The influence of experimental conditions on cylinder drag in a current. Part I and Part II: Appendices

INFLUENCE OF EXPERIMENTAL CONDITIONS ON CYLINDER DRAG IN A CURRENT.

P.3.M. LAPIDAIRE DELFI, FEBRUARY 1981.

submergence and flume blockage on cylinder drag in a current.

4 Laboratory investigations were done at Reynolds numbers between 10 and 3*lO with a cylinder of O.033 meters in diameters to check theories.

A model for the surface disturbance was found and an important parame-ter is the wave number,k,from the linear wave theory.

Near the bottem, the drag coefficient appeared to increase; this is caused by the secondary flow that is initiated by the velocity gradient of the flow near the bottom.

For flumes that had a width less then 6 times the cylinder diameter the C0 value that was measured was higher than for flumes that were wider.

List of contents

The influence of experimental conditions on cylinder drag in a current.

.1. Introduction to the total project. i

1.1 Hydrodynamic forces, generated by waves and currents. i

1.2 Motivation of this project. 2

1.3 Language. 3

2. The influence of submergence on drag forces in a current. 4

2.1 Introduction. 4

2.2 Summary of the work on the influence of submergence

described in this report. 4

3. Theoretical background of the influence of the surface disturbance.6

3.1 Literature. 6

3.2 Model for the influence of the surface disturbance. 8 4. Experimental research on the surface disturbance influence. 14

4.1 Introduction to the tests. 14

4.2 Test cylinder. 14

4.2.1 Description. 14

4.2.2 Imperfections of the cylinder. 15

4.2.3 Extra long cylinder. 15

4.3 Instrumentation. 15

4.4 Description of the flume. 16

4.4.1 The flume proportions. 16

4.4.2 The flow capacity. 16

4.4.3 The spillway. 18

4.4.4 Mounting of the cylinder and the velocity meters. 18

4.5 Measuring procedure. 18

4.5.1 Introduction. 18

4.5.2

The measurement of

the velocity.

₂₀

4.5.3 The measurement of the forces. 21

4.5.3.1 Introduction. 21

4.5.3.2 The force displacement relation and the

natural frequenccy of the rings. 21.

4.5.3.3 Amplification. 22

4.5.3.4 Determination of the average deflection

on the KWSA. 22

4.5.3.5 Calibration of the force 23

4.5.4 Daily test procedure. 23

4.6. Boundary conditions and indexing system. ₂₄

4.7- Additional measurement. ₂₄

10.4 Mathematical description of Cpa(9) and calculation of CD

for an infinitely wide flume. 56

10.5 Assumption of the distribution of Cpa(0) and calculation

of CD for a blockage ratio unequal to zero. 60

11. Experimental research on the influence of flume blockage. 62

11.1 Introduction. 62

11.2 Test cylinder. 62

11.3 Instrumentation. 62

11.4 Description of the flume. 62

11.5 Measuring procedure. 64

11.5.1 Introduction. 64

11.5.2 Daily test procedure. 64

11.6 Boundary conditions and indexing system. 65

11.7 Measuring results. 66

12 _Compilation of the measurement results of the blockage influence.68

12.1 Introduction. 68

12.2 Tabulation of the results. 68

12.3 Graphs of the results. 69

13 Analysis of the blockage test results. 70

13.1 Introduction. ₇₀

13.2 The difference in the measured and calculated values for _{CD. 70}

13.3 The momentum equation. 71

13.3.1 Derivation and criterion. 71

13.3.2 Comments upon the momentum equation. 76 13.3.3 _{Application of the momentum equation. 79} 13.3.4 _{Results of the application of the momentum equation. 84} 13.4 Calculation of corrected CD values and discussion of

the results. ₈₈

Conclusions on the influence of blockage ₉₂

Follow up of this research on the influence of blockage. 93

Final remarks. ₉₄ List of Appendices. ₉₅ List of tables. ₉₇ List of figures. ₉₈ Literature list. 99 Symbol list. 100 The Appendices can be found in part II of this report.

Chapter 1

Hydrodynamic forces, generated by waves and currents, have often been studied; much effort has been expended throughout the world. First the hydrodynamic forces, that can be generated will be descri-bed.

If a cylinder with diameter D and lenght L is placed in a constant uni-form velocity field,with its axis perpendicular to that flow field, two forces will be generated. They are defined as the drag force,

FD, and the lift force,FL.

The drag force acts in the same direction as the velocity and is pro-portional to the kinetic energy of the undisturbed flow, times the projected area obstructing the flow, times a dimensionless coefficient

the drag coefficient.

=

where: D is the diameter of the cylinder, L is the length of the cylinder, y is the undisturbed velocity,

ris

the mass density of the water,and

CD is the experimental drag coefficient.

The lift force, acting perpendicular to both the flow direction and the cylinder is proportional to the kinetic energy of the undisturbed flow, times the projected area obstructing the flow, _{times a coefficient,now} called lift coefficient,CL. _{But this is only an amplitude,now since}

the force fluctuates sinusoidal with _{a frequency equal to the frequency} with which eddies are shed behind the cylinder.

(2Tf-)

(1.2.) where: f is the frequency with which eddies are shed in the vortex

street behind the cylinder, and

CL is the experimental lift coefficient.

If the cylinder is located in _{a time depended flow, a third force is} generated; this force is defined as the inertia force,FM, and acts in the same direction as the acceleration of the undisturbed water _flow. Its magnitude is proportional _{to this accelaration, times the mass} den-sity of the displaced water, times an experimental coefficient,CM, the inertia coefficient.

(1.3.)

where: dv is the acceleration of the undisturbed flow, and dt

CM is the experimental inertia coefficient.

Since the coefficients are dependent upon many parameters, experiments are used to find values of the coefficients for a given set of para-meters.

1.2. Motivation of this project.

Investigations into hydrodynamic forces, generated by waves and currents on circular cylinders have been carried out in the Civil Engineering Department of the Deift University of Technology since 1975.

The investigations that have been carried out were done with circular cylinders containing a measurement instrument; one of the cylinders that was used was borrowed from the Deift Hydraulics Laboratory, the other was developed by M. Dronkers, a student of the Coastal Engi-neering Croup, a part of the Civil Engineering Department.

Both cylinders however, have imperfections that were found during the research that was done with them.

The desire to eliminate those imperfections has led to the develop-ment of a new measuredevelop-ment instrudevelop-ment. This instrudevelop-ment, that also has a cylindrical shape with a diameter of 80 mm.,is still under develop-ment.

Besides the imperfections of the cylinders the results of the tests showed a large spread in data inspite of the mentioned parameters; this implies that other experimental parameters are involved such as flume width.

This led to the desire to carry out further tests with carefully cali-brated equipment in order to know the influence of !hidden! parameters. Therefor a group of people,concisting of advanced students,working on this project and a research flume to do tests at any time,are needed. Combination of the desires, described above and the knowledge on the subject, at the start of this investigation, led to the following question:

How deep and wide should the flume be, that will be needed to continue the investigation in the desired way, using the new instrument so that no uncontrolled parameters are introduced?

-2-Question 1: How deep should the measurement instrument be submerged so that the surface disturbance, that will appear when the measurements are done with a free water surface,no longer influences the forces on the instrument and the calculated coefficients ?

