2.1. Deflection modulus

A bubble curtain in a ship lock used as a barrier across horizontal density stratification acts in a similar way as an air curtain installed in a doorway of a building to separate two rooms at different temperatures. The main difference is that the downwards blowing (unheated) air curtain, which is a typical installation in doorways, forms a plane air jet, whereas the rising bubbles create a multiphase line plume by entraining the ambient fluid. An important feature of multiphase plumes is that the bubbles possess the so-called slip velocity $u_s$ by means of which they can separate from the entrained fluid flow and escape the water body after reaching the surface. Here, we establish a formal correspondence between the bubble curtain and the air curtain. This will allow us to use the theoretical foundations developed for air curtains to explain many experimental observations made on bubble curtains.

The theoretical framework for air curtains was first formulated by Hayes & Stoecker (Reference Hayes and Stoecker1969b,Reference Hayes and Stoeckera). They derived the so-called deflection modulus,

(2.1)\begin{equation} D_m = \frac{\rho_0 b_0 u_0^{2}}{g \Delta\rho H^{2}}=\frac{b_0 u_0^{2}}{g'H^{2}}, \end{equation}

as the governing parameter for the air curtain performance in the doorway of a building. The deflection modulus is a dimensionless parameter. Here, $u_0$ is the discharge velocity of the plane air jet, $\rho _0$ its density and $b_0$ is the width of the rectangular thin outlet nozzle spanning the entire door width. The doorway height is denoted by $H$ and the density difference across the doorway is $\Delta \rho =\rho _d-\rho _l$, where $\rho _d$ is the density of dense fluid (at cold temperature $T_d$) and $\rho _l$ is the density of light fluid (at warm temperature $T_l$). The gravity acceleration $g$ can be combined with the density difference $\Delta \rho$ to give the reduced gravity $g'=g\Delta \rho /\bar \rho$. It is assumed that all the density differences are small and the Boussinesq approximation applies so that the reference density $\bar \rho$ can be taken as $\bar \rho =\rho _0$, $\bar \rho =\rho _l$ or $\bar \rho =\rho _d$ without introducing a significant error.

The deflection modulus represents the ratio between the momentum flux of the air curtain and the lateral pressure forces acting on it due to the horizontal density stratification across the doorway. Hayes & Stoecker (Reference Hayes and Stoecker1969b) delineated two possible operating regimes for the downwards blowing air curtain. For small values of the deflection modulus $D_m\lessapprox 0.15$, the air curtain does not possess enough vertical momentum to withstand the lateral forcing. In this breakthrough situation, it is completely laterally deflected and discharges horizontally to one side of the doorway (to the dense-fluid side for the downwards blowing air curtain). The air curtain fails to act as a separation barrier and an unhindered infiltration flow through the doorway can take place in that case. For $D_m\gtrapprox 0.15$, the air curtain stably reaches the bottom of the doorway opening and ensures the aerodynamic sealing. The infiltration flux of dense fluid across the doorway is minimised for the deflection modulus values in the range of $D_m\approx 0.2\text {--}0.4$ depending on the specific doorway configuration. If the $D_m$ is increased further, the air curtain operates in the curtain-driven regime. The infiltration flux across the doorway is now due to the entrainment and mixing induced by the air curtain, and, as will be seen in § 2.2, the infiltration flux is a function of the initial volume flux per unit length $q_0=b_0 u_0$ as well as $b_0$ and $H$.

We will study the bubble curtain acting as a separation barrier between two sides of the channel of water depth $H$ and width $W$ (see figure 1). The bubbles are injected at the bottom of the channel with the volumetric flow rate $Q^{air}$ from a pierced manifold spanning the entire channel width, and they form a multiphase line plume as they rise towards the water surface. The air flow rate per unit length is denoted by $q_{air}=Q^{air}/W$.

Figure 1. Sketch of the experimental set-up used for small-scale experiments. The false floor (shown as solid grey on the left-hand side and as grey dashes on the right-hand side) was placed on both sides of the pierced manifold generating the bubble curtain (indicated by an arrow). The false floor on the right-hand side is displayed in dashes to make visible the air inlet tubes that were running to the bubble curtain along the bottom of the tank and were hidden underneath the false floor.

Starting from (2.1), we first formulate a similar dimensionless parameter for the bubble curtain. According to the plume theory built upon the fundamental work by Morton, Taylor & Turner (Reference Morton, Taylor and Turner1956), the governing parameters for a line plume in a non-stratified environment are the vertical distance $z$ from the (virtual) origin of the plume and the source buoyancy flux per unit length $B$ which remains constant with height for single-phase plumes. Assuming a self-similar flow, the line plume characteristics scale as

(2.2a,b)\begin{equation} b \sim z, \quad u \sim B^{1/3}, \end{equation}

where $b$ is the line plume width and $u$ the upwards velocity.

In the case of a bubble plume, if we neglect the mass of the air, the vertical momentum flux per unit length (and unit density) of the rising plume is

(2.3)\begin{equation} b u^{2} \sim b B^{2/3} =b (g q_{air})^{2/3} \sim z(g q_{air})^{2/3}, \end{equation}

where $B=gq_{air}$ and the variation in the air flux with height due to the changing bubble size is neglected. We discuss in § 6 whether the assumption of a constant $B=gq_{air}$ holds for real-scale bubble plumes in ship locks and how our results are affected if $B$ varies. In order to adapt (2.1) to the case of a bubble curtain, the momentum flux of the plane jet needs to be replaced by the momentum flux of the line plume. However, a central point in the derivation of $D_m$ by Hayes & Stoecker (Reference Hayes and Stoecker1969b) was that the momentum flux of the plane jet remains constant with height whereas for a plane plume, it increases linearly with $z$ (see (2.3)). Therefore, we will use the maximum momentum flux $M$ when the plume reaches the free surface $(z=H)$ in the definition of the deflection modulus for the bubble curtain. Note that although we define $z$ as the vertical distance from the virtual origin of the plume, the offset between $z=0$ and the channel bottom is negligible compared with the water depth $H$ in practical situations, such that $z\approx H$ at the surface. This is equivalent to $b_0\ll H$. Paillat & Kaminski (Reference Paillat and Kaminski2014) measured that the (total) line plume width grows as

