2. Key results on Boussinesq fountains

Let us first discuss some notable results on turbulent Boussinesq fountains.

One of the pioneering contributions is that due to Turner (Reference Turner1966). From both experimental and theoretical approaches, he found that the mean fountain height $H_{m}$ and the transient fountain height $H_{tr}$ (both divided by the radius of the source $b_{i}$ ) scale linearly with the source Froude number $Fr$ :

(2.1a,b ) $$\begin{eqnarray}\frac{H_{m}}{b_{i}}\simeq 2.46Fr\quad \text{and}\quad \frac{H_{tr}}{b_{i}}\simeq 3.52Fr.\end{eqnarray}$$

The Froude number, which quantifies the ratio between the buoyancy and the momentum at the source, is defined as follows:

(2.2) $$\begin{eqnarray}Fr=\frac{w_{i}}{\sqrt{g{\it\eta}_{i}b_{i}}},\end{eqnarray}$$

where $w_{i}$ is the bulk velocity of the released fluid, $g$ is the gravitational acceleration and ${\it\eta}_{i}$ is the source density deficit. Here, the density deficit is defined by

(2.3) $$\begin{eqnarray}{\it\eta}_{i}=\frac{|{\it\rho}_{i}-{\it\rho}_{0}|}{{\it\rho}_{0}},\end{eqnarray}$$

where ${\it\rho}_{i}$ is the density of the released fluid and ${\it\rho}_{0}$ is the ambient density. A striking result found by Turner (Reference Turner1966) is that the ratio $H_{m}/H_{tr}$ remains constant and around 1.43.

As the range investigated by Turner (Reference Turner1966) was limited to high Froude numbers (i.e. forced fountains), Kaye & Hunt (Reference Kaye and Hunt2006) recently investigated low-Froude-number turbulent fountains. Their results exhibited the existence of two supplementary regimes: the weak fountain regime for which $H_{m}/b_{i}\propto Fr^{2}$ and the very weak fountain regime for which $H_{m}/b_{i}\propto Fr^{2/3}$ . In the same vein, Burridge & Hunt (Reference Burridge and Hunt2012) studied the dynamics of salt-water Boussinesq fountains experimentally. Following an experimental approach similar to that used by Turner (Reference Turner1966), they found respectively for very weak, weak and forced fountains

(2.4) $$\begin{eqnarray}\displaystyle \displaystyle \frac{H_{m}}{b_{i}} & = & \displaystyle 0.81Fr^{2/3}\quad \text{for }Fr<1,\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle \displaystyle \frac{H_{m}}{b_{i}} & = & \displaystyle 0.86Fr^{2}\quad \text{for }1<Fr<3,\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle \displaystyle \frac{H_{m}}{b_{i}} & = & \displaystyle 2.46Fr\quad \text{for }Fr>3.\end{eqnarray}$$

Burridge & Hunt (Reference Burridge and Hunt2013) also investigated the magnitude and the frequency of the fountain height oscillations around $H_{m}$ . In the particular case of forced fountains, they exhibited the existence of a dominant frequency $f$ which they used to build a Strouhal number ( $St=fb_{i}/w_{i}$ ). This Strouhal number was found to scale as $Fr^{-2}$ . The magnitude of the fountain fluctuations ${\it\delta}z$ (divided by the source radius) was found to scale linearly with the Froude number.

We shall return to these Boussinesq results for comparison after describing the experimental set-up and the measurement methods.

3. Experimental apparatus

Figure 2 shows a sketch of the experimental apparatus. It is composed of a thin horizontal rectangular panel ( $1.5~\text{m}\times 2.5~\text{m}$ ). The panel is located $1.5~\text{m}$ from the floor of the laboratory, a distance that is sufficiently large to avoid any vertical confinement effect. To produce the fountains, a light air/helium mixture is continuously released downward from a nozzle of radius $b_{i}$ , flush with the surface and located at the centre of the panel.

