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OSCILLATING FLOW ABOUT YAWED CYLINDERS.(U)MAR A2 D 0 TRYTTEN"F l Nll lg

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NAVAL POSTGRADUATE SCHOOL1- Monterey, California

THESIS *

OSCILLATING FLOW ABOUT YAWED CYLINDERS

by

Dean 0. Trytten

March 1982

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Thesis Advisor: T. Sarpkaya

Approved for public release, distribution unlimited.

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Oscillating Flow About Yawed Cylinders March 1982

S. PERPOR0NNG Ono. REPoRT IuMsam

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Naval Postgraduate SchoolMonterey, California 93940

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I". ORCLASSIVIOCATION/OO1NGAINOtING

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Approved for public release, distribution unlimited.

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Is. SUPPL1eMfNTARlY NOTES

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Oscillating FlowYawed Cylinders

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The forces acting on yawed smooth and rough circularcylinders in oscillating flow have been investigated for thepurpose of determining the appropriate force-transfer coeffi-

F cients and theapplicability of the "independence principle."The results have shown that the flow about each cylinder isunique and the independence principle does not hold true.In addition, a comprehensive sensitivity analysis has been

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carried out to investigate the effects of undetected environ-mental disturbances on the force-transfer coefficients andthe limits of probable errors have been delineated.

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Approved for public release, distribution unlimited.

OSCILLATING FLOW ABOUT YAWED CYLINDERS

by

Dean 0. TryttenLieutenant Commander, United States Navy

B.S., North Carolina State University, 1970

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

March 1982

Author: 6 L

Approved by:

-- Dean of science and Engineering

3

ABSTRACT

The forces acting on yawed smooth and rough circular

cylinders in oscillating flow have been investigated for

the purpose of determining the appropriate force-transfer

coefficients and the applicability of the 'independence

principle. - The results have shown that the flow about

each cylinder is unique and the independence principle

does not hold true. In addition, a comprehensive sensi-

tivity analysis has been carried out to investigate the

effects of undetected environmental disturbances on the

force-transfer coefficients and the limits of probable

errors have been delineated.,<

4

TABLE OF CONTENTS

I. INTRODUCTION---------------------------------------- 15

II. EXPERIMENTAL AP'?ARATUS AND CYLINDERS-----------------20

III. DATA ACQUISITION AND PROCESSING---------------------- 24

IV. GOVERNING PARAMETERS-------------------------------- 26

V. EXPERIMENTAL RESULTS-------------------------------- 28

A. PRESENTATION OF RESULTS-------------------------- 28

-.. DISCUSSION OF RESULTS ------- ------------- 28

1. Inertia and Drag Coefficients----------------28

2. RMS Values of the Inertia and

Drag Coefficients---------------------------- 31

3. Transverse (Lift) Force---------------------- 32

VI. SENSITIVITY ANALYSIS-------------------------------- 34

A. PHASE SHIFT EFFECTS------------------------------ 34

B. RANDOM SQUARE WAVE EFFECTS----------------------- 37

VII. CONCLUSIONS----------------------------------------- 43

FIGURES---------------------------------------------------- 45

LIST OF REFERENCES----------------------------------------- 128

INITIAL DISTRIBUTION LIST--------------------------------- 129

5.

LIST OF FIGURES

1. Schematic of Cylinder Installation andCalibration ---------------------------------------- 45

2a. Cm Versus K for 8 = 4057, a = 60 Deg., k/D = 0.00 --46b. Cm Versus K for 8 = 2271, a = 60 Deg., k/D = 0.00 -- 47

c. Cm Versus K for a = 1021, a = 60 Deg., k/D = 0.00 -- 48

d. Cm Versus K for 8 = 2322, a = 60 Deg., k/D = 0.01 -_49

e. Cm Versus K for a = 1042, a = 60 Deg., k/D = 0.01 -- 49

3a. Cd Versus K for 8 = 4057, a = 60 Deg., k/D = 0.00 __51

b. Cd Versus K for B = 2271, a = 60 Deg., k/D = 0.00 -- 52

C. Cd Versus K for 8 = 1021, a = 60 Deg., k/D = 0.00 -- 53

d. Cd Versus K for 8 = 2322, a = 60 Deg., k/D = 0.01 -- 54

e. Cd Versus K for a = 1042, a = 60 Deg., k/D = 0.01 -- 55

4a. Ca Versus K for 8 = 4057, a = 60 Deg.,

k/D = 0.00-------------------------------56

b. Ca Versus K for 8 = 2271, a = 60 Deg.,

k/D - 0.00------------------------57

c. Carms Versus K for 8 = 1021, a = 60 Deg.,

k/D = 0.00 58-----------------------------------------5

d. Ca Versus K for 8 = 2322, a = 60 Deg.,

k/D = 0.01 5-----------------------------------------9

e. C arms Versus K for a = 1042, a = 60 Deg.,k/D = 0.01 ------------------------------------------ 60

5a. Cmr Versus K for 8 = 4057, a = 60 Deg.,k/D = 0.00 -----------------------------------------

b. Cmr Versus K for 8 = 2271, a = 60 Deg.,

k/D = 0.00 ----------------------------------------- 62

6

.I ..... ....

