19 results
Turbulent wall plumes with detrainment
- Ziheng Yu, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 973 / 25 October 2023
- Published online by Cambridge University Press:
- 18 October 2023, R3
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This article presents a theoretical modelling framework for the previously unconsidered case of turbulent wall plumes that detrain continually with height in stably stratified environments. Built upon the classic turbulent plume model, our approach incorporates turbulent detrainment with a variable coefficient of detrainment. Based on a linear constitutive relationship between the ratio of the detrainment to entrainment coefficients and the ambient buoyancy gradient, it is found that for linear ambient stratifications, a dynamic quasi-equilibrium region, characterised by a near invariant local plume Richardson number, is achieved, downstream of which this equilibrium rapidly breaks down. With increasing ambient buoyancy gradient, while the plume becomes increasingly slender with weaker vertical motions, the level at which the plume breaks down to form a horizontal intrusion first decreases and then increases. Moreover, distinct from classic purely entraining plumes, a detraining wall plume can swell within the pre-equilibrium adjustment stage provided the local Richardson number is sufficiently low $({Ri\ll 6})$, behaviour which is in accordance with observations made in filling-box experiments.
General solutions of the plume equations: towards synthetic plumes and fountains
- Nick H. Wise, Gary R. Hunt
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- Journal of Fluid Mechanics / Volume 973 / 25 October 2023
- Published online by Cambridge University Press:
- 17 October 2023, R2
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Previous mathematical models of quasi-steady turbulent plumes and fountains have described the flow that results from a prescribed input of buoyancy. We offer a new perspective by asking, what input of buoyancy would give rise to a plume, or fountain, with given properties? Addressing this question by means of an analytical framework, we take a first step toward enabling a plume with specific characteristics, i.e. a synthetic plume, to be designed. We develop analytical solutions to the conservation equations that describe four kinds of turbulent flow: axisymmetric plume, starting fountain, line plume and wall plume. Crucially, our solutions do not require the buoyancy distribution to be specified, whether this be the source or off-source distribution. Key to our approach, we specify a function for the volume flux, $Q=f(z)$, and take advantage of the weak coupling between the conservation equations to uniquely express general solutions in terms of $f$. We show that any analytic function $f$ can form the basis for a set of solutions for the fluxes, local variables and local Richardson number, though $f/({\rm d}f/{\rm d}z)>0$ is a necessary condition for physically realistic solutions. As an example of plume synthesis, we show that an axisymmetric plume can have an invariant radius if there is an exponentially increasing input of buoyancy to the plume centreline. We also consider how plume synthesis could be achieved practically.
Local linear stability of plumes generated along vertical heated cylinders in stratified environments
- Ziheng Yu, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 971 / 25 September 2023
- Published online by Cambridge University Press:
- 08 September 2023, A1
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The linear temporal and absolute/convective stability characteristics of a thermal plume generated along a heated vertical cylinder are investigated theoretically under the Boussinesq approximation. Special focus is given to the uniform-wall-buoyancy-flux case whereby the cylinder surface sustains the same linear temperature gradient as the environment. A competition between the axisymmetric and helical modes is a remarkable feature of the instability, distinguishing these ‘annular plumes’ from free plumes/jets for which the helical mode is generally dominant. It is found that higher surface curvature stabilises the temporal axisymmetric mode significantly, but only has moderate effects on the helical mode. The most temporally unstable perturbation mode switches from a helical into an axisymmetric mode when the Prandtl number increases beyond a critical value. Both the roles of shear and buoyancy during the destabilisation are identified through an energy analysis which indicates that, while the shear work is usually a major source of perturbation energy, the buoyancy work manifests for long-wave axisymmetric perturbation modes, and for thin cylinders and high Prandtl numbers. For the specific temperature configuration considered herein, an annular plume is always convectively unstable whereas decreasing the cylinder radius from the planar limiting case first decreases and then increases the tendency of the flow towards being absolutely unstable. The helical mode is especially susceptible to being absolutely unstable on very thin cylinders. Several conditions for the onset of cellular thermal convection and plume detrainment are proposed based on our results and a hypothesis which connects the absolute instability to the detrainment phenomenon.
