2.1. Formulation

Here, we present the essentials of the theoretical model used in our investigation. This model served a guideline for the design and set-up of the experiments, and it is also used for comparisons with, and interpretation of, the data and effects recorded in our experiments. We note that the model has been presented before (Hatcher et al. Reference Hatcher, Hogg and Woods2000; Testik & Ungarish Reference Testik and Ungarish2016). The novelty here is the application of new influx conditions and the comparison with experiments aiming to an estimate of the power-law $\lambda$ and to measure the front position and the profile of the currents.

We use a cylindrical coordinate system, and assume axial symmetry (i.e. zero or negligibly small dependency on the angular coordinate). We consider rod-like vegetation which fills a horizontal domain of height $H$. In our system of reference, the coordinate $r$ is horizontal, while $z$ is vertically upward. The GC propagates in the $r$ direction, see figure 1(a). The densities of the current and ambient fluids are $\rho _{c}$ and $\rho _{a}$, with $\rho _{a} <\rho _{c}$, and $z=0$ and $z=H$ are rigid boundaries, see figure 1(b). For practical reasons, $z=H$ in the experiments is a free surface, see next discussions on the sensitivity of the results to this approximation. The height (thickness) of the current is $h(r,t)$, and the height-averaged speed of the current is $u(r,t)$, while that of the ambient is $u_a(r,t)$. The radius of propagation is $r_N(t)$.

Figure 1.

Sketch of the theoretical model: (a) three-dimensional schematic; (b) axial-radial plane section.

The equations of motion are derived along the lines of the thin-layer simplification detailed in Ungarish (Reference Ungarish2020). We use dimensional variables unless stated otherwise.

2.2. Equations of motion and boundary conditions

We start with the observation that the thickness of the ambient-fluid layer is $H-h$ and by continuity

(2.1)\begin{equation} u_a ={-} u h /(H - h) \end{equation}

for constant $H$. The vegetation introduces a drag force. As suggested by previous investigations and confirmed by our experiments, the effective drag per unit volume of fluid is proportional to the density of the fluid and to the speed at some power $\lambda \ge 1$. To facilitate the analysis, we assume that (i) the coefficient of proportionality is a constant, $\tilde {c}$; (ii) in a given GC system, both the current and the ambient have the same $\tilde {c}$ and $\lambda$ (as will be evident in the following steps, the accuracy of this assumption is not relevant since we will adopt a single-layer model where only the dynamics of the dense current is considered). We assume (and confirmed by order of magnitude estimates of the laboratory data) that, in the tests considered here, the viscous and turbulent stresses in the current and ambient domains are negligible as compared with the other forces. We consider flow fields in which the drag dominates the acceleration (inertia) term. Therefore, in the subsequent analysis the $\textrm {D}u/\textrm {D}t$ terms are neglected, except for the analysis of transition, see (2.7). The pressure is approximated by the hydrostatic balance, and hence the $x$-momentum balances for the ambient and current reduce to

(2.2a,b)\begin{equation} 0={-}\dfrac{\partial p}{\partial x}-\rho_a\tilde{c}u_a|u_a|^{\lambda-1},\quad 0={-}\dfrac{\partial p}{\partial x}-(\rho_c-\rho_a)g\dfrac{\partial h}{\partial r}-\rho_c\tilde{c}u|u|^{\lambda-1}, \end{equation}

where $p$ is the reduced pressure in the ambient and $g$ the gravity acceleration. Here, $\lambda$ is assumed to be constant. Elimination of the pressure and use of (2.1) yield one combined-momentum equation for $u$. Consequently, the motion can be expressed as the system of continuity and momentum balances

(2.3)$$\begin{gather} \sigma \frac{\partial h}{\partial t} + \frac{\partial h u }{\partial r} ={-}\dfrac{u h}{r}; \end{gather}$$

(2.4)$$\begin{gather}- g^{\,\prime} \frac{\partial h}{\partial r} - \tilde{c} |u|^{\lambda-1}u \left [ 1 + \frac{\rho_{a}}{\rho_{c}} \left( \dfrac{h}{H-h} \right)^\lambda \right ]=0, \end{gather}$$

where the reduced gravity is $g^{\prime } = g(1- \rho _{a}/\rho _{c})$, and $\sigma$ is the porosity of the vegetation medium in which the fluids move, always positive and smaller than $1$, assumed here constant. We also define the density of the vegetation as $\kappa = 1-\sigma$.

