Size effects in underwater granular collapses: Experiments and coupled lattice Boltzmann and discrete element method simulations

Immersed granular collapse is a common model case for the study of transient geophysical flows. This paper examines the effects of column size on granular collapses in water, with an emphasis on the granular flow mobility. Laboratory-scale experiments of underwater granular collapses with three different column sizes are carried out, together with their numerical simulations using the coupled lattice Boltzmann and discrete element method. Both experimental and numerical data show that, for an identical aspect ratio, a larger underwater granular collapse results in higher flow mobility and a longer runout distance normalized by the initial column length $L_i$ (increased by 18% on average as $L_i$ increases from 3 to 10 cm). Simulations show that as the column size increases, there is more potential energy being transferred into the kinetic energies of the fluid and the particles, and there is a positive relationship between the column size and the efficiency of energy conversion of the particle kinetic energies from vertical to horizontal directions, which contributes to a higher underwater granular flow mobility in larger cases. The reason is twofold. First, the fluid inertia scales disproportionately with the column size. A stronger eddy with high inertia is induced in the large case, which penetrates through the flowing layer of the granular phase and pushes the particles forward to reach a longer runout distance. Second, large underwater granular collapses are accompanied with more significant contact lubrication, which promotes basal slip and dissipates less energy during horizontal spreading.

Figure 1

Sketch of the experimental setup for underwater collapses of granular columns composed of glass beads (GBs): (a) initial condition, (b) final deposit.

Figure 2

Underwater granular collapses in experiments. The three columns and rows correspond to three different column sizes (small, medium, and large) and aspect ratios ($a=0.5$, 1, 2), respectively. Snapshots at three time instants, $t_{95}/3$, $t_{95}/2$, and $t_{95}$, for each case are shown. The yellows bars are the scales indicating a fixed length equal to 1.5 cm. Examples of video recordings of underwater granular collapses with small, medium, and large sizes are available in the Supplemental Material [47].

Figure 3

Direct comparison between the numerical results (sim.) and the experimental data (lab.) for the time evolution of granular free surface from granular collapses with different column sizes when $a=1$. The free surface of the granular phase at four time instants $t=t_c$, $2t_c$, $5t_c$, and $10t_c$ has been extracted and compared from (a) to (d). The inset of (d) zooms in the frontal region of the final deposit with the arrows pointing to the front positions averaged among five measurements in experiments.

Figure 4

Experimental results of the normalized front position ${\tilde{x}}_{ft}$ plotted against time $t$ when $a=1$ for the large case with $L_i=10\mathrm{cm}$. The inset compares the runout evolution between the small (S), medium (M), and large (L) cases.

Figure 5

Plot of the normalized final runout distance $(L_f-L_i)/L_i$ against the normalized column size $L_i/d_p$ for dry and underwater granular collapses.

Figure 6

(a) Plots of the normalized final runout distance $(L_f-L_i)/L_i$ against the aspect ratio $a$ for small, medium and large granular column collapses. The dashed lines are the linear fit of the experimental and numerical data with their color matching the data points. (b) Plots of the normalized residual column height $H_f/L_i$ against the aspect ratio $a$ for small, medium, and large granular column collapses. The dark gray and the light gray solid lines in (a) and (b) set the two limits characterized by the free-fall regime and the viscous regime, respectively, according to Bougouin and Lacaze [6].

Figure 7

(a) Variations of the runout duration $t_{\mathrm{run}}=t_{95}-t_{\mathrm{trig}}$ with the aspect ratio $a=H_i/L_i$. The times $t_{95}$ and $t_{\mathrm{trig}}$ are defined in Fig. 4. The dashed lines are the linear fit of the data for three different column sizes. The color of dashed lines matches the color of data points. (b) Plot of the runout duration normalized by the characteristic time $t_c=\sqrt{H_i/g^{\prime }}$ against the aspect ratio. The horizontal dashed line indicates the averaged values for $t_{\mathrm{run}}/t_c$ (from both experiments and simulations with different column sizes) when $a>0.75$.

Figure 8

Numerical results: (a), (b) Particles at their initial positions from the small ($L_i=3\mathrm{cm}$) and the large ($L_i=10\mathrm{cm}$) granular collapses when $a=1$. The particles are painted according to their displacements in the $xy$-plane: black (${\delta }_{xy}\le d_p$), red ($d_p<{\delta }_{xy}\le 0.5L_i$), green ($0.5L_i<{\delta }_{xy}\le L_i$), blue ($L_i<{\delta }_{xy}\le 1.5L_i$), yellow (${\delta }_{xy}>1.6L_i$). (c) Comparison between underwater granular collapses with different column sizes in terms of the flow mobility $\overline{L}/\overline{H}$ at various aspect ratios. The inset of (c) sketches the centers of mass of the flowing particles at their initial and final positions, and their difference in horizontal ($\overline{L}$) and vertical ($\overline{H}$) directions.

