2.1 Set-up and notation

The stratified inclined duct experiment (hereafter abbreviated ‘SID’) is sketched in figure 1(a). This conceptually simple experiment consists of two reservoirs initially filled with aqueous solutions of different densities $\unicode[STIX]{x1D70C}_{0}\pm \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/2$, connected by a long rectangular duct that can be tilted at an angle $\unicode[STIX]{x1D703}$ from the horizontal. At the start of the experiment, the duct is opened, initiating a brief transient gravity current. Shortly after, at $t=0$, an exchange flow starts and is sustained through the duct for long periods of time, until the accumulation of fluid of a different density from the other reservoir reaches the ends of the duct and the experiment is stopped at $t=T$ (typically after several minutes and many duct transit times). This exchange flow has at least four qualitatively different flow regimes, based on the experimental input parameters, as we discuss later in the paper.

Figure 1. (a) The stratified inclined duct, in which an exchange flow takes place through a rectangular duct connecting two reservoirs at densities $\unicode[STIX]{x1D70C}_{0}\pm \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/2$ and inclined at an angle $\unicode[STIX]{x1D703}$ from the horizontal. (b) Notation (in dimensional units). The $x$ and $z$ axes are, respectively, aligned with the horizontal and vertical of the duct (hence $-z$ makes an angle $\unicode[STIX]{x1D703}$ with gravity, here $\unicode[STIX]{x1D703}>0$). The duct has dimensions $L\times W\times H$. The streamwise velocity $u$ has typical peak-to-peak magnitude $\unicode[STIX]{x0394}U$. The density stratification $\unicode[STIX]{x1D70C}$ has magnitude $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, with an interfacial layer of typical thickness $\unicode[STIX]{x1D6FF}$.

Our notation is shown in figure 1(b) and largely follows that of Lefauve et al. (Reference Lefauve, Partridge, Zhou, Caulfield, Dalziel and Linden2018), Lefauve, Partridge & Linden (Reference Lefauve, Partridge and Linden2019). The duct has length $L$, height $H$ and width $W$. The streamwise $x$ axis is aligned along the duct and the spanwise $y$ axis is aligned across the duct, making the $z$ axis tilted at an angle $\unicode[STIX]{x1D703}$ from the vertical (resulting in a non-zero streamwise projection of gravity $g\,\sin \,\unicode[STIX]{x1D703}$). The angle $\unicode[STIX]{x1D703}$ is defined to be positive when the bottom end of the duct sits in the reservoir of lower density, as shown here. The velocity vector field is $\boldsymbol{u}(x,y,z,t)=(u,v,w)$ along $x,y,z$, and the density field is $\unicode[STIX]{x1D70C}(x,y,z,t)$. All spatial coordinates are centred in the middle of the duct, such that $(x,y,z,t)\in [-L/2,L/2]\times [-W/2,W/2]\times [-H/2,H/2]\times [0,T]$.

Next, we define two integral scalar quantities of particular interest in exchange flows:

1. (i) $Q$ the volume flux as the volumetric flow rate averaged over the duration of an experiment

   (2.1)$$\begin{eqnarray}Q\equiv \langle |u|\rangle _{x,y,z,t},\end{eqnarray}$$
   where $\langle |u|\rangle _{x,y,z,t}\equiv 1/(LWHT)\int _{0}^{T}\int _{-H/2}^{H/2}\int _{-W/2}^{W/2}\int _{-L/2}^{L/2}|u|\,\text{d}x\,\text{d}y\,\text{d}z\,\text{d}t$. The volume flux $Q>0$ measures the magnitude of the exchange flow between the two reservoirs. It is different from the net (or ‘barotropic’) volume flux $\langle u\rangle _{x,y,z,t}\approx 0$, since, to a good approximation, the volume of fluid in each reservoirs is conserved during an experiment (assuming the levels of the free surface in each reservoir are carefully set before the start of the experiment).

2. (ii) $Q_{m}$ the mass flux as the net flow rate of mass averaged over the duration of an experiment

   (2.2)$$\begin{eqnarray}Q_{m}\equiv \frac{2}{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}\langle (\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0})u\rangle _{x,y,z,t},\end{eqnarray}$$
   which is equivalent to a buoyancy flux up to a multiplicative constant $g$. By definition $0<Q_{m}\leqslant Q$. The first inequality holds since, in our notation, negatively buoyant fluid ($\unicode[STIX]{x1D70C}_{0}<\unicode[STIX]{x1D70C}\leqslant \unicode[STIX]{x1D70C}_{0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/2$) flows on average to the right ($u>0$) and vice versa. The second inequality would be an equality in the absence of molecular diffusion or stirring inside the duct (i.e. if all fluid moving right had density $\unicode[STIX]{x1D70C}_{0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/2$ and vice versa). In any real flow, laminar (and potentially turbulent) diffusion at the interface are responsible for an interfacial layer of intermediate density $|\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0}|<\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/2$ of finite thickness $\unicode[STIX]{x1D6FF}>0$ (figure 1b).

2.2 Non-dimensionalisation

A total of seven parameters are believed to play important roles in the SID: four geometrical parameters: $L$, $H$, $W$, $\unicode[STIX]{x1D703}$; and three dynamical parameters: the reduced gravity $g^{\prime }\equiv g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{0}$ (under the Boussinesq approximation $0<\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{0}\ll 1$), the kinematic viscosity of water ($\unicode[STIX]{x1D708}=1.05\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$) and the molecular diffusivity of the stratifying agent (active scalar) $\unicode[STIX]{x1D705}$. In this paper, we will primarily consider salt stratification ($\unicode[STIX]{x1D705}_{S}=1.50\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$), but will also discuss temperature stratification ($\unicode[STIX]{x1D705}_{T}=1.50\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$). From these seven parameters having two dimensions (of length and time), we construct five independent non-dimensional parameters below.

The first three non-dimensional parameters are geometrical: $\unicode[STIX]{x1D703}$, and the aspect ratios of the duct in the longitudinal and spanwise direction, respectively,

(2.3a,b)$$\begin{eqnarray}A\equiv \frac{L}{H}\quad \text{and}\quad B\equiv \frac{W}{H}.\end{eqnarray}$$

We choose to non-dimensionalise lengths by the length scale $H/2$, defining the non-dimensional position vector as $\tilde{\boldsymbol{x}}\equiv \boldsymbol{x}/(H/2)$ such that $(\tilde{x},{\tilde{y}},\tilde{z})\in [-A,A]\times [-B,B]\times [-1,1]$. As an exception, we choose to non-dimensionalise the typical thickness of the interfacial density layer by $H$, for consistency with other definitions in the literature on mixing in exchange flows: $\tilde{\unicode[STIX]{x1D6FF}}\equiv \unicode[STIX]{x1D6FF}/H$, such that $\tilde{\unicode[STIX]{x1D6FF}}\in [0,1]$.

The last two non-dimensional parameters are dynamical. We define an ‘input’ Reynolds number based on the velocity scale $\sqrt{g^{\prime }H}$ and length scale $H/2$

(2.4)$$\begin{eqnarray}Re\equiv \frac{\sqrt{g^{\prime }H}H}{2\unicode[STIX]{x1D708}}=\frac{\sqrt{gH^{3}}}{2\unicode[STIX]{x1D708}}\sqrt{\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{0}}}.\end{eqnarray}$$

Consequently, we non-dimensionalise the velocity vector as $\tilde{\boldsymbol{u}}\equiv \boldsymbol{u}/\sqrt{g^{\prime }H}$, and time by the advective time unit $\tilde{t}\equiv 2\sqrt{g^{\prime }/H}t$ (hereafter abbreviated ATU). We define our last parameter, the Prandtl number (or Schmidt number), as the ratio of the momentum to active scalar diffusivity

(2.5)$$\begin{eqnarray}Pr\equiv \frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D705}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ takes the value $\unicode[STIX]{x1D705}_{S}$ or $\unicode[STIX]{x1D705}_{T}$ depending on the type of stratification (salt or temperature, giving respectively $Pr=700$ and $Pr=7$). Finally, we define the non-dimensional Boussinesq density field as $\tilde{\unicode[STIX]{x1D70C}}\equiv (\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0})/(\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/2)$, such that $\tilde{\unicode[STIX]{x1D70C}}\in [-1,1]$.

We now reformulate the aim of this paper (introduced in § 1) more specifically as: exploring the behaviour of flow regimes, mass flux $\tilde{Q}_{m}$ and interfacial layer thickness $\tilde{\unicode[STIX]{x1D6FF}}$ in the five-dimensional space of non-dimensional input parameters $(A,B,\unicode[STIX]{x1D703},Re,Pr)$.

In the next section we address the dimensional scaling of the velocity in the experiment. By discussing the a priori influence of the input parameters identified above on the velocity scale in this problem, we will provide a basis for subsequent scaling arguments in the paper.

2.3 Scaling of the velocity

Having constructed our Reynolds number (2.4) using the velocity scale $\sqrt{g^{\prime }H}$, we show here that it is the relevant velocity scale to use in such exchange flows. As sketched in figure 1(b), we define the typical peak-to-peak velocity as $\unicode[STIX]{x0394}U$. This velocity scale is not set by the experimenter as an input parameter, rather it is chosen by the flow as an output parameter. From dimensional analysis, we write

(2.6)$$\begin{eqnarray}\frac{\unicode[STIX]{x0394}U}{2}=\sqrt{g^{\prime }H}\,f_{\unicode[STIX]{x0394}U}(A,B,\unicode[STIX]{x1D703},Re,Pr).\end{eqnarray}$$

In order to show that our Reynolds number (2.4) and our non-dimensionalisation of the velocity by $\sqrt{g^{\prime }H}$ are relevant (and such that $\tilde{u} \in [-1,1]$), we will show below that we indeed expect $\unicode[STIX]{x0394}U/2\sim \sqrt{g^{\prime }H}$ and the non-dimensional function $f_{\unicode[STIX]{x0394}U}(A,B,\unicode[STIX]{x1D703},Re,Pr)\sim 1$. Although some aspects of this discussion can be found in Lefauve et al. (Reference Lefauve, Partridge, Zhou, Caulfield, Dalziel and Linden2018, Reference Lefauve, Partridge and Linden2019), the importance of this dimensional analysis for this paper justifies the more detailed discussion that we offer below.