Further simplifications of the equations are possible and useful. First, we introduce the coordinate stretching of the time to $t/\sigma$; this eliminates $\sigma$ from the subsequent analysis. Second, we note that the Boussinesq system with $\rho _{c} \approx \rho _{a}$ admits the simplification of the drag term in the momentum equation (2.4) by setting $\rho _{a}/\rho _{c} = 1$ in front of the term $h/(H-h)$ which expresses the drag coupling between the current and the upper ambient fluid. Next, for a deep current $h/H \ll 1$, the coupling term can be neglected. This simplification, also called the one-layer model, is facilitated by the fact that the value of $\lambda$ is larger than 1 and typically $2$. In the range of parameters considered in the present investigation, the model based on Boussinesq and one-layer simplifications is expected to provide a fair approximation to the main behaviour (propagation and shape) of the current. The one-layer theory is an approximation that is expected to work well when the height of the current to ambient at $r/r_N \approx 0.5$ is less than 0.5 (which implies that a significant part of the current is even thinner), and when the rate of increase of the height due to influx is much smaller than the speed of propagation. (We keep in mind that $h$ decreases with $r$ and that the propagation of the current is governed by the ring of fluid close to $r_N$). Consequently, we shall use the momentum equation

(2.5)\begin{equation} 0 ={-} g^{\prime} \frac{\partial h}{\partial r} - \tilde{c} |u|^{\lambda-1}u, \end{equation}

which expresses the balance between the buoyancy driving force and the vegetation-drag hindering effects on the layer of the dense fluid. We note that the curvature terms are not present in this balance; however, the curvature effect is strongly manifested on the right-hand side continuity (2.3) and hence the present geometry needs a separate investigation from the two-dimensional Cartesian counterpart.

We emphasize that the coefficient $\tilde {c}$ in the drag term is different from the classical dimensionless ‘drag coefficient’ defined by the ratio of non-viscous opposing force on a body to dynamic pressure. The present drag coefficient is defined by the ratio of the opposing force between vegetation and fluid, per unit volume of fluid, divided by $\rho |u|^\lambda$ (the density and speed pertain to either the domain of the current or ambient), and $\tilde {c}$ is dimensional with $[\tilde {c}]=L^{1-\lambda }T^{\lambda -2}$.

In general, the volume of the current is known. This can be expressed as

(2.6)\begin{equation} \int_0^{r_N(t)} r\,h(r,t)\, \textrm{d}r =qt^{\alpha}\equiv \mathcal{V}(t), \end{equation}

where $q$ and $\alpha \ge 0$ are given constants. Note that $\mathcal {V}(t)$ is the volume per radian, and contains both fluid and vegetation.

At the nose $r= r_N(t)$, we impose the condition $h=0$ while $u_N$ is finite. The justification is as follows. The motion takes place because the buoyancy driving $g^{\prime } (- \partial h/\partial r)$ in the radial direction balances the opposing drag $\sim - |u|^{\lambda -1}u$. This means that, at any radial position of the moving current, a negative slope $\partial h/\partial r$ is present, and hence $h$ is expected to decrease to $0$ at the nose. The subsidiary assumption is that a nose of zero thickness (i.e. a contact line) is compatible with the equations of motion, and this will be confirmed later. (We note in passing that a moving contact-line condition is an abstraction of a complex $r$-thin corner region in which strong gradients are present, and hence in practice the decrease to $h=0$ is less sharp than in the theory. There is evidence from the viscous-current counterpart, see Ungarish (Reference Ungarish2020), that this singularity has little effect on the speed of propagation, because the larger $h$ creates a larger buoyancy drive from the dense material behind. We conjecture that the same behaviour occurs in the present case.) In this context we also note that it is possible to eliminate $u$ from (2.3) and (2.5) to obtain one equation for $h(r,t)$, which bears similarity with the formulation for power-law fluid GCs, see for example Sayag & Worster (Reference Sayag and Worster2013).