Figure 9

Numerical results: temporal evolution of the cumulative potential energy loss $\mathrm{\Delta }{E}_h$, the cumulative dissipated energy $E_d$, the particle kinetic energy $E_k^p$, and the fluid kinetic energy $E_k^f$ (if applicable) normalized by the initial total potential energy $E_0$ in (a) small ($L_i=3\mathrm{cm}$, underwater), (b) large ($L_i=10\mathrm{cm}$, underwater), (c) small ($L_i=3\mathrm{cm}$, dry), and (d) large ($L_i=10\mathrm{cm}$, dry) granular column collapses for $a=1$.

Figure 10

Numerical results: temporal evolution of the partial kinetic energies computed from the $x$ and $y$ components of the fluid (if applicable) and particle velocities in (a) small ($L_i=3\mathrm{cm}$, underwater), (b) large ($L_i=10\mathrm{cm}$, underwater), (c) small ($L_i=3\mathrm{cm}$, dry), and (d) large ($L_i=10\mathrm{cm}$, dry) granular column collapses for $a=1$.

Figure 11

Numerical results: plots of (a) the peak partial kinetic energies of particles ($\max E_{kx}^p$ and $\max E_{ky}^p$) normalized by the initial total potential energy $E_0$ and (b) the ratio between $\max E_{kx}^p$ and $\max E_{ky}^p$ against the aspect ratio $a$ for granular collapses with different column sizes. Trend (dashed) lines are drawn to help the comparison. The hollow symbols are the results from dry granular column collapses.

Figure 12

Numerical results: comparison between the small (left panel) and large (right panel) underwater granular collapses ($a=1$) regarding (a), (b) the spatial distribution of solid volume fraction; (c), (d) the vector field of the particle velocity; and (e), (f) the fluid streamlines and the spatial distribution of the fluid velocity magnitude, when $t=3t_c$.

Figure 13

Numerical results: relative fluid-particle velocity ($x$ component) profiles of small and large granular collapses ($a=1$) at the cross sections: (a) $x/L_i=1.3$, (b) $x/L_i=1.35$, (c) $x/L_i=1.4$, (d) $x/L_i=1.45$. The fluid and particle velocities are averaged over $0.2t_c$. The horizontal blue and red dashed lines separate the dense granular flow region and the dilute suspended particle region for the small and large granular collapses, respectively. The region with zero relative velocity is the pure water region.

Figure 14

Numerical results: spatial distribution of the locally averaged and time-averaged interparticle contact force ${\overline{f}}_c$ normalized by a characteristic force based on the hydrostatic solid pressure at four sections: $L_i=1.3$, 1.35, 1.4 and 1.45 in (a)–(d) underwater and (e)–(h) dry granular collapses with small and large sizes.

Figure 15

Numerical results: comparison between (a) the small and (b) the large underwater granular collapses for the normalized particle velocity profiles at cross sections $x/x_{ft}=0.15$, 0.3, 0.45, 0.6, 0.75, 0.9, when $a=1$ and $t=3t_c$. The particle positions are normalized by the maximum flow thickness, $y_{max}$. Plots in (c) and (d) compare the temporal evolution of the normalized basal slip velocity for underwater and dry granular collapses with different column sizes, respectively.

Figure 16

Numerical results: (a) correlation between the maximum fluid velocity $u_{max}^f$ that occurs in the eddies and the characteristic velocity $\sqrt{g^{\prime }{H}_i}$; (b) plot of the maximum number of suspended particles against the Reynolds number $\text{Re}={g^{\prime }}^{1/2}{H}_i^{3/2}/{\nu }_f$ based on the initial column height. The symbol size is proportional to the magnitude of initial aspect ratio.

Figure 17

Numerical results: comparison between the five repeated LBM-DEM simulations of underwater granular column collapse of a medium size with different random seeds ($L_i=5\mathrm{cm}$ and denoted as M1, M2, M3, M4, and M5), together with the small (S: $L_i=3\mathrm{cm}$) and large (L: $L_i=10\mathrm{cm}$) cases, when the initial aspect ratio $a=1$ for (a) the final granular free-surface profile and (b) the particle kinetic energy evolution. The minimum and the maximum values of the normalized runout distance and the normalized peak particle kinetic energy from the five repeated simulations are indicated by the solid straight lines.