2.1 Description of the fountain top

Before concentrating on the modelling of entrainment into the cap, it is helpful to first consider the general aspects of the fountain top.

In its instantaneous nature, there is no well-defined region of flow reversal (cf. figure 1 a). Instead, the flow atop a fountain appears to be highly irregular. Observations reveal that the pulsating fountain top undergoes large-amplitude fluctuations, shedding blobs or lobes of fluid. These lobes subsequently descend around the upflow to form the counterflow. A comprehensive study of the fountain top undertaken by Burridge & Hunt (Reference Burridge and Hunt2013) clarifies the picture (see also Pantzlaff & Lueptow Reference Pantzlaff and Lueptow1999). Burridge & Hunt (Reference Burridge and Hunt2013) analysed the time series of rise height produced by the motion of the fountain front. They found that there was a discernible rhythm, or periodicity, to the dominant mode of fluctuations occurring at the fountain top. For forced fountains, the dominant period of fluctuation was found to correspond approximately to the timescale of (large) eddies forming on the scale of the upflow radius at (the level of) the fountain top. Thereafter, these eddies collapse onto the counterflow, completing the fluctuation cycle. Thus, Burridge & Hunt (Reference Burridge and Hunt2013) argue that the fluctuations atop forced fountains can be assimilated as quasi-periodic cycles of formation and collapse of large-scale eddies.

To date, no attempt has been made (to the authors’ knowledge) to implement fluctuations into a model for fountains. By the same token, the potential links between these fluctuations and the entrainment of ambient fluid at the fountain top have seemingly been overlooked. In immiscible fountains, the reversing flow absorbs large pockets of surrounding fluid into the body of the fountain (Friedman et al. Reference Friedman, Vadakoot, Meyer and Carey2007). A similar phenomenon takes place in miscible fountains: as large-scale eddies spill off the fountain top, they wrap up external fluid into billow-like structures. Thereby, ambient fluid is entrained into the body of the fountain. It therefore seems essential that a faithful model for fountain-top entrainment incorporates the effects of fluctuations.

Figure 1. Structure of a turbulent forced fountain. (a) Instantaneous snapshot of an aqueous-saline fountain (source diameter 0.6 cm) with steady-state rise height and vertical extent of a characteristic large-scale fluctuation marked (cf. measurements by Burridge & Hunt (Reference Burridge and Hunt2013)). (b) Time-averaged image of the same fountain. The modelled, time-averaged fountain-top morphology is depicted in (c). The dotted lines in (b) and (c) outline the approximate boundaries of upflow, fountain top and counterflow.

Next, the question of the shape of the fountain top must be addressed. To the experimentalist who has observed these flows in controlled laboratory tests, this question is relatively straightforward to answer. This is mirrored by the fact that across the literature, schematics of fountains are depicted similarly: namely with an approximately hemispherical structure at their top (e.g. Turner Reference Turner1966; Mizushina et al. Reference Mizushina, Ogino, Takeuchi and Ikawa1982; Cresswell & Szczepura Reference Cresswell and Szczepura1993; Burridge & Hunt Reference Burridge and Hunt2012). The broadly hemispherical morphology is also apparent from figure 1(b), which was obtained by averaging over time instantaneous snapshots, such as that shown in figure 1(a). Thus, in line with a long history of observations, we adopt a time-averaged description of the cap as a hemisphere of height $z_{t}$ , as illustrated in figure 1(c). At its base, the cap is fed by an incoming volume flux of dense fluid from the fountain upflow, $Q_{u}\propto {\rm\pi}b_{u}^{2}w_{u}$ , characterised by a reduced gravity $g_{u}^{\prime }=g({\it\rho}_{u}-{\it\rho}_{a})/{\it\rho}_{a}$ . Here, $b_{u}$ , $w_{u}$ and ${\it\rho}_{u}$ denote the radius, velocity and density of the upflow, respectively, at a height corresponding to the onset of flow reversal, and ${\it\rho}_{a}$ the ambient density (figure 1 c). For convenience, we adopt a top-hat profile for $w_{u}$ , thus $Q_{u}={\rm\pi}b_{u}^{2}w_{u}$ . A local Froude number, indicative of the relative strengths of inertial and buoyant forces in the cap, can be formed as

