertia coefficients. In contrast, the industry-standard version of the Morison relative-velocity formulation cannot reproduce the variation of the measured force in time, so present practice should be regarded as inadequate for combined steady, low frequency and high frequency motion acting on obstacle arrays.
Snow chemistry across Antarctica
- N. Bertler, P.A. Mayewski, A. Aristarain, P. Barrett, S. Becagli, R. Bernardo, S. Bo, Xiao C., M. Curran, Qin D., D. Dixon, F. Ferrona, H. Fischer, M. Frey, M. Frezzotti, F. Fundel, C. Genthon, R. Gragnani, G. Hamilton, M. Handley, S. Hong, E. Isaksson, Kang J., Ren J., K. Kamiyama, S. Kanamori, E. Kärkäs, L. Karlöf, S. Kaspari, K. Kreutz, E. Meyerson, A. Kurbatov, Y. Ming, Zhang M., H. Motoyama, R. Mulvaney, H. Oerter, E. Osterberg, M. Proposito, A. Pyne, U. Ruth, J. Simões, B. Smith, S. Sneed, K. Teinilä, F. Traufetter, R. Udisti, A. Virkkula, O. Watanabe, B. Williamson, J-G. Winther, Li Y., E. Wolff, Li Z., A. Zielinski
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- Journal:
- Annals of Glaciology / Volume 41 / 2005
- Published online by Cambridge University Press:
- 14 September 2017, pp. 167-179
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An updated compilation of published and new data of major-ion (Ca, Cl, K, Mg, Na, NO3, SO4) and methylsulfonate (MS) concentrations in snow from 520 Antarctic sites is provided by the national ITASE (International Trans-Antarctic Scientific Expedition) programmes of Australia, Brazil, China, Germany, Italy, Japan, Korea, New Zealand, Norway, the United Kingdom, the United States and the national Antarctic programme of Finland. The comparison shows that snow chemistry concentrations vary by up to four orders of magnitude across Antarctica and exhibit distinct geographical patterns. The Antarctic-wide comparison of glaciochemical records provides a unique opportunity to improve our understanding of the fundamental factors that ultimately control the chemistry of snow or ice samples. This paper aims to initiate data compilation and administration in order to provide a framework for facilitation of Antarctic-wide snow chemistry discussions across all ITASE nations and other contributing groups. The data are made available through the ITASE web page (http://www2.umaine.edu/itase/content/syngroups/snowchem.html) and will be updated with new data as they are provided. In addition, recommendations for future research efforts are summarized.
Influence of a wall on the three-dimensional dynamics of a vortex pair
- Daniel J. Asselin, C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 817 / 25 April 2017
- Published online by Cambridge University Press:
- 20 March 2017, pp. 339-373
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In this paper, we are interested in perturbed vortices under the influence of a wall or ground plane. Such flows have relevance to aircraft wakes in ground effect, to ship hull junction flows, to fundamental studies of turbulent structures close to a ground plane and to vortex generator flows, among others. In particular, we study the vortex dynamics of a descending vortex pair, which is unstable to a long-wave instability (Crow, AIAA J., vol. 8 (12), 1970, pp. 2172–2179), as it interacts with a horizontal ground plane. Flow separation on the wall generates opposite-sign secondary vortices which in turn induce the ‘rebound’ effect, whereby the primary vortices rise up away from the wall. Even small perturbations in the vortices can cause significant topological changes in the flow, ultimately generating an array of vortex rings which rise up from the wall in a three-dimensional ‘rebound’ effect. The resulting vortex dynamics is almost unrecognizable when compared with the classical Crow instability. If the vortices are generated below a critical height over a horizontal ground plane, the long-wave instability is inhibited by the wall. We then observe two modes of vortex–wall interaction. For small initial heights, the primary vortices are close together, enabling the secondary vortices to interact with each other, forming vertically oriented vortex rings in what we call a ‘vertical rings mode’. In the ‘horizontal rings mode’, for larger initial heights, the Crow instability develops further before wall interaction; the peak locations are farther apart and the troughs closer together upon reaching the wall. The proximity of the troughs to each other and the wall increases vorticity cancellation, leading to a strong axial pressure gradient and axial flow. Ultimately, we find a series of small horizontal vortex rings which ‘rebound’ from the wall. Both modes comprise two small vortex rings in each instability wavelength, distinct from Crow instability vortex rings, only one of which is formed per wavelength. The phenomena observed here are not limited to the above perturbed vortex pairs. For example, remarkably similar phenomena are found where vortex rings impinge obliquely with a wall.
