Development of supercritical motion and internal jumps within lock-release radial currents and draining flows

The unsteady radial flow of a relatively dense fluid released from an axisymmetric lock is analyzed when it freely drains over an edge and when it generates a gravity current propagating over a horizontal surface. In both situations when modeled using the shallow water equations, despite initiation from rest within a lock, the flow thins, accelerates, and becomes supercritical in a region close to the symmetry axis. For free drainage, this alters the outflow, while for radial gravity currents, it leads to the formation of an internal jump connecting rapidly moving fluid to more tranquil flow at the front. Both scenarios share the same supercritical flow structure, which is related to their initiation from lock-release conditions and is a function of axisymmetry; the phenomenon is not found in analogous two-dimensional flows. Through analysis and numerical integration of the governing equations, the common onset and consequences of supercriticality are revealed in both flows, as well as, the progression of the fluid motions to self-similar states that develop at late times.

Figure 1

The configuration of (a) draining flows and (b) gravity currents. Note that the vertical scales are exaggerated; the flows under consideration are assumed to be shallow.

Figure 2

(a) Height, $h(r,t)$, and (b) velocity, $u(r,t)$, fields as functions of the radial distance from lock-release initial conditions, with $r_i=0$ at $t=1$, 1.2, 1.4, 1.6, 1.8, and 2 denoted (I)–(VI), respectively (solid lines). Insets: Plots of (a) $t^{2}h(r,t)$ and (b) $tu(r,t)$ as functions of the time at $t=1$, 2, 3, 4, and 5, (solid lines) along with the similarity solutions, (16) (dashed lines). (c) The associated characteristic plane showing $\alpha$ characteristics (dot-dashed blue lines) and $\beta$ characteristics (solid red lines). Note that there is a $\beta$ characteristic which lies along $r=0.72$ for $t>1.4$ (thick red line), which implies that critical conditions $(u=h^{1/2})$ on this curve prevail, together with supercritical conditions for $r>0.72$.

Figure 3

(a) Froude number at the outflow, $F_o$, as a function of the time for (i) $r_i=0$, (ii) $r_i=\frac{1}{10}$, (iii) $r_i=\frac{1}{6}$, (iv) $r_i=\frac{1}{3}$, and (v) $r_i=\frac{1}{2}$ (solid lines). Also plotted is the asymptotic expression for $F_o$ at early times (dashed line). (b) Outflow flux, $q$, as a function of the time (solid black line). Also plotted are the series expansion, (10), derived when $t\ll 1$ (dot-dashed red line) and the similarity solution, (16), when $t\gg 1$ (dashed blue line).

Figure 4

The arrival time at the outflow, $t_a$, of the $\alpha$ characteristic that is launched from the inner wall by the arrival of the rearmost $\beta$ characteristic from the rarefaction fan as a function of the inner annular radius, $r_i$ (solid line). Also plotted are points that correspond to the times at which the Froude number at the outflow is maximized, $t_m$ [see Fig. 3], and the asymptotic dependence of $t_a$ when $1-r_i\ll 1$ given by (12) (dashed line).

Figure 5

Trajectories in the $(F,V)$ phase plane, with arrows depicting the direction of evolution with increasing $r$. Also plotted are nullclines ($\mathrm{d}V/\mathrm{d}r=0$, dashed green line; $\mathrm{d}F/\mathrm{d}r=0$, dashed red line) and $F=1$ (dot-dashed blue line). Recall that $V\left(r_i\right)=0$ for $r_i>0$ and hence only the lower left quadrant is relevant for an annular reservoir.

Figure 6

Similarity solutions for (a) $H\left(r\right)={(rV/F)}^2$ and (b) $U\left(r\right)=rV\left(r\right)$ as functions of the radial distance for $r_i=0.1$, 0.3, 0.5, 0.7, and 0.9.

Figure 7

Froude number at the outflow as a function of $h_i$ evaluated at $t=50$ (solid line, crosses) and $t=30$ (dashed line). Note that these curves are virtually indistinguishable.

Figure 8

(a) Height and (b) velocity fields as functions of the radial distance at $t=2$, 3, 4, 5, 6, 8, 10, 12, 14, and 16 for $\mathrm{Fr}=\sqrt{2}$. Inset: The similarity solution, (16), with $F_o=1.39$ (dashed lines) is also included.

Figure 9

(a) The position of the front of the current, $r_N\left(t\right)$, and the trajectory of shock, $r_s\left(t\right)$, as functions of the time for $\mathrm{Fr}=1$ (dashed red lines), $\mathrm{Fr}=2^{1/4}$ (solid black lines), and $\mathrm{Fr}=2^{1/2}$ (dot-dashed blue lines). (b) The positions of the outward (dashed red lines)- and inward (dot-dashed blue line)-propagating shocks and the characteristic curves from the similarity solution, (20) as functions of $\mathcal{T}$ (solid black line).

Figure 10

(a) Rescaled height, $t^{2}h(r,t)$, and (b) rescaled velocity, $tu(r,t)$, fields as functions of the radial distance at $t=2$, 3, 4, 5, 6, 8, 10, 12, 14, and 16 for $\mathrm{Fr}=2$ (solid lines). Also plotted is the supercritical similarity solution, (16), with $F_o=1.39$ (dashed line). (c) Position of the front, $r_N\left(t\right)$ (solid lines), and the shock, $r_s\left(t\right)$ (dashed lines), for $\mathrm{Fr}=2^{3/4}$ (blue) and $\mathrm{Fr}=2$ (red). The asymptotic state, $r=r_b$, is also plotted (dotted black line).

Figure 11

Velocity of the similarity solution at the outflow, $V_0$, as a function of the inner annular radius, $r_i$, showing numerical computations (solid black line) and asymptotic solutions for $r_i\ll 1$ (dot-dashed red line) and $1-r_i\ll 1$ (dashed blue line).