Hydrodynamic viscous force acting on oscillating cylinders with various shapes

TECHNISCHE UNIVERSITELT Laboratorium voor Scheopshydromechanlca Archief

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TeL: 015- 75873 . Fax: 015. 781833

Report

of

Department of Naval Architecture

University of Osaka Prefecture

No.

407

January

1983

HYDRODYNAMIC VISCOUS FORCE ACTING ON

OSCILLATING CYLINDERS WITH VARIOUS SHAPES *

by

Nono TANAKA**

Yoshio IKEDA***

and

Kimio NISHINO****

* published in proceeding of 6th Symposium of Marine

Technology,

The Society of Naval Architects of Japan, Dec. 1982

** Professor,

_{University of Osaka Prefecture}

Research Assistant, ditto

HYDRODYNAMIC VISCOUS FORCE ACTING ON OSCILLATING CYLINDERS WITH VARIOUS SHAPES

by N. TANAKA, Y. IKEDA and K. NISHINO Dept. of Naval Architecture

University of Osaka Prefecture

INTRODUCTION

To predict the wave force on piles and elements of offshore structures Morison's formula has been widely used. The wave force in the formula consists of the viscous drag which is proportional to the square of the fluid velocity

and the mass force which is proportional to the fluid acceleration. The formula is very simple and convenient to use and several experimental results have shown that the formula has enough accuracy in practical usage. Then most of engineers who concern with the design of offshore structures in the field of naval architecture as well as civil engineering have become to use this formula.

In order to use Morison's formula, it is necessary to know the values of

the drag and mass coefficients. It is difficult, however, to obtain them

theoretically because of the viscous flow-separation, and then many experimental researches have been done to reveal the characteristics of unsteady viscous

forces [2-7].

There are various shaped elements of modern offshore structures, for which the drag and the mass (or added mass) coefficients are needed. In this paper,

the results of the measurements of the hydrodynamic viscous forces on cylinders

with various shapes are shown to reveal the dependency on KC number, the attack angle and the radius of rounded edge.

EXPERINENTAL APPARATUS AND ANALYSIS

Tow-dimensional cylinder models were forced to oscillate horizontally in a

small water tank, which is 4m long, O.57m broad and O.9m deep as shown in Fig.l. To avoid the effect of free water surface, the models were submerged deeply

enough (9 times depth of the model width) . The hydrodynamic force acting on the

cylinder was measured by strain gages on the supporting arms. The dimensions of the cylinders used here are O.55m long and O.05m wide, and the sections are

shown in Fig.2.

-1-The drag coefficient CD and the added mass coefficient Ca is obtained by Eqs.(l) and (2) using Fourier analysis like Keulegan et al [2] and Sarpkaya [3].

3 ,27t

F cosO

_o CD

₌

4

pULD

d 2T 27

F

sinO Ca

₌

t-

_PUmLD2

dO (2) (1)

The term F denotes the measured hydrodynamic force in the same direction as the

motion, Um the amplitude of the velocity of the motion, L the length of the cylinder, D the reference breadth of a cylinder and T the period. Note that the

Ca in Eq. (2) is the added mass coefficient of the cylinder oscillating in still

water and that it is different from the mass coefficient Cm in oscillating flow. To obtain the hydrodynamic force F from the measured output of the strain gages, the inertia force, the added mass force and the drag force which are acting

on the supporting arms, and the inertia force acting on the cylinder must be subtracted from the output. The former were obtained by the forced swaying test

for the supporting arms only, and the latter was calculated using the measured

mass of the cylinder.

3. RESULTS AND DISCUSSIONS

The forced oscillation apparatus used here makes ari accurate sinusoidal motion with amplitudes up to 0.7m and periods from 0.8 to 3 sec. In this period range, the high-frequency components are within at most 5% of the base-frequency component.

In order to check the accuracy of the present experimental results, the

results of CD and Ca for circular cylinder are compared with the Sarpkaya's

data. The agreement is so good except at low Reynolds number that it can be said that the accuracy of the present measurements is satisfactory.

In Figs.4 thru. 14, the measured values of _CD and _Ca are shown. In

Figs.15 thru. 21, the experimental results are arranged with sevaral parameters

like an attack angle, a cross-sectional aspect ratio and the radious of rounded

edge of a cylinder.

