Proceedings of the Second WEGEMT Graduate School, Advanced Aspects of Offshore Engineering, The Norwegian Institue of Technology, Module 1. Environmental Conditions and Hydrodynamic Analysis, Volume 3

West

European

Graduate

Education

Marine

Technology

VOLUME III

IMPACT FORCES

P I

-

F

Second WEGEMT Graduate School

Advanced Aspects of Offshore Engineering

_{Volume 3}

The Norwegian Institute of Technology, Jan. 1979

Module 1. Environmental Conditions and Hydrodynamic Analysis

PROF. O.M. FALTINSEN

THE NORWEGIAN INSTITUTE OF TECHNOLOGY

WAVE FORCES ON CYLINDERS

IN THE DRPG/INERTJA

REGIME

- IT

PROF. H. LUNDGREN

TECHNICAL UNIVERSITY OF DENMARK

WAVE AND CURRENT FORCES ON RISERS

- 15

PROF. H. LUNDGREN

TECHNICAL UNIVERSITY OF DENMARK

WAVE LOADS ON PIPELINES ON THE SERFLOOR

PROF. H. LUNDGREN

M.SC, STUDENT B. MATHIESEN

RES.ENP. H. GRAVFSEN

TECHNICAL UNIVERSITY OF DENMARK

SHORT-TERM

WAVE STATISTICS

J.A. BATTJES

DELFT UNIVERSITY OF TECHNOLOGY

p1979-5

Delft University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2

2628 CD Delft

Phone: +31 (0)15 2786873 E-mail: p.w.deheer(atudelft.n1 12 P ₁₄

IMPACT FORCES

0. M.

FALTINSEN

Department of Naval Architecture

Norway

Professor of

and Marine Engineering

Marine Hydrodynamics

The Norwegian Institute of Technology

PREFACE

As a contribution to the lectures by H. Lundgren the text below is

presented.

It is an excerpt from a preliminary version, dated June 5,

1978, of lecture notes 'Sea loads and motions of marine structures'

for a course at the Department of Naval Architecture and Marine

Engi-neering, Norwegian Institute of Technology.

Water impact problems might be an important problem for offshore

structures.

An 'example is water impact on horizontal jacket

trusses in the splash zone.

Due to the wave motion those members

will be in and out of the water.

Both the fatigue problem and

the extreme load case are important.

Another example is the

vertical columns of a gravity platform which can be subjected to

slamming forces due to breaking waves.

In order to better describe the physics of the problem, let

us

examine a related and more simple problem.

It is a horizontal

cylinder that is forced through an initially calm water surface

with constant velocity V.

_{We may divide the problem into two}

phases.

One

phase is when the cylinder hits the free surface

High slamming pressures are then occurring.

The other phase is

when the cylinder further proceeds through the

water.

To describe

the first phase properly the compressibility of the fluid and the

air has to be taken into account.

_{If this}

_{is not done infinite}

pressures would be predicted when the cylinder hits the water

surface.

_{If only the compressibility of the fluid}

_{is taken into}

account, it can be shown that the slamming pressure is pVCe where

Ce is

the velocity of sound in the fluid.

For a fluid with no

air content this gives an unrealistic high pnassure.

Chuang (39

and

40) has pointed out that the real pressure might be

the

order of magnitude 10 times smaller for drop tests.

The

air-content in the water may change the velocity of sound

significantly,

from 25 m/s to 800 m/s.

One of the first to take into account the influence of

the airflow

on the slamming pressure was Verhagen (41).

He considered a flat

plate and assumed the water to

be incompressible.

Oakley (42)

has later generalized Verhagen's

calculation for models with

convex geometry.

Verhagen shows in his paper reasonable

agreement between calculations and drop test experiments.

However, the applicability of drop test experiments to the

real case of slamming pressure on a structure in waves is very

much questioned.

Ochi and Motter (43) writes the

slamming pressure on a ship in waves as 0.5 pk1v2 where v is

the relative vertical velocity between wave and ship and kl is

a constant depending on

the form of the body.

In the case of a horizontal jacket truss in the splash zone one

is not particularly interested in the high pressures

occurring

when the water hits the member.

One is more interested in the

force

on the body as it hits the

cylinder and proceeds through

the water.

One of the first calculations of this type was

done

by von Karman (44).

He used the free surface condition 4)

= 0,

where y5

is the velocity potential of the fluid motion.

The

force was written as dt(A33V)

where A33 is the vertical infinite

frequency added mass coefficient for the body as a

function of

time.

Von Karman simplified the added mass

calculations by

using flat plate results.

Fahula (45) and Fabula and Ruggles (46)

did similar calculations, but their added mass

calculations are

more sophisticated than von Karman's calculations.

They were only

able to perform calculations up to the point that the

circular

cylinder bottom has proceeded a distance approximately a radius

down in the fluid.

In Faltinsen et al.

(47)

is shown a method

which is applicable for any cylinder submergence. The vertical

waterinduced force per unit length is written as

(2D) d (2D)

(2D)

A33

(t)

is the two-dimensional vertical

inEnite frequency

added mass coefficient as a function of

submergence.

pgAd is

the buoyancy force per unit length as a

function of submergence.

(2D)

i

Result of A33

non-dimensionali zed by pR

s shown in figure 16,

as a function of

h/R.

R is the cylinder-radius and h

is the

sub-mergence as defined

in the figure.

The asymptotic value of

(2DV 2

(2D)

A33 /(Jt

for large h/Ris

7.

A3 2D)

3

is calculated distributing

sources and dipoles over

the average wetted surface.

