2. Experimental method

We have carried out a series of experiments in two different flumes, each of width 0.1 m, depth 0.5 m, which are 3 and 5 m long. Silicon carbide particles ( $\rho _p = 3217$ kg m⁻³, Washington Mills) were mixed with fresh water ( $\rho _a = 998$ kg m⁻³) and then released from behind a lock gate located 0.1 m from one end of the tank. The fluid in the tank was 0.4 m deep. A series of experiments was run with different particle loads ( $g^{\prime }_0 = 0.0304{-}0.33$ m s⁻²) and sizes ( $d_p = 12.8{-}180$ ${\unicode[Arial]{x03BC}}$ m) as listed in table 2, and a single-phase aqueous saline current of the same initial buoyancy was run as a reference in each case. The experiments were repeated with the fluid behind the lock gate being dyed red or being clear so that the fate of the particles and the original suspension fluid could be followed. For the dyed experiments, 0.75 g l⁻¹ of TRS red food dye was mixed inside the lock gate, and to compensate for the change in density of this fluid, 0.6 g l⁻¹ of salt was stirred into the ambient water so the solutions had the same density. The densities of the different aqueous solutions were measured by weighing a fixed volume of each solution; these were also compared with density tables from the CRC Handbook of Chemistry and Physics. The experiments were recorded by video using Nikon D5300 cameras, with the tank being back-lit by an electroluminescent light sheet (W & Co Displays and Signs).

Figure 2.

Photographs of ( $a$ ) a saline gravity current, Exp. 1; ( $b$ ) a particle-driven gravity current with no dye in the source fluid, Exp. 8; ( $c$ ) a particle-driven gravity current with the source fluid dyed, Exp. 9. In each image, the green line, obtained by thresholding the light intensity of the image, indicates the top of the current as a function of distance along the tank.

Images were analysed using MATLAB in order to identify the volume of the current, and as far as was possible, to measure the particle load, as a function of position in the current. There are two different volumes that we measure: first, the volume of the particle-driven flow, and second, the volume of fluid into which the fluid behind the lock gate has mixed. In order to measure the volume of the particle-driven current, we do not use any dye. We take a reference image prior to the start of the experiment. Once the current is released from the lock and runs along the flume, we subtract the reference image from each of the series of images of the evolving current. We then assess the location of the edge of the current by using a small threshold value for the change in light intensity of each pixel. By varying the value of this threshold, we estimate that our measurement of the current volume is accurate to within 5 $\,\%$ for a given experiment. In order to measure the volume of fluid into which the fluid behind the lock gate has mixed, we carry out a second experiment in which we add dye to the particle-driven fluid behind the lock gate. We then use the same thresholding technique to detect the edge of the region of dyed fluid.

Figure 2 illustrates the result of the edge-detection algorithm for a photograph taken from each of three experiments. First, we show a single-phase saline current; second, we show a particle-driven current with no dye; and third, we show a particle-driven current with dye. In each case, the top boundary of the current is shown by the green line.

To measure the particle load in the current, calibration experiments were carried out in which the tank was filled with a series of suspensions of different known particle loads, and in each case, a series of photographs were taken. The particle-driven fluid becomes opaque above a certain particle concentration, so it is hard to measure the full distribution of particle load using light attenuation. However, as the concentration falls below approximately 10 $\,\%$ of the initial value in our experiments, it is possible to measure variations in the light intensity. This enables us to relate the light intensity, averaged across the width of the tank, to the width-averaged particle load, in cases in which the fluid had no dye. To measure the particle load in an experiment, we take a series of images of a current as it moves along a tank, and using the calibration curve, we convert the light intensity to particle load, averaged across the width of the tank, as the current evolves along the tank.

The turbulent fluctuations in the flow can lead to some variability in the measured volume between different experiments using the same source conditions. We have therefore repeated several of the experiments, and find that the instantaneous volume varies by up to $10 \,\%$ between otherwise identical experiments. An example of this is shown in Appendix.