Question 2: Above which ratio between cylinder diameter and flume width, denoted as blockage ratio,is the influence of the

flume walls on the forces that are generated negligible ? In addition to the laboratory work on these questions also literature studies were done to investigate the theoretical background, and cal-culations, based upon theory have been made as well; all the things will be discussed more in detail in the following chapters of this re-port.

An answer on question 1 is given in chapter 2 through 8 and an answer on question 2 is handled in chapters 9 through 15.

1.3. Language.

This report is written in English due to the fact that investiagtions similar to these are of international importance.

Chapter 2

The inluence of submergence on drag forces in a current.

2. The influence of submergence on drag forces in a current. 2.1. Introduction.

Wave and current forces on cylinders have often been measured in the laboratory using vertical, surface piercing cylinders. Forces have been measured over the entire length or on an elemental length of the cylin-der. In either case, the interaction of the cylinder with the free water surface generates a surface disturbance which disturbs the pres-sure field in the fluid near the cylinder. This prespres-sure disturbance leads to distortion of the measured forces.

The implications of such a disturbing influence should be obvious. Element forces, measured too near to a free water surface will not be representative for conditions of deeper water and will yield coeffi-cients which are not generally applicable in design circumstances. This investigation, done to determine the influence of submergence on hydrodynamic forces, is done only in a constant current and only drag forces are measured; it is assumed that the influence of the surface disturbance on lift forces is not greater than on drag forces because of the symmetry of the disturbace.

During a short preliminary investiagtion the drag coefficient appeared to increase with decreasing submergence; the increase started at a level that was dependent upon the particular flow conditions. The objective was to find this level and relate it to the flow conditions.

2.2. Summary of the work on the influence of submerqemce _{described in} this report.

Investigations were made to find the CD value as a function of heigth above the bottom in a uniform flow; the tests _{were done for five} diffe-rent flow conditions and _{every flow condition was investigated twice.} Besides the laboratory investigations, calculations _{were made and} lite-rature research was made to obtain and explain the test results; those

results were tabulated and put into graphs.

Chapter 3 of this report discusses some theoretical background of this problem and a model to describe the influence of the surface disturbance is given, too. In chapter 4 the laboratory work is described and in chapter 5 the way the results are compiled is described.

The analysis of the results is given in chapter 6 and an explanation is given for the things that _{presented themselves in the analysis; the} model, for the influence of the _{surface disturbance, explained in} chap-ter 3, is compared with the measurement results to find out the appli-cability of this model.

In chapter 7 the conclusions that were drawn for this investigation are given; more general remarks are made in chapter 16.

In chapter 8 some indications are given for a follow up of the research on the influence of submergence when a surface disturbance excists.More

influence of the surface disturbance.

3. Theoretical backgroud of the influence of the surface disturbance. 3.1. Literature.

The book 'Fluid dynamic drag" by S.F.. Hoerner, an excellent book on many aspects of drag, summarizes the results of many tests that are done with a circular cylinder that pierces the free water surface. These

tests are done however with a cylinder that did not extend over the entire flume depth; the cylinder extended into the water only over a length, h. Water was piling up at the upstream side of the circular cylinder and a hollow was formed at the downstream side of the cylinder. The Froude number, denoted as

h was defined as the undisturbed

velocity dividéd by the root of the lenght h times the acceleration of gravity, g.

where: g is the acceleration of gravity, and

h is the length of the submerged part of the cylinder.

The result of these tests is that if the CD values are calculated using equation 1.1 and are plotted afterwards against Fh , then the line given by equation 3.2 fits the data for higher Froude numbers; this can be seen on fig. 1.

c

+

(3.2.)

o

0 2 6 S Io

Fig. 1. _{Drag coefficient of surface piercing circular} cylinders as function of Froude number, _Fh

(from Hoerner )

Although these results are interesting they do not answer the question at hand, since the results include an influence of the end condition of the cylinder.

(3.1.)

6

P.d .Io t'. IO' wove and o, spray

o'.d 2

- 22

A second approach of the problem at hand was made by W.W. Massie and based upon the paper "The wove resistance of a ship" by J.H. Mitchell. In this paper of Mitchell, formulas are derived that describe the total wave resistance of a ship moving through still water at a given speed. Mitchell treated this problem as that of a stationary ship in an infi-nite wide and constant flow field with an undisturbed velocity equal to the given ship speed.

Massie applied these formulas for an infinite long vertical circular cylinder and examined the wave resistance force per unit length as a function of submergence, z z is measured from the still water level, with its axis positive downwards ). Massie's work resulted in a method to find the limit value to which CD decreased as the submergence in-creased (this work is given in Appendix 1).

The basis for this was the idea that the force, FD , on a element of a cylinder can be expressed as the sum of a "normal" drag force,

6 and a wave induced force, , Each of these forces can be reduced to

a drag coefficient by dividing by ½.?.v2.D.L , yielding two coefficients,

and respectively, whose sum is Of these coefficients only CD will be known from the measured drag. However, only

,

attributable to the surface wave, is depended on submergence.

Massie found that the coefficient _C can be related linearly to a sub-mergence parameter, S, (see fig 2. ) which is a function of the follo-wing variables;

The submergence Froude number: _(3.3.)

and

F

where: z is the submergence, measured from the still water level

CD

the diameter Froude number

C

_Cb+Cw

CD+ S

S (Ed, F5)

Fig. 2. CD as a function of the submergence parameter S.

For the function for S, Massie found

t

-k0 ( /)

-

e

k0 (L)

where: t is the reciprocal of the square of F, and

K is the modified Bessel function of the order zero, o(u)

evaluated at u.

To make this function of S a bit more convienient to apply, Massie expressed 3.8 as into:

s

=

_I

dL

_(3.9.)

and plotted a graph of the function f(F5). This graph of _{f(F5) is also} given in Appendix 1.

3.2. Model for the influence of the surface disturbance.

Since both Hoerner and Mitchell _{are suggesting that the disturbance of} the free water surface around the cylinder conforms to a wave, one is inclined to think of a linear short surface wave In the preliminary investigation the CD value _{decreased to a limit value as the submergence} increased. Massie suggested that the the surface wave influence

de-creased to zero as the submergence inde-creased.

This conformity reinforces the idea of describinçj the surface disturbance as a short surface wave.

An extra reason to think of a short surface wave is the fact that the formula for the pressure under a short wave in the linear wave theory has the form given in equation 3.10 if deep water is assumed.

(3.8.)

-8-In rormula

C=t ±w

(3.5.) with

oc.S

_(3.6.) and

s

(

Id)

(3.7.)

Ita'

where: H is the wave heigth,

k is the wave number, equal to

k

Ais

the wave length

x is the horizontal coordinate,'

z' is the vertical coordinate measured from _{still water level} + down),

is the circular frequency, equal to ) _{j, and} T is the wave period.

where: _{is the force on the layer dz, and}

C _{is a constant dependent}

on the body shape. If one would move the body _{along with the waves, wíth}

a speed equal to the wave speed, the force _{on the thin layer, caused by this wave} would be constant since cos(kx-t) _{would be independent of t since the} phase at any x is now constant. _{Now one can write for}

on a thin layer:

(3.10.)

where: C' is a constant dependent on the body shape. Looking closer at equation _{3.12 it is easy to understand}

that on a layer with a thickness dz _{decreases if the layer is}

more submerged. This is caused by the _{exponential part in the formula.}

How can all this be _{applied here ?} Here a circular cylinder _{is put into}

a constant flow; since the cylin-der pierces the free _{water surface,}

a disturbance of the water surface is generated. This disturbance has a constant position _{around the} cylin-der.