(2.4)\begin{equation} b=2\alpha_E z \approx 0.14 z, \end{equation}

where $\alpha _E\approx 0.071$ is the experimentally measured entrainment constant for single-phase line plumes. Therefore, we define the deflection modulus for a bubble curtain as

(2.5)\begin{equation} D_{m,b}=\frac{2\alpha_E H(g q_{air})^{2/3}}{g'H^{2}}=\frac{2\alpha_E(g q_{air})^{2/3}}{g'H}=2\alpha_E{\textit{Fr}}^{2}_{air}. \end{equation}

The air Froude number

(2.6)\begin{equation} {\textit{Fr}}_{air}=\frac{(g q_{air})^{1/3}}{\sqrt{g'H}}, \end{equation}

was previously used by Keetels et al. (Reference Keetels, Uittenbogaard, Cornelisse, Villars and van Pagee2011), van der Ven et al. (Reference van der Ven, O'Mahoney and Weiler2018) and Oldeman et al. (Reference Oldeman, Kamath, Masterov, O'Mahoney, van Heijst, Kuipers and Buist2020) to characterise the bubble curtain. In analogy to air curtains, we expect two operating regimes to exist for the bubble curtain as well, namely the breakthrough and the curtain-driven regime. The transition point between these two regimes should occur at a well-defined value of the deflection modulus $D_{m,b}$ and we will determine this critical value from our experimental measurements.

We note that the bubble slip velocity $u_s$ is expected to cause a separation of bubbles from the entrained water plume such that the bubbles might follow a different path than the fluid. Additionally, the entrainment coefficient $\alpha _E$ might depend on $u_s$. We will discuss this in more detail in the following sections.

2.2. Mass transfer

When the air curtain operates in the curtain-driven regime, the infiltration flux of dense fluid across the doorway is self-induced by the air curtain. The turbulent air curtain entrains fluid from both sides of the doorway, mixes it and then spills the mixed fluid back to both sides upon reaching the floor. Such an entrainment-spill mechanism should (to the leading order) be unaffected by the horizontal density stratification (see (2.11) and (2.16)). Instead, we can expect the infiltration flux to be caused by the vertical volume flux within the plane jet that is deflected laterally when the air curtain impinges on the floor. The governing parameters for a plane jet are its source momentum flux per unit length, $b_0u_0^{2}$, and the vertical distance $z$. Thus, the vertical volume flux per unit length within the air curtain close to the floor scales as

(2.7)\begin{equation} q(H)\sim\sqrt{b_0u_0^{2}H}=q_0\sqrt{\frac{H}{b_0}}, \end{equation}

where $q_0=b_0u_0$ is the initial volume flux per unit length. Therefore, we expect the infiltration volume flux per unit length of dense fluid across the doorway to scale as

(2.8)\begin{equation} q_{ac}\approx \frac{q(H)}{4}\sim\sqrt{b_0u_0^{2}H}=q_0\sqrt{\frac{H}{b_0}}, \end{equation}

where the subscript ‘ac’ stands for ‘air curtain’. The factor $1/4$ arises since $q(H)$ is assumed to divide equally between two sides when the air curtain impinges on the floor and, assuming that the fluid inside the air curtain is well-mixed, the density of the spilled fluid is $(\rho _l+\rho _d)/2$. Hence, the infiltration flux of dense fluid across the air curtain is $q(H)/4$. The mass and the heat fluxes per unit length across the doorway can be calculated from the volume infiltration flux as, respectively,

(2.9)\begin{equation} q_{mass}=q_{ac}(\rho_d-\rho_l) \end{equation}

and

(2.10)\begin{equation} q_{heat}=q_{ac}\bar\rho c_p(T_l-T_d), \end{equation}

where $c_p$ is the specific heat capacity. Conventionally, the heat and the mass transfer are described in terms of non-dimensional groups

(2.11)\begin{equation} \frac{{\textit{Nu}}}{{\textit{Re}}{\textit{Pr}}}=\frac{q_{heat}}{u_0b_0\bar\rho c_p(T_l-T_d)}=\frac{q_{ac}}{q_0}\sim\sqrt{\frac{H}{b_0}} \end{equation}

and, similarly,

(2.12)\begin{equation} \frac{{\textit{Sh}}}{{\textit{Re}} {\textit{Sc}}}=\frac{q_{mass}}{u_0b_0(\rho_d-\rho_l)}=\frac{q_{ac}}{q_0}\sim\sqrt{\frac{H}{b_0}}. \end{equation}

Here, the Reynolds, Prandtl and Schmidt numbers are

(2.13a–c)\begin{equation} {\textit{Re}}=\frac{u_0b_0}{\nu}, \quad {\textit{Pr}}=\frac{\nu}{\kappa}, \quad {\textit{Sc}}=\frac{\nu}{D}, \end{equation}

where $\nu$ is the kinematic viscosity, $\kappa$ is the thermal diffusivity and $D$ is the mass diffusion coefficient. The Nusselt number is defined as

(2.14)\begin{equation} {\textit{Nu}} = \frac{hH}{\bar\rho c_p \kappa}=\frac{q_{heat}}{\bar\rho c_p \kappa (T_l-T_d)}, \end{equation}

with $h=q_{heat}/(H(T_l-T_d))$ being the convective heat transfer coefficient. The Sherwood number is given by

(2.15)\begin{equation} {\textit{Sh}}=\frac{kH}{D}=\frac{q_{mass}}{D(\rho_d-\rho_l)}, \end{equation}

with $k=q_{mass}/(H(\rho _d-\rho _l))$ being the convective mass transfer coefficient.

It was demonstrated both numerically and experimentally that in the curtain-driven regime ${\textit {Nu}}/({\textit {Re}}{\textit {Pr}})$ and ${\textit {Sh}}/({\textit {Re}}{\textit {Sc}})$ assume a constant value for a given geometrical configuration of the air curtain and the doorway (Hayes & Stoecker Reference Hayes and Stoecker1969b; Costa et al. Reference Costa, Oliveira and Silva2006; Frank & Linden Reference Frank and Linden2014), so for large values of $D_m$,

(2.16)\begin{equation} \frac{q_{ac}}{q_0}={\rm const.} \end{equation}

for fixed $H$ and $b_0$. In particular, this confirms that the infiltration flux $q_{ac}$ is independent of the horizontal density difference across the doorway. The scaling relation (2.11) was theoretically derived by Sirén (Reference Sirén2003b). However, to the best of our knowledge, the basic scaling arguments in (2.7) and (2.8) based on a simple physical picture have not been presented in the existing literature on air curtains in this form before.

