Morphodynamics of a sediment bed in a fluid-filled cylinder during spin-down: An experimental study

This paper presents a detailed experimental study of the evolution of a sediment bed under the action of the boundary layers formed under a spin-down flow in a rotating fluid-filled cylindrical tank. Both complete and partial spin-down are considered, i.e., with the tank coming to a complete stop or being only partially decelerated. Two nondimensional numbers, the Reynolds (Re) and Rossby (Ro) numbers, govern the flow. During the spin-down, the bed morphology changes, exhibiting distinct characteristic patterns. The changes are measured using a light attenuation technique (LAT). Background rotation suppresses the emergence of instabilities and turbulence, making the patterns more regular and smoother. In spite of the differences and complexity in the patterns, radially inward transport is explained by the total radial force exerted by the flow as computed using classical expressions for the structure of laminar boundary layers.

Figure 1

Value of ${\tau }^{*}$ in the bottom boundary layer obtained numerically as a function of Ro.

Figure 2

Schematic of the experimental setup used, consisting of a rotating table (A), a base containing the light source (B), an outer square tank (C), an interior cylindrical tank (D), a layer of particles of height $h_p$ (E), and side (F) and top (G) cameras.

Figure 3

Reconstructed average bed thickness $\left(h_p\right)$ compared with the bed thickness measured at every level used for the performed calibration $\left(z\right)$. Error bars indicate the standard deviation of the reconstruction.

Figure 4

Patterns formed for four experiments with ${\text{Ro}}^{-1}=0$ and increasing Re values. (a) Spiral pattern that emerged in the granular bed with $\text{Re}=29800$. (b) A complex spiral that emerged with $\text{Re}=35700$. (c) Semiconical dome with spiral arms that formed with $\text{Re}=41700$. (d) Semiconical dome that formed with $\text{Re}=89300$. It can be observed that higher Re values lead to larger particle transport, causing a higher particle accumulation at the center of the tank.

Figure 5

Patterns formed in four experiments with ${\text{Ro}}^{-1}>1$ and $\text{Re}=17900$. (a) Crescent ripple pattern formed with ${\text{Ro}}^{-1}=2$. (b) Transition between a crescent ripple pattern and an annulus that occurred with ${\text{Ro}}^{-1}=3$. (c) Dome with ripples formed with ${\text{Ro}}^{-1}=4.3$. Spherical cap formed with ${\text{Ro}}^{-1}=7.6$. As the ${\text{Ro}}^{-1}$ value increases, particle transport also increases, and smaller features (ripples) in the emerging patterns diminish.

Figure 6

Four examples of patterns observed for experiments with ${\text{Ro}}^{-1}<1$ (weak rotation) and different Re values. (a) Simple spiral pattern that emerged with $\text{Re}=23800$ and ${\text{Ro}}^{-1}=0.16$. (b) Transition between a crescent wave pattern and a simple spiral that emerged with $\text{Re}=23800$ and ${\text{Ro}}^{-1}=0.63$. (c) Complex spiral that emerged with $\text{Re}=29800$ and ${\text{Ro}}^{-1}=0.32$. (d) Dome with ripples that emerged with $\text{Re}=29800$ and ${\text{Ro}}^{-1}=0.8$.

Figure 7

Diagram of the parameter space explored showing the six different types of patterns. $\diamond$ markers indicate the formation of spiral ripples. $⊳$ markers indicate the formation of crescent ripples. $★$ markers indicate the formation of complex spirals. $▵$ markers indicate the formation of a semiconical mound with spiral arms. $\square$ markers indicate the formation of a dome with ripples along its contour. $\circ$ markers indicate the formation of a spherical cap. Experiments with overlapping markers indicate that the emerging pattern shares characteristics of two typical patterns.

Figure 8

Photographs showing the formation of a semiconical pattern. Images are taken from the experiment with $\mathrm{\Delta }\mathrm{\Omega }=0.7\mathrm{rad}{\mathrm{s}}^{-1}$ and ${\mathrm{\Omega }}_f=0\left(\text{Re}=41700,{\text{Ro}}^{-1}=0\right)$. The formation eventually collapses into a semiconical mound surrounded by spiral arms around it.

Figure 9

Photographs showing the formation of a spherical-cap dome pattern. Images are taken from the experiment with $\mathrm{\Delta }\mathrm{\Omega }=0.4\mathrm{rad}{\mathrm{s}}^{-1}$ and ${\mathrm{\Omega }}_f=1.9\mathrm{rad}{\mathrm{s}}^{-1}\left(\text{Re}=23800,{\text{Ro}}^{-1}=4.75\right)$. Once the flow weakens, the dome flattens and small ripples emerge around it.