In order to parametrise the experiments (table 1), the initial reduced gravity is given by

(2.1) \begin{equation} g^{\prime }_0 = \phi _0 g(\rho _p-\rho _a)/\rho _a, \end{equation}

where the initial volume fraction of particles is $\phi _0$ , the particle and fluid densities are $\rho _p$ and $\rho _a$ , and $g$ is the gravitational acceleration. In each experiment, the volume fraction of particles $\phi _0$ was calculated by combining the density of the particles and the density of the solution, with the measurement of the mass of particles $m_p$ that were mixed with a mass $m_a$ of aqueous solution, so that

(2.2) \begin{equation} \phi _0 = \frac {m_p/\rho _p}{(m_p/\rho _p) + (m_a/\rho _a)}. \end{equation}

The speed of the head $u_h$ is scaled with $u_0$ in (1.4). The particle settling velocity $v_s$ for the present particles, of diameter $d_p$ , was based on data presented by Lippert & Woods (Reference Lippert and Woods2020) using the Stokes settling speed $v_s$ in (1.5) The currents are characterised in terms of the parameter $S_0$ in (1.6), which compares the scaling for the current speed in (1.4) with the particle fall speed.

The Reynolds number of the currents is defined as ${\textit{Re}}_0=(u_0 L_s)/\nu _w$ , where, ${\nu _w} = 1.0\times 10^{-6}$ m $^2$ s⁻¹ is the kinematic viscosity of water. In all the experiments, this exceeds $\sim 10\,000$ , so the flows rapidly become fully turbulent following release from the lock gate.

3. Experimental observations

In figure 3, we illustrate three series of photographs of gravity currents with the same initial buoyancy. For reference, figure 3( $a$ ) shows a series of images of a single-phase current of an aqueous salt solution, and figure 3( $b$ ) corresponds to a particle-driven gravity current. The frames are taken at time intervals of 1.8 s. As these currents advance along the base of the tank, the volume initially increases as ambient fluid ahead of the current is displaced upwards over the head of the flow, and some of this fluid mixes into the current (cf. Sher & Woods Reference Sher and Woods2015). This leads to formation of a tail of mixed fluid, and dilution of the head. In the particle-driven flow (figure 3 $b$ ) some of the particles settle out from the tail, eventually leading to a reduction in the volume of particle-driven fluid.

Figure 3.

Series of photographs showing the time evolution of: ( $a$ ) a saline gravity current, at intervals of 1.8 s, Exp. 1; ( $b$ ) a particle-driven gravity current with clear source fluid, at intervals of 1.8 s, Exp. 8; ( $c$ ) a particle-driven gravity current with the source fluid dyed red, at intervals of 1.8 s, Exp. 9. Experiments ( $a$ – $c$ ) had initial buoyancy $g^{\prime }_0=0.33$ , and the particle diameter (b,c) is 63 ${\unicode[Arial]{x03BC}}$ m, so (b,c) $S_0$ = 43.0.

A complementary understanding of the evolution of the source fluid in the particle-driven gravity current can be seen in the third set of images, in which the source fluid is dyed red (figure 3 $c$ ). On release from the lock gate, the source fluid spreads out in the tail as well as the head of the flow. As particles settle through the tail, the fluid develops a vertical particle stratification, with the upper part of the tail being nearly particle-free and red, while the lower parts are much darker owing to the presence of particles. The volume of the particle-bearing region of the current initially increases but then decreases as the particles settle from the flow. However, the volume of red-dyed fluid continues to increase. Once the particles settle from the tail, eddies in the particle-free dyed fluid appear to engulf and mix more ambient fluid. This may be contrasted with the saline current in figure 3( $a$ ), in which the depth of the fluid in the tail remains much smaller.

