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The influence of flow acceleration onstone stability

Master of Science thesis August 2004 Author: M. Dessens

Graduation committee: prof. dr. ir. M.J.F. Stive ir. H.J. Verhagen dr. ir. H.L. Fontijn ir. B. Hofland

Faculty of Civil Engineering and Geosciences Hydraulic Engineering Section

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Preface

The influence of flow acceleration on stone stability i

Preface This master of science thesis is written as the conclusion of my study at the Faculty of Civil Engineering of Delft University. The research was carried out by order of the Hydraulic engineering section. The experiments that were conducted in the scope of my thesis were carried out in the Laboratory of Fluid Mechanics of Delft University of Technology. Readers of this thesis who are mostly interested in the conclusions of the experiment are referred to chapter 6. The way the experimental data are interpreted can be found in chapter 4 and 5. During the process of writing my thesis I was supported by many people who gave helpful advise and practical assistance. I would like to express gratitude towards Mr. Verhagen, Mr. Fontijn, Mr. Stive and Mr. Hofland for their guidance and support during the project. I would also like to thank the staff members of the laboratory for their assistance in building up the experiment. I also want to mention the fact that I worked with a lot of pleasure with Maarten Tromp. In May 2003 we started our projects together after which we worked almost a year on them. Michiel Dessens, August 2004

Abstract

The influence of flow acceleration on stone stability ii

Abstract The stability of a bed of stones subject to a flow is often described in terms of a critical velocity or shear stress generated by the flow. These ‘classical design’ methods like for example Shields, do not take the influence of flow acceleration into account. In experiments and practice, it appeared that when a flow is accelerated, stones start to move at a point where the so-called critical velocity is not reached yet. The movement of stones must have a second cause beside the velocity of flow. Only a little information is known on the influence of flow acceleration on stone stability of the bed. The objective of this thesis is to obtain more insight into the influence of acceleration of flow on the stability of stones. By carrying out experiments in a flume containing a local contraction, the stone stability in an accelerated flow is investigated. In the contraction the stability of two different stone sizes, subject to different velocity-acceleration combinations, is analysed. If the hypothesis is correct, than for some velocity-acceleration combinations movement occurs while for the same velocity combined with a lower acceleration no movement occurs. The shear stress occurring in the accelerated flow is determined using the shear velocity. According to the classical Shields method the shear velocity is responsible for the movement of the stones. Movement is detected for lower shear velocities then expected. According to the hypothesis this is a result of the extra generated force on the stones due to acceleration. After analysing the data it appeared that combinations of the same velocity and different accelerations showed differences in movement. The amount of movement goes up for an increase in acceleration combined with a constant or slightly decreasing velocity. This proves that there is a relation between the stability of the stones and a combination of the velocity and acceleration generated forces. The Morison equation is used to describe the relation between the forces acting on a stone. It combines the force generated by acceleration and the force generated by the peak velocities due to turbulence, as the sum of both forces. The extra force due to acceleration appeared to be of the same order as the force due to the velocity. Therefore, when looking at the stone stability in an accelerated flow, it is important to take the force generated by the acceleration into account. The resulting Morison force acting on a stone is proved to be responsible for the stability of the stones. Finally, a unique relation, valid for both stone diameters, between the force acting on the stone and the entrainment is found. This power relation consists of a dimensionless Morison-Shields parameter representing the force on a stone and a dimensionless entrainment parameter. The relation does not depend on stone size and is therefore expected to be universal in use.

Table of contents

The influence of flow acceleration on stone stability iii

Table of contents Preface .............................................................................................................................i Abstract ..........................................................................................................................ii Table of contents.......................................................................................................... iii 1 Introduction........................................................................................................1

1.1 General...........................................................................................................1 1.2 Objectives ......................................................................................................2 1.3 Project description .........................................................................................2 1.4 Outline of report.............................................................................................2

2 Theory ................................................................................................................4

2.1 Introduction....................................................................................................4 2.2 Stone stability in an uniform flow .................................................................4

2.2.1 Forces acting on a single grain...............................................................4 2.2.2 Stone stability in a uniform flow ...........................................................7 2.2.3 Shear velocity.........................................................................................9

2.3 The Morison equation..................................................................................11 2.3.1 Force balance including the force due to acceleration.........................13 2.3.2 Velocity of the accelerated flow ..........................................................14 2.3.3 Coefficients CB en CM..........................................................................19

3 The experiment ................................................................................................21

3.1 Introduction..................................................................................................21 3.2 The flume .....................................................................................................21 3.3 The structure ................................................................................................23 3.4 Definition of the xyz-plane ..........................................................................23 3.5 Measuring instruments.................................................................................25

3.5.1 Velocity measurements........................................................................25 3.5.2 Water depth measurements ..................................................................27

3.6 The used stones ............................................................................................28 3.6.1 Measured stone parameters..................................................................28 3.6.2 Placement of the stones........................................................................28

3.7 Testing procedures .......................................................................................31 3.8 Accuracy and reliability...............................................................................33 3.9 Evaluation ....................................................................................................35

4 Flow analysis ...................................................................................................36

4.1 Introduction..................................................................................................36 4.2 The level of the bed......................................................................................37 4.3 Determination of z0 ......................................................................................39

4.3.1 z0 for small stones ................................................................................39 4.3.2 z0 for large stones.................................................................................42

4.4 Shear velocity...............................................................................................43 4.4.1 Shear velocity, small stones.................................................................44 4.4.2 Shear velocity, large stones..................................................................48 4.4.3 Shear velocity using the momentum balance.......................................49

Table of contents

The influence of flow acceleration on stone stability iv

4.5 Turbulence ...................................................................................................49 4.5.1 Turbulence in case of small stones ......................................................50 4.5.2 Turbulence in case of large stones .......................................................57

4.6 Velocity of the flow .....................................................................................62 4.6.1 Introduction..........................................................................................62 4.6.2 Difference in velocity measurement approaches .................................63

4.7 Velocity-acceleration combinations occurring in the flow..........................66 5 Threshold of motion.........................................................................................72

5.1 Introduction..................................................................................................72 5.2 Shields..........................................................................................................73 5.3 Comparing of the velocity-acceleration combination based on the

percentage of movement ..........................................................................75 5.4 Revised definition of movement..................................................................76 5.5 The threshold of motion set on different percentages of movement ...........77 5.6 CB and CM ....................................................................................................82

5.6.1 Introduction..........................................................................................82 5.6.2 The threshold of motion that will be used for determining CB and CM83 5.6.3 The critical force ..................................................................................85 5.6.4 The values for CB and CM ....................................................................85

5.7 Entrainment..................................................................................................90 5.7.1 The number-entrainment......................................................................91 5.7.2 The dimensionless force and entrainment parameters .........................92

5.8 Evaluation ....................................................................................................95 6 Conclusions and recommendations..................................................................96

6.1 The fluid mechanical results of the experiment ...........................................96 6.2 The results concerning the stability of the stones ........................................97

List of symbols.............................................................................................................99 References..................................................................................................................101 Appendix Table of contents A1 Appendix A Theory A2 Appendix B The orifice plate A13 Appendix C Dimensions of the flume A14 Appendix D Sieve curves A16 Appendix E Placement of the stones A19 Appendix F Flow profiles A20 Appendix G NEN-norm A23 Appendix H Determination of z0 A26 Appendix I Turbulence intensity A36 Appendix J Comparing the parabolic equations and the measured values A39 Appendix K Velocity of the flow versus acceleration A41 Appendix L Difference (error) between methods for calculating the acceleration A47 Appendix M Comparing v and a combinations vs. movement A51 Appendix N The bulk and inertia coefficient A53

1. Introduction

The influence of flow acceleration on stone stability 1

1 Introduction

1.1 General When a disturbance of the flow by a hydraulic structure or any other cause creates erosion, a protection of the bed is the most suitable measure. This bed protection can consist of stones, which increase the strength of the bottom. So far, scientists assign the stability of the stones to the velocity of the flow. Shields and later on van Rijn created a way to determine the threshold of motion. In these calculation methods, the horizontal flow velocity and the values of the grain size are the most important variable parameters. In experiments and practice it appeared that when a flow is accelerated, stones start to move at a point where the so-called critical velocity is not reached yet. The movement of stones must have a second cause beside the velocity of flow. In practice, numerous examples of accelerated flows can be found.

• Bowthruster propellers inducing jet wash at berthing facilities or strong bends in rivers or channels

• Waves on a slope • Contraction of flow due to hydraulic structures • Local turbulence • Gaps in a closure dam

Schokking (2002) found in an experimental model of jet wash on a slope that more damage occurred at the location of maximum acceleration and not maximum velocity of the flow. Understanding the consequences of accelerating flow on the stability of stones will have its influence on designing a bottom protection. Needless use of stones can be prevented or extra measures in the design can be taken. The section of hydraulic engineering of Delft Technical University started a research program to obtain more insight on the effect of accelerating flow on the stability of stones. It will be the subject of this master thesis.

1. Introduction


1.2 Objectives The objective of this thesis is: Obtain more insight into the influence of flow acceleration on the stability of stones. This will be done by carrying out experiments in a flume containing a local contraction. In the contraction the flow accelerates and the stability of different stone sizes will be investigated. The discharge in the flume during the experiments will be constant in time.

1.3 Project description A study of the theory on accelerated flows shows that not a lot is known on this subject. The fluid mechanical aspects as well as the influence on the stone stability in an accelerated flow have not yet been investigated thoroughly. Experiments will be carried out in a flume in which the flow will be accelerated by a contraction of the flow placed at the end of the flume. The known theory of undisturbed flows will be applied on the part of the flow just before it is accelerated. This information will be used to interpret the obtained measurements of the accelerated flow. This way the velocity and acceleration occurring in the contraction can and will be determined. Two different stone sizes will be used in the experiments. Both the determined velocity and acceleration generate a force which acts on the stones. The influence of both forces will be looked at in more detail. It will be investigated whether the extra generated acceleration force will have a distinguishable effect on the so called threshold of motion of the stones.

1.4 Outline of report The contents of the report can be described as follows: Chapter 2 deals with the theory concerning the matter of the scope of this thesis. The forces acting on a single grain and the literature dealing with the threshold of motion is summarized. Important information about the fluid mechanics used in the experiment is also given. In chapter 3 the experimental model is described as well as the measuring instruments and the testing procedures. The parameters of the stones used for the experiment are also presented in this chapter. The results of the experiments are shown in chapter 4. The turbulence and other important flow parameters such as the occurring velocities and accelerations are dealt with.

1. Introduction


In the next chapter, chapter 5, the experimental results are used for the objective of this thesis, finding a way to describe the threshold of motion for an accelerated flow. In the last chapter, the conclusions and recommendations resulting from this thesis are presented. Each conclusion contains a reference to the figure or section on which it is based.

2. Theory


2 Theory

2.1 Introduction The objective of this thesis is to get more insight into the influence of flow acceleration on the stability of stones. First a literature study has been performed to investigate the already existing theories. In order to understand these ‘classical’ theories the different forces that act on a single particle in an undisturbed flow will be looked at in detail in the first section. Next the results of the most important researches on stone stability in an uniform flow will be discussed. The ‘classical’ theories on stone stability, such as the Shields method, make use of the so called shear velocity. This shear velocity can be determined in different ways. The methods used in this thesis are then explained. In this thesis an accelerating flow will be created. The classical methods do not take acceleration into account so they need to be expanded or modified. An approach that takes both forces generated by the flow velocity and the flow acceleration into account, the Morison formula, will be introduced. First the classical force balance needs to be expanded with a force generated by the acceleration. Then the terms used in the Morison formula will be looked at in more detail, so the formula will be applicable in case of the experiments. In the experiment non-cohesive particles are used so the theory presented in this chapter only deals with such grains.

2.2 Stone stability in an uniform flow

2.2.1 Forces acting on a single grain The knowledge concerning the behaviour of a single grain exposed to a current is despite much research, still very empirical. If we want to understand the stability of stones we have to consider the forces acting on a particle. The forces that act on the grain due to the velocity of the flow can be divided in acting forces trying to move the grain and resisting forces that keep the grain in it’s place. The drag force (FD) and the lift force (FL) are both acting forces. The drag force acting in the direction of the fluid motion, is caused by pressure and viscous skin friction.

2. Theory


DwDD AuCF 2

21 ρ= (2.1)

FD: drag force [ N ] CD: drag coefficient [ - ] ρw: density of water [ kg/m3 ] u: velocity of the flow near the bottom [ m/s] AD: exposed surface area [ - ]

The lift force always acts perpendicular to the drag force. Due to the contraction of the water streamlines on all sides of the grain a lower pressure is created. This causes a lift force on the grain.

LwLL AuCF 2

21 ρ= (2.2)

FL: lift force [ N ] CL: lift coefficient [ - ] AL: exposed surface area [ - ]

current contraction

FL

Figure 2.1 Drag force on a single particle

Figure 2.2 Lift force on a single particle

current

FD

skin friction

sub pressure

2. Theory


The gravity force (FG) is the resisting force, caused by the submerged weight of the grain, acting downward.

gVF wsG )( ρρ −= (2.3) FG: gravity force [ N ] ρs: density of stone [ kg/m3 ]

g: gravitational acceleration [ m/s2 ] V: volume of the stone (dn

3) [ m3 ]

In figure 2.4 all force components acting on the grain are shown. Combining the two acting forces, gives a resulting force (FR).

22LDR FFF += (2.4)

As long as equilibrium between the acting forces and the resisting forces exists, the grain will not move. The grain will move if the momentum of forces around the point of contact (point S, figure 2.4) with the adjacent grain is positive.

0>− yFxF GR (2.5) The acting forces will always try to lift the grain so it can start rolling or sliding.

current

FG

FL

FD

FR

FG

current

S

y x

Figure 2.3 Gravity force on a single particle

Figure 2.4 All force components acting on a single particle in a uniform flow

2. Theory


2.2.2 Stone stability in a uniform flow A lot of research has been performed on the stability of stones under the influence of a uniform stationary current. The most common formula for stability of stones is the Shields formula. Shields found a relation between the initiation of motion of grains and the occurring critical shear stress. Shields developed a relation between the dimensionless critical value of the shear stress (Ψc) and the so-called particle Reynolds-number ( *Re ) to describe the initiation of motion. (Schiereck, 2001)

)(Re)( *

2* fgd

ugd

C

WS

CC =

∆=

−=Ψ

ρρτ (2.6)

Ψc: Shields parameter [ - ] τc: critical shear stress [ N/m2 ] ρs: density of stone [ kg/m3 ] ρw: density of water [ kg/m3 ] g: gravitational acceleration [ m/s2 ] d: stone diameter [ m ] u*C: critical shear velocity [ m/s ] ∆: relative density ( = (ρs-ρw ) / ρw ) [ - ]

υdu c*

*Re = (2.7)

Re*: particle Reynolds number [ - ] υ: kinematic viscosity [ m2/s ]

In figure 2.5 the Shields relation is presented. For a high particle Reynolds-number (Re* > 500), which is similar to a large grain size, a constant value for the Shields parameter of 0.055 is found.

2. Theory


Figure 2.5 indicates that for values of the Shields number above the curve movement occurs. For values below the curve no movement occurs. The initiation of movement is difficult to define and a subjective matter. It is important to define the criteria used for describing the motion of grains. For example in Schiereck (2001) a research by DHL is presented in which 7 transport stages are discerned. Figure 2.6 shows the results and the Shields curve.

0. no movement at all 1. occasional movement at some locations 2. frequent movement at some locations 3. frequent movement at several locations 4. frequent movement at many locations 5. frequent movement at all locations 6. continuous movement at all locations 7. general transport of the bed

Figure 2.5 Critical shear stress according to Shields

Figure 2.6 Different transport stages compared to the Shields relation

2. Theory


The goal of the presentation of this investigation by DHL is not to use it in this thesis but to show that initiation of motion is not an unequivocal idea. The existing, classical methods have in common that they did not take flow acceleration into account when investigating stone stability.

2.2.3 Shear velocity The difficulty that arises when for example using the Shields formula is the determination of the shear velocity, formula (2.6). There are different ways to describe the relation between the shear velocity and the mean velocity of the flow which can be measured, appendix A. The theoretical relation that will be used in this thesis as well as the empirically determined relation by Chezy are described below. An alternative method for calculating the shear stress will finally be presented in this section. First the shear stress and shear velocity will be defined. In a uniform stationary current the velocity of the flow is affected by the bottom friction. In case of a uniform current the forces acting on a volume of water are balanced. The shear stress and the downward acting gravitational force are equal. The shear velocity is used to describe the bottom shear stress. The shear stress at the bottom (τ0) is the developed shear force per unit wetted area. The definition of shear velocity is:

W

uρτ 0

* = (2.8)

*u : shear velocity [ m/s ] τ0: shear stress on the bottom [ N/m2

]

2. Theory


Theoretical approach When we consider the flow to be a log profile, the term z0 is the position above the bed where the velocity is zero. In literature the term z0 is sometimes replaced by y0 depending on the choice of the xyz-plane. In this thesis z0 will be used. The logarithmic part of the profile appears in the vertical velocity profile at a distance of 2 or 3 times the grain diameter above the bed up to at least 0.2 times the water depth (h), figure 2.7. The relation between the logarithmic part of the flow profile and the shear velocity can be written as:

0*

ln1zz

uu

κ= (2.9)

κ: von Kàrmàn constant ( = 0.4 ) [ - ]

z: height measurement above the bed [ m ] Equation (2.9) is valid for an undisturbed flow. In the case of the experiment the theory can be used in the approach section of the flume where the flow is not accelerated yet. In case of an accelerated flow this theory is not proved to be valid so can not be used groundlessly. In section 4.3 will be explained in which way this theory is going to be used for the obtained experimental data. Chezy The Chezy formula is an empirical formula for the relation between the shear velocity and the flow velocity in an undisturbed flow. In the experiment it can and will only be used in the acclimatization part of the flume. In this part of the flume the flow is not

0.2 h

2 or 3 stones

z

Figure 2.7 Logarithmic part of an undisturbed flow profile

2. Theory


accelerated and the Chezy formula can serve as a check for the determined shear velocity for the undisturbed flow using (2.9).

==

kh

uC

guu

12log75.5* (2.10)

h: water depth [ m ]

k: roughness (2 or 3 times dn50) [ m ] C: Chezy coefficient [ m1/2/s ]

In appendix A an extensive explanation of the Chezy formula is presented. Momentum balance An alternative way to calculate the shear velocity is to consider a momentum balance. When looking at a contraction of a flow, a momentum balance and a mass balance can be defined. When a momentum balance is considered for a stationary flow, the pressure and the convection contribute to the exchange of momentum. Combination of the mass and momentum balance gives an equation which can be used to solve the unknown shear stress in a flow contraction caused by the bottom and wall friction. In appendix A the steps taken for deriving the equation below are presented.

dxdB

gBu

Bh

dxdh

ghu

gh

−+

−=

22

21 ββ

ρτ (2.11)

B: width [ m ]

In section 4.4.3 it will be discussed whether it is possible to use the obtained data from the experiments for calculating the shear stress with the help of the momentum balance. The obtained data appeared to be not accurate enough to be able to calculate the shear stress. Equation (2.11) can also be derived when the momentum balance is defined using polar coordinates. In case of the experiment this method would be preferable if the data are accurate enough.

2.3 The Morison equation The objective of this thesis is to investigate whether flow acceleration has influence on stone stability. In an accelerated flow the velocity generated force (section 2.2.1) and an extra acceleration generated force act on the stones. This extra force is caused

2. Theory


by a pressure difference over each single stone. Due to turbulence the acting forces fluctuate in time but have an average value over time. Morison (1950) proposed the following formula for waves that combines both forces.

DtDuVCuAuCFFF MDonaccelaratidrag ρρ +=+=

21 (2.12)

The drag coefficient, CD, is already mentioned in section 2.1.1. The term CM is an acceleration coefficient. This equation needs to be adjusted to an accelerated flow as is the case in the experiments. The Morison formula does not include the influence caused by lift forces and turbulence. Both contribute to the force balance, as will be proved in the following sections. The drag force in the Morison equation can be replaced by a force which contains drag and lift. The following formula is proposed:

∂∂

+=xuuVCuAuCF MB ρρ

21 (2.13)

The first term on the right-hand side will be referred to as the velocity term and the second term as the acceleration term. In appendix A it is shown that when looking at a single grain in an accelerated flow the force generated by the flow acceleration can be described as:

Vxuu

dxdpVFp )(

∂∂

== ρ (2.14)

This term is represented in the acceleration term in equation (2.13). This makes it assumable to use the Morison equation when investigating the forces acting on the stones in a flow acceleration as will be the case in the experiments. In the next sections equation (2.13) will be looked at in more detail because different difficulties arise when it is used for analysing the experiments. The force generated by the acceleration will add to the acting forces on a single grain and thus to the force balance, figure 2.4. The force balance needs adjustment by adding the acceleration force. In section 2.3.1 the renewed force balance will be deduced. The velocity term in (2.13) holds twice a velocity, u. Due to flow acceleration the flow profile will change, see figure 2.10. The difficulty will be how to determine the velocity in the Morison equation using the changed flow profile and whether it is possible to measure this velocity using the instruments available. The fluid mechanical aspects need to be looked at more closely to understand what happens in the accelerated flow. Turbulence of the flow will cause peak velocities which are hold responsible for the maximum occurring force on the stones and are therefore responsible for the stone stability. Section 2.3.2 will deal with the turbulence and the peak velocities. When dealing with an accelerated flow which causes an increasing flow velocity and a decreasing water depth, it must be kept in mind that the flow

2. Theory


should not become critical. In the end of section 2.3.2 attention will be paid to the number of Froude which indicates whether a flow is critical or not. Both the velocity and acceleration term in the Morison formula contain a coefficient, CB and CM. The bulk coefficient, CB, contains the coefficients CD and CL and other terms as will be shown in section 2.3.3. There is not a lot known about the inertia coefficient, CM, because few research is carried out concerning stone stability in an accelerating flow. In section 2.3.3 one of the few researches on CM is presented.

2.3.1 Force balance including the force due to acceleration In section (2.1.1) a classical force balance is shown for a grain under the influence of a fluid motion. Now we will expand this theory by adding a force due to acceleration (FA). If we consider the most common angle of initiation of movement, the pivoting angle, we see that this is related to the angle of internal friction of the bed material. Kirchner (1990) investigated this angle to be between 30º and 45º. The angle, φ, does not dependent on the character of the initial motion (rolling or sliding). In figure 2.8 all forces acting on a single grain are drawn. Again the acting forces will try to move the grain in the direction of the flow around point S. The FR acts at an angle β compared to the FA, which works in horizontal direction. The angle β depends on the contribution of the lift, drag and turbulent forces. It is assumed that all the forces acting on the stone have their origin in the centre of mass of the stone.

The three acting forces in figure 2.8 can be resolved in forces acting in the most common direction of initiation of movement and through point S. In the figure below this is done for the case of figure 2.8. If we look at the momentum around S we see that the forces acting through point S will not contribute to the initiation of motion.

FG

φFA

FR

current

S

β

Figure 2.8 All forces acting on a particle in an accelerating flow

2. Theory


The resulting force balance, containing the two forces in the direction of the pivoting angle and the forces through S, is shown in figure 2.9.

At the point of initiation of movement the two forces in the direction of movement will stabilize each other. In formula this can be written as:

MARMG FF ;,; = (2.15)

ϕβϕϕ cos)cos(sin ARG FFF +−= (2.16) Movement of the grain occurs when:

MARMG FF ;,; < (2.17)

ϕβϕϕ cos)cos(sin ARG FFF +−< (2.18) Equation (2.18) describes the threshold of motion for a single grain. When the resultant of the gravity force, FG;M, is larger than the acting forces, FR;A;M, the stone will not move and vice versa the stone will move. This equation will be used later on in this thesis when the movement of the stones is analysed.

