mathematical modelling of the morphodynamic aspects of the 1996 flood in the Ha! Ha! river conceptual model and solution Rui M. L. Ferreira :: Joo G. B. Leal :: Antnio H. Cardoso Instituto Superior Tcnico :: september 2005 justification of the work introduction severe rainstorms scourged the Saguenay region, south Qubec, Canada, in July of 1996. 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!. the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley. 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. it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow. justification of the work introduction severe rainstorms scourged the Saguenay region, south Qubec, Canada, in July of 1996. 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!. the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic

impacts in the downstream valley. 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. it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow. justification of the work introduction severe rainstorms scourged the Saguenay region, south Qubec, Canada, in July of 1996. 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!. the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley. 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. it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow. justification of the work introduction severe rainstorms scourged the Saguenay region, south Qubec, Canada, in July of 1996. 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!.

the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley. 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. it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow. justification of the work introduction severe rainstorms scourged the Saguenay region, south Qubec, Canada, in July of 1996. 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!. the dam-break wave, superimposed to the hydrologic flood, provoked massive geomorphic impacts in the downstream valley. 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. it constitutes a very demanding boundary value problem because of the magnitude of the impacts and because of the transcritical nature of the flow. introduction objectives of the work 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. 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) introduction objectives of the work 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. 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) introduction objectives of the work 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. 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) structure of the

presentation description of the conceptual model presentation of the simulation results structure of the presentation description of the conceptual model presentation of the simulation results structure of the presentation description of the conceptual model presentation of the simulation solutions conceptual model physical system conceptual model observations suggest stratification fig 1. dam-break wave generated by instantaneous rupture; Shields parameter at dam location is 2.5. observation window located downstream the reservoir, at about 10 times the water depth on the reservoir (10h0). physical system conceptual model 0 1

2 3 cm physical system 0 1 conceptual model upper plane bed clear water/suspended sediment contact load bed (immobile particles) 2 3 cm physical system 0 conceptual model debris flow contact load bed (immobile particles) 1 2 3 cm idealised system fig 2. flow idealised as a multiple layer structure based on stress predominance. conceptual model

clear water/ suspended sediment transition region collisional region frictional region contact load layer bed (immobile grains) model development conservation equations two-dimensional conservation equations (profile) granular phase fluid phase conceptual model shallow water flow cinematic non-material boundary conditions negligible segregation between phase continuum hypothesis incompressible fluid and granular phases one-dimensional conservation equations model development one-dimensional conservation equations mass and momentum

t h Yb x uh 0 total mass t h Yb x uh 0 t smuh x scuc 2 hc us 2hs 12 g x hs 2 2hs hc sc hc 2 conceptual model total momentum sc hc hs x Yb bc ( w) (1 p)t Yb t Cc hc x Ccuc hc 0 sediment mass Cc capacity transport: net S bc 0 net S bc model development one-dimensional conservation equations mass and momentum t h Yb x uh 0 t h Yb velocity x uh inthe

0 contact load layer t smuh x scuc 2 hc us 2hs 12 g x hs 2 2hs hc sc hc 2 conceptual model thickness of the contact load layer sc hc hs x Yb bc ( w) (1 p)t Yb t Cc hc x Ccuc hc 0 capacity (equilibrium) concentration capacity transport: Cc net S bc 0 net S bc model development closure equations uc hc net S bc Cc bc

conceptual model two-dimensional conservation equations (profile) constitutive stress tensor equations flux of grain kinetic energy collisional dissipation granular phase fluid phase model development closure equations uc hc net S bc Cc bc conceptual model two-dimensional conservation equations (profile) constitutive equations granular phase fluid phase collisional region described by dense gas kinetic theory (Chapman-Enskog)

model development closure equations uc hc net S bc Cc bc conceptual model two-dimensional conservation equations (profile) constitutive equations granular phase fluid phase collisional region described by dense gas kinetic theory (Chapman-Enskog) negligible streaming component of the stress tensor (chaos molecular) model development closure equations uc hc net S bc

Cc bc conceptual model two-dimensional conservation equations (profile) constitutive equations granular phase fluid phase collisional region described by dense gas kinetic theory (Chapman-Enskog) negligible streaming component of the stress tensor (chaos molecular) quasi-elastic approximation: e1 closure equations simulation results initial value problems simulation results Riemann problem: the dam break flood wave fig 3. idealised geometry for the dambreak problem understood as a Riemann problem. b) a) 2 = 1 Wtyeyy dd xcbdc c chdfxc,njlkjncflks