This balance between the two competing forces determines the dynamics of the (time-averaged) fountain top and needs to be specified. In order to select a sensible value for $Fr_{u}$ , we may reason as follows. We know that a (large) eddy of nominal width $2b_{u}$ will reverse at the fountain top when the gravitational downward pull exceeds the upward thrust of momentum. The height at which the flow starts to reverse can therefore be defined by the ‘tipping’ point $w_{u}=\sqrt{2b_{u}g_{u}^{\prime }}$ , i.e. by $Fr_{u}=\sqrt{2}$ . A similar argument is developed by McDougall (Reference McDougall1981), who initiates his cap at the height at which conditions locally in the upflow are characterised by $Fr_{u}=\sqrt{2}$ .

2.2 Morton–Taylor–Turner entrainment closure

When faced with a flow for which (macroscopic) entrainment is a prominent feature, it is tempting to rely on the closure popularised by Morton et al. (Reference Morton, Taylor and Turner1956) (hereinafter MTT). MTT postulated that the entrainment velocity across the outskirts of a jet or a plume is directly proportional to the time-averaged centreline velocity. The constant of proportionality, the entrainment coefficient ${\it\alpha}$ , varies across applications, but typically ${\it\alpha}\approx 0.1$ . Their ‘entrainment assumption’ has since been successfully applied to many different types of flow; see Turner (Reference Turner1986) for a review. Crucially, our interest lies primarily in describing the physical phenomena responsible for entrainment into the cap. In the following, we apply MTT’s closure to the fountain cap and discuss its appropriateness with regard to the dynamics of a forced fountain.

In order to apply MTT, we seek a characteristic vertical velocity, ${\hat{w}}$ , in the cap. As relatively dense fluid from the upflow rises through the buoyancy-dominated cap, it experiences a downward acceleration $g_{u}^{\prime }$ . Some fluid elements reach the crest of the cap ( $r=0$ , $z=z_{t}$ , figure 1), where the vertical velocity has reduced to zero. Eventually, these fluid elements reverse direction and are accelerated under gravity. Assuming that pressure remains hydrostatic throughout the cap, an estimate of ${\hat{w}}$ is obtained on applying the Bernoulli equation along a streamline from the crest to a point at height $z$ . This yields

The characteristic entrainment (or inflow) velocity is therefore

In order to evaluate the entrained volume flux $Q_{top}$ , we must integrate the inflow velocity (2.3) over the cap. For a hemispherical geometry, the radius of the cap $b_{t}$ is given by $b_{t}^{2}(z)+z^{2}=z_{t}^{2}$ . Thus, integrating leads to an entrainment flux of

where ${\it\delta}=0.163$ (to three decimal places) with ${\it\alpha}=0.1$ , and $Q_{u}(={\rm\pi}b_{u}^{2}w_{u})$ is the volume flux supplied to the cap by the upflow. In (2.4), the factor multiplying $Q_{u}$ is a constant whose numerical value evidently depends on the geometry of the cap and on the characteristic local balance between inertia and buoyancy.

In a similar configuration, Shrinivas & Hunt (Reference Shrinivas and Hunt2014) (see § 2.3) obtained an expression for $z_{t}$ (namely $z_{t}=0.94b_{u}Fr_{u}^{2}$ ) based on the assumption that there was an efficient conversion of kinetic to potential energy in the cap. Although their study pertains to localised interfacial mixing induced by an impinging jet, later we motivate why it is reasonable to apply directly some of their results to a fountain cap, rather than solely to an impingement dome for which their model was developed. Therefore, for $Fr_{u}=\sqrt{2}$ , (2.4) evaluates to

