Source Froude Number Research Articles

The Influence of Source Froude Number and Turbulent Fluctuations on the Development of Turbulent Fountains in Stratified Ambient

The Influence of Source Froude Number and Turbulent Fluctuations on the Development of Turbulent Fountains in Stratified Ambient

Entrainment and dilution in a fountain top

Experiments are carried out on a turbulent fountain to investigate the entrainment of ambient fluid and the dilution of scalar concentration in the fountain top (also referred to as the ‘cap’). The source Froude number ( $Fr_o$ ), defined based on the density difference between the source and the ambient fluids, is varied between 10 and 30, while the Reynolds number ( $Re_o$ ) is set to a minimum of 3000 to ensure a fully turbulent flow at the source. High-fidelity velocity and concentration measurements are obtained using particle image velocimetry and planar laser induced fluorescence techniques, respectively. The mean concentration field indicates that the cap is approximately hemispherical and its base is characterised by the local Froude number $Fr_z \approx 1.5$ . It is observed that the ratio of the entrained ( $Q_{top}$ ) volume flux in the fountain top and the volume flux supplied ( $Q_{in}$ ) at the base of the cap varies between 1.5 and 3.5 at different $Fr_o$ , exceeding the values of $Q_{top}/Q_{in} (= 0.5\text {--}0.8)$ for a fountain developing across a density interface (Lin & Linden, J. Fluid Mech., vol. 542, 2005, pp. 25–52). The present experimental results for $Q_{top}/Q_{in}$ are in excellent agreement with published numerical simulations of turbulent fountains at similar $Fr_o$ . Lastly, a robust metric to estimate the dilution of scalar in the fountain top has been proposed. The results clearly indicate that dilution is not equal to the entrainment ratio; however, the self-similarity of non-dimensional dilution profiles at different $Fr_o$ suggests that the phenomenological model for entrainment in the fountain top put forth by Debugne & Hunt (J. Fluid Mech., vol. 796, 2016, pp. 195–210), can also be effectively used to model the dilution of scalar concentration in the cap. Further, it is found that the variation of the dilution ratio is closely linked to the local Reynolds number ( $Re_z$ ) at the base of the cap. This is verified using an analytical model that describes the dependency of $Re_z$ on the local and source parameters – $Fr_z$ , $Re_o$ and $Fr_o$ .

Characterising entrainment in fountains and negatively buoyant jets

Turbulent fountain flow consists of two distinct stages, the initial ‘negatively buoyant jet’ (NBJ) stage, and the fully developed ‘fountain’ stage. The present study investigates both stages of the flow using particle image velocimetry and planar laser-induced fluorescence, over a range of source Froude numbers, $10\lesssim Fr_o\lesssim 30$ , and Reynolds numbers, $5500\lesssim Re_o\lesssim 7700$ . While the velocity and buoyancy profiles in NBJs take similar Gaussian shapes over a wide range of axial locations, this was not observed in fountains. The changing profile shape is most evident in the outer flow (OF) region, while there is a degree of similarity in the inner flow (IF). Entrainment between IF and OF is shown to depend on the local Richardson number, $Ri$ . The fountains are found to have a negative entrainment coefficient, $\alpha <0$ , for the majority of their height, implying a net radial outflow of fluid from the IF to the OF. An alternative description of entrainment is considered, the ‘decomposed top-hat’ model, where the radial flow is separated into inflow and outflow components that are then estimated using the present experimental data. The inflow component was found to be proportional to the axial IF velocity, which is similar to the classical description of entrainment in pure jets/plumes, while the outflow depends on the local $Ri$ . Entrainment in NBJs may also be described by this framework, which, despite not having an OF, is still subject to an $Ri$ -dependent radial outflow.

Entrainment and structure of negatively buoyant jets

Abstract

The structure of a turbulent line fountain

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.

Temperature profile and flame extension length of a ceiling impinging round jet fire in an inclined tunnel

Ceiling jet fire may popularly occur due to gas fuel leakage caused by corrosion of the pipeline in underground galleries or other accidental factors, and usually inflict major heat hazards and related risks. So the characteristics of a ceiling impinging flow induced by a jet fire in an inclined tunnel are valuable to be investigated. Therefore, the temperature profile and flame extension length along an inclined tunnel ceiling in upward and downward directions under different initial fuel momentums and tunnel inclination angles are obtained and analysed. The results show that the flame extension length in both directions along the ceiling can be correlated with flame free height, inclination angles and height between the fire source and ceiling accounting for the non-symmetrical distribution of un-burnt fuel after impingement. Furthermore, the temperature profile of the ceiling jet is obviously affected by the source initial momentum comparing to the flames with negligible initial momentum. New correlations for predicting the temperature decay profiles are proposed considering the source Froude number and inclination angles. The results of this paper would be valuable for modelling of a ceiling jet fire in inclined tunnel or providing significant guidance/raw data for protection and risk assessment of such fire scenarios.

