3.1 Weakly confined regime, $Fr_{0}\lesssim 8$

Instantaneously, confined fountains produced for small $Fr_{0}$ bear a strikingly similar appearance to their unconfined, axisymmetric counterparts. The resemblance is evident on comparing figure 2(a–c) to figure 2 in Burridge & Hunt (Reference Burridge and Hunt2012). The fountain consists of an upflow–counterflow double structure that is symmetric about the centreline and which fluctuates vertically about its time-averaged rise height. Visually, the fountain appears largely unaffected by the confinement.

A straightforward check allows us to confirm that the fountain is indeed in contact with the walls: were it unconfined, we expect the fountain top to attain an external radius of $b_{t}\approx 0.5\,r_{0}Fr_{0}$ (Burridge & Hunt Reference Burridge and Hunt2013; Hunt & Debugne Reference Hunt and Debugne2016). For the fountain shown in figure 2, produced at $Fr_{0}=5.4$ , this leads to $b_{t}>2.5\,r_{0}$ , i.e. $b_{t}>W/2$ and the fountain fills the gap. Moreover, it is clear that the induced radial inflow and the sideways fluctuations will prompt the fountain to cling to the walls sooner, though how much sooner is difficult to quantify. Crucially, the distance over which the fountain has clung is not sufficient to induce significant changes in its dynamics.

When the fountain is weak ( $Fr_{0}\lesssim 2$ ), such that the gap width is larger than but comparable to its external radius, the fountain cyclically attaches to the front and back wall as it precesses around the source. We report on this behaviour in appendix B. However, we do not include ‘weak’ confined fountains as a separate regime in our classification; the effects of confinement are indeed the same as in weakly confined fountains: the walls restrict the spanwise growth, but not sufficiently to induce two-dimensional effects.

Figure 2.

Weakly confined fountains: $z_{f}(t)$ for $Fr_{0}=5.4$ and $W/r_{0}=4.7$ (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2018.194). The dashed black lines in the main image correspond to the time series snapshots shown in panels (a–c). The bulk flow is unchanging (as illustrated by the sequence a–c) and visually indistinguishable from unconfined round fountains.

3.2 Asymmetric–stable regime, $8\lesssim Fr_{0}\lesssim 12$

The effects of confinement become apparent beyond $Fr_{0}\sim 8$ . Shortly after the initial (symmetric) rise-and-fall transient, any slight asymmetry or instability trips the counterflow so as to fall to the left or right of centreline. On the side to which the counterflow falls, a large recirculation region develops which extends from source to fountain top (figure 3). This recirculation region is maintained, forcing the fountain to arch over it. Most of the fountain fluid is therefore channelled along that arc and, in part, re-entrained into the upflowing core, as is highlighted by the curved arrow in figures 3(a–c). Patches that are not re-entrained disconnect (are shed) from the top of the recirculation region to form a (somewhat intermittent) gravity current on reaching the horizontal base. On the other side of the centreline, where the flow remains slender, smaller lobes are shed; their rate of shedding however appears more regular, hence the gravity current is more continuous on the slender side. Moreover, some fluid in the recirculation region crosses the centreline near the source and feeds into the replete gravity current.

Figure 3.

Asymmetric–stable fountains: $z_{f}(t)$ for $Fr_{0}=8.9$ and $W/r_{0}=4.7$ (see supplementary movie 2). A sustained recirculation region forces the fountain to fold to one side (here, left of centreline in a–c) for the duration of the experiment, although some surges in rise height (‘spikes’) can momentarily perturb that arrangement. The curved arrow highlights the large-scale recirculation region.

The large recirculation region is to be likened to the large-scale eddies which typify confined jets and plumes. Thus, for $Fr_{0}\gtrsim 8$ we observe the first instance of (nascent) quasi-two-dimensional effects. A qualitative difference can also be noted from the time series in figure 3, where, by contrast with the weakly confined regime, departures about the mean are larger, and time scales associated with a given rise-and-fall event are longer. Strikingly, the fountain is unable to unbind from the recirculating region. Unlike jets and plumes which wind around eddies and propagate farther, for $Fr_{0}\sim 8$ the fountain is not sufficiently energetic to overcome the arched trajectory imposed by the recirculation region. Instead, on account of the arched trajectory, this implies that more dense fluid will be channelled toward and re-entrained into the folded side of the upflow, which in turn fuels the recirculation region, and so on.

The asymmetric configuration proved stable to relatively large disturbances. To illustrate this, we refer to the time series in figure 3 which showcases isolated disturbance events seen as ‘spikes’, in which a marked increase in rise height occurs (e.g. around $t=35~\text{s}$ ). During these spikes, the fountain momentarily breaks loose from the recirculation region, rises rapidly upwards and then typically returns to its original asymmetric state. Only after $O(100)$ such spikes did the fountain eventually fold to the other side of centreline (as observed in $11$ out of $30$ experiments conducted in the asymmetric regime at $W/r_{0}=4.7$ ), showing that the asymmetric state was robust. Changes in side were more frequent towards the higher end of the range $8\lesssim Fr_{0}\lesssim 12$ , when the flow, being more energetic at the source, overcomes the recirculation region more readily.

Figure 4.

Transitional fountains: $z_{f}(t)$ for $Fr_{0}=14.0$ and $W/r_{0}=4.7$ (see supplementary movie 3). Instantaneous snapshots (a–f) illustrate the transition from one stable state (a) (folded left) to another (f) (folded right). The transitions are prompted by sporadic spikes in rise height (c), which appear as peaks in the time series.

3.3 Transitional regime, $12\lesssim Fr_{0}\lesssim 16$

The fountain becomes increasingly unstable on further increasing the forcing at the source. While mostly retaining an asymmetric structure (figure 4 a), the flow now carries sufficient energy to sporadically unbind from the recirculation region (4 a,b, dotted outline). The latter is shed sideways, allowing the fountain to soar up as a near-symmetric jet (4 c). Subsequently, during collapse (4 d), instabilities prompt the flow to fold to either side of centreline and another recirculation region is rapidly re-established. The flow then locks into an asymmetric state until the next ‘spike’ (4 e,f). These spikes, which show as large peaks in time series 4, appear to have no one specific frequency associated with them; equally, intervals of lower rise heights (which mark periods of relatively stable, asymmetric flow) are also irregular in duration.

