Bed topography in shallow river bends

( , «

BED TOPOGRAPHY IN

SHALLOW RIVER BENDS

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van

de Rector Magnificus, prof .dr. J . M . Dirken, in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Dekanen o p dinsdag 22 september 1987 te 16.00 uur precies

door

Kim Wium Olesen geboren te Lunde, Denemarken

civiel ingenieur

1987

Dit proefschrift is goedgekeurd door prof.dr.ir. J.P.Th. Kalkwijk

CONTENTS

Summary

Acknowledgement

Notation

1 Introduction

1.1 Relevance of the investigation 1.2 Previous work

1.3 The present investigation

2 Steady flow computation in river bends 2.1 Introduction

2.1.1 Physical description 2.1.2 Previous work

2.2 A mathematical model of the flow 2.2.1 Co-ordinate system

2.2.2 Governing differential equations 2.3 Vertical distribution of the flow

2.3.1 The longitudinal flow velocity components 2.3.2 The transverse flow velocity

2.3.3 The vertical flow velocity 2.3.A Developing secondary flow

2.4 A depth integrated model of the flow 2.4.1 Analysis of the depth integrated flow mod 2.5 Verification of the model

2.6 Conclusions

3 Sediment transport

3.1 Introduction

3.2 3.2.1 3.3 3.4 3.5 Transport capacity Bed slope effect

Direction of the sediment transport on a sloping bed Non-uniform sediment

Conclusions and discussion

75 79 82 91 100 4 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.4 4.5 4.5.1 4.5.2 4.5.3 4.6 4.6.1 4.6.2 4.6.3 4.7

Analyses of the morphological model Introduction

The axi-symmetric solution Steady first order solution

Linear solution of the simplified model Extended analysis

Discussion of the steady linear solution Sensitivity analysis

Linear stability analysis

Alternate bars, meandering and braiding Linear bed stability analysis

Linear steady first order analysis Linear planform analysis

Further analyses

Influence of sediment gradation Influence of suspended load Influence of variable roughness Conclusions and discussion

103 103 106 108 109 114 118 125 128 134 135 138 140 146 146 148 152 154 5 Integration procedure 158 5.1 Introduction 158 5.2 Integration procedure for the flow model 158

5.3 Integration procedure for the bed level model 163

V e r i f i c a t i o n of t h e model 170

6.1 Introduction 170 6.2 Simulation of the Experiments Tl, T2 and T3 172

6.2.1 Experimental and numerical set-up 172

6.2.2 Calibration of the model 175

6.2.3 Discussion 179 6.3 Simulation of the Experiments T4 and T20 181

6.3.1 Experimental and numerical set-up 181

6.3.2 Calibration of the model 185

6.3.3 Discussion 188 6.4 Simulation of the Experiments T8, T15 and T19 189

6.4.1 Experimental and numerical set-up 189

6.4.2 Calibration of the model 194

6.4.3 Discussion 199 6.5 Simulation of the experiments with non-uniform sediment 203

6.5.1 Relevance 203 6.5.2 Experimental set-up 205

6.5.3 Numerical set-up 209 6.5.4 Simulation of the Experiment T6 213

6.5.5 Simulation of the Experiment T5 222 6.5.6 Simulation of the Experiment T7 226

6.5.7 Discussion 227 6.6 Conclusion 229 7 Conclusions 232 Appendix A 237 Appendix B 241 References 252 Summary in dutch 263

SUMMARY

A mathematical model of the flow, bed topography and grain size variation in curved alluvial rivers is presented and analyzed with the purpose of obtaining insight into the governing physical processes. The model consists of three sub-models, viz. a depth-integrated flow model, a sediment transport model, and a sediment budget model.

A sensitivity analysis of the complex three dimensional flow pattern in a river bend is carried out by applying different eddy viscosity models in the appropriate equations. The flow can be considered to consist of a longitudinal flow component (main flow) and a transverse circulation (secondary flow). This analysis leads to the conclusion that the bed shear stress due to flow curvature is very sensitive to the eddy viscosity model applied. The secondary flow gives rise to significant redistribution of the main flow. This essential three dimensional effect has been maintained in the depth-integrated model by assuming similarity of the vertical distribution of both main and secondary flow. The redistribution effect of the secondary flow is especially dominant in cases with small width-depth ratios, low alluvial roughness, and vertical side walls.

In the sediment transport model only bed load is considered. As the main purpose of the investigation is to investigate the physical mechanism only a couple of fairly simple sediment transport rate models are applied. Most of the available models for the direction of the sediment transport are tested. In case of graded sediment the transport of several grain size fractions can be simulated. The sediment budget model simulates the bed topography changes and, if transport of more fractions is considered, also the changes of the grain size characteristics.

The bed topography development in curved channels is investigated by means of the linear and the axi-symmetric solution of the model. The axi-symmetric solution is obtained by neglecting all stream wise variation in the mathematical model, i.e. this solution depends solely on local conditions. The analysis leads to several conclusions. The most important one is that the bed topography cannot be predicted from local conditions only. In parts of the curved channel a significant part of the lateral bed slope is caused by overshoot effects due to redistribution of the flow and the sediment transport. Moreover, it is concluded that suspended load has a very significant influence on the bed topography, as it gives rise to a larger axi-symmetric bed slope and an increased overshoot. It is also shown that the result of the mathematical model is very sensitive to the description of the alluvial roughness applied in the model.

The linear solution of the mathematical model is used to analyse the development of alternating bars and meanders in alluvial channels. This analysis leads to the conclusion that the linear stability analysis, which often has been used to explain the development of meanders, only applies to fast migrating bed forms of the alternating bar type. Meanders migrate with a speed negligible compared to the celerities of bars predicted by the mathematical model. This implies that a steady state analysis applies to meanders. Results of a steady state analysis of the linear mathematical model are in fair agreement with meander lengths measured in natural rivers.

Finally, computational results of the mathematical model are compared with measured data from various laboratory flumes. The flume experiments considered have been selected in such a way that a variety of hydraulic conditions and geometries are represented.

Also experiments with non-uniform sediment are simulated. The general conclusion is that the mathematical model, with proper calibration, can simulate the bed topography and flow distribution very well. Also the over-all grain size distribution of the bed can be simulated qualitatively, whereas quantitative prediction is impeted by shortcomings in the description of the sediment transport per size fractions close to initiation of motion. The main difficulty is that more combinations of calibration factors result in nearly the same solution of the mathematical model, so no unambigeous calibration can be obtained. Nevertheless, experiments with comparable hydraulic conditions can be simulated with the same calibration factors. This is very encouraging for practicable applications of the model.

ACKNOWLEDGEMENT

The present investigation has been carried out at the Laboratory of Fluid Mechanics of the Delft University of Technology, Dept. of Civil Engineering under guidance of Professor J.P.Th. Kalkwijk. I want to thank the staff of the Laboratory of Fluid Mechanics for making my stay in Delft so pleasant.

The investigation is a part of the river bend investigation project of the River Research Group in the joint hydraulics research program T.O.W. (Toegepast Onderzoek Waterstaat) in which Rijkswaterstaat, the Delft Hydraulics Laboratory and the Delft University of Technology participate. The many meetings and fruitful discussions with my friends and colleagues in the river bend task group of T.O.W., especially N. Struiksma, H.J. de Vriend and C. Flokstra, is gratefully acknowledged.

Finally, I want to thank Ineke for her encouragement and for the laborious typing work.

