Fully coupled 1D model of mobile-bed alluvial hydraulics: application to silt transport in the Lower Yellow River

Fully coupled 1D model of mobile-bed alluvial hydraulics: application to silt transport in the Lower

Yellow River

Nicolas Huybrechts August 2008

Promoteur :

Prof. Michel Verbanck

Traitement des Eaux et Pollution

Thèse présentée en vue de l’obtention du grade de docteur en Sciences Appliquées

The overall objective is to improve the one-dimensional numerical prediction of the fine and non-cohesive bed material load in alluvial rivers, especially during high intensity episodes during which sediment beds are strongly remobilized. For this reason, we attempt to reduce the major inaccuracy sources coming from the alluvial resistance and bed material load relations needed to close the mathematical system. Through a shared parameter called the control factor m, the interactions occurring in alluvial rivers are incorporated more deeply into the mathematical model and more particularly into the closure laws: bed material load (SVRD, Suction-Vortex Resuspension Dynamics) and the energy slope (Verbanck et al. 2007). The control factor m is assumedly related to the Rossiter resonance modes of the separated flow downstream the bed form crest.

To further improve the representation of the flow-sediment-morphology interactions, a fully coupled model approach has been naturally chosen. In this work the terminology fully coupled means that the three equations forming the system are solved synchronously and that the terms often neglected by more traditional decoupled models are kept.

The feasibility of the new closure methodology has been drawn up by reproducing numerically the silt-flushing experiment conducted by the Yellow River Conservancy Commission (Y.R.C.C.) in the Lower Yellow River (LYR) in Northern China. The objective of the silt flushing experiment is to reverse the aggradation trend of the Lower Yellow River which, in the last decades, has become a perched river. The numerical simulation specifically reproduces the silt-flush effects in a reach of LYR located in the meandering part of the river. This reach (around 100 km) is delimited by Aishan and Luokou hydrometric stations.

Since the SVRD formulation has been developed from flume observations, the law has first been confronted to river datasets. The confrontation has revealed that the SVRD law becomes less suitable for fine sediment fluxes (ratio of water depth over median particle size > 5000). Therefore, a modified equation SVRD-2 has been built to enlarge the validity range.

The suitability of the SVRD-2 equation to predict fine sediment fluxes has been tested on data available from several hydrometric stations located in the meandering reach of the LYR: historical observations and measures collected during the flushes. The SVRD-2 has also been compared with relations specifically calibrated for this configuration. The comparison has pointed out that the performance of the two formulas is similar, which is encouraging for the SVRD-2 approach as it has not been calibrated on those data.

The closed equation system has been written on its quasi-linear form and is solved by a Finite Volume Method combined with a linearized Riemann algorithm. The numerical

As it is not yet possible to predict dynamically the value of the control factor m, a possible solution would be to extract its value from the measured data at the inlet cross section. Unfortunately, the necessary data are not measured locally. Moreover, a uniform value of the control factor m may not suffice to reproduce the flow along the whole reach. Therefore, it has been proposed to work temporarily in the reverse way.

From the comparison between the numerical results and the experimental data, a time evolution of the control factor m has effectively been extracted and it has been shown that it varies along the reach. At Aishan, the evolution of the control factor m corresponds to the evolution expected from the data analysis previously conducted on other data sets:

the value of the control factor m decreases during the flush as it tries to reach the optimal value m=1. The time evolution at Luokou behaves differently to the one at Aishan, but remains in agreement with m evolution patterns observed historically for the river section flowing round Jinan City walls. For Luokou, the highlighted differences may come from three dimensional effects coming from the meander bend upstream the station.

Generally, the results obtained for the hydraulics, the sediment transport and bed adaptation are encouraging but still need improvements and additional feeding from the experimental data. The results for the concentration and therefore the bed elevation are very sensitive to the value of the control factor m as it influences most of the terms of the bed material load equation (SVRD-2).

The major remaining difficulties are, firstly, to deal with the rapid transients for which the model is less suitable and, secondly, to improve the prediction of the value of control factor m. Before paying more attention into the transients, enhancements concerning the flow along the reach (initial condition and discharge rates during the first days of the flush) must be conducted in priority. Indeed as the prediction of the bed or the cross section evolutions depend directly on the quality of the prediction of the sediment concentration and the hydraulics, one should first improve these aspects. To perform this study, more information about the water levels or sediment concentrations is necessary at some intermediate stations. One solution is to lengthen the studied reach, upstream to Sunkou and downstream to Lijin, totaling a river length of 456 Km.

A more entire signal of the energy slopes and the associated bed configurations at different stations would enlighten how the control factor m evolves along the reach during the silt-flush events.

Je tiens à remercier le Professeur Verbanck de m’avoir accueilli dans son service et d’avoir dirigé ce travail. Je remercie également les Professeurs Gerard Degrez et Jean- Pierre Hermand pour leurs encadrements et leurs conseils.

J’exprime ma gratitude envers Arielle Cornette, Caroline De Bodt, Didier Bajura, Jean- Pierre VanderBorght, Jerôme Harley, Laura Rebreanu, Lei Chou, Michèle Loijens, Nathalie Roevos et Vincent Carbonnel pour les différents services rendus et leurs soutiens.

Acknowledgments:

I wish to express my thankfulness to the Y.R.C.C for the fruitful collaboration and for the access to the data.

I would also like to thank Zhang Yuanfeng and Luong Giang Vu for their helps and for the several discussions we had about the control factor m.

The study contributes to the AquaTerra Project 'Integrated modelling of the river- sediment-soil-groundwater system' funded by the European 6th Framework Programme, research priority 1.1.6.3 Global change and ecosystems (European Commission, Contract No 505428-GOCE). It is part of Flux3 'Input/output mass balances in river basin:

dissolved and solid matter load', a sub-component of the AquaTerra Integrated Project.