The velocity scale $\unicode[STIX]{x0394}U$ in quasi-steady-state results from a dynamical balance in the steady, horizontal momentum equation under the Boussinesq approximation (in dimensional units)

(2.7)$$\begin{eqnarray}\underbrace{\vphantom{\frac{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0}}{\unicode[STIX]{x1D70C}_{0}}g\sin \unicode[STIX]{x1D703}}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}u}_{inertial\,(I)}=\underbrace{\vphantom{\frac{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0}}{\unicode[STIX]{x1D70C}_{0}}g\sin \unicode[STIX]{x1D703}}-(1/\unicode[STIX]{x1D70C}_{0})\unicode[STIX]{x2202}_{x}p}_{hydrostatic\,(H)}+\underbrace{\vphantom{\frac{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0}}{\unicode[STIX]{x1D70C}_{0}}}g\sin \unicode[STIX]{x1D703}(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0})/\unicode[STIX]{x1D70C}_{0}}_{gravitational\,(G)}+\underbrace{\vphantom{\frac{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0}}{\unicode[STIX]{x1D70C}_{0}}g\sin \unicode[STIX]{x1D703}}\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}u}_{viscous\,(V)}.\end{eqnarray}$$

In addition to the standard inertial (I) and viscous (V) terms, this equation highlights the two distinct ‘forcing’ mechanisms in SID flows:

(H)

a hydrostatic longitudinal pressure gradient, the minimal ingredient for exchange flow, resulting from each end of the duct sitting in reservoirs at different densities. This hydrostatic pressure in the duct increases linearly with depth $\unicode[STIX]{x2202}_{x}p=g\cos \unicode[STIX]{x1D703}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(4L)z$, driving a flow in opposite directions on either side of the neutral level $z=0$: $-(1/\unicode[STIX]{x1D70C}_{0})\unicode[STIX]{x2202}_{x}p=g^{\prime }\cos \unicode[STIX]{x1D703}/(4L)z$;

(G)

a gravitational body force reinforcing the flow by the acceleration of the positively buoyant layer upward (to the left in figure 1) and of the negatively buoyant layer downward (to the right) when the tilt angle is positive $g\sin \unicode[STIX]{x1D703}>0$ (the focus of this paper), and vice versa when the tilt angle is negative.

Rewriting (2.7) in non-dimensional form and ignoring multiplicative constants, we obtain

(2.8)$$\begin{eqnarray}\underbrace{\vphantom{\frac{(\unicode[STIX]{x0394}U)^{2}}{L}}(\unicode[STIX]{x0394}U)^{2}\,\tilde{\boldsymbol{u}}\boldsymbol{\cdot }\tilde{\unicode[STIX]{x1D735}}\tilde{u} }_{I}~\sim ~\underbrace{\vphantom{\frac{(\unicode[STIX]{x0394}U)^{2}}{L}}(g^{\prime }H\cos \unicode[STIX]{x1D703})\tilde{z}}_{H}~+~\underbrace{\vphantom{\frac{(\unicode[STIX]{x0394}U)^{2}}{L}}(g^{\prime }L\sin \unicode[STIX]{x1D703})\tilde{\unicode[STIX]{x1D70C}}}_{G}~+~\underbrace{\vphantom{\frac{(\unicode[STIX]{x0394}U)^{2}}{L}}(\unicode[STIX]{x1D708}\unicode[STIX]{x0394}U\ell ^{-2}L)\tilde{\unicode[STIX]{x1D6FB}^{2}}\tilde{u} }_{V},\end{eqnarray}$$

where $\ell$ is the smallest length scale of velocity gradients ($\ell =\unicode[STIX]{x1D6FF}$ in laminar flows, and $\ell \ll \unicode[STIX]{x1D6FF}$ in turbulent flows).

To simplify this complex ‘four-way’ balance, it is instructive to consider the four possible ‘two-way’ dominant balances to deduce four possible scalings for $\unicode[STIX]{x0394}U$ (ignoring constants and assuming $\cos \unicode[STIX]{x1D703}\approx 1$ since the focus of this paper is on small angles).

(IH)

The inertial–hydrostatic balance. First, we can neglect the gravitational (G) term with respect to the hydrostatic (H) term if $g^{\prime }H\cos \unicode[STIX]{x1D703}\gg g^{\prime }L\sin \unicode[STIX]{x1D703}$, i.e. when the tilt angle of the duct $\unicode[STIX]{x1D703}$ is much smaller than its ‘geometrical’ angle

(2.9)$$\begin{eqnarray}0<\unicode[STIX]{x1D703}\ll \unicode[STIX]{x1D6FC},\end{eqnarray}$$

where we define the geometrical angle as

(2.10)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}\equiv \tan ^{-1}(A^{-1}).\end{eqnarray}$$

Second, we can neglect the viscous (V) term if $g^{\prime }H\gg \unicode[STIX]{x1D708}\unicode[STIX]{x0394}U\ell ^{-2}L$, i.e. if the Reynolds number is larger than $Re\gg HL/\ell ^{2}$. This corresponds to

(2.11)$$\begin{eqnarray}Re\gg A\end{eqnarray}$$

in laminar flow (ignoring the case $B\ll 1$ for simplicity), and to a larger lower bound in turbulent flows. Under these conditions, balancing I and H gives the scaling $\unicode[STIX]{x0394}U\sim \sqrt{g^{\prime }H}$, i.e. $f_{\unicode[STIX]{x0394}U}\sim 1$, which corresponds to our choice in § 2.1.

(IG)

The inertial–gravitational balance. Using analogous arguments, if $\unicode[STIX]{x1D703}\gg \unicode[STIX]{x1D6FC}$ and $Re\gg HL/\ell ^{2}$, we expect the scaling $\unicode[STIX]{x0394}U\sim \sqrt{g^{\prime }L\sin \unicode[STIX]{x1D703}}$, i.e. $f_{\unicode[STIX]{x0394}U}(A,\unicode[STIX]{x1D703})\sim \sqrt{A\sin \unicode[STIX]{x1D703}}\gg 1$.

(HV)

The hydrostatic–viscous balance. If $\unicode[STIX]{x1D703}\ll \unicode[STIX]{x1D6FC}$ and $Re\ll A$, we expect $f_{\unicode[STIX]{x0394}U}(A,B,Re)\sim A^{-1}Re\ll 1$ (some dependence on $B$ being unavoidable in such a viscous flow).

(GV)

The gravitational–viscous balance. If $\unicode[STIX]{x1D703}\gg \unicode[STIX]{x1D6FC}$ and $Re\ll A$, we expect $f_{\unicode[STIX]{x0394}U}(B,\unicode[STIX]{x1D703},Re)\sim \sin \unicode[STIX]{x1D703}Re\ll A$.

Figure 2. Summary of the scaling analysis of $\unicode[STIX]{x0394}U$ based on the four two-way dominant balances of the streamwise momentum equation (2.8). In each corner of the $(\unicode[STIX]{x1D703},Re)$ plane, the IH, IG, HV and GV balances predict the scaling of $f_{\unicode[STIX]{x0394}U}\equiv \unicode[STIX]{x0394}U/(2\sqrt{g^{\prime }H})$ on either extreme side of $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}\equiv \tan ^{-1}(A^{-1})$ and $Re=A$. The region of practical interest studied in this paper is shown in blue. Although no a priori ‘two-way’ balance allows us to determine accurately the scaling of $f_{\unicode[STIX]{x0394}U}(A,B,\unicode[STIX]{x1D703},Re,Pr)$ in this region, hydraulic control requires that $f_{\unicode[STIX]{x0394}U}\sim 1$, as in the IH scaling (see text).

Figure 2 summarises the above analysis and the following conclusions:

1. (i) The parameters $A$, $\unicode[STIX]{x1D703}$ and $Re$ play particularly important roles in SID flows, since the variation of $\unicode[STIX]{x1D703}$ and $Re$ above or below thresholds set by $A$ can alter the scaling of $\unicode[STIX]{x0394}U$ (i.e. $f_{\unicode[STIX]{x0394}U}$). The parameter $B$ appears less important in this respect (except in narrow ducts where $B\ll 1$ and the $Re$ threshold becomes $AB^{-2}$).

2. (ii) At low tilt angles $0<\unicode[STIX]{x1D703}\ll \unicode[STIX]{x1D6FC}$, $f_{\unicode[STIX]{x0394}U}$ increases from $\ll 1$ when $Re\ll A$ to ${\sim}1$ when $Re\gg A$. At high enough $Re$, $f_{\unicode[STIX]{x0394}U}$ likely retains a dependence on $A,B,Re$ due to turbulence (the constant ‘IH’ scaling being a singular limit for $Re\rightarrow \infty$).

3. (iii) At high tilt angles $\unicode[STIX]{x1D703}\gg \unicode[STIX]{x1D6FC}$ and Reynolds number $Re\gg A$, $f_{\unicode[STIX]{x0394}U}$ should increase well above $1$, and likely retains a dependence on $A,B,\unicode[STIX]{x1D703},Re$ (the ‘IG’ scaling being a singular limit for $Re\rightarrow \infty$).

4. (iv) The blue rectangle in figure 2 represents the region of interest in most exchange flows of practical interest and in this paper. In this region, three or four physical mechanisms must be considered simultaneously (IHV, IGHV or IGV). Since few flows ever satisfy $\unicode[STIX]{x1D703}\ll \unicode[STIX]{x1D6FC}$ or $\gg \unicode[STIX]{x1D6FC}$, we consider that in general $f_{\unicode[STIX]{x0394}U}=f_{\unicode[STIX]{x0394}U}(A,B,\unicode[STIX]{x1D703},Re,Pr)$ (the $Pr$ dependence reflects the fact that the active scalar can no longer be neglected at high $Re$ due to its effect on turbulence and mixing). The existence and value of the upper edge of this region, i.e. the $Re$ value at which viscous and diffusive effects are negligible (the ‘practical $Re=\infty$ limit’) are a priori unknown.

Although the above ‘two-way’ balances do not allow us to confidently guess the scaling of $f_{\unicode[STIX]{x0394}U}$ in the blue region, theoretical arguments and empirical evidence of hydraulic control support $f_{\unicode[STIX]{x0394}U}(A,B,\unicode[STIX]{x1D703},Re,Pr)\sim 1$ for IHV, IGHV and IGV flows.

Hydraulic control of two-layer exchange flows dates back to Stommel & Farmer (Reference Stommel and Farmer1953), Wood (Reference Wood1968, Reference Wood1970) and was formalised mathematically by Armi (Reference Armi1986), Lawrence (Reference Lawrence1990) and Dalziel (Reference Dalziel1991). In steady, inviscid, irrotational, hydrostatic (i.e. ‘IH’) exchange flows, the ‘composite Froude number’ $G$ is unity, which using our notation and assuming streamwise invariance of the flow ($\unicode[STIX]{x2202}_{x}=0$), reads

(2.12)$$\begin{eqnarray}G^{2}=4\frac{\langle u^{2}\rangle _{x,y,z,t}}{\sqrt{g^{\prime }H}}=1\quad \Longrightarrow \quad \langle |\tilde{u} |\rangle _{x,y,z,t}=\tilde{Q}=\frac{1}{2}.\end{eqnarray}$$

Such exchange flows are called maximal: the phase speed of long interfacial gravity waves $\sqrt{g^{\prime }H}$ ‘controls’ the flow at sharp changes in geometry (on either end of the duct), and sets the maximal non-dimensional volume flux to $\tilde{Q}=1/2$.

