2.1. Governing equations

We model the two-dimensional motion of relatively dense fluid following instantaneous release from a quiescent state in a lock of dimensional depth $Z$ and length $X$ surrounded by a relatively deep and dynamically passive ambient. This two-dimensional model is equivalent to a laterally uniform three-dimensional flow along a rectangular channel. Driven by its density difference with the ambient, the released fluid flows along the underlying impermeable boundary forming a gravity current. On the assumption that the motion is shallow ($Z \ll X$) and predominantly parallel with the basal boundary, the pressure adopts a hydrostatic distribution to leading order. Neglecting mixing with the ambient and assuming the inclination of the bed is small, the dimensionless governing equations are the nonlinear shallow water equations (e.g. Peregrine Reference Peregrine1972; Ungarish Reference Ungarish2020)

(2.1a,b)\begin{equation} {\frac{\partial h}{\partial t} + \frac{\partial }{\partial x} (uh) = 0,}\quad \text{and}\quad {\frac{\partial }{\partial t}(uh) + \frac{\partial }{\partial x} {\left( u^2 h + \frac{h^2}{2} \right)} ={-}h \frac{\mathrm{d} b}{\mathrm{d} x},} \end{equation}

which represent conservation of mass and the balance of momentum respectively. We have non-dimensionalised the horizontal coordinate $x$ by $X$; the time $t$ by ${X}/{\sqrt {g'Z}}$; the depth of the fluid $h(x,t)$ and bed elevation $b(x)$ by $Z$; and the velocity of the fluid $u(x,t)$ by $\sqrt {g'Z}$, where $g' \equiv {g (\rho _f - \rho _a)}/{\rho _f}>0$ is the reduced gravity, $g$ is the acceleration due to gravity, $\rho _f$ is the density of the flowing layer and $\rho _a$ is the density of the dynamically passive ambient. When considering volumes of fluid, we scale by the width of the implied rectangular channel, and therefore calculate the area in the two-dimensional model.

In regions where $u$ and $h$ are continuous (2.1) may be written in terms of the characteristic invariants $\alpha \equiv u + 2 h^{1/2}$ and $\beta \equiv u - 2 h^{1/2}$ (e.g. Stoker Reference Stoker1957)

(2.2a)\begin{gather} \frac{\partial \alpha}{\partial t} + \lambda \frac{\partial \alpha}{\partial x} + \frac{\mathrm{d} b}{\mathrm{d} x} = 0,\quad \text{where}\ \lambda \equiv u + h^{1/2}, \end{gather}

(2.2b)\begin{gather}\frac{\partial \beta}{\partial t} + \mu \frac{\partial \beta}{\partial x} + \frac{\mathrm{d} b}{\mathrm{d} x} = 0,\quad \text{where}\ \mu \equiv u - h^{1/2}. \end{gather}

We term the curves ${\mathrm {d}x}/{\mathrm {d}t} = \lambda$ as the $\alpha$-characteristics, and ${\mathrm {d}x}/{\mathrm {d}t} = \mu$ as the $\beta$-characteristics. When $b$ is constant then $\alpha$ is constant on $\alpha$-characteristics and $\beta$ is constant on $\beta$-characteristics. The flow is subcritical when the characteristics move in opposite directions, that is $\lambda \mu < 0$ or equivalently ${\left | u \right |}< h^{1/2}$; supercritical when the characteristics move in the same direction, that is $\lambda \mu > 0$ or equivalently ${\left | u \right |}>h^{1/2}$; and critical when one of the characteristics is stationary, that is $\lambda \mu = 0$ or equivalently ${\left | u \right |}=h^{1/2}$.

Along shock curves $x = x_s(t)$ where the solution is discontinuous (but $b$ is continuous) we enforce conservation of the mass and momentum fluxes (e.g. Stoker Reference Stoker1957), given by

(2.3a,b)\begin{equation} [{(u-s) h}]_{-}^{+} =0, \quad\text{and} \quad\left[{(u-s)^2 h + \frac{h^2}{2}}\right]_{-}^{+} =0, \end{equation}

respectively, where $s \equiv {\mathrm {d}x_s}/{\mathrm {d}t}$ and $[{\,f(x,t)}]_{-}^{+} \equiv f(x_s^+(t),t) - f(x_s^-(t),t)$. In addition, we require that any shocks absorb characteristics of one family, consistent with the Lax entropy condition (Lax Reference Lax1957). Shocks for which $\lambda ^- > s > \lambda ^+$ are termed $\alpha$-shocks and satisfy $h^->h^+$, $s>u^->u^+$, whilst those with $\mu ^- > s > \mu ^+$ are termed $\beta$-shocks and satisfy $h^-< h^+$, $u^->u^+>s$.

The domain is $0 \leq x \leq x_f(t)$, and the initial conditions are $h(x,0)=1$, $u(x,0)=0$, $x_f(0)=1$. The bed is horizontal up until the barrier at $x=L_1 \geq 1$, that is $b(x) = 0$ for $0 \leq x < L_1$, and the fluid is confined to the rear by an insurmountable barrier imposing $u(0,t)=0$, which we call the back wall. The motion during $t>0$ is due to the removal of the lock at $x=1$ which produces a moving front obeying (Benjamin Reference Benjamin1968)

(2.4)\begin{equation} \dot{x}_f \equiv \frac{\mathrm{d} x_f}{\mathrm{d} t} = u = {Fr} \, h^{1/2} \quad \text{at}\ x = x_f (t), \end{equation}

where ${Fr}$ is the frontal Froude number, which is dependent on the density ratio $\rho _f/\rho _a$. By this definition of ${Fr}$, Boussinesq currents ($\rho _f/\rho _a \approx 1$) take a theoretically derived value of $\sqrt {2}$ (Benjamin Reference Benjamin1968; Ungarish & Hogg Reference Ungarish and Hogg2018) and experimentally measured value of $1.19$ (Huppert & Simpson Reference Huppert and Simpson1980), while dam-break flows operate in the limit ${Fr} \to \infty$ and the front corresponds to vanishing height $h(x_f,t)=0$. In our analysis we will assume that ${Fr}>0$ is constant. The configuration is shown in figure 2.

Figure 2. The flow configuration, showing the dimensionless variables. Panel (a) shows the early time release with a moving front, and panel (b) the flow over the barrier during subcritical overtopping. The vertical scale is exaggerated and the horizontal axis nonlinearly scaled to show the current and the barrier. In dotted is the initial release, in solid is the depth field, and dashed lines are used to indicate measurements for the barrier.

A barrier is located across the range $L_1 < x < L_2$, and over this interval ${\mathrm {d}b}/{\mathrm {d}x} \geq 0$ with the crest at elevation $B \equiv b(L_2)$, thus the overall angle of the incline is $\theta \equiv \arctan {\left ({B Z}/{\epsilon X} \right )}$ where $\epsilon \equiv L_2 - L_1$. We assume the barrier is steep relative to the scales of the current, so that $1 \ll B/\epsilon = (X/Z) \tan \theta$. Note that this does not prevent $\theta$ from being small because $Z \ll X$. For the current to be able to surmount the barrier we require that $B \lesssim 1$ (it will turn out that we require $B<2$, § 4.1), so $0 < \epsilon \ll 1$.