Direct measurement of thrust and efficiency of an airfoil undergoing pure pitching
- A. W. Mackowski, C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 765 / 25 February 2015
- Published online by Cambridge University Press:
- 26 January 2015, pp. 524-543
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We experimentally investigate the thrust and propulsive efficiency of a NACA 0012 airfoil undergoing oscillating pitching motion at a Reynolds number of $1.7\times 10^{4}$. While previous studies have computed thrust and power indirectly through measurements of momentum deficit in the object’s wake, we use a pair of force transducers to measure fluid forces directly. Our results help solidify a variety of experimental, theoretical and computational answers to this classical problem. We examine trends in propulsive performance with flapping frequency, amplitude and Reynolds number. We also examine the measured unsteady forces on the airfoil and compare them with linear theory dating from the first half of the 20th century. While linear theory significantly overpredicts the mean thrust on the foil, its prediction for the amplitude and phase of the time-varying component is surprisingly accurate. We conclude with evidence that the thrust force produced by the pitching airfoil is largely insensitive to most wake vortex arrangements.
Current blockage experiments: force time histories on obstacle arrays in combined steady and oscillatory motion
- H. Santo, P. H. Taylor, C. H. K. Williamson, Y. S. Choo
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- Journal:
- Journal of Fluid Mechanics / Volume 739 / 25 January 2014
- Published online by Cambridge University Press:
- 17 December 2013, pp. 143-178
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This paper revisits the problem of forces on obstacle arrays in combined waves and an in-line steady current. The intended application is the design and reassessment of offshore platforms. A series of experiments are performed on planar grids moved in both steady and oscillatory motion through otherwise stationary water. Detailed comparisons are made to a wave-current–structure interaction model recently presented by Taylor, Santo & Choo (Ocean Engng, vol. 57, 2013, pp. 11–24). We present new features of the model and test these against the experimental data. For relatively small current speed (${u}_{c} $) compared with oscillatory velocity amplitude (${u}_{w} $) with phase angle ($\omega t$), the drag force time history on grids with solid area ($A$) and projected frontal area (${A}_{f} $) is well approximated by a summation of the wave drag and the current drag components independently, so there is no ${u}_{w} \times {u}_{c} $ cross-term. The wave drag component is proportional to $\cos \omega t\vert \cos \omega t\vert $, while the current drag component to $\vert \cos \omega t\vert $, i.e. it is phase-locked to the oscillatory wave crests. The form of the predicted time history is new, so much of this paper is occupied in testing the adequacy of this theoretical form both in terms of an improved Morison-type formulation and also in the precise variation of the experimental drag force in time. We show that the measured crest and trough peak values of the drag force are consistent with the force peaks and troughs of the model prediction. The odd frequency harmonics of the measured drag force scale as the square of the oscillatory velocity amplitude $({ u}_{w}^{2} )$ and on the total hydrodynamic area (${C}_{d} A$). The shape of the odd harmonics is very similar to that for a pure oscillatory motion without steady current, but there are also even frequency harmonics associated with the current component. The even harmonics of the force scale as the square of the current speed $({ u}_{c}^{2} )$ and on the ${A}_{f} $, not on the ${C}_{d} A$. All of the above features are identified within the experimental data, and provide considerable support for the new current blockage model.
The new model is also shown to fit the entire force time history well for a wide range of individual cases, with different blockage ratio ($A/ {A}_{f} $) and number of grids, requiring only calibration of the Morison-type drag and inertia coefficients. In contrast, the industry-standard form of the Morison equation can only be matched at a single instant of the oscillation cycle, so present practice should be regarded as seriously inadequate for combined steady current and oscillatory flow acting on obstacle arrays.
Extensive spectroscopic and photometric study of HD 25558, a long orbital-period binary with two SPB components
- Á. Sódor, P. De Cat, D. J. Wright, C. Neiner, M. Briquet, R. J. Dukes, F. C. Fekel, G. W. Henry, M. H. Williamson, M. W. Muterspaugh, E. Brunsden, K. R. Pollard, P. L. Cottrell, F. Maisonneuve, P. M. Kilmartin, J. M. Matthews, T. Kallinger, P. G. Beck, E. Kambe, C. A. Engelbrecht, R. J. Czanik, S. Yang, O. Hashimoto, S. Honda, J.-N. Fu, B. Castanheira, H. Lehmann, N. Behara, H. Van Winckel, S. Scaringi, J. Menu, A. Lobel, P. Lampens, P. Mathias
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- Journal:
- Proceedings of the International Astronomical Union / Volume 9 / Issue S301 / August 2013
- Published online by Cambridge University Press:
- 18 February 2014, pp. 491-492
- Print publication:
- August 2013
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We carried out an extensive photometric and spectroscopic investigation of the SPB binary, HD 25558 (see Fig. 1 for the time and geographic distribution of the observations). The ~2000 spectra obtained at 13 observatories during 5 observing seasons, the ground-based multi-colour light curves and the photometric data from the MOST satellite revealed that this object is a double-lined spectroscopic binary with a very long orbital period of about 9 years. We determined the physical parameters of the components, and have found that both lie within the SPB instability strip. Accordingly, both components show line-profile variations consistent with stellar pulsations. Altogether, 11 independent frequencies and one harmonic frequency were identified in the data. The observational data do not allow the inference of a reliable orbital solution, thus, disentangling cannot be performed on the spectra. Since the lines of the two components are never completely separated, the analysis is very complicated. Nevertheless, pixel-by-pixel variability analysis of the cross-correlated line profiles was successful, and we were able to attribute all the frequencies to the primary or secondary component. Spectroscopic and photometric mode-identification was also performed for several of these frequencies of both binary components. The spectroscopic mode-identification results suggest that the inclination and rotation of the two components are rather different. While the primary is a slow rotator with ~6 d rotation period, seen at ~60° inclination, the secondary rotates fast with ~1.2 d rotation period, and is seen at ~20° inclination. Our spectropolarimetric measurements revealed that the secondary component has a magnetic field with at least a few hundred Gauss strength, while no magnetic field was detected in the primary.
