2 Theoretical analysis

We start from the general governing equation and then proceed with the depth-integrated equations, following closely the procedure given in Turner (Reference Turner1973). The flow is considered to be two-dimensional. Using the boundary layer and Boussinesq approximations, the momentum and continuity equations in cylindrical coordinates are

where $(u,w)$ are the velocity components opposite to the $\unicode[STIX]{x1D703}$ and $r$ directions respectively (see figure 1), $p$ is the pressure, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{0}$ , with $\unicode[STIX]{x1D70C}_{0}$ the density of the ambient fluid, and $\unicode[STIX]{x1D70F}$ is the wall shear stress. (For geometric reasons, we preferred to define $\unicode[STIX]{x1D703}$ and $r$ in directions opposite to the flow direction (figure 1) and then change to variables $(x,z)$ .) Assuming a stationary flow, using the boundary layer approximation, for which $w\sim u(h/R)$ , with $h$ being the current height, and using the continuity equation, the momentum equations reduce to where $g^{\prime }=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}$ and $r$ has been approximated by $r=(R-z)=R(1-z/R)\approx R$ assuming a small curvature. We now change variables to $x=R(\unicode[STIX]{x1D703}_{0}-\unicode[STIX]{x1D703})$ and $z=(R-r)$ , and integrate (2.2b ) in the $z$ direction to get the expression for the pressure,

Substitution in (2.2a ) yields

To further simplify the equations, we make use of the similarity hypothesis,

where $z^{\star }=z/h$ and $\int _{0}^{\infty }f(z^{\star })\,\text{d}z^{\star }=1$ . The similarity assumption implies that the velocity and density profiles do not depend on $x$ . This is, of course, not strictly true, as indicated by the shape factors, which vary somewhat, but we take them as constant. The depth-integrated quantities are defined as follows:

where $S_{1}$ , $S_{2}$ , $S_{3}$ are factors related to the shape of the velocity and density profiles. For currents on slopes, Ellison & Turner (Reference Ellison and Turner1959) propose $0.2$ – $0.3$ for $S_{1}$ and $0.6$ – $0.9$ for $S_{2}$ . In the current experiments, the shape factors $S_{1}$ , $S_{2}$ and $S_{3}$ were determined by integrating the velocity and density profiles at different $x$ positions in the $z$ direction according to (2.7b ). The values vary from $S_{1}=0.2$ to $0.4$ $(S_{1}=0.3\pm 0.1)$ , $S_{2}=0.6$ to $0.9$ $(S_{2}=0.75\pm 0.15)$ and $S_{3}=0.4$ to $0.6$ $(S_{3}=0.5\pm 0.1)$ . Substituting in (2.4), the depth-integrated equation is

where $c_{D}=\unicode[STIX]{x1D70F}_{0}/(\unicode[STIX]{x1D70C}U^{2})$ is the bottom friction coefficient. In order to normalize the velocity with initial conditions, we consider the buoyancy flux $B=g^{\prime }hU$ per unit width, which is constant and equal to the buoyancy flux supplied at the gate, $B_{0}=g_{0}^{\prime }h_{0}U_{0}$ , where $U_{0}$ and $h_{0}$ are respectively the velocity and height at the gate. Equation (2.8) can then be rearranged as

where $\text{d}(Uh)/\text{d}x=EU$ , with $E$ the entrainment coefficient, and where the term with $S_{3}$ has been neglected because the ratio between the second and the first terms in parentheses in (2.8) is $(S_{3}/S_{1})(1/Ri)(h/R)\approx 0.1$ . This way of normalizing the velocity, $U^{3}/B$ , also enables us to isolate the contribution of each term, i.e. bottom friction, entrainment, acceleration and variation in the Richardson number. Here, $Ri=\bar{g}^{\prime }h\cos \unicode[STIX]{x1D703}/U^{2}=B\cos \unicode[STIX]{x1D703}/U^{3}$ is the bulk Richardson number, with $\bar{g}^{\prime }\cos \unicode[STIX]{x1D703}$ the gravitational reduced acceleration.

The first two terms in the denominator of (2.9) represent the total friction due to bottom drag ( $c_{D}$ ) and interfacial entrainment ( $E$ ), while the last term in the denominator represents the pressure variation along the slope. The third term represents the acceleration along the slope. Herein, we define an acceleration parameter $T_{a}=(h/U)\,\text{d}U/\text{d}x$ , which gives the ratio between an acceleration time scale $\unicode[STIX]{x1D6FF}x/\unicode[STIX]{x1D6FF}U$ and an overturning time scale proportional to $h/U$ when the flow becomes unstable at the interface. This term along with the term of variation in $Ri$ (which represents pressure variations) along the slope can be either positive or negative. On the horizontal wall, $\unicode[STIX]{x1D703}=0$ and $E=\text{d}h/\text{d}x+T_{a}=0$ , so that (2.9) reduces to the known equation (Turner Reference Turner1973)

