Minor graduation project * * *

Report

experiments on FLOW OVER OBLIQUE WEIRS

Nguyen Ba Tuyen

Supervisors Dr.Ir. Wim S.J. Uijttewaal

December 15, 2005 Delft University of Technology

Faculty of Civil Engineering and Geosciences Section of Hydraulic Engineering

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TABLES OF CONTENT

Abstract I. Introduction 1. Weir and Oblique weir review 2. Study objective 3. Methodology 4. Domain of the study II. Physical background and theory research 1. Basics on open channel flow 2. Flow over sharp-crested weirs

2.1. Definitions 2.2. Flow regimes 2.3. Discharge equations

3. Flow over broad-crested weirs 3.1. Definitions 3.2. Flow regimes 3.3. Discharge equations

4. Flow over oblique weirs III. Experimental setup 1. Experiment design

1.1. Match the objectives 1.2. Weir design 1.3. Experimental procedure 1.4. Data collection

2. Equipments 2.1. The experimental site 2.2. Parameters – Measurement devices

3. Image processing tools – PIV & PTV 3.1. PIV - Particle Image Velocimetry 3.2. PTV - Particle Tracking Velocimetry

4. Experiments with sharp-crested weir 5. Experiments with broad-crested weir 6. Interpretation of the experimental data

6.1. Process the measurement data 6.2. Process the images

IV. Results and discussions 1. Sharp-crested weir

1.1. Flow velocity field 1.2. Water depth 1.3. Head loss – Energy loss 1.4. Discharge coefficient 1.5. Velocity variation along weir center- line

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1.6. Velocity variation along streamlines 1.7. Other phenomena

2. Broad-crested weir 2.1. Flow velocity field 2.2. Water depth 2.3. Head loss – Energy loss 2.4. Discharge coefficient 2.5. Velocity variation along weir center- line 2.6. Velocity variation along streamlines 2.7. Other phenomena

3. PIV results 4. Discussions

4.2 Comparison 4.3 Uncertainties in measurement - Tolerance 4.4 Limits

V. Conclusion and remarks 1. Conclusion 2. Feasibility of this study and further research Acknowledgement Appendix 1: Notation Appendix 2: References Appendix 3: Figures

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Abstract

This report is the conclusion of a series of experiments, which were performed on weirs placed obliquely in a shallow flow. Its purpose is to report on laboratory investigation on the flow over different types of oblique weirs, including behaviour and hydraulic characteristic of the flow, different phenomena in the neighbourhood of the weir, hydraulic parameters and physical laws that govern the process. To that end, many experiments were performed in a shallow flume under various flow conditions. Two different types of impermeable weirs are tested, including a rectangular sharp-crested weir and a rectangular broad-crested weir, both placed 450 obliquely to the flow direction. These weirs are schematically designed with vertical upstream and downstream walls, sharp edges, and two small end parts of each weir perpendicular to the flow. Flow velocities were measured using Particle Tracking Velocimetry technique, which helps gaining instantaneous whole field velocity maps. In conjunction with Matlab we can get almost necessary statistical information. By changing the flow discharge and the downstream water level, different flow behaviour and various phenomena like vortex, hydraulic jump, undulation, flow divergence, flow concentration, etc. can be observed. The data collected from measurements are used to investigate the hydraulic process and the phenomena of interest. This report also aims at a quantitative view on the energy loss and the discharge coefficient. It is shown that the flow tends to change its direction to perpendicular direction with the weir crest. This leads to the difference in water levels at two ends of the weir, the flow concentration at on one side of the flume behind weir, the variation in flow velocity distribution. In case of emerged flow condition, the flow behind weir becomes highly turbulent and very complex, which make it difficult to perform accurate measurements. This flow regime also accounts for the higher head loss and energy dissipation than in case of submerged flow condition.

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I. INTRODUCTION 1. Weir and Oblique weir review

A weir is one of the most common and simple hydraulic structures that has been used for centuries by hydraulic engineers. They can be used for various purposes like energy dissipation, flow measurement and diversion, regulation of flow depth, and many others. Besides, many obstacles in a flood plain can act as weirs. For example, the summer dike, the groyne, or the barrage. Although the definition of many different kinds of weirs is clear, and their hydraulic behaviours have been investigated for long time, few studies have been done on weirs placed obliquely to the flow direction. Probably the first rational approach to studying oblique flow over a weir was published by S.M. Borghei and his co-authors (Z. Vatannia, M. Ghodsian and M. R. Jalili), “Oblique rectangular sharp-crested weir” – Water & Maritime Engineering (June 2003). The main object of the research was to introduce the coefficient of discharge for oblique rectangular sharp-crested weir. Their research showed that when a normal (or plain) weir has limitations of water depth above the weir crest, the oblique weir can be used as a precise discharge measurement device. 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 coefficient 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 on the phenomena that happen to the flow over an oblique weir, which has hardly been described on published. By means of detailed description on what happens in the vicinity of 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.

Flow

Obliquely placed weir Flow seperation

Stream linesDeflection of flow

Flow convergence

The flume

Circu

lation

Fig.1.1: The flow over an oblique weir and its interesting phenomena

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2. Study objective

Up to now, there are not many studies on oblique weirs, most research on flow over weirs has been conducted for a weir perpendicular to the flow direction. The behaviour and the characteristics of the flow in the neighbourhood of the oblique weir are not fully understood. The parameters governing the hydraulic process have not been investigated thoroughly. Therefore, the main objective of this study is to investigate the behaviour of the flow over oblique weirs. In order to achieve this goal, we need to accurately measure the flow properties, examine the phenomena that happen in the neighbourhood of the weir, and investigate the relationships between them to draw conclusions as to what respect the flow over an oblique weir deviates from the flow over a perpendicular weir. The secondary purpose needs further research, but still is an important part of this study, is to determine what physical processes play a role, how they influence the flow over an oblique weir, and if there are common laws that govern this behaviour.

3. Methodology A study is conducted for the flow over oblique weirs on a horizontal bottom. Experiments were performed with two different types of weirs: a sharp-crested weir and a broad-crested weir. The important parameters that determine the flow are described. The behaviour and the hydraulic characteristics of the flow were investigated by means of measurements and image processing tools. In parallel with the experiments, there was also a different study on 3D computer modelling of flow over oblique weirs. Comparison of the experiment with the model is of importance in order to see if the phenomena that occurred in the flume can be reproduced in the model. It’s 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 good understanding on the underlying governing physical laws. But this is beyond the scope of this report. Experiments are needed to determine the flow conditions and characteristics. The preliminary experiment was conducted to get some ideas on the flow regimes, transition conditions, discharge levels. This is done in order to get an estimate of how many experiments should be done, what the magnitude of discharges and downstream water levels can be. Measurements were conducted for two different weir shapes as mentioned before, under different flow conditions. The results of the measurements were then investigated to find out relations between the flow conditions and the occurrence of different phenomena. PIV technique (Particle Image Velocimetry) and PTV (Particle Tracking Velocimetry) were employed to extract the flow velocity field in this stage. In that way, hypotheses that correspond to physical processes are formulated. The final step is the evaluation of the experimental data and discussion. The flow conditions are compared to each other; the experimental data were also compared to the numerical models. This results in an overview of the flow structure and comprehensive phenomena.

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4. Domain of the study Experiments were conducted in a flume 19.2m long and 2m wide. The first series of experiments was concerned with a sharp-crested weir, and the second was on a broad-crested weir. Since a weir can be used in a channel with sub-critical or super-critical flow, with different hydraulic behaviours, both flow regimes were taken into account. With a certain discharge, by changing the downstream water level, the flow regimes can be switched from free flow to submerged flow and vice versa. To get a more general view on the phenomena, different discharges were used (16l/s, 20l/s, 35l/s). In this study, the effects of gravity are included, but the effects of viscosity, surface tension, the nature of the weir crest, the roughness of the weir channel and the velocity distribution in the vertical plane are not explicitly considered.

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II. PHYSICAL BACKGROUND AND THEORY RESEARCH 1. Basics on open channel flow

The following basic materials on open channel flow were taken from the book “Environmental Hydraulics of open channel flows”, Hubert Chanson (2004).

d V

Fee surface

Total headline

yz

Datum

Channel bottom

θ

V /2g2

d.cosθ

z0z0

E

H

V

Fig.2.1: Definition sketch of open channel flow

In an open channel, the mean total head (energy head) is defined as:

2

0cos2VH d z

gθ= + + (2.1)

The specific energy is defined as: E = H – z0 (2.2) The continuity equation reads: Q = V. A = V.B.d (2.3) In a rectangular channel with the assumption of a hydrostatic pressure distribution

⇒ 2

2 2cos2

QE dgd B

θ= + (2.4)

When the velocity varies across the section, the mean total head is corrected as:

2

0cos2VH d z

gθ α= + + (2.5)

Where: H : the total head θ : the bed slope V : the depth-averaged flow velocity g : the gravity acceleration d : the water depth (d.cosθ is the pressure head) z0 : the bed elevation (z0 is the potential head) V2/2g : the kinetic energy head A : the wet area of the cross section B : the free surface width Q : the total discharge α : The kinetic energy correction coefficient.

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zo1zo2 H2H1

d1 1E

1V

Free surfaceTotal head line

V2

2E

2d

Datum

gV22 2

3E

H3

3d

Fig.2.2: Definition sketch of broad-crested weir overflow

The law of conservation of mass, or continuity equation, states that the mass within a closed system remains constant with time. For an incompressible fluid such as water, the inflow must equal the out flow: Q = V1.A1 = V2.A2 (2.6) Where subscripts: 1 : refer to the upstream flow cross-section 2 : refer to the flow cross-section above weir crest 3 : refer to the downstream flow cross-section The momentum principle states that, for a given control volume, the rate of change in momentum flux equals the sum of the forces acting on the control volume. Considering the steady overflow above a broad-crested weir placed in a horizontal, rectangular channel sketched in figure 2.2, the momentum equation applied in the horizontal direction yields:

2 23 1 1 3

1 12 2 friction weirQV QV gd B gd B F Fρ ρ ρ ρ− = − − − (2.7)

In open channel hydraulics, the energy principle is often expressed in term of the total head: H1 = H3 + ∆H (2.8) where ∆H is the sum of the head losses between sections 1 and 3.

The pressure force at a cross section is Bgd 2

21 ρ . The reaction force of the weir onto the fluid

equals exactly the horizontal component of the resultant of the pressure forces acting on the weir.

The friction loss ∆H over a distance ∆x along the flow direction in open channel flows is given by the Darcy equation:

2

H

xH=fD 2

Vg

∆∆ (2.9)

where f is the Darcy friction factor, DH is the hydraulic diameter.