Notation

1. Context and objectives 1

1.1. Introduction 1

1.2. The closure issue 1

1.3. The three components of the interactions and their associated prediction laws 2

1.3.1. Definitions of the three components 2

1.3.2. Prediction laws 5

1.3.3. Synthesis about these prediction tools 10

1.4. Proposals for improving the prediction tools 11

1.4.1. A criterion including the bed material load to delineate the bed forms occurring in the lower alluvial regime 11

1.4.2. A common parameter for the estimation of the energy slope and the bed material load 11

1.4.3. A common closure parameter for the fully coupled mathematical model 14

1.5. The control factor m as the key element of the interactions 16

1.5.1. The different links 16

1.5.2. Link between the control factor m and the bed form configuration 16

1.5.3. Remarks concerning the rippled bed configurations 18

1.5.4. Remarks concerning the Strouhal number 19

1.5.5. Link between the control factor m and the resuspension parameter 21

1.6. Procedure and operation plan 22

1.6.1. Procedure 22

1.6.2. Operation plan 23

2. A bed-material load criterion to delineate the bed forms occurring in the lower alluvial regime 24

2.1. Introduction 24

2.2. Evolution of the flow resistance with the alluvial regime 25

2.3.2. Transition zone 28

2.3.3. Remarks concerning the definition of Ev 28

2.3.4. Remarks concerning the importance of the viscosity and median diameter 29

2.4. Data sources: laboratory and field 29

2.5. Performance of P* 30

2.5.1. On flume data- dataset A 30

2.5.2. On field observation-dataset B 32

2.5.3. On other laboratory runs 32

2.6. Application domain 33

2.6.1. Range of the sediment size 33

2.6.2. Behaviour for fine sediment (d50<0,1mm) 33

2.6.3. Behaviour for coarse sand (d50>0,75mm) 34

2.7. Comparison with other criteria from the literature 35

2.8. Conclusions 37

3. Computation of the bed material load based on the Rossiter modes concept 38

3.1. Introduction 38

3.2. Comparison on the Brownlie dataset 38

3.2.1. Data used for the comparison 38

3.2.2. Formula used for the comparison 39

3.3. Comparison of the methods 40

3.3.1. Presentation of the results 40

3.3.2. Influence of the settling velocity 41

3.3.3. Application domain 42

3.3.4. Improvement of the initial equation 44

3.3.5. Remarks about the modified approach 46

3.4. Conclusions 47

4.2. The Lower Yellow River database 48

4.2.1. The Lower Yellow River 48

4.2.2. Reach simulated with the FLUXO model 50

4.2.3. Presentation of the data 51

4.2.4. Remarks concerning the cohesiveness of the material 52

4.2.5. Range of the control factor m 55

4.2.6. Remarks concerning the values of the control factor m 57

4.2.7. Remarks concerning the Luokou hydrometric station 61

4.2.8. Remarks concerning the settling velocity and the median diameter 61

4.3. Other formulas used for the comparison 62

4.3.1. The Formula of Zhang Y.F. 62

4.3.2. The modified formula of Zhang Y.F. 62

4.3.3. The formula of Zhang and Xie 1993 62

4.4. Applications of the formulas to the LYR dataset 63

4.4.1. Quantitative comparison 63

4.4.2. Qualitative comparison 64

4.4.3. Zoom on the data collected at Luokou during the flush 65

4.4.4. Zoom on the data collected at Aishan between years 1959-1965 68

4.5. Conclusions 69

5. Fully coupled mobile bed alluvial hydraulics with a closure drawn from Rossiter modes 71

5.1. Introduction 71

5.2. Mathematical model 71

5.3. Applicability of the mathematical model 72

5.3.1. Limitations of the mathematical model 72

5.3.2. Specifications for the test cases in practice 73

5.4. Study of the characteristics 74

5.4.1. Previous work: 74

5.4.2. Quasi linear form 74

5.5.1. The linearized Riemann solver 78

5.5.2. Modifications of the CLAWPACK software 78

5.6. Deposition upstream of a dam 79

5.6.1. The configuration 79

5.6.2. Numerical parameters and closure relation originally used 80

5.6.3. Numerical parameters used here 80

5.6.4. Comparison 80

5.7. Aggradation experiments 81

5.7.1. The configuration 81

5.7.2. Numerical parameters 83

5.7.3. Closure strategy 84

5.7.4. Results 84

5.7.5. Attempt to predict the control factor m value 86

5.8. Conclusions 86

6. Application of the model to the silt flushing experiment: feasibility study 88

6.1. Introduction 88

6.2. The silt flushing experiment 88

6.2.1. The flush events 88

6.2.2. The studied reach and the data available 89

6.3. The set up of the numerical model 91

6.3.1. Geometry 91

6.3.2. Physical parameters 93

6.3.3. Numerical parameters 94

6.3.4. Initial conditions 94

6.3.5. Boundary conditions 96

6.3.6. Dynamic behaviors of the cross sections 97

6.3.7. Limitations concerning the applicability of the numerical model to the current test case 97

6.4. Presentation of the results 98

6.4.3. Bed material load 102

6.4.4. Bed evolution at Aishan and Luokou 103

6.4.5. Bed elevation and cross sections along the reach after the flush 105

6.5. Conclusions 108

7. Application of the FLUXO model to the silt flushing experiment: sensitivity study 110

7.1. Introduction 110

7.2. Influence of the resuspension parameter 111

7.2.1. Presentation of the computations 111

7.2.2. Modifications brought to the results: Aishan 111

7.2.3. Modifications brought to the results: Luokou 112

7.2.4. Discussion 114

7.3. Influence of the settling velocity 114

7.3.1. Estimations of the settling velocity 114

7.3.2. Modifications brought by the linear variation of the settling velocity 115

7.4. Influence of the porosity 117

7.5. Reference test case 118

7.6. Conclusions 120

8. Conclusions 121

8.1. Modeling the whole flow-sediment-morphology system in alluvial rivers 121

8.1.1. Context and closure issue 121

8.1.2. The three efforts proposed to enhance the prediction tools 121

8.1.3. Remarks concerning the methodology 122

8.1.4. The procedure 123

lower alluvial regime 124 8.2.2. Prediction of the bed material load based on the control factor m