where $F^{2}=U^{2}/\bar{g}^{\prime }h<1$ is the squared Froude number. On the horizontal boundary, the density profile remains of top-hat shape and for this reason $S_{1}$ and $S_{2}$ are close to $1$ . The interface slope is negative and from the small-angle approximation of (2.9), the slope angle when $F=1$ is $c_{D}$ , taking $S_{2}=1$ . For the conditions of the experiments, $U_{0}\approx 4~\text{cm}~\text{s}^{-1}$ , $X=D=200$  cm, where $D$ is the length of the horizontal boundary, the value of $c_{D}$ on the horizontal wall is assumed to be of the order of ${\approx}10^{-3}$ . The squared Froude number downstream of the gate is $F^{2}=0.59$ , resulting in a mean interface slope of $\text{d}h/\text{d}x\approx -10^{-3}$ .

A developed gravity current on constant slope moves at constant velocity (Ellison & Turner Reference Ellison and Turner1959; Britter & Linden Reference Britter and Linden1980). In this case, equation (2.9) reduces to

where on a slope $(S_{1}/2)Ri\ll 1$ because $Ri=O(10^{-1})$ . The entrainment coefficient is now a function of slope angle only (Turner Reference Turner1973) and can be approximated by (Hopfinger Reference Hopfinger1983)

and when the slope angle $\unicode[STIX]{x1D703}>5^{\circ }$ , bottom friction can be neglected. Thus, since $S_{2}\sin \unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}$ varies little in the range $10^{\circ }<\unicode[STIX]{x1D703}<50^{\circ }$ , the velocity is nearly the same on slopes in this range (Hopfinger Reference Hopfinger1983). This would suggest that on a varying slope boundary, the velocity might remain constant provided that the changing buoyancy force is balanced by entrainment.

Numerical integration of the full equation (2.8) would be of interest, as it would allow a parametric study of the influence of the different parameters. When assuming $c_{D}=\text{const}.$ (which is not at all the case here, e.g. figure 6 f), there are three variables, namely $U$ , $h$ and $g^{\prime }$ , and there are in addition to (2.8) two more equations, i.e. $B_{0}=g^{\prime }Uh$ and $EU=\text{d}/\text{d}x(Uh)$ . Taking an empirical expression for $E(Ri)$ , as for instance in Parker et al. (Reference Parker, Fukushima and Pantin1986) or Turner (Reference Turner1986), integration would be possible. However, attempts to integrate the equation were unsuccessful because of the unexpected rapid variations of the parameters with $x$ , including $c_{D}$ . Therefore, this task would require an extensive parametric study.

3 Experimental apparatus

The experiments were conducted in an open glass-walled tank approximately 6 m long connecting two 800 l reservoirs, as shown in figure 1. The channel cross-section was reduced to 25 cm wide and 20 cm deep. The first section of the channel had a total length of roughly 2.3 m (with the gate at 2 m upstream), was horizontal and permitted the boundary layer of the gravity flow first to adjust and then to fully develop (Pope Reference Pope2000). The next section was a concave or linear slope and was free to pivot about $O$ (see figure 1), enabling different starting angles $\unicode[STIX]{x1D703}_{0}$ for the same radius of curvature $R=2$  m. The length $S=R\unicode[STIX]{x1D703}_{0}$ was varied in the experiments over two concave slopes of 60 cm and 110 cm, resulting in $\unicode[STIX]{x1D703}_{0}=17^{\circ }$ and $31.5^{\circ }$ respectively. On the concave wall, the slope angle $\unicode[STIX]{x1D703}$ varied from the maximum $\unicode[STIX]{x1D703}_{0}$ at the beginning of the slope to $0$ at $S$ (see figure 1). Additionally, one experiment over a linear slope with $\unicode[STIX]{x1D703}=22^{\circ }$ and a length of $S=73$  cm was performed. For each experiment C3, C1 and L2, at least three runs were conducted. Each run was analysed, and variations between them in the primary averaged quantities were less than 10 %, so that only results relative to one run for each experiment will be shown in this paper. The experimental conditions and measurement method are summarized in table 1. The $(X,Z)$ coordinate system represents horizontal and vertical directions, whereas $(x,z)$ represents the coordinate system relative to the inclined boundary, with the origin at the ridge (cf. figure 1).

Table 1.

Summary of the experimental parameters ( $D\equiv$ length of horizontal channel upstream, C1 and C3 indicate concave slopes, L2 indicates a linear slope, $S\equiv$ slope length). The flow rate $0.6~\text{l}~\text{s}^{-1}$ and the total water depth in the channel $H_{0}=19$  cm were kept equal in all of the experiments. The overall and integral scale Reynolds numbers are $Re=3500$ and $Re_{{\mathcal{L}}}=66$ respectively.