For the boundary conditions at the axis $r=0$, we must distinguish between the following cases: (i) fixed volume (or LR) admits the simple $u=0$ at $r=0$; (ii) for GCs created by influx, it is in general difficult to prescribe the precise boundary condition at the source. What we usually know is the total volume flux. Near the axis, the influx speed may be large and the thin-layer assumptions may be invalid. From the point of view of the present set of equations, this domain is a singularity. For progress, we apply the usual assumption that there is some adjustment domain in which the flux is matched with the values of $h$ imposed by the outer region. The relative volume of fluid in this singular region is expected to be small with respect to the volume of the dense fluid, (the radial geometry embeds a bottom area growth proportional to the square of the radius and we are considering the region near the axis) and, hence, of little relevance to the major flow field (see Di Federico, Archetti & Longo Reference Di Federico, Archetti and Longo2012). The generally accepted mathematical solution for the axisymmetric current created by influx imposes the conditions at the nose, but leaves open the behaviour near the axis. We shall keep this in mind during the comparisons with realistic data.

In the present simplified model, we cannot apply realistic initial conditions of $u$ and $h$ at $t=0$. The physical meaning is that the initial inertial adjustment is missed, and we assume that this adjustment is a short episode, whose details are insignificant to the subsequent motion.

2.3. Regimes of GCs

An estimate of the transition from the buoyancy–inertial to buoyancy–drag (turbulent drag) regime can be obtained by imposing that the flow inertia, drag and buoyancy terms are of the same order of magnitude

(2.7)\begin{equation} \dfrac{\partial u}{\partial t}\sim g^{\prime}\dfrac{\partial h}{\partial r}\sim\tilde{c}|u|^{\lambda-1}u, \end{equation}

while the mass conservation in integral form brings $hr^2\sim \mathcal {V}$. By assuming that $r\sim u t_c$ and $\mathcal {V}=qt^\alpha$, and defining $t_c$ as the time of transition, we get

(2.8a,b)\begin{equation} \dfrac{u}{t_c}\sim g^{\prime}\dfrac{h}{ut_c}\sim \tilde{c}u^{\lambda},\quad hu^2t_c^2\sim qt_c^\alpha, \end{equation}

which leads to the scaling

(2.9)\begin{equation} t_c\sim (g^{\prime} q \tilde{c}^{4/(\lambda -1)})^{(1-\lambda)/[\alpha (\lambda -1)-2 \lambda +6]}, \end{equation}

equal to

(2.10)\begin{equation} t_c\sim\tilde{c}^{{-}1};\quad (\tilde{c}^{4} g^{\prime} q)^{{-}1/(\alpha +2)}\quad\textrm{for}\ \lambda=1,2. \end{equation}

The critical time increases with $\lambda$ and decreases with $\alpha$. Later on a viscous–buoyancy balance dominates, at a time obtained by imposing that the viscous drag, turbulent drag and buoyancy terms are of the same order of magnitude

(2.11)\begin{equation} \nu\dfrac{\partial^2u }{\partial y^2}\sim g^{\prime}\dfrac{\partial h}{\partial r}\sim\tilde{c}|u|^{\lambda-1}u, \end{equation}

or

(2.12a,b)\begin{equation} \dfrac{12\nu u}{h^2}\sim g^{\prime}\dfrac{h}{ut_{v1}}\sim \tilde{c}u^{\lambda},\quad hu^2t_{v1}^2\sim qt_{v1}^\alpha, \end{equation}

which leads to the scaling

(2.13)\begin{equation} t_{v1}\sim \left(\dfrac{g^{\prime} q}{\tilde{c}}\right)^{1/(3-\alpha)}\left(\dfrac{\tilde{c}h^2}{12\nu}\right)^{(3+\lambda)/[(\lambda-1)(3-\alpha)]}. \end{equation}

Considering the viscous drag due to the obstacles, instead of the viscous drag due to the bottom, results in

(2.14a,b)\begin{equation} \dfrac{\kappa\nu u}{s^2}\sim g^{\,\prime}\dfrac{h}{ut_{v2}}\sim \tilde{c}u^{\lambda},\quad hu^2t_{v2}^2\sim qt_{v2}^\alpha, \end{equation}

where $s$ is a scale of the spacing of the obstacles and $\kappa$ is the density of the obstacles. The corresponding time of transition is