NOTATION Eq. a b CD d dm d50 e 8 h ±s k ki kr kuu ksn m n P Pi Pi P°i q

exponent in generalized transport direction exponent in generalized transport magnitude drag coefficient

characteristic grain size of sediment mean grain size of sediment

median grain grain size of sediment

model model

coefficient in generalized sediment transport model acceleration due to gravity

water depth

secondary flow intensity wave number in s-direction imaginary part of k

real part of k

velocity distribution coefficient secondary flow convection factor coefficient in velocity profile model transverse space co-ordinate

pressure

fraction of grain size i in active layer avaraged fraction of grain size i

fraction of grain size i in passive layer coefficient in model for decay of secondary flow

3.18 3.7 3.31 3.3 3.4 3.13 2.4 2.15 2.31 4.15 4.16 4.16 2.47 2.48 2.2 2.1 2.2 3.27 3.26 3.27 intencity 2.67 s longitudinal curvelinear space co-ordinate 2.1

t time co-ordinate 3.1 u velocity component in s-direction 2.2

u depth averaged velocity componet in s-direction 2.18

Uj friction velocity 3.36 v velocity component in n-direction 2.2

Eq.

v depth averaged velocity component in n-direction 2.43

w velocity component in z-direction 2.2

wf fall velocity of sediment 4.48

x cartesian co-ordinate 4.35 y cartesian co-ordinate 4.35 z vertical co-ordinate 2.1 z level at which the flow velocity vanishes 2.19

o

z, bed level 3.1

A eddy viscosity 2.8 B width of channel 2.60 C Chézy roughness coefficient 2.18

D numerical dispersion coefficient 6.8

E bank erosion coefficient 4.38 G cross-sectional mean value of the gravitational term 3.18

I water surface slope 2.68 L mixing length 2.15 M Manning roughness coefficient 4.54

P pressure at rigid lid (water surface) 2.12 Q river discharge

R local curvature of co-ordinate system 2.1

R streamline curvature 2.50 S sediment transport per unit width 3.3

S sediment transport per unit width in n-direction 3.1 S sediment transport per unit width in s-direction 3.1

Eq.

l I

r

I _max

/ dimensionless velocity profile in s-direction 2.18 / dimensionless velocity profile in n-direction 2.28 ^ dimensionless velocity profile in z-direction 2.36 _w

9 function describing gradual adaptation of secondary

flow 2.38

■*■ =V-\, imaginary unit 2.58

■i secondary flow intensity 2.29

k-n dimesionless wave number in n-direction A.23 k dimesionless wave number in s-direction 4.23

^ =kJkr>t relative wave number 4.24

imaginary part of £ 4,30

real part of £ 4.30

relative wave number with max. amplification in linear stability analysis

m mode number in linear analysis 4.24

•2 =z/h, dimensionless vertical co-ordinate 2.15

20 =z0/h 2.19

a =Kg/(C ) , dimensionless roughness parameter 2.30

8 =-R/(h tan6), coefficient in model for the

direction of the bed shear stress 2.31 Y numerical amplification factor B.5

~Yf dynamic friction coefficient 3.13

6 angle between streamline and bed shear stress 2.42

e =g/(C'kB) 4.25

H thickness of the transport (active) layer 3.28

< =0.4, von Karman's constant 2.15

A length scale in flow model 2.58 A , adaptation length of the bed shear stress due

to secondary flow 2.42 A adaptation length of bed topography 3.25

Eq.

A adaptation length of suspended load 4.48 A adaptation length of main flow 2.59 u ripple factor in Meyer-Peter & Muller sediment

transport model 3.8

5 =as/h 2.39 P mass density of water 2.2

tn z shear stress in the n-z plane 2.7

t shear stress in the s-z plane 2.6 T, bed shear stress in n-direction 2.31

T[jS bed shear stress in s-direction 2.31

<t> dimensionless complex celerity in linear

stability analysis 4.33 X ratio between suspended load and the total

sediment transport 4.49 <|J angle between sediment transport vector and s-lines 3.2

w polar co-ordinate 2.1

T relative pore volume 3.5 A =1.65, relative density of sediment 3.3

An space step in n-direction in numerical model As space step in s-direction in numerical model At time step in numerical model

9 Shields parameter 3.4 0' Shields parameter related to skin friction 3.8

critical Shields parameter 3.8 = Ag u sp / Aw, ratio between the adaptation lengths

of the suspended load and of the main flow 4.53 $ dimensionless sediment transport parameter 3.3 fi relaxation coefficient in integration procedure

for the flow model 5.1 c

Sub- and superscripts Eq.

u depth averaged value 2.18 u' first order (perturbation) variable 2.55

u coraplex amplitude of analytical first order

(perturbation) variable 2.58 u complex amplitude of numerical first order

(perturbation) variable B.14

u0 zero order (unperturbed) variable 2.55

Rc value in center line (axis)

9m value based on mean grain size diameter 3.30

1 INTRODUCTION

1.1 Relevance of the investigation

The use of rivers for navigation and the increased human activity along their banks generally requires river control and improvement measures. Most rivers have a natural tendency for continuous change of alignment, e.g. meandering and braiding rivers. Construction of bridges, towns, berths, etc. have required fixation of the river alignment at many places, changing the natural morphology of the rivers. This can give rise to bank erosion, erosion around bridge pillars, sedimentation of navigation channels, etc. Adequate measures against this require a reliable prediction of the morphological changes.

Physical or mathematical models are often applied in order to assess morphological changes. Like in other fields of hydraulic engineering an increasing number of studies is carried out with the help of mathematical models instead of scale models. Until now, however, most mathematical morphological models are one-dimensional. Such models are well suited for prediction of morphological changes taking place on large time and length scales. The impact of, for instance, a bend cut-off on the (local) navigability, channel alignment stability, etc. evidently requires the application of a two-dimensional (horizontal) morphological model; also at water intakes (irrigation, cooling water), outlets, confluences, bifurcations and river bends the application of two-dimensional morphological models is relevant. Especially at a bifurcation the prediction of the distribution of both the sediment transport rate and composition of the sediment over the river branches is very important due to the large impact on the long term morphological development of the river branches (cf.

Jansen, 1979).

Until recently two-dimensional morphological changes of alluvial rivers were mostly investigated with the use of (expensive) hydraulic scale models even though the complicated flow pattern and the complex interaction of the flow and the sediment unavoidably gives rise to scale effects which make the model results difficult to interpret (cf. De Vriend, 1981a and Struiksma et al., 1985). The fast improvement of electronic computer facilities during the past decade has made it feasible to develop and solve complex two-dimensional mathematical models of the flow and the bed topography in alluvial river sections. In such a model scale effects do not play any role. However, a mathematical model cannot be applied without either comprehensive field data or scale model results for calibration of it. Any model is only as good as the data used to verify and calibrate it. But in any case, the application of a mathematical model introduces a great flexibility in an investigation.

The aim of the present investigation is to develop a general two-dimensional mathematical model for morphological changes in alluvial rivers and to gain insight into the physical processes involved. The bed topography in an alluvial river is determined by the interaction between the flow and the sediment transport. One of the main problems concerning this is the fact that the flow in river bends, at bifurcations, around structures etc. is curved. A characteristic feature of curved flows is the spiral (helical) flow, which causes a difference between the direction of the bed shear stress and the direction of the depth-averaged flow. The spiral flow induces a pronounced bed slope perpendicular to the main flow direction. Most rivers are meandering. The curvature of meandering rivers changes continuously implying a fast variation of the lateral bed slope and a continuous process of

redistribution of the flow and sediment transport. Moreover, pronounced bed topography and flow variations cause spatial variation of the grain size of the river bed. A mathematical model of an alluvial river bend comprises many of the fundamental aspects of two-dimensional morphological modelling. The present investigation will focus on such a model.

