2006nguyen vertedores oblicuos tesis de maestría

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FLOWOVER OBLIQUEWEIRS

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TU Delft, October 2006

MSc. ThesisNguyen Ba TuyenSection of Hydraulic Engineering

Faculty of Civil Engineering and Geosciences

Graduation committee

Prof. Dr. Ir. M.J.F. StiveDr. Ir. W.S.J. UijttewaalProf. Dr. Ir. G.S. Stelling

Ir. H.J. VerhagenIr. R.J. Labeur


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Master of Science Thesis:

Draft report.

Name:

Nguyen Ba TuyenDate:

October 2006

Graduation committee:

Prof. Dr. Ir. M.J.F.Stive Section of Hydraulic Engineering

Dr. Ir. W.S.J. Uijttewaal Section of Environmental Fluid Mechanics

Prof. Dr. Ir. G.S.Stelling Section of Environmental Fluid Mechanics

Ir. H.J.Verhagen Section of Hydraulic Engineering

Ir. R.J. Labeur Section of Environmental Fluid Mechanics


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The experimental and theoretical study on flow over oblique weirs is the subject of myMaster of Science Thesis. This is the most important part that is required for the MSc

degree of Civil Engineering and Geosciences at Delft University of Technology. The studywas conducted at the Fluid Mechanics Laboratory (Stevin III Laboratory) TU Delft.

First of all I would like to express my deep appreciation of the thorough provision andguidance provided by Dr.Ir. W.S.J. Uijttewaal. During the last year, I have learned a lot

from his inspiring guidance and enthusiastic daily work. I wouldnt have overcome mymistakes and faults without his help. I am also grateful to Prof. Dr.Ir. M.J.F. Stive, Dr.Ir.H.L. Fontijn, and Ir. H.J. Verhagen and for their valuable advices during the first steps,which is also the foundation of this work.

I would like to thank all the members of my thesis committee for sharing their knowledgeand experiences, for their guidance and judgement. Special thank to Prof. Dr. Ir. Stellingfor the introduction of his theoretical analyses on flow over weirs in submerged condition,

on which is computational simulations are based.

Many thanks also to S. de Vree, J.A. van Duin, H. Tas and the laboratory staff; B.A. Wols,

H. Talstra, W.A. Breugem, E.A. van Blaaderen and my colleagues for providing thenecessary facilities and guidance for conducting my work. The devotion of S. de Vree andthe always-positive feedbacks from B.A. Wols greatly contributed to my final thesis.

Last but not least, I would like to acknowledge the sponsors CICAT and TUDELFT, under

the framework of CE-HWRU project, for their financial and other valuable supports during

the time I have been living and studying in Delft. Great thanks to my family and friends forbeing a firm moral support and source of encouragement.


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ABSTRACT

This report is the conclusion of a comprehensive set of experiments, which were

performed on weirs placed obliquely in an open channel. Its purpose is to report onlaboratory investigation on the flow over different types of oblique weirs, including behavior

and hydraulic characteristic of the flow, different phenomena in the neighborhood of theweir, and hydraulic parameters and physical laws that govern the process. The report alsoaims at presenting a quantitative view on the energy loss and the discharge coefficient for

oblique weirs.

To that end, many experiments were performed in a shallow flume under various flow

conditions. Three different types of impermeable weirs are tested, namely a rectangularsharp-crested weir, a rectangular broad-crested weir (both placed 45 degrees obliquely tothe flow direction) and a dike-form weir with both upstream and downstream slopes of 1:4.

The last type was tested with several oblique angles of 0, 45 and 60 degrees with the

incident flow.

By adjusting the flow discharge and the downstream water level, different flow regimesand states reveal the complex three dimensional structure of the flow with variousphenomena like hydraulic jump, undulation, flow concentration, flow divergence, gyre

formation, etc. In case of emerged flow condition, the flow behind weir becomes highlyturbulent and very complex, which make it more difficult to perform accuratemeasurements. This flow regime also accounts for the higher head loss and energy

dissipation than in case of submerged flow. Generally speaking, the hydraulic phenomenathat happen in the neighborhood of an oblique weir are equivalent for different weir form,although there are some remarkable differences such as the size of the recirculation zone

behind weir, and the amplitude of the undulation waves.

Experimental data were obtained by many instruments and techniques; most of them had

been carefully calibrated and were highly accurate and reliable. The data collected fromacoustic and electro-magnetic single point velocimeters and depth measurements were

used to investigate the hydraulic process and the phenomena of interest. Meanwhile thewhole surface flow velocity field was measured using particle tracking velocimetrytechnique, which helps obtaining instantaneous whole field velocity maps. Combine with

mathematic tools we can interpret the data and gain necessary statistical information.

When the oblique angle of the weir is altered, both the flow direction and the flow rate

change. The flow always tends to keep its direction to nearly perpendicular to the weir

crest when it reaches and passes the weir. This leads to the difference in water levels attwo ends of the weir, the flow concentration at on one side of the flume behind weir, the

variation in flow velocity distribution and other asymmetries across the flow. Increasing theoblique angle, the effective length of the weir increases significantly, whereas thedischarge coefficient Cd slightly decreases. Together they make the discharge capacity of

the oblique weir increases.

Finally, the discharge coefficient and its relations to other flow and geometry parameters

obtained from this research were compared to the available knowledge on oblique weirs,including the published researches from De Vries (1959), Borghei et al. (2003) and thenumerical models simulations from Wols (2005). The common findings between

researches enhanced each other reliability; whereas the differences are a motivation for

further studies.


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TABLE OF CONTENTS

..............................................................................................................................i

LIST OF FIGURE...............................................................................................................viiiLIST OF TABLES...............................................................................................................xiiLIST OF SYMBOLS ..........................................................................................................xiii

CHAPTER 1. INTRODUCTION ..................................................................................... 1

1.1. General................................................................................................................... 1

1.2. Problem description ............................................................................................... 2

1.3. Objective................................................................................................................ 31.4. Research method.................................................................................................... 4

1.5. Domain of the study............................................................................................... 5

1.6. Outline of the thesis ............................................................................................... 5

CHAPTER 2. PHYSICAL BACKGROUND AND THEORY ........................................ 6

2.1. Introduction............................................................................................................ 62.2. Flow in open channel ............................................................................................. 6

2.2.1. Basic concepts................................................................................................ 6

2.2.2. Resistance in the flow .................................................................................... 8

2.2.3. Turbulence and energy loss ........................................................................... 9

2.3. Flow over plain weirs........................................................................................... 102.3.1. Introduction.................................................................................................. 10

2.3.2. Flow regimes................................................................................................ 10

2.3.3. Weir geometry ............................................................................................. 11

2.3.4. Energy loss with the present of weir............................................................ 12

2.3.5. Discharge coefficient - Cd and Cdv............................................................... 122.3.6. Discharge formulas ...................................................................................... 13

2.3.7. Energy balance and Momentum balance ..................................................... 15

2.4. Flow over oblique weir ........................................................................................ 17

2.4.1. Oblique sharp crested weir........................................................................... 17

2.4.2. Oblique trapezoidal weir.............................................................................. 182.4.3. Numerical simulations on oblique weirs...................................................... 19

2.4.4. Theoretical analysis of the flow over an oblique weir................................. 20

Conclusion ................................................................................................................... 23

CHAPTER 3. SET-UP AND IMPLEMENTATION OF EXPERIMENTS ................... 24

3.1. Introduction.......................................................................................................... 24

3.2. Experimental parameters ..................................................................................... 243.2.1. Choosing experiment parameters................................................................. 24

3.2.2. Dimensional analysis, scaling and similitude .............................................. 25

3.3. Design of weirs .................................................................................................... 27

3.3.1. Sharp crested weir:....................................................................................... 273.3.2. Broad crested weir: ...................................................................................... 27

3.3.3. Dike form weirs: .......................................................................................... 27

3.4. Major experimental equipments .......................................................................... 28

3.4.1. Measurement site ......................................................................................... 28

3.4.2. Point gauges................................................................................................. 283.4.3. Manometer with flange ................................................................................ 29