The Bernoulli equation is

2

0 cos2VH z d const

gθ= + + = (2.10)

For the broad-crested weir, the Bernoulli equation may be rewritten in terms of the specific energy: E1 = E2 + (zo2 – z01) = E3 (2.11)

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2. Flow over sharp-crested weirs 2.1 Definition

The weir is termed as sharp-crested if the length of the weir in the direction of flow is such that H1/L >15 (Bos, 1976). In practice, the crest length of the sharp-crested weir is usually less than 2.0 mm (French, 1985)

H

Pw

Draw downNappe

Weir plate

Q

Fig.2.3: Sharp-crested weir geometry

2.2 Flow regimes

The main mechanisms governing the flow over a weir are gravity and inertia. Viscous and surface tension effects are usually of secondary importance, but cannot be entirely neglected. The complex nature of the flow over a weir makes it almost impossible to obtain precise analytical expression for the flow as a function of parameters such as Pw, H, b/H… Figure 2.3 illustrates the common sharp-crested weir in practice, where there ventilation tubes to ensure atmospheric pressure in the region underneath the nappe. Some performance characteristics of flow in this situation:

• Downstream of the weir, the flow springs clear off the weir body. An air pocket is formed beneath the nappe.

• As the air pressure in the pocket decreases, the curvature of the overflowing jet increases and the value of the coefficient of discharge will also increase.

• If the supply of air to the pocket is irregular, then the jet will vibrate and the flow over the weir will be unsteady.

Beside free flow over weir (fig.2.3), depending on downstream conditions, the weir operation can be submerged, or there is undulation behind weir, as illustrated in the following figure.

Weir plate Weir plate (a) (b)

Fig.2.4: Flow condition over a weir (a) Plunging nappe, (b) submerged nappe

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2.3. Discharge equations

Most of studies have been done in rectangular channel, the weir block were placed perpendicular to the direction of the flow. In case of a sharp-crested weir, the concept of critical depth is not applicable. The discharge equations are derived from writing the Bernoulli equation between the upstream section and the control section. For a rectangular sharp-crested weir:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

23

223

21

222

32

gV

gVHbgQ (2.12)

In practical situation, often the upstream velocity is negligible small, i.e. Hg

V<<

2

21 , thus Eq.

2.12 simplifies to the basic rectangular weir equation:

23

..232 HbgQ = (2.13)

In this case, the weir head, H, is the height of the upstream free surface above the crest of the weir. Note that because of the drawn down effect, H is not the distance of the free surface above the weir crest as measured directly above the weir plate (as indicated in Fig. 2.2). Numerous approximations have been used to obtain Eq. 2.13, namely the neglect of the viscous effects, turbulence, non-uniform velocity distributions, and centripetal accelerations in the derivation of this discharge equation. So that an experimentally determined correction factor must be used to obtain the actual flowrate as a function of weir head. Thus the final form, suggested by Kindsvater and Carter (1957), is:

322. 2 . .

3ws e eQ C g b h= (2.14)

where be is the effective width of the weir he is the effective water head Cws is the (dimensionless) effective discharge coefficient of sharp-crested

rectangular weirs. It’s a function of Reynold number (viscous effects), Weber number (surface tension effects), and H/Pw (geometry parameter). In most practical situation, the Reynold and Weber number effects are negligible, and the following correlation can be used (Rehbock, 1929):

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

wws P

HC 075.0611.0 (2.15)

where Pw is the height of the weir. For submerged flow, the submerged coefficient Ks should be introduced into equation 2.14 to produce submerged discharge Qs. The general formula is: QKQ ss .= (2.16) Brater and King (1976) suggested the following form for the experimental coefficient Ks (for a plain weir):

385.0

23

1⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛−=

HH

K ds (2.17)

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

⎟⎠⎞

⎜⎝⎛−+= −

HH

HH

K dds

1sin.331.1162.11 (2.18)

Where H is the upstream head, H = H1 – P Hd is the downstream head, Hd = H2 – P

3. Flow over broad-crested weirs

3.1. Definition 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):

10.08 0.50HL

≤ ≤ (2.19)

The simplest form of broad-crested weir is a rectangular block. Both upstream and downstream faces of the weir are vertical planes.

Weir block

H1

y1

Pw

y2

Lw = 40cm

V1

V2

H2

Fig. 2.5: Broad-crested weir geometry

3.2. Flow regimes As the flow passes over the weir block, it accelerates and reach critical conditions, y2 = yc and Fr2 = 1. The flow does not accelerate to supercritical conditions (Fr2>1), it remains critical across the weir block. The operation of a broad crested-weir is based on that fact. Depending on the value of the parameter H1/L, four different flow regimes over rectangular broad-crested weir can be identified:

• 1 0.08HL

< : Long weir block, the flow over the weir crest is sub-critical.

• 10.08 0.33HL

≤ ≤ : a region of parallel flow occurs in the vicinity of the mid point of the

crest. In general, CD has a constant value in this range of H1/L.

• 10.33 1.5HL

≤ : the weir should be termed as short-crested.

• 11.5 HL

≤ : the nappe may separate completely on the crest of the weir, and the

flow pattern over the crest is unstable.

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3.3. Discharge equations

Applying the Bernoulli equation between point (1) upstream of the weir and point (2) above the weir crest where the flow is critical, neglect the velocity head upstream, we obtain the equation for flow rate:

23

23

32 HgbQ ⎟⎠⎞

⎜⎝⎛= (2.20)

To account for various real world effects not included in the simplified analysis, again an empirical weir coefficient, Cwb, is used (Munson et al, 2002):

23

23

32. HgbCQ wb ⎟⎠⎞

⎜⎝⎛= (2.21)

The broad-crested weir coefficient Cwb can be obtained from the formula:

w

wb

PH

C+

=1

65.0 (2.22)

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). For a sharp leading edge broad-crested weir, commonly called a rectangular weir, the discharge depends upon the weir height Pw (represented by the experimental weir discharge coefficient Cd) (White, 1994):

2/323

21 .

2. HgbC

gV

HgbCQ dd ≈⎟⎟⎠

⎞⎜⎜⎝

⎛+= (2.23)

For a wide sharp-crested weir:w

d PHC 0846.0564.0 +≈ for

wPH

≤2 (2.24)

For a round-nosed broad-crested weir: 2/3

//*1544.0 ⎟

⎠⎞

⎜⎝⎛ −≈

LHLCd

δ (2.25)

where * 0.001 0.2 E

L Lδ

≈ + (2.26)

However, in a certain range of weir height and length, Cd is nearly constant:

Sharp-nosed broad sharp-crested weir: Cd ≈0.462 for 10.08 0.33HL

≤ ≤

56.022.0 <<wP

H

Again, equations 2.16, 2.17 and 2.18 can be used to calculate the discharge for the submerged flow.

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4. Flow over oblique weirs

The study of Borghei et al (2003), mentioned in the introduction part of this paper, is one of the few published study on oblique weirs. Authors discuss their experimental result with existing discharge formulas, particularly for rectangular sharp-crested weirs with different upstream and downstream beds.

Fig. 2.6: Oblique weir

For a general sharp-crested weir, formula 2.14 can be used for flow measurement:

23

..232. HLgCQ d= (2.27)

where L is the effective weir length, Cd is the discharge coefficient. For plan and full width weir, the general form for Cd is:

PHbaCd += (2.28)

As proposed in Eq.2.15, a = 0.611 and b = 0.075 (Rehbock, 1929). The results of the research show that for a plain sharp crested weir, Cd is only a function of H/P if the water head is large enough to minimise any surface tension effects. Therefore, it is possible to find the values of a and b in the above equation for each different oblique angle using the experimental results and determine if there is only one coefficient of discharge relationship for all angles. Experiments were carried out by Borghei on several different oblique angle, weir length (L), discharge (Q), channel width (B), weir height (P), water head (H), downstream water head (Hd). Total number of runs is more than 600 runs. The Cd results were plotted against H/P for each L/B. For free flow, it shows a trend of decreasing Cd as H/P increase. That means the oblique weir should be used for low values of H/P. The results also showed that increasing the ratio H/P will decrease the oblique effect and should not be designed. Another important result is that, for the same discharge, using an oblique weir instead of a plain weir will decrease the water head noticeably.

Fig. 2.7: Free flow For a rectangular sharp-crested weir, the standard equation of Cd is approximated as follow:

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

PH

LB

LBCd )663.1229.2121.0701.0 (2.29)

Where LB

= sinθ ; θ is the angle between oblique weir axis and the flow direction.

H

P

LFlowB

Sharp crested weir

Plan view

The flume

θ

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P

HH d

Fig. 2.8: Submerged flow

For submerged flow, the general formula for the flow discharge is (2.16) with Ks determined by (2.17) and (2.18). In his research, after analysed the experimental data, Borghei suggest the following formula for submerge flow over oblique weir:

23

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

HH

dcK ds (2.30)

Where c and d are coefficients and can be found in plots in literature (Borghei et al, 2003). And the general form of Ks will be:

23

479.0161.0985.0008.0⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +=

HH

BL

BLK d

s (2.31)

One should mention that the general formulas given in (2.26) and (2.28) have good agreement with the measured values, taken into account ± 5% accuracy.

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III. EXPERIMENTAL SETUP 1. Experiment design

1.1. Match the objectives:

In the view of the objectives of this study, the following issues had to be solved: - Energy loss. To estimate the energy loss, water depth, velocity upstream and

downstream of the weir were measured. From those data, the surface line and energy line can be drawn. Finally head loss can be calculated between different cross sections.

- Flow field. PIV and PTV technique were employed. First, a large number of consecutive images were taken, and then the PIV algorithm of Davis ver.6.2 and PTV algorithm of Kadota (running in Matlab 6.5 environment) were used to process the images collected. Finally we got the flow velocity vector field.

- Phenomena that occurred. Images were recorded using a video camera as well as a photo camera. Phenomena were also noted and illustrated by figures.

1.2. Weir design

The objects of interest are two weirs. They are made of composite, painted in white. General shape and dimension of the two weirs are shown in the following figures:

Sharp-crested weir Broad-crested weir

1.2cm

1.2cm8.8cm

10cmFlow

Flow

40cm

10cm

Fig. 3.1: Weir cross sections

Sharp crested weir

Flow

200cm

30cm

30cm

200cm

Flow

Broad crested weir

45°200cm

30cm

30cm

200cm40cm

40cm

Fig. 3.2: Weirs plan view

1.3. Experimental procedure For each type of weir, firstly the discharge was kept constant and changed the downstream water level. Then another value of discharge was chosen and the same experiments were repeated. With each experiment, the seeded flow was recorded by the camera, the water depth and flow velocity at some cross sections was measured. The camera’s frequency was set at 30Hz to

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get a good brightness in unfavoured contrast condition, and to adapt with the high velocity and highly turbulent movements behind weirs. Before recording and doing any measurements, it is necessary to wait for the flow to be stable enough. After open the valves to start one experiment, or after finish one experiment and adjust the effective flow width downstream to do the next experiment, the water level upstream and downstream gradually change, affecting each other. This feed-back effect will take place until the flow reaches an equilibrium state. Normally, it becomes stable enough for measuring after 15 minutes.

