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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.
Shear-flow dispersion in turbulent jets
- John Craske, Antoine L. R. Debugne, Maarten van Reeuwijk
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- Journal:
- Journal of Fluid Mechanics / Volume 781 / 25 October 2015
- Published online by Cambridge University Press:
- 16 September 2015, pp. 28-51
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We investigate the transport of a passive scalar in a fully developed turbulent axisymmetric jet at a Reynolds number of $\mathit{Re}=4815$ using data from direct numerical simulation. In particular, we simulate the response of the concentration field to an instantaneous variation of the scalar flux at the source. To analyse the time evolution of this statistically unsteady process we take an ensemble average over 16 independent simulations. We find that the evolution of $C_{m}(z,t)$, the radial integral of the ensemble-averaged concentration, is a self-similar process, with the front position and spread both scaling as $\sqrt{t}$. The longitudinal mixing of $C_{m}$ is shown to be primarily caused by shear-flow dispersion. Using the approach developed by Craske & van Reeuwijk (J. Fluid Mech., vol. 763, 2014, pp. 538–566), the classical theory for shear-flow dispersion is applied to turbulent jets to obtain a closure that couples the integral scalar flux to the integral concentration $C_{m}$. Model predictions using the dispersion closure are in good agreement with the simulation data. Application of the dispersion closure to a two-dimensional jet results in an integral transport equation that is fully consistent with that of Landel et al. (J. Fluid Mech., vol. 711, 2012, pp. 212–258).