The principal difference between asymmetric–stable and transitional fountains is that, after being destabilised, transitional fountains show no preferential direction for folding. In addition, transitional fountains also display prolonged periods in which the flow flaps left and right several times before settling to one side. Both behaviours are indicative of a diminished influence of the recirculation region. As the fountain becomes increasingly forced, the recirculation region forms farther from the source and, as a consequence, becomes increasingly unable to sustain itself: whereas close to the source, the recirculation region is able to deflect the fountain towards it to fuel itself, when forming farther from the source it disconnects from the fountain and loses its bind on the central core (see § 3.4).

The transitional regime marks a range of Froude numbers for which the flow is neither stably asymmetric nor meandering (§ 3.4). As such, it shares features with both regimes, making a clear delineation difficult. Qualitative differences, such as repeated changes in flopping direction (compare supplementary movies 2 and 3), were used to classify the flow as either ‘asymmetric–stable’ or ‘transitional’. This led to the threshold between both regimes falling at $Fr_{0}\sim 12$ . We later allude to more rigorous approaches in § 4, which consist in comparing the average flow left and right of the centreline. However, the thresholds predicted by these approaches coincide with those inferred from straightforward observation.

3.4 Meandering (quasi-two-dimensional) regime, $Fr_{0}\gtrsim 16$

If one extrapolated observations of quasi-two-dimensional jets and plumes to flows with negative buoyancy, the ‘meandering’ (or quasi-two-dimensional) regime is what one would naturally anticipate. Large circulatory structures form, on a scale comparable to the rise height, at or near the peak of the fountain. After they disconnect from, and thus are no longer fed by, the upflow, the circulation of these structures rapidly dissipates, typically to cease rotating within a single revolution. Gravity then prompts them to fall around the upflow, randomly to the left and right. The upflow sinuously wraps around the descending structures; their presence deflects the upflowing core, soon prompting the growth of another eddy on the opposite side of the centreline to the previous structure. These cycles, of growth and decay on one side prompting growth and decay on the other side, repeat. Thus the fountain describes a sinuous or flapping motion, reminiscent of the meandering behaviour of quasi-two-dimensional jets and plumes. One such cycle is depicted in figure 5, with the growth and decay (a–d) of a ‘right’ eddy soon followed by the growth and decay of a ‘left’ eddy (e–i). Note how eddies ‘stack’ (figure 5 a,b), and hence how the behaviour is different from the flapping motion observed in planar fountains (Zhang & Baddour Reference Zhang and Baddour1997). Also discernible from  5(a–i) (as indeed from snapshots of all previous regimes) are small-scale eddies forming on the periphery of larger structures. Although the fountain has become quasi-two-dimensional at the large scale, the smaller scales which drive dissipation remain three-dimensional in nature.

It would be ill advised to regard the meandering regime as fundamentally different from previous regimes. Once circulatory structures form on a large scale and collapse around the fountain, their presence inclines the upflow away from them (figures 3 a, 4 a,f and 5 i). Simultaneously, as these structures collapse, dense fluid is re-entrained along one side of the upflow, creating a torque. If a structure forms sufficiently close to the source, this torque inclines the tip of the fountain, causing it to fuel the structure with lobes of fluid. The structure’s circulation, being fuelled, is maintained and the fountain locks into an asymmetric state. If, on the other hand, the structure forms farther from the source, it disconnects from the fountain core: its circulation rapidly dissipates and the structure collapses around the core, prompting the cycles of growth and decay on alternate sides of the fountain described in the paragraph above. Hence, the mechanism by which large-scale structures form is the same throughout all regimes; what varies, however, is the subsequent interplay between these structures and the fountain core.

Figure 5.

Meandering fountains: $z_{f}(t)$ for $Fr_{0}=24.5$ and $W/r_{0}=4.7$ (see supplementary movie 4). Instantaneous snapshots (a–i), spaced one second apart, depict the growth and subsequent shedding of large-scale structures on alternating sides of the fountain that are typical of the meandering (‘forced’) regime.

3.5 Differences in behaviour with confinement ratio

The previous description of the four regimes applies to all values of $W/r_{0}$ explored in our experiments, albeit with different $Fr_{0}$ -thresholds (§ 4). However, some peculiarities of fountains at lower and higher $W/r_{0}$ are worth pointing out.

For strongly confined fountains (set $A$ , $W/r_{0}=2.7$ ), the asymmetric–stable and transitional regimes are bypassed. Instead, they are replaced by a sustained flapping, with short-lived recirculation regions forming and shedding on alternate sides of the core. This behaviour is reminiscent of transitional fountains on the brink of meandering, although it persists throughout the transition from weakly confined to meandering. The flapping motion, as expected, then evolves into meandering at higher  $Fr_{0}$ .

As in the asymmetric regime, the formation of large-scale structures (recirculation regions) signifies the onset of quasi-two-dimensional effects; unlike the asymmetric or the transitional regimes however, at $W/r_{0}=2.7$ these structures are not maintained. Notably, at this low confinement ratio we did not observe any significant amount of fluid crossing the centreline from one side to the other. At other confinement ratios, when a recirculation was maintained and dense fluid channelled towards the upflow, any excess fluid that could not be re-entrained crossed the centreline and fed into the opposite outflowing gravity current (§ 3.2). The narrowness of the gap at $W/r_{0}=2.7$ prevents this crossing, which might be instrumental in sustaining recirculation regions and hence, in sustaining a stable folded configuration.