Because the barrier is short relative to the length of the current, we neglect the ${O}\left ({\epsilon }\right )$ effects of its finite extent by including it as a boundary condition at $x=L$. That is, in the model we analyse in all subsequent sections, the bed elevation is set to zero across all of $0< x< L$, where $x=L$ is the location of the boundary condition which models the barrier. In principle, we may set the boundary to be at any location $L_1 < L < L_2$, but to minimise the error in measuring the volume of fluid that escapes the domain (§ 5) we ensure that the confined volume (that is, the volume of the region behind the barrier) $V_c$ is the same as in the configuration with $\epsilon$ non-vanishing, where

(2.5)\begin{equation} V_c \equiv \int_0^{L_2} B - b(x) \,\mbox{d}{x}. \end{equation}

That is, we choose $L \equiv V_c/B$.

In what follows we denote the velocity and depth of the fluid at the base of the barrier by $u_b$ and $h_b$. When considering the configuration with non-vanishing $\epsilon$ these are expressed by $u_b(t) = u(L_1,t)$ and $h_b(t) = h(L_1,t)$, whereas at the crest of the barrier the velocity and depth are $u_c(t) = u(L_2,t)$ and $h_c(t) = h(L_2,t)$. The overtopping model developed below provides a dynamical connection between the states at the base and the crest and leads to conditions on $u_b$ and $h_b$. For our simplified model of the interaction between the current and barrier, which is accurate in the limit $\epsilon \rightarrow 0^+$, we impose these as conditions at $x=L$, that is the consideration of the barrier provides constraints on $u_b(t) = u(L,t)$ and $h_b(t) = h(L,t)$. We will also utilise the flow state just upstream of the barrier, in the simplified model this is given by $u_l(t) = \lim _{x \to L^-} u(x,t)$ and $h_l(t) = \lim _{x \to L^-} h(x,t)$.

To develop a boundary condition that captures the effect of the barrier on the fluid, we use its relative shortness, $\epsilon \ll 1$, which implies the time scale of the fluid flow local to the barrier is vastly smaller than that of the flow in the bulk. Thus, the dynamics is quasi-static and steady state analysis may be applied. Following Long (Reference Long1954, Reference Long1970) and Baines (Reference Baines1995), we assume the flow to be continuous between the base and the crest (shocks are known to be unstable, see Baines & Whitehead Reference Baines and Whitehead2003). Thus, the dimensionless energy density $E \equiv \tfrac {1}{2} u^2 + h + b$ and volume flux $q \equiv uh$ are constant in space and time, with $q_b$, $E_b$ the values evaluated at the base of the barrier, and $q_c$, $E_c$ at the crest. At constant $q$ and $b$ the energy takes its minimal value at critical flow, $h = q^{2/3}$, at which $E = \tfrac {3}{2} q^{{2}/{3}} + b$. For larger energies there are two possible corresponding values of $u$ and $h$, a subcritical state and a supercritical state. For the fluid to overtop the barrier the energy at the crest $E_c$ must be at least the critical energy there,

(2.6a)\begin{equation} E_c \geq {E_c}|_{{crit}} = \tfrac{3}{2} q_c^{{2}/{3}} + B = \tfrac{3}{2} q_b^{{2}/{3}} + B, \end{equation}

thus

(2.6b)\begin{equation} \Delta E_b \equiv E_b - {E_c}|_{{crit}} = \Delta E (u_b,h_b,B) \geq 0 , \end{equation}

where

(2.6c)\begin{equation} \Delta E (\mathscr{U},\mathscr{H},\mathscr{B}) \equiv \tfrac{1}{2}\mathscr{U}^2 + \mathscr{H} - \tfrac{3}{2} (\mathscr{U} \mathscr{H})^{{2}/{3}} - \mathscr{B} \end{equation}

is the energy discrepancy. Note that this analysis assumes there is no energy dissipation as the fluid flows over the obstacle, which is justifiable for shallow sloped barriers. For steeper slopes, Skevington & Hogg (Reference Skevington and Hogg2020) showed, for open channel flows, how to account for the small amount of dissipation present. This is not pursued here.

We assume that the flow downstream of the barrier is downhill and supercritical, and conclude that there are three possible modes that can be exhibited.

1. (i) Supercritical overtopping. The flow on the incline is supercritical, $u_b > h_b^{1/2}$, and satisfies $\Delta E_b > 0$, which imposes no boundary condition on the bulk flow.

2. (ii) Subcritical overtopping. The flow on the incline is subcritical and transitions to supercritical beyond the crest. Thus, it is critical at the crest, and the appropriate boundary condition is $\Delta E_b = 0$, taking the solution in $0 \leq u_b \leq h_b^{1/2}$.

3. (iii) Blocked flow. If $\Delta E_b < 0$ then the fluid cannot overtop the barrier, and the appropriate boundary condition is $q_b = 0$.

The strict inequality on energy for the case of supercritical overtopping ensures the stability of the continuous flow on the incline. For supercritical flow with $\Delta E_b = 0$, a small perturbation to the values of $u_b$, $h_b$ may cause $\Delta E_b$ to become negative, at which point an upstream propagating bore will be generated, transitioning to one of the other two modes.

Identifying which of the three modes is selected depends on the flow local to the barrier $u_l$, $h_l$. As discussed by Cozzolino et al. (Reference Cozzolino, Cimorelli, Covelli, Morete and Pianese2014), there is a choice of whether to permit supercritical overtopping in some cases. By the experimental measurements of Greenspan & Young (Reference Greenspan and Young1978) we expect that slopes of $\theta = 60^{\circ }$, perhaps greater, are able to produce a supercritical jet of overtopping fluid for an incident dam-break flow, and to capture this flow regime we permit supercritical overtopping when $u_l > h_l^{1/2}$ and $\Delta E_l \equiv \Delta E (u_l,h_l,B) > 0$. Otherwise, if a solution exists to imposing subcritical overtopping then this boundary condition is used, else we impose blocked flow. Collectively, we term all three modes and the method of selection the critical barrier boundary condition, and we examine the consequence of its imposition in § 3.

Works by other authors (e.g. Greenspan & Young Reference Greenspan and Young1978; Rottman et al. Reference Rottman, Simpson and Hunt1985) have used a different outflow condition where a constant surface elevation is imposed rather than conservation of energy. In Appendix A we show that the surface elevation model is unphysical, and that the conservation of energy condition is the physically permitted condition that minimises the change in surface elevation.

The model is validated by comparison with experiment in § 6, showing a good correspondence. However, we first analyse the properties and predictions of the model in §§ 3–5 to give context to the comparison.

2.2. Numerical methods

We simulate using the transformed shallow water system presented in Skevington (Reference Skevington2021a). The numerical method employs the central upwind scheme by Kurganov & Levy (Reference Kurganov and Levy2002). This is a finite volume scheme which requires reconstruction in each cell, for this purpose we use the suppressed minmod limiter from Skevington (Reference Skevington2021b). Boundary conditions are implemented using techniques developed in Skevington (Reference Skevington2021a), which ensure the solution is locally continuous at the boundaries for almost all time, and that there is no drift off error in the algebraic boundary conditions.