To favour a uniform inflow, an inlet pipe facing upwards was installed at the bottom of the upstream section of the channel, and a plastic grid was positioned at the inlet to reduce interface fluctuations. The gravity current was generated by pumping salt water at a given density ( $g^{\prime }=4.5~\text{cm}~\text{s}^{-2}$ ) and flow rate ( $Q=0.6~\text{l}~\text{s}^{-1}$ ) from the first reservoir equipped with an electric agitator to maintain a uniform salt concentration. An adjustable pump with a maximum capacity of $2.5~\text{l}~\text{s}^{-1}$ was used to pump the salt water to the channel inlet via a rotameter so that flow rates could be recorded. The outlet was placed at the bottom of the downstream channel end to control the total water depth and allow discharge of the lower-layer salty flow.

Polyamide particles (Orgasol) with a mean diameter of $60~\unicode[STIX]{x03BC}\text{m}$ and a density of $1.016~\text{g}~\text{cm}^{-3}$ were added to both the salt and the fresh water as tracers for velocity measurements with particle image velocimetry (PIV). A 6 W argon-ion laser (Coherent) operating in multimode ( $\unicode[STIX]{x1D706}_{1}=488$  nm, $\unicode[STIX]{x1D706}_{2}=514$  nm) was used as a continuous light source. The beam was transmitted through a fibre optic cable to a line generator with spherical lenses (OZ Optics). The generated laser sheet spanned a length of approximately 1 m, was of 5 mm thickness and was positioned in the middle of the channel. Images of 70 cm  $\times$  50 cm were grabbed with a CCD camera (FlowMaster 3, 14 bit, $1400\times 1600$ pixels) at the maximal allowed frame rate of 23.84 Hz. With the use of an optical lens on the CCD camera, the spatial resolution was $0.435~\text{mm}~\text{pixel}^{-1}$ . With the software package DaVis (LaVision), the velocity fields were computed using a cross-correlation PIV algorithm. For this purpose, an adaptive multipass routine was used, starting with an interrogation window of $32\times 32$  pixels and a final window size of $16\times 16$  pixels, with $50\,\%$ overlap. Each vector of the resulting vector field represented an area of roughly $0.35~\text{cm}\times 0.35~\text{cm}$ . The velocity vectors were postprocessed using a local median filter. Given the velocities encountered in the experiments, the experimental error in the instantaneous velocity was estimated to be approximately $3\,\%$ and in $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ approximately $10\,\%$ .

For dye visualizations, Rhodamine 6G was added to the dense (salt) water of the gravity current (experiments C3 and C1 in table 1). The dye visualizations were used to estimate the density profiles and some averaged values of the density field by normalizing locally with the maximal value.

4 Initial conditions: current on the flat bottom

The current on the horizontal wall gives the initial conditions for the current on the slope (see figure 2 a). Using the experimental conditions as given in table 1, we estimate a Froude number at the gate of $F_{0}^{2}=q^{2}/(g_{0}^{\prime }h_{0}^{3})\approx 0.59$ for a gate opening of $h_{0}\approx 6$  cm and on the horizontal channel of $F^{2}=U/(g_{0}^{\prime }h)\approx 0.53$ using the data of figure 2. These values compare well with those given by Simpson (Reference Simpson1982) for a gravity current flowing along a horizontal surface for a depth ratio $h/H_{0}\approx 0.3$ .

Figure 2.

Initial conditions of the gravity current flowing on the initial horizontal channel. (a) Time-averaged velocity field from PIV measurements (experiment H) displaying the initial conditions of the current. (b) Mean velocity profiles at $X=20$ ( $\cdots \cdots$ ),  $85$ (– $\cdot$ ), 195 (– –) and 199 cm (–) from the gate. (c) Vertical density profile obtained from the dye visualizations at $X=195$  cm from the gate. (d) Vertical profiles of the averaged (over time and all experiments) velocity fluctuations in the streamwise direction $u^{\prime }$ at $X=195$  cm. (e) Ratio between the streamwise buoyant acceleration normalized with the initial value on the initial horizontal channel, showing that it remains constant within $\pm 4\,\%$ fluctuations.

Very close to the slope change at $x=195$  cm downstream of the gate, the velocity and the current depth have reached $u_{m}\approx 6~\text{cm}~\text{s}^{-1}$ and $h\approx 6$  cm respectively, resulting in a bulk Reynolds number based on these scales of $Re=3500$ .

The expected boundary layer thickness at the end of the horizontal wall is $\unicode[STIX]{x1D6FF}_{B\unicode[STIX]{x1D708}}=5DRe_{D}^{-1/2}=3$  cm and the interface thickness is $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}\approx 8\sqrt{\unicode[STIX]{x1D708}t}$ (see Batchelor (Reference Batchelor1967), p. 188). Herein, the time is expressed using the distance $D=200$  cm (see figure 1) and the convective velocity estimated as half of the maximal velocity $U_{m}/2$ . This gives $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}\approx 11.3DRe_{D}^{-1/2}\approx 7.9$  cm. The density interface thickness is $\unicode[STIX]{x1D6FF}_{s}=\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D70C}}Sc^{-1/2}\approx 0.3$  cm, where $Sc$ is the Schmidt number, here close to 700.