Interestingly, this conclusion finds support in the contrasting observations made on line fountains by Baines, Turner & Campbell (Reference Baines, Turner and Campbell1990) and Zhang & Baddour (Reference Zhang and Baddour1997). Baines et al. noted that line fountains were mostly symmetrical about the centreline, but sporadically folded for prolonged periods of time. By contrast, in Zhang & Baddour, line fountains were reported to continually flap. An explanation for this discrepancy that is in line with our observations may be found in their source geometry. Whilst Zhang & Baddour dispensed fluid from a thin slot, Baines et al. approximated a line source with an array of closely spaced circular sources. Hence, close to the source their flow was three-dimensional. This arrangement permits a flow across the source from the folded recirculation region and, hypothetically, might allow the recirculation to be sustained. Conversely, such cross-flow is not possible with Zhang & Baddour’s source; thus, as in our strongly confined fountains, their line fountains flapped left and right.

Figure 6.

Illustration of a ‘burst’ event observed at $Fr_{0}=63.8$ and $W/r_{0}=11.8$ . The same colour map has been applied to (a) and (b), with dark red being indicative of high density. Whereas in (a) the fountain is dense and compact, in (b) it has spread and diluted significantly, leading to an increase in rise height of ${\sim}50\,r_{0}$ . Note that (a) and (b) are not correlated in time but have been chosen to maximise the contrast between the dense and the dilute configuration.

The next distinction needs to be made at moderate-to-high $W/r_{0}\,({\geqslant}11.8)$ , i.e. for weaker confinement. Fountains at these confinement ratios exhibited a greater range of dilution in our depth-averaged visualisations. Sporadically, the increased dilution led to ‘burst’ events, during which vast clouds of dilute fluid occupied a much larger space. This process is illustrated in figure 6, where we contrast two snapshots of the same run: the fountain is compact and dense in 6(a), whilst it is large and dilute in 6(b). The associated increase in rise height from (a) to (b) is approximately $50\,r_{0}\approx 4W$ .

The ‘bursts’ depicted in figure 6 occurred rarely and only for high $Fr_{0}$ , such that the fountain was meandering. Whilst we do not have the trace of $Q_{0}(t)$ for set D, which featured most bursts, we recorded $Q_{0}(t)$ at similar $Fr_{0}$ and found fluctuations of no more than $1\,\%$ (appendix A), which is insufficient to explain the observed increases in rise height.

It is relatively straightforward to reason why increased dilution leads to increased rise heights. Descending lobes of fluid that are more dilute and more spread out interact less strongly with the upflow, hence the fountain is free to rise vertically upwards. What is less clear is how the additional dilution is made possible. One explanation is that the fountain dilutes as a result of not completely filling the gap at all time. However, such behaviour was not observed in our experiments.

4.1 Rise heights and lateral excursions

We begin by inspecting the time-averaged rise height of the fountain front on the centreline, $\overline{z}_{f}$ . In figure 7, we apply the scaling for round unconfined fountains and plot $\overline{z}_{f}$ normalised by $r_{0}$ against the source Froude number $Fr_{0}$ . The plot encompasses all experimental sets. Two observations are immediately apparent: first, within each set the rise heights scale linearly on the jet length, $\overline{z}_{f}\propto r_{0}Fr_{0}$ , across all regimes; second, the gradient of that linear relation varies with $W/r_{0}$ . Indeed, the rise heights in sets B and C (

$W/r_{0}=4.7$

$W/r_{0}$

$W$

$r_{0}$

$\overline{z}_{f}/r_{0}$

$W/r_{0}\leqslant 11.8$

$W/r_{0}\geqslant 16.6$

$r_{0}$

$Fr_{0}$

Also plotted in figure 7 (dashed line) is the rise height for unconfined round fountains, $\overline{z}_{f}=2.46\,r_{0}Fr_{0}$ . We notice that the rise heights of relatively strongly confined fountains (low $W/r_{0}$ (

A notable preliminary result is nonetheless that $\overline{z}_{f}$ scales linearly on $Fr_{0}$ (the gradients for each set can be inferred from figure 8 a). As alluded to in § 1, the internal variables of quasi-two-dimensional jets and plumes recover the scaling of their planar counterparts for sufficiently large $z$ (Landel et al. Reference Landel, Caulfield and Woods2012a ; Rocco & Woods Reference Rocco and Woods2015). Planar fountains rise as $\overline{z}_{f}\propto Fr_{0}^{4/3}$ (Baines et al. Reference Baines, Turner and Campbell1990; Zhang & Baddour Reference Zhang and Baddour1997), in clear contrast with our measurements of confined fountains (figure 7). Of note is that in jets and plumes, variables such as the velocity are measured at, or near, the core of the flow, i.e. not at the periphery of the flow where eddies form. The core, which meanders between the eddies, maintains its coherence over height and is free to develop quasi-two-dimensionality. On the contrary, the upflowing core in fountains becomes progressively weaker with height as buoyancy depletes its momentum; near the top, the fountain is swayed by eddies to the extent that its motion becomes predominantly lateral. It is undeniable that eddies have a stronger and more disruptive influence on (or feedback with) fountains than in jets and plumes. The dynamics of confined fountains, being disrupted, becomes distinct from (free) planar fountains. It is therefore not surprising that their rise heights scale differently. Likewise, experiments on rectangular sources by Vinoth & Panigrahi (Reference Vinoth and Panigrahi2014) have shown that large source ratios are required at the source to attain the two-dimensional scaling. In our experiments, it is questionable whether the upflowing core, which remains slender (e.g. figure 5), departs sufficiently from being round to form a planar fountain. Again, therefore, we would expect $\overline{z}_{f}\propto Fr_{0}$ .

The fact that we observe the same qualitative behaviour for confined fountains at different values of $Fr_{0}$ and $W/r_{0}$ strongly suggests that there is a common scaling which applies to, and governs, confined fountains such as those studied herein (at least over the range of confinement ratios considered, $2.7\leqslant W/r_{0}\leqslant 24.2$ ). We set out to identify this common scaling by collapsing the rise heights $\overline{z}_{f}$ across all confinement ratios.