Y

channel initial value problems the dam-break flood wave simulation results bed 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 the dam-break flood wave bed initially flat :: erodible banks simulation results bank erosion model 1 m m: inverse bank slope initial value problems the dam-break flood wave simulation results bed initially flat :: erodible banks evolution of the bed width at the level of the initial bed.

evolution of the inverse bank slope. a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river geometry and flood hydrograph Ha! Ha! Bay 1000.0 900.0 800.0 Photo C Eaux-mortes Discharge [cms] simulation results 700.0 600.0 500.0 400.0 300.0 Photo B 200.0 100.0 Chute Perron Boilleau 0.0 19.00 19.50

20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00 Time [days] Dam fig 5. flood hydrograph: superposition of the natural flood and the discharge released by the breached dyke. Lake outlet Photo A Cut-away dyke Lake Ha! Ha! Rive-gauche dyke Chute Baptiste fig 4. plan view of river and lake Ha! Ha!. 400 simulation of the 1996 flood in the Ha! Ha! river observed geomorphic impacts

Chute Baptiste Photo A foto A 350 300 Chute Perron Boilleau 250 zb (m) simulation results a boundary/initial value problem Photo B 200 Photo C 150 100 Eauxmortes 50 0 -50 0 4000 8000 12000

16000 20000 distance (m) 24000 28000 32000 36000 fig 6. longitudinal profile of river Ha! Ha!. fig 7. photo A: dyke location after the collapse. note the pronounced erosion (about 12 metres). 400 simulation of the 1996 flood in the Ha! Ha! river observed geomorphic impacts Chute Baptiste Photo A foto B 350 300 Chute Perron Boilleau 250 zb (m) simulation results

a boundary/initial value problem Photo B 200 Photo C 150 100 Eauxmortes 50 0 -50 0 4000 8000 12000 16000 20000 distance (m) 24000 28000 32000 36000 fig 6. longitudinal profile of river Ha! Ha!. fig 8. photo B: generalized deposition at Eaux-mortes (about 2meters deposits).

400 simulation of the 1996 flood in the Ha! Ha! river observed geomorphic impacts Chute Baptiste Photo A foto C 350 300 Chute Perron Boilleau 250 zb (m) simulation results a boundary/initial value problem Photo B 200 Photo C 150 100 Eauxmortes 50 0 -50 0

4000 8000 12000 16000 20000 distance (m) 24000 28000 32000 36000 fig 6. longitudinal profile of river Ha! Ha!. fig 9. photo C: bank erosion and channel widening at a convex reach. a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river observed geomorphic impacts 400 Chute Baptiste Photo A 350 300 Chute Perron Boilleau

250 zb (m) simulation results chute Perron Photo B 200 Photo C 150 100 Eauxmortes 50 0 -50 0 4000 8000 12000 16000 20000 distance (m) 24000 28000 32000 36000 fig 6. longitudinal profile of river Ha! Ha!.

fig 10. chute Perron: massive erosion as the flow evaded its normal fixed bed course (from Brooks & Lawrence 1999) a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river computational domain 5340000 5356000 5338000 5352000 cross-sections detailed in figure distance (m) 5350000 distance (m) simulation results 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 Herv Capart (Taiwan University).

Ha! Ha! Bay Chute Baptiste 5336000 5334000 362 361 ... 5354000 5332000 5330000 5348000 5346000 Eaux-mortes Chute Perron 5328000 Boilleau 5344000 5326000 Cut-away dyke Lake Ha! Ha! 5324000 5322000 274000

276000 278000 distance (m) 5342000 1 2... 280000 282000 5340000 274000 276000 278000 distance (m) 280000 282000 a boundary/initial value problem 300 simulation of the 1996 flood in the Ha! Ha! river computational domain section 128 Zb = 291.27 m m = 7.1 m Bf = 10.2 m z (m) z (m)

296 294 290 80.0 300 100.0 120.0 b (m) 140.0 296 294 290 80.0 160.0 Zb = 291.58 m m = 4.4 m Bf = 8.1 m section 130 298 296 294 292 290 100.0 Zb = 291.93 m m = 7.0 m Bf = 10.3 m 292 295

100.0 120.0 b (m) 140.0 160.0 Zb = 287.60 m m = 2.9 m Bf = 6.0 m section 131 293 z (m) fig 12. idealized trapezoidal sections used for computational purposes (computed from an algorithm operating over the DTM data). section 129 298 292 z (m) simulation results 298 300 291 289

