3.1 Entrainment rate with $Ri_{b}$

The entrainment rate results for all the data collected are presented in figure 3 with the three Reynolds number values indicated. It is immediately clear that within the data presented there is a single entrainment regime across the entire $Ri_{b}$ range. Within our Reynolds number range we are unable to discern a significant sensitivity of entrainment to this parameter. We obtain a regression fit of $E=CRi_{b}^{n}$ , with $n=-1.38$ and $C=0.0023\pm 0.0007$ , for the more extensive $Re=11\,300$ results. Entrainment measurements obtained using (3.2) for the same experiments give $n=-1.35$ and $C=0.0022\pm 0.0007$ (data not shown). The estimates of experimental error in the calculation of $Ri_{b}$ and $E$ are indicated by error bars in the figure. In the same figure we present the entrainment correlations of Strang & Fernando (Reference Strang and Fernando2001) obtained using an Odell–Kovasznay type re-circulating water channel. In their regime 2 over $Ri_{b}=1.5$ to 5 they observed $E\simeq CRi_{b}^{n}$ , where $C_{2}\simeq 0.22\pm 0.11$ and $n\simeq -2.63\pm 0.45$ . At higher Richardson number, in their regime 3 over $Ri_{b}=5-20$ , they found $C_{3}\simeq 0.02\pm 0.01$ and $n\simeq -1.3\pm 0.15$ . Our exponent $n=-1.38$ lies within the range of values they report for their regime 3, but our constant $C$ is approximately an order of magnitude smaller and we observe this scaling holds down to $Ri_{b}\simeq 1$ . The consistency among the exponent values between the studies suggests that similar mechanisms may be present, that is in both flows entrainment is a result of shear-driven effects at the interface rather than cavity-related mechanisms, and that the entrainment regime is related to the $Ri_{b}>5$ regime reported by Strang & Fernando (Reference Strang and Fernando2001). Also shown is the empirical result ( $C=0.007$ , $n=-1.5$ ) obtained by Narimousa & Fernando (Reference Narimousa and Fernando1987) in an Odell–Kovasznay water channel over $Ri_{b}=2$ to 15 and the relationship obtained by Christodoulou (Reference Christodoulou1986) ( $C=0.007$ , $n=-1.5$ ), who re-interpreted data from a range of earlier studies of two-layer shear flows including the buoyant wall jet data of Lofquist (Reference Lofquist1960) and screen-driven annulus data of Kantha et al. (Reference Kantha, Phillips and Azad1977). The data set included no cavity flow configurations. The exponent obtained by Christodoulou (Reference Christodoulou1986) is consistent with our measurements and those of Strang & Fernando (Reference Strang and Fernando2001), although the coefficient is again larger in this data set.

The entrainment data of Deardorff & Willis (Reference Deardorff and Willis1982) are plotted over our regression fit in figure 4. In the annulus configuration, entrainment has an additional dependency on annulus curvature (Scranton & Lindberg Reference Scranton and Lindberg1983) such that, with increasing curvature, turbulence in the mixed layer is damped, resulting in a reduction in mixing at the interface (Deardorff & Yoon Reference Deardorff and Yoon1984). Interestingly this data set is similar to our result both in terms of the magnitude and in terms of the $Ri_{b}$ range for this entrainment regime. We further investigate the origins of the reduced mixing in the present results in § 4.

Figure 5.

Coefficient $C=ERi^{1.38}$ with flow aspect ratio $D/H$ (a), $L/D$ (b) and $L/H$ (c) for $l_{c}=2.1$ (●) (black) and $l_{c}=1.6$ (▪) (red) for $Re=11\,300$ . Mean values plotted, with bars giving the range of data.

