Shallow-water equations and box model simulations of turbidity currents from a moving source

We present numerical simulations of turbidity currents generated by a moving source that model particle clouds released by underwater vehicles such as those used in deep-sea dredging and mining. We use two approaches to model the flow and resulting deposits. First, we adapt the shallow-water equations to include a moving source of both momentum and particles. Second, we modify the classical box model to include a moving source. We validate our simulations before studying the impact of the most important governing dimensionless parameters: a vehicle Froude number Fr based on the depth of sediments being resuspended and the dimensionless settling speed of those sediments, $u_s$. We find that Fr is most determinant in the current and deposit shape, with $\text{Fr}>2$ resulting in elongated shapes and $\text{Fr}<2$ resulting in rounder ones, and that $u_s$ determines the rate at which deposits grow. Another parameter, the vehicle's drag coefficient, was found not to have much effect beyond the immediate vicinity of the vehicle. Overall, both models found similar trends. The box model, much less computationally expensive, was less accurate when spatial nonuniformities developed, as happens at long times, but captured the early spreading rate well. The shallow-water equations handled nonuniformities correctly but tended to over-predict the spreading rate. We used the insight gained from those simulations to obtain criteria for the maximum extent of deposits left but such currents in both the low and high Froude number regimes.

Figure 1

Schematics of the parameters of the moving vehicle releasing particles into suspension.

Figure 2

Schematics of the box model used to approximate the gravity current spreading from a moving vehicle.

Figure 3

[(a)–(h)] Simulated spreading of a deep-sea cloud generated by a dredging vehicle modeled by the shallow-water equations. [(i)–(l)] Same current simulated using a box model. The current height $h$ is shown from the side [(a)–(d)] and from above [(e)–(h)]. In panels [(a)–(d)], the height was truncated to better show the current away from the vehicle, with the actual maximum height reaching $h=0.88$. The vehicle, shown as a red circle in panels [(e)–(l)] starts at (0,0) and moves toward the positive $x$ axis until $t=20$, when it stops. Physical parameters are ${\text{Fr}}^2=2$, $u_s=0.01$, $C_D=\pi /10$ and for the box model $C_F=1.2$. Numerical parameters for the shallow-water solver are $\mathrm{\Delta }x=0.025$, $\mathrm{\Delta }t=0.0025$ (so CFL = 0.1), $h_m=0.001$, ${\text{Re}}_n=500$, and for the box model $\mathrm{\Delta }t=0.002$, $\mathrm{\Delta }{s}_m=0.025$, and $R_0=0.1$.

Figure 4

(a) Sample calculation of the current height at various times for two choices of time step $\mathrm{\Delta }t$. We indicate the position of the vehicle with a red star. Each curve is one time-unit apart and the black curve and circle are the initial current extent and vehicle position, respectively. (b) Closeup of the front of the current. The difference in distance traveled is less than 0.1%.

Figure 5

(a) Current height and (b) resulting deposit height normalized by $D/L$ for various CFL ratios obtained by varying the time step $\mathrm{\Delta }t$. Other parameters are as shown in Table 2 and we show a cut from the center of the domain in a direction perpendicular to the vehicle motion. [(c), (d)] Closeups emphasizing the region where variations are most pronounced.

Figure 6

(a) Current height and (b) resulting deposit height normalized by $D/L$ for various values of $\mathrm{\Delta }x$. Other parameters are as shown in Table 2 and we show a cut from the center of the domain in a direction perpendicular to the vehicle motion. [(c), (d)] Closeup emphasizing the region where variations are most pronounced.

Figure 7

(a) Current height and (b) resulting deposit height normalized by $D/L$ for various values of the minimum height $h_m$. Other parameters are as shown in Table 2 and we show a cut from the center of the domain in a direction perpendicular to the vehicle motion. (c) Closeup of the deposit height at the front where variations are most pronounced.

Figure 8

(a) Current height and (b) resulting deposit height normalized by $D/L$ for various values of the numerical Reynolds number (${\text{Re}}_n$). Other parameters are as shown in Table 2 and we show a cut from the center of the domain in a direction perpendicular to the vehicle motion. (c) Closeup of the deposit height at the front where variations are most pronounced.

Figure 9

(a) Contour of the current height $h=2h_m$ for various numerical Reynolds numbers. (b) Contour of the current height $h=2h_m$ for various minimum current heights $h_m$. Other parameters are as shown in Table 2 and the vehicle is shown as a cyan circle.

Figure 10

Front position of a two-dimensional current with no source present as computed by the SWE. The slumping phase with constant velocity is recovered, with a computed velocity of 1.42.

Figure 11

Box model simulated contours of the current boundary for (a) $u_s=0.005$, (b) $u_s=0.01$, and (c) $u_s=0.02$. Here the vehicle position is shown with red circles and the black contours are the boundary until time $t=20$, at which point the vehicle stops. Green contours are the boundary location after the vehicle has stopped moving. We set ${\text{Fr}}^2=2$.