3.4.4. ADV Acoustic Doppler Velocimeter ........................................................ 293.4.5. EMF Electro Magnetic Flow meter .......................................................... 30


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3.4.6. Facilities for PTV analysis........................................................................... 31

3.5. Measurements elaboration ................................................................................... 32

3.5.1. PTV measurement........................................................................................ 32

3.5.2. Loss measurement........................................................................................ 323.5.3. Velocity measurement ................................................................................. 33

3.6. Whole field measurements with PIV and PTV.................................................... 343.6.1. General......................................................................................................... 34

3.6.2. PTV processing procedure........................................................................... 35

3.6.3. Calibration.................................................................................................... 363.6.4. Further investigations................................................................................... 37

3.6.5. Limits ........................................................................................................... 38

3.7. Accuracy and tolerance of measurements............................................................ 38

3.7.1. Water depth.................................................................................................. 38

3.7.2. Velocity........................................................................................................ 383.7.3. Discharge ..................................................................................................... 39

CHAPTER 4. RESULTS AND ANALYSIS OF............................................................ 40

LOSS MEASUREMENTS .................................................................................................. 404.1. Loss measurements with plain weir..................................................................... 40

4.1.1. Present losses by flow condition .................................................................. 404.1.2. Analysis of losses......................................................................................... 42

4.2. Loss measurements on oblique weirs .................................................................. 46

4.2.1. Present loss by flow condition ..................................................................... 46

4.2.2. Analysis of loss measurement...................................................................... 49

CHAPTER 5. RESULT AND ANALYSIS.................................................................... 56OF VELOCITY MEASUREMENTS.................................................................................. 56

5.1. Introduction.......................................................................................................... 56

5.2. Surface flow velocity field................................................................................... 56

5.2.1. Perpendicular weir ( = 00 ).......................................................................... 575.2.2. 45

0oblique weir........................................................................................... 59

5.2.3. 600 oblique weir........................................................................................... 65

5.2.4. Flow velocity far downstream of the weir ................................................... 66

5.3. Vertical profiles of velocities............................................................................... 69

5.3.1. Illustration of velocity profiles over flow depth .......................................... 69

5.3.2. Velocity measurement along the flume ....................................................... 705.3.3. Three dimensional flow structure ................................................................ 72

5.3.4. Analysis of the flow structure ...................................................................... 73

5.3.5. The recirculation zone.................................................................................. 75

5.4. Two velocity component analysis........................................................................ 765.4.1. Velocity variation along the flow ................................................................ 765.4.2. Velocity variation across the flow ............................................................... 78

5.4.3. Spatial (2D) variation of the velocity components ...................................... 78

5.5. Angle of obliqueness............................................................................................ 80

5.5.1. General......................................................................................................... 80

5.5.2. Oblique trapezoidal weir, = 450 ................................................................ 825.5.3. Oblique trapezoidal weir, = 600 ................................................................ 83

CHAPTER 6. DISCUSSION.......................................................................................... 85

6.1. General structure and behavior of the flow.......................................................... 85

6.2. Discussions on the discharge coefficients............................................................ 86

6.2.1. The compatibility of available empirical formulas for Cd ........................... 866.2.2. Comparison of results from this research with data from De Vries ............ 87


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6.2.3. Comparison of results from this research with data from Stelling .............. 89

6.3. Projected discharge coefficient ............................................................................ 89

CHAPTER 7. CONCLUSIONS...................................................................................... 93

7.1. Conclusions.......................................................................................................... 937.2. Feasibility of this study and further research ....................................................... 94

7.2.1. Accomplishments of this study.................................................................... 947.2.2. Recommendations........................................................................................ 94

REFERENCES .................................................................................................................... 96

APPENDICES ...................................................................................................................- 1 -APPENDIX A: THEORETICAL ISSUES....................................................................- 2 -

A.1. Turbulence in the flow and Reynolds stresses...................................................- 2 -

A.2. Dimensional analysis, modeling and similitude ................................................- 3 -

A.3. A simple flow model ..........................................................................................- 4 -

APPENDIX B: LOSS MEASUREMENTS AND ANALYSIS ..................................- 11 -

B.1. Trapezoidal weir, = 00

..................................................................................- 11 -

B.2. Trapezoidal weir, = 45

0

................................................................................- 13 -B.3. Trapezoidal weir, = 600 ................................................................................- 15 -APPENDIX C: VELOCITY MEASUREMENTS AND ANALYSIS........................- 17 -

C1. Processing procedure from raw image to time-averaged velocity vector field.- 17 -

C2. Analyses of the velocity vector field.................................................................- 20 -

APPENDIX D: MAJOR MATLAB SCRIPTS ...........................................................- 25 -D.1. theorymain.m ...................................................................................................- 26 -

D.2. theorypro.m......................................................................................................- 27 -

D.3. ImportDataF.m.................................................................................................- 28 -

D.5. ADVpro.m........................................................................................................- 30 -

D.6. PlotFigures.m ...................................................................................................- 31 -

D.7. weirmain.m ......................................................................................................- 33 -D.8. weirpro.m .........................................................................................................- 35 -

D.9. tuyen_calib.m...................................................................................................- 36 -

D.10. calibratePTV.m ..............................................................................................- 37 -

D.11. sline.m ............................................................................................................- 38 -D.12. extract1diagonal.m.........................................................................................- 39 -

D.13. extract1streamline.m ......................................................................................- 41 -

D.14. extract1horizontal.m ......................................................................................- 42 -

D.15. obliqueangle.m...............................................................................................- 43 -

D.16. AnglePrediction.m .........................................................................................- 45 -D.17. AnglePrediction 4560.m ................................................................................- 47 -

APPENDIX E: ARCHIVES OF DATA ......................................................................- 49 -


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LIST OF FIGURE

Figure 1.1: The flow over an oblique weir and its relating phenomena

Figure 1.2: Typical cross section of big rivers in the Netherlands

Figure 2.1: Definition sketch of open channel flow

Figure 2.2: The specific energy diagram

Figure 2.4: The convention of parameters

Figure 2.5: Four main types of flowFigure 2.6: Sharp-crested weir geometry

Figure 2.7: Broad-crested weir geometry

Figure 2.8: Definition of flow zones

Figure 2.9: The relation between C* and S, p=525.Figure 2.10: Cd as a function of Fr1.Figure 2.11: The submergence as a function of Fr1.

Figure 2.12: Plan view of an oblique weir.

Figure 2.13: Free flow

Figure 2.14: Submerged flowFigure 2.15: De Vries experiment

Figure 2.16: Streamlines show the spiral recirculation movement of flow downstream an

oblique weir (450). An illustration from simulations Wols (2005).

Figure 2.17: Velocity component analysis

Figure 2.18: Upstream energy balance

Figure 2.19: Relation between the flow direction and flow condition above the weirFigure 2.20: Relation between the flow direction and oblique angle of the weir

Figure 3.1: The experiment site in the Laboratory.Figure 3.2: Sharp crested weir

Figure 3.3: Broad crested weir

Figure 3.4: Dike-form weir

Figure 3.5: Schematized plan view of the flume and its apparatus

Figure 3.6: The instrument carriageFigure 3.7: Manometer with Pitot tube.

Figure 3.8: The Vectrino probeFigure 3.9: Typical arrangement of the ADV and the bubble generator.

Figure 3.10: The xyz coordinate of the instrument

Figure 3.11: The flow seeded with black tracers.Figure 3.12: Head loss measurement points

Figure 3.13: Positioning velocity measurement points

Figure 3.14: Measurement points over the flow depth.

