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TRANSCRIPT
Role of pore pressure gradients in geophysical flows over permeable substrates
Michel Louge, Barbara Turnbull, Cian Carroll, Alexandre Valance, Ahmed Ould el-‐Moctar, Jin Xu
Aussois, May 24, 2013
thanks to Patrick Chasle, Smahane Takarrouht, Ryan Musa, Michael Berberich,Amin Younes, Daniel BalenDne, MaEhew Pizzonia, Olivier Roche, Robert Foster
hEp://grainflowresearch.mae.cornell.edu
Thursday, May 30, 13
Shields erosionShields, A. (1936). Anwendung der
Aehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung, MiEeilungen der Preußischen Versuchsanstalt für Wasserbau 26. Berlin: Preußische
Versuchsanstalt für Wasserbau.
Thursday, May 30, 13
Shields erosionShields, A. (1936). Anwendung der
Aehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung, MiEeilungen der Preußischen Versuchsanstalt für Wasserbau 26. Berlin: Preußische
Versuchsanstalt für Wasserbau.
Shao, Y., and H. Lu (2000), A simple expression for wind erosion threshold fric[on velocity, J. Geophys. Res., 105, 22437-‐22443.
Thursday, May 30, 13
Flow over porous media
R. M. Iverson, Regula[on of landslide mo[on by dilatancy and pore pressure feedback, J. Geophys. Res. 110, F02015 (2005).
Roche, O., S. Montserrat, Y. Niño, and A. Tamburrino (2010), Pore fluid pressure and internal kinema[cs of gravita[onal laboratory air-‐par[cle flows: Insights into the emplacement dynamics of pyroclas[c flows, J.
Geophys. Res., 115, B09206.
T. Yamamoto, H. L. Koning, H. Sellmeijer, and EP Van Hijum, On the response of a poro-‐eleas[c bed to water waves, J. Fluid Mech. 87, 193-‐206 (1978).
R. M. Iverson, et al, Posi[ve feedback and momentum growth during debris-‐flow entrainment of wet bed sediment, Nature Geoscience 4, 116-‐121 (2010).
Thursday, May 30, 13
Powder snow avalanches
Sovilla, Burlando and Bartelt, JGR (2010)
Release
WSL Institute for Snow and Avalanche Research SLF
Thursday, May 30, 13
Growth rate
ddt
!ρ !H 2W( )g !H"#
$% ~ !ρ u*2 !HW( )u*
rate of poten[al energy working rate of basal shear stress
Growth rate of a cloud of height and density
due to basal shearing at a shear velocity
!H !ρ
u*
d !Hdt
~ u*3
3g !H~
0.05×50( )3
3×9.81×20~ 0.03 m/s
Thursday, May 30, 13
ErupDon current
rapid erup[on
Issler (2002) Sovilla et al (2006)
[me (s)
height (m
)
Thursday, May 30, 13
ErupDon current
rapid erup[on
Issler (2002) Sovilla et al (2006)
[me (s)
height (m
)
Erup[on current = a gravity current driven by massive frontal erup[on
Thursday, May 30, 13
ErupDon current
rapid erup[on
Issler (2002) Sovilla et al (2006)
[me (s)
height (m
)
[me (s)
sta[
c pressure (P
a)
McElwaine & Turnbull JGR (2005)
depression
Erup[on current = a gravity current driven by massive frontal erup[on
Thursday, May 30, 13
Erup[on current frontal region
rapid erup[on
Issler (2002)Sovilla et al (2006)
source
avalanche rest frame
Thursday, May 30, 13
Erup[on current frontal region
rapid erup[on
Issler (2002)Sovilla et al (2006)
source
avalanche rest frame
frontal region
Thursday, May 30, 13
Principal assump[ons
source
frontal region• Negligible basal shear stress• Negligible air entrainment• Inviscid• Uniform mixture density
Thursday, May 30, 13
Poten[al flowRankine, Proc. Roy. Soc. (1864)
€
p + ρu 2
2+ ρgz = # p + # ρ
# u 2
2+ # ρ gz
U’
H’
Thursday, May 30, 13
Poten[al flowRankine, Proc. Roy. Soc. (1864)
€
p + ρu 2
2+ ρgz = # p + # ρ
# u 2
2+ # ρ gz
€
Ri = 2" ρ − ρ( )" ρ
g " H " U 2
€
ζ ≡1− ρ / & ρ
U’
H’
Thursday, May 30, 13
€
x / " b €
p − pa
(1/2) # ρ # U 2
Sta[c pressure in the cloudpredic[on
data: McElwaine and Turnbull
JGR (2005)
€
p − pa
(1/2) # ρ # U 2=
2(x / # b ) −1(x / # b )2 + ( # h / # b )2
pressure p, air density ρ, cloud density ρ� !stagnation-source distance b� !
fluidized depth h�"
Thursday, May 30, 13
€
x / " b €
p − pa
(1/2) # ρ # U 2
Sta[c pressure in the cloudpredic[on
data: McElwaine and Turnbull
JGR (2005)
€
p − pa
(1/2) # ρ # U 2=
2(x / # b ) −1(x / # b )2 + ( # h / # b )2
pressure p, air density ρ, cloud density ρ� !stagnation-source distance b� !
fluidized depth h�"
Thursday, May 30, 13
Porous snow pack
rapid erup[onIssler (2002)
[me (s)
height (m
)
interface!