C. Cm Versus K for a8 1021, a - 60 Deg.,

rm

k/D - .00------------------------------------------ 63

d. Cmr Versus K for 8 = 2322, a = 60 Deg.,

k/D = 0.01 ---------------------------------------- 64

e. Cmr Versus K for 8 = 1042, a = 60 Deg.,

k/D = 0.01 4----------------------------------------65

6a. Cdr Versus K for B = 4057, a = 60 Deg.,

k/D = 0.00 5----------------------------------------66

b. Cdr Versus K for 8 = 2271, a = 60 Deg.,

k/D = 0.00 ----------------------------------------67

C. Cdr Versus K for 8 = 1021, a = 60 Deg.,

k/D = 0.00 7----------------------------------------68

d. Cdr Versus K for 8 = 2322, a = 60 Deg.,

k/D = 0.01 8----------------------------------------69

e. Cdr Versus K for 8 = 1042, a = 60 Deg.,

k/D = 0.01 69---------------------------------------

7a. Cdrm Versus K for 8 = 4077, a = 60 Deg.,

k/D = 0.00 ----------------------------------------71

b. Clrms Versus K for 8 = 2272, a = 60 Deg.,

k/D = 0.00 1----------------------------------------72

C. Clrms Versus K for 8 = 1022, a = 60 Deg.,

k/D = 0.00 2----------------------------------------73

d. clrms Versus K for 8 = 2322, a = 60 Deg.,

k/D = 0.01 3----------------------------------------74

e. Clrms Versus K for 8 = 1043, a = 60 Deg.,

k/DO= 0.01 ----------------------------------------75

8a. Comparison of Cm Versus K for 8 = 4040,

k/D 0.00, a = 90, 60, and 45 Deg. ---------------- 76

7I

b. Comparison of Cm Versus K for S = 2345,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 77

c. Comparison of Cm Versus K for a = 1045,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 78

d. Comparison of C Versus K for 8 = 4100,m

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 79

e. Comparison of Cm Versus K for 8 = 2325,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 80

f. Comparison of Cm Versus K for $ = 1040,

k/D = 0.01, a = 90, 60, and 45 Deg. - --------------- 81

9a. Comparison of Cd Versus K for 8 = 4040,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 82

b. Comparison of Cd Versus K for 8 = 2345,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 83

C. Comparison of Cd Versus K for 8 = 1045,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 84

d. Comparison of Cd Versus K for 8 = 4100,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 85

e. Comparison of Cd Versus K for 8 = 2325,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 86

f. Comparison of Cd Versus K for $ = 1040,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 87

10a. Comparison of Carms Versus K for B = 4040,

k/D = 0.00, a 90, 60, and 45 Deg. ---------------- 88

b. Comparison of Car.s Versus K for 8 = 2345,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 89

c. Comparison of Carms Versus K for a = 1045,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 90

8C

d. Comparison of Carms Versus K for 8 = 4100,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 91

e. Comparison of Carms Versus K for 8 = 2325,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 92

f. Comparison of Carms Versus K for 8 = 1040,

k/D = 0.01, a = 90, 60, and 45 Deg. - --------------- 93

la. Comparison of Cmr Versus K for 8 = 4040,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 94

b. Comparison of CMr Versus K for S = 2345,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 95

c. Comparison of Cmr Versus K for 5 = 1045,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 96

d. Comparison of Cmr Versus K for 8 = 4100,

k/D = 0.01, a = 90, 60, and 45 Deg. - --------------- 97

e. Comparison of Cmr Versus K for 8 = 2345,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 98

f. Comparison of Cmr Versus K for 8 = 1040,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 99

12a. Comparison of Cdr Versus K for $ = 4040,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 100

b. Comparison of Cdr Versus K for 8 = 2345,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 101

c. Comparison of Cdr Versus K for 8 = 1045,

k/D = 0.00, a = 90, 60, and 45 Deg. ---------------- 102

d. Comparison of Cdr Versus K for 8 = 4100,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 103

e. Comparison of Cdr Versus K for 8 = 2345,

k/D = 0.01, a = 90, 60, and 45 Deg. ---------------- 104

9

i

f. Comparison of Cd Versus K for 8=1040,

k/D - 0.01, = 90, 60, and 45 Deg.----------------- 105

13a. Comparison of C Irm Versus K for 3 = 4040,

k/D = 0.00, a = 90, 60, and 45 Deg.----------------- 106

b. Comparison of C rsVersus K for a = 2365,

k/D = 0.00, = 90, 60, and 45 Deg.----------------- 107

C. Comparison of C 1 Versus K for S = 1060,

k/D = 0.00, = 90, 60, and 45 Deg.----------------- 108

d. Comparison of C lrms Versus K for s = 4100,

k/D = 0.01, = 90, 60, and 45 Deg.----------------- 109

e. Comparison of C Irm Versus K for B = 2390,

k/D = 0.01, a = 90, 60, and 45 Deg.----------------- 110

f. Comparison of C Irm Versus K for B = 1060,

k/D = 0.01, = 90, 60, and 45 Deg.----------------- 111

14a. Force-Transfer Coefficients Versus PhaseAngle for K= 6.66----------------------------------- 112

b. Force-Transfer Coefficients Versus PhaseAngle for K = 10.41---------------------------------- 113

C. Force-Transfer Coefficients Versus PhaseAngle for K = 14.08---------------------------------- 114

d. Force-Transfer Coefficients Versus PhaseAngle for K = 14.51---------------------------------- 115

e. Force-Transfer Coefficients Versus PhaseAngle for K = 25.44---------------------------------- 116

15a. Normalized Force Plus a Random Disturbance ofFrequency Ratio I------------------------------------ 117

b. Normalized Force Plus a Random Disturbance ofFrequency Ratio 2----------------------------------118

c. Normalized Force Plus a Random Disturbance ofFrequency Ratio 6------------------------------------ 119

d. Normalized Force Plus a Random Disturbance ofFrequency Ratio 36----------------------------------- 120

10

e. Normalized Force Plus a Random Disturbance of

Frequency Ratio 180---------------------------------- 121

16a. Response of C m and the Corresponding C d to

Random Disturbances of Various FreguencyRatios, K = 14.08------------------------------------ 122

b. Response of Cd and the corresponding C m to

Random Disturbances of Various FrequencyRatios, K = 14.08------------------------------------ 123

17a. Response of Cm and the Corresponding Cd to

Random Disturbances of Various FrequencyRatios, K = 6.66------------------------------------- 124

b. Response of Cd and the Corresponding Cm to

Random Disturbances of Various FrequencyRatios, K = 6.66------------------------------------- 125