The two

force terms in equation (26) are also shown in figure 16 in the

case the forced velocity

is constant.

We can then write

dA33

(2D)

--

[A (20)

(t)V) .

dt

33 dh

V2

and define a C -value by equating

(2D)

dA33

2 p 2

dh

V

=

C52R V

Some of the conclusions in Faltinsen et

al. (47)

were a) The

force coefficient C. is numerically estimated to be 3.1 for a

circular cylinder at the moment of coptact between cylinder and

water.

Experimental values of the same coefficient range

from

4.1 to 6.5. b) The force coefficient Cs is a

function of

cylinder submergence.

Buoyancy forces are important.

c)

Experi-ments indicate that the force is a function

of Froude number.

Reynold's number is not thought to have influence for the

smaller submergence (i.e. Ti <

2).

The corresponding problem in waves has been studied theoretically

by Kaplan and Silbert (48), experimentally by Dalton and Nash

(49)

and Miller (50).

Kaplan and Silbert writes the vertical force pr. unit length as

d (2D)

F =

TIT{A33} + pgAD + PAD

where

t

is the wave elevation at the cylinder.

In reality there

will be a horizontal impact force too.

This becomes more and

more pronounced the steeper the waves are.

The effect of breaking

waves will be an important special case.

3.0

2 0

1.o

FIGURE 16

Slamming forces from breaking waves on large

diameter vertical

cylinders should also be considered.

Most results from slamming

pressure measurements are from model tests, although some

pro-totype measurements are available fur structures

in shallow

water.

Larsen and TOrum (51)

give a literature survey on such

forces giving their order

of magnitude as far as they are estimated

from present knowledge.

Present knowledge does not allow us to

make any accurate predictions of the slamming forces

from breaking

waves on large diameter vertical cylinders.

A major problem in

estimating the slamming force is to accurately estimate

the

kinematics in breaking waves.

Breaking waves have in limited

extent been studied from a theoretical point of view.

But rather

many investigators have studied extreme symmetric regular waves

that is close to breaking.

They have for instance worked at

determining the maximum ratio of wave height to wave length

which a wave can attain without breaking.

All correct solutions

are close to 1/7 for deep water.

As the ratio of water depth to

015 0.10 0.05 0.06 0.04 0.02 Y 0.0 -0.02 -0.04 SCHWARTZ THEORY CHAPPELE AR THEORY

CALM WAIER LINE

0.1

h /A

FIGURE 17

Maximum Wave Steepness (wave height/wave length) vs. Mean Water

Depth/Wave length, as determined theoretically by Schwartz and

by Chappelear.

Schwartz also presents the results. of Yamada and

Shiotani. (1968) whichare esSentiiAlly identical with his.

The linear regular sinusoidal wave theory cannot be used to

predict breaking waves.

This is illustrated in figure 18.

But

for a wave steepness H/A of 0.1 in deep water, the linear theory

is not to bad.

010

1 1 I

T-T

0.08

STEEPEST TRAVELING REGULAR WAVE

FROM SCHWARTZ (1971) SINUSOIDAL WAVE OF SAME HEIGHT

z

z

z

0.05

00

0.08l

I J I I l I 1 1

-0.5 -0.4

-0.3 -0.2

-01

0 0.1 0.2 0.3 0.4

05

X / X

FIGURE 18

7

-006

..

-Profiles of the Wave of Maximum Steepness in Deep Water and of

a

Sinusoidal Wave of the Same Length and Height. Note that the

verti-cal and horizontal sverti-cales differ in the figure.

For more details about breaking waves, see Longuet-Higgins,

Cohelet and Fox (52).

References

Chuang, S.L.:

"Experimental Investigation of Flat Bottom Body

Slamming". Journal of Ship Research,

March 1966.

Chuang, S.L.: "Experiments on Slamming of Wedge-Shaped Bodies".

Journal of Ship Research, September 1967.

Verhagen, J.H.G.: "The Impact of a

Flat Plate on a Water Surface".

Journal of Ship Research, December

1967, vol. 11, p. 211.

Oakley, 0.H.: "An Analytical and Experimental

Study for

Predic-tion of Ship Impact Forces in a

Seaway".

Massachusetts Institute

of Technology, Report no. 69-6, Aug. 1969.

Ochi, M.K. & Motter, L.E.: "Prediction of Slammina

Characteris-tics and Hull Response for Ship Design".

SNAME Trans. 1973.

Karman, Th. von: "The Impact on Seaplane

Floats during Landing".

NACA Tech. Note 321, 1929.

Fabula, A.G.: "Ellipse-Fitting Approximation of Two-Dimensional

Normal Symmetric Impact of Rigid Bodies on

Water".

Proceedings

of the Fifth Midwestern Conference on Fluid Mechanics,

1957.

Fabula, A.G. & Ruggles, I.D.: "Vertical broadside

water impact of

circular cylinder. Growing circular arc approximation".

U.S.Naval

Ordnance Test Station, Cinua Lake, California,

12 October 1955.

Faltinsen, O., Kjrland, O., NOttveit, A. & Vinje, T.: "Water

Impact Loads and Dynamic Response of Horizontal Circular

Cylin-ders in Offshore Structures", Paper 2741, OTC 1977.

Kaplan, P.

& Silberg, M.N.: "Impact Forces on Platform

Horizontal

Members in the Splash Zone".

Paper 2498, OTC 1976.

Dalton, C.

& Nash, J.M.: "Wave Slam on Horizontal Members

of an

Offshore Platform".

Paper 2500, OTC 1976.