4.1. Barrier close to initial release

We assume that the barrier is sufficiently close that the presence of the back wall is not felt by the front of the current prior to impact. At the instant of release, the influence of the lock's removal creates a wave that travels to the back wall, which is then reflected forward towards the front of the gravity current (Hogg Reference Hogg2006). We neglect the influence of this forward propagating reflected wave in this section, which requires that the barrier is sufficiently close to the initial release (see § 5). Formally we let $L = 1+ \tilde {\delta }$ for some $\epsilon \ll \tilde {\delta } \ll 1$, and define $\tilde {x} = {(x-1)}/{\tilde {\delta }}$, $\tilde {t} = {t}/{\tilde {\delta }}$ so that, as $\tilde {\delta } \rightarrow 0$, the initial release occupies $\tilde {x} < 0$ and the barrier is at $\tilde {x} = 1$. At the instant of release the solution is a $\beta$-fan originating at $\tilde {x}=0$ that joins two uniform regions, and given explicitly by

(4.1a)\begin{equation} u = \frac{2 F}{F + 2}, \quad h = {\left( \frac{2}{F + 2} \right)}^2, \end{equation}

where

(4.1b)\begin{equation} F = \left\{ \begin{array}{@{}ll} 0, & \text{for } \ \dfrac{\tilde{x}}{\tilde{t}} \leq{-}1, \\ \dfrac{2 + 2 (\tilde{x}/\tilde{t})}{2 - (\tilde{x}/\tilde{t})}, & \text{for } -1 \leq \dfrac{\tilde{x}}{\tilde{t}} \leq 2 \dfrac{{Fr} - 1}{{Fr} + 2},\\ Fr, & \text{for } \ 2 \dfrac{{Fr} - 1}{{Fr} + 2} \leq \dfrac{\tilde{x}}{\tilde{t}} \leq \dfrac{2 {Fr}}{{Fr} + 2} \end{array} \right. \end{equation}

is the local Froude number. We express the solution in terms of $F$ rather than directly in terms of $\tilde {x}/\tilde {t}$ (e.g. Ungarish Reference Ungarish2020) for analytical simplicity in what follows. Note that $F$ increases monotonically with $\tilde {x}/\tilde {t}$ from $0$ to ${Fr}$ across the $\beta$-fan (the middle case in (4.1b)).

The solution (4.1) is valid until the front reaches the barrier, which occurs at time $\tilde {t} = {({Fr} + 2)}/{(2 {Fr})}$. To illustrate our analysis of the subsequent dynamics we use plots produced from numerical simulation (figure 6). These simulations were performed on a mesh of $1000$ cells on $-2 \leq \tilde {x} \leq 1$ initiated at the instant of collision, with a non-reflecting boundary condition imposed at $\tilde {x} = -2$ so that this boundary does not influence the ensuing dynamics.

Figure 6. The depth, $h$, and velocity, $u$, as functions of the scaled spatial coordinate, $\tilde {x}$, at various instances of time. The pairs of plots (a,b), (c,d) (e,f) and (g,h) are at parameter values of $B$ and ${Fr}$ marked on figure 7. The times are indicated by differing colours, with values given by the legends in (b,d,f,h). In (a,b) ${Fr} = 4.5$, $B = 0.85$, the times correspond to the instant of collision (blocked flow), a time just prior to critical overtopping, and three later times. In (c,d) ${Fr} = 15$, $B = 0.85$, the times correspond to the instant of collision (supercritical overtopping), a time just prior to overtopping ceasing, a time just prior to critical overtopping starting, and four later times. In (e,f) ${Fr} = 15$, $B = 1.05$, the times correspond to the instant of collision (supercritical overtopping), a time just prior to overtopping ceasing, and four later times. In (g,h) ${Fr} = 15$, $B = 0.3$, the times correspond to the instant of collision (supercritical overtopping), a time just prior to the supercritical overtopping transitioning to subcritical overtopping, and three later times.

Figure 7. Classification of the overtopping flow when a gravity current collides with a barrier that is sufficiently close to the initial release, with labels for each regime (§ 4.1). The boundary curves (solid) are labelled with the equation solved to compute them. The parameter values used in figure 6 are marked by $\times$. On the abscissa, we employ a transformation so that ${Fr}=0$ at the left and ${Fr} \to \infty$ at the right. The vertical dotted line represents ${Fr}=\sqrt {2}$, a Boussinesq current. The overtopping regimes are: B-I, supercritical transitioning to B-Ia blocked, B-Ib blocked then subcritical, B-Ic subcritical; B-II, subcritical; B-III, blocked, in B-IIIb transitioning to subcritical; B-IV subcritical, but generating a $\beta$-fan at the barrier rather than a $\beta$-shock.

The initial interaction with the barrier is now discussed. We emphasise that our model does not include mixing between the current and ambient fluid, which may occur during the initial collision between Boussinesq flows and relatively deep barriers. We note that the dilute fluid generated by this interaction will largely be transported downstream out of the domain, and should not affect the subsequent motion. The interaction corresponds to the boundary-Riemann problem solved in § 3. In particular, since ${Fr} > 0$, we find that four initial behaviours may occur, depending on ${Fr}$ and the relative height of the barrier: supercritical overtopping (B-Ia, B-Ib, B-Ic); subcritical overtopping with a $\beta$-shock (B-II) or a $\beta$-fan (B-IV) generated at the barrier; and blocked flow (B-IIIa, B-IIIb), see figure 7. These are divided by two curves. Firstly, $\Delta \tilde {E}_i \equiv \Delta \tilde {E} ({Fr},B) = 0$ is the boundary of supercritical overtopping, with $\Delta \tilde {E}_i$ the energy discrepancy when the front arrives at the barrier. The energy discrepancy across the domain, denoted $\Delta \tilde {E}$, is derived by substituting (4.1a) into (2.6c), yielding

(4.2)\begin{equation} \Delta \tilde{E}(F,B) \equiv \Delta E(u,h,B) = {\left(\frac{2}{F+2}\right)}^2 {\left(\frac{1}{2} F^2 + 1 - \frac{3}{2} F^{2/3}\right)} - B. \end{equation}

Secondly, $u_b = 0$, $h_b = B$, which by the shock conditions (2.3) imply $\tilde {S}({Fr},B) = 0$, where

(4.3)\begin{equation} \tilde{S}(F,B) \equiv {\left[ B + {\left(\frac{2}{F+2}\right)}^2 \right]} {\left[ B - {\left(\frac{2}{F+2}\right)}^2 \right]}^2 - 2 B F^2 {\left(\frac{2}{F+2}\right)}^4 \end{equation}

divides subcritical and blocked flow. In what follows we analyse the subsequent motion that occurs and we document the results in terms of the dimensionless height of the barrier $B$ and the Froude number at the front of the flow ${Fr}$, plotting the dynamics in figure 6 and including the regimes in figure 7.

If, at the instant of collision, the flow is blocked, then initially there will be a $\beta$-shock that travels from the barrier with a constant velocity until it interacts with the $\beta$-fan from the initial conditions. From this time on, the region of fluid between the shock and barrier progressively deepens, eventually attaining unit depth (cf. Hogg & Skevington Reference Hogg and Skevington2021). Therefore, for barriers of height less than $1$ the fluid eventually overtops (figure 6a,b), which we call regime B-IIIb, whereas for a barrier of height greater than $1$ the fluid never overtops, regime B-IIIa.