Figure 2. Schematic of the experimental apparatus: $1$ , air source; $2$ , helium source; $3$ , flow meters; $4$ , hydrochloric acid; $5$ , ammoniac; $6$ , plenum chamber; $7$ , rectangular panel; $8$ , laser sheet; $9$ , CCD camera.

Figure 3. Instantaneous snapshots of (a) a stable fountain ( $Fr=1$ , $Re=570$ and ${\it\rho}_{i}/{\it\rho}_{0}=0.95$ ) and (b) a fluctuating fountain ( $Fr=14$ , $Re=1770$ and ${\it\rho}_{i}/{\it\rho}_{0}=0.44$ ).

Before being released, the mixture first fills a large plenum chamber, located above the panel, which contains honeycomb sections. The air and helium flow rates are controlled by two independent flow meters. For low and moderate flow rates, Bronkhorst flow meters type EL-FLOW, which cover the range $0{-}12~\text{m}^{3}~\text{h}^{-1}$ with an accuracy of $2\,\%$ of the measurement, are used. For higher flow rates, Bronkhorst flow meters type E-7000, which cover the range $0{-}60~\text{m}^{3}~\text{h}^{-1}$ with an accuracy of 1 % of the measuring range, are used.

To visualise the flow, the released mixture is seeded with ammonium salt obtained by chemical reaction of ammonia vapour with hydrochloric acid. It should be noted that the mass of salt added to the flow is very weak and therefore does not affect the density of the fountain. A laser light sheet (argon $2~\text{W}$ ) cuts through the central plane of the fountain perpendicularly to the horizontal panel (see figure 3). Images are recorded as $8$  bit bitmap image files at a frequency of $40$  f.p.s. with a high-speed camera type PCO $1200$ hs equipped with a $50~\text{mm}$ Nikon lens.

For the experiments exploited in this paper, nine different nozzle diameters are used: $10$ , $20$ , $30$ , $38$ , $45$ , $84$ , $112$ , $118$ and $138~\text{mm}$ . These diameters, together with the large density deficit range covered by air–helium mixtures, allow us to produce fountains with Froude numbers lying between $0.2$ and $64$ and Reynolds numbers lying between $120$ and $3500$ . It should be noted that the (source) Reynolds number is defined by

(3.1) $$\begin{eqnarray}Re=\frac{{\it\rho}_{i}w_{i}b_{i}}{{\it\mu}_{i}},\end{eqnarray}$$

where ${\it\mu}_{i}$ is the dynamic viscosity of the mixture. It should be noted that in our experiments, the ratio between the dynamic viscosity of the released fluid and that of the ambient air (i.e.  ${\it\mu}_{i}/{\it\mu}_{air}$ ) lies between $1$ and $1.12$ . According to this relatively constant ratio, it can be assumed that the entrainment process will not be affected by the variation of the viscosity (see Campbell & Turner Reference Campbell and Turner1985).

To measure the mean fountain height $H_{m}$ , recordings begin a while after the release reaches its steady-state phase. At least $2600$ successive frames are recorded for each experiment in order to ensure statistical convergence of the measurements. The digital images are post-processed following an approach similar to those used by Williamson et al. (Reference Williamson, Srinarayana, Armfield, McBain and Lin2008) and Burridge & Hunt (Reference Burridge and Hunt2012), which can be summarised as follows.

1. (1) Frames are read by using the software MATLAB^® and converted in terms of grey level matrices. It should be noted that the values of the grey levels lie between $0$ (black) and $255$ (white).

2. (2) The matrices are then line-by-line averaged (i.e. horizontally averaged), producing vertical vectors of grey levels.

3. (3) These vertical vectors are concatenated to produce an image corresponding to a time series of rise heights for each fountain.

4. (4) The time evolution of the fountain height $H(t)$ , from which both the mean height $H_{m}$ and the fluctuations are measured, is extracted by using a light intensity threshold delimiting the fountain (white) envelope and the (black) surrounding.

It should be noted that, in this method, distances are measured with an accuracy of $4$ pixels, corresponding to an actual distance of $5~\text{mm}$ in physical space.

Results are presented and compared with the Boussinesq correlations in the next section.

