2.1. Experiments

Figure 2 illustrates the experimental set-up. A perspex tank of internal dimensions $25~\text{cm}\times 25~\text{cm}\times 30~\text{cm}$ is filled with water. The tank incorporates a series of low- and high-level openings, through which clear and particle-laden fluid are pumped. Fresh water containing particles is supplied to the tank through an opening located at the centre of the tank’s top surface (vent A in figure 2). This particle-laden fluid passes through a glass nozzle of an internal diameter 4.2 mm: the vertical distance between the tip of the nozzle and the tank’s floor is 26 cm. Two holes of diameter 8.2 mm are drilled at high level in the tank (vents B in figure 2) and are used as inlets for the background flow. Through these holes, fresh water is supplied to the tank at a constant rate during the experiment. The base of the tank incorporates two outlet openings of diameter 8.2 mm (vents C in figure 2). Through these vents, fluid is extracted from the tank over the course of an experiment.

Figure 2. Schematic illustrating the laboratory set-up.

In all experiments, the particle-laden fluid is supplied to the tank at a constant volume flux of $1~\text{cm}^{3}~\text{s}^{-1}$ using a peristaltic pump (Watson Marlow). This kind of pump generates a pulsed input condition. In our experiments, the pump frequency is of order 10 Hz, and so the time lapse between two pulses produced by the pump (a fraction of a second) is very small compared with the time required for a change in the distribution of particles in the tank (several seconds or minutes). Hence, the input condition is approximated as a continuous flow of particle-laden fluid supplied to the tank. To explore the impacts of larger or smaller particle fluxes, the concentration of particles in this fluid, $C_{in}$ , is changed throughout the series of experiments (see table 1). The ventilation fluid is also pumped into the tank at different flow rates, ranging between 5 and $15~\text{cm}^{3}~\text{s}^{-1}$ , to explore the impacts of a larger or smaller ventilation of the space (see table 1).

Silicon carbide particles of mean diameter $18.3~{\rm\mu}\text{m}$ and of mean settling speed $\bar{v}_{s}=4.04\times 10^{-4}~\text{m}~\text{s}^{-1}$ are used in the experiments (Carborex F400 by Washington Mills). The distribution of settling speeds about the mean was measured using a combination of a settling experiment and an optical particle characterisation experiment (Mingotti & Woods Reference Mingotti and Woods2015). A graph showing the distribution of settling speeds in the mixture of particles used in the experiments is given in appendix A (figure 19).

We use an image-analysis technique to assess the average line-of-sight concentration of particles in suspension in the fluid. An electroluminescent light sheet is connected to the rear of the tank and provides uniform illumination (see figure 2). The light produced by this sheet is attenuated if the fluid contains a suspension of particles. During an experiment, we take photographs of the tank using a Nikon D90 digital camera, and based on the light intensity relative to a set of calibration values, we infer the line-of-sight average concentration of particles in the tank (see appendix B and Mingotti & Woods Reference Mingotti and Woods2015).

An experiment is started when a flow of particle-laden fluid is supplied to the tank through vent A. The bulk density of this fluid is larger than that of the ambient fluid, which does not contain particles. As a result, the particle-laden fluid forms a turbulent plume, which descends through the tank while entraining ambient fluid. On reaching the base of the tank, the particle-laden fluid spreads radially over the floor (see figure 3 a). A certain number of particles become suspended in the ambient fluid in the lower part of the tank (see figure 3 b,c). Over the course of the experiment, the concentration of particles in the ambient fluid increases and so observing the motion of the plume fluid in the lower part of the tank becomes increasingly difficult. Eventually, a steady state is reached (see figure 3 d–f).

Figure 3. Transient concentration of particles in suspension in the tank during experiment b (see table 1). A series of images illustrate the concentration distribution in the tank at the following times after the beginning of the experiment: 30 s and 5, 28, 43, 70 and 86 min (a–f).

Figure 4. Steady-state distribution of particles in the tank in a typical experiment (experiment e: see table 1): (a) steady-state concentration of particles in the tank using false colours; (b) horizontally averaged particle concentration profile.

Figure 5. Schematic showing the conditions used in the modelling (§ 2).

Figure 4 shows the average of 1000 frames of experiment e (see table 1) once it has reached steady state, corresponding to 500 s of experimental time. Figure 4(a) is a false-colour image which illustrates the difference in the average concentration of particles through the tank. Figure 4(b) shows the horizontal average of the concentration at each level in the tank. Figures 4(a) and 4(b) show that at steady state particles are not uniformly distributed in the tank. In the upper part of the tank the ambient fluid does not contain particles at steady state (dark blue colour in the false-colour image). The particle-laden plume originates at the top of the tank, and descends through the enclosure while entraining ambient fluid (light blue colour in figure 4 a). In the central part of the tank, at an approximate distance 0.10–0.15 m below the source of particle-laden fluid, we observe a vertical gradient in the concentration of particles in the ambient fluid (note the light blue/yellow colours in figure 4 a, and the change in the horizontally averaged particle concentration profile plotted in figure 4 b). In the lower part of the tank, the ambient fluid contains a relatively large number of particles (red colour in the false-colour image). The concentration of these particles is approximately uniform (see figure 4 b, in which the particle concentration becomes approximately uniform at a distance 0.20–0.22 m below the source of the particle-laden fluid).