Miller, B.L.: "Wave Slamming Loads on Horizontal

Circular

Ele-ments of Offshore Structures".

RINA, Springmeeting, 1977.

Larsen, P.K.: "Some consideration on shock pressures from

break-ing waves against circular fixed columns".

VHL Report no. STF

60 A 75076, Trondheim, 1975.

Longuet-Higgins, Cokelet & Fox: "The calculation of steep

gravity

waves".

Proceedings of BOSS '76, Trondheim, 1976.

H. LUNDGREN

Professor of Marine Civil Engineering

ABSTRACT

The forces on smooth and rough cylinders in purely sinusoidal flow

(model tests) are reasonably well known.

For the design of jacket

platforms there are, however, several factors of uncertainty.

The

following uncertainties must be particularly stressed:

What is the roughness in the various zones due to corrosion and

marine fouling?

What are the time series of orbital velocities (in three

dimen-sions) in connection with the highest natural waves, and how do

the time series vary from point to point in space?

Which effect has the three-dimensional structure of the waves on

the wave force coefficients?

Much research will be required in order to clarify these points.

INTRODUCTION

The literature on wave forces on cylinders is very extended - and

most of it is obsolete, particularly because of recent model tests at

high Reynolds numbers.

_{An excellent survey of the literature up to}

about 1975 is given by Ref. 3.

Therefore, only a list of newer

liter-ature has been included in the present paper.

COMMENTS

ON MORISON'S FORMULA

The Morison formula, that has been in continuous use since 1950,

reads

f=-pCDUUId +pC AdU/dt

(2.1)

WAVE FORCES ON CYLINDERS

IN THE DRAG/INERTIA REGIME

Institute of Hydrodynamics

Denmark

and Hydraulic Engineering

Technical University of Denmark

where

fx =

force in x-direction per unit

length of cylinder,

p =

mass density of sea

water,

Cd =

drag coefficient,

Cm =

mass (inertia)

coefficient,

D =

diameter of cylinder,

1

A

=

T

it

D2 = cross-sectional area of cylinder,

U =

instantaneous velocity in

x-direction of ambient flow,

dU/dt = total acceleration of U

(often replaced by the local

acceleration DU/9t).

Morison's formula is physically correct

in the two extremes:

Steady flow (dU/dt = 0):

Cd depends upon the Reynolds

number

Re = U D/v

(2.2)

and the

roughness ratio = k/D

(2.3)

For a salinity of 35°Ao the kinematic

viscosity v has the

follow-ing values:

0

10o 20o

Temperature

(in °C) 0

106 v

(in m2/s) 1.83 1.35 1.05

The roughness k is defined as the diameter of sand grains glued

onto the cylinder (in model

tests).

Acceleration from rest (U = 0):

It follows from potential theory

that

C = 2 (2.4)

Between the two extremes, empirical values

of Cd and Cm are

usu-ally determined in such a manner that Eq.

(2.1) gives a least squares

fit to the variation of fx recorded, i.e. Cd and Cm are

assumed to be

constant over the wave cycle.

Typical values used for smooth

cylin-ders are:

Cd = 0.7 and Cm = 1.5.

These coefficients depend,

how-ever, on Re, k/D and the Keulegan-Carpenter

number

KC - Um T/D

(2.5)

where Um is the maximum velocity in a sinusoidal

flow of period T.

The following criticism may be raised against the

Morison formula:

(1) For a given ambient velocity, U, the points of flow

separation on

the surface of the cylinder depend upon the pressure

gradient,

which is - p dU/dt.

This is clearly demonstrated by the

varia-tion of the separavaria-tion angle shown in Fig.

1

for a semicycle.

From studies on steady flow it is known that Cd decreases with

(slightly) increasing separation angle, the possible variation

being as much as from 1.2 to 0.3.

For oscillatory flow an even

larger variation can be imagined as a result of the large range

of separation angles.

201

.0

0 + 1600

40o

+

2

ID

4)

t::. 1 2

3 4

5 1 1_ I

III

10 1,5

20

0-3

0

REYNOLDS No 1 5-42 x 219x104 5-42 x104

Fig. 1

Separation angle versus phase of oscillation

for KC =38

(from Ref. 8)

The direct effect of a pressure gradient is a pressure and

accel-eration field corresponding to Cm = 2, but a pressure

gradient as

such can produce neither velocities nor

turbulent stresses.

Per-haps, this is best understood by

considering a cylinder

oscillat-ing in still water.

At any given time, the fields

of pressures,

velocities, accelerations and turbulent stresses are

determined

by the previous development, including the

instantaneous velocity

and acceleration of the cylinder.

If, at the specified time, an

additional force were introduced upon the

cylinder, its

accelera-tion would be discontinuous, and the discontinuity would

corre-spond to a potential pressure and acceleration

field.

The

veloc-ities and the turbulent stresses, however, are not

changed at the

very moment of the discontinuity.

Therefore, Cm = 2 for the

ac-celeration discontinuity.

This is correct however small the

dis-continuity is.

Hence, it seems natural to

a ss um e

that Cm = 2

may be applied to the full pressure

gradient for all flows.

From a physical point of view it is not reasonable that the first

term in Eq.

(2.1) depends only upon the instantaneous velocity.

(For a sinusoidal flow the dependence upon the flow history is

partly represented by KC.)

At each instant where the flow

by-CYLINDER

EQUATIONS FOR 0

,,.

VT= 0 - 0.25 VT =0-25 -0 5 SMOOTH

0

49'111/2)-°.4 126[1:}5-th 1

0 28

90f0 S-th.

