The Level Set Method Applied to Avalanches

Excerpt from the Proceedings of the COMSOL Users Conference 2007 Grenoble

The Level Set Method Applied to Avalanches E. Bovet*,1, L. Preziosi 1, B. Chiaia1 and F. Barpi1 1 Politecnico di Torino *C.rso Duca degli Abruzzi 24, 10129 Turin (Italy), [email protected] Abstract: In this paper, the level set method, suitable for free boundary problems, is applied to snow avalanches. To this aim, the system constituted by air and avalanche is considered as a domain composed by two fluids having different densities and viscosities. In particular a shear thinning fluid-like constitutive behavior is assumed for snow as well as a Newtonian one. Simulations show, for instance, the transport of avalanche mass towards the head and that the head of the avalanche is faster than its tail, as confirmed by experimental observations. An important aspect is that this model allows to describe the shape variation, the velocity and the pressure, outputs of fundamental importance to describe correctly the interaction between the avalanches and the structures located along the avalanche path. In conclusion, although these results are still qualitative, they confirm that the proposed method can be adequately refined to describe the real behaviour of avalanches. Keywords: Level set, snow avalanche, NonNewtonian Fluid, shear-thinning.

1. Introduction In this paper some simulations are presented to describe the behavior of the dense avalanche, namely a snow avalanche with a high-density core at the bottom in which the motion is dictated by the relief, in opposite to airborne avalanche, a very rapid flow of a snow cloud in which most of the snow particles are suspended in the ambient air by turbulence. Moreover different hints to study the interaction between avalanche and structures are given.

2. Theory In literature [1] different computational models to describe snow avalanche motion exist. The empirical ones include statistical and comparative models for runout distance computation, while dynamic ones describe the physics of avalanche rooting, as instance, in hydraulic theory where the moving masses are described as Newtonian or Non-Newtonian

fluids, or considering the avalanche a granular flow [2], or, simply, simulating this phenomenon as a sliding block. In this paper the avalanche is considered a deformable body, in particular an incompressible fluid with a Newtonian behaviour and a shear thinning one. These hypothesis are in agreement with a large number of literature models [1].

2. Governing Equations Since the avalanche is considered as a fluid, a generalized version of the Navier-Stokes equations, that allows for variable viscosity too (non-Newtonian fluids) is used: ∂U − ∇ ⋅ [η (∇U + (∇U )T )] + ρ (U ⋅ ∇)U + ∇p = F ∂t ∇ ⋅U = 0

ρ

where ρ is the density, U = (u; v)T is the velocity, p is the pressure, η is the viscosity, F is a volume force field such as gravity. The first equation is the momentum balance, while the second one is the equation of continuity for incompressible fluids. To describe the avalanche motion is necessary to simulate its shape variation, and hence the evolution of the interface f ( X , t ) = 0 between the snow and the air. Deriving this expression in time the equation becomes: ∂f + U ⋅ ∇f = 0 ∂t

3. Methods To describe the evolution of the avalanche shape the level set method, suitable for free boundary problems, is applied [3]. Let’s consider the system constituted by air and avalanche as a domain composed by two fluids having different densities and viscosities. The method describes the evolution of the interface between the two fluids tracing an isopotential curve of the level set function φ . The interface is described by φ = 0 , the more dense and more viscous fluid is placed in the


domain where φ > 0 and the less dense and less viscous one is situated in the zone characterized by φ < 0 . The function Φ is transported by the equation ∂φ + U ⋅ ∇φ = 0 ∂t Let’s note that this equation is the same of this one deduced before. It is important to underline that the density and the viscosity have to be described through the level set function, since they move jointly to the function φ . The aim of defining density and viscosity in the whole domain is reached using Heaviside function that in COMSOL is substituted by its mollification flc2hs, that is smooth, C2continuous, with finite derivative. Hence density and viscosity are defined in the whole domain through:

ρ = ρ1 + H (φ ) ⋅ ( ρ 2 − ρ1 ) η = η1 + H (φ ) ⋅ (η 2 − η1 ) where ρ1 (η1 ) and ρ 2 ( η 2 ) are, respectively, the density (viscosity) lower (air) and higher (avalanche) (values in Appendix).