Motivated by the outcomes of the experiments, we develop a model based on plume theory which describes the transport of small, heavy particles through a space with downward ventilation. In § 2.2, we describe the particle-laden turbulent plume which is generated at the top of the space when particles are supplied from a localised source located at high level (see figures 3 and 4). The plume descends through the space while entraining ambient fluid, and so a return flow is produced in the ambient fluid in the lower part of the space (see figure 1 c). In § 2.3, we describe how particles of different sizes are suspended in the ambient fluid by this return flow. In § 2.4, we quantify the steady-state concentration of particles in the ambient fluid at the base of the space, and we compare the predictions of our model with the results of the experiments.

2.2. Particle-laden plume

We assume that the space is of a height $H$ and of a cross-sectional area $A$ , and is ventilated at a steady rate $Q$ (figure 5). The ventilation flow is supplied to the space at high level and extracted from the space at low level. A constant flux of relatively heavy small particles is supplied to the space at high level. We denote this flux by $P_{in}$ . A turbulent particle-laden plume is generated at the top of the enclosure, and the plume fluid descends through the space driven by negative buoyancy flux $B_{p}$ , given by

where ${\it\rho}_{P}$ is the density of the particles, ${\it\rho}_{W}$ is the density of the ambient fluid, and $g$ is gravitational acceleration. While descending through the space, the plume entrains ambient fluid. If the fall speed of the particles is much smaller than the speed of the plume, the particle-laden fluid in the plume behaves analogously to that in a single-phase descending plume (see Zarrebini & Cardoso Reference Zarrebini and Cardoso2000; Woods Reference Woods2010), and so plume theory indicates that at a vertical distance $z$ from the source, the volume flux is given by

where ${\it\lambda}$ is a universal constant which depends on the plume entrainment coefficient ${\it\alpha}$ (Morton, Taylor & Turner Reference Morton, Taylor and Turner1956). When the plume volume flux $Q_{p}$ exceeds the ventilation flux $Q$ , an upward return flow is produced in the ambient fluid (see figure 1 c). So long as the plume’s cross-sectional area is much smaller than the tank’s floor area, $A$ , we estimate that the ambient fluid rises through the space at mean speed

Equations (2.1)–(2.3) are used to measure the properties of the particle-laden plume observed during the experiments (see figures 3 and 4 a). Before the beginning of an experiment, some dye is added to a layer of ambient fluid at the base of the tank. We measure the speed at which the dyed fluid rises as a result of the return flow in the lower part of the tank and use (2.3) to infer the plume flow rate $Q_{p}$ at different levels in the tank. We then use (2.2) to estimate that in our experiments the particle-laden plume has a coefficient ${\it\lambda}=0.17\pm 0.01$ , which is consistent with the estimates of Bower et al. (Reference Bower, Caulfield, Fitzgerald and Woods2008) and Kuesters & Woods (Reference Kuesters and Woods2011). The same equations are also used to estimate that the virtual origin of the plume is located at a distance $z_{0}=7.9\pm 2.0~\text{mm}$ above the tip of the glass nozzle through which the particle-laden fluid is pumped into the tank (vent A in figure 2).

2.3. Suspension of particles in the ambient fluid

If the upward speed of the return flow exceeds the speed at which particles settle, particles become suspended in the ambient fluid (see Phillips & Woods Reference Phillips and Woods2001). Equation (2.3) indicates that the speed at which the ambient fluid rises through the space, $v_{w}$ , equals the speed at which a particle settles through the fluid, $v_{s}$ , when the plume volume flux reaches the critical level

Using a combination of (2.2) and (2.4), we calculate that this happens at a vertical distance $h_{p}$ from the plume virtual origin

Equation (2.5) indicates that particles with different settling speeds are suspended at different levels in the space by the upward return flow. We use (2.5) to calculate the maximum vertical distance from the floor of the tank, $H-h_{p}(v_{s})$ , at which particles with different fall speeds are suspended in our experiments. Figure 6(a) shows that in a typical experiment in which the tank is ventilated at a rate $Q=10\times 10^{-6}~\text{m}^{3}~\text{s}^{-1}$ and a flux of particles with buoyancy $B_{p}=7\times 10^{-8}~\text{m}^{4}~\text{s}^{-3}$ is supplied at the top of the tank (see table 1), all particles used in the experiment can be suspended a distance $H-h_{p}\leqslant 5{-}20~\text{cm}$ approximately above the base of the tank, with the lower limit corresponding to the larger particles in the mixture and the upper limit corresponding to the smaller particles in the mixture (see appendix A). This is consistent with the experimental results illustrated by figure 4(a,b), which show that the concentration of particles is approximately uniform in the lower 4–5 cm of the tank, and decreases gradually to zero by a distance of approximately 15–18 cm above the floor.

Figure 6. (a) Range of distances from the floor of the tank, $H-h_{p}(v_{s})$ (m), at which particles with different fall speeds $v_{s}~(\text{m}~\text{s}^{-1})$ are suspended in the ambient fluid in a typical experiment with $Q=10\times 10^{-6}~\text{m}^{3}~\text{s}^{-1}$ and $B_{p}=7\times 10^{-8}~\text{m}^{4}~\text{s}^{-3}$ (see table 1). (b) Vertical distance between the source of the plume and the height at which particles with a mean fall speed $\bar{v}_{s}$ are suspended in the ambient fluid, $\bar{h}_{p}$ . For each of the experiments listed in table 1, we compare the distance measured in the experiment with the distance predicted by (2.5). (c) Comparison of the steady-state particle concentration profiles obtained in a number of experiments (lines) with the prediction of the model (dots). In plotting the dotted curve in the figure, we use (2.3) to calculate the mean upward speed of the ambient fluid at different levels in the tank, $v_{w}(z)$ . In our experiments, we use particles with a range of settling speeds (see appendix A). For a given ambient fluid speed $v_{w}(z)$ , we estimate the fraction of particles in the mixture which have a settling speed $v_{s}<v_{w}(z)$ (see figure 19). These particles are sufficiently small that they can be suspended by the return flow. The dotted curve in (c) illustrates how this fraction of suspended particles changes at different levels in the tank. We observe that the concentration of the suspended particles increases rapidly in the zone $\bar{h}_{p}-{\rm\Delta}h<z<\bar{h}_{p}+{\rm\Delta}h$ .

