# Propagation of Rapid Mass Movements Outcomes from : Fluid

## Transcription

Propagation of Rapid Mass Movements Outcomes from : Fluid

Propagation of Rapid Mass Movements Outcomes from : Fluid Mechanics M. Naaim et G. Chambon Alpine Risk Research Unit Plan Rapid mass moevements Fluid mechanics Contiuum mechanics framework , Mass, moementum and energy conservation and constitutive equations Shallow flows - Depth integration Flow regimes : steady, uniform, … Rheology Snow avalanche Debris flows Constitutive equations definition and criterion, constitutive equation classes Rheometry Rheology of complex fluids (suspension, clay dispersion, granular, ..) Constitutive equations and modelling of Debris flows Constitutive equations and modelling of snow avalanches Impact pressure Hazards evaluation Towards snow avalanche hazard evaluation : Combination of Historical data, terrain features and numerical model Debris flow hazard evaluation Vulnerability relations : Snow avalanches & Debris flows Protection techniques Short description of snow avalanche Avalanches starts on steep slopes It move down the slope and accelerates under a dense form It can reach a velocity of 70 m/s, height of 10 m and a volume > 1 Mm3 Snow avalanches may entrain snow from the snowpack and grow in size From filed measurements, Sovilla 2003 showed an increase of the volume by a factor of 8 to 10. The snow may also ’mix’ with the air and form a cloud This occur when the snow is cohesionless and the terrain is teep An avalanche with a cloud is known as a ‘Mixed or powder’ snow avalanche The powder cloud is a turbulent suspension of snow particles It flows as a gravity current Mixed and powder avalanches can exceed 80 m/s, 100 m and 1 Mm3) Powder part can flow for long distance along flat valley bottoms and even up-hill Even if dense snow avalanche have a moderate velocity, it develops a higher pressure field than the powder avalanches Snow avalanches - Terminology Flow zone 40°>slope>15° Starting zone slope ≈ 40° Run-out zone slope<15° Vulnerability Risk … avalanches ‘sudden’… … Catastrophic consequences. Avalanche dynamics Dense flow Film 1 Transition : formation of powder phase Film 2 Powder (mixed) flows Film 3 Evidence damages observation Evidence from field measurements : radar FMCW Vertical stratification : mixed avalanche Echo is function of density Initial snow mantle Suspension Entrainment rate very high Dense flow Two-layers : model* * See (Naaim 1998, Issler 2003, etc..) Debris flows Debris flows Rapid mass movements involving concentrated slurries of water, fine solids, rocks and boulders. Frequent phenomenon in mountainous areas, but relatively poorly known Illgraben, Switzerland (WSL) Debris flows among the other mass movements involving liquid-solid mixtures Fluid mechanics Soil mechanics cohesive material Landslides Water ion ens p s Su bed load Hyper Concentrated Debris Flows Stability Sediment Rock Falls flows granular material Water hydraulics Torrential hydraulics mechanics Rocks Damages… Valgaudemar, 2002 Contamines - Montjoie 2005 Bovernier 2006 Debris flows characteristics • Monophasic flows with very high solid concentration (φ = 50 – 90%) • Laminar viscous flows • Transient phenomena: successive surges • Characteristic deposits: frontal and lateral lobes and levées no granulometric sorting Typical composition of surges Bardou, 2002 • Muddy slurry involving particles of all sizes • Concentration increases from tail to front Typical figures Front height Speed Average value Variation range 3m 0,5 – 10 m 4 m/s 0,5 – 10 m/s (30 m/s pour some lahars) Volume 25 000 m3 10 000 – 1 000 000 m3 Density 2300 kg/m3 1900 – 2500 kg/m3 80% 50 – 90% Solid concentration Debris flow triggering factors « Permanent » factors (related to watershed geomorphology) High slope (> 15%) Presence of sediments that can be mobilized in the bed and on the side slopes Prequently, scarce vegetation Shape conducive to flow concentration « Triggering » factors Rapid and important water income: high intensity storm (also: dam break, draining of a glacier lake,…) Frequently, preliminary soil saturation (abundant rains prior to the storm, snow melt,…) Two very different time scales Formation mechanisms of debris flows DF formed in the reception basin: superficial erosion localized erosion (e.g., scree zone) DF formed in the channel linear erosion (channel scouring) localized erosion (e.g. bank rupture,…) DF formed from a slow-moving landslide Quantitatively, these mechanisms remain poorly known… Differents types of debris flows Morphology-based classification (deposits) Name Muddy debris flow Granular debris flow Lahar-type debris flow Deposit aspect Regular, with well defined boundaries, fluid when saturated with water, very cohesive when dry Chaotic with unclear boundaries, no cohesion Terrace-shaped, with enough cohesion to walk on when dry Lobes and levées Round shape Planar shape No lobes Slope of the stopping zone >5% > 15 % <1% Speed 1 – 30 m/s ? 