In order to derive the scaling for the infiltration mass flux across the bubble curtain in the curtain-driven regime, we make recourse to a similar simple physical picture (that was also used by Abraham et al. (Reference Abraham, van der Burgh and DeVos1973)): the bubble curtain entrains water from both sides of the channel, mixes it and the mixed fluid is then deflected laterally back to both sides when the bubble curtain reaches the free surface. Using (2.2a,b), the vertical volume flux per unit length of a line (multiphase) plume close to the water surface is

(2.17)\begin{equation} q(H)=bu\sim B^{1/3}H=(gq_{air})^{1/3}H. \end{equation}

Hence, the infiltration volume flux of dense water across the bubble curtain (and the associated mass flux) can be expected to scale as

(2.18)\begin{equation} q_{c}\approx\frac{q(H)}{4}\sim (gq_{air})^{1/3}H. \end{equation}

Note that the subscript ‘$c$’ denotes ‘curtain’. In their numerical simulations, Oldeman et al. (Reference Oldeman, Kamath, Masterov, O'Mahoney, van Heijst, Kuipers and Buist2020) observed that during some initial time period after the start, the rate of mixing associated with the bubble curtain and, hence, the infiltration flux increased with the air flow rate. However, they did not quantify this increase. Using our experimental measurements, we will examine whether the infiltration flux $q_c$ of dense fluid across the bubble curtain obeys the scaling (2.18).

2.3. Effectiveness

The performance of an air curtain is conventionally described in terms of the effectiveness

(2.19)\begin{equation} E=\frac{q_{open}-q_{ac}}{q_{open}}=1-\frac{q_{ac}}{q_{open}}. \end{equation}

The effectiveness $E$ quantifies the fraction by which the infiltration flux is reduced by the air curtain compared with the case of an open doorway. The buoyancy-driven flux per unit width due to the density (or, equivalently, temperature) difference across the doorway is conventionally calculated by means of the orifice equation (see e.g Rottman & Simpson Reference Rottman and Simpson1983; Wilson & Kiel Reference Wilson and Kiel1990; Shin, Dalziel & Linden Reference Shin, Dalziel and Linden2004) as

(2.20)\begin{equation} q_{open}=\frac{C_D}{3}H\sqrt{g'H}, \end{equation}

where $C_D$ is the discharge coefficient that accounts for streamline contraction and frictional losses at the doorway. The effectiveness of an air curtain is very low in the breakthrough regime, assumes a maximum when the air curtain stabilises, and then decreases again in the curtain-driven regime as $D_m$ increases since using (2.11) and (2.20) we have

(2.21)\begin{equation} \frac{q_{ac}}{q_{open}}\sim\frac{q_0\sqrt{\frac{H}{b_0}}}{H\sqrt{g'H}}=\sqrt{D_m}. \end{equation}

The definition of the effectiveness (2.19) relies on the reduction of the buoyancy-driven flow (and the associated heating costs by means of (2.10)) but does not take into account the operating costs of the air curtain device. However, the fan power consumption of an air curtain device is negligible compared with the thermal load of the infiltrating air across the doorway (Gil-Lopez et al. Reference Gil-Lopez, Galvez-Huerta, Castejon-Navas and Gomez-Garcia2013).

The definition (2.19) for the effectiveness $E$ presents a local view of the exchange process between two sides of the doorway. It is implicitly assumed that the exchange flow rates with and without the air curtain are both steady so that the effectiveness does not change with time. In particular, this means that the dimensions of the enclosures to both sides of the doorway are large enough that during the entire exchange process neither $q_{open}$ nor $q_{ac}$ are modified by the finite size effects of the room geometry. Furthermore, the initial transient processes associated with the opening of the door and the starting of the air curtain are also disregarded in the definition of the effectiveness.

To assess the performance of bubble curtains separating two sides of a channel or a ship lock, the so-called salt transmission factor has been used in the past (Abraham et al. Reference Abraham, van der Burgh and DeVos1973; Keetels et al. Reference Keetels, Uittenbogaard, Cornelisse, Villars and van Pagee2011; van der Ven et al. Reference van der Ven, O'Mahoney and Weiler2018; Oldeman et al. Reference Oldeman, Kamath, Masterov, O'Mahoney, van Heijst, Kuipers and Buist2020). It is defined as

(2.22)\begin{equation} {\textit{STF}}=\frac{V_c}{V_{open}}, \end{equation}

where $V_c$ and $V_{open}$ are the volumes of brackish dense water present in the light-fluid half of the channel with and without the operating bubble curtain, respectively. The ${\textit {STF}}$ is calculated at a given time. It is expected to change as the time progresses to reflect the effects of the finite size geometry of the channel on the exchange process.

For the time period for which the initial transient flow features are negligible and the finite size effects are not present, the effectiveness $E$ and the salt transmission factor ${\textit {STF}}$ are related as

(2.23)\begin{equation} {\textit{STF}}=1-E. \end{equation}

In our small-scale experiments, we will focus on such a time period to examine the mechanism of the infiltration flux of dense fluid into the light-fluid half of the channel across the bubble curtain.

3.1. Experiment A

In this experiment, blue food dye was added either to the light- or the dense-fluid side of the tank to visualise the flow. The flow was recorded using a Nikon D7000 video camera at 24 frames per second. After the gate was closed at the end of an experimental run, both sides of the tank were thoroughly mixed to obtain homogeneous mixtures and measure their average densities $\rho _l^{end}$ and $\rho _d^{end}$. This experiment allows for the calculation of the volume of fluid exchanged across the bubble curtain during the run. It provides a ‘global’ diagnosis of the efficiency of the bubble curtain device.

3.2. Experiment B

The aim of this experiment was to obtain instantaneous values of the density at each location in the tank (averaged in the direction normal to the camera view) during an experimental run. We performed measurements using the dye attenuation technique with methylene blue as the light absorption agent. The flow was recorded using a JAI CVM4+MCL 1.3 megapixel camera equipped with a lens and a red filter. The data acquisition was performed by means of DigiFlow (see http://www.dalzielresearch.com/). The calibration was carried out using concentrations of methylene blue that varied between $0$ and $1.7\times 10^{-1}\,\text {ppm}$ which was the range used for the subsequent experiments. The calibration data are provided in Appendix A.

