The interaction between a vertical cylinder and regular waves

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THE INTERACTION BETWEEN A VERTICAL CYLINDER AND REGULAR WAVES.

by G.. van Oortmerssen Netherlands Ship Modél Basin

Wageningen

Summary

The huge amount of offshore activities in recent years has increased the need ô re1iáblé' data conOenthg the wave loadinq on. vértical cylinders.

A review is given of dif:ferent methods, according to which the wave forces on à vertical citcular cylindér can be cal-culated. NumeriOäl results of these methods are compared with the results of modelexperiments.

The wave diffraction around a cylindrical object was calcu-lated with the potential theory.and.iscompared with mea-suréments of the wave. äínpli.tudê.

1. Introduction

The exploration Of the natural résources of th sea and sea böttòfn is now in a phase of great expansion, not only wheré thé quanti.ty of activitiés is concerned, büt also with regard to the chaactéÎ of the activities there is a tendency towards greater diversity.

This differèntiation in the náture of offshore activities is allIed tO a variety, of shapes and dimensiOns, uséd for the structures which

TECIE UIVTT

Laboratodum vaor Stheepahydrornechaitca rchtef Mekelweg 2, 2628 CD Deift TeL: 015-786813. Faxe 015.7813

G.. van Oortmerssen are required.

Ail started with the drilling and exploitation of oil wells in rather shallow water, for which mostly platforms were used, built on cylindrical piles.

Nowadays, the vertical circular cylïnder is stilJ. a frequently oc-curringshape -in the ocean industry, but now also cylinders with a

large..diaiieter are used.

Orieof.the.,most important data for the design of structures for the océan industry is the magnitude of the forces, experienced in waves. Morison (ref. 1) has given a method for the calculation of the

ho-rizontal force, exértéd by waves upon a. pile. However, th-is method can only be uséd if the diameter of the pile is small compared to the wave length.

For cylindrical objects of large diameter a diffraction theory must be used for the calculation of wave excited forces. Using a

diffrac-tion theory it is also possible to calculate the constant resistance añd the deformation of the incident- wave.

Sorne methods will be discussed for the determination of the diffe-rent aspects of the interaction between a vertical cylinder and regular waves, while sorte numerical results will be compared with the results o-f model experiments.

2.: Wave excited forces and, moments on a vertical cylinder

A vertical circulai cylinder with -radius a, dráugt h in water of depth.d (see Fig. 1), in waves experiences a horizontal force in the direct-ion of thé wave propagation, a vertical force and a horizontal -möment.

In òert.ain conditions also forces can occur in thé direction per-pendicular to the wave d-irection, induced by the shedding of eddies, but these forces will be neglected now.

For the wave excited forces and moment three aspects are of impor-tance: 1.- inertia

gravity viscosity

The relative importance. of each of these aspects depends on the ratio of wave amplitude to cylinder radius (i/a) and- on the ratio

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of wave lerigth.to cylinderradius (ka;k is the wave number 2ir/X) as indicated in Fig. 2.

The influence of gravity effécts due to the deformation of the in-cident waves is of growing importañce for increasing values of ka. Gravity effects result in ocillating hOrizontal and vertical for-ces and in a constant resistance förce in the direction of. wave propagation.

The valué of Zia, above which the effect of viscosity is not negli-gible, depends upon the values of the drag coefficient CD, the

ra-tion of, wave length tO waterdepth kd and the ratio of cylinder draft tO waterdepth hid.

In Fig.. 3 the limit of

Z/a

is given, at which the relative impor-tancé of the horizontal visçous force. amoünts to 5 per-cent of th amplitude of the total horizontal force. This limit was calculated

assuming à value of the drag coefficient being 0.8.

The influence of viscosity on the vertical force is negligible The total wave excited forces can be found by adding the viscous force

(if necessary) to the inertia and gravity forces according to the potential theory:

F(t)

=

cin

-4- c 4- ' -4' _{' COS wt}

xp'

xc xd

F(t)

F

sin (wt +)

N(t)

=

sin (t +

i) -

lcFxc_ldixd cos ().t. ¡cos w

in which: is the amplitude .of the oscillating hor.izonta-1 force according to the potential theory.

e is the phase lag between wave and

F is the constant resistance force.