2.3.2 Velocity of the accelerated flow In order to determine the forces as described in the previous sections the fluid mechanical aspects that occur in the flume have to be investigated. A lot of research is done for undisturbed flows. In the experiment the flow is undisturbed in the part of the flume where it is not accelerated yet. For this part of the flume the flow

φFA

FR,A; M

FG,A; S

current

S FG; M

FR; S

Figure 2.9 Force balance of a particle in an accelerating flow in the direction of the pivoting angle φ

2. Theory


parameters can be determined using the already existing theory. For accelerated flows less information is available.

The flow profile changes when it is accelerated (figure 2.10) so the theory for an undisturbed flow can not be used groundlessly in the part of the flume where the flow is accelerated. The challenge is to try to use the information obtained from the undisturbed flow parameters for the accelerated part of the flow. Turbulence Turbulence is very important for the stability of stones. In turbulent flow, the velocity and pressure show irregular fluctuations. This can be seen in figure 2.11.

1.2

1.3

1.4

1.5

0 10 20 30 40 50 60

Time (s)

Vel

ocity

(m/s

)

Measured velocity (m/s)

Average velocity (m/s)

Figure 2.11 Irregular fluctuations of the velocity in a turbulent flow

Velocity (m/s)

Wat

erhe

ight

(m)

Undisturbed logaritmic flow profile

Accelerated 'full' flow profile

Figure 2.10 Difference between an accelerated flow profile and an undisturbed flow profile

2. Theory


An attempt to describe turbulence by Hinze in 1975 (Booij, 2002) is as follows: “Turbulent fluid motion is an irregular condition of flow in which the various quantities show random variation with time and space coordinates, so that statistically distinct in average values can be discerned.”

To do so the velocity and pressure measurements are averaged over a certain period of time. This can be written as:

'uuu += 'vvv += 'www += 'ppp += (2.19)

To obtain statistically distinct average values, a measure of the intensity of the velocity fluctuations needs to be defined. The root-mean-square value, which is equal to the standard deviation, provides a good way of describing the intensity of the turbulence.

x

xx u

ur

2'= ,

y

yy u

ur

2'= ,

z

zz u

ur

2'= (2.20)

The r indicates the turbulence intensity, the index the direction of the flow. The turbulence intensity can only be calculated using (2.20) when the velocity of the flow is not close to zero. Another way of describing turbulence is using the next expression in which k represents the total kinetic energy in a turbulent flow.

( )222 '''21

zyx uuuk ++= (2.21)

Turbulence results in a loss in kinetic energy. When a flow is accelerated by for example a contraction, the fluctuations in the flow direction (ux’) will decrease, because the flow concentrates. The fluctuations perpendicular to the flow (uy’) will increase. The total amount of kinetic energy will remain approximately constant. The relative turbulence, rx, will decrease due to the increased velocity in the contraction, see figure 2.12. (Schiereck 2001)

Figure 2.12 Decrease of the turbulent fluctuations in an accelerating flow

2. Theory


We distinguish two different types of turbulence, depending on the way they are created.

1) Wall turbulence 2) Free turbulence

1) The boundary layer is defined as the region in which the presence of the wall has a influence on the flow. In stationary, uniform flow, the boundary layer is fully developed and takes up the entire water depth. This leads to a logarithmic velocity distribution over the depth. When a boundary layer develops in a flow with no accelerations the shear stress along the bottom slows down the flow and the exchange of momentum will lead to the growth of the boundary layer. When a flow is accelerated due to a pressure gradient in the flow direction, the thickness of the boundary layer will reduce. 2) Free turbulence is a result of a difference in velocity of two connecting flows. Because of friction between the two, there will be transmission of momentum. A boundary layer between the two flows develops in which the velocity will change.

The number of Reynolds tells something about the type of flow. A flow with a Reynolds number < 2300 is considered to be laminar and Re > 4000 is turbulent.

υud

=Re (2.22)

Re: number of Reynolds [ - ] d: water depth [ m ] υ: kinematic viscosity [ m2/s ]

In figure 2.13 a velocity profile is given for a flow with walls on both sides (Battjes, 1997). Two turbulent and one laminar flow distribution are drawn.

Figure 2.13 Flow profile for laminar and turbulent flow with walls on both sides

2. Theory


Due to turbulence a wake behind the stone is created. This causes a lower pressure behind the stone than in front of it. The pressure gradient will create a force acting on the grain in the direction of the flow. Peak velocities Peak velocities due to fluctuations of the flow near the bottom as described in the previous section, are considered to be responsible for stability and erosion of the bed material. The peak velocities are related to the relative turbulence. Written in formula:

xxx uru )1(max, λ+= (2.23) This equation can also be written as:

2'max, uuu x λ+= (2.24)

This way of describing the peak velocity is preferred to (2.23), because the turbulence intensity can not be calculated using (2.23) when the flow velocity is close to zero. The λ in the equations above has a theoretical value of 3, as described in Schiereck (2001). Froude The Froude number is an indicating number for free open channel flow. The dimensionless number represents the ratio of inertial to gravitational forces or kinetic versus potential flow. High Froude numbers can be indicative of high flow velocity and scour potential.

ghuFr = (2.25)

Fr < 1: sub-critical flow Fr = 1: critical flow Fr > 1: super-critical flow Fr: Froude number [ - ] u: flow velocity [ m/s2 ] h: water depth [ m ]

In case of the experiment, the flow must stay sub-critical in the flume. As the flow accelerates in the contraction, the Froude number will increase but the flow may not become critical.

2. Theory


2.3.3 Coefficients CB en CM

Literature on the bulk and inertia coefficient (CB and CM) in equation (2.13) as such can not be found easily. The CB, a result of primarily the drag and lift coefficient that are subject to a lot of discussion in literature. A lot of research has been carried out to determine the values for CD and CL. Still only rough estimations are found. These estimations depend on many flow and particle parameters, so the estimations vary considerable. When looking at Shields formula, the question whether a particle moves depends on the shear stress caused by the flow. Because approximately 30 percent of the top layer of stones is influenced by this shear stress, it will be related to the drag force as follows:

3.02

1 2022 dduCF DD

τρ ≈= (2.26)

Combining this equation with the equation for the shear stress at the bottom (2.8) and the Shields formula (2.6), will provide a rough estimation for determining the drag coefficient.

215.0 ugdCD

Ψ∆≈ (2.27)

The lift force and hence the lift coefficient can be estimated using (2.27) because the lift force is approximately half the drag force. Most estimations for CD make use of a velocity measured close to the bed. In this experiment the available instruments make measurements close to the bed impossible. Estimations found for CD can not be used groundlessly for the experimental results. The values found for CD are derived for an undisturbed flow profile. The changed flow profile of the accelerated flow does not encourage the use of the same values for the coefficients. Smaller values are expected because a velocity further away from the bed will be used. The bulk coefficient, CB, contains the lift and drag coefficient, but also the term cos (φ – β), equation (2.18). The angle φ is almost similar to β, so this term will be close to one. The inertia coefficient CM can be discussed meaningfully as the sum of two terms caused by the pressure gradient required to accelerate the flow and the so called added mass. (Dean and Dalrymple, 1991)

mM kC +=1 (2.28) The unity term in equation (2.28) caused by the pressure gradient is a so called “buoyancy” force on the particle. An additional local pressure force occurs to accelerate the neighbouring fluid around the particle. The force necessary for the acceleration of the fluid around the particle yields the added mass term, km.

2. Theory


For a flow around a particle the added mass term can according to Dean and Dalrymple shown to be:

abkm = (2.29)

Figure 2.14 demonstrates the occurrence of a small contribution of the added mass on the inertia coefficient for a streamlined body. The CM is always larger than 1 and for the oval shaped stones which will be used in the experiment a value between approximately 2 and 7 for the inertia coefficient seams a reasonable first estimation.

It must be noted that Dean and Dalrymple investigated larger objects than the particles that will be looked at in this experiment. The values found by Dean and Dalrymple give an indication for the values that should be found for this experiment. It needs to be mentioned that the inertia term, CM, holds the angle factor (cos φ) which follows from equation (2.18).

Figure 2.14 The inertia coefficient for a cylinder of ellipsoid cross section

3. The experiment


3 The experiment

3.1 Introduction The objective of this thesis as mentioned in section 1.2 is to obtain more insight in the influence of acceleration of the flow on the stability of stones. This will be done by carrying out experiments in a flume containing a local contraction. In the contraction, where the flow accelerates, the stability of two different stone sizes will be investigated. In the previous chapter the forces on a single grain in an accelerated fluid motion are explained. In the experiment different combinations of velocity and acceleration of flow have to be created. All relevant information for carrying out the experiments is given in this chapter. The testing procedure, a description of the flume and the used structure are given. In the last section a summary of the most important information following from this chapter can be found.

3.2 The flume The experiments are carried out in the Fluid Mechanics Laboratory of the department of Hydraulic Engineering of the Faculty of Civil Engineering, Delft University of Technology. The experiments are carried out in a flume in which a fluid motion can be created. An impression of the flume and it’s dimensions is given in figure 3.1. Note that the dimensions are not drawn to scale.

3. The experiment


When the flow enters the flume, Qin, the water is very turbulent. To reduce the turbulence a package of tubes, through which the water flows, is placed in the flume as can be seen in the figure above. The discharge of water in the flume can be regulated using an orifice plate in the water supply pipe. By measuring the difference in water pressure before and after the orifice plate, one can determine the discharge. In the graph shown in appendix B, the discharge is plotted versus the difference in piezometric level in the pressure gauge. The discharges used in the experiment will be between 30 and 70 litres per second. This corresponds with a head difference between 15 and 82 centimetres as can be seen in appendix B. At the end of the flume a gate is installed to regulate the water level in the flume. The water height can be lowered by lowering the gate and visa versa. If the discharge is kept constant, different velocities can be created by changing the height of the gate. The gate works as a Rehbock-gate although discharge calculations based on this theory will not be used in the experiment. High turbulence and the irregularities of the profile at the end of the flume make this impossible. One more thing that can be said about a Rehbock-gate, is that after two times the height of the gate there will be no influence on the water height, upstream. This will be of use when positioning the contraction. After passing the gate the water will be discharged in a large water basin.

Figure 3.1 Dimensions of the flume used in the experiment

Q in

Q out 0.70 m 14.00 m

0.70

m

gate Flow stabilizer

Sideview

Topview

0.50

m

11.80 m

3. The experiment


3.3 The structure In the experiment different velocity-acceleration combinations have to be created as mentioned in the introduction of this chapter. This will be done by a symmetrical contraction of the flume. The symmetry is important because the water streamlines will form a symmetrical flow pattern in a cross-section of the flume and the velocity in y-direction in the middle of the flume will be zero. For the experiments two contractions are made. The dimensions of the structures can be found in appendix C. The most important dimensions are listed in table 3-1. The width of the outflow will from now on be called Be and the width of the inflow of the contraction Bb. The angle α, mentioned in the table, is the angle between the wall of the flume and the structure. In figure 3.2 and 3.3 the dimensions are indicated.

L1.50 L2.00

Length (m) 1.50 2.00 Bb (m) 0.50 0.50 Be (m) 0.15 0.15 Height (m) 0.70 0.70 Angle α 6.65º 5.00º

The wooden plates covering the bottom of the flume are 1.90 centimetres thick and are glued to the bottom over the entire length of the flume. When these wooden plates are exposed to water they will expand. In the y-direction the plates can expand 5mm on both sides before they get into contact with the glass wall of the flume. The very small expansion in the z-direction will be neglected. The area between the stabilizing structure and the beginning of the contraction is called the approach section or the acclimatization area of the flume. In this area the flow will stabilize and form a logarithmic velocity profile. The end of both contractions is placed at exactly the same position in the flume. The distance between the outflow and the gate is 1.70m. Considering the maximum height of the gate in the experiment is 0.30m, the outflow is located more than 5 times that height upstream. This is more than twice the distance required, see section 3.2.

3.4 Definition of the xyz-plane In the experiment it is very important to define a local fixed xyz-plane. The position of the origin will be at the end and the bottom of the contraction in the middle of the flume. The outflow, 0.15m wide, of the used contractions has to be placed at the same position in the flume.

Table 3-1 Important dimensions of the contractions used in the experiment

3. The experiment


Definition of the level of the bed is complicated because the layers of stones on the wooden bottom. In section 4.2 and 4.3 the bottom level for the experiment is determined. For now the level of the bed will be located on top of the wooden plates as indicated in the next figures. The only thing that will change when the actual level is determined is the height of the origin and hence the x-axis. The xy-plane used in the experiment can be found in the following figure. The walls and bottom of the flume are indicated in green. Used wood for the structure is indicated with the colour brown. Note that the discharge of the flow is in negative x-direction. The xz-plane used in the experiment can be found in figure 3.3, which is a cross-section in the middle of the flume. Notice the thin wooden plates on the bottom of the flume. The orientation of the y-plane is determined by considering a single vortex being positive from x to z. The xyz-plane forms a Cartesian system which fixes the y-plane as indicated in figure 3.2.

Figure 3.2 Top view of the contraction, the x-y plane

Figure 3.3 Side view of the contraction, the x-z plane

x

y

xz xz

QBe Bb

L

α

x

z

Q h

3. The experiment


3.5 Measuring instruments In the experiment the important parameters need to be measured accurate enough so it will be possible to analyse and interpret them correctly. The measured parameters are categorized in four different groups. For each parameter the instrument used for measurements is given. When necessary an explanation is given in the second part of this section. Geometrical measurements [ m ]:

• Dimensions of the flume tapeline • Dimensions contraction tapeline • Positions for measurements in the flume tapeline • Water depths point gauge • Gate height point gauge • Stone sizes ruler and balance • Expansion of wooden plates on bottom point gauge

Fluid motion measurements:

• Fluid velocities [ m/s ] EMS • Fluid discharge [ m/s3 ] orifice plate

Transport measurements [ - ]:

• Number of grains in coloured strip • Number of grains moved from strips during experiment

Other measurements:

• Duration experiment [ min ] stopwatch • Water temperature [ ºC ] thermometer • Water density [ kg/m3 ] balance

The list of geometrical measurements contains the position for measurements in the flume. In the next section this will be explained. The method for measuring the fluid discharge is explained in section 3.2.

3.5.1 Velocity measurements During the experiments the velocity of the flow needs to be measured at different locations in the contraction. These locations vary in x and z position, but they are all measured in the middle of the contraction (y = 0.00m), where the velocities in y-direction are zero. The velocity measurements are carried out with an EMS (Electro Magnetic velocity Sensor). The principle of the EMS is based on Faraday’s Induction Law through a magnetic field. The EMS can measure the velocity of the flow in x and y direction with a frequency of 50Hz. In the experiment a measurement with a duration of 60 seconds is carried out which results in 3000 values for each position in two directions.

3. The experiment


The output of the EMS is recorded and displayed using a software program called DAISY Lab 5.02. The x-positions of the measurements in the contraction are situated on the borderline of each coloured strip of stones (see the next section) and at the beginning of the contraction. These positions are for both contractions (L1.50 and L2.00) the same. The positions for the EMS-measurements are marked on the side of the flume. The z-positions of the measurements differ for the two grain sizes. The z-positions mentioned in this paragraph are with respect to the top of the wooden plates. Again this is not the actual level of the bed. Experiments are carried out using two different stone diameters referred to as large and small stones (see section 3.6). For the small grains the lowest measurement is taken at z = 0.04m. The z-position of the highest measurement depends on the water level. The EMS can measure properly up to 0.07m underneath this level. In figure 3.4 the locations are indicated. For the large grains the lowest measurement is taken at z = 0.075m. For the z-position the same condition as for the small stones holds. The locations can also be found in figure 3.4.

The EMS is sensitive for the temperature of the water, the distances from wall and bottom and the surface of the water level in the flume. Because in the contraction the distance from the wall varies in x-direction, ‘zero-measurements’ have been carried out, for calibration of the EMS. Such measurements are taken in still-water at all the locations indicated in the figures above. The values are subtracted from the velocities measured. The EMS has also certain other limitations. Measurements can only be taken in two directions. In the case of the experiment this will be in x and y-direction in the experiment the last one being equal to zero. Velocities in z-direction can not be

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2

x-position (m)

z-po

sitio

n (c

m)

Small stonesLarge stones

Figure 3.4 Side view contraction, positions velocity measurements, large and small stones

3. The experiment


measured. Secondly, the presence of the EMS disturbs the flow. Most important is the minimum size of the vortices that the EMS can detect. Vortices smaller than 0.01m can not be detected and are not taken into account when describing the turbulence intensity. Hofland (2004) found that vortices with a diameter between 1.5 and 2 times the stone diameter are most effective for movement. For both used stone sizes this value is larger than 0.01m and therefore it is believed that these vortices are taken into account.

3.5.2 Water depth measurements In the contraction the declination of the water depth needs to be measured. This is done with a point gauge. The difference between the bottom level (z = 0.00m) and the surface is measured. The measurements are carried out at different x-locations in the contractions. In figure 3.5 these locations are indicated for the contraction L2.00. For the contraction L1.50 the measurement at x = 2.00m will not be carried out.

Figure 3.5 Top view contraction, positions water depth measurements

-0.25

-0.15

-0.05

0.05

0.15

0.25

0 0.5 1 1.5 2

x-position (m)

y-po

sitio

n (m

) .

3. The experiment


3.6 The used stones

3.6.1 Measured stone parameters The experiments are carried out with two different grain sizes. To get a good insight in the influence of accelerations on the threshold of motion more grain sizes need to be tested. The fact that this experiment is carried out for two sizes is because of limited time and availability of the grains. For the two stone sizes is chosen because they were available in a large enough amount. In the table below the measured stone parameters are given.

Large stones Small stones Dimension# stones 139 250 [ - ] M cum. 2845.6 361.77 [ gr ]

dn50 0.0200 0.0082 [ m ] ρ av. 2682.7 2682.2 [ kg/m3 ] W50 21.422 1.490 [ gr ]

d85/d15 1.18 1.22 [ - ] ∆ 1.67 1.67 [ - ]

In the table the number of stones represent the number of stones measured. The total mass of the stones is given as M. cum. The average density is the mean value of the dataset. In appendix D the sieve curves for the mass- and dn-distributions are given. From now on the stones with dn50 = 0.02m will be referred to as large stones. The stones with dn50 = 0.0082m will be referred to as small stones.

3.6.2 Placement of the stones Approach channel The wooden plates in the approach channel are covered with a single layer of stones with a dn50 of 0.015m. These stones are glued to the plates using aqua silicones. For tests with large stones the single layer is covered with a second layer of stones with a dn50 = 0.02m. For experiments with small stones only the single layer of glued stones covers the wooden plates.

Table 3-2 Important measured stone parameters

3. The experiment


Contraction For both experiments with large and small stones the wooden bottom of the contracting is covered with a single layer of stones glued to the plates. This layer is covered with a second layer of stones. In the testing area, the last 0.40m of the contraction, coloured stones are used as the second layer. The stones are placed in strips of 0.10m by dumping them. In figure 3.6 a schematically drawing is given for the placement of the stones in the contraction. In appendix E pictures of the actual placement of the stones can be found.

The number of stones used in each coloured strip (second layer) is counted for each experiment. As mentioned before, the stones are placed by dumping. Also a

AA AA

Top view

Cross section AA

Bottom flume Wooden

plates

Measuring area

First layer of stones, glued to the bottom

Second layer of stones placed loosely on first layer

Figure 3.6 Schematically drawing of the placement of the stones

3. The experiment


calculation is made how many stones should fit in a strip. In the calculation, the assumption is made that the stones have a square shape and that they will cover one layer of the entire area between the walls of the contraction. For this calculation the next formula is used.

250

#n

strip

dA

stones = (3.1)

Then, for both stone sizes, the number of dumped stones used in the top-layer is divided by the number of stones calculated. In tables 3.3 and 3.4 this is listed.

Area L1.50 Area L2.00 # stones (L1.50) # stones (L2.00) (cm2) (cm2) # Used # Cal. #U / #C # Used # Cal. #U / #C Yellow 161.67 158.75 225 239 0.94 224 235 0.95 Blue 185.00 176.25 250 274 0.91 245 261 0.94 Green 208.33 193.75 277 308 0.90 272 287 0.95 Pink 231.67 211.25 306 343 0.89 271 313 0.87 Average 0.91 Average 0.93

Area L1.50 Area L2.00 # stones (L1.50) # stones (L2.00) (cm2) (cm2) # Used # Cal. #U / #C # Used # Cal. #U / #C Yellow 161.67 158.75 24 40 0.59 25 40 0.63 Blue 185.00 176.25 28 46 0.61 25 44 0.57 Green 208.33 193.75 34 52 0.65 31 48 0.64 Pink 231.67 211.25 34 58 0.59 31 53 0.59 Average 0.61 Average 0.61

In these tables we can see that for the large stones the number of used stones divided by the number calculated is between 0.57 – 0.65. For the small stones this is between 0.87 – 0.95. The next formulas for the second dumped layer for respectively small and large stones can be made:

250

92.0n

stripu d

AS = (3.2)

Table 3-3 Comparing the number of calculated and used small stones in the experiment

Table 3-4 Comparing the number of calculated and used large stones in the experiment

3. The experiment


250

61.0n

stripu d

AS = (3.3)

Su: amount of one layer of stones dumped in a strip [ - ]

3.7 Testing procedures For a good test result the procedure for a single experiment needs to be defined so differences in the results can not be assigned to an inconsequent measuring procedure. Besides the experiments that need to be carried out for the objectives of this thesis, the flow profile in the contraction is measured for L2.00. The velocities in the contraction have been measured for 176 different locations, varying the x,y and z coordinates, appendix F. The measurements are used to produce 3-dimensional flow profiles at different locations in the flume. During the experiments the velocities are measured in the middle of the flume (y = 0.00m). To determine the stone parameters a large number of stones are investigated as mentioned in paragraph 3.6. This procedure is described below and varies for the two stone diameters. The first four steps are the same for both stone sizes: Step 1 Sieve the stones using two sieves, one a with a gabs a little larger than

the other. Step 2 Wash the collected stones carefully. Step 3 Dry the stones. Step 4 Collect a random large amount of stones from the sieved washed and

dried stones. ( ≥ 144) Now a distinction has to be made for larger (dn50 = 0.020m) and smaller stones. First the procedure for large stones will be given. The procedure is according to the NEN norm 5186, see appendix G. Step 5 Weigh each stone separately using an accurate balance. Weigh each

stone dry, moist and under water and register the values. Step 6 Measure the largest length and the smallest length of each stone using

an accurate ruler. Step 7 Make the following calculations for determining the stone parameters:

3. The experiment


wm

drws MM

M−

= ρρ (3.4)

( )drss M

V/1

ρ= (3.5)

3

sn Vd = (3.6) In which: Mdr: dry mass [ kg ] Mm: moist mass [ kg ] Mw: under water mass [ kg ] Vs: volume stone [ m3 ] dn: nominal diameter [ m ] The procedure for the smaller stones: Step 5 Weigh each stone separately using an accurate balance. Weigh each

stone dry and under water and register the values. Step 6 Measure the largest length and the smallest length of each stone using

an accurate ruler. Step 7 Make the following calculations for determining the stone parameters:

w

drws M

Mρρ = (3.7)

( )drss M

V/1

ρ= (3.8)

3

sn Vd = (3.9) All the necessary stone parameters can now be calculated. For the actual experiments this procedure is followed: Step 1 Place a contraction of a certain length in the flume and make sure it is

fixed in the flume. Step 2 Glue one layer of stones with a certain dimension to the bottom. Step 3 Construct a second layer of stones by regular dumping under water

over the entire area of the contraction. In the measuring area this second layer needs to consist of coloured stones dumped in strips with a length in x-direction of 0.10m. Create 4 strips as shown in section 3.6.1. Count the number of stones used in each strip.