]016

SMOOTH 376[½Y°47 47 ,-ROUGH +

47.5[t103

95[0-5-tir ]C 2 0.4

_{0-5 VT}

passes the

cylinder, vortices are

shed into the wake.

These

vor-tices contain shear stresses parallel to

the direction of the

am-bient flow, thus

generating suction on

the lee side of the

cylin-der.

As

the vortices move away

from the cylinder their shear

stresses dissipate at a rate that

is related to their

'vortex

time', D/U.

In addition, the

first term in Eq.

(2.1), must now

be understood as the difference between the

actual force fx and

the intertia term,

with Cm = 2.

This first term is

character-ized by being dependent

upon 'velocity

stresses'

(cf. p U2), in

particular the turbulent

stresses on the lee side (whereas the

total load on the upstream side is normally only a

fraction of

the total drag).

There is no reason to believe that the first

term vanishes when the

ambient velocity U is

nil.

On the

con-trary, when U = 0 at

the end of the stroke,

there is still so

much turbulence left in the wake, cf. also the

small separation

angles in Fig. 1, that the suction on the lee

side could hardly

vanish.

Hence, the first term should be replaced by a

suitable

integral over the past.

Just how this integral

could be

con-structed is far from

evident at present.

As

dimensionless

param-eters it will contain Re,

k/D and the

dU

pressure gradient

ratio D --/U-

_dt (2.6)

In addition, the

integral should contain a

turbulent 'memory

function'.

(4) Eq. (2.1)gives smoothed

values, disregarding

the oscillations

connected with the alternating

vortex shedding.

These force

os-cillations are of significance for vibrating structures

and their

fatigue problems.

(This statement applies to a much higher

de-gree to the

oscillations of the transverse

force, fr.)

3. PHYSICAL

ASPECTS OF

OSCILLATORY FLOW

AROUND A CYLINDER

Fig.

2 illustrates the average

motion of a sinusoidal

flow

around a cylinder.

The normal stroke of the water that is not

dis-turbed by the cylinder goes to a2 with the trough

of the wave and to

al with the wave crest.

For both trough and crest

the maximum

veloc-ity occurs when the water considered by-passes the

cylinder.

The

pressure gradient is directed towards the right

from trough to crest,

and towards the left from crest

to trough.

With the trough flow a wake is formed to the left

of the cylinder.

Since the velocities in the wake are smaller than in

the undisturbed

STROKE

)--CREST VELOCITY

TROUGH

Fig. 2

Sinusoidal motion around cylinder

gradient before the undisturbed flow is stopped at a2.

Hence,

the pressure gradient directed towards the right will accelerate the

water around d2 toa higher velocity than that of the undisturbed

water.

The water near d2 will thus pass around the cylinder on both

sides with a velocity that is increased due to the circumstance that

it originates from the trough wake.

Since the velocity in the

bound-ary layers at the points of separation has a decisive influence on

the suction on the lee side (right side), it can be expected that the

wave force, at least under certain circumstances, exceeds the force

from a steady flow.

The water that starts its right-going motion near d2 is

acceler-ated to a high velocity at the cylinder not only by the wave pressure

gradient directed towards the right but also by the flow contraction

at the cylinder, this contraction being accompanied by large negative

pressures.

Because of the high velocity at the cylinder, the water

in question will not be stopped until cl by the wave pressure

gradi-ent directed towards the left, i.e. this water will continue beyond

the normal end, al, of the stroke, and it will be stopped later

than the undisturbed water at al.

Because of this phase shift and the increased stroke at cl, the

water that starts its left-going motion near cl will be stopped at b2

b ef or e

the undisturbed water is stopped at a2, the amplitude b2

being smaller than a2.

Naturally, the shear stresses generated in the wake (and at the

surface of the cylinder) play an important role in reducing the stroke

variations to the values indicated by the points d,

_{c, b, a.}

Only the average motion around the cylinder has been mentioned

deviations from the average motion.

In particular, the

vortices shed

to the left by the trough

flow may by-pass the cylinder

in a very

un-symmetrical manner during the crest

flow.

An example of this is

shown in Ref. 25 as a result of mathematical modelling of the

vortex

shedding (see also Ref. 4).

The main contribution to the

suction on the lee side originates,

presumably, from the shear stresses

along the dotted lines on Fig. 2.

4. STATUS OF

MORISON'S FORMULA FOR SINUSOIDAL MOTION

In the laboratory large Reynolds

numbers can be obtained in two

manners:

By use of an oscillating water

tunnel (Refs.

18 -23).

By oscillating the cylinder in still

water (Refs. 6 and 15).

The most accurate tests with sinusoidal

planar motion have been

made by Sarpkaya (Refs. 18- 23) .

For smooth cylinders the main

results of his tests are shown in Fig. 3, where K

indicates the

Keulegan-Carpenter number KC.

The highest Reynolds numbers suggest

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.2 0.4 20 30 01 0.2 \ K =15 1 1 1

I1II

1 1 _1 1 1 1 0.5 1 2 1 1 1 1

11111

80 Rexto

5

1 1 1 1 1 1

Fig. 3

Drag coefficients for smooth cylinders

a value of Ca = 0.6 - 0.7.

The corresponding mass coefficient is

around

Cm = 1.8.

Ref. 15 gives the results of one test with Re = 2.7

105

and

KC = 18,

indicating

Cd

= 0.53 and

Cm = 1.9.

This test shows a

trans-verse force coefficient as high as

CL

- 1.3, i.e. a maximum

trans-1

verse force of 1.3

T

p

D, where

Um

is the maximum velocity.