We define the depth of the head using a light attenuation technique in the region just behind the front of the current, $x_h-L_0\lt x\lt x_h$ . In particular, in this region, we find the maximum depth above the base of the tank at which the light attenuation, relative to the background (no current) value, exceeds a threshold. The threshold has been varied to ensure that our measurements are consistent within an error of $5\,\%$ . In virtually all the experiments, the depth of the head appears to remain close to $(0.4\pm 0.05) H_0$ , where $H_0$ is the depth of the water in the flume. This observation is consistent with previous observations of single-phase aqueous gravity currents (cf. Sher & Woods Reference Sher and Woods2015).

Indeed, in figure 4(a), we present a graph of the evolution of the volume of the current, and in figure 4(b), the depth of the head of a typical current as a function of the distance downstream. As expected, it is seen that the volume increases owing to the entrainment and mixing, while the depth of the head of the current remains close to $0.4H_0$ . Eventually, the volume reaches a maximum then decreases to zero, but the data suggest that regardless of whether the volume increases or decreases, the depth of the head remains approximately constant until just before reaching the maximum run-out distance. In the next section, we explore this in more detail for a series of currents with different values of $S_0$ .

Figure 4.

Variation of ( $a$ ) the volume and ( $b$ ) the height of the current head as a function of time. Regardless of whether the volume increases or decreases, the current continues to propagate while maintaining an approximately constant head height until just before the run-out distance of the current. ( $c$ ) Series of false colour images from Exp. 99, with $d_p = 63.0$ ${\unicode[Arial]{x03BC}}$ m, $S_0 = 31.9$ , which show the evolution of the regions of different particle concentration as a function of distance downstream $x/L_s$ . Particle concentration is shown in the legend. ( $d$ ) Variation of the volume of particle-laden fluid with concentration in excess of different fractions of the initial particle concentration as a function of the position of the head of the current. The colour categories correspond to different concentrations shown in the legend, and the volumes are scaled by $L_s^2$ .

Figure 4 Long description

Panel A: A line graph shows the variation of the volume of the current head as a function of time. The x-axis is labeled ‘t/√(L_s/g_0’)‘ and the y-axis is labeled ‘V/L_s^2’. The data points form a curve that rises to a peak and then gradually declines. Panel B: Another line graph displays the height of the current head as a function of time. The x-axis is labeled ‘t/√(L_s/g_0’)’ and the y-axis is labeled ‘h_h/H_0’. The data points fluctuate around a relatively constant value with a slight downward trend. Panel C: A series of false color images show the evolution of regions with different particle concentrations as a function of distance downstream. The x-axis is labeled ‘x_h/L_s’ and the color legend indicates different concentration ranges from less than 2 percent to over 10 percent. Panel D: A scatter plot illustrates the variation of the volume of particle-laden fluid with concentration in excess of different fractions of the initial particle concentration as a function of the position of the head of the current. The x-axis is labeled ‘x_h/L_s’ and the y-axis is labeled ‘V/L_s^2’. Different colored data points represent different concentration ranges as indicated in the legend.

To illustrate the interior structure of the current, we have used the light attenuation technique to assess the evolution of the particle load in the current in more detail. In figure 4( $c$ ), we present a series of images in which we show the spatial distribution of the particle load as the current advances along the flume. The different colours in each image denote different concentration values as a fraction of the initial concentration, as shown in the colour bar. The figure illustrates that a significant vertical and lateral stratification of particle concentration develops in the current, with the fluid in the head being of the highest concentration. Using the particle concentration data in the current, we have calculated the volume of fluid in the current in which the concentration exceeds a series of different values as a function of the position of the head of the current. This is shown in figure 4( $d$ ). As the current advances along the flume, the volume exceeding each particle concentration initially increases but eventually reaches a maximum, and then decreases to zero.

The time and hence position of the head at which the maximum volume is reached increases as the concentration falls, eventually leading to a plot similar to figure 4( $a$ ) of the total volume.

The data also show that the volumes of fluid with concentrations in excess of $6\,\%$ and $10\,\%$ of the initial concentration evolve in a very similar way. This corresponds to the region of relatively high particle load, which is located mainly in the head. As the particles settle through the current, and more ambient fluid is mixed into the current, a region of progressively smaller concentration forms, especially in the upper part of the flow (see the first five rows of figure 4 $c$ ). In part, this results from the separation of the fluid and particles at the top of the particle-driven flow, as a result of particle settling.

