mathematical modelling of the mathematical modelling of the morphodynamic aspects of the morphodynamic aspects of the 1996 flood in the Ha! Ha! river1996 flood in the Ha! Ha! river

conceptual model and solution

Instituto Superior Técnico :: september 2005

Rui M. L. Rui M. L. FerreiraFerreira :: João G. B. Leal :: António H. Cardoso :: João G. B. Leal :: António H. Cardoso

severe rainstorms scourged the Saguenay region, south Québec, Canada, in July of 1996.

intr

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nin

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njustification of the work

sound yet simple simulation tools to predict and analyse the consequences of catastrophic events must be perfected. river Ha! Ha! disaster is well documented, thus suitable to be used in model validation.

the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley.

overtopping and sequent failure of an earth fill dyke in lake Ha! Ha! caused a significative increase in the peak flood discharge in River Ha! Ha!.

it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow.

severe rainstorms scourged the Saguenay region, south Québec, Canada, in July of 1996.

intr

odu

ctio

nin

trodu

ctio

njustification of the work

sound yet simple simulation tools to predict and analyse the consequences of catastrophic events must be perfected. river Ha! Ha! disaster is well documented, thus suitable to be used in model validation.

the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley.

overtopping and sequent failure of an earth fill dyke in lake Ha! Ha! caused a significative increase in the peak flood discharge in River Ha! Ha!.

it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow.

severe rainstorms scourged the Saguenay region, south Québec, Canada, in July of 1996.

intr

odu

ctio

nin

trodu

ctio

njustification of the work

sound yet simple simulation tools to predict and analyse the consequences of catastrophic events must be perfected. river Ha! Ha! disaster is well documented, thus suitable to be used in model validation.

the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley.

overtopping and sequent failure of an earth fill dyke in lake Ha! Ha! caused a significative increase in the peak flood discharge in River Ha! Ha!.

it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow.

severe rainstorms scourged the Saguenay region, south Québec, Canada, in July of 1996.

intr

odu

ctio

nin

trodu

ctio

njustification of the work

sound yet simple simulation tools to predict and analyse the consequences of catastrophic events must be perfected. river Ha! Ha! disaster is well documented, thus suitable to be used in model validation.

the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley.

overtopping and sequent failure of an earth fill dyke in lake Ha! Ha! caused a significative increase in the peak flood discharge in River Ha! Ha!.

it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow.

severe rainstorms scourged the Saguenay region, south Québec, Canada, in July of 1996.

intr

odu

ctio

nin

trodu

ctio

njustification of the work

sound yet simple simulation tools to predict and analyse the consequences of catastrophic events must be perfected. river Ha! Ha! disaster is well documented, thus suitable to be used in model validation.

the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley.

overtopping and sequent failure of an earth fill dyke in lake Ha! Ha! caused a significative increase in the peak flood discharge in River Ha! Ha!.

it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow.

to develop a conceptual model suitable to tackle the difficulties posed by the simulation of catastrophic floods, namely the important geomorphic changes and the existence of shocks and critical points.

objectives of the work

to develop a robust solution technique based on a finite difference discretization.

to validate the model with the data of the 1996 river Ha! Ha! Flood (EU funded IMPACT project benchmark data) in

trodu

ctio

nin

trodu

ctio

n

to develop a conceptual model suitable to tackle the difficulties posed by the simulation of catastrophic floods, namely the important geomorphic changes and the existence of shocks and critical points.

objectives of the work

to develop a robust solution technique based on a finite difference discretization.

to validate the model with the data of the 1996 river Ha! Ha! Flood (EU funded IMPACT project benchmark data) in

trodu

ctio

nin

trodu

ctio

n

to develop a conceptual model suitable to tackle the difficulties posed by the simulation of catastrophic floods, namely the important geomorphic changes and the existence of shocks and critical points.

objectives of the work

to develop a robust solution technique based on a finite difference discretization.

