Gravity current escape from a topographic depression

An inertial gravity current released within a topographic depression may climb up the incline from a lower to an upper plateau if it is sufficiently energetic and then continue to flow unsteadily away from the step while simultaneously draining back into the depression. This density-driven motion is investigated theoretically using the shallow-water equations to simulate the flow up a smooth step from a lower to an upper horizontal plane and to compute the volume of fluid that escapes from the depression. It is shown that it is possible for all of the fluid to drain back down the step and that the volume of the escaped fluid diminishes with a power-law dependence on time. This phenomenon is explained by analyzing the unsteady flow of a gravity current along a semi-infinite horizontal plane along which the front advances but simultaneously fluid drains from the rear edge of the plane. The dynamics of this motion at early times is calculated both numerically and analytically. The latter exploits the hodograph transformation of the shallow water equation, a technique which allows rapid and precise evaluation of flow features such as reflections and the onset of bores. The flow at later times becomes self-similar, and the self-similarity is of the second kind. It features an anomalous exponent that provides the power-law dependence of the volume of fluid on time, and this exponent is a function of the imposed Froude number at the front of the current. In this way the diminishing volume of fluid that escapes from a topographic depression is explained quantitatively. Eventually the flow may transition to a regime in which drag becomes non-negligible; the model of simultaneous propagation and draining is extended to this regime to show that the escaped volume of fluid also decays temporally and ultimately vanishes.

Figure 1

The configuration of the flow problems and the dimensionless variables: (a) gravity current flow within a topographic depression with step height $b_m$, located between $x=L-\epsilon$ and $x=L+\epsilon$; and (b) simultaneous spreading and draining along a semi-infinite plane. In (a) and (b) the dotted lines depict the initial lock of fluid of unit dimensionless height and length and the front of the flow is at $x_f\left(t\right)$.

Figure 2

The elevation of the free surface of the current, $h+b$, and the velocity, $u$, as functions of distance downstream at various instances of time (shown in each legend) for (a) $b_m=0.6$, (b) $b_m=0.7$, (c) $b_m=0.8$, and (d) $b_m=0.9$. The elevation of the step, $b$, is also plotted (black solid line), as is $u=0$ (black dotted line).

Figure 3

The dimensionless volume of gravity current fluid escaped from the topographic depression, $\mathcal{V}\left(t\right)$, as a function of time on (a) linear and (b) logarithmic axes for (i) $b_m=0.6$, (ii) $b_m=0.7$, (iii) $b_m=0.8$, and (iv) $b_m=0.9$. The simulations were run with Froude number, $\mathrm{Fr}=1.2$, while the topography was additionally specified by $L=2$ and $\epsilon =0.1$. Also plotted in (b) are power-law relations, $\mathcal{V}\left(t\right)\propto t^{-0.139}$, fitted to the data at late times (dashed-dot lines).

Figure 4

(a) Depth, $h(x,t)$, and (b) velocity, $u(x,t)$, of the gravity current as functions of distance downstream at $t=1,2,3,5,10,20,30$, and 50 for $\mathrm{Fr}=1.2$. (c) The characteristic plane, showing the position of the front, $x_f\left(t\right)$ (thick black line) and some $\alpha$ characteristics (solid blue lines) and $\beta$ characteristics (dot-dashed red lines). The regions of the plane are denoted as simple wave regions $(S_1,S_2,S_3)$ and complex wave regions $(C_1,C_2)$. Also plotted are some key locations in the plane, positions $P_1$, $P_2$, and $P_3$ and the times at which the behavior of the outflow changes $(T_1,T_2,T_3)$.

Figure 5

(a) Depth, $h(x,t)$, and (b) velocity, $u(x,t)$, of the gravity current as functions of distance downstream at $t=2,3,4,5,10,20,30,40$, and 50 for $\mathrm{Fr}=4$. (c) The characteristic plane, showing the position of the front, $x_f\left(t\right)$ (thick black line) and some $\alpha$ characteristics (solid blue lines) and $\beta$ characteristics (dot-dashed red lines). Some of the regions of the plane are labeled as a simple wave regions $\left(S_3\right)$ and complex wave regions $(C_1,C_2)$. Also plotted are some key locations in the plane, positions $P_1$, $P_2$, and $P_3$ and the time at which the behavior of the outflow changes $\left(T_1\right)$.