The detailed analysis and results of this study will be published elsewhere.
Determining the stability of steady two-dimensional flows through imperfect velocity-impulse diagrams
- P. Luzzatto-Fegiz, C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 706 / 10 September 2012
- Published online by Cambridge University Press:
- 13 July 2012, pp. 323-350
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In 1875, Lord Kelvin stated an energy-based argument for equilibrium and stability in conservative flows. The possibility of building an implementation of Kelvin’s argument, based on the construction of a simple bifurcation diagram, has been the subject of debate in the past. In this paper, we build on work from dynamical systems theory, and show that an essential requirement for constructing a meaningful bifurcation diagram is that families of solutions must be accessed through isovortical (i.e. vorticity-preserving), incompressible rearrangements. We show that, when this is the case, turning points in fluid impulse are linked to changes in the number of the positive-energy modes associated with the equilibria (and therefore in the number of modes likely to be linearly unstable). In addition, the shape of a velocity-impulse diagram, for a family of solutions, determines whether a positive-energy mode is lost or gained at the turning point. Further to this, we detect bifurcations to new solution families by calculating steady flows that have been made ‘imperfect’ through the introduction of asymmetries in the vorticity field. The resulting stability approach, which employs ‘imperfect velocity-impulse’ (IVI) diagrams, can be used to determine the number of positive-energy (likely unstable) modes for each equilibrium flow belonging to a family of steady states. As an illustration of our approach, we construct IVI diagrams for several two-dimensional flows, including elliptical vortices, opposite-signed vortex pairs (of both rotating and translating type), single and double vortex rows, as well as gravity waves. By also considering an example involving the Chaplygin–Lamb dipole, we illustrate how the stability of a specific flow may be determined, by embedding it within a properly constructed solution family. The stability data from our IVI diagrams agree precisely with results in the literature. To the best of our knowledge, for a few of the flows considered here, our work yields the first available stability boundaries. Further to this, for several of the flows that we examine, the IVI diagram methodology leads us to the discovery of new families of steady flows, which exhibit lower symmetry.
Instability of secondary vortices generated by a vortex pair in ground effect
- D. M. Harris, C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 700 / 10 June 2012
- Published online by Cambridge University Press:
- 18 April 2012, pp. 148-186
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In this work, we investigate the approach of a descending vortex pair to a horizontal ground plane. As in previous studies, the primary vortices exhibit a ‘rebound’, due to the separation of secondary opposite-sign vortices underneath each primary vortex. On each side of the flow, the weaker secondary vortex can become three-dimensionally unstable, as it advects around the stronger primary vortex. It has been suggested in several recent numerical simulations that elliptic instability is the origin of such waviness in the secondary vortices. In the present research, we employ a technique whereby the primary vortices are visualized separately from the secondary vortices; in fact, we are able to mark the secondary vortex separation, often leaving the primary vortices invisible. We find that the vortices are bent as a whole in a Crow-type ‘displacement’ mode, and, by keeping the primary vortices invisible, we are able to see both sides of the flow simultaneously, showing that the instability perturbations on the secondary vortices are antisymmetric. Triggered by previous research on four-vortex aircraft wake flows, we analyse one half of the flow as an unequal-strength counter-rotating pair, noting that it is essential to take into account the angular velocity of the weak vortex around the stronger primary vortex in the analysis. In contrast with previous results for the vortex–ground interaction, we find that the measured secondary vortex wavelength corresponds well with the displacement bending mode, similar to the Crow-type instability. We have analysed the elliptic instability modes, by employing the approximate dispersion relation of Le Dizés & Laporte (J. Fluid Mech., vol. 471, 2002, p. 169) in our problem, finding that the experimental wavelength is distinctly longer than predicted for the higher-order elliptic modes. Finally, we observe that the secondary vortices deform into a distinct waviness along their lengths, and this places two rows of highly stretched vertical segments of the vortices in between the horizontal primary vortices. The two rows of alternating-sign vortices translate towards each other and ultimately merge into a single vortex row. A simple point vortex row model is able to predict trajectories of such vortex rows, and the net result of the model’s ‘orbital’ or ‘passing’ modes is to bring like-sign vortices, from each secondary vortex row, close to each other, such that merging may ensue in the experiments.