287 120.0 140.0 b (m) 160.0 180.0 285 100.0 120.0 140.0 b (m) 160.0 180.0 a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river computational domain 120 100 b (m) 60 40 20 0 0 4000 8000

12000 16000 20000 distance (m) 24000 28000 32000 36000 0 4000 8000 12000 16000 24000 28000 32000 36000 30 25 m (m) simulation results 80 fig 13. bed width (top) and inverse bank slope (bottom)

for computational purposes. 20 15 10 5 0 20000 distance (m) a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river computational domain simulation results S=0 S < Scrit Lups 1 2 3 Ldwn L NF1 NF 1 2 ... ... NS1 NS N1 N fig 14. extended computational domain featuring two

virtual reaches at the upstream and downstream ends for computational purposes. a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river computational domain b) a) simulation results t t x t1 x t1 (+) (Q,t) = 0 t () t0 (Q,A;S,) = 0 t t0

0 x1 dx x xN xN-1 x dx 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! river numerical solution 40 3 375 370 35 365 30 3802000 2500 30003.0 2.5

375 2.0 370 1.5 1.0 365 0.5 360 2000 discharge (m /s) elevation (m) 45 360 elevation (m) simulation results 380 0.0 2100 2200 2300 2400 2500 2600 distance (m) 2700

2800 2900 depth (m); Froude (-) a boundary/initial value problem 3000 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 results a boundary/initial value problem evolution of the bed elevation variation. evolution of the Froude number. simulation of the 1996 flood in the Ha! Ha! river results of the numerical simulation a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river results of the numerical simulation t = t0 critical flow

subcritical flow geomorphic hydraulic jump simulation results supercritical flow t = t1 > t2 subcritical flow subcritical flow critical flow geomorphic hydraulic jump supercritical flow fig 17. model for the evolution and disappearing of supercritical reaches, associated to pronounced convex bed profiles, as the bed morphology evolves. t = t2 > t1 subcritical flow geomorphic discontinuity simulation results a boundary/initial value problem evolution of the water depth.

evolution of the bed width. simulation of the 1996 flood in the Ha! Ha! river results of the numerical simulation a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river results of the numerical simulation simulation results longitudinal profile: reaches downstream the eroded dyke. longitudinal profile: Chute Baptiste (fixed bed). simulation results a boundary/initial value problem longitudinal profile: Chute Perron. simulation of the 1996 flood in the Ha! Ha! river results of the numerical simulation a boundary/initial value problem simulation of the 1996 flood in the Ha! Ha! river results of the numerical simulation

220 bed elevation (m). simulation results 210 200 190 180 170 160 150 18000 20000 22000 distance (m) 24000 26000 fig 18. final bed profiles at Chute 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). contributions of the present work: conclusions 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; numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction. contributions of the present work: conclusions 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; numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction. contributions of the present work: conclusions 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;

numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction. contributions of the present work: conclusions 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; numerical problems arise in the steady state computations for the initial condition. artificial viscosity proved a better solution than a TVD correction. conclusions acknowledgements: the authors wish to acknowledge the financial support offered by the European Commission for the IMPACT project under the fifth framework programme (19982002), Environment and sustainable Development thematic programme, for which Karen Fabri was the EC Project Officer. conclusions complementary slides leito plano superior equaes de conservao unidimensionais -massa e quantidade de movimento

profundidade do escoamento t h Yb x uh 0 massa total t h Yb x uh 0 t smuh x scuc 2 hc us 2hs 12 g x quantidade de movimento velocidade mdia do escoamento total conceptual model hs 2 2hs hc sc hc 2 sc hc hs x Yb bc concentrao na camada de transporte t Cc hc x Cc uc hc Snetbc massa de sedimentos na camada de transporte (1 p )t Yb net S bc 0

cota do fundo massa de sedimentos no fundo transporte em desequilbrio net S bc Cc ( w ) leito plano superior equaes de conservao unidimensionais -massa e quantidade de movimento t h Yb x uh 0 t h Yb velocidade x uh 0na camada de transporte por 2 2 1 g h 2 2h h s h 2 t smuh x scuarrastamento h u h c c s s x s s c

c c 2 conceptual model espessura da camada de transporte por arrastamento sc hc hs x Yb bc tenso de arrastamento net t Cc hc x Cc uc hc S bc fluxo vertical de sedimentos (1 p )t Yb net S bc 0 Cc net S bc transporte em desequilbrio

