Non-self-similar viscous gravity currents

Lock-release experiments are performed focusing upon the evolution of near-pure glycerol flowing into fresh water. If the lock height is sufficiently tall, the current is found to propagate for many lock lengths close to the speed predicted for energy-conserving moderately non-Boussinesq gravity currents. The current then slows to a near stop as the current head ceases to be elevated relative to its tail and the current as a whole forms a wedge shape. By contrast, an experiment of near-pure glycerol advancing under air exhibits the well-known slowing of the current such that the front position increases as a one-fifth power of time. The evolution of a viscous gravity current in water is also qualitatively different from that for a high-Reynolds number gravity current which transitions smoothly from a constant speed to self-similar to viscous regime. The reason a viscous gravity current flowing under water moves initially at near-constant speed is not due to a lubrication layer forming below the current. Rather it is due to the return flow of water into the lock establishing a current with an elevated head that is taller than the viscous boundary layer depth near the current nose. The flow near the top of the head advances to the nose where it comes into contact with the tank bottom. Meanwhile the ambient fluid is pushed up and over the head rather than being drawn underneath it. The front slows rapidly to a near stop as the head height reduces to that comparable to the boundary layer depth underneath the head. The initial speed and entrainment into the current are shown to depend upon the ratio, $R_{\ell }$, of the starting current height to the characteristic boundary layer depth. In particular, entrainment via the turbulent shear flow over the head is found to increase the volume by less than $10\%$ during its evolution if $R_{\ell }\lesssim 10$ but increases by as much as $100\%$ for high-Reynolds number gravity currents. A conceptual model is developed that captures the transition from an inertially driven current to its sudden near stop by viscous forces.

Figure 1

Snapshot from an experiment just before gate is extracted. For this experiment the dyed lock fluid is $99.2\%$ glycerol situated in a lock of width $\ell =8.3\mathrm{cm}$. The ambient fluid to the right of the gate is fresh water (the superimposed dashed line indicates its surface). The height of both lock and ambient fluids is $19.9\mathrm{cm}$. The angled mirror above the tank gives top views of the ensuing flow between approximately 50 and $130\mathrm{cm}$ from the left end of the tank.

Figure 2

Snapshots taken from an experiment with $99.2\%$ glycerol collapsing into air [17]. Initial conditions are $H_{\ell }=19.9\mathrm{cm}, L_{\ell }=8.3\mathrm{cm}, {\rho }_{\ell }=1.2590\text{g}/{\text{cm}}^3$, and ${\nu }_{\ell }=10.40{\text{cm}}^2/\text{s}$. Only the leftmost $97\mathrm{cm}$ of the tank is shown, and all but the top image shows the bottom $9\mathrm{cm}$. The times of each snapshot are indicated at the upper right of each frame. The dotted red line superimposed on the bottom image plots the shape predicted for a self-similar viscous current given by (7).

Figure 3

(a) Horizontal time series constructed from successive horizontal slices taken $0.1\mathrm{cm}$ above the bottom of the tank for the experiment with snapshots shown in Fig. 2. The superimposed white dashed line indicates the measured front position. (b) Log-log plot of corresponding front position versus time. The offset dashed line with one-fifth slope is that predicted for viscous gravity currents [4, 5]. Note that the horizontal banding in (a) is due to a stroboscopic effect between the camera's frame rate and the 60 Hz background lighting used in this experiment.

Figure 4

As in Fig. 2 but showing snapshots taken from an experiment of near-pure glycerol collapsing into water having initial condition shown in Fig. 1 [17]. The time $t=0$ is taken to be when the gate has been lifted halfway out of the lock and just before the viscous fluid begins to slump out of the gate. Only the leftmost $130\mathrm{cm}$ of the tank is shown, and all but the top image show the bottom $10\mathrm{cm}$. The dotted red line superimposed on the bottom image plots the shape predicted for a self-similar viscous current given by (7).