FwCc

I

(3.12.)

The first term on the rigth side of equation 3.10 is the term that gives the hydrostatic pressure, the second term gives _{the pressure}

fluc-tuation caused by the waves. The magnitude _{of this second term} decrea-ses with increase of z.

If a body is placed into the waves, a force will result from the pressure distribution, only because of the fact that the body has dimensions in the x- direction and the progress in time. Thus _{a thin} layer dz of the body, z meter under _{the surface will experience a force,} since the x coordinate changes along _{the body surface. One could write} this force on the thin layer as:

and

A-

_-

Z.?)

7L

Combination of equation 3.14 and 3.15 gives:

À

=

c2-.

=

In the short wave theory the formula for the wave number can àlso be found.It is given in 3.17.

(3.15.)

(3.16.)

-

10

-If one would move along with the water however, one would see a cylin-der approaching with a speed equal to the current speed and a distur-bance of the water surface, approaching with that same speed.

If one considers the disturbance as a wave, then the wave speed is equal to the current speed.

Now the question is: Can the pressure caused by the wave around the cylinder be described by equation 3.12 ? The answer is yes, because one knows from Fluid Mechanics that is does not make any difference whether the observer is moving along with the water or the observer is placed on one specific place.

The only thing that will have to change is C; some other influences are hidden in this coefficient as well. For instance, the fact that the wave is not constant in form in the y direction, or better, the problem

at hand is not two dimensional.

Since those influences are brought into the calculation by the con-stant C' , the idea just presented seems to be a good model.

How can one check if this model fits the physical phenemena ? The wave 'moves' with a speed equal to the current speed, thus:

(3.13.)

In the short wave theory for deep water the formulas for the wave speed and the wave length as function of the wave period can be found.

They are:

=

(3.14.)

ì:

I

v2-xzî) V

This equation for the wave number can be substituted into equation 3.12; this leads to:

This is the force generated by the surface wave and on a layer with thick-ness dx and a distance z under the surface. There is another force

acting however,too; this will be the normal drag fprce as described in equation Ll ( now L is equal to dz and is denoted as ).

This formula for is repeated outhere, with C as drag coefficient.

The term v.LvI is written as the square of y since the velocity is always positive.

The equation for the total resulting force is:

/

-- vL.

*

_.

d' (3.21.)

If one divides this force, and its components by ½.F.v .D.dz, the result will be drag coefficients. In formula:

=

Fo

(3.22.)

C.D' = c1

I _(3.23.) and

cw=

/

-f

e

i/pvL. 2

on dz,

is the drag coefficient caused _{by the current, and} is the drag coefficient, that results from the wave.

(3.18.) (3.24.)

iul

=

yZ

.da'

(3.19.)

7j'_ .Lf.va7.cI.da

(3.20.)

2'

CD

/

c0e

(3.25.)

If one looks at equation 3.25 one can see that if z increases, which means that the submergence increases, CD will decrease to the minimum value C and this decrease is exponential.

How can this theoretical model be verified ?

This model can be verified by measuring CD values over the entire

flume depth, and plot CD versus z, the submergence; the submergence axis has the same zero level and thus:

In the graph of CD versus z the value to which CD decreases, C

with increasing z has to be choosen. Then C CD - C can be plotted versus z on semi-logaritmic paper and by a root mean square line

through the data one can find a coefficient km but now is a value that fits the measurement data.

This procedure is repeated for different flow conditions; for all these conditions k is calculated.

Than a value of km found for a flow condition is compared with k, cal-culated with equation 3.18, using a value for y that is valid for that flow condition. If k and km are equal or nearly equal, the theoretical model is acceptable.

Another thing that can be done is to make a graph of k versus y and km versus y. The graph of k versus y shows a relation, which is

lo-gical since equation 3.18 gives a _{relation between those two} para-meters; if the model is valid the plot of km versus y should be the

same or nearly the same as of k versus y.

Experiments to verify this model are described in the next chapter.

(3.26.)

-Ch2pter 4

Experimental research on the surface disturbance influence.

C)

C)

L-I

4. Experimental research on the surface disturbance influence. 4.1. Introduction to the tests.

As already explained these experiments are concentrated on drag forces. In these experiments, the water depth and the flow velocity were varied as well as the elevation of the measurement instrument in the cylinder. For every flow condition the value of _CD was determined for levels over the whole depth of the flume. More about the boundary conditions is written in part 4.7.

The experiments are done in the Laboratory of Fluid Mechanics at the Department of Civil Engineering and the main instrument, the test cylinder, was borrowed from the Deift Hydraulics Laboratory.

4.2. Test cylinder. 4.2.1. Description.

As already written the cylinder that was used, was owned by the Deift Hydraulics Laboratory. This cylinder has a circular shape, a diameter of 33.33 mm. and a length of 800 mm. It is drawn in fig 3.

electric wire

electric wire

ETAIL A

measuring ring

strain gauges

s p rings

r

L

detail of element (dimensions in mm. and scale 1:1 ). In this slender pole there are three measurement e]1.ements,spaced

200 mm. apart and each with an heigth of 20 mm. These elements are made of synthetic material and are fixed to the pole, irside of it, by two springs in such a way that makes translation perpendicular to the pole axis possible in one direction _only.

cylinder axis

r'

4/ ' J

displacement /

(not in scale

₃₃₃₃

k

Fig. 3 _{Measuring cylinder (dimensions in mm. and scale 1:10 ) and} k.

14

-measuring

r i ng Cr)

,

ç -, C) _C) nng C) c'. - rrng

gc,

_OD

ring

OD

r-Strain gauges are mounted on the springs and covered with wax to protect them against water; those strain gauges make it possible to measure the displacement of the element. This displacement is a measure for the total force on the element acting in the direction of the

dis-placement. When the springs are bent the strain gauges will change in length and thus will change their electrical resistance, which, in turn, can be measured by connecting the strain gauges via a Wheatstone bridge.

4.2.2. Imperfections of the cylinder.

This cylinder has two major imperfections of which one is not so impor-tant in the tests that are done in these investigations.

The first one is that the cylinder is a bit flexible in bending; it is possible to bend the cylinder a bit without using very big forces. This bending can cause one of the measurement rings to stick. This possibility of sticking is even greater when the pole is not totaly clamped at both ends. This imperfection did not give complications during the experiments, however.

The second imperfection, which possibly influences the results, is the fact that the cylinder is not water thight. At some places water can flow in and out of the cylinder especially at the end of the measurement elements; this might in turn influence the boundary layer around the cylinder and the measurement elements, thus influencing the generated

force.

4.2.3. Extra lonq cylinder.

To make the cylinder longer, either to fix it at the bottom or the top of the flume, some pieces of cylinder with the same diameter were already excisting when the instrument was borrowed; in the workshop of the Laboratory of Fluid Mechanics _{some extra cylinders were made, which} made it possible to have a constant cylinder diameter over the entire flume depth, so that no other extra influences were introduced.

