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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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Symbol Parameter Unit
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
Symbol Parameter Unit
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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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