Figure 2(b) displays velocity profiles at $X=20$  cm, $85$  cm, and two profiles very close to the ridge, $X=195$  cm and 199 cm from the gate (cf. also figure 2 a). With increasing distance, the shape of the profile approaches that of a wall jet. The boundary layer thickness, defined as the distance from the bottom where the velocity becomes 99 % of the maximal velocity, decreases after the initial adjustment close to the gate to $\unicode[STIX]{x1D6FF}_{B\unicode[STIX]{x1D708}}=2$  cm at 195 cm from the gate. The interface thickness is $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}=7$  cm at 195 cm from the gate. These values are in good agreement with those evaluated above.

The shape of the density profile can be estimated from the dye visualizations with Rhodamine 6G, and is displayed in figure 2(c) at 195 cm from the gate, showing a top-hat shape with a thin interface $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D70C}}=2$  cm. This value is almost one order of magnitude larger than the value estimated from molecular diffusion only ( ${\approx}0.3$ ), but is still thin compared with $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}$ . The observation of Holmboe waves confirms the existence of a locally thin density compared with the shear interface.

We stress the fact that in our experiments, the interfacial Richardson number increases on the horizontal boundary according to $J\propto \unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}$ , and so the initial condition at the ridge is very different from that of previously performed gravity current experiments, as in Pawlak & Armi (Reference Pawlak and Armi2000), with already favourable conditions for the instability to develop due to a very thin $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}$ . Indeed, in our experiments, $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}=10$  cm at $x=10$  cm, compared with the $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}=1{-}2$  cm reported in the study of Pawlak & Armi (Reference Pawlak and Armi2000), so that the interface will become unstable only after a certain distance during the descent on the slope due to the increase of the velocity.

Figure 2(d) shows a vertical profile of the streamwise velocity fluctuations along the vertical direction typical for these experiments. Using $u^{\prime }\approx 0.5~\text{cm}~\text{s}^{-1}$ from the PIV measurements, an estimate for the integral-scale Reynolds number is given by $Re_{{\mathcal{L}}}=u^{\prime 2}\sqrt{15/(\unicode[STIX]{x1D708}\unicode[STIX]{x1D716})}\approx 66$ (the spatial resolution of the PIV of ${\sim}3~\text{mm}$ is very close to the Kolmogorov length scale $\unicode[STIX]{x1D702}_{K}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716})^{1/4}\sim O(1~\text{mm})$ , estimated using $\unicode[STIX]{x1D716}\sim u^{\prime 3}/h\sim 0.01~\text{cm}^{2}~\text{s}^{-3}$ (Tennekes & Lumley Reference Tennekes and Lumley1972)). This Reynolds number is high enough to assure a weakly turbulent flow.

The ratio $\overline{g}^{\prime }/g_{0}^{\prime }=q_{0}/(Uh)$ along the flow direction is plotted in figure 2(e). It remains constant along the horizontal wall within $\pm 4\,\%$ variations, showing that entrainment is negligible there.

5 Currents on the concave bottom

Approximately 50 s after opening the gate, the head of the current reaches the slope. Similarly to a potential flow over a weir, acceleration starts already approximately 10 cm before the ridge, as is evident from the fast increase in velocities over the short distance of 4 cm (195–199 cm from the gate in figure 2 a). Although the slope change is abrupt from horizontal ( $\unicode[STIX]{x1D703}=0$ ) to $\unicode[STIX]{x1D703}_{0}$ , the streamlines do not follow the abrupt change of bottom but are curved like in potential flow.

After an initial unsteady phase with the passage of the gravity current head, a steady two-layer flow establishes with an internal hydraulic control $F=1$ at the ridge (Armi Reference Armi1986) and a supercritical flow ( $F>1$ ) on the slope. Once the current has terminated the descent, it may return again to subcritical conditions, passing through an internal hydraulic jump or spreading out smoothly. On a straight slope that returns abruptly to the horizontal, an internal hydraulic jump is expected, but on the concave slope, a smooth transition seems to take place. This will be investigated elsewhere.

Figure 3.

Instantaneous images from dye visualizations of (a,b) the head descending the slope from experiment C3 with time intervals of $4$ and $6$  s, and of the steady current in experiments C1 (c) and C3 (d) taken 100 s after the gate opening. The vertical scale of the images is approximately 35 cm, while the horizontal scale is approximately 70 cm. As depicted in (d), three zones can be defined: the initial stable interface region with no entrainment, the central region with KH instabilities and the final region of collapse to a turbulent shear layer. The dashed rectangle in (d) highlights the boundary layer separation coupled with a KH billow which will be discussed in §§ 5.2 and 6.