Our entrainment measurements are binned into bands of the geometric ratios $D/H$ , $L/D$ and $L/H$ , plotted in figure 5(a–c), respectively. Here the arithmetic mean of the coefficient $C$ for the data within each bin and the range of data are indicated by the horizontal bars in the figure. The vertical bars indicate the range of $C$ in each bin. The data for the $l_{c}=2.1~\text{m}$ and $l_{c}=1.6~\text{m}$ experiments are plotted separately in the figure. Initially we note that in figure 5(a) the variation in the coefficient $C$ with $D/H$ is shown to be small, confirming that entrainment has no significant dependency on the upstream channel depth or the cavity depth $K$ , recalling that $(K+D)/H$ is a constant. The variation in the mean value of $C$ with $L/D$ or $L/H$ , plotted in figure 5(b) and 5(c), is generally within the scatter of the data at each geometric ratio. The mean value of $C$ in the $l_{c}=1.6~\text{m}$ data set is somewhat higher than that of the $l_{c}=2.1~\text{m}$ data set across all common values of $L/D$ or $L/H$ , but these differences are less than the ranges within each bin. At large values of $L/D$ and $L/H$ , the data set is smaller, so the reduction in the mean value of $C$ and the scatter should be treated with caution. The figures show that at least until $L/D=14$ , a single bulk Richardson number and set of constants $C$ and $n$ are effective parameters to characterise the flow along the entire length of the interface. This may be interpreted as evidence that the entrainment across the interface varies insignificantly along the length of the interface over this range. We make two inferences from this observation.

Firstly, the insensitivity of bulk entrainment to interface length suggests similar insensitivity of interfacial mixing at the interface over its length. The excitation of the interface by turbulence is expected to enhance or initiate interfacial mixing and entrainment in combination with local shear and wave activity in the cavity. Turbulence in the mixed layer is both transported from the upstream channel flow and generated locally in the cavity section by shear or buoyancy production. We may infer that these mechanisms are insensitive to cavity length or distance from the upstream channel.

Secondly, Strang & Fernando (Reference Strang and Fernando2001) demonstrated a clear relationship between entrainment and interface thickness. Our observations that entrainment is insensitive to length imply that in our experiments the interface properties are varying insignificantly over this length. As interface thickness develops in the streamwise direction, starting at the upstream slope with close to zero thickness, this suggests that the interface is very thin over the entire length of the cavity.

This observation is qualitatively confirmed by dye visualisation at the interface and by LIF measurements in § 4. Both the sharp nature of the interface and the $n=-1.38$ exponent on the entrainment rate suggest that the flow is behaving similarly to the $Ri_{b}>5$ regime of Strang & Fernando (Reference Strang and Fernando2001), where their interface is characterised by Holmboe waves. In our setting, however, this regime appears to extend over our entire experimental range, from $Ri_{b}=1$ to 19.

Finally we also note that while the density of the mixed layer increases slightly in the direction of flow, as it receives entrained fluid from the cavity, this variation is negligible, as the low entrainment rate results in a very small dilution of the mixed layer over $L/D=14$ .

4.1 Characterisation of interface $\unicode[STIX]{x0394}U$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ and $L_{21}$

Initially we use the local PIV/LIF measurements to determine how the bulk parameters characterise the flow near the interface. This is in part motivated by the different $Ri_{b}$ ranges reported in Strang & Fernando (Reference Strang and Fernando2001) for the Holmboe wave regime and also their higher constant $C\simeq 0.02$ shown in figure 3. The time-averaged density and velocity profiles are shown in figure 7(a) at $Ri_{B}=3.3$ , with the vertical location plotted relative to the mean interface depth $Z_{i}$ and normalised by the interface thickness $\unicode[STIX]{x1D6FF}_{b}$ , defined in (4.4). In § 4.2 we later show conditionally averaged statistics of the same quantities.

The mean density profile clearly illustrates the two-layer structure, defined by the upper turbulent ‘mixed layer’, a thin interface region and a lower dense layer. Density stratification within the mixed layer and lower layer is very low, so normalisation of the density field by $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}_{0}$ is reasonable. The density fluctuations in the mixed and lower layers are very low outside the narrow interface region, as shown in figure 7(b). The mean velocity decreases rapidly below the density interface, and within the lower layer is close to zero. This structure persisted at all the interface depths examined. The normalisation of the velocity field by the bulk velocity $U_{b}$ , as shown in figure 7(a), gives a velocity rise of $\unicode[STIX]{x0394}U/U_{b}\sim 1$ across the shear layer. Both the bulk scales $U_{b}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}_{0}$ used in the bulk Richardson number are well defined to characterise the flow, so the differences between our results and those of Strang & Fernando (Reference Strang and Fernando2001) for the mapping of the entrainment regime with $Ri_{b}$ are not the result of a poor choice in outer scales. At this point we also note that the streamwise turbulence intensity (given in figure 7 b) in the upper layer $u_{rms}/U_{b}\simeq 0.14$ is similar to the values present in Strang & Fernando (Reference Strang and Fernando2001). We also estimated the integral length scale of turbulence in the mixed layer using

