Dynamics of migrating sand dunes interacting with obstacles

Wind- and water-driven migrating sand dunes frequently interact with elevated natural and artificial topographical features. The dune-obstacle interaction can alter the migrating behavior of the dune and, depending on the nature of the obstacle, it may generate various societal and technological risks. Here we study the problem of dune-obstacle interaction in a paradigmatic quasi-two-dimensional domain realized in a subaqueous laboratory experiment. Generically, dunes interact with obstacles either by crossing over the obstacle or by being trapped. We describe how the selection of these two distinct dynamical behaviors depends on the size and shape of the obstacle, focusing in particular on the fluid flow in the immediate vicinity of the obstacle. Specifically, we perform a modal decomposition of the measured flow field and we discover that the outcome of the dune-obstacle interaction is closely related to the flow structure above the obstacle.

Figure 1

(a) Experimental setup. Annular flume with a $W=9\mathrm{cm}$ wide working section filled with water and sediment. The motion of water is induced by an assembly of paddles rotating counterclockwise at angular speed ${\mathrm{\Omega }}_p$ and the dominant direction of the flow is from the dune (red) to the obstacle (blue). The flume rests on a turntable, which rotates clockwise at speed ${\mathrm{\Omega }}_t$. The data are primarily collected with a stationary camera (A), but the sediment dynamics near the obstacle can be visualized also with an additional corotating camera (B) mounted to the turntable. (b) Obstacles: ridge, the cross section is a rectangle with aspect ratio 1:3; ramp, ridge with an upstream isosceles right triangle extension; and slide, mirror image of the ramp. In practice, both the ramp and the slide are composed of two adjoined detachable blocks. (c) System setup. In interpreting out results, we abstract the system as quasi-2D with a migrating dune of cross-section area $A$ encroaching on an obstacle of height $H_o$. The origin of the coordinate system is chosen so that the centroid of the obstacle is at $x=0$.

Figure 2

Example experiment with a cylinder of diameter $H_o=6\mathrm{cm}$ located at $x=0$. (a) Space-time diagram showing a migrating dune interacting with and crossing the solid obstacle. The inset allows us to compare the dune speed $c$ at different points of the experiment (to be precise, the arrows correspond to appropriate linear fits for $x\in [-0.75,-0.25]\pi R$ or $x\in [0.25,0.75]\pi R$). Note that a vertical arrow would correspond to $c=0$. (b) Schematic showing the progress of a crossing interaction.

Figure 3

Close-up photographs of a crossing dune-obstacle interaction taken every 12 s with a camera mounted to the turntable [camera B in Fig. 1]. For visualization purposes, the obstacle is masked in blue and the rest of the image is printed in false color. (a) The dune approaches the obstacle. Note that a small amount of sediment swept away from the dune has already accumulated in the downstream recirculation region. (b) The downstream region gradually fills with sediment as the dune crosses the obstacle. (c) At some point, the obstacle is fully buried by the dune. (d)–(f) The dune's crest moves away from the field of view. Upstream and downstream deposits form and gradually shrink.

Figure 4

Example PTV measurements. (a) Ensemble-averaged azimuthal velocity $\overline{U}$ around a cylinder with $H_o=6\mathrm{cm}$. Note the flow speedup above the obstacle and the momentum deficit in the wake. (b) Each inset shows the velocity profile $\overline{U}\left(y\right)$ averaged for $x\in [x_0-5\text{cm},x_0+5\text{cm}]$, where $x_0\in \{-60\text{cm},-40\text{cm},-20\text{cm},0\text{cm},20\text{cm},40\text{cm},60\text{cm},80\text{cm}\}$. For the region corresponding to the obstacle, we set $\overline{U}=0$. The red box of size $10\times 2{\text{cm}}^2$ corresponds to the region near $x=-40$ cm where we define $u_{\mathrm{incid}}$.

Figure 5

Incident speed. (a) Incident speed measured for different obstacles, under flow conditions defined in Table 1, but without any sediment in the tank. Note that apart from ${\mathrm{\Omega }}_{\mathrm{tot}}, u_{\mathrm{incid}}$ is strongly affected by the size and shape of the obstacles. For fixed ${\mathrm{\Omega }}_{\mathrm{tot}}$, the figure can be interpreted as a ranking of obstacles, from least disruptive (small, blunt) to most disruptive (big, sharp). (b) The incident speed is monotonically related to the dune approach speed. Here we fit an empirical law (4). The symbols in (b) are colored with the rotation speeds as shown in (a), with symbol shape and size being related to the particular obstacle used in the relevant experiment.