This

transverse force appears together with an in-line

coefficient of 0.7,

giving a total maximum force coefficient of 1.5.

Ref. 6 discusses various methods of determining the empirical

val-ues of

_Ca

and

Cm

and concludes that the least-squares method gives

higher force coefficients than the maximum value method.

The latter

is based on the forces measured when the acceleration or the

veloc-ity, respectively, are zero.

If the cylinder and the sinusoidal motion form an

a ng le a

dif-ferent from

90°,

it is customary in the Morison formula to apply the

perpendicular components of velocity and acceleration.

This method

seems to be supported by Ref. 13 in a tentative manner.

In the same

reference it is proposed that the force for a steady current be

mul-tiplied by sin1-5a.

For rough cylinders the drag coefficients are considerably

larger than for smooth cylinders.

The results of Sarpkaya's tests

(Ref. 20) are shown in Fig. 4, giving only the asymptotic Ca-values

at high Reynolds numbers.

A

Cd

1.9

1.8 1.7 1.6 1.5 1.4 1.3 1.2

k/D

1/50

1/100

1/200

1/400

1/800

KC I I I I I 1.10 1

50

100

Fig. 4

Drag coefficients for rough cylinders

at high Re-values

The increase of Cd from a smooth cylinder to a cylinder with a

roughness as small as k/D = 1/800 is remarkable.

For KC = 100, Fig.

5 gives the full variation with Re.

According to Ref. 24, a smooth

1 1 1

1111'

I

I1

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.5 1 ,... ... _{Rex 1-1:55} I 1 I 1

1111

5 W

Fig. 5

_{Drag coefficients for rough cylinders at KC}

_{= 100}

cylinder in supercritical steady flow (very high

Reynolds numbers)

may be considered to have a roughness ratio of k/D

= 3.5

10-5. (In

the paper referred to, the asymptotic value is

_{Ca = 0.9 for extreme}

values of the roughness Reynolds number.)

_{The asymptotic Cd-values}

in Fig. 5 have been extrapolated to this

_{roughness ratio, as}

indi-cated by a circle.

_{Thus, it must be concluded that}

_{a smooth}

cylin-der is consicylin-derably 'smoother' in sinusoidal

_{than in steady flow.}

For Reynolds numbers less than 105, Ref. 14

indicates that a

roughness ratio k/D = 1/50 increases the

_{wave force by a factor of}

1.5 relative to that of

a smooth cylinder.

5. FORCES FROM NATURAL WAVES

The general use of the directional

_{energy spectrum in ocean}

engi-neering has also had its influence on the calculation of wave forces

on cylinders (Ref. 27).

_{Three simplifications are normally}

used in

this application:

(a) The coefficients in Morison's

_{formula are assumed to be}

constant,

independent of frequency, Re, KC,

_{and the three-dimensional}

struc-ture of the waves.

The drag term is linearized.

No force in the wave crest above still water level is included.

Points (b) and (c)

lead, of course, to an underestimate of the

largest wave forces (Refs. 26, 5).

Nonlinear random waves have been considered in Refs. 10 and 16.

Ref.

7 presents the so-called 'hybrid method', where the wave loading

due to nonlinear random waves is represented as the product of a

non-linear wave force (for a high regular wave) and a non-linearized force

transfer coefficient, representing the effect of the directional

spectrum.

It follows from the discussion of Fig. 2 that the coefficients in

Morison's formula will depend on the 'flow history', in particular

on the three-dimensional character of the orbital velocities.

Hence,

thorough studies will be required with respect to both the influence

of the 'flow history' on the wave forces and the three-dimensional

description of the orbital velocities in natural waves.

For two dimensions and under the assumption of linearization, a

deterministic description of natural waves is simple (Ref. 12).

The

extension to three dimensions (with linearization) is being studied

by various scientists.

The extension of the deterministic

descrip-tion to nonlinear waves will present great difficulties.

The special effect near the water surface is discussed in Ref. 9.

A description of the complications of the design problems in

prac-tice is given in Ref.

1.

Eventually, the wave forces on cylinders in natural waves can be

found only by extensive support from field data (Refs. 2 and 17).

REFERENCES

Bea, R.G. and N.W. Lai (1978): Hydrodynamic loadings on offshore

platforms.

Offshore Techn. Conf., Houston, Texas, Paper 3064,

Vol. I.

Bishop, J.R.

(1978):

RMS force coefficients derived from

Christ-church Bay wave force data.

IAHR-Symposium on Mechanics of

Wave-Induced Forces on Cylinders, Bristol, UK.

BSRA (1976): A critical evaluation of the data on wave force

coefficients.

Contract Report W 278 (with App. I-II), The

Brit-ish Ship Research Association, Wallsend Research Station,

Walls-end, Tyne and Wear, UK.

Bullock, G.N., P.K. Stansby and J.G. Warren (1978): Loading and

response of cylinders in waves.

Proc. 16th Coastal Engrg. Conf,

Hamburg 1978, Amer. Soc. Civ. Engrs. 1979.

Burrows, R.

(1978): Probabilistic description of the response of

offshore structures to random wave loading.

IAHR-Symposium on

Mechanics of Wave-Induced Forces on Cylinders, Bristol, UK.

Chan, Yau Wai (1978): Large displacement, high Reynolds number

oscillating cylinder.

M.Sc.-thesis, Oregon State Univ.

Dean, R.G. (1977): Hybrid method of computing wave loading.

Offshore Techn. Conf., Houston, Texas, Paper 3029, Vol. IV.

Grass, A.J. and P.H. Kemp (1978): Flow visualisation studies of

oscillatory flow past smooth and rough circular cylinders.