Since all particles in the mixture are suspended in the ambient fluid at the base of the tank, we can identify the maximum height at which particles with a mean settling speed $\bar{v}_{s}$ are suspended in the ambient fluid by detecting the level at which the horizontally averaged mean concentration of particles in the tank is reduced by 50 % relative to the maximum concentration at the base of the tank. We let $\bar{h}_{p}$ denote the vertical distance between the source of the plume and this height. For each of the experiments listed in table 1, we compare the height $\bar{h}_{p}$ measured during the experiment with that predicted by (2.5) for $v_{s}=\bar{v}_{s}$ (figure 6 b). We observe that there is relatively good agreement between model predictions and experimental results, with errors of less than 5 % in most experiments.

In figure 6(c) we compare the horizontally averaged mean particle concentration profiles obtained in a number of experiments (solid lines) with the profile predicted by the model (black dots, plotted using (2.5)). The profiles are plotted in dimensionless form: particle concentration is scaled by the maximum concentration at the base of the tank, $C_{a}$ , while height is scaled by $\bar{h}_{p}$ (see figure 6 b). The graph shows that the model based on plume theory provides a leading-order prediction of the horizontally averaged mean concentration of particles at different levels in the tank, with average errors of order 5–10 % approximately. However, we observe that the magnitude of errors increases to 15–20 % at the base of the tank. In § 3 we will demonstrate that the particle-laden plume transitions into a momentum-driven fountain in the lower part of the tank. As a result of the difference between the descending volume fluxes in the plume and in the fountain, the effective speed of the return flow in the lower part of the tank will be different from that predicted by (2.3). This explains why the model prediction based on plume theory (black dots in figure 6 c) is less accurate in this region.

In figure 6(c) we also observe that the distribution of settling speeds in the mixture of particles used in the experiments is sufficiently narrow that the fraction of particles suspended in the ambient fluid rapidly increases from 10 % to 90 % in a relatively small region in the tank, $\bar{h}_{p}-{\rm\Delta}h<z<\bar{h}_{p}+{\rm\Delta}h$ , with ${\rm\Delta}h\approx 3{-}4~\text{cm}$ (see appendix A).

2.4. Concentration of particles in the ambient fluid at the base of the space

After quantifying the maximum height at which particles are suspended in the ambient fluid, we now develop a model based on plume theory to quantify the concentration of particles in the suspension. For convenience, we first consider the simpler case in which all particles have uniform diameter $D$ and settling speed $v_{s}$ . We then consider the more general case in which particles with different diameters and settling speeds are supplied to the space.

Equation (2.5) indicates that if all particles have a uniform fall speed $v_{s}$ , an interface is formed at a distance $h_{p}$ from the plume source (see (2.5)): the interface separates the ambient fluid in the upper part of the space, which does not contain particles, from the ambient fluid in the lower part of the space, which contains a suspension of particles (figure 5). The concentration of particles in the plume fluid at the level of the interface is given by

To calculate the concentration of particles in the ambient fluid below the interface, we note that at steady state the flux of particles supplied to the space, $P_{in}$ , is balanced by the combination of the flux of particles vented from the space, $P_{out}$ , and the flux of particles which sediment over the floor, $P_{floor}$ (figure 5):

Motivated by the experimental results (e.g. figure 4 a) we assume that particles are well mixed horizontally near the base of the space, and we let $C$ denote the uniform concentration of particles at the level of the floor. The flux of particles vented from the space is then given by

If particles sediment uniformly over the entire floor area at a speed $v_{s}$ , the flux of particles which deposit onto the floor at steady state is given by

A combination of (2.7)–(2.9) indicates that if particles are well mixed in the ambient fluid below the interface and sediment uniformly over the entire floor area, then the concentration of particles at the base of the space reaches the equilibrium value at steady state

In this case, the concentration of particles in the ambient fluid below the interface equals the concentration of particles in the plume fluid at the level of the interface (see (2.6) and (2.10)).

We now extend the model to consider the more general case in which particles of different sizes and settling speeds are supplied to the space. We assume that all particles in the mixture are suspended in the ambient fluid at the base of the space (see figure 6 a). In this case the flux of particles vented from the tank, $P_{out}$ , and the flux of particles which sediment onto the floor, $P_{floor}$ , are given by

and

respectively. Here $D$ denotes particle diameter, which ranges between $D_{min}$ and $D_{max}$ (see figure 6 a), and $c_{b}(D)$ represents the particle concentration density function just above the base of the tank, so that the total concentration of all the particles suspended just above the base of the tank is

For each particle size $D$ , conservation of particles requires that (see (2.7))

where $p_{in}(D)$ is the density function of the concentration flux supplied to the space such that

Using (2.11)–(2.15), we calculate the mean concentration of particles at the base of the tank, $C_{eq}$ , at which the flux of particles supplied to the space is balanced by the combination of the flux of particles vented from the space and the flux of particles which sediment over the entire floor area (see (2.10))

For each of the experiments listed in table 1, we compare this concentration with the mean concentration of particles which we measured during the experiment at a vertical distance 2 cm from the base of the tank, $C_{a}$ . Figure 7 shows that in all our experiments the measured concentration at the base of the tank exceeds the concentration predicted by (2.16) by 20–40 %. We observe that the difference between $C_{a}$ and $C_{eq}$ increases when the vertical distance between the floor of the tank and the height at which particles with a mean settling speed $\bar{v}_{s}$ are suspended, $H-\bar{h}_{p}$ , is reduced. We now develop the model to account for this difference by modelling the interaction of the plume fluid with the floor of the tank.