3. The experiment


Step 4 Determine the discharge and height of the gate for the experiment. Step 5 Bring the gate up to this height. Step 6 Measure the height of the wooden plates on the bottom of the

contraction to calibrate the water depth measurements and calibrate the EMS as explained in section 3.5.1.

Step 7 Let different discharges flow through the flume for the duration of an

hour, but make sure the discharge is lower than the one that is going to be used in the actual experiment. Start with a low discharge and gradually built up the discharge. This step is required because it will cause the second layer of stones placed by hand, to move into stable and favourable positions. Doing this will prevent stones laying in an unstable position to have influence on the experimental results by contributing to the measured entrainment.

Step 8 Create the discharge determined for the experiment. Step 9 Measure and register the water profile. Step 10 Stop the experiment after 15 minutes and go to step 12 or go to step 11

once during each dataset. Step 11 Measure the velocity profile with the EMS. Do this once for each

dataset (see section 3.9 for explanation) and stop the experiment. Count and register the number of grains that moved out of their strips before starting the EMS measurements. Go to step 13.

Step 12 Count and register the number of grains that moved out of their strip,

see section 5.1 for the definition of movement. Step 13 Restore the coloured strips and return to step 4.

3.8 Accuracy and reliability When preparing an experiment it is important to know the accuracy needed to make a decision on which instruments to use. In the case of this experiment it appeared to be more of a question which instruments were available. During the experiment it is important to realize that the precision of the instruments also depends on the user’s skill. Other errors like reading errors and incorrect instrument usage can only be eliminated through careful and continual checking of the experiments. (Tromp, 2004) It must be noted that systematic differences are inevitable in the case of the experiment. This is due to the fact that for every experiment the stones are placed differently although an attempt is made to create the same testing conditions.

3. The experiment


Accuracy is also important when analysing the values found in the experiment. In case of the experiment a test-retest reliability procedure is carried out. The measurement is repeated under conditions as similar as possible. First the number of tests taken for obtaining a dataset of values is analysed. If a standard normal deviation is expected from the obtained data, the student t-test can be used to determine the precision of the mean value of the dataset. The precision P can be defined as:

( ),m pP tn

σ= ± (3.10)

σ: standard deviation n: number of tests In case of this experiment the number of tests taken for obtaining a dataset is 6. In the table below the value for t can be found for different values of α (0.5α = p), the maximum admissible risk. The value of m (number of degrees of freedom) is equal to the number of tests minus 1 (m = n -1).

m\p 0.1 0.05 0.025 0.01 4 1.533 2.132 2.776 3.747 5 1.476 2.015 2.571 3.365 6 1.440 1.943 2.447 3.143 7 1.415 1.895 2.365 2.998

When assuming a probability p of 2.5%, t will be 2.571. Equation (3.10) can now be written as:

1.050P σ= ± (3.11) The accuracy of reading for the different instruments used is listed in table 3-6.

Instrument Error Balance ± 0.01 g Thermometer ± 1º C Stopwatch ± 1 s EMS ± 0.02 m/s Vernier calliper ± 0.1 mm Ruler ± 0.5 mm Tapeline ± 0.5 mm

Table 3-5 The precision of the mean value versus the number of degrees of freedom in case of a student t-test

Table 3-6 The accuracy of reading of the different instruments used in the experiment

3. The experiment


3.9 Evaluation In this chapter all the relevant information concerning the experiments is dealt with. The most important findings will be summarised in this section. First needs to be mentioned that an experimental dataset or just dataset, consists of 6 successive experiments. They share the same discharge and water depths. They are carried out in a contraction with a certain length and position in the flume. In paragraph 3.4 the xyz-plane is defined. In the next chapter this plane is used as an important guidance when describing a lot of parameters and flow characteristics. Two grain sizes will be used in the experiments. Stones with a grain size of dn50=0.0082m will be called small stones in the text and stones with a grain size of dn50=0.02m will be referred to as large stones. The experiments are carried out in two contractions with a different length but the same size of starting and ending width. The contraction with a length of 2.00m will be referred to as L2.00 and the contraction with a length of 1.50m as L1.50. The parameters which will be varied for the different datasets are listed in table 3-7. The dimension, the range and the parameters that will change as a consequence of the variation are also mentioned.

Parameters Dimension Range Changing parameters Discharge Q [ m3/s ] 0.03 - 0.07 h, u resulting in a change of a Gate height hgate [ m ] 0.10 - 0.25 h, u resulting in a change of a Length contraction L [ m ] 1.50 and 2.00 a Median nominal diameter dn50 [ m ] 0.0082 and 0.02 # stones moved and # stones in strip

In section 3.7 the experimental procedure is described. The calibration of the EMS and the point gauges should be noticed. The procedure for gaining the grain parameters is also part of this chapter.

Table 3-7 Varying parameters in the experiment

4. Flow analysis


4 Flow analysis

4.1 Introduction In this chapter the obtained fluid mechanical data will be analysed for both large and small stones. The flow parameters such as z0 and u* are being determined in the following paragraphs. In the analysis of the data, the turbulence plays an important role and will also be treated in this chapter. The objective of this chapter is to determine the different occurring velocity-acceleration combination which can be used when determining the forces on a stone with the Morison equation. This will be done in the next chapter. In the table below the number of experiments that have been carried out are mentioned. The amount of obtained datasets and the resulting combinations of velocities and accelerations can also be found in the table. As mentioned in section 3.9, 6 identically executed experiments form a dataset.

large stones small stones# of experiments L2.00 28 58 # of experiments L1.50 28 67 # of experiments used 46 114 # datasets 8 19 # combinations u & a 32 76

In the previous chapter the xyz-plane is defined. It became clear that the height of the bed has to be determined in order to fix the x-axis. The height of the two layers of stones is measured at different locations. These measurements are used to determine the bed height that is going to be used when analysing the experiments. The measured height will be compared to a calculation method for fixing the level of the bed. Next, in section 4.3, the undisturbed flow at the end of the approach section will be analysed. The logarithmic part of the flow will be used to determine the z0, see section 2.2.3 which covers the theory on this subject. The shear velocity for the beginning of the contraction (x = L) can be determined and compared to the Chezy formula. Determination of the shear velocity in the accelerated flow will be the next matter of this chapter. The z0 for the undisturbed flow is assumed to be the same in the accelerated flow. The measured flow profile and the z0 will be used to determine the shear velocity. In chapter 5 the shear velocity will be used to employ the Shields formula to an accelerated flow. The velocity and the acceleration of the flow are responsible for the force acting on the stones. The Morison formula uses a velocity and an acceleration to determine the

Table 4-1 The number of carried out and useful experiments

4. Flow analysis


total force on a stone. These two variable flow conditions have to be looked at in more detail to find the applicable velocity-acceleration combinations for the Morison formula. The relation between the peak velocity, which is important when looking at the stone stability, and the mean velocity can be determined using the turbulence intensity as mentioned in section 2.3.2. When this relation is found, the flow profile of the accelerated flow will be looked at in more detail. The occurring velocity can be determined in different ways. The velocity profile measured using EMS will be compared to the average velocity through a passage. This average velocity is calculated using the discharge, the water depth and the width of the contraction. The water depth will be calculated using the earlier in the chapter determined level of the bed and the water level measurements. In the final section of this chapter the accelerations that occur in the contraction will be calculated. In this chapter it will become clear that there are different ways to calculate the occurring velocity and acceleration. At the end of the last section the different methods will be compared with each other and the most favourable method for calculating the velocity-acceleration combination for a certain position in the contraction will be used in the next chapter when the threshold of motion is analysed. In each of the following sections all data concerning the small stones will be analysed first.

4.2 The level of the bed For a first estimation of the water depth the bed height of the two layers of stones for both small and large stones are measured. With the help of a point gauge the height of the bed at 30 different arbitrary locations is determined. The point gauge was lowered until it reached the bed, figure 4.1. This implicates that not only the tops of the stones in the top layer of the bed were measured, but also lower points of the bed, figure 4.1. Subtracting this height from the depth of the wooden bottom plates and averaging the obtained values will give an indication for the height of the two layers called the perpendicular layer thickness, tm.

tm

Measured depths using a point gauge Average level of the top of the bed

Figure 4.1 Determination of the perpendicular layer thickness, tm

4. Flow analysis


The CIRA/CUR (1991) gives the following definition for the perpendicular layer thickness of randomly placed armour:

50ntc dknt ⋅⋅= (4.1) tc: calculated layer thickness [ m ] n: number of layers [ - ] kt: layer thickness coefficient [ - ] The layer thickness coefficient is according to Bosma (2001) between 0.75 and 0.9. In table 4-2 the values for the calculated layer thickness are compared to the measurements using kt = 0.75. The measured values are almost similar to the calculated ones. Theories on the determination of the level of the bed state that the flow will reach between 0.15 and 0.30 times the dn50 under the average measured level of the bed, tm. In the next figure this will be indicated.

In the next table the results found during the experiments will be presented. dn50 (m) # layers tm (m) tc (m) tm - 0.15 dn50 (m) tm - 0.30 dn50 (m)small stones 0.0082 2 0.0142 0.0123 0.0130 0.0117 large stones 0.0200 2 0.0295 0.0300 0.0265 0.0235

A first estimation for the bed level (z = 0.00m) for small stones will be between 1.17x10-2m and 1.30x10-2m above the wooden plates. For large stones the bed level will be between 2.35x10-2m and 2.65x10-2m above the plates. It should be mentioned that the level of the top of the bed is measured using a point gauge with a sharp end. To make sure that holes in the bed do not contribute to this level, a hemispherical probe (with a size of 0.5dn50) should be used. (Bosma, 2001) In the calculations made in this thesis the measured thickness of the layers, tm, is used.

Table 4-2 First estimation for the level of the bed

Figure 4.2 First estimation for determining the level of the bed

0.15 – 0.30 dn50

tm

Z = 0

4. Flow analysis


4.3 Determination of z0 As mentioned in section 2.2.3, figure 2.7, the undisturbed flow has a logarithmic profile from 2 or 3 stone diameters from the bed up to 0.2 times the water depth. The velocities of the flow are measured at several locations in the contraction using EMS. For both contraction L2.00 and L1.50, the velocity profiles are measured at x = L. Objective of this section is to determine the height above the bed where the velocity is zero according to the fact that the flow profile in the lower part of the water depth is logarithmic. This point, which lies above the bed (z = 0.00m), is called z0. At the end of the contraction, where the flow is accelerated, the profile will no longer consist of a logarithmic form, figure 2.10. The z0 for the undisturbed flow calculated in the next sections, will be used in the accelerated flow (section 4.4.1). It is assumed that it will have the same value through the entire contraction.

4.3.1 z0 for small stones First a bed level where z = 0.00m needs to be determined. In the previous section it is found that this level will be between 1.17x10-2m and 1.30x10-2m above the wooden plates. For both heights the measured velocities are plotted versus the logarithmic values of the measuring height above the bottom level, see appendix H. For each discharge and contraction length a value for z0 is found. When these different values are averaged a common value for z0 can be determined for the two different bed levels, see figure 4.3.

Figure 4.3 Common value of z0 for the two different levels of the bed found in section 4.2

Z = 0

0.01

30m

(tm

- 0.

15d n

50)

0.01

17m

(tm

- 0.

30d n

50)

Z = 0

z0 z0

4. Flow analysis


For this experiment the level of the bed will be defined at 1.30x10-2m above the plates for reasons that will be explained later on in this section. In the next figure (figure 4.4) the velocity measurements are plotted versus the height above the bed level. The average value of z0 is found to be 1.17x10-3m above z = 0.00m. It should be mentioned that two datasets (L1.50, 40l/s and L2.00, 50l/s) are not plotted in the figure because no logarithmic profile could be found. The dashed lines in the figure belong to L1.50 and the solid lines to L2.00.

The following equation is given for a logarithmic flow profile:

)ln(ln1ln10

0*

zzzz

uu

−==κκ

(4.2)

The straight lines through z0 in the figures can be written as:

buaz +=ln (4.3) Equation (4.2) can also be written in this form:

0*

0*

ln1lnln zuu

zuuz +=+= κκ (4.4)

Figure 4.4 The velocity of the undisturbed flow of 6 different datasets versus the height of the measurement above the bed with a common value for z0, for experiments using small stones

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.00 0.05 0.10 0.15 0.20 0.25

Velocity (m/s)

ln (z

+ ∆

z)

L=1.50m; Q=30l/sL=1.50m; Q=50l/sL=1.50m; Q=60l/sL=2.00m; Q=30l/sL=2.00m; Q=40l/sL=2.00m; Q=60l/s

4. Flow analysis


So:

*ua κ

= and 0ln zb = (4.5 a and b)

The shear velocity can also be determined using Chezy’s formula, equation (2.10), to compare the result found for the shear velocity calculated using equation (4.5 a). In the table below all values are listed.

L (m) Q (m3/s) h (m) ū(m/s) a (1/ m/s) u* (m/s) u* Chezy(m/s) 1.50 0.03 0.257 0.246 18.07 0.022 0.021 2.00 0.03 0.260 0.243 17.96 0.022 0.020 2.00 0.04 0.324 0.258 18.18 0.022 0.021 1.50 0.05 0.377 0.275 17.46 0.023 0.021 1.50 0.06 0.434 0.285 18.66 0.021 0.021 2.00 0.06 0.438 0.282 18.69 0.021 0.021

The reason for choosing the level of the bed at 1.30x10-2m above the wooden plates can now be explained. There are three reasons for choosing this level. The value of z0 should be approximately:

mdz n 3500 1082.0

100082.0

10−×==≈ (4.6)

The z0 for the level of the bed at 1.30x10-2m (z0 = 1.17 x 10-3m) is closer to this value than for the level of the bed at 1.17x10-2m (z0 = 1.56 x 10-3m). The second reason is that the calculated values for the shear velocity are closer to the Chezy values for a bed level at 1.30x10-2m. In appendix H this is illustrated. As mentioned in section 4.2 a hemispherical probe should have been used instead of the point gauge. This would have resulted in a higher average surface level of the bed, so the choice of the highest level of the bed (tm – 0.15dn50) will be favourable. In the next figure the different layer thickness approaches (tc and tm – 0.15dn50) and the z0 are illustrated. The level of the bed used in this thesis, z = 0.00m (tm – 0.15dn50), is located at 1.30x10-2m above the wooden plates.

Table 4-3 Measured and calculated flow parameters of the undisturbed flow for experiments using small stones. z=0.00m is located at 1.30x10-2m above the plates and z0 = 1.17 x 10-3m. The shear velocity found in the experiments is compared with the shear velocity calculated using the Chezy formula.

4. Flow analysis


4.3.2 z0 for large stones The level of the bed in case of the large stones lies between 2.35x10-2m and 2.65x10-2m above the plates. For both cases the average z0 can be determined. The level is fixed at 2.65x10-2m above the plates for the same reasons as for small stones. In the figure below the velocities are plotted versus the height above the bottom level with z0 = 2.80 x 10-3m. The values for L2.00 and a discharge of 50l/s can not be found in the figure because the measurements did not show a logarithmic profile. The dashed lines belong to L1.50 and the solid lines to L2.00.

Figure 4.6 The velocity of the undisturbed flow of 7 different datasets versus the height

of the measurement above the bed with a common value for z0, for experiments using large stones

z =

0.00

m (t

m -

0.15

dn5

0)

t m

t c

z =

0.00

m +

z0

Figure 4.5 Different layer thickness approaches used for the experiment

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Velocity (m/s)

ln (z

+ ∆

z)

L=1.50m; Q=40l/sL=1.50m; Q=50l/sL=1.50m; Q=60l/sL=1.50m; Q=70l/sL=2.00m; Q=40l/sL=2.00m; Q=60l/sL=2.00m; Q=70l/s

4. Flow analysis


For the value of z0 the following approximation can be made:

mdz n 3500 1000.2

10020.0

10−×==≈ (4.7)

This value is closer to the value of z0 for the level of the bed at 2.65x10-2m above the plates (z0 = 2.80 x 10-3m) than to the z0 for the bed level at 2.35x10-2m above the plates (z0 = 3.40 x 10-3m). In the next table the values calculated for large stones in an undisturbed flow are presented.

L (m) Q (m3/s) h (m) ū(m/s) a (1/ m/s) u* (m/s) u* Chezy (m/s) 1.50 0.04 0.293 0.301 12.78 0.031 0.030 2.00 0.04 0.299 0.294 11.63 0.034 0.029 1.50 0.05 0.340 0.319 12.20 0.033 0.031 1.50 0.06 0.386 0.333 13.03 0.031 0.031 2.00 0.06 0.392 0.328 12.12 0.033 0.031 1.50 0.07 0.428 0.349 11.13 0.036 0.032 2.00 0.07 0.433 0.344 10.60 0.038 0.031

The values found using Chezy differ slightly from the values found using (4.5a). In appendix H the method used for determining the flow parameters is illustrated. In the same appendix calculations for the stones used in the acclimatization area can be found.

4.4 Shear velocity The shear velocity can be calculated in different ways. The relation between the shear velocity and mean velocity for an undisturbed (not accelerated) flow can be found in literature. For an accelerated flow there is no literature which covers this relation. In the next section an attempt will be made to find this relation for the case of this experiment. The shear velocity is necessary for the calculation of the Shields parameter. The calculated z0 in the previous section for the undisturbed flow will be used in case of the accelerated flow and is assumed to be equal to the value found for the undisturbed flow. The EMS measurements carried out for the experiments using small stones are investigated first.

Table 4-4 Measured and calculated flow parameters of the undisturbed flow for experiments using large stones. z = 0.00m is located at 1.30x10-2m above the plates and z0 = 1.17 x 10-3m. The shear velocity found in the experiments is compared with the shear velocity calculated using the Chezy formula.

4. Flow analysis


4.4.1 Shear velocity, small stones As explained before a full profile is measured and a solution has to be found which fits the EMS measurements. It is assumed that the flow profile close to the bed will consist of a logarithmic form. The profile will reach a velocity of zero at z0. The aim of this section is to find a profile which fits the velocity measurements, shows a logarithmic profile close to the bed and is equal to zero at z0. If such a profile is found, the area between the y-axis and the profile, see figure 4.7, can be calculated and multiplied by the width of the contraction at that location. Doing so, the value found must approximately be equal to the discharge measured. In the next figure an example will be given of a calculated flow profile which satisfies all the requirements mentioned above.

When investigating the flow profile for a x-value, close to the end of the contraction, also a full profile in y-direction is found. (Appendix F) In the next figure this will be illustrated. In section 2.3.2, this is already mentioned, figure 2.13.

Figure 4.7 Measured velocities (L1.50; Q=60l/s; x=0.00m; y=0.00m; small stones and the ‘fitting’ flow profile

z0 0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100 120

Velocity (cm/s)

Hei

gth

abov

e th

e w

oode

n pl

ates

(cm

)

.

Calculated flow profile

Measured velocities using EMS

z=0.00m

4. Flow analysis


The figure shows that the flow profile is constant between approximately 0.95W and 0.98W, indicated as Weff, the effective width. For each time the volume of the profile is calculated, the effective width needs to be found to compare the adjusted discharge and the calculated discharge. The logarithmic profile will be used to determine the shear velocity. The slope of the profile has a linear relation with the u* and can be calculated. For each experiment the following steps are taken:

1) The calculated profile has to fit the measured points. 2) The volume of the calculated profile is equal to the discharge.

effprofileWAQ = (4.8)

3) The slope of the logarithmic flow profile near the bed is calculated to

determine the shear velocity u*. The same method is used as in section 4.3 where the z0 was determined.

W

Weff

Q

Figure 4.8 The effective width of a ‘full’ flow profile. The figure shows a top view of the contraction

4. Flow analysis


In table 4-5 the average Weff calculated for the different datasets and x-values (step 2) will be presented.

L1.50; Q = 30, 40, 50, 60l/s L2.00; Q = 30, 40, 50, 60l/s Position W (m) Weff (m) perc. W (m) Weff (m) perc.

x = 0.00m 0.150 0.144 96.1% 0.150 0.145 96.7% x = 0.10m 0.173 0.164 94.7% 0.168 0.159 95.0% x = 0.20m 0.197 0.187 94.9% 0.185 0.176 95.3% x = 0.30m 0.220 0.210 95.4% 0.203 0.194 95.8% x = 0.40m 0.243 0.235 96.7% 0.220 0.214 97.4%

Step 3, determining the slope of the logarithmic profile, can now be executed. For each dataset the occurring shear velocities can be calculated. The result is plotted in figure 4.9 and compared to the mean velocity calculated using the water depth (Q=uBh). The values found in section 4.3.1, table 4-3 are also plotted in the figure. These values are calculated for x = L.

The first thing that is notable when looking at the figure is the difference between the two contractions. This can be explained by the fact that during the lowest

Table 4-5 The effective width at different locations in both contractions using small stones

Figure 4.9 The calculated shear velocity versus the mean velocity calculated using Q=uhB for both contractions and small stones

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.00 0.20 0.40 0.60 0.80 1.00 1.20

mean velocity (m/s)

shea

r ve

loci

ty (m

/s)

.

L=1.50m

L=2.00m

x = L

4. Flow analysis


measurements for L1.50 the EMS measured closer to the bottom than intended so the bottom has influenced the measurement. The fact that a parabolic line can be drawn through the red measurements (L2.00) the measurements in x = L and the origin (when the mean velocity is zero the shear velocity is equal to zero as well) makes it plausible that these measurements and calculations are correct. The shear velocity calculated for L2.00 yields the following equation:

hh uuu 073.0031.0 2* += (4.9)

In section 4.5.2 it will be proved that the velocity measured using EMS is 1.09 times larger than the mean velocity calculated using h, so equation (4.9) can also be written as: EMSEMS uuu 066.0026.0 2

* += (4.10) Equation (4.9) can be used to recalculate the shear velocities for L1.50 with the use of the mean velocity, figure 4.10.

Figure 4.10 The recalculated shear velocity versus the mean velocity. The values for L1.50 are obtained by multiplying the mean velocity by equation (4.9)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 0.20 0.40 0.60 0.80 1.00 1.20

mean velocity (m/s)

shea

r ve

loci

ty (m

/s)

.

L=1.50m

L=2.00m

x = L

4. Flow analysis


4.4.2 Shear velocity, large stones The same method for calculating the shear velocity as described for the case of small stones will be carried out for large stones. The measurements show a less visible bend of the flow profile. This is due to the fact that the measurements close to the bottom could not be carried out. It occurred that it is impossible to find a fitting flow profile through the measured points. The relation between the shear velocity and the calculated velocity found for the case of small stones will be used. (4.9) The values found for the shear velocity are plotted in the next figure. The values found in section 4.3.2, table 4-4, can also be seen in the figure. Because of the use of equation (4.9) the values are all on the same line. The values for x = L are of the same order of magnitude as would have been found using (4.9) and the mean velocity at x = L. The values for the shear velocity will be used in the next chapter for an estimation of the Shields parameter.

Figure 4.11 Relation between the mean and the shear velocity using large stones

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

mean velocity (m/s)

shea

r ve

loci

ty (m

/s)

.

L=1.50m

L=2.00m

x = L

4. Flow analysis


4.4.3 Shear velocity using the momentum balance The data obtained can also be used in another way as in the previous sections. In section 2.2.3 the momentum balance is given as:

dxdB

gBu

Bh

dxdh

ghu

gh

−+

−=

22

21 ββ

ρτ (4.11)

This equation can be used to calculate the unknown shear stress created by the bottom and the wall friction. All the variables are known because they have been measured and by filling them in, the shear stress and hence the shear velocity can be calculated. The data appeared to be not accurate enough to find a satisfying result. The decrease due to loss of energy will be in the order of less then a millimetre. The measured differences (errors) of the bottom level and water level along the flume are of the same order of magnitude which makes it impossible to use them in the momentum balance. More accurate measurements are required to be able to use the balance in order to calculate the shear stress and hence the shear velocity. It must also be mentioned that this is only a first approximation for the momentum balance. Other methods have to be used to produce the actual momentum balance. For now the balance found is a good approximation.