( w ) model development closure equations Cc bc uc hc net S bc b) 16 14 14 12 12 12 2 3 10 1 8 y /ds y /ds 16 14 10 conceptual model

c) 16 2 1 8 6 6 4 4 4 2 2 2 0 0 0 5 10 15 0 velocity 1 8 6

0 3 2 10 3 y /ds a) 0.2 0.4 solid fraction 0.6 0 0.5 1 1.5 granular temperature 2 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. (g) ux 34 y 5 2 u* ds 3 4

depth integration uc 10 7 g ( s 1)d s 1 4 hc ds 3 4 model development closure equations Cc bc uc hc net S bc Integration the t h Ybof x equation uh 0 of conservation of particle fluctuation kinetic energy conceptual model

(g) y Qy Tyx y u x( g ) 0 diffusion productio n hc hc , h; d s , e, hc 1.7 5.5 ds dissipation hc production Q model development closure equations Cc bc uc hc net S bc conceptual model integration of the vertical momentum equation in the frictional sub-layer t Yb gw ( s 1)Cc tan(b )hc wC f u 2 net S bc (1 p ) cu x y Y

friccional sub-layer f g w ( s 1)Cc tan(b )hc w C f u 2 c ux y Y f model development closure equations Cc bc uc hc net S bc integration the t h Ybof x vertical uh 0momentum equation in the frictional sub-layer conceptual model t Yb 0 (1 p ) Cc g w ( s 1)Cc tan(b )hc wC f u 2 cu x y Y C f u2 tan(b ) g ( s 1)hc f 0

friccional sub-layer model development closure equations Cc bc uc hc net S bc 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. 2 (ds u ) / (h ws ) u (-) conceptual model 0.5 cb w C f u 2 w s 3 /u * = 1.0 0.4 0.3 w s 4 /u * = 1.0 C f C fa 0.2 0.1 w s 2 /u * = 1.0 w s 1 /u * = 1.0 0 0 0.005 0.01

0.015 0.02 plastic 2 acrylic 0.025 / (-) predicted instability zone plastic 1 sand u ds h ws C f C fb u 2 C fa u ds h ws fig 4. flow resistance. sheet flow data from Sumer et al. (1996). conceptual model numerical experiments uniform flow of a mixture of water and a granular material conservation equations, granular phase (g) d y Tyy ( w ) s 1 g momentum equation, vertical direction

momentum equation, horizontal (g) d y Tyx C D ( g ) U u ( g ) ( g ) g sin() dir. ds d y Q T ( g ) d y u x( g ) 0 conservation fluctuation energy conservation equations, fluid phase ( w) d y Tyx momentum equation, horizontal C D ( w ) U u ( g ) ( w ) g 1 sin() dir. ds 63 conceptual model numerical experiments constitutive equations, granular phase (g)

(g) Tij 4 1 2 (g) (g) T12 T21 T (g) 8 1 5 2 8 1 3 2 1 (g) 16 1d s ui ,i ij 1 1 3 d s 2 Dij particle stress 5 2 (g) (g) 1 2 tensor (g)

1 3 d s u1,2 shear stresses normal stresses (isotropic pressure) (g) T11( g ) T22 P ( g ) 4( g ) 1 2 (g) Q d y 1 2 ( g ) ( g ) flux of fluctuating energy 4 1 2 1 1 4 d s 2 collisional thermal difusivity 3 24 1 dissipation of fluctuation ( gw ) 2 1 e

1 1 energy: collisional and d 2 s viscous constitutive equations, fluid phase (g) ( gw ) (g) ( w) Tyx 2 y 2 1 f(Ri )d y U 2 fluid shear stress; (mixing length) 64 conceptual model numerical experiments overall: 8 ordinary differential equations, 8 unknowns the thickness of the contact load layer is also unknown; thus the system must be solved iteratively (shooting method); an extra boundary condition is necessary (it is physically and mathematically well posed): Q (y=hc) = 0.0; a value of hc is proposed; the equations are solved and a new hc is computed from the above condition; other boundary conditions: u ( g ) (0) u ( w ) (0) 0 (g) (g) Txy (0) Pyy tan(b )

no slip frictional stresses (0) 0.55 Q (0) 2 reciprocal of bed porosity 1 2 (g) ( gw ) (0) P (0) e , tan(b ) 1/ 2 (fluctuations persist in the bed) 65 conceptual model 16 16 14 14 12

12 10 10 y /ds y /ds data from the numerical experiments 8 6 2 4 2 3 3 2 different from zero 8 6 1 4 1 2 3 1 3 2