18a. Response of C m and the Corresponding C d to

Random Disturbances of Various FreqencyRatios, K = 25.44------------------------------------ 126

b. Response of C d and the Corresponding C I to

Random Disturbances of Various FrequencyRatios, K = 25.44------------------------------------ 127

TABLE OF SYMBOLS AND ABBREVIATIONS

A Coefficients of sin ne in Fourier Seriesn

B Coefficient of cos ne in Fourier Seriesn 2

C Normalized force coefficient, C = 2F/(ODUM 2

C Actual RMS value of Carms

CF(spp) Semi-Peak-to-Peak force coefficient

C d Drag coefficient

Cdr Drag coefficient normalized by the RMS value of Cd

Clrms RMS value of the normalized lift force

C Inertia coefficientm

Cmr Inertia coefficient normalized by RMS value of C

D Diameter of the test cylinder

F In-line force

F Measured in-line forcem

f sFrequency of square wave disturbanceSW"

ft Frequency of the tunnel, ft = l/T

K Keulegan-Carpenter number, K = U mT/D = 21A/D

k Mean sand-roughness height

k/D Relative roughness

L Cylinder length

Re Reynolds number, Re = U mD/

t Time

T Period of flow oscillations

U Instantaneous velocity, U = -Um cosO

12

Um Maximum velocity in a cycle

OL Yaw angle, measured between ambient flow directionand cylinder axis

Frequency parameter, 8 = Re/K = D2 /vT

a Angle, 8 = 2nt/T

v Kinematic viscosity of water

Density of water

13

ACKNOWLEDGEMENT

The author wishes to thank Professor T. Sarpkaya for

his continual guidance and patience throughout this

investigation. It has been both a privilege and a most

worthwhile experience to work with a man of such profes-

sional stature.

Great appreciation is also expressed to Mr. J. McKay of

the Machine Shop in the Department of Mechanical Engineering

who is truly an artisan in every sense of the word.

Also, the author wishes to thank the United States Navy

for providing him the opportunity to further his pro fes-

sional development.

Finally, the author wishes to thank his wife for the

continual understanding that she provides regardless of 11

the circumstances.

14

I. INTR~ODUCTION

The study of time dependent flow about yawed cylinders is

a topic which presently has great interest, both theoretically

and practically. Of the great variety of time-dependent flows

about yawed cylinders, the wave motion and sinusoidally oscil-

lating flow are of such importance as to warrant immediate

investigation. For lack of adequate information concerning

time-dependent flows, industry is often forced to adopt

steady flow relations to time-dependent flow situations.

This research proposes to investigate the forces acting upon

yawed circular cylinders immersed in sinusoidally oscillat-

ing uniform flow and to examine the validity or limitations

of the present methods of analysis.

In 1950, Morison, et al. [11 introduced an equation for

calculating the in-line force due to unbroken surface waves

acting on a vertical pile. For a cylinder of diameter D,

the force per unit length is expressed as:

1 2 DUF = p D C d UlU I + 0. 25 p ir D Cm It (1

where U represents the incident flow velocity; C d' the drag

coefficient; and Cm, the inertia coefficient. The coeffi-

cients Cm and Cd were considered to be time-invariant and

constant along the length of the cylinder. Morison's pacer

15

was intended as a preliminary report with follow-on studies

to be done on other structures in various wave actions.

Morison, et al. did not consider the contributions of

transverse forces and vortex shedding in the calculation

of the in-line forces.

Attempts have been made to extend this rather simple

relation to far more complex situations such as combined wave

and current flow, hydro-elastic oscillations, and yawed

cylinders. Heideman, et al. [2] studied Morison 's equation

using ocean data and concluded that Morison's equation was

satisfactory for normal cylinders. The utilization of an

equation unproven even under ideal flow situations to

correlate ocean data so as to prove the validity of the

equation is not a very meaningful exercise. Sarpkaya and

Isaacson £ 31 reviewed in detail the numerous studies regard-j

ing the application of the Morison equation.

It is obvious that there is a great need for an ideali-

zation of the problem, or an experiment which is more

manageable (e.g., a sinusoidally oscillating uniform flow).

only in this way can all the complex interactions be

separately taken into account.

Engineers faced with the problem of dealing with wave

forces on yawed members, having no other recourse, drew

upon previous work with steady flow. Hoerner [4] proposed

the "independence principle", which stated that the normal

pressure forces are independent of the tangential velocity

16

for subcritical values of Ren, where Ren is the Reynolds

number based upon the component of the flow velocity normal

to the cylinder. This principle allowed the neglecting of

the tangential components. Bursnall and Loftin [5] found

that the independence principle does not apply to the critical

and transcritical flow regimes. Norton, et al. [6] found that

the independence principle does apply to the postcritical as

well as subcritical flow, but not to the critical and trans-

critical regions in between. Thus, recent research has shown

that the independence principle applies when the boundary

layer is wholly laminar (Hoerner) or wholly turbulent (Norton),

but its use in the critical and transcritical regions is

uncertain. F

The designers of offshore structures were thus led to

adopt the independence principle for the wave force calcula-

tions and, in doing so, to generalize the Morison equation.

This has been accepted practice in industry for lack of a

better relation. It can be asked what would be anticipated

for oscillating flow or waves on the basis of what is known

for steady flow. The instantaneous Reynolds number in such

a flow will vary from -Remax to +Re max during a complete flow

cycle. It could be postulated that the boundary layer would,

at times, be fully laminar; at other times fully turbulent;

and the rest of the time be in transition. In light of this,

it is rather doubtful that the independence principle applies

at all to oscillating or wave flow.