IAHR-Symposium on Mechanics of Wave-Induced Forces on Cylinders,

Bristol, UK.

Hallermeier, R.J.

(1976): Nonlinear flow of wave crests past a

thin pile.

Proc. Amer. Soc. Civ. Engrs. 102, WW4, 365-377.

Hudspeth, R.T.

(1975): Wave force predictions from nonlinear

random sea simulations.

Offshore Techn. Conf., Houston, Texas,

Paper 2193, Vol. I.

Hutchinson, R.S. (1978): An experimental investigation of wave

induced pressures on a vertical cylinder.

IAHR-Symposium on

Mechanics of Wave-Induced Forces on Cylinders, Bristol, UK.

Lundgren, H. and S.E. Sand (1978): Natural wave trains:

Descrip-tion and reproducDescrip-tion.

Proc. 16th Coastal Engrg. Conf., Hamburg

1978, Amer. Soc. Civ. Engrs. 1979.

Massie, W.W.

(1978): Hydrodynamic forces in waves and currents.

IAHR-Symposium on Mechanics of Wave-Induced Forces on Cylinders,

Bristol, UK.

Matten, R.B.

(1977): The influence of surface roughness upon the

drag of circular cylinders in waves.

Offshore Techn. Conf.,

Houston, Texas, Paper 2902, Vol. III.

Matten, R.B., N. Hogben and R.M. Ashley (1978): A circular

cylin-der oscillating in still water, in waves and in currents.

IAHR-Symposium on Mechanics of Wave-Induced Forces on Cylinders,

Bristol, UK.

Raman, H. og P. Venkatanarasaiah (1976): Forces due to nonlinear

waves on vertical cylinders.

Proc. Amer. Soc. Civ. Engrs. 102

WW 3, 301-316, Aug. 1976 and

104, WW1, 91-92, Feb. 1978.

Roy, F.E., C. Der and W. Gibson (1978): Stress due to wave forces

in an offshore research platform.

Offshore Techn. Conf., Houston,

Texas, Paper 3108, Vol. I.

Sarpkaya, T.

(1976a): Vortex shedding and resistance in harmonic

flow about smooth and rough cylinders at high Reynolds numbers.

Report NPS-59SL76021, Feb. 1976, Naval Postgraduate School,

Monterey, Ca., USA.

Sarpkaya, T.

(1976b): In-line and transverse forces on cylinders

in oscillatory flow at high Reynolds numbers.

Offshore Techn.

Conf., Houston, Texas, Paper 2533, Vol. II.

Sarpkaya, T.

(1976c): In-line and transverse forces on smooth

and sand-roughened cylinders in oscillatory flow at high

Rey-nolds numbers.

Report NPS-69SL76062, June 1976, Naval

Post-graduate School, Monterey, Ca., USA.

Sarpkaya, T.

(1976d): Vortex shedding and resistance in harmonic

flow about smooth and rough circular cylinders.

_{Proc. 1st Int.}

Conf. Behaviour Off-Shore Structures (BOSS), Trondheim, Aug. 1976,

Vol. I, pp. 220-235.

Sarpkaya, T.

(1977): The hydrodynamic resistance of roughened

cylinders in harmonic flow.

Roy. Inst. Nay. Arch.

Spring

Meet-ings 1977.

Sarpkaya, T., N.J. Collins and S.R. Evans (1977): Wave forces on

rough-walled cylinders at high Reynolds numbers.

Offshore Techn.

Conf., Houston, Texas, Paper 2901, Vol. III.

Szechenyi, E.

(1975): Supercritical Reynolds number simulation

for two-dimensional flow over circular cylinders.

J. Fluid Mech.

70,

3, 529-542.

Stansby, P.K.

(1978): Mathematical modelling of vortex shedding

from circular cylinders in planar oscillatory flows, including

effects of harmonics and response.

IAHR-Symposium on Mechanics

of Wave-Induced Forces on Cylinders, Bristol, UK.

Tickell, R.G. and M.H.S. Elwany (1978): A probabilistic

descrip-tion of forces on a member in a short-crested random sea.

IAHR-Symposium on Mechanics of Wave-Induced Forces on Cylinders,

Bristol, UK.

White, J.K. and P.L. Carr (1978): On the estimation of Morison

coefficients in irregular waves.

IAHR-Symposium on Mechanics

of Wave-Induced Forces on Cylinders, Bristol, UK.

H. LUNDGREN

Professor of Marine Civil Engineering

ABSTRACT

Most of the work that has been done with respect to the forces

on

risers, consisting of a group of cylinders, is classified.

Below are presented:

A paper written by Professor T. Sarpkaya for the Symposium

on

Mechanics of Wave-Induced Forces on Cylinders by Internat.

Assoc.

Hydr. Res. at the University of Bristol, September

_1978.

Results of riser tests at Danish institutions.

1.

WAVE LOADING IN THE DRAG/INERTIA REGIME

WITH PARTICULAR REFERENCE

TO

GROUPS

OF

CYLINDERS

by PROFESSOR T. SARPKAYA, NAVAL POSTGRADUATE

_SCHOOL,

MONTEREY, CA., USA

Summary

The interference effect in steady and time-dependent flows is briefly reviewed and the drag and inertia coefficients for two particular riser configurations subjected to harmonic flow are presented.