Figure 4( $c$ ) also illustrates that the tail of the current is composed of a series of eddy-type structures, with relatively low particle concentration. These structures do not advance very far along the flume before the particles settle to the base of the flume, releasing particle-free fluid into the ambient. Meanwhile, the head continues advancing down the flume. The key processes observed in the experiments are sketched in figure 1( $d$ ), illustrating the mixing, settling and sedimentation.

Figure 5.

Experimental measurement of: ( $a$ ) the volume of a particle-driven gravity current as a function of distance of the head from the source; ( $b$ ) the distance of the head from the source as a function of time; and ( $c$ ) the height of the head as a function of the position of the head, scaled by the position of the head when the volume is a maximum. In ( $a$ ), the open symbols correspond to the particle-driven fluid in experiments using undyed source fluid, while the solid symbols correspond to the volume of dyed fluid in an equivalent experiment in which the source fluid was dyed ( $d_p = 12.8{-}180$ ${\unicode[Arial]{x03BC}}$ m, $g^{\prime }_0=0.33$ m s⁻², Exp. 1–17). The dashed line in ( $a$ ) corresponds to the rate of increase of volume of a single-phase gravity current (Sher & Woods Reference Sher and Woods2015). In ( $c$ ), curves are shown for a range of values of $S_0$ as indicated in the legend in (a).

4. Quantitative measurements

In order to understand the relative rates of the different processes controlling the evolution of the flow, we now explore the impact of variations in $S_0$ on the evolution of the volume, the speed, and the height of the head of the current.

In figure 5( $a$ ), we present a series of experimental measurements of the volume of the particle-driven fluid (open symbols) and of the fluid containing dye that originates behind the lock gate (solid symbols). Both are shown as a function of the distance of the head of the current from the rear of the lock gate. The volume is normalised by the initial volume behind the lock gate, $L_s^2$ , and the position of the head, $x_h$ , is normalised by the length scale of the flow, $L_s$ . The different experiments correspond to different particle sizes, as shown in the legend for the plot, and for a fixed total initial current buoyancy. The currents all have different values of $S_0$ . The dashed line corresponds to the rate of change of the volume of a saline gravity current, as measured by Sher & Woods (Reference Sher and Woods2015).

During the initial phases of the flow, the volumes of both the dyed fluid and the particle-driven fluid increase at a similar rate to the reference saline gravity current. However, eventually, the volume of the particle-driven fluid reaches a maximum, and as the flow continues downstream, this volume begins to fall. We interpret this as the point at which settling of particles from the top surface of the flow exceeds the rate of mixing of ambient fluid into the head, and a layer of particle-free dyed fluid develops above the current (cf. figure 2 $c$ ). This fluid retains its forward momentum, but loses the buoyancy associated with the particle load. It is then able to mix more readily with the overlying ambient fluid, so the volume of fluid that contains red dye continues to increase with time. The data for the single-phase saline current shown by the black crosses show the same increase in volume with distance for $x_h/L_s \lt 10$ , but as the flow travels further from the lock gate, there is a gradual decrease in the rate of increase of volume; this may be the result of bottom friction (cf. Sher & Woods Reference Sher and Woods2015).

In figure 5( $b$ ), we illustrate the position of the head of the current as a function of time, scaled with the characteristic time for the gravity current $\sqrt {L_s/g'_0}$ for currents with a range of values of $S_0$ . This figure illustrates that the currents all initially have a comparable speed, but for smaller values of $S_0$ , the particles settle through the fluid progressively more quickly, and the flow does not travel as far along the flume before slowing down and waning. As a result, the maximum volume and the overall run-out length of the current decrease.