5. Experimental Results In this section the results of different simulations are reported. It’s important to underline that the results obtained are still qualitative, because the model should be refined and calibrated with experimental data. The first case considered describes the avalanche through a Newtonian fluid with the real values of density and viscosity. By the analysis of the density (Fig. 1) we can note that the mass of the avalanche is transported towards the head, as in nature happens, while the velocity representation (Fig. 2) shows that the head of the avalanche is faster than the queue, as confirmed by experimental observations. Besides, the introduction of friction forces allows the avalanche to stop itself (Fig.3). Indeed only on the horizontal ground, for simplicity’s sake, two resistive forces are considered: a Coulomb force and a viscous one. The first term is proportional to the normal force through a friction coefficient assumed to decrease with velocity as µ = µ C (1 − e − u ) .

4. Numerical Model COMSOL simulations are carried through the multiphysics mode of convection and diffusion (CD) in the transient analysis, to which it is coupled the incompressible Navier-Stokes (NS) of the Multiphysics option. The expressions of the subdomain setting are related in Appendix. As initial conditions, velocity and pressure are both set to zero, while the concentration is determined as a circumference. As boundary settings concerns, along the slope a slip condition (NS) is imposed, to allow the slipping of the snow mass along the slope, as well as an insulation/symmetry condition (CD). Hence stabilization technique are used in both CD and NS multiphysics modes. Besides mesh is refined close to the avalanche path. Finally, let’s note that in the area of the obstacle the NS and CD are been deactivated. Figure 1. Simulation 1: snapshots showing the density (avalanche is the red zone, air is the blue one) at time t=0, 1, 2 and 4 s.


Figure 2. Simulation 1: snapshot shows the velocity field (surface) and the interface between the two phases (solid red line) at time 1.2 s.

Figure 4. Simulation 2: Snapshot shows the viscosity at time t=2 s.

The third and the fourth cases consider the interaction between an avalanche, described for simplicity’s sake as a Newtonian Fluid, and a rigid structure. Fig.5 shows the effect that a dam has of containing the snow. On the other hand, Fig.6 shows as an obstacle open snow wide and the behaviour of the pressure: on the windward side the pressure is positive, on the sides parallel

Figure 3. Simulation 1: snapshot shows the density (avalanche is the red zone, air is the blue one) at time t=12 s. Avalanche is at rest.

In this way the Coulomb force is absent when the avalanche is at rest, while when it runs the force is opposed to the motion. In particular the value of µ C is supposed equal to 0.1763=sin 10, since experimentally the avalanche begins to decelerate when inclination is equal to 10 degrees. As the viscous force concerns, it is proportional to the velocity, through a coefficient estimated [1, 4] to range from 10-3 to 104 kg/s. Finally to concentrate the force only close to the avalanche path an Heaviside function is used. The second case considered describes the avalanche through a Non-Newtonian fluid [5], in particular with a shear thinning behaviour. The viscosity values are deduced by [6]. Fig.4 shows that the lower part of the avalanche has a lower viscosity than the upper one, confirming that in the upper area the deformation is more important. In addition velocity profile deduced in laboratory experiments shows that the avalanche is constituted by two layers: the upper one behaves as a rigid body and the lower one is characterised by an high velocity gradient [4].

Figure 5. Simulation 3: Snapshots showing the density at time t=0, 1, 2.5 and 5 s.


to the flux is negative, and leeward is negative and lower in absolute value. Although these results are qualitative they evidence the possibility, by using a calibrated model, of describing the interaction with structures as houses, avalanche protections, piles, and so on. This aspect is not negligible in the mountain regions.