(4.2) $$\begin{eqnarray}L_{21}(z)=\frac{1}{R_{21}(0,z,0)}\int _{0}^{X}R_{21}(x,z,r)\,\text{d}r.\end{eqnarray}$$

Here the cross-correlation coefficient $R_{21}(x,z,r)=\overline{w^{\prime }(x,z)w^{\prime }(x+r,z)}$ , where $w^{\prime }$ is the fluctuation in the vertical velocity component, $x$ is the distance in the streamwise direction, $X$ is the width of the PIV velocity field data, and the overline denotes an average over the image sequence. In this range the integral length scale as defined, normalised by mixed layer depth $L_{21}/D\simeq 0.2$ , is also similar to the value of $L_{11}/D\simeq 0.18$ obtained by Strang & Fernando (Reference Strang and Fernando2001). The Reynolds number ranges of the two studies are also of the same order $Re\sim O(10^{4})$ .

Figure 7.

Mean streamwise velocity profile, density (a) and r.m.s. density fluctuation $(\unicode[STIX]{x1D719}^{\prime }\unicode[STIX]{x1D719}^{\prime })^{1/2}/(\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}_{0})$ and turbulence intensity (b) at $Ri_{b}=3.3$ , $Ri_{g}=19$ , averaged over $\unicode[STIX]{x1D70F}=224~\text{s}$ .

4.2 Characterisation of interface $\unicode[STIX]{x1D6FF}_{b}$ and $\unicode[STIX]{x1D6FF}_{s}$

The instantaneous interface thickness properties were interrogated using a conditional average of the flow field which has been shifted relative to the local interface height. The instantaneous interface location vector $[x_{i},z_{i}]$ was obtained using an interface tracking tool that traces the contour of $0.5(\unicode[STIX]{x1D719}_{1}-\unicode[STIX]{x1D719}_{0})$ across the image, where $\unicode[STIX]{x1D719}_{1}$ is the scalar concentration of the mixed layer and $\unicode[STIX]{x1D719}_{0}$ that in the lower layer. The interface is tracked such that there is one contiguous interface between the two layers. This interface does not include secondary interfaces formed where wholly detached regions of dense flow are entrained into the overflow or light fluid is engulfed into the lower dense layer by overturns. In the automated procedure the interface is located on one side of the image and tracked across the image until the opposing edge is reached. The interface coordinates are stored as a vector $\boldsymbol{C}_{i}=[x_{i},z_{i}]$ . Where there is an overturn in the interface, multiple values of $x_{i}$ will exist in the $\boldsymbol{C}_{i}$ for that image.

Figure 8.

Conditionally averaged density (a) and velocity (b), PDF of the interface location $z_{i}$ (thin solid line) with a Gaussian fit (thick solid line) (c), and the $Ri_{g}$ profile (d). Data are presented for a single image sequence with $Ri_{b}=3.3$ , $Ri_{g}=19$ and $\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{b}=10.7$ .

We obtained $\unicode[STIX]{x1D6FF}_{s}$ and $\unicode[STIX]{x1D6FF}_{b}$ by conditionally averaging the velocity and density fields across all images within an image sequence. This was achieved by taking each interface coordinate in the vector $\boldsymbol{C}_{i}$ and shifting the flow field at that $x_{i}$ location vertically so that $z$ is centred on $Z_{i}$ . The profiles are then averaged and plotted against displacement from the interface $z-Z_{i}$ . Conditional averaging across all images within an image sequence is indicated by $\langle \cdot \rangle$ . In this way velocity and density profiles were obtained relative to the local interface height at each location along the interface. These profiles are given in figures 8(a) and 8(b). We define the shear layer thickness as