(2.15)\begin{equation} t_{v2}\sim \left(\dfrac{g^{\prime} q}{\tilde{c}}\right)^{1/(3-\alpha)}\left(\dfrac{\tilde{c} s^2}{\kappa\nu}\right)^{(3+\lambda)/[(\lambda-1)(3-\alpha)]}. \end{equation}

Note that, for $\lambda =1$, the two times $t_{v1}$ and $t_{v2}$ are undefined since viscous resistance and drag scale with $u$ and are indistinguishable. In conclusion, the obtained self-similar solution is intermediate asymptotic in the interval $t_c< t< t_v$ where $t_v=\min (t_{v1},t_{v2})$. The condition $t_{v2}< t_{v1}$ corresponding to viscous drag due to vegetation dominating with respect to the viscous drag due to the bottom, for $\alpha <3$ requires that $s< h\sqrt {\kappa /12}$, which for $\kappa =0.10$ and $h=1\ \textrm {cm}$ indicates $s\approx 1\ \textrm {mm}$, a condition generally not satisfied. A waxing inflow rate with $\alpha >3$ is unrealistic and, moreover, we expect it to occur for a short time interval, not sufficient to allow viscosity to dominate the balance with buoyancy. For this reason, we limit our analysis to a viscous–buoyancy balance regime due to the floor of the tank and not to the vegetation. The previous results for $\lambda =2$ correspond to values given in Hatcher et al. (Reference Hatcher, Hogg and Woods2000).

Equation (2.13) can be also expressed as a function of the parameters of the flow process as

(2.16)\begin{equation} t_{v1}\sim\left(\dfrac{g^{\prime} q}{\tilde{c}}\right)^{(3 \lambda +1)/[7 \lambda -3-\alpha (3 \lambda +1)]}\left(\dfrac{\tilde{c}^3}{12 \nu g^{\prime 2}}\right)^{(\lambda +3)/[7 \lambda -3-\alpha (3 \lambda +1)]}, \end{equation}

where $h$, retained in (2.13) only for comparing $t_{v1}$ and $t_{v2}$, has been eliminated. Equating (2.9) and (2.16) yields a time scale $t^*$ and volume coefficient scale $q^*$

(2.17)$$\begin{gather} t^*=\left[\dfrac{1}{(12 \nu g'^2)^{({\lambda-1 })}\tilde{c}^{4}}\right]^{1/(\lambda+3)} , \end{gather}$$

(2.18)$$\begin{gather}q^*=\left[\dfrac{(12 \nu g'^2)^{\alpha (\lambda -1)-2 \lambda +6} }{ g'\tilde{c}^{4 (3-\alpha)}}\right]^{1/(\lambda+3)}, \end{gather}$$

that allow us to express the two times in dimensionless form

(2.19)\begin{equation} \tilde{t}_c\sim \tilde{q}^{(1-\lambda)/[\alpha (\lambda -1)-2 \lambda +6]}, \end{equation}

and

(2.20)\begin{equation} \tilde{t}_{v1}\sim\tilde{q}^{(3 \lambda +1)/[7 \lambda -3-\alpha (3 \lambda +1)]}, \end{equation}

where $\tilde {t}=t/t^*$ and $\tilde {q}=q/q^*$. The exponent in (2.19) is always negative for $1<\lambda <2$ and $\alpha >0$, and $\tilde {t}_c$ monotonically decreases with $\tilde {q}$. The exponent in (2.20) is positive if $\alpha <\alpha _c\equiv (7\lambda -3)/(3\lambda +1)$ and it is negative otherwise. Figure 2(a) shows the theoretical dimensionless times vs the volume coefficient, as a function of $\alpha$ and for $\alpha <\alpha _c$. The hatched area refers to $\alpha =0$ and $\lambda =2$ and is the domain of existence of an intermediate asymptotic similarity solution, which is possible only for $\tilde {q}>1$: the initial stage of the GC is dominated by inertia, which progressively becomes less important leaving space to turbulent drag of the vegetation. The turbulent drag–buoyancy balance is finally substituted by a viscous–buoyancy balance. The domain of existence of the turbulent drag–buoyancy balance widens with $\alpha$ and also for reduced $\lambda$, and for $\alpha \to \alpha _c$ viscosity never dominates over turbulent drag.