'xd is the amplitúdéof thé .hôrjzòntal viscous force.

is the ainplitüde:Of the oscillating vertical force according to the pOtential theory.

n is the phase lag between wave and

is the amplitude of the oscillating momént according to the potential theory.

p _{is the phase lag between wave and M.}

is the distance between the point of application of the constant Ìesistance force and the still water level.

XI 3

g.

w

cosh k(d + z) cosh kd

= incident wave amplitude

w = circular frequency

g = acceleration of gravity

k = wave number = 2Tr/A

A = ate length

w and k are related by the dispersion equation:

w2 =gktanhkd

is the potential of the scattering waves, deséribing the dis-turbance of the flow by the presence of the cylinder.

is a solution of the Laplace equation 05(x,y,z,t) = O

which satisfies the following boundary conditions:

at the sea bottOm: _z=-d = O

in the free surface: g()

₌

=0

at the surface of the cylinder: = O

If the velocity potential is known, the pressure in any point at. the surface of the cylinder can be calculated with Bernoulli's theorem:

p = p(c(t) ½V2

t

G. van Oortmerssen

is the distance between thé point of application of the drag force and the still water level.

2.1. Oscillating wave forces accordin9 to the oténtial theory The f:low around, a vertical circular cylinder in regular sinusoidal waves can be characterized by the velocity potential:

ø=.øi+øs

øi is the potential of the incident waves: sin (kx - wt)

in which

=

Now the osciliating horizontal and vertical force and the

oscilla-ting horizOntal moment on the cylinder can be found from the

linea-rized pressure:

F(t) = .rr_p}

cos (n,x)

₌

sin. (wt+c)

F(t) =

cos (n,z) ds

₌

_zp

(wt+fl)

M(t)

=

íj-4

z.cos(n,x)

x.cos(n,z)

_ds

sin (wt+u)

At the NSMB a computer program has been developed for the nuìnerical

calculation of the velocity potential of the flow around an object

of arbitrary shape in waves, using a source distribution for the

representation of the object.

Sorne results of calculations of horizontal and vertical forces

on

a cylinder are given in Figs; 5 and 6.

Havelock (see Ref. 2) has given añ analytical solution of the velo

city potntial for the flow arOund an infinitely long cylinder in

unrestricted water.

MacCarny and Fuchs (see Ref. 3) and Fiokstra (Ref. 4) have adapted

this. solution for a cylinder extending to the. bottom in

_hallow

water.

hey give for the oscillatIng horizontal forçe

on the cylinder:

F(t)

4pg

_{tanh kd}

k2

_{'J12,r(ka) + Yi2,r(ka)'}

Yi,r(ka)-Ø) 2

30)2

r;

(3 .

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the oscillating horizontal force on a circular cylinder with draught

h can be approximated by

sin (wt+c)

Since F

(t) is found by integration of

_-

_{and since the}

distribution of

ovèr the waterdepth is given by

cosh .k

(z±d)

cosh kd

cosh k(z+d)dz

xp

cosh k(z+d)dz

or F(t)

sinh

cöshkd

k(d-h)

'xpsin(wt+c)

in which

_xp

is the amplitude of the horizontal force on an

infinit-ely long cylinder iñ unrestricted water. Fxp is given in Fig. 4 on

a base of ka. According to this approximation the point of

applica-tion of the oscillating part of the horizontal force can be found

from

which results in:

kh sinh k(d-h) - cosh kd + cosh k(d-h)

x

k{sinh kd

sinh k(d-h))

It is not possible to calculate the vertical force oi the

cylinder.

with this method.

In ref. 5, Garret deduces an approximation of the velocity potential

of the flàw around thé cylinder, using variational principles. He

gives ntirnérical results of the calculation of the vertical

and

hori-zontal force and the moment on the cylinder.

He claims that the error in his results is less than one, percent.

Fig. 5

shows the oscillating horizontal wave force for hid = 0.67

and h/a = 1, according to the source theory (NSMB corüputer prograrn,

the approximation using Havelocks theory ánd the values, given by

Garret. In thïs Fig. also the values are given, which were

obtained

from cross-fairing of the results of a great number of measurements,

which were performed at the NSMB with different cylinders.

