2.1 General arrangement

Experiments were conducted in a water-filled visualisation tank of dimensions 0.7 m (height) $\times$ 1.0 m $^{2}$ (base) made of clear acrylic. A nozzle, positioned centrally on the base, steadily released a saline solution (NaCl) vertically upwards through a slot of length $L=\{300,150\}$  mm and half-width $b_{0}=0.5$  mm. The source aspect ratios were thus $L/b_{0}=\{600,300\}$ . Precautions were taken to ensure that the release conditions were uniform so that, as confirmed in §§ 3 and 4, the flow might reasonably be regarded as two-dimensional at the source. A detailed account of our bespoke nozzle design is given in appendix A. Notably in our arrangement, since the nozzle was positioned far from adjacent walls, the fountain produced was free to propagate in all directions.

Saline solution was delivered to the nozzle by means of a high-precision gear pump: an Ismatec MCP-Z Process at low to moderate flow rates, and a Pompe Cucchi HCX41 at high flow rates. These pumps were connected to an Atrato ultrasonic flow meter which logged the trace of the source volume flow rate $Q_{0}$ during experiments. This trace allowed us to verify that $Q_{0}$ was satisfactorily constant, with standard deviations within $1\,\%$ of the base flow rate. The density of the saline solution $\unicode[STIX]{x1D70C}_{0}$ and environment $\unicode[STIX]{x1D70C}_{a}$ were measured with an Anton Paar DMA5000 densitometer to within an accuracy of $5\times 10^{-5}~\text{g}~\text{cm}^{-3}$ . Densities ranged from $\unicode[STIX]{x1D70C}_{0}=1.0001$ to $1.0210~\text{g}~\text{cm}^{-3}$ , i.e. $(\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x1D70C}_{a})/\unicode[STIX]{x1D70C}_{a}=1.4\times 10^{-3}$ to $0.022$ , well within the recommended range for Boussinesq flows (Mehaddi, Vauquelin & Candelier Reference Mehaddi, Vauquelin and Candelier2015). The range of experimental parameters we investigated is summarised in table 2 together with our estimates of the associated errors. Additionally, we estimate that relative errors on our rise height and frequency measurements are within 2 %; see appendix B.

Table 2.

The range of experimental parameters covered in this study for measurement sets A and B. The vast majority of experiments were run for $Fr_{0}\geqslant 6.5$ , producing what may be regarded as ‘forced’ line fountains (1.1). The source reduced gravity is defined as $g_{0}^{\prime }=g(\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x1D70C}_{a})/\unicode[STIX]{x1D70C}_{a}$ . The mean source velocity $w_{0}$ (used in the estimate of $Fr_{0}$ ) was estimated as the ratio of the source volume flux to the source area, $w_{0}=Q_{0}/2b_{0}L$ . Errors on $Fr_{0}$ and $Re_{0}$ are upper-bound estimates (see appendix B).

To commence an experiment, a valve (blocking return flow from the nozzle and providing back-pressure for the flow meter) was opened and the flow rate of saline solution increased to the target value $Q_{0}$ . Recordings at $24$ frames per second (f.p.s.) were started once the fountain had reached a quasi-steady state and continued either until the flow had been captured for a sufficient number of characteristic time scales (see § 4.1), or until the saline layer, which accumulated at the base of the tank, deepened to the level of the flange into which the slot was set (figure 1). Crucially, our results, e.g. of rise height in figures 7 and 13, confirm that the flow was captured over a sufficiently long duration.

2.2 Flow visualisation

To aid flow visualisation, the source solution was stained with methylene blue and the flow uniformly backlit by an array of closely spaced high-frequency fluorescent tubes behind an opalescent screen. Two JAI Spark 5000-M CMOS cameras recorded the experiment: one, facing the long side of the nozzle, provided a side view ( $x{-}z$ plane); the other, facing a plane mirror suspended above the tank at $45^{\circ }$ to the horizontal, provided a top view ( $x{-}y$ plane). This arrangement is depicted in figure 1 with typical images of the fountain as viewed from these perspectives.

Figure 1.

Schematic of the general experimental set-up. Cameras simultaneously recorded (a) the top view as seen through a $45^{\circ }$ mirror and (b) the side view. For the light-induced fluorescence experiments (§ 2.3) the backlight was replaced with a vertical light sheet.

Beyond lending a qualitative picture of the flow, dye visualisations are commonly coupled with edge detection techniques to extract quantitative information (e.g. Burridge & Hunt Reference Burridge and Hunt2013). In the side view (figure 1 b), the sharp contrast between the dyed fountain fluid and the clear ambient environment guaranteed the success of such techniques. In the top view (figure 1 a) however, contrasts are low as the dyed fountain fluid is set against approximately equally dyed gravity currents that laterally traverse the flange; thus, traditional edge detection failed. Even on applying alternative techniques, including edge-aware contrast enhancement, wavelet filtering, and partial differential equation based de-noising (Mathworks 2017), we found no reliable method for identifying the outline of the fountain in top views. Nevertheless, we show in § 3 that observing the spanwise motion from a top view is central to understanding fountain structure in three dimensions. Moreover, although the outline of the fountain viewed from above may not be traced accurately, the frequency of the lateral fluctuations (i.e. those in the $y$ -direction) can be confidently inferred from top views without relying on edge detection (§ 4.4). Analysing the flow in the $(y{-}z)$ plane provided a robust means of identifying the local outline.

2.3 Refractive index-matched light-induced fluorescence (LIF) visualisations

Flow visualisations were also conducted to study the fountain in the $(y{-}z)$ plane at a section cutting perpendicularly across the centre of the slot. In this case, the source solution was stained with fluorescein sodium and the internal structure of the fountain made visible by means of a vertical light sheet. A high-power (480 W) LED light strip Metaphase UL line light encased behind a 1 m vertical array of cylindrical lenses produced the light sheet. To improve beam quality, the edges of the beam were trimmed with a rectangular aperture slot before the beam entered the visualisation tank. This allowed for a $d=10$  mm thick light sheet with low divergence (spreading by 2 mm over a distance of 600 mm) so that $d/L\approx \{1/60,1/30\}$ .