Figure 3.15: Sliding image process

Figure 3.16: Cross-correlation makes instantaneous velocity vector field imageFigure 3.17: Velocity vector field

Figure 3.18: The streamlines

Figure 3.19: The velocity variation along a streamline


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Figure 4.1: Relation between head loss and Froude number above the weir, = 00

Figure 4.2: Relation between head loss and the relative downstream water depth, = 00

Figure 4.3: Cd, C* and fit curves for C* by VillemonteFigure 4.4: Relation between Cd and submergence, =00, Q=30l/sFigure 4.5: Relation between fit parameter and discharge

Figure 4.6: Relation between discharge coefficient and Froude number above the weir, =0Figure 4.7: Relation between submergence and Froude number above the weir, = 00

Figure 4.8: Relation between discharge coefficient and the relative upstream head, = 00

Figure 4.9: Relation between discharge Cdv and other parameters, = 00

Figure 4.10: Relation between head loss and Froude number above the weir

Figure 4.11: Relation between Fr1 and the relative downstream water depth

Relation between head loss and the relative downstream water depth

Figure 4.12: Measurement data for Cd, C* and Ks with different oblique weirs.Figure 4.13: Relation between discharge coefficient and Froude number above the weir

Figure 4.14: Relation between submergence and Froude number above the weir

Figure 4.15: Relation between discharge coefficient and the relative upstream head

Figure 4.16: Discharge coefficient with different oblique angles

Figure 5.1: Flow velocity vector field, =00, Q=30l/s, Emerged flowFigure 5.2: Flow velocity vector field, =00, Q=30l/s, Undulated flowFigure 5.3: Flow velocity vector field, =00, Q=25l/s, Submerged flowFigure 5.4: Variation of the total velocity, =00, Q=25l/s, Submerged flowFigure 5.5: Flow velocity vector field, trapezoidal weir, =450, Q=30l/s, Emerged flowFigure 5.6: Flow velocity vector field, trapezoidal weir, =450, Q=30l/s, Undulated flowFigure 5.7: Flow velocity vector field, =450 , Q=30l/s, Submerged flowFigure 5.8: Variation of the total velocity, =450, Q=30l/s, Undulated flowFigure 5.9: Flow velocity field for sharp crested weir, =450, Q=35l/sFigure 5.10: Velocity along the middle streamline, =450, Q=35l/s, undulated flowFigure 5.11: Flow velocity field for broad crested weir, =450, Q=35l/sFigure 5.12: Velocity along the middle streamline, =450, Q=35l/s, emerged flowFigure 5.13: Velocity along the middle streamline, =450, Q=35l/s, undulated flowFigure 5.14: Standing wave on top of a broad-crested weir, =450, Q=35l/s

Figure 5.15: Flow velocity vector field, =600, Q=40l/s, Emerged flowFigure 5.16: Flow velocity vector field, =600, Q=35l/s, Undulated flowFigure 5.17: Flow velocity vector field, =600, Q=40l/s, Submerged flowFigure 5.18: Variation of the total velocity, =600, Q=20l/s, Undulated flowFigure 5.19: Downstream flow velocity vector field, =450, Q=30l/s, emerged flowFigure 5.20: Downstream flow separation, =450, emerged flowFigure 5.21: Downstream velocity distribution across the flume, =450, Q=30l/s, emergedflow.

Figure 5.22: Downstream velocity distribution across the flume, =450, Q=30l/s,submerged flow.

Figure 5.23: Velocity distribution over the flow depthFigure 5.24: Relation between U and z


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Figure 5.25: Positioning the measurement points

Figure 5.26: Surface flow velocity along the center streamline

Figure 5.27: Vertical velocity profiles at several sections downstream of the weir

Figure 5.28: Vertical velocity profiles, V and WFigure 5.29: Velocity variations along the streamline and over the depth.

Figure 5.30: Velocity profiles on the downstream and on the slope of the weir.Figure 5.31: The left-right difference in water depth above the weir

Figure 5.32: The recirculation zone behind a trapezoidal weir, q=0.15m2/s

Figure 5.33: The recirculation zone behind a sharp-crested weir, q=0.15m2/sFigure 5.34: Velocity components analysis for different flow regimes.

Figure 5.35: Velocity variation with different weir configurations

Figure 5.36: Velocity variation along the weir crest

Figure 5.37: Spatial velocity distribution

Figure 5.38: Spatial distribution of velocity components

Figure 5.39: Spatial distribution of oblique angle, =450, Q=30l/s, submerged flowFigure 5.40: Spatial distribution of oblique angle, =450, Q=30l/s, undulated flowFigure 5.41: Chaos downstream of an oblique angle, =450, Q=30l/s, emerged flowFigure 5.42: Spatial distribution of oblique angle, =600, Q=40l/s, emerged flowFigure 5.43: Result of oblique angle analysis for the 450 oblique weir

Figure 5.44: Relation among , and flow conditions.Figure 5.45: Result of oblique angle analysis for the 60

0oblique weir

Figure 5.46: Relation among , and flow conditions.

Figure 6.1: Relation between Cd and S by different formulas, = 450

Figure 6.2: Comparison of formulas from Borghei and Villemonte

Figure 6.3: Comparison of discharge coefficientsFigure 6.4: Comparison of Cdv with several oblique angles from De Vries (1959).

Figure 6.5: Figure 6.4: Comparison of Cdv for plain weir

Figure 6.6: Velocity component parallel to the weir, =450, Q=30l/s, submerged flow.Figure 6.7: Variation of the perpendicular velocity component along the weir crest,

=450 , submerged flow.

Figure A1: The stress tensorFigure A2: Flow model with its parameters

Figure A3: Cd(R) with and without bottom friction

Figure A4: Cd(R) for different specific dischargesFigure A5: Comparison of Cd(R) calculated by the complex method

and by the empirical formula of G.S.StellingFigure A6: Differences between flow parameters calculated by different methods

Figure C1: The flow seeded with black tracers (original image, captured by the camera)

Figure C2: Ensemble-averaged picture (background picture)Figure C3: The subtracted picture

Figure C4: Instantaneous velocity vector field

Figure C5: Fix-grid-points velocity vector field

Figure C6: Fix-grid-points, time-averaged velocity vector field


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Figure C7: Velocity vector field with streamlines

Figure C8: Velocity variation a long one streamline (the center streamline)

Figure C9: Variation of a velocity component a long the weir crest

Figure C10: Variation of different velocity components a long a cross-sectionFigure C11: Spatial distribution of the total velocity

Figure C12: Contours of different velocity componentsFigure C13: Contours of the angle of obliqueness of the flow


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LIST OF TABLES

Table 3.1: Experiment variables

Table 4.1: Measurement from experiment with Q = 40l/s, = 00.Table 4.2: Cdv with different discharges and different oblique angle

Table 5.1: Measurements of velocity profiles

Table 5.2: Measurements points by the ADV

Table 5.3: Extra measurements for oblique angle, = 450

Table 5.4: Extra measurements for oblique angle, = 600

Table B1: Loss measurements for trapezoidal weir, = 00

.Table B2: Loss measurements for trapezoidal weir, = 450.Table B3: Loss measurements for trapezoidal weir, = 600.

Table D1: List of presented Matlab scripts


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LIST OF SYMBOLS

Symbol Parameter Unit

A area of the wet section of the flow m2

aw weir height m

B flume width (channel width) m

C Chezy resistance coefficient m1/2/s

C* discharge reduction coefficient -

Cd discharge coefficient -

Cdv dicharge coefficient for perfect weir (free flow or emerged flow) -

d water depth w.r.t the channel bottom md flow depth m

E specific energy m

f Weisbach resistance coefficient

Fr Froude number -

g gravitational acceleration m/s2

H total head m

h the water depth w.r.t. the weir crest m

i bottom slope -

K calibration factor (PTV analysis) -

Ks submergence coefficient -

L length of the weir m

Lw length of the weir crest (in the flow direction) m

m weir slope -

mk discharge coefficient for the weir -

ms discharge coefficient for the channel -

n Manning resistance coefficient

p

fit parameter (Villemonte) [-]; atmospheric pressure [m]; reduction

factor of the discharge coefficient due to the obliqueness of the flow

(De Vries) [-]

-

P wet perimeter of the flow m

Q total discharge m3/s

q specific discharge m2/s

R hydraulic radius m

Re Reynolds number -

S submergence; flow resistance slope (the energy slope or friction slope) -

Uflow velocity component perpendicular to the flume axis (PTV

analysis)m/s

u flow velocity m/su* the shear stress velocity m/s


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V mean flow velocity 0

V flow velocity component parallel to the flume axis (PTV analysis) m/s

V depth-averaged flow velocity m/s

x longitudinal coordinate (of the flume) my flow depth m

y lateral coordinate (of the flume) m

z vertical coordinate (of the flume) m

z0 bed elevation m


kinetic energy correction coefficient - angle of obliqueness of the flow streamline 0