17

The study of forces acting on yawed cylinders can be

investigated either by oscillating a yawed cylinder in a

tank, using small amplitude waves in a laboratory channel,

oscillating the flow about a fixed cylinder, or by using

ocean data if available. Oscillating the cylinder has

proven to be impractical because of difficulties in account-

ing for the inertial force acting on the body, producing

repeatable oscillations, controlling spurious vibrations in

the system, and in measuring the in-line and traverse forces

simultaneously. Waves have relatively more-complex flow4

kinematics due to the orbital motion of particles and the

decay of wave amplitude with depth. Ocean data are not

available for such a study. oscillation of the flow past

the cylinder has proved to be the most practicable method.

By using a sinusoidally oscillating uniform flow (which can

be fairly easily produced) the kinematics of the flow are

accurately known. Sarpkaya and Isaacson (31 discuss the

similarities and differences between such a flow and the

flows described by wave theory.

The primary objective of this investigation was to study

the forces exerted by a sinusoidally oscillating uniform flow

on yawed circular cylinders to determine whether or not the

independence principle is applicable. If the principle holds,

the force transfer coefficients calculated by Fourier analysis

using the normal velocity compont.'t. should reduce identically

to the normal cylinder case at corresponding values of K, Re,

18

and k/D. If the independence principle does not apply, it

is desired to determine what the coefficients are as functions

of yaw angle, roughness ratio k/D, Re, and K. It would also

be necessary to determine how well Morison's equation works

with the new coefficients.

With the foregoing objectives in mind a detailed investi-

gation was undertaken using smooth and rough cylinders of

nominal 6-inch, 4-inch, and 3-inch diameters at yaw angles

of 45 degrees, 60 degrees, and 90 degrees (as measured between

the ambient flow direction and the cylinder axis). Part of

this effort has already been reported (7]. The present study

extends the previous work, accomplishes the original objec-

tives of the investigation, and takes a much closer look at

the sensitivity of the modified Morison equation and of the

measured forces to random external disturbances encountered

in nature.

19

.. . . . ..I.. .. .... .. .... . .mm lf , Il lnll l ll " , ,.,, ,,. ,... . .. .

II. EXPERI4ENTAL APPARATUS AND CYLINDERS

The experiments were carried out in the large oscillating-

flow tunnel at the Naval Postgraduate School. The tunnel was

basically unchanged from the configuration described in (8

and 9]. Mounting pads for the strain gage housings were

placed on opposite sides of the tunnel in such a manner so

as to accommodate the yawed cylinders (Fig. 1). The mounting

pad on one side of the tunnel was fitted with a slide so that

the pad could be adjusted small distances to exactly match

the cylinder length. The force transducers and housings were

unchanged from those described in [9].

The cylinders also were similar to those previously

reported, except that the ends were cut exactly at the

desired angle of yaw. The cylinder ends were parallel to

the tunnel walls with a 1/32-inch gap at each end. This gap

was filled with a soft foamy material glued to the ends of

the cylinder. Ball bearings were mounted in the ends of the

cylinder with the outer bearing faces flush with the face of

the cylinder.

The cylinders mounted in the tunnel were supported by

the pins in the force transducers. The pins were perpendicu-

lar to the tunnel wall. Rubber washers were placed behind

the pins so that the lateral motion of the cylinders (normal

to the tunnel wall) was restrained, but without compressing

20

3I

the cylinder. The cylinders were free to rotate, but a small

clip was placed on one end of the cylinder to restrict the

rotation to a very small amount so that the yawed end of the

cylinder did not strike the tunnel wall as it rotated. Thus,

as mounted, the cylinder was laterally constrained, but free

to rotate within a small range. Careful calibration and

extensive testing proved that this mounting system allowed

accurate and repeatable recording of the normal and trans-

verse forces acting on the cylinder.

Calibration was first conducted in a vertical direction,

since yaw would not affect the total force in this direction.

Weights were hung from the center of the horizontal cylinder,

and calibration factors were obtained by converting the sum

of the signals of two force transducers to pounds/mm and

pounds/volt. The strain gages were found to be linear

throughout the range of expected forces. The horizontal

forces normal to the cylinder are related to the streamwise

force by the sine of the yaw angle at (as measured between the

cylinder axis and ambient flow). A system of supports and

pulleys allowed applying horizontal force at the center of

the cylinder and normal to it (Fig. 1). The sum of the

outputs of the two transducers was recorded on the strip

chart recorder, establishing the necessary relationship

between the normal in-line force* F and the total electrical

The force in the plane defined by the velocity vector andthe cylinder axis.

21

output of the gages. The same gages were used to measure

the lift force or the in-line force by rotating the gages

90 degrees. It is easy to show that the calibration factor

of the lift force (here in the vertical direction, up or

down), expressed in terms of pounds or millimeter deflection,

is sina times the calibration factor for the in-line force.

This proved to be true experimentally also and demonstrated

independently the validity of the measurement technique.

Raines (71 carried out experiments with the following

cylinders:

90 degrees, 5.905 inch, rough and smooth (nominalsize 6 inches)

60 degrees, 6.005 inch, rough

45 degrees, 5.923 inch, rough and smooth

45 degrees, 4.42 inch, rough and smooth (nominalsize 4.5 inches)

45 degrees, 3.00 inch, rough and smooth (nominalsize 3 inches)

The present experiments were carried out with the

following cylinders:

60 degrees, 5.965 inch, smooth (nominal size6 inches)

60 degrees, 4.450 inch, smooth and rough(nominal size 4.5 inches)

60 degrees, 2.985 inch, smooth and rough(nominal size 3 inches)

Relative sand roughness of kID = 1/100 was used for all

rough cylinders.

22

In addition, the foregoing data were complemented with

data reported in [10). These data, taken using 3 inch and

4.5 inch nominal diameter cylinders at 90 degree yaw angle,

were used in the analysis and comparison of the entire data.

-2I1

23I

III. DATA ACQUISITION AND PROCESSING

Data acquisition was accomplished using an HP3052A

automatic data acquisition system. The system was configured

as described in [9].