Fig. 3.3: The flume and its elements

1.4. Data collection For each experiment, 401 instantaneous images were taken. They need to be analysed by PIV and PTV methods before we got 400 instantaneous velocity vector field images. With Matlab (having Image Processing Toolbox) we can take the average, mean value, standard deviation and all kind of statistical information we need from that set of result. Graphs and charts can be derived from the statistical results. The above number of images was chosen because it is sufficiently large to produce a good average velocity vector field. The flow upstream has a low velocity and a smooth character, it can be well represented on the vector image. Near the weir, the flow accelerates and reaches critical state. Behind the weir, the flow velocity is relatively high and it becomes highly turbulent in some cases. That fact requires a large number of images from each experiment to get a good statistical view. The highest possible frequency of the video camera was used (30Hz). That helps to trace the particles movement and gain a good brightness pictured with sufficient contrast. A larger number of images was taken, normally was 2400 pictures (it took approximate 80 seconds). And then a best series comprising 401 consecutive images, with good density and well distribution of particles, was chosen from that set to be stored on hard disk. Still there were some big time scales, which may require much larger times of recording, such as the time scale of fluctuations in the hydraulic jump behind weir, time scale of vortices downstream of the weir. They are out of the scope of this study. Thus there was no need to record longer and process more images.

2. Equipments 2.1. The experimental site

The study was conducted in the Stevin III Lab (The Hydraulic Laboratory of Delft Technical University). Following are some of the main facilities that were used.

- A two meter wide glass flume. The flume bottom consists of 32 segments of glass, each with 0.6 meter wide. The two parallel side walls are glass also. At the end of the flume, there was a 7cm high sill, its width is adjustable with several removable concrete blocks.

- Water supply: Pipes and valves connect the flume with the water circulation No.2 in the laboratory, provided the flume with constant discharge in each experiment.

- Computer, camera (video recorder), lighting, power supply.

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A general arrangement of the flume is given in the following figure:

Water inlet

Area of interest

Oblique weir

CameraPlatform Flow

Particles supplier

Water outlet

End sill

Concrete blocks

Flow

Computer

Fig. 3.4: Flume arrangement

2.2. Parameters – Measurement devices • Water depth

A pointer-gauge was used to measure the level of the free surface and of the bed level to ±0.1 mm.

Fig. 3.5: The point gauge • Velocity

The value of upstream and downstream overall velocities can be roughly estimated using a stopwatch. It measured the travelling time of buoys between certain cross sections. Knowing distance and time, the surface velocity was then calculated. Measures were taken for several sections both upstream and downstream in each experiment.

• Discharge To measure the flow discharge, a sill at the end of the flume was used. In combination with the discharge estimation for the shallow flow in the flume, the discharge measured was although not highly accurate, but acceptable. For more details, please refer to the next section in this paper.

• Pressure Different experiments were taken at different times. A pressure meter was used to make sure the pressure in the supply pipe didn’t varied much between experiments, thus maintaining the flow discharge more or less constant in each series of experiments.

• Other instruments Other modern devices to measure the above parameters more precisely will be available for further research on this topic.

3. Image processing tools – PIV & PTV The terms Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) denote two established classes of image processing methods for extracting the underlying velocity fields from particle images. PTV methods determine the flow field by tracking individual tracers and give a higher resolution than PIV methods, as it is not necessary to average over sub-regions in the image (i.e. interrogation windows). Here in this study, a one mega pixel CCD camera and the software Video Savant 4.0 were used to capture flow images. The seeding used to visualise the surface velocity field is floating black polystyrene particles with 3mm diameter in size. They follow the flow satisfactory and reflect enough light to be captured by the camera.

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The number of particles in the flow is rather important in obtaining a good signal peak in the cross-correlation of the PIV technique. A rule of thumb is 10 to 25 particles images should be seen in each interrogation area.

3.1. PIV - Particle Image Velocimetry PIV is a whole-flow-field technique providing instantaneous velocity vector maps in a cross-section of a flow. In PIV the velocity vectors are derived from sub-sections of the target area by measuring the movement of the particle patterns between two consecutive images. V = dx/dt

The particle seeded flow is illuminated in the target area with a good light condition. The camera lens images the target area on to the sensor. It is possible to capture images in sequence of separate image frames. The images are then divided into small subsections called interrogation areas (IA). The IA from each of two consecutive image frames, I1 and I2, are cross-correlated with each other, pixel by pixel. The correlation of the two results in the particle displacement dx, represent by a signal peak in the correlation C (dx). And the average particle displacement dx is identified. Sub-pixel interpolation then employed to give an accurate measurement of displacement, and thus also the velocity. A velocity vector map over the whole target area is obtained by repeating the cross-correlation for each interrogation area over the two image frames captured by the CCD camera. In order to gain good data statistics, hundreds of picture frames of one target area were recorded. The correlation procedure gives hundreds of velocity vector maps. Using Davis 6.2, the velocity vector maps can be extracted from the picture frames. Before processing, the images (stored in *.tif files) were first averaged to get a background, and then they were subtracted from the background image to get the *.imx picture files. This process was carried out by two scripts running in Davis environment, namely “background.cl” and “subtract.cl”. After all the imx pictures have been created, PIV image processing will be run in Davis. Finally we got the desired number of velocity vector maps, stored in *.vec files. 3.2. PTV - Particle Tracking Velocimetry PTV methods are based on binary-image cross correlation (two frames), or on nearest-neighbour search with geometrical constrains (using four or more consecutive frames). It computes the cross correlation between regions around particles in the first and in the second frame. There are two basic assumptions: small displacements and smoothness of motion.

Fig. 3.6: The Illustration of PIV method

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PTV returns the displacement of individual, randomly located particles (tracers). For post-processing purpose, a re-mapping of the velocity vector map onto a regular grid is required. For the data processing of this study, Kadota (*) Matlab scripts combined with the Matlab Image toolbox were employed to process the data. The image frames (stored in image files) were first averaged to get a (nice) averaged background, and then subtract each frames from the background to get images with white tracers on black ground (executed by means of Matlab script named sliding_image.m). After running the main_program_ptv.m script, we got numbers of vector maps stored in matrix form in matlab files. With inter_grid.m script, data from randomly distributed velocity vectors can be interpolated and reproduced into velocity vectors that are fixed to a desired grid. Finally, post_proc.m script helps us to get an average picture of all vector maps from numbers of initial picture frames. (*) Kadota is a Japanese researcher, who formerly did the data processing using PTV technique in TU Delft. He made three Matlab scripts, namely sliding_image.m, main_program_ptv.m, and post_proc.m, to implement this task. These scripts are used by some researchers in Hydraulic Laboratory of TU Delft.

4. Experiments with sharp-crested weir Two different discharges were used: 16l/s and 35l/s. For each discharge level, there were 4 experiments each with a different downstream water level and correspondingly different flow regimes over weir. These are submerged flow, emerged (or free, plunging) flow, and submerged flow with undulation behind the weir. The downstream water level was adjusted by changing the number of concrete blocks that block the flow over the end sill (see figure).

Fig.3.7: The concrete blocks to adjust downstream water level

The data from experiments were tabulated and used to investigate the hydraulic process and the phenomena observed. Hereafter, a brief description of those experiments will be presented:

• Q = 55 l/s Exp.1: Submerge &smooth flow (Fig. 3.8.a) Exp.2: Undulation behind weir (Fig. 3.8.b) Exp.3&4: Perfect weir, emerge flow (Fig. 3.8.c)

(a) Submerged flow (b) Undulation (c) Emerge flow

Fig. 3.8: Flow regimes, sharp-crested weir, Q = 35l/s

- 21 -

• Q = 20 l/s Exp.5: Submerge &smooth flow (Fig. 3.9.a) Exp.6: Submerge flow with undulation behind weir (Fig. 3.9.b) Exp.7&8: Perfect weir, emerge flow with hydraulic jump right after weir (Fig. 3.9.c)

(a) Submerged flow (b) Undulation (c) Emerge flow

Fig. 3.9: Flow regimes, sharp-crested weir, Q = 16l/s

5. Experiment with broad-crested weir For the broad-crested case, three discharges 16l/s, 20l/s and 35l/s were used. For the discharge of 35l/s, three experiments with a different downstream water level, and correspondingly different flow regimes over the weir, were carried out. They are submerged flow, emerged flow and submerged flow with undulation behind weir. For the discharge of 16l/s, there was one experiment corresponding with the submerged flow case. For the discharge of 20l/s, there was one experiment corresponding with the case of emerged (plunging nappe) flow. The downstream water level was also adjusted at the out-flow section.

(a) Submerged flow (b) Undulation (c) Emerge flow

Fig. 3.10: Flow regimes, broad-crested weir, Q = 35l/s

6. Interpretation of the experimental data 6.1. Process the measurement data

The water surface profile was measured and the flow surface line was plot against the distance along the flume axis. As the water depth was being measured at several points along the weir itself, it is interesting to see if the water depth changes from left to right hand side along the weir length. This is shown individually for the sharp-crested weir and the broad-crested weir in the next two paragraphs.

- 22 -

Analysis yields a discharge coefficient for the flow over oblique weir for different conditions. Results were tabulated. The variation of the discharge coefficient Cd with the upstream Froude number for different types of weir was also investigated. The velocity profile near the two side walls and at two ends of the weir could not be measured and didn’t have much meaning, because we are interested in the oblique part of the weir. At the two ends, weirs are designed like perpendicular weir, and the flow over these parts is a combination of flow over oblique and plain weir. Together with the effect by the proximity of the wall, flow patterns are difficult to interpret.

6.2. Process the images The main part of processing images is to run PTV with Matlab. Before processing, 410 images recorded in each experiment (picture size 1008 x 1018 pixels) were averaged to make a background, and then each instantaneous image was subtracted from the background image to get an image of white pixels on black background (all are same size). The procedure is illustrated as follow:

Fig. 3.11: Sliding image process After all the images were subtracted from the back ground, PTV algorithm was applied. Cross-correlation of each couple of images gave us one image of the instantaneous velocity vector field. Thus, at the end we got 400 instantaneous vector field images.

Fig. 3.12: Cross-correlation makes instantaneous velocity vector field image Each instantaneous vector image was then recalculated and interpolated to calculate the vectors at fixed grid points. Grid size was chosen at 20x20 pixels. Then all those fix-grid-points vector-images were averaged to make the final vector field which looks like this:

- 23 -

The averaged velocity vector field represents the surface velocity field of the flow in the period of recording. It gives us an overview of the flow field as well, because the flow in experimental flume is a shallow flow with assumed logarithmic velocity distribution profile. The image size now is 1000x1000 pixels, and the position of the weir can be determined. Generally speaking, this image shows clearly the main behaviour of the flow: flow direction, the relative velocity magnitude, different zones in the flow. Some time it even shows the interesting phenomena observed in the flume, such as the surface standing waves on top of weir, the vortex and hydraulic jump behind weir, the flow separation and acceleration near and on top of the weir. Precise values of the velocity at each grid point can also be known. They were calculated and stored in several different matrices, upon requirement. Two main matrices are U and V, store horizontal (perpendicular) and vertical (longitudinal) components of the velocity vectors. Their normal sizes are 51x51 (in accordance with grid size 20 pixels). The calibration is also important. Figures like fig. 3.13 only give the vector dimensions in pixels per frame (movement magnitude in pixels between two consecutive frames). It’s necessary to calibrate this dimension to meter per second. Providing two certain points in the flume, we know the distance in reality (meter), and the distance in image (pixel). This ratio was then multiply with 30 (the chosen frequency of the camera). Then we got the correction factor for each case. Each element of the above matrixes was then multiplied with their correlative correction factor. For the sharp-crested weir, the factor K1 = 0.046; for the broad-crested weir, K2 = 0.060. The streamlines in the flow can be determined and plot on the vector field image (figure 3.14).