Figure 6

Dune-cylinder interaction for (a) $H_o=3\mathrm{cm}$ (experiment 1), crossing; (b) $H_o=6\mathrm{cm} \approx H_d$ (experiment 2), crossing; and (c) $H_o=9\mathrm{cm}$ (experiment 3), trapping. (d)–(f) PTV-measured field for the corresponding experiments showing that the size of the wake region increases with $H_o$.

Figure 7

Dune-square interaction for (a) $H_o=3\mathrm{cm}$ (experiment 4), crossing; (b) $H_o=6\mathrm{cm} \approx H_d$ (experiment 7), crossing; and (c) $H_o=9\mathrm{cm}$ (experiment 9), trapping. (d)–(f) PTV-measured field for the corresponding experiments showing that the size of the wake region increases with $H_o$ (cf. Fig. 6).

Figure 8

Shape effects. (a) Ridge of height $H_o=6\mathrm{cm}$ traps the migrating dune (experiment 11). The outcome can be changed by appending (b) a ramp (experiment 13) or (c) a slide (experiment 14). See Fig. 1 for the definition of the shapes. The gap in data in (c) near $x=0.5\pi R$ indicates that the transient bedform was so flat that it fell below our noise level $h=1.6\mathrm{cm}$. (d)–(f) The differences between the three experiments can be related to the PTV measurements. The turbulent wake forming downstream of the ridge is larger than the ones behind the ramp or the slide.

Figure 9

Flow speed effects. In all the panels, the obstacle is a square with $H_o=6\mathrm{cm}$. The return time $t_{\mathrm{return}}$ increases as we increase counterrotation rate ${\mathrm{\Omega }}_{\mathrm{tot}}$ from (a) 10.5 rpm (experiment 5) through (b) 11.5 rpm (experiment 6) to (c) 15.5 rpm (experiment 8). Note that an intermediate case (13.5 rpm) has already been presented in Fig. 7. (d)–(f) PTV measurements confirm that the average azimuthal velocity of the fluid increases with ${\mathrm{\Omega }}_{\mathrm{tot}}$. This increase correlates with the increase in the dune migration rate and so $t_{\mathrm{project}}$ is approximately 14 times larger in (a) than in (c).

Figure 10

Return time for the dune crossing the square with $H_o=6\mathrm{cm}$ (experiments 5–8) plotted as a function of the incident flow speed $u_{\mathrm{incid}}$. Small values of $t_{\mathrm{return}}/t_{\mathrm{project}}$ correspond to strong interruption of the migrating dune. Note that $t_{\mathrm{return}}$ is always lower than $t_{\mathrm{project}}$ and there is an approximate lower bound $t_{\mathrm{return}}>\frac{1}{2}{t}_{\mathrm{project}}$ (note that by construction the dune collides with the obstacle at $t\approx \frac{1}{2}{t}_{\mathrm{project}}$).

Figure 11

Singular values ordered by magnitude. The leading singular value is larger by an order of magnitude as the data have not been centered.

Figure 12

First two principal modes (a), (c), and (e) ${\widehat{u}}_1$ and (b), (d), and (f) ${\widehat{u}}_2$. (a) The first principal mode is strictly positive and is closely related to the mean flow field. (b) The second principal mode contains a marked flow structure of negative velocity localized near the obstacle. (c) The weights of the first principal component are all positive. (d) The weights of the second principal component correlate with the outcome of the dune-obstacle interaction with high values for the trapping (red outlined symbols) and low values for the crossing events (blue outlined symbols). The symbols are colored with the rotation speeds as shown in Fig. 5, with symbol shape and size being related to the particular obstacle used in the relevant experiment. (e) The weights for the first principal component correlate with the incident speed $u_{\mathrm{incid}}$. (f) For the crossing experiments, ${\alpha }_{i2}$ is negatively correlated with the ratio of the actual return time $t_{\mathrm{return}}$ to the projected return time $t_{\mathrm{project}}$. Note that $t_{\mathrm{return}}$ is not defined for the trapping events.