H energy loss mh head loss m dynamic viscosity kg/(m.s)

oblique angle of the weir 0

Von Karman constant -

kinematics viscosity, =/ m2/s bed slope 0

mass density of water kg/m3

shear stress N/m2

relative weir crest length -

Frequently used subscripts:

Subscript Description

0 upstream of the weir

1 on top of the weir crest

2 downstream of the weir

bw for broad-crested weir

c critical flow condition

d downstream of the weir

d discharge

dv discharge for perfect weir

e effective

for oblique weirl parallel component

max maximum value

min minimum value

p perpendicular component

s submerged flow condition

sw for sharp-crested weir


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MSc. Thesis Flow over Oblique weirs

Page 1

CHAPTER 1. INTRODUCTION

1.1. GeneralWeirs are among the most popular and simple hydraulic structures. They can be used forvarious purposes, from flow measurement and diversion, energy dissipation to regulation

of flow depth, and many others. Weirs can be encountered in real life in various types and

shapes, and they can sometimes be placed obliquely to the incident flow in order to

increase the efficiency. Besides, many obstacles in a flood plain can act as weirs during a

high water period, for example the summer dikes, roads with high foundation, groynes orbarrages. In fact, those weir-like obstacles can hardly be considered as plain

(perpendicular) weirs.

The characteristics and hydraulic behavior of plain weirs or standard weirs have been

studied for a long time and the understanding on them is rather deep. We can easilycalculate the parameters of the flow using the conventional weir equations and formulae.

However, few studies have been done on weirs placed obliquely in the flow. Sufficient

information on the discharge coefficient, energy loss and other behaviors of the flow over

an oblique weir is still not available. Up to now, there is no commonly accepted design

discharge equation for an oblique weir.

Probably the first rational approach to studying flow over oblique weirs was published by

De Vries in his report in Dutch (Scheef aangestroomde overlaten, April 1959). The main

objective of the research was to examine the influence of the obliqueness of the weir to the

flow. Experiments were done on trapezoidal weirs.

More recently, Borghei (2003) and his co-authors published their study on the discharge

coefficient for oblique rectangular sharp-crested weir. Their research demonstrated the

influence of the oblique angle to the discharge coefficient Cd, resulting in a single formula

for Cd, and correction coefficients for submerged flow.

Since then, many researchers have investigated the weir discharge coefficient with the

main channel upstream Froude number. However, sufficient information on the variation of

the coefficients used in their equations is still not available. Up to now, there is no

commonly accepted design discharge equation for an oblique rectangular sharp crested

weir.

Furthermore, we need some more understanding of the physical processes that are

predominant in a flow over oblique weirs in various forms, which has hardly been

described or published. By means of qualitative description on what happens in the vicinityof oblique weirs, on the weir crest, and with flow structures down stream, this report tries

to bring a fresh and further look for those interesting hydraulic phenomena. It also aims at

quantitatively describing the absolute energy loss and discharge coefficients for the flow

over oblique weirs. Experiments were performed on physical models of some common

types of weir, including a sharp crested weir, a broad crested weir and trapezoidal weirsunder several different oblique angles. This research was done in the laboratory of Fluid

mechanics, faculty Civil Engineering and Geosciences, Delft University of Technology.


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Page 2

Flow

Obliquely placed weir Flow seperation

Stream linesDeflection of flow

Flow convergence

The flume

Circ

ulatio

n

Figure 1.1: The flow over an oblique weir and its relating phenomena

1.2. Problem descriptionThe recent high waters in the Netherlands, for example the flood in 1995, have raised a

number of matters in river flow and the concern of public about the security against

flooding. In 1996, the policy Room for the river had become effective, which aimed at

increasing the capacity of the large rivers and reducing the impact of high waters. Thetypical cross section of a river in the Netherlands often comprises of winter dike, summer

dike and other constituents (see figure 1.2). Summer dikes are part of the foreland as

mentioned in the dike law. Their main functions are water control (protection from

inundation), restriction of morphological impacts by the outer flow, and accessibility

(playing the role of roads). During the flooding periods, summer dikes however reduce thedischarge capacity of the river. The summer dikes will be overflowed under such extreme

discharges, and part of the river discharge flow through the winter bed with several weir-like obstacles mentioned earlier. River managers are now seeking for the understanding of

discharge capacity, energy loss and other characteristics of the flow concerning the summerdike and those obstacles. Generally speaking it has the form of a flow over an oblique weir

because of the randomness in their orientation. However, the influence of the obliqueness

on energy loss and flow rate has hardly been examined.

Built up areaAgriculture

Primari river dikeWinter dike Winter dike

Sidechannel

Summer dike

Groynes

Summer dike

Figure 1.2: Typical cross section of big rivers in the Netherlands

The above problem is also essential for Vietnam as our country has several big river

systems with their social-economically important flood plains. Especially in the South hub,where the great Mekong river has hardly any primary river dike protection, but a lot of

regional dikes, roads, structures and other obstacles.


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What is also important for Vietnam in the current situation is the safety of sea dikes and the

flooding problems caused by storm surges. Researches on the damage of the Damrey

typhoon, which landed in September 2005 to the Northern provinces of Vietnam, has

revealed a fact that the design height of the sea dikes was too low that their failure was notonly due to wave run-up and over topping, but also because of over flowing. The

combination of the shape of the local coast line, the current, storm surge, and tide make theflow over sea dikes, which can be interpreted as flow over weir, not always perpendicular

to the dike axis. Research on flow over oblique weirs is an urged need from practice.

1.3. Objective

The first purpose of this research is to qualitatively describe the structure of the flow over

some main types of oblique weirs and other complex phenomena related to it. The flowsbehaviors, hydraulic characteristics, and their inter-relation are therefore the objectives of

investigation. We will also examine the physical processes which play a role, how they

influence the flow over an oblique weir, and if there are common laws that govern this

behavior.

Secondly we need to quantitatively determine the energy loss and discharge coefficient

concerning the flow in those cases, i.e. to determine the absolute size of losses and theirs

relation with other flow and geometry parameters, especially the oblique angle. Those to

serve drawing conclusions as to what respect the flow over an oblique weir deviates from

the flow over a perpendicular weir. Finally the result will be checked with availableknowledge on flow concerning oblique weirs.

To express the purposes more specifically, the discharge coefficient Cd and its relation to

other parameters is one of the main objectives. Variations of Cd under different oblique

angles need to be investigated. The experimentally determined correction factor Cd is usedto account for the energy loss not included in the simplified analysis to get the actual

discharge (Q) as a function of the water head upstream (H):3

22

23

dQ C B g H

The oblique angle of the flow streamlines over a weir is also the main object of the

experiments. The change in the direction of the flow in different flow regimes under the

influence of several parameters is to be investigated. The relation between the oblique

angle of the flow streamlines (), the oblique angle of the weir (), and the Froude number

(Fr) of the flow is especially interesting.

= f(, Fr0, Fr1)

where Fr0 is the upstream Froude number, Fr1 is the Froude number above the weir crest.

Velocity components analysis contributes to the understanding of the physical process that

governs flows over an oblique weir. Earlier studies on oblique weirs have revealed

interesting facts that need to be checked and studied more detailed where possible. Besides,

the flow downstream of an oblique weir has a complex three dimensional structures. Itshows many turbulence properties that contribute to the transfer and dissipation of energy.

One example is the flow with a recirculation zone behind weir. These phenomena need to

be described qualitatively and their effects determined qualitatively.


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1.4. Research method

The flow structure in the presence of a weir is generally three dimensional and

complicated. In case the weir is placed obliquely, it becomes even more complex with a

variety of interesting phenomena and their underlying physical laws. An approach to this

problem is found through the use of a combination of analysis and experiments.