3.1 CYLINDER A SERIES

In Fig.4, the effects of Reynolds number ( Umax'(D/\)Umax the maximum speed) on CD and Ca of the cylinders of A series are shown. Both of CD and Ca are almost constant, which indicates that the Reynolds number effect of thos

cylinders can hardly be seen in the Reynolds number range of the present

experiments. In Fig.5 thru. 14, therefore, the mean values of CD and Cal which are measured at several different Reynolds numbers, are shown.

-2-Fig.5 shows the effect of Kc number ( umax*T/D2TrYa/D, where denotes sway amplitude) on _CD

and

Ca The drag coefficient CD decreases as K number increases. The CD for cylinders with small radius of rounded edge decreases rapidly as Kc number increases in the region of low number, and then becomes almost constant for greater Kc numbers. On the contrary, the _{CD values} for

cylinders with large radius of rounded edge decrease gradurally as Kc number increases in the whole range of K number. The added mass coefficient Ca for a

cylinder with a small radius of rounded edge has a maximum value near Kc=2O. As

the rounded-edge radius increases, the CD curve becomes more flat, and Ca for a

circular cylinder has a minimum near KC=20.

In Fig.6, the effect of attack angle on CD

and

Ca for square cylinder is

shown. Note that the constant length h is used as the reference breadth D Lin the difinition of _CD

and

_Ca for each attack angle. The pattern of Ca curves is different with the attack angle O . The Ca with attack angle has minimum points,

and the KC number at the minimum Ca becomes larger as the attack angle

approaches to O deg or 90 deg. This tendency of Ca was pointed out by Bearman et al [5]. The reason why the Ca shows such a different trend with the change of

attack angle is not clear at present.

3.2. CYLINDER

B, C, D,

E SERIES

The measured CD

and

Ca for cylinders with various aspect ratios and various radii of rounded edge are shown in Figs.7 thru. 14. The constant length h is

also used as a reference length D in the definitions of CD

and

Ca

Figs.7 thru. 10 show the results for cylinders with sharp edge. The CD

decreases as KC number increases except for the case of 0 =90 deg for cylinders

Bi

and

Ci.

The Ca varies complicatedly with hump and hollow as shown in Figs.7 thru. 10.

The results for the cylinder with various radii of rounded edge are shown

in Figs.11 thru. 14. The CD value at large KC number decreases as the radius of rounded edge increases, those values for Cylinders

_{B and C}

at low KC number have not always such a tendency. As was pointed out previously by the authors

[6], this may be due to the existence of separation bubbles at the side of the cylinder.

3.3. EFFECT OF ATTACK ANGLE ON CD

In Figs.15 thru. 17, the measured values of CD are arranged in terms of the angle of attack O

-3-Consider that a cylinder oscillates sinusoidally with the velocity V(Vm*CoSWt) and with the angle of attack 6 as shown in Fig.18. Then, the

viscous drag F in the same direction as the motion and with the same phase as

the volocity of motion can be described as,

F

=

4c0

(Ka, O)SIvIV

(3)

where _CD is a function of and O - Linearlizing Eq. (3) by an equivarent linearizing technique, the following force components in the directions of x and y can be obtained respectively,

F =-PCD

(K,O ) S V V (4)

Fy=_PCD(Kc,6)SVmVy

(5)

where, V and V, are the x and y component of the velocity vector V

respectively,and V the amplitude of V.

In the linear ship-motion theory based on the assumption of small amplitude of motion, it is generally assumed that the viscous forces are proportional to

the square of the velocity components. Then, the force components are described as follows in equivalent linearized forms,

F

_L

P CD (Ka, ?) S Vxm V (6)

Fy ±. P CD (Ka, 90°) S 2Vym Vy (7)

3r

where _Vxm and V denote the amplitude of V and V, respectively. These forms

are derived from the point of view of a cross-flow-drag assumption. If the

constant length h is used as a reference length D in the definition of CD , the CD on the basis of the assumption becomes the form,

CD=CD(Kc,0°)cos3 O+CD

(K 90°) sin3 O (8)

The values predicted by Eq. (8) are shown in Figs.l5 thru. 17. The predicted

values are different from the experiment, and the deviations between them increase with the decrease of K number and as the cross-sectional aspect ratio becomes close to 1. As mentioned above, the cross-flow-drag concept sometimes brings much smaller viscous drag of cylinders as shown in the figures. It is better to use Eq. (3) or a set of Eqs. (4) and (5) with the resultant velocity

amplitude Vm, and with CD which depends on angle of attack.