In figure 5( $c$ ), we show the scaled height of the head of the current as a function of position of the head scaled by $x_{V_{\textit{max}}}$ , the position of the head at which the maximum volume is reached. Curves are shown for a large range of values of $S_0$ . This figure illustrates that for all currents, the depth of the current remains close to $(0.4\pm 0.05)H$ as the current advances to points beyond approximately $x_h = 1.2x_{V_{\textit{max}}}$ , but then it begins to decrease to zero as the current wanes and all the particles sediment to the base.

We have measured the maximum volume of the particle-driven current, and the dimensionless position of the head at which the current reaches this volume; the data are shown in figure 6. As $S_0$ increases, the current travels further before reaching the maximum volume (figure 6 a). Also, there appears to be a strong, and nearly linear, relationship between the maximum volume and the distance at which this is attained for a range of values of $S_0$ , $30 \lt S_0\lt 300$ (figure 6 b). With larger values of $S_0$ , the current reaches the distal end of the flume prior to reaching the maximum volume. As $S_0$ increases, the settling slows down, and the current is able to entrain more fluid prior to reaching the maximum volume. Figure 6(c) shows the position of the distal end of the current as a function of $S_0$ , illustrating that the maximum run-out distance increases with $S_0$ .

Figure 6.

Experimental measurement of ( $a$ ) the position of the head when the current reaches the maximum volume as a function of $S_0$ , and ( $b$ ) the maximum volume as a function of the position of the head when the current reaches the maximum volume. Points are shown for a range of values of $S_0$ in (a). ( $c$ ) The position of the head at the distal end of the current as a function of $S_0$ . The curves are predictions from the box model, and error bars denote a 10 % error in distances.

Figure 6 Long description

Three scatter plots compare experimental data with model predictions for particle-driven gravity currents. Panel A: A scatter plot shows the position of the head when the current reaches the maximum volume as a function of S0. The x-axis is labeled S0, and the y-axis is labeled xVmax. The plot includes experimental data points and a model prediction line. Panel B: A scatter plot shows the maximum volume as a function of the position of the head when the current reaches the maximum volume. The x-axis is labeled xVmax, and the y-axis is labeled Vmax. The plot includes experimental data points and a model prediction line. Panel C: A scatter plot shows the position of the head at the distal end of the current as a function of S0. The x-axis is labeled S0, and the y-axis is labeled xterm. The plot includes experimental data points and a model prediction line. Error bars denote a 10 percent error in distances.

5. Box model development

In order to illustrate the competing effects of entrainment and settling on the current volume, as a function of $S_0$ , it is helpful to use a ‘box model’ approach to describe the evolution of the current (cf. Huppert & Simpson Reference Huppert and Simpson1980; Sher & Woods Reference Sher and Woods2015). Although such box models are simplified and do not describe all details of the flow, they do have value in providing insight. In this section, we motivate a new box model from the physical processes that we have described, which influence the boundary conditions, coupled with the underpinning equations for fluid and particle conservation. Figure 7 provides a reference for the properties of the current defined in the model.

Figure 7.

Schematic of the particle-driven current to illustrate the definitions of the variables used in the box model.

If we consider a small element of fluid in the current lying between $x$ and $x+\delta x$ , and spanning the depth of the current, $0\lt y\lt h(x,t)$ , then the rate of change of the volume of this element of the current in some interval of time, ${\partial h(x,t)}/{\partial t}$ , can be expressed as

(5.1) \begin{equation} \frac {\partial h}{\partial t} =- \frac {\partial (h\overline {u})}{\partial x} - v_s, \end{equation}

where the first term on the right-hand side denotes the difference between the horizontal flux into and out of the volume, with

(5.2) \begin{equation} h\overline {u} = \int _0^h u\, \text{d}y, \end{equation}

and the second term on the right-hand side represents the loss of fluid associated with any settling of the particles at the top surface of the flow, assumed to occur with speed $v_s$ . We note that since the flow is turbulent, there may be some resuspension of particles in the flow that suppresses the particle settling (cf. Blanchette & Bush Reference Blanchette and Bush2005; Guazzelli & Hinch Reference Guazzelli and Hinch2011; Lippert & Woods Reference Lippert and Woods2020). This effect is likely to depend on the local ratio of the current speed compared to the particle fall speed. Near the head of the current where the turbulence is very intense, many of the particles may remain suspended, but in the growing tail of the current, behind the advancing head, the flow is less intense, and the capacity for the current to resuspend particles gradually decreases. We therefore assume, for simplicity, that the particles settle with their fall speed $v_s$ along the top surface of the current, noting that this approximation should become progressively more accurate as the lateral extent of the current becomes larger than the length scale of the head of the current.

