2.1. Global conservation equations

For an incompressible flow, the volume flux $Q_{P_{i}}$ supplied to the upper layer by the plume at the interface is exactly balanced by the volume flux leaving this layer, i.e. via the sum of that entrained $Q_{e}$ and driven through the top opening $Q_{t}=Q_{f_{0}}$ . The buoyancy flux supplied to the upper layer comprises $B_{P}$ and the flux of buoyancy $g_{1}^{\prime }Q_{P_{i}}$ entrained by the plume from the lower layer. The impinging fountain entrains a flux of buoyancy $g_{2}^{\prime }Q_{e}$ from the upper layer and the outflow from the box expels buoyant fluid at a rate $g_{2}^{\prime }Q_{t}$ . Thus, for the upper layer, conservation of volume and buoyancy require, respectively,

Inspired by the long history of simplified models on turbulent plumes and fountains, top-hat profiles are adopted for the time-averaged horizontal variation of the vertical velocity and buoyancy of the plume and fountain. Following the classic plume theory of Morton, Taylor & Turner (Reference Morton, Taylor and Turner1956),

where ${\it\alpha}_{P}$ is the top-hat entrainment coefficient for the plume. Following Turner (Reference Turner1986) we take ${\it\alpha}_{P}=0.117$ . As the plume is merely an artefact in the problem, in (2.4) we have neglected the effects of the secondary flow on plume entrainment. Although the secondary flow may modify the entrainment coefficient ${\it\alpha}_{P}$ , the choice of ${\it\alpha}_{P}$ does not influence the scaling of $Q_{e}$ , which is the quantity of primary interest to us.

The source buoyancy flux of the fountain $B_{f_{0}}$ and the buoyancy flux turbulently entrained via the impingement dome from the upper layer $g_{2}^{\prime }Q_{e}$ are supplied to the lower layer. Due to entrainment into the plume, a flux of buoyancy $g_{1}^{\prime }Q_{P_{i}}$ is removed from the lower layer. Thus, conservation of buoyancy for the lower layer requires

Denoting $b_{f_{0}}$ and $w_{f_{0}}$ as the radius and vertical velocity, respectively, of the fountain at its source,

The conservation equations, (2.3) and (2.5), are reduced to their simplest form by scaling the layer buoyancies on the source buoyancy $g_{f_{0}}^{\prime }=|B_{f_{0}}|/Q_{f_{0}}$ of the fountain and scaling the interface height on the box height  $H$ . Accordingly, we introduce the dimensionless variables

To proceed, we substitute (2.4) into (2.3a ) and rearrange for $h$ . We then substitute for $g_{1}^{\prime }$ from (2.5) into (2.3b ). Non-dimensionalising the resulting equations yields closed-form solutions for the dimensionless interface height and the dimensionless layer buoyancies:

The three resulting non-dimensional parameters (2.9) of our system are the buoyancy flux ratio  ${\it\psi}$ , the fountain source Froude number $\mathit{Fr}_{0}$ and the dimensionless radius $\mathscr{L}$ of the fountain source:

At this stage, $E=Q_{e}/Q_{f_{0}}$ is the single unknown dimensionless quantity. In order to determine $Q_{e}$ and hence close the problem, we now model the fountain–interface interaction.

2.2. Secondary flow in the environment

The impingement of the fountain produces a ‘local’ upward deflection of the interface. To produce a local deflection, we require the radius of the impinging fountain $b_{i}\ll \sqrt{S}$ . Although the two-layer fluid remains close to hydrostatic equilibrium, the ‘tilting’ of the interface indicates that the surfaces of constant pressure and constant density are no longer parallel. The action of the resulting baroclinic torque reduces the horizontal density gradients eventuating from the interfacial deflection and restores the interface back to the horizontal. The sustained forcing of the interface by the fountain gives rise to a near-continuous cycle of vertical deflections from the horizontal. This perpetual process is confirmed by several experimental studies (Kumagai Reference Kumagai1984; Shy Reference Shy1995; Hunt & Coffey Reference Hunt and Coffey2010). Figure 5 shows shadowgraph images from Hunt & Coffey’s (Reference Hunt and Coffey2010) experiments; in figure 5(a), the interface between two saline layers of different density is deflected upwards due to the localised impingement of a fountain from below. A short time later, the interface has been temporarily restored to its stable horizontal orientation (figure 5 b).

When the interface tilts, (heavy) fluid of density ${\it\rho}_{1}$ within the lower layer is displaced upwards and (light) fluid of density ${\it\rho}_{2}$ within the upper layer is displaced downwards. Thence, as the baroclinic torque acts to restore the interface, the potential energy of the displaced fluids is released and a fraction of this energy is converted into kinetic energy that drives fluid motions. As a direct consequence of the near-continuous release of potential energy from the perpetual cycle of interfacial deflections, a quasi-steady secondary flow is established and maintained in the upper and lower layers (see appendix A for arguments on why the secondary flow is quasi-steady). Crucially, therefore, fluid with non-zero velocity is entrained into the impingement dome. Ching et al. (Reference Ching, Fernando and Noh1993) observed secondary flows in the environment with velocities in the upper layer of up to 30 % of the plume’s vertical velocity at the interface (figure 3). Accordingly, to account for the non-zero flux of momentum entrained into the dome, we consider a time-averaged picture and seek an expression for the mean velocity $u_{2}$ of the secondary flow in the upper layer.

Figure 6. Schematic showing the modelled maximum deflection of the interface resulting from the impingement of the fountain. The position of the undisturbed interface is $\acute{z} =0$ . Due to the deflection, fluid is displaced in shaded regions $\text{A}$ and $\text{B}$ . Solid arrows in the upper and lower layers indicate the secondary flow induced by interfacial tilting.