First, we explore how the rise heights vary with the confinement ratio by extracting, for each value of $W/r_{0}$ , the gradients $k_{s}$ (where ‘ $s$ ’ reads ‘set’) of the linear relationship $\overline{z}_{f}/W=k_{s}Fr_{0}$ from figure 7. Here, we normalise the rise heights by $W$ rather than by $r_{0}$ since the fountain experiences the length scale $W$ as soon as it has expanded to reach the walls; suitably highly forced confined fountains are therefore expected to scale on $W$ . Since $W/r_{0}$ is constant within each set, the relationship $\overline{z}_{f}/r_{0}\propto Fr_{0}$ (figure 7) also implies that $\overline{z}_{f}/W$ scales linearly on $Fr_{0}$ . The gradients $k_{s}$ are plotted against the confinement ratio $W/r_{0}$ in figure 8(a). The rate of decrease of $k_{s}$ as a function of $W/r_{0}$ is notably steeper at low values of $W/r_{0}$ than at high values of $W/r_{0}$ , which implies that the rise heights are more sensitive to small changes in confinement when confinement is relatively strong (low $W/r_{0}$ ) than when confinement is relatively weak (high $W/r_{0}$ ). Most of the variation of $k_{s}$ with $W/r_{0}$ is therefore occurring at low values of $W/r_{0}$ . Of note is that the value of $k_{s}$ at $W/r_{0}=2.7$ (

$13$

$W/r_{0}=24.2$

Figure 8.

(a) Gradients $k_{s}\equiv \unicode[STIX]{x0394}(\overline{z}_{f}/W)/\unicode[STIX]{x0394}Fr_{0}$ , obtained from linear fits to the rise heights in figure 7, against $W/r_{0}$ . The black line represents the best fit (4.1) to the gradients as a function $W/r_{0}$ . In contrast to this best fit, grey lines highlight the discrepancy that ensues on adopting a simplified scaling for $k_{s}$ based on the assumption that, prior to attachment, the flow behaves like an unconfined fountain (appendix C). The shaded region demarks $W/r_{0}<2$ , which is not physical. (b) Time-averaged centreline rise height $\overline{z}_{f}$ normalised by $W$ against the confined Froude number $Fr_{c}\equiv Fr_{0}(W/r_{0})^{-5/4}$ (4.3). Error bars in light grey. Symbols as in table 1.

Next, attempting to derive the trend observed for $k_{s}$ in figure 8(a) on theoretical grounds is difficult in the absence of detailed measurements for the evolution of inner variables (e.g. the local velocity or buoyancy). We therefore opt to calculate, and adopt, a best fit through the gradients $k_{s}$ instead. The resulting power law, drawn in figure 8(a) as a black line, follows

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle k_{s}=3.32\left(\frac{W}{r_{0}}\right)^{-5/4}. & \displaystyle\end{eqnarray}$$

Equation (4.1) suggests that confined fountains reach a time-averaged rise height of

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{z}_{f}/W=3.32\,Fr_{0}\left(\frac{W}{r_{0}}\right)^{-5/4}+m_{s}, & \displaystyle\end{eqnarray}$$

where $m_{s}\sim 1$ denotes intercepts at $Fr_{0}=0$ . The normalised rise heights $\overline{z}_{f}/W$ are plotted against the suggested scaling $Fr_{0}(W/r_{0})^{-5/4}$ in figure 8(b). The collapse is excellent (with a mean coefficient of determination of $R^{2}=0.99$ ), especially considering the range of parameters explored in this study. There remains a slight offset at the source, $m_{s}$ , which varies across sets, but typically $m_{s}\sim 1$ . Crucially, neither the scaling nor the pre-factor of the fit to $k_{s}$  (4.1) depend on $m_{s}$ , so long as $m_{s}$ is not a function of $Fr_{0}$ (as is indeed supported by the linear dependencies in figure 7, and by the collapse in figure 8 b).

Whilst our data do not extend beyond $W/r_{0}=24.2$ , and as such, we cannot verify the suitability of (4.2) at greater confinement ratios, the gentle slope of $k_{s}$ at higher values of $W/r_{0}$ (figure 8 a) implies that most of the variability of $k_{s}$ with $W/r_{0}$ has been captured by (4.1). Therefore, it seems likely that (4.2) should apply, at least in an approximate sense, at greater values of $W/r_{0}$ too. Equation (4.2) will not apply, however, in the limit of no confinement ( $W/r_{0}\rightarrow \infty$ ), where the rise heights scale as $\overline{z}_{f}/r_{0}=2.46\,Fr_{0}$ (Turner Reference Turner1966) and where accordingly, there can be no dependence on $W$ ; see the discussion in § 5.

Figure 9.

Regime diagram for the different flow regimes observed in § 3. Each experiment, run at a particular $\{Fr_{0},W/r_{0}\}$ , is colour-coded according to the observed flow behaviour of that experiment, with red (dark grey) representing $Fr_{c}=1.13+/-0.18$ , yellow (light grey) representing $Fr_{c}=1.86+/-$ 0.33 and turquoise (mid-grey) representing $Fr_{c}=2.58+/-0.16$ . Horizontal lines, which represent average threshold values for $Fr_{c}$ (table 2), demarcate approximate boundaries for the symmetric–asymmetric transition (bottom line, red) and the transitional–meandering transition (top line, turquoise), respectively. The vertical dotted line denotes the limit of fully confined fountains, $W/r_{0}=2$ . The symbols, as in previous plots, represent different sets of experiments (table 1), although for the purpose of this plot their colour has been changed to refer to the flow regimes (see annotation).