Fig. 3.14: The streamlines

Fig. 3.13: Velocity vector field

- 24 -

It is helpful to take a closer look into what happened along a stream line. Another script can help us to draw the streamlines, extract the velocity values along that line, and plot it on graphs (figure 3.15).

It is also helpful to know the actual variation of flow velocity along certain lines. As for the plain perpendicular weir, ignore the side wall effects, the flow is uniform along the weir center line. So it is interesting to extract the velocity (components) along that line and plot on graph, this can be done by a Matlab script. Examples were presented in figure 4.10 and 4.11 in the next part of the report. Another important part is processing the data with PIV method, using the software Davis version 6.2, available at the Laboratory. In this study, PIV method was only used to produce the instantaneous velocity vector field images for all cases, in order to compare with PTV result. This is the verification for Kadota’s PTV algorithm running in Matlab, which was later on used in most image processing work of this report.

Fig. 3.15: The velocity variation along a streamline

- 25 -

IV. RESULTS AND DISCUSSIONS 1. Sharp-crested weir

The flow structure is far from one-dimensional and has a variety of interesting phenomena. For each type of weirs (sharp-crested and broad-crested weir), to present all these phenomena clearly, they will be divided in to seven small parts as follow:

o Flow velocity field o Water depth o Head loss – Energy loss o Discharge coefficient o Velocity variation along weir center- line o Velocity variation along streamlines o Other phenomena

1.1. Flow velocity field

The following figures illustrated velocity field of the flow in the vicinity of the sharp crested weir in different flow conditions. It covers an area approximates 1.5m x 1.5m as illustrated in figure 4.1.

Each vector shows the direction and relative magnitude of the velocity vector at a certain point. The absolute values of velocity vectors (in m/s) in certain area of interest will be shown in many plots later on. The lines in between the vectors show the streamlines. They also help to clearly see the direction of vectors. The underlying velocity vector field of the flow in different flow conditions were obtained with PTV methods. Results from PIV method will be presented later and were used only to make a comparison. Please refer to appendix 3 to view full size figures for each experiment.

o Q = 35 l/s:

In general, the flow shows a different behaviour and velocity field for each experiment. Exp.1: Submerge &smooth flow (Fig.4.2.a) Exp.2: Undulation behind weir (Fig.4.2.b) Exp.3: Perfect weir, emerge flow (Fig.4.2.c) Exp.4: Perfect weir, emerge flow (Fig.4.2.d)

(a) Submerged flow (b) Undulation

Fig. 4.1: Measured area

- 26 -

(c) Emerged flow (d) Emerged flow

Fig.4.2: Flow field, Q = 35l/s

o Q = 16 l/s With a discharge much smaller in comparison with the above cases, the flow shows different behaviour. In emerged flow condition, the hydraulic jump developed less dominant than the previous case, and end up somewhere in the middle of the flume width. To the left part of the weir, the flow tends to have a smoother characteristic, with less turbulence.

Exp.5: Submerge &smooth flow (Fig.4.3.a) Exp.6: Submerge flow with undulation behind weir (Fig.4.3.b) Exp.7: Perfect weir, emerge flow with hydraulic jump right after weir (Fig.4.3.c) Exp.8: Perfect weir, emerge flow (Fig.4.3.d)

(a) Submerged flow (b) Undulation

(c) Emerged flow with hydraulic jump (d) Emerged flow

Fig.4.3: Flow field, Q = 16l/s

- 27 -

1.2. Water depth The flume comprises of 32 equal parts, divided by 33 cross sections. They are counted from the starting point of the flume (cross section 0). The weir was located from 15cm upstream of cross section 15 to 15cm downstream of cross section 17. It is symmetrical about cross section 16 (see figure 4.4).

Fig.4.4: The flume arrangement Upstream of cross section 11, i.e. far enough from the weir, the flow is nearly uniform and can be considered a constant depth flow. The same assumption applied for the downstream part of the flow, from cross section 21 to the end sill. Keeping in mind that the height of the crested weir is 10cm, water depth can be measured and tabulated as follow:

Table 4.1: Water depth, Q = 35l/s

Cross section 11 14 14.5 15 15.5 16 16+1.2cm 21 30

Distance (cm) -210.0 -120.0 -90.0 -60.0 -30.0 0.0 1.2 300.0 840.0

Exp1 18.02 18.35 18.17 18.17 17.98 17.56 17.54 17.71 17.78

Exp2 14.01 14.20 14.17 14.08 13.92 13.08 12.95 12.10 12.20

Exp3 13.56 13.68 13.70 13.56 13.41 12.63 12.14 9.65 9.65

Water depth (cm)

Exp4 13.40 13.71 13.69 13.51 13.45 12.64 12.34 7.79 7.80

Fig. 4.5: Flow surface profile, Q = 35l/s

The flow

- 28 -

From cross section 17 to 21, i.e. the area just behind the weir, the flow became highly turbulent. It was almost impossible to measure the water depth and flow velocity in these part of the flume also because there stood the camera platform. In case discharge was reduced to 16l/s, the values of water depth are shown in table 4.2.

Table 4.2: Water depth, Q = 16l/s

Cross section 11 14 14.5 15 15.5 16 16+1.2cm 21 30

Distance (cm) -210.0 -120.0 -90.0 -60.0 -30.0 0.0 1.2 300.0 840.0

Exp5 16.73 16.86 16.84 16.72 16.53 16.18 16.16 16.29 16.40

Exp6 13.40 13.62 13.61 13.50 13.39 12.90 12.83 12.79 12.96

Exp7 12.20 12.40 12.38 12.25 12.11 11.57 11.16 6.74 6.91

Water depth (cm)

Exp8 12.22 12.42 12.37 12.28 12.15 11.56 11.14 8.87 9.06

And the water surface profile is as follow:

Fig. 4.6: Flow surface profile, Q = 16l/s

Again, the real water surface profile form x=0 to x=300cm cannot be shown here. 1.3. Head loss – Energy loss

The head loss was calculated by subtracting the downstream water depth from the upstream water depth in both cases of submerged and emerged flow. 3 1H H H∆ = − (4.1) The energy head of the flow at the sections far enough from the weir (upstream and downstream) can be calculated by equation 2.1. The energy loss was calculated by subtracting the downstream energy head from the upstream energy head:

2 23 1

3 1 3 0 1 0cos cos2 2V VE E E d z d z

g gθ θ

⎛ ⎞ ⎛ ⎞∆ = − = + + − + +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

2 2

3 13 12 2

V VE H Hg g

⎛ ⎞ ⎛ ⎞∆ = + − +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ (4.2)

The flow

- 29 -

Results are shown in the following table:

Table 4.3: Head loss and energy loss

Experiment No. 1 2 3 4 5 6 7 8

Q (l/s) 35 35 35 35 16 16 16 16

H1 (cm) 18.02 14.01 13.56 13.4 16.73 13.4 12.2 12.22

V1 (m/s) 0.1 0.115 0.14 0.148 0.043 0.059 0.070 0.074 Up-stream

Fr1 0.075 0.098 0.121 0.129 0.034 0.051 0.064 0.068

H3 (cm) 17.78 12.2 9.65 7.8 16.4 12.96 6.91 9.06

V3 (m/s) 0.105 0.144 0.17 0.261 0.043 0.048 0.11 0.067 Down-stream

Fr3 0.080 0.132 0.175 0.298 0.034 0.043 0.134 0.071

Head loss (cm) 0.24 1.81 3.91 5.60 0.33 0.44 5.29 3.16

Energy loss (cm) 0.23 1.77 3.86 5.36 0.33 0.45 5.25 3.17

It’s important to determine the Froude number value and relate it to the head loss and discharge coefficient. The Froude number is dimensionless, it proportional to the square root of the ratio of the inertia force over the weight of fluid. For a horizontal rectangular channel, the Froude number is defined as:

gdVFr = (4.3)

In horizontal, rectangular channels, the Froude number is unity at critical flow conditions. It may be written as:

2/3

⎟⎠⎞

⎜⎝⎛=

dd

Fr c (4.4)

Where d is the flow depth and dc is the critical flow depth. In open channel flow, Fr = 1 at critical flow condition (d = dc), Fr < 1 for sub-critical flow (d >dc). The relation between the upstream Froude number and the energy loss can be illustrated by the following plot:

Upstream Froude number & Head loss, Energy loss

0

1

2

3

4

5

6

0.000 0.050 0.100 0.150

Fr1

Head

loss

, Ene

rgy

loss

(cm

) Head lossEnergy loss

Fig. 4.7: Head loss and energy loss against Fr1

- 30 -

1.4. Discharge coefficient

Consider the weir as a whole, comprised of the oblique part and the perpendicular part. Its effective width is 2.6m, larger than the channel width = 2m, also larger than the effective width of the perpendicular weir placed in that channel. Using the discharges as in experiments 1 to 8, we can calculate the values of the discharge coefficient for our oblique weir in different cases. The general formula to calculate the discharge is (2.14):

322. 2 . .

3ws e eQ C g b h=

Where be : effective width of the weir = 2*0.3 + 2.0 = 2.6m (see fig.3.2). he : effective head = h1 + Kn = (H1 – P) + Kn (P = 10cm).

Kn : represents the effects of the flow phenomena caused by viscosity and surface tension. Kn is generally considered to be constant =0.001m.