Equation (2.5) is promising in that it confirms our intuition that $Q_{top}$ should scale on $Q_{u}$ . That said, one should be mindful of applying MTT to the fountain top. In plumes, a typical entrainment event consists of peripheral eddies engulfing ambient fluid into the plume (Turner Reference Turner1986). The MTT entrainment assumption gives a time-averaged representation of this engulfment process. As outlined in § 2.1, we indeed expect ‘engulfment’ to be the primary driver of fountain-top entrainment. There is a difference, however, between ‘engulfment’ in the context of slender flows such as plumes, and in the context of a fountain top. Slender flows have eddies on their perimeter that entrain patches of external fluid into the bulk flow as they propagate. Significantly, these eddies do not occupy the whole cross-section of the flow (e.g. plate 166 in Van Dyke (Reference Van Dyke1982)). By contrast, observations of the dominant fluctuation mode suggest that the instantaneous fountain top consists of single large-scale eddies shed in succession (e.g. Burridge & Hunt (Reference Burridge and Hunt2013)). It is therefore unclear a priori that the same closure should hold in a flow which, like the fountain top, comprises one such notional eddy. A related concern is the requirement within the MTT closure to define a meaningful time-averaged velocity. In the fluctuating fountain top, where the velocity changes sign over the course of a fluctuation, the existence of such a time-averaged velocity is questionable.

2.3 Entrainment by baroclinic vortices

An alternative account of the time-averaged entrainment process in fountain tops may be drawn from the work of Shrinivas & Hunt (Reference Shrinivas and Hunt2014) (hereinafter SH14). In the context of a two-layer stratified system, SH14 present a closure for the entrainment caused by a jet weakly impinging on a density interface. This localised impingement penetrates the interface as a recirculating bulge (or ‘dome’) which, as we shall see, shares notable similarities with a fountain top. In their account, inspired from experiments by Shy (Reference Shy1995) and Cotel et al. (Reference Cotel, Gjestvang, Ramkhelawan and Breidenthal1997), external fluid is entrained into a dome through the action of small baroclinic vortices located on the dome’s periphery (figure 2). Herein, we briefly recount some of the SH14 arguments to motivate why their closure warrants inspection and to illustrate the fundamental differences between their conceptual picture and an MTT-based approach.

Interested in the mixing dynamics of a jet, Shy (Reference Shy1995) performed a series of experiments in which a turbulent jet impinged on an interface across which there was a buoyancy jump ${\rm\Delta}g^{\prime }$ . When the impingement was weak, the interface distorted locally into a dome-like structure. For sufficiently energetic impingements, the jet penetrated the interface into a region of lower density fluid as a fountain. Using laser-induced fluorescence, Shy was able to distinguish fractions of the fountain which had mixed with the surroundings from fractions which had merely been ‘stirred’. In his flow visualisation, a fluorescent layer, indicative of mixing, persisted around the dark bulk silhouette of the fountain. Around the fountain top specifically, this fluorescent layer was thin, hinting that eddies responsible for the entrainment of ambient fluid were mostly confined to a thin region around its periphery. Later, Cotel et al. (Reference Cotel, Gjestvang, Ramkhelawan and Breidenthal1997) confirmed that for moderately energetic impingements (again forming a dome) this narrow layer was comprised of small, sustained baroclinic vortices. Performing several rotations in place before dislodging, these persistent eddies were responsible for drawing ambient fluid into the dome. A diagram of this conceptual picture is shown in figure 2(a).

Figure 2. Diagram of the shear-driven entrainment model. Confined to a thin vortical layer (shaded region), baroclinic vortices, shown in (a), drive a volume flux $Q_{top}$ of external fluid of density ${\it\rho}_{a}$ into the dome. In (b), mean flow streamlines through the vortical layer are illustrated. After Shrinivas & Hunt (Reference Shrinivas and Hunt2014, figure 5).

Significantly, this picture suggests a physical origin for entrainment that is different from MTT: the primary entraining mechanism observed is a near-steady drawing in of external fluid effectuated by the sustained rotation of small vortices, rather than a quasi-periodical engulfment event averaged over time (as in MTT).

There is, however, more than a passing resemblance between the ‘entrainment dome’ formed when a jet-like flow impinges on a density interface and the time-averaged fountain top in uniform surroundings: both flows are dominated by their buoyancy and are approximately hemispherical in shape. Their close connection is reinforced on considering that, if one increases the forcing at the interface (or decreases the buoyancy jump), the entrainment dome naturally transitions into a fountain top. This is apparent in the experiments of Shy (Reference Shy1995) and, to a lesser extent, in those of Lin & Linden (Reference Lin and Linden2005). Thus, in light of these similarities and seeking a time-averaged entrainment closure, it seems natural to inspect the rate of entrainment which baroclinic eddies drive into the fountain top.