Figure 2.

Evolution of the balance regime for a radial GC in the presence of vegetation as a function of the time exponent of the volume of denser fluid. (a) Diagrams for $\alpha <\alpha _c\equiv (7\lambda -3)/(3\lambda +1)$ and (b) for $\alpha >\alpha _c$. The continuous curves refer to $\lambda =2$, while the dashed curves refer to $\lambda =1.5$. The hatched areas represent the domain where the intermediate asymptotic solution exists with turbulent drag–buoyancy balance; the intermediate asymptotic solution regime occurs after the initial phase, dominated by inertia, ending before the phase when viscosity is dominant.

Figure 2(b) shows the same data of figure 2(a) for $\alpha >\alpha _c$, with the hatched area referring to $\alpha =2$ and $\lambda =2$. The intermediate asymptotic similarity solution is allowed only for $\tilde {q}<1$, the domain of existence becomes smaller for increasing $\alpha >\alpha _c$.

This analysis is approximate because it refers to homogeneous and invariant assumptions for $h$, $u$ and for $\lambda$, $\tilde {c}$, which in real GCs are variable in space and time. In particular, near the front the current is always in the viscous–buoyancy balance regime. However, it provides useful insights to the discussion of the deviations of the theoretical model from experiments and to framing of the validity limits of a self-similar solution.

2.4. Solution

When $\tilde {c}$ and $\lambda$ are constant and the volume is of the form $\mathcal {V} = q t^\alpha$ (including the fixed-volume LR case $\alpha = 0$) analytical solutions of similarity type can be obtained, as follows.

Let $y = r/r_N(t)$. The equations of motion can be expressed in terms of the variables $h(y,t)$ and $u(y,t$). The domain of solution is $y \in [0,1]$ and the nose of the current is at $y=1$. We seek a similarity solution of the form

(2.21a–c)\begin{equation} r_N = \mathcal{K} t^\gamma, \quad h = t^\delta \mathcal{H}(y), \quad u = \dot{r}_N \mathcal{U}(y), \end{equation}

where the upper dot denotes time derivative and $\mathcal {K}, \gamma, \delta$ are constants. The task is to obtain the values of $\mathcal {K}, \gamma, \delta$ and the profiles $\tilde {\mathcal {H}}(y), \mathcal {U}(y)$ subject to the boundary conditions. The boundary conditions at the nose ($y = 1$) are $\mathcal {U}(1) = 1, \mathcal {H}(1) = 0$. The global volume balance (per radian) is

(2.22)\begin{equation} \mathcal{V} = \int_0^{r_N} h r\,\textrm{d}r = \mathcal{K}^2t^{2\gamma} t^\delta\int_0^1 \mathcal{H}(y) y \,\textrm{d} y = q t ^{\alpha}; \end{equation}

and yields

(2.23a,b)\begin{equation} 2 \gamma + \delta = \alpha; \quad \mathcal{K}^2 J = q, \end{equation}

where $J = \int _0^1 \mathcal {H}(y) y \,\textrm {d} y$.

Substitution of the similarity form (2.21a–c) into the momentum equation (2.5) yields, after some algebra

(2.24)$$\begin{gather} \delta ={-} \lambda + \gamma (\lambda + 1); \end{gather}$$

(2.25)$$\begin{gather}|\mathcal{U}(y)|^{\lambda-1}\mathcal{U}(y) ={-}\tilde{\mathcal{H}}'(y) , \end{gather}$$

where

(2.26a,b)\begin{equation} \tilde{\mathcal{H}}(y) = \frac{ \mathcal{H}(y)}{C}; \quad C = \frac{\tilde{c}}{g^{\prime}} \gamma ^\lambda \mathcal{K}^{1 + \lambda}. \end{equation}

Combining (2.23a,b) with (2.24) we find

(2.27)\begin{equation} \gamma = \frac{\alpha + \lambda}{3 + \lambda}. \end{equation}