to validate the model with the data of the 1996 river Ha! Ha! Flood (EU funded IMPACT project benchmark data) in

trodu

ctio

nin

trodu

ctio

n

structure of the structure of the presentationpresentation

description of the conceptual model

presentation of the simulation results

description of the conceptual model

presentation of the simulation results

structure of the structure of the presentationpresentation

description of the conceptual model

presentation of the simulation solutions

structure of the structure of the presentationpresentation

conceptual modelconceptual model

physical systemphysical system

observations suggest stratification

fig 1. dam-break wave generated by instantaneous rupture; Shields parameter at dam location is é ≈ 2.5. observation window located downstream the reservoir, at about 10 times the water depth on the reservoir (10h0).

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0 1 2 3 cm

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physical systemphysical system

0 1 2 3 cm

upper plane bed

bed (immobile particles)

clear water/suspended sediment

contact load

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physical systemphysical system

0 1 2 3 cm

debris flow

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cep

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contact load

bed (immobile particles)

physical systemphysical system

bed (immobile grains)frictional region

collisional region

transition region

contact load layer

clear water/ suspended sediment

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idealised systemidealised system

fig 2. flow idealised as a multiple layer structure based on stress

predominance.

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conservation equationsconservation equations

two-dimensional conservation equations (profile)

granular phase

fluid phase

one-dimensional conservation equations

shallow water flow

cinematic non-material boundary conditions

incompressible fluid and granular phases

negligible segregation between phase –

continuum hypothesis

model model developmentdevelopment

one-dimensional conservation one-dimensional conservation equationsequations

mass and momentummass and momentum

0t b xh Y uh

( )

2 2 2 212

2

w

t m x c c c s s x s s c c c

c c s x b bc

s uh s u h u h g h h h s h

s h h Y

(1 )

0

t b

t c c x c c c

p Y

C h C u h

0t b xh Y uh total mass

total momentum

sediment massCc

capacity transport:capacity transport: 0netS bc

netS bc

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model model developmentdevelopment

0t b xh Y uh

( )

2 2 2 212

2

w

t m x c c c s s x s s c c c

c c s x b bc

s uh s u h u h g h h h s h

s h h Y

(1 )

0

t b

t c c x c c c

p Y

C h C u h

0t b xh Y uh

Cc

0netS bc

thickness of the contact load layer

velocity in the contact load layer

capacity (equilibrium) concentration

capacity transport:capacity transport:con

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one-dimensional conservation one-dimensional conservation equationsequations

mass and momentummass and momentum

model model developmentdevelopment

netS bc

closure equationsclosure equations

two-dimensional conservation equations (profile)

granular phase

fluid phase

stress tensor

flux of grain kinetic energycollisional dissipation

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model model developmentdevelopment

cu ch netS bc cC bc

constitutive equations

collisional region described by dense gas kinetic theory (Chapman-Enskog)

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closure equationsclosure equations

model model developmentdevelopment

two-dimensional conservation equations (profile)

granular phase

fluid phaseconstitutive equations

cu ch netS bc cC bc

negligible streaming component of the stress tensor (chaos molecular)

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cep

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collisional region described by dense gas kinetic theory (Chapman-Enskog)

two-dimensional conservation equations (profile)

granular phase

fluid phaseconstitutive equations

closure equationsclosure equations

model model developmentdevelopment

cu ch netS bc cC bc

closure equations

quasi-elastic approximation: e≈1

con

cep

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granular phase

fluid phase

negligible streaming component of the stress tensor (chaos molecular)

collisional region described by dense gas kinetic theory (Chapman-Enskog)

closure equationsclosure equations

model model developmentdevelopment

cu ch netS bc cC bc

constitutive equations

two-dimensional conservation equations (profile)

simulationsimulation resultsresults

initial value problems

Wtyeyy dd xcbdc c chdfxc,njlkjncflks

<sddsmnjnmvnmvb cfbnv dvb fgb

Riemann problem:

the dam break flood wave

YbL1

YL1

hR1

hL1

YL2

hR2 = hR1

hL2

’2 = ’1 a) b)