The recorded images from the camera were converted into instantaneous density maps inside the tank by means of the technique described in detail in Appendix A. The experiment B allows us to understand in more detail the temporal and spatial evolution of the mixing process between the dense fluid and the light fluid. It provides a ‘local’ and instantaneous diagnosis of the exchange across the bubble curtain.

5.1. Scaling law for the cell size

We recall the scaling law for the line plume velocity

(5.1)\begin{equation} u\sim B^{1/3}\sim (g q_{air})^{1/3}, \end{equation}

that was discussed when we defined the deflection modulus $D_{m,b}$ (2.5). We assume that the velocity of the horizontally deflected fluid after reaching the free surface will obey the same scaling (which was verified experimentally by Bulson (Reference Bulson1961)). The rising bubble plume mixes water from both sides, so the density difference between the horizontal flow next to the surface and the light-fluid side of the tank should be ${\sim}\Delta \rho$. Thus, the horizontally deflected fluid is subjected to the reduced gravity ${\sim}g'$ due to which it falls towards the bottom of the tank. The time for this process can be estimated as

(5.2)\begin{equation} t_{cell} \sim \sqrt{\frac{2H}{g'}}. \end{equation}

Combining (5.1) and (5.2), we obtain a scaling prediction for the horizontal extent $L_{cell}$ of the recirculation cell in the light-fluid half

(5.3)\begin{equation} L_{cell}\sim (g q_{air})^{1/3} \sqrt{\frac{2H}{g'}}. \end{equation}

The experimentally measured values for the horizontal extent $L_{cell}$ of the recirculation cell are plotted against the scaling (5.3) in figure 6. We observe a very satisfactory data collapse for $L_{cell}\lessapprox 0.5\,{\rm m}$, which is half the length of the light-fluid side. For higher values of $L_{cell}$, we expect the finite length of the channel to start affecting the flow. Furthermore, higher values of $(g q_{air})^{1/3} \sqrt {{2H}/{g'}}$ are mainly achieved by reducing $\Delta \rho$ in our experiments which renders an accurate measurement of $L_{cell}$ difficult since for small $\Delta \rho$ its boundaries may be blurred due to secondary flows and turbulent effects.

Figure 6. The measured horizontal extent of mixing cells on the light-fluid side of the tank is shown as a function of the scaling law (5.3). The black fitted line is given by (5.4).

For $L_{cell}\lessapprox 0.5\,{\rm m}$, we perform a linear fit of the data and obtain

(5.4)\begin{equation} L_{cell} \approx(0.62\pm 0.02)(g q_{air})^{1/3} \sqrt{\frac{2H}{g'}}. \end{equation}

We note here that the error in (5.4) is based on the 95 % confidence interval for the coefficient estimate of the linear regression model whereas an individual data point for the extent of the recirculation cell is determined within the accuracy of $2\text {--}3\,{\rm cm}$ (see error bars in figure 6).

This can be rewritten as

(5.5)\begin{equation} \frac{ L_{cell}}{H} \approx(0.62\pm 0.02)\sqrt{2}\left(\frac{B}{(g'H)^{3/2}}\right)^{1/3}, \end{equation}

which is the functional relationship predicted by (4.9).

The recirculation cells around bubble curtains were studied by Fanneløp, Hirschberg & Küffer (Reference Fanneløp, Hirschberg and Küffer1991) and Riess & Fanneløp (Reference Riess and Fanneløp1998). Their experiments, however, were conducted in a homogeneous environment and did not include any density variations. McGinnis et al. (Reference McGinnis, Lorke, Wøest, Stöckli and Little2004) observed circulation patterns around bubble curtains in their field experiments in lakes that were vertically stratified. Keetels et al. (Reference Keetels, Uittenbogaard, Cornelisse, Villars and van Pagee2011) noted the presence of recirculation cells in their small-scale experiments on bubble curtains separating two sides at different densities but did not study the variation of the horizontal extent of the recirculation cells and how it depends on other parameters of the system.

The proportionality coefficient in (5.4) provides a good fit to our measured extents of recirculation cells in small-scale experiments. However, as will be discussed in § 6, it might be affected by the water depth $H$ or the slip velocity $u_s$ and thus, might need to be adjusted for real-scale bubble curtains.

5.2. Model for the exchange through the curtain

Qualitative descriptions of the curtain-driven regime provided in § 4.1.2 indicate very clear features of the flow. A schematic of the flow is shown in figure 7.

Figure 7. Schematic of the steady flow in the curtain-driven regime. The right-hand side is initially filled with denser fluid and the left-hand side with lighter fluid. The notations used in the theoretical model are provided.

As before, $\rho _l$ and $\rho _d$ are the initial densities of the light and dense fluids, respectively. We denote by $\rho _d'$ and $\rho _l'$ the densities of the weak quasi recirculation cell on the dense-fluid side and the recirculation cell on the light-fluid side, respectively, as is indicated in figure 7, and by $Q_H$ the volume flow rate of mixed fluid that the curtain discharges horizontally to each side upon reaching the water surface. The volume flow rate per unit width of the tank $W$ is denoted as $q_H$. To account for the recirculation, we denote the volume flow rate per unit width that leaves the cell on the light-fluid side as a gravity current as $\alpha q_H$ (or volumetric flow $\alpha Q_H$), on the dense-fluid side, as $\beta q_H$ (or $\beta Q_H$), where $\alpha <1$, $\beta <1$.

We have implicitly assumed that symmetry implies that the same volume flow of liquid per unit width $q_H$ is discharged to each side. Similarly, we assume that the bubble curtain entrains the fluid equally from both sides, which is again $q_H$ in a steady state. We further postulate that the fluid in both recirculation cells (especially also in the quasi recirculation cell on the dense-fluid side) is well-mixed and that the gravity currents originating from them possess the same density: $\rho _d'$ or $\rho _l'$. We refer to figure 7 for a schematic of the involved flows.