Vortex-induced vibration of a rising and falling cylinder
- M. HOROWITZ, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 662 / 10 November 2010
- Published online by Cambridge University Press:
- 19 August 2010, pp. 352-383
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In this study, we investigate the dynamics of a freely rising and falling cylinder. This is, in essence, a vortex-induced vibration (VIV) system comprising both transverse (Y) and streamwise (X) degrees-of-freedom (d.o.f.), but with zero spring stiffness and zero damping. This problem represents a limiting case among studies in VIV, and is an extension of recent research of elastically mounted bodies having very low spring stiffness, as well as bodies with very low mass and damping. We find that if the mass ratio (where m* = cylinder mass/displaced fluid mass) is greater than a critical value, m*crit = 0.545, the body falls or rises with a rectilinear trajectory. As the mass ratio is reduced below m*crit = 0.545, the cylinder suddenly begins to vibrate vigorously and periodically, with a 2P mode of vortex formation, as reported in the preliminary study of Horowitz & Williamson (J. Fluids Struct. vol. 22, 2006, pp. 837–843). The similarity in critical mass between freely rising and elastically mounted bodies is unexpected, as it is known that the addition of streamwise vibration can markedly affect the response and vortex formation in elastically mounted systems, which would be expected to modify the critical mass. However, we show in this paper that the similarity in vortex formation mode (2P) between the freely rising body and the elastically mounted counterpart is consistent with a comparable phase of vortex dynamics, strength of vortices, amplitudes and frequencies of motion and effective added mass (CEA). All of these similarities result in comparable values of critical mass. The principal fact that the 2P mode is observed for the freely rising body is an interesting and consistent result; based on the previous VIV measurements, this is the only mode out of the known set {2S, 2P, 2T} to yield negative effective added mass (CEA < 0), which is a condition for vibration of a freely rising body. In this paper, we deduce that there exists only one possible two degree-of-freedom elastically mounted cylinder system, which can be used to predict the dynamics of freely rising bodies. Because of the symmetry of the vortex wake, this system is one for which the natural frequencies are fNX = 2fNY. Although this seems clear in retrospect, previous attempts to predict critical mass did not take this into account. Implementing such an elastic system, we are able to predict vibration amplitudes and critical mass (m*crit = 0.57) for a freely rising cylinder in reasonable agreement with direct measurements for such a rising body, and even to predict the Lissajous figures representing the streamwise–transverse vibrations for a rising body with very small mass ratios (down to m* = 0.06), unobtainable from our direct measurements.
Steady, unsteady and transient vortex-induced vibration predicted using controlled motion data
- T. L. MORSE, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 649 / 25 April 2010
- Published online by Cambridge University Press:
- 13 April 2010, pp. 429-451
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In this study, we represent transient and unsteady dynamics of a cylinder undergoing vortex-induced vibration, by employing measurements of the fluid forces for a body controlled to vibrate sinusoidally, transverse to a free stream. We generate very high-resolution contour plots of fluid force in the plane of normalized amplitude and wavelength of controlled oscillation. These contours have been used with an equation of motion to predict the steady-state response of an elastically mounted body. The principal motivation with the present study is to extend this approach to the case where a freely vibrating cylinder exhibits transient or unsteady vibration, through the use of a simple quasi-steady model. In the model, we use equations which define how the amplitude and frequency will change in time, although the instantaneous forces are taken to be those measured under steady-state conditions (the quasi-steady approximation), employing our high-resolution contour plots.
The resolution of our force contours has enabled us to define mode regime boundaries with precision, in the amplitude–wavelength plane. Across these mode boundaries, there are discontinuous changes in the fluid force measurements. Predictions of free vibration on either side of the boundaries yield distinct response branches. Using the quasi-steady model, we are able to characterize the nature of the transition which occurs between the upper and lower amplitude response branches. This regime of vibration is of practical significance as it represents conditions under which peak resonant response is found in these systems. For higher mass ratios (m* > 10), our approach predicts that there will be an intermittent switching between branches, as the vortex-formation mode switches between the classical 2P mode and a ‘2POVERLAP’ mode. Interestingly, for low mass ratios (m* ~ 1), there exists a whole regime of normalized flow velocities, where steady-state vibration cannot occur. However, if one employs the quasi-steady model, we discover that the cylinder can indeed oscillate, but only with non-periodic fluctuations in amplitude and frequency. The character of the amplitude response from the model is close to what is found in free vibration experiments. For very low mass ratios (m* < 0.36 in this study), this regime of unsteady vibration response will extend all the way to infinite normalized velocity.