In a given cycle of interfacial deflection, the interface attains a maximum vertical displacement of $\acute{z} ={\it\eta}$ from its equilibrium position at $\acute{z} =0$ , as depicted in figure 6. Dense lower-layer fluid is displaced upwards above $\acute{z} =0$ in region A and an equal volume of light upper-layer fluid is displaced downwards below $\acute{z} =0$ in region B (of horizontal extent $l_{B}$ ). Guided by observations (cf. figure 5), we assume that the horizontal extent, $l_{A}$ , of the upward deflection is equal to the horizontal length scale, $b_{i}$ , of the forcing that produces the deflection. The deflection of the interface is thereby assumed to occur within a circular region of radius $r_{id}=b_{i}+l_{A}+l_{B}=2b_{i}+l_{B}$ , which we refer to as the ‘zone of interfacial deflection’ (the subscript ‘ $id$ ’ meaning ‘interfacial deflection’). Whilst the deflected interface typically exhibits a relatively smoothly varying profile (figure 5 a), the gradient of our assumed profile is discontinuous at $r=r_{id}$ . Nevertheless, we will see that our model provides a simple yet effective starting point for quantifying the velocity $u_{2}$ resulting from the deflections. For the impingement of an axisymmetric fountain, the deflected regions are characterised by an annular prism whose outer radius and inner radius are $\{2b_{i},b_{i}\}$ for region A and $\{2b_{i}+l_{B},2b_{i}\}$ for region B. Accordingly, in the displaced state

Rearranging (2.10) gives a quadratic in $l_{B}/b_{i}$ , which has a single positive root:

When the interface attains its maximum displacement ( $\acute{z} ={\it\eta}$ ), fluid in regions A and B momentarily comes to rest. Subsequently, as the interface is restored, the displaced fluids are set in motion. Thus, at the instant when maximum tilting occurs there is no flux of momentum into, or out of, region A. Hence, conservation of vertical momentum for this fluid volume requires

i.e. the interface is brought momentarily to rest when the upward excitation force (left-hand side) causing the interfacial deflection is balanced by the downward restoring buoyancy force (right-hand side) acting on the displaced fluid of volume $3{\rm\pi}b_{i}^{2}{\it\eta}/2$ (2.10). Here, ${\it\rho}_{i}$ denotes the density of the fountain at the level of the undisturbed interface. Under the Boussinesq approximation, the maximum vertical displacement of the interface is, from (2.12),

As we shall see in § 2.3, the penetration depth of the impingement dome is $z_{d}/b_{i}=0.94\mathit{Fr}_{i}^{2}$ . Evidently, the scalings of ${\it\eta}$ and $z_{d}$ on $\mathit{Fr}_{i}$ are identical, which is not surprising given that both the origin of the interfacial deflection and the shallow-penetration dome is the vertical forcing by the fountain. We note that ${\it\eta}/z_{d}<1$ , a result one would expect on physical grounds. Moreover, our prediction that ${\it\eta}/z_{d}\approx 0.7$ is consistent with observations (cf. figure 5 a).

Upon calculating the potential energy of the two-layer fluid in the displaced state $\text{PE}_{dis}$ and the potential energy of the equilibrium state $\text{PE}_{eq}$ (see appendix B), we find that the surplus potential energy available for conversion into kinetic energy is

As the action of the baroclinic torque temporarily restores the tilted interface from the point of maximum deflection $(\acute{z} ={\it\eta})$ to the equilibrium position $(\acute{z} =0)$ , the available potential energy (2.14) is converted into kinetic energy for driving fluid motion in the upper and lower layers primarily over the radial extent $r_{id}$ . A flow towards the impingement dome is established within the upper layer, whereby light fluid of density ${\it\rho}_{2}$ replaces heavy fluid of density ${\it\rho}_{1}$ in region A (figure 6). We note that the hydrostatic pressure variations established by the tilting, and indeed the entrainment into the dome, induce a flow towards the dome. Similarly, a flow towards the side walls of the box is established within the lower layer, whereby fluid of density ${\it\rho}_{1}$ replaces fluid of density ${\it\rho}_{2}$ in region B. In our time-averaged conceptualisation, the mean kinetic energy of the secondary flow prevailing in the upper layer is

Given that we seek a local velocity for the bulk-flow-induced adjacent to the dome, in (2.15) we have calculated the kinetic energy of the flow within the region of interfacial deflection (i.e. for $r\leqslant r_{id}$ ). Continuous vertical oscillations of the interface typically contribute to the generation of interfacial gravity waves that extract a fraction of the potential energy released from the interfacial deflections. Some of the available potential energy may also be dissipated due to viscous effects. Thus, a fraction $c$ of the available potential energy remains after dissipation. For simplicity, and indeed in the absence of data to support or justify a more complex energy balance, we apportion the available potential energy (after dissipation) equally between the kinetic energies of the flows in the upper and lower layers, i.e. $\text{KE}_{2}=c(\text{APE}/2)$ . Seeking an upper-bound solution for $u_{2}$ , we neglect local energy losses so that $c=1$ . Under the Boussinesq approximation, the dimensionless mean velocity $\tilde{u} _{2}$ of the secondary flow in the upper layer is then

where the ‘confinement’ parameter

Evidently, ${\it\lambda}_{i}$ characterises the length scale of interfacial turbulence relative to the vertical extent $(H-h)$ of the confinement. Accordingly, hereafter ‘confinement’ refers to ‘vertical confinement’. For a given $b_{i}$ and $\mathit{Fr}_{i}$ , a relatively weak secondary flow ( $\tilde{u} _{2}\ll 1$ ) is established in the upper layer when the impingement occurs at low elevations ( $h\ll H$ ) in a tall box such that ${\it\lambda}_{i}$ is relatively small. In this limit, we expect our prediction of $u_{2}$ to be less reliable as the local interfacial deflections are unlikely to induce fluid motions over the depth of a relatively deep upper layer. However, we note that in the absence of confinement ${\it\lambda}_{i}\rightarrow 0$ and $\tilde{u} _{2}\rightarrow 0$ (2.16), which is entirely consistent with the unconfined case where interfacial entrainment occurs in the absence of a secondary flow (Shrinivas & Hunt Reference Shrinivas and Hunt2014).

