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Thèse de doctorat/ PhD Thesis Citation APA:
Huybrechts, N. (2008). Fully coupled 1D model of mobile-bed alluvial hydraulics: application to silt transport in the Lower Yellow River (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences appliquées – Mécanique, Bruxelles.
Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/210484/4/3afc64da-db07-4b9d-bce3-65b547016283.txt
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uiNivm^ûiir, un dis ^ u XELLES
Faculté des Sciences Appliquées ULB
Fully coupled ID 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
Université Libre de Bruxelles
Faculté des Sciences Appliquées ULB
Fully coupled ID 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 prédiction of the fine and non-cohesive bed material load in alluvial rivers, especially during high intensity épisodes during which sédiment beds are strongly remobilized. For this reason, we attempt to reduce the major inaccuracy sources coming from the alluvial résistance 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 résonance modes of the separated flow downstream the bed form crest.
To further improve the représentation 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 équations forming the System are solved synchronously and that the tenus 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 sût flushing experiment is to reverse the aggradation trend of the Lower Yellow River which, in the last décades, 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 fïrst been confronted to river datasets. The confrontation has revealed that the SVRD law becomes less suitable for fine sédiment fluxes (ratio of water depth over médian particle size > 5000). Therefore, a modified équation SVRD-2 has been built to enlarge the validity range.
The suitability of the SVRD-2 équation to predict fine sédiment fluxes has been tested on data available from several hydrometric stations located in the meandering reach of the LYR: historical observations and measures collected dming 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 équation 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 évolution of the control factor m has effectively been extracted and it has been shown that it varies along the reach. At Aishan, the évolution of the control factor m corresponds to the évolution 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=l. The time évolution at Luokou behaves differently to the one at Aishan, but remains in agreement with m évolution patterns observed historically for the river section flowing round Jinan City walls. For Luokou, the highlighted différences may corne from three dimensional effects coming from the meander bend upstream the station.
Generally, the results obtained for the hydraulics, the sédiment 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 hed élévation are very sensitive to the value of the control factor m as it influences most of the terms of the bed material load équation (SVRD-2).
The major remaining diffîculties are, firstly, to deal with the rapid transients for which the model is less suitable and, secondly, to improve the prédiction of the value of control factor m. Before paying more attention into the transients, enhancements conceming 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 prédiction of the bed or the cross section évolutions dépend directly on the quality of the prédiction of the sédiment concentration and the hydraulics, one should first improve these aspects. To perform this study, more information about the water levels or sédiment 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 Gérard 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.
Ackno wledgments :
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, Contact 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.
Abstract 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 prédiction
laws 2
1.3.1. Définitions of the three components 2
1.3.2. Prédiction laws 5
1.3.3. Synthesis about these prédiction tools 10 1.4. Proposais for improving the prédiction tools 11 1.4.1. A criterion including the bed material load to delineate the bed forms occurring in the lower alluvial régime 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 conceming the rippled bed configurations 18 1.5.4. Remarks conceming 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 régime 24
2.1. Introduction
2.2. Evolution of the flow résistance with the alluvial régime
24
25
2.3.1. Définition 26
2.3.2. Transition zone 28
2.3.3. Remarks conceming the définition ofEv 28 2.3.4. Remarks conceming the importance of the viscosity and médian
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 sédiment size 33
2.6.2. Behaviour for fine sédiment (d50<0,1mm) 33 2.6.3. Behaviour for coarse sand (d50>0,75mm) 34 2.7. Comparison with other criteria fi'om the literature 35
2.8. Conclusions 37
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. Présentation 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 équation 44
3.3.5. Remarks about the modified approach 46
3.4. Conclusions 47
4.1. Introduction 48
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. Présentation of the data 51
4.2.4. Remarks conceming the cohesiveness of the material 52
