Moving boundary problems in earth-surface dynamics Damien Kawakami, Vaughan R. Voller, Chris Paola
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Transcript of Moving boundary problems in earth-surface dynamics Damien Kawakami, Vaughan R. Voller, Chris Paola
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Moving boundary problems in earth-surface dynamics
Damien Kawakami, Vaughan R. Voller, Chris PaolaNSF, National Center for Earth-surface Dynamics,
University of Minnesota, USA.
![Page 2: Moving boundary problems in earth-surface dynamics Damien Kawakami, Vaughan R. Voller, Chris Paola](https://reader035.fdocuments.in/reader035/viewer/2022062516/56812bc6550346895d900e45/html5/thumbnails/2.jpg)
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
What is NCED?
NCED develops integrated models of the physical and ecological dynamics of the channel systems that shape Earth’s surface through time, in support of river management, environmental forecasting, and resource development
A National Science Foundation Science and Technology Center
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
1km
Examples of Sediment Fans
How does sediment-basement interfaceevolve
Badwater Deathvalley
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Sediment mass balance gives
Sediment transported and deposited over fan surface
xxt
From a momentum balance anddrag law it can be shown thatthe diffusion coefficient is a function of a drag coefficientand the bed shear stress
when flow is channelized = constant
when flow is “sheet flow”
A first order approx. analysis indicates 1/r
(r radial distance from source)
Sediment Transport on a Fluvial Fan
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
An Ocean Basin
Swenson-Stefan
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Limit Conditions: Constant Depth Ocean
q=1
L
A “Melting Problem” driven by a fixed flux with Latent Heat L
s(t)
angle of repose
Enthalpy solution
0if,LH
2
2
xt
H
Track of Shore Line
05101520
25303540
0 100 200
time
sh
ore
line
NOT
t~s
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Limit Conditions: A Fixed Slope Ocean
q=1
A Melting Problem driven by a fixed flux with SPACE DEPENDENT
Latent Heat L = s
s(t)
0if,LH
2
2
xt
H
Enthalpy Sol.
dt
dss
x)t(sx0,
xt s2
2
similarity solution
22/1 2
)(erf2e2
)(erf21,t2s 2
0
5
10
15
20
25
0 100 200 300
Time
shoreline
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
The Desert Fan Problem -- A 2D Problem
xxt )t,s(,0x s
A Stefan problem with zero Latent Heat
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
A two-dimensional version (experiment)
• Water tight basin -First layer: gravel to allow easy drainage-Second layer: F110 sand with a slope ~4º.
• Water and sand poured in corner plate
• Sand type: Sil-Co-Sil at ~45 mm• Water feed rate:
~460 cm3/min• Sediment feed rate: ~37cm3/min
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
The Numerical Method
-Explicit, Fixed Grid, Up wind Finite Difference VOF like scheme
Flux out of toe elements =0Until Sediment height >Downstream basement
fill point
P
)qq(t
out2PnewP in
E
The Toe Treatment
EPq
Square grid placed onbasement
At end of each time stepRedistribution scheme is requiredTo ensure that no “downstream” covered areas are higher
r
Determine height at fill : Position of toe
.05 grid size
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
• Pictures taken every half hour– Toe front recorded
• Peak height measure every half hour
• Grid of squares 10cm x 10cm
Experimental Measurements
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Observations (1)• Topography
– Conic rather than convex– Slope nearly linear across position and time – bell-curve shaped toe
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Observations (2) • Three regions of flow– Sheet flow– Large channel flow– Small channel flow
• Continual bifurcation governed by shear stress
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
y – (x,t) = 0
0y)t,x(,0xW
),y,x(Qxxxxt
),x(,0 s n
On toe0
0.10.20.30.40.50.60.7
00.511.5
x-location (m)
y-location (m
)
r
k
0
0.05
0.1
0.15
0 100 200 300
time (min)
feed
hig
ht
(m) height at input
fan with time
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Moving Boundaries on Earth’s surface
A number of moving boundary problems in sedimentary geology have beenidentified.
It has been shown that these problems can be posed as Generalized Stefan problems
Fixed grid and deforming grid schemes have been shown to produce results inReasonable agreement with experiments
Improvements in model are needed
Utilize full range of moving boundary numerical technologies to arrive at a suite of methods with geological application
Use large scale general purpose solution packages
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National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Full sim sol
t2
xerfC1
x
)(erf2e2
2C
))t2
x(erf
t
xe2(Ctx)t,x(
2
2
t4
x
22/1
2
2
Will give a q=-1 at x =0 a consrant q on s=2lam t to ½And eta = 0 at s