Turbulent plumes above a heated plate
- Jamie P. Webb, Nick H. Wise, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 959 / 25 March 2023
- Published online by Cambridge University Press:
- 22 March 2023, A29
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Our focus concerns the turbulent convective flow above a uniformly heated high-aspect-ratio rectangular plate on $z=0$. Prior to this study, classic plume theory could not be applied directly as the Richardson number is ill-defined at the plate. Guided by observation, conservation equations are posed for the near-plate region where the attached buoyant flow is predominantly horizontal. Analytical solutions under the Boussinesq approximation reveal this to be a dynamically invariant region where the attached ‘plumes’ grow linearly toward the plate centreline, their merger forming an ‘apparent’ source for the vertical plume above. Coupling predictions with data from flow visualisation and temperature measurement, we deduce the half-width $0.72b_0$, height ${z=0.46b_0}$ and finite Richardson number $({\sim }18)$ of an apparent source from which plume theory can be applied to model the plume above a heated plate of width $2b_0$. Finally, practical implications of this advancement to the analytical theory of turbulent plumes are noted.
What is the entrainment coefficient of a pure turbulent line plume?
- James Richardson, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 934 / 10 March 2022
- Published online by Cambridge University Press:
- 14 January 2022, A11
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Despite its pivotal role in the classic theory of turbulent line plumes, there has been no consensus on the value of the entrainment coefficient $\alpha$ suitable for a pure plume. Reported measurements vary by 100 %, from $\alpha =0.1$ to $\alpha =0.2$, hindering the predictive capabilities of plume theory. Following our theoretical developments, measurements of plume entrainment using a new approach and a rigorous assessment of reported values for $\alpha$, we conclude that ${\alpha =0.11\pm 15\,\%}$ should be adopted as the consensus value. Our theoretical framework demonstrates how $\alpha$ is determined from underlying plume measurements, and places an emphasis on the link between measurement uncertainty and uncertainty in $\alpha$. This framework inspired our experimental design, intentionally conceived to precisely determine $\alpha$. From measurements of the plume scalar width and the entrainment velocity outside the plume, we determine that $\alpha =0.108\pm 2\,\%$ ($95\,\%$ confidence interval). Complementing our experiments is an evaluation of the historical data which, after we explain why some reported values of $\alpha$ are erroneous, supports the range $0.095\lesssim \alpha \lesssim 0.13$. The proposed consensus value thus represents both our precisely determined value and the variation in the published data. The significance of a consensus value for $\alpha$ can be summarised as follows: (i) it enhances confidence in the application of plume theory to practical situations and (ii) it permits more detailed comparison of entrainment between pure line plumes and related turbulent flows, including forced and lazy line plumes and wall plumes.
Two-dimensional buoyant plumes in a uniform co-flow
- Gary R. Hunt, Jamie P. Webb
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- Journal:
- Journal of Fluid Mechanics / Volume 932 / 10 February 2022
- Published online by Cambridge University Press:
- 14 December 2021, A41
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The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.