The influence of spanwise confinement on round fountains

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.

Generalization of the radiative fraction correlation for hydrogen and hydrocarbon jet fires in subsonic and chocked flow regimes

The radiative fraction is one key parameter to characterize the jet flame combustion dynamics and to calculate the thermal radiant heat emitted from jet fire. A theoretical analysis is conducted to clarify the key parameters that dominate the radiative fraction of jet fires, with discussion of the limitation of previous radiative fraction correlations. A completely new dimensionless group, consisting of the mass fraction of fuel at stoichiometric conditions, the density ratio of fuel gas to ambient air and the flame Froude number, is proposed to correlate the radiative fraction of jet fires. The current up-to-date experimental data are used to build the radiative fraction correlation that covers orifice exit diameters from one to hundreds of millimeter, hydrogen, methane and propane fuels, vertical and horizontal jets, buoyance- and momentum-controlled releases, subsonic, sonic and supersonic jets. It is found that the source Froude number can fit the radiative fraction of a particular fuel jet fire. However, the new dimensionless group can correlate the radiative fractions of fuel-different jet fires. The predictive capability of the new correlation exceeds that of previously published work based on the source Froude number only or the global residence time with/without correction factors.

Propagation and entrainment of buoyancy-driven flows in a narrow horizontal space and implications for buoyant contaminant transport under natural ventilation

Contaminant transport induced by buoyancy-driven flows is crucial for proper ventilation design in buildings. Compared to conventional single-zone spaces, considerably less research has been conducted on the dynamics of buoyant contaminated flows in narrow horizontal spaces with large length-to-width ratios. Experiments were performed to explore the characteristics of natural convection produced by a brine source in a narrow horizontal tank immersed in fresh water. The source was located at the center of the tank ceiling; hence, the buoyancy-driven flow consisted of a plume region and a symmetrical horizontal current. A light attenuation technique was used to measure the density distribution and vertical profile of the horizontal velocity of the current. The current density was vertically stratified, but declined very slowly in the horizontal direction. The head of the horizontal current advanced toward the two ends of the tank at a constant velocity that depended only on the source buoyancy flux. The average thickness of the horizontal current was correlated to the tank height and the source Froude number. In our experiments, the interface between the tail of the horizontal current and the fresh water was very stable, and nearly no mass exchange through the interface was observed. However, mixing between the fresh water and brine occurred in both the plume region and the rearward side of the horizontal current head. Both the total entrainment and the proportion of the entrainment occurring in the head were determined. The latter increased with the source Froude number.

Turbulent bubble fountains

We explore the dynamics of turbulent bubble fountains produced when a descending stream of fresh water and air bubbles issues from a nozzle submerged in a tank of water. The bubbles have diameters of 2 to 5 mm and the fountains have source Froude numbers ranging from 10 to 240. The Reynolds numbers of the bubbly fountains range from 4000 to 24 000. The bubbles, carried into the tank by the downward jet of water, lead to a buoyancy force which reduces the downward momentum of the jet, thus producing a fountain. We find that $H_{F}$, the downward penetration distance of the bubbles into the water reservoir, may be characterised by two parameters: $\unicode[STIX]{x1D6EC}$, the ratio of the bubble rise speed to the characteristic fountain velocity, $u_{F}=f_{0}^{1/2}/m_{0}^{1/4}$, and $Fr_{0}$, the source Froude number, given by $m_{0}^{5/4}/(q_{W_{0}}f_{0}^{1/2})$, where $q_{W_{0}}$, $m_{0}$ and $f_{0}$ are the source volume, momentum and buoyancy fluxes. As $\unicode[STIX]{x1D6EC}$ increases, $H_{F}$ decreases, a result which is directly analogous to the height of rise of particles in a particle-laden fountain (Mingotti & Woods, J. Fluid Mech., vol. 793, 2016, R1). Also, we find that $H_{F}$ increases as $Fr_{0}$ increases, a result directly analogous to single-phase fountains (Turner, J. Fluid Mech., vol. 26, 1966, pp. 779–792). We present a model for the conservation of volume, momentum and buoyancy fluxes and use this to predict the penetration distance of the bubbles corresponding to that point at which the fountain liquid velocity equals the bubble rise speed. Using the best-fit value for the entrainment coefficient, $\unicode[STIX]{x1D6FC}=0.04\pm 0.004$, we find that our experimental measurements of the bubble penetration distance are in good accord with the model predictions for $10<Fr_{0}<240$ and $2<\unicode[STIX]{x1D6EC}<15$. In our experiments the bubble rise speed, $u_{slip}$, is large compared to the entrainment velocity of the descending fountain. Thus, only a small fraction of the rising bubbles are re-entrained, and so the buoyancy flux of the fountain is approximately independent of depth. Flow-visualisation experiments also show that the liquid momentum flux is not exhausted at the point of bubble separation and so the liquid in the fountain continues to travel downward, separated from the bubbles. We use the new theoretical model to estimate the flux of air entrained into plunging water jets.