The collapse of the rise height data in figure 8(b) across all values of confinement ratios tested herein provides convincing evidence that the scaling $Fr_{0}\,(W/r_{0})^{-5/4}$ encapsulates the dynamics of confined fountains. More evidence is provided later, e.g. in figures 9–11. Consequently, we define the ‘confined’ Froude number

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle Fr_{c}\equiv Fr_{0}\,(W/r_{0})^{-5/4} & \displaystyle\end{eqnarray}$$

as the governing parameter for confined fountains. In physical terms, identical values of $Fr_{c}$ can be achieved either on increasing the forcing $Fr_{0}$ for fixed $W/r_{0}$ , or on narrowing the dimensionless gap $W/r_{0}$ for fixed $Fr_{0}$ . Since $Fr_{0}$ is indicative of the height attained by the fountain, ‘confinement’ is therefore expressed in $Fr_{c}$ in terms of a ratio of the streamwise extent of the fountain to its spanwise extent (albeit not one to one).

If the confined Froude number $Fr_{c}$ is to encapsulate the effects of confinement, then bands of $Fr_{c}$ should coincide with the flow regimes observed in § 3. This notion is pursued in figure 9, where we colour-code every experiment according to their observed regime, and in table 2, where we cite the thresholds for the regimes in terms of $Fr_{0}$ for each set and calculate corresponding values of $Fr_{c}$ . In figure 9, horizontal lines (i.e. lines of constant $Fr_{c}$ ) conclusively separate the colour-coded regimes. Likewise in table 2, we notice that $Fr_{c}$ takes a nominally constant value across all sets: $Fr_{c}\sim 1.1$ in the asymmetric regime and $Fr_{c}\sim 2.6$ in the meandering regime. Agreement in the transitional regime is poorer: by its very nature this regime marks a transition from one distinct behaviour to another, hence it is not surprising that transitional fountains are difficult to classify. Nevertheless, for threshold values of $Fr_{0}$ varying by a factor of $10$ , thresholds of $Fr_{c}$ vary by approximately 6 %–17 % only, a variation only slightly above our error margin. The overall ability of $Fr_{c}$ to delineate regimes, therefore, is good, reinforcing the suitability of the scaling (4.3).

Table 2.

Thresholds for the flow regimes (§ 3) for all confinement ratios considered in terms of source Froude number $Fr_{0}$ and confined Froude number $Fr_{c}$  (4.3). A blank entry (—) indicates that a regime was not observed in our experiments. The thresholds given correspond to the first value of $Fr_{0}$ for which the features characteristic of a particular regime first appeared. For a graphic representation of the onset of flow regimes as a function of $Fr_{c}$ and $W/r_{0}$ , the reader is referred to figure 9.

To readers familiar with work on unconfined fountains, the threshold values in table 2, and in general values of $Fr_{c}$ achieved in our experiments ( $Fr_{c}\lesssim 7.0$ , figure 8 b), may appear low in magnitude. It is important to point out that, unlike unconfined fountains for which low values of $Fr_{0}$ do imply weak releases, in confined fountains low values of $Fr_{c}$ cannot be viewed as giving rise to momentum-poor fountains. The weakly confined fountain in figure 2, for instance, was formed at $Fr_{c}=0.78$ , yet it clearly retains the slender upflow–counterflow structure characteristic of highly forced (unconfined) releases. ‘Low’ values of $Fr_{c}$ (‘low’ relative to $Fr_{0}$ in unconfined fountains) should therefore not be mistaken for low levels of forcing. Weak confined fountains, such as shown in appendix B (where $Fr_{c}=0.08$ ), probably occur at much lower values of $Fr_{c}$ and lie outside the scope of this paper.

Figure 10.

(a) Time-averaged lateral extent of the fountain left of centreline ( $\overline{x}_{l}$ , various symbols) and right of centreline ( $\overline{x}_{r}$ , dots) normalised by $W$ against the confined Froude number $Fr_{c}$ . For clarity, error bars (in light grey) are plotted for $\overline{x}_{l}/W$ only. The dotted lines delineate the limits for the asymmetric–stable and meandering regimes (table 2). The inset provides a zoomed view onto the interval $0.7\lesssim Fr_{c}\lesssim 2.9$ to highlight the formation of two distinct branches in the asymmetric–stable and transitional regimes. Set A is not plotted in the inset, because fountains in set A do not fold asymmetrically (and hence, their lateral extents to not splay). (b) Aspect ratio of the fountain: ratio of the rise height $\overline{z}_{f}$ to the lateral extent of the fountain $\overline{x}_{l,r}$ against $Fr_{c}$ for forced releases, $Fr_{0}\geqslant 3.0$ . Symbols and dotted lines as in (a).

Figure 11.

Representative sizes for large-scale eddies, as inferred from the lateral fluctuations. (a) Time- and height-averaged eddy size $\overline{\unicode[STIX]{x1D706}}_{l,r}$ , and (b) time-averaged maximal eddy size $\unicode[STIX]{x1D6EC}_{l,r}$ , both normalised by $W$ , against $Fr_{c}$ . For clarity, error bars (in light grey) have been plotted for $\overline{\unicode[STIX]{x1D706}}_{l}$ and $\unicode[STIX]{x1D6EC}_{l}$ only. Symbols as in figure 10.

Next, we inspect a salient feature of confined flows, namely their considerable lateral excursions (as seen, for example, in figure 5 c). In order to analyse their extent, we consider separately the farthest points left ( $x_{l}$ ) and right ( $x_{r}$ ) of centreline which lie above $\overline{z}_{f}/2$ (figure 1). The minimal height of $\overline{z}_{f}/2$ is imposed only to avoid the lateral outflows and has little bearing on the results. Rather than tracking a specific point in the Lagrangian sense, $x_{l}(t)$ and $x_{r}(t)$ follow the extreme outer edge of the fountain regardless of the height at which this peak excursion occurs (so long as it exceeds $\overline{z}_{f}/2$ ). Hence, $x_{l}(t)$ and $x_{r}(t)$ are susceptible to discontinuities or ‘jumps’. Whilst these discontinuities do not affect time averages, they will have to be taken into account later when estimating velocities based on the gradients of $x_{l}(t)$ and $x_{r}(t)$ .