Introduction and Review

A body's resistance to flow is strongly affected by what surrounds it. When two bodies are in close proximity, not only the flow about the down-stream body but also that about the updown-stream body may be influenced. Examples include condenser tubes in heat transfer, variety of columns in

WAVE AND CURRENT FORCES

ON RISERS

Institute of Hydrodynamics

and Hydraulic Engineering

Technical University of Denmark

pressure suppression pools of nuclear reactors, risers, piles, and other tubular structures in offshore engineering, turbine and compressor blades in mechanical or aerospace engineering, and high-rise buildings, cooling towers, and transmission lines in civil engineering. The quantification of the interference effects in terms of the pressure distribution, lift and drag forces on individual members, vortex shedding frequency, and the dynamic response of members of the array in terms of the governing flow and structural parameters constitute the essence of the problem.

There are infinite numbers of possible arrangements of two or more bodies positioned at right or oblique angles to the approaching flow

direction. Numerous experiments, often prompted by the need to solve

problems of immediate practical interest, provide data on the force transfer coefficients. But the intrinsic nature of the phenomenon still remains a mystery.

A careful review of flow interference between two circular cylinders in various arrangements in steady flow has been presented by Zdravkovich

(1977) where an extensive list of references may be found. Numerous studies have shown that the changes in drag, lift and vortex shedding are not necessarily continuous. In fact the occurrence of a fairly abrupt

change in one or all flow characteristics at a critical spacing is one of the fundamental observations of flow interference in cylinder arrays.

For the tandem arrangement (one cylinder behind the other), it has been shown that at relative spacings S/D < 3.5 there is a strong mutual interference between the two cylinders. This critical distance increases with the bluffness of the cylinders. For two plates in tandem, the

critical spacing is about 10 times the plate width (Ball & Cox, 1978). In general, the tandem arrangement has a strong effect on drag and is

sensitive to spacing. The upstream cylinder takes the brunt of the burden and the total drag for the group is smaller than the sum of the drag

forces acting on each cylinder in isolation in a tunnel with the same

partly by shielding and partly by the occurrence of earlier transition in the boundary layers due to the turbulence in the approaching flow.

The side-by-side arrangement exhibits a discontinuity in the flow and measured forces for spacing ratios smaller than a critical value (S/D < 2.2 for two circular cylinders and about 4 for two plates). The observed discontinuity ( a switching phenomenon and bistable nature of the flow between the plates) results in two values of the drag coefficient. This results from the mutual interference of the vortices on the adjacent

sides of the vortex streets. The bistable nature of the flow may be avoided by increasing the size of one of the cylinders. The sum of the bistable high and low drag is always less than twice the drag of the

single cylinder. For two plates in side-by-side arrangement, the said sum may be about 10 percent larger than twice the drag of the single

plate. This increase strongly depends on the width of the flow field

relative to the total width of the bodies, i.e., on the blockage ratio. For relatively large blockage, the flow is forced through the openings between the cylinders rather than around the total configuration.

Consequently, the use of ordinary shielding and blockage factors for groups of cylinders is meaningless. Experiments must be conducted in channels or tunnels with very small blockage in order to obtain valid force-transfer coefficients. There is at present no means to separate the flow interference effects from the blockage effects for groups of

cylinders.

Additional work with tube arrays in steady flows has been reported by a number of people (Crua 1967, Hammeke et al. 1967, Laird et al. 1967,

Chen 1972, Sachs 1972, Arita et al. 1973, ESDU 1975, Dalton and Szabo 1976). Ross (1959) conducted large scale wave tank tests for the case

of one cylinder on each side of the test cylinder in the range of

critical Reynolds numbers. His results appear to indicate that the wave force increases significantly only when the spacing between two cylinders Is less than about one diameter.

The foregoing studies, conducted mostly in steady flows, show that the sum of the individual drag forces for isolated cylinders is greater than the total force acting on the group within or outside the critical spacing as long as the local boundaries do not constrain the flow. They further show that the results obtained from the tandem and side-by-side arrangements cannot be generalized to predict the combined interference resulting from the staggered arrangement.

In wavy or time-dependent flows one needs the lift, drag and inertia coefficients for all members of the array. Evidently, the members of the array may not be all parallel and normal to the flow.

The quantification of the flow interference on lift, drag, and inertia coefficients for cylinders with relative inclinations and spacings in a design-wave environment is the designer's dream but the researcher's

nightmare. In the absence of data on the interaction between drag and

inertia coefficients in wavy or harmonic flows for cylinder groups one is tempted to use steady-flow results for the drag coefficient and the unseparated potential flow results for the inertia coefficient. The

inertia coefficient for a group of cylinders in inviscid unseparated flow may be obtained through the use of the method of images (e.g., Dalton and Helfinstine 1971, Yamamoto 1976) or through the use of the linear potential theory including the wave diffraction (e.g., Spring and Monkmeyer 1974, Chakrabarti 1978). These analyses do not deal with the effects of separation and vortex shedding. Consequently, the

results are more appropriate to the determination of earthquake forces

an wave forces on large bodies rather than to the evaluation of the

inertial component of the force in the drag/inertia dominated regime. Gibson and Wang (1977) carried out two different experiments to determine the added mass of a series of tube bundles. The bundles consisted of tubes of uniform diameter d arranged either in a square configuration or a circular configuration. In the first series of experiments, they towed the model of pile cluster under linear

natural frequency. For both cases, they have calculated the added mass through the use of the measured force and acceleration and plotted them as a function of the 'solidification ratio' defined by Edi/ITD where D is the pitch diameter of the bundle. Their results have shown that the added mass increases sharply after the solidification ratio reaches the value of 0.4 to 0.5. Beyond this value, the volume enclosed rather than the volume displaced by the structure becomes important. This result is disputed (Chakrabarti 1978b) and the results of both series of tests are no more applicable to separated wavy or oscillatory flows about tube bundles than those predicted from the potential theory with or without diffraction effects.