4.5 Turbulence The velocity at a certain point in the contraction is not constant due to fluctuations caused by turbulence. The velocity at different locations is measured during one minute with a frequency of 50Hz, as mentioned in section 3.5.1. When the mean velocity for a certain point is calculated, the difference between the actual and the mean value for each measurement can be obtained. Also the turbulence intensity can be calculated as mentioned in section 2.3.2. This dimensionless parameter provides a good way of describing the turbulence and is defined in equation (2.20) First it will be investigated whether the theory provided in section 2.3.2 is valid for this experiment to get a first impression whether the results from the obtained measurements are correct. The theory says that in an accelerated flow the turbulence intensity, rx, will decrease due to the increased velocity in the contraction. Next the turbulence intensity will be used to determine the occurring peak velocities. Before starting the analyses of the obtained data it should be mentioned again that all measurements have been carried out in the middle of the flume, y = 0.00m. In the next sections different calculated flow parameters obtained from the measurements for both contractions are combined in one figure. To do so, the x-position of the

4. Flow analysis


measurement is divided by the length of the contraction thus the x-axis used in the graphs becomes equal to x/L.

4.5.1 Turbulence in case of small stones The turbulence intensity varies in the contraction. Its value depends on the location and the height of the measurement. Both contractions, L1.50 and L2.00, show the results plotted in figures 4.12 and 4.13. In these figures it can be seen that for x = L, the beginning of the contraction, the relative turbulence strongly depends on the height of the measurement. Further downstream, as the width of the contraction decreases, the velocity increases and the relative turbulence decreases which fits the theory. In the narrowest part of the contraction, where the experiments take place, the turbulence intensity shows a constant value over the depth with exception of the measurement closest to the bottom.

Figure 4.12 Relative turbulence at different locations and heights in the contraction L1.50 for varying discharges (30, 40, 50 and 60l/s) in experiments using small stones. The rx decreases as the velocity increases due to contraction of the flow and becomes constant over the water depth with exception of the measurement closest to the bottom

0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Relative turbulence (-)

Hei

ght z

(cm

)

x=0.00mx=0.20mx=0.40mx=1.50m x=1.50m; Q=50l/sNezu & Nakagawa Q=50l/s; x=L

4. Flow analysis


The turbulence intensity is in the beginning of the contraction clearly more dependant on the discharge. As the velocity increases due to the decreasing width, the rx becomes less dependent on the discharge. Both contractions show the same development of the turbulence intensity. In the area where the experiments take place they are almost identical. The results of the experiments will not depend on differences between both contractions in rx because there are none.

Nezu and Nakagawa (1993) found a relation between the turbulence intensity and the height of the measurement above the bed in an undisturbed flow.

hz

euu −

= 30.2)'(

*

σ (4.12)

This can also be written as:

ueu

uur

hz

x

−

== *30.2)'(σ (4.13)

0

5

10

15

20

25

30

35

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Hei

ght z

(cm

)

x=0.00m

x=0.20m

x=0.40m

x=2.00m

Figure 4.13 Relative turbulence at different locations and heights in the contraction L2.00 for varying discharges (30, 40, 50 and 60l/s) in experiments using small stones

4. Flow analysis


In figure 4.12 this equation is plotted for a discharge of 50l/s. The shear velocity used in this equation can be found in table 4-3, section 4.3.1. The measured points for this discharge are the filled green points. The curve found using Nezu and Nakagawa shows slightly larger values for the relative turbulence. The form of the curve is similar to the measurements. The values found for the rx using EMS seem to be quite accurate. According to the existing theory and the results found in the figures above, the turbulence intensity, rx, decreases as the velocity increases. The questions is whether a relation can be found between the position in the contraction and the turbulence intensity. The values of rx found for a certain location are divided by the value of rx found for the same height in x = L. The values are averaged over the different heights and discharges, see appendix I for more details. In the next figure, figure 4.14, the final result can be seen. The turbulence intensity appears to be independent on the discharge and the height when it is divided by it’s value in x = L. The figure below shows that the results do not depend on the length of the contraction.

The experimental results in figure 4.14 yield:

11.043.045.0)()( 2

+

+

=

Lx

Lx

Lrxr

x

x (4.14)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

rx(x

) / r

x(L

) (-

)

1.50 m

2.00 m

Figure 4.14 The relative turbulence versus the position of the measurement in the contraction. The values are independent of the discharge and the height of the measurement

4. Flow analysis


The relative turbulence can be used to determine the peak velocities, responsible for movement of the stones. Formula (2.23) will be repeated here:

xxx uru )1(max, λ+= (4.15) This can also be written as:

2'max, uuu x λ+= (4.16)

To obtain a relation between the peak and the mean velocity, the measured peak velocity for each dataset, the calculated mean velocity and the turbulence intensity will be investigated. First λ will be calculated. In literature the term λ is always the mean value calculated over the entire water depth. The velocities at the bottom are important in this experiment. The mean value of λ will be compared to the value of λ near the bottom, figure 4.15, to see whether the use of an average λ is justified.

The graph shows that all the values of λ are close to the value of 3, which is common in literature. The average value shows for both L1.50 and L2.00 a small increase in the narrow part of the contraction. The value of lambda near the bottom fluctuates much more than the average value, for both contractions. The use of a mean value for λ seems correct.

Figure 4.15 Comparing of the mean value of λ over the water depth and the value of λ near the bottom versus the position in the contraction for L1.50 and L2.00

2.00

2.20

2.40

2.60

2.80

3.00

3.20

3.40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

λ (-

)

L=1.50 m; average value lambda over water depth

L=1.50 m; lambda at z=0.052m

L=2.00 m; average value lambda over water depth

L=2.00 m; lambda at z=0.052m

4. Flow analysis


Formula (4.15) describes the relation between the peak velocity and the mean velocity as (1+λrx). This relation is given in the graph below. For λ, the mean value is used because it shows less fluctuation. The same mean value for the turbulence intensity is used as in figure 4.14.

For both contractions a similar smooth decreasing relation between the mean velocity and the peak velocity can be seen. The parabolic relation between both velocities can be written as:

03.111.010.02

+

+

=

Lx

Lx

uupeak (4.17)

This equation suits both contractions. In appendix J this equation is compared with two datasets (L1.50, Q=40l/s and L2.00, Q=40l/s) of the measured peak velocities divided by the calculated mean velocities. Equation (4.17) fits the measured values well. Because the peak velocities are responsible for the movement of the stones, its relation with the mean velocity is important. A decrease in width shows a decrease in the relation between the mean velocity and the peak velocity. In the area where the experiments take place the peak velocity is equal to a value between 1.03 and 1.06

Figure 4.16 The calculated relation between the peak velocity and the mean velocity versus the position in both contractions. The peak velocity relatively decreases as the mean velocity increases

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

(1+r

x*λ)

= U

peak

/ U

mea

n (-

) .

L=1.50 m L=2.00 m

03.111.010.02

+

+

=

Lx

Lx

uupeak

4. Flow analysis


times the mean velocity. This is much smaller than the value of 1.24 for the undisturbed flow. Now it is clear that the difference between the occurring peak velocity and the calculated average velocity in the contraction decreases as the width decreases and hence the velocity of the flow increases. In the following figures the actual differences between the two velocities, can be seen. The values are measured at z = 0.052m, relatively close to the bed. The figures show for both contractions a similar downward course as the location of the measurement is closer to x = 0.00m.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

(upe

ak -

umea

n) (

m/s

)

Q=30L

Q=40L

Q=50L

Q=60L

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

(upe

ak -

umea

n) (

m/s

)

Q=30L

Q=40L

Q=50L

Q=60L

Figure 4.17 Actual measured decrease of the difference between the mean and the peak velocity as the mean velocity increases, measured for varying discharges and L1.50 close to the bed (z = 0.052m)

Figure 4.18 Actual measured decrease of the difference between the mean and the peak velocity as the mean velocity increases, measured for varying discharges and L2.00 close to the bed (z = 0.052m)

4. Flow analysis


The different discharges create similar velocities at the same locations in the contraction because of variations in the water depth. Both contractions yield the same linear equation, which is drawn in both figures:

036.0045.0 +=−Lxuu meanpeak (4.18)

The numbers in this equation have dimensions. Equation will not be applicable for any situation. Experiments using different contraction lengths and the same stone diameter seem to give similar results. The same plots are made in the next section for large stones. The results for both stone diameters will be compared at the end of the next section.

4. Flow analysis


4.5.2 Turbulence in case of large stones The same figures and equations can be made for the experiments using large stones. The calculation methods to achieve the results we are looking for used in this section are similar to the ones used in the previous section. When looking at the turbulence intensity or the relative turbulence as it is indicated in the figures, a decrease in rx is found for a decrease in width of the contractions. This is indicated in the two figures below for L1.50 and L2.00. The results are similar to the results found in case of the experiments using small stones and to the expectations following from the theory.

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Hei

ght z

(cm

)

x=0.00m

x=0.20m

x=0.40m

x=1.50m

Figure 4.19 Relative turbulence at different locations and heights in the contraction L2.00 for varying discharges (40, 50, 60 and 70l/s) in experiments using large stones. The rx decreases as the velocity increases due to contraction of the flow and becomes constant over the water depth with exception of the measurement closest to the bottom

4. Flow analysis


A relation between the position in the contraction and the turbulence intensity is found as can be seen in the next figure. The values of rx found for each location are divided by the value found in x = L, see appendix I. The values are averaged over the different heights and discharges. The results found in the figure below do not depend on the length of the contraction. The results shown in figure 4.21 yield:

09.040.051.0)()( 2

+

+

=

Lx

Lx

Lrxr

x

x (4.19)

Equation (4.14) found in case of the small stones is also drawn. A small but significant difference can be seen between both lines. Because of the small difference both lines can be combined for practical use.

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Hei

ght z

(cm

)x=0.00m

x=0.20m

x=0.40m

x=2.00m

Figure 4.20 Relative turbulence at different locations and heights in the contraction L2.00 for varying discharges (40, 50, 60 and 70l/s) in experiments using large stones

4. Flow analysis


For the calculation of the peak velocity, the term λ, see equation (4.15), is calculated in two different ways. In figure 4.22 the mean value of λ over the water depth and the value at z = 0.049m, near the bottom, can be seen. Figure 4.22 shows that all values are again close to 3. The average value of lambda shows for both L1.50 and L2.00 a small increase in the narrowest part of the contraction. The value near the bed fluctuates more than the average λ, as is the case in experiment using small stones. The current use of an average λ in literature seems also in case of the use of large stones justified.

Figure 4.21 The relative turbulence versus the position of the measurement in the contraction for large stones. The values are independent of the discharge and the height of the measurement

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

rx(x

) / r

x(L

)1.50 m2.00 mSmall stones

4. Flow analysis


The relation between the peak and the mean velocity is given in figure 4.23. For λ, the mean value is chosen because it gives less fluctuation. The turbulence intensity used in the next figure is the same mean value as used in figure 4.21. The parabolic equation found in case of the small stones is also drawn in the next figure so both equations can be compared. For L1.50 an extra measurement at x = 1.00m is taken. The figure shows that the parabolic equation fits this extra measurement well. It makes it even more acceptable to assume that the following equation describes the relation between the mean velocity and the peak velocity for large stones.

03.109.015.02

+

+

=

Lx

Lx

uupeak (4.20)

For small values of x, the area where the experiments take place, the equations for both small and large stones are identical as can be seen in figure 4.23. For larger values they slightly differ. In appendix J equation 4.20 is compared to two dataset (L1.50, Q = 50l/s and L2.00, Q = 50l/s) of the measured values of the peak velocities divided by the calculated mean velocities. The equation fits the measured values well for the area where the experiments take place. For the beginning of the contraction (x = L) the measurements differ from the calculated points in contradiction to the small stone measurements.

Figure 4.22 Comparing of the mean value of λ over the water depth and the value of λ near the bottom versus the position in the contraction for L1.50 and L2.00, using large stones in the experiments

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

Lam

bda

(-)

.L=1.50 m; average value alpha over water depth

L=1.50 m; alpha at z=0.049m

L=2.00 m; average value alpha over water depth

L=2.00 m; alpha at z=0.049m

4. Flow analysis


The following figures show the actual differences between the two velocities for the experiments using large stones.

Figure 4.23 The calculated relation between the peak velocity and the mean velocity versus the position in both contractions using large stones. The peak velocity relatively decreases as the mean velocity increases

Figure 4.24 Actual measured decrease of the difference between the mean and the peak velocity as the mean velocity increases, measured for varying discharges, large stones and L1.50 close to the bed (z = 0.049m)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

(1+r

x*la

mbd

a) =

Upe

ak /

Um

ean

(-)

.

L=1.50 m

L=2.00 m

Large stones:

Small stones:

03.109.015.02

+

+

=

Lx

Lx

uu peak

03.111.010.02

+

+

=

Lx

Lx

uu peak

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

upea

k - u

mea

n (m

/s)

Q=40L

Q=50L

Q=60L

Q=70L

Small stones

4. Flow analysis


The values are measured at z = 0.049m, relatively close to the bottom and almost similar to the height of the measurements in case of the small stones. The figures show for both contractions a similar downward course as the location of the measurement comes closer to x = 0.00m. Both contractions yield the same linear equation, which is drawn in both figures:

05.0062.0 +=−Lxuu meanpeak (4.21)

Even tough these measurements are carried out at the same height above the bed as in the case of small stones they show a different linear relation between both velocities, see figure 4.24. Different contraction lengths and similar stone diameters give also for large stones similar results. Nevertheless variation in stone diameter does not show the same result in the difference between the peak and the mean velocity near the bottom.

4.6 Velocity of the flow

4.6.1 Introduction There are a lot of different ways to deal with the in the experiment occurring velocities. The velocity can be measured in different ways. In the experiment the water depth at different locations in the contraction is measured. For each location, the width and the discharge are known so the mean velocity through that passage can be calculated.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

upea

k - u

mea

n (m

/s)

Q=40L

Q=50L

Q=60L

Q=70L

Figure 4.25 Actual measured decrease of the difference between the mean and the peak velocity as the mean velocity increases, measured for varying discharges, large stones and L2.00 close to the bed (z = 0.049m)

4. Flow analysis


uBhQ = (4.22) Another way that the velocity is measured is by use of an EMS as explained in section 3.5.1. The velocity profile can be determined for different locations in the contraction. The problem when using EMS is that no measurements can be taken close to the bed. At the end of the contraction where the experiments take place the velocity profile is what is called full. This means that the profile is constant until a point close to the bed from which the profile will fast go to zero. In the next figure a profile for an undisturbed flow will be compared to an accelerated flow profile. For both profiles the average profile is also plotted. Notice that the discharges for the different profiles are not equal. The figure is mentioned to serve as an indication of the form of the profile of the flow.

The EMS measures a full velocity profile as indicated in figure 4.26 in the last 0.40m of the contraction. For the small stones the bend in the profile can just be seen when looking at the results from the EMS measurements. In the measurements for large stones this is more difficult to observe. This is due to the fact that close to the bed the EMS could not measure. Because of the more whimsical profile of the bed in case of the larger stones, measurements less close to the bed were taken.

4.6.2 Difference in velocity measurement approaches When dealing with the different methods of calculating and looking at the occurring velocities, first the mean velocity using equation (4.22) will be compared with the EMS-measurements. For the latter the average velocity is calculated for the measurements where the flow profile showed a constant value for the velocity. These measurements are carried out in the middle of the flume, where y = 0.00m.

Velocity (m/s)

Wat

erhe

ight

(m)

Undisturbed logaritmic flow profile


Average flow velocity of undisturbed flowprofile

Average flow velocity of accelerated flowprofile

Figure 4.26 Velocity profiles of an undisturbed and an accelerated flow resulting average velocities

4. Flow analysis


For the average velocity using the water depth, the in section 4.2 determined height of the bed is used. From the theory it became clear that the force acting on a stone in an accelerating flow is a combination of velocity and acceleration generated forces. The actual occurring velocity of the flow (uEMS) will create this force. The EMS sometimes gives inaccurate results which appeared in previous experiments in which the instruments were used. The simple but reliable method of calculating the water depth and hence the average velocity (uh) can be used to find a relation between both in different ways derived velocities. The uh can be multiplied by this relation to find the actual velocity which generates the force on the stone. This way the sometimes inaccurate EMS-measurements are not taken into account. In the first figure, figure 4.27, the mean velocity using the water depth (h) and the measured velocity (EMS) for experiments using small stones are compared.

It appears that the velocity measured using EMS is 1.09 times larger than the mean velocity calculated using h, or:

EMSh uu 916.0= (4.23) This is the case for both contractions L1.50 and L2.00.

Figure 4.27 Relation in case of small stones between the average velocity calculated using (4.22) and the measured average velocity using EMS

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

Average velocity h (m/s)

Ave

rage

vel

ocity

EM

S (m

/s) 1 : 1

4. Flow analysis


For large stones the same graph is produced, figure 4.28.

For both contractions, L1.50 and L2.00, the relation between both velocities is:

EMSh uu 904.0= (4.24) This is of the same order as the relation in case of the smaller stones. When equation (4.23) and (4.24) are combined the following equation can be made:

EMSh uu 91.0= (4.25) For an accelerated flow where a full profile is measured, equation (4.25) describes the relation between the part of the profile where its value remains constant and the average velocity. (figure 4.29)

Figure 4.28 Relation in case of large stones between the average velocity calculated using (4.22) and the measured average velocity using EMS

1 : 1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

Average velocity h (m/s)

Ave

rage

vel

ocity

EM

S (m

/s)

.

4. Flow analysis


Finally it needs to be mentioned that the velocity near the bottom can be measured using a Laser-Doppler instrument. This is a time and money consuming method so for this experiment other more simple instruments are used.

4.7 Velocity-acceleration combinations occurring in the flow The forces that act on the grains are not only caused by velocity but also by acceleration as shown in section 2.3.1. The acceleration causes a pressure gradient that acts on the grain and tries to move it. The acceleration is given by:

dxduua = (4.26)

In this equation the ū is the average velocity over the two positions between which the acceleration of the flow is calculated. In the experiments coloured strips of stones are used to determine the movement of stones in a certain area. The accelerations that occur due to the increasing velocity in the contraction can be calculated in two different ways. Both methods require a velocity in the beginning and the end of each strip. The velocities used in both methods are derived in the following way:

=

h

EMS

startstartstart u

uhB

Qu

=

h

EMS

endendend u

uhB

Qu (4.27 a, b)

1) The acceleration in the middle of a coloured strip am. The velocity at the beginning and end of each strip is known. The acceleration can be calculated assuming the fact that the velocity increases linear. The

Figure 4.29 Relation between the measured and the calculated velocity and their flow profiles

uh = 0.91 uEMS

uEMS

Velocity (m/s)

Wat

er d

epth

(m)


Average flow velocity of accelerated flowprofile

4. Flow analysis


actual increase is not linear but for the small length of the strip this can be assumed. The acceleration can be calculated using equation (4.26). The average velocity, ū, is the velocity in the middle of the strip.

2) The acceleration over the last stone in a coloured strip ae.

The accelerations can also be calculated for a different location. The acceleration between the end of each strip and the length of one stone size (dn50) in front of the end of the strip can be determined. This can only be done when the increase of the velocity is considered to be linear between the beginning and the end of each strip. The actual occurring acceleration will even be a little larger because the velocity does not increase linear but its increase is of a higher order.

The velocity of the flow and the water depth at the beginning and at the end of each strip are measured. There are three different ways to deal with the velocity measurements:

1) The average velocity through a passage. The average velocity is calculated and multiplied with the value found in sections 4.5.1 and 4.5.2 for the relation between the peak and the mean velocity.

=

mean

peak

uu

BhQu (4.28)

u sta

rt

u end

am

u sta

rt

u end

ae

Figure 4.31 The acceleration over the last stone in a colored strip

Figure 4.30 The acceleration in the middle of a colored strip

4. Flow analysis


2) The average of the measured velocities where the velocity in the flow profile is constant. In the next figure, a velocity profile for an accelerated flow is given. The average EMS velocity is indicated in the figure. This velocity is multiplied with the ratio between the peak and the mean velocity, equations (4.17) and (4.20).

=

mean

peakEMS

uu

uu (4.29)

3) The average velocity calculated in 1) can be multiplied with the ratio between

the measured velocities and the calculated velocities. In section 4.6.2 the ratio between the ūEMS and the average velocity used in 1) is calculated. The average velocity is multiplied with this ratio and the ratio between the peak and the mean velocity:

=

mean

peak

h

EMS

uu

uu

BhQu (4.30)

Velocity (cm/s)

Hei

ght a

bove

bot

tom

(cm

)

.

ūEMS

Figure 4.32 Velocity profile for an accelerated flow, indicating the average EMS velocity

4. Flow analysis


In the next figure the combinations of velocity and acceleration for small stones are shown. There are three different ways to calculate the velocity and two different ways to calculate the acceleration.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

) .

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

Figure 4.33 Velocity versus acceleration in case of the use of small stones calculated in six different ways. For the figures on the left acceleration is calculated in the middle of a colored strip, on the right over the last stone. The top figures are made using velocity method 1, the middle plots using method 2 and the bottom ones using velocity method 3

4. Flow analysis


In the next figure the combination between velocity and acceleration for large stones are shown.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.L=1.50mL=2.00m

Figure 4.34 Velocity versus acceleration in case of the use of large stones calculated in six different ways. For the figures on the left acceleration is calculated in the middle of a colored strip, on the right over the last stone. The top figures are made using velocity method 1, the middle plots using method 2 and the bottom ones using velocity method 3

4. Flow analysis


For L2.00 the acceleration that goes along with a certain velocity is smaller than in case of L1.50, as can be expected. In appendix K larger versions of the same plots can be found. Only one of these calculation methods for both large and small stones can be used to determine the forces acting on a stone. First the best acceleration method is chosen:

• The understanding of the forces acting on a stone and the occurring movement of the stones are the main goal of the experiment. The peak velocity in combination with the highest occurring acceleration for a certain strip of stones has to be considered in order to be able to describe the forces acting on a stone. It represents the most extreme occurring situation for the stones in the area that is investigated. This is the reason why the second method for calculating the acceleration is chosen.

The plots on the left will not be considered for the reason mentioned above when the most favourable method for determination of the occurring velocity is chosen. The following facts are considered:

• The average velocity through the passage (method 1) is lower than the occurring velocity in the middle of the flume at the end of each coloured strip. The highest occurring velocities, the peak velocities, are responsible for the movement of the stones. As mentioned when determining the most favourable acceleration method, the most extreme occurring situation is representative for the stability of the stones. For this reason the first method is not sufficient.

• There are more data for the two combinations in which the velocity is

calculated using the water depth than for the measured velocities (EMS) because for some locations the velocity was not measured.

• The EMS measurements show irregularities for some data as can be seen in

appendix L. The accelerations, sensitive to small deviations in the velocity, are looked at in more detail in the appendix. The error for the accelerations calculated using the EMS-measurements show a larger error in parabolic increase through the contraction, than the method using the water depth. These deflections can be caused by errors in measuring (wrong position) or by errors of the equipment.

The threshold of motion for both large and small stones will be analysed using the third velocity and the second acceleration method. The plots on the bottom right in figures 4.34 and 4.35 correspond with this combination of methods.