In the arguments that follow we distinguish between ${\rm\Delta}g^{\prime }$ , the reduced gravity across a density interface and pertinent to the interfacial dome, and $g_{u}^{\prime }$ , the reduced gravity at the onset of flow reversal in a fountain (in an unstratified environment), and characteristic of the buoyancy in the cap. Similarly, we use ‘dome’ to designate the interfacial dome, depicted in figure 2, and ‘cap’ to refer to the top of a fountain rising through a uniform environment.

If there were not a sustained flux of vorticity into the dome, the baroclinic torque, a torque resulting from the misalignment of pressure and density isolines, would ‘unwrap’ the dense dome into a horizontal surface. The upflow provides the dome with a sustained influx of vorticity that maintains the persistent rotation of eddies, or vortices, within the vortical layer. The steady vorticity equation for a baroclinic vortex revolving at mean radial and vertical velocity, $u_{v}(z)$ and $w_{v}(z)$ , respectively, is given by

where $P(z)$ designates the pressure in the environment, assumed hydrostatic, and ${\it\Omega}_{v}(z)$ the vorticity of a baroclinic vortex. In (2.6), the left-hand side represents the rate of rotation induced by the upflow. This is counteracted by the right-hand side, the baroclinic torque acting on the dome. Vortex stretching is absent because the time-averaged morphology of the dome is unchanging. SH14 established that $u_{v}/w_{v}\ll 1$ , such that (2.6) simplifies to

without significantly affecting the analysis: neglecting the contribution of $u_{v}$ modifies their final estimate of the entrained volume flux by less than 2 % (Shrinivas & Hunt Reference Shrinivas and Hunt2014).

The small baroclinic vortices, driving entrainment at a mean velocity $u_{e}(z)$ , are fuelled by an incident vorticity flux from the upflow over a length scale $b_{u}$ . Thus, their mean circulation scales as ${\it\Gamma}_{v}\propto u_{e}b_{u}$ and their vorticity ${\it\Omega}_{v}$ scales as

The baroclinic vorticity is generated at a mean rate

Substituting for ${\it\Omega}_{v}$ from (2.8) and for the baroclinic torque from (2.9) into the vorticity equation (2.7) yields

It remains to determine $w_{v}$ . To that end, consider the mean flow within the dome, illustrated in figure 2(b). As in § 2.2, the ascent of fluid in the cap is stifled by its negative buoyancy. Due to the combined action of the weak inflow engendered by the entrainment flux and of the internal recirculation, some fluid is steered into the vortical layer and begins rotating at a velocity $\{u_{v},w_{v}\}$ ; cf. streamline $R$ on figure 2(b). Therefore, applying the Bernoulli equation along streamline $R$ , we approximate $w_{v}$ as

since the radial velocity $u_{v}$ has negligible magnitude. Using (2.11), equation (2.10) becomes

where ${\it\alpha}$ denotes a constant of proportionality.

Equation (2.12), which states that the mean entrainment velocity is proportional to the mean vertical velocity of a vortex, is formally identical to the entrainment velocity (2.3) found in § 2.2: remarkably, the MTT entrainment closure is recovered. This assertion comes despite the different entrainment mechanism at the start of the argument. Thus, if one were to assume that the SH14 dome is an appropriate representation of the time-averaged fountain top, with reference to (2.5) we retrieve $Q_{top}=0.525Q_{u}$ (taking, for consistency, ${\it\alpha}=0.1$ ).

Here too, caution is advisable. Some aspects of the entraining mechanism proposed by SH14 for the dome appear to be incompatible with the flow atop a fountain. Pertinently, the SH14 closure, or at least route to closure, would rely on the presence of sustained baroclinic vortices surrounding the fountain top. This requirement may seem somewhat contrived, even in a time-averaged sense, in a fountain top that is perpetually fluctuating. As a whole, the entrainment process presumed in SH14 (a steady, regular drawing in of external fluid), though successful in describing entrainment in an interfacial dome, does not comply well with the engulfing events that are observed in forced fountains (e.g. Burridge & Hunt Reference Burridge and Hunt2013).