Optical distortions, arising from differences in the refractive indices of saline and freshwater, prevented the clear visualisation of the flow in section due to the relatively lengthy (circa 200 mm) path of non-uniform saline concentration the fluoresced light rays travelled through. These distortions were most significant close to the source where the fountain upflow is bounded by laterally flowing currents. To minimise distortions, the density of the source solution was varied by adding sodium nitrate (NaNO $_{3}$ ) salt and the environment by adding sodium chloride (NaCl) salt in such a way as to obtain different densities of solution with approximately equal refractive indices. The refractive indices of the chloride and nitrate solutions, and , were measured using a Hanna HI-96821 refractometer and matched to within .

3.1 Breakup of liquid sheet

Initial transients were recorded for a few typical experiments at 60 fps in order to assess that the nozzle design achieved what may be regarded as uniform release conditions (figure 2). In figure 2(a,b), a uniform dense liquid sheet rises upwards from the source, with roll-up vortices forming symmetrically at either end of the slot. As the sheet rises further, a counterflow begins to form (seen as a bright horizontal band in figure 2 c,d) which descends around the upflow. Simultaneously, instabilities begin to appear along the span and grow into larger billows (figure 2 c,d), eventually leading to the turbulent breakdown of the flow (figure 2 e–g). Subsequently, the fountain adopts and maintains a quasi-steady structure. Significantly, images (a–g) provide convincing evidence for the uniformity of the release conditions. Further evidence is provided in § 4.1.

Whilst we were able to achieve a target value of $Fr_{0}$ after only a relatively short ‘ramp up’ time $t_{r}$ (achieving, at best, $t_{r}=0.1$  s), our rig was not designed to additionally explore the initial rise height. Exploratory observations we made indicated that evaluating the initial rise height is not as straightforward as it may seem given what has been previously reported. We observed that this height varies considerably depending on the impulsiveness of the injection that forms the fountain, i.e. with $t_{r}$ . The initial transients are therefore considered no further and the remainder of this article focuses on the quasi-steady structure.

Figure 2.

The initial formation of a line fountain under near-impulsive start-up conditions from a slot of aspect ratio $600:1$ at $Fr_{0}=18.9$ highlighting the uniformity of the release conditions. Images (a–f), spaced $\unicode[STIX]{x0394}t=167$  ms apart, illustrate the breakdown of an initially laminar liquid sheet of dense fluid issuing vertically upwards from the source. The formation of the descending counterflow can clearly be seen from (c) onwards as a deepening horizontal band. Image (g), taken $60$ frames (i.e.  $1$  s) after image (f), depicts the turbulent breakdown. Movie 1 available online (available at https://doi.org/10.1017/jfm.2019.514).

Figure 3.

Synchronous side views ( $z/b_{0}$ versus $x/b_{0}$ , instantaneous rise heights $z_{v}(x,t=\text{cst}.)$ superimposed as a black line) and top views ( $y/b_{0}$ versus $x/b_{0}$ ) showing: (a) non-uniform flapping; (b) uniform flapping; and (c) disorganised flapping. Top views show: for case (a) a non-uniform oscillation along the span in the form of a sinusoidal-like wave, with the central section out of phase with the ends ( $y$ -scale magnified by a factor of two); for (b) that the fountain flaps uniformly along the span: ‘below’ ( $y<0$ ) the slot as seen in (i), then ‘above’ ( $y>0$ ) in (iii); and for (c) that whilst the flapping is uniform overall, the fountain top is dominated by large-scale structures which distort the regularity of this flapping. For (a) $\{L,b_{0}\}=\{300,0.5\}$  mm;  $Fr_{0}=10.2$ ( $Re_{0}\approx 53$ );  $\unicode[STIX]{x0394}t=2.0$  s ( $\cong 1.0~w_{0}/g_{0}^{\prime }$ ); for (b): $\{L,b_{0}\}=\{150,0.5\}$  mm;  $Fr_{0}=26.4$ ( $Re_{0}\approx 114$ );  $\unicode[STIX]{x0394}t=3.0$  s ( $\cong 1.0~w_{0}/g_{0}^{\prime }$ ); and for (c) $\{L,b_{0}\}=\{300,0.5\}$  mm;  $Fr_{0}=51.3$ ( $Re_{0}\approx 114$ );  $\unicode[STIX]{x0394}t=7.5$  s ( $\cong 2.0~w_{0}/g_{0}^{\prime }$ ). Movies 2–4 available online.

3.2 Spatial structure and flapping patterns

Sample images of that quasi-steady structure are given in figure 3(a–c) in order to reveal the role of $Fr_{0}$ (and, to a lesser extent, of $L/b_{0}$ ). Although our results herein may be regarded as $Re_{0}$ -independent, as reasoned in appendix C, the approximate value of $Re_{0}$ is given in the captions for completeness. The images shown were chosen specifically to highlight the distinct behaviours observed, behaviours we refer to as non-uniform flapping (figure 3 a), uniform flapping (figure 3 b) and disorganised flapping (figure 3 c) on account of the swaying (or flapping) motions of the fountain about a vertical plane through the source, hereafter referred to as the centre plane.

Judging first from the side views alone, the flow instantaneously resembles a wall of fluid with irregular wave-like crenellations formed by distinct turbulent structures, or eddies, that populate the fountain top. The motion of these structures, which extend in the vertical ( $z$ ) and spanwise ( $x$ ) directions, results in the localised vertical fluctuations. Neither the height of rise, nor the size of these structures appears to vary noticeably along the span for given release conditions. On increasing $Fr_{0}$ , the fountain rises to a greater height. The scale of structures at, or near, the fountain top also increases both in the vertical and spanwise directions. For example, at $Fr_{0}\approx 10$ (figure 3 a) one counts $O(20)$ structures over a length $L=600b_{0}$ (i.e. above the source), whereas at $Fr_{0}\approx 50$ (figure 3 c) one discerns only three to four structures over the same length. Thus, whilst the absolute size of large-scale structures increases with $Fr_{0}$ , the relative fraction of space they occupy along the span is evidently a function of both $Fr_{0}$ and $L/b_{0}$ ; the fountain in figure 3(b), $L/b_{0}=300$ , though rising only half as high (approximately) as the fountain in figure 3(a), $L/b_{0}=600$ , is comprised of a similar number of structures because its aspect ratio is half as large. As the structures become larger on increasing $Fr_{0}$ , their presence increasingly dominates the flow. As a result, the fountain loses some of its homogeneity along the span, e.g. figure 3(c).