Cws: the discharge coefficient for sharp crested weir. In each experiment, knowing the discharge, the effective width and the effective head, Cws can be calculated as:

3/ 2

32 2 . .WS

e e

QCg b h

= (4.5)

To compare, Cws was also calculated by Rehbock’s formula (2.15)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

wws P

HC 075.0611.0

The values of Cws for the cases of emerged flow were shown in table 4.4. For the cases of submerged flow ( QKQ ss .= ), the submerged coefficient Ks must be known first. Ks can be calculated by Brater and King’s formula (2.17) or Wu and Rajaratnum’s formula (2.18) or Borghei’s formula (2.31). Cws was then calculated as follow:

3/ 2

32 2 . .WS

S e e

QCK g b h

= (4.6)

For all the experiments, the discharge coefficient values using different formula were calculated and tabulated as follow:

Table 4.4: Discharge coefficient

Experiment No. 1 2 3 4 5 6 7 8

Flow condition Submerged Undulation Emerged Emerged Submerged Undulation Emerged Emerged

Q (m3/s) 0.035 0.035 0.035 0.035 0.016 0.016 0.016 0.016

H (m) 0.080 0.040 0.036 0.034 0.067 0.034 0.022 0.022

Hd (m) 0.078 0.022 - - 0.064 0.0296 - -

CWS Rehbock 0.671 0.641 0.638 0.637 0.661 0.637 0.628 0.628

KS Brater&K.(1) 0.302 0.818 1 1 0.364 0.525 1 1

KS Wu&R.(2) 0.363 0.865 1 1 0.433 0.606 1 1

KS Borghei (3) 0.561 0.904 1 1 0.583 0.668 1 1

CWS (1) 0.665 0.694 0.679 0.727 0.328 0.633 0.639 0.630

CWS (2) 0.553 0.657 0.679 0.727 0.276 0.549 0.639 0.630

CWS (3) 0.358 0.628 0.679 0.727 0.205 0.497 0.639 0.630

- 31 -

The 1st and 5th experiments (the completely submerged cases) show a noticeable difference between Cws (determined by Ks calculated with Borghei’s formulas) and others. Other experiments show small difference between Cws calculated by different formulae. Cws is highest when Ks determined by Brater & King’s formula and lowest when Ks determined by Borghei’s formula. With the same condition (discharge, channel width, weir height…), the oblique weir gives higher discharge than perpendicular weir because of the higher effective width. In case oblique angle = 450, the difference is 1.4 time higher. To put it another way, with the same discharge over weir, the water head above oblique weir is much smaller than the water head above plan weir. The discharge coefficient was then plotted against the ratio Hd/H to see the relation between the discharge coefficient and the submergence.

DISCHARGE COEFFICIENT

φ = 450

0.0000.1000.2000.3000.4000.5000.6000.7000.800

0.0 0.2 0.4 0.6 0.8 1.0 1.2Hd/H

Cw

s

RehbockBrater & K.Wu & RBorghei

Fig. 4.8: Discharge coefficient and submergence To investigate the relation between the discharge coefficient and the flow regimes, represented by the upstream Froude number, they were plotted in figure 4.9.

Table 4.5: Discharge coefficient

Experiment No. 1 2 3 4 5 6 7 8

Fr1 0.075 0.098 0.121 0.129 0.034 0.051 0.064 0.068

CWS Rehbock 0.671 0.641 0.638 0.637 0.661 0.637 0.628 0.628

CWS Brater&K. 0.665 0.694 0.679 0.727 0.328 0.633 0.639 0.630

CWS Wu&R) 0.553 0.657 0.679 0.727 0.276 0.549 0.639 0.630

CWS Borghei 0.358 0.628 0.679 0.727 0.205 0.497 0.639 0.630

- 32 -

Discharge coefficient & Upstream Froude number

0.0000.1000.2000.3000.4000.5000.6000.7000.800

0.000 0.050 0.100 0.150

Fr1

Cw

s

RehbockBrater&K.Wu & R.Borghei

Fig. 4.9: Discharge coefficient and the upstream Froude number

Rehbock formula leads to a linear relation between Cws and Fr1, Cws remains more or less constant when Fr1 changes. Other formulae lead to a non-linear relationship between the two. Brater & King’s formula has a Cws higher than others, while Borghei’s formula has the lowest Cws of all.

1.5. Velocity variation along weir center- line

Along the center line of the sharp-crested weir, we can see some dominant trends in velocity variation:

- The perpendicular component of flow velocity (U) was always positive, that means all velocity vectors in this area have the direction from left hand side of the flume to the right (direction right to left in fig. 4.10).

- The longitudinal component of flow velocity (V) was also always positive, that means the all velocity vectors in this area have the direction from upstream to downstream (up to down direction on paper).

- The magnitude of the total velocity C (where C2 = U2 + V2) tends to increase from right hand side of the flume to the left in case of free flow (perfect weir) and vice versa for the case of submerged flow (imperfect weir).

These trends are illustrated in the following figures. In figure 4.10, the total velocity and its components tend to increase from the right hand side to the left. This figure is taken from result of experiment No.4, the emerged flow.

(a) – Total velocity (b) – Illustration of U and V components

The flow V

U

- 33 -

(c) – The perpendicular velocity component (d) - The parallel velocity component

Fig. 4.10: Velocity variation along weir center-line, free flow

To illustrate the opposite trend, figure (4.11) from experiment 1 was presented below. In this fully submerged flow over the weir, the total flow velocity decrease from the right hand side of the weir to the left. Note that the direction in these figures is the direction in reality, in the flume. It is opposite with the direction on paper. More about other experiments can be found in Appendix 3.

Fig. 4.11: Velocity variation along weir center-line, submerged flow

1.6. Velocity variation along streamlines To see the variation along a streamline, we should investigate the center streamline, far enough from the side walls. This is the typical streamline representative for the whole velocity vector field. Besides, we should not look over the streamlines near two side of the flume, for they may reveal interesting phenomena not yet researched. The center streamline is numbered 13, while the streamline on the right marked as number 5 to 7 depends on certain situation. The left-flume streamline is numbered 19 to 21. The point on the streamline and other velocity lines stand for the point where the streamline cross the weir center line. It’s interesting to see the variation along a streamline. Generally speaking, the variation in velocity magnitude follows quite well the variation in water surface. The flow tends to

- 34 -

accelerate on the way it reaches the weir crest, then decrease when it leaves the crest. But there is a big difference between the submerged and emerged case. The flow reaches its highest velocity (both component velocities and unified velocity) before it crosses the weir crest in the case of an emerged flow. On the contrary, he flow reaches its highest velocity after it crosses the weir crest in the case of a submerged flow. These trends are illustrated in figure 4.12 and 4.13.

Fig. 4.13: Velocity variation along center streamline, submerged flow

Fig. 4.12: Velocity variation along center streamline, emerged flow

- 35 -

Another phenomenon worth seeing is that in case of emerged flow with undulation behind weir, there the water surface has form of “waves”, at the same time the flow velocity also varies in a similar manner, like a sin curve, or a “wave”. Following is example from experiment 3 to illustrate this.

More information about other streamlines (to the left and right of the flume) of these experiments and all other experiments can be found in Appendix 3.

• Velocity analysis

Fig. 4.15: Velocity components

In the above paragraph, we analyse a velocity vector from the point of view XOY coordinate. To get an insight in the physical process play a role in the velocity variation along the flow, we decompose a velocity vector into two components perpendicular and parallel to the oblique weir, namely VP and VL. Using Matlab to draw graphs on these components for different experiments and assess the results, one conclusion can be drawn is that the parallel velocity

Fig. 4.14: Velocity variation along center streamline, submerged flow

X

Y

The streamline i

j

- 36 -

component hardly changes, whereas the perpendicular component mainly accounts for the change in total velocity. This can be illustrated by one graph taken from experiment 1:

In this case, the velocity component VL has an almost constant value (only a small variation, 0.01 m/s) in comparison with a dramatic change in the value of VP.

Velocity components

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2

Distance (m)

Velo

city

(m/s

)

VP

VL

Total Velocity

Fig. 4.17: Velocity components

This can be explained by the change in water level in the direction perpendicular to the weir, whereas the water level in the direction parallel to the weir is almost constant. In this case, the potential energy has changed into the kinetic energy.

VL

VP Total Velocity

Fig. 4.16: Variation in velocity components

- 37 -

1.7. Other phenomena The first noticeable phenomenon is that the flow depth decrease along the flow direction (above crest) and decrease along the weir length from right to left.

7.20

7.30

7.40

7.50

7.60

7.70

7.80

7.90

8.00

0 0.5 1 1.5

Weir crest (cm)

Dept

h (c

m)

RightCenterLeft

1.50

1.70

1.90

2.10

2.30

2.50

2.70

2.90

3.10

0 0.5 1 1.5

Weir crest (cm)

Dep

th (c

m)

RightCenterLeft

(a) (b)

Fig. 4.18: Flow depth above weir crest, Q = 35l/s (a) Experiment 1 (b) Experiment 3

This result from experiment can contribute in weirs designing, especially in determining the appropriate weir crest height and side wall height. The other phenomenon should not be forgotten is the change in flow direction. As can be seen in most figures on flow velocity vector fields, the flow is nicely uniform far upstream of the weir, start to turn to the right (with this angle and direction of weir), and turn almost perpendicular to the weir axis. Behind the weir, depends on the certain case, the flow pattern is more complex, but still we can see the trend of flow biases to the left. The change in the water level is not noticeable in the fully submerged case, where the water surface is almost level over the weir. If we pay attention to an individual particle and trace its movement in case of an emerged flow condition, we can observed a vorticular motion be hind weir. Underneath the surface, the particle moves vortically from upper part to lower part, right to left side of the flume. This movement ends up somewhere in the left part of the flume when the particle meets a strong hydraulic jump and the flow separation. This can be well illustrated in figure 1.1. Although the above mentioned behaviour is out of the scope of this study, this is still an interesting hydraulic phenomenon and needs a more detailed investigation. There are many other phenomena of interest, such as the large horizontal vortices downstream of weir, the convergence of the flow to the right hand side of the flume behind weir, and the divergence of the flow on the left hand side. This may cause lots of differences in the flow pattern downstream of the weir. The experimental results show that the velocity at the right hand side of the channel is often three times higher than at the left, some times even more. Thus the right bank of the channel could suffer from more severe erosion and need much better protection.

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2. broad-crested weir

2.1. Flow velocity field The following figures illustrated velocity field of the flow in the vicinity of the broad crested weir in different flow conditions. It covers an area of 2m x 2m as illustrated in figure 4.19 below. Please refer to appendix 3 to view full size figures.

Each vector shows the direction and relative magnitude of the velocity vector at a certain point. The absolute values of velocity vectors (in m/s) in certain area of interest will be shown in many plots later on. The red lines in between the vectors show the streamlines. They also help to clearly see the direction of vectors. Exp.1: Q = 35l/s, submerged &smooth flow (Fig. 4.20.a) Exp.2: Q = 35l/s, submerged flow with undulation behind weir (Fig. 4.20.b) Exp.3: Q = 35l/s, emerged flow with hydraulic jump behind weir (Fig. 4.20.c) Exp.4: Q = 20l/s, emerged flow (Fig. 4.25.d) Exp.5: Q = 16l/s, submerged flow (Fig. 4.20.e)

(a) Q = 35l/s, submerged &smooth (b) Q = 35l/s, submerged flow with undulation

(c) Q = 35l/s, emerged flow with hydraulic jump (d) Q = 20l/s, emerged flow

Fig.4.19: Measured area

- 39 -

(e) Q = 16l/s, submerged flow Fig. 4.20: Flow field, Q = 55l/s

In all cases, the flow velocity reaches the highest value above weir crest. Depend on the downstream water level, the flow showed different behaviour after passing the broad crested weir. In the case of a completely submerged weir, with a smooth flow (Fig. 4.20.a), the streamlines kept parallel to each other, then converged to the left side of the flume. The flow velocity field seem to be uniformed. When the water level downstream decreases a little bit, undulated flow appears behind the weir making the flow more turbulent. Further decreasing the water depth, the flow slowly changes to an emerged regime with a hydraulic jump behind weir. The circulation zone in Fig. 4.20.c and 4.20.d are about 50 cm wide. In this part of the flow, the surface velocity is directed towards the weir, and the bottom velocity towards the downstream direction. Illustrated streamlines are interrupted in these areas.