Alternatively, if at the instant of collision the fluid supercriticality overtops then no shock is generated and instead the solution for early $\tilde {t}$ is valid until $\Delta E_b = 0$ at the barrier. Thus the supercritical outflow will cease while the $\beta$-fan from the initial conditions is adjacent to the barrier. Specifically it will cease when $\Delta \tilde {E}(F,B)=0$ (for a given $B$), which can be solved to find $F$ which is the Froude number adjacent to the barrier. At this instant we must solve another boundary-Riemann problem, but it is substantively the same as the one solved above for the initial interaction, with $\alpha =2$ and the Froude number now being $F$. Consequently, the division between transition to subcritical overtopping (B-Ic, see figure 6g,h) and transition to blocked flow (B-Ia, B-Ib) is a line of constant $B$ which can be found by solving $\Delta \tilde {E}(F,B)=\tilde {S}(F,B)=0$, yielding $(F,B) = (4.47,0.661)$ (3 s.f.). As for flows that initially do not overtop (B-IIIa, B-IIIb), flows that transition from supercritical overtopping to blocked flow can subsequently transition to subcritical overtopping (B-Ib, see figure 6c,d) if $B<1$, and do not (B-Ia, see figure 6e,f) if $B>1$. On the dividing line between these regimes the supercritical overtopping ceases when $\Delta \tilde {E}(F,1)=0$, thus $F = 7.12$ (3 s.f.). We note that the given values of $F$ correspond to the values of ${Fr}$ on the bounding curve of supercritical overtopping $\Delta E_i=0$, and therefore mark the locations at which the dividing curves meet.

We note that as $\tilde {t} \rightarrow \infty$ the solution limits to the same solution as the boundary-Riemann problem with $\hat {h}_i = 1/B$, $\hat {u}_i = 0$ (§ 3). Therefore with $B<1$ we find subcritical overtopping, whereas with $B>1$ the flow is blocked at late times. For this reason, once these modes of overtopping have been reached, we expect no further changes.

The classification of dynamics is shown in figure 7. One surprising result is that for ${Fr} > 7.12$ the initial collision can only produce supercritical overtopping or blocked flow, and that supercritical is possible for $B>1$. That is to say, fluid can overtop a barrier taller than the initial release, and for ${Fr} \rightarrow \infty$ it can overtop a barrier twice the height of the release, as a consequence of energy conservation.

We emphasise that these results are valid before the finite extent of the release influences the overtopping through the arrival at the barrier or shock of the characteristic reflected from the back wall (formally $\tilde {\delta } \rightarrow 0$). The effects of a finite extent will modify the boundaries between the regimes in a way that depends on $L$.

4.2. Barrier far from initial release

We now assume that the barrier is sufficiently far from the initial release that the current is in similarity form when it reaches the barrier, which is the late time limit for all finite values of the Froude number, ${Fr}$, at the front of the current (Gratton & Vigo Reference Gratton and Vigo1994). We express the similarity solution in terms of the variables

(4.4a)\begin{equation} \breve{x} = \frac{x}{L}, \quad \breve{t} = \frac{t-t_0}{L^{3/2}}, \quad \xi = \frac{K \breve{x}}{\breve{t}^{2/3}}, \end{equation}

where $t_0$ is some arbitrary time offset and

(4.4b)\begin{equation} K = {\left(\frac{2}{3}\right)}^{{2}/{3}} {\left( \frac{1}{{Fr}^2} + \frac{\xi_D^3 - 1}{6} \right)}^{{1}/{3}}, \quad \xi_D = \max{\left( 1 - \frac{4}{{Fr}^2} , 0\right)}^{1/2}. \end{equation}

The similarity solution is, in terms of the variables $\breve {u} \equiv L^{1/2} u$, $\breve {h} \equiv L h$,

(4.4c)\begin{gather} \breve{u} = \frac{2 \xi}{3 K \breve{t}^{{1}/{3}}}, \quad \breve{h} = \frac{1}{9 K^2 \breve{t}^{{2}/{3}}} {\left( \xi^2 + \frac{4}{{Fr}^2} - 1 \right)}, \quad \text{for}\ \xi_D \leq \xi \leq 1, \end{gather}

(4.4d)\begin{gather}\breve{h} = 0,\quad \text{for}\ 0 \leq \xi < \xi_D. \end{gather}

In this rescaled system the remaining parameters are $V_c = BL$ (the confined volume (2.5)) and ${Fr}$ (the front condition (2.4)), and we determine the dynamical regimes after collision in terms of them. The similarity solution is valid until the front reaches the barrier, which occurs at time $\breve {t} = K^{{3}/{2}}$. At this time a boundary-Riemann problem must be solved at $\breve {x} = 1$ to determine the dynamics immediately after the collision (see § 3). This initial local interaction divides parameter space into four regimes corresponding to: blocked flow (C-III); subcritical overtopping with a $\beta$-shock (C-II) or a $\beta$-fan (C-IV) generated at the barrier; and supercritical overtopping (C-Ia, C-Ib, C-Ic) (see figure 8). These regimes are divided by two curves. Firstly, the curve $\Delta \breve {E}_i \equiv \Delta \breve {E} (K^{3/2},V_c) = 0$ (energy discrepancy at impact), where

(4.5)\begin{align} \Delta \breve{E} (\breve{t},V_c) &\equiv \left. \Delta E(\breve{u},\breve{h},V_c) \right|_{\breve{x} = 1} \nonumber\\ &= \frac{1}{9 K^2 \breve{t}^{2/3}} {\left[ \frac{3 K^2}{\breve{t}^{4/3}} + \frac{4}{{Fr}^2} - 1 - \frac{3 K^{2/3}}{2^{1/3} \breve{t}^{4/9}} {\left( \frac{K^2}{\breve{t}^{4/3}} + \frac{4}{{Fr}^2} - 1 \right)}^{2/3} \right]} - V_c \end{align}

(energy discrepancy computed at $\breve {x} = 1$ across $\xi _D \leq \xi \leq 1$) is the boundary of supercritical overtopping. Secondly, $\breve {u}_b = 0$, $\breve {h}_b = V_c$, which by the shock conditions (2.3) imply $\breve {S}(K^{3/2},V_c) = 0$, where

(4.6)$$\begin{align} \breve{S} (\breve{t},V_c) &= {\left[ V_c + \frac{1}{9~K^2 \breve{t}^{2/3}} {\left(\frac{K^2}{\breve{t}^{4/3}} + \frac{4}{{Fr}^2} - 1\right)} \right]} {\left[ V_c - \frac{1}{9 K^2 \breve{t}^{2/3}} {\left(\frac{K^2}{\breve{t}^{4/3}} + \frac{4}{{Fr}^2} - 1\right)} \right]}^2\nonumber\\ &\quad - \frac{2^3 V_c}{3^4~K^2 \breve{t}^{8/3}} {\left(\frac{K^2}{\breve{t}^{4/3}} + \frac{4}{{Fr}^2} - 1\right)}, \end{align}$$

is the boundary between subcritical overtopping and blocking. In what follows we analyse the subsequent transitions of overtopping behaviour that occur within each of the regimes.

Figure 8. Classification of the overtopping flow for a gravity current in similarity form (§ 4.2); (b) is the same as (a) except for a rescaling of the axes. Each regime is labelled, and the boundary curves (solid) are labelled with the equation solved to compute them. The circle marks the intersection of two dividing curves at ${Fr} = 7.03$, $V_c = 984$. On the abscissa, we employ a transformation so that ${Fr}=0$ at the left of (a) and ${Fr} \to \infty$ at the right, while in (b) we show $2 < {Fr} < \infty$. The vertical dotted line in (a) represents ${Fr}=\sqrt {2}$, a Boussinesq current. The overtopping regimes are: C-I, supercritical transitioning to C-Ia subcritical then blocked, C-Ib blocked, C-Ic dry; C-II, subcritical then blocked; C-III blocked; C-IV subcritical, but generating a $\beta$-fan at the barrier rather than a $\beta$-shock, transitioning to blocked.