Figure 6

The drainage flux, $q\left(t\right)$, as a function of time, $t$, on (a) linear and (b) logarithmic axes for (i) $\mathrm{Fr}=1.2$ (red dot-dashed line), (ii) $\mathrm{Fr}=\sqrt{2}$ (blue dot-dashed line), and (iii) $\mathrm{Fr}=1.6$ (purple dot-dashed line), along with the universal curve for $\mathrm{Fr}\ge 2$ (black solid line). Also plotted at the values of $q$ at $t=T_1$ (cross), $t=T_2$ (circle), and $t=T_3$ (square). The first time, $T_1$, is the same for all values of Fr, but the latter two vary with Fr.

Figure 7

The volume of fluid per unit width in the flowing current, $V\left(t\right)$, as a function of time for (i) $\mathrm{Fr}=1.2$, (ii) $\mathrm{Fr}=\sqrt{2}$, (iii) $\mathrm{Fr}=1.6$, and (iv) $\mathrm{Fr}=2$. Also plotted for (i)–(iii) are the volume at $t=T_3$ (squares) and the self-similar solution of the second kind that captures the late time evolution (dashed line).

Figure 8

The self-similar exponent, $\gamma$, as a function of the Froude number $\mathrm{Fr}$. Note that $\gamma =0$ for $\mathrm{Fr}\ge 2$, indicating that at late times, volume is conserved for these currents.

Figure 9

(a) The self-similar height, $H\left(\xi \right)$, and (b) velocity, $U\left(\xi \right)$ fields as functions of the similarity variable, $\xi$, for $\mathrm{Fr}\in \{1.2,1.4,1.6,1.8,2\}$ [labelled (i)–(v), respectively in (a)].

Figure 10

The time at which a bore forms, $t_s$ (blue) and its position, $x_s$ (red), as functions of the Froude number at the front of the current, Fr.

Figure 11

The rescaled depth and velocity fields as a function of the similarity variable $\xi =x{t}^{-(2-\overline{\gamma })/3}$ for $\mathrm{Fr}=4$ and at late times after release $t=80,120,160,200,240,280$, and 320, determined by the numerical integration of the shallow water equations (solid lines). Also plotted are the self-similar forms upstream of the bore (black dot-dashed line).

Figure 12

(a) The similarity solution for the depth of the flow, $H\left(\eta \right)$, and flux, $H\left(\eta \right)U\left(\eta \right)$, as functions of the similarity variable, $\eta$, when the drag depends quadratically on the velocity field. (b) The drainage flux as a function of time for $\mathrm{Fr}=1.2$ and $C_D=0.1$ (blue line). Also plotted is a power-law relation, $t^{-(1+\sigma )}$, with $\sigma =0.2678$.

Figure 13

The similarity solution for the depth of the current, $H\left(\eta \right)$, and flux, $H\left(\eta \right)U\left(\eta \right)$, as functions of the similarity variable, $\eta$, when the current is viscously dominated [18, 24].

Figure 14

The hodograph plane for $\mathrm{Fr}=1.2$, showing regions $C_1$ and $C_2$. Characteristics corresponds to lines of constant $\alpha$ (solid) and constant $\beta$ (dot-dashed). The simple wave regions, $S_1$, $S_2$, and $S_3$, correspond to characteristic lines and uniform regions, and $U_1$ and $U_2$ to points in the hodograph plane. The outflow boundary corresponds to the line $\beta =-3\alpha$ and the front to $\beta =(\mathrm{Fr}-2)\alpha /(\mathrm{Fr}+2)$ (both plotted with thick black line).The points $ABDE$ form a closed curve that is used to construct the solution.