Visually, it is difficult to discern from the side views whether the number of structures (and, by extension, the source aspect ratio $L/b_{0}$ ) has a significant effect on the flow overall. Therefore, we retain from the side views that, as confirmed in § 4.1, the flow may be regarded as approximately uniform in the sense that there is no apparent variation along the span either in the extent of the vertical fluctuations or in the underlying rise height.

To gain a more complete picture of line fountain behaviour, we now turn to the top views of the flow (bottom rows in figure 3(a–c) taken synchronously with the side views. The vertical arrows (far right, figure 3 a–c) indicate that, instantaneously, fountain fluid collapses to a single side (

$y>0$

$y<0$

$x$

The fundamental patterns of flow observed in figure 3(a–c) were characteristic of all experiments. Non-uniform flapping systematically occurred at lower values of $Fr_{0}$ ( $Fr_{0}\lesssim 27$ for $L/b_{0}=600$ and $Fr_{0}\lesssim 20$ for $L/b_{0}=300$ ), followed at higher $Fr_{0}$ by uniform flapping and eventually, more disorganised, albeit relatively uniform, flapping. The non-uniform flapping takes the form of a sinusoidal oscillating wave with approximately one and a half wavelengths, or winds (figure 3 a), maintained across the range of forcing. The threshold values of $Fr_{0}=\{20,27\}$ are compared with estimates based on a physical argument in § 4.6. At lower $Fr_{0}$ , an increase in the number of winds was observed (appendix D), however, we do not examine these cases further as the fountains were laminar or transitional in character. A preliminary conclusion is nonetheless that line fountains adopt different spanwise behaviours depending on the values of $Fr_{0}$ and $L/b_{0}$ . The dependence on both $Fr_{0}$ and $L/b_{0}$ was to be expected since $Fr_{0}$ governs the size of structures populating the fountain top, whilst $L/b_{0}$ determines the fractional length of the fountain span which any one of these structures occupies. The presence of many, smaller structures would impose less uniformity on the flow than that of fewer, larger structures, explaining the decreasing wavelength with decreasing $Fr_{0}$ .

A final observation from the top views concerns the extent of the lateral excursions (i.e. those in $y$ ) that constitute the flapping. Identifying an outline to the fountains as seen from above is not straightforward, as discussed in § 2.2, even by eye. However, notwithstanding their precise extent, it is evident from figure 3(a–c) that the lateral excursions are significant at between two thirds to one unit of rise height.

3.3 Internal structure and flapping mechanism

Flow visualisations of the internal structure of line fountains, achieved by light-induced fluorescence with refractive index matching of the $(y{-}z)$ plane at $x=0$ , i.e. the central section, are shown in figures 4–6 for the non-uniform, uniform and disorganised flapping behaviours, respectively. As we shall see, in each case, flapping entails that locally the fountain counterflow forms predominantly to one side of the slot at any given instant.

Viewed in a vertical section, it is evident that these fountains do not resemble the classic upflowing core and symmetric counterflow structure that is characteristic of round fountains. The structure is instead reminiscent of quasi-two-dimensional confined fountains (Debugne & Hunt Reference Debugne and Hunt2018) in that it consists of a slender upflowing core that undergoes a quasi-periodic swaying to each side of the slot. In the proximity of the fountain top, the core wraps up into vortex sheets that extend in the $x$ -direction (into the page, figures 4–6). Lateral excursions of the fountain are substantially greater than the spread in width of the upflowing core due to the coiling of these vortex sheets; lateral excursions appear to scale on the rise height as may be independently inferred from the top view on the flow (§ 2.2). These large-scale vortical structures are subsequently shed and collapse under gravity. It is the regular impingement of these structures with the horizontal flange, and subsequent interaction of the unsteady lateral current they create with the fountain core, that is instrumental to inducing the flapping behaviours.

The flapping mechanism is most clearly appreciated in figure 5 which has been annotated so as to link the lateral motion of the core with the shedding, left then right, of vortical structures from the fountain top. The flapping of the fountain to either side of the slot appears to be intrinsically linked to feedback between the fountain upflow and the (quasi-)regular shedding of large-scale vortical structures away from the fountain. The periodicity in the resulting flapping is less stark at larger rise heights (see figure 6 and movie 7) in which, for limited periods of time, the fountain is shrouded by a counterflow on both sides.

Figure 4.

Non-uniform flapping. LIF visualisation of the central section for $\{L,b_{0}\}=\{300,0.5\}$  mm and $Fr_{0}=12.1$ ( $Re_{0}\approx 41$ ). Images spaced at time intervals of $\unicode[STIX]{x0394}t=2.3$  s ( $\cong 1.0~w_{0}/g_{0}^{\prime }$ ). Movie 5 available online.

Figure 5.

Uniform flapping. LIF visualisation of the central section for $\{L,b_{0}\}=\{150,0.5\}$  mm and $Fr_{0}=24.1$ ( $Re_{0}\approx 84$ ). Images spaced at time intervals of $\unicode[STIX]{x0394}t=1.7$  s ( $\cong 0.5~w_{0}/g_{0}^{\prime }$ ). Movie 6 available online.

Figure 6.

Disorganised flapping. LIF visualisation of the central section for $\{L,b_{0}\}=\{300,0.5\}$  mm and $Fr_{0}=57.6$ ( $Re_{0}\approx 194$ ). Images spaced at time intervals of $\unicode[STIX]{x0394}t=4.7$  s ( $\cong 1.0~w_{0}/g_{0}^{\prime }$ ). Movie 7 available online.

4.1 Uniformity

To assess the uniformity of the fountains along the length of the slot, in figure 7 the instantaneous time-averaged rise height $\left\langle z_{v}\right\rangle _{t}$ normalised by the source jet length $b_{0}Fr_{0}^{4/3}$ (Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks1979) is plotted against the spanwise coordinate $x/b_{0}$ . These profiles were obtained by averaging over a time interval $[0,T]$ the local rise height $z_{v}(x,t)$ at every $x$ -location, i.e.