The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres
- M. HOROWITZ, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 651 / 25 May 2010
- Published online by Cambridge University Press:
- 29 March 2010, pp. 251-294
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In this paper, we study the effect of the Reynolds number (Re) on the dynamics and vortex formation modes of spheres rising or falling freely through a fluid (where Re = 100–15000). Since the oscillation of freely falling spheres was first reported by Newton (University of California Press, 3rd edn, 1726, translated in 1999), the fundamental question of whether a sphere will vibrate, as it rises or falls, has been the subject of a number of investigations, and it is clear that the mass ratio m* (defined as the relative density of the sphere compared to the fluid) is an important parameter to define when vibration occurs. Although all rising spheres (m* < 1) were previously found to oscillate, either chaotically or in a periodic zigzag motion or even to follow helical trajectories, there is no consensus regarding precise values of the mass ratio (m*crit) separating vibrating and rectilinear regimes. There is also a large scatter in measurements of sphere drag in both the vibrating and rectilinear regimes.
In our experiments, we employ spheres with 133 combinations of m* and Re, to provide a comprehensive study of the sphere dynamics and vortex wakes occurring over a wide range of Reynolds numbers. We find that falling spheres (m* > 1) always move without vibration. However, in contrast with previous studies, we discover that a whole regime of buoyant spheres rise through a fluid without vibration. It is only when one passes below a critical value of the mass ratio, that the sphere suddenly begins to vibrate periodically and vigorously in a zigzag trajectory within a vertical plane. The critical mass is nearly constant over two ranges of Reynolds number (m*crit ≈ 0.4 for Re = 260–1550 and m*crit ≈ 0.6 for Re > 1550). We do not observe helical or spiral trajectories, or indeed chaotic types of trajectory, unless the experiments are conducted in disturbed background fluid. The wakes for spheres moving rectilinearly are comparable with wakes of non-vibrating spheres. We find that these wakes comprise single-sided and double-sided periodic sequences of vortex rings, which we define as the ‘R’ and ‘2R’ modes. However, in the zigzag regime, we discover a new ‘4R’ mode, in which four vortex rings are created per cycle of oscillation. We find a number of changes to occur at a Reynolds number of 1550, and we suggest the possibility of a resonance between the shear layer instability and the vortex shedding (loop) instability. From this study, ensuring minimal background disturbances, we have been able to present a new regime map of dynamics and vortex wake modes as a function of the mass ratio and Reynolds number {m*, Re}, as well as a reasonable collapse of the drag measurements, as a function of Re, onto principally two curves, one for the vibrating regime and one for the rectilinear trajectories.
Prediction of vortex-induced vibration response by employing controlled motion
- T. L. MORSE, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 634 / 10 September 2009
- Published online by Cambridge University Press:
- 26 August 2009, pp. 5-39
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In order to predict response and wake modes for elastically mounted circular cylinders in a fluid flow, we employ controlled-vibration experiments, comprised of prescribed transverse vibration of a cylinder in the flow, over a wide regime of amplitude and frequency. A key to this study is the compilation of high-resolution contour plots of fluid force, in the plane of normalized amplitude and wavelength. With such resolution, we are able to discover discontinuities in the force and phase contours, which enable us to clearly identify boundaries separating different fluid-forcing regimes. These appear remarkably similar to boundaries separating different vortex-formation modes in the map of regimes by Williamson & Roshko (J. Fluids Struct., vol. 2, 1988, pp. 355–381). Vorticity measurements exhibit the 2S, 2P and P + S vortex modes, as well as a regime in which the vortex formation is not synchronized with the body vibration. By employing such fine-resolution data, we discover a high-amplitude regime in which two vortex-formation modes overlap. Associated with this overlap regime, we identify a new distinct mode of vortex formation comprised of two pairs of vortices formed per cycle, where the secondary vortex in each pair is much weaker than the primary vortex. This vortex mode, which we define as the 2POVERLAP mode (2PO), is significant because it is responsible for generating the peak resonant response of the body. We find that the wake can switch intermittently between the 2P and 2PO modes, even as the cylinder is vibrating with constant amplitude and frequency. By examining the energy transfer from fluid to body motion, we predict a free-vibration response which agrees closely with measurements for an elastically mounted cylinder. In this work, we introduce the concept of an ‘energy portrait’, which is a plot of the energy transfer into the body motion and the energy dissipated by damping, as a function of normalized amplitude. Such a plot allows us to identify stable and unstable amplitude-response solutions, dependent on the rate of change of net energy transfer with amplitude (the sign of dE*/dA*). Our energy portraits show how the vibration system may exhibit a hysteretic mode transition or intermittent mode switching, both of which correspond with such phenomena measured from free vibration. Finally, we define the complete regime in the amplitude–wavelength plane in which free vibration may exist, which requires not only a periodic component of positive excitation but also stability of the equilibrium solutions.