The time-averaged lateral excursions $\overline{x}_{l}$ and $\overline{x}_{r}$ are normalised by the gap width $W$ and plotted against $Fr_{c}$ in figure 10(a). In order to not clutter the figure, the symbols (cf. table 1) were retained for the ‘left’ variables, but replaced by dots for the ‘right’ variables. As expected, the fountain grows laterally as $Fr_{c}$ increases. We note that left and right excursions generally overlap, but splay out in the asymmetric regime ( $Fr_{c}\sim 1.1$ ) where the fountain stably folds to one side (figure 3). The upper branch corresponds to the side on which the fountain has folded; the lower branch corresponds to the opposite side where, by comparison, the fountain is slender. The two branches then rejoin at the onset of the meandering regime ( $Fr_{c}\gtrsim 2.6$ ) and follow a linear scaling $\overline{x}_{l,r}\propto W\,Fr_{c}$ . A linear scaling is also observed in the weakly confined regime, albeit with a lower prefactor. These observations are reinforced in figure 10(b), where we plot the height-to-width aspect ratio of the fountain $\overline{z}_{f}/\overline{x}_{l,r}$ . For all values of confinement, the bulk structure of the fountain appears to be similar in $Fr_{c}$ . Moreover, we notice that the trends in figure 10(b) reflect the classification into regimes given in table 2: in the weakly confined regime, the aspect ratio is large ( $\overline{z}_{f}/\overline{x}_{l,r}\sim 5{-}6$ ) and similar to the aspect ratio of unconfined round fountains (Burridge & Hunt Reference Burridge and Hunt2013; Hunt & Debugne Reference Hunt and Debugne2016); then, as the asymmetric–stable regime is established, large circulatory structures begin to form, causing the aspect ratio to decrease; finally, in the meandering regime, the aspect ratio asymptotes to a constant value of approximately two, meaning that the fountain is as wide as it is high (adding $\overline{x}_{l}$ and $\overline{x}_{r}$ ). Thus, figure 10(b) offers a new interpretation of the regimes in terms of the fountain’s height-to-width ratio.

4.2 Fluctuations

Having examined the time-averaged excursions $\overline{x}_{l,r}$ , we now turn to their fluctuating part. In particular, suitably chosen fluctuations about the mean excursion should be representative of the size of the large-scale eddies which typify confined flows.

We first construct the ‘fluctuating’ part of our instantaneous snapshots by subtracting from each snapshot the experiment-averaged picture of the flow. In the resulting frames, the large-scale eddies are thereby made distinct from the fountain core and their outline can then be traced by applying a luminosity threshold (see the black lines in figure 12 a). For each (fluctuating) snapshot, we then record the lateral extent of eddies at every height to the left and right of centreline as $\unicode[STIX]{x1D706}_{l,r}(z,t)$ . Two key quantities are inferred: first, the average eddy size $\overline{\unicode[STIX]{x1D706}}_{l,r}$ , calculated by averaging in $z$ and $t$ ; and second, the average size of large eddies $\unicode[STIX]{x1D6EC}_{l,r}$ , calculated by averaging in $t$ the instantaneous maxima in $z$ . As a remark, we note that many entries in $\unicode[STIX]{x1D706}_{l,r}(z,t)$ are zeroes, since eddies form intermittently and rarely at great heights. Therefore, in order to obtain representative eddy sizes, we average $\unicode[STIX]{x1D706}_{l,r}(z,t)$ over non-zero entries only.

Figure 12.

Illustration of our ‘eddy counting’ approach for $W/r_{0}=11.8$ and $Fr_{0}=63.8$ . (a) An instantaneous snapshot from which the time-averaged luminosity has been subtracted. The black lines encircling eddies were used to calculate $\unicode[STIX]{x1D706}_{l,r}(z,t)$ (§ 4.2). The dashes outline the ‘windows’, left and right of the centreline, within which the luminosity is averaged (in $x$ ) to produce the time series (b) and (c), respectively. In (b) and (c), crosses mark the detected peaks and the dashed line indicates the instant at which (a) was taken. The dashed line has just passed a peak in (c) (right window), and a remnant of the associated eddy is visible in (a). Similarly, the dashed line is approaching a peak in (b) (left window), and the nascent eddy can also be observed in (a). An animated version is available online (see supplementary movie 5).

The results are shown in figure 11. Average (large-scale) eddies $\overline{\unicode[STIX]{x1D706}}_{l,r}$ and large (large-scale) eddies $\unicode[STIX]{x1D6EC}_{l,r}$ clearly follow the same trends, with $\unicode[STIX]{x1D6EC}_{l,r}$ being approximately a factor two greater. Similarly to the average excursions $\overline{x}_{l,r}$ in figure 10, three phases of growth can be distinguished: modest growth in the symmetric regime, followed by a phase of nonlinear growth in the asymmetric and transitional regimes as quasi-two-dimensional effects become prominent and finally linear (but accelerated) growth in the meandering regime. However, the measures $\overline{\unicode[STIX]{x1D706}}_{l,r}$ and $\unicode[STIX]{x1D6EC}_{l,r}$ are distinct from $\overline{x}_{l,r}$ in that they do not include the width of the upflow; thereby making them a more accurate measure of the scale of intermittent eddies. Landel et al. (Reference Landel, Caulfield and Woods2012a ), who studied the properties of large-scale eddies in quasi-two-dimensional jets, also noted a linear growth in the far field – here, that far field would correspond approximately to the meandering regime, $Fr_{c}\gtrsim 2.6$ . Over distances from the source comparable to the heights attained by our fountains, they reported eddy sizes between $4W$ and $4.5W$ , consistent in magnitude with our findings. Crucially, it is evident that the large-scale eddies occupy a significant proportion of the flow: ratios of $\overline{\unicode[STIX]{x1D706}}_{l,r}$ to $\overline{x}_{l,r}$ lie slightly above $1:2.5$ , whilst ratios of $\unicode[STIX]{x1D6EC}_{l,r}$ to $\overline{x}_{l,r}$ lie closer to $3:4$ . It is therefore not surprising that the dynamics of confined fountains is closely linked to the behaviour of large-scale eddies.