1 – 30 m/s rare in the Alps Fluid mechanics framework Continuum mechanics Framework Navier-Stokes Equations r ∂ρ + div ( ρv ) = 0 ∂t r r r r ∂ v ρ + v .∇v = ρg + div(σ ) ∂t Suitable for propagation and stopping phases Hypothesis : Incompressible Adiabatic Shallow flows h/L<<1 Hydrostatic pressure distribution Equations : ∂h ∂hu + = ϕ (u , h) ∂t ∂x ∂hu ∂hu 2 +α = ∂t ∂x LL gh cos θ (tan θ − µ (u , h) ) α ?, µ (u, h,...) ?,ϕ (u, h,...) ? Outline Constitutive law of an incompressible fluid Rheometry Rheology of some model complex materials suspensions muddy materials granular materials Constitutive law of an (incompressible) fluid Local balance equations (Navier Stokes) Local constitutive law links the local stress tensor to the strain rate tensor : 1 σ = Fτ <t (D ) where D = ( ∇v + ∇v ) 2 Intrinsic material property : rheology Depth-integrated equations : long wave approximation (shallow water) Relationship between bottom shear stress and average velocity and height : ( ) τ b = f u, h Boussinesq coefficient : Effective (integrated) properties α = u /u 2 2 T Stress tensor decomposition In a fluid at rest, the stress tensor is isotropic and depends on body forces and boundary conditions More generally, in an incompressible fluid, an isotropic stress tensor does not produce any work The constitutive law determines the stress tensor only up to an unknown isotropic component : the pressure p Stress tensor decomposition : σ = − p I + τ where τ, the extra-stress tensor, is governed by the constitutive law : τ =F D τ <t ( ) Usual assumption : p = −tr σ ( ) and τ is the deviatoric part of σ General principles Objectivity principle: constitutive law must be invariant under any change of reference frame Constitutive law only depends on deformation rate D Energy dissipation rate (τ : D = Fτ <t (D ) : D ) only depends on the 3 invariants of D Causality principle Local action principle Other classical hypotheses (simple fluids): Homogeneity Isotropy « Vanishing memory » : τ = f D , where f is a function ( ) (counterexamples: thixotropy, viscoelasticity) Fluid versus solid Definitions Solid: behavior depends on a reference state = « long memory » : σ = H 0<τ <t σ , ε (t ,τ ) [ 0 ] Fluids : behavior rapidly forgets any previous state = « short memory » : σ = Hτ <t [ε (t ,τ )] = Fτ <t (D ) Most materials can exhibit solid or liquid behaviors depending on the considered time scale Rapid gravitary flows Involved materials: mud, snow, grains, suspensions, mixtures,… Complex fluids: microscopic length scale λ >> λmolec Monophasic, continuum mechanics approach: Lmacro >> λ Jammed materials: behavior intermediate between fluids and solids (existence of yield stresses) Very complex constitutive laws (simple fluid hypotheses must sometimes be released) Newtonian fluids Linear constitutive law: τ ij = 2η d ij = η (∂ i v j + ∂ j vi ) η : dynamic viscosity (Pa.s) ν = η/ρ : cinematic viscosity (m2.s-1) Behavior of most molecular fluids Water (η = 10-3 Pa.s under ambient conditions) Alcohol Oils (η = 1-10 Pa.s typically) Depth-integrated approach: basal stress: τ b = 3η Boussinesq coefficient: α= 6 5 u h How to measure viscosity? Thought experiment: simple shear flow 0 γ& / 2 with γ& = u / H D = γ& / 2 0 viscosity: τ η= γ& y x linear velocity profile: constant shear stress: u τ = τ xy = T / S H Non-newtonian fluids Apparent viscosity (simple shear): flow curve τ η a = = η a (γ& ) γ& Classical behaviors for complex geophysical fluids shear-thinning: η a (γ& ) ↓ shear-thickening η a (γ& ) ↑ yielding: γ& ≠ 0 only for τ ≥ τ c thixotropy: (rare) η a (γ&, λ ) where λ is a state variable governed by a cinetic equation (reversible time evolutions) viscoelasticy normal stresses Rheometry Principle: create a stationary flow in which both stress and strain rate distributions are known Macroscopic measurements (force, velocities) allow to infer the local constitutive law In most rheometers, flows