4.3. Instrumentation.

The instrumentation consisted of: carrier wave amplifier.

velocity meters. intergrators. x-t pen recorder point gauge.

16

-The carrier wave amplifier that was used was a Hottingen Baldwin type KWSA 6T-5, with six channels, of which three were used.

During the tests a full Wheatstone bridge was used; the KWS ampli-fier, further in this report called KWSA, gave the translation of the measurement ring, from its neutral position, in scale parts. Before the start of the measurements the KWSA was balanced.

The scale of the KWSA is from ±100 to -100 scale parts, correspon-ding to an output signal of +2 Volts to -2 Volts.

The amplifiaction of the incomming signal can be changed; this amplification was choosen as high as possible to achieve the greatest accuracy.

Ott current meters were used to measure the velocities. It is

assu-med that the reader is familiar with this type of current meter. -e- The output voltage of the KWSA was fed into a linear intergrator,

that performs an intergration with respect to time during a choosen time interval; the average signal over that interval is determined by dividing the intergrator result by the time interval.

An x-t penrecorder was used to make it possible to examine the force signal coming from the KWSA. It was also used to determine the

natural frequency of the measurement rings.

It is assumed that the reader is familiar with the use of a point gauge.

4.4. Description of the flume. 4.4.1. The flume proportions.

The biggest flume in the Laboratory of Fluid Mechanics, the large research flume, was used for these experiments. This flume is 45 meters long,

0.8 meters wide and has a depth of 1 _{meter. 0f the 45 meters length} part is used for an irregular wave generator, sand lifts and for the flow inlet and outlet. The actual _{open flume length is 32.90 meters}

( see fig. 4 ).

4.4.2. The flow capacity.

The flow of the flume is regulated by a valve in the inlet. The water is drawn from a constant head _{supply system. The maximum flow capacity} of the flume is 0.5 m3/s. When there are obstacles in the flume that capacity will not be reached.

J__---\_J

/ ,P777

/

_Valve

7/

32.90 Fit]. 4

The large research flume at the Laboratory of Fluid Mechanics. scale 1:50

dimensions in meters. Wave

Inlet

Wire

Sand

Plastic

Sand

generator

max. flow

capacity

mesh.

screens

lift

and

sheet on

top of

lift

0.5

m3/

sec.

honeycomb

_of

_coruated

plastic

pLates

flow

18

-4.4.3. The spil].way.

The spillway was used to create a constant water level and a good velo-city profile in the flume; it was made by stoplogs of different heigths,

that could be placed in rabbets at the end of the flume.

The spillway was not ventilated, but in all tests there was a consistent flow pattern near it. Ventilation of the spillway might have increased the accuracy a bit, even so.

4.4.4. Mounting of the cylinder and the velocity meters.

The cylinder was fixed in the flume 13.80 meters upstream of the spill-way. The top of the cylinder was fixed to the flume with an aluminum

beam and some clamp screws (see fig. 5 ).

Through the bottom of the flume a construction was made to make it possible to move the cylinder up and down while still maintaining a constant diameter over the entire flume depth. By this construction, a measurement ring could be placed at any desired elevation in the flume. The construction consisted of a watertight cylinder fixed in the

bottom to make a well; inside the well some rings, with an inside diameter of 34.00 mm. were made, to support the instrumented cylinder in its

position. At the bottom _{level of the flume a flat rinq, with an inside} diameter of 34.00 mm. was put on top of the well; this was done to be

sure that the velocity profile around the cylinder was not disturbed by the well.

By loosening the clamp at the top of the flume the cylinder could be moved into a desired vertical position after which the clamp was screwed

again.

The Ott current meters were fixed on beams at the top of the flume by a construction that made it possible to move the meters up and down rather easily; the point gauge was fixed in the same way, too.

4.5. Measuring procedure. 4.5.1. Introduction.

To calculate a drag coefficient it is necessary to have either instanta-neous force and velocity signals, or an average velocity and an average force over a certain period. In both _{cases the velocity and the force} should be measured simultaneously.

Other methods to calculate the drag _{coefficient are discussed in} chap-ter 6.

glassi

,30 33.3

drain

00

clamp screws

N

plastic ring

/

30

trovidu r

pvc pipe

Fig. 5

aluminum

1000

Mounting of the cylinder in

the flume and bottom construction. scale 1:10

dimensions in mm.

3

In this investigation Lhe velocity and force were measured 2S an average over a certain period. This was done because of the fact that the velo-city and the force were not measured atthe same location so phase dif-ferences caused by small variations in the flow were unavoidable. Also Ott current meters which register only by a counter are not suited to giving instantaneous values.

The period over which the average was made was 25 seconds. This period was choosen to be long enough to give acceptable average values and

short enough to allow a reasonable time scedule.

4.5.2. The measurement of the velocity.

The undisturbed velocity has to be substituted into the formula to de-termine the drag coefficient. If the velocity is measured with _{a current} meter placed in the flume, in front of the cylinder, this instrument will, in general, disturb the flow and might influence the forces on the

cylinder. One would like to have a measuring procedure that does not disturbe the flow in front of the cylinder.

Using a Laser Doppler current meter is a good possiblity. However, since at the time the tests were done the Laser Doppler meter was not fully operatinal it was not used. Another reason why it was not used was the desire to do the submergence tests and the experiments on blockage with the same measuring procedure. In the experiments on flume width,wooden sheets were used to narrow the flume; this made it impossible to use a Laser Doppler current meter.

Before each test series time averaged velocities, _{in front of and} behind the cylinder, were measured; this was done 3 meters in front of

and 3 meters behind the cylinder. The velocity ratio, defined as the velocity in front divided by the velocity behind, and denoted as br was determined at given levels; between those levels _{the ratio was}

de-termined by graphical interpolation.

During the force tests, the velocity behind the cylinder was measured and used to calculate the velocity in front, by multiplying the velocity measured behind, without a meter in front of the cylinder in the flume, by the velocity ratio valid for that level on which the velocity was measured.

A short investigation was made to determine the error in the velocity that could result from this procedure. It turned out that the error was always smaller than 1 %.

The velocities themselves were calculated using the indicated frequency and the calibration formula _{valid for that specific meter.}

-4.5.3. The measurement of the forces. 4.5.3.1. Introduction.

As already explained in 4.3. the KWSA indicated a translation of the measuring ring in scale parts. The deflection from the neutral position

on the KWSA was dependent upon the amplification of the signal. To find the force that created the displacement of the ring the relation between translation and that force had to be found for all three rings. The ca-libration was done before the experiments were started and at that time also the natural frequency of the rings in air and in water was deter -mined.

4.5.3.2. The force-displacement relation and the natural frequency of the rings.

The calibration of the rings was done while the cylinder axis was hori-zontal and the cylinder was oriented in such a way that the rings dis-placement direction was also horizontal; in that position the KWSA was balanced. When that was done the cylinder was rotated 90 degrees about its axis to allow displacement of the rings by gravity so that the ring translated under its own weight. The deflection on the KWSA was recorded. The weights of the rings, as determined during earlier tests were:

After the deflection on the KWSA was noted together with the force that had caused that deflection, extra weight was hung on the ring and

again the deflection was noted. This was done for several weights. Next the cylinder was turned 180 degrees about its axis and the same procedure was repeated.