4.2.5. Range of the control factor m 55
4.2.6. Remarks conceming the values of the control factor m 57 4.2.7. Remarks conceming the Luokou hydrometric station 61 4.2.8. Remarks conceming the settling velocity and the médian 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 ffom 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. Spécifications for the test cases in practice 73
5.4. Study of the characteristics 74
5.4.1. Previous work:
5.4.2. Quasi linear form
74
74
5.5. Discretization of the équations 78
5.5.1. The linearized Riemann solver 78
5.5.2. Modifications of the CLA WP ACK software 78
5.6. Déposition 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
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 conceming the applicability of the numerical model to
the current test case 97
6.4. Présentation of the results 98
6.4.3. Bed material load 102 6.4.4. Bed évolution at Aishan and Luokou 103 6.4.5. Bed élévation 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. Présentation 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 ri vers 121
8.1.1. Context and closure issue 121
8.1.2. The three efforts proposed to enhance the prédiction tools 121
8.1.3. Remarks conceming the methodology 122
8.1.4. The procedure 123
8.2.1. The prédiction of the bed form configurations occurring in the
lower alluvial régime 124
8.2.2. Prédiction 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 conceming 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 ffom experimental
data 130
8.3.2. Improvements of the model and further validations 131
Référencés
Appendix: detailed tables
ai
coefficients of a polynom
c propagation celerity of the gravity
[m/s]
waves
dso
médian diameter of the particle [m]
di diameter of the particle of a grain
size class [m]
da
médian diameter of the particle
devided by the gradation factor [m]
t detacfaed
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/s^]
h water depth [m]
ki,k2,k3 coefficient of the energy slope
équation (Correia) [-]
ks
roughness [m]
m control factor m [-]
[s/m"^]
n Manning coefficient
P pressure [N/m^]
Pi percentage [-]
q flow rate per unit width [mVs]
qs sédiment flow rate per unit width [mVs]
r discrepancy ratio [-]
q
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]
Wff
settling velocity corrected by the
gradation factor [m/s]
X
streamwise coordonate [m]
average length of the séparation zone [m]
y coordinate along the water depth [m]
A wet area
[m^]Ar Archimedes number [-]
B coefficient B [-]
C Chezy coefficient []
Ci estimated concentration [kg/m^*]
Cm measured concentration [kg/m^]
Q
transport capacity [-]
Cv depth average volumétrie
concentration [-]
Cw depth average massive
concentration [kg/m’]
C. non dimensional Chezy [-]
coefficient related to the grains [-]
Ev viscous power [W/m"]
Fr simplified Fronde number [-]
Frg generaiized Fronde nnmber [-]
G génération of the tnrbnlent energy [W/m^]
Gs, power needed for the snspension [W/m^]
H total energy head [m]
L length of the reach [m]
L. period of the sinns curve nsed for
the control factor m predictor [-]
N. criterion of Karim [-]
P wet perimeter [m]
P. criterion P. [-]
Q flow rate [mVs]
Qs sédiment flow rate [mVs]
R hydranlic radins [m]
Re. Reynolds nnmber related to the
grains [-]
S energy slope [-]
s„ channel slope [-]
Sr Stronhal nnmber
Srg generaiized Stronhal number
T Transport stage parameter of Van
Rijn [-]
U depth average velocity [m/s]
W channel width [m]
Y bed élévation [m]
Z Zanke average [-]
/3 Rouse number [-]
X porosity [-]
X
bfbed form wavelength [m]
Vsl suspension efficiency [-]
Vs\ c* resuspension parameter [-]
P density [kg/m’i
Ps sédiment density [kg/m"i
a gradation factor [-]
V
kinematic viscosity [mVs]
T
shear stress [N/m^]
A excess of relative density [-]
n stream power [W/nP]
1. Context and objectives
1.1. Introduction
The overall objective of the thesis is to improve the prédiction of non-cohesive particle fluxes in alluvial rivers, especially during high intensity épisodes during which sédiment beds are strongly remobilized. The knowledge of sédiment 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 présent study is focused especially on the prédiction of the fine sédiment particles fluxes that are causing the highest environmental concem in terms of pollutant transfer. More attention is thus paid at the sédiments with grain size smaller than 0,25 mm.
The prédiction of the sédiment 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 inaecuracy sources affecting the results quality (Cao and Carling 2002), we investigate ffom where they can origin. From this examination, it is pointed out that the interactions characterizing the alluvial rivers hâve 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 équations (St. Venant-Exner): the mixture continuity équation (water and sédiment), the sédiment continuity équation and the mixture momentum équation. The main variables are represented in the following diagram.
Figure 1-1 Stream wise profile and main variables (redrawn from Cao and Egashira 2000)
^ d{hU) ^
dt dx dt (1-1)
(1-2)
(1-3) While the bed porosity X is considered as input data, the équations System holds five unknowns: the water depth h [m], the depth-averaged velocity U [m/s], the bed élévation Y [m], the energy slope S and the bed material load per unit width qs [mVs].