2.2. Water depth The broad crested weir was located from 40cm upstream of cross section 15 to 40cm downstream of cross section 17, symmetrical about cross section 16. Upstream of cross section 11, i.e. far enough from the weir, the flow is nearly uniform and can be considered as a constant depth flow. The same assumption applied for the downstream part of the flow, from cross section 21 to the out flow weir.

Table 4.6: Water depth

Cross section 11 14 14.5 15 15.5 16 16.5 18 30

Distance (cm) -210.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 120.0 840.0

Exp1 17.90 18.31 18.25 18.22 17.36 17.30 17.41 17.91 17.83

Exp2 14.66 14.78 14.73 14.63 13.64 12.51 11.89 11.91 12.11

Exp3 14.64 14.74 14.71 14.59 13.61 12.51 11.70 10.06 9.94

Exp4 12.89 12.94 12.91 12.82 12.10 12.06 11.53 7.42 7.31

Water depth (cm)

Exp5 16.94 17.19 17.15 16.98 16.86 16.37 16.42 17.05 16.89

In these experiments, water depth behind weir was measured at cross section 18 to help drawing the water surface profile. The accuracy of measurement at these points is of order 0.25cm. From cross section 18 to 21, it was impossible to measure the water depth because of the camera platform. Far downstream, the water depth was measured at cross section 30.

- 40 -

Fig. 4.21: Flow surface, Q = 35l/s

Upstream of the weir, the flow depth decreases significantly because of the draw down effect. The flow is accelerated at the same time its depth decreases. Highest velocity can be observed above the weir crest. This can be clearly illustrated in all cases in appendix 3. After passing the weir, the flow depth tends to recover for the case of a submerged and smooth flow. Whereas the depth shows a much weaker trend to recovery as it continues to decrease in case of an emerged flow.

2.3. Head loss – Energy loss Head loss and energy head were calculated by subtracting the downstream values from the value upstream values, using equation 4.1 and 4.2. Results are shown in the following table:

Table 4.7: Head loss and energy loss, Q = 16l/s

Upstream Downstream Exp. No.

H1 (cm) V1 (m/s) Fr1 H3 (cm) V3 (m/s) Fr3

Head loss (cm)

Energy loss (cm)

1 17.9 0.137 0.103 17.83 0.105 0.079 0.07 0.109

2 14.66 0.137 0.114 12.11 0.144 0.132 2.55 2.540

3 14.64 0.137 0.114 9.94 0.171 0.173 4.70 4.647

4 12.89 0.066 0.059 7.31 0.089 0.105 5.58 5.562

5 16.94 0.04 0.031 16.89 0.056 0.044 0.05 0.042

The results from table clearly show that the energy loss (and also head loss) is highest in the case of an emerged flow, and lowest in the case of a submerged and smooth flow. The turbulent flow behind weir accounts for the energy dissipation. There is a slight difference between sharp-crested weir and broad-crested weir. The friction between flow and weir-crest is much more significant in case of broad-crested weir than in case of sharp-crested weir (almost negligible).

- 41 -

In comparison with a perfect weir, the imperfect weir with submerged flow always has a higher water head above weir and a smaller loss in water head. The following figure shows the relation between the upstream Froude number and the head loss.

Upstream Froude number & Head loss, Energy loss

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.000 0.050 0.100 0.150

Fr1

Hea

d lo

ss, E

nerg

y lo

ss

(cm

) Head lossEnergy loss

Fig.4.22: Head loss with upstream Froude number, Q = 35l/s

The values of the velocity were not measured at exactly those points where water depth was measured. These values are averaged over a certain area. Therefore it is difficult to draw an exact energy line. Following is the water surface line and the energy line from experiment number 2 to illustrate the water head loss and energy loss along the flow.

Water surface profile

10.00

11.00

12.00

13.00

14.00

15.00

16.00

-300.0 -200.0 -100.0 0.0 100.0 200.0

Distance from weir center (cm)

Wat

er d

epth

(cm

) Exp.2Energy line

Fig.4.23: Head loss and energy loss, Q = 35l/s, submerged flow.

2.4. Discharge coefficient As for the sharp-crested weir, we consider the weir as a whole, comprised of the oblique part and the perpendicular part. Its effective width (be) is 2.6m. Denote Cwb as the discharge coefficient for the oblique broad-crested weir. Using the discharges as in experiments 1 to 5, we can calculate the values of the discharge coefficient for our oblique weir in different cases. The general formula to calculate the discharge is (2.20):

- 42 -

33222. . .

3wb e eQ C g b h⎛ ⎞= ⎜ ⎟⎝ ⎠

In each experiment, knowing the discharge, the effective width and the effective head, Cwb can be calculated as:

32 3/ 22 . .

3

wb

e e

QC

g b h

=⎛ ⎞⎜ ⎟⎝ ⎠

(4.7)

To compare, Cwb was also calculated by Munson’s formula (2.21)

0.65

1wb

w

CHP

=+

For the cases of submerged flow ( QKQ ss .= ), the submerged coefficient Ks must be known first. Ks can be calculated by Brater and King’s formula (2.17) or Wu and Rajaratnum’s formula (2.18) or Borghei’s formula (2.31). Cwb was then calculated as follow:

32 3/ 22 . .

3

wb

s e e

QC

K g b h

=⎛ ⎞⎜ ⎟⎝ ⎠

(4.8)

The values of Cwb were shown in table 4.8.

Table 4.8: Discharge coefficient for oblique broad-crested weir

Experiment No. 1 2 3 4 5

Flow condition Submerged Submerged Emerged Emerged Submerged

Q (m3/s) 0.035 0.035 0.035 0.020 0.016

H (m) 0.079 0.047 0.046 0.029 0.069

Hd (m) 0.078 0.021 - - 0.069

Hd/H 0.991 0.453 - - 0.993

Cwb Munson et al 0.486 0.537 0.537 0.573 0.499

KS Brater&K.(1) 0.189 0.869 1 1 0.175

KS Wu&R.(2) 0.238 0.901 1 1 0.223

KS Borghei (3) 0.537 0.942 1 1 0.535

Cwb (1) 1.878 0.903 0.790 0.918 1.129

Cwb (2) 1.492 0.871 0.790 0.918 0.886

Cwb (3) 0.662 0.834 0.790 0.918 0.369

Using the formulae for calculating Ks for a plain weir (Brater & King; Wu & Rajaratnum) gives very high discharge coefficient. In these experiments, the Borghei’s formula to calculate Ks for oblique weir seems to be the most reliable formula. For a broad crested weir, with the same condition (discharge, channel width…), the oblique weir gives higher discharge than perpendicular weir because of the higher discharge coefficient and higher effective width. In the case of an oblique angle = 450, the difference is 1.41 times higher.

- 43 -

To investigate the relation between the discharge coefficient and the flow regimes, represented by the upstream Froude number, they were plotted in figure 4.24

Table 4.9: Discharge coefficient

Experiment No. 1 2 3 4 5 Fr1 0.103 0.114 0.114 0.069 0.036

Cwb Munson et al 0.486 0.537 0.537 0.573 0.499 Cwb Brater&K.(1) 1.878 0.903 0.790 0.918 1.129 Cwb Wu&R.(2) 1.492 0.871 0.790 0.918 0.886

Cwb Borghei (3) 0.662 0.834 0.790 0.918 0.369

Discharge coefficient & Upstream Froude number

0.000

0.500

1.000

1.500

2.000

0.000 0.050 0.100 0.150

Fr1

Cw

b

MunsonBrater&K.Wu & R.Borghei

Fig. 4.24: Discharge coefficient and the upstream Froude number

2.5. Velocity variation along weir center-line Along the center line of the broad-crested weir, we can see some dominant trends in velocity variation:

- The longitudinal component of the flow velocity (V) was always positive, that means all velocity vectors in this area have the direction from upstream to downstream (up to down direction on paper).

- The perpendicular component of flow velocity (U) was mostly positive, that means that velocity vectors in this area usually have the direction from the left hand side of the flume to the right.

- The magnitude of total velocity C is calculated from: C2 = U2 + V2. General speaking, C increases from right hand side of the flume to the left. In case of submerged flow (imperfect weir), there is a significant fall in velocity magnitude, thus the velocity in the left hand side of the flume is more or less equal to the velocity on the right side.

These trends are illustrated in the following figures. Figure 4.25a is taken from result of experiment 1 (the submerged flow) whereas figure 4.25b is taken from result of experiment 3 (the emerged flow).

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Fig. 4.25: Velocity variation along weir center-line (a) submerged flow (b) emerged flow

2.6. Velocity variation along streamlines Upstream of the weir, velocity is more or less constant. It increases sharply when the flow reaches weir-crest, and decreases when it leaves the crest. Peak value usually obtained somewhere in the middle of the crest. This can be illustrated by figures in appendix 3. Following is one of the figures to illustrate this general behaviour.

Fig. 4.26: Velocity variation along a streamline, emerged flow, Q =35l/s

• Velocity analysis

Similarly to the velocity analysis process for sharp-crested weir, we decompose a velocity vector into two components perpendicular and parallel to the oblique weir, namely VP and VL. Using Matlab to draw graphs on these components for different experiments and assess the results, one conclusion can be drawn is that the parallel velocity component has only a small

- 45 -

variation in comparison with the change in the value of VP, whereas the perpendicular component mainly accounts for the change in total velocity. This can be illustrated by a graph taken from experiment 5:

Velocity components

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.05 0.1 0.15 0.2

Distance (m)

Velo

city

(m/s

)

VPVLTotal Velocity

Fig. 4.27: Variation in velocity components, submerged flow, Q =16l/s

2.7. Other phenomena In these experiments, standing waves were observed on top of the weir crest. With low downstream water level, there was one dominant wave with wave height is about 1.5cm (in the case of discharge = 35l/s). Particles passing this wave showed a jump in its movement, sometimes they got stuck in this wave for a while. Maintaining the discharge and increasing the downstream water level, the number of waves increases and wave height decreases. Increasing the downstream water level further, the waves disappeared and were replaced by a smooth flow over the weir and undulations behind the weir.

