The August 27, 2014, rock avalanche and related impulse water waves in Fuquan, Guizhou, China

Abstract

At about 8:30 p.m. on 27 August 2014, a catastrophic rock avalanche suddenly occurred in Fuquan, Yunnan, southwestern China. This landslide and related impulse water waves destroyed two villages and killed 23 persons. The impulse waves occurred after initiation of the landslide, caused by the main part of the slide mass rapidly plunging into a water-filled quarry below the source area. The wave, comprising muddy water and rock debris, impacted the opposite slope of the quarry on the western side of the runout path and washed away three homes in Xinwan village. Part of the displaced material traveled a horizontal distance of about 40 m from its source and destroyed the village of Xiaoba. To provide information for potential landslide hazard zonation in this area, a combined landslide–wave simulation was undertaken. A dynamic landslide analysis (DAN-W) model is used to simulate the landslide propagation before entering the quarry, while Fluent (Ansys Inc., USA) is used to simulate the impulse wave generation and propagation. Output data from the DAN-W simulation are used as input parameters for wave modeling, and there is good agreement between the observed and simulated results of the landslide propagation. Notably, the locations affected by recordable waves according to the simulation correspond to those recorded by field investigation.

Appendix—model validation

Numerical model

Governing equations

For 2D transient flow, a number of governing equations are required, including continuity (Eq. 4a) and conservation of momentum (Eqs. 4b, 4c), in which the velocity and pressure of fluid are unknown.

$$ \rho \left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)={S}_m $$

(4a)

$$ \rho \left(\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\mu \left(\frac{\partial^2u}{\partial {x}^2}+\frac{\partial^2u}{\partial {y}^2}\right)+{S}_x $$

(4b)

$$ \rho \left(\frac{\partial v}{\partial t}+\frac{\partial v}{\partial x}+\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial x}-\rho g+\mu \left(\frac{\partial^2v}{\partial {x}^2}+\frac{\partial^2v}{\partial {y}^2}\right)+{S}_y $$

(4c)

where ρ is fluid density, u and ν are the x component and y component of fluid velocity, respectively, μ is the dynamic viscosity coefficient, S _m is an additional mass source in the x direction, S _x is an additional momentum source in the x direction, and S _y is an additional source in the y direction. In the present study, S _m , S _x , and S _y are equal to zero.

Turbulence model

The renormalization group (RNG) k-ε model is derived from the instantaneous N–S equations using the renormalization group theory developed by Yakhot and Orszag (1986). Although similar to the standard k-ε model, the RNG k-ε is more accurate and reliable for a wider class of flows than the standard k-ε model, especially for swirling flows. The transport equations for the RNG k-ε model are expressed as follows:

$$ \frac{\partial }{\partial t}\left(\rho k\right)+\frac{\partial }{\partial {x}_i}\left(\rho k{u}_i\right)=\frac{\partial }{\partial {x}_i}\left({\alpha}_k{\mu}_{eff}\frac{\partial k}{\partial {x}_i}\right)+{G}_k+{G}_b-\rho \varepsilon -{Y}_M+{S}_k $$

(5a)

and

$$ \frac{\partial }{\partial t}\left(\rho \varepsilon \right)+\frac{\partial }{\partial {x}_i}\left(\rho \varepsilon {u}_i\right)=\frac{\partial }{\partial {x}_j}\left({\alpha}_{\varepsilon }{\mu}_{eff}\frac{\partial \varepsilon }{\partial {x}_j}\right)+{C}_{1\varepsilon}\frac{\varepsilon }{k}\left({G}_k+{C}_{3\varepsilon }{G}_b\right)-{C}_{2\varepsilon}\rho \frac{\varepsilon^2}{k}-{R}_{\varepsilon }+{S}_{\varepsilon } $$

(5b)

In Eq. 5a, 5b, G _k represents the generation of turbulent kinetic energy from the mean velocity gradients. The factor G _b represents the generation of turbulent kinetic energy from buoyancy. The contribution of fluctuating dilatation in compressible turbulence to the overall dissipation rate is represented as Y _M . The quantities α _k and α _ε are the inverse effective Prandle numbers for k and ε, respectively. S _k and S _ε are user-defined source terms. In this model, the model constants are assigned as shown in Table 2.