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{s}=\frac{\mathscr{D}[\langle U\rangle ]}{\displaystyle \left.\frac{\text{d}\langle U\rangle }{\text{d}z}\right|_{MAX}},\end{eqnarray}$$

where $\text{d}\langle U\rangle /\text{d}z|_{MAX}$ is the peak velocity gradient in the interface region. The velocity jump across the interface $\mathscr{D}[\langle U\rangle ]$ is the velocity difference between the upper and lower extents of the shear layer. These are obtained from the intersection of three linear fits to the velocity profile, as indicated in figure 8(b). The upper limit of the shear layer is located from the intersection of a linear fit to the velocity profile at the centre of the shear layer $z_{s}$ , where $z_{s}$ is the location of $\text{d}\langle U\rangle /\text{d}z|_{MAX}$ , with a linear fit to the velocity profile in the upper layer. The lower limit of the shear layer is obtained from the intersection of the linear fit through the centre of the interface and $\langle U\rangle =0$ . The mixing layer thickness is similarly obtained by

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{b}=\frac{\mathscr{D}[\langle B\rangle ]}{\displaystyle \left.\frac{\text{d}\langle B\rangle }{\text{d}z}\right|_{MAX}},\end{eqnarray}$$

where $\mathscr{D}[\langle B\rangle ]$ is the difference in buoyancy in the upper and lower layers measured at the upper and lower extents of the flow field. With these definitions we obtain $\unicode[STIX]{x1D6FF}_{s}=8.4~\text{mm}$ and $\unicode[STIX]{x1D6FF}_{b}=0.78~\text{mm}$ for the data sequence given in figures 8(a) and 8(b). Across this experimental range from $Ri_{b}=2$ to 10 we find relatively small variation in interface thickness with ranges of $\unicode[STIX]{x1D6FF}_{b}=0.76~\text{mm}$ to $\unicode[STIX]{x1D6FF}_{b}=1.5~\text{mm}$ and with $\unicode[STIX]{x1D6FF}_{s}=8.2~\text{mm}$ to $\unicode[STIX]{x1D6FF}_{s}=12.8~\text{mm}$ . The centre of the shear layer and density interface are nearly coincident, with the offset distance being $z_{s}-Z_{i}=\unicode[STIX]{x1D6FF}_{d}=0.04~\text{mm}$ for this data set, which is within the resolution of our PIV measurements. Across all results we find the offset varies between $\unicode[STIX]{x1D6FF}_{d}/\unicode[STIX]{x1D6FF}_{s}\simeq 0.01$ and $-0.05$ . The data shown in figure 9 indicate a similar trend to those shown in Strang & Fernando (Reference Strang and Fernando2001), although extended to higher $Ri_{g}$ values in our study.

Figure 9.

Ratio of $\unicode[STIX]{x1D6FF}_{d}/\unicode[STIX]{x1D6FF}_{s}$ versus the gradient Richardson number at the centre of the interface $Ri_{g}|_{z_{i}}$ .

Figure 10.

Ratio of length scales $\unicode[STIX]{x1D6FF}_{s}/D$ (●) and $\unicode[STIX]{x1D6FF}_{b}/\unicode[STIX]{x1D6FF}_{s}$ ( $+$ ) with $Ri_{g}|_{z_{i}}$ .