If the fluid is unable to supercritically overtop at the instant of collision (i.e. blocked flow or subcritical overtopping) then two cases are possible. One case is when a $\beta$-fan is generated at the subcritically overtopping barrier (C-IV), so that the depth at the barrier $\breve {h}_b$ is smaller than the depth that arrived at the front. This continues until $\breve {h}_b$ reduces to $V_c$ at which time the flow is blocked. The other case is when an upstream propagating shock is formed. We find from simulation that, as time passes, the fluid depth between the shock and wall decreases. This can be rationalised as follows. Initially, the distance between the shock and barrier is small, and therefore the dynamics in this regime is quasi-static and has constant velocity throughout. Using the perturbed shock condition (C3), and the fact that upstream of the shock both $u$ and $h$ are decreasing, we deduce that $h$ downstream of the shock must decrease also. At later times the fluid is not quasi-static, and therefore we make use of the expressions involving the invariants, (C7), remembering that the shock sets the changes in $\alpha$ immediately downstream of the shock, these changes are then reflected off the barrier resulting in changes of opposite sign to $\beta$. The large coefficient of the perturbations to $\alpha$ suggests that $\alpha$ is decreasing and, therefore, $\beta$ is increasing, corresponding to a decreasing depth. Because the fluid adjacent to the barrier is becoming shallower, we deduce that the fluid will cease its initial overtopping (if any), and not overtop again. Thus, in regime C-III, no fluid makes it over the barrier, and in regime C-II the subcritical overtopping ceases after finite time.

For supercritical outflow we find that there are three possible types of subsequent motion. Firstly, it is possible that all of the gravity current may overtop the barrier. If $\min _{\breve {t}} \Delta \breve {E}(\breve {t},V_c) \geq 0$ (equivalent to $\min _{\breve {t}} \Delta \breve {E}(\breve {t},0) \geq V_c$) the energy of fluid reaching the barrier is sufficient to allow supercritical overtopping for all time, thus all of the fluid flows out of the domain supercritically (C-Ic). We note that for ${Fr} \geq 7.03$ (3 s.f.) the time of minimal $\Delta \breve {E} (\breve {t},0)$ is at the instant of collision, $\breve {t} = K^{3/2}$ (at ${Fr}=7.03$, $\Delta \breve {E}(K^{3/2},0) = 984$), thus if there is supercritical overtopping then it will certainly dry the domain. For supercritical overtopping that is not a part of C-Ic, we identify the time at which $\Delta \breve {E}(\breve {t},V_c) = 0$, and at this instant the fluid can either transition to subcritical overtopping (C-Ia) or blocked flow (C-Ib), following the boundary-Riemann problem at that instant. The dividing curve is when $\breve {h} = V_c$ and $\breve {u} = 0$ to the right of the $\beta$-shock generated at the barrier, and to find it we solve $\Delta \breve {E}(\breve {t},V_c) = \breve {S}(\breve {t},V_c) = 0$ as a coupled system for $\breve {t}$ and $V_c$. This curve will meet the corresponding dividing curve for subcritical outflow at the boundary of the supercritical regime, that is $\Delta \breve {E}(K^{3/2},V_c) = \breve {S}(K^{3/2},V_c) = 0$ which has solution ${Fr} = 4.47$, $V_c=133$. In addition, it can be shown that the $V_c = 0$ solution is found at ${Fr} = 2$. Note that, as in region C-II, the subcritical outflow of C-Ia will eventually transition to no-overtopping.

Classification of the dynamics is shown in figure 8. For ${Fr} \gg 1$ it is found that the current is able to surmount barriers that are much taller than the average depth of the current, that is $V_c$ is able to be very large and still permit outflow. This is because the current has most of its volume at the front, moving forwards with very high momentum. For more modest ${Fr}$ this effect is weakened.

5.1. Boussinesq gravity current: ${Fr} = \sqrt {2}$

We begin by establishing how tall the barrier must be to prevent outflow. To find this we run simulations imposing an insurmountable barrier at $x=L$ (that is impose the condition $u=0$ when the current reaches the barrier, which is equivalent to the limit $B \rightarrow \infty$), the shortest barrier required to prevent outflow will be equal to the maximum depth of fluid at the barrier over all time.

The variation of the maximum depth of fluid, $h_m$, with downstream distance to the barrier, $L$, arises from the interaction of two key flow features: the upstream moving shock that is generated by the arrival of the flow at the barrier, and the trajectory of the ‘reflected’ $\alpha$-characteristic from the back wall of the lock $(x=0)$ in response to the arrival of the rearmost $\beta$-characteristic formed by the removal of the lock-gate. It is this latter characteristic that ‘communicates’ the finite extent of the flow. If the barrier is sufficiently distant from the lock, then the reflected $\alpha$-characteristic catches up with the front of the motion before the gravity current reaches the barrier; however, for a closer barrier this characteristic intersects the shock. In figure 9, we plot the dependence of the maximum depth, $h_m$, on $L$ and observe that there are two values of $L$ at which the behaviour changes. In what follows we demonstrate how to evaluate these transitions.

Figure 9. The maximum depth of fluid at the barrier, $h_m$, as a function of the location of the barrier, $L$, for a (fully reflected) Boussinesq current (${Fr} = \sqrt {2}$). Crosses show numerical data; dotted line the maximum depth when the current is uniform at the front; and solid line the exact solution (Hogg Reference Hogg2006).

In § 4.1 we showed that the initial motion of the gravity current features a uniform region, leading a $\beta$-fan in which the depth and velocity vary (see (4.1)). On reflection from the barrier, the frontal uniform region develops a uniform reflection in a region adjacent to the barrier within which the fluid is motionless and the shock moves steadily upstream (see Hogg & Skevington Reference Hogg and Skevington2021). If the barrier is very close to the initial release $(L-1\ll 1)$ then the fluid progressively deepens at the barrier after the uniform state (reaching the asymptotic values $h_m\to 1$ as $L\to 1$). The deepening stops when the reflected $\alpha$-characteristic emanating from $x=0$ intersects the shock and then propagates further to the barrier. The time between the first arrival of the front of the gravity current and this $\alpha$-characteristic is reduced with increasing $L$ in this regime, and so the maximum depth, $h_m$, also reduces with $L$ (see figure 9 with $1< L<2.02$).