are viscosimetric Constitutive law can be expressed using only one parameter 0 1 0 & γ D = 1 0 0 2 0 0 0 Practically : Difficult to design perfect rheometrical experiments (theoretical simple shear configuration is not feasible) Difficult to determine the complete, tensorial form of the constitutive law Parallel-plate rheometer Principle: Shear rate distribution is known, but shear stress is unknown Impose rotation velocity Ω and mesure torque Γ, or vice-versa γ& (r ) = Ωr h R Γ = 2π ∫ r 2τ (r )dr 0 still possible to infer the constitutive law: τ= d ln Γ Γ + 3 3 & d ln γ 2πR variant: cone and plate geometry Numerous experimental artifacts (difficult experiments) Limited to small-scale samples (h ~ 1mm) Coaxial cylinders (Couette) Principle: Impose rotation velocity Ω and mesure torque Γ, or vice-versa Shear stress distribution is known, but shear rate is unknown Γ τ= 2πr 2 h R2 γ& ( r ) Ω=∫ dr R1 r Procedures to infer the constitutive law involve strong approximations, except in the small gap limit (R2-R1 << R1) Possibility to test large-scale samples Field Couette rheometer with 20 cm gap Inclined plane Principle : Impose flow height h and measure average velocity u (or discharge q), or vice-versa Shear stress distribution is known but shear rate is unknown v(y) y g θ τ = ρg (h − y ) sin θ h y still possible to infer the constitutive law from q(h) relationship: 1 dq γ& = h dh q = ∫ ∫ γ& ( y′)dy′dy 0 0 direct velocity profile measurements are possible Geometry well adapted to geomaterials: possible to test « real » materials allow direct determination of SW-type rheology ( ) τ b = f u, h Other rheometrical techniques Flow in a pipe Sedimentation experiments Practical engineering techniques example: slump test for yield stress fluids hc = τc ρg sin θ hc θ … No perfect rheometers: always account for experimental biases! Stability of shear flows For simple fluids, shear flows are stable if and only if the flow curve is a monotonically increasing function: dτ ≥0 dγ& Experimental flow curves displaying decreasing parts cannot represent the constitutive law of an homogeneous simple fluid Such flow curves frequently occur in case of shear localization They can also indicate thixotropy Suspensions Suspensions of particles in a Newtonian fluid: Dilute regime (φ < 4%) for non-colloidal particles: effective Newtonian behavior with a viscosity given by Einstein formula: 5 2 η = η S 1 + φ When φ increases: complex interactions between particles: −α φ • hydrodynamic regime: Krieger-Dougherty formula η = η S 1 − φm • direct (lubricated contacts) Multiplicity of regimes (as a function of φ, γ, particle size,…), with soft transitions Ancey & Coussot, 1999 Muddy materials (1) Natural materials: mixture of water and particles of all sizes, from clays to metric rock blocks Numerous studies concerning clay dispersions in water Above a given clay concentration: shear-thinning yield stress fluid Coussot, 1995 Muddy materials (2) Viscoplastic rheology: mechanical behavior intermediate between fluid and solid τ Bingham model: τ τ Herschel – Bulkley model: τ < τ c ⇒ γ& = 0 ≥ τ c ⇒ τ = τ c + η Bγ& < τ c ⇒ γ& = 0 ≥ τ c ⇒ τ = τ c + K γ& sign (γ& ) n Tensorial (3D) formulations exist Depth-integrated approach: for n≠1, τb(u,h) is given as the solution of a non-algebraic equation coefficient α depends on h (usually neglected) Muddy materials (3) Addition of non-colloidal particles in a clay dispersion As long as the amount of colloidal particles remains sufficient, the global behavior remains viscoplastic: φ τ c = τ c , 0 1 − φm −α Coussot et al., 1998 Above a given concentration of non-colloidal particles, direct contact become important: granular-type behavior Coussot et Ancey, 1999 Muddy materials (4) Complex behavior close to the yield stress Existence of a critical strain rate • viscosity bifurcation (creep tests) • shear localization (instability) « Catastrophic » fluidification • yielding is closely related to thixotropy Coussot et al., 2004 Coussot et al., 2002 Dense granular materials (1) Behavior governed by inter-granular friction Quasi-static regime relatively well described (soil mechanics) Elasto-plastic constitutive laws Critical state theory Constitutive law for the flowing regime ? Usual assumption (simple shear): local Coulomb law τ = µσ n • Coupling between shear and normal stresses • Evolution of the friction coefficient with shear rate = friction law? µ (γ& ) ? Dense granular materials (2) Inclined plane experiments Determination of the height – discharge relationship in steady uniform regime: u h =β hstop (θ ) gh Effective friction law (depth-integrated approach) ( ) µeff u , h = f ( Fr ) Pouliquen, 1999 Dense granular materials (3) Derivation of a local constitutive law Dimensional postulate: µ depends on a non-dimensional number I Local shear rate and hence local constitutive law can be derived from the experiments µ (I ) = µ1 + µ 2 − µ1 I / I0 +1 Tensorial (3D) extension Cassar et al., 2005 I= γ&d p/ρ Dense granular materials (4) Numerous unsolved issues extrapolation of the law towards high shear rates (collisional regime)? no description of the flow threshold • avalanche angle hysteresis • avalanche angle depends on grain layer thickness necessity of a thixotropic and non-local constitutive law around the threshold? GdR MiDi, 2004 how to account for shear rate-dependent dilatancy? • release incompressibility hypothesis? Rheology of the different types of debris flows Proposed constitutive laws For muddy DF: derived from conventional and large scale rheometry experiments For granular and lahar-type DF: more speculative γ τ Name Muddy debris flow Granular debris flow Lahar-type debris flow Rheology Viscoplastic (yield stress fluid) Frictional (Coulomb) at low shear rates, collisional at high shear rates Frictional (Coulomb) at low shear rates, Newtonian at high shear rates Granulometry very wide fine matrix ++ clay ++ very wide fine matrix – clay -- narrower fine matrix ++ clay - silt + Rheophysical approach DF material = matrix (water+fine particles) + large grains Fine particles (< 40 µm) have colloidal properties that confer viscoplastic properties to the matrix. Depending on the proportion of fine particles in the solid fraction, the complete material remains viscoplastic or not. • Coussot criterion: • fine particle proportion > 10% • fine particle proportion < 1% • notion of granulometric spindle (Bardou et al., 2003) muddy DF granular DF Rheology of muddy debris flows (1) Herschel - Bulkley constitutive law: τ < τ c ⇒ γ& = 0 n τ > τ c ⇒ γ& ≠ 0 et τ = τ c + K γ& 3 constitutive parameters: Name (unit) τc yield stress (Pa) K consistency (Pa.sn) n flow index Typical value for debris flows 500 – 10 000 Pa (highly dependent on concentration) K/τc = 0.1 – 0.3 0.3 Rheology of muddy debris flows (2) Consequences of the yield stress hc • Existence of deposits on non-zero slopes hc = τc ρg sin θ θ Way to determine τc from field measurements… • Existence of an unsheared plug zone in uniform flows hc v(y) y θ Application: steady uniform regime • Height discharge relationship for a 2D steady uniform flow of a Herschel-Bulkley fluid down a slope : n ρ g sin θ q = uh = n +1 K n • Allow to assess overflowing hazard in channel sections n h h − 2n + 1 c h c ( n +1 ) / n Application: shallow water model Result produced by LAVE2D model (Cemagref) extension observée hauteurs des dépôts (m) d’après simulation bâtiments Input parameters: hydrogram, density, rheological parameters, DTM Application: laboratory experiments model viscoplastic fluids : kaolin slurry, carbopol gel,… Scaling governed by 3 non-dimensional numbers (n=1/3): 2 Fr = ρ g h sin θ τc h u , G= , Hb = g h cos θ K u τc 1/ 3 Rheology of muddy debris flows : limits The homogeneous viscoplastic rheology constitutes a strong approximation: • during the propagation phase, surges generally present a complex structure with a granular front followed by a muddy body: rheological heterogeneities inside the surges • during the propagation phase, surges frequently undergo strong interactions with the torrent bed (erosion, deposit) temporal evolution of the rheological properties • the connection between the mechanical properties of the static soil and those of the flowing material represents a largely unsolved issue (predetermination of the rheological parameters) Nevertheless, the viscoplastic rheology yields good results, in particular during the stopping phase: • compute the spreading of debris flows on alluvial fans • hazard zoning Introduction to dense snow avalanches Dense snow avalanches cause extensive damage in mountainous areas. Understanding of snow flow is crucial for risk mitigation, via a better prediction of hazard zones and optimization of costly defence structures Despite these practical needs, the characteristics of the dense snow flow are still largely ignored This due to : the strong difficulties inherent