If the depth-averaged velocity U, the bed élévation 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 définitions 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 prédiction of the bed form configuration can also be necessary to estimate these quantifies, the different bed forms and associated prédiction criteria are detailed too.
1.3. The three components of the interactions and their associated prédiction laws
1.3.1. Définitions of the three components
Alluvial rivers are characterized by interactions between the flow, the sédiment transport and the bed morphology: the sédiment transport is controlled by the flow structure whereas the flow structure dépends on the bed geometry that is itself influenced by the sédiment 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 résistance
The hydraulic losses are commonly represented by the energy slope S ftom the Bernoulli équation;
S = dH
dx H = Y +
Pg
(1-4)
where H is the total energy head [m] and p the pressure [N/m^].
(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 signifîcant quantifies 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 continuons contact with the bed; the particles maintain in the water column by lift effects ftom the turbulence.
For the treated particle sizes, the sédiments 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 sédiment load that can be carried by a flow in equilibrium (dqj/dx
= 0). Moreover, we ideally focus on configurations without cohesive effects. The cohesive sédiments are characterized by dominant effects due to electrochemical répulsive and attractive forces between particles. The cohesive sédiment particles do not behave as individual particles anymore and rather tend to collapse and to form flocs. The border between cohesive and non cohesive sédiments is not clearly defined and is generally site-specific. In practice, the sédiments 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 (sût) are usually reckoned as between cohesive and non-cohesive sédiments.
(c) Bed morphology adaptation
For the non cohesive sand bed we presently consider, different type of bed-forms (BF)
occurs through the alluvial régime. They are usually categorized by the code defined by
Simons and Richardson 1966: BF 1 lower régime plane bed, BF 2 ripple, BF 3 dune, BF
4 transition ftom lower to upper alluvial régime, BF 5 upper régime plane bed, BF 6 in-
phase waves (often called antidunes), BF 7 breaking waves (or breaking antidunes) and
BF 8 chute and pool.
watcr surface
boil boil
water surface
bf 5 Upper régime 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 régime is characterized by large alluvial résistance and small sédiment transport whereas small flow résistance and large sédiment transport typify the upper régime. Two kinds of bed forms are observed for the lower alluvial régime, namely ripples and dunes. The geometry of these bed-forms is quite similar: an as)mmetric shape with a gentle stoss slope, a sharp crest and a steeper lee side (Simons and Richardson 1966). Nevertheless, they présent different geometrical dimensions: the length and the height of the ripples scale with the sédiment 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 accélération of the velocity favorable to the sédiment érosion process. The lee face is marked by the flow séparation that cornes from an abrupt expansion of the sections. The low velocities of the séparation are associated with sédiment déposition. In the dune case, the séparation 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 (Beimett and Best 1996). Consequently, the flow perturbations stay confined near the bed and the energy losses coming from the séparation zones are less important than for the dunes.
For the upper alluvial régime, 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 atmosphère is in-phase with the bed forms and their shapes look both as sinusoïdal 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. Prédiction laws
(a) Prédiction of the energy slope
Several laws hâve 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, ail the laws are written as a product of a multiplicative coefficient ki and the two ratios:
S = k,
*2
U j U^J
ki
(1-5) Where R is the hydraulic radius and di is the sédiment size (generally the médian diameter dso).
The médian diameter dso 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 sédiment size and the second is the square of the Froude number (Fr). The influences of these ratios are weighted by the coefficients kz and IC3 whose values are given in table 1-1.
Formula Validity k, k2 kî
Brownlie Lower régime Fr<l 0,02050-°-'"" -0,075 1,286
1981 Upper régime Fr>l -0,281 1,086
0,088<d5o(mm)<2,8 0,01250- ’
Kishi and Dunes I: ¥,<0,02 0,0052 1 3
Kuroki Dunes II: 0,02 0,013 0 1
1974 Trans: 0,02 Z‘"^<Ys:^,02 0,018 (pjpf^ -3/7 1/7 Plane beds: 0,02 Z^^<Ys<0,07
7 2/5
0,021 -1/3 1
Antidunes:
Ys>0,07 Zj^^0,375<d5o(mm)<3,6
0,021 (pjpy 3,46-'( g ,7G3)
1/5 3
Garde and Dunes and Ripples: Gi Fr<0,33 -1/3 1
Ranga-Raju
1966 Transition 0,33 ^iFr<l [3,46-2,83 log(3G,Fr)]-' xkM)
-1/3 1
Antidunes Gi Fr > 1 -1/3 1
0,01 l<d5o(mm)<5,2 Engelund Dunes: Fr<l
qm
{
p!