The development of a mathematical model of the flow and the bed topography in curved alluvial rivers is the main project of the task group on river bends 'of the River Project of the joint hydraulic research program T.O.W, (Toegepast Onderzoek Waterstaat) in which Rijkswaterstaat (Dutch Governmental Department of Public Works), the Delft Hydraulics Laboratory and the Delft University of Technology participate. The present investigation has been carried out at the Laboratory of Fluid Mechanics of the Delft University of Technology, Department of Civil Engineering within the framework of the T.O.W. task group on river bends.

1.2. Previous work

The morphology of a river bend is one of the key problems in river engineering and geomorphology, and it has had the attention of scientists for a long time. The first theoretical work concerning this topic was an analysis of curved laminar flow published by Boussinesq already in 1868. Since then a large number of publications have appeared; in the beginning only on the flow in a bend but later also on the sediment movement.

In a morphological model three aspects are important, viz. the computation of the flow distribution, the sediment transport and the bed level variation (cf. Jansen, 1979). In the following 'chapters modelling of the flow and of the sediment transport will

be discussed separately and specific contributions to these topics will be mentioned when the modelling is treated, (viz. contributions specificly on the flow model will be discussed in Chapter 2 and on the sediment model in Chapter 3 ) . In this introduction only contributions to an integrated approach, i.e. models directly resulting in a bed level prediction, will be mentioned.

Two groups of publications can be distinguished among the theoretical work on bed level prediction in alluvial river bends, viz. one concerning the transverse bed slope in an infinitely long bend (i.e. streamwise variations are disregarded)and one in which also transition phenomena are considered. The methods that belong to the first group are all based on the assumption that the depth-averaged flow and the sediment transport are parallel to the river axis, whereas the investigations in the second group also account for the redistribution of the flow and the sediment transport.

The pioneering work of Van Bendegom (1947) belongs to the first group of publications . He developed a model for the computation of the turbulent spiral flow in a curved flow and by considering various forces on a grain travelling parallel to the river axis at the alluvial bed he obtained an expression for the transverse bed slope. The contributions by, among others, Kikkawa et al. (1976), Odgaard (1981), Engelund (1981) and Koch (1981) are essentially based on the same principle as the Van Bendegom model (see also Chapter 3 ) .

Engelund (1974) was the first investigator who considered the redistribution effects of the flow and sediment transport, i.e. neither the main flow nor the sediment transport are considered parallel to the river axis. He investigated the equilibrium flow

and bed topography in channels with sinusoidal curvature variation using a simple flow model that accounts for bed friction and main flow inertia. By means of a mainly linear solution procedure of the model he obtained good agreement between the theoretical and the experimental results. Later, Struiksma et al. (1985) presented computational (numerical) results of a mathematical model comprising essentially the same elements as the model of Engelund. They also analysed the model and quantified the influence of the redistribution of the flow and of the sediment transport on the bed topography.

The cause of river meandering has fascinated scientists for a long time (cf. Callander, 1978). Until recently, it was generally accepted that the initiation of meanders can be explained by a linear stability analysis of a mathematical model of the flow and the bed topography in a straight channel. Such analyses have been presented by Hansen (1967), Callander (1969), Engelund & Skovgaard (1973), Parker (1976), Fredsoe (1978) and many others. Struiksma (1983), Olesen (1983) and Struiksma et al. (1985) have suggested that a steady state analysis may yield a more adequate explanation for the meander initiation. This suggestion is based upon the fact that the bed disturbance celerity is much larger than the meander migration velocity (low bank erodibility, see also Chapter 4 ) . A most interesting approach to the initiation and development of meanders has been suggested by Ikeda et al. (1981), Parker et al. (1982) and Parker (1983). This approach is based on the mathematical model of the flow and bed topography in curved alluvial channels as suggested by Engelund (1974) in combination with a simple bank erosion model. The above mentioned analyses have contributed considerably to the understanding of two-dimensional bed level development in (curved) alluvial channels.

Finally, the large number of publications on the results of laboratory experiments in curved flumes and of measurements in natural river bends have provided a large number of data for verification of morphological models. The often thorough discussion and interpretation of the measured data have contributed considerably to the understanding of the morphological processes in river bends. Especially, the contributions of the geophysicists Hooke (1975), Bridge and Jarvis (1976) and Dietrich et al. (1979 and 1983) should be mentioned in this context.

1.3 The present investigation

The present investigation deals with a mathematical model for two-dimensional morphological computation in alluvial river bends. The model is primarily used for research purposes. So some simplifications of the model and limitations of the geometries considered have been accepted, however, only those that are not relevant for the physical processes involved.

The planform of the rivers considered is submitted to two restrictions, viz. the river must be mildly curved and the width must be (approximately) constant. In mildly curved flows the modelling of the spiral flow can be based on a simple similarity hypothesis (see Chapter 2 ) . The limitation to constant width is not an essential one. Due to this limitation a number of small terms in the mathematical model drops out and a uniform transverse step size in the numerical integration procedure can be applied. Variable width can be introduced without altering the basic principle of the integration procedure.

also subject to some limitations. The width-depth ratio must be large so that wall effects are insignificant and lateral fluid friction can be neglected. Moreover, the Froude number must be small to allow a rigid lid approximation and to ensure that the celerity of bed disturbances is much smaller than the celerity of flow disturbances. This implies that a quasi-steady flow model can be applied, so that the integration of the mathematical model can be split up into alternate steady flow, sediment transport and bed level computation (see De Vries, 1981 and Jansen et al., 1979). In addition, the restrictions permit the application of a relatively simple (steady) flow model. As a further simplification the side walls of the considered river are assumed vertical. This enables a simple structure of the solution algorithm for the sediment transport and bed level computation.

Finally, only bed load transport is considered, which allows a sediment transport prediction with local hydraulic parameters only. The behaviour of suspended load is fundamentally different from that of bed load. In Chapter 4 the influence of suspended load on the bed topography is discussed qualitatively. The influence seems to be very significant, so this is an obvious topic for further research.

In Chapter 2 a system of partial differential equations describing the three dimensional flow pattern is presented. The system is simplified and, based on a similarity hypothesis, the vertical distribution of the three velocity components is computed using several eddy viscosity models. Subsequently, the flow model is integrated over depth to yield a two dimensional horizontal mathematical model in which some three-dimensional effects are maintained. Computational results are compared with data from two flume experiments and with data from a large natural river.

Next, the bed level model is presented. Some fundamental aspects of the sediment transport mechanism will be discussed briefly and a short review of the most important models will be given. The bed level model will be extended to account for grain sorting effects and non-uniform sediment. This is achieved by considering the sediment transport of a number of grain size fractions. Moreover, a transport direction formula for a grain size fraction in a sediment mixture will be derived.

In Chapter h the mathematical model will be analysed by

introducing several simplifying assumptions. By means of a sensitivity analysis of the steady linear solution of the mathematical model the most important parameters of the model are determined and the influence of two important length scales is discussed. Furthermore, it is investigated whether the mathematical model' can be simplified, and, partially based on the meander deformation model of Ikeda et al. (1981), a new theory for the initiation of meanders in alluvial rivers is presented.