In ‘plug-like’ hydraulic flows ($Re\rightarrow \infty$), the velocity in each layer $\unicode[STIX]{x0394}U/2$ is equal to its layer average $Q$, giving an upper bound $f_{\unicode[STIX]{x0394}U}=\tilde{Q}=1/2$. By contrast, in real-life finite-$Re$ flows, the peak $\unicode[STIX]{x0394}U/2$ is larger than the average $Q$ (typically by a factor ${\approx}2$), such that the upper bound is $f_{\unicode[STIX]{x0394}U}\approx 2\tilde{Q}\approx 1$. This upper bound remains approximately valid throughout the blue region of figure 2. We thus answer the question motivating this section: our choice of non-dimensionalising $\boldsymbol{u}$ by $\sqrt{g^{\prime }H}\approx \unicode[STIX]{x0394}U/2$ in order to have $|\tilde{\boldsymbol{u}}|\lesssim 1$ is indeed relevant to SID flows.

Henceforth, we drop the tildes and, unless explicitly stated otherwise, use non-dimensional variables throughout.

3.1 Current state of knowledge

The flow regimes have been observed and classified in a relatively consistent way in the literature. Throughout this paper, we adopt the nomenclature of Meyer & Linden (Reference Meyer and Linden2014): $\mathsf{L}$ (laminar flow with flat interface), $\mathsf{H}$ (interfacial Holmboe waves), $\mathsf{I}$ (intermittently turbulent), $\mathsf{T}$ (fully turbulent). The consensus is that the flow becomes increasingly disorganised and turbulent with increasing $A$, $\unicode[STIX]{x1D703}$ and $Re$. At a fixed $\unicode[STIX]{x1D703}\geqslant 0^{\circ }$, all flow regimes ($\mathsf{L},\mathsf{H},\mathsf{I},\mathsf{T}$) can be visited by increasing $Re$, and vice versa at fixed $Re$ and increasing $\unicode[STIX]{x1D703}$ (Macagno & Rouse Reference Macagno and Rouse1961; Wilkinson Reference Wilkinson1986; Kiel Reference Kiel1991; Meyer & Linden Reference Meyer and Linden2014; Lefauve et al. Reference Lefauve, Partridge and Linden2019) (hereafter MR61, W86, K91, ML14 and LPL19, respectively). Both K91 and ML14 observed regime transitions scaling with $A\tan \unicode[STIX]{x1D703}=\tan \unicode[STIX]{x1D703}/\tan \unicode[STIX]{x1D6FC}$ (or $A\unicode[STIX]{x1D703}$ for small angles), i.e. $A$ controls the $\unicode[STIX]{x1D703}$ scaling. However, the scaling in $Re$ is subject to debate, and may change on either side of $\unicode[STIX]{x1D703}\approx \unicode[STIX]{x1D6FC}$ (LPL19). These conclusions are illustrated schematically in figure 3(a) (the interrogation marks denote open questions).

The mass flux has a complex non-monotonic behaviour in $A,\unicode[STIX]{x1D703},Re$ sketched in figure 3(b). While the dependence on $Re$ is clear at $Re<500A$ (MR61, W86, ML14, LPL19) due to the influence of viscous boundary layers, it is still debated at $Re>500A$: Mercer & Thompson (Reference Mercer and Thompson1975) (hereafter MT75) and ML14 argued in favour of this dependence on $Re$ even above $500A$ whereas Leach & Thompson (Reference Leach and Thompson1975) (hereafter LT75) and K91 argued against it. The mass flux reaches a maximum $Q_{m}\approx 0.4$–$0.5$ at $\unicode[STIX]{x1D703}\approx \unicode[STIX]{x1D6FC}/2$ and ‘high enough’ $Re$ (MT75, K91, ML14, LPL19) and decays for smaller/larger $\unicode[STIX]{x1D703}$ and $Re$ (W86, LPL19) in a poorly studied fashion.

The interfacial layer thickness has only been studied experimentally in K91, who observed monotonic increase of $\unicode[STIX]{x1D6FF}$ with both $A$ and $\unicode[STIX]{x1D703}$, good collapse with $A\tan \unicode[STIX]{x1D703}$ (reaching its maximum $\unicode[STIX]{x1D6FF}=1$ at $\unicode[STIX]{x1D703}\gtrsim 2\unicode[STIX]{x1D6FC}$) and independence of $Re$ (figure 3c). The behaviour of $\unicode[STIX]{x1D6FF}$ at low $Re<500A$ remains unknown.

Figure 3. Illustration of the current state of knowledge on the idealised behaviour of the (a) flow regimes, (b) mass flux and (c) interfacial layer thickness with respect to $A,\unicode[STIX]{x1D703},Re$ (the axes have logarithmic scale). Interrogation marks refer to open questions. For more details, see the literature review in appendix A.

3.2 Limitations of previous studies

Many aspects of the scaling of regimes, $Q_{m}$ and $\unicode[STIX]{x1D6FF}$ with $A,B,\unicode[STIX]{x1D703},Re,Pr$ remain open questions. For example, the effects of $Re$ on $\unicode[STIX]{x1D6FF}$, and the effects of $B$ and $Pr$ on all three variables have not been studied at all. Moreover, despite our efforts to unify their findings in § 3.1 and appendix A, these past studies of the SID experiment inherently provide a fragmented view of the problem due to the following limitations (made clear by table 2):

1. (i) they used slightly different set-ups and geometries (e.g. presence versus absence of free surfaces in the reservoirs, rectangular ducts versus circular pipes) and slightly different measuring methodologies (e.g. for $Q_{m}$);

2. (ii) only one study (K91) addressed the interdependence of the three variables of interest (regime, $Q_{m}$, $\unicode[STIX]{x1D6FF}$), while the remaining studies measured either only regimes (MR61), only $Q_{m}$ (LT75, MT75) or both (ML14, LPL19);

3. (iii) they focused on the variation of a single parameter (MR61), two parameters (W86, K91, LPL19), or at most three parameters (MT75, ML14) in which case the third parameter took only two different values;

4. (iv) they studied limited regions of the parameter space, and it is difficult to confidently interpolate results obtained by different set-ups in different regions (such as $Re<500A$ and ${>}500A$).

The experimental results and models in the next two sections attempt to overcome the above limitations by providing a more unified view of the problem.

4.1 Data sets

All experimental data presented in the following were obtained in the stratified inclined duct set-up sketched in figure 1. We used four different duct geometries and two types of stratification (salt and temperature) to obtain the following five distinct data sets, listed in table 1:

LSID

(L for large) with height $H=100~\text{mm}$, and $A=30$, $B=1$;

HSID

(H for half) which only differs from the LSID (the ‘control’ geometry) in that it is half the length: $A=15$ (highlighted in bold in table 1);

mSID

(m for mini) which only differs from the LSID in its height $H=45~\text{mm}$, but keeps $A,B,Pr$ identical (this is done by scaling down $H,W,L$ by the same factor $100/45$ such that the mSID and LSID ducts remain geometrically similar). Note that the mSID and LSID configurations should yield identical data at identical $Re$ since $H$ should only play a role through the non-dimensional parameters $A,B,Re$. However, we will see in §§ 4.2–4.4 that this hypothesis is challenged by our data.

tSID

(t for tall) which differs from the HSID primarily in its tall spanwise aspect ratio $B=1/4$ (and, secondarily, in a marginally smaller height $H=90~\text{mm}$);

mSIDT

(m for mini and T for temperature) which differs from the mSID in that the stratification was achieved by different reservoir water temperatures (hence $Pr=7$), as opposed to different salinities in the above data sets (where $Pr=700$). This limited the density difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ achieved, reflected in the lower $Re$.

Table 1 also lists, for each data set, the range of variation of $\unicode[STIX]{x1D703}$ and $Re$, and the number of data points, i.e. distinct $(\unicode[STIX]{x1D703},Re)$ couples for which we have data on regime, $Q_{m}$ and $\unicode[STIX]{x1D6FF}$.

Note that the regime and $Q_{m}$ data of the top three data sets have already been published in some form by Meyer & Linden (Reference Meyer and Linden2014) (ML14, denoted by $^{\ast }$) and Lefauve et al. (Reference Lefauve, Partridge and Linden2019) (LPL19, denoted by $^{\dagger }$), as discussed in appendices A.1–A.2. However, ML14 plotted their LSID and HSID data together (see their figures 7–8) and did not investigate their potential differences, while LPL19 only commented in passing on a fit of the $Q_{m}$ data in the $(\unicode[STIX]{x1D703},Re)$ plane (see their figure 9). The individual reproduction and thorough discussion of these data alongside more recent data using a unified non-dimensional approach will be key to this paper. All five data sets have been used in the PhD thesis of Lefauve (Reference Lefauve2018) (especially in chapters 3 and 5, and the detailed parameters of all experiments are tabulated in his appendix A). Most of the raw and processed data used in this paper are available on the repository doi:10.17863/CAM.48821 (more details in appendix B).

Our focus on long ducts, evidenced by our choice of $A=15$ and $30$, reflects our focus on flows relevant to geophysical and environmental applications, which are typically largely horizontal ($\unicode[STIX]{x1D703}\approx 0^{\circ }$) and stably stratified in the vertical (as opposed to the different case of vertical exchange flow with $\unicode[STIX]{x1D703}=90^{\circ }$). The SID experiment conveniently exhibits all possible flow regimes, including high levels of turbulence and mixing, between $\unicode[STIX]{x1D703}=0^{\circ }$ and a few $\unicode[STIX]{x1D6FC}$ at most (§ 3.1). In long ducts (large $A$), $\unicode[STIX]{x1D6FC}\equiv \tan ^{-1}(A^{-1})$ is therefore small enough to allow us to study all the key dynamics of sustained stratified flows while keeping $\unicode[STIX]{x1D703}$ small enough for these flows to remain largely horizontal, and thus geophysically relevant.

As a result of this focus on long ducts, in the remainder of the paper we make the approximation that

(4.1a,b)$$\begin{eqnarray}\cos \unicode[STIX]{x1D703}\approx 1\quad \text{and}\quad \sin \unicode[STIX]{x1D703}\approx \unicode[STIX]{x1D703}.\end{eqnarray}$$

This approximation is accurate to better than $2\,\%$ for the angles considered in our data sets ($\unicode[STIX]{x1D703}\leqslant 10^{\circ }$). Unless explicitly specified, $\unicode[STIX]{x1D703}$ will now be expressed in radians (typically in scaling laws).

4.2 Flow regimes

The $\mathsf{L},\mathsf{H},\mathsf{I},\mathsf{T}$ flow regimes were determined following the ML14 nomenclature as in § B.1 (except for a new regime which we discuss in the next paragraph). Figure 4 shows the resulting regime maps in the $(\unicode[STIX]{x1D703},Re)$ plane corresponding to the five data sets.

Figure 4. Regime diagrams in the $(\unicode[STIX]{x1D703},Re)$ plane (linear–log scale) using the five data sets of table 1 (the scaled cross-section of each duct is sketched for comparison in the top right corner of each panel). The error in $\unicode[STIX]{x1D703}$ is of order $\pm 0.2^{\circ }$ and is slightly larger than the symbol size, whereas the error in $Re$ is much smaller than the symbol size, except in (e) at small $Re$.