The continuity equation reads

(2.28)\begin{equation} \frac{\delta}{\gamma} \, \tilde{\mathcal{H}}(y) + (\mathcal{U} - y) \tilde{\mathcal{H}}'(y) + \tilde{\mathcal{H}}(y) \mathcal{U}'(y) ={-} \frac{\mathcal{U}}{y} \tilde{\mathcal{H}}(y). \end{equation}

For given $\lambda$ and $\alpha$ we obtain $\gamma$ and $\delta$ by (2.27) and (2.24), then solve (2.25) and (2.28). Note that $\mathcal {U}(y)$ and $\tilde {\mathcal {H}}(y)$ are dimensionless. An analytical solution of (2.25) and (2.28) can be obtained for $\delta /\gamma = -2$

(2.29a,b)\begin{equation} \mathcal{U} =y; \quad \tilde{\mathcal{H}}(y) = \frac{1}{1+ \lambda} (1 - y^{1+ \lambda}). \end{equation}

This is valid when

(2.30a–c)\begin{equation} \gamma = \frac{\lambda}{3 + \lambda}; \quad \delta ={-}2 \gamma; \quad \alpha = 0. \end{equation}

The validity of this result can be verified by direct substitution into the equations of motion, and it is evident that the boundary conditions $\mathcal {U}(0) = 0, \tilde {\mathcal {H}}(1) =0$ are satisfied. This describes the propagation of a GC of fixed volume.

For influx values of $\alpha >0$ the similarity profiles can be calculated by numerical integration. Typical results are shown in figure 3. We recall that the solution is irrelevant for small $y$, because a realistic source must have a finite radius. On the other hand, the solution is well behaved about the nose. Using an expansion in powers of $1-y$, we find that for small values of $1-y$ the solution of (2.25) and (2.28) can be approximated by

(2.31a,b)\begin{equation} \tilde{\mathcal{H}} = (1-y) + \lambda \frac{\delta}{4 \gamma }(1-y)^2 + \cdots, \quad \mathcal{U} = 1 + \frac{\delta}{2 \gamma }(1-y) + \cdots . \end{equation}

Consequently, for given $\alpha$ and $\lambda$ we calculate the profiles $\mathcal {U}(y), \, \tilde {\mathcal {H}}(y)$ by numerical integration from $y = 1- \varDelta$ to smaller values of $y$. Here, $\varDelta$ is some small interval which allows the application of the approximation (2.31a,b) as the initial condition for the integration, thus avoiding the problem of the zero coefficient of $\mathcal {U}'(1)$ in (2.28).

Figure 3.

Theoretical profiles (a) $\tilde {\mathcal {H}}$ and (b) $\mathcal {U}$ as functions of $y$ for $\alpha = 0.5 \text {--} 3$, $\lambda = 1$ (dotted lines) and $\lambda = 2$ (solid lines).

For $\alpha = 3$ we find $\gamma = \delta = 1$ independent of $\lambda$; the aspect ratio of the current does not change with $t$. Typical solutions with influx conditions relevant to the present investigation are illustrated in figure 3.

The front position in dimensional variables is

(2.32)\begin{equation} r_N=\left(\dfrac{qg^{\prime}}{\tilde{c}\,\gamma^{\lambda}\tilde{J}}\right)^{1/(3+\lambda)}t^{\gamma}, \end{equation}

where $\tilde {J}=\int _0^1\tilde {\mathcal {H}}(y)y\,\textrm {d} y$, which for $\alpha =0$ becomes

(2.33)\begin{equation} r_N=\left(\dfrac{qg^{\prime}}{\tilde{c}}\right)^{1/(3+\lambda)}\left[\dfrac{2(3+\lambda)^{\lambda+1}}{\lambda^{\lambda}}\right]^{1/(3+\lambda)}t^{\lambda/(3+\lambda)}. \end{equation}

Figure 4 shows the numerical value of $\tilde {J}$ for different $\alpha$ and $\lambda$.

Figure 4.

Value of the integral $\tilde {J}$ as a function of $\alpha$ for $\lambda =1.5,2$.