In Chapter 5 the non-linear numerical integration procedure is presented, and the accuracy and the stability of this procedure are investigated.

In Chapter 6 the mathematical model for uniform sediment is verified by comparing its results with measured data from flume experiments with a large variety of planforms and hydraulic conditions, as well as by comparing computational results of the model for non-uniform sediment with measured data from flume experiments. The performance of the models is evaluated.

Finally, in Chapter 7, the most important conclusions will be summarized and their practical implications will be discussed. Moreover, suggestions for further research will be given.

2 Steady flow computation in river bends

2.1 Introduction

In the present morphological model the main purpose of the flow computation is to obtain a bed shear stress distribution from which the bed load sediment transport rates (in two directions) can be estimated sufficiently accurately. In view of the strongly non-linear dependence of the sediment transport rate on the bed shear stress an accurate flow predictor seems necessary. On the other hand, the flow distribution strongly depends on the alluvial roughness distribution, which cannot be predicted very accurately and, in addition, the sediment transport models are not very reliable (see Section 3.2). So the application of a very refined flow model will not necessarily result in a more reliable sediment transport prediction.

Selecting a flow model the computational effort involved is an important aspect which should not be underestimated. In view of the large number of steady flow computations which will be necessary in the morphological model the application of a fully three-dimensional model is not feasible. Furthermore, three-dimensional flow models for rivers with arbitrary cross-sectional shapes, as would be necessary in the morphological model, have not been reported yet. This points to the application of a two-dimensional depth integrated flow model, although the flow pattern in a river bend is essentially three-dimensional. The main problem of the flow modelling is to incorporate the three-dimensional effects into a two-dimensional flow model in an adequate way, so that the additional computational costs will be minimized.

In the following a physical description of the flow is given and previous work on mathematical modelling of curved flow is discussed. Subsequently, the fully three-dimensional Navier-Stoke equations in a curved co-ordinate system are presented. The system of partial differential equations is simplified and, based on a similarity hypothesis, the vertical distribution of the three velocity components are computed for three different eddy viscosity models. Next the system of equations is depth averaged and the resulting two-dimensional model is analysed. Finally, computational results are compared with flow data from a curved flume with uneven bed, a curved flume with rectangular cross-section and a curved natural river.

2.1.1 Physical description

Mathematical modelling of the flow in a river bend requires some insight into the physics of the water motion. For this purpose a physical explanation for the flow (re-)distribution in a bend is given. In the first place fully developed bend flow is discussed, whereafter redistribution effects are treated.

The centripetal acceleration of the water particles in a river bend is established by a transverse slope of the water surface. This implies that the centripetal acceleration is uniformly distributed over the vertical (hydrostatic pressure distribution). As the longitudinal flow velocity increases from zero at the bed until its maximum at or close to the water surface the flow close to the bottom will follow a path with a smaller radius of curvature than the flow closer to the water surface, i.e. the flow will form a spiral motion. This spiral (or helical) flow pattern can be considered as the sum of a longitudinal flow component (main flow) and a circulation in a plane perpendicular to the main

flow direction (secondary flow). The secondary flow is directed towards the center of curvature near the bottom and outwards in the upper part of the cross-section.

The secondary flow has a significant influence on the main flow distribution due to convection of (main flow) momentum. In the central part of the flow horizontal convection is dominant, whereas close to the banks vertical convection is dominant. The net influence of secondary flow convection is an increase of the main flow velocity in the outer bend and a decrease in the inner part.

In a fully developed curved flow the longitudinal water surface slope is inversely proportional with the distance to the center of curvature. This implies that the water surface slope is largest in the inner bend, which causes an increase of the main flow velocity there. In general this effect is less important than secondary flow convection.

In case of changing curvature and bed topography the flow distribution is more complicated. Considering, for instance, an alluvial river, consisting of a straight reach with uniform depth proceeded by a curved reach with non-uniform depth distribution (alluvial bed), the flow field undergoes several changes which can be attributed to different effects. Around the entrance of the bend the transverse water surface slope will grow rapidly from zero until its final value in the bend. So, along the inner bank the longitudinal water surface slope will increase rapidly and cause an acceleration of the flow there. Along the outer bank the slope decreases and the flow decelerates. In turn this ensures a gradual growth of the streamline curvature around the entrance. The secondary flow will also grow gradually and it will establish a transverse bed slope (see Section 3.3). The main flow will adapt

to the changing bed topography (bed friction) and to the influence of secondary flow convection, i.e. the main flow velocity will increase in the outer bend. Due to the inertia of the main flow this adaptation will take place gradually, i.e. the flow distribution will lag the bed topography. The relevant length scales of the adaptation of the secondary and main flow are discussed in Sections 2.3.4 and 2.4.1, respectively.

2.1.2 Previous work

The first work on mathematical modelling of flow in curved channels is based on the assumption of laminar flow (e.g. Boussinesq (1868), Dean (1927) and many others). These analyses have also contributed to the understanding of the complex flow pattern in curved channels with turbulent flow, but the quantitative description is not good enough for most engineering purposes.

A mathematical analysis of the secondary flow in a uniform turbulent flow was carried out by Van Bendegom (1943 & 1947; the latter Engl. transl. 1963). This analysis and the introduction of perturbation techniques (with the depth-radius of curvature ratio as small parameters) in the analysis of curved flow by Ananyan (1957; Engl. transl. 1965) and Rozowskii (1957; Engl. transl. 1961) ment a large improvement of the understanding and mathematical description of the secondary flow and partly also its interaction with the main flow. Later on many solution methods of mathematical models of curved flows based on perturbation techniques have been published; for instance Yen (1965), De Vriend (1973, 1976 & 1977), Ikeda (1975), Gottlib (1976) and Falcon (1979). In these perturbation methods it is assumed that the main flow distribution is unaffected by the secondary flow. De Vriend

(1981a) avoids this assumption, but his two-dimensional model only works well in mildly curved flows with vertical side walls.

Qualitatively, the influence of the secondary flow on the main flow (secondary flow convection) has been known for quite a while, but mathematical modelling of this effect, without too high computational costs, has been lacking. Only recently Kalkwijk & De Vriend (1980) incorporated this effect into a two-dimensional depth integrated model for rivers with gently curved alignment and mildly sloping banks. For the case of channels with steep banks and of rectangular channels no simple two-dimensional model which includes secondary flow convection has been developed yet.

Due to the fast improvement of computer capabilities fully three-dimensional flow computations have become feasible in the past decade. For curved channel flow the models can (so far) only cope with rectangular channels. Pratap & Spalding (1975), Leschziner & Rodi (1979), De Vriend & Koch (1981) report such computations. In combination with a thorough analysis these computations may contribute to an improved understanding of the flow in a bend. Presently, the three-dimensional models are not applicable in a morphological model, because the computational costs are far too high.

2.2 A mathematical model of the flow

2.2.1 Co-ordinate system

A mathematical model of the flow in rivers with curved alignment can be described most conveniently in a curvilinear co-ordinate system. The co-ordinate system is chosen as follows: the s-axis coincides with the channel axis, the n-axis is horizontal and

perpendicular to the s-axis and the z-axis is vertical and positive upwards. In this co-ordinate system the three-dimensional mathematical description of the flow is very comprehensive, but it can be considerably simplified if only rivers of constant width are considered. In that case the n-axis will be straight. As a consequence of this simplification a number of small inertia and friction terms vanish in the mathematical model. It does not qualitatively or quantitatively influence the result of the analysis carried out in the following. Kalkwijk et al. (1980) give a three-dimensional mathematical description of the flow in a curvilinear co-ordinate system in which both horizontal co-ordinate axis are curved, but they avoid the comprehensive description of the friction terms.