In general, the volume of the current can be written in the form

(5.3) \begin{equation} V = \int _0^{x_h(t)} h(x,y)\, \text{d}x, \end{equation}

where $x_h$ is the location of the nose of the flow. Using the integral expression for $V$ in (5.3), we find that

(5.4) \begin{equation} \frac {\text{d}V}{\text{d}t} = h_h \frac {\text{d}x_h}{\text{d}t}- \int _0^{x_h(t)} \frac {\partial h}{\partial t}\, \text{d}x . \end{equation}

Substituting (5.1) then leads to the relation

(5.5) \begin{equation} \frac {\text{d}V}{\text{d}t} = h_h \frac {\text{d}x_h}{\text{d}t}- h_h \overline {u_h}- v_s x_n. \end{equation}

Following Sher & Woods (Reference Sher and Woods2015), we note that the entrainment of fluid in the head of the flow leads to the relation (their (4.17))

(5.6) \begin{equation} \overline {u_h} = (1-\alpha ) \frac {\text{d}x_h}{\text{d}t}, \end{equation}

so that in the present case,

(5.7) \begin{equation} \frac {\text{d}V}{\text{d}t} = \alpha h_h \frac {\text{d}x_h}{\text{d}t} - v_s x_h . \end{equation}

This equation reduces to the result of Sher & Woods (Reference Sher and Woods2015) in the limit $v_s=0$ , corresponding to a single-phase saline aqueous gravity current.

We now consider the local conservation of particles:

(5.8) \begin{equation} \frac {\partial g'}{\partial t} + \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }g' = -v_s\frac {\partial g'}{\partial y} . \end{equation}

If we integrate over the depth of the flow, then we obtain the relation

(5.9) \begin{equation} \frac {\partial }{\partial t} (\hat {g'}(x,t)\,h) + \lambda \frac {\partial }{\partial x} (\overline {u} \hat {g'}(x,t)\,h) = -\beta \hat {g'}(x,t)\, v_s, \end{equation}

where

(5.10) \begin{equation} \hat {g'}h = \int _0^h g'\,\text{d}y,\quad \lambda \overline {u} \hat {g'}h = \int _0^h u g'\,\text{d}y,\quad \beta \hat {g'} = g'(0,x,t) . \end{equation}

Here, $\lambda$ is a constant associated with any vertical variations in the horizontal velocity and/or the particle load, while $\beta$ relates the concentration at the base of the flow to the mean concentration. Equation (5.9) is equivalent to that presented by Bonnecaze et al. (Reference Bonnecaze, Huppert and Lister1993), but in that earlier work, it was assumed that $\lambda = 1$ and $ \beta = 1$ , corresponding to a well-mixed flow. We also adopt this simplification in the present idealised box model, and test the model with our experimental data that span the range $30 \lt S_0 \lt 300$ .