Figure 7. The measured concentration of particles at the base of the tank, $C_{a}$ , compared with the concentration $C_{eq}$ given by (2.16). We observe that in our experiments $C_{a}$ is consistently larger than $C_{eq}$ .

3.1. Particle-laden fountain in the lower part of the space

We assume that at a distance $z=\bar{h}_{p}$ below the source the plume fluid enters a layer of ambient fluid with a mean concentration of particles $C_{a}>C_{p}(\bar{h}_{p})$ . On entering this layer of relatively dense ambient fluid, the plume fluid has an upward buoyancy flux $B_{f}$ relative to the surrounding ambient fluid,

where

is the fractional difference between the mean concentration of particles in the ambient fluid and that in the plume fluid. Equation (2.4) indicates that at this level the plume fluid has momentum flux

where

is the radius of the plume at a distance $\bar{h}_{p}$ below the source (Morton et al. Reference Morton, Taylor and Turner1956). As a result of the upward buoyancy flux $B_{f}$ combined with the momentum flux $M_{f}$ , the plume transitions into a momentum-driven fountain. Literature on turbulent fountains, including Turner (Reference Turner1966), Mizushina et al. (Reference Mizushina, Ogino, Takeuchi and Ikawa1982), Bloomfield & Kerr (Reference Bloomfield and Kerr1998, Reference Bloomfield and Kerr2000) and Burridge & Hunt (Reference Burridge and Hunt2012), indicates that in a deep layer of fluid the maximum distance travelled by such a fountain, $h_{f}$ , is given by

where ${\it\gamma}$ is a constant of order 1. In a set of experiments on turbulent fountains, Bloomfield & Kerr (Reference Bloomfield and Kerr2000) measured ${\it\gamma}=1.70\pm 0.17$ ; similar results were obtained by the other researchers cited above.

In our experiments, with ${\hat{C}}\approx 0.2{-}0.4$ (see figure 7), the upward buoyancy flux of the fountain fluid relative to the surrounding particle-laden ambient fluid is of order $B_{f}\approx 10^{-8}~\text{m}^{4}~\text{s}^{-3}$ (see (3.1) and table 1). For a typical plume radius at the level of the interface $r_{p}\approx 2{-}3~\text{cm}$ and a typical rate of ventilation of the tank $Q\approx 10\times 10^{-6}~\text{m}^{3}~\text{s}^{-1}$ , the momentum flux of the fountain fluid at the interface is of order $M_{f}\approx 10^{-6}~\text{m}^{4}~\text{s}^{-2}$ (see (3.3)). Consequently, (3.5) indicates that $h_{f}\approx 0.3{-}0.5~\text{m}$ . This length exceeds the typical depth of the layer in which particles with a mean settling speed $\bar{v}_{s}$ are suspended in the ambient fluid in the tank, $H-\bar{h}_{p}\approx 0.1{-}0.2~\text{m}$ . Hence, the fountain fluid reaches the bottom of the tank and impinges on the floor in our experiments (figure 8 a).

3.2. Motion of the fountain fluid at the level of the floor

When the fountain fluid reaches the bottom of the tank, it initially spreads radially over the floor, but then lifts off from the floor surface at a radial distance $R_{L}$ from the fountain vertical axis as a result of its buoyancy relative to the particle-laden ambient fluid (figure 8 a). Cooper & Hunt (Reference Cooper and Hunt2004, Reference Cooper and Hunt2007) ran a series of experiments to explore the behaviour of a turbulent fountain containing fresh water in a relatively dense environment containing a solution of water and salt. They observed that when the fountain impinges perpendicularly on a floor, the critical radius $R_{L}$ is given by

where ${\it\varepsilon}=0.03$ is a coefficient which Cooper & Hunt (Reference Cooper and Hunt2007) obtained by an empirical fit of their experimental results. In our experiments, the fountain fluid contains particles, which are subject to gravitational settling and may fall out of the fountain fluid as it spreads over the floor. Equation (3.6) can be used to estimate the critical radius at which the particle-laden fountain fluid lifts off from the floor surface only if the fraction of particles which fall out of the fluid as it spreads over the floor is small. If this is the case, then in our experiments we estimate $R_{L}\approx 5{-}7~\text{cm}$ (see (3.6) and table 1).