In contrast, the secondary flow is relatively strong when the impingement occurs at high elevations in a short box such that ${\it\lambda}_{i}$ is relatively large. In this limit, we expect our model to capture well the dynamics of the secondary flow as the interfacial deflections are likely to excite fluid motions over the full depth of a relatively shallow upper layer, as indicated in figure 3(b). Indeed, it is this limit that is of primary interest to us. Moreover, we will see that for previous experimental studies ${\it\lambda}_{i}$ varies in the range $0.1<{\it\lambda}_{i}<1$ , with the majority of the values of ${\it\lambda}_{i}$ exceeding ${\it\lambda}_{i}=0.3$ , i.e. ${\it\lambda}_{i}$ was relatively large for previous experiments. Given that we focus on $\mathit{Fr}_{i}\lesssim 1$ , $\tilde{u} _{2}\sim O(1)$ (2.16). Thus, the velocity $\tilde{u} _{2}$ of the secondary flow cannot be reasonably ignored. In § 3.1, we establish the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’.

2.3. Impingement dome

Having determined $u_{2}$ , our attention turns to the volume flux $Q_{e}$ of buoyant fluid, with velocity $u_{2}$ , turbulently entrained from the upper layer into the axisymmetric semi-ellipsoidal impingement dome atop the incident fountain (figure 7). To model the entrainment into the dome, we employ the theoretical framework developed in Shrinivas & Hunt (Reference Shrinivas and Hunt2014). The entrained fluid is transported across the interface through an annulus of width $(b_{d}-b_{i})$ . Adopting top-hat profiles for the variation of the vertical downward velocity $\grave{w} _{d}<0$ and density ${\it\rho}_{d}$ across this annular region, conservation of volume for the impingement dome requires

where $w_{d}=|\grave{w} _{d}|$ . The dome receives a flux of momentum ${\rm\pi}{\it\rho}_{i}b_{i}^{2}w_{i}^{2}$ from the fountain and a flux of momentum ${\rm\pi}{\it\rho}_{d}(b_{d}^{2}-b_{i}^{2})w_{d}^{2}$ is transported out of the dome through the annulus. Guided by the measurements of Ching et al. (Reference Ching, Fernando and Noh1993) (figure 3), we assume that the secondary flow in the vicinity of the dome boundary is horizontal. Accordingly, the flux of vertical momentum entrained into the dome is zero. Therefore, conservation of vertical momentum for the dome requires

where $g_{i}^{\prime }$ is the mean buoyancy experienced by dense fluid within the dome of volume $2{\rm\pi}b_{d}^{2}z_{d}/3$ . Substituting for $w_{d}$ from (2.18) into (2.19) gives

for Boussinesq flows, where

are the dimensionless penetration depth, the aspect ratio of the dome and the dimensionless buoyancy of the fountain at the interface, respectively. We will see (§§ 3.2, 3.3) that solutions of (2.8) are realised for fountains with source Froude numbers $\mathit{Fr}_{0}\gg 1$ and source buoyancy flux ratios ${\it\psi}\gg 1$ , i.e. the impinging fountain is highly forced at its source. For these source conditions, the fountain behaviour is jet-like over a significant fraction of the rise height, and hence its density ${\it\rho}_{i}$ at the interface is comparable with ${\it\rho}_{1}$ for Boussinesq flows. Accordingly, we invoke the approximation that ${\check{g}}_{i}^{\prime }\approx 1$ , which considerably simplifies the analysis whilst having little effect on the solution. By considering conservation of buoyancy for the dome, we find that our approximation ${\check{g}}_{i}=1$ leads to a marginal overprediction (by no more than 4 %) of $Q_{e}$ (see appendix C).

Figure 7. Schematic showing (a) the small- $\mathit{Fr}_{i}$ axisymmetric semi-ellipsoidal impingement dome atop the impinging fountain and (b) the modelled vortex layer (shaded area) of the dome. In (b), the dashed box is the control volume considered for the inflow into the dome due to turbulent entrainment.

To close the problem, it is necessary to describe the mechanism by which external fluid is entrained into the dome. The representation of turbulence, necessary to close the problem mathematically, forms the cornerstone of the mechanistic entrainment model developed by Shrinivas & Hunt (Reference Shrinivas and Hunt2014). Shy (Reference Shy1995) and Cotel et al. (Reference Cotel, Gjestvang, Ramkhelawan and Breidenthal1997) identified that the interaction between the incident vorticity within the impinging flow and the baroclinic vorticity generated at the interface, due to its tilting, results in the formation of strong persistent vortices within a relatively thin layer around the perimeter of the dome. These rotational motions are predominantly responsible for the entrainment into the dome. By modelling this peripheral region of strong vorticity as a finite-thickness vortex layer (the shaded region of the schematic of figure 7 b), Shrinivas & Hunt (Reference Shrinivas and Hunt2014) related the generation of baroclinic vorticity to the resulting mean entrainment velocity $u_{e}$ and showed that

i.e. the entrainment velocity is proportional to a local buoyancy velocity. Here, ${\it\alpha}_{d}$ is an entrainment coefficient, an appropriate value for which is ${\it\alpha}_{d}=0.1$ (Turner Reference Turner1986).

