Multi-phase flow simulation of impulsive waves generated by a sub-aerial granular landslide on an erodible slope

Abstract

This study uses a multi-phase flow model to examine the impulsive waves generated by collapse of steep slopes at a water body. Possible effects of slope’s erodibility on the motion of the landslide and the resulting waves are investigated by comparing a sub-aerial landslide on a rigid slope to that on an erodible slope. Key parameters that are relevant to the generation of the impulsive waves are examined, including the time series of the elevation and location of first wave crest, the location and thickness of granular flow front, submerged granular volume, and potential and kinetic energy for water and granular landslide. Before it is free from the influence of the granular front, the generated wave is a forced wave traveling at the speed of the granular flow front. The first wave crest is about to leave the granular flow front when the speed of the granular front is close to the celerity at which a free long wave can propagate in the water above the granular front, and the maximum height of the first wave crest is reached at the time when the first wave has completed its transition from a forced wave to a free wave. It appears that the erodible slope can increase (1) the volume of the granular material plunged into water, (2) the thickness of granular flow front, and (3) the height of the first wave generated by the landslide. However, the total volume of the granular material plunged into water is not directly related to the height of the first wave.

Appendix. Model formulas

In the present multi-phase flow model, the air and water are regarded as a single fluid (air-water mixture) with variable density and viscosity. For the air-water mixture, the water saturation s, defined by the volumetric ratio of the water to the void (pore space) in a computational cell, is used to track air-water interface. Only one velocity vector, u^f, is required to describe both the air and the water velocities because of the no-slip condition at the air-water interface.

The density ρ_f and the viscosity ν_f of the air-water mixture are determined by

$$ {\rho}_f=s{\rho}_w+\left(1-s\right){\rho}_a, $$

(1)

and

$$ {\nu}_f=s{\nu}_w+\left(1-s\right){\nu}_a, $$

(2)

where, and hereinafter, the subscripts a and w refer to as air and water, respectively.

Rearranging the mass balance equations for the sediment, water, and air lead to the following three equations.

$$ \frac{\partial c}{\partial t}+\nabla \cdotp \left[c{\mathrm{u}}^s\right]=0, $$

(3)

$$ \frac{\partial s}{\partial t}+\nabla \cdotp \left[s{\mathrm{u}}^f\right]-s\nabla \cdotp {\mathrm{u}}^f=0, $$

(4)

and

$$ \nabla \cdotp \left[\left(1-c\right){\mathrm{u}}^f+c{\mathrm{u}}^s\right]=0, $$

(5)

where, and hereinafter, the superscripts s and f refer to the solid and fluid phase, respectively, c is the volumetric concentration of the sediment phase, and u is the velocity. The momentum balance equations for the sediment and fluid phases are

$$ {\displaystyle \begin{array}{ll}\frac{\partial {\rho}_sc{u}^s}{\partial t}+\nabla \cdotp \left[{\rho}_sc{\mathrm{u}}^s{\mathrm{u}}^s\right]=& {\rho}_s cg-c\nabla {p}_f-\nabla \left(c{p}_s\right)+\nabla \cdotp \left[c{\mathrm{T}}^s\right]\\ {}& +\left\{c{\rho}_s\frac{\left({u}^f-{u}^s\right)}{\tau_p}-\frac{\rho_s}{\tau_p}\frac{\left(1-c\right){\nu}_{ft}}{\sigma_c}\nabla c\right\},\end{array}} $$

(6)

and

$$ {\displaystyle \begin{array}{ll}\frac{\partial {\rho}_f\left(1-c\right){u}^f}{\partial t}& +\nabla \cdotp \left[{\rho}_f\left(1-c\right){\mathrm{u}}^f{\mathrm{u}}^f\right]={\rho}_f\left(1-c\right)g-\left(1-c\right)\nabla {p}_f\\ {}& +\nabla \cdotp \left[\left(1-c\right){\mathrm{T}}^f\right]-\left\{c{\rho}_s\frac{u^f-{u}^s}{\tau_p}-\frac{\rho_s}{\tau_p}\frac{\left(1-c\right){\nu}_{ft}}{\sigma_c}\nabla c\right\}.\end{array}} $$

(7)

In Eqs. (6) and (7), g is the gravitational acceleration, p the pressure, T the stress tensor, τ_p the particle response time, ν_ft the eddy viscosity of the fluid phase, and σ_c the Schmidt number. The first term in {·} of Eq. (6) or (7) denotes the drag force and the second term represents the turbulent dispersion; these two terms parameterize the momentum exchange between the sediment and fluid phases.