3.4. EFFECT OF ASPECT RATIO OF CROSS SECTION

Fig.l9 shows the effect of aspect ratio of cross section(=d/D) on _CD and

Ca Note that the projected breadth of a cylinder is used as the reference

length D in the definition of CD and _Ca in the figure. As shown in the

figure, the CD increases with the increase of d/D at first and reaches a maximum

-4-value near d/D=O.3, then decreases to a constant value. The maximum value of CD near d/D=0.3 increases with the decrease of K0 number.

The Ca decreases as d/D increases and reaches a minimum in the range of d/t0.5 to 1.0, and then increases gradually. The effects of K0 number on CD and Ca are significant when d/D is smaller than 0.5. In Fig.19, the drag coefficient in the steady flow measured by Nakaguchi et al [8] is also shown, in which the

fact that the CD has a peak at d/D0.7 is often called the critical geometry [9]. Comparing the present result with the steady-flow result, they are farily

good agreement when the d/D larger than 1, while the value of d/D where CD has a maximam value is shifted to a smaller d/D value in the present unsteady flow.

3.5. EFFECT OF RADIUS OF ROUNDED EDGE

Figs.20 and 21 show the effect of radius of rounded edge of bluff

cylinders on CD and Ca

The CD for square cylinders in Fig.20 decreases rapidly with the increase

of radius of rounded edge, and this tendency is similar to those in steady flow [8]. The Ca also decreases with the increase of radius of rounded edge, and the

decreasing rate depends on _KC number. The CD and Ca for cylinders in Flg.20 generally decrease with the increase of radius of rounded edge. At _Kc=l0,

however, CD has a little different tendency. As was pointed out by the authors [6], the drag coefficient CD of bluff cylinders varies in a complicated manner with a slight change of the cross-sectional aspect ratio at low KC number. It may be necessary to study on the viscous forces at low K0 number in more

detales.

4. CONCLUSION

The drag and the added mass coefficients were measured by forced

oscillation tests for submerged bluff cylinders with various shape. The

effects of the aspect ratio, of the radius of rounded edge and of the angle of

attack on the drag and the added mass coefficients are shown. The

experimental results presented in this paper will be useful in the design

stage of offshore structures. However, many problems on the viscous forces

acting on the oscillating bluff bodies are left to be studied in the future. The authors should like to express their gratitudes to Prof. Yoji Himeno

for his valuable suggestions and discussions.

A part of this study was supported by the Grant-In-Aid for Scientific

Research, The Ministry of Education, Japan.

-5-REFE RENCE

Iwagaki,Sawaragi : Coastal Engineering, Kyoristu-Syuppan, 1978, (in JapanE

Keuleqan and Carpenter : Forces on Cylinders and Plates in Oscillat

Fluid, J.R.N.B Standard, 1958

Sarpkaya : The Hydrodynamic Resistance of Roughened Cylinders in Harmc

Flow, The Naval Architect, No.2, 1978

Koderayama : A State of The Art in The Resent Research on The Hydrodyn

Ocean Engineering in Japan (III), Bulletin of The Soc. of Naval Arch. Japan, No. 625, 1981, (in Japanese)

Rearman, Graham and Singh Forces on Cylinders in Harmonical]

Oscillating Flow, Mechanics of Wave-Induced Forces on Cylinders, Pitman Advanced Publication, 1979

N.Tanaka, Y.Ikeda, Y.Himeno and Y.Fukutomi : Experimantal Study on

Hydrodynamic Viscous Force Acting on Oscillating Bluff Body, Jour. of The Kansal Soc. of Naval Arch., Japan, No.179, 1980, (in Japanese)

K.Kudo, A.Kinoshita and M.Nakato Experimental Study on Hydrodynamic Forces Acting on the Oscillating Rectangular Cylinders, Jour. of The Kansai Soc.

of Naval Arch., Japan, No.177, 1980, (in Japanese)

H.Nakaguchi, K.Hashimoto and S.Muto : An Experimental Study on Aerodynamic

Drag of Rectangular Cylinders, Jour. of The Japan Soc. for Aeronautical

and Space Sciences, Japan, No.168, 1968, (in Japanese)

H.Nakaguchi : Critical Geometry, Nagare, Japan, Vol.10 No.4, 1979, (in

Japa-nese) -6-se) ing

flic

mi c of y

forscillation

L:

1maximum sway amp.