Defining the ‘modified Griffin plot’ in vortex-induced vibration: revealing the effect of Reynolds number using controlled damping
- R. N. GOVARDHAN, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 561 / 25 August 2006
- Published online by Cambridge University Press:
- 09 August 2006, pp. 147-180
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In the present work, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We employ a technique to accurately control the structural damping, enabling the system to take on both negative and positive damping. This permits a systematic study of the effects of system mass and damping on the peak vibration response. Previous experiments over the last 30 years indicate a large scatter in peak-amplitude data ($A^*$) versus the product of mass–damping ($\alpha$), in the so-called ‘Griffin plot’.
A principal result in the present work is the discovery that the data collapse very well if one takes into account the effect of Reynolds number ($\mbox{\textit{Re}}$), as an extra parameter in a modified Griffin plot. Peak amplitudes corresponding to zero damping ($A^*_{{\alpha}{=}0}$), for a compilation of experiments over a wide range of $\mbox{\textit{Re}}\,{=}\,500-33000$, are very well represented by the functional form $A^*_{\alpha{=}0} \,{=}\, f(\mbox{\textit{Re}}) \,{=}\, \log(0.41\,\mbox{\textit{Re}}^{0.36}$). For a given $\mbox{\textit{Re}}$, the amplitude $A^*$ appears to be proportional to a function of mass–damping, $A^*\propto g(\alpha)$, which is a similar function over all $\mbox{\textit{Re}}$. A good best-fit for a wide range of mass–damping and Reynolds number is thus given by the following simple expression, where $A^*\,{=}\, g(\alpha)\,f(\mbox{\textit{Re}})$: \[ A^* \,{=}\,(1 - 1.12\,\alpha + 0.30\,\alpha^2)\,\log (0.41\,\mbox{\textit{Re}}^{0.36}). \] In essence, by using a renormalized parameter, which we define as the ‘modified amplitude’, $A^*_M\,{=}\,A^*/A^*_{\alpha{=}0}$, the previously scattered data collapse very well onto a single curve, $g(\alpha)$, on what we refer to as the ‘modified Griffin plot’. There has also been much debate over the last three decades concerning the validity of using the product of mass and damping (such as $\alpha$) in these problems. Our results indicate that the combined mass–damping parameter ($\alpha$) does indeed collapse peak-amplitude data well, at a given $\mbox{\textit{Re}}$, independent of the precise mass and damping values, for mass ratios down to $m^*\,{=}\,1$.
Three-dimensional wake transition
- C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 328 / 10 December 1996
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- 26 April 2006, pp. 345-407
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It is now well-known that the wake transition regime for a circular cylinder involves two modes of small-scale three-dimensional instability (modes A and B), depending on the regime of Reynolds number (Re), although almost no understanding of the physical origins of these instabilities, or indeed their effects on near-wake formation, have hitherto been made clear. We address these questions in this paper. In particular, it is found that the two different modes A and B scale on different physical features of the flow. Mode A has a larger spanwise wavelength of around 3–4 diameters, and scales on the larger physical structure in the flow, namely the primary vortex core. The wavelength for mode A is shown to be the result of an ‘elliptic instability’ in the nearwake vortex cores. The subsequent nonlinear growth of vortex loops is due to a feedback from one vortex to the next, involving spanwise-periodic deformation of core vorticity, which is then subject to streamwise stretching in the braid regios. This mode gives an out-of-phase streamwise vortex pattern.
In contrast, mode-B instability has a distinctly smaller wavelength (1 diameter) which scales on the smaller physical structure in the flow, the braid shear layer. It is a manifestation of an instability in a region of hyperbolic flow. It is quite distinct from other shear flows, in that it depends on the reverse flow of the bluff-body wake; the presence of a fully formed streamwise vortex system, brought upstream from a previous half-cycle, in proximity to the newly evolving braid shear layer, leads to an in-phase stream-wise vortex array, in strong analogy with the ‘Mode 1’ of Meiburg & Lasheras (1988) for a forced unseparated wake. In mode B, we also observe amalgamation of streamwise vortices from a previous braid with like-sign vortices in the subsequent braid.