All data of one ring were used to determine a relation between force and deflection on the KWSA for that ring. Thís was done by finding the least square line through the data.

The calibration was done at _{an amplification of 2000; the results were:}

where: F is in Newtons and the KWSA deflection is in scale parts.

The natural frequency was determined by giving the ring a static trans-lotion and than letting it vibrate freely.

ring 1 : 21.0 * 10_2 _N

ring 2 21.0 * io_2 _N ring 3 : 21.0 * io_2 N

ring 1 : F 6.28 * l0 * deflection on KWSA ring 2 : F z 6.17 * l0 * deflection on KWSA ring 3 F z 6.23 * 1O _{* deflection on KWSA}

4.5.3.3. Amplification.

The calibration of the inq5 was done with an amplification of 2000; during the tests other amplifications were used to increase the output

signal. To find the forces, the influence of the other amplifications had to be brought into calculation.

Since the amplification was linear, the influence could be brought in by dividing the amplification used during a test by the one used during the calibration.

Now the force can be found using the following formula:

ring 1 : F 6.28 * l0 * test amplification * deflection on KWSA

2000

The x-t recorder was used to register the deflection as function of time. Afterwards the number of periods in the registration was counted over a certain time. By dividing the number of periods by the time the

natural frequency was found. The natural frequency of each ring was:

in water in air ring i 17.5 Hz 23.3 Hz ring 2 17.5 Hz 22.3 Hz ring 3 16.5 Hz 21.8 Hz * _l0 * * 2000

4.5.3.4. Determination of the averaqe deflection on the KWSA.

Since the deflection of the KWSA was not constant in time, it was not possible to make a reading from the KWSA directly. Therefor the signal was

electronicly intergrated for 25 seconds and the result was recorded. To get the average deflection of the KWSA in scale parts, as needed in the formula to calculate the force, some additional calibrations were done first. Before the test series were started the maximum signal of the KWSA, 2 Volts, was put into the intergrator for 25 seconds and was

recorded.

Since a signal of 2 Volts corresponded to 100 scale parts, the average deflection during the series could be found with the following formula: deflection on KWSA intergration result KV/SA test siqnal * 100

intergration result KV/SA at 2 Volts

22 -ring 2 ring 3 : : F F 6.17 6.23 * l0 * test amplification * deflection in KWSA 2000 test amplification

4.5.3.5. íalibration of the force.

Substitution of the expression for the deflection on the KWSA, from 4.5.3.4 into the formula for the force given in 4.5.3.3 leads to; ring i

F 6.28*i0* test ampi. * interqr. result KWSA test signal * 100 2000 intergr. result KWSA at 2 Volts

ring 2

F = 6.l7*l0* test ampi. * interqr. result KWSA test signai * 100 2000 intergr. result KWSA at 2 Volts

ring 3

F 6.23*iO* test ampi. * intergr. result KWSA test signal * 100 2000 intergr. result KWSA at 2 Volts

4.5.4. Daily test procedure.

The test procedure that was followed each day is described below; the things that had to be done are put down in sequential order.

Procedure

close the outlet.

open the inlet, fill the flume and shut the inlet again when the water level is above the highest ring.

let the water come to rest and lubricate the Ott current meters in the meantime.

balance the KWSA, with the cylinder in water at rest ( three rings). intergrate the 2 Volts signal of the KWSA over 25 seconds (three rings). re-open the inlet and the outlet and bring the valve to the desired position.

install the spillway with the desired heigth. determine the velocity ratio at certain levels.

remove the Ott current meter from in front of the cylinder.

measure the water depth at the flume wall adjecent to the cylinder. li. make additional measurement _{on the water surface ( see section 4.7).} 12. conduct the test series;

bring the Ott current meter and one of the rings to a desired ela-vation above the flume bottom.

intergrate the force signal and count the Ott current meter signai over the same period of 25 seconds and record the results; this is done 5 times.

move the Ott current meter and the cylinder into a new position as described under 12a and go on with 12b.

close the inlet and the outlet, after all the positions, in which measurementsare desired are completed.

take out the spillway

move the cylinder downwards into a position in which the highest ring is under the water surface.

let the water come to rest.

intergrate the 2 Volts signal for 25 seconds again.

note the zero shift on the three channels of the KWSA, when the rings are still under water.

open the outlet draining the flume.

4.6. Boundary conditions and indexing system. The boundary conditions that could be varied were: - spiliway height,

- valve position, and

- heights at which measurements were done.

The spiliway height was : 0.23 meters, 0.32 meters and 0.40 meters.

The valve position was : 50 % , 60 % and 75 % ( relative to full open). The heights at which measurements were done were every 0.01 m. over the total water depth of the flow in every first test series done with a combination of valve position and spillway height and every 0.02 m. in the second series with exception of the upper 0.10 m. of the flow where it was again 0.01 meters.

Only one valve position and spillway height were used on a given day; all tests on a given day were indexed with a letter followed by a sequence nunter on that day. Further, a single test ( for example B7 ) represents values determined from 5 measurements, each made during a period of 25 seconds.

The constant parameters and the index code for each specific day are listed in table 1 along with the average velocities and depths.

4.7. Additional measurement.

One additonal measurement was done for every set of boundary conditions. The measurement of the surface elevation in front of the cylinder and the drawndown behind the cylinder with the flume axis as section line was done with the point gauge. The test series that included this

addi-tional measurement are: A, C, F, H and L; the results are plotted in Appendix 2.

-Table i Index code, constant and related parameters for each day. (for surface disturbance tests)

4.8. Measurement results.

After the measurements were done the velocities, forces, and with their values, the CD coefficients and the Reynolds numbers were calculated. Only the average results ( from 5 individual tests ) are listed in this report in Appendix 4. The more complete data are kept by the Coastal Engineering Group at the Department of Civil Engineering.

date main spiliway valve water average

code height position depth velocity

800310 A 0.32 60 0.610 0.534 800311 B .32 60 .615 .538 800312 C .32 75 .705 .704 800312 D .32 75 .700 .695 800409 F .40 60 .690 .464 800410 G .40 60 .710 .444 800414 H .40 50 .630 .367 800415 1 .40 50 .630 .367 800416 K .23 60 .515 .611 800417 L .23 60 .525 .622

(m)

(o,

_(m)

_(m/s)

Chapter 5

Compilation of the measurement results for the submergence influence.

5. Compilation of the measurement. results of submerqence influence. 5.1. Introduction.

As already explained in chapter 4 the CD values were calculated using the velocity and force average over a time period of 25 seconds. Since 5 tests were done with the same boundary conditions, it was possible to calculate an average CD value over those five tests.

Calculation of the average CD value could be done in two ways;

- take the time average of the force and the velocity after adding the results of the five tests of 25 seconds, over 125 seconds, denoted as CD12S. This is equivalent to one sirìgle measureient lasting 125 seconds. - take the group average over the CD values of the five tests of 25

seconds, denoted as

D' using the folowing formula:

ç

=

-

/

(/z)

(5.1.)

In the compilation of the results CD12S was used; calculation of the average values over the five tests using both techniques was done for some tests. The resulting values were almost the same, which is not supri-sing since values of force and velocity remain nearly constant during the five tests.