(4.1) $$\begin{eqnarray}\left\langle z_{v}\right\rangle _{t}(x^{\prime })=\frac{1}{T}\mathop{\sum }_{t=0}^{T}z_{v}(x^{\prime },t),\quad x^{\prime }\in [-L/2,L/2].\end{eqnarray}$$

The duration $T$ was typically $O(10^{2})$ – $(10^{3})$ multiples of the buoyancy time scale $w_{0}/g_{0}^{\prime }$ , a time scale characteristic of the vertical fluctuations (§ 4.3), and the start time $t=0$ chosen to be well after the initial period of formation (§ 3.1). Shorter averaging times were unavoidable at high $Fr_{0}$ due to the increased rate at which the saline layer in the tank deepened. The profiles are convincingly uniform despite some irregularities at high $Fr_{0}$ . Judging from these profiles alone, the fountains might reasonably be regarded as two-dimensional. End effects are only evident as a marginal reduction in rise heights near the ends of the slot.

Figure 7.

Normalised time-averaged rise height profiles. The profile for each individual fountain is plotted as a thin line and the average across all experiments is plotted as a thick line. (a) Set A ( $L/b_{0}=600$ ), plotting 97 experiments. (b) Set B ( $L/b_{0}=300$ ), plotting 28 experiments. For consistency with the fits (4.3), only profiles for $Fr_{0}\geqslant 10$ are included. (c,d) Normalised difference between the experiment-averaged and the centreline rise heights plotted against $Fr_{0}$ for sets A and B, respectively.

4.2 Rise height

In figure 8, the double-averaged (spanwise and time) rise height $\overline{\overline{z}}_{v}$ scaled on $b_{0}$ is plotted against $Fr_{0}$ , where

(4.2) $$\begin{eqnarray}\overline{\overline{z}}_{v}=\frac{1}{T}\frac{1}{L}\mathop{\sum }_{t=0}^{T}\mathop{\sum }_{x=-L/2}^{L/2}z_{v}(x,t).\end{eqnarray}$$

Hereafter, $\overline{\overline{z}}_{v}$ is referred to as the experiment-averaged rise height. Also plotted are the (corrected) rise heights $\overline{z}_{v}$ from Baines et al. (Reference Baines, Turner and Campbell1990) and those from Zhang & Baddour (Reference Zhang and Baddour1997). Although rise heights are not the primary focus of our work, making the comparison between end-on, $\overline{z}_{v}$ , and spanwise, $\overline{\overline{z}}_{v}$ , rise heights is informative, and it is evident that our data for $\overline{\overline{z}}_{v}$ lie in close agreement with previous measurements of $\overline{z}_{v}$ . Lines of best fit to our data for $Fr_{0}\geqslant 10$ suggest that

(4.3) $$\begin{eqnarray}\overline{\overline{z}}_{v}/b_{0}=\unicode[STIX]{x1D6FE}_{z}\,Fr_{0}^{4/3},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}_{z}=2.00$ for $L/b_{0}=600$ (blue circles) and $1.62$ for $L/b_{0}=300$ (orange diamonds). Both of our sets of rise height measurements fall predominantly between the fits of Baines et al. (Reference Baines, Turner and Campbell1990) and Zhang & Baddour (Reference Zhang and Baddour1997). The near absence of end effects (figure 7) explains, in part, why different boundary conditions at $x=\pm L/2$ , such as confining walls (Zhang & Baddour Reference Zhang and Baddour1997; Srinarayana et al. Reference Srinarayana, Williamson, Armfield and Lin2010) or damping screens (Baines et al. Reference Baines, Turner and Campbell1990), appear to have no adverse effect on the average rise heights. The difference in pre-factor of $19\,\%$ between our datasets is within experimental error (estimated as $7\,\%$ on $\overline{\overline{z}}_{v}/b_{0}$ and $14\,\%$ on $Fr_{0}$ , appendix B), and should not be interpreted as a dependence of rise height on $L/b_{0}$ . The influence of $L/b_{0}$ , both in terms of flapping behaviour and rise height, is discussed below.

At higher $Fr_{0}$ , the rise heights we measured fall marginally below the fits (4.3). This trend (for $Fr_{0}\gtrsim 30$ , figure 8 b) is expected to be a consequence of the free ends of the slot. Indeed, while we inspect the rise heights $z_{v}(x,t)$ above the slot only, i.e. for $x\in [-L/2,L/2]$ (4.2), as the fountain rises it spreads beyond $x=\pm L/2$ (e.g. figure 3 c). This spanwise expansion, absent from set-ups in which walls confine the ends, effectively represents a reduction in kinetic energy for the vertical motion above the slot. Therefore, fountains for which a spanwise spread is possible do not rise as high.

To make this argument more tangible, let us suppose that the fountain spreads at a rate $c_{y}$ in $y$ , and $c_{x}$ in $x$ at the ends. The total volume of fluid above the slot, $V_{in}$ , can be approximated as a trapezoid of length $L$ , so that $V_{in}=\left(4b_{0}+2c_{y}\overline{\overline{z}}_{v}\right)\,\overline{\overline{z}}_{v}L/2$ . Thus at high Froude numbers, $V_{in}\approx c_{y}\,\overline{\overline{z}}_{v}^{2}L$ . Similarly, the total volume of fluid lying beyond the ends, $V_{out}$ , can be approximated as two halves of an elliptic cone with axes $c_{x}\overline{\overline{z}}_{v}$ and $c_{y}\overline{\overline{z}}_{v}$ , so that $V_{out}\approx \unicode[STIX]{x03C0}c_{x}c_{y}\overline{\overline{z}}_{v}^{3}/3$ . Alternative geometric shapes can be proposed for $V_{in}$ and $V_{out}$ , yet it seems inevitable that $V_{in}\propto \overline{\overline{z}}_{v}^{2}L$ and $V_{out}\propto \overline{\overline{z}}_{v}^{3}$ . It follows that