The mass balance yields the density of the fluid rising within the plume as

(5.6)\begin{equation} \frac{1}{2}(\rho_l'+\rho_d'). \end{equation}

We can establish the mass balance equations in the recirculation cell in the light-fluid half as

(5.7)\begin{equation} \alpha q_H \rho_l + q_H \frac{1}{2}(\rho_l'+\rho_d') - \alpha q_H \rho_l' - q_H \rho_l' = 0 \end{equation}

and in the dense-fluid half as

(5.8)\begin{equation} \beta q_H \rho_d + q_H \frac{1}{2}(\rho_l'+\rho_d') - \beta q_H \rho_d' - q_H \rho_d' = 0. \end{equation}

This is equivalent to

(5.9) \begin{equation} \left(\begin{array}{cc} -\tfrac{1}{2}-\alpha & \tfrac{1}{2}\\ \tfrac{1}{2} & -\tfrac{1}{2}-\beta \end{array}\right) \left(\begin{array}{c} \rho_l' \\ \rho_d' \end{array}\right) = \left(\begin{array}{c} -\alpha\rho_l \\ -\beta\rho_d \end{array}\right). \end{equation}

Therefore,

(5.10)\begin{equation} \left. \begin{aligned} \rho_l' & = \dfrac{\left(\tfrac{1}{2}+\beta\right)\alpha\rho_l+\tfrac{1}{2}\beta\rho_d}{\Delta} ,\\ \rho_d' & = \dfrac{\left(\tfrac{1}{2}+\alpha\right)\beta\rho_d+\tfrac{1}{2}\alpha\rho_l}{\Delta} , \end{aligned} \right\} \end{equation}

with

(5.11)\begin{equation} \Delta = \left(\frac{1}{2}+\alpha\right)\left(\frac{1}{2}+\beta\right)-\frac{1}{4} =\frac{\alpha+\beta}{2}+\alpha\beta. \end{equation}

Then, the equations for the gravity currents fluxes in both halves give (see (2.20))

(5.12)\begin{equation} \left. \begin{aligned} \alpha q_H & = \frac{C_d}{3}H^{3/2}g^{1/2}\left(\frac{\rho_l'-\rho_l}{\bar{\rho}}\right)^{1/2},\\ \beta q_H & = \frac{C_d}{3}H^{3/2}g^{1/2}\left(\frac{\rho_d-\rho_d'}{\bar{\rho}}\right)^{1/2}. \end{aligned} \right\} \end{equation}

Solving the system (5.10) and (5.12) of four equations and four unknowns $\rho _l'$, $\rho _d'$, $\alpha$ and $\beta$ yields

(5.13)\begin{gather} \alpha^{3}+\alpha^{2} = \frac{C_d^{2}}{18}\frac{H^{3} g'}{q_H^{2}}, \end{gather}

(5.14)\begin{gather}\alpha =\beta, \end{gather}

(5.15)\begin{gather}\rho_l'= \frac{\left(\frac{1}{2}+\alpha\right)\rho_l+\frac{1}{2}\rho_d}{\alpha + 1}, \end{gather}

(5.16)\begin{gather}\rho_d'= \frac{\left(\frac{1}{2}+\alpha\right)\rho_d+\frac{1}{2}\rho_l}{\alpha + 1}. \end{gather}

The only obstacle with respect to the comparison of this model with our experiments is the unknown value of the volume flux per unit length $q_H$. The dimensional scaling inspired by Morton et al. (Reference Morton, Taylor and Turner1956) that has been previously discussed is

(5.17)\begin{equation} q_H \sim (g q_{air})^{1/3} H. \end{equation}

It is important, however, to determine the order of magnitude of the coefficient that goes with the scaling since the model we have developed is not a scaling law but aims at predicting the values as precisely as possible. Abraham et al. (Reference Abraham, van der Burgh and DeVos1973) suggest that the depth of the outflowing current is $H/4$ and that the velocity increases linearly from a zero value at the lower boundary of the current to a maximum value at the water surface. Bulson (Reference Bulson1961) measured that the maximum surface velocity $V_{max}$ of the horizontal current created by a line bubble plume scales as

(5.18)\begin{equation} V_{max} \approx C_v (g q_{air})^{1/3} \end{equation}

with the proportionality factor $C_v\approx 1.46$.

This results in

(5.19)\begin{equation} q_H \approx \frac{1}{2}V_{max}\frac{H}{4}=\frac{C_v}{8}(g q_{air})^{1/3} H\approx 0.18 (g q_{air})^{1/3} H. \end{equation}

We conducted some experiments to examine whether the scaling (5.18) is appropriate in our case (at our small scale, in particular) that are briefly presented in Appendix C.

5.3. Comparison between model and experiments

We now use the model developed in the previous section in order to predict the infiltration flux $q_c^{th}$ of dense fluid across the bubble curtain during an experimental run and compare it with the experimentally measured flux $q_c$ (4.3). Note that the superscript ‘th’ stands for ‘theoretical’.

The procedure to calculate $q_c^{th}$ is as follows. First, we use (5.13) to determine the value of the parameter $\alpha$ for each experimental run. The right-hand side of (5.13),

(5.20)\begin{equation} \frac{C_d^{2}}{18}\frac{H^{3}g'}{q_H^{2}}=\frac{64C_d^{2}}{18C_v^{2}}\frac{Hg'}{(gq_{air})^{2/3}}=\frac{64C_d^{2}\alpha_E}{9C_v^{2}}\frac{1}{D_{m,b}}\approx\frac{0.072}{D_{m,b}}, \end{equation}

where we use (5.19) for $q_H$, contains the known initial condition $D_{m,b}$ along with proportionality coefficients measured in the previous literature and can be calculated directly for every experimental run. Equation (5.13) is then solved for $\alpha$. This yields one well-defined positive root $0<\alpha <1$ for any choice of initial conditions for which the bubble curtain is operating in the curtain-driven regime. A short proof of this statement is provided in Appendix D.