125

8.2.3. Application of the SVRD-2 formulation to the LYR dataset

126

8.2.4. Remarks concerning the relationships of the control factor m

126

8.2.5. Development of the FLUXO model 127 8.2.6. Application of the FLUXO model to the silt flushing experiment:

feasibility 128

8.2.7. Application of the FLUXO model to the silt flushing experiment:

sensitivity 129

8.3. Perspectives 130

8.3.1. Improvements of the methodology and feeding from experimental

data 130

8.3.2. Improvements of the model and further validations 131

References

Appendix: detailed tables

c propagation celerity of the gravity

waves [m/s]

d50 median diameter of the particle [m]

di diameter of the particle of a grain

size class [m]

dσ median diameter of the particle

devided by the gradation factor [m]

f detached frequency of the vortex leaving

the bed forms crest [1/s]

f attached occurrence frequency of the bed

form assending stoss slope [1/s]

g gravity [m/s²]

h water depth [m]

k1,k2,k3 coefficient of the energy slope

equation (Correia) [-]

ks roughness [m]

m control factor m [-]

n Manning coefficient [s/m^1/3]

p pressure [N/m²]

pi percentage [-]

q flow rate per unit width [m²/s]

qs sediment flow rate per unit width [m²/s]

r discrepancy ratio [-]

ri eigenvector

si eigenvalue [m/s]

t time [s]

u* shear velocity [m/s]

u’* shear velocity related to the

grains [m/s]

u*c critical shear velocity [m/s]

ws settling velocity [m/s]

wi settling velocity of the grain size

class [m/s]

wσ settling velocity corrected by the

gradation factor [m/s]

x streamwise coordonate [m]

xr average length of the separation

zone [m]

y coordinate along the water depth [m]

A wet area [m²]

Ar Archimedes number [-]

B coefficient B [-]

C Chezy coefficient []

Ci estimated concentration [kg/m²]

Cm measured concentration [kg/m³]

Cs transport capacity [-]

CV depth average volumetric

concentration [-]

Cw depth average massive

concentration [kg/m³]

C* non dimensional Chezy [-]

Ev viscous power [W/m²]

Fr simplified Froude number [-]

Frg generalized Froude number [-]

G generation of the turbulent energy [W/m²]

Gsl power needed for the suspension [W/m²]

H total energy head [m]

L length of the reach [m]

L* period of the sinus curve used for

the control factor m predictor [-]

N* criterion of Karim [-]

P wet perimeter [m]

P* criterion P* [-]

Q flow rate [m³/s]

Qs sediment flow rate [m³/s]

R hydraulic radius [m]

Re* Reynolds number related to the

grains [-]

S energy slope [-]

So channel slope [-]

Sr Strouhal number

Srg generalized Strouhal number T Transport stage parameter of Van

Rijn [-]

U depth average velocity [m/s]

W channel width [m]

Y bed elevation [m]

Z Zanke average [-]

β Rouse number [-]

λ porosity [-]

λBF bed form wavelength [m]

ηsl suspension efficiency [-]

ηsl C* resuspension parameter [-]

ρ density [kg/m³]

ρs sediment density [kg/m³]

σ gradation factor [-]

ν kinematic viscosity [m²/s]

τ shear stress [N/m²]

Δ excess of relative density [-]

Π stream power [W/m²]

1. Context and objectives

1.1. Introduction

The overall objective of the thesis is to improve the prediction of non-cohesive particle fluxes in alluvial rivers, especially during high intensity episodes during which sediment beds are strongly remobilized. The knowledge of sediment fluxes is essential for river training, flood control, navigation maintenance and environmental protection. Typically, the treated range of the particle grain size is between 0,025 mm and 1 mm. Within the EU integrated project Aquaterra Flux3, the present study is focused especially on the prediction of the fine sediment particles fluxes that are causing the highest environmental concern in terms of pollutant transfer. More attention is thus paid at the sediments with grain size smaller than 0,25 mm.

The prediction of the sediment fluxes is performed through a one-dimensional numerical model. As discussed in this introductive chapter, the mathematical system requires additional relations for the closure. Since the choice of these relations is one of the major inaccuracy sources affecting the results quality (Cao and Carling 2002), we investigate from where they can origin. From this examination, it is pointed out that the interactions characterizing the alluvial rivers have not been sufficiently integrated into these relations.

To enhance the incorporation of the interactions and thus the quality of the numerical results, three solutions are suggested. Finally, the methodology and the test cases are presented.

1.2. The closure issue

The mathematical model is formed by three equations (St. Venant-Exner): the mixture continuity equation (water and sediment), the sediment continuity equation and the mixture momentum equation. The main variables are represented in the following diagram.

Y

h U

x

Figure 1-1 Stream wise profile and main variables (redrawn from Cao and Egashira 2000)

( )

∂ +∂

∂ +∂

∂

∂

t Y x

hU t

h (1-1)

(

)

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ +∂

∂

− ∂

U q t x q t

Y _{s s}

λ (1-2)

( )

(

)

x h g Y

x U U t

U = −

∂ + + ∂

∂ + ∂

∂

∂

0 (1-3)

While the bed porosity λ is considered as input data, the equations system holds five unknowns: the water depth h [m], the depth-averaged velocity U [m/s], the bed elevation Y [m], the energy slope S and the bed material load per unit width qs [m²/s].

If the depth-averaged velocity U, the bed elevation Y, the water depth h are chosen as primary variables, additional relations are then needed for the energy slope S and the bed material load qs. The definitions of the energy slope and the bed material load are given in the next section. There, the laws traditionally used to predict their values are analyzed.

As the prediction of the bed form configuration can also be necessary to estimate these quantities, the different bed forms and associated prediction criteria are detailed too.

1.3. The three components of the interactions and their associated prediction laws

1.3.1. Definitions of the three components

Alluvial rivers are characterized by interactions between the flow, the sediment transport and the bed morphology: the sediment transport is controlled by the flow structure whereas the flow structure depends on the bed geometry that is itself influenced by the sediment transport (Nezu and Nakagawa 1993). These interplays can be schematized as illustrated in the following figure.