Figure 3 shows dye visualizations of the steady flow for the experiments C1 (c) and C3 (d) with a concave slope. As depicted in figure 3(d), we can distinguish three zones during the steady phase. The first zone (zone I) is characterized by a strong initial acceleration of the flow and a reduction of the layer depth with relatively thin velocity and density shear layers and a stable interface with respect to the KH instability. The intermediate zone (zone II) is characterized by the onset of KH billows due to the increased velocity shear which cause significant entrainment of upper-layer fluid into the lower layer. The final zone (zone III) is characterized by the collapse of the KH billows which generate an intermediate upper layer with an intermediate density. As is evident from figures 3(c) and 3(d), a significant change in the development of the interfacial shear layer occurs during the descent down the slope. In zone III, the stratified shear layer collapses and subsequently restratifies, which is a well-documented process examined by Koop & Browand (Reference Koop and Browand1979) (see also, e.g., Corcos & Sherman (Reference Corcos and Sherman1976)). They observed an initial rapid growth through vortex pairing of the shear layer, and as the Richardson number increased past a critical value, the pairing mechanism was suppressed, the development of the layer ceased and production of turbulence in the shear layer died out, with the exception of turbulence generated by the restratification process. When the slope angle at this location is still high enough, the current accelerates and the Richardson number will drop again to unstable values so that further mixing can occur within the already restratified shear layer.

We observed that during one experiment the limits between these three zones slowly shift upstream. Indeed, with time the density increases in the lower part of the upper layer and so does the return flow caused by the strong dilution in zone III. Kelvin–Helmholtz billows can thus appear further upstream for lower velocities.

Two ranges of acceleration have been considered by varying the initial concave slope angle $\unicode[STIX]{x1D703}_{0}$ , as described in § 3. Time-averaged velocity (a,c) and density (b,d) profiles at four different downstream positions ( $x=3$  cm, $x=22$  cm, $x=35$  cm, $x=57$  cm) are plotted in figure 4 for the experiments C1 (a,b) and C3 (c,d). These sections are characteristic of the three zones I, II and III discussed above, with the latter two profiles being at the beginning and at the end of zone III. We see that initially the velocity increases at the beginning and the profile approaches that of a wall jet, and smooths subsequently in zone III. Density profiles (see figure 4 for experiments C1 (b) and C3 (d)) relative to the maximal value are stepwise at the beginning of the descent and smooth, similarly to the velocity profiles, while moving down the slope.

In the following sections, we will present time-averaged quantities versus the downstream position measured from the ridge. This average is made over 65 s, from $t=20$  s after gate opening, excluding thus the initial passage of the head, until $t=85$  s. Results are presented for the two concave slopes, first for experiment C1 with a smoother transition on the slope and then increasing the initial slope angle as for experiment C3. For comparison, a linear sloping boundary is used in § 6. These quantities are not non-dimensionalized as is customary for shear flows, since there is no single non-dimensional parameter for downstream distance in a spatially accelerating flow.

Figure 4.

Time-averaged velocity (a,c) and density (b,d) profiles at $x=3$  cm,  $x=25$  cm,  $x=43$  cm and $x=57$  cm (from dark to light grey respectively) from the ridge for experiment C1 (a,b) and for experiment C3 (c,d).

6 Comparison with steep straight slopes

Figure 9.

Time-averaged quantities for experiment L2. (a) Variation of the depth-integrated velocity compared with the predictions of (2.11) (— ⋅ —); ( $\cdots \cdots$ ) free fall solution; head velocity ( $+$ ). (b) Richardson numbers $Ri$ ( ▹) and $J$ ( ▸). (c) Current depth $h$ fitted with $\text{d}h/\text{d}x=0.1631$ for $20~\text{cm}<x<55$  cm (–) and the relation of Ellison & Turner (Reference Ellison and Turner1959),  $\text{d}h/\text{d}x=0.025$ (– –). The data for experiment C3 in (a–c) are shown in light grey. (d) Entrainment coefficients $E=-w_{h}/U$ , (– –) $E=9(5+\unicode[STIX]{x1D703})\times 10^{-4}$ . (e) Bottom friction $(c_{D})$ determined from (2.9) using experimental data. (f) Selected vertical velocity profiles at $x=23$  cm (empty symbols) and $x=35$  cm (filled symbols); ( $\cdots \cdots$ ) plane wall jet from Zhou et al. (Reference Zhou, Heine and Wygnanski1996).