'In this case the viscous force can be neglected, so that. a direct

comparison can be made. The vertical forces for the same case

ac-cording to Garret., the NSMB computer program and as obtained from

experiments are given in Fig. 6.

From Figs.

5 and 6, it becomes obvious that the agreement between

the measured values and the values, calculated with the

potential

o

f cosh k(z+d).z dz

-h

f°cosh k(z.+d) dz

-h

G.van Oortmerssen

4gh

tanhkd

k2

/J2 ,r(ka) + y2

r(ka)

I

XI, 6

sin(wt-4-c)

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The approximation of the horizontal force,using Havelock's theory, approaches the exact solution for ka

In the particular casé, given in Fig.. 5,the difference between this approximation on the one side and the results of Garrets theory and the source theory on the other side, is already very small for ka>1. 2.2. The constant resistance force

Besides the oscillating wave force ari object in regular waves ex-periences a constant resistance in the directión Of propagation of the wavê. The manitude of this force is proportiOnal to the square of the wave height.

In most cases this force is small in comparison wïth the oscillating force, but especially for big anchored structures it may be of

do-rninan.t interest.

Since the constant resistance force chànges with the wave frequency, this resistance force is no more constant in irregular seas and is knòwn then as the slowly oscillating drifting force, which has a, period of oscillating of the order of magnitude of ten timés the mean wave period.

According to Bernoulli, the total pressure upon the surface of the c'linder amounts to

p.= p(c(t) ½v2

-As was stated in 2.1, the oscillating wave forces can be found by integration of

Now the constant resistance force can be calculated by integration òf the term -½ V2 over the surface of the object:

= f5f-½p (__)2 +

(Ø)2

+ Cos (n,x) dS

Havelock has calculated the constant resistance force in regular waves of small amplitude for an infinitely long circular cylinder

in water of unrestricted depth.

He found the following expression for this force:

yZ

_(ka))

n+1,r n(n+1) }2 : =° k2a

_{n,rn2,r (ka)}J1(ka)}

XI, 7 + F

4pg2

îr2k 9a2

Since the error, involved in the above mentioned _assumption, dimi-nishes for great values of _{ka, and since the constant resistance} becomes Only of importance for great values of ka, this formula _will give satisfactory results _{for practical applications.}

The constant resistance force is applying on à distance lc béneath

i

hfsinh 2k(d-h) -kh) - 2k {cosh 2kd _{- cosh 2k(d-h)}}

c _{sixTh 2kd}

sirih 2k(d-h) + 2kh

Model tests have been performed at the NSMB in order to verify _the theoretical values of theconstarit _resistance.

From registrations on magnetic tape of the total resulting horizon-tal wave force, récorded _{during model experiments with different} cylinders, the average value was determined with the aid of _a computer.

In general, this measured mean force agrees well with the computed value, as can be seen from the example, given in Fig. 7.

2.3. The viscous horizontal force

The viscous horizontal _{force on a vertical cylinder can be estimated}

by:

Fxd(t) = _PCd.Vx.lVxj. dz

XI 8 the water surface, which _{is given by:}

In Fig. 4 F<

is given on a base of ka.

Flokstra has adapted Havelock's Solution for shallow _water. Accor-ding to him, for a cyliñder extenAccor-ding to the bottOm _{n water of} depth d, the constant _{resistance amounts to}

F = (1 + 2 kd

xc _s:inh2kd ) F

If it is assumed, that in the general case of a cylinder with draught h in water of dèpth d, the velöcity potential _{is equal} to the potential, given by Flokstra for a cylinder standing _{on the} bottom, it can be deduced _{that in this case the}

cnstänt resistancé can be approximated by:

F = xc G. van Oortmerssen sinh 2 kd - sinh 2k(d-h) + 2 kh sinh 2 kd

.F

xc

LI

-

,.2

22

O

FA=PC

j

X.4 d

w2 -h

which results in:

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Cd is thedrag còefficient in a uniform steady flowand is assumed constant, över the length of the cylinder.