To account for the horizontal momentum $(\propto u_{2})$ of the fluid entrained from the upper layer, we consider a control volume of height $\text{d}\acute{z}$ around the dome perimeter (figure 7 b). Owing to the secondary flow, a volume flux $q_{2}$ per unit height with fluid of velocity $u_{2}$ enters the control volume. A volume flux $q^{\ast }$ per unit height leaves the control volume and enters the dome at velocity $u^{\ast }$ . Thus, conservation of horizontal momentum for the flow near the dome boundary requires

where $F_{e}$ denotes the inertial force of the peripheral energy-containing vortices that drive entrainment into the dome and ${\it\rho}^{\ast }$ denotes the density of the fluid entering the vortex layer. Over a vertical extent $\text{d}\acute{z}$ of the vortex layer, a volume $\text{d}V_{e}$ of external fluid is engulfed at velocity $u_{e}$ in a time $\text{d}t$ . Therefore, the inertial force associated with the energy-containing vortices that gives rise to and maintains the steady inflow into the dome is

Under the Boussinesq approximation, substituting for $F_{e}$ (2.24) into (2.23) and given that conservation of volume for the steady flow requires $q_{2}\,\text{d}\acute{z} =q_{e}\,\text{d}\acute{z} =\text{d}V_{e}/\text{d}t$ , we obtain

i.e. the mean horizontal inflow velocity $u^{\ast }$ across the boundary between the turbulent flow and the external environment is given by the sum of the entrainment velocity of the peripheral vortices and the velocity of the secondary flow in the upper layer. To further support this entrainment formulation, we may draw a direct analogy between plumes rising in a weak crossflow and our dome in the midst of a predominantly horizontal secondary flow. The net entrainment into a plume rising in a crossflow is the sum of two parts, namely, the entrainment driven by the vertical shear and the entrainment of the crossflow (Hoult & Weil Reference Hoult and Weil1972; Lee & Chu Reference Lee and Chu1979; Devenish et al. Reference Devenish, Rooney, Webster and Thomson2010). Similarly, the net entrainment into the dome in the presence of a secondary flow comprises two components, namely, the entrainment driven by baroclinic vortices and the entrainment of the secondary flow. Thus, (2.25) is consistent with well-established entrainment models for plumes in a crossflow.

The total volume flux entrained into the dome over the height $0\leqslant \acute{z} \leqslant z_{d}$ is

where $b_{p}$ is the radius of the dome at height $\acute{z}$ . The dome perimeter is the semi-ellipse:

Substituting for $u_{2}$ (2.16), $u_{e}$ (2.22) and $b_{p}$ (2.27) into (2.26) gives

To complete our solution we require the penetration depth $z_{d}$ of the dome. For the unconfined case, wherein interfacial entrainment occurs in the absence of a secondary flow (i.e. $\tilde{u} _{2}\rightarrow 0$ as ${\it\lambda}_{i}\rightarrow 0$ ), Shrinivas & Hunt (Reference Shrinivas and Hunt2014) considered conservation of energy for the dome and showed that

For the confined problem of interest here, entrainment of fluid into the dome results in an inflow of kinetic energy $\propto u_{2}^{2}$ and potential energy $\propto {\rm\Delta}g^{\prime }z_{d}$ . Whilst both can be accounted for in the energy budget, an elegant approximation becomes evident on examining the relative magnitudes of the kinetic and potential energies entrained. Given that $z_{d}/b_{i}\propto \mathit{Fr}_{i}^{2}$ , the ratio of the entrained energies is $u_{2}^{2}/({\rm\Delta}g^{\prime }z_{d})\propto u_{2}^{2}/w_{i}^{2}$ . Noting that $u_{2}^{2}/w_{i}^{2}\ll 1$ (2.16) for the range of $\{\mathit{Fr}_{i},{\it\lambda}_{i}\}$ considered, the kinetic energy entrained into the dome is relatively small and thus may reasonably be neglected. Hence, the secondary flow has a relatively minor influence on the energy budget for the dome. Thus, to a good approximation, the penetration depth of the confined dome is given by (2.29).

With the problem fully closed, we substitute for $u_{2}$ (2.16) and $z_{d}$ (2.29) into (2.28). The resulting expression is rearranged for the aspect ratio $k$ and substituted into (2.20). This yields a cubic in $E_{i}$ as a function of the interfacial Froude number and the confinement parameter:

where

This cubic has a single positive root, namely

For $\mathit{Fr}_{i}<1$ and ${\it\lambda}_{i}<1$ , a two-step procedure (see appendix D) reduces (2.32) to the simple analytic solution

which closely approximates the full solution. In (2.33),

The first term of (2.33) is the contribution to the total volume flux into the dome due to entrainment driven by peripheral vortices and the second term is the volume flux into the dome due to entrainment of the secondary flow.

When interfacial entrainment occurs in an unconfined environment, ${\it\lambda}_{i}\rightarrow 0$ and $\tilde{u} _{2}\rightarrow 0$ (2.16). In this limit $E_{i}=E_{i}(\mathit{Fr}_{i})$ and we recover our solution $E_{i}=A\mathit{Fr}_{i}^{2}$ for unconfined entrainment (Shrinivas & Hunt Reference Shrinivas and Hunt2014). When interfacial entrainment occurs within the confines of a box so that ${\it\lambda}_{i}>0$ , a secondary flow persists ( $\tilde{u} _{2}>0$ ) and $E_{i}=E_{i}(\mathit{Fr}_{i},{\it\lambda}_{i})$ . Thus, the second term of (2.33) solely accounts for the role of confinement. To quantify the influence of confinement on $E_{i}$ , we consider the total entrainment flux $E_{i}$ relative to the entrainment flux in the absence of confinement (i.e. the unconfined component in (2.33)). Accordingly, we introduce the ratio

is the rate of entrainment of the secondary flow (the subscript ‘ $c$ ’ meaning ‘confined’). When ${\it\phi}\sim 1$ , the contribution of $E_{i,c}$ to $E_{i}$ is small and box confinement does not significantly influence the rate of interfacial entrainment. When ${\it\phi}\gg 1$ , $E_{i,c}$ provides a significant contribution to $E_{i}$ , and hence confinement plays an instrumental role in increasing the total entrainment flux $E_{i}$ .