In order to achieve the main objectives of this study, we need to accurately measure the

flow properties, examine the phenomena that happen in the neighborhood of the weir, and

investigate the relationships between them. To effectively perform those tasks, we can take

refuge in modern instruments and techniques. One of those is the PTV (Particle Tracking

Velocimetry) method with the capability of extracting the underlying flow velocity field. Ithelps giving us the instantaneous as well as the time averaged surface velocity vector maps

of the whole flow area concerning the weir. For further investigation into the flow

structure, an ADV (Acoustic Doppler Velocimeter) has been used for point measurements

of all three components of the velocity vector.

The PTV analysis mainly gives an overview of the free surface flow velocity field.

However, the weir configurations and arrangements in the flume cause the flow to show

three-dimensional complexity, particularly downstream of the weir and under emerged

flow conditions. In order to understand the flow characteristic and be able to describe its

structure, we need additional information from other experimental techniques andmeasurement methods.

The flow can be represented by simple models, based on physic and basic equations: the

energy balance, momentum balance and continuity equations. Given the input parameters,

the flow parameters of interest can be analytically determined. These outputs can be the

water depth, specific discharge, oblique angle, and many others. This kind of analysisproved to be indispensable during the course of this research. As mentioned earlier, there

was also a more extensive and sophisticated study on 3D computer modeling of flow over

oblique weirs carried out by Wols (2005).

The comparison of experimental results with model outcomes is of importance in order to

see if the phenomena that occurred in the flume can be reproduced in the model, and to

determine experimental correction coefficient for the calculation of model. Its also

important to use the model to verify the data obtained from the experiments, and to use

these to adapt and improve the model. The comparison of the two helps us to gain goodunderstanding on the underlying governing physical laws.

In this research, a number of experiments have been carried out to fulfill the objectives:

- Measuring loss with several weir geometries and configurations, followed by

inferring the relation between the loss and flow regimes as well as otherparameters. These data were also used to determine the discharge coefficient, an

important and widely used parameter, and its relations to other. Part of these data

was also helpful when comparing with available knowledge on plain weirs to verify

the reliability and accuracy of the physical model, with which we performed further

investigations.- Measuring the surface velocity field for the free, submerged and transition flow

regimes over all those weirs by means of PTV analysis. For these cases we have an

overview of the surface flow and can reveal some large scale flow features.


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- Measuring the velocity distribution over the depth and along the flow for a limited

number of cases. This helps depicting the flow structure and checking some other

parameters such as the roughness of the bottom, the discharge of the flow, the

agreement between different measurement instruments and methods (ADV andPTV). In some cases, this is the only available method to investigate the

recirculation zone downstream of the weir.

The final step is the evaluation of the experiments, analysis of the data and discussion. The

flow conditions are compared to each other; results from experiment were also compared tothose from the numerical simulations. This results in an overview of the flow structure and

other phenomena.

1.5. Domain of the study

In order to make the goals clear, some limits for the experiments in this research will be

listed:

- The research will mainly focus on measuring and analyzing the energy loss

(including discharge coefficient) under different weir configurations and flowconditions.

- During the research, only a limited number of parameters were varied, namely the

oblique angle, the discharge, the downstream water level, the weir form (sharp

crested, broad crested and dike form). Other parameters were kept unchanged, for

examples the weir height, the bottom roughness and slope, the crest width and slopeof the dike-form weir.

- For the rectangular weirs, only the oblique angle of 45 degrees is examined. For

the trapezoidal weir, the oblique angle was varied between 0, 45 and 60 degrees.

1.6. Outline of the thesis

In chapter 2, background information is given. In this chapter we can find the basicdefinitions for parameters and phenomena, equations and formulae on open channel flow,

flow over a weir and an overview of the published research on flow over oblique weirs.

Chapter 3 provided us with the experimental setup, the choice of parameters to investigate

and the most important experimental equipment. It also introduces the experimental

elaboration, data collection and techniques used for analyzing. Result and analysis of lossmeasurement and velocity measurement is presented in chapter 4 and 5 respectively.

Chapter 6 compares the experimental data and results with the available knowledge on

oblique weirs. Finally the conclusions and recommendations are presented in chapter 7.


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CHAPTER 2. PHYSICAL BACKGROUND AND THEORY

2.1. IntroductionThis chapter presents the theories and analysis that are important to this study. Experimentswere carried out on physical models in a horizontal rectangular flume, thus the basic

knowledge on open channel flow is indispensable. Energy concepts and the way energy is

dissipated, is briefly presented subsequently. We then take a closer look on flow over

special types of weirs, including definitions of parameters that characterize the flow

phenomena that occurred, as well as the basic equations. Last but not least, the availableknowledge on flow over oblique weirs from published researches and an analytical study

on oblique weirs are briefly introduced. This information on oblique rectangular sharp

crested weir and trapezoidal weir will later be the reference for this study results.

2.2. Flow in open channel

2.2.1. Basic concepts

The flow in a prismatic (or rectangular) open channel with constant bed slope can be

schematized as follow:

V

H

Ed.cos

d

Channel bottom

Datum

z0 0z

V /2g2

z y

Total headline

V

Fee surface

Figure 2.1: Definition sketch of open channel flow (Chanson, 2004)

In such a flow, the mean energy head is defined as:2

0cos 2

V

H d z (2.1)When the velocity varies across the section, the mean total head is corrected as:

g

VdzH

2cos

2

0 (2.2)

Where H : total head

: bed slopeV : depth-averaged flow velocity

g : gravitational acceleration

d : water depth (d.cos is the pressure head)z0 : bed elevation(z0 is the potential head)

: The kinetic energy correction coefficient.V2/2g : the kinetic energy head


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The Bernoulli equation describing energy conservation along a streamline is:

constg

VdzH

2cos

2

0 (2.3)

The specific energy:

The specific energy is the flow energy with regard to the channel bed as datum. It reads:

g

VzHE

2

2

0 (2.4)

If we consider a rectangular channel of width B and horizontal bottom, the specific energycan be written in terms of flow rate per unit width, q = Q/B, as:

2

2

2gy

qyE (2.5)

In which Q is the total discharge, y is the flow depth.

Flow states:For the channel in our experiment (constant width), with a given discharge Q, the flow rate

q remains constant along the flow meanwhile the depth y varies. The so called specificenergy diagram, for E = E(y) with a fixed value of q is shown in the following figure.

For a given value of flow rate and specific energy,

the flow may have two possible depths, which are

termed alternate depths. These two depths, whichare presented on two branches of the E = E(y)

curve, are characteristics of two different regimes of

the flow, namely super-critical flow (at small depth

y1) and sub-critical flow (at larger depth y2). Point C

in the diagram represents the transition state of theflow between the two regimes - the critical state. It

can be defined as the state at which the specific

energy is minimum for a given flow rate.

Figure 2.2: The specific energy diagram

Critical state of the flow:

By differentiation to find the minimum of the specific energy (Emin) from equation (2.4),

we obtain the value of the critical depth. Then substituting this value of yc back into the

equation we obtain Emin.

3

2

g

qyc (2.6) cyE

2

3min (2.7)

This means at the critical state of flow, the velocity head is equal to half the hydraulic

depth. The critical velocity is:

c

c

c ygy

qV (2.8) 1

yg

V, this dimensionless parameter is

defined as the Froude number.

Froude number:

The Froude number is important in open channel flows. It is defined as the ratio of the low

velocity (V) to the wave velocity (c).


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yg

V

c

VFr

(2.9)

The flow state is super critical when Fr > 1 and sub critical when Fr 1, i.e. V < c, the flow velocity is smaller than the wave velocity, its fluid

moving in a tranquil manner, the waves or disturbances at downstream can propagate

upstream. Thus the upstream locations are in hydraulic communication with the

downstream locations. On the other hand, when Fr > 1, i.e. V > c, the stream is moving

rapidly so that the flow velocity is greater than the wave speed. In this case, nocommunication between upstream locations and downstream locations is possible

(Munson, 2003); the flow is not influenced by the downstream water depth.