’ = (YbL YbR)/YL sim

ula

tion

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lts

Non-dimensional parameters: ’ = hR/YL

fig 3. idealised geometry for the dam-break problem understood as a Riemann problem.

the dam-break flood wavebed initially flat :: fixed banks :: prismatic channel

evolution of the longitudinal flow profile.si

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initial value problems

sim

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tion

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initial value problems the dam-break flood wave

bed initially flat :: fixed banks :: prismatic channel

evolution of the longitudinal flow profile. comparison

between observations and computed results.

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the dam-break flood wavebed initially flat :: erodible banks

evolution of the longitudinal flow profile.

evolution of the bed width at the level of the initial bed.

initial value problems

bank erosion model

m

1

m: inverse bank slope

sim

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tion

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lts

initial value problems the dam-break flood wave

bed initially flat :: erodible banks

sim

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tion

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initial value problems

evolution of the bed width at the level of

the initial bed.

evolution of the inverse bank slope.

the dam-break flood wavebed initially flat :: erodible banks

Lake outlet Dam

Chute á Perron

Chute á Baptiste

Eaux-mortes

Boilleau

Photo C

Photo B

Photo A

Lake Ha! Ha!

“Cut-away” dyke

Ha! Ha! Bay

“Rive-gauche” dyke

0.0

100.0

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19.00 19.50 20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00

Time [days]

Dis

char

ge [

cms]

fig 4. plan view of river and lake Ha! Ha!.

fig 5. flood hydrograph: superposition of the natural flood and the discharge released by the breached dyke.

a boundary/initial

value problem

sim

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simulation of the 1996 flood in the Ha! Ha! rivergeometry and flood hydrograph

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0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)

z b (m

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Boilleau

Eaux-mortes

Chute á Baptiste

Chute á Perron

Photo A

Photo C Photo B

fig 6. longitudinal profile of river Ha! Ha!.

fig 7. photo A: dyke location after the collapse. note the pronounced erosion (about 12 metres).

foto A

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simulation of the 1996 flood in the Ha! Ha! riverobserved geomorphic impacts

a boundary/initial

value problem

foto B

fig 8. photo B: generalized deposition at Eaux-mortes (about 2meters deposits).

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0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)

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Boilleau

Eaux-mortes

Chute á Baptiste

Chute á Perron

Photo A

Photo C Photo B

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fig 6. longitudinal profile of river Ha! Ha!.

simulation of the 1996 flood in the Ha! Ha! riverobserved geomorphic impacts

a boundary/initial

value problem

foto C

fig 9. photo C: bank erosion and channel widening at a convex reach.

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0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)

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Eaux-mortes

Chute á Baptiste

Chute á Perron

Photo A

Photo C Photo B

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fig 6. longitudinal profile of river Ha! Ha!.

simulation of the 1996 flood in the Ha! Ha! riverobserved geomorphic impacts

a boundary/initial

value problem

fig 10. chute á Perron: massive erosion as the flow evaded its normal fixed bed course (from Brooks & Lawrence 1999)

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0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)

z b (m

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Boilleau

Eaux-mortes

Chute á Baptiste

Chute á Perron

Photo A

Photo C Photo B

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tion

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fig 6. longitudinal profile of river Ha! Ha!.

chute á Perron

simulation of the 1996 flood in the Ha! Ha! riverobserved geomorphic impacts

a boundary/initial

value problem

5322000

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distance (m)

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ance

(m

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362361

...

“Cut-away” dyke

Chute á Perron

Chute á Baptiste

Eaux-mortes

Boilleau

Lake Ha! Ha!