A history of high-power laser research and development in the United Kingdom
- Part of
- Colin N. Danson, Malcolm White, John R. M. Barr, Thomas Bett, Peter Blyth, David Bowley, Ceri Brenner, Robert J. Collins, Neal Croxford, A. E. Bucker Dangor, Laurence Devereux, Peter E. Dyer, Anthony Dymoke-Bradshaw, Christopher B. Edwards, Paul Ewart, Allister I. Ferguson, John M. Girkin, Denis R. Hall, David C. Hanna, Wayne Harris, David I. Hillier, Christopher J. Hooker, Simon M. Hooker, Nicholas Hopps, Janet Hull, David Hunt, Dino A. Jaroszynski, Mark Kempenaars, Helmut Kessler, Sir Peter L. Knight, Steve Knight, Adrian Knowles, Ciaran L. S. Lewis, Ken S. Lipton, Abby Littlechild, John Littlechild, Peter Maggs, Graeme P. A. Malcolm, OBE, Stuart P. D. Mangles, William Martin, Paul McKenna, Richard O. Moore, Clive Morrison, Zulfikar Najmudin, David Neely, Geoff H. C. New, Michael J. Norman, Ted Paine, Anthony W. Parker, Rory R. Penman, Geoff J. Pert, Chris Pietraszewski, Andrew Randewich, Nadeem H. Rizvi, Nigel Seddon, MBE, Zheng-Ming Sheng, David Slater, Roland A. Smith, Christopher Spindloe, Roy Taylor, Gary Thomas, John W. G. Tisch, Justin S. Wark, Colin Webb, S. Mark Wiggins, Dave Willford, Trevor Winstone
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- Journal:
- High Power Laser Science and Engineering / Volume 9 / 2021
- Published online by Cambridge University Press:
- 27 April 2021, e18
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The first demonstration of laser action in ruby was made in 1960 by T. H. Maiman of Hughes Research Laboratories, USA. Many laboratories worldwide began the search for lasers using different materials, operating at different wavelengths. In the UK, academia, industry and the central laboratories took up the challenge from the earliest days to develop these systems for a broad range of applications. This historical review looks at the contribution the UK has made to the advancement of the technology, the development of systems and components and their exploitation over the last 60 years.
On the stratification and induced flow in an emptying–filling box driven by a plane vertically distributed source of buoyancy
- Ziheng Yu, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 912 / 10 April 2021
- Published online by Cambridge University Press:
- 04 February 2021, A1
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A theoretical model is presented for the steady multi-layered flow induced by a plane vertically distributed buoyancy source producing a turbulent wall plume in a ventilated box. While aspects of the stratification and rate of fluid exchange between box and exterior have been studied previously, the streamline pattern and velocity field have not been considered until now, despite having potentially important practical implications for achieving comfort in naturally ventilated buildings and for the indoor spread of airborne contagions. The boundary condition at the wall for each layer is established by deducing the turbulent entrainment rate. Using conformal mapping techniques and Poisson's integral theorem, closed-form solutions for the streamfunction of the induced flow in each layer are established. While the flow near the ceiling was overlooked in the classic model for the multi-layered stratification, after considering the possible flow scenarios, the stratification is re-evaluated herein by incorporating an entraining ceiling current. With a markedly thinner top layer, the refined stratification matches well with the available experimental observations, the restrictions we place on the applicability of the model overcoming the previous over-prediction in the number of interfaces. The magnitude of the dimensionless flow velocity, independent of the wall buoyancy flux and physical scale of the box, decreases significantly with the number of layers. Three types of layer, each with a distinct induced flow pattern, are distinguished and their implications for room airflow considered. Notably, the flow in the base layer represents a continual and smooth flushing of air between the inlet opening and the wall plume, whereas an intermediate layer is almost entirely comprised of near-stagnant air.