Figure 12

Shallow-water equation simulated time evolution of the current height for [(a)–(d)] $u_s=0.005$, [(e)–(h)] $u_s=0.01$, and [(i)–(l)] $u_s=0.02$. The vehicle is shown as a red circle and stops at time 20. Other parameters are ${\text{Fr}}^2=2$ and $C_D=\pi /10$.

Figure 13

Using the BM, (a) current area and (b) suspended mass as a function of time for various settling speeds. Using the SWE, (c) current area and (d) suspended mass as a function of time. Here the vehicle is moving until time $t=20$, at which point it stops. Other parameters are ${\text{Fr}}^2=2$ and, for SWE, $C_D=\pi /10$.

Figure 14

[(a)–(c)] BM simulated deposit heights at time $t=26$ for various particle settling speeds $u_s$. [(d)–(f)] Corresponding SWE simulated deposit heights at time $t=26$. Other parameters are ${\text{Fr}}^2=2$ and, for SWE, $C_D=\pi /10$.

Figure 15

Deposit height at time $t=26$ at the level immediately to the side of the vehicle, $y=1$ for various particle settling speeds $u_s$ obtained (a) by the BM and (b) by the SWE. Other parameters are ${\text{Fr}}^2=2$ and, for SWE, $C_D=\pi /10$.

Figure 16

Deposit heights at time $t=26$ for various settling speeds, $u_s$, at the three points along the progression of the vehicle: $x=2.5$, $x=10$, and $x=17.5$. Results are obtained [(a)–(c)] using the BM and [(d)–(f)] using the SWE. Other parameters are ${\text{Fr}}^2=2$ and, for SWE, $C_D=\pi /10$.

Figure 17

Box model simulated contours of the current boundary for (a) ${\text{Fr}}^2=0.5$, (b) ${\text{Fr}}^2=2$, and (c) ${\text{Fr}}^2=8$. Here, the vehicle position is shown with red circles and the black contours are the boundary until time $t=20$, at which point the vehicle stops. Green contours are the boundary location after the vehicle has stopped moving. We set $u_s=0.01$.

Figure 18

Shallow-water equation simulated time evolution of the current height for [(a)–(d)] ${\text{Fr}}^2=0.5$, [(e)–(h)] ${\text{Fr}}^2=2$, and [(i)–(l)] ${\text{Fr}}^2=8$. The vehicle is shown as a red circle and stops at time 20. Other parameters are $u_s=0.01$ and $C_D=\pi /10$.

Figure 19

Snapshots of the shape of a gravity current at time $t=18$ for various Froude numbers as simulated by the shallow-water equations. The vehicle is shown as a red circle and the estimated opening angle with tangent $\sqrt{2}/\text{Fr}$ is shown in pink. Other parameters are $u_s=0.01$ and $C_D=\pi /10$.

Figure 20

Using the BM, (a) current area and (b) suspended mass as a function of time for various Froude numbers. Using the SWE, (c) current area and (d) suspended mass as a function of time. Here the vehicle is moving until time $t=20$, at which point it stops. Other parameters are $u_s=0.01$ and, for SWE, $C_D=\pi /10$.

Figure 21

[(a)–(c)] BM simulated deposit heights at time $t=26$ for various Froude numbers Fr. [(d)–(f)] Corresponding SWE simulated deposit heights at time $t=26$. Other parameters are $u_s=0.01$ and, for SWE, $C_D=\pi /10$.

Figure 22

Deposit heights at time $t=26$ at the level immediately to the side of the vehicle, $y=1$, for various Froude numbers Fr obtained (a) by the BM and (b) by the SWE. Other parameters are $u_s=0.01$ and, for SWE, $C_D=\pi /10$.

Figure 23

Deposit heights at time $t=26$ for various Froude numbers, Fr, at the three points along the progression of the vehicle: $x=2.5$, $x=10$, and $x=17.5$. Results are obtained [(a)–(c)] using the BM and [(d)–(f)] using the SWE. Other parameters are $u_s=0.01$ and, for SWE, $C_D=\pi /10$.

Figure 24

(a) Spanwise deposit heights at time $t=26$ for various settling speeds $u_s$ at high Froude number, ${\text{Fr}}^2=8$, at streamwise location $x=2.5$. (b) Maximum extent of the deposits, defined, over all values of $x$, as the greatest value of $y$ where the deposit exceeds 0.001, as a function of settling speed. Results are compared to Eq. (23), shown as the red line.

Figure 25

(a) Maximum deposit extent, $R_m$, in meters for two different settling speeds corresponding to silt (top) and sand (bottom). The solid lines are the relevant maximum deposit curves based on the Froude number and the dashed lines are their continuations. Computations are made at high Froude number for $U_v>1.4\mathrm{m}$/s using Eq. (24), and at low Froude number for $U_v<1.4\mathrm{m}$/s using Eq. (27). Other parameters are $\mathrm{\Delta }{\rho }_s=1.5$, $L=2\mathrm{m}$ and $D=10\mathrm{cm}$, so that $\text{Fr}\approx U_v$ if $U_v$ is in m/s. (b) Maximum deposit extent in meters in the high Froude number regime, where $R_m$ is independent of the vehicle speed, as a function of particle settling speed. Other parameters are as in panel (a).