Curved slopes start steep and then rapidly decrease in steepness to the horizontal plane. The observations above suggest that the strong acceleration at the ridge leads to the formation of KH billows and a related large increase in the entrainment rate $E$ and drag $c_{D}$ , whereas the downhill deceleration seems to have no influence. In order to confirm that it is indeed only the steep part of the slope that determines the essential part of the downslope dynamics, we consider a gravity current on a steep but straight slope of comparable steepness. For comparison, we superimposed in figure 9(a–c) the velocity, Richardson numbers and current depth variations obtained on straight slopes on those obtained on concave slopes. It is seen that the spatial development of the current on the linear slope is qualitatively similar to that observed on the concave bottom in experiment C3 with an abrupt topography change. As described in § 5, the three zones of first acceleration of the flow (zone I), development of KH billows (zone II) and their subsequent collapse (zone III) are also found here. A close-up view of the onset and development of KH billows (zone II) is shown in figure 8. There is a strong effect of the KH instability on the bottom boundary layer. Moreover, the averaged flow quantities for the straight slope (experiment L2) shown in figure 9 are similar to those for the concave slope (experiment C3), with the head velocity ( $+$ symbols in figure 9 a) increasing steadily until they reach roughly the constant equilibrium-state velocity and then dropping off when the horizontal bottom is reached. Detailed comparison shows that the depth-integrated velocity (figure 9 a) increases rapidly up to a maximal velocity of $9~\text{cm}~\text{s}^{-1}$ at $x\approx 23$  cm, which is close to the values of $8.5~\text{cm}~\text{s}^{-1}$ at $x\approx 17.5$  cm of experiment C3. Then, the current decelerates and reaccelerates again, finally reaching the horizontal bottom at $x=73$  cm, where $U\approx 8~\text{cm}~\text{s}^{-1}$ . The scatter in the velocity data of experiment L2 for $x>60$  cm is due to the sudden passage to the horizontal boundary, which causes significant velocity fluctuations throughout the density current layer. The velocity is always much less than the predicted equilibrium state velocity of $11.2~\text{cm}~\text{s}^{-1}$ (dotted line in figure 9 a).

The overall Richardson number $Ri$ and the overall interfacial Richardson number $J$ displayed in figure 9(b) (empty and filled symbols respectively) decrease strongly with increasing $x$ in the initial accelerating region and drop to values of $0.13$ and $0.1$ respectively, corresponding to the appearance of KH billows. These values are in the same range as on steep curved slopes (figure 6 b). After a readjustment, $Ri$ drops again to an unstable regime just before reaching the horizontal bottom, while $J$ settles at a constant value of roughly $0.25$ .

Figure 9(c) displays the variation of the current depth in the downstream direction. As for the concave slope, $h$ decreases until $x\approx 25$  cm, then increases as the current decelerates and subsequently increases slowly for $x>50$  cm. For 25 cm  ${<}x<50$  cm, the data have been fitted with a linear relation for which $\text{d}h/\text{d}x=0.163$ , while for $x>50$  cm, the relation of Ellison & Turner (Reference Ellison and Turner1959) predicts $\text{d}h/\text{d}x=0.025$ (dashed line in figure 9 d).

The variation of the entrainment rate $E=-w_{h}/U$ with the downstream position $x$ is plotted in figure 9(d). We see that the entrainment rate reaches a maximal value of approximately $E\approx 0.08$ during the first deceleration phase, corresponding to the generation and collapse of the KH billows (see also figure 9), which is very close to the value of the current on the concave slope (experiment C3, $E_{max}=0.07$ ). For comparison, on this slope, the constant velocity value is $\text{d}h/\text{d}x=0.025$ . The entrainment drops quickly for $60~\text{cm}<x<80$  cm to nearly zero when the horizontal bottom is reached. This value is close to the value of $\text{d}h/\text{d}x=0.025$ predicted by Ellison & Turner (Reference Ellison and Turner1959) for $50~\text{cm}<x<73$  cm (cf. figure 9 c).

Finally, we observe the same behaviour also for the bottom friction coefficient, which presents similar very high values to those reported in the experiments with a concave topography, of the order of $10^{-1}$ (figure 9 e). We observe similar high values of the Reynolds stresses in those regions (8 % of $U_{m}^{2}$ for experiment L2 compared with 5 % for experiment C3; cf. figure 9 f). Figure 9(e) gives $c_{D}\approx 0.13$ , where $c_{D}=\overline{u^{\prime }w^{\prime }}/U^{2}$ has been evaluated using $u_{m}=1.3U$ as determined from the experimental data, which is in good agreement with the maximal value of $c_{D}=0.1$ reported in figure 9(e). For experiment C3, $c_{D}=\overline{u^{\prime }w^{\prime }}/U^{2}=0.1$ using $u_{m}=1.4U$ from the experimental data, which is also in very good agreement with the values of $c_{D}$ reported in figure 6(f).

Figures 8(a) and 8(b) clearly show bottom billows, the generation of which is closely related to the development of the KH instabilities and turning in the opposite direction, and the consequent boundary layer separation beneath the KH billows causing a large drag similar to the drag reported already in the experiment over a curved bottom (experiment C3).

