Equations of Motion of Hydronamically Driven Granular Surfaces

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Equations of Motion of Hydronamically Driven
Granular Surfaces
Hans Herrmann
ETH Zürich
Collaborators

• Eric Parteli
• Orencio Duran
• Gerd Sauermann
• Veit Schwämmle
• Klaus Kroy
• Haim Tsoar

IPAM Workshop II Random Curves,
Surfaces and Transport Los Angeles April
16, 2007
Departamento de Física, Univ. Federal do Ceará
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Temet, Sahara
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Merzouga, Marocco
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Um el-Maa, Libya
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Road blocked
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Nouakchott, Mauritania
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Algeria
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Shape of a barchan dune
Crest and brink might coincide or not
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Barchan field in Peru(Pampa de La Joya)
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Martian dunesMars Global Surveyor July 26, 1998
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Aim of this work
The height of the free granular surface be
described by h(x,y) We need an equation of motion

where u is the wind strength, d is the grain
diameter, g is gravity, etc..
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Wind is a fluid motion
and are velocity and pressure field
of the fluid, ? and ? its density and
dynamic viscosity.
Incompressible Navier-Stokes equation
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Wind field over a dune
schematic diagram
release of smoke
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Velocity field over Barchan
Using FLUENT
Longitudinal cut in symmetry plane (in wind
direction)
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Trajectories of test particles
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Turbulent velocity field
Atmospheric boundary layer profile is logarithmic
with
z0 roughness length t shear stress u
shear velocity ? von Kármán constant
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Shear stress at the ground

• H and L, characteristic height and length of the
  hill
• t0, shear velocity far upwind of the hill
• A(L/z0) 3.2 and B(L/z0) 0.3
• Depression before and after the hill
• Asymmetry, upwind shift of the maximum

P. S. JACKSON and J. C. R. HUNT, Q. J. R.
Meteorol. Soc. 101, 929 (1975)
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The full equations in 2d
K0 and K1 are modified Bessel functions, u(l) is
the normalized velocity profile, kx and ky are
the components of the wave vector, means
Fouriertransformation and z0 1mm.
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Ralph A. Bagnold
Aeolian transport of sand
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Types of aeolian sand transport

• Creep
• Large grains (d 0.4 mm)
• Rolling over the surface
• Saltation
• Typical dune grains (0.1 mm
• Surface load
• Suspension
• Small grains (d
• Over long distances in the air

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Aeolian transport
wind tunnel measurements
(K. Pye 1990 using data from Bagnold and Chepil)
either erosion or deposition depending on the
grain diameter and the wind velocity
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Saltation
Grains are pulled out of the granular bed and
accelerated by the wind.
M.P. Almeida, J.. Soares Andrade Jr, H.J.H.,
Phys. Rev. Lett. 96, 018001 (2006)
They impact again on the ground with higher
energy ejecting in this way several new grains.
Through this cascade of splashes more and more
grains are added to the saltation process.
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Saltation
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Saltation
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Saturated sand flux qs
Bagnold (1941)
Lettau Lettau (1978)
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One particle in fluid
e.g. pull sphere through fluid
no-slip condition
moving boundary condition
fluid
G
create shear in fluid exchange momentum
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Drag force
drag force
(Bernoullis principle)
stress tensor
? ?? is static viscosity
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Homogeneous flow
Re for Re 0)
R
v
Re 1 Newtons law FD 0.22p ? R2 v2
general drag law
CD is the drag coefficient
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Drag coefficient CD
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Schematic saltating trajectory
mobile wall at top
? ejection angle up particle
velocity ux(y) wind velocity
profile
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Two types of transient behaviour
force on particle
threshold velocity ut ? 0.35
Solve fluid with k-? model using FLUENT.
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Saturated flux
Bagnold (1941)
q
Lettau and Lettau (1978)
fit of solid line
Physical Review Letters, 96, 018001 (2006)
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Wind velocity profile
difference between disturbed and
undisturbed velocity profile
u 0.51
collapse when normalizing with flux q
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Height of saltation layer
ymax height of maximum loss of velocity
linear increase with u
becomes zero at
ut 0.33?0.01
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Continuous saltation model
F be the vertical flux and n the number of
ejected grains
is the saltation length
Saturation means n 1, make Taylor expansion
around ta / tt
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Sand flux q
K. KROY, G. SAUERMANN, H.J. HERRMANN, Phys. Rev.
Lett. 88, 054301 (2002)
Sand flux evolution
Logistic equation
Saturation length
Saturated flux qs
Lettau Lettau (1978)
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Evolution of flux
Saltation model including saturation transients
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Full set of equations
Variable fields
Jackson Hunt
flux
with
mass conservation
extra condition
T is angle of repose
K. KROY, G. SAUERMANN, H.J. HERRMANN, Phys.
Rev.E, 66 031302 (2002)
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Slip face
The surface is relaxed until the angle of repose
is obtained everywhere
? angle of repose
Here we use ua0 0.25 m/s and C4 10.
Inspired from BCRE Model (J. P. Bouchaud, M. E.
Cates, J. R. Prakash and S. F. Edwards, Phys.
Rev. Lett. 74, 1982 (1995))
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Separation bubble