0.7m i

cover plate

1. 0m

supporting aim

cylinder

strain-gage

V

1.Om

4. 0m

Fig.l

Apparatus of Experiment

S.XpUT1AD JO s[n3T1d TPdTDUT.Id

5-rj

10 00

0 T0 00 E0

O 1.0 00

0

0 10 00 0.1 80

000

0io 1oOOoo

rj T Ecj ci 1 fi EH f 6 B L 9

weu

puriçû

00

OO 0

0.0 X 100

00 X

1UO

0

O X

Q Q

O.0

X ÇQIJ

(w)

q x p

FO

0

£0

01

r.

U

k-u11oJ JPUTTÄD

Ç

q

Ca

A

A.

AA

5.0

_R

_{X 10} -L

10.0

10.0

Fig.3

_{CD and Ca of circular cylinder}

-

measured

Sarpkaya

present

40

Ca o 3 2 i o

- - -

__

-circular cyl. (Ao)

I I I I I I

4.0 5.0 6.0 7.0 8.0 9.0 IO

--

-square cyl. (A1)

- -

-:

-"circular cyl.

I I

I

I

t

I

4.0 5.0 6.0 7.0 8.0

Fiq.4

Effect of Re-No. on CD and Ca

9.0 flxl04

line

name

2r/h

CYLINDER

A

series

_{A1 0.0} 5 A2 0.05 A3 0.1 A4 0.15 CD _{A5 0.2} 4 A6 A7 0.3 0.5 h

[<>1

A8 0.8 d Ay 1.0 3

characteristic length D = h

K=40

2 i -

_{quare cyl.}

i

circular cyl.

O I I I I I 0 _{10 20} 30 40 50 60 _Kc

Fig.5

_{CD and. Ca}

-CYLINDER /\ series

D=h

t

me

name

2r/h

A1

0.0

A2

0.05

A3 0.1 A4

0.15

A5

0.2

A6

0.3

A7

0.5

A8

0.8

A9

1.0

R

for A9 = 42000

square cyl. (A

-circular cyl. (A9)

I I

0 ₁₀

20 30 40 50 60 I

3-2

square cyl. (A1)

h

2

d 3 2 i 0

i

CYLINDER A1

D

A

4

A

®=O,9ûdeg.

h

Bearman et. al.(O9Odeg.)

o t I I t t I O 10 20 30 40 50 60

1k

e=40 , 5odeg. .1 0=20, 7Odeg.

5üdeg.

A

Bearman et.al. (O=9Odeq.

I I I I t 0 10 20 30 40 50 60

Fig.6

_{CD and Ca}

line

O (deg.) 40 50 70 , 90 , , A.

-

20 s O 3 2 i o

CD 5 4 2

i

o o

CYLINDER B1

e

D=h

--

---a

-Fig.7

_{CD arid Ca}

line

10 20 30 _{40 50} e

(deg.)

90

60

40

20

O

O=9Odeg.

C=6Odeg.

e=4 o cTe 60

e=2Odeg.

e=Odeg.

I I I I t I 30

₄₀

_{50 60}

--

0-6Odeg.

0=4ûdeg

-

---0=2Odeg.

- _____ /

_O=Odeg.

I t I I '<C 0

10

20

3 Ca 2

-'

I

i

- -%

o,

I

CD 5 4 3 2 S%

_d

s'

1

/

s'

CYLINDER

C1 4---5% 5%

D= h

s-

5-line

O (deg.) 90 60 40 20 o

-de

--. -

O=40deg.

--. ..O=

9e.

%.__.5____

-i - -

- -

O=2üdeq.

e=odeq.

I I I I I I 0

10

20

30

40

50 60 Kc -4---.

'-4-.

.---4-

O=60deg.

-5. j

i..--

le = 20 d e 0 10 20

3r

40

50 60

Fig.8

_{CD and Ca}

t I ---..' o 3 Ca 2 1

R

O=6üdeg.

OOdeg.

f I t t t t 0 10 20 30 40 50 60

4_ --K

.*-A

-.-H=40deg.

\Y

----' p..s*5%.

/'

0=2odeg.

- e'

-0deq

D= h

Ficj.9

_{CD and Ca}

line

0(deg.)

90 60

..--

40 20 o 10 20 30 40 50 60 Kc -'-II e-._-

0=90deg.

I

4-

_'s

Us.

'.5

'

R 3 2 . i

AO6Odeg.

fj2eg.

A

R

CYLINDER

I-3 Ca 2 1 o

\%

\

CYLINDER

'

u -.