To estimate the number of particles which sediment from the fountain fluid at the level of the floor, we compare the speed at which particles settle through the fountain fluid with the radial speed at which the fountain fluid spreads over the floor. We assume that on impinging on the floor, the fountain has a radius $r_{0}$ and a volume flux $Q_{f0}$ (figure 8 b). Mizushina et al. (Reference Mizushina, Ogino, Takeuchi and Ikawa1982) indicate that the radius of a fountain grows linearly with distance so that

where $r_{p}$ is the radius of the fountain at a distance $H-\bar{h}_{p}$ above the floor (see (3.4)). At the level of the floor, the volume flux of the fountain, $Q_{f0}$ , is given by

where $k$ is a constant of order 1 (Cooper & Hunt Reference Cooper and Hunt2007). On reaching the base of the space the fountain fluid starts spreading radially over the floor at an initial speed

where $h_{0}$ denotes the initial depth of the spreading flow (figure 8 b). Using (3.7) and (3.8), we estimate that in our experiments the typical radius of the descending fluid in the fountain at the level of the floor is of order $r_{0}\approx 0.03{-}0.04~\text{m}$ , and that the typical volume flux is of order $Q_{f0}\approx 50{-}70\times 10^{-6}~\text{m}^{3}~\text{s}^{-1}$ . If we assume that $h_{0}$ is of the same order as the fountain fluid radius, $h_{0}=r_{0}\approx 0.03{-}0.04~\text{m}$ (figure 8 b), we obtain that the initial speed at which the fountain fluid spreads over the floor of the tank is of order $u_{0}\approx 8{-}12~\text{mm}~\text{s}^{-1}$ , i.e. 20–30 times larger than the mean speed at which particles settle through the fluid in the tank, $\bar{v}_{s}$ (see § 2.1 and appendix A). Since $R_{L}\approx 2{-}3\;r_{0}$ , we expect the mass of particles which sediment onto the floor before the fountain fluid lifts off to be small, and therefore we assume that we can use (3.6) to calculate the critical lift-off radius $R_{L}$ .

It can be envisaged that after lifting off from the floor surface, the fountain fluid keeps propagating radially while rising through the ambient fluid in the lower part of the tank. Hence, we expect a portion of the floor area just outside the lift-off radius to be shielded and to receive a limited flux of sedimenting particles from the ambient fluid above (figure 8 a). We let ${\rm\Delta}R_{L}$ denote the radial width of this additional area of reduced sedimentation.

For $r>R_{L}+{\rm\Delta}R_{L}$ , we expect particles to sediment onto the floor of the tank. Consequently, the effective area $A_{s}$ over which particles sediment at the level of the floor is given by

Equation (3.10) indicates that as a result of the fountain flow the effective area over which particles sediment at the level of the floor is reduced, and so the flux of particles which sediment over the floor of the space is also reduced, and (2.12) should be revised to the value

Thus at steady state we expect the concentration of particles in the ambient fluid at the base of the space to be given by

The model developed in §§ 3.1 and 3.2 is tested in figure 9 by comparing our model predictions with experimental results. For each of the experiments listed in table 1 with $Q=10~\text{cm}^{3}~\text{s}^{-1}$ , we measure the dimensionless concentration ${\hat{C}}$ in the ambient fluid at a distance 2 cm from the floor of the tank (3.2). We then use (3.12) to infer the effective area $A_{s}$ over which particles sediment in the tank, and the radius $R_{L}+{\rm\Delta}R_{L}$ of the portion of the floor area over which particles do not sediment. We compare the results of the experiments (dots) with the predictions of the model (lines). If we assume that ${\it\varepsilon}(1+{\it\Delta})$ is a constant (3.10), and that ${\it\varepsilon}=0.03$ as indicated by Cooper & Hunt (Reference Cooper and Hunt2007), we observe in figure 9 that small errors of less than 2–3 % are obtained for ${\it\Delta}=0.4$ (see (3.10)).

Figure 9. Comparison of experimental results with model predictions. For each of the experiments listed in table 1 with $Q=10~\text{cm}^{3}~\text{s}^{-1}$ , we measure the dimensionless concentration ${\hat{C}}$ of particles in the ambient fluid at the base of the tank. Based on ${\hat{C}}$ , we calculate the fraction of the floor area over which particles do not sediment at steady state (dots). We compare the experimental results with the predictions of the model for ${\it\Delta}=0$ (dashed line) and ${\it\Delta}=0.4$ (solid line). Error bars of $\pm 5\,\%$ are plotted.

In the model we have assumed that the fountain fluid reaches the base of the tank, $h_{f}>H-\bar{h}_{p}$ . This is consistent with there being a region of the floor in which sedimentation is suppressed, leading to the increase in particle concentration observed in the experiments, $C_{a}>C_{eq}$ (see figure 7). If there was a steady state in which the fountain did not reach the floor of the tank, this would imply that particles can sediment over the whole floor area; therefore, the concentration of suspended particles in the ambient fluid would equal the concentration of particles in the plume fluid at the level $\bar{h}_{p}$ ((2.10) and (2.6) respectively). In this case, we would then expect that the plume fluid would descend below the level $z=\bar{h}_{p}$ as a neutrally buoyant jet, and hence it would eventually reach the floor.

It should be noted that in our experiments we could only investigate the case in which $R_{L}^{2}\ll A$ . When a small flux of particles is supplied to the space, the buoyancy flux of the plume decreases and we expect the critical lift-off radius $R_{L}$ to increase (3.6). However, at low particle concentrations the image analysis technique used in our experiments (see § 2.1) becomes less accurate as the signal-to-noise ratio of the particle concentration decreases, and so we have not been able to investigate this limit quantitatively (see appendix B).

As noted in § 2.1, in the present paper we only consider the transport of small particles through the space. If the particles were larger, so that a significant fraction were predicted to sediment as the fountain fluid spreads over the floor, then the flow regime would become rather different. This may be seen by noting that the time to sediment through the radial flow is ${\it\tau}_{s}\sim h_{0}/\bar{v}_{s}$ , while the time for the radial flow to reach the tank’s walls is ${\it\tau}_{f}\sim Ah_{0}/Q_{f0}$ , where $Q_{f0}$ is the flux in the fountain (see (3.8)). With fast sedimentation ${\it\tau}_{s}<{\it\tau}_{f}$ , and so we find that $A\bar{v}_{s}>Q_{f0}$ . This implies that particles would not be suspended by the upward return flow (2.4). In this case, the particle-laden plume would descend to the base of the space and sediment particles over the floor, and the fluid vented from the space would be largely depleted of particles.