The total particle load in the current is given by

(5.11) \begin{equation} \overline {g'}V = \int _0^{x_h} \hat {g'}(x,t)\,h\,\text{d}x , \end{equation}

so differentiating with time, we find

(5.12) \begin{equation} \frac {\text{d}(\overline {g'}V)}{\text{d}t} =\frac {\text{d}x_h}{\text{d}t} \left [ \int _0^{h(x_h)} g'|_{x_h}\,\text{d}y \right ] + \int _0^{x_h} \frac {\partial (\hat {g'}h)}{\partial t}\,\text{d}x . \end{equation}

If we substitute the expression for ${\partial (\hat {g'}h)}/{\partial t}$ in the last term on the right-hand side, as given by (5.9), we find that this equation becomes

(5.13) \begin{equation} \frac {\text{d} (\overline {g'} V) }{\text{d}t} = \frac {\text{d}x_h}{\text{d}t} \left [ \int _0^{h(x_h)} g'|_{x_h}\,\text{d}y \right ] - \left [ \int _0^{h(x_h)} u g'|_{x_h}\, \text{d}y \right ]- \overline {g'} x_h v_s, \end{equation}

where for the last term, for simplicity, we assume that the particle load is uniform along the length of the flow, so that $\int _0^{x_h} \hat {g'}\, \text{d}x = \overline {g'} x_h$ . The first two terms on the right-hand side are associated with the entrainment of ambient fluid at the nose of the flow; this does not change the particle load of the current, so these terms add to zero (cf. Sher & Woods Reference Sher and Woods2015). This leads to the relation

(5.14) \begin{equation} \frac {\text{d}(\overline {g'} V)}{\text{d}t} = - \overline {g'} x_h v_s. \end{equation}

This equation is equivalent to that in the box models of Bonnecaze et al. (Reference Bonnecaze, Huppert and Lister1993) and Dade & Huppert (Reference Dade and Huppert1995).

To complete the model, we draw on the existing literature and use (1.1) to specify the speed of the head of the flow:

(5.15) \begin{equation} \frac {\text{d}x_h}{\text{d}t} = \textit {Fr}\, (\overline {g'}h_h)^{1/2} . \end{equation}

For convenience, we scale the system so that the dimensionless volume, distance, buoyancy, depth of the head, and time (denoted by subscript $s$ ) are given by $ V_s = V/L_s^2$ , $ x_s = x_h/L_s$ , ${g^{\prime }_s}=\overline {g^{\prime }}/g^{\prime }_0$ , $h_s=h_h/L_s$ and $ t_s = t/(L_s / g^{\prime }_0)^{1/2}$ .

Combining the equation for the speed of the head (5.15) with that for the rate of change of the volume with time (5.7) leads to the dimensionless equation for the rate of change of volume as a function of the position of the head,

(5.16) \begin{equation} \frac {\text{d}{V_s}}{\text{d}{x_s}} = \alpha h_s - \frac { x_s }{ S_0 {g'_s}^{1/2}h_s^{1/2} }, \end{equation}

and the rate of change of the product of the buoyancy and volume with position of the head,

(5.17) \begin{equation} \frac {\text{d}(g'_sV_s)}{\text{d}{x_s}} = - \frac { x_s {g'_s}^{1/2} } { S_0 {h_s}^{1/2} }, \end{equation}

while the position of the head increases with time according to the relation

(5.18) \begin{equation} \frac {\text{d}x_s}{\text{d}t_s} = \textit {Fr}\, {g'_s}^{1/2} h_s^{1/2}. \end{equation}

The three equations (5.16)–(5.18) determine the evolution of the current volume $V_s$ , nose position $x_s$ , and particle load $g'_s$ , as a function of time. The main parameter controlling the evolution of the current is $S_0$ , which appears in (5.16). The initial conditions for our experiments are $x_s=L_0/L_s=0.5$ , $V_s=1$ and $g'_s =1$ . Motivated by the data in figure 5( $c$ ), we assume that the depth of the head is constant and given by $h_h=0.4H_0$ . We now compare the solutions of (5.16)–(5.18) with our data.