2.2.2. Resistance in the flow

Energy loss in open channel flow is due to hydraulic resistance, including wall roughness,viscosity, cross-sectional shape, boundary non-uniformity, the unsteadiness of the flow,

and turbulences. General speaking, the resistances can be divided into two distinction types

so called surface resistance and form resistance. The surface resistance is caused by the

roughness of the bottom, the side walls and the weir surface. The form resistance is caused

by the local acceleration, deceleration and stagnancy of the flow. Normally the formresistance is much larger than the surface resistance, thus the surface resistance can often

be neglected (Bloemberg, 2001).

In an open channel flow, energy is continually dissipated, whether the flow is laminar (at

low Reynolds number, Re < 1000) or turbulent (Re >1000), eventually the energy is

converted into heat due to the effect of viscosity.

There are several formulas to calculate the energy loss in an open channel flow without a

weir. The most frequently used formulas relating the mean flow velocity to resistance

coefficient are:

V C RS (Chezy) (2.10)2/13/249.1

SRV (Manning) (2.11)

f

gRSV

8 (Darcy Weisbach) (2.12)

Where

C, n, f : the Chezy, Manning and Weisbach resistance coefficients respectively.

R : the hydraulic radius of the flow, R = A/P is the ratio between the cross

sectional area and the wet perimeter of the flow.

S : the flow resistance slope (the energy slope or friction slope)

Here in this study, because of some objective difficulty, the loss in the water depth of the

flow due to resistances cannot be measured. Hence the result from an earlier research of

Uijttewaal and Booij (2001) on the same flume and typical flow conditions will be used.

For both low upstream water depth (42mm) and high upstream water depth (67mm), the

loss over five meter distance approximates 1mm.


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The bottom shear stress:

Starting from the momentum balance for a control volume in a schematized straight open

channel flow, we come up with a general expression for the shear stress at a vertical

position below the free surface:

izhg (2.13)where i is the bottom slope and h is the flow depth. At the bottom of the channel, z = 0, thebottom shear stress then is:

2

0

1. .

2fh i c u (2.14)

thus define a friction coefficient.

2.2.3. Turbulence and energy loss

Turbulent is one of the most prominent properties of the flow because most of the flows

encountered in nature and technical installations are turbulent flows. It is the manner of

flow in which various flow quantities show a random variation with time and space. The

two most important types of turbulence are free turbulence and wall turbulence. Turbulenceis capable of transfer energy of the flow from the mean motion to the turbulent fluctuations.Those fluctuations will then induce turbulent shear stresses, and energy will eventually be

dissipated into heat via the viscosity. (Uijttewaal, 2006)

Reynolds number:

Turbulence will appear in flows with high Reynolds number (Re > 1000) with the presence

of a velocity gradient. In the experiments of this study, surface resistance and formresistance were present due to the bottom friction, wall friction, and the presence of a weir

in the flow. The flow in these experiments was completely turbulent due to a high

Reynolds number,Re always higher than 2104.

The Reynolds number by definition is:

LU.Re (2.15)

Where U : the characteristic velocity differenceL : characteristic length

: kinematics viscosity, =/ : dynamic viscosity : mass density of the fluid (water).

Energy dissipation:

There are two main ways for the dissipation of energy in the flow, one is the viscousdissipation induced by the mean motion, and the other is the dissipation via the turbulent

motion. By considering the energy equations of the flow, we can see that the loss of energy

from the mean motion is dominated by turbulence, and the work done by the Reynoldsstresses is much bigger than the direct dissipation due to viscous. Beside, for a flow at high

Reynolds number the turbulent shear stresses are generally much larger than the viscous

shear stresses. That means the dissipation by work done against turbulent shear stresses

also has a much larger contribution to the energy loss in the flow (Uijttewaal, 2006).


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2.3. Flow over plain weirs

2.3.1. Introduction

This paragraph deals with the question of how the flow over a weir is described and

calculated. First the flow over a weir will be classified in to different regimes for an easier

interpretation. Then different discharge formulas for those flow regimes will be introducedfor different weir geometries and flow regimes. Lastly, the equations to evaluate the energy

loss will be presented.

The figure below provides a typical view of the flow over a weir. It is followed by the

convention of the relevant flow and geometrical parameters.

i=0

Figure 2.4: The convention of parameters

Notation explanation:H : energy height

h : the water depth with relative to (w.r.t.) the weir crest

d : water depth w.r.t the channel bottom

m : weir slope

Lw : weir crest length (in the direction of the flow)aw : weir height

H : the energy lossQ : total discharge

i : bottom slopeSubscripts: 0: upstream; 1: above the weir crest; 2: downstream.

2.3.2. Flow regimes

The classification of flow regimes is really necessary and helpful for the experiment

process. An early classification by Escande (1939) divides the flow into four types.Although his classification refers to the cylindrical crested weir but it can be equally

applied to all other overflow structures. Recently a study by Fritz and Hager (1998) has

strengthened this classification by extending it with a trapezoidal weir with the downstream

slope of 1:2.

Depending on the downstream water depth, the flow regimes can be divided into four main

types (Escande, 1939). In the order of raising downstream water depth, they are:

- (A) Hydraulic jump: with a clear bore (roller) located at or downstream of the weir

structure. In this case, the flow is not influenced by the downstream water depth.

- (B) Plunging flow: with the main stream following the downstream slope of theweir and with a clear surface roller. The flow is almost independent from the

downstream water depth.

- (C) Surface wave flow: with the main stream following the free and wavy surface

and a recirculation zone near the bottom.


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- (D) Surface jet flow: the flow depth is larger; the surface is nearly horizontal and

smooth. The flow strongly depends on the downstream water depth.

(A) Hydraulic jump (B) plunging flow

(C) Surface wave f low (D) surface jet flow

Figure 2.5: Four main types of flow

For the sake of simplicity, the flow is divided in to three regimes. According to Kolkman(1989), they are:

(1) Free flow: the flow is entirely independent of the downstream water depth. Aroller can be observed. (A combination of types A and B)

(2) Submerged flow: the flow is influenced by the downstream water depth. Theflow surface is nearly horizontal. (Type D)

(3) Transition regime: analogous to type C. It is also called the Undulated flow.

As we will discuss later, the transition flow regime also has the clearest backward

movement behind weir, the so called recirculation, in comparison with the two other flow

regimes. In other words, the size of the recirculation zone in the case of transition state

flow has the biggest size.

2.3.3. Weir geometry

Sharp crested weir:

The weir is termed as sharp-crested if the

length of the weir in the direction of flow (L)

is such that H/L >15 (Bos, 1976). [In practice,the crest length of the sharp-crested weir is

usually less than 2.0 mm (French, 1985),

requiring H to be > 30mm].

Fig.2.6: Sharp-crested weir geometryBroad crested weir:By definition, a broad-crested weir is a

structure with a horizontal crest above which

the fluid pressure may be considered

hydrostatic (Fig.2.5). The following inequality

must be satisfied for such weirs (Bos, 1976):

5.008.0 1 wL

H

Fig.2.7: Broad-crested weir geometry

Dike form weir (trapezoidal weir):

This type of weir is chosen to be investigated further than the two types above because of

the problems related to summer dikes and other obstacles in the flood plain, as described

H

.

.

Nappe

Weir plate

Q

Drawn down

a w

d2aw

d0

L

h0

V0

Weir block

h1V1

w


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earlier. The upstream and downstream slopes of the weir are chosen as 1:4 according to the

Dutch norm for dike design.

2.3.4. Energy loss with the present of weir

The energy loss in the flow over a weir is due to incomplete conversion of kinetic energy

into potential energy (Ref.1). Upstream of the weir, in the acceleration area, part of thepotential energy of the flow is converted into its kinetic energy. While the water is flowing,

it continuously losing energy against surface resistance, but this loss is negligible small

with regard to the loss in form resistance. The mechanism of form resistance here is: part ofthe kinetic energy of the flow is converted into energy of vortices of all scales, and

turbulent fluctuations; then this supply of energy will eventually be dissipated into heat via

the viscosity.

The following figure illustrates different zones in the flow over a weir according toHoffmans (1992).