5.2. Velocity profiles.

As already explained in 4.5.2. before the force tests series,the velocities behind and in front of the cylinder were measured at certain levels.

Those values were plotted versus water depth and a graphical interpolation was made. Also the velocity ratio was plotted versus water depth after it was determined for the given levels, and graphically interpolated. Graphs of the velocity profiles and the velocity ratio versus height above the bottom can be found in Appendix 3.

5.3. Tabulation of the results.

In Appendix 4, tables are presented _{in which the values of the following} items can be found for each test series:

- the main index of the total series,

- the spiliway height used for the series, and - the valve position used for the series

are listed at the top of the Appendix sheet, and below that - the index,

- the measurement ring that was used,

- the height of the middle of the ring above the bottom in meters ( denoted as h ),

- the submergence of the middle of the ring in meters (denoted as z _), - the velocity ratio ( denoted as Vf/Vb ,

- the 125 seconds average of the velocity in front of the cylinder in me-ters per second ( denoted as Vf ),

- the 125 seconds average of the force in Newtons ( denoted as FD ,

- the 125 seconds average CD value ( denoted as CD125 ) , and

- the Reynolds number, based upon cylinder diameter and the 125 seconds average velocity

are listed in sequence for each testO

In the tables the results of the series are listed in order of increasing height above the bottom, which is the same as in order of decreasing submergence.

As an example one line, A20, is taken out of the Appendix _{and described} further here.

This test was done in series A, _{in which a spiliway height of 0.32 meters} and a valve position of 60 % were used.

The test was done with ring 2, and the distance between the bottom and the middle of the ring was 0.27 meters; _{since the water depth was 0.61} meters this gives a submergence of 0.34 meters.

The velocity ratio for this height above the bottom was found by graphical interpolation and can be found in Appendix 3. The result, 1.08 is noted. The average velocity over 125 _{seconds determined using the results of the} 5 tests of 25 seconds, _{was 0.561 meters per second. This was done for} the force, too, and the result _{of that measurement was 0.114 Newtons.} The value of

Dl25 was calculated after that,using the usual formula ( as given below ) _{and the Reynolds number was calculated with the other} formula given below

F0

'/z.p.vÇ. z. dL

where; _{dEi. is the height of the}

measuring ring, D _{is the diameter of the ring, and}

) is the kinematic viscosity of _{the water.}

(5.2.)

(5.3.)

-5.4. Graphs of the results.

The resulting 0D125 values, that were tabulated in Appendix 4 were plot-ted in three ways.

- First, CD12S was plotted versus height above the botlom over the entire water depth.

- Second, 0Dl25 was plotted versus height above the bottom again, now only for the lowest 0.20 meters of the water depth using an enlarged scale.

- Third 0Dl25 was plotted against submergence over the upper 0.20 meters of the water depth,also using an enlarged scale.

Since two test series were done with one set of boundary conditions, the results of both the series were plotted in one graph, with exception of the graph of 0D125 versus height above the bottom over the entire depth, where the series were plotted seperate.

Chapter 6

Analysis of the submergence test results.

6. Analysis of the submergence test results.

6.1. Introduction.

The graphs in Appendix 5, CDJ25 versus height above the bottom, h, give the following impressions;

- In the middle part of the flow the

Dl25 value is essentially constant; the surface wave or the bottom seem to have no influence on these values. - Although not expected at first, the

D125 values near the bottom

are increasing first with decreasing height above the bottom, but close to the bottom the values are decreasing; an explanation for this path of the lines near the bottom is given in part 6.2.

- Near the surface the

Dl25 values increase first with increase of height above the bottom; very close to the water surface the values are decrea-sing again; an explanation for this path of the line is given in part 6.3 and the model for the influence of the surface disturbance is verified there, too.

The graphs in Appendix 6 and 7, are in fact, just different presentations of the graphs in Appendix 5; the two series done with the same flow con-ditions are presented in one graph. Additional information now presents itself:

- The lines for two series, done with the same flow conditions show a systematic deviation. An explanation for this systematic variation is given in part 6.4.

6.2. CDÌ2S versus h near the bottom; the influence of secondary flow. 6.2.1. Introduction.

In the graphical presentation of

D125 versus the height above the bottom, h, it appears that the CD value increased with decreasing h ( except

for very small values of h, then CD decrased again ).

Since the velocity is the only parameter that _{changes when the height} above the bottom is decreasing, this change in velocity has _{to be the} cause of the increase in CD values.

In the literature some experiments andtheir results are described that support this idea. Since those experiments on drag have been done with another method to find values of _{CD some methods that can be used to}

lind CD values are listed in 6.2.2. The question that has to be answered is:

Has the flow pattern near, but not at, the level at which the measure-ment of CD is done influence on the value of CD.

These two types of velocity fields are:

- a velocity field without velocity gradients in the z- direction.

- a velocity field with velocity gradients in the z- direction;

this field is called a velocity gradient flow and sometimes shear flow. The answer to the question is given in 6.2.3 for the first velocity field and in 6.2.4 for the second.

In 6.2.5 the results of measurements done with a velocity gradient flow, are given and discussed; in 6.2.6 an interpretation is given of the results from 6.2.5 for the investigation described in chapter 4.

6.2.2. Methods to find values by experiments.

Four methods that can be used to find values by experiments are described briefly

Method 1.

The force, , on an elemental part of the cylinder, with length dL, is

measu-red together with the undisturbed velocity; the velocity is measumeasu-red at the same level as where the middle of the elemental cylinder part is. Then the local value is found with the formula given by equation 6.1.

(6.1.)

This method is the one that is used in the experiments that are described in chapter 4.

Method 2.

The total force on the whole cylinder,

Dl' is measured together with the average velocity of the flow. Now the value can be calculated with the formula given by equation 6.2.

(6.2.)

-where: L is the submerged lenght of the cylinder and thus equal to the water depth.

-

30

-Method 3.

The pressure distribution, p(9) , around the cylinder is measured at a

certain level; at this level the undisturbed velocity is measured as well. Now the force on the cylinder, per unit length is equal to:

where:

D the drag force per unit length r is the radius of the cylinder,

g is the angle between the point where the pressure is measured and the flow axis, and

p(g) is the local pressure on the cylinder.

If p(0) is only known in a discreet number of points around the cylinder surface the force can be found by equation 6.4.

ÍD =

p(9).(Oil.

(6.4.)

where: 8 is the angle between the adjecent measurement points on

cylinder =

is determined either using equation 6.3 or 6.4, the local CD value can be found with equation 6.5.

l'ö

CD

-

_h/a.p.VL.zr

Method 4

When it is possible to measure the pressure coefficient, C() , defined as the difference between local pressure and the undisturbed pressure, divided by ½.J.v2 , then the CD can be found by using equation 6.7;

in equation 6.6 the formula for C(o) is given first. p(-e) _Pc,

(6.5.)

(6.6.)

(6.7.)

When C(Q) is only known in a discreet number of points on the cylinder surface then CD can be determined analogous to the method as given in equation 6.4 for

D yielding:

(o).

(6.8.)

6.2.3. Correlation between CD at a level and the flow pattern near that level _{in n velocity field without velocity gradients in z direction.} When a circular cylinder is put into a velocity field without velocity

gradients it is very easy to understand that it does not matter at which level CD is determined, since the data, necessary to calculate CD values are the same on all levels.