From this comparison, it is evident that the slope curvature has practically no effect on the spatial flow development; it is rather the sudden slope change and the initial flow conditions (high $Ri$ ) that cause the current to pass through acceleration–deceleration cycles, with locally high bottom drag coefficients $c_{D}$ .

7 Scaling analysis of the results

As discussed in §§ 5.2 and 6, there is a high shear stress in the bottom boundary layer in the experiments over steep slopes. This is due to the particular development of the KH billows (merging process) which cause locally a significant decrease of the lower layer depth. Coupled with this phenomenon, we have also observed the generation of bottom billows in the bottom boundary layer rotating in the opposite direction as compared with the KH billows. This is shown in figure 8, which displays zoomed images of the dye visualizations in the regions of maximal bottom friction for experiment L2 over the linear slope. Taking a first KH billow as in figure 3(d) in the dashed square and more clearly in figure 8(a) on the left, the local velocity below the billow core is directed in the same direction as the flow. At the same time, the depth of the lower layer, as mentioned above, is reduced so that the flow experiences a fast increase in velocity due to the buoyancy force and the rotation of the KH billow, very close to the bottom boundary. An important velocity gradient is thus present in the bottom boundary layer, which decreases suddenly on the back of the KH billow, where the depth is larger and the local vertical velocity is opposed to the bottom. This development enables finally detachment of the boundary layer, producing there a counter-rotating billow. The typical dimension of the boundary billows can reach the total depth of the lower layer, as is evident from the images in figure 8.

In order to generalize the results, notably those of the velocity, it is appropriate to represent the development in terms of non-dimensional quantities. From (2.9), we get the velocity scale $(B_{0}\sin \unicode[STIX]{x1D703})^{1/3}$ . An appropriate length scale is the distance $x_{c}$ where KH instability is first observed and the current velocity reaches its maximum. This is related to a decrease in Richardson number down to a critical value $Ri_{c}$ ,

Using for $U$ the free fall velocity $U_{free}=\sqrt{2g_{0}^{\prime }x_{c,th}\sin \overline{\unicode[STIX]{x1D703}}+U_{0}^{2}}$ , where $U_{0}$ is the velocity at the start of the ramp, we get an equation for $x_{c,th}$ as a function of the known initial parameters $B_{0}$ , $g_{0}^{\prime }$ , $U_{0}$ and $\overline{\unicode[STIX]{x1D703}}$ ,

For the critical Richardson number, we take the experimental value of $Ri_{c}=0.13$ at onset of KH instability, and on the concave boundary, we take for the slope angle $\unicode[STIX]{x1D703}=\overline{\unicode[STIX]{x1D703}}$ ; that is, a mean slope between $x=0$ and $x$ at the onset of the KH instability. The theoretical predictions for $x_{c,th}$ are in good agreement with the experimental observations of $x_{c,exp}$ for all experiments (see table 1).

In figure 10(a), the depth-integrated velocities are plotted for all experiments using $x_{c}\equiv x_{c,exp}$ to normalize the downstream direction $x$ and using $(B_{0}\sin \overline{\unicode[STIX]{x1D703}})^{1/3}$ to normalize the velocity $U$ . We see that the velocity data nicely collapse onto a single curve $x/x_{c}\leqslant 1$ , with a non-dimensional velocity maximum of approximately $2.6$ . Different decays are observed for $x>x_{c}$ for the different experiments, including experiment C1, for which a near overlap between the equilibrium-state velocity and the experimental velocity has been highlighted in § 5 (cf. figure 5 a) close to the end of the slope, where bottom drag starts to be dominant. The first velocity minimum is at $x/x_{c}\approx 2.5$ for all experiments.

Figure 10.

(a) Depth-integrated velocity normalized by $(B_{0}\sin \overline{\unicode[STIX]{x1D703}})^{1/3}$ , (b) entrainment rates $E=-w_{h}/U$ along $x$ normalized by $x_{c}\equiv x_{c,exp}$ for all experiments, (c) the ratio $x_{c}/h_{i}$ , (d)  $\overline{T_{a}}$ versus the slope $\tan \unicode[STIX]{x1D703}$ and (e) estimated bottom drag coefficients $c_{D}$ versus $\overline{T_{a}}$ ; ▫, experiment C1; ○, C3; ▹, L2; $\star$ , from Pawlak & Armi (Reference Pawlak and Armi2000) with $\unicode[STIX]{x1D703}=14.5^{\circ },10.8^{\circ },7.2^{\circ }$ .

Using the same scale to non-dimensionalize the downstream direction, we also plot the entrainment rates $E=-w_{h}/U$ for all experiments in figure 10(b) and we see that the values collapse onto a curve for $x<2.5x_{c}$ , while there is a spread when $x>2.5x_{c}$ . The maximal values are obtained for experiments C3 and L2, with an abrupt slope change at the ridge with $E_{max}\approx 0.07{-}0.08$ .

Table 2.