(4.4) $$\begin{eqnarray}\frac{V_{out}}{V_{in}}\sim c_{x}\frac{\overline{\overline{z}}_{v}}{L}.\end{eqnarray}$$

Thus, the greater the ratio of $\overline{\overline{z}}_{v}$ to $L$ , the greater the volume and, by extension, kinetic energy that is imparted to the spanwise motion for $|x|>L/2$ . Since greater rise heights are achieved on increasing $Fr_{0}$ , the ratio (4.4) is consistent with the view that the rise heights will depart from (4.3) at higher $Fr_{0}$ . Moreover, the onset of that departure is expected to be a function of the ratio $\overline{\overline{z}}_{v}/L$ . The relative difference between the centreline (taken to be the central 1 % span, $x\in [-0.005L,0.005L]$ ) and the experiment-averaged rise height is shown in figure 7(c,d). Evidently, the centreline of the fountain rises marginally higher in most cases than the span average for $L/b_{0}=600$ (figure 7 c) and is marginally lower for $L/b_{0}=300$ (figure 7 d). Differences in rise height are within 10 % of the span-averaged rise height and are smaller than the scale of typical vertical fluctuations (see § 4.3).

For the vast majority of our experiments, the rise heights follow (4.3); even at high $Fr_{0}$ , the observed departure from (4.3) is relatively small. Therefore, we proceed on the premise that our measurements of line fountain rise height lie in good agreement with previous measurements with the mean across our datasets being $\overline{\overline{z}}_{v}/b_{0}=1.81Fr_{0}^{4/3}$ .

Figure 8.

(a) Experiment-averaged rise height $\overline{\overline{z}}_{v}/b_{0}$ against $Fr_{0}$ from our side view $(x{-}z)$ measurements. Also plotted are the fits by Zhang & Baddour (Reference Zhang and Baddour1997) and Baines et al. (Reference Baines, Turner and Campbell1990) (1.1) from their end view $(y{-}z)$ measurements. Shaded regions bound our experimental error. The dark red shaded symbols indicate data extracted from our LIF visualisations. (b) Experiment-averaged rise height data from (a) shown on a compensated plot. The trends shown in the data indicate a departure from a 4/3 power-law scaling at higher $Fr_{0}$ .

4.3 Vertical fluctuations

Fountains are typified by quasi-periodic fluctuations whose scaling, both in amplitude and frequency, is often revelatory of the underlying dynamics that govern their behaviour (Williamson et al. Reference Williamson, Srinarayana, Armfield, McBain and Lin2008; Srinarayana et al. Reference Srinarayana, Williamson, Armfield and Lin2010; Burridge & Hunt Reference Burridge and Hunt2013). For the fountains studied herein, there are two principal fluctuating components: one vertical, of smaller amplitude and shorter period, and one lateral, of larger amplitude and longer period. Viewed from the side, vertical fluctuations appear as a local phenomenon in that there is no obvious coherence between fluctuating motions along the span (cf. movies 2–4 supporting figure 3 a–c). By contrast, lateral fluctuations (as viewed from the top) appear as a global phenomenon in that they form organised patterns, for example, with the fountain swaying back and forth as a whole.

The amplitude of vertical fluctuations, $\unicode[STIX]{x1D6FF}z_{v}$ , calculated as twice the average of the standard deviations of the experiment-averaged rise heights $\overline{\overline{z}}_{v}$ , is plotted against $Fr_{0}$ in figure 9. Two standard deviations provided a good measure for the peak-to-trough amplitude of typical fluctuations (based on visual inspection), which motivated our choice of definition for $\unicode[STIX]{x1D6FF}z_{v}$ . Moreover, deviations about the mean rise height profile $\left\langle z_{v}\right\rangle _{t}(x)$ were uniform along $x$ in the time average and distributed normally, so that the single measure $\unicode[STIX]{x1D6FF}z_{v}$ is representative of the whole fountain.

The fluctuations echo our observations made about the rise heights (figure 8). First, $\unicode[STIX]{x1D6FF}z_{v}$ scales on $Fr_{0}^{4/3}$ , with the fluctuations being slightly lower for set B. This scaling is also evident from figure 9(b) where, for $Fr_{0}\lesssim 30$ , the fluctuations represent a near constant fraction of the experiment-averaged rise height; at circa 20 %, discounting a few outliers, this fraction is somewhat higher than for round fountains ( ${\sim}15\,\%$ , Burridge & Hunt Reference Burridge and Hunt2013). Second, $\unicode[STIX]{x1D6FF}z_{v}$ departs from the $4/3$ -scaling at higher values of $Fr_{0}$ . This departure is likely to be due to the reasons laid out in § 4.2, the fountain expanding beyond the ends of the source so that a fraction of kinetic energy is unavailable to drive the vertical motion over $x\in [-L/2,L/2]$ .

Figure 9.

Amplitude of vertical fluctuations $\unicode[STIX]{x1D6FF}z_{v}$ against $Fr_{0}$ , normalised (a) by the slot half-width $b_{0}$ and (b) by the experiment-averaged rise height. Symbols as in figure 8.

Figure 10.

Characteristic frequency of vertical fluctuations $f_{v}$ against $Fr_{0}$ , (a) dimensional frequency and (b) frequency normalised by the inverse of the time scale $b_{0}/w_{0}$ . Vertical error bars are smaller than the symbol size. In (b), lines of best fit (4.5) have a slope of $-2$ and are offset for ease of reading. Symbols as in figure 8.

Next, we inspect the typical (or dominant) frequencies for the vertical motion, $f_{v}$ . By applying a fast Fourier transform to the signal $z_{v}(x=x^{\prime },t)$ , the dominant frequency was extracted at each location $x=x^{\prime }$ . The distribution of the resulting frequencies (one for each $x^{\prime }$ ) was typically bimodal, with the larger mode corresponding to the first dominant frequency, and a distinctly lower mode corresponding to the second. Therefore, to obtain a characteristic frequency $f_{v}$ representative of the whole fountain, we define $f_{v}$ as the mode of the individual frequencies sampled at each $x$ -location.