5. Threshold of motion


5 Threshold of motion

5.1 Introduction The objective of this thesis is to obtain more insight into the influence of acceleration on the threshold of motion of stones dumped on a bed. In the previous chapters research on the fluid mechanical aspects of an accelerated flow has been done. In this chapter these findings will be used to analyse the influence of flow acceleration on stone stability. There are a lot of different ways to describe the movement of a stone. In this thesis a stone moves when it moves out of a coloured strip. Stones that move but do not leave the coloured strip do not contribute to the actual movement. In practise it occurred that as soon as a stone starts moving it keeps rolling until it passes the end of the contraction and the flow velocity decreases. In the first section the values found for the shear velocity will be used for the determination of the Shields parameter. If movement occurs for a ΨS when no movement was expected, the explanation can be the extra acceleration force acting on the grain. In the next section the velocity and the acceleration of the flow will be compared with the movement of the stones. If there is a relation between the acceleration force (FA), the velocity force (FV) and the threshold of motion, combinations of the same velocity and different accelerations will show differences in movement. The amount of movement should go up for an increase in acceleration combined with a constant or slightly decreasing velocity, when flow acceleration generates an extra force on a stone. The threshold of motion will be set on different percentages of moved stones to show that there are more ways of interpreting the obtained data. For both stone diameters only one threshold will be used for further analysis of the force acting on a stone due to a combination of velocity and acceleration of the flow. In the theory chapter it became clear that a critical force is required to move a stone. The individual influence of FV and FA on the total force acting on a stone will be determined in the shape of respectively a bulk and an inertia coefficient. Finally, a useful universal relation between the stability of the stones and the combination of velocity and acceleration of the flow for designing in practice will be looked at. This relation should be independent on the diameter of the stones. In the previous chapter a lot of combinations between velocity and acceleration have been obtained from the experiments that were carried out in the scope of this thesis. The determination of the values used in this chapter for the velocity-acceleration combinations can be found in section 4.6.



5.2 Shields The most common formula for the stability of stones, the Shields formula, can now be used for the accelerated flow. The shear velocity, determined in the previous chapter is used to calculate the dimensionless shear stress.

gdu

gdWSS ∆

=−

=Ψ2*

)( ρρτ

(5.1)

For each experiment the amount of stones moved out of their strip is counted. Because the different strips do not consist of the same amount of stones due to the sloping walls, movement per strip cannot just be compared to each other. By dividing the amount of stones moved by the total amount of stones in a strip, the percentage of moved stones per strip can be determined. Each dataset consists of six experiments and the results are averaged for each dataset. In the next figure the Shields parameter is plotted versus the percentage of moved small stones.

In the figure it can be seen that the first movement occurs for a Shields parameter lower than 0.03. The higher the Shields parameter the more movement can be seen. The experiments were carried out for 15 minutes so more movement can be expected when the execution time is enlarged, Forschelen (1999). Shields theory is based on the fact that only the velocity of the flow contributes to the total acting force on the stone. The figure above shows that movement of the stones

Figure 5.1 Movement of the small stones using the Shields parameter

0%

5%

10%

15%

20%

25%

30%

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Shields parameter (-)

Perc

enta

ge o

f mov

ed st

ones

.

L = 1.50m

L = 2.00m

Shields = 0.055 [-]



can be seen for values of ΨS lower than 0.055. There must be another extra acting force on the stones that explains the movement of the stones. This extra contributing force can possibly be generated by the acceleration of the flow. This assumption is being supported by the fact that the values obtained from the shorter contraction (L1.50) show a higher percentage of movement for similar values of ΨS considering the fact that the accelerations are larger in the shorter contraction. The percentage of movement seems to have a higher order relation with the Shields parameter. This will be dealt with later in the text. For the experiments using large stones, the same figure can be made, figure 5.2. It must be noted that only ten percent of the amount of stones used in the small stone experiments, is used in case of large stones. The amount of stones lies between 24 and 34 stones per strip, see table 3-4. The same percentage of movement does not imply the same amount of moved stones. In case of the large stones there were ten times fewer possibilities of stones to be moved.

Also for large stones the first transport occurs around ΨS = 0.025. This is lower than expected when only a force acting on the stone generated by the velocity of the flow is considered. The extra force that moves the stone is possibly generated by the acceleration of the flow. The values for L1.50 show a higher percentage of movement which can be explained by the increase of the acceleration.

Figure 5.2 Movement of the large stones using the Shields parameter

0%

5%

10%

15%

20%

25%

30%

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Shields parameter (-)

Perc

enta

ge o

f mov

ed st

ones

.

L = 1.50m

L = 2.00m

Shields = 0.055 [-]



5.3 Comparing of the velocity-acceleration combination based on the percentage of movement

In the previous section it was shown that the stones started to move for a lower value of the Shields parameter than expected. The possible explanation is the influence of the extra generated force due to the acceleration of the flow. In this section it will be investigated whether the amount of stones moved during the experiments depends on the acceleration. If acceleration has influence on the movement of the stones it can be expected that when two different velocity-acceleration combinations can be found with a similar occurring velocity but difference in acceleration, differences in percentage of movement should occur. A higher acceleration should cause an increase in movement. It would even be better to find a higher percentage of movement when the velocity decreases and the acceleration increases. In the next figure the velocity versus the acceleration for the experiments using small stones is given.

The average percentage of movement for each dataset is calculated. When there was no movement at all during the experiments the velocity-acceleration combination is indicated with a red mark. The combinations of velocity and acceleration that are compared with each other can be found in the figure. The dashed lines between two circles indicate that the two combinations inside the circles are compared to each other. It must be noted that not all combinations are used for the analysis. Only the

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Acc

eler

atio

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.

No movementMovement

Figure 5.3 Combinations of velocity and acceleration for which the percentage of movement is compared (small stones)



combinations with a clearly distinguishable difference in acceleration and a constant or decreasing velocity are looked at. It appears that for all the points that are compared the percentage of movement is higher for an increasing acceleration and a constant or decreasing velocity. In appendix M the values derived from this figure are shown. The same analysis for experiments using large stones can be done, figure 5.4. Less data are available but there are still enough datasets to compare as can de seen in the figure. In appendix M the obtained values can be found. Also for the large stones the percentage of movement increases for a constant or decreasing velocity.

Now it is found that the percentage of movement increases as the velocity stays constant or slightly decreases and the acceleration increases for both large and small stones. The actual influence of the acceleration on the total force on a stone must now be investigated, but first the definition of movement is revised in the next section.

5.4 Revised definition of movement The definition of movement of stones is not unequivocal. A lot of different methods have been introduced in all the investigations on the stability of stones. How many

Figure 5.4 Combinations of velocity and acceleration for which the percentage of movement is compared (large stones)

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stones have to move before actual movement of the stones takes place and how long were the stones subject to the flow? In case of this experiment an extra difficulty arises because the areas of the coloured strips of stones are not the same size and there is also a difference in size between the two contractions. This means that one strip contains more stones than the other and the more stones in a strip the more possibilities there are that one stone moves. In the previous section it is already mentioned that in this thesis we will speak of percentages of moved stones to be able to compare the movement for the different strips. In the previous section the average percentage of moved stones from the six experiments that form a dataset is used. The disadvantage of this method is that if in one experiment for whatever reason considerable more stones move than in the other five experiments the average percentage will increase. For a first estimation as used in the previous section the method is correct. When looking more in detail, as will be done in the next sections, the following method is used. Each dataset consists of six identically carried out experiments. From now on movement of the stones will be categorized into the following classes:

• If for a certain dataset, for a certain location, for less than 33.3% of the experiments movement occurs, it will be classified as no movement.

• Between 33.3% and 66.7% will be referred to as sometimes movement. • When movement occurs in more than 66.7% of the experiments, it will be

classified as movement. Movement of a stone will be defined as a stone that leaves its coloured strip. A stone that moves but stays in the strip of its colour will not contribute to the movement. In the following sections attention will be paid to the definition of the threshold of motion. Is the movement of one stone a good indication for the threshold of motion, or should another definition be used?

5.5 The threshold of motion set on different percentages of movement

In this section the threshold of motion will be set on different percentages of movement. The graphs that belong to these percentages will be presented and discussed. In the next section the visible analyses from this section will be looked at in a more quantifying way. For both stone diameters one of the criteria for movement determined in this section will then be chosen and used for further analysis. The relation between the forces generated by the velocity and the acceleration and their influence on the threshold of motion will be determined. All figures in this section have to be interpreted in the following manner:

• If a vertical line can be drawn between the red and the green points in the figure, a critical velocity for movement exists. This means that when a certain



velocity is exceeded the stones start to move. There is no influence on the stability of the stones caused by the acceleration.

• A horizontal line would suggest that a critical acceleration exists and the velocity has no influence on the stability of the stones.

• If a linear downward sloping line can be drawn between the red and the green points a combination between a critical acceleration and a critical velocity exists. Both the velocity and the acceleration have influence on the stability of the stones.

It is already known from many previous studies that velocity of the flow influences the stability of the stones. This is the reason why a horizontal line between the red and the green points will not be investigated. This line would suggest that the velocity of the flow has no influence on the stability. So far there are not enough velocity-acceleration combinations obtained for both large and small stones so in most cases a horizontal line can be drawn (figures 5.5 up to 5.10). This does not imply that a critical acceleration exists. The combinations of velocity and acceleration found in this experiment are not appropriate to be able to draw an unequivocal downward sloping line. More data have to be obtained for this purpose. The first criterion that will be used is the movement of at least one stone. In the next figures the velocity is plotted versus the acceleration. The first plot shows the movement in case of small stones the second in case of large stones. In the previous section it was mentioned that the movement will be expressed in percentages. The criterion for movement will be that movement occurs when the amount of stones moved is larger than 0%.

Figure 5.5 Threshold of motion 0%, small stones

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For a velocity of the flow around 0.80m/s the stones start to move in combination with an acceleration higher than 0.60m/s2. For large stones, figure 5.6, the same can be observed as in case of the small stones. Only the values for the velocity and the acceleration are higher because a larger force is needed to move the stones. For a combination of v and a (1.11m/s; 1.21m/s2) no movement occurs and for a lower velocity and higher acceleration combination (1.05m/s; 1.55m/s2) movement occurs. The disadvantage of the criterion of a threshold of motion of 0% for small stones, used to produce figure 5.5 is the fact that when a small stone is placed in an unfavourable position it unjustly contributes to the actual movement. There are ten times more small stones than large stones used to cover the same area. The chance of movement of one large stone because it is placed in an unfavourable position, is therefore much smaller. The criterion of 0% for large stones can be used when looking for the threshold of motion in contradistinction to small stones. First a higher percentage of movement in case of small stones will be investigated. In a coloured strip there are at least 224 stones dumped. (see section 3.6) When a threshold of motion is set on 2%, more than 4 stones have to move, to speak of actual movement. For the largest strip (pink) more than 5 stones have to move. In the next figure the threshold of motion for small stones is set on 2%. It can be seen that for a velocity of the flow around 0.90m/s no movement occurs for an acceleration lower than 0.84m/s2. For a higher acceleration movement occurs in combination with a slightly decreasing velocity. It is clearly visible that the

Figure 5.6 Threshold of motion 0%, large stones

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acceleration generates an extra force which acts on the stone. This extra force causes the stone to move earlier than expected when only the velocity generated force is considered.

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The threshold of motion can be set on 10%, figure 5.8. In the figure it can be seen that also in this case movement occurs for a constant or decreasing velocity with a higher acceleration than in the case where no movement occurs. For large stones two plots are made for a threshold of 6% and 15%. These percentages differ from the percentages used in case of small stones. 2% of movement in case of large stones gives the same result as 0% because ten times less stones are used. The three different percentages are chosen so clearly different definitions for the threshold of motion can be distinguished. No stones, at least two stones and more than four stones are distinguished this way. In figure 5.9 the threshold is set on 6% and in figure 5.10 on 15%. In both plots the same as in case of the small stones can be seen. For some velocity-acceleration combinations no movement can be seen while for combinations with the same or a lower velocity and a higher acceleration movement occurs.


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5.6 CB and CM

5.6.1 Introduction In this section one of the different threshold of motion cases for each stone diameter, presented in the previous section, will be elaborated. This is done with help of the Morison-like equation given in section 2.3 and repeated here:

+=

xuuVCuAuCF MB δ

δρρ21 (5.2)

The coefficients, CB and CM, will be determined in this section. The method that is used to determine the velocity and acceleration, is presented in section 4.7. In the previous section it was shown that different thresholds of motion can be used when looking at the stability of stones. In this section, one of these thresholds for both large and small stones will be looked at in more detail. First, it will be determined which threshold will be used. Then the critical force, necessary to move a stone, is calculated. This force and other conditions, which the coefficients should satisfy, will be used to calculate CB and CM. A pivoting angle between 30º and 45º gives a range of possible combinations for the coefficients. At the end of this section both coefficients for one certain angle will be determined satisfying both stone diameters.


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5.6.2 The threshold of motion that will be used for determining CB and CM

In section 5.5 it became clear that many possibilities exist to describe the threshold of motion. For both stone sizes, three different thresholds have been chosen. In this section, one threshold for each stone size will be chosen to use for determination of CB and CM. The threshold of motion of 2% for small stones and 0% for large stones will be used to determine the inertia and bulk-coefficient. In the next figures (figure 5.11 and 5.12) these criteria and detailed figures of the change between the velocity-acceleration combinations that cause movement or no movement are shown. The combinations of velocity and acceleration are going to be used in the Morison formula. This equation calculates the force on a stone generated by the accelerated flow.

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Figure 5.11 Threshold of motion 2% including a detailed plot in which the combinations used for determining CB and CM, are indicated (small stones)

Figure 5.12 Threshold of motion 0% including a detailed plot in which the combinations used for determining CB and CM, are indicated (large stones)

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12

3



There are different reasons for choosing these criteria. In previous sections some are already mentioned and will be repeated here shortly. In case of small stones, the chance of movement of one stone is 10 times larger than in case of the large stones. One moved small stone does not imply the same as one moved large stone. The high possibility of movement of one small stone because of an unfavourable position makes the use of this criterion (0%) for small stones not preferable. It has to be considered which thresholds of the different stone diameters can be compared with each other. If one large stone moves the total amount of movement is already larger than 2%. This implicates that for large stones the result found in case of 0% of movement is equal to the result found for 2% of movement. The threshold set on 2% for small stones is therefore compared with the result found for a threshold set at 0% for large stones. The higher thresholds of motion of 10% and 15% for respectively small and large stones will not be used for determination of both coefficients. The percentages imply frequent movement and do not represent the point where movement of the stones starts. When the velocity-acceleration combinations are used to determine the bulk and inertia coefficients and finally to calculate the Morison force generated by the accelerated flow, the points where ‘sometimes movement’ occurs are not taken into account. The amount of these occurrences should be kept as low as possible to be able to use all the data. For both stone sizes, the thresholds chosen show the fewest of these points. Finally, a last remark following from the previous section will be repeated. In section 5.5 is given how the figures 5.5 up to 5.10 have to be interpreted. It is already mentioned that the available data imply the fact that possibly a critical acceleration exists. If this critical value is exceeded according to the data, stones start to move. In the same section is mentioned that it is proved in previous investigations that the velocity-generated forces have their influence on the stability of the stones. There are not enough data to confirm this. More contraction lengths should be investigated to be able to clearly distinguish the influence. The bulk and inertia coefficient for both stone sizes are going to be calculated and compared. Both stone sizes should give similar results. To prevent misunderstanding the threshold for large stones will be set at 2%. This does not change anything concerning the data but makes the text more comprehensive when both stone sizes are compared. Instead of the threshold of motion of 0% for large stones, 2% will from now on be used.



5.6.3 The critical force The force balance found in section 2.1.4 will be used for calculating the acting force necessary to move a stone, Fcrit. Movement of a grain occurs when:

MARMG FF ;,; < (5.3)

ϕβϕϕ cos)cos(sin ARG FFF +−< ;

ϕϕβϕ cossin ARG FFF +<⇒≈ (5.4) The pivoting angle φ lies between 30º and 45º, so: Small stones: NFF Gcrit

31043.655.4sin −×−≈= ϕ Large stones: NFF Gcrit

21033.960.6sin −×−≈= ϕ When the force acting on a stone exceeds the critical force, it will move. The detailed plots in section 5.6.2 show velocity-acceleration combinations that are on the edge of movement or no movement. The following equation should satisfy these combinations: F1(movement) ≥ Fcrit ≥ F2(no movement) (5.5) Combination of equation (5.2) and (5.5) gives:

)()(21)()(

21

22

212

1 aVCuACFaVCuAC MBcritMB ρρρρ +≥≥+ (5.6)

In which:

ϕcos*MM CC = (5.7)

This equations can be used for calculating CB and CM. The points in the detailed plots in figures 5.11 and 5.12 provide the velocity-acceleration combinations required in (5.6). This will be the subject of the next section.

5.6.4 The values for CB and CM The Morison equation can be used when the values of both coefficients CB and CM are determined. There are different ways to obtain these values. The first approximation is by the use of equation (2.27) for determining the drag coefficient, CD and then the CL. The bulk coefficient will be a result of the lift and drag coefficient. Then, both coefficients will be determined using the critical force necessary to move the stone.



Finally it will be considered whether one preferable combination of CB and CM can be found. First, it needs to be considered that CB and CM have to meet the requirements following from the theory and the previous sections of this chapter:

• The bulk coefficient is larger then zero. In previous investigations is found that the velocity generated forces influences the stability of the stones

• The bulk coefficient, CB, is primarily a result of the lift and drag coefficient • The inertia coefficient is larger than one, see equation (2.28) and will not

exceed a value of about 7 (section 2.3.3)

mM kC +=1 1 ≤ CM ≤ 7 (5.8)

• Both coefficients, CB and CM, should remain constant for differences in stone diameter

First a rough estimation for the value of the bulk coefficient will be calculated. This will be done using equation (2.27) which is a combination of the drag force, the definition of the Shields parameter and the shear stress at the bottom.

215.0 ugdCD

Ψ∆≈ (5.9)

The lift force is approximately half the drag force and hence the CL is half the CD. The CB can be written as:

222 25.1 DLDB CCCC ≈+≈ (5.10) The lift coefficient only gives a rough first indication. The velocity normally used in this equation is a velocity close to the bed, where in this experiment the instruments available could not measure. The velocities used in the Morison formula, are also not close to the bed. The relation between the velocity used in the Morison formula and the velocity near the bed is the same as for the velocity in (5.10). So, when the velocities found in figure 5.11 and 5.12 are used in (5.10) a first estimation for CB can be found. This estimation is only valid for this experiment or experiments with the same velocity approach. It cannot be compared to values found for the drag and lift coefficients in experiments where the velocity close to the bed was measured. The velocities that will be used in (5.10) are for both stone sizes in the area where movement starts. The average Shields parameter calculated in section 5.2 for the change from no movement to movement (figures 5.11 and 5.12) is used.

dn50 (m) ΨS (-) u (m/s) CD (-) CB (-) 0.0082 0.040 0.89 0.045 0.051 0.0200 0.033 1.08 0.062 0.070

Table 5-1 First rough estimation bulk coefficient for both stone sizes



The estimation in table 5-1 can serve as a check for the order of magnitude of the bulk coefficient calculated below. This second method for determining both coefficients uses the critical force and equation (5.6) from the previous section. For small stones the velocity-acceleration combinations following from the detailed plot in figure 5.11 will be used to obtain CB and CM. The combinations that represent the change between movement and no movement with equal or decreasing velocity and increasing acceleration can be found in table 5-2. From the theory it became clear that the pivoting angle lies between 30º and 45º. The combination of the bulk and inertia coefficient will be calculated for an angle of 30º (Fcrit = 4.55x10-3N) for reasons given at the end of this section. The calculations made for the whole range of pivoting angles can be found in appendix N. In this appendix a detailed calculation for an angle of 45º can also be found. The velocity-acceleration combinations, the critical force (4.55x10-3N) and equation (5.6) are used to obtain different possible combinations for both coefficients. The shaded area in figure 5.13 represents the possible combinations for CB and CM that fulfil all the requirements.

Comb. v (m/s) a (m/s2)1 No mov. 0.89 0.83 Mov. 0.89 0.92 2 No mov. 0.89 0.83 Mov. 0.89 1.05 3 No mov. 0.89 0.83 Mov. 0.88 0.92 4 No mov. 0.90 0.74 Mov. 0.89 0.92 5 No mov. 0.90 0.74 Mov. 0.89 1.05 6 No mov. 0.90 0.74 Mov. 0.88 0.92

Figure 5.13 needs to be interpreted as follows. The grey area indicates that for those combinations of CB and CM:

Fmov > Fcrit > Fno mov and φ = 30º

The same can be done for large stones. The velocity-acceleration combinations following from the detailed plot in figure 5.12 will be used to obtain CB and CM. The velocity-acceleration combinations, the critical force (6.60x10-2N) and equation (5.6) are used to obtain different possible combinations for both coefficients. Again, the shaded area in figure 5.14 represents these combinations.

Figure 5.13 Possible combination of CB and CM for a pivoting angle of 30º and experiments using small stones

Table 5-2 Velocity-acceleration combinations used for calculating CB and CM

0

1

2

3

4

5

6

7

0.00 0.10 0.20Cb

Cm

2%; small stones; 12%; small stones; 22%; small stones; 32%; small stones; 42%; small stones; 52%; small stones; 6



Comb. v (m/s) a (m/s2)1 No mov. 1.11 1.21 Mov. 1.05 1.55 2 No mov. 1.11 1.21 Mov. 1.11 1.71 3 No mov. 1.06 1.27 Mov. 1.05 1.55

The bulk and inertia coefficient should be equal for both stone sizes. Figures 5.13 and 5.14 can be combined to find the area for which the combination of both coefficients satisfies this requirement. In the next figure the grey area represents the combinations of both figures.

For this angle, more combinations can be found that satisfy the requirements. It is expected that when more stone sizes are investigated the possible combinations for CB and CM can also be found in this area and will finally lead to one combination.

Figure 5.14 Possible combination of CB and CM for a pivoting angle of 30º and experiments using large stones

Figure 5.15 Possible combination of CB and CM for a pivoting angle between 30º and 45º, large and small stones

Table 5-3 Velocity-acceleration combinations used for calculating CB and CM

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Cm

2%; large stones; 12%; large stones; 22%; large stones; 3

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Cm

pivoting angle 30º



Formula (5.11 a and b) are used to determining the best combination for both coefficients, table 5-4. The in appendix N determined values for 45º are also given in the table.

MMM C

CC=

+

2min,max, and B

BB CCC

=+

2min,max, (5.11 a and b)

CB CM

pivoting angle 30º 0.10 3.92 pivoting angle 45º 0.14 5.55

The coefficients for φ = 30º and the velocity-acceleration combinations (figure 5.11 and 5.12) and the Morison equation (5.2) can now be used to calculate the force acting on the stones generated by the accelerated flow. In the next figures, the force is plotted versus movement or no movement. It can be seen that there is a clearly distinguishable change in movement for the critical force.