A change in dependence of $h_m$ with $L$ occurs if the $\alpha$-characteristic intersects the shock when the fluid beyond it is still in a uniform state. From (4.1), the depth and velocity of the front of the gravity current are given by $u_f=2Fr/(Fr+2)$ and $h_f=4/(Fr+2)^2$. On reflection from the barrier the fluid is motionless and of depth $h_b$, which by (2.3) satisfies

(5.1)\begin{equation} 2u_f^2 h_f h_b=(h_f-h_b)^2(h_f+h_b). \end{equation}

For example, when $Fr=\sqrt {2}$ we find $h_b = 0.930$ (3 s.f.). The speed of the shock is $s=u_fh_f/(h_f-h_b)$ when the gravity current front first arrives at the barrier at $t=(L-1)/u_f$. The position of the shock is therefore given by

(5.2)\begin{equation} x_s = L + \frac{u_fh_f}{h_f-h_b} {\left( t-\frac{L-1}{u_f} \right)}. \end{equation}

The reflected $\alpha$-characteristic meets the uniform region when

(5.3a,b)\begin{equation} x=1+(Fr-1)\left(\frac{Fr+2}{2}\right)^{1/2}\quad\text{and}\quad t=\left(\frac{Fr+2}{2}\right)^{3/2} \end{equation}

(see Hogg Reference Hogg2006). Thus the distance from the lock, $L=L_A$, at which the change of behaviour occurs is determined by substituting (5.3a,b) into (5.2). For $Fr=\sqrt {2}$, we find $L_A=2.02$ (3 s.f.; see figure 10a).

Figure 10. Plots of the characteristic trajectories for the collision of a gravity current with an insurmountable barrier for (a) $L = L_A$ and (b) $L=L_B$, both for ${Fr} = \sqrt {2}$. The dot dash lines are $\beta$-characteristics, dashed lines are $\alpha$-characteristics, solid lines are the front and shock and bold lines are the location of the barrier.

For $L_A< L< L_B$, the $\alpha$-characteristic intersects the shock while the reflected motion is uniform, and the maximum depth, $h_m$, is constant and equal to $h_b$ at the instant of collision. The next change of behaviour is at $L=L_B$ when the characteristic catches up with the front at the instant of collision (figure 10b). Using the results in Hogg (Reference Hogg2006), this occurs for

(5.4)\begin{equation} L_B = 1 + 2Fr {\left(\frac{{Fr} + 2}{2}\right)}^{{1/2}}, \end{equation}

and thus $L_B=4.70$ (3 s.f.) for $Fr=\sqrt {2}$.

For a more distant barrier, $L>L_B$, the fluid depth at the front of the current begins to diminish before it reaches the barrier. In this scenario, the maximum depth at the barrier, $h_m$, is determined by the depth of the arriving current, which may be evaluated directly from Hogg (Reference Hogg2006). The results are plotted in figure 9, which shows how $h_m$ diminishes with $L$.

For barriers which permit overflow we may expect to see some supercritical overtopping, but for ${Fr} = \sqrt {2}$ this has been found to only occur for very short barriers indeed. Using the results of § 3 and the early time solution (4.1) we find that supercritical overtopping only occurs for $B \lesssim 0.038$. We do not consider $B$ this small, and instead focus on barriers which cause subcritical overtopping or blocked flow.

In figure 11 we plot the results from our numerical computations for $L=4$, $B=1/4$ and $L=6$, $B=1/6$. On reaching the barrier, both flows overtop subcritically but the nature of the interaction at early times differs between the two cases. Following the analysis above for insurmountable barriers, when $L<4.70$, the overtopping is not initially affected by the finite extent of the release (through the reflection of the characteristic from the back wall, figure 11e), whereas for $L>4.70$ it is (figure 11f). This means that the former case exhibits a short period during which the overtopping flux is constant (figure 11a,c) and thereafter it varies temporally.

Figure 11. (a,b) The depth, $h$, as a function of $x$ at various instances of time as indicated in the legend, with $h=B$ in grey dotted line. The first time plotted is when the current reaches the barrier. (c,d) The velocity, $u$, as a function of $x$ at the times in the legend of (a,b) respectively, with $u=0$ in grey dotted line. (e,f) The characteristic plane, with gradient discontinuities in dashed lines, and the front and shock locations in solid lines. (g,h) The depth at $x=0$ (dashed) and $x=L$ (solid), with $h=B$ in grey dotted line. Panels (a,c,e,g) are for $L = 4$, $B = 1/4$, while (b,d,f,h) are for $L=6$, $B=1/6$.

After the initial overtopping event, subsequent overtopping can occur as the flow oscillates in the basin, a shock being reflected between the back wall and the barrier. These oscillations may cause the depth at the barrier to exceed $B$ again at later times, indeed in figure 11(g,h) both simulations exhibit a secondary overtopping when the shock collides with the barrier. It is possible for the oscillations to result in many overtopping events.

During this oscillating phase of motion, we may be concerned that mixing becomes important. The mixing at the upper shear layer can be shown to be small by an analysis of mixing rate reported in Strang & Fernando (Reference Strang and Fernando2001). Across the shock, however, the depth approximately doubles during the first passage of the shock across the domain (figure 11a,b), becoming substantially weaker after the first reflection due to energy dissipation. As shown by Borden, Meiburg & Constantinescu (Reference Borden, Meiburg and Constantinescu2012) for moving shocks and Wood & Simpson (Reference Wood and Simpson1984) and Lawrence & Armi (Reference Lawrence and Armi2022) for stationary shocks, shocks of this size do not cause substantial mixing, and instead the principal difference between our model and real currents is in the balance of momentum flux in the single layer model (2.3b) due to the energy dissipation and inertia in the upper layer. We do not expect this effect to be significant in our case of interest, flows under a deep ambient, but certainly would need to be included if the ambient and current were of a similar depth.

To characterise the overtopping, we calculate the dimensionless volume of fluid that has escaped at late times

(5.5)\begin{equation} \Delta V = 1 - V_\infty,\quad \text{where}\ V_\infty = \lim_{t \rightarrow \infty} \int_{0}^L h \,\mbox{d}{x}. \end{equation}

Whilst formally $V_\infty$ should be evaluated at $t \rightarrow \infty$, we compute it numerically by stopping the simulation when there is less than $0.001$ change in volume over $2$ oscillations of the fluid trapped behind the barrier. Specifically, at the instant of collision the volume of fluid in the domain is calculated as $V_0 = 1$, then the simulation is run for a period $L^{3/2}/V_0^{1/2}$, which approximates the time for a characteristic to cross the domain. This is then repeated, $V_n$ being the volume after $n$ runs, and the duration of the $(n+1){\rm th}$ run being $L^{3/2}/V_n^{1/2}$. If, after $V_n$ is calculated, we have $\max _{m \in \{n-4 , \ldots, n\}}(V_m) - \min _{m \in \{n-4 , \ldots, n\}}(V_m) < 0.001$, then we cease performing further computations, and approximate $V_\infty \approxeq V_n$. We simulate for $L \in \{1,1.5, \ldots,10\}$, $V_c \equiv BL\in \{0.2,0.4, \ldots,3\}$ with a resolution of 1000 cells, and include in our plots that all the fluid escapes for $B=0$, and that the fluid drains to a height $h=B$ for $L=1$ as discussed in Skevington & Hogg (Reference Skevington and Hogg2020). We now explore the overtopping that occurs for $L>1$, for which the escaped volume is shown in figure 12(a).

Figure 12. (a) The escaped volume $\Delta V_\infty$ (the colour bar) as a function of $L$ and $V_c$ for ${Fr} = \sqrt {2}$. The black lines are the contours $\Delta V_\infty \in \{ 0.1,0.2,\ldots,1 \}$. We also plot in a white solid curve the maximum $V_c$ for overtopping from figure 9. The crosses mark the parameter values used in figure 11. (b) The difference between the trapped volume and the value predicted by a purely geometric argument, $\min (V_c,1) - V_\infty$. We plot in a white solid curve the maximum $V_c$ for overtopping from figure 9, and in a white dashed curve the function (5.10) with $R=8$ which approximately bounds the region in which $V_\infty = V_c$.