Figure 5

Top views of the experiment with side-views shown in Fig. 4 as seen at $t=2$, 3, and $4\mathrm{s}$ from the angled mirror above the tank between $50\mathrm{cm}$ and $140\mathrm{cm}$ from the lock end of the tank. Superimposed black arrows point to the location of the front at midspan in the tank.

Figure 6

(a) Horizontal time series and (b) front position as in Fig. 3 but shown for a gravity current moving into water, with corresponding snapshots shown in Fig, 4. The dashed line with one-fifth slope is drawn to indicate that the front does not enter the classical viscous regime for any appreciable time.

Figure 7

Snapshots at times $t=1,2,3\mathrm{s}$ from experiments with salt water advancing into water (left column) and nearly pure glycerol advancing into water (right column) [17]. Each frame shows a window extending between 20 and $90\mathrm{cm}$ from the lock end of the tank and extending $10\mathrm{cm}$ from the bottom in full-depth lock-release experiments with total depth $20\mathrm{cm}$. Potassium permanganate crystals are placed along the bottom of the tank over the range indicated by the dotted lines in the top two panels. The experiments compare how dye from the partially dissolved crystals is carried by the currents in each circumstance. In all images the intensity has been adjusted to enhance the contrast between the current (medium gray) and potassium permanganate (black).

Figure 8

For the experiment with snapshots shown in Fig. 4. (a) Time series of measured current height as function of time and distance from the gate, (b) height profile of the current at $t=6.7\mathrm{s}$, and (c) area of the current as function of time. In (c) the dotted line indicates times where the computed area is not reliable due to difficulty in tracking the current height in the lock at early times.

Figure 9

Front position versus time in experiments with $H_{\ell }\simeq 20\mathrm{cm}$ and $L_{\ell }=8.3\mathrm{cm}$ and with lock fluids of different densities and viscosities. The horizontal position ahead of the gate is given relative to the lock length, $L_{\ell }$, and time is given relative to the time for a full-depth lock-release energy-conserving gravity current with speed $U_H$ to travel one lock length. The diagonal dashed black line shows the predicted front position for a current moving at constant speed $U_H$. The meaning of the colored lines are indicated at the upper left of the plot, and the corresponding viscosities and values of Re and $R_{\ell }$ are indicated toward the right of each curve. Note that the dotted curve, corresponding to $64\%$ glycerol, underlies the blue curve for salt water almost everywhere along its length.

Figure 10

As in Fig. 9 but showing the front position for experiments with nearly pure glycerol released from locks with different heights and lengths as indicated at the top. In experiments with $H_{\ell }\simeq 16\mathrm{cm}$, the ambient fluid height is $H=20\mathrm{cm}$; in experiments with with $H_{\ell }\simeq 7.9\mathrm{cm}$, the ambient fluid height is $H=10\mathrm{cm}$. The values of $R_{\ell }$ are indicated near the end points of each corresponding plot. Time is normalized using the predicted speed, $U_h$, for energy-conserving partial-depth lock-release gravity currents.

Figure 11

Measured initial speed of the current front relative to the predicted energy-conserving speed plotted against the current-to-boundary layer height parameter, $R_{\ell }$. Circles (squares) denote full-depth (partial-depth) lock-release experiments.

Figure 12

Measured near-stopping position of the current front relative to the estimated near-stopping scale (11) plotted against $R_{\ell }$.

Figure 13

Net entrainment into current measured through the relative increase in area of the current at the end of an experiment compared with the initial lock area, $A_0\equiv L_{\ell }{H}_{\ell }$. Circles (squares) denote full-depth (partial-depth) lock-release experiments.

Figure 14

Horizontal velocity of current and return flow computed from the conceptual model with initial conditions determined from those of the experiment shown in Fig. 4. In each plot, the height of the current is indicated by the solid white line and the white dashed lines indicate the height of the viscous boundary layer. Only half the vertical extent of the domain is shown. The color scale for horizontal speed in all plots is given in the upper-right corner of (a). Color is plotted only between the lock end of the tank and the current nose. Beyond, the ambient fluid is assumed to be stationary and not included in the model.