The combination for CB and CM satisfies all the requirements made in the beginning of this section. Finding a universal equation for the relation between the force acting on a stone and the movement of the stones will be the topic of the next section. The choice to set the pivoting angle on 30º will now be explained. The minimum and maximum pivoting angle are investigated to base the choice on. The results for both coefficients can be found in table 5-4 and appendix N. There is not enough information available to make a profound choice between the combinations. The relation between CB and CM for both pivoting angles is the same. This makes it

0 0.005 0.01 0.015 0.02

Generated force by accelerated flow (N)

MovementSometimes movementNo movementFcrit = 0.00455 NFcrit = 0.00643 N

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

Generated force by accelerating flow (N)


Figure 5.16 Movement versus no movement for small stones (left figure) and large stones (right figure) due to the force generated by the accelerated flow, calculated using the Morison equation in which CB = 0.10 and CM = 3.92 (φ = 30º)

Table 5-4 CB and CM for two different pivoting angles



impossible to compare both combinations by plotting the Morison force versus the logarithmic percentage of movement. (appendix N). The line that can be drawn through the points shows the same deviation for both large and small stones. This method does not encourage the use of one combination over the other. In table 5-1 a rough estimation for the bulk coefficient is presented. Both values for the bulk coefficient are closest to the CB found for a pivoting angle of 30º. This is the first reason to use the values for the coefficients following from this angle. Tromp (2004) also found the pivoting angle to be more in the region of 30º than of 45º. The inertia coefficient found in case of a pivoting angle of 45º (CM = 5.55) seems, when looking at figure 2.14 and the calculated relation between the length and width of the stones, rather high. Another reason for choosing a pivoting angle of 30º is the fact that the water-working of the bed was not done completely because this is time consuming and difficult in the experiment. This makes it assumable that there are still stones sticking out a little above the bed. These stones start to move for a ‘low’ pivoting angle so 30º is used. Equation (5.2) can now be written as:

+=

xuuVCuAuCF MB δ

δρρ21 (5.12)

In which: CB = 0.10 CM = 3.92 φ = 30º CM

* = 4.53 (see equation (5.7)) To make a more profound choice for the coefficients CB and CM more research is required. For both stone sizes the force generated by the acceleration (FA) appeared to be of the same order as the force generated by the velocity (FV). When the accelerated flow generated the critical force on a stone the relation between both forces can be expressed as:

Small stones: 76.0≈V

A

FF

Large stones: 54.1≈V

A

FF

5.7 Entrainment So far it became clear that the accelerations generate an extra force on the stones and hence influence the threshold of motion. It was also shown that the total force on the stone is a combination of velocity and acceleration multiplied by respectively the



coefficients CB and CM. The purpose of finding a relation between the total force and the stability of the stones can be found in practice. When designing an hydraulic structure where besides velocity of the flow also accelerations occur (section 1.1), it can be important to be aware of these influences. Finding a useful universal relation between the stability and the combination of the velocity and the acceleration of the flow for designing in practice will be the objective of this section.

5.7.1 The number-entrainment During the test, the number of stones moved are registered. The running time of each experiment is also registered, see section 3.7. Using these data, a time dependant parameter can be produced. This will lead to a number-entrainment rate defined as:

ATnE = (5.13)

E: number-entrainment [ 1/(m2s) ] n: number of stones moved [ - ]

A: area of the stones [ m2 ] T: duration experiment [ s ] The entrainment can be calculated for the carried out experiments. The entrainment is expected to increase as the total force on the stone increases. The total force on the stones can be calculated using the Morison forces found in the previous section. The advantage of the use of entrainment is that its definition is clear. The entrainment can be made dimensionless so it is valid for all stone sizes. It does not depend on the experimental set up and makes it possible to predict damage of the bed in time. Two figures can now be produced to verify the expected increase of the entrainment. As the force caused by the accelerated flow increases, the entrainment will also increase. In both figures, logarithmic values of the number-entrainment have been used for the y-axis. The difference in values for the force between both figures can be explained by the fact that the force necessary for moving a large stone is larger than in case of a small stone. The difference in number-entrainment is due to the fact that less large stones fit in a square metre compared to small stones. In the next section figures 5.17 and 5.18 will be combined. When both parameters can be made dimensionless, a unique relation between the force and the entrainment can be found. This relation does not depend on the stone diameter and is therefore practical and universal in use.



5.7.2 The dimensionless force and entrainment parameters The entrainment can be even more useful when both parameters used in the two previous figures are made dimensionless. This will enable to find a relation between

Figure 5.17 The entrainment of small stones versus the Morison force (CB = 0.10 and CM = 3.92)

Figure 5.18 The entrainment of large stones versus the Morison force (CB = 0.10 and CM = 3.92)

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Morison force (N)

Ent

rain

men

t (1/

(m^2

s)

.

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0.10

1.00

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Morison force (N)

Ent

rain

men

t (1/

(m^2

s)

.



the force and the entrainment that is independent of the stone diameter. The force found with the use of the Morison equation is dependent on the diameter of the stone, as can be seen in equation (5.13). When dimensionless parameters are used, the data for small and large stones should collapse into one single curve. The Shields parameter provides a good indication on how to make the force parameter dimensionless. The Shields parameter divides the shear stress by ρ∆gd. The shear stress is the force caused by the combined drag forces on stones in a certain area. The dimensionless ‘Morison-Shields’ parameter becomes:

gd

daCuC

gd

adCudC MBMB

MS ∆

+=

∆

+=Ψ

2

3

322

21

21

ρ

ρρ [ - ] (5.14)

The entrainment also has to be made dimensionless. This results in an entrainment parameter.

gdEdE ∆

=Φ 2 [ - ] (5.15)

It is expected that a plot of both dimensionless parameters and both stone diameters shows an increasing trend for an increasing force parameter. The values found for both stone sizes should overlap each other when both dimensionless parameters and the obtained data are correct. In figure 5.19 can be seen that the data for both stone sizes meet all the expectations. The entrainment parameter seems to have a power relation of with the force parameter which yields:

73.46106 MSE Ψ⋅=Φ − 0.2 ≤ ΨMS ≤ 1.4 (5.16) Both stone sizes collapse into the same curve as can be seen in the figure. More research is required to improve and verify this relation. The entrainment is probably underestimated because of the small area of the coloured strips. When a stone moves but does not leave the strip of the stones colour it does not contribute to the actual movement. This creates an underestimation for the entrainment. On the other hand it can be noticed that an overestimation of the entrainment is made due to the fact that the water-working of the bed was not completely as was already mentioned in the previous section. This makes it assumable that there are still stones sticking out a little above the bed. These stones start to move easy and unjustly contribute to the entrainment which creates an overestimation.



The Morison-Shields parameter is the result of the force generated by velocity and acceleration. To finally verify the fact that the acceleration has influence on the stability of the stones, figure 5.20 is made. In this figure the inertia coefficient, CM, in equation (5.14) is set on zero. This way only the velocity contributes to the Shields-Morison parameter which will in the figure be referred to as Ψv, in which v indicates the velocity, to prevent misunderstanding with figure 5.19.

Figure 5.19 The dimensionless entrainment and ‘Morison-Shields’ parameter for both large and small stones

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

0.1 1 10

Ψms (-)

ΦE (-

)


1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

0.1 1 10

Ψv (-)

ΦE (-

)


Figure 5.20 The dimensionless entrainment and velocity parameter for both large and small stones. (CB = 0.10 and CM = 0)



Figure 5.20 shows, when comparing the figure to figure 5.19, that the acceleration contributes to the Morison-Shields parameter. It is even more remarkable that the data for both stone diameters do not overlap when only the velocity is taken into account. It appears that the acceleration has more influence on the large stones than on the small stones. This can be explained when looking at the Morison formula or at the Morison-Shields parameter. The velocity term holds the stone diameter to the second power. The acceleration term includes the stone diameter to the third power. An increasing stone diameter results in relatively more influence of the acceleration on the total force acting on the stone. It can also be seen that the points in figure 5.20 show more scatter than in figure 5.19. This implicates that due to the extra acceleration force the points for both stone diameters collapse more into one line described by equation 5.16.

5.8 Evaluation In this chapter it became clear that flow acceleration influences the stability of stones and how this influence can be described. To obtain this conclusion different steps have been taken. First it is proved that both investigated stone sizes start to move for a lower Shields parameter then expected in case of a uniform (not accelerated) flow. This can be explained when looking at the acceleration of the flow. Classical methods (Shields) do not take the extra force generated by flow acceleration into account. The different obtained datasets showed that combinations of the same velocity and different accelerations result in variations in movement. The amount of movement goes up for an increase in acceleration combined with a constant or slightly decreasing velocity. Using the Morison equation makes it possible to find the proportions in which the acceleration force (FA) and the velocity force (FV) contribute to the total force on a stone, responsible for the stability of the stones. The critical force required for movement of the stones is used for determining these proportions valid for both stone diameters. Finally, a stability relation is designed and applied on the available data for both stone sizes. Both acceleration and velocity of the flow are present in this stability relation. The established dimensionless entrainment and Morison-Shields parameter show a similar increase in entrainment for an increasing force on both stone diameters. It needs to be mentioned that the flow velocity was not measured close to the bed because this was not possible using the instruments available in the experiment. Most existing methods used for calculating the force on a stone subject to flow use a velocity just above the bed. For further use of the results of this experiment it has to be considered that the velocity-acceleration combinations used in the Morison formula are not measured close to the bed.

6. Conclusions and recommendations


6 Conclusions and recommendations In this chapter the conclusions and recommendations following from the research presented in the previous chapters are given. First the objective of the thesis as described in section 1.2, will be repeated: Obtain more insight into the influence of flow acceleration on the stability of stones. This will be done by carrying out experiments in a flume containing a local contraction. In the contraction the flow accelerates and the stability of different stone sizes will be investigated. The discharge in the flume during the experiments will be constant in time. In this chapter it must become evident whether the objective is reached. The conclusions and recommendations are divided into two categories:

• The fluid mechanical results of the experiment (chapter 4) • The results concerning the stability of the stones (chapter 5)

6.1 The fluid mechanical results of the experiment Conclusions:

• The EMS can not measure close enough to the bed to find the actual change near the bed of the flow profile as the flow accelerates. Assuming the fact that the flow profile shows a logarithmic form close to the bed provides a good first indication for the occurring shear stress. (section 4.4.1 and 4.4.2)

• The measurements of the water depth taken during the experiment are not accurate enough for using a momentum balance to calculate the shear stress. (section 4.4.3)

• The relative turbulence decreases as the velocity increases. A relation for both stone diameters can be found for the position in the contraction and the relative turbulence. The relation differs slightly for both stone diameters. (figure 4.14 and 4.21)

• For both stone diameters the relation between the peak velocity and the mean velocity decreases as the velocity increases. (figure 4.16 and 4.23) Near the bottom the absolute difference between both velocities also shows a visible decrease. (figures 4.17, 4.18, 4.24 and 4.25)

• The measured velocities using EMS show a linear relation with the calculated velocity using Q = uBh. (figure 4.27 and 4.28)

Recommendations:

• The dimensionless fluid mechanical relations found in the experiments need to be reviewed when more tests are performed.



• More accurate measurements near the bottom are required to verify or adjust the method used for determining the shear velocity. The shear velocity in the accelerated flow is calculated using the assumption that the z0 for the accelerated flow is equal to the value found for the undisturbed flow. The validity of this assumption needs to be investigated. The term z0 is the position above the bed where the velocity is zero, when we consider the flow to be a log profile.

• The water depth needs to be measured more accurately to be able to use a momentum balance to calculate the shear stress. Another approach using polar coordinates should be considered when actually using the momentum balance.

• Different methods for accurate measurements of the shear stress exist. It should be considered whether the use of these instruments would benefit the experimental results.

• The level of the bed (tm) should be measured using a hemispherical probe with a size of 0.5dn50 instead of using a point gauge as was done in this experiment.

6.2 The results concerning the stability of the stones Conclusions:

• Stones in an accelerating flow start to move for lower values of the Shields parameter than expected. (section 5.2)

• For both stone diameters used in the experiment, combinations of the same or slightly smaller velocities and different accelerations show a clearly distinguishable increase in movement as the acceleration increases. (section 5.3)

• For each stone diameter a critical force exists, balanced by the gravitation force. If this force is exceeded stones will start to move. (section 5.6)

• The Morison equation provides a satisfying method to deal with the forces generated by the velocity and the acceleration. (section 5.6.4)

• On the threshold of motion it appears that for both stone sizes the force due to the acceleration of the flow is of the same order as the force due to the velocity of the flow. (section 5.6.4)

• The relative share of the acceleration force to the Morison equation increases as the stone diameter increases. (section 5.7.2)

• A universal relation between the entrainment and the force on a stone is found. This power relation does not depend on the diameter of the stone and consists of a dimensionless Morison-Shields parameter and a dimensionless entrainment parameter. (section 5.7.2)

Recommendations:

• More experiments have to be carried out in order to determine entrainment of stones for more velocity-acceleration combinations. Experiments in contractions with other dimensions or more tests in the existing contractions can be carried out in order to obtain more data.



• Other stone diameters should be used in the experiments in order to verify the relation found between the force and the entrainment. The bulk and inertia coefficient in the Morison equation should also be verified to be the same for different stone sizes.

• Despite the difference in flow profile, the bulk coefficient for a uniform flow should be determined and compared to the values found in this experiment.

• More accurate velocity measurements using Laser-Doppler or other techniques have to be carried out. The velocity can also be measured closer to the bed this way.

• The relation between the velocities used in this thesis and the velocities measured close to the bed (Laser-Doppler) has to be determined.

• The influence of turbulence is left out of the analysis concerning the stability of the stones. The use of more advanced and accurate instruments can result in a better understanding of the influence of turbulence.

• The underestimation of the entrainment due to the limited area of the coloured strips should be looked at in more detail. A correction factor or a variable size of the strips should be introduced. On the other hand the overestimation due to the incomplete water-working of the bed has to be investigated.

• Video images can be made to be able to investigate the movement of the stones in the time.

• Experiments in which waves on a slope are created also show the influence of acceleration on the stability of stones. (Tromp, 2004) Results from these experiments can be compared to results obtained from this thesis. Similarities between both experiments will result in more understanding of the subject.

List of symbols


List of symbols Symbol Definition Unit

a: acceleration [ m/s2 ] ae: acceleration over last stone in a col. strip [ m/s2 ] am: acceleration in the middle of a col. strip [ m/s2 ] A: area of the stones [ m2 ] AD: exposed surface area (drag force) [ - ] AL: exposed surface area (lift force) [ - ] Aprofile: area of the flow profile [ m2 ] Astrip: area of a strip of stones [ m2 ] B: width of the flume [ m ] Bb: width of the inflow of the contraction [ m ] Be: width of the outflow of the contraction [ m ] C: Chezy coefficient [ m1/2/s ] CB: bulk coefficient [ - ] CD: drag coefficient [ - ] CL: lift coefficient [ - ] CM: inertia coefficient [ - ]

d: 1. stone diameter [ m ] 2. water depth [ m ] dn: nominal diameter [ m ] dn50: median nominal diameter [ m ]

E: entrainment [ 1/(m2s) ] F: force [ N ] FA: acceleration force [ N ] FD: drag force [ N ] FG: gravity force [ N ] FL: lift force [ N ] FM: Morison force [ N ] FR: resulting force (drag and lift) [ N ] FV: velocity force [ N ] Fr: Froude number [ - ] g: gravitational acceleration [ m/s2 ] h: water depth [ m ] hgate: gate height [ m ] k: 1. roughness (2 or 3 times dn50) [ m ] 2. total kinetic energy in a turbulent flow [ Nm ] km: added mass term [ - ] kt: layer thickness coefficient [ - ] L: contraction length [ - ] Mcum: cumulative mass [ kg ] Mdr: dry mass [ kg ] Mm: moist mass [ kg ]

Mw: under water mass [ kg ] n: 1. number of stones moved [ - ] 2. number of tests [ - ] 3. number of layers [ - ]

List of symbols


p: 1. pressure [ N/m2 ] 2. probability [ - ]

Q: discharge [ m3/s ] rx: turbulence intensity [ - ] Re: number of Reynolds [ - ] Re*: particle Reynolds number [ - ] Su: amount of one layer of dumped stones [ - ] tc: calculated layer thickness [ m ] tm: measured layer thickness [ m ] T: duration experiment [ s ] u: flow velocity in x-direction [ m/s2 ] ū: mean or average velocity [ m/s ] ub: velocity of the flow near the bottom [ m/s] uh: calculated velocity using the water depth [ m/s ] uEMS: velocity measured using EMS [ m/s ] umax: maximum velocity [ m/s ] upeak: peak velocity [ m/s ]

*u : shear velocity [ m/s ] u*C: critical shear velocity [ m/s ] v: flow velocity in y-direction [ m/s ] Vs: volume stone [ m3 ] V: volume of the stone (dn

3) [ m3 ] w: flow velocity in z-direction [ m/s ] Weff: effective width [ - ] W50: median nominal weight [ kg ] y0: height above bed where velocity is zero [ m ]

z: height of the measurement above the bed [ m ] z0: height above bed where velocity is zero [ m ]

α: angle between flume wall and structure [ - ] β: angle between lift and drag force [ - ] ∆: relative density ( = (ρs-ρw ) / ρw ) [ - ] κ: von Kàrmàn constant ( = 0.4 ) [ - ] λ: relaxation length in turbulence [ - ] ρs: density of stone [ kg/m3 ] ρw: density of water [ kg/m3 ] σ: standard deviation [ - ] τc: critical shear stress [ N/m2 ] τ0: shear stress on the bottom [ N/m2

] υ: kinematic viscosity [ m2/s ] φ: pivoting angle [ - ] ΦE: entrainment parameter [ - ] ΨC: critical Shields parameter [ - ] ΨMS: Morison-Shields parameter [ - ] ΨS: Shields parameter [ - ] Ψv: Morison-Shields parameter depending only

on the velocity [ - ]

References


References BATTJES, J.A., 1997. Lecture notes CT2100 Fluid Mechanics. Delft: Delft Faculty of civil engineering1

BATTJES, J.A., 2000. Lecture notes CT3310 Stroming in waterlopen. Delft: Delft Faculty of civil engineering1 BOOIJ, R., 2002. Lecture notes CT5312 Turbulentie in de waterloopkunde Delft: Delft Faculty of civil engineering1 CIRA/CUR, 1991. Manual on the use of rock in coastal and shoreline engineering. CUR report 154. Rotterdam: Balkema CUR REPORT 169, 1995. Manual on the use of Rock in Hydraulic Engineering. Gouda: CUR DEAN, R.G., DALRYMPLE, R.A., 1991. Water wave mechanics for engineers and scientist. Singapore: World Scientific Publishing Co. Pte. Ltd. DHL, 1969. Begin van bewegen bodemmateriaal. Report S159-1 FORSCHELEN, P., DEC. 1999. M.Sc. Thesis Transport van granulair bodemmateriaal. Delft: University of Technology2

FRAAIJ ET AL., 2002. Experiment manual A CT1120 Materiaalkunde. Delft: Delft Faculty of civil engineering1 HENDRIKS, CH.F. ET AL, 1997. Lecture notes CT4030 Onderzoeksmethodologie. Delft: Delft Faculty of civil engineering1

HINZE, J.O., 1975. Turbulence. New York: McGraw-Hill

1 Delft University of Technology, the Netherlands. Main library of the Department of Civil Engineering 2 Delft University of Technology, the Netherlands. Library Hydraulic Engineering, Department of Civil Engineering

References


NEZU, I., NAKAGAWA, H., 1993. Turbulence in open-channel flows. Rotterdam: A.A. Balkema RIJN, L., VAN, 1989. Handbook sediment transport by currents and waves. Delft: Delft Hydraulics SCHIERECK, G.J., 2001. Introduction to Bed, bank and shore protection. Delft: Delft University Press SCHOKKING, L.A., 2002. M.Sc. Thesis Bowthruster-induced Damage. Delft: University of Technology (http://www.waterbouw.tudelft.nl/index.php?menu_items_id=65) SIMONS, D.B., Senturk, F., 1992. Sediment transport technology. Colorado: Water Resources Publications TROMP, M.M.A., 2004. M.Sc. Thesis The influence that fluid accelerations have on the threshold of motion. Delft: University of Technology (http://www.waterbouw.tudelft.nl/index.php?menu_items_id=65) VELDEN, E.T.J.M., VAN DER, 2000. Lecture notes CT5309 Coastal Engineering. Delft: Faculty of civil engineering1 Articles BOSMA, C.F., VERHAGEN, H.J., d’Angremond, K., Sint Nicolaas, W., march 2002. Void porosity measurements in coastal structures, proc. ICCE 2002, Cardiff, pages 1411-1423 (http://www.waterbouw.tudelft.nl/index.php?menu_items_id=19) CARDOSO, A.H., GRAF, W.H. AND GUST G., 1991. Steady gradually accelerating flow in a smooth open channel. Journal of Hydraulic Research, vol. 29, 1991, No. 4 pages 525–543 HOEFEL, F. AND ELGAR, S., march 2003. Wave induced sediment transport and bar migration Science. vol. 299 No. C9 21 march 2003 pages 1885-1887 HOFLAND, B., BATTJES, J.A., BOOIJ, R., to be published 2004. Measurement of fluctuating pressures on course bed material. Journal of Hydraulic Engineering

References


HOFLAND, B. AND BOOIJ, R., 2004. Measuring the flow structures that initiate stone movement. River flow 2004 vol 1 Napels, Italy. Rotterdam: A.A. Balkema Editors: GRECO, M., CARRAVETTA, A. AND DELLA MORTE, R. KIRCHNER, J.W., DITTRCH, W.E., IKEDA, H. AND ISEYA F., 1990. The variability of critical shear stress, friction angle and grain protrusion in water-worked sediments. Sedimentology, 37, pages 647-672 MORISON, J.R., O’BRIAN, M.P., JOHNSON J.W. AND SCHAAF, S.A., 1950. The force exerted by surface Waves on piles. Petrol. Trans. AIME vol. 189 pages 149-154 SONG, T. AND CHIEW, Y.M., march 2001. Turbulence measurement in non-uniform open-channel flow using acoustic Doppler velocitymeter (ADV) Journal of engineering mechanics, Delft: Delft Faculty of civil engineering, pages 219-231 STIVE, M.J.F. AND RENIERS A.J.H.M., march 2001. Sandbars in motion. Science. vol. 299, 21 march 2003, pages 1855-1856 Internet September 19th 2003: http://www.nen.nl; NEN 5186

Table of contents

The influence of flow acceleration on stone stability A1

Table of contents Table of contents.........................................................................................................A1 Appendix A Theory .................................................................................................A2 Appendix B The orifice plate ................................................................................A13 Appendix C Dimensions of the flume...................................................................A14 Appendix D Sieve curves ......................................................................................A16 Appendix E Placement of the stones.....................................................................A19 Appendix F Flow profiles .....................................................................................A20 Appendix G NEN-norm.........................................................................................A23 Appendix H Determination of z0 ...........................................................................A26 Appendix I Turbulence intensity..........................................................................A36 Appendix J Comparing the parabolic equations and the measured values ..........A39 Appendix K Velocity of the flow versus acceleration...........................................A41 Appendix L Difference (error) between methods for calculating the acceleration...... ...........................................................................................................A47 Appendix M Comparing v and a combinations vs. movement ..............................A51 Appendix N The bulk and inertia coefficient ........................................................A53

Appendix A


Appendix A Theory A.1 Shear velocity In a uniform stationary current the velocity of the flow is affected by the bottom friction. In case of a uniform current the forces acting on a volume of water are balanced. The shear stress and the downward acting gravitational force are equal. The shear velocity is used to describe the bottom shear stress. The definition of shear velocity is:

W

uρτ 0

* = (A.1)

*u : shear velocity [ m/s ]

τ0: shear stress on the bottom [ N/m2 ]

There are different ways to describe the relation between the shear velocity and the mean velocity of the flow. Three for this experiment important descriptions will be dealt with in this section. The first way to relate both velocities is by the use of the friction factor. Equation (A.1) can also be written in the form:

22*0 ucu wfw ρρτ == (A.2)

The shear stress at the bottom (τ0) is the developed shear force per unit wetted area. The relation between the velocities in equation (A.2) is given in the form of the already mentioned friction coefficient, cf.

fcuu 1

*

= (A.3)

An other way to describe the shear stress on the bottom is:

wgRiρτ =0 (A.4) In this equation R represents the hydraulic radius and iw the water level gradient. For a two dimensional flow the next equation for the discharge is given:

∫ ∫≅=h h

z zzuudzq

0 0

*

0

lnκ

(A.5)

κ: von Kàrmàn constant ( = 0.4 ) [ - ]

Appendix A


With ∫ −= xxxxdx lnln this becomes

−≈

−−= 1ln1ln

0

*0

0

*

zhhu

hz

zhhuq

κκ (A.6)

This can be written as

=

−== −

0

1*

0

* ln1lnzheu

zhu

hqu

κκ (A.7)

We find a second way to relate the shear and the mean velocity:

=

−== −

0

1

0*

ln11ln11zhe

zh

cuu

f κκ (A.8)

When we consider the flow to be a log profile, the term z0 is the position above the bottom where the velocity is zero. In literature the term z0 is sometimes replaced by y0 depending on the choice of the xyz-plane. In this thesis z0 will be used. The logarithmic part of the profile appears in the vertical velocity profile at a distance of 2 or 3 times the grain diameter above the bottom up to at least 0.2 times the water depth (h), figure A.1. The third way of describing the relation between the velocity and the shear velocity is given by:

s

z

kzz

uu )(ln1 0

*

+=

κ (A.9)

0.2 h

2 or 3 stones

z

Figure A.1 Logarithmic part of an undisturbed flow profile

Appendix A


The term (z + z0) is equal to the height of the occurring velocity (uz) above the bottom. When looking at the kinematic turbulent viscosity (υt) we find:

ρηκυ t

t zu == * (A.10)

υt: kinematic turbulent viscosity [ m2/s ]

ηt: dynamic turbulent viscosity [ m2/s ] z: distance from wall [ m ]

For the occurring shear stress we can write:

zu

dzdu

dzduzu

dzdu

dzdu

tt1*

* κκυ

ρτητ =⇒==⇒= (A.11)

This formula can be written as a logarithmic flow profile by integrating over z. Equation (2.31) can now be written as:

0*

ln1zz

uu

κ= (A.12)

All the theory as described in this section is valid for an undisturbed flow. In the case of the experiment the theory can be used in the approach section of the flume where the flow is not accelerated yet as mentioned in the beginning of this section. In case of an accelerated flow this theory is not proved to be valid so can not be used groundlessly. In section 4.3 will be explained in which way the theory is going to be used for the obtained experimental data. A.2 Chezy The Chezy formula is an empirical relation for an undisturbed flow. In the experiment it can and will only be used in the approach or acclimatization part of the flume. In this part of the flume the flow is not accelerated and the Chezy formula can serve as a check for the determined shear velocity for the undisturbed flow. In the previous section the determination of the shear velocity using the logarithmic flow profile is explained. Chezy’s relation of velocity resistance to flow is based on two assumptions.