The horizontal velôity is taken equal to the, velocity in the un-disturbed wave:

30i gk cash k(,d+z). cos wt

-x cosh kd

So the amplitude of theviscous force becomes: coshk(d+z) }2dz

côh kd

-d

=pga

sinh2k&-sinb 2k(d-h)-+ 2kh

x d

sïn2kd

The distance from thé point of application of Fxd to the water

level is:.

i

h(sinh 2k(d-h) - kh} - 2k cosh 2kd - cosh 2k(d-h))

d _{sinh 2kd - sihh 2k(d-h) +2kh}

In the last few years a greät number. of experiments was performed at the NSMB in order to determine, the drag coefficient of circular cylinders as a function of the variables h/a and h/d

The measurements were conducted in _{tèady flow conditions. From the} results of these. measurement. the follot.lincT _{regression equation for} Cd was obtained:

= 0.794 + 0.397() + 0.0028 () (h)2 -

O.062()2

This expression is valid for values of the Reynolds number rànging from 3.i0 up to 5.l0 and can be used under the following condi-tions:

0.8 <h/a < 10 and

O.4<h/d<

i

3. Wave diffraction

Another aspect of the interaction between a vertical cylinder and waves is the deformation of the waves by the _{presence of the cylinr.} Up till now not much attention was paid to this _{phenomenon. Hôwever,}

since thère is a development towards bigger structures in the ocean

XI, 9

(h

G. van Oortmerssefl

industry, knowledge of the wave pattern around a structure can be of interest. Such a sttucture, for instance, can bean oil stôragé arid.produòtiOrl plant, standing on the sea bottoirt in shallow water, to which a tanker can be moored to be loaded.

In this case it is of importance to know if there is a region be-hind the structure with lower waves, so that the motions of the

tanker could be favourably affected.

The wave diffraction around an object of arbitrary shape can be calculated by mears of the potential theory.

When the pötential

0 (x,y,z,t)

describing the flow around the ob-ject is known, the resulting wave elevation can be obtained from:

1 3 Ø(x,y,z,t)

(x,y,t) =

--r _{3t z=o}

At the NSMB a computer program is available, for the calculation of the wave pattern around a, circular cylinder standing on the sea bottom in deep or shallow vater.

For this particular case the analytical solution of the potential is given by Flokstra in ref. 4 in cylindrical coordinates:

Ø(r,O,z,t)

_{- ¿j"öh kd}

cosh k(-z+d).e

no

cC(i)cos nO

in which: _{Jn(kr)Yn,rka) - r,r(ka.) Y(kr)} cn=i - Jn,rU '+ ï =1 for n=o n

En2 for nO

In the Figs. 8 and 9, examples are given of computer plots of the wäve pattern 'aroünd a circular cylinder.

The lines in these. Figures connect the points with equal values of the ratio of resultIng wave height to incident wave height.

From these Figures it appears that there is a shadow field behind the cylinder, and that there are also regions where the waves are. higher than the incident wave.

In order to check the results of the theoretical computations the wave pattern around a cylinder was determined experïmefltallY at the

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NSMB. T this end the wave height was measured in several points around a circular cylinder for different wave periods, while the incident wave heights wée measùred before the cylinder was placed in the basin.

The model of the cylindér had a diameter of 0.96 meters, while the waterdepth amounted to 0.80 meters. The results of the experiments were in good agreement withthe calculated values.

As an example in Figs. 10 ad 11 the calculated and measured values of the wave height behind an in front of the cylinder are given for

G. van Oortinersseíi

References

Mbrison, J.R.

:"The forces exerted by surfacéwaves. on

Johnson, J.W.

piles"

O'Brien, M.P and

J..

Petrol.Technol.Arn.Inst. Mining

Engin-Schaaf,. S.A.

eers, 189 p.149, (1950).

Bavelock, T.H.

:"The pressure of water wavçs upon a fixed

obstacle"

Proc. of the

oyal Society of London.

XI, 12

Series A no. 963 vol. 175 p 409-421,

(JUly, 1940)

3

MacCainy, .R.C. and

:"Wave forces on piles: a diffraction

theo-Fuchs, R.A.

Technical Nernorandthii No.. 69, Beach ErosiOn

Board,(1954).

4

Flokstra, C.

:"Wave forces on a vertical cylinder in

finite waterdepth"

NSMB report no. 69-107-WO, Wagenïngen,

(September, 1969)

5

Garret, C.J.R.