4.1. Experiments

We now introduce a second series of experiments, in which we explore the transport of particles through a ventilated space which contains a localised source of particles and negatively buoyant fluid (see figure 1 c,d). Table 2 contains a list of the experiments described in this section. These experiments are performed using the same laboratory apparatus described in § 2.1 and illustrated by figure 2. In this second series of experiments, we use the same silicon carbide particles and the same image analysis technique described in § 2.1 to quantify the concentration of particles in the tank during an experiment. As in the previous experiments, fresh water is used as the ambient fluid in the tank. However, in this second group of experiments, a particle-laden solution of water and NaCl (20 % in weight) is used as the plume fluid. The salinity of this solution is checked by refractometry. The saline fluid is denser than fresh water, and so it forms a turbulent, dense plume which descends through the tank while entraining ambient fluid.

For convenience, we usually start our experiments by supplying a continuous flow of saline solution which does not contain particles. This fluid is pumped into the tank through vent A (see figure 2), and forms a dense turbulent plume which descends through the tank. A ventilation flow is supplied to the tank through vent B and extracted from the tank through vent C, as described in § 2.1. Over time, the system evolves to an equilibrium, at which the fluid in the tank is stratified: the upper part of the tank contains fresh water, while the lower part of the tank contains a solution of water and salt. The two layers are separated by an interface (Linden et al. Reference Linden, Lane-Serff and Smeed1990). In our experiments, this equilibrium is usually obtained 20–30 min after the beginning of the experiment. Once the saline stratification is established, we add particles to the saline fluid and observe the transport of the particles through the tank. This experimental procedure minimises the amount of time required for the steady-state stratification of particles to develop in the tank; however, our observations indicate that the details of how an experiment approaches the steady state do not affect the steady state, and so the same steady-state distribution of particles in the tank would be obtained if both the particles and the saline fluid were supplied from the start.

Over the course of the experiment, the particle-laden plume fluid descends through the tank while entraining ambient fluid. On reaching the base of the tank, it spreads over the floor, in an analogous fashion to that described in § 2.1. In the lower part of the tank, an upward return flow is produced in the ambient fluid, as described in § 3.1. This flow suspends particles in the ambient fluid (figure 1 d). The concentration of particles in the suspension increases over time, until a steady state is reached.

Figure 10. Steady-state distribution of particles in the tank in a typical experiment (experiment h: see table 2): (a) steady-state concentration of particles in the tank using false colours; (b) horizontally averaged particle concentration profile.

Figure 11. Detrainment of particle-laden plume fluid at the level of the interface between layers 1 and 2.

At steady state, we observe that the fluid in the tank is stratified, and that three layers can be identified. The layers are characterised by different concentrations of particles and salt in the ambient fluid (figure 10). In the upper layer (layer number 1 in figure 10), the ambient fluid contains fresh water. In this layer, no particles are suspended in the ambient fluid which surrounds the plume. At the base of this layer, we observe a sharp interface which separates fresh water in the upper layer from a solution of water and salt in the lower layers in the tank. In the central part of the tank (layer number 2 in figure 10), the ambient fluid contains a solution of water and salt and a very small number of particles, which are detrained from the plume fluid at the level of the interface between layers 1 and 2 (see figure 11). In the lower part of the tank (layer number 3 in figure 10), the ambient fluid contains both salt and a large number of suspended particles. In our experiments, a continuous stratification in particle concentration is observed rather than a sharp interface between layers 2 and 3. This is the result of the distribution of grain sizes and settling speeds in the mixture of particles used for the experiments.

Over the course of a typical experiment, most of the plume fluid descends from the top of the tank to the level of the floor. However, we observe that from time to time a small amount of particle-laden fluid is detrained from the plume into the surrounding ambient at the level of the interface which separates layer 1 and layer 2 (figure 11). The concentration of particles in the ambient fluid at this level does not increase significantly during the experiment: this suggests that as new particles are intermittently detrained from the plume at the level of the interface, previously detrained particles fall from the level of the interface towards the base of the tank.

We observe that in some of our experiments (see table 2) the three-layer stratification illustrated by figure 10 is subject to intermittent instabilities. These instabilities develop over a long time scale, and they modify the distribution of the particles in suspension in the tank for a limited amount of time. In the next sections we first explore the three-layer stratification illustrated by figure 10, and develop a model which quantifies the height of the interface between layer 1 and layer 2, the maximum height at which particles are suspended in the ambient fluid, and the concentration of particles in the ambient fluid in layer 3. A description of the large-scale instabilities observed in some of the experiments is given in § 5.

4.2. Three-layer stratification

We investigate the transport of heavy, small particles through an enclosed space with downward ventilation which contains a source of heavy particles and of relatively dense fluid located at the top of the space (figure 12). We let $B_{p}$ denote the negative buoyancy flux associated with the presence of particles in the plume fluid (see (2.1)), and let $B_{s}$ denote the negative buoyancy flux associated with the presence of relatively dense fluid in the plume. We assume that the settling speeds of the particles are small compared to the plume speed, so that the plume can be modelled as a single-phase flow (see §§ 2 and 3).

Figure 12. (a) Schematic showing the conditions used in the modelling (§ 4). (b) The measured concentration of particles at the base of the tank, $C_{a}$ , compared with the expected concentration $C_{eq}$ given by (2.16).