4. Observations and results

In this study $88$ fountains have been investigated over the ranges $0.2<Fr<64$ , $120<Re<3500$ and $0.13<{\it\rho}_{i}/{\it\rho}_{0}<0.96$ . These ranges allow us to cover transitional turbulent fountains (weak and forced) and laminar transitional fountains (weak and very weak). As expected, two different behaviours have been observed according to the values of the Froude number and the Reynolds number.

1. (1) For $Fr\lesssim 1$ and for a sufficiently small Reynolds number (typically $Re\lesssim 1000$ ), the fountain exhibits a stable aspect (see figure 3 a). In this regime, a steady toroidal recirculating cell surrounding the base of the up-flow of the fountain is observed.

2. (2) For $Fr>1$ and for a sufficiently high Reynolds number (typically $Re\gtrsim 1000$ ), the fountain exhibits a turbulent aspect (see figure 3 b). In these cases, vertical oscillations of the fountain height around $H_{m}$ are observed.

4.1. Fountain mean heights

We now focus on the influence of the density contrast (between the fountain and the ambient) on the mean fountain height. Figure 4 shows the experimental results together with the Boussinesq correlations (2.4), (2.5) and (2.6) corresponding to very weak, weak and forced fountain regimes respectively. In addition, in this figure, the data points are also shaded according to the value of the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ (light dots for weak density contrast and dark dots for strong density contrast). It is seen that some points deviate markedly from the Boussinesq correlations, especially when the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ is small (i.e. large density contrast). This is particularly true for forced fountains. This suggests non-Boussinesq effects.

Figure 4. Comparisons between the present experimental results for the mean height $H_{m}$ and the correlations by Burridge & Hunt (Reference Burridge and Hunt2012): solid line, $H_{m}/b_{i}=2.46Fr$ ; dashed line, $H_{m}/b_{i}=0.86Fr^{2}$ ; dash-dotted line, $H_{m}/b_{i}=0.81Fr^{2/3}$ . The experimental data points $(\circ )$ are shaded according to the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ .

Figure 5. Determination from experimental data $(\circ )$ the corrective function associated with the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ . The solid line represents the best fit, with a slope of $1/2$ , and the dashed line, whose slope is $3/4$ , represents the solution derived from the Morton-like model (see appendix A). The dash-dotted line corresponds to the Boussinesq case.

To quantify these effects more precisely, let us first consider the experimental data corresponding to forced fountains ( $Fr>3$ ). To do so, the ratio $H_{m}/(2.46b_{i}Fr)$ (bearing in mind that $2.46b_{i}Fr$ corresponds to the mean height in the Boussinesq case) is plotted on a linear scale and on a log–log scale in figure 5 with respect to ${\it\rho}_{i}/{\it\rho}_{0}$ . As the ratio $H_{m}/(2.46b_{i}Fr)$ is expected to be equal to $1$ for Boussinesq fountains, any deviation from unity therefore provides insight into the role of non-Boussinesq effects. As expected, for weak density contrasts (i.e.  ${\it\rho}_{i}/{\it\rho}_{0}\rightarrow 1$ ), the experimental results are close to unity. However, for greater density contrasts (thus increasing the non-Boussinesq effects), the fountain reaches a mean height significantly weaker than that predicted by the Boussinesq correlation. In particular, the deviation of the ratio $H_{m}/(2.46b_{i}Fr)$ from unity clearly depends on the value of ${\it\rho}_{i}/{\it\rho}_{0}$ . It turns out that the data points (see figure 5 b) are scattered about a straight best-fit line (solid line) whose slope is close to $1/2$ . This best-fit relation reads as

(4.1) $$\begin{eqnarray}\frac{H_{m}}{2.46b_{i}Fr}\simeq 1.05\left(\frac{{\it\rho}_{i}}{{\it\rho}_{0}}\right)^{1/2}\quad \text{for }\frac{{\it\rho}_{i}}{{\it\rho}_{0}}<0.8.\end{eqnarray}$$

This is one of the main results of the present experimental study.