pT 0,043 2,15
1967 Antidunes: Fr>l -1/3 1
0,19<d5o(mm)<0,93 0,021
Griffiths Gravel bed: Fr<l 0,026 -0,43 0,66
1981 12<d5o(mm)< 152
Table 1-1 Example of energy slope laws (extracted from Correia 1992)
The dimensionless parameters in table 1-1 are defined by (Yalin 1977):
K = = Rjd,^ , G,Fr = q/^jAgR^ ■
is the sédiment density (equal to 2650 kg/m^ for quartz sands, as is generally the case in this monograph), A is the relative excess of density, Gi and G2 are fünctions of djo defined by Garde and Ranga-Raju 1966.
In this table, it is observed that the values of the parameters ki, k2, k3 vary according the alluvial régime and the bed form configurations. The laws differ ffom one author to another and their domains of application are restricted. The energy slope is assumed to dépend only on the flow, the sédiment grain size and the bed forms configurations: the sédiment transport has not been included into this analysis. A parameter which is also often met is the roughness ks, generally estimated ffom either the grain roughness or a roughness induced by the bed form geometry (Karim 1999; van Rijn 1984b; Yang and others 2005).
For a spécifie application the difficulty is thus to select the most convenient law ffom the list.
(b) Prédiction of the bed material load
As many formulas hâve been developed for the transport capacity, the présent review only focuses on some équations représentative of the different families.
The first family is formed by formulas involving the Einstein bed load parameter and the Shields stress parameter, as the équations 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 (UVgRws), Yang C.T. 1973 or Bagnold 1966 (US/Ws) and van Rijn 1984a transport-stage parameter T=[(u*)^-(u*cr)^]/(u*cr)^. More recently, Yang S.Q. 2005 has built a formula up which combines the different parameters met in the other families. In these équations Wg 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 sédiment is about to move (équivalent to the so-called Shields condition for incipient transport of non-cohesive sédiment 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 équations hâve been written in their volumétrie concentration form to highlight their similarities or différences:
(1-6)
Authors Family Engelund
and Hansen 1967
Engelund
Molinas and Wu 2001
Velikanov
Celik and Rodi 1991
Bagnold
Zhang and Xie 1993
Velikanov
Yang S.Q.
2005
combined
Formula
C. = 0,05 U ul
C, =
Q = 0,034
1,43 (o,86 + 7i^) 2650 (0,016+
(k \
0,06 “ T1- Kh]
U
c.=^
P.
( w
iPs - p)gh W,
C =12,5 1
ghw^j
•2 2
r
p^-pgUh w„
Table 1-2 Examples of some bed material load formulas
where ks is the roughness [m], T is the bed shear stress [N/m^], Ws the settling velocity [m/s] and u’* is the shear velocity related to the grains [m/s].
The \J/ parameter intervening into the Molinas and Wu 2001 formula is calculated:
V^ =
Aghw^
r U V log — V ^50 y
(1-7)
The gradation factor, which quantifies the non homogeneity of the sédiments, is given by (Chien and Wang 1999):
U (1-8)
The shear velocity related to the grains is computed ffom (van Rijn 1984c):
-^ = 2,51n
U,
( 117?'
Id
a /(1-9)
The settling velocity intervenes into most of the équations. Its value Wi for a sédiment
group size dj is estimated ffom (Zhang and Xie 1993):
V d, J d;
where v is the kinematic viscosity [mVs]. The value of the coefficients in équation (1-10) cornes from a calibration on natural sédiments performed recently in the department.
For uniform sédiment distribution, the settling velocity calculation is usually based on the médian diameter. In the formula of Yang S.Q. 2005, the influence of non-homogeneity is merged by dividing the médian diameter divided by the gradation factor (d<,, w<,). When the sédiment size distribution is available, it is préférable to compute the effective settling velocity with a weighted average of the settling velocity of each sédiment group (Bagnold 1966):
/
where pi is the percentage of each group.
As for the energy slope prédictions, 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 sédiments 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 spécifie application and need adjustments or new calibrations for another application.