The data acquisition system consisted of an HP3495A

20 channel scanner and an HP3437A system voltmeter, both

controlled by an HP9845B desk top computer. The in-line

or transverse signal, originating as a voltage from the

strain gages, was amplified and sent to the strip chart

recorder. This amplified signal was also sent to the scan-

ner as one channel of data. The flow amplitude signal

originated from a differential pressure transducer, and was

similarly amplified and sent to both the strip chart recorder

and the scanner as another channel of data. Strip chart data

was maintained as a visual record of the flow amplitude and

force. The scanner switched between the two channels at

designated intervals when triggered by the voltmeter. The

voltmeter read each one of these voltage values and trans-

ferred them to the computer for immediate calculation of

the governing parameters.

At least six cycles of in-line force data, recorded

aoproximately 70 times per cycle, were acquired with the

system. The data were then averaged to give one cycle from

which the governing parameters were calculated. Ten cycles

24

,, ... . - ... ,i i.4 '': - , , , - ' - "" . .. ,, ,

of transverse force data, recorded at the same rate, were

acquired with the system. The transverse data was not

averaged prior to calculating the governing parameters.

The use of the HP3052A system proved highly beneficial

for a number of reasons. The calculated parameters were

available on a real time basis, allowing for detailed study

of anomalies in the force patterns. Taking data over a

large number of cycles removes more of the randomness of

the flow forces. This was particularly valuable for

analyzing the transverse force data.

25

IV. GOVERNING PARAMETERS

Data reduction for the in-line forces is based on

Morison's equation using the normal component of velocity.

The derivation of the governing parameters is described in

detail in (3 and 9]. Here only a brief description of the

parameters will be given.

For a sinusoidally varying flow, Eq. (1) may be

written:

2F -

= K Cm sin8 - Cdlcoselcose (2)o DU

where the flow is represented by U = -U cos5 with 9 = 27t/T.

The Fourier averages of Cm and Cd are given as [11:

2,m f Fmsine

K 2 de (3)m D Um

0 in

and

Cd 2 f m Cs de (4)0 pD U2

where Fm represents the measured force.

26

As discussed in [9], it would be beneficial to remove

the cycle-to-cycle variations in the force. This was accom-

plished by using the normalized RMS value of the actual force

(Camrs) Two new coefficients were defined from this:

Cdr Cd/Crms Cdr = Cd/Camrs

and

Cr = Cm/Crms ,Cmr = Cm/Carms (6)

Using Eq. (2), the rms value of the calculated force

can be shown to be:

!4 C2

3 = 2 + (7)2K2

Transverse force data were represented by the normalized

rms value of the measured lift force (Clrms). Previously,

some of the lift data were analyzed [7] using a Fourier

transform to obtain frequency information. In the oresent

study, this effort was discontinued because the frequency

information so far obtained was more than sufficient to

draw general conclusions regarding the frequency content of

the lift force.

27

V. EXPERIMENTAL RESULTS

A. PRESENTATION OF THE RESULTS

The force-transfer coefficients (Ca, C d' Carms Cri Cdr'

C lms) are presented in graphical form in terms of the

governing parameters K, $, a, and k/D in Figs. 2 through 7.

The new as well as the previously-obtained data [7 and

10] have been cross-plotted in Figs. 8 through 13 in order

to exhibit the effects of the governing parameters.

B. DISCUSSION OF THE RESULTS

1. Inertia and Drag Coefficients

The first indication of the fact that the independence

principle is not applicable to yawed cylinders in oscillating

flow is seen in Fig. 8a where C m is plotted as a function of

K for S = 4040, k/D = 0.00, and a = 90, 60, and 45 degrees.

Evidently, had the independence principle been valid, the

three sets of data corresponding to three yaw angles would

have collapsed into a single set, thus rendering the C m

versus K relationship independent of the yaw angle (at least

within the range of the test parameters and experimental errors) .

This conclusion is more universal than it may appear at first

sight on the basis of Fig. 8a. In fact, Figs. 8b through 8f,

presenting C mversus K for both smooth and rough cylinders of

6, 4.5, and 3 inch nominal diameters, show that C m varies with

the yaw angle as it did in Fig. 8a. Thus, one may conclude

28

that the independence principle is not valid at least for the

inertia coefficient. The recognition of this fact is impor-

tant for two reasons. Firstly, the inertial force is only

part of the total force. Secondly, the significance of the

inertial force depends on K. Thus, the failure of the

inertia coefficient to obey the independence principle may

not by itself be very important if the inertial force is

small relative to the total force for the particular case

under consideration. Consequently, it is important to

examine the variation of the drag coefficient and its

relation to the total force and the independence principle.

Figure 9b shows the variation of C d as a function of

Kfor 3 = 2345, k/D = 0.00, and a = 90, 60, and 45 degrees.

Clearly, the variation of C d with K and the yaw angle is far

more complex than that of C m Figures 9a through 9c,

depicting C d versus K for the three smooth cylinders, show

that the independence principle does not hold true except at

relatively large values of K. The implication of this result

is that when the vortex motion about the body becomes ex-

tremely turbulent and incoherent, as it would at high K

values, the overall effect of yaw becomes less significant.

It is also in this region that the force is drag dominated

and the drag force may be calculated in terms of the force

acting on the projected area. Though the inertia coefficient

exhibits strong variation with yaw even at high K values,

this variation is of academic interest only since the

29

contribution of the inertial force to the total force is

rather insignificant.

Roughness plays an important and, oddly enough, a

unifying role on the drag coefficient. This is somewhat

anticipated on the basis of the fact that roughness precip-

itates earlier transition and greater incoherence in the

flow about a yawed cylinder, thereby creating conditions

more favorable to the independence principle. Figures 9d

through 9f show that the independence principle for rough

cylinders is almost valid with the exception of the drag-

inertia dominated regime (8<K<20).