1. The motion of water in a channel is caused by the resultant of the external forces acting in the direction of the motion.

2. The resistance to flow is proportional to the square of the velocity.

Appendix A


These two assumptions lead to the following formula of Chezy:

bRiCu = (A.14) C: Chezy coefficient [ m1/2/s ] R: hydraulic radius [ m ] ib: slope [ - ]

The Chezy coefficient can be written as:

khC 12log18= (A.15)

h: water depth [ m ] k: roughness (2 or 3 times dn50) [ m ] The Chezy formula also describes the relation between the shear velocity and the occurring velocity of the flow. In fact, equation (2.35) is an other way of writing equation (2.30). This can also be seen when looking at (2.37). The smoothness coefficient, C, can be used to describe the shear velocity as follows:

==

kh

uC

guu

12log75.5* (A.16)

A.3 Momentum balance When the energy level of the flow is not uniform through a certain passage because of wall-friction as is the case in the experiment, still one energy-level for that passage can be defined.

gQPH

ρ>=< (A.17)

<H>: mean energy level in cross section passage [ m ] P: energy flux [ W ] Q: discharge [ m/s3 ]

H is defined for every single position, while <H> is a flow characteristic for a passage.

Appendix A


∫∫∫∫>=<

dAu

dAHuH

N

N (A.18)

When the flow is uniform, <H> becomes H in equation (A.18). When we substitute H = h + u2/2g in the equation and the streamlines are straight in the passage, equation (A.18) will become:

udA

dAu

ghH ∫∫+>=<

3

21 (A.19)

The denominator in the last term is equal to ūA. When the local velocity is not constant the numerator is not equal to ū3A, but larger. The ratio between the mean velocity and the local velocity will be defined as

∫∫∫∫ == dAuuAdAu

dAu3

3

3

)/(1α (A.20)

and

guhH2

2

α+>=< (A.21)

When a non-uniform flow is the result of wall-friction in a turbulent flow, the value of α is a little larger than 1. When looking at a contraction of a flow, a momentum balance and a mass balance can be defined. When a momentum balance is considered for a stationary flow, the pressure and the convection contribute to the exchange of momentum. In formula this can be written as:

mvp FFF += (A.22) In which

∫∫= dAepF Np (A.23) and

∫∫= dAeuF Nmv2ρ (A.24)

This gives

∫∫ += dAeupF N)( 2ρ (A.25)

Appendix A


We consider the streamlines to be straight (constant water level) and parallel (e constant) in a passage.

∫∫ ∫∫ === NNNmv eAudAuedAeuF βρρρ 22 (A.26) The factor β can be defined as

∫∫= dAuuA

2)/(1β (A.27)

The ratio between α and β is in the order of α-1 = 3(β-1). For the total amount of momentum through a passage with parallel streamlines the following equation can be used

∫∫ += uQpdAF βρ (A.28) Combination of the mass and momentum balance gives an equation which can be used to solve the unknown flow parameters Momentum balance for the contraction:

[ ] [ ] 021

022 =−+

τβρρ BBuh

dxdBgh

dxd (A.29)

[ ] [ ]βρρτ BuhdxdBgh

dxdB 22

0 21

+

= (A.30)

Solving the three equations separately: a: τB

b: dxdBgh

dxdhghB 2

21

ρρ +

c: dxdB

hudxdh

uBdxdu

uhB 222 ρββρβρ ++

Combining a, b and c now gives:

dxdBhu

dxdhuB

dxduuhB

dxdBgh

dxdhghBB 222 2

21 ρββρβρρρτ ++++= (A.31)

flow pressure Shear stress

a b c

Appendix A


dxduuhB

dxdBhu

dxdBgh

dxdhuB

dxdhghBB 2

21 222 βρρβρβρρτ ++++= (A.32)

( ) ( )dxduuhB

dxdBhugh

dxdhghBuBB 2

21 222 βρρβρρβρτ +

+++= (A.33)

Mass balance for the contraction:

0=dxhBud (A.34)

0=++dxdBhu

dxdhBu

dxudhB (A.35)

Combination of equation (A.33) and equation (A.35) (momentum and mass balance) gives:

dxduhuB

dxdBgh

dxdhghBB βρρρτ ++= 2

21 (A.36)

The fraction containing the velocity must be replaced by terms of the width (B) and water depth (h). The constant discharge can be written as:

UBhQ = (A.37)

Using this expression in the term containing the velocity we find:

dxBhQd

huBdxduhuB

= βρβρ (A.38)

dxBhQd

huBdxdBgh

dxdhghBB

++= βρρρτ 2

21 (A.39)

dxBh

dhuQB

dxdBgh

dxdhghBB

++=

1

21 2 βρρρτ (A.40)

Appendix A


( )dxdB

hBdxdh

BhdxBhd

dxBh

d

22

1 111

−−==

−

(A.41)

Combining equation (A.40) and (A.41) gives:

dxdBuQB

dxdhuQh

dxdBgh

dxdhghBB βρρβρρτ 112

21 −− −−+= (A.42)

( )dxdB

Bhu

Bgh

dxdhugh

−+−=

222

2ρβρρβρτ (A.43)

The terms in equation (A.43) can be divided by ρgh:

dxdB

gBu

Bh

dxdh

ghu

gh

−+

−=

22

21 ββ

ρτ (A.44)

The second term on the right hand side of equation (A.44) represents Fr2 multiplied by the factor β. A.4 Force generated by acceleration In the conducted experiment, an attempt is made to investigate the influence of a combination of the already mentioned velocity and the acceleration of the flow on the stability of stones. The acceleration will induce an extra force due to pressure on the stones. The acceleration of the fluid around a single grain will result in a horizontal pressure gradient. In the figure below a particle in an accelerating fluid is considered.

Appendix A


The pressure difference can be determined using Bernoulli’s theory, which is given by:

=+=++=g

uhg

ugpzH

22

22

ρconstant (A.45)

For the case in figure A.2 equation (A.45) can be written as:

222

211 2

121 upup ρρ +=+ (A.46)

This is equivalent to:

)(21 2

122 uudp −= ρ (A.47)

The grains are assumed to be very small compared to the variations in the flow, which leads to:

=dxdp constant (A.48)

Hence:

dp= dxdxdp (A.49)

P1

dx

P2

U1 U2=U1+dU

dP/dx

Acceleration

Figure A.2 Pressure difference due to acceleration

Appendix A


The force on the area of the grain can now be expressed as follows:

dxdxdpdydzAdpF == (A.50)

This force due to difference in pressure caused by acceleration of the flow is equal to:

∫∫∫ ==dxdpVdxdydz

dxdpFP (A.51)

In the equation the V represents the volume of the stone and the pressure gradient is linear. The Euler equation is given by:

)( gzpDt

uD ρρ +−∇= (A.52)

When considering the dynamical pressure, this can be written as:

dpDt

uD∇=ρ (A.53)

The acceleration term in this formula is:

dtdx

xu

tu

DtuDax ∂

∂+

∂∂

== + dtdz

zu

∂∂ (A.54)

We can neglect the last term so the equation will become:

xuu

tuax ∂

∂+

∂∂

= (A.55)

Using the Euler equation:

dxdp

xuu

tu

DtuD

=∂∂

+∂∂

= )(ρρ (A.56)

The pressure force, due to acceleration now becomes:

Vxuu

tu

dxdpVFp )(

∂∂

+∂∂

== ρ (A.57)

For the case of an accelerating stationary flow the first term is equal to zero because the mean velocity does not change with time.

Appendix A


Equation (A.57) can for a stationary flow be written as:

Vxuu

dxdpVFp )(

∂∂

== ρ (A.58)

Appendix B


Appendix B The orifice plate In figure B.1 the discharge of the flume is plotted the difference in piezometric level in the pressure gauge. By turning the wheel (figure B.2) the discharge can be varied. The pressure gauge used in the experiment can also be seen in figure B.2.

Meetflens 170/120; beta=0.8; kleurcode zwart

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

0 10 20 30 40 50 60 70 80 90 100

Discharge Q (L/s)

h (c

m)

Figure B.1 Discharge of the flume versus the difference in pressure height

Figure B.2 Wheel for regulating the discharge (left), pressure gauge for measuring the discharge (right)

Appendix C


Appendix C Dimensions of the flume In the figure below the flume is drawn. The important dimensions are given as well as the names of the structures in the flume. Notice that the flume is not drawn to scale, especially in the length of the flume. In figures C.2 and C.3 a pictures of the actual situation is given to compare it with the schematic drawing, figure C.1.

Q in

Q out 0.70 m

14.00 m

0.70

m

gate Flow stabilizer

Sideview

Topview

0.50

m

1.85 m 1.50 / 2.00 m

contraction

0.90 m

0.15

m α

acclimatization/approach area

Figure C.1 Top and side view of the flume indicating the dimensions

Figure C.2 Pictures of the actual situation

Appendix C


Figure C.3 Pictures of the actual situation

Appendix D


Appendix D Sieve curves The sieve curves of both grain sizes used in the experiment are shown in this appendix. In order to get a good test result the diameters of the grains can not differ much. By using sieves with a small gap difference this can be established. The curve has to show a steep profile. For both grain sizes a distribution of the dry mass is plotted versus the percentage of the cumulative weight. Also a distribution of the diameter of the grains (dn) is plotted versus the cumulative weight. In the dry mass plot, the curve is not smoothly increasing, but shows points that visible differ. This can be explained by the fact that the whole set of stones exists of grains with a different density. The set is sorted on their nominal diameter, dn, which explains the deviations. For the determination of the stone parameters of the small stones 250 grains were measured. In section 3.6.1 a table containing the important parameters can be found. In case of the small stones, painted as well as non-painted stones are measured. No difference was found between the two.

Mass distribution smaller stones

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Mass dry (gr)

Perc

enta

ge o

f cum

ulat

ive

wei

ght (

%)

.

Appendix D


For the determination of the stone parameters of the large stones 139 grains were measured. In section 3.6.1 a table containing the important parameters can be found.

dn distribution smaller stones

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0.002 0.004 0.006 0.008 0.01 0.012

dn (m)

Perc

enta

ge o

f cum

ulat

ive

wei

ght (

%)

.

Mass distribution larger stones

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 5 10 15 20 25 30 35 40

Mass dry (gr)

Perc

enta

ge o

f cum

ulat

ive

wei

ght (

%)

.

Appendix D


dn distribution larger stones

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0.005 0.01 0.015 0.02 0.025

dn (m)

Perc

enta

ge o

f cum

ulat

ive

wei

ght (

%)

.

Appendix E


Appendix E Placement of the stones On the bottom of the contraction of the flume one layer of stones is glued as shown in the pictures below. The picture on the left shows the structure before the stones were glued to the bottom, the picture on the right after.

Before one can carry out an experiment the second layer of coloured stones has to be placed in strips of 0.10m. In the pictures below is shown how this was carried out.

In the first picture a prefabricated form was placed in the contraction. The coloured stones were placed by dumping as can be seen in the second picture. Then the shape was removed.

Figure E.1 Pictures of the contraction before and after gluing the stones to the bottom

Figure E.2 Dumping the colored strips for an experiment using large stones

Appendix F


Appendix F Flow profiles Velocity measurements are taken using EMS at different locations in the flume. The contraction L2.00 was placed in the flume during the measurements. At the end of the contraction (x = 0.00, 0.25 and 0.50m) the EMS could not measure close to the wall. The velocity near the wall is set on zero and the figures are plotted. The velocities measured over the width at the end of the contraction are almost identical. This makes the assumption made in section 4.4.1 that the flow profile is ‘full’ over the width, assumable.

Figure F.1 3-Dimenional flow profile at x = 2.00m, B = 0.5m and Q = 50l/s


Appendix F




Appendix G


Appendix G NEN-norm On the next pages NEN-norm 5186 can be found. This norm is used for determining the parameters of the large stones. The ‘small stone procedure’ is already explained in section 3.7.

Appendix G


Appendix H


Appendix H Determination of z0 Small stones In the following graphs the mean velocity measurements are plotted versus the logarithmic value of the height. Through each dataset a straight line is drawn using the measurements in the lowest part of the flow profile to determine the value of z0, where the velocity is equal to zero. The discharge for each measurement is different, but because the water depth also differs for each dataset the velocities will be of the same order of magnitude. First a level of he bed at 1.17x10-2m above the bottom plates is assumed. In the next figures the measurements of the velocities measured for L1.50 and L2.00 are given. The values found for z0 are for each dataset of the same order of magnitude.

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Velocity (m/s)

ln (z

+ d

elta

z)

Q=30l/sQ=40l/sQ=50l/sQ=60l/s

Figure H.1 Determination of z0 for different discharges, L1.50, x = 1.50m, small stones and the level of the bed (z = 0.00m) at 1.17x10-2m above the wooden plates

Appendix H


Now a bottom level at 0.01297m above the wooden plates is assumed. Again for both contractions L1.50 and L2.00 the velocities are plotted versus the logarithmic value of the height of the measurement.

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Velocity (m/s)

ln (z

+ d

elta

z)


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Velocity (m/s)

ln (z

+ d

elta

z)




Appendix H


All values that can be determined from these graphs can now be listed in the next two tables. The first table gives the results for a bottom level at 0.01174m above the wooden plates, the second table for 0.01297m. As mentioned before two datasets where not used because their values diverge from the other datasets.

L (m) Q (m3/s) delta z (m) z0 (m) h (m) ū (m/s) a (1/ m/s) u* (m/s) cf u* Chezy (m/s)1.50 0.03 0.01174 0.0012 0.258 0.243 18.07 0.022 0.0083 0.020 2.00 0.03 0.01174 0.0019 0.261 0.240 15.83 0.025 0.0111 0.020 1.50 0.04 0.01174 0.0022 0.323 0.257 14.36 0.028 0.0117 0.021 2.00 0.04 0.01174 0.0016 0.325 0.255 16.67 0.024 0.0088 0.020 1.50 0.05 0.01174 0.0006 0.378 0.273 21.21 0.019 0.0048 0.021 2.00 0.05 0.01174 0.0027 0.383 0.270 13.29 0.030 0.0125 0.021 1.50 0.06 0.01174 0.0031 0.436 0.283 17.12 0.023 0.0068 0.021 2.00 0.06 0.01174 0.0010 0.440 0.281 19.44 0.021 0.0054 0.021

average 0.00156

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Velocity (m/s)

ln (z

+ d

elta

z)



Table H-1 Determination of z0, and other flow parameters with a level of the bed at 1.17x10-2m above the wooden plates and small stones

Appendix H


Using the results from these tables the average z0 can be determined and all the lines can be drawn trough the measurements and the vertical axis crossing at the same point as can be seen in figure 4.4, section 4.3.1. Large stones For the large stones the same graphs and calculations can be made. First a level of the bed at 2.35x10-2m above the bottom plates is assumed. In the next figures the measurements of the velocities measured for L1.50 and L2.00 are given.

L (m) Q (m3/s) delta z (m) z0 (m) h (m) ū (m/s) a (1/ m/s) u* (m/s) cf u* Chezy (m/s)1.50 0.03 0.01297 0.0011 0.257 0.246 18.58 0.022 0.0077 0.021 2.00 0.03 0.01297 0.0017 0.260 0.243 16.28 0.025 0.0102 0.020 1.50 0.04 0.01297 0.0019 0.321 0.259 14.74 0.027 0.0109 0.021 2.00 0.04 0.01297 0.0015 0.324 0.258 17.15 0.023 0.0082 0.021 1.50 0.05 0.01297 0.0005 0.377 0.275 21.83 0.018 0.0045 0.021 2.00 0.05 0.01297 0.0024 0.381 0.271 13.63 0.029 0.0117 0.021 1.50 0.06 0.01297 0.0015 0.434 0.285 17.59 0.023 0.0064 0.021 2.00 0.06 0.01297 0.0009 0.438 0.282 19.93 0.020 0.0051 0.021

average 0.00117

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

Velocity (m/s)

ln (z

+ d

elta

z)


Table H-2 Determination of z0, and other flow parameters with a level of the bed at 1.30x10-2m above the wooden plates and small stones

Figure H.5 Determination of z0 for different discharges, L1.50, x = 1.50m, large stones and the bottom level (z = 0.00m) at 0.0235m above the wooden plates

Appendix H


In the next figure the discharge of 50l/s (L2.00) is not plotted because no logarithmic profile could be found.

Now a level of the bed at 2.65x10-2m above the wooden plates is assumed. Again for both contractions L1.50 and L2.00 the velocities are plotted versus the logarithmic value of the height of the measurement.

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400

Velocity (m/s)

ln (z

+ d

elta

z)

Q=40l/sQ=60l/sQ=70l/s

Figure H.6 Determination of z0 for different discharges, L2.00, x = 2.00m, large stones and the level of the bed (z = 0.00m) at 2.35x10-2m above the wooden plates

Appendix H


-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

Velocity (m/s)

ln (z

+ d

elta

z)


-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400

Velocity (m/s)

ln (z

+ d

elta

z)

Q=40l/sQ=60l/sQ=70l/s

Figure H.7 Determination of z0 for different discharges, L1.50, large stones and the level of the bed (z = 0.00m) at 2.65x10-2m above the wooden plates

Figure H.8 Determination of z0 for different discharges, L2.00, large stones and the level of the bed (z = 0.00m) at 2.65x10-2m above the wooden plates

Appendix H


In the next two tables all the values obtained for the large stones are listed. In the first table the level of the bed is set at 2.35x10-2m above the plates, in the second at 2.65x10-2m.

Using the results from these tables the average z0 can be determined and all the lines can be drawn trough the measurements and the vertical axis crossing at the same point as can be seen in figure 4.6, section 4.3.2.

L (m) Q (m3/s) delta z (m) z0 (m) h (m) ū (m/s) a (1/ m/s) u* (m/s) cf u* Chezy (m/s)1.50 0.04 0.0235 0.0034 0.296 0.294 12.22 0.033 0.0124 0.029 2.00 0.04 0.0235 0.0015 0.302 0.288 13.89 0.029 0.0100 0.029 1.50 0.05 0.0235 0.0055 0.343 0.313 9.82 0.041 0.0169 0.030 1.50 0.06 0.0235 0.0035 0.389 0.328 12.35 0.032 0.0098 0.031 2.00 0.06 0.0235 0.0038 0.395 0.323 11.12 0.036 0.0124 0.030 1.50 0.07 0.0235 0.0017 0.431 0.343 13.05 0.031 0.0080 0.031 2.00 0.07 0.0235 0.0044 0.436 0.339 9.34 0.043 0.0159 0.031

average 0.0034

L (m) Q (m3/s) delta z (m) z0 (m) h (m) ū (m/s) a (1/ m/s) u* (m/s) cf u* Chezy (m/s)1.50 0.04 0.0265 0.0028 0.293 0.301 12.77 0.031 0.0109 0.030 2.00 0.04 0.0265 0.0013 0.299 0.294 14.38 0.028 0.0089 0.029 1.50 0.05 0.0265 0.0047 0.340 0.319 10.24 0.039 0.0150 0.031 1.50 0.06 0.0265 0.0030 0.386 0.333 12.79 0.031 0.0088 0.031 2.00 0.06 0.0265 0.0031 0.392 0.328 11.68 0.034 0.0109 0.031 1.50 0.07 0.0265 0.0014 0.428 0.349 13.64 0.029 0.0071 0.032 2.00 0.07 0.0265 0.0037 0.433 0.344 9.73 0.041 0.0143 0.031

average 0.0028

Table H-3 Determination of z0, and other flow parameters with a bed level at 2.35x10-2m above the wooden plates and large stones


Appendix H


Acclimatization area For the single layer of stones on the bottom of the flume in the acclimatization area, the value of y0 can also be determined using the extra measurement taken at x = 2.50m. The height of a single layer of stones is 1.50x10-2m above the plates. The layer of the bed will be set on 0.15 times the dn50 of the stones, which is equal to 1.28x10-2m above the plates. In the next figure the velocity measurements are plotted versus the logarithmic height of the measurement above the bottom level. Only the measurements near the bottom are used as explained in section 2.2.3 and 4.3.

In the next table all values obtained from figure H.9 are listed and the average value for z0 is calculated.

L (m) Q (m3/s) delta z (m) z0 (m) h (m) ū (m/s) a (1/ m/s) u* (m/s) cf u* Chezy (m/s)2.00 0.03 0.0128 0.0008 0.260 0.242 19.92 0.020 0.0069 0.023 2.00 0.04 0.0128 0.0019 0.324 0.257 15.82 0.025 0.0097 0.023 2.00 0.05 0.0128 0.0025 0.382 0.271 18.94 0.021 0.0061 0.024 2.00 0.06 0.0128 0.0009 0.439 0.282 19.08 0.021 0.0055 0.024

average 0.0015

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Velocity (m/s)

ln (z

+ d

elta

z)


Table H-5 Determination of z0, and other flow parameters with a bed level at 1.28x10-2m above the wooden plates in the acclimatization area

Figure H.9 Determination of z0 for different discharges, x = 2.50m, z = 0.00m at 1.28x10-2m above the wooden plates in the acclimatization area

Appendix H


The average value for the z0 can be determined. According to the theory the value should approximately be:

mdz n 3500 105.1

10015.0

10−×==≈

The calculated average value for z0 is equal to 1.50x10-3m, the same as according to the theory. In the next figure all lines go through z0 and these lines will be used in table H-6 for calculation of the important flow parameters of the stones used in the acclimatization area.