Ha! Ha! Bay

cross-sections detailed in figure

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simulation of the 1996 flood in the Ha! Ha! rivercomputational domain

fig 11. computational domain: discretization of river Ha! Ha! between the lake and Ha! Ha! bay.original data converted to a DTM by Benoit Spinewine ( UCL) and Hervé Capart (Taiwan University).

a boundary/initial

value problem

section 128

290

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80.0 100.0 120.0 140.0 160.0

b (m)

z (m

)

section 131

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z (m

)

section 129

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80.0 100.0 120.0 140.0 160.0b (m)

z (m

)

Zb = 291.27 m m = 7.1 m Bf = 10.2 m

Zb = 291.93 m m = 7.0 m Bf = 10.3 m

Zb = 291.58 m m = 4.4 m Bf = 8.1 m

Zb = 287.60 m m = 2.9 m Bf = 6.0 m

section 130

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100.0 120.0 140.0 160.0 180.0b (m)

z (m

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simulation of the 1996 flood in the Ha! Ha! rivercomputational domain

fig 12. idealized trapezoidal sections used for computational purposes (computed from an algorithm operating over the DTM data).

a boundary/initial

value problem

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simulation of the 1996 flood in the Ha! Ha! rivercomputational domain

fig 13. bed width (top) and inverse bank slope (bottom) for computational purposes.

a boundary/initial

value problem

S = 0

S < Scrit

Lups Ldwn L

1 2 3 NF1 NF 1 2

... ... N1 NNS1 NS

sim

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tion

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simulation of the 1996 flood in the Ha! Ha! rivercomputational domain

fig 14. extended computational domain featuring two virtual reaches at the upstream and downstream ends for computational purposes.

a boundary/initial

value problem

t

x

(+)

xN-1xN

t0

t1

dx

(Q,A;S,…) = 0

x

t

t

x

()

0 x1

t0

t1

dx

(Q,t) = 0

x

t

a) b)

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tion

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fig 15. stencil of the characteristics at the upstream and downstream reaches. boundary conditions at the virtual reaches function in the subcritical regime. the actual dam location is a critical flow point.

simulation of the 1996 flood in the Ha! Ha! rivercomputational domain

a boundary/initial

value problem

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elev

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)

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disc

harg

e (m

3 /s)

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)

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3.0

dept

h (m

); F

roud

e (-

)

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fig 16. step discontinuity at critical flow points in steady flow. TVD algorithm is unable to fix the problem. artificial viscosity of the Von Neuman type is used to correct the problem.

simulation of the 1996 flood in the Ha! Ha! rivernumerical solution

a boundary/initial

value problem

sim

ula

tion

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lts evolution of the

bed elevation variation.

evolution of the Froude number.

simulation of the 1996 flood in the Ha! Ha! riverresults of the numerical simulation

a boundary/initial

value problem

sim

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tion

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simulation of the 1996 flood in the Ha! Ha! riverresults of the numerical simulation

a boundary/initial

value problemcritical flow

subcritical flow

subcritical flow

geomorphic hydraulic jump

t = t0

t = t2 > t1

geomorphic discontinuity

supercriticalflow

geomorphic hydraulic jump

t = t1 > t2 subcritical flow critical flow

supercritical flow

subcritical flow

fig 17. model for the evolution and disappearing of supercritical reaches, associated to pronounced convex bed profiles, as the bed morphology evolves.

sim

ula

tion

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lts evolution of the

water depth.

evolution of the bed width.

simulation of the 1996 flood in the Ha! Ha! riverresults of the numerical simulation

a boundary/initial

value problem

sim

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tion

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longitudinal profile: reaches downstream the eroded dyke.

longitudinal profile: “Chute á Baptiste” (fixed bed).

simulation of the 1996 flood in the Ha! Ha! riverresults of the numerical simulation

a boundary/initial

value problem

sim

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tion

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simulation of the 1996 flood in the Ha! Ha! riverresults of the numerical simulation

a boundary/initial

value problem

longitudinal profile: “Chute á Perron”.

fig 18. final bed profiles at “Chute á Perron”. initial bed; field data expressing the final bed profile. Results of scenario HaHaF03 (Ks = 24 m1/3s-1 and ac = 0.0019 s2m-1) are: t = 26 h ( ), t = 32 h ( ), t = 40 h ( ), t = 67.5 h ( ). stands for the results of NTU (Taiwan). stands for the results of the model of Cemagref. Results from Cemagref and NTU taken form Zech et al. (2004).