It is clear from the foregoing discussion of the

inertia and drag coefficients that the normal force acting

on a yawed cylinder is significantly underestimated through

the use of Morison' s equation, independence principle, and

the drag and inertia coefficients appropriate to the normal

cylinder. only a narrow range of K values (from about 9 to

17), in the drag-inertia dominated regime, that the drag

coefficient for a normal cylinder exceeds that for a yawed

cylinder. However, the large increase in the inertia

coefficient of the yawed cylinder in the same region of K

values more than offsets the relatively small decrease in

its drag coefficient.

The absence of the inertia crisis, the large increase

in the inertia coefficient, the relative decrease in the drag

and lift coefficients, all in the drag-inertia dominated

30

regime, of yawed cylinders are a consequence of the decrease

of the spanwise coherence of vortices. In general, the

smaller the spanwise coherence, the larger the Cm in this

region. It appears that the independence principle or the

cosine law, as it is sometimes called, is a gross simplifica-

tion of the behavior of flow in the near wake. One may,

therefore, conclude that the Fourier-averaged drag and inertia

coefficients, based on Morison's equation, are unique for

each angle of yaw, Reynolds number, Keulegan-Carpenter number,

and the relative roughness.

2. EMS Values of the Inertia and Drag Coefficients

It has already been shown [71 that the normalized rms

values of the inertia or drag coefficients remove signifi-

cantly the scatter from the data and establish clearer

relationships between these coefficients and the governing

parameters. The same rms coefficients are used here to

examine the effect of yaw and experimental error. Figures

lla through llf show Cmr = Cm/Carms as a function of K.

Clearly, Cmr is independent of Reynolds number (or 3), yaw

angle, and the roughness for K smaller than about 8. For

smooth cylinders, Cmr is nearly independent of yaw for K

larger than about 20. Additionally, for rough cylinders Cmr

is least dependent on yaw for relatively low values of Re.

The drag coefficient Cdr exhibits similar character-

istics (see Figs. 12a through 12f). The scatter in Cdr is

considerably less than that in Cd primarily because of the

31

f act that the variations in C d as a consequence of the random

disturbances imposed on the measured force are smoothened out

by the variations in the corresponding C am values. This

fact has been substantiated through numerous examples by

examining the relative variations of Cd' Cm, and Carms- As

K increases C dr reaches its theoretical value of 1.63, as

seen from Eq. (7). Finally, roughness tends to decrease the

dependency of C dr on Re (see Figs. 12d through 12f).

3. Transverse (Lift) Force

The lift force data are presented in terms of the

normalized rms values of the measured lift force in Figs. 13a

through 13c for the three smooth cylinders. These figures

show that C is largest for the 90-degree cylinders, asIrms

one would expect, and decreases rapidly with yaw. However,

the variation of C 1rm with the yaw angle (60 and 45 degrees)

is not as large as that with the Reynolds number. The lift

coefficient increases rapidly as K increases from about 6

to 12. Then, Clrm again decreases rapidly due to the

increased lack of coherence of the vortices. Similar

observations may be made for rough cylinders (see Figs. 13d

through 13f). It suffices to note that the yaw has an

immediate effect on the coherence of vortices and on the lift

coefficient, particularly in the drag-inertia dominated

regime. For large values of X or Re, the irregularity of

the vortex shedding even for a 90-degree cylinder obscures

the yaw effect. It is then rather difficult to distinguish

32

between the lift coefficients of cylinders at various yaw

angles. In theocean environment, one is more likely to find

a confused vortex shedding with very small coherence length.

Thus, one may safely state that the inclined members subjected

to the wave motion in the ocean environment will experience

considerably less lift force than those normal to the waves.

I;

33

VI. SENSITIVITY ANALYSIS

A. PHASE SHIFT EFFECTS

It has been known since the inception of Morison' s

equation in 1950 that the phase change between the maximum

force and the maximum velocity may lead to varying degrees

of error, depending on the governing parameters. This phase

shift may arise due to electronic or numerical filtering of

the force and velocity data or due to variations in the

coherence of the vortices along a given separation line.

Even though it is not yet possible to attribute a given

fraction of the estimated error to a specific cause, it is

most desirable to examine in detail the isolated effects of

such things as an imposed phase shift and/or a random

disturbance of a given amplitude and frequency. Eventually,

such information will show the region of sensitivity of the

force coefficients and Morison's equation to the disturbances

imposed by the environmental conditions and will lead to a

more rational interpretation of the causes of large scatter

observed in the results of ocean tests.

The e f fects o f phase sh if t on the fo rce- trans fe r

coefficients have been examined as follows. For a given

K, $, k/D, and yaw angle the drag and inertia coefficients

have been calculated from Eqs. (3) and (4). These

coefficients correspond to cases where there was no phase

34

shift as ascertained by the manner in which the experiments

were carried out. Then the force was recalculated through

the use of Morison 's equation and the above coefficients.

The calculated force was then given a phase shift and a new

set of force coefficients were calculated through the use of

Eqs. (3) and (4). The method was repeated for representa-

tive values of K and reasonable phase angles.

Figures 14a through 14e show the results for five

coefficients. Clearly, the phase angle has significant

effect on all of these coefficients with the exception of

Carms. Concentrating attention on Cm and Cd it is noted

that C mdecreases and C d increases with increasing phase

angle (where the force curve is shifted forward in time).

The relative rate of decrease or increase of these two

coefficients certainly depends on the particular values of

the governing parameters. For example, for K = 6.66 (Fig.

14a) C m decreases rather gradually whereas C d increases

rapidly with increasing phase angle. The reason for this

is that for values of K less than about 8 the fluid force

is inertia dominated and the inertia coefficient is not

very sensitive to small disturbances imposed on it. In

mathematical terms the foregoing simply means that the force

curve is like a sine curve and that when it is shifted right

or left by some amount and then multiplied with sine, as in

Eq. (3), the integral could not be too sensitive to the

phase difference. The drag coefficient, on the other hand,

35

is extremely sensitive to small phase shifts in the inertia

dominated region simply because a small phase shift in a

"sine-like" curve multiplied with icoselcosO, as in Eq. (4),

produces large changes.