A transport capacity prédiction procedure is generally judged as efficient when it gives aroimd 70% of calculated concentrations within 0,5 - 2 times the measured values (van Rijn 1984a; Zhang and others 1999). It illustrâtes by itself the difficulty of estimating accurately the actual concentration of sédiment transported in rivers.
(c) Prédiction of the bed form configurations
The prédiction of bed configuration has also already interested many researchers. Most of these studies hâve an experimental basis although some theoretical backgrounds hâve also been developed (see Graf 1971 for a review). The varions contributors hâve defined prédiction criteria that can be classified according to the fact that they hâve 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 = is the Archimedes number, C’* = U/u’* is the non-dimensional Chezy
coefficient related to the grains and Re»=u*d5o/ï' is the Reynolds number related to the
grain, ô [m] is the thickness of the laminar boundary layer. Il [W/m^] is the spécifie
stream power. Conceming the parameters of Southard (Uio and hio), they correspond to the équivalent flow velocity and water depth taken at T=10° C.
The spécifie steam power (Bagnold 1960) is defined as:
n = pgRUS [W/m^] (1-12)
Authors Axis X Axis Y
Langbein 1942 Fr UR
Znamenskaya 1969 Fr U/Ws
Vanoni 1974 Fr h/dso
Southard and Boguchwal 1990
^10
Liu 1957 Re* U*/Ws
Garde and Ranga 1963 R/dso S/A
Bogardi 1965 g djo/u.^ dso
Athaullah 1968 h S/(A dso) S
mil 1971 U*cr’ dso/r gdsoVr^
van Rijn 1984b T ^■vAr
Gilbert 1914 UR S
Garde and Albertson 1959 Fr R S/(A dso)
Simons and Richardson 1966 n dso
Engelund 1965 Fr C.’
Brownlie 1981 US/vAgdso d50/ô’
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 régime together. Table 1-3 contains fifteen criteria. The first four ones only use quantities relating to the grains (dso, Wj) 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 conceming the séparation between the
ripples and the other bed-forms hâve 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 sédiment 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:
N, = Re. U
yj S^^so
<80 (1-13)
From table 1-3, it still appears that only quantities related to the grains or the hydraulics hâve been used by both approaches. The parameter Re* is widely used by several criteria.
The inconvénient of relying on this parameter is that shear velocity must be known beforehand. This quantity is obtained for instance ffom vertical profile of the velocity or the energy slope. If the final objective is to détermine the energy slope, it can be problematic.
1.3.3. Svnthesis about these prédiction tools
Many formulas for the energy slope, the transport capacity and bed form configuration hâve up to now been proposed in the literature. They involve several different parameters and coefficients. The laws generally corne from calibration and hâve 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 prédiction of bed form configurations or the energy slope.
These observations probably resuit ffom the décomposition 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 hâve adopted this philosophy. Another illustrative example, which will not be developed further in this thesis, concems the several analyses of the turbulent structure conducted on flume with fixed concrète bed forms. By freezing the bed forms, the interactions coming ffom 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
représentation of the interactions into these prédiction tools. It should improve their
accuracies and thus the ID numerical quantification of sédiment fluxes.
1.4. Proposais for improving the prédiction tools
1.4.1. A criterion includins the bed material load to delineate the bed forms occurrins in the lower alluvial resime
The fïrst 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 régime: ripples and dunes. The effort focused on the transition from ripples and dunes as this séparation is particularly crucial for the energy slope prédiction 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 résistance.
1.4.2. A common parameter for the estimation of the enerev 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 ffom flume observations. It introduces the concept of Rossiter résonance 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 créâtes pressure fluctuations and acoustic waves that influence the flow at the séparation point.
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 extemal flow. According
to Levi (1983a, 1983b), the Strouhal number “Sr” is equal to l/2;r for several aerodynamic applications (such as autorotating wings, jet flows, wakes induced by a cylinder,...) and therefore is stated as “universal Strouhal law”:
Sr = fLr
Ur iTt (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: séparation 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/2;r ).
/c
m U
detached 2 k —
X r with m>\.
(1-15)
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 sédiment transport, such as in-phase waves configurations. With such configuration, the séparation 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 harmonies (m = 2) is assumedly associated to 2D fiilly developed dune
configuration (Verbanck 2004b; Verbanck 2008) for which the séparation zone is more
wide-spread. These harmonies 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;
s^p
^ f YJ àçXached\ f
attached j(1-16)
The frequency
(fdetached)of the vortices escaping ffom the dune crest is associated to the séparation 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 résistance.