The PDF (probability density function) has been obtained from the concatenated $z_{i}$ location data, obtained from each image within an image sequence. The PDF given in figure 8(c) for $Ri_{b}=3.3$ shows the function is well approximated by a Gaussian profile, $p(z)=ae^{(z-Z_{i})/2c^{2}}$ , where $Z_{i}$ is the mean interface height and $c$ the standard deviation of $z_{i}$ . Using this result we can form a length scale $\unicode[STIX]{x1D6FF}_{w}=6c$ , which is an indicator of maximum interface oscillation amplitude. For the $Ri_{b}=3.3$ result, shown in figure 8(d), $\unicode[STIX]{x1D6FF}_{w}\simeq 20~\text{mm}$ , which is much larger than either the shear layer thickness $\unicode[STIX]{x1D6FF}_{s}$ or interface thickness $\unicode[STIX]{x1D6FF}_{b}$ . The interface oscillation amplitude $\unicode[STIX]{x1D6FF}_{w}$ decreases with increasing $Ri_{b}$ , as shown in figure 11. In the same figure $\unicode[STIX]{x1D6FF}_{s}$ is shown to be relatively constant and at $Ri_{b}\simeq 6$ the shear layer thickness and the wave amplitude are of similar magnitude, $\unicode[STIX]{x1D6FF}_{s}\simeq \unicode[STIX]{x1D6FF}_{w}$ . In turbulent shear free entrainment flows, the interface wave peak height has been shown to be related to the kinetic energy of the impinging eddies (Xuequan & Hopfinger Reference Xuequan and Hopfinger1986). In the present context, however, we are unable to see any consistent relationship between $\unicode[STIX]{x1D6FF}_{w}$ and $u_{rms}^{2}/\unicode[STIX]{x0394}b$ . Furthermore we do not see any correlation between $Ri_{b}$ and a Richardson number based on mixed layer turbulence scales $Ri_{L}=\unicode[STIX]{x0394}bL_{12}/u_{rms}^{2}$ in these results.

Figure 11.

Length scales $\unicode[STIX]{x1D6FF}_{w}/D$ (●) and $\unicode[STIX]{x1D6FF}_{s}/D$ ( $+$ ) with $Ri_{b}$ .

4.3 Entrainment with $Ri_{g}$

The average gradient Richardson number is obtained as

(4.5) $$\begin{eqnarray}Ri_{g}=\frac{\displaystyle \frac{g}{\unicode[STIX]{x1D70C}_{0}}\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}z}\right\rangle }{\displaystyle \left\langle \left(\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}z}\right)^{2}\right\rangle }.\end{eqnarray}$$

The vertical profile of $Ri_{g}$ given in figure 8(d) indicates the maximum value of $Ri_{g}$ is located on the interface at height $z_{i}$ , with rapid reduction over ${\sim}10\unicode[STIX]{x1D6FF}_{b}$ . This profile is characteristic of the Holmboe mode (e.g. Smyth & Winters Reference Smyth and Winters2003). We denote the gradient Richardson number at the centre of the interface $Ri_{g}|_{z_{i}}$ . An interesting dynamic in the present results is that the ratio of the shear layer thickness and the mixing layer thickness is nearly constant, with $\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{b}\simeq 10$ , as shown in figure 10 over $Ri_{b}=2$ to 10. Hogg & Ivey (Reference Hogg and Ivey2003) obtained the scaling $\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{b}\sim Sc^{1/3}$ by considering the contribution of molecular diffusion to the growth of the interface thickness and shear layer thickness over a fixed development length. The authors found good agreement with this scaling using laboratory measurements of a stratified exchange flow. The comparison is also close for our flow, where this scaling predicts a ratio of $\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{b}\simeq 9$ .

Figure 10 also suggests that $\unicode[STIX]{x1D6FF}_{s}/D$ is constant for the present results, as seen in other studies (Narimousa & Fernando Reference Narimousa and Fernando1987; Strang & Fernando Reference Strang and Fernando2001). These quantities are related to $Ri_{g}|_{z_{i}}$ as

(4.6) $$\begin{eqnarray}Ri_{g}|_{z_{i}}\sim \frac{\unicode[STIX]{x0394}b/\unicode[STIX]{x1D6FF}_{b}}{(U_{b}/\unicode[STIX]{x1D6FF}_{s})^{2}},\end{eqnarray}$$

which implies $Ri_{g}|_{z_{i}}\sim Ri_{b}$ . This relationship is observed in figure 12 with a regression fit of $Ri_{g}|_{z_{i}}=5.3Ri_{b}^{1.06}$ . In the same figure the results from Strang & Fernando (Reference Strang and Fernando2001) are presented together with a regression line of $Ri_{g}=0.038Ri_{b}^{2}$ . Across the entire experimental range our interface remains sharper with a high gradient Richardson number. The significant variation between the studies is notable; however, exact comparison is difficult, as our results are conditionally averaged about the interface to remove effects of large-scale interfacial waves.

Figure 12.