It should be noted that the present experimental results are in qualitative agreement with the recent experiments by Ahmad & Baddour (Reference Ahmad and Baddour2015) carried out for upward turbulent fountains. Indeed, for a fixed value of the Froude number, these authors have shown that an increase of the density difference leads to a decrease of the fountain height.

Such a behaviour of the fountain height with respect to the density ratio seems to be not so obvious. In particular, it cannot be recovered by using a Morton-like theoretical approach in which the entrainment coefficient is modified to account for non-Boussinesq effects (as proposed by Ricou & Spalding (Reference Ricou and Spalding1961) or Rooney & Linden (Reference Rooney and Linden1996) for turbulent plumes). To show this more clearly, the conservation equations applied to a rising fountain are solved in appendix A, leading us to a fountain height corrected with a density ratio to the power of $3/4$ . The Morton-like solution, given by (A 12), is then plotted in figure 5. It is seen that this approach overestimates the density effects. However, it is important to mention that the theoretical model presented in appendix A does not take into account the influence of the fountain down-flow. This is a possible explanation for the differences observed.

For convenience, the dependence on ${\it\rho}_{i}/{\it\rho}_{0}$ can be included in the definition of the non-Boussinesq Froude number,

(4.2) $$\begin{eqnarray}Fr_{NB}=Fr\left(\frac{{\it\rho}_{i}}{{\it\rho}_{0}}\right)^{1/2}=\frac{w_{i}}{\sqrt{gb_{i}\bar{{\it\eta}_{i}}}},\quad \text{where }\bar{{\it\eta}_{i}}=\frac{{\it\rho}_{0}-{\it\rho}_{i}}{{\it\rho}_{i}},\end{eqnarray}$$

where the density difference $\bar{{\it\eta}_{i}}$ is made dimensionless by using ${\it\rho}_{i}$ instead of ${\it\rho}_{0}$ (Boussinesq case). Introduction of such a non-Boussinesq Froude number indeed allows the dependence on both the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ and the Boussinesq source Froude number $Fr$ to be captured. In particular, (4.1) can be simply rewritten as follows:

(4.3) $$\begin{eqnarray}\frac{H_{m}}{b_{i}}\simeq 2.58Fr_{NB}.\end{eqnarray}$$

It should be noted that to study turbulent plumes with large density contrasts, this non-Boussinesq Froude number has already been used by several authors, such as, for instance, Crapper & Baines (Reference Crapper and Baines1978).

Figure 6. Ratio between the dimensionless fountain height and its corresponding correlation as a function of the source Reynolds number.

Relation (4.3) thus extends (2.6) to fountains with large density differences. In addition, we have also verified that in this regime, the fountain height does not depend on the source Reynolds number $Re$ . Indeed, figure 6 shows the ratio $H_{m}/(2.58b_{i}Fr_{NB})$ as a function of $Re$ . We observe that all of the experimental points are close to unity despite a weak scattering, not exceeding $10\,\%$ for the lowest Reynolds number values. These results corroborate the findings of Philippe et al. (Reference Philippe, Raufaste, Kurowski and Petitjeans2005), who have exhibited experimentally the independence of the mean fountain height $H_{m}$ on the source Reynolds number for $Re>200$ .

We now propose to plot in figure 7, for all of the experiments (i.e. including the very weak and weak regimes), the mean fountain height as a function of the non-Boussinesq Froude number $Fr_{NB}$ . As expected, we observe that the experimental points in the forced regime align along the straight line corresponding to (4.3). It is remarkable, however, that for very weak and weak fountains, the experimental points also align with two straight lines (on a log–log scale) whose slopes are respectively $2/3$ and $2$ . These two exponents are the same as those found by Burridge & Hunt (Reference Burridge and Hunt2012) for Boussinesq fountains. This leads us to the conclusion that the classical Boussinesq correlations concerning the mean fountain height can be straightforwardly generalised to the non-Boussinesq case by using $Fr_{NB}$ . These new correlations are provided in table 1 with their specific ranges of validity given in terms of $Fr_{NB}$ .