Figure 1-2 Interactions occurring in alluvial rivers (taken from Verbanck 2004b)

(a) Alluvial resistance

The hydraulic losses are commonly represented by the energy slope S from the Bernoulli equation:

dx

S = dH ,

g U g h p Y

H 2

+ ² + +

= ρ ^(1-4)

where H is the total energy head [m] and p the pressure [N/m²].

(b) Bed material discharge

In a river, the particle flux can be divided in three modes (Colby 1963):

• The wash load: fine particles transported by the water but not found in significant quantities in the composition of the bed.

• The bed load: movement of the particle more or less in contact with the bed (rolling or sliding).

• The suspension load: particles moving without continuous contact with the bed; the particles maintain in the water column by lift effects from the turbulence.

For the treated particle sizes, the sediments are mostly transported through its bed material load modes (the bed load and the suspension load).

In this work, the bed material load is assumed to be always equal to its transport capacity, which is the maximum sediment load that can be carried by a flow in equilibrium (dqs/dx

= 0). Moreover, we ideally focus on configurations without cohesive effects. The cohesive sediments are characterized by dominant effects due to electrochemical repulsive and attractive forces between particles. The cohesive sediment particles do not behave as individual particles anymore and rather tend to collapse and to form flocs. The border between cohesive and non cohesive sediments is not clearly defined and is generally site-specific. In practice, the sediments finer than 0,002 mm (clay) are however often reckoned as cohesive and those coarser than 0,063 mm (sand) as non-cohesive. The materials between 0,002 – 0,063 mm (silt) are usually reckoned as between cohesive and non-cohesive sediments.

(c) Bed morphology adaptation

For the non cohesive sand bed we presently consider, different type of bed-forms (BF) occurs through the alluvial regime. They are usually categorized by the code defined by Simons and Richardson 1966: BF 1 lower regime plane bed, BF 2 ripple, BF 3 dune, BF 4 transition from lower to upper alluvial regime, BF 5 upper regime plane bed, BF 6 in- phase waves (often called antidunes), BF 7 breaking waves (or breaking antidunes) and BF 8 chute and pool.

bf 2 Ripples bf 3 Dunes

bf 5 Upper regime plane bed bf 6 in phase waves

Figure 1-3 The different bed forms occurring in rivers (Modified from Simons and Richardson 1966) According to Simons and Richardson 1966, the lower alluvial regime is characterized by large alluvial resistance and small sediment transport whereas small flow resistance and large sediment transport typify the upper regime. Two kinds of bed forms are observed for the lower alluvial regime, namely ripples and dunes. The geometry of these bed-forms is quite similar: an asymmetric shape with a gentle stoss slope, a sharp crest and a steeper lee side (Simons and Richardson 1966). Nevertheless, they present different geometrical dimensions: the length and the height of the ripples scale with the sediment size (Yalin 1985; Baas 1999) and those of the dunes scale rather with the water depth (Kostaschuk and Church 1993), at least in the typical shallow river environment case regarded here (Flemming 2000). The stoss face is associated with a converging section and an acceleration of the velocity favorable to the sediment erosion process. The lee face is marked by the flow separation that comes from an abrupt expansion of the sections. The low velocities of the separation are associated with sediment deposition. In the dune case, the separation domain usually influences the whole flow from the bed to the free surface.

The vortex partially obstructs the flow section and consumes a part of the stream power for maintaining its oscillation. In the case of ripples, the variations of the flow section and the velocities are slighter (Bennett and Best 1996). Consequently, the flow perturbations stay confined near the bed and the energy losses coming from the separation zones are less important than for the dunes.

For the upper alluvial regime, only one remarkable kind of bed form is observed: the in- phase waves. The term “in-phase waves” (as suggested by Cheel 1990) indicates that the deformation of the free surface between the water and the atmosphere is in-phase with the bed forms and their shapes look both as sinusoidal curves. Contrarily, the bed forms of the dunes are in opposition of phase with the free surface waves. On the ascending stoss side, the velocity is increasing which tends to reduce the water depth.

As discussed later, the configuration of the in-phase waves is particularly interesting as high stream power flows naturally develop this configuration (Gilbert 1914), which is thus often viewed as an optimum configuration (Nanson and Huang 2008).

1.3.2. Prediction laws (a) Prediction of the energy slope

Several laws have been published to estimate the value of the energy slope, as for instance the laws of Manning and Chezy widely used by engineers (Yen 1991). In Table 1-1, some of the most widespread laws are summed up. To differentiate the formulations more clearly, all the laws are written as a product of a multiplicative coefficient k1 and the two ratios:

2 3 1

2 k

k

i gR

U d

k R

S ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛ (1-5)

Where R is the hydraulic radius and di is the sediment size (generally the median diameter d50).

The median diameter d50 is determined from sieve analysis. It refers to the diameter of the filter for which 50% of the grains by weight pass through. The first ratio is the hydraulic radius by the sediment size and the second is the square of the Froude number (Fr). The influences of these ratios are weighted by the coefficients k2 and k3 whose values are given in table 1-1.

Formula Validity k₁ k₂ k₃

Brownlie

1981 Lower regime Fr<1 Upper regime Fr>1 0,088<d₅₀(mm)<2,8

4131 ,

0205 0

,

0 σ

2786 ,

0125 0

,

0 σ

-0,075 -0,281

1,286 1,086 Kishi and

Kuroki 1974

Dunes I: Y_s<0,02 Z_s^1/2 Dunes II: 0,02 Z_s^1/2

Trans: 0,02 Z^1/2<Y_s≤ 0,02 Z_s^5/9 Plane beds: 0,02 Z^5/9<Y_s≤ 0,07 Z_s^2/5

Antidunes: Y_s>0,07 Z_s^2/5 0,375<d₅₀(mm)<3,6

0,0052 0,013

( )^{6 7}

018 ,

0 ρ_s ρ 0,021

( )²

021 ,

0 ρ_s ρ

1 0 -3/7 -1/3 1/5

3 1 1/7 1 3

Garde and Ranga-Raju 1966

Dunes and Ripples: G₁ Fr<0,33 Transition 0,33 ≤ G₁ Fr≤ 1

Antidunes G₁ Fr > 1 0,011<d₅₀(mm)<5,2

(

)