3.1. Set-up for the GCs

We ran a series of experiments at the Hydraulic Laboratory of the University of Parma. Our tank was a poly-methyl-methacrylate circular sector, with a radius of $R = 180 \ \mathrm {cm}$ and an angle $\beta = 30^{\circ }$ as sketched in figure 5(a). The vegetation was modelled by (i) 200 vertical plastic rods, with a density $\kappa = 0.050$, as sketched in figure 5(a), and (ii) 320 rods, with a density $\kappa = 0.081$. The rods, with diameter $D = 1.6$ cm, were positioned according to an equilateral triangular mesh grid, with the edge of the triangles equal to 5.6 cm for $\kappa = 0.050$ and 5.0 cm for $\kappa = 0.081$.

Figure 5.

Experimental set-up with $\kappa = 0.050$. (a) Top view; (b) lateral view of a LR experiment; and (c) lateral view of a continuous flux experiment. Units are in centimetres. Images are not corrected for distortion.

For the LR experiments, a stainless steel planar gate 0.2 cm thick was positioned at $r_0 = 55 \ \mathrm {cm}$ and manually lifted just before the start of the experiment, see figure 5(b). The opening of the gate took approximately 0.2 s, a negligible time when compared with the duration of the experiments, $t = 40\text {--}60 \ \mathrm {s}$. A criterion for establishing the maximum gate opening time was formulated by Lauber & Hager (Reference Lauber and Hager1998) for dam breaks on a dry horizontal bottom with a vertical lift gate: if the opening occurs in a time greater than $t_{min}=\sqrt {2h_0/g^{\prime }}$, the initial flow field is distorted by the presence of the flat gate surface, increasing the discrepancy between theory and experiments. Applying this criterion, the critical condition for this experimental activity corresponds to $t_{min}=0.8\ \textrm {s}$, more than the opening time of 0.2 s for the present experiments.

For the experiments with a continuous source of flux, a home-made diffuser was inserted at $r_0 = 25 \ \mathrm {cm}$, see figure 5(c). The diffuser was connected to a pump controlled in feedback by a LabView software. Different experiments were conducted with a constant, waxing or waning volume inflow rate.

The lateral view of the experiments was recorded by a full HD video-camera ($1920\ \textrm {pixels} \times 1080\ \textrm {pixels}$, iPhone 7, Apple Inc.), working at 30 frames per second (fps). A 5 cm square grid stuck on the inner side of the front lateral wall was used to convert pixels into laboratory coordinates, with the aid of a proprietary Matlab code. The algorithm preliminary requires (i) the acquisition of the pixels of the knots of the grid; (ii) the computation of a conversion polynomial function for the two horizontal and vertical directions, mapping pixels into laboratory coordinates. The order of the polynomial is automatically chosen to minimize the root-mean-square deviation between the grid knots in laboratory coordinates and the reconstructed values starting from the pixels in the image. The conversion polynomials allow us to compensate for defects in the optics and for the position of the camera (Longo et al. Reference Longo, Ungarish, Di Federico, Chiapponi and Maranzoni2015,  Reference Longo, Ungarish, Di Federico, Chiapponi and Addona2016), and are used to transform the pixels corresponding to the instantaneous profile of the current into laboratory coordinates.

The top view was recorded by a 4 k video camera (3840 pixels $\times$ 2160 pixels, iPhone 11, Apple Inc.), working at 30 fps. The time evolution of the position of the nose of the current was measured by means of three radial grids glued at the base of the tank. High-frequency neon lamps provided stable and homogeneous illumination at the back wall of the tank. Softened tap water, with density $\rho _a = 1000 \ \mathrm {kg}\ \mathrm {m}^{{-3}}$, was always used as ambient fluid, although sometimes $\rho _a = 1001 \text {--}1002 \ \mathrm {kg}\ \mathrm {m}^{-3}$ because of salty traces left from previous experiments. The dense current was made of tap water, sodium chloride (NaCl) and red aniline dye, well mixed before the use.

For practical reasons, the fixed top condition is replaced by the free-surface condition, with no lid on the top of the tank. There is theoretical and experimental evidence that for Boussinesq systems (as considered in this paper) the difference between fixed and open top has negligible influence on the flow of the current, see Ungarish (Reference Ungarish2020).