Effects of a Full Scale Aircraft Immersed within a Large Aviation-Fuel Fire in Crosswind on the Flame Shape - A Numerical Study

Large Eddy Simulation combined with an Eddy Dissipation Concept (EDC) combustion model shows the feasibility for simulations of a full scale aircraft immersed within a large aviation-fuel fire in a moving fluid medium. Coupling between the condensed fuel and gas phases allows investigating the roles of the wind speed on the burning rate of condensed fuel, and consequently, the heat release rate. The predicted combustion efficiency of a large scale liquid pool fire remains approximately constant with a value of 85%. The predicted burning rate of a liquid pool fire increases with wind speed up to critical value of about 3-4 m/s, then decrease strongly with a further increase in the wind speed. It appears that the primary flame zone is not significant affected by the presence of an aircraft for the wind speed below 2 m/s. The prediction indicates that interaction between the aircraft and fire environment combined with the influence of medium wind speed of 5 m/s affects dramatically the flame shape. Presence of an aircraft leads to a factor of about 1.8 increases in the elongation of turbulent boundary layer diffusion flame for the wind speed beyond 5 m/s. globally, the flame height in large-scale pool fire is mainly associated with the fire dynamics at low source Froude number and the flame length with the wind speed.

Mixing in continuous gravity currents

We present measurements of the entrainment of ambient fluid into high-Reynolds-number gravity currents produced by a steady flux of buoyancy. The currents propagate along a horizontal channel and the mixing is measured using a light attenuation technique to obtain the cross-channel average of the density throughout the current. The total volume of the current increases linearly with time, at a rate in the range $(1.8{-}2.1)Q_{o}$ for source Froude numbers, $Fr_{o}$, in the range $0.1{-}3.7$, where $Q_{o}$ is the source volume flux per unit width. Most mixing occurs either immediately downstream of the inflow or near the head of the flow, with an increasing proportion of the entrainment occurring in a mixing zone near the inflow as $Fr_{o}$ increases. A vertical gradient in the density and horizontal velocity develops in this mixing zone. This enables relatively dense fluid at the base of the current to catch up with the head, where it rises and mixes with the ambient fluid which is displaced over the head. The mixed fluid continues forward more slowly than the head, forming the relatively dilute fluid in the upper part of the current. Our data show that the depth and the depth-averaged buoyancy are primarily dependent on the position relative to the front, with the speed of the front being $\unicode[STIX]{x1D706}(Fr_{o})B_{o}^{1/3}$, where $B_{o}$ is the source buoyancy flux per unit width. Here, $\unicode[STIX]{x1D706}(Fr_{o})$ increases from 0.9 to 1.1 as $Fr_{o}$ increases from 0.1 to 3.7, while the Froude number at the head of the flow has a value of $1.1\pm 0.05$.