in experimenting with snow. and the complexity of the material: not only does snow belong to the wide variety of cohesive granular materials but its microstructure evolves in time as function of thermo dynamical conditions. As a consequence, snow grains may exhibit various shape, size and cohesive interaction. Full scale experiments Snow flows properties were investigated through two complementary approaches: Full scale studies : artificial release (La sionne, Lautaret, Ryggfun, ) Small scale experiments Several full-scale avalanche test sites provided interesting information concerning velocity and impact pressure of snow avalanches which are useful for engineering purpose (Dent & Lang, 1983 ;Naaim & Naaim-Bouvet, 2001; Vallet et al., 2001). Such studies also lead to distinguish three types of flows: powder, mixed and dense. Powder flows are made of snow grains suspended in air (with a very low density around 1 to 20 kg/m3), and move very fast (10 to 100 m/s). In contrast, dense flows (density between 100 and 500 kg/m3) follow the slope and are made of a continuous network of grains in contact. Natural snow avalanches usually contain a basal dense flow above which a powder flow develops. Although they give access to crucial information, full-scale experiments are not controlled and impossible to reproduce, and measurements within flows are difficult to obtain. Laboratory scale experiments Small scale experiments can be much more easily controlled, and are therefore more appropriate for a rheological investigation. The typical approach consists in performing dense snow flow down an inclined channel, either set into a cold room (Nishimura & Maeno, 1989) or at high altitude (Bouchet et al., 2003, 2004 ; Tiefenbacher & Kern, 2004; Kern et al., 2004;), and then analyzing internal velocity profiles in rheological terms. Despite the strong differences in experimental procedures (channel size, snow preparation, etc), the velocity profiles were found to be remarkably consistent with the velocity profile measured on full-scale avalanches by Gubler (1987); Dent et al. (1998): snow flows are strongly sheared in a thin basal layer and much less sheared in the upper thick part. Such a velocity profile evoked the behaviour of a yield stress fluid for which the free surface flow over an inclined plane exhibits a plugged region above a critical depth. In this context the Bingham (Nishimura & Maeno, 1989), the HerschelBulkley model (Kern et al., 2004), and a bi-viscous model (Dent & Lang, 1983) were used to represent the snow behaviour. Scale model with granular material Since snow is made of grains, several studies indirectly investigated snow avalanche behaviour through extensive experiment with granular materials such as glass beads. Friction laws that describe granular flow down an inclined plane (Savage, 1979; Savage & Hutter, 1989; Pouliquen, 1999; Louge & Keast, 2001) are often used into full-scale avalanche simulation using Saint-Venant approach (Naaim et al., 1997; Mangeney et al., 2003). Rapid granular chute-flow experiments : pointed out the formation of a dilute layer at the free surface of dense flow, (Barbolini et al., 2005b), and also investigated the strong interaction between the flowing material and an erodible bed which is a crucial process for snow avalanches (Naaim et al., 2004; Barbolini et al., 2005a), (Naaim et al., 2004). However, how the granular flows are similar to the snow flows is still an open question, since the comparison would require many more experiments with natural snow. While granular experiments generally involve cohesionless grains, one can expect that intergranular cohesive force between snow grains play an important role in the rheological properties. And since this cohesion significantly evolve throughout snow metamorphism, it is not clear whether there exists a generic behavior for dry snows Dense avalanche : shallow flows is suitable theoretical framework Such as many geophysical phenomena, the snow avalanches have geometrical dimensions that allow adopting the shallow flows framework, (h<<L) Mass and momentum conservation averaged over the depth Velocity profiles, constitutive equations, entrainment are required ∂h ∂hu = ϕ (u , h) + ∂t ∂x ∂hu ∂hu 2 +α = ∂t ∂x LL gh cos θ (tan θ − µ (u , h) ) α ?, µ (u, h,...) ?,ϕ (u, h,...) ? Bibliography Snow / Snow Snow / Ski, Metal, Ice, … 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Référence bibilographique 0.01 Effective friction coefficient Coefficient fe frottement effectifµ: µ 0.1 1 Empirical analysis (and model) of Voellmy 1955 Hypothesis : avalanche is a sliding box of a constant height Snow is Coulombian material τ (u , h,.) = µ (u , h,.)