The applied co-ordinate system is depicted in Figure 2.1. It is necessary to introduce a sign convention for the radius of curvature of the s-axis, viz. R is negative if the n-axis points towards the centre of curvature.

\ \ R < 0 \ \ R > 0 \ \ \

3 3A 3n 3s 3 1 3A 3R R 3w " 1 32A " R 3R3w " 1 3A " R7 3w " 3 3A 3s 3n 1 3A " R 3s

The co-ordinate system has one disadvantage, namely the "n" and "s" are not independent. This implies, for instance, that in case of cross-differentiation the order of . differentiation is important. This can easily be demonstrated by introducing a polar co-ordinate system, i.e. s=Rw and n=R-R (R is the radius of curvature in the channel axis), where R and ware independent.

(2.1)

2.2.2 Governing differential equations

Steady incompressible turbulent flow can be described by four partial differential equations. Three dynamical equations (one for each direction) and one equation representing the conservation of mass. According to Rozowskii (1957) the equations read

3u , 3u , 3u uv 1 3D 3s 3n 3z R p 3s s \*-*£) 3v 3v , 3v u 1 3p „ u - r - + v ^ - + w — - - — + - - ^ - = F (2.3) 3s 3n 3z R p 3n n v. * ■ • - > , / 3w 3w 3w 1 3p u -5— + v -5— + w -5— + g + — -7f- = F , „ . N 3s 3n 3z & p 3z z (2.4) 3u _3_y_ _3w v_ 3s + 3n + 3z + R = ° (2.5) in which

g = acceleration due to gravity p = pressure

u,v,w = velocity component in s-, n- and z-direction, respectively ,F = friction te

density of fluid. F„ ,F ,F = friction terms

The solution of the mathematical model is facilitated by introducing the rigid lid approximation for the water surface boundary condition. This implies that the water surface is considered as a rigid impermeable and shear stress free plate only with normal stresses (pressure). The error.introduced by the rigid lid approximation will be small when the deviation between the local water surface level and the "average" water surface level (i.e. the level of the rigid lid) is small. This is the case when the Froude number and the ratio of waterdepth-radius of curvature (h/R) are small.

The friction terms are very complex (cf. Rozowskii, 1957 and De Vriend, 1981a). Neglecting wall effects and assuming a scalar eddy viscosity coefficient Rozowskii shows that the friction terms of Equations (2.2) and (2.3) can be divided into terms of the relative order of magnitude of unity and of ( h / R )3, where h is the

water depth. As the depth radius of curvature ratio is very small in natural rivers the 0 [(h/R)]J -term can be omitted. In this case the right hand side of Equations (2.2) and (2.3) can be approximated by 1 3 T F

s = i " # (2.6)

where T „ and T are shear stresses. In the following the second

sz nz ° index will be omitted when that does not give rise to confusion.

In fact, these approximations imply that all lateral exchange of momentum due to friction in the fluid will be neglected (shallow water approximation).

The friction terms can be further elaborated. Introducing the Reynold's stress concept and the Prandtl mixing length hypothesis and assuming that the viscous (laminar) friction is much smaller than the turbulent friction the shear stresses in the fluid can be expressed by (2.8) T S P T n P A A 3u 3z

a

in which A is the turbulent coefficient of viscosity or the eddy viscosity coefficient and L is the mixing length, a function of z. The applied formulation implies that the eddy viscosity is assumed isotrop.

Rozowskii (1957) shows that Equation (2.4) can be simplified to

S + ^ = ° (2.11)

i.e. hydrostatic pressure. The error introduced is of the order of magnitude (h/R) close to the side walls. In the central regions of the flow the error is much smaller. Integration of this equation, 'assuming z=0 at the channel bed and applying the rigid lid

approximation, yields

p = P + pg(h-z) (2.12)

where P denotes the pressure at the rigid lid and h, rather than the depth, the distance from the rigid lid to the channel bed.

Combining the Equations (2.2), (2.3), (2.6) through (2.9) and (2.12) yields the simplified horizontal dynamical equations for the flow, viz.

3u 3u 3u uv 1 3P 3 ,.3us U T — + V T - + W — + — + — — = — ( A T — ) (2.13) 3s 3n 3z R p 3s 3z 3z v.■'•■'■->>' 3v , 3v 3v u2 1 3P 3 ,.3v. ,„ , , N u ^ — + V - 5 — + W - 5 — -•=— + — - 5 — = 3—(A~—) (2.14) 3s 3n 3z R p 3s dz 3z v '

Summarizing: these equations are approximations of the exact horizontal momentum equations. If the Froude number is small and the regions close to the side walls (banks) are disregarded then the introduced error is of the order of magnitude (h/R)2.

Consequently, the simplified flow model only applies to mildly curved and shallow rivers with small Froude number.

2.3 Vertical distribution of the flow

This analysis of the mathematical model of the flow is based on the technique of asymptotic expansion. The approach is quite similar to the one applied by (among others) Rozowskii (1957) and De Vriend (1981a) in their extensive analyses. The analysis is facilitated by introduction of a natural co-ordinate system, i.e.

a co-ordinate system where the s-lines follow the streamlines of the depth averaged flow. If the flow is not fully developed (i.e. all derivatives in s-direction equals zero), the streamlines will either diverge or converge, so developing flow cannot be described using a natural curvilinear co-ordinate system without curvature of the normal axis. So, for a start, it is assumed that the flow is fully developed.

Solution of the flow model demands a choice of the mixing length in order to determine the eddy viscosity coefficient (cf. Equation 2.10). The analysis of the mathematical model is also ment as a sensitivity analysis so three different mixing length models will be applied, viz.

L = K z / ü ï h (2.15)

L =

K z

~

/T=ï h (2.16)

L = 2 K (1 _ /Ï^I) /T=i h (2.17)

where < is the Von Karman constant, m is a factor depending on the bed roughness and the dimensionless vertical co-ordinate z=z/h has been introduced for convenience. In straight uniform shear flow the mixing length given by Equation (2.15) results in the well known logarithmic velocity profiles, Equation (2.16) results in a power profile and Equation (2.17) in the Von Karman velocity profile (cf. Jansen 1979). So, Equations (2.15) and (2.16) represent the two most frequently used mixing length models for river flows. Attempts to verify the mixing length models have mainly been based on flow measurements in uniform shear flow, but in this case the models predict nearly the same velocity distribution; so based on performance it is hard to give a 'preference for one of the models. However, it is physically

reasonable to assume that the mixing length close to the bottom grows linear with the distance to the bottom, so, Equations (2.15) and (2.17) probably yield the most realistic shear stress and flow distribution in this region.

The vertical distribution of the flow velocities can be obtained by asymptotic expansion. In the first place, the zero order approximation of the longitudinal flow velocity is obtained from Equation (2.13) assuming v and w equal zero (and 3u/3s=0, fully developed). Next, the transverse velocity is computed from Equation (2.14) with the zero order longitudinal flow velocity inserted and with terms of the order of magnitude ( h / R )2, i.e. the

terms v3v/3n and w3v/3z, disregarded. The vertical flow velocity is obtained directly from the equation of continuity, i.e. Equation (2.5).