Once the value of $\alpha$ is found, the infiltration flux $q_c^{th}$ is calculated as

(5.21)\begin{equation} q_c^{th}=\left(\alpha q_H+\frac{L_{cell}H}{2t}\right)\frac{1}{2(\alpha+1)}, \end{equation}

where $L_{cell}$ can be computed using the correlation (5.4). Figure 8 elucidates the two terms inside the brackets contributing to the infiltration flux. The first term, $\alpha q_H$, is due to the gravity current propagating along the channel bottom. The second term, $L_{cell}H/2t$, arises because of the presence of the recirculation cell in the upper half of the channel next to the bubble curtain. It is calculated as the volume in the upper half of the recirculation cell $L_{cell}H/2$, which is fixed once the recirculation cell is established and is shown in the blue dashed square in figure 8, divided by the time $t$. We recall that we consider only those times $t$ which are longer than the initial transient time during which the recirculation cell is establishing. Note that this correction due to the recirculation cell decreases with an increasing duration $t$ of an experiment. The fluid inside the dyed volume in the light fluid possesses the density

(5.22)\begin{equation} \rho_l'=\frac{(1+2\alpha)\rho_l+\rho_d}{2(\alpha+1)}. \end{equation}

This means that a fraction of $(1+2\alpha )/(2(\alpha +1))$ of fluid inside the dyed volume originates from the light-fluid half whereas a fraction of $1/(2(\alpha +1))$ is the original dense fluid that has infiltrated the light-fluid half across the bubble curtain. This explains the presence of the factor $1/(2(\alpha +1))$ in the calculation of the theoretical infiltration flux (5.21).

Figure 8. Sketch of the light-fluid half of the channel illustrating two volume contributions of the dyed fluid in the calculation of the theoretical infiltration flux $q_c^{th}$ in (5.21).

Figure 9 shows the comparison between the experimentally measured flux $q_c$ across the bubble curtain in experiments A and the theoretically expected infiltration flux $q_c^{th}$ calculated using (5.21) with (5.19) and (5.4). We observe a near perfect collapse of the data around the line $q_c=q_c^{th}$ which shows that our theoretical model makes an excellent prediction of the infiltration flux. We can calculate the mean deviation of the experimentally measured values from the bisectrix of the first quadrant as $\sqrt {{\sum }_i(q_c(i)-q_c^{th}(i))^{2}/N}\approx 2.1\,\textrm {cm}^{2}\,\textrm {s}^{-1}$, where the index $i$ runs through our experiments and $N$ is the total number of our measurements. We note here, however, that our model does not take into account any reflections of the intruding gravity current from the end wall of the tank.

Figure 9. Experimentally measured values of the infiltration flux per unit length $q_c$ of dense fluid across the bubble curtain plotted against the theoretically expected infiltration flux per unit length $q_c^{th}$. We use (5.21) with (5.19) and (5.4) to calculate $q_c^{th}$ where $\alpha$ is determined for every set of initial conditions by (5.29). The black line depicts $q_c=q_c^{th}$.

We note that (5.21) presents the functional relationship given by (4.6). The time $t$ can be linked to the horizontal propagation distance $L$ of the intruding gravity current as

(5.23)\begin{equation} t = \frac{3 L}{2C_d\sqrt{\dfrac{g(\rho_l'-\rho_l)}{\bar\rho}H}}=\frac{3L\sqrt{2(\alpha+1)}}{2C_d\sqrt{g'H}}, \end{equation}

where we apply (5.12) and model the intruding volume as $\alpha q_H t = LH/2$. Then, using (5.19) and (5.4) we obtain

(5.24)\begin{align} \frac{q_c^{th}}{(gq_{air})^{1/3}H}&=\left(C_1\alpha+C_2\frac{1}{\sqrt{2(\alpha+1)}}\times\sqrt{\frac{H}{g'}}\times\frac{\sqrt{g'H}}{L}\right)\frac{1}{2(\alpha+1)}\nonumber\\ &=\left(C_1\alpha+C_2\frac{1}{\sqrt{2(\alpha+1)}}\times\frac{H}{L}\right)\frac{1}{2(\alpha+1)}, \end{align}

where $\alpha = \alpha (D_{m,b})$ and $C_1$ and $C_2$ are constant coefficients. This is exactly the form of the functional relationship (4.6) expected from the dimensional analysis.

For an infinitely long horizontal propagation distance $L$ (or, equivalently, for $t\to \infty$)

(5.25)\begin{equation} \frac{q_c^{th}}{(gq_{air})^{1/3}H}\to C_1\frac{\alpha}{2(\alpha+1)}, \end{equation}

which is equivalent to

(5.26)\begin{equation} q_c^{th}\to \frac{\alpha}{2(\alpha+1)}q_H. \end{equation}

We see that in this case, the non-dimensionalised infiltration flux is a function only of the deflection modulus, so that the scatter observed in figure 3(c) is an artefact of the finite length of our tank. We also recover the scaling proposed in (2.18) with the proportionality factor depending on $D_{m,b}$ as argued in (5.20). Thus, in contrast to the relation given by (2.16) for air curtains, the ratio $q_c/q_H$ for a bubble curtain is not constant with $D_{m,b}$ for a fixed opening geometry in the curtain-driven regime. However, as figure 3(c) indicates, the variation of $q_c/q_H$ with $D_{m,b}$ in the curtain-driven regime is relatively slow.

Figure 8 allows us to estimate the relative contributions of the terms $\alpha q_H$ and $L_{cell}H/2t$ to $q_c^{th}$ in (5.21). The first term dominates if the volume $\alpha q_H t$ is larger than the volume $L_{cell}H/2$, that is for those times $t$ when the recirculation cell is already established and the gravity current starts to flow out of the recirculation cell. Nevertheless, the correction due to the second term can be significant for a fixed finite length of the channel. Recall that we stopped our experiments when the gravity current reached the end wall of the light-fluid side. If, for example, the horizontal extent of the recirculation cell is half the length of the light-fluid side, then the correction due to the recirculation cell will be 50 %. We can assess the relative contribution of the correction due to the recirculation cell in our experiments by defining

(5.27)\begin{equation} q_c^{th,approx}=\frac{\alpha q_H}{2(\alpha+1)}, \end{equation}

which neglects the second term of (5.21). Figure 10 shows the experimentally measured flux $q_c$ against the approximate infiltration flux $q_c^{th,approx}$ (calculated using (5.27) with (5.19)). The horizontal distance between the axis of ordinates and a point is the contribution due to the term $\alpha q_H$, the horizontal distance between a point and the bisectrix of the first quadrant (shown as a black line) is the missing contribution due to the term $L_{cell}H/2t$ in (5.21).