σ = µ (u, h,.) ρgh cos θ Avalanche stop on θ > θ 0 or tan θ > tan θ 0 Powder avalanches Dry dense avalanches Moist snow Field measurements U p hi ⇒ U p h Steady and uniform flow : µ=tan(θ θ) u = C (tan θ − tan θ 0 )h Analogy with hydraulic model u = C (tan θ − tan θ 0 )h u2 ⇒ tan θ = (tan θ 0 ) + 2 (Chezy ) C h g µ = tan θ = µ o + 2 F 2 C dv g = g cos θ (tan θ − µ 0 − 2 F 2 ) dt C Uniform flow : u max = C (tan θ − µ 0 )h Rheometry (1)* µ = µ0 + 0.0013V 2 µ (V ) µ0 = 0.45+ 0.012T * G. Cassassa 1991 µ 0 (T ) Idea : permanent & Uniform flows Conditions : • Permanent & uniform flow regime • Measuring for a large slope range : • Velocity profiles (v(z)) • Flow height (h) • Stresses at the bottom (σ, σ, τ) • And deduce the relation that link the slope to h and v µ = f (v, h,...) Dense snow rheology Experimental procedure : The snow flows have been performed over 3 years at the experimental in-situ test site situated at the col du Lac Blanc, a pass near the Alpe d’Huez ski resort in the French Alps. The high altitude (2830 m) enables the access to large amounts of natural snow between January and April. Flow geometry and feeding system The flow geometry is a 10 m long channel. Its width and height are 20 cm. The slope can be set from 27° to 45°. In order to avoid wall slip the channel bottom was covered with sand paper with a roughness of the order of the snow grain size (0.4 mm). In contrast the lateral walls were smooth (PVC) and the material could slip. Effectively our measurements showed that the velocity at the free surface of the flow was almost constant in a cross-section, which suggests that the lateral walls negligibly affected the flow characteristics, probably due to wall slip Feeding system The feeding system is an important feature of the experimental set-up. It is made of a hopper which can store up to 5 m3 of snow and an Archimedean screw 4 m long with a 0.6 m diameter. The screw injects the snow into the channel at a constant flow rate which can be adjusted up to 0.1 m3/s by varying the rotational frequency (up to 1 Hz). Although the flow rate averaged over a period of rotation is constant, it slightly depends on the orientation of the screw, leading to periodical variations of the order of 20%. In order to limit the effect of such variations on the flow characteristics we set up at the beginning of the chute a system that deviates outside the flume the upper part of the flow and thus ensures a constant flow rate downstream Instrumentation Normal and shear stresses: bidirectional piezoelectric sensors Height: Distance optic sensor Velocity: Pairs of optoelectronic sensors Signal correlation (Dent, 98) Film Velocity measurements D component 1 component 2 component = LED + phototransistor Cross-correlation function: ρ j = ∑ ( x ) i ( y )( i + i dt V=D dt j) Position of the sensors on the channel Velocity profiles P3 P2 P1 Position of the sensors on the channel Height sensors H3 H2 H1 Main characteristics of the flows Rounded grains, typical size: 0,2 – 0,4 mm Snow density : 210 kg/m3 to 270 kg/m3 Slopes investigated : 31° to 45° Height : 4 cm to 15 cm Snow is sieved Permanent and uniform flows 1 S S1 1 2 S 1 H H 1 Vs y 2 S l Permanent and uniform 0 10 3 ρ (kg/m ) 5 0 10 5 0 180 160 140 200 180 5 10 15 t(s) dH dH = =0 dx dt 20 25 140 120 100 160 10 0 160 τ, −τg(Pa) P(Pa) 5 θ Accelerated 41° 200 10 2 Vs Vs l l 33° H2 1 Vs Vs 2 Decelerated H1 2 S H 2 Vs x H3 (cm) H2 1 H H S 11 12 13 14 15 16 17 18 10 12 Permanent and uniform : stresses measured (Constant effective friction coefficient) 16 t(s) t(s) P=ρgHcosθ 14 τ=|τg| =ρgHsinθ µ * ( y) = τ ( y) P( y ) = tan θ 18 Presence of aggregates z x Ecoulement Flow 100 5 Vx (m/s) Population 80 60 40 4 3 20 2 4 6 8 10 12 14 16 18 20 2 -10 Mesure Symétrie -5 0 z (cm) taille (cm) Size distribution Flow Free surface velocity Presence of large aggregates without lateral shear rate 5 10 Velocity profile 6 10 y=4.8cm 5 8 V y (cm) 4 3 2 1 0 6 4 2 0 2 4 6 0 t(s) 0 1 4 5 Bi-sheared profile 10 10 8 8 y (cm) y (cm) Sintering in the hopper Sieving Times of ‘repose’ in the hopper 3 V(m/s) Constant velocity Preparation effects 2 6 4 2 1 min 10 min 15 min 6 4 2 0 0 0 1 2 3 V (m/s) Reproducibility 4 5 0 1 