Figure 2.8: Definition of flow zones

As mentioned earlier, the energy loss of the flow over a weir, or form resistance, is much

larger than the loss in surface resistance. In the existing theories, discharge relations are

often used to express the energy loss. The discharge value can be used to calculate thewater depth and velocity of the flow. Knowing the water depth and velocity of the flow up

and downstream of the weir, the decrease in energy height can be calculated. The discharge

itself strongly depends on the downstream water level (depends on which flow regimes)

and the weir geometry. Later in chapter 4 of this report, the discharge coefficient and itsrelations with other flow parameters will be the subject of investigation.

2.3.5. Discharge coefficient - Cd and Cdv

Because of the complex nature of the flow, it is impossible to obtain a clear expression for

the flow rate as a function of other relevant parameters. An experimentally determined

correction factor Cd was then used to account for the various real world effects not included

in the simplified analysis to get the actual flow rate as a function of the water headupstream.

We denote Cd as the discharge coefficient for submerged flow and Cdv as the dischargecoefficient for the free flow (Cd perfect). For the physical meaning, Cdv represents the

influence of the weir geometry on the maximum discharge over weir, meanwhile Cdrepresents the combined influences of the weir geometry and the downstream water head

on the discharge over weir.

The discharge coefficient in case of a free flow (or so called the perfect weir) Cdv is

independent of the downstream water level and depends on the upstream flow condition

only. It is given by:


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003

2.

3

2gHHB

QCdv (2.22)

For an imperfect weir, the discharge coefficient depends on the downstream flow water

head, it decreases when the water head increases. Also there is no explicit relation betweendischarge and water head for the case of submerged flow and flow in transition state. A

discharge reduction coefficient C* is introduced to account for the reduction in thedischarge coefficient of the perfect weir.

dv

d

C

CC * (2.23)

C* itself is a function of the weir geometry and

submergence (S=H2/H0). There are many formulas to

determine C*, for example the analytical formula

applies for sharp weir by Seida and Quarashi(Kolkman, 1989), the semi-empirical relation by

Villemonte (Kolkman, 1989), Varshney and Mohanty

(Kolkman, 1989). Nowadays, the most suitable and

widely used formula is an entirely empirical formula

by Villemonte (1947):P

SC 1* (2.24)Figure 2.9: The relation between C* and S, p=525.

The fit parameter p depends on the geometry and configuration of the weir. The above

figure illustrates the relation between C* and Sfor different values of p.

The study of Bloemberg (2001) has given some relations of Cd and Cdv with otherparameters, i.e. Cdv decrease with either increase the weir downstream slope m2, or increase

the relative weir height (a/H1), or decrease the relative weir length (H1/Lw). (Ref.1). A

function for Cdv is given by Hager (1998):

55.0sin06.043.0

3

1

3

2 dv

C(2.25)

Where wLH

H

1

1 is the relative weir crest length. When 0.1


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Sharp crested weir

The energy conservation equation for a weir is:

2

3

..23

2HBgQ (2.26)

Numerous approximations have been used to obtain the above equation, namely the neglectof the viscous effects, turbulence, non-uniform velocity distributions, and centripetalaccelerations in the derivation of this discharge equation. To obtain the actual flow rate as a

function of weir head, an experimentally determined correction factor (Cws) must be used.

The final form, suggested by Kindsvater and Carter (1957) is:

2

3

22

. eews hgBCQ (2.27)

where Be : the effective width of the weir

he : the effective water head

Cws : the (dimensionless) effective discharge coefficient.Cws is a function of Reynold number (viscous effects), Weber number (surface tension

effects), and H/Pw (geometry parameter) (where aw is the height of the weir). In most

practical situation, the Reynold and Weber number effects are negligible, and the following

formula can be used (Rehbock, 1929):

w

wsa

HC 075.0611.0 (2.28)

Broad crested weir

The equation for discharge is again (2.26). To account for various real world effects not

included in the simplified analysis, again an empirical weir coefficient, Cwb, is used

(Munson, 2002):

2

3

3

2

3

2. HgBCQ wb (2.29)

The broad-crested weir coefficient Cwb can be obtained from the formula (Munson, 2002):

w

wb

a

HC

1

65.0(2.30)

The broad-crested weir is considerably more sensitive to geometric parameters in

comparison with the sharp-crested weir. Those parameters include for example the surface

roughness of the crest and the shape of the leading-edge nose (sharp or rounded).

Submerged flow

For the submerged flow, the submerged coefficient Ks should be introduced into equation2.27 and 2.29 to produce submerged discharge Qs. The general formula is:

QKQ ss . (2.31)

Since 2/303

2

3

2. HgBCQ ds and

2/30

3

2

3

2. HgBCQ

vd

dv

ds

C

CK (2.32)

Thus KS can be interpreted as the discharge reduction coefficient C* in the formula ofVillemonte. They can be compared to each other for the same flow condition.


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Brater and King (1976) suggested the following form for the experimental coefficient Ks(for a plain weir):

385.0

2

3

0

2

1

H

HK

s (2.33)

Wu and Rajaratnum (1996) also suggest a formula (for plain weir):

0

21

0

2sin.331.1162.11

H

H

H

HKs (2.34)

Where H0 and H2 is the upstream head and downstream head respectively. Here the

downstream condition starts playing a role.

2.3.7. Energy balance and Momentum balance

The main mechanisms governing the flow over a weir are gravity and inertia; other effectslike viscous and surface tension are of secondary importance. As mentioned earlier, for a

steady flow, the Bernoulli equation, which is an energy equation, can always be applied

taking into account certain portion of energy losses. Providing proper allowance for all

forces acting, the momentum equation always holds true as well.

Generally speaking, in the analysis of a flow, the energy and momentum equations play a

complementary role for each other. Usually the information which is not supplied by one

equation is provided by the other. Together with the continuity equation, they provided the

analytical means to assess the energy loss in the flow over a weir.

Energy balance:

Upstream of the weir, the energy is usually assumed to be constant. If we consider a reach

of only two meters from weir center, usually the loss due to bed friction is small, and the

constriction of the flow when it reaches the weir causes almost no loss. The Bernoulli

equation for two cross sections one upstream and one above the weir, with the level of the

weir crest as the reference level, reads:

1

2

110

20

022

pg

uhp

g

uh (2.35)

Usually the atmospheric pressure is constant in the laboratory, p0 = p1. The upstream total

head then can be written as a function of flow condition above the weir:

1

21

1

21

1022

hFr

hg

uhH (2.36)

21

21

10

FrhH (2.37)

Downstream from the weir, the energy is no longer conserved. Energy loss (and also headloss) is sometimes significant, especially for free flow over weir. The widening and

deceleration of flow and the turbulence downstream of the weir account for the main part

of losses. A parameter representing these losses is the discharge coefficient (Cd), given in

equation 2.22. Substitute H0 into this equation leads to a relation of the dischargecoefficient and the flow condition (Fr1 is unique for each flow state):


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2/32

11

1

2/32

12/3

1

2/3

11

2/3

0

233

2

2

3

2

.

.3

2.

3

2 Frgh

u

Frhg

hu

HgB

QCdv

2/3211

2

33

Fr

FrCdv

(2.38)

The above relation can be used for imperfect weir, when the flow on top of the weir is sub-

critical, as well. It will be used to check with measurement data later. The following figure

illustrates this relation.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fr1

(m)

Cd

Cd

~ Fr1

Figure 2.10: Cd as a function of Fr1.

It should be note that here the energy conservation is assumed and Cdv is only governed bythe conditions above the weir crest.

Momentum balance:

Downstream of the weir, the momentum equation is of great importance when the energy

equation breaks down in the presence of unknown energy losses. With complement results

from the momentum equation, the energy loss can be estimated.

The submergence is an indicator for the water head loss over the weir, and the Froude

number above the weir is an important dimensionless parameter, which determines the

flow regimes. The link between these two parameters is thus of interest. With the

assumption of momentum conservation in the downstream part of the weir, this relation can

be found by an iterative process.