Figure 10. Experimentally measured values of the infiltration flux per unit length $q_c$ of dense fluid across the bubble curtain plotted against the approximate infiltration flux per unit length $q_c^{th,approx}$ given by (5.27). The black line depicts the bisectrix of the first quadrant.

5.4. Effectiveness revisited

After establishing the model (5.13)–(5.16) allowing us to calculate the theoretical infiltration flux $q_c^{th}$ in (5.21), we revisit the effectiveness measurements presented in § 4.2.2 and can now explain the observed scatter in the experimental data for large $D_{m,b}$.

For $t\to \infty$, we can deduce that theoretically

(5.28)\begin{align} E&= 1-\frac{q_c^{th}}{q_{open}}\nonumber\\ &\to 1-\frac{\frac{\alpha}{2(\alpha+1)}\frac{C_v}{8}(gq_{air})^{1/3}H}{\frac{C_d}{3}H\sqrt{g'H}}\nonumber\\ &= 1-\frac{3\alpha C_v}{16C_d(\alpha+1)}\frac{\sqrt{D_{m,b}}}{\sqrt{2\alpha_E}}, \end{align}

where $\alpha$ is given by

(5.29)\begin{equation} \alpha^{3}+\alpha^{2}=\frac{64C_d^{2}\alpha_E}{9C_v^{2}}\frac{1}{D_{m,b}}. \end{equation}

This demonstrates that for $t\to \infty$, the effectiveness $E$ can be expressed as a function of $D_{m,b}$ and we would expect the collapse of our measured data. The data scatter observed in figure 4 arises from the presence of the recirculation cell and the fact that our channel is not long enough to achieve the asymptotic scaling for the infiltration flux. Thus, it is an artefact of the finite dimensions of our channel. Indeed, if we superimpose the curve $E(D_{m,b})$ as given in (5.28) and (5.29) on our data as shown in figure 11, we observe that it constitutes the upper envelope of the measured data points. The data points for $H=9\,\textrm {cm}$ (diamonds) are closer to our theoretical limit than for $H=20\,\textrm {cm}$ (squares), which is the expected behaviour because of the smaller aspect ratio $H/L$ for $H=9\,\textrm {cm}$.

Figure 11. Theoretical curve $E(D_m)$ in the limit $t\to \infty$ for an infinitely long channel calculated by means of (5.28) and (5.29). The experimentally measured effectiveness values (cf. figure 4) are below the theoretical curve due to the presence of the recirculation cell next to the bubble curtain and the finite length of the channel.

We stress here, in particular, that our theoretical model enables us to provide a theoretical upper limit on the effectiveness values of the bubble curtain by means of (5.28) and (5.29). The optimum effectiveness at $D_{m.b}\approx 0.12$ can be theoretically calculated as $E\approx 0.83$. We also observe that for $D_{m.b}\to \infty$, we obtain $E\to 1-1/2\sqrt {2}\approx 0.65$.

5.5. Instantaneous density measurements in experiments B

Our set of model equations (5.13)–(5.16) allows not only the calculation of the infiltration flux $q_c^{th}$ but also the prediction of the asymptotic values of densities $\rho _l'$ and $\rho _d'$.

Experiments B provide instantaneous and local density measurements in the curtain-driven regime by means of dye attenuation as explained in § 3.2. In particular, processing the data from an experiment B (see Appendix A) shows the temporal evolution of $\rho _{l,exp}'(t)$ and $\rho _{d,exp}'(t)$ in the recirculation cells. This can confirm that a steady state has indeed been reached and allows for a comparison between the predicted final values of $\rho _l'$ and $\rho _d'$ given in (5.15) and (5.16) and their measured final counterparts $\rho _{l,exp}'$ and $\rho _{d,exp}'$.

The quantitative results for the final densities $\rho _{l,exp}'$ and $\rho _{d,exp}'$ in experiments B are shown in table 1 while figure 12 shows a few examples of how $\rho _{l,exp}'(t)$ and $\rho _{d,exp}'(t)$ evolve over time. The examples provided in figure 12 show that the densities $\rho _{l,exp}'(t)$ and $\rho _{d,exp}'(t)$ in each half of the channel tend towards a constant value. The evolution $\rho _{l,exp}'(t)$ is steep near the beginning and quickly reaches its constant, asymptotic value. In contrast, the decrease of $\rho _{d,exp}'(t)$ is more stable and slower. An explanation for this asymmetry is that the flow itself is initially highly asymmetric around the bubble curtain. Indeed, on the dense-fluid side, the bubble curtain impinging on the surface immediately creates an outflowing gravity current which propagates in the upper half of the channel. Over time, some mixing occurs and some part of this mixed fluid is re-entrained by the bubble curtain giving rise to a weak quasi recirculation cell. In the light-fluid side, however, the fluid spilled laterally by the bubble curtain first sinks to the bottom and creates a recirculation cell so that this fluid can be re-entrained by the bubble curtain. Based on this physical picture, we indeed expect that the transient phase of $\rho _{d,exp}'(t)$ in the dense-fluid side will be longer because mixed fluid will require a longer time to recirculate.

Figure 12. Examples of the temporal evolution of densities $\rho _{l,exp}'$ (red line) and $\rho _{d,exp}'$ (blue line) in experiments B. The dashed lines indicate their respective theoretical predictions $\rho _l'$ and $\rho _d'$. The time $t$ is real time with the $t=0\,\textrm {s}$ corresponding to the gate removal at the start of an experiment. Here (a) $Q^{air}=10\,\textrm {l}\,\textrm {min}^{-1}$, $H=9\,\textrm {cm}$, $\Delta \rho =12\,\textrm {kg}\,\textrm {m}^{-3}$; (b) $Q^{air}=30\,\textrm {l}\,\textrm {min}^{-1}$, $H=15\,\textrm {cm}$, $\Delta \rho =12\,\textrm {kg}\,\textrm {m}^{-3}$; (c) $Q^{air}=10\,\textrm {l}\,\textrm {min}^{-1}$, $H=9\,\textrm {cm}$, $\Delta \rho =25\,\textrm {kg}\,\textrm {m}^{-3}$; (d) $Q^{air}=30\,\textrm {l}\,\textrm {min}^{-1}$, $H=15\,\textrm {cm}$, $\Delta \rho =25\,\textrm {kg}\,\textrm {m}^{-3}$.