Figure 10

Example of a displacement field $\mathrm{\Delta }h$ (a) and its corresponding transmitted light intensity $I$ recording (b). The experiment used in this example has parameter values of $\text{Re}=23800$ and ${\text{Ro}}^{-1}=0.625$.

Figure 11

Temporal behavior of the net displaced particle volume $\left(V_{\mathrm{disp}}\right)$ of two experiments with equal Re values but different Ro values. Circle markers indicate the experiment performed with $\text{Re}=23800$ and $\text{Ro}=1.0$. Square markers denote the experiment performed with $\text{Re}=23800$ and $\text{Ro}=0.27$.

Figure 12

Maximum net radially displaced particle volume $\left(V_{\mathrm{disp}}\right)$ as a function of the dimensionless total radial force exerted by the spin-down flow $F^{*}$ [Eq. (7)]. Circle markers indicate complete spin-down experiments $\left({\text{Ro}}^{-1}=0\right)$ and square markers partial spin-down experiments $\left({\text{Ro}}^{-1}>0\right)$. Error bars indicate the difference between the total particle volume when the maximum $V_{\mathrm{disp}}$ occurs and the initial total particle volume $V_T$.

Figure 13

Measurement of the error in time for three experiments. Each of them has a different cause of error. $▿$ markers indicate the experiment where the largest amount of particle suspension occurred $\left(\text{Re}=89300,{\text{Ro}}^{-1}=0\right)$. $\circ$ markers indicate one of the experiments where small particles (debris) increased the error in the measurements $\left(\text{Re}=17900,{\text{Ro}}^{-1}=0.37\right)$. $\square$ markers indicate an experiment where particle suspension and small particle accumulation do not occur $\left(\text{Re}=23800,{\text{Ro}}^{-1}=1.0\right)$.

Figure 14

Plot of the force balance presented in Eq. (16). Markers indicate the rotation difference in each experiment: $\mathrm{\Delta }\mathrm{\Omega }=0.3\mathrm{rad}{\mathrm{s}}^{-1}(\square$ markers), $0.4\mathrm{rad}{\mathrm{s}}^{-1}(♦$ markers), and $0.5\mathrm{rad}{\mathrm{s}}^{-1}(▵$ markers). The fitted curve (dashed red line) follows $\alpha x+\mu =\mathrm{\Gamma }$, where $\alpha =(1.477\pm 0.100)\times {10}^{-6},{\mu }_c=-0.02\pm 0.03,x={\nu }^2/g^{\prime }{(d/2)}^3{F}^{*}$, and $\mathrm{\Gamma }=3V_{cn}/\left(\pi R_{\min }^3\right)$. The prediction bounds are indicated by the dotted blue line.

Figure 15

Minimal radius of a particle mound for the partial spin-down experiments as a function of the dimensionless force $F^{*}$ [Eq. (7)]. The marker's shape indicate the same experiments as in Fig. 14. Error bars indicate the standard deviation in the measured sediment bed radius. The fitted curve (dashed red line) follows Eq. (17), where $\mu =0$.

Figure 16

Diagram of the explored parameter space showing the experiments where particles were suspended or not. Filled symbols are experiments for which particle suspension occurred during spin-down. Experiments denoted with empty symbols showed no particle suspension. The marker's shape indicate the same regimes as in Fig. 7.

Figure 17

Side view of the particle layer illuminated by a vertical laser sheet for an experiment $\mathrm{\Delta }\mathrm{\Omega }=0.6\mathrm{rad}{\mathrm{s}}^{-1}$ and ${\mathrm{\Omega }}_f=0\mathrm{rad}{\mathrm{s}}^{-1}\left(\text{Re}=35700,{\text{Ro}}^{-1}=0\right)$. The side camera is placed at a shallow angle for an oblique view. Only half of the tank is shown, with the side wall located on the left of the image. A mix of sediment particles and tracer particles is used to visualize the flow. Three distinct regions are observed: (A) the surface of the particle bed that is not completely illuminated, (B) sediment particles illuminated by the vertical laser sheet, and (C) tracer particles illuminated by the laser. In region (C) isolated patches of high particle concentration can be observed, indicating the presence of turbulent structures. These structures are only observed for the experiments with a low ${\text{Ro}}^{-1}$ value.