It is deduced that the large scatter in previous measurements concerning mode A is due to the presence of vortex dislocations. Dislocations are triggered at the sites of some vortex loops of mode A, and represent a natural breakdown of the periodicity of mode A instability. By minimizing or avoiding the dislocations which occur from end contamination or which occur during wake transition, we find an excellent agreement of both critical Re and spanwise wavelength of mode A with the recent secondary stability analysis of Barkley & Henderson (1996).
Wake transition is further characterized by velocity and pressure measurements. It is consistent that, when mode-A instability and large-scale dislocations appear, one finds a reduction of base suction, a reduction of (two-dimensional) Reynolds stress level, a growth in size of the formation region, and a corresponding drop in Strouhal frequency. Finally, the present work leads us to a new clarification of the possible flow states through transition. Right through this regime of Re, there exist two distinct and continuous Strouhal frequency curves: the upper one corresponds with purley small- scale instabilities (e.g. denoted as mode A), while the lower curve corresponds with a combination of small-scale plus dislocation structures (e.g. mode A*). However, some of the flow states are transient or ‘unstable’, and the natural transitioning wake appears to follow the scenario: (2D→A*→B).
Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers
- C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 206 / September 1989
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- 26 April 2006, pp. 579-627
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Two fundamental characteristics of the low-Reynolds-number cylinder wake, which have involved considerable debate, are first the existence of discontinuities in the Strouhal-Reynolds number relationship, and secondly the phenomenon of oblique vortex shedding. The present paper shows that both of these characteristics of the wake are directly related to each other, and that both are influenced by the boundary conditions at the ends of the cylinder, even for spans of hundreds of diameters in length. It is found that a Strouhal discontinuity exists, which is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode. This transition is explained by a change from one mode where the central flow over the span matches the end boundary conditions to one where the central flow is unable to match the end conditions. In the latter case, quasi-periodic spectra of the velocity fluctuations appear; these are due to the presence of spanwise cells of different frequency. During periods when vortices in neighbouring cells move out of phase with each other, ‘vortex dislocations’ are observed, and are associated with rather complex vortex linking between the cells. However, by manipulating the end boundary conditions, parallel shedding can be induced, which then results in a completely continuous Strouhal curve. It is also universal in the sense that the oblique-shedding Strouhal data (Sθ) can be collapsed onto the parallel-shedding Strouhal curve (S0) by the transformation, S0 = Sθ/cosθ, where θ is the angle of oblique shedding. Close agreement between measurements in two distinctly different facilities confirms the continuous and universal nature of this Strouhal curve. It is believed that the case of parallel shedding represents truly two-dimensional shedding, and a comparison of Strouhal frequency data is made with several two-dimensional numerical simulations, yielding a large disparity which is not clearly understood. The oblique and parallel modes of vortex shedding are both intrinsic to the flow over a cylinder, and are simply solutions to different problems, because the boundary conditions are different in each case.
A new mechanism for oblique wave resonance in the ‘natural’ far wake
- C. H. K. Williamson, A. Prasad
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- Journal:
- Journal of Fluid Mechanics / Volume 256 / November 1993
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- 26 April 2006, pp. 269-313
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There has been some debate recently on whether the far-wake structure downstream of a cylinder is dependent on, or ‘connected’ with, the precise details of the near-wake structure. Indeed, it has previously been suggested that the far-wake scale and frequency are unconnected with those of the near wake. In the present paper, we demonstrate that both the far-wake scale and frequency are dependent on the near wake. Surprisingly, the characteristic that actually forges a ‘connection’ between the near and far wakes is the sensitivity to free-stream disturbances. It is these disturbances that are also responsible for the regular three-dimensional patterns that may be visualized. Observations of a regular ‘honeycomb’-like three-dimensional pattern in the far wake is found to be caused by an interaction between oblique shedding waves from upstream and large-scale two-dimensional waves, amplified from the free-stream disturbances. The symmetry and spanwise wavelength of Cimbala, Nagib & Roshko's (1988) three-dimensional pattern are precisely consistent with such wave interactions. In the presence of parallel shedding, the lack of a honeycomb pattern shows that such a three-dimensional pattern is clearly dependent on upstream oblique vortex shedding.
