Dynamics and scaling laws of underwater granular collapse with varying aspect ratios

We perform coupled fluid-particle simulations to understand the granular collapse in an ambient fluid (in particular, water) with a wide range of initial aspect ratios. We observe both similar and distinct features in underwater collapses compared to their dry counterparts. As aspect ratio $a$ increases, the normalized runout distance follows a piecewise power-law growth, transitioning at $a=2.5$. We associate this transition with the different growth rates of kinetic energy (with $a$) in vertical and horizontal directions. The ability of utilizing available energy for horizontal motion becomes limited when $a>2.5$. Moreover, the front propagation during underwater collapses can be well scaled by using the initial column height as length scale and considering a reduced gravity (due to buoyancy) in timescale. Under the reduced gravity, the initial fall of tall columns is found to be ballistic, consistent with dry collapses. On the other hand, underwater collapses (especially for large $a$) exhibit unique dynamics due to the presence of water. The eddies generated in water, which may carry considerable fluid inertia, tend to erode the surface of the granular layer, thus modifying the deposit morphology. The energy conversion is also affected by the ambient fluid. While water obviously consumes energy from the granular phase through fluid-particle interactions, it actually increases the efficiency of energy conversion from vertical to horizontal directions. The latter effect compensates the difference of runout distance between underwater and dry collapses.

Figure 1

Setup and primary notation. (a) The three-dimensional model in CFD-DEM, with a computational domain of size $L\times W\times H$. Periodic boundaries are imposed in the width direction. The bottom is roughened by a layer of fixed particles. (b) Initial configuration with initial length $l_i$ and height $h_i$. (c) Front position $x_f\left(t\right)$ and top height $y_t\left(t\right)$ at an arbitrary time $t$. (d) Final configuration with final length $l_f$ and height $h_f$ (measured at $x=0$). The middle deposit thickness $h_f^m$ is measured at $x=0.5l_f$.

Figure 2

Snapshots of granular collapse with $a=1$. From left to right: $t=0.05,0.10,0.20,0.50$ s. (a)–(d) Dry. (e)–(h) Underwater. The color map indicates the magnitude of velocity of individual particles ($|{\mathbf{u}}_i|$), while the arrows in the lower panels represent the fluid velocity field of a central slice. The fixed layer of base particles are not shown.

Figure 3

Snapshots of granular collapse with $a=8$. From left to right: $t=0.05,0.15,0.25,0.50$ s. (a)–(d) Dry. (e)–(h) Underwater. The color map indicates the magnitude of velocity of individual particles ($|{\mathbf{u}}_i|$), while the arrows in the lower panels represent the fluid velocity field of a central slice. The fixed layer of base particles is not shown.

Figure 4

Transport of particles on the deposit surface by the fluid. The arrows indicate the velocity vectors of the fluid, while the points with dark to bright colors represent particles with low (zero) to high velocities.

Figure 5

Front propagation of dry (left panels) and underwater (right panels) cases, with varying aspect ratios ($a=0.5,1,2,4,8$). Circular markers indicate the stoppage time when the change of $x_f$ is smaller than $0.01d_p$. (a), (b) The front position $x_f$ as a function of time $t$ with physical units. The gray thick lines show the acceleration of $0.75g$ and $0.75g^{\prime }$, respectively. (c), (d) The normalized front position $(x_f-l_i)/l_i$ against the normalized time $t/\sqrt{h_i/g}$ and $t/\sqrt{h_i/g^{\prime }}$ for dry and underwater cases, respectively.

Figure 6

Normalized front position and frontal velocity as a function of time for dry (left panels) and underwater (right panels) cases, with varying aspect ratios ($a=0.5,1,2,4,8$). (a), (b) Normalized front position $(x_f-l_i)/h_i$ against normalized time $t/\sqrt{h_i/g}$ and $t/\sqrt{h_i/g^{\prime }}$, respectively. The gray thick lines show the acceleration of 0.75 in its dimensionless form. Circular markers indicate the stoppage time when the change of $x_f$ is smaller than $0.01d_p$. (c), (d) Normalized frontal velocity $u_f/\sqrt{g{h}_i}$ and $u_f/\sqrt{g^{\prime }{h}_i}$ for dry and underwater cases, respectively.

Figure 7

Evolution of the height of top surface ($y_t$) measured at $x=0$, for dry (left panels) and underwater (right panels) cases with varying aspect ratios ($a=1,2,4,8$). (a), (b) $y_t$ as a function of $t$ with physical units. (c), (d) The normalized top height $y_t/h_i$ against the normalized time $t/\sqrt{h_i/g}$ and $t/\sqrt{h_i/g^{\prime }}$ for dry and underwater cases, respectively. The gray thick lines imply the free-fall acceleration of 1 in its dimensionless form.

Figure 8

Normalized runout distance $(l_f-l_i)/l_i$ as a function of the initial aspect ratio $a=h_i/l_i$. The laboratory data (indicated by “lab”) are the results of dry granular collapse from the work of Lajeunesse et al. [8]. The dashed lines are power-law fittings for dry cases [see Eq. (1)]. Error bars obtained from five repeated simulations with varying random seeds are smaller than the symbol size (hence not shown) for cases $a=0.5,0.6,0.8,1,6$ under both dry and underwater conditions.

Figure 9

Final thickness as a function of the initial aspect ratio $a=h_i/l_i$. (a) Normalized deposit thickness $h_f/l_i$ measured at $x=0$. The dashed lines are power-law fittings for dry cases [see Eq. (2)]. The laboratory data (indicated by “lab”) are the results of dry granular collapse from the work of Lajeunesse et al. [8]. (b) Normalized middle deposit thickness $h_f^m/l_i$, which is measured at $x=0.5l_f$. Error bars obtained from five repeated simulations with varying random seeds are smaller than the symbol size (hence not shown) for cases $a=0.5,0.6,0.8,1,6$ under both dry and underwater conditions.

Figure 10

Comparison of deposit morphology between dry and underwater cases. From top to bottom: $a=1,2,4,6,8$, respectively.

Figure 11

Evolution of normalized energy for $a=1$ under dry (left panels) and underwater (right panels) conditions. (a), (b) Total kinetic energy of particles (and fluid, if applicable), compared with the energy converted from total potential energy. (c), (d) Partial kinetic energy of particles (and fluid, if applicable) computed from the velocity components in $x$ and $y$ directions. The triangular markers indicate the time when kinetic energy becomes negligible (i.e., smaller than ${10}^{-4}{E}_0$).

Figure 12

Evolution of normalized energy for $a=8$ under dry (left panels) and underwater (right panels) conditions. (a), (b) Total kinetic energy of particles (and fluid, if applicable), compared with the energy converted from total potential energy. The square markers indicate the time when granular dissipation starts. (c), (d) Partial kinetic energy of particles (and fluid, if applicable) computed from the velocity components in $x$ and $y$ directions. The triangular markers indicate the time when kinetic energy becomes negligible (i.e., smaller than ${10}^{-4}{E}_0$).

Figure 13

Effective coefficients of restitution as a function of aspect ratio. (a) Dry. (b) Underwater.

Figure 14

Peak kinetic energy of particles as a function of aspect ratio. (a) Dry. (b) Underwater. (c) The ratio between peak kinetic energies in horizontal and vertical directions.