€
∇2p = 0
pore pressure p!
€
" h
€
" b
Pore pressure gradients "defeat cohesion!
u = −Kµ∇p
∇⋅u = 0
∇2p = 0
Thursday, May 30, 13
Force balance in the snowpack
€
∂σ x
∂x+∂τ∂y
= −F ⋅ ˆ x = ˆ x ⋅ ∇p
€
∂τ∂x
+∂σ y
∂y= −F ⋅ ˆ y − ρcg
= ˆ y ⋅ ∇p − ρcg
shear stress τ, normal stress σ, gravity g, snowpack density ρc!
air-‐cloud interface
Thursday, May 30, 13
Snowpack failure
€
tanα = µEinternal friction!
α!
€
ˆ n α!
€
ρcg
€
tanβ ≡ F ⋅ ˆ y + ρcgF ⋅ ˆ x
α
€
ˆ n
β!
€
F ⋅ ˆ x
€
F ⋅ ˆ y + ρcg
Thursday, May 30, 13
Snowpack failure
€
tanα = µEinternal friction!
α!
€
ˆ n α!
€
ρcg
€
tanβ ≡ F ⋅ ˆ y + ρcgF ⋅ ˆ x
α
€
ˆ n
β!
€
F ⋅ ˆ x
€
F ⋅ ˆ y + ρcg
€
π2−α −β
€
π − 2α − 2β
Thursday, May 30, 13
Rela[on amongst stresses at fluidiza[on
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ!
τ!
α!
2α�2β+π!
σy!σx!
τ!
τs > 0!
σ1!σ2! σ1!
σ2!
€
σ x =σ y1− sinα sin 3α 2β( )1+ sinα sin 3α 2β( )
&
' (
)
* +
€
σ i =σ y1− −1( )i sinα
1+ sinα sin 3α 2β( )
&
' ( (
)
* + +
€
τ = ±σ ysinα cos 3α 2β( )1+ sinα sin 3α 2β( )
&
' (
)
* +
failure!
failure!
Thursday, May 30, 13
Principal stress variations ⊥ failure surface
€
∂σ i
∂x= −F ⋅ ˆ x
1− −1( )i sinα1− sin2α
'
( ) )
*
+ , ,
f±
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
failure!
air-‐cloud interface
Thursday, May 30, 13
Principal stress variations ⊥ failure surface
€
∂σ i
∂x= −F ⋅ ˆ x
1− −1( )i sinα1− sin2α
'
( ) )
*
+ , ,
f±
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
failure!
air-‐cloud interface
€
f± ≡cos(α β)cosβ
cosα + 2sinα sin 2 α β( )[ ]{ }
Thursday, May 30, 13
Principal stress variations ⊥ failure surface
€
∂σ i
∂x= −F ⋅ ˆ x
1− −1( )i sinα1− sin2α
'
( ) )
*
+ , ,
f±
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
failure!
air-‐cloud interface
€
f± ≡cos(α β)cosβ
cosα + 2sinα sin 2 α β( )[ ]{ }
sustainable failure:!
€
∂σ i
∂x> 0
Thursday, May 30, 13
Principal stress variations ⊥ failure surface
€
∂σ i
∂x= −F ⋅ ˆ x
1− −1( )i sinα1− sin2α
'
( ) )
*
+ , ,
f±
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
failure!
air-‐cloud interface> 0!
> 0! sign of f±?!
€
f± ≡cos(α β)cosβ
cosα + 2sinα sin 2 α β( )[ ]{ }
sustainable failure:!
€
∂σ i
∂x> 0
Thursday, May 30, 13
Sign of f± sets avalanche sustainability
€
tanα = µEinternal friction!
€
tanβ =F ⋅ ˆ y + ρcg
F ⋅ ˆ x ≥ −
1µe
Condition for sustainable failure,"whence fluidized snow pack does "
not de-fluidize.!
f± > 0
Thursday, May 30, 13
Fluidiza[on interface!
€
∇2p = 0
pore pressure p!
€
" h
€
" b
Pore pressure gradients "defeat cohesion!
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
Mohr-Coulomb failure!