Gradient Richardson number at the centre of the interface $Ri_{g}|_{z_{i}}$ versus $Ri_{b}$ ( $+$ ) presented with the measurements of Strang & Fernando (Reference Strang and Fernando2001) (●) (taken from their figure 5). Regression lines of $Ri_{g}|_{z_{i}}=mRi_{b}^{n}$ are given with $m=5.3$ and $n=1.06$ for our data (solid line) and $m=0.038$ , $n=2.0$ for the data of Strang & Fernando (Reference Strang and Fernando2001) (dashed line).

With a shear Richardson number $Ri_{s}=\unicode[STIX]{x0394}b\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x0394}U^{2}>0.071$ , $\unicode[STIX]{x1D6FF}_{d}/d=0$ and $\unicode[STIX]{x1D6FF}_{b}/\unicode[STIX]{x1D6FF}_{s}\rightarrow \infty$ , Holmboe (Reference Holmboe1962) showed using inviscid linear stability analysis that the flow was not unstable to K–H waves but that another mode of instability, the Holmboe wave, was amplified. This mode has since been seen experimentally and numerically (Smyth, Klaassen & Peltier Reference Smyth, Klaassen and Peltier1988; Carpenter et al. Reference Carpenter, Lawrence and Smyth2007; Tedford et al. Reference Tedford, Pieters and Lawrence2009; Carpenter et al. Reference Carpenter, Tedford, Rahmani and Lawrence2010; Salehipour et al. Reference Salehipour, Caulfield and Peltier2016). In the present study, $\unicode[STIX]{x0394}U\simeq U_{b}$ and $\unicode[STIX]{x1D6FF}_{s}\simeq 0.07D$ , so we obtain $Ri_{s}\simeq 0.07Ri_{b}$ . Over our experimental range of $1<Ri_{b}<19$ the flow is above the $Ri_{s}=0.07$ limit for K–H waves. The infrequent overturns and lack of clear K–H billows in our flow visualisation is then consistent with stability analysis, not withstanding the highly perturbed and turbulent nature of the cavity flow and variation in velocity and density profiles. The extent of this regime for $Ri_{b}<5$ is in contrast to the studies of Narimousa & Fernando (Reference Narimousa and Fernando1987) and Strang & Fernando (Reference Strang and Fernando2001). We have shown that the bulk parameters $U_{b}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}_{0}$ correctly characterise the problem and that turbulence in the mixed layer is similar, i.e. the magnitude of the turbulence intensity and the integral length scale are comparable.

One difference is the configuration, with the cavity setting being a spatially developing flow where the mixing layer has a finite growth time, leading to a flow where $\unicode[STIX]{x1D6FF}_{b}$ remains thin and is not correlated with the bulk parameters $D$ in the same way as a more developed flow such as is achieved in the re-circulating water channel results obtained by Kantha et al. (Reference Kantha, Phillips and Azad1977), Narimousa & Fernando (Reference Narimousa and Fernando1987) and Strang & Fernando (Reference Strang and Fernando2001). The same behaviour was shown by Salehipour et al. (Reference Salehipour, Caulfield and Peltier2016) using direct numerical simulations. The authors simulated two flows with an initial $Ri_{b}=0.16$ but with differing $\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{s}$ . The flow with $Ri_{g}=0.16$ developed into K–H waves while the $Ri_{g}=0.16\sqrt{8}$ simulation developed into Holmboe waves.

Both our measurements and those of Strang & Fernando (Reference Strang and Fernando2001) (at $Ri_{b}>5$ ) indicate entrainment is via Holmboe waves, the occurrence and development of which is related to the stability properties of the interface, as indicated by the gradient Richardson number. In figure 13 we plot our data with $Ri_{g}|_{z_{i}}$ . Using the empirical fit to the data of Strang & Fernando (Reference Strang and Fernando2001) $Ri_{g}=0.038Ri_{b}^{2.0}$ and their result of $E=0.0023Ri_{b}^{-1.3}$ we obtain $E\simeq 0.0024Ri_{g}|_{z_{i}}^{-0.65}$ . The comparison of this inferred result is close to the regression fit to our data of $E=0.0021Ri_{g}|_{z_{i}}^{-0.63}$ . We note that Strang & Fernando (Reference Strang and Fernando2001) report measurements over $Ri_{g}<10$ whereas our results extend to $Ri_{g}|_{z_{i}}\simeq 50$ and lower values of $E$ , but the correlation for both sets of data appears consistent.