In light of these concerns, it is advisable to search for different methods to quantify entrainment. Ideally in the context of forced fountains, an entrainment closure should replicate the characteristic large-scale fluctuations at the fountain top. This forms the focus of § 2.4.

2.4 Entrainment by vortical engulfment

So far, we have considered two separate approaches to evaluate fountain-top entrainment: MTT and SH14, each based on very distinct conceptualisations of the bulk flow. A noteworthy preliminary result is that they both predict $Q_{top}=0.525Q_{u}$ .

There remain, however, open questions regarding the pertinence of either approach when applied to the fountain top. These questions centre around how meaningful it is to adopt a time-averaged representation of the fountain top. Indeed we argue that fluctuations play a fundamental role in engulfing external fluid into the fountain body. It then seems difficult to reconcile this fluctuating behaviour with a time-averaged view of the flow.

Seeking a time-averaged entrainment closure has the obvious benefit that entrainment through the cap may be expressed as an entrainment rate, $Q_{top}$ , rather than an entrained volume, $V_{top}$ . However, for the purposes of developing the arguments that follow (and indeed to alleviate some of the previous concerns), it is useful to shift our mindset from a time-averaged to a periodical representation of the cap.

Once we adopt a periodical framework, the fluctuations can readily be implemented into an entrainment model. To achieve this, we represent fluctuations conceptually as periodical cycles of ‘filling’ (first half-cycle) and ‘draining’ (second half-cycle) of the cap, during which the fountain top forms and collapses, respectively. Simultaneously, the cap rises during the first half-cycle and subsequently falls back in the second half-cycle. The two stages are depicted schematically in figures 3 and 4, respectively.

Figure 3. First half-cycle, $0\leqslant t\leqslant T_{p}/2$ : schematic depicting three stages in the formation of a fountain top by a filling process. (a) Initially, the cap is empty. (b) The cap gradually forms due to the inflow $Q_{u}$ from below. In (c), the cap has reached the peak fluctuation height $z_{m}$ . The dashed line depicts the outline of the cap at maximal volume.

Figure 4. Second half-cycle, $T_{p}/2\leqslant t\leqslant T_{p}$ : schematic depicting three stages in the collapse of a fountain top by a draining process. (a) The cap has reached a maximal volume at the peak height of a fluctuation cycle and starts to drain. (b) As the cap collapses, vortical motion entrains ambient fluid into the counterflow by engulfment. The cap is fully depleted in (c). Dashed outline as in figure 3.

We now consider, in turn, how external fluid is absorbed into the cap during formation and collapse. Suppose that, in the first instance, the cap gradually ‘fills up’ via the volume flux supplied by the upflow. The kinetic energy provided by this influx is stored as potential energy in the growing cap; simultaneously, it enables the cap to rise to the maximal quasi-steady fluctuation height $z_{m}=z(T_{p}/2)$ (figure 3 c), where the time $t=T_{p}/2$ denotes the end of the first half-cycle. Based on the assumption that this raising of the cap results from a conversion of kinetic to potential energy, we may write

The rise to a height $z_{m}$ concludes the first half of a single oscillation cycle. At this stage, the amount of fluid entrained (i.e. for $0\leqslant t\leqslant T_{p}/2$ ) by engulfment is expected to be negligible. In practice, we expect some fluid to spill weakly from the fountain top and to roll up into tiny billows at its rear, although their contribution to entrainment is likely to be small. The actual entrainment is probably occurring at a smaller scale, akin to that in SH14. However, as discussed, the applicability of their model to a periodical framework is questionable. Therefore, it seems reasonable to neglect entrainment altogether during the first half-cycle. We will see later that our conclusions are largely unaffected by this assumption.