Various definitions for $\unicode[STIX]{x1D706}_{l,r}$ were considered before settling on the most intuitive, namely that associated with tracking the eddies’ outline. One alternative would have been to define average-sized fluctuations as twice the standard deviation $\unicode[STIX]{x1D70E}$ , or as a measure closely related. Landel et al. (Reference Landel, Caulfield and Woods2012a ), for instance, follow the centroid of eddies using particle image velocimetry measurements and infer typical eddy sizes as two standard deviations about the centroid’s location. Burridge & Hunt (Reference Burridge and Hunt2013) calculate the extent of vertical fluctuations in unconfined fountains as the difference between ‘peaks’ and ‘troughs’, which represent averages over time intervals when the rise height $z_{f}(t)$ is greater or lower than $\overline{z}_{f}\pm \unicode[STIX]{x1D70E}$ , respectively. Such definitions are meaningful when the underlying distribution is approximately symmetric; in that case, quantities above and below one standard deviation have a similar physical interpretation (e.g. ‘eddy present’ versus ‘eddy absent’). By contrast, the lateral fluctuations of confined fountains have a skewed distribution: there are more small-to-moderate lobes than there are large-scale eddies. Consequently it becomes less straightforward to interpret ‘one standard deviation’. We therefore resorted to the more natural approach of tracking the widths of eddies via $\unicode[STIX]{x1D706}_{l,r}(z,t)$ .

One drawback of this approach is that it cannot easily be extended to the vertical fluctuations in rise height. However, except in the weakly confined regime, the latter are not relevant dynamically. Only occasionally does a large eddy form at or near the centreline. Rather, the vertical fluctuations constitute a damped response to the left–right meandering of the fountain, which is dictated by the lateral fluctuations $\overline{\unicode[STIX]{x1D706}}_{l,r}$ . Hence, we do not further investigate the vertical fluctuations herein.

4.3 Frequencies

Based on figure 11 we established that the large-scale eddies grow larger with increasing $Fr_{c}$ . A natural continuation is to inspect the time scales over which these eddies form.

There are many techniques available to extract temporal information from a series of snapshots or signals. Decomposition methods, for instance, have successfully separated different oscillating modes in low- $Re_{0}$ fountains in Vinoth & Panigrahi (Reference Vinoth and Panigrahi2014). Burridge & Hunt (Reference Burridge and Hunt2013), in their study of fluctuations at the (unconfined) fountain top, performed Fourier transforms of the rise height signal $z_{f}(t)$ to identify dominant frequencies. Both methods are valid, but tend to require some tuning. In this section, we present a novel, intuitive approach in which eddies are ‘counted’ as they form. We demonstrate the robustness of this approach by comparing the resulting frequencies to those identified with Fourier transforms in appendix D.

Figure 12 depicts our ‘eddy counting’ approach. The main idea consists in identifying traces left by large-scale eddies in suitably constructed time series. Specifically, we wish to construct a $(z,t)$ time series by averaging the flow horizontally over a region where large structures typically form. By doing so, the large structures will appear in the time series as dense streaks separated by intervals of lighter ambient. These streaks are then counted, and their total is divided through the recording interval to calculate the formation frequency associated with large eddies.

Figure 12(a) shows an instantaneous snapshot from which the mean flow has been subtracted in order to increase the contrast between the intermittent eddies and the ever-present core. Then, as described above, the time series in figure 12(b,c) were obtained by averaging the flow horizontally over the ‘windows’ delimited by white dashed lines in figure 12(a). One clearly discerns streaks, which correspond to eddies forming and collapsing. These streaks were counted by identifying luminosity peaks in the time series. To ensure that our algorithm selected only large-scale eddies, we stipulated that the peaks exceed the median luminosity and had a width (at half-prominence) of at least half the median. These settings led to good detection properties while minimising omissions and double counting. The peaks thus detected are indicated by crosses in figures 12(b,c). We note very good agreement between separate bright streaks and detected peaks. In particular, for the time instant shown in figure 12(a) (marked as a dashed line in 12 b,c), our algorithm is about to successfully identify the formation of an eddy to the left of the centreline. Our results were not sensitive to the size of the windows, provided they were sufficiently large so as to enclose the majority of eddies. We opted for windows spanning $([-1.8\,\overline{x}_{l},-0.5\,\overline{x}_{l}],[0.5\,\overline{z}_{f},1.5\,\overline{z}_{f}])$ and $([0.5\,\overline{x}_{r},1.8\,\overline{x}_{r}],[0.5\,\overline{z}_{f},1.5\,\overline{z}_{f}])$ left and right, respectively.

Figure 13.

Formation frequencies $f_{l,r}$ of large-scale eddies to the left and right of centreline identified by our ‘eddy counting’ approach: (a) in dimensional form and (b) normalised by the source frequency $w_{0}/r_{0}$ . In (b), solid lines denote best fits for $f_{l}$ and $f_{r}$ over $Fr_{0}\geqslant 3$ (shifted for ease of reading). Symbols as in figure 10.

The formation frequencies $f_{l,r}$ are plotted against $Fr_{0}$ in dimensional form in figure 13(a). The error in $f_{l,r}$ was estimated as $\pm 2/\unicode[STIX]{x0394}T$ , where $\unicode[STIX]{x0394}T$ is the recording interval. The resulting error bars were mostly smaller than the symbol size and are omitted for clarity. The formation frequencies decrease as $Fr_{0}$ is increased, consistent with the notion that eddies grow larger and more persistent with $Fr_{0}$ (figure 11). At moderate $Fr_{0}$ , the ‘eddy counting’ approach predicts formation frequencies in the range $0.1{-}0.2~\text{s}^{-1}$ , i.e. one eddy being formed (left or right) every $2.5{-}5~\text{s}$ , which agrees well with our observations. We notice that, although there is some variation across the experimental sets, the formation frequencies generally lie close together, which suggests that the gap width $W$ has little influence on the formation of eddies. This is confirmed in figure 13(b), where we normalise $f_{l,r}$ by the source frequency $w_{0}/r_{0}$ . All sets fall onto one line, regardless of regime or confinement ratio. Lines of best fit yield the relations