O=9odeg.

-

_{- -L}

O6Odeg.

-

_-.--..----

-

s---

O = 40de g.

_----4

D= h

ft

/

---

?...2 (H =

z'

A 5- .5

line

-

A

--O (deg.) 90 60

40

20 o

s

O=90deg.

_J

-..*-'

O=60deg.

O=2ûdeg.

0= Ode

O=40deg.

-.$---______

s 0

10

20

-

30 --

40

Too

-Fig.l0

_{CD and Ca}

60 _Kç u

-u 0 10 20 30

40

50 5 CD 4 3 2 1

CD 4 2 i o 2 i o

---

--// /

--

_E_3. 0 10

D= h

-

-20 30

Fig.11

_{CD and Ca}

40 50 60 B2 60 B4 Kc

CYLINDER

B

series

line

name

2r/ h

0=Yüdeg.

_B1 O B2 0.1 B3 0.2 B4 0.5 o lo 20 30 40 50 3 Ca - .B1

2 i o 3 2 i o

f

CYLINDER

C

series

- - - - -. - - .

-e=9odeg.

D=h

-Fig.12

_{CD and Ca}

line

name

2r/h

Cl O

cl

C2

-

--

.

-I 60 0.1 0.2 0. 3

____ -

C2

---- C3

- --- C4

I

I Kc I 60 40 30 50 20 O 10 o

lo

20 30 40 50

CD 5 4 3 1 o 3 2 i

CYLINDER

D

series

I I 10 20 30

40

D2

-- .-- -- .--

-

D3

---I 50 D1 D2 C

8=9odeg.

_line

_name

_2r/h

D=h

D1 O 0.1 0.2 D3

0'

I t I I I I o 10 20 30 40 50 60 K1.

ig.l3

CD and Ca

2 D3

o 2 i O 10 20 30 40 50 60 _Kc

CYLINDER

E

series

t'

sharp-edged flat plate

measured by Bearman et.al.

-8=9ûdeg.

D=h

- - -E

E2

sharp-edged flat plate

measured by Bearman et.al.

- ---E2 I I I I I I 10 20 30 40 50 60 Kc

Fig.14

_{CD and Ca}

o O

name

2r/h

E1 O E2 0.1

line

3

0

45 ₉₀

O (deg.)

Fig.15

Effect of Attack angle

o

CYLINDER

A1

Kc

measured

cal.(eq.(8))

A

40 70

RI

e

e

R

5

A A

CD 4

A

A

5

CD

4

i

CYLINDER

_Bi

KC

measured

cal. (eq. (8))

10

A

40

U

__--70

A

U

0.

I 0

₄₅

_0(deg.)

₉₀

F'ig.16

Effect of Attack angle

3

2

cai.(eq. (8))

CYLINDER

C1

A

,/.

.

-o 45

Odegj

₉₀

Fig.17

Effect of Attack an1e

Kc

measured

io

A

40

I

motion

(velocity=V)

4 2 1 0 3

RECTANGULAR CYLINDERS

2r/h=0

,

0=0

D= projected width

O

.

.

R

ARR

-A

A

CD in steady flow

measured by Nakaguchi

et. al. D

KC

mesaured

10

A

40

_R

70 d

A

Oa I t i I _J 0 1.0 d/D 2.0

Fig.19

Effect of Aspect ratio

On CD

and Ca

3 3

I

A

A

a

.

AA

A

R

_u

R

_A

RO. .

1.0 d/D 2.0

R

.

I

0

o

CYLINDER

A

series

A

'ii

CD in steady flow

a

I

_I

.

by Hoerner

\/by

Nakaguchi et.al.

A

a

-a

0.5

RC

measured

io

A

40 70

A

a

a

e

2r/h

A

a

.

i t 1.0 L I . L 0.5 _2r/b 1.0

Fig.20

_{Effect of Rounding radius On CD and Ca}

2-A

A1A

a

a

A

lu

I

1.

o 0

o

CYLINDER

B series

Kc

measured

io

A

40

u

70

A

u

A

_u

*

_.

0.2 0.4

_2r/h

A

A

I,

_u

A

0

02

0.4 2r/h CD Ca

CYLINDER

C series

u

O

A

u

I

A

Fig.21

Effect of Rounding radius on CD and Ca

CYLINDER

D

series

A

A

O

0.2

A

A

u

A

A

u

u

_R

0.2 0.3 2r/h o