2.4. Impinging fountain

To predict $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ , it is necessary to model the fountain. Assuming that the downflow of the fountain is fully incorporated into the lower layer, conservation of volume flux $({\rm\pi}b_{f}^{2}w_{f})$ , momentum flux $({\rm\pi}b_{f}^{2}w_{f}^{2})$ and buoyancy flux $(-{\rm\pi}b_{f}^{2}w_{f}g_{f}^{\prime })$ for the fountain upflow requires (Morton et al. Reference Morton, Taylor and Turner1956)

where ${\it\alpha}_{f}$ is the top-hat entrainment coefficient for the fountain. Following Kaye & Hunt (Reference Kaye and Hunt2006), we take ${\it\alpha}_{f}=0.085$ . Given that the fountain behaviour is jet-like for the majority of its rise height, the vertical velocity $w_{f}$ of the fountain is large compared with the velocity of the secondary flow in the lower layer. Therefore, it is reasonable to assume that entrainment into the fountain is not significantly influenced by the secondary flow in the lower layer. Whilst the secondary flow may modify the entrainment coefficient ${\it\alpha}_{f}$ , it is reassuring that the choice of ${\it\alpha}_{f}$ does not influence the scaling of $E_{i}$ , the quantity of primary interest to us.

Scaling quantities of interest on their values at the fountain source, we seek the vertical variation of the dimensionless radius ${\it\beta}_{f}=b_{f}/b_{f_{0}}$ , vertical velocity ${\it\omega}_{f}=w_{f}/w_{f_{0}}$ and local Froude number $\mathit{Fr}_{f}=w_{f}/\sqrt{b_{f}g_{f}^{\prime }}$ of the fountain. The non-dimensional governing equations for the fountain are

where $\mathscr{Z}=(6{\it\alpha}_{f}/5)z/b_{f_{0}}$ . The source conditions of the fountain are ${\it\beta}_{f}(\mathscr{Z}=0)=1$ , ${\it\omega}_{f}(\mathscr{Z}=0)=1$ and $\mathit{Fr}_{f}(\mathscr{Z}=0)=\mathit{Fr}_{0}$ .

The non-dimensional parameters of our system (2.9) are the source Froude number $\mathit{Fr}_{0}$ of the fountain, the buoyancy flux ratio ${\it\psi}$ and the fountain source radius $\mathscr{L}$ . For a given $\{\mathit{Fr}_{0},{\it\psi},\mathscr{L}\}$ , solutions were obtained using an iterative procedure. First, by making an initial estimate of $E=Q_{e}/Q_{f_{0}}$ , the interface position $({\it\xi})$ and mean layer buoyancies $({\it\delta}_{1},{\it\delta}_{2})$ were calculated from (2.8a−c ). Subsequently, the fountain equations (2.37) were solved to obtain the dimensionless vertical velocity ${\it\omega}_{i}$ and dimensionless radius ${\it\beta}_{i}$ of the fountain at the interface. The interfacial Froude number and the confinement parameter were determined from

We then calculated the dimensionless entrainment flux $E_{i}$ (2.33) and re-estimated $E=E_{i}{\it\beta}_{i}^{2}{\it\omega}_{i}$ , the interface position and the layer buoyancies. This procedure was repeated until $E_{i}$ , ${\it\xi}$ , ${\it\delta}_{1}$ and ${\it\delta}_{2}$ converged to fixed values.

3.1. Power laws for unconfined and confined entrainment

As an illustrative example of the role of confinement, figure 8(a) plots $E_{i}$ as a function of ${\it\lambda}_{i}$ for $\mathit{Fr}_{i}=0.7$ . Also plotted is the power law $E_{i}=A\mathit{Fr}_{i}^{2}$ for unconfined interfacial entrainment. As ${\it\lambda}_{i}$ increases, and thus the confinement is enhanced, the velocity $\tilde{u} _{2}(\propto \sqrt{{\it\lambda}_{i}})$ of the secondary flow increases (see § 3.3 for a further discussion on the role of ${\it\lambda}_{i}$ ). Therefore, the rate of entrainment $E_{i,c}(\propto \tilde{u} _{2})$ of the secondary flow and hence $E_{i}$ increase with ${\it\lambda}_{i}$ (figure 8 a). Notably, when the radius of the impinging fountain equals the depth of the upper layer (i.e. ${\it\lambda}_{i}=1$ ), $E_{i}$ exceeds the rate of interfacial entrainment in an unconfined environment by a factor of three.

Figure 8. Dimensionless entrainment flux $E_{i}$ across the interface: (a) as a function of the confinement parameter ${\it\lambda}_{i}$ for $\mathit{Fr}_{i}=0.7$ and (b) as a function of the interfacial Froude number $\mathit{Fr}_{i}$ for ${\it\lambda}_{i}=\{0,0.5\}$ .

Figure 8(b) plots $E_{i}$ as a function of $\mathit{Fr}_{i}$ for ${\it\lambda}_{i}=\{0,0.5\}$ and further emphasises the disparities between entrainment in unconfined and confined environments. As $\mathit{Fr}_{i}$ increases, and thus the vertical forcing of the interface is strengthened, larger deflections ${\it\eta}$ $(\propto \mathit{Fr}_{i}^{2})$ are produced. Hence, for ${\it\lambda}_{i}>0$ a greater amount of potential energy is released and converted into kinetic energy of the secondary flow. Therefore, $\tilde{u} _{2}$ and $E_{i}$ increase with $\mathit{Fr}_{i}$ (figure 8 b). Significantly, when the diameter $(2b_{i})$ of the impinging fountain equals the upper-layer depth (i.e. ${\it\lambda}_{i}=0.5$ ), doubling $\mathit{Fr}_{i}$ from $\mathit{Fr}_{i}=0.5$ to $\mathit{Fr}_{i}=1$ gives rise to a sevenfold increase in $E_{i}$ ; this compares with a fourfold increase in the unconfined case.