(4.4a,b ) $$\begin{eqnarray}\displaystyle \frac{f_{l}}{w_{0}/r_{0}}=0.126\,Fr_{0}^{-1.58}\quad \text{and}\quad \frac{f_{r}}{w_{0}/r_{0}}=0.131\,Fr_{0}^{-1.60}, & & \displaystyle\end{eqnarray}$$

with coefficients of determination $R^{2}=0.97$ and $0.98$ , respectively. A similar scaling is retrieved when extracting $f_{l,r}$ via Fourier transforms of $x_{l,r}(t)$ (appendix D), hence (4.4) is not unique to our approach. Moreover, our attempts to include $W/r_{0}$ in the frequencies inevitably separate the data points into their respective sets, unless the scaling is chosen such that $(W/r_{0})^{-5/4}$ cancels out (i.e. $f_{l,r}\,(W/r_{0})^{2}\propto (w_{0}/r_{0})\,Fr_{c}^{-1.60}$ ). We conclude that the gap width does not affect the formation frequencies.

The absence of a dependence on $W$ from (4.4) is counter-intuitive and not well understood. Based on our observations, the quasi-steady flapping appears to stem from the interplay between the large-scale eddies and the fountain core (see § 3.4), rather than, say, from instabilities propagating from the source. Following this reasoning, the formation of eddies, being a large-scale process, should accordingly scale on $W$ . An argument, however, which supports the absence of $W$ in the frequency scalings centres on the volume of large-scale eddies $V_{e}$ , which follows $V_{e}\sim W\overline{\unicode[STIX]{x1D706}}_{l,r}^{2}\propto r_{0}^{3}Fr_{0}^{2}$ (from $\overline{\unicode[STIX]{x1D706}}_{l,r}\propto WFr_{c}$ , figure 11). Thus, $V_{e}$ is independent of $W$ . Although this still poses constraints on the volume flux supplying the eddies $Q_{e}$ , it is at least conceivable that $f_{l,r}\sim Q_{e}/V_{e}$ is independent of $W$ . Likewise, it is worth recalling that, within each set, the rise height $\overline{z}_{f}\propto r_{0}Fr_{0}$ ; the gap width entering only when a common scaling is sought across the sets. It therefore seems plausible that $W$ indeed cancels out in (4.4).

Finally, it is traditional for studies on confined flows to recast the formation frequencies of large-scale eddies into Strouhal numbers. Herein, we define the lateral Strouhal numbers

(4.5) $$\begin{eqnarray}\displaystyle St_{l,r}\equiv \frac{f_{l,r}}{\hat{u} _{l,r}/\overline{\unicode[STIX]{x1D706}}_{l,r}}, & & \displaystyle\end{eqnarray}$$

where $\hat{u} _{l,r}$ represents the characteristic velocity of large-scale eddies, defined later. Equation (4.5) is equivalent to rescaling the formation frequencies $f_{l,r}$ on length and velocity scales that pertain to the large-scale eddies themselves, rather than to the fountain. In quasi-two-dimensional jets, Landel et al. (Reference Landel, Caulfield and Woods2012a ) found Strouhal numbers ranging between $0.07$ and $0.25$ , approximately, at downstream distances of $20W$ to $60W$ . Specifically, the Strouhal number associated with a single eddy increased with downstream distance until merger with another eddy, at which point it consistently dropped to $0.07$ . These observations echo those of Dracos et al. (Reference Dracos, Giger and Jirka1992). Rocco & Woods (Reference Rocco and Woods2015) reported that the core meandered at an average Strouhal number of $0.16$ in quasi-two-dimensional plumes, which should broadly reflect the Strouhal numbers of lateral eddies.

Unfortunately, we do not have direct measurements of the velocities. Nevertheless, we may gain an estimate of the velocities representative of large-scale eddies by differentiating in time the position of the eddies’ edge, $x_{l,r}(t)$ . The characteristic velocities $\hat{u} _{l,r}$ are then calculated as the root mean square (r.m.s.) of $\text{d}x_{l,r}/\text{d}t$ . Some precaution is necessary given that $x_{l,r}(t)$ is not a smooth or continuous signal (see § 4.1). The details of how we calculated $\hat{u} _{l,r}$ are laid out in appendix E.

Figure 14.

Strouhal numbers $St_{l,r}$  (4.5) against $Fr_{c}$ . Faded vertical lines denote the error bars. Symbols as in figure 10.

Left and right Strouhal numbers $St_{l,r}$  (4.5) are plotted against $Fr_{c}$ in figure 14. Note that we evaluated (4.5) based on the values of $f_{l,r}$ , $\hat{u} _{l,r}$ and $\overline{\unicode[STIX]{x1D706}}_{l,r}$ for each experiment, i.e. we did not use fitted power laws. Despite some scatter, the Strouhal numbers for our experiments convincingly fall within the band prescribed by Landel et al. (Reference Landel, Caulfield and Woods2012a ) and, to some extent, Rocco & Woods (Reference Rocco and Woods2015). For sets A and B (

$St_{l,r}$

$0.12{-}0.16$

$St_{l,r}$

As an aside, time series such as shown in figure 12(b,c) can be used to develop rigorous methods for separating the asymmetric–stable, transitional and meandering regimes. The streaks in figure 12(b) and (c), for instance, stagger, which is indicative of flapping, and hence of the meandering regime. In the asymmetric regime, where large-scale eddies form primarily on one side, one time series showcases many streaks, whereas the other showcases barely any. The transitional regime features a mixture of both behaviours. Thus, diagnostics can be built by subtracting one time series from the other and inspecting the distribution of peaks in the resulting matrix. Yet another method consists in tracking the centre of gravity $\{z_{g}(t),x_{g}(t)\}$ from ‘fluctuating’ snapshots such as figure 12(a) and inferring the regime from its trajectory. As stated in § 3 however, in the vast majority of cases, visual inspection suffices to distinguish regimes.