The cavity flow considered here then has the same entrainment characteristics as seen in the temporally developing water channel flow when compared using local characteristics, which indicate the state of development of the flow more accurately than the bulk quantities.

Figure 13.

Entrainment with gradient Richardson number at the centre of the interface $Ri_{g}|_{z_{i}}$ . A regression fit of $E=a(Ri_{g}|_{z_{i}})^{b}$ with $a=0.0021$ and $b=-0.63$ is given (solid line) together with $a=0.0024$ and $b=-0.65$ (dashed line) derived from the data of Strang & Fernando (Reference Strang and Fernando2001).

4.4 Higher Reynolds number and Richardson number

Finally we consider the possible extension of the entrainment regime we have observed to higher Richardson number. With increasing interface stability, entrainment by molecular diffusion increasingly dominates total mass transport over shear-induced turbulent interfacial mixing (Narimousa et al. Reference Narimousa, Long and Kitaigorodskii1986; Narimousa & Fernando Reference Narimousa and Fernando1987). We can quantify the molecular entrainment present in our experiments by inspection of the density gradients. The molecular flux of salinity through the interface is given by $q_{i}=\unicode[STIX]{x1D705}\text{d}\unicode[STIX]{x1D70C}/\text{d}z=(\unicode[STIX]{x1D708}/Sc)(\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D6FF}_{b})~\text{kg}~\text{m}^{-2}~\text{s}^{-1}$ . For the experiments performed, $Sc\simeq 700$ and $\unicode[STIX]{x1D708}\simeq 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ . The flux can be written in terms of an entrainment coefficient as $q_{i}=u_{e}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=E_{m}U_{b}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}_{0}$ . Combining the relationships we obtain

(4.7) $$\begin{eqnarray}E_{m}=\frac{1}{ScRe_{\unicode[STIX]{x1D6FF}}},\end{eqnarray}$$

where $Re_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x1D6FF}_{b}U_{b}/\unicode[STIX]{x1D708}$ . For our $Re=7100$ experiments we obtain $Re_{\unicode[STIX]{x1D6FF}}\simeq 70$ and $E_{m}=2\times 10^{-5}$ . The lowest entrainment rate measured in our $Re=7100$ experiments was $E=10^{-4}$ at $Ri_{b}=10$ . The ratio of total entrainment $E$ to entrainment resulting from molecular diffusion $E/E_{m}=5$ indicates that our measurements approach a regime where the total entrainment rate is comparable to molecular diffusion.

Salehipour et al. (Reference Salehipour, Caulfield and Peltier2016) performed direct numerical simulations of the development of the Holmboe wave instability in a periodic domain and identified substantial variation in mixing behaviour with Reynolds number over $Re_{\unicode[STIX]{x1D6FF}}=200$ to 12 000. At higher Reynolds number the authors observed the formation of secondary instabilities at the interface, the growth of which is Reynolds-number-dependent up until a critical Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}\simeq 8000$ . At high $Re$ the authors further observed vigorous long-lived turbulence at the interface which acts to both scour and transport fluid away from the interface and irreversibly mix the fluid. At lower $Re$ , mixing was shown to scale with the molecular diffusion rate, and weak ‘wisps’ of fluid on either side of the interface were observed, similar to the flow seen in the present study, as shown in figure 6. In our experiments, there is additional forcing of the interface by turbulence in the overflow, and there is a larger range of wavelengths present in our channel. Neither of these features appear to change this basic low- $Re$ mixing characteristic of the flow. Despite the relatively large Reynolds number in the overflow, up to $Re=15\,000$ in this study, the shear at the interface remains sufficiently weak that Reynolds numbers at the interface are in the weakly turbulent regime identified by Salehipour et al. (Reference Salehipour, Caulfield and Peltier2016). The extension of this work to environmental scale occurrence of this flow in river systems where $Re>10^{5}$ , $Sc=700$ and $Ri_{b}=10$ to 150 remains a challenge.