Crucially, the scaling of $E_{i}$ on $\mathit{Fr}_{i}$ is determined by ${\it\lambda}_{i}$ . From (2.33), our entrainment law follows

In other words, when ${\it\lambda}_{i}\ll {\it\lambda}_{i,c}$ , the secondary flow is sufficiently weak such that the contribution of $E_{i,c}$ to $E_{i}$ is small, and thus $E_{i}$ is quadratic in $\mathit{Fr}_{i}$ as per ‘unconfined’ entrainment (Shrinivas & Hunt Reference Shrinivas and Hunt2014). When ${\it\lambda}_{i}\gg {\it\lambda}_{i,c}$ , the secondary flow is sufficiently strong such that $E_{i,c}$ provides the dominant contribution to $E_{i}$ and thus, $E_{i}$ is cubic in $\mathit{Fr}_{i}$ (2.33). We refer to interfacial entrainment in these small- ${\it\lambda}_{i}$ and large- ${\it\lambda}_{i}$ limits as weakly confined and strongly confined, respectively.

Figure 9. (a) Interface height ${\it\xi}$ against the source Froude number $\mathit{Fr}_{0}$ of the fountain and (b) velocity $\tilde{u} _{2}$ of secondary flow against the confinement parameter ${\it\lambda}_{i}$ for buoyancy flux ratio ${\it\psi}=25$ and fountain source radius $\mathscr{L}=0.1$ .

This result provides a plausible physical explanation for why some experimental studies support a $E_{i}\propto \mathit{Fr}_{i}^{2}$ power law, whereas others report $E_{i}\propto \mathit{Fr}_{i}^{3}$ . Our model (3.1) indicates that measurements of $E_{i}$ following a cubic power law (Baines Reference Baines1975; Coffey & Hunt Reference Coffey and Hunt2010) may have been influenced by a secondary flow owing to relatively large values of  ${\it\lambda}_{i}$ . Inferring values of ${\it\lambda}_{i}$ from previous works, we find that for the experiments of Coffey & Hunt (Reference Coffey and Hunt2010), ${\it\lambda}_{i}$ was as large as unity, as discussed in § 3.3. Conversely, measurements of $E_{i}$ supporting a quadratic power law (Cardoso & Woods Reference Cardoso and Woods1993; Ching et al. Reference Ching, Fernando and Noh1993) were not significantly influenced by a secondary flow as ${\it\lambda}_{i}$ was sufficiently small. Indeed, for the experiments of Ching et al. (Reference Ching, Fernando and Noh1993), a value of ${\it\lambda}_{i}=0.2$ was never exceeded. We will see (§ 3.3) that for previous studies, ${\it\lambda}_{i}$ exhibits wide variation, spanning the range $0.1<{\it\lambda}_{i}<1$ . Hence, these studies capture $E_{i}$ not only within the weakly confined and strongly confined regimes, but also in the transitional regime where comparable contributions from the quadratic and cubic components are expected.

3.2. Entrainment for varying $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$

Thus far, we have focused on entrainment for constant $\mathit{Fr}_{i}$ or constant ${\it\lambda}_{i}$ (figure 8). The difficulty in achieving this in practice is exemplified by the significant variation of $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ in previous experiments (see § 3.3). To examine $E_{i}$ when both $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ vary, we increase the source Froude number $\mathit{Fr}_{0}$ of the fountain, whilst holding the source radius $\mathscr{L}$ and buoyancy flux ratio ${\it\psi}$ constant. As this produces more energetic interfacial impingements, the rate of entrainment $(E_{i})$ of fluid from the upper layer increases and hence the interface height ${\it\xi}$ increases (figure 9 a). Consequently, the radius ${\it\beta}_{i}$ of the fountain at the interface increases. Therefore, increasing $\mathit{Fr}_{0}$ results in larger values of ${\it\lambda}_{i}\propto {\it\beta}_{i}/(1-{\it\xi})$ . Enhancing the confinement in this way gives rise to stronger secondary flows, i.e. $\tilde{u} _{2}$ increases with ${\it\lambda}_{i}$ , as shown in figure 9(b). Notably, $u_{2}$ is approximately 25 % of the fountain’s vertical velocity $w_{i}$ at the interface when ${\it\lambda}_{i}\sim 1$ .

Figure 10. Entrainment flux ratio ${\it\phi}$ (2.35): (a) against the interfacial Froude number $\mathit{Fr}_{i}$ and (b) against the confinement parameter ${\it\lambda}_{i}$ for source buoyancy flux ratio ${\it\psi}=25$ and fountain source radius $\mathscr{L}=0.1$ .

Figures 10(a) and 10(b) plot the entrainment flux ratio ${\it\phi}$ (2.35) against $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ , respectively. Two limiting cases can be identified: one in which $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ are small and a second in which $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ are large. For $\mathit{Fr}_{i}\ll 1$ a weak forcing of the interface by the fountain produces relatively small deflections as ${\it\eta}\propto \mathit{Fr}_{i}^{2}$ , and for ${\it\lambda}_{i}\ll 1$ the zone of interfacial deflection $(r_{id}\propto b_{i})$ is small compared with the depth of the upper layer, $r_{id}\ll D$ . Thus, a relatively small amount of potential energy $({\sim}b_{i}^{2}{\it\eta}^{2})$ is released from the deflections and converted into kinetic energy of a relatively large mass of upper-layer fluid. As a result, the secondary flow is weak ( $\tilde{u} _{2}\ll 1$ ), and hence the rate of entrainment $E_{i,c}$ of the secondary flow provides only a small contribution to $E_{i}$ . For example, ${\it\phi}\sim 1$ when $\{\mathit{Fr}_{i},{\it\lambda}_{i}\}\ll 1$ (figure 10). When $\mathit{Fr}_{i}$ and ${\it\lambda}_{i}$ are both large, the vertical excursions of the interface are relatively large ( ${\it\eta}$ is comparable with $b_{i}$ ) and the zone of deflection is comparable with the upper-layer depth. Thus, a significant amount of potential energy is released and converted into kinetic energy of a relatively shallow mass of upper-layer fluid. As a result, the secondary flow is strong and hence, $E_{i,c}$ provides a significant contribution to the total entrainment flux $E_{i}$ . For example, ${\it\phi}\approx 4$ when $\{\mathit{Fr}_{i},{\it\lambda}_{i}\}\sim 1$ , i.e. with identical forcing, the entrainment flux in the confined environment exceeds that in an unconfined environment by a factor of almost four (figure 10).