First, we note that the equation for the evolution of the volume includes the term proportional to $\alpha$ that parametrises the entrainment through the head of the flow. Near the source, the current volume increases nearly linearly with the position of the head (figure 5 $a$ ). The best-fit value for the rate of increase of this volume with position of the head, $\alpha h_s$ , in the region $0\lt x_s\lt 0.5 x_{V_{\textit{max}}}$ , suggests that $\alpha \sim 0.6 \pm 0.1$ . This is consistent with the value of $\alpha$ found for a saline current (cf. Sher & Woods Reference Sher and Woods2015), as also shown in the figure. The second term in (5.16), corresponding to the settling of particles through the top surface of the current, is new, and becomes progressively more important as the flow travels further along the flume. As a result, the volume of particle-driven fluid reaches a maximum and then begins to decrease. We have measured the maximum volume of the current $V_{\textit{max}}$ , and the position of the head at this maximum volume $x_s(V_{\textit{max}})$ , for a number of experimental currents, covering a large range of values of $ S_0$ , $30 \lt S_0 \lt 300$ . The model predictions for the maximum volume and the head position at the maximum volume as a function of $ S_0$ are compared with the experimental data in figures 6(a) and 6(b). The model captures the trend in the experimental data.

Our experiments show that once the current has reached the maximum volume, the height of the head remains at value $0.4H_0$ for some distance, but then begins to decrease, and falls to zero at the end of the flow, when all the particles have separated from the fluid; as a result, we do not expect the model to be fully accurate in this region. However, we have compared the prediction of the model for the final run-out distance with the experimental data for those experiments that did not reach the end of the experimental flume. This comparison is shown in figure 6( $c$ ) as a function of $S_0$ . The trend of increasing run-out distance with $ S_0$ in the data is also seen in the model, but we note that there is some scatter in the experimental data.

Figure 8.

Experimental observations from a range of particle-laden gravity currents ( $d_p = 12.8{-}106.0$ ${\unicode[Arial]{x03BC}}$ m, $g'_0 = 0.030{-}0.33$ m s⁻², $S_0 = 10.3{-}1041.6$ , Exp. 2, 34, 55, 76, 93) illustrating ( $a$ ) the volume as a function of distance, and ( $b$ ) the nose position as a function of time. The symbols correspond to measurements from the experiments, and the solid lines correspond to the idealised box model.

Figure 8 Long description

Two line graphs illustrate experimental observations of particle-laden gravity currents. Panel A shows the volume as a function of distance, while Panel B shows the nose position as a function of time. In Panel A, the x-axis represents the normalized horizontal distance (xh/Ls) and the y-axis represents the normalized volume (V/Ls^2). Different colored lines and circles represent various initial conditions (S0 values). The solid lines correspond to the idealized box model, while the circles represent measurements from the experiments. Panel B shows the x-axis as normalized time (t/√(Ls/g’0)) and the y-axis as normalized nose position (xh/Ls). Similar to Panel A, different colored lines and circles represent various initial conditions, with solid lines for the model and circles for experimental measurements. The graphs show how the volume and nose position evolve over time and distance, highlighting the competition between entrainment of ambient fluid and particle settling.

In the experiments, for values of $S_0$ smaller than approximately 30, the particles tend to settle from the flow more rapidly, within a distance of a few lock lengths from the source, and this coincides with the initial slumping phase of the flow (cf. Huppert & Simpson Reference Huppert and Simpson1980); for this range of values, the model is less applicable.

In figure 8, we compare the predictions of the idealised box model with data from five example experiments, with different values of $S_0$ , in which we have measured the volume of the flow as a function of distance from the source, and the nose position as a function of time. The figure demonstrates how the simplified box model, accounting for entrainment, settling and sedimentation, captures some of the complexity of the temporal evolution of the flow, including the increase and then decrease in volume, and the slow-down of the flow. It also captures some of the variation in the rates of these different processes as $S_0$ changes.

Although the model captures the processes and the trend in the behaviour with $S_0$ , it is important to stress that the detailed dynamics of each experiment is more complex than this simplified box model; however, for the range of values of $S_0$ in our study, the approximate model provides some insight into the evolving balance between the different controlling processes as the current moves along the flume. We have repeated many of the experiments, and found that there is approximately 5–10 $\,\%$ variability in the data of both the volume and the total run-out distance between repeated realisations of the same experiment owing to some of the natural turbulent fluctuations in the flow (e.g. see Appendix).