A first order approximation of the longitudinal flow velocity can now be obtained by introducing the (first order) secondary flow velocities into Equation (2.13). This was done by De Vriend (1981a) and De Vriend & Struiksma (1983). They showed that the form of the first order solution only differs a little from the zero order solution. A second (or higher) order approximation for the flow velocities can in principle be determined from Equations (2.13) and (2.14) in largely the same way as outlined above. However, that does not make sense as the equations already are second order approximations of the complete mathematical model.

2.3.1 The longitudinal flow velocity components

After a few manipulations the zero order version of the longitudinal momentum equation (Equation 2.13) can be written as

Me>V>] =ï?sk = -b

(2

-

18)

where

u is the depth-averaged flow velocity

/ =u/u (the shape of the longitudinal flow velocity) C is the Chézy roughness coefficient

The boundary conditions for this differential equation are vanishing shear stress at the water surface and no slip at the bottom or rather at z0 , where za =exp(-l-KC/v/g) for Equation

(2.15), -z0 =0 for Equation (2.16) and za =exp(-l.U-KC/,/g) for Equation (2.17). So, the zero order solution for the longitudinal flow velocity profile reads

4

u = f [ï°-&

]

(2.19)

■ Z oL J

The velocity profiles are obtained by inserting the models for the mixing length, Equations (2.15) through (2.17) and after integration, viz.

I = a In f- = 1 + a(l + In z.) (2.20)

/u = «"> ^1 / m (2.21)

/

= a[/T=S -

SÏ=T

+ In

J Z ^ ] (2.22)

in which a is a dimensionless roughness coefficient given by a=/g/(KC).

integration of Equation (2.21) over the depth a relation between the "m" and the roughness coefficient can be derived. This relation reads

m+1

~ m ~ (2.23)

Equation (2.23) is given by, among others, Jansen (1979). Other relations between m and a are known from the litterature, e.g. Zimmermann & Kennedy (1978) and Nunner (1956) These relations do not necessarily violate the definition of ^ ; they simply imply that the mixing length given by Equation (2.16) is defined in another way.

The zero-order viscosity coefficient for the logarithmic model can be obtained by combining the Equations (2.10), (2.15) and (2.21), viz.

A = K2 a z (1-z) h Ü (2.24)

The zero order eddy viscosity for the power model and the Von Karman model are obtained in a similar way, viz.

, 2 1—1/m /, s , —

A = KA a z (1-z) h u (2.25)

A = 2 K2 a ( l - / n ) ( l - z ) h Ü (2.26)

LEGEND:

logarithmic Top : C = 30 m1/3/s

power Bottom: C = 6 0 m"Vs

2.3.2 The transverse flow velocity

The truncated (first order) version of the transverse momentum equation, (2.14), reads

1

¥■ - f = -iMr) (2.27)

p 3n R 3zv 3z'

In an appropriate dimensionless form this equation can be written as

3-[—V ^ 1 +

K /

= c (2.28)

in which

/ =v/i (the shape of the secondary flow) v s

c is a constant (independent of z) which is proportional with the transverse water surface slope and

^ = " R (2.29)

the secondary flow intensity (cf. De Vriend, 1981a).

The boundary conditions for Equation (2.28) are the usual ones, i.e. vanishing velocity and shear stress at 2 - z0 and 2 = 1 ,

respectively. For the determination of the right hand side (i.e. the transverse water surface slope) the auxiliary condition

f1vdz=0 is used (i.e no net flow in transverse direction).

The introduction of the boundary condition at 2 = z0 instead of at

2=0 makes analytical treatment of the logarithmic and the Von Karman model laborious. On the other hand, it makes a numerical

approach much easier because the "tail" of the logarithm is disregarded, i.e. there is no problem with / ->-oo and 3/ /9z-xx> at

u u the bottom.

The analytical solution of Equation (2.28) and the remaining secondary flow quantities discussed in this chapter in case of the power model are derived in Appendix A by means of Taylor series expansion. A numerical model has been used to obtain the various secondary flow quantities for the two other eddy viscosity models.

In case of the power model Equation (2.28) can be integrated analytically without applying Taylor series expansion for two values of m. Rozowskii (1957) gives the explicit solution for m=6 and m=8. If the logarithmic tail is not omitted the analytical solution of Equation (2.28) reads (cf. Rozowskii, 1957 and De Vriend, 1981a)

/

= p"[2F!(2) + aF

(z) - 2(l-a)/J (2.30)

in which

In Figure 2.2 the vertical distribution of eddy viscosity, the main flow velocity (J ) and secondary flow {j ) for the three

u ^v mixing length models are depicted for two different values of the Chezy roughness coefficient assuming K = 0 . 4 . No very pronounced difference between the flow profiles of the different mixing length models can be observed in the figure, although the logarithmic model seems to result in a somewhat larger secondary flow.

The model for the bed shear stress direction in a curved flow plays an important role in a bed topography model for river bends. The logarithmic model as obtained by Rozowskii (1957) and others (i.e. with the logarithmic tail) yields a bed shear stress direction given by

B = p - (1-a) (2.31)

in which 3 = -R/h tan 6g

and tan (6 ) = T, /T,

s bn bs

The power model yields (cf. Van Bendegom, 1947 and Appendix A)

6 = ^ (m+2)(m+3) ( 2-3 2 )

The quantity g, according to Equations (2.31) and (2.32) and according to the numerical results, are depicted as a function of the Chézy roughness coefficient in Figure 2.3. The differences, concerning bed shear stress direction, between the models are surprisingly large in view of the close similarity of vertical distribution of eddy viscosity and transverse as well as longitudinal flow velocity. Consequently, the bed shear stress direction (which is of great importance in morphological models) is extremely sensitive to the mixing length model applied.

It is hardly feasible to verify a bed shear stress direction model experimentally in case of an alluvial bed. . According to Figure 2.2, the direction of the flow varies rapidly close to the bottom, so measurements of the flow direction should be carried out

12.5 Legend : Logorithmic Power - v.Karman Log.with "tail"

Figure 2.3 Variation of the coefficient in the model for the direction of the bed shear stress in a curved flow.

very close to the bed in order to estimate the bed shear stress direction. This will be nearly impossible due to the migrating bed forms. By means of morphological models the bed shear stress direction is estimated indirectly (cf. Struiksma et al., 1985 and Olesen, 1985b), suggesting that in case of laboratory flumes the logarithmic model (with "tail") overestimates the bed shear stress direction with a factor of about two. In large natural rivers the model seems to perform well (Struiksma, 1985). It is likely that this difference can be explained by the fact that in a laboratory flume the relative bed form height is generally significantly larger than in natural rivers (cf. Chapter 3 and Van Rijn, 1982). This causes different flow and eddy viscosity distributions in the flume, which, according to Figure 2.3, have a significant influence on the bed shear stress direction.

10 7.5 CO. A 2.5 20 30 40 50 60 70 ► C(m"2/s)

The determination of the transverse flow velocity and the bed shear stress component as outlined above has a long history. It started with Boussinesq (1868) who obtained the secondary flow velocity distribution in a bend with laminar flow. Later on Van Bendegom (1947), Ananyan (1957), Rozowskii (1957) and many others applied the same approach for turbulent flow using different eddy viscosity models. Recently also another approach to determine the transverse (bed) shear stress component has been introduced. The first publications on this approach were Zimmermann et al. (1978) and Falcon (1979), after which Falcon & Kennedy (1983) made an extension of the former analyses. In all three publications the results, concerning transverse (bed) shear stress deviate more or less from the result used in this thesis. Below the new approach is described based on the analysis of Falcon et al. (1983).