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18000 20000 22000 24000 26000distance (m)

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).

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simulation of the 1996 flood in the Ha! Ha! riverresults of the numerical simulation

a boundary/initial

value problem

con

clusi

on

s

a mathematical model for the simulation and analysis of floods featuring intense sediment transport and important morphologic impacts was developed;

the model was validated with the data of the 1996 flood in the river Ha! Ha!;

although the modelling exercise is of great difficulty, the scales of the phenomena were well reproduced. quantitatively, erosion was not as well reproduced as aggradation;

contributions of the present work:

numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction.

con

clusi

on

s

a mathematical model for the simulation and analysis of floods featuring intense sediment transport and important morphologic impacts was developed;

the model was validated with the data of the 1996 flood in the river Ha! Ha!;

although the modelling exercise is of great difficulty, the scales of the phenomena were well reproduced. quantitatively, erosion was not as well reproduced as aggradation;

contributions of the present work:

numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction.

con

clusi

on

s

a mathematical model for the simulation and analysis of floods featuring intense sediment transport and important morphologic impacts was developed;

the model was validated with the data of the 1996 flood in the river Ha! Ha!;

although the modelling exercise is of great difficulty, the scales of the phenomena were well reproduced. quantitatively, erosion was not as well reproduced as aggradation;

contributions of the present work:

numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction.

con

clusi

on

s

a mathematical model for the simulation and analysis of floods featuring intense sediment transport and important morphologic impacts was developed;

the model was validated with the data of the 1996 flood in the river Ha! Ha!;

although the modelling exercise is of great difficulty, the scales of the phenomena were well reproduced. quantitatively, erosion was not as well reproduced as aggradation;

contributions of the present work:

numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction.

con

clusi

on

s

acknowledgements:

the authors wish to acknowledge the financial support offered by the European Commission for the IMPACT project under the fifth framework programme (1998-2002), Environment and sustainable Development thematic programme, for which Karen Fabri was the EC Project Officer.

complementary slides

con

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leito plano leito plano superiorsuperior

equações de conservação equações de conservação unidimensionaisunidimensionais-massa e quantidade de movimento-massa e quantidade de movimento

0t b xh Y uh

( )

2 2 2 212

2

w

t m x c c c s s x s s c c c

c c s x b bc

s uh s u h u h g h h h s h

s h h Y

(1 ) 0nett b S bcp Y

nett c c x c c c S bcC h C u h

0t b xh Y uh profundidade do escoamento

velocidade média do escoamento

cota do fundo

massa totalmassa total

quantidade de movimento quantidade de movimento totaltotal

massa de sedimentos na camada de massa de sedimentos na camada de transportetransporte

massa de sedimentos no fundomassa de sedimentos no fundo

Cc

concentração na camada de transporte

transporte em desequilíbriotransporte em desequilíbrio

netS bc

con

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leito plano leito plano superiorsuperior

equações de conservação equações de conservação unidimensionaisunidimensionais-massa e quantidade de movimento-massa e quantidade de movimento

0t b xh Y uh

( )

2 2 2 212

2

w

t m x c c c s s x s s c c c

c c s x b bc

s uh s u h u h g h h h s h

s h h Y

(1 ) 0nett b S bcp Y

nett c c x c c c S bcC h C u h

0t b xh Y uh

Cc

transporte em desequilíbriotransporte em desequilíbrio

velocidade na camada de transporte por arrastamento

tensão de arrastamento

espessura da camada de transporte por arrastamento

netS bc

fluxo vertical de sedimentos

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closure equationsclosure equations