Fig. 4.27: Standing wave characteristic

The broad-crested weir

The standing wave

VL

VP Total Velocity

- 46 -

Over one third of the weir on the left, the standing waves on top of the weir disappear, but right behind this part of the weir, strong flow causes undulations (in comparison with hydraulic jump observed behind the right part of the weir). And from velocity field analysis, the results show that the velocity on top of the left part of the weir is higher than the velocity on the right. This is illustrated in the following figure:

The vorticular motion behind the weir in case of an emerged flow condition is still visible in these experiments. The same comment can be made on the behaviour of particle movements underneath the flow surface. Looking from right side of the flume to the left, we can see a clock-wise rotation in the particle’s vortical movement. This can be well illustrated in figure 1.1. Apparently the water head difference (higher in the right hand side) induces a secondary flow from right to left, which can be observed in most of cases under the development and dissipation pattern of standing waves on top of the weir. The following figure illustrates the difference in water levels in the two side of the flume above the weir.

Water depth above weir crest

00.5

11.5

22.5

33.5

44.5

0 10 20 30 40 50

Distance (cm)

Wat

er d

epth

(cm

)

Right

Center

Undisturbed

Left

(a) – experiment 1

Water depth above weir crest

-3-2-1012345

0 10 20 30 40 50

Distance (cm)

Wat

er d

epth

(cm

)

RightCenterUndisturbedLeft

(b) - experiment 2

Fig. 4.28: Flow depth above weir crest, Q = 35l/s What can be seen from the experiments is that the flow over the oblique part of the weir mostly turns to the right, thus making the water layer on top of the right part of weir thicker than on the left. This is true for most cases. The detailed figure of the flow surface is quite unusual because of some other factors. They cause standing waves on top of weir and cause different behaviour behind weir from right to left. Usually over two third of the weir length from

- 47 -

the right, the standing waves disappeared. The cross section beyond that position is denoted as the undisturbed cross section, because the flow at that section shows a normal behaviour as described before in all papers (flow over perpendicular broad crested weir).

3. PIV results In parallel with the image processing procedure using PTV methods, the results were also obtained by means of using PIV methods. Here after are some figures from experiment 6 and 7 were demonstrated to illustrate the underlying velocity field in particle images. Figure 4.33 shows the velocity vector fields from experiment 6 for sharp-crested weir, Q = 16l/s, submerged flow with undulation behind weir. It shows the velocity vector field on the background of the mean velocity value (4.29a), on the background of the average picture of original particle images (4.29b), and the velocity vector field with streamlines (4.29c). For the same experiment, we had the velocity vector field by PTV method shown in figure 4.3b.

(a) (b) (c) Fig. 4.29: Velocity vector field from experiment 6

Similarly, figure 4.30 shows the velocity vector fields from experiment 7 for sharp-crested weir, Q = 16l/s, emerged flow with hydraulic jump behind weir. It is comparable with figure 4.3c.

(a) (b) (c)

Fig. 4.30: Velocity vector field from experiment 7 For the reason of similarity, only the pictures of velocity vector fields using PTV technique were presented in the above paragraphs.

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4. Discussion

4.1. Comparison

The experiments on oblique weirs were performed for both sharp-crested and broad-crested weir. The results presented in the above paragraphs show some common phenomena and similar behaviour but also and some differences. For both types of oblique weirs, the flow always turns its direction towards a perpendicular orientation when it reaches and passes the weir. The flow over the oblique part of the weir mostly turns to the right. The flow accelerates when it reach the weir and decelerates when it leaves the weir. Maximum velocity can be observed above the weir crest for the case of a sharp crested weir. For the case of a broad crested weir, the maximum velocity was observed just behind the weir. From one side of the flume to the other, there are many differences in flow characteristic. Downstream of the weir, the two side walls force the flow to its initial direction, thus the flow converges to one side of the flume and a separation zone can be observed on the other side. In these experiments, where the oblique angle is 450 and the weir alignment is as mentioned before, the water lever at the right hand side is higher than the water level at the left side, whereas the flow velocity at the left side is usually higher than at the right. Changing the downstream water level will lead to changes in the flow regime and the behaviour of the flow over the oblique weir. With low downstream water level, there is usually a classical hydraulic jump. The hydraulic jump is dominant in the right side of the flume. To the left, the flow has a smoother pattern. Increasing the downstream water level further, the hydraulic jump will change into an undular jump. From the energy point of view, the energy dissipation of both oblique sharp-crested weir and broad-crested has its maximum value for the case of a hydraulic jump behind weir, and minimum value for the case of completely submerged flow. The discharge coefficient of an oblique weir (both sharp-crested and broad crested weir, with oblique angle equals 450) is much higher than the discharge coefficient of a perpendicular weir with the same channel width and water depth. That implies a lower water head in front of an oblique weir than in front of a perpendicular weir when the channel width and the discharge do not change. For a certain type of weir, there is also different behaviour for different cases. With sharp-crested weir, the magnitude of total velocity C tends to increase from right hand side of the flume to the left in case of free flow (perfect weir) and vice versa for the case of submerged flow (imperfect weir). For the case of a broad crested weir, the noticeable standing wave on top of the weir crest appears in most of the cases with low discharge, imperfect flow regimes. The result from experiments also show that for a broad crested weir, in case of submerged flow (imperfect weir), there is a significant fall in velocity magnitude, thus the velocity in the left hand side of the flume is more or less equal to the velocity on the right hand side.

4.2. Uncertainties in measurement - Tolerance • Water depth

Pointer-gauges were used to measure the height of the free surface and the bed level to ±0.1 mm. In the supercritical flow, the flow surface became unsteady behind the weir, and there was air entrainment, these aspects together decreased the accuracy to ± 0,5mm.

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• Velocity

The method used to measure the mean flow velocity was measuring the travel time of a light floating object over a known distance. This method can be well applied far upstream of the weir and gives an approximation for the average flow velocity. Down stream of the weir, the flow is absolutely not uniform, and some times highly turbulent. Thus this method can hardly give an exact result of the flow velocity. In fact, the average flow velocity downstream of the weir was estimated by taking three measurements over the flume width.

• Discharge The discharge was measured at the end of the flume, where a sill exists. Using the formula of flow over the perpendicular sharp-crested weir, the flow discharge can be calculated, provided that water depth in front of the sill is known. That means, the accuracy of the discharge is in order of power 3/2 of water depth accuracy. Another problem with the discharge measurement is that there are some holes in the sill body itself. Thus, water can flow through the holes, underneath the flow over sill, but cannot be measured. It can only be estimated as some percent of the total discharge, depend on the certain situation. Measured value of the discharge can then be compared with the discharge in the upstream part of the flume. If we assume an uniform velocity distribution in the vertical cross-section of the flow (due to shallow condition of the experiment), and uniform distributed velocity across the flume section, then the flow discharge equals: Q = V(m/s).H(m).B(m) (m3/s) These two method of discharge measurement although not accurate enough, but can be complimentary. The result is acceptable with the difference not greater than 5%.

4.3. Limits For each experiment, 401 instantaneous images were taken. They need to be analysed before we got 400 velocity vector field images. Although the computers’ storage and simulation capacity is sufficiently large, it still needs ten to thirty hours to process all data of one experiment. That restriction in time induces a restriction in the resolution of the final vector images. With both PIV and PTV algorithms, we cannot increase the number of iteration to an infinity large value, or cannot reduce the grid size of those vector images (also the size of interrogation windows) to too small values. That is also not extremely necessary from the point of view that we only need pictures with certain resolution to view and interpret the phenomena. The water surface itself also varied considerably, thus a much finer resolution of the instantaneous velocity vector field image will be of little use.

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V. CONCLUSION AND REMARKS

1. Conclusion

Although the weir is a common and standard engineering structure and there is a wide knowledge on perpendicular weirs, few studies have been done on weirs placed obliquely to the flow direction. The experiments in this study have provided us with some further information regarding the behaviour and the characteristic of the flow in the neighbourhood of the oblique weir. The experiment was conducted in the flume for shallow open channel flow. It suits the purpose of the study well. During the course of the experiments, data was collected and processed by both classic instruments and modern techniques, such as the PTV image processing method. The hydraulics of oblique weirs has been investigated for both sharp-crested and broad-crested weirs. There are some common phenomena and similar behaviour but also and some differences. Following is the main conclusions that were obtained:

• The flow always turns its direction when it reaches and passes the weir, towards a perpendicular orientation.

• The flow accelerates when it reach the weir and decelerates when it leaves the weir. The peak value of flow velocity over a sharp-crested weir was observed just behind the weir crest in the case of a submerged flow and just in front the weir crest in the case of an emerged flow. For a broad-crested weir, the maximum value of the flow velocity is located above the weir crest.

• The velocity component perpendicular to the weir accounts for the main change in the value of velocity whereas the parallel component stays almost unchanged.

• Downstream of the weir, the flow converges to one side of the flume and a separation zone can be observed on the other side. Underneath the flow surface behind weir, vorticular flow movement from right-upper part to left-lower part of the flume can be observed.

• From one side of the flume to the other, there are many differences in flow characteristic. In these experiments, the water level at the right hand side is higher than the water level at the left side, whereas the flow velocity at the left side is usually higher than at the right (with respect to figure 5.1).

• With low downstream water level, there is usually a classical hydraulic jump. The hydraulic jump is dominant in the right side of the flume. To the left, the flow has a smoother pattern. Increasing the downstream water level further, the hydraulic jump will be replaced by an undular jump.

• The energy dissipation of both oblique sharp-crested weir and broad-crested has its maximum value for the case of a hydraulic jump behind weir, and minimum value for the case of completely submerged flow. The turbulent flow behind weir accounts for the energy dissipation.

• The discharge of an oblique weir (with oblique angle equals 450) is much higher than the discharge of a perpendicular weir with the same channel width and water depth because of the higher effective length and a high discharge coefficient. That implies a lower water head in front of an oblique weir than in front of a perpendicular weir when the channel width and the discharge do not change.

Please refer to the previous part, chapter IV.1 and IV.2 to see the derivation of these conclusions.

Fig. 5.1: Directions in the flume

- 51 -

2. Feasibility of this study and further research

In this study, a number of experiments were performed on two different types of weir. It suits the preliminary purposes well. The chosen constant angle between the flow direction and the weir center line, 450, can be considered representative for various values of angles, in a short period of time. There were 14 runs for sharp-crested weir and 12 runs for broad-crested weir in total. Hereafter, some suggestions will be given for further research on the same or related topics. It will cost plenty of time, but the advantage is accuracy, generality and reliability in the results and conclusions.

* Experiments first will be carried out on the main straight flume to determine the actual discharge coefficient and different effects of the viscous force, roughness of the flume, the head loss and energy loss.

* To investigate the energy loss more precisely, velocity values should be measured at several locations where water depths are measured. Together they provide a better impression of the energy line above the water surface profile. This could be done by means of a device applying the Doppler effect.

* Previous experiments on plain weirs (both sharp-crested and broad-crested if possible) will be taken to compare the presented discharge coefficients with oblique weirs.