3.3. Confinement in previous experimental studies

Figure 11 plots contours of constant $E_{i}$ in $\{{\it\lambda}_{i},\mathit{Fr}_{i}\}$ space, thereby encompassing the dynamics of interfacial entrainment in a confined environment. For a given $\mathit{Fr}_{i}$ , $E_{i}$ can vary by up to 80 % depending on  ${\it\lambda}_{i}$ . For example with $\mathit{Fr}_{i}=0.7$ , $E_{i}=0.18$ when ${\it\lambda}_{i}=0.1$ , whereas $E_{i}=0.32$ when ${\it\lambda}_{i}=0.8$ . Given that ${\it\lambda}_{i}$ plays an instrumental role in the entrainment dynamics, it is informative to examine the confinement in previous experimental studies.

A wide range of box geometries have been used in experiments to measure $E_{i}$ , with the box height  $H$ differing by more than a factor of two: see table 1. Accordingly, it is instructive to examine the effect of the vertical extent $H$ of the confinement on $E_{i}$ . To this end, we vary the box height  $H$ for constant $\{b_{f_{0}},\mathit{Fr}_{0},{\it\psi}\}$ , equivalent to varying $\mathscr{L}\propto b_{f_{0}}/H$ , and calculate $E_{i}$ . Figure 12 plots the entrainment flux ratio ${\it\phi}$ against $\mathscr{L}$ . The values of $\mathscr{L}$ we consider span those of the experimental studies (table 1). Reducing $H$ (i.e. increasing $\mathscr{L}$ ) strengthens the confinement and hence results in stronger secondary flows. Consequently, the rate of entrainment $E_{i,c}$ of the secondary flow increases, and thus ${\it\phi}$ increases with $\mathscr{L}$ . For a typical source radius of $b_{f_{0}}=0.6$  cm, the dashed lines in figure 12 indicate the values of ${\it\phi}$ corresponding to the minimum and maximum box heights used in experiment (table 1). Evidently, the enhancement of $E_{i}$ due to the confinement increases by more than 50 % when the box height is decreased from $H=50$  cm to $H=23$  cm.

Figure 11. Contours of constant entrainment flux $E_{i}$ in $\{{\it\lambda}_{i},\mathit{Fr}_{i}\}$ space.

Figure 12. Entrainment flux ratio ${\it\phi}$ against dimensionless source radius $\mathscr{L}$ for source Froude number $\mathit{Fr}_{0}=20$ and buoyancy flux ratio ${\it\psi}=10$ . The dashed lines indicate ${\it\phi}=\{2.75,1.8\}$ for box height $H=\{23,50\}~\text{cm}$ and source radius $b_{f_{0}}=0.6$  cm.

These results highlight that confinement has a significant effect on the strength of the secondary flow and thus on the entrainment flux. Of course, this variation is entirely unwanted from a practical perspective as experiments have sought to determine a universal entrainment law.

To establish whether the role of confinement can shed new light on the significant spread in the existing measurements of $E_{i}$ (figure 2), figure 13 plots ${\it\lambda}_{i}$ for the experiments of Kumagai (Reference Kumagai1984), Lin & Linden (Reference Lin and Linden2005) and Coffey & Hunt (Reference Coffey and Hunt2010) in $\{\mathit{Fr}_{i},{\it\lambda}_{i}\}$ space. We note that Baines (Reference Baines1975) does not provide sufficient information to calculate ${\it\lambda}_{i}$ for his experiments. Evidently, the experimental values of ${\it\lambda}_{i}$ vary by up to an order of magnitude for a given $\mathit{Fr}_{i}$ . Notably, the range of experimental values of ${\it\lambda}_{i}$ $(0.1<{\it\lambda}_{i}<1)$ corresponds to the range of values of ${\it\lambda}_{i}$ predicted by our model (figure 10 b). Moreover, there is no systematic variation of ${\it\lambda}_{i}$ with $\mathit{Fr}_{i}$ , thus affirming that ${\it\lambda}_{i}$ is an independent parameter. Based on this evidence and given that the influence of ${\it\lambda}_{i}$ on $E_{i}$ has not been previously considered, it is plausible and indeed we assert that the spread in the existing data of $E_{i}$ is attributed, at least in part, to the dependence of $E_{i}$ on ${\it\lambda}_{i}$ . To reinforce this assertion, also shown in figure 13 are our theoretical curves of constant ${\it\phi}=\{1.2,4\}$ . Clearly, almost all the data points lie within the shaded region bounded by these curves. In other words, we predict that for the majority of existing measurements, at least 20 % of $E_{i}$ may be attributed to the rate of entrainment $E_{i,c}$ of the secondary flow, and that for strongly confined flows, the entrainment flux inferred from measurements is increased by a factor of four over that which may be expected for truly unconfined interfacial entrainment. This is compelling evidence that previous measurements have been influenced by a secondary flow and that the disparities in $E_{i}$ are likely to be associated with variations in ${\it\lambda}_{i}$ .

