moving boundary problems in earth-surface dynamics , vaughan r. voller
DESCRIPTION
Moving boundary problems in earth-surface dynamics , Vaughan R. Voller NSF, National Center for Earth-surface Dynamics, University of Minnesota, USA. Input From Chris Paola, Gary Parker, John Swenson, Jeff Marr, Wonsuck Kim, Damien Kawakami. What is NCED?. A National Science Foundation - PowerPoint PPT PresentationTRANSCRIPT
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
, Vaughan R. VollerNSF, National Center for Earth-surface Dynamics,
University of Minnesota, USA.
Input From
Chris Paola, Gary Parker, John Swenson, Jeff Marr,Wonsuck Kim, Damien Kawakami
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
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
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Fans Toes Shoreline
Two Problems of Interest
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
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
An Ocean Basin
Swenson-Stefan
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
05
101520
25303540
0 100 200
time
sh
ore
line
NOT
t~s
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),x(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
sh
ore
line
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Limit Conditions: Sea-Level Change Very Steep Angle of Repose
q=1
s(t)
t
dt
dL)s(
,dt
dss
x)t(sx0,
xt s2
2
tdt
dLif,sH
2
2
xt
H
Enthalpy Sol.
Reaches Steady State Position s = 1/(dL/dt)
0
2
4
6
8
10
12
0 100 200 300 400 500
time
sh
ore
line
dL/dt = 0.1
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
nvq
2
sx
1
zzwith,
dt
dZ
dt
ds
xq
Limit Conditions: Sea-Level Change Finite Angle of Repose
v
n
2
2
xt
An enthalpy like fixed gridSolution can be constructed
s(t)
L(t)
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
The concept of an “Auto-Retreat”
To stay in one place the flux to the shore frontNeeds to increase to account for the increase in the accommodation increment with each time step
NOT possibleFor flux to increaseSo shoreline moves landwardAuto-retreat
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Shoreline Projection
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10
Horizonal Position
Ver
tica
l P
osi
tio
n
5.0,2,05.0dt
dL
s(t)
L(t)
Stratigraphy and Shoreline
-30-20-10
0102030405060
0 10 20 30
0
2
4
6
8
10
0 200 400 600 800 1000
time
sh
ore
line
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
“Jurassic Tank” A Large Scale Exp.
~1m
Computer controlled subsidence
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
XES basin (“Jurassic Tank”) Subsidence Mechanism
pressurizedwater reservoir
to water supply
solenoidvalve
stainless steelcone
to gravel recycling
transport surface
gravel basement
rubber membrane
experimental deposit
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
xxt
How does shore line move in response to sea-level changes
Swenson et al can be posed as a generalized Stefan Problem
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Base level
Measured and Numerical results ( calculated from 1st principles)
Numerical Solution1-D finite difference deforming grid
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
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
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
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
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
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
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 toe00.10.20.30.40.50.60.7
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
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Example shows a “numerical experiment”
of sediment filling of a deep constant depthocean with persistent (preferred) channelization
Solution of Exner with
Simplified Swenson-Stefan condition and
Spatially changing diffusion coefficient
Front Perturbations: An Initial Model
Next change Diffusion field with time
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
National Center for Earth-surface DynamicsModeling physical and ecological dynamics of channel systems that shape Earth’s
surface
Full sim sol
21
2
2
2
2
t2s
et
Cx
et
Ct
t2
xerfC1
x
)(erf2e2
2C
))t2
x(erf
t
xe2(Ctx)t,x(
t4
x
22
2
t4
x
2
2
2
t4
x
22/1