link between moisture dynamics and microbial activity...

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.


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.


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).


Two examples



Powder snow avalanches

Sovilla, Burlando and Bartelt, JGR (2010)

Release

WSL Institute for Snow and Avalanche Research SLF


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


ErupDon current

rapid erup[on

Issler (2002) Sovilla et al (2006)

[me (s)

height (m

)


ErupDon current

rapid erup[on


[me (s)

height (m

)

Erup[on current = a gravity current driven by massive frontal erup[on


ErupDon current

rapid erup[on


[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


Erup[on current frontal region

rapid erup[on

Issler (2002)Sovilla et al (2006)



rapid erup[on


source

avalanche rest frame



rapid erup[on


source

avalanche rest frame

frontal region


Principal assump[ons

source

frontal region• Negligible basal shear stress• Negligible air entrainment• Inviscid• Uniform mixture density


Poten[al flowRankine, Proc. Roy. Soc. (1864)

€

p + ρu 2

2+ ρgz = # p + # ρ

# u 2

2+ # ρ gz

U’

H’


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’


€

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�"


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


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


Snowpack failure

€

tanα = µEinternal friction!

α!

€

ˆ n α!

€

ρcg


Snowpack failure

€


α!

€

ˆ n α!

€

ρcg

€

tanβ ≡ F ⋅ ˆ y + ρcgF ⋅ ˆ x

α

€

ˆ n

β!

€

F ⋅ ˆ x

€

F ⋅ ˆ y + ρcg


Snowpack failure

€


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


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!


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!




€

∂σ i

∂x= −F ⋅ ˆ x


'

( ) )

*

+ , ,

f±

σ!

τ!

α!

2α+2β�π!

σy!

σx!τ!

τs < 0!

σ1!σ2!

failure!


€

f± ≡cos(α β)cosβ

cosα + 2sinα sin 2 α β( )[ ]{ }



€

∂σ i

∂x= −F ⋅ ˆ x


'

( ) )

*

+ , ,

f±

σ!

τ!

α!

2α+2β�π!

σy!

σx!τ!

τs < 0!

σ1!σ2!

failure!


€



sustainable failure:!

€

∂σ i

∂x> 0



€

∂σ i

∂x= −F ⋅ ˆ x


'

( ) )

*

+ , ,

f±

σ!

τ!

α!

2α+2β�π!

σy!

σx!τ!

τs < 0!

σ1!σ2!

failure!

air-‐cloud interface> 0!

> 0! sign of f±?!

€



sustainable failure:!

€

∂σ i

∂x> 0


Sign of f± sets avalanche sustainability

€


€

tanβ =F ⋅ ˆ y + ρcg

F ⋅ ˆ x ≥ −

1µe

Condition for sustainable failure,"whence fluidized snow pack does "

not de-fluidize.!

f± > 0


Fluidiza[on interface!

€

∇2p = 0

pore pressure p!

€

" h

€

" b


σ!

τ!

α!

2α+2β�π!

σy!

σx!τ!

τs < 0!

σ1!σ2!

Mohr-Coulomb failure!



€

∇2p = 0

pore pressure p!

€

" h

€

" b


σ!

τ!

α!

2α+2β�π!

σy!

σx!τ!

τs < 0!

σ1!σ2!


€

R ≡2ρcg $ b µe

$ ρ $ U 2

snowpack density ρc, friction µe!



€

∇2p = 0

pore pressure p!

€

" h

€

" b


σ!

τ!

α!

2α+2β�π!

σy!

σx!τ!

τs < 0!

σ1!σ2!


€

R ≡2ρcg $ b µe

$ ρ $ U 2

snowpack density ρc, friction µe!€

" h " b

€

R€

h'b'≈1Ra1

€

a1 ≈ 0.42


References


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).


Sand ripples

wind

crest

trough


Measurements


Surface density probe


Ripple elevaDon profile

Dah Ould AhmedouAhmed Ould el-‐Moctar

Ahmedou Ould Mahfoud

Alexandre Valance


ElevaDon and volume fracDon

h0 / λ ≈ 0.02

h / λ

x / λ


Seepage, moisture and dust

∇p = 0

trough crest trough

low pfast uhigh p

slow uhigh pslow u


Wind tunnel experiments

Amin Younes

Brian MiEereder

Smahane Takarrouht

Scanning valvePressure gauge


Pressure field

p = pa + p0 cos 2πxλ+ϕ

!

"#

$

%&

p = pa

Hy

wind

∇2p = 0


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)%


Pressure field


Pressure field

h0 / λ = 3%


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


Gong et al, JFM (1996)

p− pa( )ρu*2

λh 0

"

#$

%

&' x / λ

troughcrest

troughu* = 0.41 m/sh0 / λ ≈ 7.9%

roughsurface

smoothsurface


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%


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


References


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.


Jin Xu

hEp://grainflowresearch.mae.cornell.edu/index.html


http://grainflowresearch.mae.cornell.edu/index.html

http://grainflowresearch.mae.cornell.edu/index.html

link between moisture dynamics and microbial activity...

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