Summary of the key parameters for the present experiments and those of Pawlak & Armi (Reference Pawlak and Armi2000) with $\unicode[STIX]{x1D703}=14.5^{\circ }$ (L14), $10.8^{\circ }$ (L10) and $7.2^{\circ }$ (L07).

It is clear that a smooth transition to an equilibrium flow governed by (2.11) requires that the initial acceleration of the current, expressed by an acceleration parameter, remains below a certain value. The local value of $T_{a}$ , as shown in figures 5(e) and 6(e), is not appropriate because it depends on the local flow parameters. We can introduce an overall acceleration parameter in the form

where $x_{c}$ is the position of maximum velocity $U_{m}$ , $\unicode[STIX]{x0394}U=U_{m}-U_{i}$ and $h_{i}$ is the initial current height. All values are summarized in table 2, giving $\overline{T_{a}}=0.052$ for experiment C1, $\overline{T_{a}}=0.07$ for experiment C3 and $\overline{T_{a}}=0.06$ for experiment L2. These values are close to the local $T_{a}$ (figure 6(e) for experiment C3) at midposition, $x=x_{c}/2$ . An alternative and quantitative estimation of the overall acceleration parameter can be made using the definition of $Ri$ , from which we get $U_{i}=(B\cos \unicode[STIX]{x1D703}_{i}/Ri_{i})^{1/3}$ and $U_{m}=(B\cos \unicode[STIX]{x1D703}_{m}/Ri_{c})^{1/3}$ . This gives

From figures 5(b), 6(b) and 9(b), the critical bulk Richardson number is taken as $Ri_{c}\approx 0.13$ and $Ri_{i}\approx 0.45$ at $x=0$ . Pawlak & Armi (Reference Pawlak and Armi2000) obtain a value of $J_{c}\sim 0.2$ in their experiments defined with $\unicode[STIX]{x1D6FF}_{I\unicode[STIX]{x1D708}}$ and maximum velocity. We take here for their experiments $Ri_{c}=0.15$ and $Ri_{i}=0.65$ . With these values, we get for the present experiments $\unicode[STIX]{x0394}U/U_{m}\sim 0.35$ , which is very close to the values obtained by taking the velocity measurements directly; for their experiments, $\unicode[STIX]{x0394}U/U_{m}\sim 0.37$ . It seems thus that this ratio varies very little (10 % maximum) depending on the initial acceleration.

In figure 10(c), we plot $x_{c}/h_{i}$ as a function of the initial slope $\tan \unicode[STIX]{x1D703}_{0}$ , including also the results of Pawlak & Armi (Reference Pawlak and Armi2000) ( $\star$ ), with errorbars coming from the uncertainty in estimating the values of $x_{c}$ , $h_{i}$ , $\unicode[STIX]{x0394}U$ and $U_{m}$ from their plots. In figure 6, it is seen that in the present experiments $\overline{T}_{a}\sim 0.07$ for the conditions of experiment C3 and decreases to $\overline{T}_{a}\sim 0.052$ in C1 (cf. figure 5). In the latter, a weak oscillation would be observed if the slope did not decrease rapidly to zero beyond the maximum velocity. Figure 10(d) plots $\overline{T_{a}}$ as a function of the slope, making $x_{c}/h_{i}$ independent of the initial conditions of the flow $\unicode[STIX]{x0394}U/U_{m}$ . We see in both plots 10(c) and (d) that, in general, as the slope increases, the acceleration is larger and the position in which the Richardson number becomes critical and the maximal velocity is reached decreases. The data of Pawlak & Armi (Reference Pawlak and Armi2000) are shifted upwards and are higher for the same slope because the interface is unstable from the start, which adds additional drag via entrainment, thus reducing the acceleration. When the slope angle decreases and approaches zero, the distance required to reach the equilibrium-state velocity will tend to be very large, as we can extract from figure 10(c). We can conclude that a gradual acceleration up to maximum velocity without oscillations requires that $\overline{T}_{a}<0.05$ , as in the experiments of Pawlak & Armi (Reference Pawlak and Armi2000) and marginally for our experiment C1.

Finally, we plot the bottom drag coefficients $c_{D}$ as a function of the overall acceleration parameter $\overline{T_{a}}$ in figure 10(e). We see that for small acceleration parameters $\overline{T_{a}}<0.05$ , as in the experiments of Pawlak & Armi (Reference Pawlak and Armi2000), values remain low. As $\overline{T_{a}}$ approaches the value of $0.05$ , $c_{D}$ increases rapidly to reach a much larger value. For the experiments of Pawlak & Armi (Reference Pawlak and Armi2000), the values of $c_{D}$ have been evaluated using (2.9), taking $E=0.02$ and taking the local acceleration parameter $\overline{T_{a}}=0$ at maximum velocity. This gives $c_{D}<0.02$ . Errorbars are relative to the uncertainty in evaluating their final entrainment $E$ , velocity $U$ and current depth  $h_{0}$ .