* Experiments will be done with several different oblique angles and many more discharge levels to investigate the effect of oblique angle to the discharge and head loss. The geometry parameter (H/P) should also be evaluated by means of changing P as well as H. The upstream vertical wall of the weir should be change to a rounded nose or a slope with more streamlined shape. * The three-dimensional structure and behaviour of the flow in general and the velocity distribution and vorticular movement underneath the flow surface in particular were only roughly described in this study. They are interesting topics that need further research. For all of these research topics a fully 3D measurement technique is of importance. The use of multiple cameras and laser combines with a three-dimensional particle tracking velocimetry algorithm (3D-PTV) can be a solution.

- 52 -

Acknowledgement

Thanks to the study on flow over oblique weirs, my preliminary objective of this period, namely “Minor graduation project”, was successfully done. I have gotten used to the facilities in Laboratory, learned how to operate a specific flume, use different instruments, etc. The most important thing is I could learn how to turn a practical issue in to an experiment, how to carry it out, and what should be done in addition to make it better. The study was supported by Hydraulics Laboratory – TU Delft. I deeply appreciate the provision and guidance provided by Dr.Ir. Wim S.J. Uijttewaal. I am grateful to Dr.Ir. Henri L. Fontijn and Ir. H.J. Verhagen for their valuable advices during the course of this work. Many thanks to Bas A. Wols, Harmen Talstra, Jaap van Duin, Hans Tas and the laboratory staff for providing the necessary facilities and guidance for conducting my work.

* * *

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Appendix 1: Notation The following symbols are used in this report:

g : the gravity acceleration ρ : the fluid density

E : the specific energy H : the total head

h : the depth of the flow A : the wet area of the cross section B : the free surface width

α : The kinetic energy correction coefficient. he : the effective water head P : the height of the weir

b : the width of the weir L : the length of the weir

be : the effective width of the weir φ : the oblique angle of the weir

d : the water depth Hd : the downstream head

θ : the bed slope z0 : the bed elevation V : the depth-averaged flow velocity v : the main velocity flow Q : the total discharge q : the discharge per unit width

KS : the submerged coefficient Cd : the discharge coefficient

Cws : the effective discharge coefficient of the sharp-crested weir Cwb : the effective discharge coefficient of the broad-crested weir U : the flow velocity component perpendicular to the flow direction V : the flow velocity component parallel to the flow direction VP : the flow velocity component perpendicular to the weir VL : the flow velocity component parallel to the weir C : the total flow velocity IA : the interrogation area of a particle image dt : the time interval between two consecutive images (frames) dx : the movement of the particle patterns between two consecutive images K1 : the conversion coefficient for the flow velocity over a sharp-crested weir K2 : the conversion coefficient for the flow velocity over a broad-crested weir

The subscripts: 1 : refer to the upstream flow cross-section 2 : refer to the flow cross-section above weir crest 3 : refer to the downstream flow cross-section

- 54 -

Appendix 2: References

1. Chow, V.T., “Open-channel hydraulics”, McGraw-Hill International editions, Civil

Engineering series, 1959. 2. French, Richard H., “Open-channel hydraulics”, McGraw-Hill Book Company, 1985. 3. Munson, Young & Okiishi, “Fundamentals of Fluid Mechanics” 4th edition, John Wiley &

Sons, Inc., 2002. 4. S.M. Borghei, Z. Vatannia, M. Ghodsian and M. R. Jalili, “Oblique rectangular sharp-

crested weir” – Water & Maritime Engineering 156, 2003. 5. Frank M.White, “Fluid Mechanics”, Third edition, Mc. Grao-Hill, Inc, 1994. 6. S.M. Borghei, M. R. Jalili and M.Ghodsian, “Discharge coefficient for sharp-crested side

weir in subcritical flow”, Water & Maritime Engineering 156, June 2003 Issue WM2, pages 185-191.

7. M.A.Sarker, D.G.Rhodes, “Calculation of free-surface profile over a rectangular broad-

crested weir”, Sciendirect Journal, February 2004, pages 215-219. 8. Hubert Chanson, “Environmental Hydraulics of open channel flows”, Elsevier Butterworth

Heinemann, 2004.

9. Kadota’s Matlab scripts. (*) Kadota is a Japanese researcher, who formerly did the data processing using PTV technique in TU Delft. He made three Matlab scripts to implement this task. These scripts are used by some researchers in Hydraulic Laboratory of TU Delft.

Minor graduation project

* * *

experiments on FLOW OVER OBLIQUE WEIRS

Nguyen Ba Tuyen

APPENDIX 3

Supervisors Dr.Ir. Wim S.J. Uijttewaal

December 15, 2005 Delft University of Technology

Faculty of Civil Engineering and Geosciences Section of Hydraulic Engineering

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LIST OF EXPERIMENTS Experiment with the sharp crested weir

Q = 35l/s (04-11-05) Exp1: a2.5_Su30_051104 Exp2: b2.5_Wa30_051104 Exp3: c2.5_Hy30_051104 Exp4: d2.5_Em30_051104

Q = 16l/s (07-11-05) Exp1: a1.5_Su30 Exp2: b1.5_Wa30 Exp3: c1.5_Hy30 Exp4: d1.5_Em30

Experiment with the broad crested weir.

Q = 35l/s (18/11/05) Exp1: A2.5su30_231105 Exp2: D2.5wa30_241105 Exp3: B2.5em30_231105 Q = 20l/s (23/11/05) Exp4: E2em30_181105

Q = 16l/s (24/11/05) Exp5: C1.5su30_241105

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List of figures

SHARP CRESTED WEIR Q = 35 l/s EXP. 1: SUBMERGED AND SMOOTH FLOW

Figure 1: Velocity vector field Figure 2: Velocity vector field with streamlines

Figure 3: Velocity variation along weir-center-line Figure 4: Velocity variation along center streamline (number 13)

EXP.2: UNDULATION BEHIND WEIR Figure 5: Velocity vector field with streamlines

Figure 6: Velocity variation along weir-center-line Figure 7: Velocity variation along center streamline (number 13)

EXP.3: PERFECT WEIR, EMERGED FLOW Figure 8: Velocity vector field with streamlines

Figure 9: Velocity variation along weir-center-line Figure 10: Velocity variation along center streamline (number 13)

EXP.4: PERFECT WEIR, EMERGED FLOW Figure 11: Velocity vector field with streamlines

Figure 12: Velocity variation along weir-center-line Figure13 : Velocity variation along center streamline (number 13)

Q = 16 l/s EXP.5: SUBMERFED & SMOOTH FLOW

Figure14 : Velocity vector field with streamlines Figure 15: Velocity variation along weir-center-line

Figure 16: Velocity variation along center streamline (number 12) Exp. 6: Submerge flow with undulation behind weir

Figure 17: Velocity vector field with streamlines Figure 18: Velocity variation along weir-center-line

Figure 19: Velocity variation along center streamline (number 13) Exp. 7: Perfect weir, emerge flow with hydraulic jump right after weir

Figure 20: Velocity vector field with streamlines Figure 21: Velocity variation along weir-center-line

Figure 22: Velocity variation along center streamline (number 13)

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Exp. 8: Perfect weir, emerge flow Figure 23: Velocity vector field with streamlines

Figure 24: Velocity variation along weir-center-line Figure 25: Velocity variation along center streamline (number 13)

BROAD CRESTED WEIR

Q = 35 l/s Exp1: Submerged flow

Figure 26: Velocity vector field with streamlines Figure 27: Velocity variation along weir-center-line

Figure 28: Velocity variation along center streamline (number 13) Exp2: Submerged flow with undulation behind weir

Figure 29: Velocity vector field with streamlines Figure 30: Velocity variation along weir-center-line

Figure 31: Velocity variation along center streamline (number 13) Exp3: Emerged flow

Figure 32: Velocity vector field with streamlines Figure 33: Velocity variation along weir-center-line

Figure 34: Velocity variation along center streamline (number 13) Q = 20 l/s Exp4: Emerged flow

Figure 35: Velocity vector field with streamlines Figure 36: Velocity variation along weir-center-line

Figure 37: Velocity variation along center streamline (number 13) Q = 16 l/s Exp5: submerged & Smooth flow

Figure 38: Velocity vector field with streamlines Figure 39: Velocity variation along weir-center-line

Figure 40: Velocity variation along center streamline (number 13)

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SHARP CRESTED WEIR

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o Q = 35 l/s

EXP. 1 – SUBMERGED AND SMOOTH FLOW

Figure 1: Velocity vector field

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Figure 2: Velocity vector field with streamlines

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Figure 3: Velocity variation along weir-center-line

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Figure 4: Velocity variation along center streamline (number 13)

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EXP.2: UNDULATION BEHIND WEIR

Figure 5: Velocity vector field with streamlines

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Figure 6: Velocity variation along weir-center-line

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Figure 7: Velocity variation along center streamline (number 13)

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EXP.3: PERFECT WEIR, EMERGED FLOW

Figure 8: Velocity vector field with streamlines

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Figure 9: Velocity variation along weir-center-line

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Figure 10: Velocity variation along center streamline (number 13)

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EXP.4: PERFECT WEIR, EMERGED FLOW

Figure 11: Velocity vector field with streamlines

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Figure 12: Velocity variation along weir-center-line

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Figure 13 : Velocity variation along center streamline (number 13)

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Q = 16 l/s

EXP.5: SUBMERFED & SMOOTH FLOW

Figure14 : Velocity vector field with streamlines

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Figure 15: Velocity variation along weir-center-line

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Figure 16: Velocity variation along center streamline (number 12)

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Exp. 6: Submerge flow with undulation behind weir

Figure 17: Velocity vector field with streamlines

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Figure 18: Velocity variation along weir-center-line

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Figure 19: Velocity variation along center streamline (number 13)

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Exp. 7: Perfect weir, emerge flow with hydraulic jump right after weir

Figure 20: Velocity vector field with streamlines

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Figure 21: Velocity variation along weir-center-line

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Figure 22: Velocity variation along center streamline (number 13)

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Exp. 8: Perfect weir, emerge flow

Figure 23: Velocity vector field with streamlines

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Figure 24: Velocity variation along weir-center-line

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Figure 25: Velocity variation along center streamline (number 13)

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BROAD CRESTED WEIR

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Exp1: Submerged flow

Figure 26: Velocity vector field with streamlines

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Figure 27: Velocity variation along weir-center-line

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Figure 28: Velocity variation along center streamline (number 13)

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Exp2: Submerged flow with undulation behind weir

Figure 29: Velocity vector field with streamlines

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Figure 30: Velocity variation along weir-center-line

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Figure 31: Velocity variation along center streamline (number 13)

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Exp3: Emerged flow

Figure 32: Velocity vector field with streamlines

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Figure 33: Velocity variation along weir-center-line

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Figure 34: Velocity variation along center streamline (number 13)

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Q = 20l/s Exp4: Emerged flow

Figure 35: Velocity vector field with streamlines

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Figure 36: Velocity variation along weir-center-line

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Figure 37: Velocity variation along center streamline (number 13)

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Q=16l/s

Exp5: submerged & Smooth flow

Figure 38: Velocity vector field with streamlines

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Figure 39: Velocity variation along weir-center-line

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Figure 40: Velocity variation along center streamline (number 13)