:"Wave forces on a circular dock"

Journal, of Fluid MechanIcs 46, p .129,

(1971).

6

Miles, J.W.

:"A nöte on variational principles for

sur-face wave scattering"

Journal of Fluid Mechanics 46, p 141,

(1971)

7

Miles, LW. and

:"Scattering of gravity waves by a circular

Gilbert, J.F.

dock",

Journal of Fluid Mechanics 34, 70.783,

(1968).

8

Black, J.L.

:"Radiatiori and scattering of watér waves

Mel, C.C. and

by rigid bodies"

Special: Projeòts Nomenclature

radius of cylsinder

Cd = drag coefficient d = waterdepth

F = constant resistance fotcè

Fxd _{= amplitude of the horizontal viscous force}

= arnpitude of the oscillating horizontal forcé according to the potential theory

= anpIitude of the oscillating vertical force according to the potential theory

g= acceleration due to gravity

h = draught of c'linder

Bessel function of thé first kind of order n

j _{= derivative with respect to r of J}

n,r n

k = wave number

distance between point of application of and the still water level

= distance between point of application òf _{and the} still water level

1d = distance between point of application of Fxd and the still water level

oscillating moment according to the potential

_theoy

n = normal on surface of thé body

G. van. Oortznerssen

p pressure

total velocity

Weber's Bessel function of the second kind of order n

n,r derivative with respect to r of

£ phase lag between wave and

n = phase lag between wave and p = phase lag between wave and

X wave length

w = circular frequency

p fluid densi-ty

0 = velocity pçtèntial

velocity potential of the incident waves

= velocity potential of the scattering waves = amplitude of the incident wave

L. = amplitude of the resulting wave

s, = surface. of cylinder

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z

direction of wave propagation

XI,15

L

h

(

y

-/

_t

Fig.

i

Definitions.

10 C 4-u w

w

w

00

_u G. van Oortinerssen t t I t t

I--)

0.1 0.2 0.3 0.5 0.6 0.7

ka.

Fig. 2 The reach of the viscous-, inertia- and gravity-influence on wave excited forces.

XI, 16

Line of maximum

-wave steepness

=0.1

Gravitation

Inertia

_{effects become}

dominant

.10

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calculatéd with CD:08

Fig. 3 _{Limit of /a for the neglect of viscosity.}

XI.,.17 kd h,d 1 0.75

0.50

0.25

3 2

i

G. vän Oortmerssen Fxpoo

Fxc

2 Ka

Fig. 4 The constant part arid the amplitude of the priödical part of the horizontal wave force for infinitely waterdepth.

XI, 18 3 1.0 0.6 0.4 0. Fxc 0.8

*P2a

1'

FxP

p ga2 6

Fp

pga2 =O.67 a Special Projects

experimental

NSMB computer program

Garret

approximation

using

Havelock's pctential

0.2 0.4 0.6 0.8 1.0

Fig. 5. Oscillating horizontal wave force.

N

N

=Ò.67 h

\\

pga G. van Oortirerssen

experimental

0.2 _{Q4 0.6 0.8 10}

Fig. 6 Oscillating vertical wave force.

XI, 20

NSMB computer

prOgram

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xI,21

computed

0.3

4

Q5

Fig. 7 Constant resistance force on a cylïnder. a3 . Q2 0.1 0.6 Q5 Fx c

2p2a

OA

2-o o

G. van Oortrnerssen

Q5 to 15 2.0

Distance before the cylinder in meters

Fis. 11 Wave heïght in front of a circular cylinder.

XI,24

resulting wave amplitude amplitüde. of the incident wave

ka 4 hid =1 h,a=i.67 2a 096m calculated

G. van Oortmerssen

Fig. 8

Fig. 9

XI,22

WAVE PATTERN ARJD A CYLINDER

DIRECTION OF WAVE

PROGAT ION

h/d.

ka j4

WAVE TTERN AROUND A CILINDER

DIRECTION (W WAVE

PROPAGATION

h/d. i

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resultiog wave amp'itude

= an,plitude of the incident wave

ka'4

hid :1 h,a:1.672àC.96 m

cälcutate d

o measured

Distañcè behind the cylinder in meters

Fig. 10 Wave height behind a circuÏar cy1inde.