The new approach is based on the power model and a moment-of-momentum consideration with respect to a longitudinal axis on the bed using the power model. The three most important terms of the moment-of momentum equation used in the new approach are

i gp i u2 l

■f ^ Tz d z ~ Ï &o~zAz + ƒ T dz = 0 (2 331

o on o K o nz K^.OJJ

As the turbulent stress tensor is symmetrical: T = T ( = T ) : so nz zn n

i r -ii l 3 T i 3 T

"'. V* = "Mo

'„ 3]>- =

a T * * ' (2.34)

Consequently, Equation (2.33) is not an extra equation, but it is merely Equation (2.27) multiplied by z and integrated between the

bed and the bottom. A consistent analysis should therefore not result in another result.

The reason why Falcon et al. (1983) obtain a deviating result is that they omit the last terra in Equation (2:33). From this truncated version of the moment-of-momentum equation the water surface slope can be obtained directly. Subsequently, they insert the water surface slope into the momentum equation, i.e. Equation (2.27), after which the bed shear stress can be obtained by integration. Consequently, they use the same equation twice. The omission of the last term in Equation (2.33) implies that they introduce T =0 (where the bar implies depth averaged value)

nz _ instead of the correct auxiliary condition, i.e. v=0, used in the

traditional approach.

2.3.3 The vertical flow velocity

The vertical secondary flow velocity component can be obtained by integration of the continuity equation for fully developed flow, viz.

z. d± i

w = -J (IÏT + R ^ ^ (2.35)

The shape of the vertical velocity, / , is given by

z

^ = ! /v d 2 (2-36)

2.3.4 Developing secondary flow

In regions of changing curvature the secondary flow will adapt gradually. The inertia of the secondary flow has been investigated analytically (using similarity hypothesis) by among others Rozowskii (1957) and Nouh & Townsend (1979). Numerical investigations of this topic have been carried out by De Vriend (1981a), Booij & Kalkwijk (1982) and Kalkwijk & Booij (1986). All these investigations have been based on Equation (2.14) with the second order term omitted, viz.

3v 1 3P u2 3 ,A3v. ,„ n^N

u 37 + p 3n" - ÏT = 3i<A3i> (2-37)

The first important question is as to whether a similarity hypothesis suffices. In order to investigate this a schematized case is considered: a straight river, at s=0, entering a long bend with constant curvature. Similarity is a priori assumed, so the following approximations apply

v = is £v gr(s) ( g(0) = 0 , 9M = 1 )

(2.38) 3P 3 P l . , 9 P*

3n"= 3 n ~+ ^( s ) 3ÏT

i.e. the transverse water surface slope is split up into a part (Pj) balancing the centrifugal force and a part (P2) due to the

bed shear stress and #(s) is a function describing the adaptation of the secondary flow.

Equation (2.37) can be made dimensionless in the same way as Equation (2.28), viz.

V v ^ + t ' - ^

^]---hK (2.39)

in which Equations (2.28) and (2.38) and the dimensionless

longitudinal co-ordinate E, =as/h has been introduced. The

integration constant in the equation C = : ( ^U) / K2 (cf. Equation 2.28). So, Equation (2.39) can be elaborated to (see also Booij et al. 1982)

Az) ^f1 + <?(€) = 1 (2.40)

#z) = K

l^K^l - r

) (2.41)

where the bar denotes depth averaging.

Strict similarity requires that Equation (2.41) is not a function of z. This is clearly not the case. For a certain value of z, /(Z)-MO, implying an infinite large adaptation length and f(0)=0 implies instantaneous adaptation at the bed. Anyhow, some kind of averaged adaptation length could be obtained from Equation (2.41) by depth averaging, but the result is sensitive to the made assumptions (e.g. the approximation for c ) . So, an analytical approach based on similarity is not likely to provide an accurate description of the gradual adaptation of the secondary flow.

Consequently, numerical investigation, as carried out by De Vriend (1981a), Booij et al. (1982) and Kalkwijk et al. (1986) seems to be the best way. The numerical investigations of De Vriend and Booij et al. more or less confirm the results of the analytical analyses concerning the growth of the secondary flow (secondary flow profile). Only close to the bottom the numerical computation gives a considerably faster adaptation of the

secondary flow (and the bed shear stress due to the secondary flow). According to the numerical results the adaptation of the bed shear stress is not exponential. So strictly speaking the process of adaptation cannot be characterized by one length scale only. Anyhow a characteristic length scale is obtained from the numerical computations. De Vriend obtained an adaptation length for the secondary flow intensity of about 1.3 hC/^g" and about the half for the bed shear stress. Booy et al. (1983) obtains for C=50 m^/s a somewhat smaller adaptation length, namely 15h and for the bed shear stress, Ah for the initial adaptation until about 60% of the final value and ca. 15h for the remaining adaptation. Kalkwijk & Booij (1986) obtain quite similar results. In Figure 2.4 the computational result concerning adaptation of the bed shear stress of Booij et al. is depicted and it is compared with a purely exponential adaptation with a length scale of X ,=0.6 hC/i/g.

l.O t 0.5 0 0 10 20 30 40 50 60 » s/h Legend:

computation by Booy et all 1982)

Figure 2.4 Adaptation of the transverse bed shear stress after a sudden change in main flow curvature.

The adaptation length of the secondary flow is small compared with other length scales in the morphological model (cf. Chapters 3 and 4 ) . So, it may be sufficiently accurate to assume instantaneous adaptation of the secondary flow, but this gives rise to stability

1

//

/ /

/ /

//

1/

If

-problems in the mathematical model (see Olesen, 1982a) . In any case no great accuracy of the adaptation length is required. So, in the morphological model, where the bed shear stress direction is of major interest, an adaptation length of 0.6 hC//g~ will be used.

As a consequence the direction of the bed shear stress in case of a continuously varying curvature will be calculated with

9tan 6 ,

X , — r - + tan 6 = -3 £- (

-

>

sf 3s s R s

where 6 can be obtained from Figure 2.3.

2.4. A depth integrated model of the flow

In developing flow the transverse velocity averaged over the depth may differ from zero. In order to carry out the depth integration of the mathematical model, i.e. the Equations (2.5), (2.13) and (2.14), it is important to be able to distinguish between transverse flow velocity due to flow curvature (secondary flow) and transverse flow velocity due to redistribution of the (main) flow. Therefore the transverse velocity is expressed by

v

= * 4

^ 4 (2.43)

The depth integration of the mathematical model is simplified if it is assumed that the streamlines nearly coincide with the s-lines of the co-ordinate system (i.e. v/S'is assumed small, which almost always applies in alluvial channels) and that the water depth/radius of curvature ratio is small. These assumptions

imply, amongst others, that quadratic terms of the transverse flow velocity can be neglected (i.e. v , i and Vi are very small) and, furthermore, secondary flow components in longitudinal direction can be omitted.