L (m) Q (m3/s) delta z (m) z0 (m) h (m) ū (m/s) a (1/ m/s) u* (m/s) cf u* Chezy (m/s)2.00 0.03 0.0128 0.0015 0.260 0.242 16.51 0.024 0.0100 0.023 2.00 0.04 0.0128 0.0015 0.324 0.257 17.03 0.023 0.0083 0.023 2.00 0.05 0.0128 0.0015 0.382 0.271 22.00 0.018 0.0045 0.024 2.00 0.06 0.0128 0.0015 0.439 0.282 16.89 0.024 0.0071 0.024

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300

Velocity (m/s)

ln (z

+ d

elta

z)



Figure H.10 The velocity of the undisturbed flow of 4 different datasets versus the height of the measurement above the bed with a common value for z0

Appendix H


The values for the shear velocity calculated using the slope of the lines through z0 and the measurements in the figure above are almost the same as the values calculated with Chezy.

Appendix I


Appendix I Turbulence intensity In section 4.5.1, figure 4.14, the turbulence intensity is shown for both contractions using small stones. The following relation between the turbulence intensity for a arbitrary location in the contraction and the rx at x = L is found:

11.043.045.0)()( 2

+

+

=

Lx

Lx

Lrxr

x

x (I.1)

The following steps have been taken to obtain this relation. First the turbulence intensity is plotted versus x/L, figure I.1. It can be seen that rx close to the bottom is higher than further away from the bottom. This was already found in section 4.5.1, figures 4.12 and 4.13.

It is also clear that there is no difference between both contractions. For each location four datasets differing in discharge are plotted. When we want to find a relation between the turbulence intensity and the position in the contraction independent on the height above the bottom, the values for each dataset can be divided by its value at x = L, figure I.2. The figure shows that the turbulence intensity does not depend on the height anymore. When the average value of the points for the two different contractions is calculated we find figure 4.14 and equation (I.1).

Figure I.1 The relative turbulence versus the position of the measurement in the contractions at two different heights (small stones)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

rx (-

)

L=1.50m; z=0.027mL=1.50m;z=0.152mL=2.00m; z=0.027mL=2.00m; z=0.152m

Appendix I


This is also done for the experiments using large stones. The relation between the turbulence intensity for an arbitrary location in the contraction and the rx at x = L is:

09.040.051.0)()( 2

+

+

=

Lx

Lx

Lrxr

x

x (I.2)

Figure 4.21 shows the turbulence intensity versus x/L. The value for rx depends on the height of the measurement. In the next figure, figure I.4, the rx(x) is divided by rx(L). The figure shows that the turbulence intensity does not depend on the height anymore. When the average value of the points for the two different contractions is calculated we find figure 4.21 and equation (I.2).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

rx(x

) / r

x(L)

(-)

L=1.50m;z=0.027mL=2.00m;z=0.027mL=1.50m;z=0.152mL=2.00m; z=0.152m

Figure I.2 The relative turbulence divided by the value for that same height at x = L, versus the position of the measurement in the contractions (small

stones)

Appendix I


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

rx (-

)L=1.50m; z=0.049m

L=1.50m; z=0.17m

L=2.00m; z=0.049m

L=2.00m; z=0.17m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

rx(x

) / r

x(L)

L=1.50m; z=0.049mL=1.50m; z=0.17mL=2.00m; z=0.049mL=2.00m; z=0.17m

Figure I.3 The relative turbulence versus the position of the measurement in the contractions at two different heights (large stones)

Figure I.4 The relative turbulence divided by the value for that same height in x = L, versus the position of the measurement in the contractions (large stones)

Appendix J


Appendix J Comparing the parabolic equations and the measured values In section 4.5.1 a parabolic equations is found that described the measured values of the peak velocities divided by the calculated mean velocities. For experiments using the small stones the following equation is found:

03.111.010.02

+

+

=

Lx

Lx

uupeak (J.1)

In the figure below this equation is drawn and compared to the actual measured values. This is done for two datasets at different heights with a discharge of 40l/s, for each contraction one.

The measured values fit the parabolic equations rather well as can be seen in the figure above.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

Upe

ak /

Um

ean

(-)

L=1.50m; Q=40l/s; z=7.7cm

L=1.50m; Q=40l/s; z=15.2cm

L=2.00m; Q=40l/s; z=7.7cm

L=2.00m; Q=40l/s; z=15.2cm

03.111.010.02

+

+

=

Lx

Lx

uu peak

Figure J.1 Equation (J.1) compared to actual measurements (small stones)

Appendix J


For the large stones the following equation is found:

03.109.015.02

+

+

=

Lx

Lx

uupeak (J.2)

In the figure below this equations is drawn and compared to the actual measured values. This is done for a two datasets at different heights with a discharge of 50l/s, for each contractions one. For L2.00, Q=50l/s and z=9.9cm no measurements were carried out.

In the narrow part of the contractions where the experiments take place, the measured values are almost similar to the parabolic equations. In the wider part of the contractions the measured values differ slightly from equation (J.2). The average value of the different measurements seems to fit the parabolic equation at x/L = 0.7.

Figure J.2 Equation (J.2) compared to actual measurements (large stones)

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/L (-)

Upe

ak /

Um

ean

(-)

L=1.50m; Q=50l/s; z=9.9cm

L=1.50m; Q=50l/s; z=17.4cm

L=2.00m; Q=50l/s; z=17.4cm

03.109.015.02

+

+

=

Lx

Lx

uu peak

Appendix K


Appendix K Velocity of the flow versus acceleration The average velocity through a passage. (1) The acceleration in the middle of a colored strip am. (1)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

Figure K.1 Small stones, v1 and a1

Figure K.2 Large stones, v1 and a1

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

Appendix K


The average velocity through a passage. (1) The acceleration over the last stone in a colored strip ae. (2)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m



0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

Appendix K


The average of the measured velocities where the velocity in the flow profile is constant. (2) The acceleration in the middle of a colored strip am. (1)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m



Appendix K


The average of the measured velocities where the velocity in the flow profile is constant. (2) The acceleration over the last stone in a colored strip ae. (2)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m



Appendix K


The average velocity calculated in 1 can be multiplied with the ratio between the measured velocities and the calculated velocities. (3) The acceleration in the middle of a colored strip am. (1)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m



0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

Appendix K


The average velocity calculated in 1 can be multiplied with the ratio between the measured velocities and the calculated velocities. (3) The acceleration over the last stone in a colored strip ae. (2)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m



0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60Velocity (m/s)

Acc

eler

atio

n (m

/s^2

)

.

L=1.50mL=2.00m

Appendix L


Appendix L Difference (error) between methods for calculating the acceleration The Morison formula requires a velocity and an acceleration for determination of the force acting on a stone. In section 4.7 different methods for determining the velocity-acceleration combination necessary in the Morison formula are presented. Between the two most favourable methods a choice has to be made. The two methods are described in section 4.7 and will be repeated here shortly:

• Velocity and acceleration determined using EMS: The average of the measured velocities where the velocity in the flow profile is constant. (2) The acceleration over the last stone in a colored strip ae. (2)

• Velocity and acceleration determined using the water depth: The average velocity calculated in 1 can be multiplied with the ratio between the measured velocities and the calculated velocities. (3) The acceleration over the last stone in a colored strip ae. (2)

When looking at the data the first impression is that the data obtained using EMS show small fluctuations and irregularities in the velocity which result in larger irregularities in the acceleration. The relative acceleration in the last part of the contraction (where the experiments take place) should increase in a higher order than linear (parabolic). This can be seen when looking at the following example. The velocity and acceleration in the last 0.40m of the contraction can be calculated using the fact that the energy level does not change. In practise it will change because of wall friction, but these changes are so small (already shown when looking at the momentum balance) that this assumption can be made. The velocity and acceleration can now be calculated for fictive starting values at x = 0.40m. The only thing that will be proved here is that the relative acceleration should increase parabolic. The following formula is used for calculation of the relative acceleration:

30.040.0 −

∆=

aaar ( L.1)

ar: relative acceleration [ - ] ∆a: increase of acceleration over a strip [ m/s2 ] a0.30-0.40: acceleration in first pink strip between x = 0.40m and x = 0.30m [ m/s2 ] The starting values used for the fictive example are: L = 1.50m; Q = 50l/s; h = 0.34m

Appendix L


The following velocity acceleration combinations are calculated:

x (m) v (m/s) a (m/s2)0.40 - 0.30 0.750 0.577 0.30 - 0.20 0.857 0.862 0.20 - 0.10 1.009 1.420 0.10 - 0.00 1.261 2.855

49.0577.0

577.0862.03; =

−=ra

97.0577.0

862.0420.12; =

−=ra

49.2577.0

420.1855.21; =

−=ra

The following figure can be made:

The figure shows that the increase of the acceleration is parabolic.

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (0

.40-

0.30

)

L=1.50m; Q=50l/s; v (H)

Figure L.1 Parabolic increase of the relative acceleration in the contraction in case of a fictive example in which the energy level is assumed to be constant

Table L-1 Velocity-acceleration combinations used in the fictive example

Appendix L


When the same calculations for the actual measurements are carried out for both methods described above the following figures can be made. Measurements for experiments using the large stones are used. It is assumed that the same holds for the experiments using small stones.

The figures above show the irregularities in the increase of the acceleration for the EMS-measurements. When the error of this increase is calculated for both contractions and both velocity methods, the plots in figure L.3 can be made.

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=1.50m; Q=40l/s; a(h)

L=1.50m; Q=40l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a /

a (b

egin

)

L=1.50m; Q=40l/s; a(h)

L=1.50m; Q=40l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=1.50m; Q=60l/s; a(h)

L=1.50m; Q=60l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=1.50m; Q=70l/s; a(h)

L=1.50m; Q=70l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=2.00m; Q=40l/s; a(h)

L=2.00m; Q=40l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=2.00m; Q=50l/s; a(h)

L=2.00m; Q=50l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=2.00m; Q=60l/s; a(h)

L=2.00m; Q=60l/s; a(EMS)

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2x/L

∆a

/ a (b

egin

)

L=2.00m; Q=70l/s; a(h)

L=2.00m; Q=70l/s; a(EMS)

Figure L.2 Parabolic increase of the relative acceleration in the contraction in case of the experiments using large stones

Appendix L


It is expected that the increase in acceleration will be of the second order (parabolic). When the following formula is used for the obtained data, a parabolic course of the measured points is expected.

30.040.0 −

=a

aap (L.2)

ap: parabolic increase of the relative acceleration [ - ]

The plots show that the different calculation methods show clearly distinguishable differences in the increase of the acceleration for the same experiments. A parabolic equation is drawn through the measurements. The relative increase of the acceleration should be the same for experiments carried out in the same contraction. The error in case of the EMS-measurements is larger than in case of the acceleration calculated using the water depth as can be seen in the figures. The method that uses the water depth for determination of the velocity will be used in the Morison formula for the reasons following from this appendix.

y = 41.563x2 - 24.145x + 4.5453R2 = 0.9817

0

1

2

3

4

5

6

0 0.1 0.2 0.3x (m)

a / (

a 0.

40 -

0.30

)

.

L=1.50m; a(h)

y = 30.328x2 - 19.561x + 4.139R2 = 0.9439

0

1

2

3

4

5

6

0 0.1 0.2 0.3x (m)

a / (

a 0.

40 -

0.30

)

.

L=1.50; a(EMS)

y = 24.245x2 - 16.192x + 3.6833R2 = 0.9889

0

1

2

3

4

5

6

0 0.1 0.2 0.3x (m)

a / (

a 0.

40 -

0.30

)

.

L=2.00m; a(h)

y = 12.649x2 - 10.938x + 3.1075R2 = 0.9327

0

1

2

3

4

5

6

0 0.1 0.2 0.3x (m)

a / (

a 0.

40 -

0.30

)

.

L=2.00m; a(EMS)

Figure L.3 Error in parabolic increase of the acceleration in the contraction for experiments using large stones.

Appendix M


Appendix M Comparing v and a combinations vs. movement In the next tables the average percentage of moved stones for different velocity-acceleration combinations are compared to each other, section 5.3.

L (m) Q (m/s3) h (m) ∆ h (m) strip v (m/s) a (m/s2) av. perc. mov. 2.00 0.05 0.359 0.004 pink 0.788 0.510 0.0% 1.50 0.04 0.300 0.005 green 0.774 0.661 0.4% 2.00 0.04 0.302 0.004 pink 0.748 0.463 0.1% 1.50 0.03 0.237 0.005 green 0.736 0.619 0.4% 2.00 0.06 0.415 0.004 pink 0.818 0.530 0.2% 1.50 0.04 0.324 0.006 blue 0.806 0.804 0.3% 2.00 0.04 0.268 0.006 pink 0.845 0.643 0.1% 1.50 0.04 0.310 0.007 blue 0.843 0.901 0.6% 2.00 0.05 0.353 0.005 green 0.870 0.690 0.5% 1.50 0.04 0.347 0.009 yellow 0.864 1.096 1.4% 2.00 0.06 0.409 0.006 green 0.902 0.743 0.7% 1.50 0.04 0.347 0.009 yellow 0.864 1.096 1.4% 2.00 0.06 0.409 0.006 green 0.902 0.743 0.7% 1.50 0.04 0.292 0.008 blue 0.895 1.047 2.5% 2.00 0.06 0.409 0.006 green 0.902 0.743 0.7% 1.50 0.04 0.347 0.009 yellow 0.864 1.096 1.4% 2.00 0.04 0.260 0.007 green 0.945 0.912 3.0% 1.50 0.05 0.346 0.009 blue 0.944 1.158 3.9% 2.00 0.06 0.400 0.009 blue 1.013 1.093 2.8% 1.50 0.04 0.313 0.011 yellow 0.958 1.416 4.5% 2.00 0.06 0.400 0.009 blue 1.013 1.093 2.8% 1.50 0.04 0.298 0.012 yellow 1.007 1.602 6.0% 2.00 0.04 0.264 0.010 blue 1.021 1.234 6.7% 1.50 0.04 0.256 0.008 blue 1.020 1.409 13.9% 2.00 0.04 0.250 0.011 blue 1.081 1.422 11.8% 1.50 0.04 0.278 0.014 yellow 1.078 1.914 12.9% 2.00 0.05 0.332 0.012 yellow 1.128 1.631 11.6% 1.50 0.05 0.333 0.013 yellow 1.126 1.993 11.6%

Appendix M


2.00 0.06 0.387 0.012 yellow 1.161 1.672 9.4% 1.50 0.06 0.388 0.013 yellow 1.160 2.043 13.5% 1.50 0.04 0.258 0.015 yellow 1.164 2.294 22.9% 2.00 0.04 0.250 0.014 yellow 1.198 2.049 32.9% 1.50 0.04 0.252 0.016 yellow 1.189 2.441 60.9% 2.00 0.04 0.235 0.015 yellow 1.275 2.391 65.5% 1.50 0.04 0.240 0.016 yellow 1.247 2.730 72.9%

It can be seen that for a constant or decreasing velocity and an increasing acceleration the average percentage of movement increases. The first table contains values obtained in experiments using small stones, the second table is made for large stones.

L (m) Q (m/s3) h (m) ∆ h (m) strip v (m/s) a (m/s2) av. perc. mov. 2.00 0.07 0.400 0.006 pink 1.002 0.852 0.0% 1.50 0.06 0.352 0.008 green 1.002 1.148 0.5% 2.00 0.06 0.352 0.008 green 1.062 1.110 1.7% 1.50 0.04 0.253 0.010 blue 1.048 1.545 4.3% 2.00 0.07 0.391 0.009 green 1.114 1.206 0.6% 2.00 0.04 0.250 0.013 blue 1.093 1.517 2.8% 1.50 0.05 0.297 0.011 blue 1.114 1.709 7.4% 2.00 0.05 0.295 0.013 blue 1.160 1.637 2.8% 1.50 0.06 0.341 0.011 blue 1.163 1.809 5.4% 2.00 0.07 0.377 0.015 blue 1.270 1.907 4.5% 1.50 0.07 0.376 0.012 blue 1.230 2.022 12.9% 2.00 0.05 0.277 0.018 yellow 1.371 2.760 11.5% 1.50 0.04 0.234 0.019 yellow 1.300 3.110 21.0% 2.00 0.06 0.321 0.017 yellow 1.421 2.801 16.5% 1.50 0.06 0.322 0.020 yellow 1.416 3.415 26.9% 2.00 0.07 0.360 0.016 yellow 1.475 2.886 20.9% 1.50 0.07 0.360 0.017 yellow 1.477 3.494 37.8%

Table M-1 Small stones, comparing velocity and acceleration combinations with the amount of movement

Table M-2 Large stones, comparing velocity and acceleration combinations with the amount of movement

Appendix N


Appendix N The bulk and inertia coefficient The determination of the two coefficients CB and CM will be explained in this chapter. There appear to be a lot of different combinations which satisfy the range of pivoting angles as will be shown in this appendix. In section 5.6 and 5.7 only one pivoting angle and the resulting critical force is used. First the small stones will be investigated. The velocity-acceleration combinations (table N-1) , the critical force (4.55 – 6.43x10-3N) and equation (5.6) are used to obtain different possible combinations for both coefficients. The shaded area in figure N.1 represents the possible combinations for CB and CM that fulfil all the requirements (see section 5.6.4). The value for the pivoting angle (φ) lies between 30º and 45º, which explains the range of possible combinations.

Comb. v (m/s) a (m/s2)1 No mov. 0.89 0.83 Mov. 0.89 0.92 2 No mov. 0.89 0.83 Mov. 0.89 1.05 3 No mov. 0.89 0.83 Mov. 0.88 0.92 4 No mov. 0.90 0.74 Mov. 0.89 0.92 5 No mov. 0.90 0.74 Mov. 0.89 1.05 6 No mov. 0.90 0.74 Mov. 0.88 0.92

Figure N.1 needs to be interpreted as follows. The left side of the grey area indicates that for those combinations of CB and CM:

Fmov > Fcrit > Fno mov and φ = 30º

The right side of the grey area indicates that for those combinations CB and CM: Fmov > Fcrit > Fno mov and φ = 45º The down side indicates that:

Fmov = Fcrit = Fno mov and 30º ≤ φ ≤ 45º

Figure N.1 Possible combination of CB and CM for a pivoting angle between 30º and 45º and experiments using small stones

Table N-1 Velocity-acceleration combinations used for calculating CB and CM

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.10 0.20 0.30Cb

Cm

2%; small stones; 12%; small stones; 22%; small stones; 32%; small stones; 42%; small stones; 52%; small stones; 6

Appendix N


Below the grey area between the lines indicates that:

Fmov < Fcrit < Fno mov and 30º ≤ φ ≤ 45º The same can be done for large stones. The velocity-acceleration combinations following from the detailed plot in figure 5.12 will be used to obtain CB and CM. The velocity-acceleration combinations, the critical force (6.60 – 9.33x10-2N) and equation (5.6) are used to obtain different possible combinations for both coefficients. Again, the shaded area in figure N.2 represents these combinations.

Comb. v (m/s) a (m/s2)1 No mov. 1.11 1.21 Mov. 1.05 1.55 2 No mov. 1.11 1.21 Mov. 1.11 1.71 3 No mov. 1.06 1.27 Mov. 1.05 1.55

The bulk and inertia coefficient should be equal for both stone sizes. Figures N.1 and N.2 can be combined to find the area for which the combination of both coefficients satisfies this requirement. The pivoting angle is still variable. In the next figure the shaded area represents the combinations of both figures.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.10 0.20 0.30 0.40Cb

Cm

2%; large stones; 12%; large stones; 22%; large stones; 3

Figure N.2 Possible combination of CB and CM for a pivoting angle between 30º and 45º and experiments using large stones

Table N-2 Velocity-acceleration combinations used for calculating CB and CM

Appendix N


The maximum and minimum value for φ, respectively 30º and 45º, will be used to determine the coefficients. In the next figure the possible combinations for 30º are indicated in grey, the possible combinations for 45º in blue.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.00 0.10 0.20 0.30 0.40Cb

Cm

2%; small stones; 12%; small stones; 22%; small stones; 32%; small stones; 42%; small stones; 52%; small stones; 62%; large stones; 12%; large stones; 22%; large stones; 3

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.00 0.05 0.10 0.15 0.20Cb

Cm

pivoting angle 45º

pivoting angle 30º

Figure N.4 Possible combinations for the bulk and inertia coefficient for a pivoting angle of 30º and 45º and both stone sizes

Figure N.3 Possible combination of CB and CM for a pivoting angle between 30º and 45º, large and small stones

Appendix N


For both angles, more combinations can be found as was already seen in section 5.6.4 that satisfy the requirements. It is expected that when more stone sizes are investigated the possible combinations for CB and CM can also be found in these areas and will finally lead to one combination. Formula (5.11 a and b) is used to determine the combination for both coefficients, table N-4.

CB CM

pivoting angle 30º 0.10 3.92 pivoting angle 45º 0.14 5.55

The coefficients (table N-3) and the velocity-acceleration combinations (figure 5.11 and 5.12) and the Morison equation (5.2) can no be used to calculate the force acting on the stones generated by the accelerated flow. In section 5.6.4 this is done for 30º, in the next figure it is done for 45º. In the next figure, the force is plotted versus movement or no movement. It can be seen that there is also a clearly distinguishable change in movement for the critical force due to the pivoting angle of 45º.

To compare both combinations of the coefficients found in section 5.6.4, the Morison force is plotted versus the logarithmic percentage of movement. A line can be drawn through the points. The best solution for CB and CM should give the least deviation of the points from the line. It appears that both combinations show the same deviation for both large and small stones because the relation between the coefficients is for both combinations the same. This method does not encourage the use of one combination over the other. As can be seen in the next figures.

0 0.005 0.01 0.015 0.02

Generated force by accelerated flow (N)


0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

Generated force by accelerating flow (N)


Figure N.5 Movement versus no movement for small stones (left figure) and large stones (right figure) due to the force generated by the accelerated flow, calculated using the Morison equation in which CB = 0.14 and CM = 5.55 (angle 45º)

Table N-3 CB and CM for two different pivoting angles

Appendix N


The error (R2) of the points and both lines in the figures above is exactly the same. This method doe not encourage the use of one combination of CB and CM over the other. For large stones the same figures are produced. These figures show the same as the plots for small stones. The error of the points and the drawn line does not provide a way to distinguish between the coefficients.

R2 = 0.9461

0.1%

1.0%

10.0%

100.0%

0.001 0.01 0.1

Morison force (N)

Perc

enta

ge o

f mov

emen

t ,

R2 = 0.9461

0.1%

1.0%

10.0%

100.0%

0.001 0.01 0.1

Morison force (N)

Perc

enta

ge o

f mov

emen

t ,

Figure N.6 Deviation of the percentage of movement versus the Morison force in which CB = 0.10 and CM = 3.92 (angle 30º) and small stones

Figure N.7 Deviation of the percentage of movement versus the Morison force in which CB = 0.14 and CM = 5.55 (angle 45º) and small stones

Appendix N


R2 = 0.8596

0.1%

1.0%

10.0%

100.0%

0.01 0.1 1

Morison force (N)

Perc

enta

ge o

f mov

emen

t ,

R2 = 0.8594

0.1%

1.0%

10.0%

100.0%

0.01 0.1 1

Morison force (N)

Perc

enta

ge o

f mov

emen

t ,

Figure N.9 Deviation of the percentage of movement versus the Morison force in which CB = 0.14 and CM = 5.55 (angle 45º) and large stones

Figure N.8 Deviation of the percentage of movement versus the Morison force in which CB = 0.10 and CM = 3.92 (angle 30º) and large stones

the influence of flow acceleration on stone stability - core · 2016. 12. 16. · the influence of...

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