First, we note the introduction of a ‘new’ $\mathsf{W}$ regime in the tSID and mSIDT data (panels d,e). This $\mathsf{W}$ (wave) regime is similar to the $\mathsf{H}$ (Holmboe) regime, but describes interfacial waves which were not recognised as Holmboe waves in shadowgraphs. These waves were of two types. First, in the tSID geometry at positive angles $\unicode[STIX]{x1D703}>0$, the waves did not exhibit the distinctive ‘cusped’ shape of Holmboe waves and the waves appeared to be generated at the ends of the duct and to decay as they travel inside the duct. Second, in the mSIDT larger-amplitude, tilde-shaped internal waves were observed across most of the height of the duct, contrary to Holmboe waves which are typically confined to a much thinner interfacial region. Further discussion of these waves falls outside the scope of this paper, but can be found in Lefauve (Reference Lefauve2018, §§ 3.2.3–3.2.4) (hereafter abbreviated L18). This new observation highlights the richness of the flow dynamics in the SID experiment. However, for the purpose of this paper, the $\mathsf{H}$ and $\mathsf{W}$ regimes are sufficiently similar in their characteristics (mostly laminar flow with interfacial waves) that we group them under the same regime for the purpose of discussing regime transitions.

The main observation of figure 4 is that the transitions between regimes can be described as simple curves in the $(\unicode[STIX]{x1D703},Re)$ plane that do not overlap (or ‘collapse’) between the five data sets. The slope and location of the transitions varies greatly between panels: the difference between the LSID and HSID data (panels a,b) is due to $A$, the difference between the HSID and tSID data (panels b,d) is due to $B$ and the difference between the mSID and mSIDT data (panels c,e) is due to $Pr$.

However, one of the most surprising differences is that between LSID and mSID data (panels a,c), due to the dimensional height of the duct $H$ (already somewhat visible in LPL19, figure 2). It is reasonable to expect that this $H$-effect is responsible for the main differences between the LSID/HSID/tSID data and the mSID/mSIDT data. In other words, it appears that the dimensional $H$ is the main reason why the LSID/HSID/tSID transitions curves lie well above those for mSID/mSIDT, i.e. the same transitions occur at higher $Re$ for larger $H$. The factor of ${\approx}2$ quantifying this observation suggests that a Reynolds number built using a length scale identical in all data sets (rather than $H/2$) would better collapse the data. However, such a length scale is missing in our dimensional analysis (§ 2.2) because we are unable to think of an additional length scale (such as the thickness of the duct walls or the level of the free surfaces in the reservoirs) that could play a significant dynamical role in the SID experiment.

We conclude that the transitions between flow regimes can be described by hyper-surfaces depending on all five parameters $A,B,\unicode[STIX]{x1D703},Re,Pr$ because their projections onto the $(\unicode[STIX]{x1D703},Re)$ plane for different $A,B,Pr$ do not overlap. This dependence of flow regimes on all five parameters is interesting because it was not immediately obvious from our dimensional analysis which concerned the scaling of the velocity $f_{\unicode[STIX]{x0394}U}$ alone (§ 2.3 and figure 2). Furthermore, the existence of another non-dimensional parameter involving $H$ and a ‘missing’ length scale is a major result that could not be predicted by physical intuition, and which this paper unfortunately does not elucidate.

Let us now investigate in more detail the scaling of regime transitions with respect to $\unicode[STIX]{x1D703}$ and $Re$, for which we have much higher density of data than for $A,B,Pr$. In figure 5, we replot the $\unicode[STIX]{x1D703}>0$ data of figure 4 using a log–log scale (each panel corresponding to the respective panel of figure 4). To guide the eye to the two main types of regime transition scalings observed in these data, we also plot two families of lines: dashed lines with a $\unicode[STIX]{x1D703}Re=$ const. scaling, and dotted lines with a $\unicode[STIX]{x1D703}Re^{2}=$ const. scaling. We also show using grey shading special values of interest: $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}$ and $Re=50A$. The former was highlighted as particularly relevant in our scaling analysis (§ 2.3) and literature review (§ 3.1), notably as the boundary between lazy and forced flows (LPL19, § A.1). Although W86 and K91 quoted $Re=500A$ as a threshold beyond which the effects of viscosity should be negligible on the turbulence in the SID, we believe that $Re=50A$ is a physically justifiable threshold beyond which the influence of the top and bottom walls of the duct becomes negligible. In the absence of turbulent diffusion, laminar flow in the duct is significantly affected by the top and bottom walls if the interfacial and wall 99 % boundary layers overlap in the centre of the duct ($x=0$), which occurs for $Re<50A$ (L18, § 5.2.3). If, on the other hand, $Re\gg 50A$ ($Re=500A$ being a potential threshold), the top and bottom wall laminar boundary layers (as well as the side wall laminar boundary layers, assuming that ) do not penetrate deep into the ‘core’ of the flow (however, at these $Re$, we expect interfacial turbulence to dominate the core of the flow). Note that black contours representing a fit of the $Q_{m}$ data are superimposed in panels (a–d); these will be discussed in § 4.3.

Figure 5 shows that regime transitions scale with $\unicode[STIX]{x1D703}Re^{2}=$ const. (dotted lines) in LSID, tSID and mSIDT (panels a,d,e), and with $\unicode[STIX]{x1D703}Re=\text{const.}$ (dashed lines) in HSID (panel b). In mSID (panel c), these two different scalings coexist: $\unicode[STIX]{x1D703}Re^{2}$ for $\unicode[STIX]{x1D703}\lesssim \unicode[STIX]{x1D6FC}$ (lazy flows) and $\unicode[STIX]{x1D703}Re$ for $\unicode[STIX]{x1D703}\gtrsim \unicode[STIX]{x1D6FC}$ (forced flows), as previously observed by LPL19, who physically substantiated the $\unicode[STIX]{x1D703}Re$ scaling in forced flows, but not the $\unicode[STIX]{x1D703}Re^{2}$ scaling in lazy flows. Furthermore, these five data sets show that this dichotomy in scalings between lazy and forced flows in mSID does not extend to all other geometries: lazy flows in the HSID exhibit a $\unicode[STIX]{x1D703}Re$ scaling and forced flows in the mSIDT exhibit a $\unicode[STIX]{x1D703}Re^{2}$ scaling. These observations further highlight the complexity of the scaling of regime transitions with $A,B,\unicode[STIX]{x1D703},Re,Pr$.

Figure 5. Regime and $Q_{m}$ in the $(\unicode[STIX]{x1D703},Re)$ plane (log–log scale, thus only containing the regime and $Q_{m}$ data of figure 4 for which $\unicode[STIX]{x1D703}>0^{\circ }$). The dashed and dotted lines represent the power law scalings $\unicode[STIX]{x1D703}Re=$ const. and $\unicode[STIX]{x1D703}Re^{2}=\text{const.}$, respectively. The grey shadings represent the special threshold values of interest $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}$ and $Re=50A$. The ML14 arrow in panel (a) denotes the $\mathsf{I}\rightarrow \mathsf{T}$ transition curve identified by ML14. Black contours in panels (a–d) represent the fit to the $Q_{m}$ data (see § 4.3), representing (a) 20 data points (coefficient of determination $R^{2}=0.56$), (b) 34 points ($R^{2}=0.81$), (c) 162 points ($R^{2}=0.80$) and (d) 92 points ($R^{2}=0.86$).

Figure 6. Mass flux for the mSID data set (full symbols) and tSID data set (open symbols) for as a function of $Re$ for various $\unicode[STIX]{x1D703}\in [-1^{\circ },3.5^{\circ }]$ by $0.5^{\circ }$ increments (a–j). The symbol colour denotes the regime as in figures 4 and 5. The mass flux $Q_{m}$ is computed using the average estimation of the run time, and the error bars denote the uncertainty in this estimation (see § B.2).

4.3 Mass flux

Mass fluxes were determined using the same salt balance methodology as ML14 described in § B.2.

In figure 6 we plot the $Q_{m}$ data for mSID (full symbols) and tSID (open symbols) as a function of $Re$ for all the available $\unicode[STIX]{x1D703}$ (from $\unicode[STIX]{x1D703}=-1^{\circ }$ in panel a to $\unicode[STIX]{x1D703}=3.5^{\circ }$ in panel j). The shape and colour of each symbol denote the regime as in figures 4–5 and the error bars denote the uncertainty about the precise duration $T$ of the ‘steady’ flow of interest in an experiment (used to average the volume flux and obtain $Q_{m}$, as explained in § B.2). We do not plot the LSID and HSID data in this figure because they are sparser and do not have error bars (these data were collected by ML14 prior to this work).

At low angles $\unicode[STIX]{x1D703}\lesssim 1^{\circ }<\unicode[STIX]{x1D6FC}$ (where $\unicode[STIX]{x1D6FC}\approx 2^{\circ }$ in mSID and $4^{\circ }$ in tSID) we observe low values $Q_{m}\approx 0.2$–0.3 in the $\mathsf{L}$ and $\mathsf{H}$ regimes. At intermediate angles $\unicode[STIX]{x1D703}\approx \unicode[STIX]{x1D6FC}-2\unicode[STIX]{x1D6FC}$ we observe convergence to the hydraulic limit $Q_{m}\rightarrow 0.5$ (denoted by the dashed line), as discussed in § 2.3, which coincides with the $\mathsf{I}$ and $\mathsf{T}$ regimes. We also note that this hydraulic limit is not a strict upper bound in the sense that we observe values up to $Q_{m}=0.6$ in some experiments (some error bars even going to $0.7$). At higher angles $\unicode[STIX]{x1D703}\gtrsim \unicode[STIX]{x1D6FC}\approx 2^{\circ }$, $Q_{m}$ drops with $Re$ while remaining fairly constant with $\unicode[STIX]{x1D703}$.

As in the regime data, the mSID and tSID $Q_{m}$ data do not collapse with $Re$: all the tSID data (open symbols) are shifted to larger $Re$ compared to the mSID data (full symbols) suggesting again that a Reynolds number based on a ‘missing’ length scale independent of $H$ would better collapse the data.