The simplest limiting case is that $\Delta V = 0$ when no fluid is able to overtop, thus $V_\infty = 1$. For smaller barriers which permit overtopping, it is possible for the limiting volume to be exactly that required by volume conservation, $V_\infty = V_c \leq 1$, thus $\Delta V = 1 - V_c$. For this to happen the limiting behaviour must be that discussed in Skevington & Hogg (Reference Skevington and Hogg2020), i.e. at late times

(5.6a,b)\begin{equation} h \sim h_a \equiv B + \frac{27}{2 L^2 \tau^2} \quad uh \sim q_a \equiv \frac{27 x}{L^4 \tau^3}, \end{equation}

where $\tau$ is an offset time. For the proper ordering of terms the difference between the leading order solution above and the exact solution cannot be too large. We suppose that, at the instant the current reaches the barrier, $h_a L = 1$ which determines $\tau$, thus

(5.7)\begin{equation} q_a = {\left( \frac{2 (1 - V_c)}{3} \right)}^{3/2} \frac{x}{L^{5/2}}. \end{equation}

We then compare the total inertia of the flow at the instant of collision $Q$ (from the simulation of a lock-release current) with that of the asymptotic $Q_a$, producing the ratio

(5.8)\begin{equation} R \equiv \frac{Q}{Q_a}, \end{equation}

where

(5.9a,b)\begin{equation} Q \equiv \int_0^L uh \,\mbox{d}{x}, \quad Q_a \equiv \int_0^L q_a \,\mbox{d}{x} = {\left(\frac{2}{L}\right)}^{1/2} {\left( \frac{(1 - V_c)}{3} \right)}^{3/2}, \end{equation}

thus

(5.10)\begin{equation} V_c = 1 - \frac{3 L^{1/3} Q^{2/3}}{2^{1/3} R^{2/3}}. \end{equation}

Our simulations suggest that, so long as $R \lesssim 8$ then $V_\infty \approx V_c$, see figure 12(b). We see then that the requirement for limiting to the draining discussed in Skevington & Hogg (Reference Skevington and Hogg2020) is that the inertia is not too much greater than that of the asymptotic solution, $Q \lesssim 8 Q_a$. This bound is surprisingly weak, a priori it may be anticipated that ${\left | R - 1 \right |} \ll 1$ is required; it seems the asymptotic solution is unusually robust.

This leaves an intermediate regime, where the inertia is large enough to cause overtopping in excess of that required by volumetric arguments, and the barrier insufficiently tall to prevent outflow. In this regime the inertia is enough to decrease the remaining volume below that which is required by simple volumetric arguments, $V_\infty = \min (V_c,1)$. Indeed, for $V_c \approx 1$, $L \gtrsim 3$ the outflow volume is over $0.3$ greater than the minimal value, which shows the inertia has a significant effect. In this regime the fluid oscillates in the basin, and during each oscillation some portion of the fluid overtops (figure 11).

Beyond $L=10$, indeed largely beyond $L=5$, the current is approximately in similarity form at collision and, to leading order, $\Delta V$ is a function of $V_c$. This is because the characteristic reflected off the back wall has caught up with the front, see figure 11. This means that, so long as $V_c$ remains constant, the barrier may be placed any distance downstream and the same behaviour will be observed.

Our analysis shows that a barrier constructed so that $V_c = 1$ is not capable of containing the fluid. That is, if the confined volume is equal to the volume of the current then we should expect approximately 30 % of the fluid to escape. From our simulations, we expect that a barrier confining around 2–3 times the volume of the current is required to stop it, and even then a small portion of the current, around 5 %–10 %, will still overtop.

5.2. Dam-break flow: ${Fr} \rightarrow \infty$

In this case the current does not evolve into similarity form, and the front is not caught up by the reflected characteristic from the back wall (Hogg Reference Hogg2006). Thus the $\beta$-fan (4.1) generated by the initial release persists for all time. Hence, the initial overtopping behaviour is independent of $L$ and only depends on $B$ as shown in figure 7 for ${Fr} \to \infty$. Specifically, supercritical overtopping occurs for $B < 2$, although the volume that escapes may be quite small prior to the transition to other modes.

The escaped volume as measured using (5.5) is plotted in figure 13(a) from numerical simulations. We observe that, for $L \gtrsim 5$ and $B \lesssim 0.5$, $\Delta V_\infty$ is only weakly dependent on $L$. This is because the reflected characteristic from the back is advancing slowly across the $\beta$-fan. For $L < 5$ the effect of the back wall is more pronounced, thus the dependence on $L$ is stronger. Similarly to the Boussinesq case, the boundary of the region in which $V_\infty \approx V_c$ is given by (5.10) with $R=8$ ($Q$ evaluated for a dam-break current), see figure 13(b). However, in this case the region is much smaller. Indeed, across a wide range of parameters the actual escaped volume is substantially underestimated by volumetric considerations; in excess of an additional $60\,\%$ of the initial volume overtops the barrier for $L=10$, $B=1/10$.

Figure 13. (a) The escaped volume $\Delta V_\infty$ (the colour bar) as a function of $L$ and $V_c$ for ${Fr} \to \infty$. The black lines are the contours $\Delta V_\infty \in \{ 0.1,0.2,\ldots,1 \}$. (b) The difference between the trapped volume and the value predicted by a purely geometric argument, $\min (V_c,1) - V_\infty$. We plot in a white dashed curve the function (5.10) with $R=8$ which approximately bounds the region in which $V_\infty = V_c$.

6.1. Dam-break flow

The boundary condition used in this study is very similar to that developed by Cozzolino et al. (Reference Cozzolino, Cimorelli, Covelli, Morete and Pianese2014), only differing in that they chose to suppress supercritical overtopping. They compared the predictions of their model with a dam-break (${Fr}\to \infty$) experiment performed by Hiver (Reference Hiver2000) with $L=1.74$, $B=0.53$, $\epsilon =0.19$, $\theta =7.6^{\circ }$, $H/X=0.048$, and found a good agreement. However, we note that the overtopping predicted for this set of parameter values features only a very short period of supercritical overtopping, and so does not clearly differentiate between the two boundary conditions.

Instead, we compare our model with the experiments of Greenspan & Young (Reference Greenspan and Young1978). In particular, their data evidence an initial period of supercritical overtopping, as is modelled by our boundary condition. Their figure 9 shows a dam-break experiment with $L=2$, $B=0.5$, $\epsilon =0.26$, $\theta =60^{\circ }$, $H/X=0.89$, a jet can be seen that initially leaps the barrier prior to the formation of an upstream bore.

The dam-break experiments of Greenspan & Young (Reference Greenspan and Young1978) only explored the initial period of overtopping for the case $V_c \geq 1$. In this parameter regime the fluid is below the elevation of the barrier when the surface is level, and consequently overtopping is driven by the oscillatory sloshing motion depicted in figure 11. The initial period of overtopping is the time up to the instant when the depth drops from $h>B$ to $h \leq B$ at $x=L^-$ (if $h< B$ for all $t$ then then initial period extends to $t\to \infty$). Both the volume that overtops during the initial period and the volume that has escaped by late times are plotted in figure 14(a) from simulations of our model, showing that there is little difference between these two measures (at least, for the range of $B$ and $L$ plotted). Additionally, as noted by Greenspan & Young (Reference Greenspan and Young1978), the curves for different $L$ are very similar. We understand this as the effect of the $\beta$-fan in the frontal region of the dam-break that persists over long distances (Hogg Reference Hogg2006). The extent of this frontal region relative to the length of the overall current varies slowly as it is caught up by the reflection of the $\beta$-fan from the back wall.