2 3 4 V (m/s) Effect of time in the hopper 5 Velocity profiles shape 14 34° 35.5° 37° 38° 39.5° 40.7° 8 6 H 12 4 8 8 6 4 0 1 2 3 4 5 6 0 1 2 Slope Fluide Fluid Fluide Pente d’arrêt Stop atd’arrêt slope: Pente Profils deprofiles vitesse Velocity Profils vitesse Neige Snow Neige ~33° ~33° bicisaillé Bi-sheared bicisaillé Pente d’arrêt nulle 0 Pente d’arrêt nulle Seuil Yield stress Seuil Constitutive equation τ Thresh old 4 5 Non nulle Non>0 nulle = f (γ& ) τ = ηγ& τ = τ c + ηγ& si τ > τ c si τ < τ c γ& = 0 Partout cisaillé Sheared Partout cisaillé y/H y/H Profil Bouchon 1 2 3 4 5 6 Type of snow 1.0 1.0 ProfilPlug Bouchon 0 V (m/s) Height Newtonien Newton 3 V (m/s) V (m/s) Newtonian Newtonien 4 0 0 0 6 2 2 2 Campagne : 7 10 11 14 10 10 y (cm) y (cm) 10 12 y (cm) 12 14 θ θ= 0.5 0.5 Neige Neige V0 =0 V0 =0 V V00 libre libre Velocity profile on an incline τ ( y ) = ρg ( H − y ) sin θ γ& ( y ) = f −1 (τ ( y )) y V ( y ) − V0 = ∫ γ&( y' )dy' y '= 0 0.0 0.00.0 0.0 Threshold Newton 0.5 0.5 V/V V/VHH 1.0 1.0 Interpretation Bilinear fit of velocity profile Couche inférieure 800 H 12 y (cm) 10 -1 γp (s ) 14 600 . 400 8 200 6 0 4 Symbole 34 yp 2 0 1 2 3 4 36 38 θ(°) 40 4 6 8 10 12 H(cm) Pas d’effet du type de neige 5 Couche supérieure V (m/s) si y > y p 20 γn (s ) si y < y p -1 V ( y ) = γ& p y V ( y ) = γ&n y + B = campagne θ − θ c ∝ (γ& p − γ& cp ) Vp 0 θ = 37° H =10cm 15 . 10 θ = 37° H =10cm 5 Consequence for the constitutive equation Symbole 34 1) Difference between layers 2) For each layer: shear rate constant γ& ( y ) = cte y cos θ P( y ) ⇒ = g ∫ ρ ( y ' )dy ' τ ( y) = = tan θ µ * sin θ τ ( y ) y '=0 P ( y ) Constitutive equation of frictional type µ * ( y ) = tan θ = f p (γ& p ) = f n (γ&n ) = campagne 0 36 θ 38 40 θ − θ c ∝ ( γ& n − γ& nc ) 4 6 8 10 H(cm) Effet du type de neige Questions: 1) Why the two layer behave differently ? 2) For the upper layer why the shear rate decreases when H increase ? (Answer from Numerical Simulation DM) 12 Snow rheology comparison with granular flow Grains Spheres No interstitial fluid No distant forces No cohesion Grains Solicitation Masse m Pressure P Diameter d Shear rate Shear strength: τ γ& Level of shear measured by the inertial number I I = γ& m P Inertia Imposed force 0.05 0.01 • I<10-3 quasi-static I 0.2 regime (solid) 0.1 0.025 • 10-3<I<0.3 dense regime (liquid) • I<10-3 inertial regime (gaz) Friction law in dense regime 0.5 µ* = P τ P N 0.4 µ* 0.3 0.2 0.0 * µ * ≈ µ min + bI 0.1 I 0.2 0.3 V 0.3 Snow rheology comparison with granular flow Incline : Permanent and uniform flows for θstop<θ θ<θ θacc Stress profile µ * = tan θ P = ρg ( H − y ) cos θ + Constitutive equation 18° 14° 16° θ 22° 20° * µ * ≈ µ min + bI I = γ& mP 24° Velocity is well described by Bagnold profile ( * V ( y ) − V0 ∝ (tan θ − µ min ) H 3 / 2 − ( H − y )3 / 2 1.0 Comparison with snow neige V0=0 • Permanent and uniform regime for large range of slopes (+) V0 libre • Stop at non null slope and velocity profile sheared all over the depth (+) y/H 0.5 • No difference between the bottom layer and the upper layer (-) Granular flow = interesting framework 0.0 0.0 0.5 1.0 V/VH ) Simulation method Molecular dynamics Material Spheres Non interstitial fluid Interaction by direct contact Flows : Boundary conditions (velocity, stress) For each grain et for each time step Forces at contact (+ gravity) New position, new velocity and new contacts Integration of Newton equations System size Choice 14 12 Small systems (~1000 grains) Two dimensions (25*40 grains) Jour 10 8 6 Limits 4 2 0 0 5000 10000 15000 20000 nombre de grains Typical calculation time for one flow Quantitative comparison with the experience is impossible Possibilities Control and variation of parameters Access to the microstructure Comprehension of mechanisms Effect of cohesion –Adhesion model Without cohesion h • Elastic repulsion (Hertz) • Energy dissipation • Friction Hertz N N N = k n h 3 / 2 + g n h& Origin of cohesive forces • van der Waals (powder, small grains) • Capillarity (humidity) • Sintering h Adhesion between the ice grains non well known Adhesion normal force Used model : generic form • Interaction at contact • Possibility to break the contact during the flow • The contact resist to traction force Nc 2 1 N/Nc 0 0.5 N = k n h + g n h& − 4k n N c h -1 1.0 h/hc Cohesion effect : simple shearing With boundary One imposes : P V H(t) L Pressure P (Height is a result) . Shear rate γ (Velocity is a result) The system