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0 5 10 15velocity

y/d

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0 0.2 0.4 0.6solid fraction

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0 0.5 1 1.5 2granular temperature

y/d

s

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fig 3. profiles of: a) velocity; b) solid fraction; c) granular temperature. results for = 1.74, = 2.49 and = 3.07. granular material with s = 1.5, ds = 0.003 m and e = 0.82.

a) b) c)

341

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( 1) cc s

s

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u d

depth integrationdepth integration

ch bccu netS bc cC

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model model developmentdevelopment

0t b xh Y uh

( ) ( ) 0g g

y y yx y xQ T u

dissipationdissipationdiffusiondiffusion productioproductionn

Integration of the equation of Integration of the equation of conservation of particle fluctuation conservation of particle fluctuation kinetic energykinetic energy

ch Q

productionproduction

, ; , ,c c sh h h d e

1.7 5.5c

s

h

d

bccu netS bc cCch

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closure equationsclosure equations

model model developmentdevelopment

integration of the vertical momentum integration of the vertical momentum equation in the frictional sub-layerequation in the frictional sub-layer

2( 1) tan( )

f

w c b c w ft b

c x y Y

g s C h C uY

u

2( 1) tan( )(1 )

f

w c b c w fnetS bc

c x y Y

g s C h C up

u

bccu cCch netS bc

friccional sub-layerfriccional sub-layer

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closure equationsclosure equations

model model developmentdevelopment

0t b xh Y uh

0t bY

2

tan( ) ( 1)f

cb c

C uC

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2( 1) tan( )(1 ) 0

f

w c b c w f

c x y Y

g s C h C up

u

bccu ch netS bc cC

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model model developmentdevelopment

closure equationsclosure equations

integration of the vertical momentum integration of the vertical momentum equation in the frictional sub-layerequation in the frictional sub-layer

friccional sub-layerfriccional sub-layer

0

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0.4

0.5

0 0.005 0.01 0.015 0.02 0.025

/ (-)

(ds

u) /

(h w

s) u

2 (-

)

predicted instability zone plastic 1 plastic 2 acrylic sand

w s 1/u * = 1.0w s 2/u * = 1.0

w s 3/u * = 1.0

w s 4/u * = 1.0

2cb w fC u

sf fa

s

duC C

h w

2 sf fb fa

s

duC C u C

h w

figfig 4. flow resistance. sheet 4. flow resistance. sheet flow data from Sumer flow data from Sumer etet alal. . (1996).(1996).

Shear stresses depend on the square of the shear rate; hence, bed shear stress is expected to depend on the square of the flow velocity.

cu ch netS bc cC bc

con

cep

tual m

od

el

model model developmentdevelopment

closure equationsclosure equations

•uniform flow of a mixture of water and a granular material

6363

numerical experimentsnumerical experiments

( ) ( )d 1g w

y yyT s g

( )

( ) ( ) ( )d sin( )g

g g gDy yx

s

CT U u g

d

( )

( ) ( ) ( )d 1 sin( )w

w g wDy yx

s

CT U u g

d

momentum equation, vertical directionmomentum equation, vertical direction

momentum equation, horizontal momentum equation, horizontal dir.dir.

conservation fluctuation energyconservation fluctuation energy

momentum equation, horizontal momentum equation, horizontal dir.dir.