Figure 14. Comparison between our theoretical predictions (§ 5.2) and the experimental results of Greenspan & Young (Reference Greenspan and Young1978). In all plots the escaped volume is plotted as a function of barrier height; the curves are our simulations, and the markers are experiments. In (a,b) the different distances to the barrier $L$ correspond to the plot colours and markers: red $\times$, $4/3$; green $\bigcirc$, $2$; blue $+$, $3$; magenta $\triangle$, 4; cyan $\ast$, 5. In (a) we plot only numerical results, the solid curves being the volume that overtops during the initial period of overtopping, the dashed representing the late time cumulative escaped volume. In (b) we plot the same solid curves, along with experimental results for a barrier at $\theta =90^{\circ }$. In (c) we plot the same solid curves, but now experiments are shown for different angles $\theta$: $\bigcirc$, $30^{\circ }$; $+$, $60^{\circ }$; $\square$, $90^{\circ }$. In (d) the experiments are for $\theta =90^{\circ }$, but in this panel $L=1/B$.

We next consider the experiments with a barrier at $\theta = 90^{\circ }$. There may be concerns that the deviation of the critical barrier boundary condition precludes application to such steep slopes, however, we only require that there is no energy difference between the ‘base’ and ‘crest’. These are not necessarily at the locations immediately adjacent to the incline, but rather the first locations up- and down-stream where the assumptions of the shallow water equations are satisfied, thus the model may be applied with a small change in interpretation. The experiments show good agreement with our simulations (figure 14b), with only small discrepancies. For the barriers that are close and tall (red) the experiments show a slightly larger escaped volume than predicted by our simulations, and we suspect that this discrepancy is because the region around the barrier where the shallow water assumptions are violated is large in comparison with the domain ($\epsilon \not \ll 1$). For the shorter, distal barriers (cyan and magenta) our simulations over-predict the escaped volume, and it is likely that this is in part because of drag reducing the inertia of the physical current, which will reduce the escaped volume. Indeed, comparing our simulations with the experiments of Hiver (Reference Hiver2000) we find that our initial overtopping exceeds that found experimentally, while the example simulation with drag reported by Cozzolino et al. (Reference Cozzolino, Cimorelli, Covelli, Morete and Pianese2014) showed better agreement. In the regime $V_c=1$ ($L=1/B$), however, our simulations agree remarkably well with experiment (figure 14d), indicating that the small errors cancel in this important case.

Investigating the effect of different barrier angles $\theta$, we find that our simulations slightly underestimate the escaped volume for $\theta =30^{\circ }$ and $\theta =60^{\circ }$. These experiments were not performed at the scales at which our model was derived (see § 2), in particular $\epsilon \not \ll 1$ and $Z \not \ll X$, thus higher-order corrections to the model may be required to fully capture this non-shallow dynamics.

6.2. Boussinesq gravity current

Our analysis applies to overtopping currents, propagating under relatively deep ambient fluid so that the momentum of the latter may be neglected. This condition is particularly important when interpreting data from laboratory and numerical simulations of Boussinesq currents, many of which feature currents that are not initially shallow relative to the ambient and are potentially strongly influenced by the motion of the ambient fluid. Notably, however, Gonzalez-Juez & Meiburg (Reference Gonzalez-Juez and Meiburg2009) report two-dimensional DNS of gravity currents driven from a sustained source and propagating under a deep ambient towards an obstacle that spans the channel. They present data on the depth and velocity of the gravity current motion downstream of the obstacle (their figure 12), from which we can compute the overtopping flux of fluid. We present their simulation results in figure 15, along with the outflow flux predicted by our solution of the boundary-Riemann problem in § 3, in which we use the Froude number of the incident fluid, $F_i\equiv u_i/h_i^{1/2}$, determined by Gonzalez-Juez & Meiburg (Reference Gonzalez-Juez and Meiburg2009). The data series with no-slip and slip basal boundary conditions have different reported frontal Froude numbers. In both cases, we observe in figure 15 that there is quite close agreement between the simulated results and the predictions from our boundary-Riemann calculation. We also note that the analysis in § 3 gives a formal justification for the simplified modelling used by Gonzalez-Juez & Meiburg (Reference Gonzalez-Juez and Meiburg2009); their box model analysis upstream of the barrier is identical to the full solution of a shallow water model with the critical barrier boundary condition derived in § 2, and their simulations all fall into regime A-II of the boundary-Riemann problem of § 3 (see figure 5).

Figure 15. The overtopping flux, $q_c/h_i^{3/2}$, as a function of barrier height, $B/h_i$, both scaled relative to the depth of the incident fluid, $h_i$. Potted are data from the simulations of Gonzalez-Juez & Meiburg (Reference Gonzalez-Juez and Meiburg2009) ($\square$, ${Re}=3535$; $\bigcirc$/$\triangle$ with error bar, ${Re}=707$) and the predictions of the analysis in § 3 (solid lines). In (a) the simulations have no-slip boundary conditions and ${Fr}=0.83$, while (b) have slip boundaries and ${Fr}=1.00$. Also plotted in both are the predictions with ${Fr} = \sqrt {2}$ (dashed lines).

More complete model comparisons with data from simulations and experiments would enable us to test other aspects of our predictions, such as flow depths and velocities and shock speeds and to probe dynamical effects that we have not included. The three-dimensional, large eddy simulations of gravity currents flowing over a triangular obstacle performed by Tokyay & Constantinescu (Reference Tokyay and Constantinescu2015) report some of these features. However, we find that the depth computed above the crest of the obstacle differs substantially from the critical depth, as calculated using their reported data. It is possible that this discrepancy comes from how the depth and velocity of the current have been calculated: the former is actually the depth integral of concentration (normalised by the concentration at release), and the latter the concentration flux per unit ‘depth’. Consequently, any mixing present prior to overtopping has a substantial effect on the measurement, and the initial conditions used to generate these currents are likely to cause a substantial mixed layer to form. Some recent laboratory experiments have been performed by De Falco, Adduce & Maggi (Reference De Falco, Adduce and Maggi2021) and Adduce, Maggi & De Falco (Reference Adduce, Maggi and De Falco2022) using lock release of saline water propagating through fresh water and overtopping an initially distant triangular obstacle. The width-averaged density of the the flowing current is measured, as well as the position of the front and the proportion of the released fluid that is retained behind the barrier. The currents are released from a lock that occupies the full depth of the ambient; therefore the motion is strongly influenced by flow within the ambient. Evidence of this is provided from the measurement of the speed of the front, which is approximately constant and close to the predictions of two-layer hydraulic models (see Appendix B of Hogg et al. (Reference Hogg, Nasr-Azadani, Ungarish and Meiburg2016) for the analytical evaluation of the front speed in a two-layer model of slumping). The effects of the upper layer are not included in the model of overtopping developed here and this precludes comparison with this dataset.