Pool fire flame base drag behavior with cross flow in a sub-atmospheric pressure

This paper investigates the effect of cross flow of air on flame base drag in pool fires in sub-atmospheric pressure, where no data is available, by combining the effects of these two factors: sub-atmospheric pressure and cross flow. Experiments were carried out using heptane and acetone as fuels for 10–25cm square pool fires where the cross flow was provided by a wind tunnel built in Lhasa city-Tibet (64kPa) as well as correspondingly by a wind tunnel in Hefei city (100kPa). Additional larger scale experiments were also conducted with 45cm, 60cm and 70cm square pool fires in Hefei city. The evolution of flame base drag length as a function of cross flow air speed was quantified. The flame base drag length was found to be larger in the sub-atmospheric pressure than that in normal pressure, indicating higher risk of causing fire spread to nearby fuel at the ground level. Previous correlations based on a wind Froude number are not in agreement with each other. In addition, they do not include all the physics and they do not account for the effect of the pressure change on flame base drag length. A new theoretical approach is proposed accounting for the relation of wind force with the buoyancy of the flame and the interaction of wind with the total heat release rate from the fire. It follows that three independent dimensionless quantities are introduced: the (inverse) fire source Froude number owing to flame buoyancy, the total buoyancy–wind interaction number and the ratio of fuel vapor density near the fuel surface to the air density. This theory allows us to develop and validate a correlation for flame base drag length to account for pressure effect and for generalizing the existing relations for flame base drag.

Forced fountains

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.

Tilt angle of turbulent jet diffusion flame in crossflow and a global correlation with momentum flux ratio

The flame tilt angle is an important phenomenological parameter of the jet fire in crossflow. Previous work focused on buoyant-controlled flames with small jet-to-crossflow momentum flux ratio (RM) or momentum-controlled flames with large RM. The present study investigated coupling influences of buoyancy, shear force, and inertia force on tilt angles of turbulent flames with continuous variation of RM. A global flame tilt angle formula employing buoyancy and momentum was derived and simplified. Experiments of turbulent jet propane flames with medium RM of 0.1–10 were conducted in a wind tunnel. Based on the results, at lower RM, flames presented a three-zone structure: the flame attached to the leeward side of the nozzle, the long symmetric tail of the flame, and the junction that connects these two zones. With increasing RM, the zone attached to the leeward side of the nozzle became small and eventually disappeared. Three dominated regimes are identified combining RM and source Froude numbers (Fr): crossflow-dominated (RM < 0.01, Fr < 0.1), transitional (0.01 < RM < 10, 0.1 < Fr < 103), and low Mach number jet-dominated regime (RM > 10, 103 < Fr < 105). In a specific dominating mode, the theoretical tilt angle decreases with increasing RM. When the domination mode changes from crossflow to transition and jet flow, the mode bandwidth of the experiments becomes narrower in the log coordinate of RM. Correspondingly, the flame tilt angle (θ in degrees) generally decreases as θ=74.4RM−0.006, θ=61.2RM−0.050, and θ=73.2RM−0.115. For the crossflow-dominated regime, most of the jet fluid is entrained into the down-wash area, causing a large tilt angle. For the transitional regime, the vortex in the leeward side of the nozzle moves upward and becomes small, causing a medium angle. For the jet-dominated regime, the flow near the leeward sides of both the nozzle and the jet points upward, causing a small angle and accompanying the backward-rolling jet-type vortices with coherent structure.

Sustained gravity currents in a channel

Gravitationally driven motion arising from a sustained constant source of dense fluid in a horizontal channel is investigated theoretically using shallow-layer models and direct numerical simulations of the Navier–Stokes equations, coupled to an advection–diffusion model of the density field. The influxed dense fluid forms a flowing layer underneath the less dense fluid, which initially filled the channel, and in this study its speed of propagation is calculated; the outflux is at the end of the channel. The motion, under the assumption of hydrostatic balance, is modelled using a two-layer shallow-water model to account for the flow of both the dense and the overlying less dense fluids. When the relative density difference between the fluids is small (the Boussinesq regime), the governing shallow-layer equations are solved using analytical techniques. It is demonstrated that a variety of flow-field patterns are feasible, including those with constant height along the length of the current and those where the height varies continuously and discontinuously. The type of solution realised in any scenario is determined by the magnitude of the dimensionless flux issuing from the source and the source Froude number. Two important phenomena may occur: the flow may be choked, whereby the excess velocity due to the density difference is bounded and the height of the current may not exceed a determined maximum value, and it is also possible for the dense fluid to completely displace all of the less dense fluid originally in the channel in an expanding region close to the source. The onset and subsequent evolution of these types of motions are also calculated using analytical techniques. The same range of phenomena occurs for non-Boussinesq flows; in this scenario, the solutions of the model are calculated numerically. The results of direct numerical simulations of the Navier–Stokes equations are also reported for unsteady two-dimensional flows in which there is an inflow of dense fluid at one end of the channel and an outflow at the other end. These simulations reveal the detailed mechanics of the motion and the bulk properties are compared with the predictions of the shallow-layer model to demonstrate good agreement between the two modelling strategies.