The structure of a turbulent line fountain
- Gary R. Hunt, Antoine L. R. Debugne, Francesco Ciriello
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- Journal:
- Journal of Fluid Mechanics / Volume 876 / 10 October 2019
- Published online by Cambridge University Press:
- 06 August 2019, pp. 680-714
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Line fountains form when heavy miscible fluid is ejected steadily upwards as a jet from a high-aspect-ratio rectangular slot, of length $L$ and half-width $b_{0}$, into lighter quiescent surroundings. Viewed along the slot from one end, previous observations reveal that the ejected fluid mixes with the environment and reaches a peak height before partially collapsing back downward under gravity to form a fountain whose top thereafter fluctuates vertically about a mean height. While the motion as perceived from this single view has provided insights that have successfully guided theoretical predictions for the initial rise height, until now a wider understanding of line fountains, and corresponding predictive capability, has been limited to this single prediction due to a lack of any other observational data. Indeed, the general behaviour of line fountains, including the structure internally and along the spanwise length $L$ of the slot, has not been reported previously. To address this, flow visualisations and comprehensive measurements of saline fountains in an aqueous environment are presented here that reveal their complex overall structure and behaviours. After establishing the uniformity of the source conditions from slots of aspect ratio $600:1$ and $300:1$, we first show that double-averaged (spanwise and time) rise heights $\overline{\overline{z}}_{v}/b_{0}$ scale on $Fr_{0}^{4/3}$, $Fr_{0}$ being the source Froude number, with vertical fluctuations being circa 20 % of these heights. Then, simultaneously interrogating the flow as viewed from above and from the side onto the spanwise dimension, we identify three distinct patterns of behaviour. Instrumental to distinguishing these behaviours were the contrasting signatures we observed in the time series of rise height departures from the mean which led us to the following classification: (i) non-uniform flapping for $0.05\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.30$, in which the lateral motion of the fountain takes the form of an oscillatory wave with a wavelength of $2L/3$ (approx.); (ii) uniform flapping for $0.30\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.45$, in which the entire fountain sways to the left and then to the right side of the slot; and (iii) disorganised flapping for $\overline{\overline{z}}_{v}/L\gtrsim 0.45$. Regarding the internal structure, we show that unlike a classic round fountain, eddying structures comparable in scale with the rise height form towards the top of the fountain, and the counterflow forms predominantly to one side of the jet. We then identify the single dominant mechanism driving the flapping motions, successfully linking the wave-like behaviour observed along the span to the internal structure and vertical oscillations. Quantifying the oscillatory motions, both the vertical and flapping frequencies scale as $Fr_{0}^{-2}$, and we demonstrate and explain a robust coupling between these frequencies that follows a ratio of 2:1.
The influence of spanwise confinement on round fountains
- Antoine L. R. Debugne, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 845 / 25 June 2018
- Published online by Cambridge University Press:
- 26 April 2018, pp. 263-292
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We study experimentally the effects of spanwise confinement on turbulent miscible fountains issuing from a round source of radius $r_{0}$. A dense saline solution is ejected vertically upwards into a fresh-water environment between two parallel plates, separated by a gap of width $W$, which provide restraint in the spanwise direction. The resulting fountain, if sufficiently forced, rapidly attaches to the side plates as it rises and is therefore ‘confined’. We report on experiments for five confinement ratios $W/r_{0}$, spanning from strongly confined ($W/r_{0}\rightarrow 2$) to weakly confined ($W/r_{0}\approx 24$), and for source Froude numbers $Fr_{0}$ ranging between $0.5\leqslant Fr_{0}\leqslant 96$. Four distinct flow regimes are observed across which the relative importance of confinement, as manifested by the formation and growth of quasi-two-dimensional structures, varies. The onset of each regime is established as a function of both $W/r_{0}$ and $Fr_{0}$. From our analysis of the time-averaged rise heights, we introduce a ‘confined’ Froude number $Fr_{c}\equiv Fr_{0}(W/r_{0})^{-5/4}$, which encompasses the effects of confinement and acts as the governing parameter for confined fountains. First-order statistics extracted from the flow visualisation, such as the time-averaged rise height and lateral excursions, lend further insight into the flow and support the proposed classification into regimes. For highly confined fountains, the flow becomes quasi-two-dimensional and, akin to quasi-two-dimensional jets and plumes, flaps (or meanders). The characteristic frequency of this flapping motion, identified through an ‘eddy counting’ approach, is non-dimensionalised to a Strouhal number of $St=0.12{-}0.16$, consistent with frequencies found in quasi-two-dimensional jets and plumes.
