Moving boundary problems in earth-surface dynamics Damien Kawakami, Vaughan R. Voller, Chris Paola

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.

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

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

05101520

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,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

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 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

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

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