Physical modeling of the dam-break flow of sedimenting suspensions

We develop a physical model of the dam-break flow of fine noncohesive particles initially fluidized by a gas. By revisiting previous experiments, we show that the dynamics of such flows involves two uncoupled phenomena. On the one hand, the settling of the particles is the same as that of a nonflowing suspension, so that the mass flux of particles that deposit can be related solely to the properties of the suspension. On the other hand, the flow of the gas-particle mixture is similar to that of an equivalent fluid of constant density and negligible viscosity. The momentum lost by the flowing mixture is equal to the product of the deposited mass flux and the longitudinal velocity. These properties allow us to model the time duration of the flow as the time taken by the particles to settle and the slope of the final deposit as the ratio between the growth rate of the deposit height and the velocity of the front of the dam-break flow. Finally, these findings lead to the formulation of consistent shallow-water equations involving specific terms of mass and momentum transfer at the bottom wall, which can be used to compute the dense lower layer of ash flows generated by a volcanic eruption. They also provide tools for the interpretation of field measurements by geologists.

Figure 1

Schemes of the various experimental configurations: (a) packed state, (b) homogeneously fluidized and expanded suspension, (c) nonflowing defluidization process, (d) simultaneous flow and defluidization process performed through the dam-break flow, and (e) final packed deposit after flow has ceased.

Figure 2

Picture of the dam-break flow of an expanded suspension of volcanic ash heated at $180{}^{\circ }\mathrm{C}$. Note that the picture is not a pure side view, but taken from a point above the surface. The whitish zone corresponds to the surface the suspension seen in perspective. The two inserts are pictures of volcanic ash (on the left) and FCC particles (on the right).

Figure 3

Experimental ratio between the average front velocity, $L/T$, and the constant velocity of the second phase of the dam-break-flow, $U_{{\mathcal{F}}_2}$, as a function of ${\mathrm{\Phi }}_s$/${\mathrm{\Phi }}_{\mathrm{pack}}$, for all experiments.

Figure 4

Profiles of the final deposit within the channel ($x_0\le x\le L$). Black curves, experimental data; blue lines, model prediction. (a) $As{h}^2$ deposits obtained for different values of ${\mathrm{\Phi }}_s$/${\mathrm{\Phi }}_{\mathrm{pack}}$; (b) ${}^1$, $As{h}^2$, and FCC deposits obtained at a given value of ${\mathrm{\Phi }}_s$/ ${\mathrm{\Phi }}_{\mathrm{pack}}$.

Figure 5

Assessment of the model regarding the deposit slopes. (a) Measured deposit slopes $|h_{d_{\infty }}/L|$ (black symbols) and values predicted by Eq. (6) (blue symbols) as a function of ${\mathrm{\Phi }}_s$/${\mathrm{\Phi }}_{\mathrm{pack}}$, for all experimental cases. (b) Ratio between measured and predicted slopes.

Figure 6

Assessment of the model regarding the total duration $T$. (a) Measured values of $T$ as a function of ${\mathrm{\Phi }}_s$/${\mathrm{\Phi }}_{\mathrm{pack}}$, for all experiments. (b) Ratio between measured values of $T$ and values predicted by Eq. (8).

Figure 7

Analysis of the wall friction ${\tau }_p$ and of its contribution to the dissipation from experimental data. (a) Shields number comparing wall friction to particle weight; (b) potential energy $E_p$ released during the total flow duration; (c) energy $D_{\mathrm{sed}}$ dissipated by the sedimentation flow; (d) energy $D_{db}$ lost by dam-break flow due to ${\tau }_p$, calculated under the assumption of a linear longitudinal velocity profile, Eq. (22).