• Phenomenological model separation bubble
• 3rd order polynomial function
• s(0)h(x0),s(0)h(x0),s(L)0,s(L)0

The envelope of h and s is used to calculate the
shear stress t.
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Flow diagram
h(x,y)
initial surface
v(x,y)
calculation of wind shear
q(x,y)
calculation of sand flux
calculation of the change due to sand flux
h(x,y)
avalanches
final surface
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Different initial conditions
Time evolution of the maximum height
Initial shape
Final shape
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Different initial conditions
small initial hill
big initial hill
final states
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Jericoacoara
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Jericoacoara
Studied barchan

• height 34 m
• width 600 m
• stoss side 200 m

• correlated measurements of wind, sand flux and
  profile

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Jericoacoara

• Wind velocity

• max. speed-up of 1.4
• depression of 0.8
• long range interaction
• two anemometers to normalize wind speed

G. SAUERMANN et al, Geomorphology 1325, 1-11
(2003)
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Jericoacoara

• Sand flux

• non-equilibrium flux model
• equilibrium flux relation by Lettau and Lettau
• problems at the dunes foot, boundary
• measurement of the sand flux during the same
  time period at different positions of the profile.

G. SAUERMANN et al, Geomorphology 1325, 1-11
(2003)
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Dunes in Morocco
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Dunes at Laayoune
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Scaling of profiles
Profile along the symmetry plane
Profile perpendicular to the wind
G. SAUERMANN et al, Geomorphology 36, 47-62
(2000)
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Small and large dunes
large
small
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longitudinal profile
height h in normalized units
length l in normalized units
field measurement
calculation
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transverse profile
hight h in normalized units
width w in normalized units
calculation
field measurement
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Evolution of a field
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Formation of a dune field from a beach
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Solitary behavior
V. SCHWÄMMLE, H.J. HERRMANN, Nature 426, 619-620
(2003)
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Snapshots of solitary behavior
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Breeding
V. SCHWÄMMLE, H.J. HERRMANN, Nature 426, 619-620
(2003)
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Breeding in nature
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Lençóis Maranhenses, Brazil
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Lençóis Maranhenses, Brazil
Profiles of transversal dunes
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Lençóis Maranhenses, Brazil
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Lençóis Maranhenses, Brazil
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Barchanoids
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Hummock dune (Brazil)
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Dying Vegetation
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Vegetation Growth Model
Vegetation growth model
Shear stress partitioning model
Parameters
maximum vegetation height, vegetation
characteristic growth time, vegetation
parameters.jjj
M. R. RAUPACH et al. J. Geo. Res.96 (1993)
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Different stages of evolution
Numerical simulation
Real images
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Simulation vs observation
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Planet Mars
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Barchan dune field (South Pole)
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Parameters of the model
g gravity
rair air density
rgrain density of the grains
d grain diameter
u shear velocity of the wind ? shear stress t
rair (u)2
ut threshold shear velocity of the wind
h viscosity of the air (fluid that transport
the grains)
qr angle of repose
g parameter associated with the splash gives
the efficiency of the wind in carrying grains
into saltation.
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The unknown parameter on Mars
How many grains n enter saltation after splash?
The efficiency of the wind in carrying grains
into saltation is given by the model parameter g
.
n
dn
g
g
d(t/tt)
1.0
t/tt
1.0
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known parameters
Earth
Mars
ut ? 2.0 m/s
ut ? 0.2 m/s
Greeley and Iversen (1985)
h ? 1.1 ? 10-6 kg/s?m
h ? 1.8 ? 10-6 kg/s?m
(Viscosity of CO2 at -100oC)
- u on Earth is 0.4 m/s and on Mars, Pathfinder
Mission 1997 found u close to threshold.
Further, it has been found that the angle of the
slip face of martian dunes is the same as of
terrestrial dunes.
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g must be much larger!
If we use for Mars g gEarth in simulations...
But if we use for Mars g 40 gEarth in
simulations...
... no martian dune appears.
... then we find martian dunes!
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Dunes on Mars
Simulation
Real image
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Dunes on Earth
Real image
Simulation
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Transverse dunes on Titan
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Outlook

• Study other types of dunes
• Change direction of wind coupled to meteorology
• Simulate techniques of dune destruction (Meunier)
• Dunes under water
• Interaction with vegetation

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Mathematical outlook

• Stability analysis for transverse dunes
• Analog to KP equation ? solitons?
• How generic is Barchan shape?
• Singular behaviour at separation point