1

46 2

,

3 ⁻ G G

( )

[ ]

(

)

1

2 1

1

3 log 83 , 2 46 , 3

G G

Fr G

×

− ⁻

(

)

1

1 2

,

2 ⁻ G G

-1/3 -1/3

-1/3 1 1

1

Engelund 1967

Dunes: Fr<1 Antidunes: Fr>1 0,19<d50(mm)<0,93

( )

01 ,

0 ρ ρ_s 0,021

0,043 -1/3

2,15 1 Griffiths

1981

Gravel bed: Fr<1 12<d50(mm)<152

0,026 -0,43 0,66

Table 1-1 Example of energy slope laws (extracted from Correia 1992)

The dimensionless parameters in table 1-1 are defined by (Yalin 1977):

) ( )

( RS d₅₀

Y_s = ρ ρ_s , Z_s = R d₅₀ , G₁Fr = q ΔgR³ .

ρs is the sediment density (equal to 2650 kg/m³ for quartz sands, as is generally the case in this monograph), Δ is the relative excess of density, G1 and G2 are functions of d50

defined by Garde and Ranga-Raju 1966.

In this table, it is observed that the values of the parameters k1, k2, k3 vary according the alluvial regime and the bed form configurations. The laws differ from one author to another and their domains of application are restricted. The energy slope is assumed to depend only on the flow, the sediment grain size and the bed forms configurations: the sediment transport has not been included into this analysis. A parameter which is also often met is the roughness ks, generally estimated from either the grain roughness or a roughness induced by the bed form geometry (Karim 1999; van Rijn 1984b; S-Q. Yang and others 2005).

For a specific application the difficulty is thus to select the most convenient law from the list.

(b) Prediction of the bed material load

As many formulas have been developed for the transport capacity, the present review only focuses on some equations representative of the different families.

The first family is formed by formulas involving the Einstein bed load parameter and the Shields stress parameter, as the equations of Ackers and White 1973; Yalin 1977;

Bagnold 1973; Einstein 1942; Meyer-Peter and Muller 1948. The other widely used parameters and their associated families are those of Velikanov 1954 (U³/gRws), C.T.

Yang 1973 or Bagnold 1966 (US/ws) and van Rijn 1984a transport-stage parameter T=[(u*)²-(u*cr)²]/(u*cr)². More recently, S-Q.Yang 2005 has built a formula up which combines the different parameters met in the other families. In these equations ws is the settling velocity, R the hydraulic radius (wet area by wet perimeter) and the critical shear velocity, u*cr, corresponds to the threshold value of shear velocity for which the sediment is about to move (equivalent to the so-called Shields condition for incipient transport of non-cohesive sediment layers).

To illustrate the disparity between the different approaches, some of the simplest formulas to apply for practical engineer works are expressed in more details in table 1-2.

The equations have been written in their volumetric concentration form to highlight their similarities or differences:

Uh

C_s = q^s (1-6)

Authors Family Formula Engelund

and Hansen 1967

Engelund

50 3

*

²

² 05 , 0

hd g

u C_s U

= Δ Molinas and

Wu 2001

Velikanov

( )

(

)

ψ ψ

+

= +

016 , 0 2650

86 , 0 43 ,

1 ^1,5

Cs

Celik and Rodi 1991

Bagnold

(

)

s

s w

U gh h

C k

ρ ρ

τ

⎥ −

⎥⎦

⎤

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

−⎛

=

06 , 0

1 034 , 0 Zhang and

Xie 1993

Velikanov ⁿ

s s

s ghw

U

C k ⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛ ³

ρ S.Q. Yang

2005

combined

σ

τ ρ

ρ w

u u

C gUh ^c

s s

2

*, 2 '

1 *

5 ,

12 −

= −

Table 1-2 Examples of some bed material load formulas

where ks is the roughness [m], τ is the bed shear stress [N/m²], ws the settling velocity [m/s] and u’* is the shear velocity related to the grains [m/s].

The ψ parameter intervening into the Molinas and Wu 2001 formula is calculated:

2

50 3

log ⎟⎟⎠

⎜⎜ ⎞

⎝ Δ ⎛

=

d ghw h

U

s

ψ (1-7)

The gradation factor, which quantifies the non homogeneity of the sediments, is given by (Chien and Wang 1999):

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

=

50 84 16 50

2 1

d d d

σ d (1-8)

The shear velocity related to the grains is computed from (van Rijn 1984c):

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

dσ

R u

U

2 ln 11 5 ,

' 2

*

(1-9)

The settling velocity intervenes into most of the equations. Its value wi for a sediment group size di is estimated from (Zhang and Xie 1993):

i i

i

i g d d

w d ν 13,86ν

096 , 86 1

,

13 ⎟⎟⎠² + Δ −

⎜⎜ ⎞

⎝

= ⎛ (1-10)

where ν is the kinematic viscosity [m²/s]. The value of the coefficients in equation (1-10) comes from a calibration on natural sediments performed recently in the department.

For uniform sediment distribution, the settling velocity calculation is usually based on the median diameter. In the formula of S.Q.Yang 2005, the influence of non-homogeneity is merged by dividing the median diameter divided by the gradation factor (dσ, wσ). When the sediment size distribution is available, it is preferable to compute the effective settling velocity with a weighted average of the settling velocity of each sediment group (Bagnold 1966):

∑

=

i i i

s p w

w (1-11)

where pi is the percentage of each group.

As for the energy slope predictions, a great disparity in the choice of the parameters, multiplicative coefficients and exponents is observed. The parameters are also related to the hydraulics or the sediments grain size. The bed forms influence is not taken into account, or implicitly if the shear velocity or the energy slope is estimated from the relations given in the previous section. Generally, the coefficients are calibrated for a specific application and need adjustments or new calibrations for another application.