To gain more insight into the scaling of $Q_{m}$ and its relation to the flow regimes, we superimpose on the regime data of figure 5(a–d) black contours representing the least-squares fit of our four $Q_{m}$ data sets using the following quadratic form:

(4.2)$$\begin{eqnarray}\displaystyle & & \displaystyle Q_{m}(\unicode[STIX]{x1D703},Re)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6E4}_{00}+\unicode[STIX]{x1D6E4}_{10}\log \unicode[STIX]{x1D703}+\unicode[STIX]{x1D6E4}_{20}(\log \unicode[STIX]{x1D703})^{2}+\unicode[STIX]{x1D6E4}_{01}\log Re+\unicode[STIX]{x1D6E4}_{02}(\log Re)^{2}+\unicode[STIX]{x1D6E4}_{11}\log \unicode[STIX]{x1D703}\log Re\nonumber\\ \displaystyle & & \displaystyle \quad =\left[\begin{array}{@{}ccc@{}}\log \unicode[STIX]{x1D703} & \log Re & 1\end{array}\right]\underbrace{\left[\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D6E4}_{20} & \unicode[STIX]{x1D6E4}_{11}/2 & \unicode[STIX]{x1D6E4}_{10}/2\\ \unicode[STIX]{x1D6E4}_{11}/2 & \unicode[STIX]{x1D6E4}_{02} & \unicode[STIX]{x1D6E4}_{01}/2\\ \unicode[STIX]{x1D6E4}_{10}/2 & \unicode[STIX]{x1D6E4}_{01}/2 & \unicode[STIX]{x1D6E4}_{00}\\ \end{array}\right]}_{\unicode[STIX]{x1D71E}}\left[\begin{array}{@{}c@{}}\log \unicode[STIX]{x1D703}\\ \log Re\\ 1\end{array}\right].\end{eqnarray}$$

This is the general equation of a conic section, where $\unicode[STIX]{x1D71E}$ is commonly referred to as the matrix of the quadratic equation. It is well suited to describe the non-monotonic behaviour observed above, despite the fact that the non-monotonicity in $\unicode[STIX]{x1D703}$ (i.e. the decay of $Q_{m}$ at large $\unicode[STIX]{x1D703}$ widely observed in the literature) cannot be clearly confirmed by our data.

These contours describe hyperbolas ($\det \unicode[STIX]{x1D71E}<0$) for LSID, HSID and mSID (panels a–c), and concentric ellipses ($\det \unicode[STIX]{x1D71E}>0$) for tSID (panel d). The hydraulic limit $Q_{m}\approx 0.5$ is reached either at the saddle point of the hyperbolas (panels a–c), or at the centre of the ellipses (panel d), and, encouragingly, no $Q_{m}=0.6$ contour exists here.

We again note that these four data sets do not collapse in the $(\unicode[STIX]{x1D703},Re)$ plane. For example, the angle at which this maximum $Q_{m}$ is achieved is a modest $\unicode[STIX]{x1D703}=0.3\unicode[STIX]{x1D6FC}$ in mSID (panel c) but appears much larger in tSID. The eigenvectors of $\unicode[STIX]{x1D71E}$ for each data set reveal that the major axis of these conic sections has equation $\unicode[STIX]{x1D703}Re^{\unicode[STIX]{x1D6FE}}$ where $\unicode[STIX]{x1D6FE}=2.6,0.3,1.5,1.2$ respectively for panels (a–d) (a larger exponent $\unicode[STIX]{x1D6FE}$ represents a larger dependence on $Re$, hence a more horizontal axis).

The exponent $\unicode[STIX]{x1D6FE}$ characterising the slope of the horizontal major axis is roughly of the same order as the exponent characterising the lines of regime transition (which is 1 for the $\unicode[STIX]{x1D703}Re$ scaling, and 2 for the $\unicode[STIX]{x1D703}Re^{2}$ scaling), suggesting that both phenomena (regime transition and non-monotonic behaviour of $Q_{m}$) are linked. However, this agreement is not quantitative except in mSID (panel c) where $\unicode[STIX]{x1D6FE}=1.5$ is precisely the average of the two different regime transition exponents. This general lack of correlation suggests that the relationship between regimes and $Q_{m}$ is not straightforward and is dependent on the geometry.

4.4 Interfacial layer thickness

Interfacial layer thickness was determined using the non-intrusive shadowgraph imaging technique (in salt experiments only). Shadowgraph is particularly suited to detect peaks in the vertical curvature of the density field $|\unicode[STIX]{x2202}_{zz}\unicode[STIX]{x1D70C}|$ which we define as the edges of the interfacial density layer, as explained in § B.3.

Figure 7. Interfacial density layer thickness $\unicode[STIX]{x1D6FF}(Re)$ in salt experiments for three selected angles $\unicode[STIX]{x1D703}=1^{\circ },2^{\circ },3^{\circ }$ (only a fraction of the available data) and for the four duct geometries: (a–c) LSID, (d–f) HSID, (g–i) mSID, (j–l) tSID. Symbol shape and colour denotes flow regime as in previous figures.

Figure 8. Interfacial density layer thickness $\unicode[STIX]{x1D6FF}$ in salt experiments fitted from (a) LSID: 115 points ($R^{2}=0.88$), (b) HSID: 58 data points ($R^{2}=0.97$), (c) mSID: 91 data points ($R^{2}=0.80$), (d) tSID: 87 data points ($R^{2}=0.75$). Symbol denotes location of the $\unicode[STIX]{x1D6FF}$ data and colour denotes flow regime. Grey shading denotes $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}$ and $Re=50A$.

In figure 7 we plot $\unicode[STIX]{x1D6FF}$ for our four duct geometries (rows) and three particular angles (representing only a subset of our data) $\unicode[STIX]{x1D703}=1^{\circ },2^{\circ },3^{\circ }$ (columns). In figure 8 we plot a quadratic fit (black contours) to all the available $\unicode[STIX]{x1D6FF}$ data (represented by the symbols) in the $(\log \unicode[STIX]{x1D703},\log Re)$ plane following (4.2). We also added in grey shading the $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}$ and $Re=50A$ values of interest for comparison between panels. In both figures, the shape and colour of the symbol denote the flow regimes as in figures 4–6.

In figures 7 and 8, $\unicode[STIX]{x1D6FF}$ monotonically increases with both $\unicode[STIX]{x1D703}$ and $Re$, starting from values as low as $\unicode[STIX]{x1D6FF}\approx 0.05$ in the $\mathsf{L}$, $\mathsf{H}$ and $\mathsf{W}$ regimes (see figure 13a for an illustration with $\unicode[STIX]{x1D6FF}=0.069$), and ending with values as high as $\unicode[STIX]{x1D6FF}\approx 0.8$ in the $\mathsf{T}$ regime (see figure 13c for an illustration with $\unicode[STIX]{x1D6FF}=0.47$). The upper bound corresponds to the turbulent mixing layer filling 80 % of the duct height, with unmixed fluid only filling the remaining top and bottom 10 %. We substantiate the lower bound by the thickness of the 99 % laminar boundary layer resulting from the balance between streamwise advection and vertical diffusion of an initially step-like density field. This calculation gives, at any point in the duct, $\unicode[STIX]{x1D6FF}_{99}\approx 10A^{1/2}(Re\,Pr)^{-1/2}\approx 0.03-0.1$ in the range $Re\in [300,6000]$ where the $\mathsf{L},\mathsf{H},\mathsf{W}$ regimes are found.

Figure 7 also shows a greater scatter of data points in the $\mathsf{I}$ and $\mathsf{T}$ regimes than in the $\mathsf{L}$ and $\mathsf{H}$ regime. This scatter cannot be attributed to measurement artefacts caused by turbulent fluctuations in the streamwise or spanwise position of the mixing layer (§ B.3), but rather demonstrates the inherent physical variability and limited reproducibility of $\mathsf{I}$ and $\mathsf{T}$ flows.

Both figures show the role of the dimensional parameter $H$ in ‘shifting’ the LSID/HSID/tSID data to higher $Re$ than the mSID data and hindering their overlap, hinting at a ‘missing’ length scale, as already discussed in the regime and $Q_{m}$ data. Note that $A$ and $B$ play additional, more subtle roles as shown by the differences between the LSID and HSID data and between the HSID and tSID data, respectively.

Finally, figure 8 shows good agreement between iso-$\unicode[STIX]{x1D6FF}$ contours and ‘iso-regime’ curves, or regime transitions curves (not shown for clarity, but easily visualised by the different symbols). This suggests that $\unicode[STIX]{x1D6FF}$ is more closely correlated to regime than $Q_{m}$ is.

5.1 Volume-averaged energetics

The simultaneous volumetric measurements of the density and three-component velocity fields of LPL19 confirmed their theoretical prediction that, in forced flows, ($\unicode[STIX]{x1D703}\gtrsim \unicode[STIX]{x1D6FC}$) the time- and volume-averaged norm of the strain rate tensor (non-dimensional dissipation) followed the scaling $\langle \unicode[STIX]{x1D668}^{2}\rangle _{x,y,z,t}\sim \unicode[STIX]{x1D703}Re$ (see § A.1 for a review). They further decomposed the dissipation into:

1. (i) a ‘two-dimensional’ component $\unicode[STIX]{x1D668}_{2d}^{2}$ (based on the $x-$averaged velocity $\boldsymbol{u}_{2d}\equiv \langle \boldsymbol{u}\rangle _{x}$). LPL19 measured flows in the mSID geometry at $Re<2500$, i.e. $Re\not \gg 50A=1500$, in which case the viscous interfacial and top and bottom wall boundary layers are well or fully developed and $\unicode[STIX]{x1D668}_{2d}^{2}\sim \langle (\unicode[STIX]{x2202}_{z}u_{2d})^{2}\rangle _{x,y,z,t}=O(1)$. They indeed observed that $\langle \unicode[STIX]{x1D668}_{2d}^{2}\rangle _{x,y,z,t}$ plateaus at ${\approx}4$ in the $\mathsf{I}$ and $\mathsf{T}$ regimes due to the hydraulic limit;

2. (ii) a complementary ‘three-dimensional’ part $\unicode[STIX]{x1D668}_{3d}^{2}=\unicode[STIX]{x1D668}^{2}-\unicode[STIX]{x1D668}_{2d}^{2}$ which, as a consequence of the plateau of $\unicode[STIX]{x1D668}_{2d}^{2}$, takes over in the $\mathsf{I}$ and $\mathsf{T}$ regime and explains the $\unicode[STIX]{x1D703}Re$ scaling of regime transitions for forced flows in mSID.

In flows at $Re\gg 50A$ (well above the horizontal grey shading in figures 5, 8) we expect the 99 % viscous boundary layers to be of typical thickness ${\sim}10A^{1/2}Re^{-1/2}\ll 1$, and therefore volume-averaged two-dimensional dissipation to be higher $\unicode[STIX]{x1D668}_{2d}^{2}\sim \langle (\unicode[STIX]{x2202}_{z}u_{2d})^{2}\rangle _{x,y,z,t}\sim 10^{-1}A^{-1/2}Re^{1/2}\gg 1$. Therefore, we extend the prior results of LPL19 that regime transitions correspond to threshold values of

(5.1)$$\begin{eqnarray}\langle \unicode[STIX]{x1D668}_{3d}^{2}\rangle _{x,y,z,t}\sim \unicode[STIX]{x1D703}Re\quad \text{for }Re<50A,\end{eqnarray}$$

by conjecturing that they correspond to threshold values of

(5.2)$$\begin{eqnarray}\langle \unicode[STIX]{x1D668}_{3d}^{2}\rangle _{x,y,z,t}\sim \unicode[STIX]{x1D703}Re-A^{-1/2}Re^{1/2}\quad \text{for }Re\gg 50A,\end{eqnarray}$$

which introduces $A$ and a different exponent to $Re$ into the scaling.