In order to have an estimate of the mixing between the current and the ambient, some samples of the dense fluid were collected during some of the experiments listed in table 1, at the bottom of the current and a few millimetres below the interface, at $r = 90$, 170, 175 cm. The samples were manually collected with a syringe connected to a 2 mm brass pipe, and then analysed with a refractometer.

Table 1.

Parameters of the experiments with rods of diameter $D=1.6\ \textrm {cm}$: $\rho _{a,c}$ is the density of the ambient/current fluid, $g^{\prime }$ is the reduced gravity, $H$ is the initial height of the ambient fluid, $h_0$ is the height of the current in the LR experiments, $q'\equiv q(1-\kappa )$ is the inflow rate coefficient referring to the $30^\circ$ sector tank, $\kappa$ is the density of vegetation, $\alpha$ and $\gamma$ are constants; $\gamma _{th}$ is the theoretical value evaluated for $\lambda = 1, 2$, while $\gamma _{exp}$ is the experimental value; $U = \sqrt {g^{\prime } H}$ is the velocity scale, $T = H/U$ is the time scale. The abbreviations in the ‘Remark’ column stand for LR, LR full depth/partial depth (f/p), constant inflow rate (CF), waxing (WX) and waning (WN) inflow rate, respectively. Note that, in the comparison between experiments and theory, the fact that the volume in (2.26) includes both fluid and vegetation has been taken into account.

Appendix A describes further experiments for measuring the drag coefficient of a set of rods, in the same configuration adopted for the GC experiments.

3.2. The uncertainty in variables and parameters

We considered the instrument accuracy and the sequence of operations during tests to estimate the uncertainty of the variables and parameters. Hydrometers with an accuracy of $10^{-3}\ \mathrm {g}\ \mathrm {cm}^{-3}$ were used to measure density, hence the corresponding uncertainty for the reduced gravity $g^{\prime } = (1-\rho _a/\rho _c)g$ is $\Delta g^{\prime }/g^{\prime } \le 0.2\,\%$. The level of the dense fluid in the lock and the level of the ambient fluid were measured by a ruler with an accuracy of $0.2\ \textrm {cm}$. The relative uncertainty is $\Delta h_0/h_0 \le 2\,\%$. The volumetric flow rate of the pump was controlled within 1 % of accuracy.

The Reynolds number of the current has an uncertainty $\Delta Re/Re \le 4\,\%$, also based on the assumption of an uncertainty of $1\,\%$ in estimating the kinematic viscosity of salty water. The resolution in grabbing the lateral profiles of the dense current is approximately 0.1 cm pixel$^{-1}$, while the overall uncertainty due to parallax errors is approximately 0.3 cm.

The velocity of water during drag coefficient measurements has an uncertainty $\Delta V/V = 2\,\%$ at high inflow rates and $\Delta V/V = 4\,\%$ at low inflow rates. Reynolds number of the vegetation has an uncertainty $\Delta Re_{{rod}}/Re_{{rod}} \le 6\,\%$ and the arrangement for measuring the drag force gives results with an uncertainty less than 10 %.

The inflow produces an increase of the level of the ambient fluid. The underlying idea of the thin-layer model is that the vertical velocity is much smaller than the horizontal velocity, $w\ll u$. If we consider a constant inflow rate $q$, the rate of increase of level can be expressed as $w=2q/(\beta R^2),\beta =30^\circ$, which for the largest value $q=400\ \textrm {cm}^3\ \textrm {s}^{-1}$ of our experiments (exp. 33 in table 1) results in $w=0.04\ \textrm {cm}\ \textrm {s}^{-1}$, compared with a typical radial speed of $u=10\ \textrm {cm}\ \textrm {s}^{-1}$, hence the condition $w\ll u$ is satisfied. Over 25 s of experiment, the variation of the ambient fluid level equals $\Delta H=1\ \textrm {cm}$, representing 5.2 % of the initial ambient fluid $H=19\ \textrm {cm}$. This value is reached at the end of the experiment, and on average, $\Delta H/H=2.6\,\%$ which is in the range of the other experimental errors. In most of the other experiments $\Delta H/H$ is much smaller because $q$ is smaller and even null for LR experiments.