Motivated by the experimental observations described in § 4.1, we assume that the dense, particle-laden fluid which is supplied at the top of the space produces a turbulent plume which descends through the ambient fluid in layer 1 driven by a combination of the negative buoyancy fluxes $B_{p}$ and $B_{s}$ . In § 4.3, we explore the entrainment of ambient fluid into the plume in this layer, and we show that at a distance $h_{s}$ from the plume source an interface is formed which separates layer 1 and layer 2 (figure 12 a).

Our experiments indicate that below this interface the ambient fluid is saline, and therefore its density is larger than the density of the ambient fluid in layer 1 as expected (see Linden et al. Reference Linden, Lane-Serff and Smeed1990). In § 4.4 we show that the particle-laden plume fluid descends through layer 2 driven by negative buoyancy flux $B_{p}$ . In this layer, a return flow is produced in the ambient fluid which surrounds the plume. In the lower part of the space, particles are suspended by this flow and form a layer of particle-laden fluid (layer 3 in figure 12 a: see § 2.4).

Using the same approach described in § 2.4, for each of the experiments listed in table 2 we use (2.16) to calculate the mean concentration of particles in the ambient fluid, $C_{eq}$ , at which the flux of particles supplied to the space, $P_{in}$ , is balanced at steady state by the combination of the flux of particles which are vented from the space, $P_{out}$ , and the flux of particles which sediment over the entire floor area, $P_{floor}$ (see figure 12 a and (2.11) and (2.12)). In figure 12(b) we compare this concentration with the effective concentration $C_{a}$ which we measured at the base of the tank in each experiment (see figure 7). Figure 12(b) shows that in all our experiments the measured concentration at the base of the tank exceeds by 25–45 % the concentration given by (2.16). As described in § 3, we therefore expect the particle-laden plume to transform into a momentum-driven fountain in layer 3. We explore the behaviour of the fountain fluid in § 4.5.

4.3. Plume containing particles and dense fluid in layer 1

In the upper layer the plume fluid has a negative buoyancy flux $B_{s}+B_{p}$ relative to the ambient fluid (see table 2). The plume entrains ambient fluid while descending through the layer. Plume theory indicates that the volume flux at a distance $z$ from the source is given by (Linden et al. Reference Linden, Lane-Serff and Smeed1990)

When the plume volume flux equals the ventilation flux, $Q_{p}(z)=Q$ , all of the ventilation flow has been entrained by the plume and so the ambient fluid which surrounds the plume is not subject to any vertical motion (Linden et al. Reference Linden, Lane-Serff and Smeed1990). At this level, an interface is formed, which separates layer 1 and layer 2 (see figure 12 a). The vertical distance between the plume source and this interface is given by

At the level of this upper interface, the plume has a radius $r_{s}$ (Morton et al. Reference Morton, Taylor and Turner1956),

so the momentum flux of the plume fluid at the level of the upper interface, $M_{s}$ , is given by

4.4. Plume containing particles in layer 2

On crossing the interface between layers 1 and 2, the plume fluid experiences a change in the density of the surrounding fluid. The density difference between the ambient fluid in layer 1 and the ambient fluid in layer 2 is associated with the change in salinity, and can be given in terms of the reduced gravity, $g_{2}^{\prime }$ (Linden et al. Reference Linden, Lane-Serff and Smeed1990):

Plume theory indicates that at the level of the interface, the time-averaged mean salinity of the plume fluid equals that of the ambient fluid below the interface. However, the plume fluid contains particles, and so the bulk density of the plume fluid is larger than the density of the ambient fluid in layer 2, and the mean negative buoyancy flux of the plume in this layer is given by $B_{p}$ (figure 13). It is worth noting that, in practice, during our experiments plumes are effectively composed of transient or intermittent eddies. It can be envisaged that as a result of the entrainment of ambient fluid in layer 1, these eddies may have a range of salinities. Hence, it is likely that from time to time a certain amount of plume fluid might be slightly less dense than the surrounding ambient fluid in layer 2. When this happens, the plume fluid detrains just below the interface between layer 1 and layer 2, as illustrated by figure 11. It can be envisaged that the presence of particles in the plume fluid may change the frequency of such detrainment events. Since the ambient fluid in layer 2 does not contain particles, the detrainment events will be more frequent when the plume fluid contains a small concentration of particles.

Figure 13. Virtual source of the particle-laden plume which descends through layer 2.

Cooper & Linden (Reference Cooper and Linden1996) explored how the properties of a plume change when the plume crosses a density interface and enters a layer of fluid in which its buoyancy is reduced. Using their approach, we assume that the particle-laden plume which descends through layer 2 with reduced negative buoyancy flux $B_{p}$ originates from a virtual source of finite momentum flux but zero volume flux (figure 13). This virtual source is located above the interface between layers 1 and 2, at a distance $z_{v}<h_{s}$ from the top of the space: (Cooper & Linden Reference Cooper and Linden1996)

where

is the ratio between the plume buoyancy flux below the level of the interface which separates layer 1 and layer 2 and the plume buoyancy flux above the level of that interface. In our experiments, this ratio is typically of order $\hat{B}\approx 10^{-2}$ , so the virtual origin of the particle-laden plume which descends through layer 2 is typically located 2–3 cm above the interface which separates layer 1 and layer 2.

Figure 14. Comparison of experimental results with model predictions: (a) distance $h_{s}$ between the plume source and the interface between layer 1 and layer 2; (b) distance $\bar{h}_{p}$ between the plume source and the level at which particles with a mean settling speed $\bar{v}_{s}$ are suspended in the ambient fluid.