adapts itself: V(t) Without boundary Resistance T (or µ*=T/P) Solid fraction, microstructure … H(t) y Objective x -V(t) Without gravity Homogeneous shearing With boundary Homogeneous Shearing state ( stress and shear rate) Without boundary 1.0 0.8 y/H 0.6 0.4 0.2 0.0 0.0 0.2 . γ 0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 P=σxx=σyy 0.2 0.4 τ 0.6 0.8 0.0 0.2 0.4 0.6 ν 0.8 Cohesion effect : dimensional analysis Objective : determination of the constitutive equation T P One imposes I V(t) The system resists : µ* H(t) -V(t) y x By applying different I Behaviour without cohesion * µ * = µ min + bI L Homogeneous shearing I ( x, y, t ) = cte µ * ( x, y, t ) = cte What is the effect of cohesion Nc ? Two dimensional numbers can be obtained - Shearing state I = γ& - Cohesion intensity Nc η= Pd m P Inertia imposed force Resistance to traction Imposed force 10-3<I<0.1 : dense regime (liquid) 0<η<100 (η =0 no cohesion) Non cohesive I = 0.1, η = 0 Cohesive I = 0.1, η = 40 Effect of cohesion Cohésion 0,80 1 0,70 0,65 µ* Compacité 0,75 0,60 0,55 0,50 Cohésion 0,45 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,0 0,45 0,1 I Compacity : φ(I, Cohesion) 0,3 0,4 I Friction : µ*(I, Cohesion) = τ/P P (γ , Φ , C ), τ (γ , Φ , C ) . Constitutive equation 0,2 . I = γ& Remarque : Formation of aggregates, how it influence the flow ? m P EFFECT OF COHESION ON THE CONSTITUTIVE EQUATION µ =µ * * min (η ) + b(η ) I Origin of friction P V P With cohesion V 2.5 Répulsion 2.0 Without cohesion 1.5 µ* 1.0 * µ * = µ min + bI 0.5 0.0 0.1 I = γ&I 40 0.2 m P 20 0.3 0 100 80 60 N η = ηc Pd µ* << η µ* ∝ η Without aggregates With aggregate 5 2.0 3.0 4 1.5 2.5 3 ∗ µmin1.0 2 0.5 0.0 0.1 Attraction 1 1 η 10 η>10 : friction increase 0 100 b 2.0 l/d 1.5 1.0 0.5 0.0 0.1 1 η 10 Typical length of correlation of the compacity field : I=0.1 100 Effect of cohesion : incline Permanent and uniform flows Number of granular bonds Bog = Nc = mg Resistance to traction weight of grains One imposes Bog et µ*(y)=tanθ L And I(y) is a result 40 10 30 y P( y) ∝ ( H − y) x Bog Nc ∝ η= P( y) ( H − y) H g θ y/d 60 100 20 Bog 10 0 0 10 20 The cohesion is higher close to the free surface Velocity profile : plug 40 Increase of the stopping slope 10 Bog 100 60 35 30 1.0 y/H η 30 θstop 25 20 0.5 15 0.0 0.0 10 0.5 1.0 0.5 1.0 0.5 1.0 Without cohesion V/VH With cohesion 0 50 100 Bog 150 200 50 Effect of poly-dispersity : studied system • Nd small grains : d • ND big grains : D • Roughness : small grains N T h θ Normal force : visco-elastic Coulonb Friction H Two parameters: Two mixture parameters : - Slope θ ∈[12°;30°] - Size ratio : Dr = D/d ∈[1;8] - Flow height H ∈[10d;50d] - Proportion of big grains Sr = NDD²/(NDD²+Ndd²) ∈[0;1] Bi-disperse flow Sr = ¾ Dr = 4 Upper layer big grains Intermediate layer: mixture Bottom layer small grains Comparison of velocity profiles Cohesive Grains Resistance to traction 1.0 y/H 0.5 Monodisperse and non cohesive grains 0.0 0.0 1.0 0.5 1.0 0.0 y/H 0.5 1.0 0.0 0.5 1.0 V/VH 0.5 bi-disperse grains Proportion of big grains 0.0 0.0 0.5 1.0 1.0 V/VH y/H 0.5 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 V/VH In the snow flow, the cohesion intervenes through the formation of ‘breakable’ aggregates 0.5 1.0 Two layer vision of dense snow flow Two-layer Vision : • bottom layer made of isolated small grains • upper layer made of mixture of aggregates Confirmed by observation : • aggregates at the free surface • a thin layer of small grains remains after the end of the flow Couche inférieure -1 γp (s ) 800 . Isolated grains : d ~ 0.1mm θ = 37° H =10cm 600 Shear rate linked to gravity : γ&g = 400 g ≈ 300s −1 d 200 34 36 38 θ(°) 40 4 6 8 10 12 H(cm) Explain why no effect of snow type is observed Aggregates of : H~10cm Upper layer -1 γn (s ) 20 15 . 10 θ = 37° H =10cm Shear rate linked to gravity: γ&g = 5 0 34 36 θ 38 40 4 6 8 10 H(cm) g ≈ 15s −1 H 12 Explains why the dimension of shear rate when H increase Explains the difference of behaviour between the two layers Comparaison to Voellmy model Steady flows impossible FR = V µ = tan θ = tan θ0 + 0.0067 gh 2 How the dense snow behave when F<2 ? g ξ = 0.0067 ⇒ ξ = 1490 ( Shearer / Roger Pass ) V gh Predetermination: assembling knowledge Distribution de l’avalanche centennale Montroc, 9 Feb 1999 Reference hazard ! Topography of the site Historical data Expert knowledge Distribution Of input variables Operator: physics of propagation Distribution Of output variables