Figure 13. Confinement parameter ${\it\lambda}_{i}$ against the interfacial Froude number $\mathit{Fr}_{i}$ for the experiments of Kumagai (Reference Kumagai1984) (▫), Lin & Linden (Reference Lin and Linden2005)  $(+)$ and Coffey & Hunt (Reference Coffey and Hunt2010) (▵). Solid lines are our theoretical curves of constant entrainment flux ratio ${\it\phi}=\{1.2,4\}$ .

Baines (Reference Baines1975), Kumagai (Reference Kumagai1984) and Coffey & Hunt (Reference Coffey and Hunt2010) inferred $E_{i}$ by tracking the position of a moving interface and, as a consequence, ${\it\lambda}_{i}$ was time-dependent in their experiments. By using a wide tank, Ching et al. (Reference Ching, Fernando and Noh1993) measured the entrainment rate before the interfacial gravity current reached the side boundaries. Thus, the interface position and ${\it\lambda}_{i}$ were invariant for the duration of interest. It is therefore unsurprising that their measurements of the entrainment rate exhibit considerably less scatter than the data sets of Baines (Reference Baines1975) and Kumagai (Reference Kumagai1984), for which ${\it\lambda}_{i}$ differs by almost an order of magnitude (figure 13).

3.4. Implications for the design of an experiment

It is not immediately clear for a given experiment (or indeed from the wealth of previous measurements) how the individual components of $E_{i}$ (2.33) may be accurately determined. What is clear is that in search of a universal law it is the entrainment flux in the absence of confinement that has been, and arguably remains, the true goal. Our analysis raises the question of what is an appropriate box height $H$ for measuring the entrainment flux. As a secondary flow persists for $H>0$ , the influence of confinement on $E_{i}$ is inexorable. However, through careful selection of the box height, it is possible to tailor the contribution of $E_{i,c}$ to $E_{i}$ in an experiment so that ${\it\phi}=(A\mathit{Fr}_{i}^{2}+E_{i,c})/A\mathit{Fr}_{i}^{2}$ does not exceed an acceptable threshold, ${\it\phi}_{T}$ . Based on the accuracy to which $E_{i}$ is measured, experimentalists can specify ${\it\phi}_{T}$ . By ensuring that ${\it\phi}<{\it\phi}_{T}$ , the effect of confinement on $E_{i}$ can, in principle, be restricted solely to the realms of experimental uncertainty.

With a view to guiding future experiments, we seek the box height required to achieve a given ${\it\phi}={\it\phi}_{T}$ . While the plan area $S$ of the box is needed to fully define the geometry, its precise value is not of concern, and implicit in the following discussion is that $S$ is sufficiently large that both the plume and fountain are free to entrain and their sources sufficiently well separated that they may be considered to be non-interacting. Moreover, the sources are sufficiently small in area that they approximate to localised sources (e.g. $b_{f_{0}}\ll h$ ). As the fountain exhibits jet-like behaviour below the interface (see § 2.3), its radius scales approximately linearly with height (Kaye & Hunt Reference Kaye and Hunt2006), and hence we take $b_{i}=c_{1}h$ , where $c_{1}$ is a constant. Recalling that ${\it\lambda}_{i}=b_{i}/(H-h)$ , we substitute for $b_{i}$ into (2.35) and rearrange for $h/H$ to give

Taking $c_{1}=0.17$ (Mizushina et al. Reference Mizushina, Ogino, Takeuchi and Ikawa1982), figure 14 plots $(h/H)_{design}$ (3.2) as a function of $\mathit{Fr}_{i}$ for ${\it\phi}_{T}=\{1.2,1.5,2.0,2.5\}$ . The region beneath a given curve of constant ${\it\phi}_{T}$ encompasses the values of $(h/H)_{design}$ for which ${\it\phi}<{\it\phi}_{T}$ . Hence, for a given interface position $h$ , figure 14 indicates the box height $H$ required to limit the influence of the confinement on $E_{i}$ to ${\it\phi}={\it\phi}_{T}$ . For example, if the interface is established 12 cm above the base of the box and $\mathit{Fr}_{i}=0.3$ , we would require $H>47$  cm to restrict the rate of entrainment $E_{i,c}$ of the secondary flow to no more than 20 % of $E_{i}$ . Lower box heights could be used, but with the penalty of higher thresholds ${\it\phi}_{T}$ .

Figure 14. Interface height $h$ scaled on box height $H$ as a function of $\mathit{Fr}_{i}$ for a constant entrainment flux ratio ${\it\phi}_{T}=\{1.2,1.5,2.0,2.5\}$ .

3.5. Summary of key modelling assumptions

Guided by previous experimental results and observations, a number of key modelling assumptions and simplifications have been made and these are as follows.

1. (i) The interfacial deflections occur within a circular region of finite radius.

2. (ii) The fountain, the dome and the plume occupy only a small fraction of the total volume of the box, so we may ignore the potential energies contained in these regions.

3. (iii) Interfacial deflections induce a flow predominantly within the zone of interfacial deflection.

4. (iv) Energy losses associated with the generation of interfacial waves and the energy dissipated due to viscous effects may be neglected in our analysis of the secondary flow.

5. (v) Persistent baroclinic vortices around the periphery of the dome are primarily responsible for entrainment into the dome.

6. (vi) The entrainment coefficients for the plume and the fountain are not significantly influenced by the secondary flow in the lower layer.

7. (vii) The available potential energy is apportioned equally between the kinetic energies of the secondary flows in the upper and lower layers.

Whilst the interfacial Froude number and the confinement parameter are instrumental in determining the entrainment flux across the interface, their values here are not specified a priori but span the range $0.1\lesssim {\it\lambda}_{i}\lesssim 1$ , $0.1\lesssim \mathit{Fr}_{i}\lesssim 1.4$ as a consequence of the system we consider (see figures 10 and 13).