-

32

-This was already found by Hoerner; in his work it is stated that there is a perfect correlation between the CD value at a level and the flow pattern in the adjecent levels.

A more mathematic formulation is:

Since the flow has no velocity gradients in the z direction, all partial differentials to z are zero thus there will be a perfect correlation between the CD value and the flow pattern around the level at which CD is determined.

6.2.4. Correlation between CD at a level arid the flow pattern near that level level in a velocity gradient flow.

When a cylinder is placed in a velocity gradient flow there is a correla-tion between CD at a level and the flow pattern near that level, but the correlation is not perfect.

F.D. tasch & W.L. Moore and later C. Dalton & F.D. Masch found this by experiments, that will be described in 6.2.5 ; only the cause of the not

perfect correlation wil be discussed here and some parameters that might describe the correlation are given.

When a vertical cylinder is placed into a flow the stagnation point is defined as: the _{point where the horizontal velocity is equal to zero.} The stagnation pressure is equal to the mass density of the fluid times the square of the undisturbed velocity,at the level the pressure is cal-culated or measured, divided by two; in formula this is: ½.f'.v2

Further, the vertical line through the stagnation points _{on different} levels is defined as the stagnation line.

When the flow field in which the cylinder is place is a velocity gradient flow it is easy to understand that the stagnation pressure changes along the stagnation line since the undisturbed velocity changes with changing heigth above the bottom.

In fig 6 and 7 two velocity gradient flows _{are given together with the} stagnation pressure versus heigth above the bottom.

h Cm)

U

...1

T

F ₂

V(m/s)

(N/rn

Fiq. 6 _{Velocity gradient flow and stagnation pressure versus height above} the bottom.

V ( m/s)

Fig. 7 Velocity gradient flow and stagnation pressure versus heigth above the bottom.

In both fig 6 and 7 the graph for the stagnation pressure versus heigth above the bottom shows an increase in the stagnation pressure with

in-crease in h; also it can be seen that the the distibution of the pressure is deviating from the hydrostatic pressure. This deviation will cause a

secondary flow along the stagnation line towards the area of lower pressure. The secundary flow will deform into a horse shoe vortex when it meets the bottom; this horse shoe vortex can be seen in fig 8.

C.d JocT'f

T

FLGW DEPTH d J-Fig. 8 HCÑSESHOE P4 PAT'O/ L.WP

Horse shoe vortex near the bottom.

On the back of the cylinder a similar phenomena appears, but only in the opposite direction, because the pressure reduction in the wake is

propor-tional to the pressure in the seperation point. This pressure is negative and just like the stagnation pressure, depended upon the square of the undisturbed velocity. Thus for a value of the velocity on a certain level the pressure in the wake is smaller than the pressure in the wake a little bit closer to the bottom.

As a consequence of the velocity gradient flow, _{a secondary flow will be} generated on the stagnation line and in the wake behind the cylinder; in front towards the cylinder end where the undisturbed velocity is the lo-west ans in the wake towards the end where the undisturbed velocity is the highest.

p4pV

The secondary flow will influence the pressure that would be round if one would try to find C values by using method 4, given in 6.2.2 and thus the resulting

D value.

One of the parameters that seems to be important is the velocity gradient since that causes the secondary flow. Thus v/4z will be a parameter. Masch & Moore and Dalton & Masch found this too in their experiments, that will be discussed in the next part. They used a different notation

however; instead of nv/Az they used ¿v/iy, since their vertical axis was defined as the y-axis.

6.2.5. The results of Masch & Moore and Dalton & Masch.

Masch and Moore used method 4 to find C values and a velocity profile like that drawn in fig 6. They found two important things.

The first one was that there were pressure differences along the cylinder if that was placed in the velocity gradient flow. They showed that in the graph that is presented here as fig 9. In this figure a higher number of the lines means that the distance from the piezometer to the flume bottom was larger than for the distance between the bottom and a piezometer with _a lower number. The distance between to adjecent piezometer was constant.

.30 .20 00 360° 34 -NUMBERS INDICATE ON CURVES

PIE ZOMETER Y V -Q-O077

7

i

I

_/>--

i,

t i

4Q0 80° 20° 60° 200° 2400 2800 320°

ANGLE IN CEGREES

Fig. 9 Pressure distribution along the cylinder. ( accordong to Masch & Moore )

What Masch and Moore found, too, _{was the influence of the parameter}

v/y.D/v

; this parameter is dimensionless and represents the velocity

gradient times the cylinder diameter divivded by the average velocity. Masch & Moore found that if the value of _{.v/y.D/v increased the value}

of ¿C/y increased too. This

can be seen on fig. 10.

Dalton & Masch also used method 4 to find

C values and a velocity profile

given by fig 6. They found _{a stronger increase in CD values ( which can} be seen by comparing the maximum measured values of CD from fig 10 and

.6 .4 .2 .0 'B

p-.6 H

s.

LL .0 IL

o

8-I

H O-tuJ

o.

.2 .0 .8 220 2.00 1.80 l.60 .40 .20 CO LOO .80 2.40 S. 220 200 180 1.60 -L40 ¡.20 I 00 -30 O 20

T-1

-

f AV D_-006' AY V

-¿Vuo77

AYV .0 _L

I

._.. 0.5 I'D .5 2.0 2.5 3.0 3.5 VELOCITY-FT. PER. SEC.

.40 .60 y/ L Run IO Run 2

N

_.80 LOO .6

4

.0 .8 r .6 H u_ 4 o .0

0.8

4L

_no

uc.o77H lyv 1 0 , I 0.6 07 06 09 1.0 '.1 .2 I 3 L0CL DRAG C0EFFICENT

Fig. lo Local drag coefficient along the cylinder as function of the velocity profiles.

( according to Masch & Moore )

220 2.00 I. 80 GO ¡.40 (.0 LOO .80 0 .20 O 60 AV D_-0.058 o.

--

-R 22 80

( according to Dalton & Masch ) r

-:OII7

--AYV

1.00

Fig. 11 Variation of the local drag coefficient.

---t--What Da1ton & Masch also found was the answer to the question: How does the secondary flow disturb the pressure coefficient ?

To answer that question Dalton and Masch used a flow without velocity gradients and the velocity gradient flow given in fig 6. They measured the

pressure coefficient distribution around the cylinders for both flow condition on the level where the local velocity was the same in both profiles; this was at half depth.

They found that the pressure coefficient at the stagnation side of the cylinder was larger for the linear gradient flow than for the velocity field without velocity gradient in the z- direction, and that the coeffi-cient was smaller for the linear gradient flow than for the velocity field without velocity gradients in the z- direction on the wake side. This can be seen in fig 12, where the results of Dalton and Masch are shown. 003 002 R I.36nlO4 i 001 00 -c'o' 002 0C3 002 00? 0.0 cP -00 -0.02 003 60 90 ¡20 50 ¡1 R. 30 CO 90 ¡20 ¡50 ¡80 Sin decrte, 0.0 00 Fig. 12

Comparison of the pressure coeffi-cient for a flow without velocity gradients in the z- direction, plotted with the mark , and velocity gradient flow, plotted with the mark o.