Table 1. Quantitative comparison of $\rho _l'$ and $\rho _d'$ with their measured final counterparts $\rho _{l,exp}'$ and $\rho _{d,exp}'$ in experiments B. The measurement error in the density using the dye attenuation is approximately $1\,\textrm {kg}\,\textrm {m}^{-3}$.

Figure 13 shows the ratios $(\rho _{l,exp}'-\rho _l')/\Delta \rho$ and $(\rho _{d,exp}'-\rho _d')/\Delta \rho$ of the 17 experiments listed in table 1. Given the measurement errors using the dye attenuation technique explained in § 3.2 and Appendix A, we observe that predicted values of $\rho _l'$ in (5.15) are very close to experimental measurements for $\rho _{l,exp}'$: the discrepancy is always within 10 % of the initial density difference $\Delta \rho$ and is in almost all experiments within $1\,\textrm {kg}\,\textrm {m}^{-3}$. There is, however, a bigger discrepancy when it comes to $\rho _d'$, with differences between the model and the experiments that sometimes reach $4\,\textrm {kg}\,\textrm {m}^{-3}$. In all experiments $\rho _{d,exp}'$ is underestimated by the theoretical values for $\rho _d'$ in (5.16). This may be a consequence of the fact that a steady state on the dense-fluid side is not quite reached in our experiments, as argued below.

Figure 13. The relative differences $(\rho _{l,exp}'-\rho _l')/\Delta \rho$ (red diamonds) and $(\rho _{d,exp}'-\rho _d')/\Delta \rho$ (blue circles) of the 17 experiments listed in table 1.

As a side remark, we note that the asymptotic values of $\rho _l'$ and $\rho _d'$ (or $\rho _{l,exp}'$ and $\rho _{d,exp}'$, respectively) are not the same: there is still a density difference between the recirculation cells on either side of the bubble curtain in the steady state. The difference between $(\rho _d-\rho _l)/2$ and both $\rho _l'$ and $\rho _d'$ gives an idea of how imperfectly the curtain acts as a mixing device. Indeed, if it were a perfectly efficient mixer we would have a fluid of perfectly homogeneous density around it. With our model $\rho _d'-\rho _l'=\alpha (\rho _d-\rho _l)/(\alpha +1)$, which goes to 0 as $D_{m,b}\to \infty$.

The results provided in table 1 and figure 12 are obtained by averaging the information collected in experiments B over the recirculation cells (see Appendix A). In order to try to understand the quasi-systematic discrepancy between the predicted values of $\rho _d'$ and the measured values $\rho _{d,exp}'$, it is useful to consider the density maps inside the tank.

Figure 14 shows a density map for $Q^{air}=30\,\textrm {l}\,\textrm {min}^{-1}$, $H=15\,\textrm {cm}$ and $\Delta \rho =25\,\textrm {kg}\,\textrm {m}^{-3}$. The quantitative data for this particular experiment B15 are listed in table 1 and the evolution of the density in both cells is shown in figure 12. At the end of the experiment, there is a noticeable volume of fluid at the bottom of the tank below the recirculation cell in the dense-fluid half (right-hand side of the tank), that is denser than the cell. This phenomenon was not taken into account in the theoretical model (see § 5.2) which could explain why there is a discrepancy between the model and the experiments in terms of $\rho _d'$.

Figure 14. Density map in $\textrm {kg}\,\textrm {m}^{-3}$ obtained with data from experiment B15 for $Q=30\,\textrm {l}\,\textrm {min}^{-1}$, $H=15\,\textrm {cm}$ and $\Delta \rho =25\,\textrm {kg}\,\textrm {m}^{-3}$.

It is an interesting question where this bottom layer of fluid which is denser than the recirculation cell density $\rho _d'$ comes from. It could either consist of the dense fluid $\rho _d$ that is attracted along the bottom of the channel by the entrainment flow created by the bubble curtain and, thus, would continue to exist in the final steady state. Alternatively, it could be the original boundary layer of pure dense fluid that slowly disappears as the quasi recirculation cell slowly builds up, but would in our case take more time to disappear than the experiment lasts. In this latter case, we expect $\rho _{d,exp}'(t)\to \rho _d'$ as $t\to \infty$, but the time for the final steady state in the dense-fluid side to be reached would be longer than the duration of our experiments.

A simple experiment was conducted to understand what happens at the bottom of the channel in the dense-fluid side. The set-up was the same as in experiment A (see § 3.1) except that the dense fluid was filled into the half of the channel where the water was still before the removal of the sealing gate, i.e. the left-hand side of the channel in the recorded videos. Some dyed dense fluid was carefully injected at the bottom of the dense-fluid half with the least possible momentum in order to create a stable bottom layer of dyed fluid in the dense-fluid half of the channel. This dyed layer at the bottom allows the visualisation of the flow at the bottom of the tank in the dense-fluid half. We qualitatively observed that the bottom dyed layer in the vicinity of the bubble curtain was slowly disappearing by mixing into the recirculation cell and its colour was gradually fading out. We did not observe any significant flow of dense dyed fluid along the channel bottom caused by the entrainment of the bubble curtain.

Based on this qualitative experiment, it appears that the layer of denser fluid underneath the recirculation cell observed in figure 14 is most likely the initial bottom layer of the dense fluid that is still in the process of being slowly mixed. In particular, based on our experimental observations we can estimate that the time it takes for the initial bottom layer of dense fluid to disappear and the steady state to be reached is longer than the duration of our dye attenuation experiments: in figure 15, we can observe that the dyed fluid from the bottom layer is still not entirely mixed after $t=40\,\textrm {s}$ which is a typical duration of experiments B. Summarising, we expect that in the steady state $\rho _{d,exp}'(t)\to \rho _d'$. However, the times needed to reach this steady state are longer than the running times we were able to achieve in our dye attenuation measurements in experiments B. This is also corroborated by the observation of the blue curves in figure 12 that still appear to slowly decrease.

Figure 15. Time sequence of an experiment where only a thin layer of dense fluid is dyed at the bottom (the dense fluid is on the left and the light fluid is on the right) with $Q=20\,\textrm {l}\,\textrm {min}^{-1}$, $H=15\,\textrm {cm}$ and $\Delta \rho =25\,\textrm {kg}\,\textrm {m}^{-3}$.