Figure 5. Illustration of the modelling approach whereby the engulfment is driven by a thin vortex ring of circulation ${\it\Gamma}$ with core at height $z_{c}(t)$ (the italicised subscript ‘c’ reading ‘core’) across a cylindrical stem of radius $b_{u}$ . To avoid ill-posedness, the vortex-induced velocity $U_{v}$ follows the velocity of a Rankine vortex.

In the second half-cycle, the collapsing cap wraps around and thereby engulfs ambient fluid as it falls. To describe this engulfment mechanism, a new model is needed.

Figure 5 illustrates the modelling approach that we have taken. Whilst its application to the fountain top is novel, a similar approach has been used to describe entrainment in the near-field of fires (Cetegen Reference Cetegen1998) and, qualitatively, in jets (Sreenivas & Prasad Reference Sreenivas and Prasad2000). The swirling motion induced by the inflow of ambient fluid and simultaneous collapse of the fountain top is modelled by a thin vortex ring to which we assign a diameter $2b_{u}$ and a constant strength ${\it\Gamma}$ ( $\text{m}^{2}~\text{s}^{-1}$ ). This vortex ring induces a lateral velocity $U_{v}$ which drives a volume flux across an extension of the upflow. For simplicity, this extension is modelled as a cylindrical stem of perimeter $2{\rm\pi}b_{u}$ and of height $z_{c}$ . The position of the core of the vortex ring, $z_{c}(t)$ , varies over the course of one half-cycle from $z=z_{m}$ at $t=T_{p}/2$ to $z=0$ at $t=T_{p}$ .

Since we only consider $U_{v}$ along the perimeter of a vertical cylinder, we can express $U_{v}$ as $U_{v}(z)$ , with $0\leqslant z\leqslant z_{c}(t)$ . The profile of $U_{v}(z)$ is shown qualitatively on figure 5. To bypass the difficulty of the velocity tending to infinity on the vortex core, we specify a length scale $z_{sb}$ over which the flow rotates as a solid body. This is formally known as a Rankine vortex (Saffman Reference Saffman1992, p. 22), where $z_{sb}$ is generally introduced to account for viscous effects. We shall see later that the choice of $z_{sb}$ is immaterial to the final solution, provided that it is small. The velocity induced by a vortex ring with a viscous core of extent $z_{sb}$ follows

where $\mathscr{Z}(t)=z_{c}(t)-z$ is a local coordinate pointing radially outward from the vortex core (figure 5). The total volume $V_{top}$ entrained over a duration $t$ is therefore

Substituting for $U_{v}$ from (2.14), we evaluate the inner integral of (2.15) as

where

represent contributions to $I$ (2.15) over the solid body of rotation ( $\mathscr{F}$ ) and over the free vortex ( $\mathscr{G}$ ), respectively. Denoting the vertical velocity of the core as $w_{c}$ and making the substitution of variables $\text{d}t^{\prime }=\text{d}z_{c}/w_{c}$ transforms the integral (2.15) into

where $z_{c,max}$ designates the maximal height attained by the core of the cap over one complete cycle. Given that the collapsing cap experiences a reduced gravity $g_{u}^{\prime }$ , we may express its vertical velocity as

Substituting for $w_{c}$ from (2.19) and for $\mathscr{F}$ and $\mathscr{G}$ from (2.17) into (2.18), on integrating we obtain

with

and

It remains to estimate the bounds of integration $z_{sb}$ and $z_{c,max}$ , and to specify the form of the vortex circulation ${\it\Gamma}$ . Over one draining cycle, as depicted on figure 4, we readily recover

As alluded to previously, the precise form of $z_{sb}$ is inconsequential. For convenience, we may assume that it scales on the length scale pertinent to the fountain top, i.e.

with $B\ll A$ to ensure that $z_{sb}$ is suitably small ( $z_{sb}/z_{m}\ll 1$ ). Further, in order to determine ${\it\Gamma}$ , we use the knowledge that the upflow velocity $w_{u}$ (at level $z=0$ ) generates circulation over an area ${\rm\pi}b_{u}^{2}$ . It follows that ${\it\Gamma}\propto {\rm\pi}b_{u}w_{u}$ , i.e.