Forced fountains
- Gary R. Hunt, Antoine L. R. Debugne
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- Journal:
- Journal of Fluid Mechanics / Volume 802 / 10 September 2016
- Published online by Cambridge University Press:
- 03 August 2016, pp. 437-463
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We present a three-region model for the time-averaged behaviour of established turbulent axisymmetric fountains at high source Froude numbers $(Fr_{0})$ in which we uniquely account for entrainment of ambient fluid both laterally and at the fountain top. High-$Fr_{0}$ ‘forced’ fountains, as originally investigated experimentally by Turner (J. Fluid Mech., vol. 26 (4), 1966, pp. 779–792), are characterised by an upflow, a counterflow and a fountain top where the flow reverses direction. Through the inclusion of the flow-reversal region and by accounting for fountain-top entrainment, which is neglected in all existing models, close agreement is achieved between our solutions and existing experimental data. Moreover, our predictions of the fluxes within the fountain are in accord with scaling arguments deduced in recent studies. Our model reveals five key ratios that characterise the fountain asymptote to constant values in the high-$Fr_{0}$ limit. These are the ratios of the (1) initial and mean rise heights, (2) vertical extents of the fountain top and upflow regions, (3) fluxes of volume entrained into the fountain top and entrained laterally into the counterflow, (4) forces of inertia and buoyancy acting on the counterflow at the level of the source and (5) average times taken for fluid to rise through the upflow and fall through the counterflow. Attributing the invariance of these ratios to the global self-preserving behaviour of the fountain, we propose a threshold source Froude number for which a continuous negatively buoyant release may be regarded as giving rise to a ‘forced’ fountain.
A phenomenological model for fountain-top entrainment
- Antoine L. R. Debugne, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 796 / 10 June 2016
- Published online by Cambridge University Press:
- 28 April 2016, pp. 195-210
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In theoretical treatments of turbulent fountains, the entrainment of ambient fluid into the top of the fountain, hereinafter fountain-top entrainment $Q_{top}$ ($\text{m}^{3}~\text{s}^{-1}$), has been neglected until now. This neglect, which modifies the energetic balance in a fountain, compromises the predictive ability of existing models. Our aim is to quantify $Q_{top}$ by shedding light on the physical processes that are responsible for fountain-top entrainment. First, estimates for $Q_{top}$ are obtained by applying, in turn, an entrainment closure in the vein of Morton et al. (Proc. R. Soc. Lond., vol. 234, 1956, pp. 1–23) and then of Shrinivas & Hunt (J. Fluid Mech., vol. 757, 2014, pp. 573–598) to the time-averaged fountain top. Unravelling the assumptions that underlie these approaches, we argue that neither capture the dynamical behaviour of the flow observed at the fountain top; the top being characterised by quasi-periodic fluctuations, during which large-scale eddies reverse and engulf parcels of ambient fluid into the fountain. Therefore, shifting our mindset to a periodical framework, we develop a new phenomenological model in which we emphasise the role of the fluctuations in entraining external fluid. Our model suggests that $Q_{top}$ is similar in magnitude to the volume flux supplied to the fountain top by the upflow ($Q_{u}$), i.e. $Q_{top}\sim Q_{u}$, in agreement with experimental evidence. We conclude by providing guidance on how to implement fountain-top entrainment in existing models of turbulent fountains.