The continuity Equation (2.5) integrated over the depth reads

3hu 3hv hv . ,„ ,,N

9r

3n-

i T

<

-

The longitudinal and transverse momentum equation, i.e. Equations (2.13) and (2.14), can be elaborated into (see also Kalkwijk et al. 1980) 1 3P T- 9Ü - 3Ü vul Tbs , p 3s uuL 3s 3n K J n . , .

r a -

s

1

k /h f-(ui h) + 2 - # - = 0

k = j'

£ f. èz (

-

>

In Equation (2.46) the second order terms have been neglected and in Equation (2.13) the vertical velocity component has been eliminated by partial integration and application of the continuity equation (2.5) in order to facilitate the depth integration.

1.12 .1.10 4 1.08 1.06 1.0 A 1.02 -1.00 Legend : Logarithmic Power v Ka'rmdn Log. with " t a i l " 50 60 ► C(m,/2/s)

Figure 2.5 Main flow velocity distribution coefficient.

The velocity distribution coefficients, k and k , are depicted

' uu sn

in Figures 2.5 and 2.6, respectively. They have been obtained with the numerical model for the three different mixing length models and in case of the power model also analytically as outlined in Appendix A. In case of the logarithmic model with the "logarithmic tail" included the variation of k has been obtained from Kalkwijk et al. (1980) and ku u by direct integration of Equation

(2.20), viz.

1 + a2 (2.49)

In most cases ku u=l will be sufficiently accurate. So, even though

the computational effort involved is small, this approximation is applied.

.__ according to Kalkwijk etal.(1980) J° 0.6 0.2 0.0 Legend: Logarithmic Power v. K a r m a n Log. w i t h " t a i l " 20 30 40 50 60 70 ► C(m, / 2/s)

Figure 2.6 Secondary flow velocity distribution coefficient or secondary flow convection factor.

The velocity distribution coefficient ks n, which is related to convection of momentum by the secondary flow, is very sensitive to the "logarithmic tail" (cf. Figure 2.6) in case of rough bed. The

the logarithmic model and k based on the other two models.

6 sn

The secondary flow convection term in the depth integrated transverse momentum equation is very small (cf. Section 2.3.4) compared to the dominant inertia term ( uJ/ R ) , S O this term may be

If v/u«l the streamline curvature can be approximated by (cf. De Vriend, 1978 and Olesen, 1982b)

^

l _ ^ 3 v

R R u 3n

After substitution of this expression into Equation (2.46) and neglecting the term due to secondary flow convection the depth integrated transverse momentum equation reads

u "2 1 3P . Tbn

^

+

^

+

^rT=° <

2

-

51

>

s

The bed shear stress (s and n components) and the direction can be approximated by

T

= p f r ü

(2.52)

Tbn = P C7 u^v + u t a n 5s ' (2.53)

tan 6 = =■ + tan 6 (2.54) u s

in which tan(6 ) must be obtained from Equation (2.42) and again quadratic terms of the transverse flow velocity have been disregarded. In addition, in the expression for the longitudinal bed shear stress component small terms describing the influence of main flow acceleration and deformation of the flow profiles have been neglected (cf. De Vriend et al., 1983). Equation (2.52) is a good approximation in slowly varying and mildly curved flows.

2.4.1 Analysis of the depth-integrated flow model

Some of the most important features of developing flow can be displayed by means of a linear analysis of a strongly simplified version of the flow model; namely the curvature of the co-ordinate system, all influence of secondary flow convection and the transverse bed shear stress are neglected. Furthermore, it is assumed that the depth is uniform (rectangular channel). In this case the linear approximation of the flow model reads

1 3P' - 3u' 2g_ i n rn cc\ + u0 ^ — + T^T u0u' = 0 (2.55) p 3s ° 3s hC 1 3 P' +üo| ^ = 0 (2.56) p dn ds f ^ + | ^ = 0 (2.57) 3s 3n

in which the prime indicates first order perturbation variable and subscript 0 refers to the unperturbed situation.

From this linear system some relevant length scales can be obtained by assuming that the dependent perturbation variables can be expressed as

P' | I P

u u } exp(-sA) exp(iTTn/B) (2.58)

in which P, u, v are complex amplitudes of the perturbations, 1

The term "exp(-s/A)" indicates an exponential adaptation, with an adaptation length A in longitudinal direction (if X appears to be

complex the solution is harmonic or mixed harmonic/exponential in longitudinal direction). The term "exp(_tTm/B)" indicates a sinusoidal shape of the perturbation in transverse direction. In fact, Equation (2.58) is the first term of a Fourier series. It appears that this first term is dominant in alluvial rivers (smallest damping and largest amplitude, cf. Chapter 4 ) .

It appears convenient to distinguish between two separate cases in the linear analysis, viz. a case where the influence of pressure variations can be disregarded and a case where the influence of bed friction can be disregarded.

In the first case the pressure term in Equation (2.55) drops out and the corresponding length scale can then directly be obtained from this equation. The length scale reads

A = £ - h (2.59) w 2g

Obviously this length scale is associated with processes governed by bed friction; for instance the gradual adaptation of the main flow to a changing bed topography.

In the second case the bed shear stress term in Equation (2.55) drops out (i.e. potential flow). From Equations (2.55) en (2.56) the pressure can be eliminated by cross differentiation, whereafter the longitudinal flow velocity can be eliminated by means of the first order equation of continuity, i.e. Equation

(2.57). The length scale from the potential flow model can now be obtained directly by substitution of Equation (2.58). The length scale reads

1 (2.60) X = ±

ï ï

These length scales are related to adaptation of the flow due to pressure variation (the same as variation of the streamline curvature, cf. Equation 2.51). The entrance of a bend can be considered as a discontinuity of the (zero order) pressure. This implies that the streamline curvature (according to this simplified linear solution) in the transition region between a straight and a curved reach will vary in longitudinal direction as

i_ jk-p^f)

s < 0 (2

.

61)

R ) i i

S / 1 1 / „SN „ n

[ R - 2R ^ P ^ V s ' °

where s=0 at the bend entrance. So, the transverse water surface slope adapts very rapidly (e.g. exp(-TT)=0.043).

The length scales given by Equations (2.59) and (2.60) can also be obtained from the compound system, i.e. without distincting between the two cases. In this case the length scales should be obtained from the polynomial

+ A A2 - A (-)2 = 0 (2.62)

W W IT

Equations (2.59) and (2.60) are good approximations of the roots of this polynomial for most relevant values of A and B (i.e.

Xw> B> '

The influence of secondary flow convection can be investigated by combining the depth integrated momentum equations. Kalkwijk et al. (1980) showed that, with a few simplifications (namely, k =1,

T, =0 and R =R) the Equations (2.45) and (2.51) can be combined to yield

holding along the characteristics

i ü

ü

±s

ds v sn u

Terms due to secondary flow convection figure at two places in this set of equations. In Equation (2.64) the secondary flow convection term causes a deviation between the flow direction and the direction of the characteristics, viz. the characteristics gradually shift towards the concave bank. This effect is due to the net outwards transport of momentum by the horizontal component of the secondary flow. A relevant length scale for this horizontal convection of momentum is the distance needed for the characteristics to cross the channel (with v=0). In a channel with rectangular cross-section this distance is given by

R £ (2.65) 2k i 2k h

sn s sn

This length scale is very large compared to the arc length of a bend. For instance, in a bend of 180° in a river with a width / depth ratio of 50 the length scale is about 75 times as long as the arc length. Obviously this effect may be omitted. In this case Kalkwijk et al. (1980) showed that the longitudinal momentum equation may be approximated by

1 3P , - 35 - 35 ïïv ïï'rs 3 ish 1