where $C$ is an order one constant. Crucially, knowing the form of ${\it\Gamma}$ enables us to evaluate the total volume entrained by engulfment into the fountain top over one half of an oscillation cycle of duration $T_{p}/2$ . Substituting (2.25) and the bounds (2.23) and (2.24) into (2.20) yields our estimate

Two comments need to be made on (2.26) so as to ascertain its validity. First, ${\it\beta}>0$ for all positive values of $B<A$ , so that the entrained volume $V_{top}$ is always positive. Second, we recall that the length scale $z_{sb}$ was introduced as a modelling artefact which, ideally, should approach zero. Whilst the factor ${\it\beta}$ is unbounded in the limit as $B\rightarrow 0$ , for small but finite $B$ , ${\it\beta}$ is well-behaved. Indeed, $\text{arctanh}(x)$ asymptotes to infinity slowly as $x\rightarrow 1$ . These two points are illustrated in figure 6, where we plot ${\it\beta}$ against $B/A$ . Moreover, for small $B$ , $V_{top,\mathscr{F}}\rightarrow 0$ (2.21). Given $Q_{u}={\rm\pi}b_{u}^{2}w_{u}$ , we may approximate the entrained volume as

In (2.27) the ratio $w_{u}/g_{u}^{\prime }$ represents a timescale $T_{u}$ that is indicative of the fluctuation period. In terms of average entrainment rates therefore, we recover that

Equation (2.28) shows that the rate of entrainment through the fountain top is a constant fraction of the volume flux provided by the upflow. Thus, our model for vortical entrainment predicts the same functional dependence as models based on MTT or SH14. In the case of MTT, this functional agreement should perhaps not come as a surprise. The entraining mechanism which forms the basis of (2.27) resembles MTT: large eddies, whose size and velocity scale on the mean flow, engulf ambient fluid. The main novelty and indeed primary outcome of our approach is that, by coupling the engulfing motion with the large-amplitude fluctuations that typify a forced fountain top, equation (2.27) faithfully replicates the phenomenology that is observed in practice. Significantly, one key implication of this agreement is that a time-averaged description of entrainment successfully captures the dynamic processes in the fountain top. This result enables the direct implementation of a mechanistic approach for fountain-top entrainment into a (three-region) model for a forced fountain. First, one replaces McDougall’s (Reference McDougall1981) cap with a hemispherical fountain top (figure 1 c) diluting at a rate $Q_{top}$ . Then, following McDougall’s lead, but crucially including the additional dilution, one obtains a modified set of outflow conditions for fluid leaving the fountain top. Given that these conditions drive the counterflow, the effects of fountain-top entrainment are incorporated into a complete description for a forced fountain. The details of this implementation, which lie beyond the desired scope of this paper, can be found in Hunt & Debugne (Reference Hunt and Debugne0000).

Figure 6. Numerical value of the factor ${\it\beta}$ (thick line) normalised by $Fr_{u}\sqrt{b_{u}}\sqrt{A}$ against different ratios $B/A$ for $A=0.9$ . Also plotted are the contributions to ${\it\beta}$ of the individual terms in (2.26) (thin lines, annotated).

We can gain some insight into the magnitude of $Q_{top}$ by giving estimates for $A$ and $T_{u}$ . The relative lack of internal fountain statistics prevents us from deducing these estimates purely from the literature. In appendix A, we show how data from Burridge & Hunt (Reference Burridge and Hunt2013) can be coupled with our model for forced fountains (presented in a complementary paper, Hunt & Debugne (Reference Hunt and Debugne0000)) to yield $A\approx 0.9$ and $T_{u}\approx 0.1T_{p}$ . From these estimates and taking ${\it\beta}\approx 10Fr_{u}\sqrt{b_{u}}\sqrt{A}$ (figure 6), we can deduce the approximation $Q_{top}/Q_{u}\approx 0.81C$ . Given that the magnitudes of ${\it\beta}$ and $C$ are independent in the limit of a vanishingly thin vortex ring ( $B\rightarrow 0$ ), it is difficult to assign a precise value to $C$ , however, by construction we expect $C\sim 1$ (2.25). Therefore, our model for vortical engulfment suggests $Q_{top}/Q_{u}\sim 1$ .