Unfortunately, we have little regime data for forced flows at $Re\gg 50A$ (upper right quadrants of each panel in figure 5) except in LSID (panel a). Nevertheless, it does not appear that this conjectured scaling would be able to explain the observed $\unicode[STIX]{x1D703}Re^{2}$ scaling. Detailed flow measurements would be required in this geometry to confirm or disprove the above two assumptions that two-dimensional dissipation follows a different scaling, and that regime transitions are tightly linked to three-dimensional dissipation.

Furthermore, we recall that the under-determination of the energy budgets of lazy flows ($\unicode[STIX]{x1D703}<\unicode[STIX]{x1D6FC}$, see LPL19 figure 8a) does not allow us to predict the rate of energy dissipation ($\unicode[STIX]{x1D668}^{2}$) from the rate of energy input (${\sim}\unicode[STIX]{x1D703}Re$) and therefore to substantiate the transition scalings in lazy flows (left two quadrants of each panel in figure 5).

5.2 Frictional two-layer hydraulics

We introduce the fundamentals of this model in § 5.2.1, before examining the physical insight it provides in § 5.2.2, and its implications for the scaling of regime transitions and mass flux in § 5.2.3.

5.2.1 Fundamentals

The two-layer hydraulic model for exchange flows (figure 9a) assumes two layers flowing with non-dimensional velocities $u_{1}(x)>0$ (lower layer) and $u_{2}(x)<0$ (upper layer), and separated by an interface of non-dimensional elevation $\unicode[STIX]{x1D702}(x)\in [-1,1]$ above the neutral level $z=0$.

In the idealised inviscid hydraulic model (i.e. in the absence of viscous friction) the conservation of volume and of Bernoulli potential, and the requirement of hydraulic control yield a horizontal and symmetric interface $\unicode[STIX]{x1D702}(x)=0$ for $x\in [-A,A]$ and a volume flux $Q=u_{1}=-u_{2}=1/2$ as already mentioned in § 2.3 (see § C.1 for more details).

Figure 9. Schematics of the (a) hydraulic model set-up and notation and (b) the frictional model with stresses acting on the top and bottom walls $\unicode[STIX]{x1D70F}_{1,2}^{Z}$ (in blue), sidewalls $\unicode[STIX]{x1D70F}_{1,2}^{Y}$ (in green) and interface $\unicode[STIX]{x1D70F}^{I}$ (in red) of an infinitesimally small slab of fluid $\text{d}x$.

The frictional hydraulic model is of more relevance to SID flows at finite $Re$. This model parameterises the effects of viscous friction while retaining the hydraulic assumptions (hydrostatic, steady, two-layer flow with uniform velocities $u_{1,2}(x)$). Dating back to Schijf & Schönfled (Reference Schijf and Schönfled1953), Assaf & Hecht (Reference Assaf and Hecht1974) and Anati, Assaf & Thompson (Reference Anati, Assaf and Thompson1977), it was formalised by Zhu & Lawrence (Reference Zhu and Lawrence2000), Gu (Reference Gu2001) and Gu & Lawrence (Reference Gu and Lawrence2005), who considered the effects of friction at the interface and bottom wall only, with applications to wide, open, horizontal channels. Here we further develop this model to add the effects of gravitational forcing ($\unicode[STIX]{x1D703}>0$) and friction at the top and sidewalls. The full derivation of this model can be found in (L18, § 5.2) and we offer a summary in appendix C. Some of its conclusions were briefly discussed in LPL19 (their § 4.3.1), e.g. the distinction between lazy/forced flows. Below we provide a self-contained presentation of the key results of this model regarding the particular problem of the scaling of regimes and $Q_{m}$.

As sketched in figure 9(b), we consider that each infinitesimal duct sub-volume $\text{d}x\times 2B\times 2$ centred around $x$ is subject to horizontal, resistive stresses at the bottom wall $\unicode[STIX]{x1D70F}_{1}^{Z}(x)$, top wall $\unicode[STIX]{x1D70F}_{2}^{Z}(x)$ (in blue), sidewalls $\unicode[STIX]{x1D70F}_{1,2}^{Y}(x)$ (respectively in the bottom and top layers, in green) and interface $\unicode[STIX]{x1D70F}^{I}$ (in red). The inclusion of these stresses in the evolution of Bernoulli potential along the duct (see § C.1) yields a nonlinear differential equation for the slope of the interface along the duct of the form

(5.3)$$\begin{eqnarray}\unicode[STIX]{x1D702}^{\prime }(x)=\unicode[STIX]{x1D702}^{\prime }(\unicode[STIX]{x1D702},Q,\unicode[STIX]{x1D703},Re,f_{Z},f_{Y},f_{I})\end{eqnarray}$$

(see (C 24) for the full expression). Here, $f_{Z},f_{Y},f_{I}$ are constant friction factors parameterising, respectively, the top and bottom wall stresses, the sidewall stress and the interfacial stress (they can be determined a posteriori from any finite-$Re$ flow profile, see § C.2 and (C 21)). For any set of parameters $\unicode[STIX]{x1D703},Re,f_{Z},f_{Y},f_{I}$, this dynamical equation can be combined with the hydraulic control condition and solved numerically using an iterative method to yield a unique solution for $Q$ and $\unicode[STIX]{x1D702}(x)$ (§ C.3). The volume flux $Q$ generally increases with the forcing $\unicode[STIX]{x1D703}Re$, and decreases with friction $f_{Z},f_{Y},f_{I}$ and $A$.

5.2.2 Physical insight

We now consider the mid-duct slope $\unicode[STIX]{x1D702}^{\prime }(x=0)$, whose simplified expression shows the balance between the forcing $\unicode[STIX]{x1D703}Re$ and the ‘composite friction parameter’ $F$

(5.4)$$\begin{eqnarray}\unicode[STIX]{x1D702}^{\prime }(0)=\frac{\unicode[STIX]{x1D703}Re-2QF}{Re(1-4Q^{2})}\quad \text{where }F\equiv f_{Z}(1+2r_{Y}+8r_{I}),\end{eqnarray}$$

and $r_{Y}\equiv B^{-1}f_{Y}/f_{Z}$ and $r_{I}\equiv f_{I}/f_{Z}$ are respectively the sidewall friction ratio and the interfacial friction ratio.

We further note that our model has three properties: (i) the interface must slope down everywhere ($\unicode[STIX]{x1D702}^{\prime }(x)<0$) since the lower layer accelerates convectively from left to right ($u_{1}u_{1}^{\prime }(x)>0$) and vice versa ($u_{2}u_{2}^{\prime }(x)<0$); (ii) the interface must remain in the duct $|\unicode[STIX]{x1D702}(x=\pm A)|<1$; (iii) $\unicode[STIX]{x1D702}^{\prime }$ always reaches a maximum ($|\unicode[STIX]{x1D702}^{\prime }|$ reaches a minimum) at the inflection point $x=0$.

From these properties we deduce that the existence of a solution requires the mid-duct interfacial slope to satisfy

(5.5)$$\begin{eqnarray}-A^{-1}<\unicode[STIX]{x1D702}^{\prime }(0)<0,\end{eqnarray}$$

and, therefore, using (5.4), we obtain the following bounds:

(5.6)$$\begin{eqnarray}\unicode[STIX]{x1D703}Re<2QF<(1+b)\unicode[STIX]{x1D703}Re\quad \text{where }b(A,\unicode[STIX]{x1D703},Q)\equiv \frac{1-4Q^{2}}{A\unicode[STIX]{x1D703}}.\end{eqnarray}$$

The first inequality in (5.5) comes from property (ii) and means that the mid-duct interfacial slope must not be too steep compared to the duct geometrical slope $A^{-1}\approx \unicode[STIX]{x1D6FC}$. The second inequality comes from (i) and (iii) and means that the mid-duct interfacial slope must be negative for $\unicode[STIX]{x1D702}(x)$ to be negative everywhere.

Figure 10. Predictions of the frictional hydraulic model as the ‘forcing parameter’ $\unicode[STIX]{x1D703}Re$ is increased: (a) $2QF$ is bounded above and below by (5.6); (b) volume flux $Q$ and (c) composite friction parameter $F$ (a and b in the $\mathsf{T}$ regime denote two possible scenarios). We conjecture that regime transitions correspond to threshold values of $F$.

When suitably rescaled by $2Q\in [0,1]$, the combined friction parameter $F$ must therefore follow a $\unicode[STIX]{x1D703}Re$ scaling, strictly bounded from below. The upper bound in (5.6) is loose ($b>0$) in lazy flows, and tight ($b\rightarrow 0$) in forced flows ($A\unicode[STIX]{x1D703}\approx \unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FC}\gg 1$ and $Q\rightarrow 1/2$).

5.2.3 Implications for regimes and $Q_{m}$

Combining the above physical insight with our experimental observations, we conjecture the following behaviour about regimes and $Q_{m}$, summarised in figure 10:

1. (i) At zero or ‘low’ $\unicode[STIX]{x1D703}Re$ (i.e. at $\unicode[STIX]{x1D703}\approx 0$, since $Re$ must be large for hydraulic theory to hold) due to the inevitable presence of wall and interfacial friction ($F>0$) and the looseness of the upper bound $b$, $2QF$ is typically well above the forcing $\unicode[STIX]{x1D703}Re$. The friction $F$ is independent of $\unicode[STIX]{x1D703}Re$ and the flow is typically laminar ($\mathsf{L}$ regime). The interface has a noticeable slope all along the duct $\unicode[STIX]{x1D702}^{\prime }(0)\ll 0$, associated with a small volume flux $Q\ll 1/2$ (see (C 25)). Such lazy flows are underspecified, and the scaling of $Q$ and $F$ with $\unicode[STIX]{x1D703}Re$ is therefore impossible to predict a priori.

2. (ii) At moderate $\unicode[STIX]{x1D703}Re$ ($\unicode[STIX]{x1D703}>0$): $2QF$ increases above its ‘default’ $\unicode[STIX]{x1D703}=0$ value. This is achieved, on one hand, through an increase in $Q$ (and therefore $Q_{m}$), making the flow approach the hydraulic limit (panel b), and on the other hand, through an increase in $F$, in particular through laminar interfacial shear ($r_{I}$), rendering the flow unstable to Holmboe waves above a certain threshold ($\mathsf{L}\rightarrow \mathsf{H}$ transition, panel c). The phenomenology of this transition agrees with that proposed by the energetics of LPL19 (see their §§ 6.2–6.3). The fact that the $\mathsf{L}\rightarrow \mathsf{H}$ (or $\mathsf{L}\rightarrow \mathsf{W}$) transition exhibits different scalings in our different data sets is not presently understood. It may come from the complex, individual roles of $Q$ and $F$ in the precise flow profiles $u(y,z),\unicode[STIX]{x1D70C}(z)$ responsible for triggering the Holmboe instability, and the different scalings of $Q$ and $F$ that could allow the product $2QF$ to follow a $\unicode[STIX]{x1D703}Re$ scaling.