'■BF f.
_g_
2n tanh iTài
''BF attached
^BF
Where X bf is the wavelength of the bed forms.
(1-17)
A ratio appear if équations (1-15) and (1-17) are introduced into (1-16). To by-pass the inhérent difficulty of evaluating the séparation length, Verbanck 2008 has therefore proposed to work with an alternative frequency;
/;<
s 27di A„.^tanh —
2;r ''BF
attached
(1-18)
An alternative solution would hâve been to select the bed form wavelength rather than the séparation length as length scale into the Strouhal number.
The coefficients of the équation (1-16) are ohtained 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 équation is then written as:
S =
Fr10
y
(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).
c = (1-20)
y ^
bfFor shallow flows (h «X bf ), the équation (1-20) tends to c =
This définition 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 ffom the more traditional form-drag law.
The présent approach relates sédiment resuspension to the activity of vortices as well. For the transport capacity, the control factor m intervenes into the resuspension parameter t J si
C» (suggested by Bagnold 1966). C* is the nondimensional Chezy coefficient. The suspension efficiency ï/si is the ratio of the power available for maintaining the particles in suspension by the spécifie stream power (W/m^). 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 fimetion of the control factor m (Verbanck and others 2007):
ul r
b^w^gR 1-^1 V M. y
TlüC. =
13
7on 7
(1-21)
(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 fullv coupled mathematical model
Since the seventies, numerous ID models hâve 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 équations 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 classifïed 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. Altematively, semi-coupled approaches where this process is repeated in an itérative way hâve been suggested by Kassem and Chaudhry 1998. These approaches assume that the changes in bed élévation are negligible within a time step. The second family of two- equations assumes that for a time step for which the bed élévation becomes important, the water surface will hâve already reached its equilibrium. Therefore the équation (1-1) is transformed into a constant flow rate condition.
For Systems with at least three équations (Cui and others 1996; Holly and others 1990), some terms are ignored: the third terms in the continuity équations (1-1) and (1-2), the fourth term in the sédiment continuity équation (1-2). Lyn 1987 keeps the terms of équation (1-2) but still ignores the term of équation (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 régime: low Froude number and low concentration level, when one celerity of the équation System is significantly smaller than the two others. Importantly, these simplifications hâve caused the failure of the traditional decoupled approaches to model, for instance, the Lower Yellow River (Cao et al. 2002).
We shall corne 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 équations with the sédiment équation are kept and that the équations System is solved in a synchronous way.
To our knowledge, the last fully coupled model based on three équations 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 équation replaces the analytical relation used for the transport capacity. The advantage of the latter is to allow dealing with non equilibrium sédiment transport. However, this method requires supplementary empirical équations for the sédiment flux exchanges (déposition 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 équations (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
présent objective is to test the feasibility of the new closure methodology, the equilibrium
approach is still judged préférable as it better represents the interaction and it avoids introducing disturbances from different empirical inputs.
1.5. The control factor m as the kev eiement 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 eiement representing the interactions. The connection of the control factor m with the turbulence level is essential as it constitutes the central and common eiement of these interplays (as illustrated in figure 1-2):
• The level of turbulence activity increases or decreases the alluvial résistance.
• 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 dérivés from the définition of the Vortex-Drag équation. 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 confïsuration
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 répartitions 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 1989are added
whereas only Culbertson and others 1972 has been added for the field data. The field
measurement dataset is more restricted because many runs hâve no indications about the
bed-form phases présent 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 régime 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 forai 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 ffom équation 1- 19), is calculated from the simplified Froude number (Fr = U/-y[^) 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 ffom dunes “BF3” to breaking waves “BF7” (figure 1-6). The hydraulic radius is directly computed as the ratio of the wet area and the wet perimeter. In sédiment transport analysis, application of the Einstein-Vanoni séparation technique is generally recommended to avoid the influence of latéral 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 Maiming) détermination of energy slope. Since we propose an alternative to this concept, the séparation technique is considered not essential in our case. Use of a flill 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 sédiment bed.
-*-Bf3 -X-Bf4 ^-Bf5 -*-Bf6 -«-Bf? -A-bf3-X-bf4-e-bf5