conceptual modelconceptual model

conservation equations, granular conservation equations, granular phasephase

conservation equations, fluid phaseconservation equations, fluid phase

( ) ( )d d 0g g

y y xQ T u

6464

numerical experimentsnumerical experiments

flux of fluctuating energyflux of fluctuating energy

dissipation of fluctuation dissipation of fluctuation energy: collisional and energy: collisional and viscousviscous

conceptual modelconceptual model

particle stress particle stress tensortensor

constitutive equations, granular constitutive equations, granular phasephase

1 1( ) ( ) ( ) ( )2 2

1 12 2

8 161 2 1 , 1 3

3 54g g g g

ijij s i i ij sT d u d D

1( ) ( ) ( ) ( ) ( )2

12

81 312 21 1,2

5

g g g g g

sT T T d u

( ) ( ) ( ) ( )

1 211 22 4g g g gT T P

3

( ) ( ) ( ) ( ) 212

124 1

1g gw g gw

s

ed

1( ) ( ) 2

12

41 4

g g

sd

( )dg

yQ

shear stressesshear stresses

normal stresses (isotropic pressure)normal stresses (isotropic pressure)

collisional thermal collisional thermal difusivitydifusivity

( ) 22 2 1 f( )dw

yx yT y Ri U fluid shear stress; (mixing length)fluid shear stress; (mixing length)

constitutive equations, fluid phaseconstitutive equations, fluid phase

6565

numerical experimentsnumerical experiments

conceptual modelconceptual model

overall: 8 ordinary differential equations, 8 unknownsoverall: 8 ordinary differential equations, 8 unknowns

• the thickness of the contact load layer is also unknown; thus the the thickness of the contact load layer is also unknown; thus the system must be solved iteratively (shooting method);system must be solved iteratively (shooting method);

• an extra boundary condition is necessary (it is physically and an extra boundary condition is necessary (it is physically and mathematically well posed): mathematically well posed): Q Q ((yy==hhcc) = 0.0) = 0.0;;

• a value of a value of hhcc is proposed; the equations are solved and a new is proposed; the equations are solved and a new hhcc is computed from the above condition;is computed from the above condition;

other boundary conditions:other boundary conditions:( ) ( )(0) (0) 0g wu u ( ) ( )(0) tan( )g g

xy yy bT P

no slipno slip

frictional frictional stressesstresses

(0) 0.55 reciprocal of bed porosityreciprocal of bed porosity

1

( ) ( )21 / 22(0) (0) (0) , tan( )g gw

bQ P e

(fluctuations persist in the (fluctuations persist in the

bed)bed)

0

2

4

6

8

10

12

14

16

-8 -6 -4 -2 0 2flux

y/d

s

3 2 1

6666

conceptual modelconceptual model data from the numerical experimentsdata from the numerical experiments

figfig 3. relative magnitude of the 3. relative magnitude of the terms of the equation of terms of the equation of conservation of the granular conservation of the granular temperature. profiles of diffusion, temperature. profiles of diffusion, dissipation and production.dissipation and production.

figfig 4. flux of the fluctuation energy. 4. flux of the fluctuation energy.

different from zero

0

2

4

6

8

10

12

14

16

-5 -4 -3 -2 -1 0 1 2 3 4 5dissipation, diffusion and production

y/d

s

3

21

1 2 33 2 1

0

2

4

6

8

10

12

14

16

0 5 10 15velocity

y/d

s 1

2 3

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6solid fraction

y/d

s

12 3

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2granular temperature

y/d

s

1

2

3

6767

data from the numerical experimentsdata from the numerical experiments

conceptual modelconceptual model

figfig 5. a) velocity profiles; b) profile of the solid fraction; c) profile of the 5. a) velocity profiles; b) profile of the solid fraction; c) profile of the granular temperature. granular temperature. results for results for = 1.74, = 1.74, = 2.49 and = 2.49 and = 3.07. granular material with = 3.07. granular material with ss = 1.5, = 1.5, ddss = 0.003 m and = 0.003 m and ee = 0.82. = 0.82.

a)a) b)b) c)c)

6868

data from the numerical experimentsdata from the numerical experiments

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15u /u *

y/h

c

u = y 3/4

u = y 3/2

granular phase

fluid phase

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 (-)

y/h

c(-

)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5non-dimensional gran. temperature

y/h

c

conceptual modelconceptual model

b)b)

c)c)

best fit

figfig 6. choice of the power law to express 6. choice of the power law to express the velocity profile; exponent 3/4 (as in the velocity profile; exponent 3/4 (as in Sumer Sumer et alet al. 1996) was considered the . 1996) was considered the best.best.