With the deductions above as a starting point, we describe a new mechanism for the resonance of oblique waves, as follows. In the case of two-dimensional waves, corresponding to a very small spectral peak in the free stream (fT) interacting (quadratically) with the oblique shedding waves frequency (fK), it appears that the most amplified or resonant frequency in the far wake is a combination frequency fFW = (fK–fT), which corresponds physically with ‘oblique resonance waves’ at a large oblique angle. The large scatter in (fFW/fK) from previous studies is principally caused by the broad response of the far wake to a range of free-stream spectral peaks (fT). We present clear visualization of the oblique wave phenomenon, coupled with velocity measurements which demonstrate that the secondary oblique wave energy can far exceed the secondary two-dimensional wave energy by up to two orders of magnitude. Further experiments show that, in the absence of an influential free-stream spectral peak, the far wake does not resonate, but instead has a low-amplitude broad spectral response. The present phenomena are due to nonlinear instabilities in the far wake, and are not related to vortex pairing. There would appear to be distinct differences between this oblique wave resonance and the subharmonic resonances that have been previously studied in channel flow, boundary layers, mixing layers and airfoil wakes.
Acoustic forcing of oblique wave resonance in the far wake
- C. H. K. Williamson, A. Prasad
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- Journal:
- Journal of Fluid Mechanics / Volume 256 / November 1993
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- 26 April 2006, pp. 315-341
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In this paper, we investigate to what extent the far-wake ‘signature’ of the near-wake vortex dynamics of a nominally two-dimensional bluff body is affected by the character of the free-stream noise. We confirm the existence of an oblique wave resonance (at frequency, fK–fT), which is caused by nonlinear ‘quadratic’ interactions between primary oblique shedding waves (fK) and secondary two-dimensional waves (fT), which are amplified from free-stream disturbances. In this work, oblique wave resonance is induced by acoustic forcing of two-dimensional waves. The use of acoustic forcing reveals a set of higher-order oblique wave resonances corresponding to frequencies (fK–nfT), where n is an integer. We find from visualization that, even when the secondary two-dimensional waves have the same frequency as the oblique waves, it is the oblique waves that are preferentially amplified. Oblique wave angles up to 74° have been observed. The response of the wake to a large range of forcing frequencies shows a broad region of peak response, centred around F = (fT/fK) = 0.55, and is in reasonable agreement with predictions from linear stability analysis. A similar broad response is found for each of the higher-order oblique wave modes. Simple equations for the oblique waves yield approximate conditions for maximum wake response, with a frequency for peak response given by Fmax = 1/2n = 1/2, 1/4, 1/6,…, and an oblique wave angle given by θmax = 2θK, where θK is the angle of oblique vortex shedding. An increase in forcing amplitude has the effect of bringing the nonlinear wave interactions, leading to oblique wave resonance, further upstream. Paradoxically, the effect of an increase in amplitude (A) of the two-dimensional wave forcing is to further amplify the oblique waves in preference to the two-dimensional waves and, under some conditions, to inhibit the appearance of prominent two-dimensional waves where they would otherwise appear. With a variation in forcing amplitude, the amplitude of oblique wave response is found to be closely proportional to A½. In summary, this investigation confirms the surprising result that it is only through the existence of noise in the free stream that the far wake is ‘connected’ to the near wake.
The natural and forced formation of spot-like ‘vortex dislocations’ in the transition of a wake
- C. H. K. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 243 / October 1992
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- 26 April 2006, pp. 393-441
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The three-dimensional transition of the flow behind a bluff body is studied, with an emphasis placed on the evolution of large-scale structures in the wake. It has previously been found that there are two fundamental modes of three-dimensional vortex shedding in the wake of a circular cylinder (each mode being dependent on the range of Reynolds number), with a spanwise lengthscale of the same order as the primary streamwise wavelength of the vortex street. However. it is shown in the present study that the wake transition also involves the appearance of large-scale spot-like ‘vortex dislocations’, that grow downstream to a size of the order of 10–20 primary wavelengths. Vortex dislocations are generated between spanwise vortex-shedding cells of different frequency. The presence of these dislocations explains the large intermittent velocity irregularities that were originally found by Roshko (1954) and later by Bloor (1964) to characterize transition. The presence of these vortex dislocations in wake transition is largely responsible for the break-up to turbulence of the wake as it travels downstream.
In order to study their evolution in detail, dislocations have been (passively) forced to occur at a local spanwise position with the use of a small ring disturbance. It is found that ‘two-sided’ dislocations are stable in a symmetric in-phase configuration, and that they induce quasi-periodic velocity spectra and (beat) dislocation-frequency oscillations in the near wake. Intrinsic to these dislocations is a mechanism by which they spread rapidly in the spanwise direction, involving helical twisting of the vortices and axial core flows. This is felt to be a fundamental mechanism by which vortices develop large-scale distortions in natural transition. As the wake travels downstream, the energy at the low dislocation frequency decays slowly (in contrast to the rapid decay of other frequencies), leaving the downstream wake dominated by the large dislocation structures. Distinct similarities are found between the periodic forced dislocations and the intermittent dislocations that occur in natural transition. Further similarities of dislocations in different types of flow suggest that vortex or phase dislocations could conceivably be a generic feature of transition in all shear flows.