On turbulent particle fountains

We describe new experiments in which particle-laden turbulent fountains with source Froude numbers $20&gt;Fr_{0}&gt;6$ are produced when particle-laden fresh water is injected upwards into a reservoir filled with fresh water. We find that the ratio $U$ of the particle fall speed to the characteristic speed of the fountain determines whether the flow is analogous to a single-phase fountain ($U\ll 1$) or becomes a fully separated flow ($U\geqslant 1$). In the single-phase limit, a fountain with momentum flux $M$ and buoyancy flux $B$ oscillates about the mean height, $h_{m}=(1.56\pm 0.04)M^{3/4}B^{-1/2}$, as fluid periodically cascades from the maximum height, $h_{t}=h_{m}+{\rm\Delta}h$, to the base of the tank. Experimental measurements of the speed $u$ and radius $r$ of the fountain at the mean height $h_{m}$, combined with the conservation of buoyancy, suggest that $Fr(h_{m})=u(g^{\prime }r)^{-1/2}\approx 1$. Using these values, we find that the classical scaling for the frequency of the oscillations, ${\it\omega}\sim BM^{-1}$, is equivalent to the scaling $u(h_{m})/r(h_{m})$ for a fountain supplied at $z=h_{m}$ with $Fr=1$ (Burridge &amp; Hunt, J. Fluid Mech., vol. 728, 2013, pp. 91–119). This suggests that the oscillations are controlled in the upper part of the fountain where $Fr\leqslant 1$, and that they may be understood in terms of a balance between the upward supply of a growing dense particle cloud, at the height where $Fr=1$, and the downward flow of this cloud. In contrast, in the separated flow regime, we find that particles do not reach the height at which $Fr=1$: instead, they are transported to the level at which the upward speed of the fountain fluid equals their fall speed. The particles then continuously sediment while the particle-free fountain fluid continues to rise slowly above the height of particle fallout, carried by its momentum.

Entrainment by turbulent fountains

Experimental measurements of entrainment by turbulent fountains from circular sources in quiescent uniform environments are presented. Our results span almost four orders of magnitude in the source Froude number ($0.004\leqslant \mathit{Fr}_{0}\leqslant 25$) and thereby encompass the entrainment across all classes of fountain behaviour identified to date. We identify scalings for the entrained volume flux $Q_{E}$, in terms of $\mathit{Fr}_{0}$ and the source volume flux $Q_{0}$, within a number of distinct Froude-number bands corresponding to each class of fountain. Additionally we identify a distinct class of new behaviour, as yet unreported, for $\mathit{Fr}_{0}\lesssim 0.1$.

Experimental non-Boussinesq fountains

Laboratory experiments involving downward air–helium fountains are presented. The large density differences between these releases and the ambient allow us to investigate how non-Boussinesq effects modify fountain heights and fountain fluctuations in comparison with the Boussinesq case (i.e. marginal density differences). In these experiments, the source Froude number is varied over a wide range covering (i) the very weak, (ii) the weak and (iii) the forced fountain regimes. It is shown that the classical Boussinesq correlations can be extended to the non-Boussinesq case provided that the Froude number is multiplied by the square root of the ratio between the released fluid density and that of the ambient. In the range investigated, no influence of the source Reynolds number is observed.

The effect of source Reynolds number on the rise height of a fountain

The focus of this paper is to establish the effect of source Reynolds number, Re0, on the mean rise heights of Boussinesq miscible fountains and thereby the range of Re0 for which rise heights are independent of Re0. We present results from experiments on aqueous-saline fountains spanning source Reynolds numbers of 15 ≤ Re0 ≤ 4000 and source Froude numbers of 0.3 ≤ Fr0 ≤ 40. We provide threshold values for the source Reynolds number, i.e., values above which fountain rise height may be regarded as independent of Re0. For Reynolds numbers immediately beneath these threshold values, the fountain rise height increases as Re0 decreases—in agreement with Lin and Armfield [“The Reynolds and Prandtl number dependence of weak fountains,” Comput. Mech. 31, 379–389 (2003)]. At even lower Re0, we show that the rise heights reach a maximum and that further decreases in Re0 decrease the rise height attained—in agreement with Phillipe et al., [“Penetration of a negatively buoyant jet in a miscible liquid,” Phys. Fluids 17, 053601 (2005)]. As such, we resolve an apparent inconsistency within the existing fountain literature.