3. (iii) At high $\unicode[STIX]{x1D703}Re$: the hydraulic limit is reached, the upper bound is tight ($b\approx 0$), the interface is mostly flat ($\unicode[STIX]{x1D702}(x)\approx 0$ everywhere) and the inequality (5.6) becomes $2QF\approx F\approx \unicode[STIX]{x1D703}Re$. In such forced flows, the friction parameter $F$ alone must precisely balance the forcing. Arbitrarily large $\unicode[STIX]{x1D703}Re$ requires arbitrarily large $F$, which we conjecture is largely achieved by turbulent interfacial friction (increase in $r_{I}$ responsible for the $\mathsf{H}\rightarrow \mathsf{I}$ and eventually the $\mathsf{I}\rightarrow \mathsf{T}$ transition).

From implication (iii), it is natural to conjecture that these two transitions are also caused by threshold values of the interfacial friction ratio $r_{I}$, which, as we explain in § C.2, can be written $r_{I}=1+K_{I}$, where $K_{I}$ is a turbulent momentum diffusivity (non-dimensionalised by the molecular value $\unicode[STIX]{x1D708}$) parameterising interfacial Reynolds stresses (see (C 19)). Assuming that all wall shear stresses are similar ($r_{Y}\approx 1$), and that interfacial Reynolds stresses eventually dominate over laminar shear ($K_{I}\gg 1$), we have $K_{I}\approx F/(8f_{Z})$. For $Re<50A$, fully developed boundary layers yield $f_{Z}\sim 1$, implying regime transitions scaling with (ignoring pre-factors)

(5.7)$$\begin{eqnarray}K_{I}\sim \unicode[STIX]{x1D703}Re\quad \text{for }Re<50A.\end{eqnarray}$$

For $Re\gg 50A$, thin top and bottom wall boundary layer arguments similar to those of § 5.1 yield $f_{Z}\sim A^{1/2}Re^{-1/2}$, implying regime transitions scaling with

(5.8)$$\begin{eqnarray}K_{I}\sim A^{1/2}\unicode[STIX]{x1D703}Re^{1/2}\quad \text{for }Re\gg 50A.\end{eqnarray}$$

Comparing (5.7)–(5.8) to (5.1)–(5.2) we see that the $Re<50A$ scaling obtained with frictional hydraulics is identical to that obtained by the energetics. However, the $Re\gg 50A$ scaling is different, and unfortunately it does not allow us to explain the regime transitions data (a $\unicode[STIX]{x1D703}Re^{1/2}$ or $\unicode[STIX]{x1D703}^{2}Re$ scaling is never observed). In addition, direct estimations of friction coefficients using three-dimensional, three-component velocity measurements in all flow regimes (L18, § 5.5) suggest a posteriori that the assumption that $K_{I}\gg 1$ might only hold beyond the $\mathsf{I}\rightarrow \mathsf{T}$ transition, undermining its usefulness to predict the $\mathsf{H}\rightarrow \mathsf{I}$ and $\mathsf{I}\rightarrow \mathsf{T}$ transitions.

From implication (ii), we understand why the volume flux $Q$, and hence the mass flux $Q_{m}$, both increase with $\unicode[STIX]{x1D703}$ and $Re$ in the $\mathsf{L}$ and $\mathsf{H}$ regimes, as observed in § 4.3. However, lazy flows are under-specified; only one equation governs both the volume flux and friction ($2QF\sim \unicode[STIX]{x1D703}Re$), which does not allow us to obtain the value of the exponent $\unicode[STIX]{x1D6FE}$ in the scaling $Q\sim \unicode[STIX]{x1D703}Re^{\unicode[STIX]{x1D6FE}}$. From implication (iii), we conjecture two potential reasons for the decrease of the mass flux $Q_{m}$ in the $\mathsf{T}$ regime (labelled ‘a’ and ‘b’ in figure 10b,c). In scenario ‘a’, $Q_{m}$ decreases due to increasing mixing despite the volume flux $Q$ staying relatively constant ($2QF\sim F\sim \unicode[STIX]{x1D703}Re$). In scenario ‘b’, $Q_{m}$ decreases partly due to mixing, and partly due to a decrease in $Q$ (compensated by $F$ increasing faster than $\unicode[STIX]{x1D703}Re$). Accurate $Q$ and $Q_{m}$ data obtained by volumetric measurements of velocity and density in L18 (figure 5.12b) support scenario ‘b’ up to $\unicode[STIX]{x1D703}Re=132$, but additional data are required to draw general conclusions.

The above frictional hydraulic model assumes a two-layer flow without any form of mixing, and thus ignores the behaviour of the interfacial thickness $\unicode[STIX]{x1D6FF}$, which is the subject of the next section.

5.3 Mixing models

The importance and difficulty of modelling interfacial mixing in exchange flows have long been recognised (Helfrich Reference Helfrich1995; Winters & Seim Reference Winters and Seim2000). However, despite the existence of hydraulic models for multi-layered or continuously stratified flows (Engqvist Reference Engqvist1996; Lane-Serff, Smeed & Postlethwaite Reference Lane-Serff, Smeed and Postlethwaite2000; Hogg & Killworth Reference Hogg and Killworth2004), to date there exists no ‘three-layer’ hydraulic model allowing for the exchange of momentum or mass between the two counter-flowing layers suitable to our problem (which would violate most hydraulic assumptions). Below we review some experimental, numerical and theoretical work most relevant to the scaling of $Q_{m}$, $\unicode[STIX]{x1D6FF}$, and their relation to fundamental stratified turbulence properties such as diapycnal diffusivity and mixing efficiency.

5.3.1 Turbulent diffusion models

Cormack, Leal & Imberger (Reference Cormack, Leal and Imberger1974a) tackled natural convection in a shallow ($A\gg 1$) cavity with differentially heated walls. This problem is analogous to SID flows in the limit of maximum ‘interfacial’ thickness ($\unicode[STIX]{x1D6FF}=1$) in which turbulent mixing dominates to such an extent that the exchange flow is only weakly stratified in the vertical (i.e. $\langle |\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D70C}|\rangle _{z}<1$ because $|\unicode[STIX]{x1D70C}(z=\pm 1)|<1$) and becomes stratified in the horizontal (i.e. $|\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D70C}(z=\pm 1)|>0$ and mean isopycnals are no longer horizontal). In their model, the horizontal hydrostatic pressure gradient is balanced only by a uniform vertical turbulent diffusion with constant $K_{T}$. Using the terminology of § 2.3, this balance could be called the hydrostatic-mixing (or ‘HM’) balance where ‘mixing’ plays a similar role to ‘viscosity’ in the ‘HV’ balance. Cormack et al. (Reference Cormack, Leal and Imberger1974a) solved this problem analytically and found

(5.9a)$$\begin{eqnarray}\displaystyle & \displaystyle Q={\textstyle \frac{5}{384}}(AK_{T})^{-1}, & \displaystyle\end{eqnarray}$$

(5.9b)$$\begin{eqnarray}\displaystyle & \displaystyle Q_{m}=4AK_{T}+{\textstyle \frac{31}{1451\,520}}(AK_{T})^{-3}, & \displaystyle\end{eqnarray}$$

where we assumed a turbulent Prandtl number of unity for simplicity (i.e. the density equation has the same turbulent diffusivity). The above equations are adapted from (19) and (20) of Hogg, Ivey & Winters (Reference Hogg, Ivey and Winters2001) (in their review of the results of Cormack et al. (Reference Cormack, Leal and Imberger1974a)) to match our slightly different definitions of $Q,Q_{m},A$ and especially our definition of $K_{T}$ as being non-dimensionalised by the inertial scaling $\sqrt{g^{\prime }H}H/2$ (giving $K_{T}=(4Gr_{T})^{-1/2}$ where $Gr_{T}$ is their ‘turbulent Grashof number’). We also contrast the uniform diffusivity $K_{T}$ in this model and the interfacial diffusivity $K_{I}$ in the frictional hydraulics model of § 5.2.3, which have different roles and different non-dimensionalisation ($\sqrt{g^{\prime }H}H/2$ versus $\unicode[STIX]{x1D708}$, hence ‘$K_{T}=K_{I}/Re$’). The predictions (5.9) were verified numerically and experimentally in two papers of the same series (Cormack, Leal & Seinfeld Reference Cormack, Leal and Seinfeld1974b; Imberger Reference Imberger1974), but only hold in the ‘high-mixing’ limit of $AK_{T}>1/15$ below which inertia becomes noticeable and the assumptions start to break down (at $AK_{T}<1/25$, $Q$ and $Q_{m}$ even exceed the hydraulic limit).

Hogg et al. (Reference Hogg, Ivey and Winters2001) built on the above results and developed a model with linear velocity and density profiles within an interfacial layer of thickness $\unicode[STIX]{x1D6FF}<1$ and a uniform turbulent momentum and density diffusivity $K_{T}$. This models the ‘IHM’ balance, i.e. the transition between the Cormack et al. (Reference Cormack, Leal and Imberger1974a) $AK_{T}>1/15$ high-mixing limit (the ‘HM’ balance where turbulent diffusion dominates over inertia, $\unicode[STIX]{x1D6FF}=1$ and (5.9) holds) and the $AK_{T}\rightarrow 0$ hydraulic limit (the ‘IH’ balance where inertia dominates over mixing, $\unicode[STIX]{x1D6FF}=0$ and $Q=Q_{m}=1/2$ holds). Hogg et al. (Reference Hogg, Ivey and Winters2001) argued that $\unicode[STIX]{x1D6FF}$ would increase diffusively during the ‘duct transit’ advective time scale $A$, and integrated the linear velocity and density profiles across the interfacial layer to find

(5.10a)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}\approx 5(AK_{T})^{1/2}, & \displaystyle\end{eqnarray}$$

(5.10b)$$\begin{eqnarray}\displaystyle & \displaystyle Q\approx {\textstyle \frac{1}{2}}-{\textstyle \frac{5}{4}}(AK_{T})^{1/2}, & \displaystyle\end{eqnarray}$$

(5.10c)$$\begin{eqnarray}\displaystyle & \displaystyle Q_{m}\approx {\textstyle \frac{1}{2}}-{\textstyle \frac{5}{3}}(AK_{T})^{1/2}, & \displaystyle\end{eqnarray}$$

where the prefactors $5,5/4,5/3$ come from the imposed matching with the high-mixing solution (5.9). Hogg et al. (Reference Hogg, Ivey and Winters2001) validated these predictions with large eddy simulations and found good quantitative agreement for $Q,Q_{m},\unicode[STIX]{x1D6FF}$ across the range $AK_{T}\in [1/2000,1/15]$, below which convergence to the inviscid hydraulic limit was confirmed.