The frequency $f_{v}$ ( $\text{s}^{-1}$ ) of the vertical fluctuations is plotted against $Fr_{0}$ in figure 10(a) and normalised by the inverse of the time scale $b_{0}/w_{0}$ in figure 10(b). Evidently, vertical frequencies decrease with increasing $Fr_{0}$ , consistent with the view that the amplitudes of characteristic fluctuations increase with $Fr_{0}$ (figure 9). The frequencies in both measurement sets are close in magnitude and follow the same trend. Imposing an exponent of $-2$ as the nearest integer power, lines of best fit follow

(4.5) $$\begin{eqnarray}f_{v}/(w_{0}/b_{0})=\unicode[STIX]{x1D6FE}_{v}\,Fr_{0}^{-2},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}_{v}=0.249$ for $L/b_{0}=600$ (blue circles) and $0.299$ for $L/b_{0}=300$ (orange diamonds). Coefficients of determination for the fits (4.5) are $R^{2}=0.95$ and $0.96$ for sets A and B, respectively. We reason in appendix B that the differences in pre-factor in (4.5) are within experimental error. Figure 10 and expressions (4.5) therefore indicate that the vertical frequencies scale on $w_{0}Fr_{0}^{-2}/b_{0}=g_{0}^{\prime }/w_{0}$ , the inverse of the buoyancy time scale. This scaling is characteristic of fountains that are sufficiently highly forced.

The same scaling was identified by Burridge & Hunt (Reference Burridge and Hunt2013) for the frequencies of a fluctuating, forced and round fountain top. Their scaling was developed by considering the relevant length and velocity scales for vertical fluctuations from a theoretical standpoint, and the relevant time scale deduced by combining them appropriately. Proceeding in reverse, we may infer a scale for the characteristic velocities $w_{v}$ of vertical fluctuations. Since $\unicode[STIX]{x1D6FF}z_{v}/b_{0}\propto Fr_{0}^{4/3}$ (figure 9) and $f_{v}/(w_{0}/b_{0})\propto Fr_{0}^{-2}$ (figure 10), we infer $w_{v}/w_{0}\propto Fr_{0}^{-2/3}$ , which corresponds to the theoretical scaling for two-dimensional fountains (van den Bremer & Hunt Reference van den Bremer and Hunt2014b ). Although $w_{v}$ was not measured directly in our experiments, an estimate of $w_{v}$ can be gained from the time rate of change of $z_{v}$ . In appendix G we define $w_{v}$ as the root-mean-square of $\text{d}z_{v}/\text{d}t$ and confirm that $w_{v}$ scales on $w_{0}Fr_{0}^{-2/3}$ , in accord with two-dimensional theory.

4.4 Lateral fluctuations

As raised in § 2.2, the irregular and weak contrasts between the fountain and the background precluded us from reliably tracking the lateral excursions from top views. Whilst measuring the amplitude of lateral fluctuations requires a well-defined outline, measuring the frequency of these fluctuations does not. If one could count the number of times $n$ a fountain flaps over the duration $T$ of an experiment, then an estimate for the flapping frequency would be given by $n/T$ . In Debugne & Hunt (Reference Debugne and Hunt2018), we implemented an ‘eddy-counting’ approach which rested on a similar principle, and through which the formation frequency of large eddies in meandering fountains was identified. We adapted this approach, not to count eddies, but to count flapping events.

Proceeding as described in appendix E, $(y,t)$ -time series were constructed for each experiment, the dominant frequencies extracted at each $y$ -location through Fourier transforms and their mode calculated to determine $f_{h}$ , the dominant frequency of the lateral fluctuations. The resulting frequencies are plotted against $Fr_{0}$ in figure 11(a). Akin to the vertical frequencies (figure 10), the lateral frequencies closely follow an inverse square power law. Imposing an exponent of $-2$ , lines of best fit are

(4.6) $$\begin{eqnarray}f_{h}/(w_{0}/b_{0})=\unicode[STIX]{x1D6FE}_{h}\,Fr_{0}^{-2},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}_{h}=0.125$ for $L/b_{0}=600$ (blue circles) and $0.148$ for $L/b_{0}=300$ (orange diamonds), with $R^{2}=0.98$ for both fits. The pre-factors $\unicode[STIX]{x1D6FE}_{h}$ in (4.6) are almost exactly half the pre-factors $\unicode[STIX]{x1D6FE}_{v}$ in (4.5) for both sets. Plotting the ratio of $f_{v}$ to $f_{h}$ (figure 11 b) confirms this observation: the ratio $f_{v}/f_{h}$ scatters around a value of two, with a few outliers falling mostly on harmonics.

Figure 11(b) reveals a key dynamic feature of line fountains, namely, that the largest (dominant) vertical fluctuations are caused by lateral swaying, i.e. by the flapping motions. Based on this premise, a ratio of $f_{v}/f_{h}=2$ indeed makes sense, since for every lateral sway to one side of the source, the fountain passes above the source (i.e. through the central plane) twice. This finding explains why the vertical fluctuations (figure 13) arrange into patterns reminiscent of the lateral motion: at the largest scale, the lateral motion dictates the vertical fluctuations. Note that the relationship $f_{v}/f_{h}=2$ remains valid at large values of $Fr_{0}$ , despite the rather disorganised appearance (to the naked eye) of images such as figure 3(c).

Figure 11.

(a) Characteristic frequencies for the lateral fluctuations $f_{h}$ normalised by $w_{0}/b_{0}$ . Lines of best fit (4.6) are shifted for ease of reading. (b) Ratio of the vertical frequencies $f_{v}$ to the lateral frequencies $f_{h}$ . Symbols as in figure 8.