The present model differs from the model used by Si et al. (2018) mainly in the pressure p^s and stress T^s of the sediment phase. In the present model, the pressure of the sediment phase is a superposition of three components

$$ {p}_s={p}_{sr}+{p}_{se}+{p}_{st}, $$

(8)

where p_sr accounts for the rheological characteristics of dense granular flows, p_se considers the elastic effects associated with static granular materials, and p_st reflects the effects related to the turbulent motion of sediment phase.

According to the characteristics of granular materials proposed by Boyer et al. (2011) and Trulsson et al. (2012), Lee et al. (2016) suggested computing p_sr by

$$ {p}_{sr}=\frac{2{b}^2c}{{\left({c}_c-c\right)}^2}\left({\rho}_f{\nu}_f+2a{\rho}_s{d}^2{D}^{\mathrm{s}}\right){D}^s. $$

(9)

where c_c, a, and b are the model parameters, and D^s is the second invariance of the strain rate tensor.

The pressure component p_se is computed using the following formula proposed by Hsu et al. (2004),

$$ {p}_{se}=K{\left[\max\ \left(c-{c}_o,0\right)\right]}^{1.5}\left\{1+\sin\ \left[\max\ \left(\frac{c-{c}_o}{c_{rcp}-{c}_o},0\right)\pi -\frac{\pi }{2}\right]\right\}, $$

(10)

where c_rcp is the random close packing fraction, c_o is the random loose packing fraction, and K is related to Young’s modulus. Hinze (1959) suggests that p_st can be expressed as

$$ {p}_{st}=\frac{2}{3}{\rho}_s\alpha k. $$

(11)

where k is the turbulent kinetic energy and the parameter α reflects the correlation between the sediment motion and the turbulent motion of the fluid phase.

For the sediment stresses T^s, Lee et al. (2016) used a regularization technique that considers the static sediment layer as a very viscous fluid and modeled T^s by

$$ {\mathrm{T}}^s=-\left(\frac{2}{3}{\rho}_s{\nu}_s\nabla \cdotp {\mathrm{u}}^s\right)+2{\rho}_s{\nu}_s{\mathrm{D}}^s, $$

(12)

where viscosity ν_s is a superposition of two components:

$$ {\nu}_s={\nu}_{sv}+{\nu}_{st} $$

(13)

Physically, ν_sv and ν_st account for the rheological characteristics (visco-plastic effect) and turbulence effects, respectively. The viscosity component ν_sv is determined by

$$ {\nu}_{sv}=\frac{\left({p}_{sr}+{p}_{se}\right)\eta }{2{\rho}_s{D}^s}, $$

(14)

where η is given by

$$ \eta ={\eta}_1+\frac{\eta_2-{\eta}_1}{1+{I}_o/{I}^{1/2}}, $$

(15)

Here, η₁ and η₂ as well as I_o are the constants and I is a dimensionless number. According to Trulsson et al. (2012), $I={I}_v+a{I}_i^2$ where I_v = 2ρ_fν_fD^s/cp_s and ${I}_i=2d{D}^s/\sqrt{c{p}_s/{\rho}_s}$. Hinze (1959) suggests that ν_st can be expressed as

$$ {\nu}_{st}=\frac{2}{3}\alpha {\nu}_{f\mathrm{t}}. $$

(16)

The k − ϵ turbulence model is applied to determine T^f and ν_ft with a correction for low Reynolds number (Lee et al. 2016). The formula to compute τ_p proposed by (Lee et al. 2018) is adopted in the present multi-phase flow model. The values of the main model parameters adopted in the present study are listed in Table 2.

Table 2 Model parameters

The values of random loose packing c_o = 0.57 and critical concentration c_c = 0.6 were recommended by Lee et al. (2015). The friction coefficient η₁ is related to the angle of repose and η₁ = 0.48 is adopted here. Other model parameters are identical to those used in Yu et al. (2018).