Confined turbulent entrainment across density interfaces
- Ajay B. Shrinivas, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 779 / 25 September 2015
- Published online by Cambridge University Press:
- 14 August 2015, pp. 116-143
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In pursuit of a universal law for the rate of entrainment across a density interface driven by the impingement of a localised turbulent flow, the role of the confinement, wherein the environment is within the confines of a box, has to date been overlooked. Seeking to unravel the effects of confinement, we develop a phenomenological model describing the quasi-steady rate at which buoyant fluid is turbulently entrained across a density interface separating two uniform layers within the confines of a box. The upper layer is maintained by a turbulent plume, and the localised impingement of a turbulent fountain with the interface drives entrainment of fluid from the upper layer into the lower layer. The plume and fountain rise from sources at the base of the box and are non-interacting. Guided by previous observations, our model characterises the dynamics of fountain–interface interaction and the steady secondary flow in the environment that is induced by the perpetual cycle of vertical excursions of the interface. We reveal that the dimensionless entrainment flux across the interface $E_{i}$ is governed not only by an interfacial Froude number $\mathit{Fr}_{i}$ but also by a ‘confinement’ parameter ${\it\lambda}_{i}$, which characterises the length scale of interfacial turbulence relative to the depth of the upper layer. By deducing the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’, we shed new light on the effects of confinement on interfacial entrainment. We establish that for small ${\it\lambda}_{i}$, a weak secondary flow has little influence on $E_{i}$, which follows a quadratic power law $E_{i}\propto \mathit{Fr}_{i}^{2}$. For large ${\it\lambda}_{i}$, a strong secondary flow significantly influences $E_{i}$, which then follows a cubic power law $E_{i}\propto \mathit{Fr}_{i}^{3}$. Drawing on these results, and showing that for previous experimental studies ${\it\lambda}_{i}$ exhibits wide variation, we highlight underlying physical reasons for the significant scatter in the existing measurements of the rate of interfacial entrainment. Finally, we explore the implications of our results for guiding appropriate choices of box geometry for experimentally and numerically examining interfacial entrainment.
Contributors
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- By Francesco Acerbi, Ayca Akgoz, Matthew R. Amans, Ramsey Ashour, Mohammed Ali Aziz-Sultan, H. Hunt Batjer, Donnie Bell, Bernard R. Bendok, Giovanni Broggi, Morgan Broggi, Charles A. Bruno, Steven D. Chang, In Sup Choi, Omar Choudhri, Douglas J. Cook, William P. Dillon, Peter Dirks, Rose Du, Travis M. Dumont, Tarek Y. El Ahmadieh, Najib E. El Tecle, Mohamed Samy Elhammady, Paolo Ferroli, Alana M. Flexman, John C. Flickinger, Kai U. Frerichs, Sasikhan Geibprasert, Adrian W. Gelb, Y. Pierre Gobin, Bradley A. Gross, Seunggu J. Han, Tomoki Hashimoto, Juha Hernesniemi, Roberto C. Heros, Steven W. Hetts, Randall T. Higashida, Joshua A. Hirsch, Nikolai J. Hopf, L. Nelson Hopkins, Maziyar A. Kalani, M. Yashar S. Kalani, Hideyuki Kano, Syed Aftab Karim, Robert M. Koffie, Douglas S. Kondziolka, Timo Krings, Aki Laakso, Giuseppe Lanzino, Michael T. Lawton, Elad I. Levy, L. Dade Lunsford, Adel M. Malek, Michael P. Marks, George A. C. Mendes, Philip M. Meyers, Jacques Morcos, Nitin Mukerji, Christian Musahl, Ludmila Pawlikowska, Matthew B. Potts, Ross Puffer, James D. Rabinov, Jonathan J. Russin, Mina G. Safain, Duke Samson, Marco Schiariti, R. Michael Scott, Jason P. Sheehan, Paul Singh, Edward R. Smith, Scott G. Soltys, Robert F. Spetzler, Gary K. Steinberg, Philip E. Stieg, Hua Su, Karel terBrugge, Kiron Thomas, Tarik Tihan, Babu Welch, Jonathan White, H. Richard Winn, Chun-Po Yen, Jacky T. Yeung, Byron Yip, Samer G. Zammar
- Edited by Robert F. Spetzler, Douglas S. Kondziolka, Randall T. Higashida, University of California, San Francisco, M. Yashar S. Kalani
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- Book:
- Comprehensive Management of Arteriovenous Malformations of the Brain and Spine
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- 05 January 2015
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- 08 January 2015, pp x-xiv