Thursday, May 30, 13
Fluidiza[on interface!
€
∇2p = 0
pore pressure p!
€
" h
€
" b
Pore pressure gradients "defeat cohesion!
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
Mohr-Coulomb failure!
€
R ≡2ρcg $ b µe
$ ρ $ U 2
snowpack density ρc, friction µe!
Thursday, May 30, 13
Fluidiza[on interface!
€
∇2p = 0
pore pressure p!
€
" h
€
" b
Pore pressure gradients "defeat cohesion!
σ!
τ!
α!
2α+2β�π!
σy!
σx!τ!
τs < 0!
σ1!σ2!
Mohr-Coulomb failure!
€
R ≡2ρcg $ b µe
$ ρ $ U 2
snowpack density ρc, friction µe!€
" h " b
€
R€
h'b'≈1Ra1
€
a1 ≈ 0.42
Thursday, May 30, 13
References
hEp://grainflowresearch.mae.cornell.edu
Carroll, C. S., M. Y. Louge, and B. Turnbull (2013), Frontal dynamics of powder snow avalanches,
J. Geophys. Res., in press.
Louge, M. Y., C. S. Carroll, and B. Turnbull (2011), Role of pore pressure gradients in sustaining frontal par[cle entrainment in erup[on currents: The case of powder snow avalanches, J. Geophys. Res., 116, F04030.
C. S. Carroll, B. Turnbull, and M. Y. Louge, Role of fluid density in shaping erup[on currents driven by frontal par[cle blow-‐out, Phys. Fluids 24, 066603
(2012).
Thursday, May 30, 13
Ripple elevaDon profile
Dah Ould AhmedouAhmed Ould el-‐Moctar
Ahmedou Ould Mahfoud
Alexandre Valance
Thursday, May 30, 13
Seepage, moisture and dust
∇p = 0
trough crest trough
low pfast uhigh p
slow uhigh pslow u
Thursday, May 30, 13
Wind tunnel experiments
Amin Younes
Brian MiEereder
Smahane Takarrouht
Scanning valvePressure gauge
Thursday, May 30, 13
Pressure field
Hy
wind
p = pa + p0 cos 2πxλ+ϕ
!
"#
$
%&exp −2π y / λ( )− exp −4πH / λ( )exp 2π y / λ( )
1− exp −4πH / λ( )
(
)**
+
,--
f(x)% g(y)%
Thursday, May 30, 13
High amplitude to wavelength raDo
h0 / λ = 6%! !,!
= !! + !! cos2!"! + !!
exp − !!"! − exp − !!"
! ∗ exp !!"!
1− exp − !!"!
+ !! cos4!"! + !!
exp − !!"! − exp − !!"
! ∗ exp !!"!
1− exp − !!"!
+ !! cos6!"! + !!
exp − !!"! − exp − !"!"
! ∗ exp !!"!
1− exp − !"!"!
!
p = pa
trough crest trough
∇2p = 0Pressure field
Thursday, May 30, 13
Gong et al, JFM (1996)
p− pa( )ρu*2
λh 0
"
#$
%
&' x / λ
troughcrest
troughu* = 0.41 m/sh0 / λ ≈ 7.9%
roughsurface
smoothsurface
Thursday, May 30, 13
Flow regimesKuzan, J. D., T. J. HanraEy, and R. J.
Adrian, Turbulent flows with incipient separa[on over solid waves,
Experiments in Fluids 7, 88-‐98 (1989).
Instantaneouslyreversed flows
Sinusoidal shear stress,non-‐separated
Time-‐averaged separated flows
2πµρu*λ
2h0λ
Gong, et al, 1996
Non-‐sinusoidal shear stress,non-‐separated
h0 / λ = 6%h0 / λ = 3%
Thursday, May 30, 13
Toward fluidizaDon
trough
crest
trough
∂p /∂yρsνg
against gravityh0 / λ = 6%
u* ≈1 m/sν = 0.67
ρs = 2630 kg/m3
h0 / λ = 3%
fluidized
Thursday, May 30, 13
References
hEp://grainflowresearch.mae.cornell.edu
Louge, M. Y., A. Valance, H. Mint Babah, J.-‐C. Moreau-‐Trouvé, A. Ould el-‐Moctar, P. Dupont, and D. Ould Ahmedou (2010),
Seepage-‐induced penetra[on of water vapor and dust beneath ripples and dunes, J. Geophys. Res., 115, F02002.
Louge, M. Y., A. Valance, A. Ould el-‐Moctar, and P. Dupont (2010), Packing varia[ons on a ripple of nearly monodisperse dry sand,
J. Geophys. Res., 115, F02001.
Thursday, May 30, 13
Jin Xu
hEp://grainflowresearch.mae.cornell.edu/index.html
Thursday, May 30, 13