6.2.6. Interpretation of the results of Masch & Moore and Dalton & Masoh for the experiments described in chapter 4.

It is very clear that the velocity gradients cause the increase in values near the bottom and the horse shoe vortex will decrease the values for very small values of the heigth above _{the bottom.}

To have an idea which parameters might be of influence on the increase in CD.some operations have been carried out with the results of Dalton and Masch as well _{as with the results presented in chapter 5.}

First the results of Masch & Moore out of fig 10 were used to plot

ACD/ey versus &v/y and

v/y.D/v; this plot is shown in Appendix 8;

there seems to be a correlation between

_{v/y and ¿v/4y.D/v;}

-it seems that new investigations should be carried out to study these correlations more in detail.

The results presented in chapter 5 were used to find out the correlation between CD and ¿tv/Ay and ¿v/Ay.D/v; note that y is used instead of z for this presentation.

Therefor CD was plotted versus ¿i v/ y for the five flow conditions; the operations and results are shown in Appendix 9. For every flow condition there seems to be a correlation between CD and v/y; the correlations

however are not the same for different flow conditions.

Even the correlation between CD and y.D/v will not be the same for the different flow conditions. This can be seen by immagining what would happen if CD was plotted versus v/ y.D/v for flow conditions A and C. Since ¿tv/y.D/v means only another horizontal axis, the plotted points of CD versus ¿v/y will only translated horizontally. To make the plotted points of flow conditions A and C match, the data of flow

condi-tion C should be moved more along the horizontal axis than the data of flow condition A; since the factor D/v in case C is smaller than in case A the opposite happens and plotting of

_{6C/óy versus}

_{v/y.D/v will}

not yield a correlation that is the same for all the flow conditions.

Still the conclusion that the measured increase in CD values is caused by the velocity gradient is valid, even though the present result can not be complétely correlated with earlier work by others.

6.3. Verification of the model of _{the surface disturbance influence.} At first glance the model suggested in 3.2 in which CD is decreasing exponential with z, does not seem to match the experimental data at all. In this model _{CD does not decrease for small values of the} sub-mergenze z,if this submergence decreases; in the model the CD value would increase.

However, a simple explanation for that can be given. Since the model is based on pressure distribution under a wave, it is not possible to apply the model if a part of the measurement ring _{comes out of the water;}

thus the data points that are found with the _{measurement ring only} partial-ly submerged should not be taken into account when the model is verified. Those data are the points to the left of _{the line that shows the depth} where the rings are totaly submerged , in Appendix 7.

If the measurement ring had been shorter _{this line would be closer to the} line of lowest drawndown; _{now however CD decreases s soon as the ring} comes out of the water, which seams logical since instead of the negative wake pressure the atmospheric pressure starts working on the part of the ring that is out of water.

The data that are not used in the verification of the model, for the reason described above, or because it can be seen in Appendix 7 that the point does not fit the line for versus z at all, is plotted in Appendix 10 as and in the tables they are indicated with , too.

Now the decrease in CD value is explained and the reason why those data should not be used in the verification of the model are given, the model can be verified.

First for the five flow conditions, the wave number k ( equal to 237/A) is computated using equation 3.18; the results of that computation are

shown in table 2.

Table 2. Calculation of k with the formula for k, k 9.81/ y2

To be able to plot C vesus z an estimation has to be made for C ; the value to which CD is decreasing in the plot of CD versus z is choosen for C6. With that value C is calculated and plotted versus z; through the data the least square line is found. This is done on Appendix 10; the results are given in table 3 together with the estimated value of C and the k value from table 2.

index _{Cw k}

m k

Table 3. The results for C and k

W m 38 -index

v;c

A 0.535 34.30 C .700 20.00 F .465 45.41 H .365 73.70 L .620 25.50 A 1.16 2.18 exp(-48.8 z ) 48.8 34.40 C 1.07 0.62 exp(-30.4 z ) 30.4 20.00 F 1.11 0.46 exp(-34.3 z ) 34.3 45.41 H 1.02 0.12 exp(-34.7 z ) 34.7 73.70 L 1.02 0.90 exp(-22.6 z ) 22.6 25.50 -1 -1 m m

The results in table 3 show that the model is not valid at first glance, since the values of km and k are not the same at all. Also the graph of km versus y shows little correlation; this graph is given in Appendix il together with the plot of k versus y.

Before it is concluded that the model is not valid since it does not match _{the measurements, some attention has to be spend on the estimation}

of C.

To do that the theory of Massie, shortly described in chapter 3 and totaly given in Appendix 1, is used. Whit that theory a new estimation is made; this is done in table form and the relation between C _{and the} submergence parameter, S, is found _{with a mean square line fit; the data} and the lines that are found are also plotted; the tables are presented in Appendix 12, the graphical presentation in Appendix 13.

The equation for CO3 the total drag coefficient, as function of and are given below,too, together with the correlation coefficient, r2, in table 4.

as:

Table 4. _{Formulas for CD and values of C based upon the work} of Massie.

Now that a different estimation of C, is found this value is used again to calculate C _{and plot that again versus z als proposed in 3.2 ,for the} measured CD as well as the calculated CD wherefor the equation out of the second column from table 4 is _used.

This new plot is presented in _{Appendix 14; the lines are calculated as} a root mean square line again and the results are tabulated in table 5, together with the values of k out of table 2.

Now it turs out that the values of k and km are nearly the same; the pro-posed model seems to be valid; its application is very depended upon a good estimation of C, which can be done with the work of Massie.. Now also the graph of km versus y should show an obvious correlation. This is _{checked in Appendix 15. There also the line is drawn that is} found after estimating that _{the curve of k versus y should have the} same shape as k versus _{V; thus it was assumed that}

km could be written index CD = C _{± C} C ± X .S C rL A _{1.13 ± 2.02 5 1.13 0.99} C 1.00 + 1.69 S

_LOO

_0.94 F 1.10 + 0.94 S 1.10 0.97 H _{1.04 + 0.16 5 1.04 0.93} L _{1.06 + 1.84 S 1.06 0.93}

Table 5 The results for C and k based upon an

W m

estimation with the work of Massie.

(6.9.)

which gives the following formula for w:

w=

Then five values of w were calculated and the group average was found; the result was:

W)

lo. k in m k m (6.10.) (6.11.)

with a standard deviation of 1.06.

The conclusion can be that the model is valid, but a good estimation for C is needed to apply the model.

6.4. Systematic variations between tests done with the same flow conditions. 6.4.1. Introduction

Some _{Fluid Mechanical calculations were made firts, after the systematic} variation in the results presented itself in graphs; this was done to verify whether the flow conditions used for two series that were compared, were indeed the same. Those calculations are described in 6.4.2.

Since ño deviations between flow conditions were found, considerable time has been spent to explain the _{systematic variation; this explanation} is given in 6.4.3.

6.4.2. Fluid mechanical calculations.

When the velocity profiles that belong to _{the different series done with}

40 -A 1.28 exp ( -35.1 z ) 35.1 34.30 C 0.72 exp ( -23.4 z ) 23.4 20.00 F 0.51 exp ( -41.2 z ) 41.2 45.41 H 0.16 exp ( -69.3 z ) 69.3 73.70 L 0.99 exp ( -27.8 z ) 27.8 25.50 index C