A transport capacity prediction procedure is generally judged as efficient when it gives around 70% of calculated concentrations within 0,5 – 2 times the measured values (van Rijn 1984a; Zhang and others 1999). It illustrates by itself the difficulty of estimating accurately the actual concentration of sediment transported in rivers.

(c) Prediction of the bed form configurations

The prediction of bed configuration has also already interested many researchers. Most of these studies have an experimental basis although some theoretical backgrounds have also been developed (see Graf 1971 for a review). The various contributors have defined prediction criteria that can be classified according to the fact that they have brought in a graphical tool or an analytical formulation more suitable for computational purposes.

The set of graphical criteria and their associated parameters found in the literature are summarized in table 1-3. The different physical terms met in table 1-3 are as follows:

(

)

Ar = Δ is the Archimedes number, C’* = U/u’* is the non-dimensional Chezy coefficient related to the grains and Re*=u*d50/ν is the Reynolds number related to the grain, δ [m] is the thickness of the laminar boundary layer, Π [W/m²] is the specific

stream power. Concerning the parameters of Southard (U10 and h10), they correspond to the equivalent flow velocity and water depth taken at T=10° C.

The specific steam power (Bagnold 1960) is defined as:

ρgRUS

=

Π [W/m²] (1-12)

Authors Axis X Axis Y

Langbein 1942 Fr U R Znamenskaya 1969 Fr U/ws

Vanoni 1974 Fr h/d50

Southard and Boguchwal 1990

h10 U₁₀

Liu 1957 Re* U*/ws

Garde and Ranga 1963 R/d50 S/Δ Bogardi 1965 g d50/u*² d50

Athaullah 1968 h S /(Δ d50) S Hill 1971 u*cr’ d50/ν g d50³/ ν² van Rijn 1984b T ³√Ar Gilbert 1914 U R S

Garde and Albertson 1959 Fr R S/ (Δ d50) Simons and Richardson 1966 Π d50

Engelund 1965 Fr C*’ Brownlie 1981 U S/ √Δ g d50 d50/δ’

Table 1-3 Parameters found in the literature for the classification of bed-forms in graphical mode

Generally, these tools treat the bed-forms of the lower and the upper alluvial regime together. Table 1-3 contains fifteen criteria. The first four ones only use quantities relating to the grains (d50, ws) and hydraulics (U, R or h). The following six ones do not use the depth-averaged velocity but rather the energy slope either directly or through the shear velocity. The last five ones utilize both. The criterion of van Rijn 1984b can also be used directly in an analytical way. According to the author, this criterion is constructed with a large number of reliable flume and field data. Distinct zones for ripples, dunes and transition (BF 4) are observed by the author: ripples disappear for T > 3 or Ar > 1000 and dunes are observed until T = 15.

For the analytical approach, two other criteria concerning the separation between the ripples and the other bed-forms have been found in the literature. The first one was proposed by Gyr and Schmid 1989. According to them, the ripple appearance is achieved when: a sediment transport already exists and Re* < 13. The condition on the Reynolds number physically means that the bed must be smooth or in transition. This criterion has also been used by Raudkivi 1997 but with a larger interval Re* = 10 ~ 20. The second criterion has been defined by Karim 1999 as the product between the particle Reynolds and the particle Froude numbers. The criterion for the ripple occurrence is given by Karim 1999 as:

80 Re

50

*

* ≤

= Δ

d g

N U _(1-13)

From table 1-3, it still appears that only quantities related to the grains or the hydraulics have been used by both approaches. The parameter Re* is widely used by several criteria.

The inconvenient of relying on this parameter is that shear velocity must be known beforehand. This quantity is obtained for instance from vertical profile of the velocity or the energy slope. If the final objective is to determine the energy slope, it can be problematic.

1.3.3. Synthesis about these prediction tools

Many formulas for the energy slope, the transport capacity and bed form configuration have up to now been proposed in the literature. They involve several different parameters and coefficients. The laws generally come from calibration and have restricted application domains.

For the energy slope, the influence of the bed forms is sometime included in a static way through adapted value of the coefficients. Moreover, the bed forms influence is neglected into the transport capacity formulation. Inversely, the bed material load is ignored for the prediction of bed form configurations or the energy slope.

These observations probably result from the decomposition of the alluvial system into several small elemental systems treated separately with the purpose to deal with one problem at a time. During several years, researchers and engineers have adopted this strategy. Another illustrative example, which will not be developed further in this thesis, concerns the several analyses of the turbulent structure conducted on flume with fixed concrete bed forms. By freezing the bed forms, the interactions coming from the adaptation of the bed morphology are ignored. Consequently, the flow studied does not correspond to a flow met in the nature and the results obtained are then difficult to be transferred to real configurations (Bennett and others 1998; Mazumder 2000).

For these reasons, it is planned in this work to treat the alluvial river as a whole system rather than small disconnected systems with the objective to enhance the dynamic representation of the interactions into these prediction tools. It should improve their accuracies and thus the 1D numerical quantification of sediment fluxes.

1.4. Proposals for improving the prediction tools

1.4.1. A criterion including the bed material load to delineate the bed forms occurring in the lower alluvial regime

The first effort proposes to include the bed material load into a new criterion P* to delineate between the two bed form types occurring under lower alluvial regime: ripples and dunes. The effort focused on the transition from ripples and dunes as this separation is particularly crucial for the energy slope prediction and for engineers in charge of river maintenance. Indeed, they appear within the same stream power range and generate significantly different flow structures and associated alluvial resistance.

1.4.2. A common parameter for the estimation of the energy slope and the bed material load

The second effort proposes to combine the estimation of the energy slope and the transport capacity through a common closure parameter: the control factor m. This parameter is a novel concept developed mostly from flume observations. It introduces the concept of Rossiter resonance modes into alluvial hydraulics.

With the sudden diverging section at a dune crest, vortices periodically leave the brink point, move downstream and impinge on the ascending face of the following dune. The vortex impact creates pressure fluctuations and acoustic waves that influence the flow at the separation point.