As the particle-laden plume descends through layer 2, it entrains ambient fluid in this layer. The volume flux in the plume at a distance $z^{\prime }$ from the virtual source (see figure 13) is given by (Cooper & Linden Reference Cooper and Linden1996)

Equation (4.8) is solved numerically, using $Q_{p}(z^{\prime }=h_{s}-z_{v})=Q$ as a boundary condition, to calculate the plume volume flux at different levels in layer 2. This volume flux is larger than the ventilation flux, so a return flow is generated in the ambient fluid which surrounds the plume in layer 2. The mean speed at which the ambient fluid rises through the space, $v_{w}(z^{\prime })$ , is calculated using a combination of (2.3) and (4.8). When this speed equals the speed at which a particle settles through the ambient fluid, $v_{s}$ , the particle becomes suspended in the ambient fluid. As a result of the distribution of settling speeds in the mixture of particles used in the experiments, a continuous stratification in particle concentration develops in the lower part of the space (see § 2.3). We let $\bar{h}_{p}$ denote the vertical distance between the source of the plume and the level at which particles with a mean settling speed $\bar{v}_{s}$ are suspended in the ambient fluid.

We use the model developed in §§ 4.2–4.4 to calculate the distance $h_{s}$ between the plume source and the interface between layer 1 and layer 2, and the distance $\bar{h}_{p}$ between the plume source and the level at which particles with a mean settling speed $\bar{v}_{s}$ are suspended in the ambient fluid. In figure 14(a) we consider the vertical distance $h_{s}$ between the source of the plume and the interface between layers 1 and 2: for each of the experiments listed in table 2, we compare the distance predicted by the model (4.2) with the distance which we measured during the experiment (black dots). In figure 14(b) we consider the distance $\bar{h}_{p}$ between the source of the plume and the level at which particles with a mean settling speed are suspended in the ambient fluid. For each experiment, we solve (4.8) numerically, and calculate the height at which $Q_{p}=Q+A\bar{v}_{s}$ (see (2.4)). We then compare this height with the height at which the concentration of particles in the ambient fluid is reduced by 50 % relative to the maximum concentration which we measure at the base of the tank (black dots). For comparison, in figure 14(b), we also plot the results of the experiments described in § 2.1 (plume containing particles and neutrally buoyant fluid) using white dots. Both figures 14(a) and 14(b) show that there is relatively good agreement between model predictions and experimental observations, with errors of less than 3–5 % in most experiments.

4.5. Fountain in layer 3

Motivated by the experimental results illustrated by figure 12(b), we use the same approach described in § 3 and assume that the particle-laden fluid transitions into a fountain at the level at which particles with a mean settling speed $\bar{v}_{s}$ are suspended in the ambient fluid. We assume that at this level $z=\bar{h}_{p}$ the particle-laden plume fluid has an upward buoyancy flux $B_{f}$ relative to the surrounding ambient fluid (3.1) and a momentum flux $M_{f}$ (3.3). In our experiments the fountain fluid reaches the base of the space and propagates radially over the floor until it separates from the surface of the floor at a distance $R_{L}$ from the plume vertical axis (see (3.6) and figure 8 a). As described in § 3.2, the radial speed at which the fountain fluid spreads over the floor is large compared with the mean particle settling speed in our experiments, so we expect the sedimentation of particles to be suppressed for $r<R_{L}$ . After lifting off from the floor surface, the fountain fluid propagates radially while rising through the ambient fluid (figure 8 a). Consequently, we expect a portion of the floor area just outside the lift-off radius $R_{L}$ to be shielded and to receive a limited flux of sedimenting particles from the ambient fluid above. We let ${\rm\Delta}R_{L}$ denote the radial width of this additional area of reduced sedimentation, and based on the experimental results described in § 3.2 we expect ${\it\Delta}\approx 0.4$ .

Figure 15. Comparison of experimental results with model predictions. For each of the experiments listed in table 2 with $Q=10~\text{cm}^{3}~\text{s}^{-1}$ , we measure the dimensionless concentration ${\hat{C}}$ of particles in the ambient fluid at the base of the tank. Based on ${\hat{C}}$ , we calculate the fraction of the floor area over which particles do not sediment at steady state (black dots). We compare the experimental results with the predictions of the model for ${\it\Delta}=0$ (dashed line) and ${\it\Delta}=0.4$ (solid line). The outcomes of the experiments described in § 2.1 are plotted for comparison (white dots, compare with figure 9). Error bars of $\pm 5\,\%$ are added to the graph.

Figure 16. Transient horizontally averaged mean concentration of particles in the tank in experiments k and h (see table 2). In experiment k, a steady three-layer stratification is observed (see § 4). In experiment h, we observe intermittent instabilities.

In figure 15, the model developed in §§ 4.3–4.5 is tested by comparing the results of our experiments with the prediction of the model. Using the same approach described in § 3.2, for each of the experiments listed in table 2 with $Q=10~\text{cm}^{3}~\text{s}^{-1}$ we measure the dimensionless concentration ${\hat{C}}$ of particles in the ambient fluid at the base of the tank. We then use (3.10) to infer the fraction of the floor area over which particles sediment effectively and the radius $R_{L}+{\rm\Delta}R_{L}$ of the portion of the floor area over which particles do not sediment. We compare the results of the experiments (black dots in figure 15) with the prediction of the model developed in §§ 4.3–4.5 (lines). Figure 15 shows that for ${\it\Delta}=0.4$ the model prediction lies within 5 % of the experimental data for all measurements. Similar errors were obtained when analysing the results of the experiments described in § 2.1 (white dots, which are plotted in figure 15 for comparison).