6. Discussion The most important aspect of using this model is that it allows to describe in each avalanche point the values of velocity and of pressure. Besides, along the whole slope it gives the height of the flux. For practical users these outputs are fundamental to describes correctly the interaction between the avalanches and the structures located along the path. Hence, the deduction of the runout distance, coupled with the determination of velocity and pressure, is the basis to create a risk map. Unfortunately the results are still qualitative. To can be used, the model should be refined. Firstly a real slope, with a drop also of hundreds meters, should be introduced. Secondly the slab should have the shape and the correct dimensions. Thirdly the friction force should be calibrated with the object that values of velocity, pressure, height of flux and runout distance are in agreement with experimental ones. Besides some other physical processes, such as the snow entrainment, could be taken into account. Furthermore an extension to a 3D model could allow to develop an instrument to understand the interaction with complex structures. Finally, it is necessary to choose correctly the parameters peculiar of the numerical model, like the stabilization technique for instance, or a mesh that minimizes the error or even a faster solver for the problem considered.

7. Conclusions In conclusion, although results are still qualitative, they confirm that the proposed method can be adequately refined to describe the real behaviour of avalanches.

8. References

Figure 6. Simulation 4: Snapshots showing the pressure (surface) and the interface between the two phases (solid red line) at time t=0, 1.5, 3 and 5 s. An airborne avalanche density is simulated (10 kg/m^3).

1. C. Harbitz, A survey of computational models for snow avalanche motion, pp. 128, NGI report, Oslo (1998) 2. W. Eckart, S.Faria, K.Hutter, N.Kirchner, S.Pudasiani and Y. Wang, Continum description of granular materials, Part II, pp. 221, Polytechnic Institute, Turin (2002)


3. Comsol. Comsol Multiphysics: rising bubble modelled with the level set method, (2005) 4. K.Nishimura and N. Maeno, Contribution of viscous forces to avalanche dynamics, Annals of Glaciology, 13, pp 202-206 (1989) 5. C. Macosko. Rheology: principles, measurements and application, pp. 568, Wiley, (1994) 6. J.D.Dent and T.E.Lang, A biviscous modified Bingham model of snow avalanche motion, Annals of Glaciology, 4, pp 42-46 (1983)

9. Appendix

Expression -9.81 0.001

eta1

30

eta2 rho1 rho2

30/(1+5abs(uy+ +vx)) 0.0000181 300 1.295

Quantity ρ η Fx Fy

Expression rho eta Fx Fy

Description Density Dynamic viscosity Volume force x-dir Volume force, y-dir

Initial values: u(t0) = v(t0) = p(t0) = 0 Artificial diffusion : Isotropic diffusion δid=0.5 Table 4: Subdomain settings: Convection and Diffusion (CD)

Table 1: Options/Constants

Name gy h_scale

Table 3: Subdomain settings: Incompressible Navier Stokes (NS)

Description gravity constant Scale for resolution of step function Avalanche viscosity (Newtonian) (Shear thinning) Air viscosity Avalanche density Air density

Table 2: Scalar expression

Name H

Expression Flc2hs(ph,h_scale)

rho eta Fy

rho1+H*(rho2-rho1) eta1+H*(eta2-eta1) gy*rho*(1-H)

ph_0

sqrt((x-6)2+(y-6)2)-4

Fx

flc2hs(0.5,h_scale)* *gy*rho*0.1763*(1+ -exp(-u)+3*u)

Description Step function switching at phi=0 Density Viscosity Volume force in y-direction Example of an initial value for phi Friction Force

Quantity δts

Expression 1

D isotropic

0

R u v

0 u v

Description time scaling coefficient Diffusion coefficient reaction rate x-velocity y-velocity

Initial values : ph(t0) = ph_0 Artificial diffusion: streamline diffusion: Petrov-Galerkin/Compensated δsd = 0.25