Figure 1-4 Separated flow downstream a bed form

The separated flow corresponds to a self excited oscillating system with a feedback loop, similar to a cavity flow. As suggested by Kiya and others 1997, the system has different oscillating modes. It is assumed that the different oscillating modes are associated with entire values of the control factor m (Verbanck 2004a).

The non-dimensional Strouhal number Sr links the frequency of an oscillation “f” with a characteristic length “Lr” and the velocity “Ur” of the exciting external flow. According

to Levi (1983a, 1983b), the Strouhal number “Sr” is equal to 1 2π for several aerodynamic applications (such as autorotating wings, jet flows, wakes induced by a cylinder,...) and therefore is stated as “universal Strouhal law”:

π 2

= 1

≡ Ur Lr

Sr f (1-14)

For the oscillating system downstream of the bed forms, Verbanck (2004a; 2008) has suggested to relate the frequency of the vortex leaving the brink point “f detached” to the following reference scales: separation length “xr” and depth averaged velocity.

Simultaneously, Verbanck (2004a; 2008) has also defined a generalized non-dimensional Strouhal number “Srg” via the control factor m (Srg=m 2π ).

xR

m U

f_detached = _{2π (1-15)}

with m≥1.

The fundamental mode m = 1 corresponds to the universal Strouhal number and is assumed to be associated to an optimal flow configuration in term of turbulence level, energy losses and sediment transport, such as in-phase waves configurations. With such configuration, the separation zones stay bounded and there is some relaminarization of the streamlines, developing a ‘film flow’ appearance.

Figure 1-5 Film flow appearance of the in phase waves configuration in Middle Yellow River (source Yellow River Conservancy Commission - YRCC)

The first harmonics (m = 2) is assumedly associated to 2D fully developed dune configuration (Verbanck 2004b; Verbanck 2008) for which the separation zone is more wide-spread. These harmonics are similar to those met in structure acoustics ( Zima and Ackermann 2002; Rossiter 1962).

As defined by Verbanck 2008, the energy slope relation is expressed as the ratio of two frequencies:

α

β _⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

attached ached

f

S f^det (1-16)

The frequency (fdetached) of the vortices escaping from the dune crest is associated to the separation zone and energy losses. The second frequency (fattached) is associated to the local forcing of the streamlines due to the ascending stoss slope of the bed forms. The flow is locally accelerated and this effect is assumed to decrease the alluvial resistance.

BF BF BF

attached

h g

f λ

λ π λ π tanh²

= 2 (1-17)

Where λBF is the wavelength of the bed forms.

A ratio λ_BF x_r will appear if equations (1-15) and (1-17) are introduced into (1-16). To by-pass the inherent difficulty of evaluating the separation length, Verbanck 2008 has therefore proposed to work with an alternative frequency:

r

BF BF

attached

x

h g

f λ

π λ π tanh²

= 2

′ (1-18)

An alternative solution would have been to select the bed form wavelength rather than the separation length as length scale into the Strouhal number.

The coefficients of the equation (1-16) are obtained from a calibration performed by Verbanck 2008 on the laboratory data of Guy and others 1966; Znamenskaya 1969;

Kennedy 1961; Willis and others 1972.

The equation is then written as:

3 10

2 ⎟

⎠

⎜ ⎞

⎝

=⎛ m Fr_g

S π (1-19)

where Frg is the generalized Froude number defined as the ratio of U/c, with c the celerity of the gravity wave (Airy’s law).

BF BF

h c g

λ π λ π tanh ²

= 2 (1-20)

For shallow flows (h <<λBF), the equation (1-20) tends to c= gh

This definition of the energy slope is an alternative to Manning or Chezy approaches based more on the roughness of the bed or on the detailed geometry of the bed forms.

Therefore, the relation of the energy slope is referred as Vortex-Drag law to dissociate itself from the more traditional form-drag law.

The present approach relates sediment resuspension to the activity of vortices as well. For the transport capacity, the control factor m intervenes into the resuspension parameter ηsl

C* (suggested by Bagnold 1966). C* is the nondimensional Chezy coefficient. The suspension efficiency ηsl is the ratio of the power available for maintaining the particles in suspension by the specific stream power (W/m²). Whereas Bagnold 1966 assumes the resuspension parameter as a constant equal to 0,266 (for plane bed essentially), it is suggested here to write it rather as a function of the control factor m (Verbanck and others 2007):

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

= Δ ₂

* 2 3 *,

*

* 1

u u gR w C u

C ^cr

s sl

s η (1-21)

13 15

*

2 ⎟

⎠

⎜ ⎞

⎝

=⎛ C m

sl π

η (1-22)

The relation, called SVRD (Suction-Vortex Resuspension Dynamics), considered to reflect the bed material load (suspended load and bed-load) by introduction of the term between brackets in Eq 1-22, has been calibrated by Verbanck et al. 2007 on the flume data of Guy et al. 1966. For some test cases, the bed load contribution is often considered as marginal (and thus neglected) in the total particle flux, where the suspended mode of transport is largely dominant.

1.4.3. A common closure parameter for the fully coupled mathematical model

Since the seventies, numerous 1D models have been published in the literature. The influence of interactions occurring in alluvial rivers has been reckoned progressively by increasing step by step the coupling degree of the St Venant-Exner equations system. The earlier and the less coupled models neglect some terms and solve the system asynchronously.

They are often referred as two-equations models and they can be classified in two types.

The first type is the decoupled approach where the flow continuity and flow momentum are first solved treating the bed as fixed for a time step. The calculated flow variables are used to update the bed level. Then, the flow properties are computed for the next time step. Alternatively, semi-coupled approaches where this process is repeated in an iterative way have been suggested by Kassem and Chaudhry 1998. These approaches assume that the changes in bed elevation are negligible within a time step. The second family of two- equations assumes that for a time step for which the bed elevation becomes important, the water surface will have already reached its equilibrium. Therefore the equation (1-1) is transformed into a constant flow rate condition.