From the LIF visualisations, the lateral excursions were quantified and a complementary suite of frequency measurements for the lateral and vertical fluctuations acquired. The technique we implemented to extract these quantities is illustrated in appendix F for an example experiment. Regarding the vertical motion, mean rise heights $\overline{z}_{v,(x=0)}$ and rise height fluctuations $\unicode[STIX]{x1D6FF}z_{v,(x=0)}$ measured in the illuminated central section are superimposed onto figures 8 and 9, and again indicate good agreement between the LIF and dye-attenuation approaches. Regarding lateral excursions, a minimum representative value might reasonably be anticipated to be comparable with the scale of the vertical fluctuations (i.e. circa 20 % of the rise height, figure 9) on account of these fluctuations being on the scale of a notional eddy. Flapping enhances this lateral excursion as the upflowing core then leans to either side of the slot (e.g. figure 4); based on our observations, a maximum representative value for a lateral excursion is expected to be comparable with the rise height. The actual magnitude of the time-averaged peak lateral excursions acquired from the LIF data, relative to the slot width and rise height, is plotted in figure 12(a,b) against $Fr_{0}$ . Consistent with our observations, whilst lateral excursions increase in scale as the fountain rises higher (increasing $Fr_{0}$ , figure 12 a), as a fraction of the mean rise height measured in the illuminated plane, these excursions decrease (figure 12 b). At lower $Fr_{0}$ , for which flapping occurs at a well-defined frequency, the lateral excursions are comparable to the rise height. At higher $Fr_{0}$ , the disorganised flapping results in the magnitude of the lateral excursions decreasing and approaching what would be expected if the fountain was symmetric (Hunt & Burridge Reference Hunt and Burridge2015).

As an aside, we attempted to inspect frequencies associated with secondary peaks in the Fourier transforms of the vertical motion. Since the frequencies of the most energetic fluctuations are those associated with the global lateral motion, we questioned whether secondary peaks were indicative of smaller, locally driven fluctuations. In a number of time records for the instantaneous rise height (e.g. figure 20 c), peaks could be identified in super-harmonics of the base frequency. These peaks are most likely to result from the nonlinear interaction of the fluctuations with the base frequency. Identifying these super-harmonics consistently for each experiment, however, proved to be inconclusive owing to the turbulent nature of the fountain top.

Figure 12.

Magnitude of mean lateral excursions versus $Fr_{0}$ as obtained from the LIF experiments: (a) scaled on the source half-width and (b) scaled on the time-averaged rise height measured on the illuminated central section.

4.5 Spatio-temporal contour maps for the vertical fluctuations

Figure 13.

Signatures for the lateral behaviours of turbulent forced line fountains. Contour maps of the vertical fluctuations, $z_{v}(x,t)-\left\langle z_{v}\right\rangle _{t}(x)$ . (a) Diagonal banding (chevrons) characteristic of non-uniform flapping ( $L/b_{0}=600$ , $Fr_{0}=10.2$ ); (b) vertical banding of uniform flapping ( $L/b_{0}=300$ , $Fr_{0}=26.4$ ); and (c) broken banding of disorganised flapping ( $L/b_{0}=600$ , $Fr_{0}=51.3$ ). Dotted lines demark the time instants which correspond to the images in figure 3(a–c). In each contour map, the colour scale ranges between minus and plus the largest vertical fluctuation (in absolute value).

Figures 9 and 10 each plot a single amplitude and a single frequency per experiment, respectively. As such, the two figures separate the spatial and temporal aspects of the vertical fluctuations. An alternative format for visualising the vertical fluctuations in space and time simultaneously is presented in figure 13. Each of the three contour maps therein plots, at location $x^{\prime }$ and instant $t^{\prime }$ , the difference between the instantaneous rise height $z_{v}(x=x^{\prime },t=t^{\prime })$ and the time-averaged rise height at that location, $\left\langle z_{v}\right\rangle _{t}(x=x^{\prime })$ . Thus, these maps give an $(x,t)$ , local and instantaneous, representation of the vertical fluctuations. To enable direct comparison between the vertical fluctuations and observations made in § 3, the contours in figure 13(a–c) are calculated for the experiments shown in figure 3(a–c) that typify the three distinct lateral behaviours.

Plotted thus, it becomes evident that the vertical fluctuations that occur along the span of the fountain are correlated. The fluctuations are arranged into regular patterns of bands with alternating positive and negative fluctuations. The width of each band (i.e. in time) corresponds to a fluctuating period (cf. the frequencies in figure 10), whilst its elevation corresponds to a fluctuating amplitude. The pervasiveness of these patterns indicates that, at the largest scale, the vertical fluctuations are not a local process, but instead are governed by a regular, global process. This assertion becomes clearer on recognising that the patterns in figure 13(a–c), respectively, are signatures for the non-uniform, uniform and disorganised flapping lateral behaviours noted earlier in our observations of the flow in the top views:

1. (i) Figure 13(a): the chevrons imply that, at any given time, the rise height fluctuations alternate sign along the span $x$ in an organised manner. For example, when fluctuations are positive in the central region, they are negative in neighbouring regions. The presence of chevrons, or indeed of any regular diagonal pattern, therefore indicates that the rise height fluctuations along the span are alternately in and out of phase. This is the same observation as we made for the lateral fluctuations in figure 3(a), where the top views clearly show the fountain oscillating about the slot.

2. (ii) Figure 13(b): the vertical banding indicates that the fluctuations are uniform along the span. Correspondingly, in the top views of figure 3(b), the fountain flaps uniformly.

3. (iii) Figure 13(c): the regular banded structure is lost and the contours depict less organised, broken bands. This signature is consistent with figure 3(c), where large-scale structures at the fountain top appear to dominate the flow.

We conclude that the vertical fluctuations bear the imprint of the lateral swaying of the fountain, i.e. the motions in the vertical and lateral planes are connected. Finally, we note that each band is comprised of a number of distinct structures whose spanwise extent is constant (approximately) in a given regime, but clearly increases across the regimes in figure 13(a–c) – a trend consistent with the observations made regarding the scale of eddying structures in § 3.2.

4.6 Parameter space for flapping patterns

In § 3, we established that different lateral behaviours occur at different $Fr_{0}$ -thresholds depending on the value of $L/b_{0}$ . Intuitively, we would expect the fountain to behave more uniformly along the span if the characteristic length scale of structures at the top were comparable with, rather than small compared with, the span. In the hypothetical limiting case of the fountain consisting of a single coherent structure, for instance, the flow sways back and forth as one.