6969

average velocity in the contact load layer - average velocity in the contact load layer - uucc

conceptual modelconceptual model

341

4107

( 1) cc s

s

hu g s d

d

3( ) 43

452

*

g

x

s

u y

u d

depth depth

integrationintegration

• note that the exponent 3/4 was postulated (cf. Sumer et al. 1996).

0

2

4

6

8

10

12

14

16

0 2 4 6 8granular stresses

y/d

s

321 32 1

7070

conceptual modelconceptual model

figfig 7. a) profiles of shear and normal stresses; b) profile of the ratio 7. a) profiles of shear and normal stresses; b) profile of the ratio shear toshear to

normal stress; c) profile of shear efficiency ratio normal stress; c) profile of shear efficiency ratio . .

data from the numerical experimentsdata from the numerical experiments

a)a)

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1T /P

y/d

s

1

2

3

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2R

y/d

s

12

3

( )

12

d g

s y xd u

b)b) c)c)

predominance of collisional stresses is a sound hypothesis

7171

343

0 4

2504

3 30 0

3(1 ) tan ( )0.49

tan ( )

bK c

sb

e M h

dsG

32 7 3

4 4c c

s

h h

d h

12

12

( )

34

262.5 11 0

tan( )cC

b

Ge s

N

conceptual modelconceptual modelthickness of the contact load layer - thickness of the contact load layer - hhcc

• the solution for hc/ds is obtained numerically.

• a good approximation is

1.7 5.5c

s

h

d

• depth integration of the equation of conservation of fluctuating energy:

7272

0

5

10

15

20

25

30

0 1 2 3 4 5 (-)

h c/d

s (

-)

0

5

10

15

20

25

30

0 1 2 3 4 5 (-)

h c/d

s (

-)

0

5

10

15

20

25

30

0 1 2 3 4 5 (-)

h c/d

s (

-)

0

5

10

15

20

25

30

0 1 2 3 4 5 (-)

h c/d

s (

-)

figfig 10. thickness of 10. thickness of contact load layer; contact load layer; a) influence of the a) influence of the restitution coefficient; restitution coefficient; b) influence of the flow b) influence of the flow discharge; discharge; c) influence of the c) influence of the value of the maximum value of the maximum solid fraction; solid fraction; d) influence of the type d) influence of the type of sediment (density of sediment (density and fall velocity)and fall velocity)

thickness of the contact load layer - thickness of the contact load layer - hhcc

a)a) b)b)

c)c) d)d)

conceptual modelconceptual model

Independent of the type of sediment?

1.7 5.5c

s

h

d

problemas de valor inicial onda originada pela ruptura de uma

barragemsolução teórica do problema de Riemann

demonstra-se que existem dois tipos de soluções:

x

t

Undisturbed L-state Undisturbed R-state

shock associated to (1)

shock associated to (2) rarefaction wave

associated to (3)

constant state (2)

constant state (1)

x

t

Undisturbed L-state Undisturbed R-state

shock associated to (1)

rarefaction wave

associated to (2)

rarefaction wave

associated to (3)

constant state (2) constant

state (1)

x

aplic

açõ

es

tipo A: dois choques e uma onda de expansão.

tipo B: duas ondas de expansão e um choque.

aplic

açõ

es

problemas de valores na

fronteiraonda originada pela ruptura de uma barragemleito com descontinuidade inicial

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25X ' (-)

Z ' (-)

t = 8t 0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25X ' (-)

Z ' (-)

t = 8t 0