Modelling and numerical simulation of two-phase debris flows

Abstract

Gravity-driven geophysical mass flows often consist of fluid–sediment mixtures. The contemporary presence of a fluid and a granular phase determines a complicated fluid-like and solid-like behaviour. The present paper adopts the mixture theory to incorporate the two phases and describe their respective movements. For the granular phase, a Mohr–Coulomb plasticity is employed to describe the relationship between normal and shear stresses, while for the fluid phase, the viscous Newtonian fluid is taken into account. At the basal topography, a Coulomb sliding condition for the solid phase and a Navier’s sliding condition for the fluid phase are satisfied, while the top free surface is traction-free for both the phases. For the interactive forces between the phases, the buoyancy force and viscous drag force are included. The established governing equations are expressed in a curvilinear coordinate system embedded in a curvilinear reference basal surface, above which an arbitrary shallow basal topography is permitted. Taking into account the typical length characteristics of such geophysical mass flows, the “thin-layer” approximation is assumed, so that a depth integration can be performed to simplify the governing equations. The resulting strongly nonlinear partial differential equations (PDEs) are first simplified and then analysed for a steady state in a travelling coordinate system. We find the current model can reproduce the characteristic shape of some flow fronts. Additionally, a stability analysis for steady uniform flows is performed to demonstrate the development of roll waves that means instabilities grow up and become clearly distinguishable waves. Furthermore, we numerically solve the resulting PDEs to investigate general unsteady flows down a curved surface by means of a high-resolution non-oscillatory central difference scheme with the total variation diminishing property. The dynamic behaviours of the granular and fluid phases, especially, the effects of the drag force and the fluid bed friction are discussed. These investigations can enhance the understanding of physics behind natural debris flows.

Appendix: Comparisons with the models of Pudasaini [32]

In the main text, we show that our model is an extension of Pitman and Le [28] by taking into account fluid viscosity and employing coordinate transformation. The theoretical analysis and numerical simulation prove the necessity to include fluid viscosity. While comparing with the model of Pudasaini [32], we find that there exists differences not only in different basal boundary condition for the fluid phase and coordinate system, but also in fundamental balance equations.

Pudasaini [32] starting from the governing equations proposed by Drew [8] developed a general depth-integrated two-phase model. The original governing equations are shown as follows:

$$\begin{aligned}&\partial _t(\widetilde{\rho}_s\phi _s)+\nabla \cdot (\widetilde{\rho} _s\phi _s\varvec{u}_s)=0, \end{aligned}$$

(104)

$$\begin{aligned}&\partial _t(\widetilde{\rho }_f\phi _f)+\nabla \cdot (\widetilde{\rho }_f\phi _f\varvec{u}_f)=0, \end{aligned}$$

(105)

$$\begin{aligned}&\partial _t(\widetilde{\rho} _s\phi _s\varvec{u}_s)+\nabla \cdot (\widetilde{\rho}_s\phi _s\varvec{u}_s\varvec{u}_s) =-\nabla \cdot (\phi _s\widetilde{\varvec{T}}_s)+p\nabla \phi _s+\widetilde{\rho} _s\phi _s\varvec{g}+\varvec{f}_d, \end{aligned}$$

(106)

$$\begin{aligned}&\partial _t(\widetilde{\rho }_f\phi _f\varvec{u}_f)+\nabla \cdot (\widetilde{\rho }_f\phi _f\varvec{u}_f\varvec{u}_f) =-\phi _f\nabla p+\nabla \cdot (\phi _f\widetilde{\varvec{\tau }}_f)+\widetilde{\rho }_f\phi _f\varvec{g} - \varvec{f}_d. \end{aligned}$$

(107)

where $\varvec{f}_d$ includes viscous drag force and virtual mass force.

For (104)–(107), by means of mixture theory, we can understand $-(\phi _s\widetilde{\varvec{T}}_s)$ as partial granular stress $\hat{\varvec{T}}_s$ and $p\nabla \phi _s$ as buoyancy force in (106). The fluid partial stress is taken as $\hat{\varvec{T}}_f=-\phi _fp\varvec{I}+\phi _f\widetilde{\varvec{\tau }}_f$ in (107). By comparing the model equations (106) and (107) of Pudasaini [32] with current ones (5) and (6), we find the differences in granular momentum equations, especially in buoyancy force. To compare the different expressions on buoyancy force, a particular case is considered as follows.

• For the two phases in static state, the equality of the densities of the two phases is further assumed. That means the solid particles suspend in the fluid and the partial solid stress $\nabla \cdot \hat{\varvec{T}}_s$ vanishes, which can also be justified by the expressions (47) and (51). Therefore the momentum equations (5) in the present model and (106) in Pudasaini [32] will become the following forms, respectively,

  $$\begin{aligned}&-\phi _s\nabla p+\widetilde{\rho}_s\phi _s\varvec{g}=0, \end{aligned}$$

  (108)
  $$\begin{aligned}&p\nabla \phi _s+\widetilde{\rho} _s\phi _s\varvec{g}=0. \end{aligned}$$

  (109)

  We find the equation (108) derived from the present model corresponds to the force balance of the solid particles, however Eq. (109) does not. This is a reason that the expression of the buoyancy force $-\phi _s\nabla p$ is adopted in the present model.