Table 2 Constants in RNG k-ε turbulence model

Volume of fluid model

Tracking of the water free surface is one of the key problems for water wave modeling. The volume of fluid (VOF) method is used to trace the free surface in Fluent (Hirt and Nichols 1981). The VOF technique can model two or more immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each of the fluids through the domain. Therefore, it is suitable for describing water wave generation and propagation phenomena. In the VOF model, a single set of mass, momentum, and energy equations is shared by the fluids. For each additional phase beyond air, an additional variable, the volume fraction of the phase, is introduced and is tracked throughout the domain. This means that, when using VOF, one transport equation for each variable additional air is added. In other words, considering an infinitesimal volume if the volume fraction of the q-th fluid is denoted as α _q , the following three conditions are possible:

• α _q  = 0: The infinitesimal volume is empty of the q-th fluid;

• α _q  = 1: The infinitesimal volume is full of the q-th fluid;

• 0 < α _q  <1: when the infinitesimal volume contains the interface between the q-th fluid and one or more other fluid.

The tracking of the interface between phases is accomplished by the solution of a continuity equation for the volume fraction of multi-phase flow. For the q-th phase, the equation has the following form:

$$ \frac{\partial {\alpha}_q}{\partial t}+{u}_i\frac{\partial {\alpha}_q}{\partial {x}_i}=0 $$

(6)

In each infinitesimal volume, the volume fractions of all phases sum to unity and the fields for all variables and properties are shared by the phases and represent volume-averaged values, as long as the local value of the volume fraction α _q of each of the phases is known at each location.

Numerical benchmark

Problem setup

The present numerical model is used for simulating waves generated by a rigid wedge sliding into water along an inclined plane replicating a laboratory experiment performed by Heinrich (1992). In the experiment, water waves were generated by allowing a wedge to slide freely down a plane inclined at 45° on the horizontal (Fig. 9). The wedge was triangular in cross-section (0.5 m × 0.5 m), with a density of 2000 kg/m³. The water depth was 0.4 m, and the bottom of the box was initially just above the still-free surface. In the numerical simulation, the motion of the rigid sliding block is entirely governed by the motion experimentally measured by Heinrich (1992) (Fig. 10).

Fig. 9

Schematic diagram of the experimental setup for subaerial landslide-generated waves

Fig. 10

Vertical displacement–time history of the experimental box

Numerical setup

The geometric model created in Gambit and the computational domain is discretized using an unstructured triangular grid, and the denser grid is used near the rigid sliding block. The grid size is 6 mm and 84,484 grids are generated.

No-slip boundary conditions were used at the bottom of the water pool and on the slopes. The pressure outlet boundary condition was used at the top boundary, and the static pressure was kept constant at the boundary. The free surface boundary condition requires that the normal velocity at the free surface be zero and the stress meets the equilibrium condition at the water–air interface. The velocity is set to zero in the initial flow field. The value of the water volume fraction was imposed and was equal to 1 in the water pool and equal to 0 in air.

A two-dimensional unsteady separation pressure–implicit splitting of operator algorithm is adopted in this simulation example. Air is the primary phase and water is the secondary phase in the VOF-based two-phase flow, and a geo-reconstruction scheme is used to capture the free surface. In the present work, the material properties of air and water are defined as follows: ρ _a  = 1.225 kg/m³, ρ _w  = 988.2 kg/m³, μ _a  = 17.9 × 10⁻⁶ Pa ⋅ s, and μ _w  = 0.001 Pa ⋅ s. The rigid sliding block sliding into the water is regarded as a rigid body motion, controlled by user-defined functioning programming and the dynamic mesh method. The time step is selected by a Courant number defined as follows:

$$ \varDelta t\le \mathrm{Courant}\frac{\varDelta {x}_{\mathrm{cell}}}{v_{\mathrm{fluid}}} $$

(7)

where Δx _cell is the minimum cell size and ν _fluid is the maximum fluid velocity. In our model, the Courant number is 0.25 and the time step is 0.001 s.

Analysis of numerical results

The numerical results obtained are compared with the experimental data available in Heinrich (1992). Figure 11 shows a comparison between the experimental and the simulated height of the free surface at t = 0.6 and 1.0 s. Good agreement between experimental data and numerical results demonstrates the ability of the present method to successfully simulate such flows. The numerical model has also been used to simulate wave generation caused by a landslide of granular material reproduced in laboratory conditions by Biscarini (2010).

Fig. 11

Comparison between numerical (continuous line) and experimental (dots) water surface elevation at t = 0.6 and 1.0 s

Figure 12 presents some snapshots of the fluid domain at different time steps. As the rigid sliding block moves into the water, an initial solitary wave is formed near the right end of the block. It is observed that the wave reaches the top of the sliding block at t = 0.484 s, the water surface has a maximum height at t = 0.6 s, and the wave water pitches up at t = 0.95 s.

Fig. 12

Snapshots at different time steps