In the drag-inertia dominated regime (K ranging from

about 8 to 20) both C d and C m are affected, as seen in Figs.

14c and 14d. In the drag dominated regime, however, C d is

hardly affected by phase shifts; whereas C m shows a dramatic

drop with increasing phase angle (Fig. 14e). The reason for

this is easily explainable in a manner similar to that

previously presented in connection with Fig. 14a.

The overall conclusion to be drawn from the foregoing

is that there is no flow regime where one or the other

coefficients is not effected by the phase shift. In fact,

if a constant phase shift of ±5 degrees were to be randomly

imposed on a series of data obtained with K values ranging

from 5 to 100 one would have observed large scatter in Cd

and small scatter in CM for K smaller than about 8; large

scatter in both Cd and Cm for K from about 8 to 20; and

large scatter in CM and small scatter in C d for K larger

than about 20. These scatter patterns are invariably

observed in the data obtained in the ocean environment.

As noted earlier, not all of the phase shift is due to

electronic filtering of the data. Some of the phase shift

comes from the reduction of the coherence length of the

vortices by the disturbances ever-present in the wave field.

36

Thus the purpose of the foregoing is not to suggest that

all phase shifts could be avoided or eliminated but rather

to quantify the effects of the phase shift in terms of the

expected variation of the force coefficients in order to have

a better grasp of the meaning of the observed scatter.

B. RANDOM SQUARE WAVE EFFECTS

The experiments conducted under controlled laboratory

conditions are immune to practically all of the disturbances

encountered in the ocean environment. Consequently, the

vortices have larger coherence lengths and these give rise

to maximum values of the semi-peak-to-peak force coefficient

C F(spp).* In the ocean environment, however, the variable

wave direction, change of wave velocity with depth, effect

of secondary waves with all sorts of harmonics, irregular

surface roughness, and numerous other stochastic variables

give rise to varying degrees of scatter in the force

coefficients based on ocean data. It is important from the

engineering point of view to examine the collective impact

of these disturbances on the force coefficients. It is under-

stood that very high frequency disturbances imposed on the

measured force cannot significantly affect C d and C M' On

the other hand, disturbances which simulate either in part

This does not imply that perfect coherence leads tomaximum values Of Cd and Cm. In fact, a force traceresulting from a perfect coherence can be renderedincoherent by a random square wave. The new forcetrace yields a larger C m and a smaller C d.

37

or in combination a "sine like" or a "cosine like" wave are

most effective in causing large changes in C d and Cm . This

follows from the very definition of the inertia and drag

coefficients as in Egs. (3) and (4). Clearly one must

examine the effect of randomly imposed disturbances of

given frequency and phase shift on the variation of the

,force coefficients. Such an extensive analysis has been

carried out in a manner described as follows. A frequency

ratio of f sw /ft was chosen where f wis the frequency of

the square wave and ftis the frequency of the periodic

oscillation of the flow in the tunnel. Then the square wave

was assigned a random amplitude and initial phase shift

through the use of a built-in random number generator. The

maximum of the disturbance was taken to be the absolute of

C F(spp) (almost equal to the absolute maximum of the measured

force). This procedure varied the amplitude of each disturb-

ance by multiplying a random number between +1 and -1 with

C F(spp). Calculations have been performed 1000 times for a

given frequency ratio and force trace. Each calculation

started with a randomly selected phase angle and followed

with square wave fluctuations whose amplitude varied randomly

as noted above and whose frequency was fixed by the frequency

ratio. Figures 15a through 15e show sample plots of the

original force, the disturbed force, and the square wave

disturbance. These figures are presented here for the

purpose of illustration and not necessarily for the

38

quantification of the boundaries of the sensitivity

envelope.

Figure 16a shows the results obtained in the manner

described previously. The two symbols in this figure

represent the relative changes in C m and C d' The computer

program was directed to identify the particular cases in

1000 runs which gave the largest changes in Cm. and the

corresponding changes in C d. It is important to emphasize

that the change in C d in Fig. 16a is not the maximum

possible change that could have taken place with a suitable

square wave disturbance. Thus, Fig. 16a shows that there

are certain frequency ratios which cause a large change in

C m with very little change in the corresponding C d' As

expected, disturbances with a frequency ratio of 1.0 yield

the largest AC m/C.n since this nearly corresponds to a sine

function. Surprisingly enough, high frequency disturbances

have very little effect on the inertia coefficient, at least

in the drag-inertia dominated regime (see Fig. 16a) . Thus,

one can filter out such high frequency disturbances without

materially affecting the determination of the force-transfer

coefficients.

A similar calculation has been performed with the force

trace corresponding to Fig. 16a to determine the maximum

change in C d and the corresponding change in C m for

representative frequency ratios. Figure 16b shows the

results obtained in this manner. Several facts are

~39

immediately apparent. Cd is not as sensitive as Cm to the

randomly imposed disturbances in the drag-inertia regime

even though ACd/Cd is quite appreciable. This is directly

related to the stability of the integral representing Cd,

i.e., Icos61cos6 times asquare wave.

Figures 17a and 17b show the effect of the disturbances

on Cm and Cd in the inertia dominated regime and Figs. 18a

and 18b illustrate the similar effects in the drag-inertia

dominated regime. It is clear from all of the figures

pertaining to the sensitivity of the drag and inertia

zoefficients that it is not the dominance of the drag or

inertia in a given regime that determines the sensitivity of

a given coefficient. To be sure, the said effect is present

in Figs. 16a through 18b. However, the most significant

elements of a disturbance are its frequency and relative

magnitude. Disturbances imposed on the force at a frequency

ratio of about one or unity will cause major changes in

both Cd and Cm irrespective of the flow regime. As to the

amplitude or the disturbances, they may depend on a number

of parameters such as K, Re, turbulence in ambient flow,

interference of other bodies, etc. In the present analysis

the maximum amplitude of the disturbances at any frequency

were taken to be equal to CF(spp) of the measured force.