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Unconfined turbulent entrainment across density interfaces
- Ajay B. Shrinivas, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 757 / 25 October 2014
- Published online by Cambridge University Press:
- 23 September 2014, pp. 573-598
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We present theoretical models describing the quasi-steady downward transport of buoyant fluid across a gravitationally stable density interface separating two unbounded quiescent fluid masses. The primary transport mechanism is turbulent entrainment resulting from the localised impingement of a vertically forced high-Reynolds-number axisymmetric jet with steady source conditions. The entrainment across the interface is examined in the large-time asymptotic state, wherein the interfacial gravity current, formed by the fluid entrained from the upper layer and the jet, becomes infinitesimally thin and a two-layer stratification persists. Characterising flows with small interfacial Froude numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}({{\mathrm{Fr}}}_i)$ as an axisymmetric semi-ellipsoidal impingement dome, we combine conservation equations with a mechanistic model of entrainment and reveal that, in this regime, the dimensionless entrainment flux $E_i$ across the interface follows the power law $E_i = 0.24{{\mathrm{Fr}}}_i^2$. For large-${{\mathrm{Fr}}}_i$ impingements, modelled as a fully penetrating turbulent fountain, we show that $E_i$ no longer scales with ${{\mathrm{Fr}}}_i^2$, but linearly on ${{\mathrm{Fr}}}_i$, following $E_i = 0.42{{\mathrm{Fr}}}_i$. We establish the intermediate range of ${{\mathrm{Fr}}}_i$ over which there is a transition between these quadratic and linear power laws, thus enabling us to classify the dynamics of entrainment across the interface into three distinct regimes. Finally, the close agreement of our solutions with existing experimental results is illustrated.
Transient ventilation dynamics induced by heat sources of unequal strength
- Ajay B. Shrinivas, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 738 / 10 January 2014
- Published online by Cambridge University Press:
- 02 December 2013, pp. 34-64
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We examine theoretically the transient displacement flow and density stratification that develops within a ventilated box after two localized floor-level heat sources of unequal strengths are activated. The heat input is represented by two non-interacting turbulent axisymmetric plumes of constant buoyancy fluxes ${B}_{1} $ and ${B}_{2} \gt {B}_{1} $. The box connects to an unbounded quiescent external environment of uniform density via openings at the top and base. A theoretical model is developed to predict the time evolution of the dimensionless depths ${\lambda }_{j} $ and mean buoyancies ${\delta }_{j} $ of the ‘intermediate’ $(j= 1)$ and ‘top’ $(j= 2)$ layers leading to steady state. The flow behaviour is classified in terms of a stratification parameter , a dimensionless measure of the relative forcing strengths of the two buoyant layers that drive the flow. We find that $\mathrm{d} {\delta }_{1} / \mathrm{d} \tau \propto 1/ {\lambda }_{1} $ and $\mathrm{d} {\delta }_{2} / \mathrm{d} \tau \propto 1/ {\lambda }_{2} $, where $\tau $ is a dimensionless time. When $\hspace{0.167em} \hspace{0.167em} \ll \hspace{0.167em} \hspace{0.167em} $1, the intermediate layer is shallow (small ${\lambda }_{1} $), whereas the top layer is relatively deep (large ${\lambda }_{2} $) and, in this limit, ${\delta }_{1} $ and ${\delta }_{2} $ evolve on two characteristically different time scales. This produces a time lag and gives rise to a ‘thermal overshoot’, during which ${\delta }_{1} $ exceeds its steady value and attains a maximum during the transients; a flow feature we refer to, in the context of a ventilated room, as ‘localized overheating’. For a given source strength ratio $\psi = {B}_{1} / {B}_{2} $, we show that thermal overshoots are realized for dimensionless opening areas $A\lt {A}_{oh} $ and are strongly dependent on the time history of the flow. We establish the region of $\{ A, \psi \} $ space where rapid development of ${\delta }_{1} $ results in ${\delta }_{1} \gt {\delta }_{2} $, giving rise to a bulk overturning of the buoyant layers. Finally, some implications of these results, specifically to the ventilation of a room, are discussed.
Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. 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Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- Book:
- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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The ventilated filling box containing a vertically distributed source of buoyancy
- PAUL COOPER, GARY R. HUNT
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- Journal of Fluid Mechanics / Volume 646 / 10 March 2010
- Published online by Cambridge University Press:
- 10 February 2010, pp. 39-58
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This paper describes the fluid mechanics within a box containing a vertical plane distributed source of buoyancy. A theoretical analysis is presented that models the development of plumes from such sources in an unconfined ambient of uniform density. Two extensions are considered. The first concerns a sealed box and the second involves the more general situation where the box is ventilated by openings at top and bottom. In the sealed box the stratification develops in much the same way as for a ‘filling box’ containing a single-point source of buoyancy on the floor. An initial front descends from the ceiling of the box and an asymptotic stratification eventually develops which is continuous in the vertical direction. In the case of the ventilated box it is found that a complex stratification develops where one or more horizontal intrusions are formed by detachment of the plume/boundary layer from the vertically distributed source where the buoyancy of the plume is less than, or equal to, that of the stratified ambient at a given height. Experimental results are presented to demonstrate the validity of the theory. The findings are relevant to both forced and naturally ventilated buildings containing non-adiabatic vertical surfaces.
Analytical solutions for turbulent non-Boussinesq plumes
- PIERRE CARLOTTI, GARY R. HUNT
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- Journal of Fluid Mechanics / Volume 538 / 10 September 2005
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- 17 August 2005, pp. 343-359
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Analytical solutions are developed for non-Boussinesq turbulent plumes rising from horizontal area sources in unconfined quiescent environments of uniform density. The approach adopted follows and extends an earlier approach for Boussinesq plumes and replaces the non-Boussinesq area source of interest and located at $z\,{=}\,0$ with an idealized point source located at a virtual origin $z\,{=}\,z_v$ such that the flow above the idealized source approximates that from the actual source. Asymptotic analytical expressions are developed for the location of the virtual source that are valid for large vertical distances above the non-Boussinesq source. The non-Boussinesq source is characterized by a non-dimensional parameter $\Gamma_{\hbox{\scriptsize{\it nb}}}$ which is a measure of the relative strengths of the mass, momentum and density deficit fluxes at, or at a specified height above, the source. The vertical distance between the actual and virtual sources scales on the length scale $\ell$ that characterizes the height over which the flow is non-Boussinesq and expressions for $z_v/\ell$ are developed for lazy ($\Gamma_{\hbox{\scriptsize{\it nb}}}\,{>}\,1$) and forced plume ($\Gamma_{\hbox{\scriptsize{\it nb}}}\,{<}\,1$) sources. For pure-plume source conditions $\Gamma_{\hbox{\scriptsize{\it nb}}}\,{=}\,1$, and the virtual source provides an exact representation of the actual plume above $z\,{=}\,0$. The limiting cases of a nearly pure lazy plume and of a highly lazy plume are also explored analytically. For fire plumes, $\Gamma_{\hbox{\scriptsize{\it nb}}}$ is determined from the balance of fluxes immediately above the combustion region and a procedure for estimating these fluxes is given. Solutions expressing the dependence of the mass flux with height are also developed for the near-field flow regions and thereafter an approximation for the mass and momentum fluxes valid for all heights and for source conditions yielding $0\,{<}\,\Gamma_{\hbox{\scriptsize{\it nb}}}\,{<}\,\infty$ is deduced. Applications of the model may include plumes above fires and forced releases of highly buoyant gas into the atmosphere, for example, following the rupturing of a pressurized container vessel.