Transient behavior of through-flowing gravity currents interacting with a roughness array

We present laboratory experiments that investigate the structure and flow characteristics of gravity currents traveling through an array of roughness elements. The roughness elements are of comparable height to the gravity current such that the current flows through the roughness array rather than over it. The frontal velocity and density structure are measured as the current transitions from flowing along a smooth bed to flowing through the roughness array and then back to a smooth bed. We find that, upon entering the roughness array, the gravity current decelerates and the density structure changes from the head-and-tail structure typical of smooth bed gravity currents to a wedge shape. A model is presented that explains the deceleration and change in shape based on a dynamic balance between a pressure gradient within the current tail and a drag force associated with individual roughness elements. This model accurately predicts the deceleration of the gravity current, supporting the proposed dynamic balance.

Figure 1

Schematic of the experimental apparatus. Initially, dense fluid in the lock on the left is separated from lighter, ambient fluid on the right by a gate. Upon removal of the gate, a dense gravity current forms that either immediately interacts with the roughness or develops as a smooth bed current before encountering the roughness. Roughness elements marked in blue were only used for the ‘gate’ experiments and roughness elements marked in black were only used for the standard configuration experiments. Four cameras observe the flow across the viewing window marked by dashed lines.

Figure 2

The two roughness configurations used in the experiments. The sparse configuration is shown on the left panel and the plunging configuration on the right panel. Not to scale.

Figure 3

Width-averaged density field for experiment S25 at times of $\widehat{t}=-2$, 4, 10, 16 and 22. The black line shows the current envelope, $h_c$, and the blue line shows the buoyant height, $h_B$. Black cylinders represent locations where the flow was obscured from the camera, either due to roughness elements or, at $\widehat{x}\approx 2$, a vertical brace on the side of the tank.

Figure 4

Width-averaged density field for experiment S25G at times of $\widehat{t}=16$, 21, 26, and 31. The black line shows the current envelope, $h_c$, and the blue line shows the buoyant height, $h_B$. Black cylinders represent locations where the flow was obscured from the camera, either due to roughness elements or, at $\widehat{x}\approx 0$ and 7, vertical braces on the side of the tank.

Figure 5

Width-averaged density field for experiment P25 at times of $\widehat{t}=-2$, 5, 12, 19, and 26. The black line shows the current envelope, $h_c$, and the blue line shows the buoyant height, $h_B$. Black cylinders represent locations where the flow was obscured from the camera, either due to roughness elements or, at $\widehat{x}\approx 2$, a vertical brace on the side of the tank.

Figure 6

Width-average density and centerline velocity measurements for experiment P25 as the current nose reached the second row of roughness. False color shows fluid density as measured by LA and arrows show velocity vectors as measured by PTV. Note that the two fields are from different realizations of the same experiment.

Figure 7

Width averaged density fields and velocity vectors between the third and fourth cylinders for the P50 experiment at $\widehat{t}=8$, 9, 10, 11, 12, and 13. False color shows fluid density as measured by LA and arrows show velocity vectors as measured by PTV. Note that the two fields are from different realizations of the same experiment. The black line shows the current envelope, $h_c$, and the blue line shows the buoyant height, $h_B$.

Figure 8

Front locations (a, b) and Froude numbers (c, d) for sparse (a, c) and plunging (b, d) experiments with time. Froude numbers measured by Ref. [18] are marked on panel (c) by circles with the total measurement window shown by the horizontal black lines. The dashed lines on panels (c, d) show the average Froude number before encountering the roughness array. The vertical lines on panels (c, d) show the time when each current exited the roughness array.

Figure 9

Froude number as a function of time for experiments with the roughness array starting at the gate and experiments with the roughness array starting downstream. The horizontal dashed and vertical lines have the same meaning as in Fig. 8.

Figure 10

Froude numbers for sparse configuration experiments after emerging from the roughness array. $t=t_2$ refers to the time when each current first exited the roughness.

Figure 11

Vertical density profiles (solid lines) and $h_B/h_c$ at two locations and times for (a) the S25 experiment shown in Fig. 3, and (b) the P25 experiment shown in Fig. 5. Solid lines show the measured vertical density profiles and dashed lines show the top-hat model of the current with a density of $h_B/h_c$ to a height of $h_c$. Plunging data is not shown for $\widehat{t}=12$ and $x_F-x=4$ as that location was upstream of the current encountering the roughness array.

Figure 12

Buoyant height, $h_B$, for sparse (a) and plunging (b) experiments as a function of $x$ at the time when the current in experiment S18 and P18 reached the final row of roughness ($\widehat{t}=11$ and $\widehat{t}=14$, respectively). $x_F$ refers to the current front at the time when measurements were made.

Figure 13

Buoyant height (a, b), current envelope (c, d), and an estimate of the average density (e, f) with time. The left column shows sparse configuration experiments and the right column shows plunging configuration experiments. Data are measured a distance of 1 behind the front of the current. As before, the vertical lines show the time when each current exited the roughness array.

Figure 14

Buoyant height a distance of 4 behind front with time for sparse configuration experiments (a) and plunging configuration experiments (b). $t_1^{\prime }=0$ corresponds to the measurement location first encountering the roughness array.

Figure 15

$h_B$ (a) and $h_c$ (b) as a function of time for sparse currents at the standard head location after the currents exited the roughness.

Figure 16

A schematic of the key model variables. The buoyant height and current envelope are equuivalent due to the fluid within the current being undiluted.

Figure 17

Measured and predicted Froude numbers with time for the sparse (a) and plunging (b) roughness configurations. Predicted Froude numbers are based on Eq. (11) with $\mathcal{B}=0.40$ for the sparse roughness configuration and both $\mathcal{B}=0.40$ and $\mathcal{B}=0.16$ for the plunging roughness configuration.