Since this coefficient decreases with increasing K, it is

easy to understand that at high K values ACm/Cm or ACd/Cd

40

will result in values as large as those encountered at

smaller K values (see Figs. 16a through 18b).

The significance of these results will be illustrated

with a numerical example. Assume K = 14.08, Beta = 4040

(as in Fig. 16a), for which Cm = 0.61, Cd = 1.63 and CF(spp) =

1.66. For a frequency ratio of fsw/ft = 1.0, one has

ACm/Cm = 2.7 and ACd/Cd = 0.91 from Figs. 16a and 16b.

Assuming a disturbance amplitude equal to 10 percent of

CF(spp) , one has:

C, (perturbedvalue) = (l±ACd/Cd * %CF(spp)) * Cd(original

value)

= (1±0.91 * 0.10) * 1.63 1.781.48

and

Cm(perturbed = (l±AC /Cm %C (spp)) * C (originalvalue) m m F m value)

0.77 ,(1±2.7 * 0.10) * 0.61 = 00.45

The foregoing examples show that a random disturbance of

frequency ratio 1.0, imposed on a particular force, may

have resulted in relatively large cycle to cycle variations

of the force coefficients. Here Cd varies from a minimum

41

value of 1.48 to a maximum value of 1.78 and C m varies from

a minimum value of 0.45 to a maximum value of 0.77. These

are relatively large changes and show clearly the importance

of the following two factors. First, it is not possible

to obtain scatter-free data in the ocean environment even

for a single K and Re because of the ever-present random

disturbances emanating from various sources. Secondly, a

structure cannot be designed to withstand the maximum forces

predicted on the basis of idealized laboratory experiments

because the maximum forces may occur only in a small fraction

of the lifespan of the structure. Clearly, beyond this point

one has to make subjective decisions regarding the probability

of the occurrence of maximum force. Figures similar to 16a

through 18b enable one to predict the force coefficients

which would have prevailed under less favorable conditions

(reduced coherence, directional waves, body interference,

etc.).

44

VII. CONCLUSIONS

The forces acting on yawed smooth and rough circular

cylinders in sinusoidally oscillating uniform flow have been

extensively investigated, and the following conclusions have

been reached:

1. The independence principle does not apply over signif-

icant portions of the range of K values investigated. The

drag and inertia coefficients differ significantly from those

deduced from the 90-degree case for all 45-degree and 60-

degree smooth and rough cylinder cases.

2. The effect of roughness is to bring the yawed and

unyawed cylinder force-coefficients closer, except in the

drag-inertia dominated regime.

3. Yaw has a very strong effect on the coherence of

vortices and hence on the transverse force coefficient,

particularly in the drag-inertia dominated regime. The lift

coefficient is largest for the 90-degree cylinder, as might

have been expected, and decreases sharply with yaw.

4. An extensive sensitivity analysis has shown that the

force-transfer coefficients are affected by the phase shift

between force and velocity to various amounts dependent on

the flow regime and the magnitude of the phase shift.

5. Finally, the results have shown that the force

coefficients and the validity of the Morison equation are

43

sensitive to disturbances imposed on the in-line force,

particularly in the frequency range where the disturbance

frequency matches the wave frequency. The stability envelope

has been delineated through extensive calculations to enable

one to interpret the scatter of the data obtained in the

ocean environment.

4

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LIST OF REFERENCES

1. Morison, J. R., O'Brien, M. D., Johnson, J. W., andSchaat, S.A., "The Force Exerted by Surface Waves onPiles", Petroleum Transactions AIME, Vol. 189, 1950.

2. Heideman, J. C., Olsen, 0. A., and Johnansson, P. I.,"Local Wave Force Coefficients", Civil Engineering inthe Oceans IV, ASCE, Vol 1, pp. 684-699, 1979.

3. Sarpkaya, T., and Isaacson, M., Mechanics of Wave Forceson Offshore Structures, Van Nostrand Reinhold, New York,1981.

4. Hoerner, S. F., Fluid Dynamic Drag, 3rd Ed., BookPublished by the Author, New Jersey, 1965.

5. Bursnall, W. J. and Loftin, L. K., "Experimental Investi-gation of the Pressure Distribution About a Yawed CircularCylinder in the Critical Reynolds Number Range", NACATechnical Note 2463, 1951.

6. Norton, D. J., Heideman, J. C., and Mallard, W. W., "WindTunnel Tests of Inclined Cylinders", 1981. OffshoreTechnology Conference Proceedings, Vol. IV, pp. 67-75,1981.

7. Raines, T., "Harmonic Flow about Yawed Cylinders",M.E. Thesis, Naval Postgraduate School, Monterey,California, 1981.

8. Rajabi, F., "Hydroelastic Oscillations of Smooth andRough Cylinders in Harmonic Flow", Ph.D. Thesis, NavalPostgraduate School, Monterey, California, 1980.

9. Bakmis, C., "Harmonic Flow about Cylinders", M.E. Thesis,Naval Postgraduate School, Monterey, California, 1980.

10. Sarpkaya, T., "Vortex Shedding and Resistance in HarmonicFlow about Smooth and Rough Circular Cylinders at HighReynolds Numbers", NPS-59SL76021, Naval PostgraduateSchool, Monterey, California, June 1976.

11. Keulegan, G. H. and Carpenter, L. H., "Forces on Cylindersand Plates in an Oscillating Fluid", National Bureau ofStandards Report No. 4821, 1956.

128

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22314

2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, California 93940

3. Professor T. Sarpkaya, Code 69SL 5Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

4. Chairman, Department of Mechanical 2Engineering, Code 69Naval Postgraduate SchoolMonterey, California 93940

5. LCDR Dean 0. Trytten, USN 225 Shubrick Rd.Monterey, California 93940

129

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