In order to use these models to explain the scaling of $Q_{m}$ and $\unicode[STIX]{x1D6FF}$ with $A,B,\unicode[STIX]{x1D703},Re,Pr$, we need to (i) extend them to the more complex ‘IHGM’ balance of SID flows in the $\mathsf{I}$ and $\mathsf{T}$ regimes in which gravitational forcing is present ($\unicode[STIX]{x1D703}>0$); (ii) have a model for the scaling of $K_{T}$ on input parameters (the above models prescribed $K_{T}$ as an input parameter, but it is a priori unknown in the SID). To do so, we propose to use insight gained by the energetics and frictional hydraulics models.

First, following the results of LPL19 and § 5.1 on the average rate of turbulent dissipation, it is tempting to model $K_{T}$ using a turbulence closure scheme like the mixing length or $K$–$\unicode[STIX]{x1D716}$ model, yet these require either a length scale or the turbulent kinetic energy, which are both unknown (only the rate of dissipation is known, see (5.1)–(5.2)).

Second, borrowing from the frictional hydraulics results of § 5.2, it seems natural to conjecture that the ‘Reynolds stresses’ interfacial diffusivity $K_{I}$ in the $\mathsf{I}$ and $\mathsf{T}$ regimes may play a similar role to the uniform turbulent diffusivity in the present model. Recalling that by definition $K_{T}=K_{I}/Re$, combining the scalings (5.7)–(5.8) with (5.10) would suggest

(5.11a)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}\sim {\textstyle \frac{1}{2}}-Q_{m}\sim (A\unicode[STIX]{x1D703})^{1/2}\quad \text{for }Re<50A, & \displaystyle\end{eqnarray}$$

(5.11b)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}\sim {\textstyle \frac{1}{2}}-Q_{m}\sim (A^{3}\unicode[STIX]{x1D703}^{2}Re^{-1})^{1/4}\quad \text{for }Re\gg 50A. & \displaystyle\end{eqnarray}$$

Unfortunately, these scalings are not consistent with the observations of figures 5–8 because $\unicode[STIX]{x1D6FF}$ is clearly a function of $Re$ for $Re<50A$ (less so at high $Re$ where the $A\unicode[STIX]{x1D703}$ scaling has indeed been observed by K91), and because $\unicode[STIX]{x1D6FF}$ is clearly not a decreasing function of $Re$ for $Re\gg 50A$.

5.3.3 New mixing efficiency model

To address this, we propose a different model of mixing based on the energetics framework of LPL19. As sketched in figure 11(a), we consider that the duct is composed of three volume-averaged energy reservoirs: potential energy $P$, kinetic energy $K$ and internal energy $I$ (heat). We further decompose the potential energy reservoir into an available potential energy $P_{A}$, and a background potential energy $P_{B}$ (such that $P=P_{A}+P_{B}$), as is customary in the study of mixing (see e.g. Winters et al. Reference Winters, Lombard, Riley and D’Asaro1995).

As explained in LPL19 (see their §§ 4.1–4.3 and figure 8b), forced flows have, to a good approximation, the following quasi-steady-state energetics: the external fluid reservoirs provide an advective flux of potential energy into the duct, which we identify here as being an advective flux of available potential energy $\unicode[STIX]{x1D6F7}_{P_{A}}^{adv}\approx Q_{m}\unicode[STIX]{x1D703}/8$, which is then converted to kinetic energy by the horizontal buoyancy flux $B_{x}$, and to heat by the viscous dissipation $D\approx (2/Re)\langle \unicode[STIX]{x1D668}^{2}\rangle _{x,y,z,t}$. When turbulent mixing is neglected, all these fluxes have equal magnitude, and $D\approx (1/8)Q_{m}\unicode[STIX]{x1D703}$. When turbulent mixing is included, a net vertical buoyancy flux $B_{z}$ converts part of $K$ back to $P_{A}$, and a net irreversible diapycnal flux $\unicode[STIX]{x1D6F7}^{d}$ converts part of $P_{A}$ to $P_{B}$, at a steady-state rate equal to the advective flux of $P_{B}$ out of the duct, back into the external reservoirs $|\unicode[STIX]{x1D6F7}_{P_{B}}^{adv}|=|\unicode[STIX]{x1D6F7}^{d}|$. The mixing efficiency quantifies the percentage of total time- and volume-averaged power throughput $\unicode[STIX]{x1D6F7}_{P_{A}}^{adv}$ that is spent to irreversibly mix the density field inside the duct

(5.12)$$\begin{eqnarray}{\mathcal{M}}=\frac{\unicode[STIX]{x1D6F7}^{d}}{D+\unicode[STIX]{x1D6F7}^{d}}=\frac{\unicode[STIX]{x1D6F7}_{P_{B}}^{adv}}{\unicode[STIX]{x1D6F7}_{P_{A}}^{adv}}.\end{eqnarray}$$

It is expected that ${\mathcal{M}}\ll 1$ in such flows, as represented by the respective thickness of the arrows in figure 11(a), representing the order of magnitude of the fluxes.

Figure 11. Mixing model for SID flows. (a) Time- and volume-averaged energetics model developing on that in LPL19 (their figure 8b) by subdividing the potential energy reservoir as $P=P_{A}+P_{B}$. We also show the kinetic energy $K$, internal energy $I$, and all relevant fluxes: horizontal buoyancy flux $B_{x}$, vertical buoyancy flux $B_{z}$, viscous dissipation $D$, diapycnal flux $\unicode[STIX]{x1D6F7}^{d}$ and advective fluxes with the external reservoirs $\unicode[STIX]{x1D6F7}_{P_{A}}^{adv},\unicode[STIX]{x1D6F7}_{P_{B}}^{adv}$. The direction of the arrows denotes the net (time-averaged) transfer, and the thickness of the arrows denotes the expected magnitude of the fluxes (with the expectation that $\unicode[STIX]{x1D6F7}_{P_{A}}^{adv}\approx B_{x}\approx D$ and $B_{z}\approx \unicode[STIX]{x1D6F7}^{d}\approx \unicode[STIX]{x1D6F7}_{P_{B}}^{adv}$). (b) Simplified flow model in the duct to estimate the mixing rate from $\unicode[STIX]{x1D6F7}_{P_{B}}^{adv}$ and link it to $\unicode[STIX]{x1D6FF}$. The in-flow of unmixed fluids from the external reservoirs and the out-flow of mixed fluid back into them are modelled by the broken line profiles $u(z)=\unicode[STIX]{x1D70C}(z)$ drawn at the left and right ends of the duct (consistent with the typical mid-duct profile drawn, equal to $u=\unicode[STIX]{x1D70C}=\pm 1$ above and below the mixing layer and $u=\unicode[STIX]{x1D70C}=-2z/\unicode[STIX]{x1D6FF}$ in the mixing layer, assumed elsewhere in the literature).

As sketched in figure 11(b), we propose piecewise-linear flow profiles $u(z)=\unicode[STIX]{x1D70C}(z)$ at either end of the duct as a minimal model to estimate the magnitude of $\unicode[STIX]{x1D6F7}_{P_{B}}^{adv}$ as a function of the interfacial layer thickness $\unicode[STIX]{x1D6FF}$, and eventually link it to input parameters $A,\unicode[STIX]{x1D703},Re$. We consider that fluid comes from the external reservoirs into the duct unmixed below (respectively above) the interfacial mixing layer at the left (respectively right) end of the duct, and leaves the duct mixed with a linear profile, going from 0 at the bottom (respectively top) edge of the mixed layer to $-1$ (respectively 1) at the top (respectively bottom) edge of the mixed layer. (In more central sections of the duct, mixing smooths out the discontinuities at the edges of the mixing layer present at the ends, and we expect the continuous linear profile drawn in the centre, but it is irrelevant to the following calculations.) The outflow of mixed fluid creates the following net flux of background potential energy out of the duct:

(5.13)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{P_{B}}^{adv}=\frac{1}{4A}\langle z\unicode[STIX]{x1D70C}u\rangle _{z}|_{L-R}=\frac{2}{4A\unicode[STIX]{x1D6FF}}\int _{-\unicode[STIX]{x1D6FF}/2}^{\unicode[STIX]{x1D6FF}/2}z\left(z+\frac{\unicode[STIX]{x1D6FF}}{2}\right)^{2}\,\text{d}z=\frac{\unicode[STIX]{x1D6FF}^{3}}{24A},\end{eqnarray}$$

where $|_{L-R}$ denotes the difference between the values at the left and right boundary, and the prefactor $1/(4A)$ comes from the non-dimensionalisation of the energy budget equations (see LPL19, equation (4.14a)). From (5.12)–(5.13) and $\unicode[STIX]{x1D6F7}_{P_{A}}^{adv}\approx Q_{m}\unicode[STIX]{x1D703}/8$, we now deduce

(5.14)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}\approx (3A\unicode[STIX]{x1D703}Q_{m}{\mathcal{M}})^{1/3}.\end{eqnarray}$$

Encouragingly, this estimation has the potential to be consistent with our data in the SID. Assuming that $Q_{m}\approx 0.5$ throughout most of the $\mathsf{I}$ and $\mathsf{T}$ regimes, we conjecture that most of the dependence on $Re$ observed in the $\unicode[STIX]{x1D6FF}$ data of figure 7 is due to the underlying monotonic increase of ${\mathcal{M}}(Re)$, which remains a priori unknown, but consistent with the observations of Prastowo et al. (Reference Prastowo, Griffiths, Hughes and Hogg2008) and Hughes & Linden (Reference Hughes and Linden2016). The observation of K91 and figure 7(a,b) that $\unicode[STIX]{x1D6FF}$ primarily scales with the group $A\unicode[STIX]{x1D703}$ at $Re\gg 500A$ (as sketched in figure 3c) would suggest that ${\mathcal{M}}$ asymptotes to a constant value at high $Re$, which is also consistent with the observations of Prastowo et al. (Reference Prastowo, Griffiths, Hughes and Hogg2008) and Hughes & Linden (Reference Hughes and Linden2016) at $ARe>10^{5}$ and $Re>10^{5}$ respectively. Assuming their high-$Re$ asymptotic value of ${\mathcal{M}}\approx 0.1$, we obtain

(5.15)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}\rightarrow 0.5\left(\frac{\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D6FC}}\right)^{1/3}.\end{eqnarray}$$

This gives, for example, $\unicode[STIX]{x1D6FF}\approx 0.4$ when $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FC}\approx 1/2$. This value agrees with the K91 data (see § A.3 and figure 12f,g) and our LSID data (figure 7a, at $Re>10^{4}$ and $ARe>10^{5}$). However, this value does not agree well with our HSID, tSID and mSID $\unicode[STIX]{x1D6FF}$ data (figure 7b–d), in which $\unicode[STIX]{x1D6FF}$ remains dependent on $Re$. This is presumably due to the lower values of $A$ and/or $Re$ in these data sets, which remain below the asymptotic values of $Re>10^{5}$ and $ARe>10^{5}$. In other words, we believe that our $\unicode[STIX]{x1D6FF}$ data and (5.15) are consistent and provide further (albeit indirect) evidence for the monotonic increase of ${\mathcal{M}}$ with $Re$.