Mathematical modelling of the morphodynamic aspects of the 1996 flood in the Ha! Ha! river...
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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
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.
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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el
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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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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cep
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od
el
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
con
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od
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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
cep
tual m
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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
con
cep
tual m
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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)
con
cep
tual m
od
el
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)
con
cep
tual m
od
el
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
tual m
od
el
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
resu
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
mula
tion
resu
lts
initial value problems
sim
ula
tion
resu
lts
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.
sim
ula
tion
resu
lts
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
ula
tion
resu
lts
initial value problems the dam-break flood wave
bed initially flat :: erodible banks
sim
ula
tion
resu
lts
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
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
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
ula
tion
resu
lts
simulation of the 1996 flood in the Ha! Ha! rivergeometry and flood hydrograph
-50
0
50
100
150
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0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)
z b (m
)
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
sim
ula
tion
resu
lts
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).
-50
0
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250
300
350
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0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)
z b (m
)
Boilleau
Eaux-mortes
Chute á Baptiste
Chute á Perron
Photo A
Photo C Photo B
sim
ula
tion
resu
lts
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.
-50
0
50
100
150
200
250
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350
400
0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)
z b (m
)
Boilleau
Eaux-mortes
Chute á Baptiste
Chute á Perron
Photo A
Photo C Photo B
sim
ula
tion
resu
lts
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)
-50
0
50
100
150
200
250
300
350
400
0 4000 8000 12000 16000 20000 24000 28000 32000 36000distance (m)
z b (m
)
Boilleau
Eaux-mortes
Chute á Baptiste
Chute á Perron
Photo A
Photo C Photo B
sim
ula
tion
resu
lts
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
5324000
5326000
5328000
5330000
5332000
5334000
5336000
5338000
5340000
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distance (m)
dist
ance
(m
)
12...
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5356000
274000 276000 278000 280000 282000
distance (m)
dist
ance
(m
)
362361
...
“Cut-away” dyke
Chute á Perron
Chute á Baptiste
Eaux-mortes
Boilleau
Lake Ha! Ha!
Ha! Ha! Bay
cross-sections detailed in figure
sim
ula
tion
resu
lts
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
292
294
296
298
300
80.0 100.0 120.0 140.0 160.0
b (m)
z (m
)
section 131
285
287
289
291
293
295
100.0 120.0 140.0 160.0 180.0b (m)
z (m
)
section 129
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296
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300
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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292
294
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300
100.0 120.0 140.0 160.0 180.0b (m)
z (m
)
sim
ula
tion
resu
lts
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
0
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0 4000 8000 12000 16000 20000 24000 28000 32000 36000
distance (m)
b (m
)
0
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0 4000 8000 12000 16000 20000 24000 28000 32000 36000
distance (m)
m (m
)
sim
ula
tion
resu
lts
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
ula
tion
resu
lts
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)
sim
ula
tion
resu
lts
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
360
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2000 2500 3000
elev
atio
n (m
)
30
35
40
45
disc
harg
e (m
3 /s)
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365
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2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000distance (m)
elev
atio
n (m
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
dept
h (m
); F
roud
e (-
)
sim
ula
tion
resu
lts
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
resu
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
ula
tion
resu
lts
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
resu
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
ula
tion
resu
lts
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
ula
tion
resu
lts
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).
150
160
170
180
190
200
210
220
18000 20000 22000 24000 26000distance (m)
bed
elev
atio
n (m
).
sim
ula
tion
resu
lts
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.
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0t b xh Y uh
( )
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w
t m x c c c s s x s s c c c
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w
t m x c c c s s x s s c c c
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y/d
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16
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y/d
s
12 3
0
2
4
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8
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14
16
0 0.5 1 1.5 2granular temperature
y/d
s
1
2
3
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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integration of the vertical momentum integration of the vertical momentum equation in the frictional sub-layerequation in the frictional sub-layer
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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
0.1
0.2
0.3
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