1 2 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 dissipation, diffusion and production fig 3. relative magnitude of the terms of the equation of conservation of the granular temperature. profiles of diffusion, dissipation and production. -8 -6 -4 -2 0 2 flux fig 4. flux of the fluctuation energy. 66 conceptual model data from the numerical experiments 16 16 a) 12 2

3 y /ds 2 1 8 8 6 6 4 4 4 2 2 2 0 0 0 5 10 velocity 15 2 0

1 10 3 6 0 3 12 10 1 8 c) 14 12 10 y /ds b) 14 y /ds 14 16 0.2 0.4 solid fraction

0.6 0 0.5 1 1.5 granular temperature 2 fig 5. a) velocity profiles; b) profile of the solid fraction; c) profile of the 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. 67 conceptual model data from the numerical experiments fig1 6. choice of the power law to express the 0.8 velocity profile; exponent 3/4 (as in b) Sumer et al. 1996) was considered the 0.6 best. 0.4 1 0.9 3/4 y /hc (-) u = y 0.8 best fit 0.7

3/2 0.2 granular phase 0.6 0 0 0.5 0.4 0.2 (-) 0.4 0.6 1 fluid phase 0.3 0.2 0.6 0.4 0.1 0.2 0 0 0 5 u /u *

10 c) 0.8 y /hc y /hc u = y 15 0 0.5 1 1.5 non-dimensional gran. temperature 68 conceptual model average velocity in the contact load layer - uc (g) ux 34 y 5 2 u* ds 3 4 depth integration uc 10 7 g ( s 1)d s

1 4 hc ds note that the exponent 3/4 was postulated (cf. Sumer et al. 1996). 69 3 4 conceptual model data from the numerical experiments a) 14 14 y /ds 12 10 y /ds 16 16 8 6 b) 3 12

2 12 10 1 10 8 6 4 4 2 2 0.2 0.4 0.6 T /P 1 0 0 2 2 3 1 2 3

4 6 granular stresses 8 2 1 0 0 2 3 8 6 0 4 c) 14 y /ds 16 0.8 1 0 0.5 1 R

1.5 2 predominance of collisional stresses is a sound hypothesis fig 7. a) profiles of shear and normal stresses; b) profile of the ratio shear to d s d y u x( g ) normal stress; c) profile of shear efficiencyratio 1 2 . 70 conceptual model thickness of the contact load layer - hc depth integration of the equation of conservation of fluctuating energy: 0.49 3(1 e) 45K tan 2 (b ) 0 sG0 03 tan 3 (b ) M 0 3 4

3 2 hc hc 7 hc 3 d 4 h 4 d s s 3 4 262.5 1 2 G (Cc ) 1 12 0 1 e s 3 tan(b ) N 4 the solution for hc/ds is obtained numerically. a good approximation is hc 1.7 5.5 ds 71 conceptual model thickness of the contact load layer - hc 30 30

a) 20 15 10 5 20 15 10 5 0 0 0 1 2 (-) 3 4 5 0 30 1 2 (-) 3 4 5

30 c) 25 20 15 10 20 15 10 5 5 0 0 0 d) 25 h c /ds (-) h c /ds (-) b) 25 h c /d s (-) h c /d s (-) 25 1 2

(-) 3 4 5 0 1 Independent of the type of sediment? 2 3 4 (-) 5 fig 10. thickness of contact load layer; a) influence of the restitution coefficient; b) influence of the flow discharge; c) influence of the value of the maximum solid fraction; d) influence of the type of sediment (density and fall velocity) hc 1.7 5.5 ds 72 problemas de valor inicial

onda originada pela ruptura de uma barragem soluo terica do problema de Riemann rarefaction wave (3) associated to aplicaes demonstra-se que existem dois tipos de solues: t constant state (2) shock associated to (2) constant state (1) shock associated to tipo A: dois choques e uma onda de expanso. Undisturbed L-state (1) Undisturbed R-state x rarefaction wave (3) associated to t constant state (2) rarefaction wave (2) associated to

constant state (1) shock associated to tipo B: duas ondas de expanso e um choque. Undisturbed L-state (1) Undisturbed R-state x x problemas de valores na fronteira onda originada pela ruptura de uma barragem leito com descontinuidade inicial 1.0 t = 8t 0 0.8 0.6 aplicaes Z ' (-) 0.4 0.2 0.0 -0.2 -1.25 -1.00 -0.75

-0.50 -0.25 0.00 X ' (-) 0.25 0.50 0.75 1.0 1.00 1.25 t = 8t 0 0.8 0.6 Z ' (-) 0.4 0.2 0.0 -0.2 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 X ' (-) 0.25 0.50

0.75 1.00 1.25