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
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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

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