Figure 7. Comparisons between the experimental results obtained for the fountain mean height $H_{m}$ and the following correlations: solid line, $H_{m}/b_{i}=2.58Fr_{NB}$ ; dashed line, $H_{m}/b_{i}=1.4Fr_{NB}^{2}$ ; dash-dotted line, $H_{m}/b_{i}=1.1Fr_{NB}^{2/3}$ . The data points $(\circ )$ are shaded according to the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ .

4.2. Fountain fluctuations

We now discuss the results concerning the fountain height fluctuations around $H_{m}$ . Indeed, in some cases, the fountain height fluctuates due to self-sustained instabilities associated with the dynamics of the fountain collapsing on itself. For all experiments in which the fountain fluctuates, we determine first the standard deviation ${\it\delta}z$ and second the emerging frequency $f$ from a Fourier transform analysis of the fountain height time signal $H(t)$ .

Figure 8. Variation of the standard deviation ${\it\delta}z$ as a function of the non-Boussinesq Froude number $Fr_{NB}$ . The best-fit line (solid line) is given by (4.4). The experimental data points $(\circ )$ are shaded according to the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ .

Table 1. Flow regime and the associated correlations for the mean fountain height as a function of the non-Boussinesq Froude number.

Figure 8 shows the standard deviation ${\it\delta}z$ versus $Fr_{NB}$ in the case of forced fountains. The experimental data are best-fitted by the following linear relation:

(4.4) $$\begin{eqnarray}\frac{{\it\delta}z}{b_{i}}\simeq 0.18Fr_{NB}.\end{eqnarray}$$

Again, this relation straightforwardly extends the Boussinesq correlation found by Burridge & Hunt (Reference Burridge and Hunt2012) (i.e.  ${\it\delta}z/b_{i}\simeq 0.14Fr$ ) by using the non-Boussinesq Froude number.

To analyse the results concerning the emerging frequency $f$ , let us reintroduce the Strouhal number $St$ based on the source radius and the source velocity,

(4.5) $$\begin{eqnarray}St=\frac{fb_{i}}{w_{i}}.\end{eqnarray}$$

Figure 9 shows that, in the case of highly forced fountains ( $Fr_{NB}\gg 1$ ), the Strouhal number is proportional to $Fr_{NB}^{-2}$ . This dependence is similar to that found by Clanet (Reference Clanet1998) (i.e.  $St=0.66Fr_{NB}^{-2}$ ) for non-miscible fountains (water in ambient air) as well as that found by Burridge & Hunt (Reference Burridge and Hunt2013) and Williamson et al. (Reference Williamson, Srinarayana, Armfield, McBain and Lin2008) (i.e.  $St\propto Fr^{-2}$ ) for Boussinesq fountains (salt-water into fresh water).

To analyse this dependence, let us recall that usually the initial source momentum $M_{i}$ and the source buoyancy flux $B_{i}$ are defined by

(4.6a,b ) $$\begin{eqnarray}M_{i}=\frac{{\it\rho}_{i}}{{\it\rho}_{0}}w_{i}^{2}b_{i}^{2}\quad \text{and}\quad B_{i}=\frac{g({\it\rho}_{0}-{\it\rho}_{i})}{{\it\rho}_{0}}w_{i}b_{i}^{2}.\end{eqnarray}$$

According to the relation $St\propto Fr_{NB}^{-2}$ , it can be inferred that

(4.7) $$\begin{eqnarray}\frac{1}{f}\propto \frac{M_{i}}{B_{i}}.\end{eqnarray}$$

As a result, this time scale is found to be independent of the radius of the fountain source. This is a common feature of elongated fountains (i.e.  $H_{m}\gg b_{i}$ ), for which a radius-independent ‘bobbing’ behaviour is observed (see Vinoth & Panigrahi Reference Vinoth and Panigrahi2014).

Figure 9. Variation of the Strouhal number $St$ as a function of the non-Boussinesq Froude number $Fr_{NB}$ . The experimental data points $(\circ )$ are shaded according to the density ratio ${\it\rho}_{i}/{\it\rho}_{0}$ .