The momentum balance equation for the downstream part of the weir reads:

2

2

2

2

21

2

1

2

12

1)(

2

1hughhuahg w (2.39)

Combine with the simplified form of the continuity equation:

2211 huhu (2.40)Equation 2.39 can be rewritten as follows:

022121 2

21

2

222

22

122

3

122

FrhFr

h

ah

h

ah

h(2.41)

Knowing the value of h2, h1 can be found as a root of the above polynomial. Other

dependent parameters can be calculated as well. The relation between S (S=H2/H0 is the


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submergence) and Fr1 can be found by repeating the above process for a certain range of

h2. It can be illustrated in figure 2.11.

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

Fr1

SQ = 20l/s

Q = 30l/s

Q = 40l/s

Figure 2.11: The submergence as a function of Fr1.

2.4. Flow over oblique weir

As mentioned earlier, not many studies on oblique weirs have been published. In the

following, studies of Borghei, De Vries and Wols will be briefly introduced. And then the

theoretical analysis of the flow over an oblique weir will be presented. The angle between

the longitudinal axis of the weir and the direction normal to the flow (angle in the figurebelow) is referred to as oblique angle from now on.

2.4.1. Oblique sharp crested weir

The study of Borghei et al (2003) discusses theirexperimental result with existing discharge

formulas, particularly for rectangular sharp-crested

weirs with different upstream and downstream beds.

Figure 2.12: Plan view of an oblique weir.

The general formula for flow measurement with a small amendment (using the effective

weir length L instead of channel width B) can be used:

2

3

0..23

2. HLgCQ d (2.42)

For a plan and full width weir, Cd can be calculated by equation (2.28). The general form is

w

da

HbaC

0

(2.43)The results of the research show that for a plain sharp

crested weir, Cd is only a function of H0/aw if the water

head is large enough to minimize any surface tension

effects. Therefore, it is possible to find the values of a

and b in the above equation for each different obliqueangle using the experimental results and determine if

there is only one coefficient of discharge relationshipfor all angles.

BL

Flow

Oblique weir

Q

Figure 2.13: Free flow

H

aw

0Q


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For free flow, it shows a trend of decreasing Cd as the ratioH0/aw increases. That means the

oblique weir should be used for low values of H0/aw. The results also showed that

increasing the ratio H0/aw will decrease the convergence capacity. Another important result

is that, for the same discharge, using an oblique weir instead of a plain weir will decreasethe water head noticeably.

For a rectangular sharp-crested weir, the standard equation of Cd is approximated as:

w

da

H

L

B

L

BC )663.1229.2121.0701.0 (2.44)

Where (B/L) = cos ; is the oblique angle.

For submerged flow (figure 2.14), Borghei

suggested the following formula for submerge

coefficient:23

0

2

HHdcKs (2.45)

Figure 2.14: Submerged flow

Where c and d are coefficients and can be found in plots in the report of Borghei, (2003).And the general form ofKs is:

23

0

2479.0161.0985.0008.0

H

H

B

L

B

LKs (2.46)

It should be note that the above formulas are only based on observations of small H0/Pwconditions.

2.4.2. Oblique trapezoidal weir

Another published research on oblique weirs is the report of De Vries, TU Delft 1959. He

had performed a large number of experiments to examine the loss of the flow over dike-

form weirs under different oblique angles and flow conditions.

He considered the two equations to calculate thecapacity of the flow over a plain weir (2.47) and

flow over an oblique one (2.48):

gHBmQ s3

2

3

2... 2/3 (2.47)

gHLmQ k3

2

3

2... 2/3 (2.48)

Figure 2.15: De Vries experiment

Where Bs is the channel width and Bk is the effective width of the weir.

Because of cos.LB (2.49)

Then a relation for the two discharge coefficients can be deduced:

cos

k

s

mm (2.50)

Finally he introduced a reduction factorp to account for the effect of the length of the weir,the submerged flow condition, the limited width, and the energy height which should be

calculated with the velocity component perpendicular to the weir only:

a

HHd

w

0

Q


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cos

. 0ks

mpm (2.51)

p then can be interpreted as the reduction factor of the discharge coefficient due to the

obliqueness of the flow concerning a restricted width of the channel:

0k

k

m

mp (2.52)

De Vries has performed several experiments on physical scale model with the geometrical

scaling of 25. Some parameters including the channel width (Bs), the specific discharge (q),

the weir height (aw), and specially the oblique angle (), were altered; whereas the crest

width (Lw) and the slopes (m) of weir were kept constant Lw = 3m and m = 1:4.

Some overall conclusions from his research are:

- The reduction factorp decreases when increasing the oblique angle .- The reduction factorp decreases when the submergence (S = H2/H0) increases.

- With a given value of p, the variation ofB causes no significant changes in the flow

width.

- When the submergence Sreaches the limit of 1, p converges to the value of cos.

De Vries also further investigated the relation between the discharge coefficient m with

other parameters, specially the submergence S, for different oblique angle 30, 45 and 600

and produced graphs which are very interesting to be compared with results from this study

(Flow over oblique weirs).

2.4.3. Numerical simulations on oblique weirs

The numerical simulations from Wols (2005) were performed with the k- turbulence

model and the non-hydrostatic finite elements model FINLAB. He has done extensivesimulations on different types of weirs, and tried to reproduce some of the experiments onphysical scale models. Those include perpendicular and oblique weirs, sharp and long

crested weirs, triangular weir, trapezoidal weirs, and rectangular weirs. The results of

simulations were then compared to the results of experiments. Thanks to the competence of

the computer models, he could run more simulations than physical model tests, investigate

the three dimensional structure of the modeled flows, and thus gained further informationon flow over different types of weirs.

For perpendicular weirs, the discharge coefficients can be obtained on the basis of the

upstream flow energy conservation as long as the streamline is horizontal. For the sharp

weirs and in free flow (perfect weir), the streamlines could no longer be considered ashorizontal, the non-hydrostatic impact was strong and the discharge coefficients showed

values larger than one. Energy loss can be obtained on the basis of downstream flow

momentum balance, and was described by the relation between the Froude number above

the weir and the ratio H2/H0.

The simulations for flow over oblique weirs showed similarities with the analytical analysis

in the case of a dike form and sharp crested weirs. The direction of the flow streamlines

and the energy loss depends on the Froude number above and upstream of the weir. The

flow over an oblique weir can be treated as flow over a plan weir if we consider thevelocity component normal to the weir. The velocity component parallel to the weir is

more or less constant in the flow field. This analysis gave good result for discharge


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coefficient in the case of imperfect weir, while it showed some deviations in the case of

perfect weir with oblique angle of 600.

Wols attributed this deviation to the recirculation zone behind weir. The strong influencecame from the geometry of the weir and the backward velocity component. The size of the

recirculation zone in the case of dike form weir seems to be extended for an oblique weir incomparison with a plain weir, when the size in the situation with a sharp weir stays almost

unchanged.

Figure 2.16: Streamlines show the spiral recirculation movement of flow downstream an

oblique weir (450). An illustration from simulations of Wols (2005).

2.4.4. Theoretical analysis of the flow over an oblique weir

Velocity component analysis

The flow velocity can be decomposed into two component parallel (V) and perpendicular

(U) to the flume axis. It can also be decomposed into two component parallel ( UL) and

perpendicular (UP) to the weir axis. The changes of these components will be considered

together with the change in the flow velocity. This analysis proved to be very helpful. Thenotation of the velocity components can be seen in figure 2.17 below.

U

U

U

U

U

U

Figure 2.17: Velocity component analysis

Theoretically, there is an assumption can be made for the flow over an oblique weir, that is

the velocity component parallel to the weir crest doesnt change its values when the flow

reaches and passes the weir (U0L = U1L). The acceleration force (cause by gravitation) only

acts on the velocity component perpendicular to the weir crest, and similarly reasoningapplies to the deceleration process. This assumption can be verified later by the result from

experiments.

Flow


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Flow direction and flow condition above an oblique weir

The oblique angle of the flow is understood to be the deviation angle of the flow streamline

from the normal direction to the weir. It is called (see figure 2.17). For a plain weir (=

00), the flow streamlines are perpendicular to the weir crest, thus = 0 (this also implies

that U1L = 0). For an oblique weir ( < 900 ), the value of can be predicted using the

energy conservation and continuity assumptions of the flow.

Figure 2.18: Upstream energy balance

With weir crest as the reference level, the en

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