Although we do not have a direct measure for the size of individual structures from our visualisations, it is reasonable to assume that they scale on the length relevant to forced line fountains, i.e. on $b_{0}Fr_{0}^{4/3}$ and, by extension, on $\overline{\overline{z}}_{v}$ (§ 4.3). The ratio of the size of structures to the length of the fountain should therefore be proportional to $\overline{\overline{z}}_{v}/L$ . This suggests thresholds for the different flapping behaviours in terms of the height-to-length ratio $\overline{\overline{z}}_{v}/L$ .

To determine the flapping behaviour, we constructed contour maps of the vertical fluctuations, such as those shown in figure 13, for each experiment and classified the flow into the following categories according to the signature that emerged: (i) non-uniform flapping with more than one and a half winds, i.e. wavelengths less than $2L/3$ , as is characteristic of laminar fountains, (appendix D), and for turbulent fountains, either (ii) non-uniform flapping with diagonal bands or chevrons (figure 13 a), (iii) uniform flapping with vertical bands (figure 13 b) or (iv) uniform flapping with broken bands (figure 13 c). The resulting regime diagram, plotting $\overline{\overline{z}}_{v}/L$ against $L/b_{0}$ , is shown in figure 14 where each experiment has been colour coded according to its flapping behaviour. Although we observe some variability, the flapping behaviours can be delineated by constant values of the height-to-length ratio $\overline{\overline{z}}_{v}/L$ as follows:

1. (i) $\overline{\overline{z}}_{v}/L\lesssim 0.05$ or $L/b_{0}Fr_{0}^{4/3}\gtrsim 36$ , non-uniform flapping, $L_{w}\lesssim 2L/3$ ;

2. (ii) $0.05\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.30$ or $6\lesssim L/b_{0}Fr_{0}^{4/3}\lesssim 36$ , non-uniform flapping, $L_{w}\approx 2L/3$ ;

3. (iii) $0.30\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.45$ or $4\lesssim L/b_{0}Fr_{0}^{4/3}\lesssim 6$ , uniform flapping; and

4. (iv) $\overline{\overline{z}}_{v}/L\gtrsim 0.45$ or $L/b_{0}Fr_{0}^{4/3}\lesssim 4$ , disorganised flapping,

$L_{w}$ being the longitudinal wavelength of the flapping.

Figure 14.

(a) Regime diagram for classification of lateral flapping behaviour in terms of $\overline{\overline{z}}_{v}/L$ and $L/b_{0}$ , colour coded for the characteristic signatures in the contour maps of vertical fluctuations (figure 13). Colour code as follows: (i) red, non-uniform with wavelengths less than $2L/3$ (appendix D); (ii) orange, diagonal bands, including chevrons (figure 13 a); (iii) green, vertical bands (figure 13 b); and (iv) turquoise, disorganised broken vertical bands (figure 13 c). (b) The inset provides a zoomed view onto $\overline{\overline{z}}_{v}/L\leqslant 0.15$ . (c) An alternative regime diagram in which the ratio of the slot length to the normalised predicted rise height is plotted against $Re_{0}$ .

Interestingly, this delineation entails that $\overline{\overline{z}}_{v}/L$ governs which flapping pattern is adopted and how two-dimensional a flow should be regarded. Regarding the latter, we recall from § 4.2 that rise heights begin to depart from the two-dimensional theoretical scaling for larger $Fr_{0}$ , and by extension for larger $\overline{\overline{z}}_{v}/L$ . Thus, despite their instantaneous uniformity along the span, uniformly flapping line fountains, which form only for the larger values of $\overline{\overline{z}}_{v}/L$ considered, cannot be regarded as the more two-dimensional of the fountains examined (though in a time-averaged sense, cross-sections through uniformly flapping and non-uniformly flapping line fountains would be identical).

Proceeding on the basis that the spanwise and vertical length scales of the largest-scale structures are comparable, so that specifically the spanwise extent of structures scales as $x_{e}\sim b_{0}Fr_{0}^{4/3}$ , then the number $m$ of structures along the span of a fountain top of length $L_{t}$ would be anticipated to scale as $m\sim L_{t}/b_{0}Fr_{0}^{4/3}$ . If we now postulate that the basic arrangement that is required to form a lateral flapping pattern with wavenumber $k$ comprises of structures vertically above the source and structures deflected to either side, then a minimum of $m=m_{min}=7$ is required to achieve $k=3/2$ . Such an arrangement is shown schematically in figure 15 with the largest-scale structures numbered. Introducing $c_{e}$ as a constant of proportionality, so that $x_{e}=c_{e}~b_{0}Fr_{0}^{4/3}$ , and noting that the minimum number of structures $m=m_{min}(k)$ corresponds to the maximum (or threshold) source Froude number $Fr_{0}=Fr_{0,max}$ for the given behaviour, then we have

(4.7) $$\begin{eqnarray}Fr_{0,max}=\left(\frac{L_{t}}{b_{0}}\right)^{3/4}\left(\frac{1}{c_{e}~m_{min}}\right)^{3/4}.\end{eqnarray}$$

The length $L_{t}$ of a fountain whose ends are free increases with $Fr_{0}$ . However, for the modest increases relative to $L$ observed in the non-uniform flapping regime, it is not unreasonable to approximate the length of the fountain top by the length of the slot. Substituting $L_{t}/b_{0}=L/b_{0}$ and $m_{min}=7$ in (4.7), we predict $Fr_{0,max}=\{16.8,28.2\}$ for $L/b_{0}=\{300,600\}$ with $c_{e}=1$ . In our experiments, the corresponding threshold values of the source Froude number at which non-uniform flapping was maintained (§ 3.2) were $Fr_{0,max}=\{20,27\}$ for $L/b_{0}=\{300,600\}$ . Given the approximations made and the simple form proposed for the basic arrangement of structures (figure 15), these predictions show remarkably good agreement with the experimental values.

Figure 15.

Plan view of a fountain top, of length $L_{t}$ , depicting the minimum number ( $m_{min}=7$ ) of large-scale structures each with spanwise length $x_{e}$ that arrange to form a pattern with wavenumber $k=3/2$ , as is characteristic of non-uniform flapping. Structures numbered $\{1,3,5,7\}$ are in the vertical plane of the source and structures $\{2,4,6\}$ are deflected to either side.