For systems with at least three equations (Cui and others 1996; Holly and others 1990), some terms are ignored: the third terms in the continuity equations (1-1) and (1-2), the fourth term in the sediment continuity equation (1-2). Lyn 1987 keeps the terms of equation (1-2) but still ignores the term of equation (1-1). As shown by Cao and Egashira 1999; Cao and others 2002; Lyn 1987; Morris and Williams 1996 these simplifications are valid only in a small range of the alluvial regime: low Froude number and low concentration level, when one celerity of the equation system is significantly smaller than the two others. Importantly, these simplifications have caused the failure of the traditional decoupled approaches to model, for instance, the Lower Yellow River (Cao et al. 2002).

We shall come back to this in chapters 4 – 6. The fully coupled models built by Cao and Egashira 1999; Correia and others 1992; Lai 1991; Sloff 1993, enlarge the validity range to flows with higher concentration level or Froude number. The “fully coupled”

terminology refers to the fact that the neglected terms linking the flow equations with the sediment equation are kept and that the equations system is solved in a synchronous way.

To our knowledge, the last fully coupled model based on three equations available in the literature has been proposed by Choi and Han 2003, whereas the models published afterwards (Cao 2004; Cao and others 2006; Cao and others 2007; Singh and others 2004) involve a four-equations approach. The fourth equation replaces the analytical relation used for the transport capacity. The advantage of the latter is to allow dealing with non equilibrium sediment transport. However, this method requires supplementary empirical equations for the sediment flux exchanges (deposition and entrainment), adaptation time and the bed load. The bed load relation may be necessary if the bed material load is split into suspended and bed loads. If it is not split or if only the suspension load is considered, the four-equations system becomes mathematically decoupled as shown by Cao and Egashira 2000.

Compared to the previous fully coupled models, the closure laws based on the control factor m introduce an additional coupling step, as will be elaborated in chapter 5. If the calibrations of the coefficients in equations (1-19 and 1-22) were robust enough, such that new adjustments are not required, the control factor m would stand for the sole closure parameter. This would constitute a noticeable advantage compared to the several coefficients needing to be tuned with the more traditional closure approaches. As the present objective is to test the feasibility of the new closure methodology, the equilibrium

approach is still judged preferable as it better represents the interaction and it avoids introducing disturbances from different empirical inputs.

1.5. The control factor m as the key element of the interactions

1.5.1. The different links

Since the control factor m is linked to the turbulence level and the three components, it is reckoned as the key element representing the interactions. The connection of the control factor m with the turbulence level is essential as it constitutes the central and common element of these interplays (as illustrated in figure 1-2):

• The level of turbulence activity increases or decreases the alluvial resistance.

• The turbulent fluctuations lift the particles and maintain them in suspension.

• The turbulence burst events erode the bed material and the particles deposit in separated zones.

However, the relationship between the energy slope and the control factor m directly derives from the definition of the Vortex-Drag equation. The link between the control factor m and the two other components of the interactions relies on an assumption for the bed form configuration and on a calibration for the bed material load. As two propositions feature the control factor m, these links are illustrated and discussed further in this section.

1.5.2. Link between the control factor m and the bed form configuration To calibrate the value of the coefficient in the Vortex drag law, Verbanck (2008) assumes that 2D fully-developed dunes are associated with the value m = 2 and in-phase waves with m = 1.

In a way to quantify and discuss the validity of this assumption, the repartitions of the control factor are computed for the Brownlie dataset (laboratory and field measurements).

For laboratory runs, the data of Wang and Zhang 1990; Mantz 1983, Julien and Raslan 1998, Hong and others 1984 Tanaka 1970; Jopling and Forbes 1979; Yin 1989 are added whereas only Culbertson and others 1972 has been added for the field data. The field measurement dataset is more restricted because many runs have no indications about the bed-form phases present in the riverbed at the time of the measurements. The numbers of data for each bed form code are summed up in the following table.

Bed forms laboratory field

Dunes BF3 801 106

Transition BF4 106 37

upper regime plane bed Bf5 288 62

in-phase waves BF6 41 0

breaking waves BF7 114 0

Table 1-4 Numbers of observed bed forms

The bed form wavelength has only been measured for few data. Therefore to carry out the analysis on a sufficient amount of runs, the control factor m (extracted from equation 1- 19), is calculated from the simplified Froude number (Fr = U/ gR) rather than the generalized one (Frg = U/c). As this simplification is only acceptable for shallow flows, the density distributions are plotted for the bed form from dunes “BF3” to breaking waves “BF7” (figure1-6). The hydraulic radius is directly computed as the ratio of the wet area and the wet perimeter. In sediment transport analysis, application of the Einstein-Vanoni separation technique is generally recommended to avoid the influence of lateral side walls (Knight and McDonald 1979). Moreover, the correction brought for the side walls is rather linked to the concept of boundary roughness ks, used as an intermediate parameter in the traditional (Chezy or Manning) determination of energy slope. Since we propose an alternative to this concept, the separation technique is considered not essential in our case. Use of a full hydraulic radius also corresponds to the logic of total particle mass flux, which mobilizes the total wetted area of the stream flow, and not only a fraction of it specifically associated with the sediment bed.

0 10 20 30 40 50

0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 Control factor m

Density %

Bf3 Bf4 Bf5 Bf6 Bf7

0 10 20 30 40 50

0,5 0,75 1 1,25 1,5 1,75 2 2,25 2,5 2,75 3 Control factor m

Density %

bf3 bf4 bf5

3a 3b

Figure 1-6 Distribution of the control factor m for the flumes (3a) and for the rivers (3b)

For the laboratory in-phase and breaking waves, the distribution has effectively a dominant mode around m = 1: about 80 and 90% of the data are respectively within m = [0,875 – 1,125]. For upper regime plane beds and the transitions, a part of the distributions for the laboratory data is also concentrated on both sides of m = 1: about 50% and 30% within m = [0,875 – 1,125]. The dunes are distributed on a larger range of the control factor m and reach higher value. From the flume distribution, one can effectively associate in-phase or breaking waves configuration with the fundamental mode m = 1. Inversely, the value m = 1 could not be directly associated to these configurations as it could also correspond to an upper regime plane bed.