Relatively few studies have been carried out with oscillating tube bundles (Tanida et al. 1973, Bushnell 1977). Bushnell determined the

interference effects on the drag and transverse forces acting on a single member of two cylinder configurations through the use of a pulsating water tunnel. He did not evaluate the drag and inertia coefficients through the use of a suitable method, e.g., Fourier averaging. Instead, he picked out the maximum force values which occurred in each half cycle and averaged them over ten consecutive values so as to obtain a mean maximum force for each flow direction. The results have shown that the

presence of neighbouring cylinders significantly affects the forces on an individual cylinder of an array and the interference effect increases with increasing relative flow displacement. The maximum drag force on shielded cylinders was reduced relative to an exposed cylinder by up to

50 percent. Bushnell has suggested that a design using a high Reynolds

number single-cylinder drag coefficient applied throughout the array would have an extra margin of safety against maximum drag loading due to

interference effects. The transverse force could be 3 to 4 times larger for interior array positions than that for a single cylinder. Thus, a cylinder array, such as a riser, supported at regular intervals may exhibit very complex dynamic behaviour. Some of the members between

supports may undergo in-line oscillations while the others may undergo violent transverse oscillations.

Drag and Inertia Coefficients for Two Particular Tube Bundles Two arrays consisting of 12 outer pipes and one central pipe (see Fig. 1) have been subjected to harmonically oscillating flow.

The equipment used to generate the harmonically oscillating flow has been used extensively at this facility over the past five years

(Sarpkaya 1976). Only the salient features, most recent modifications,

as well as the adaptation for this work, are briefly described here. The length of the U-shaped vertical water tunnel has been increased from 10 m to 12 m and its height from 4.8 m to 6.8 m. Previously, a butterfly-valve arrangement at the top of one of the legs was used to

initiate the oscillations. During the past year, the tunnel has been modified so that the oscillations can be generated and maintained

indefinitely at the desired amplitude. For this purpose the output of a 2 Hp fan was connected to the top of one of the legs of the tunnel with a large pipe. A small butterfly valve, placed in a special housing between the top of the tunnel and the supply line, oscillated harmonically at a frequency equal to the natural frequency of the oscillations in the

tunnel. The oscillation of the valve was perfectly synchronized with

that of the flow through the use of a feedback control system. The

output of a pressure transducer (sensing the instantaneous acceleration of the flow) was connected to an electronic speed-control unit coupled

to a DC motor oscillating the valve plate. The circuit maintained the period of oscillations of the valve within 0.001 seconds. The fluid oscillated with a period of 6.000 seconds. The amplitude of oscillations was varied by constricting or enlarging an orifice at the exit of the

fan. The flow oscillated at a given amplitude as long as desired.

The analysis of the in-line force was based on Marison's equation written as

2

F = -0.5pCdU2 ED.Icosalcnswt +_{m 1}

0.25

npLCmED. Um u) sinwt

2

Denoting Da = EDi and De = EDi and inserting in Eq. (1), one has

Da 1

¶2

----Icoswticoswt + _Cm sinwt

0.5pDeL4

_De UmT/De

For an oscillating flow represented by U = -Um coswt, the Fourier averages of Cd and Cm are given by

2n

2

Cd

= -(3/4)(De/Da)f[Fmcoswt/(PLDeUm)] dwt

2w

C = _{(2UmT/n3De)f[fr sinwt/(pL0e m}U2)j dwt 0

in which Fm represents the measured force.

The experimental results are shown in Fig. 2 as a function of the Keulegan-Carpenter number defined by K = UmT/De. The drag coefficient decreases gradually with increasing K and reaches an almost constant value for K larger than about 90. The inertia coefficient increases with increasing K and reaches a terminal value of about 6. The average

inertia coefficient defined by

2 2

Cm = (EC_{mi 1}

was found to be 2.17 for the two arrays through the use of the potential

flow theory. The comparison of this value with that obtained experimentally

shows that some fluid mass is entrapped within the array and that neither the potential theory nor the diffraction theory can adequately describe the behaviour of the complex separated flow through the bundle.

The data for two different values of Re/K show that the force coefficients are independent of the Reynolds number within the range of Re and K values reported herein.

and ( 1 )

(5)

(2) ( 3 ) (4)

The reason for the dependence of Cd and Cm on K is thought

to be

the dependence of the interaction of the wakes of the outer and inner

pipes. The vortices in the wake of a given

cylinder loose about 70

percent of their strength within 10 cylinder diameters.

Thus, for small values of K the vortices generated by a small tube at the center

front of the bundle arrive at the central tube as very weak vortices. Consequently, each tube behaves more or less as if it were independently subjected to a turbulent harmonic flow. As K increases, not only the

turbulence level but also the interaction between the wakes of the

various cylinders increases. There is a certain amplitude of oscillation beyond which neither the interaction of the wakes nor the increase of the turbulence level affects the overall force acting on the bundle. A comparison of the total drag force acting on the bundle with the

sum of the drag forces acting on each cylinder in isolation in harmonic flow (at the corresponding K and Re values appropriate to each

tube)

shows that the former is about 10 percent smaller.

The transverse force and the forces acting on the individual

members of the bundle will not be presented herein. A detailed discussion of the interference effects in steady and time-dependent flows with

particular reference to riser configurations is presented in Sarpkaya and Isaacson (1978).

Concluding Remarks

Interference effects in steady and time-dependent flows can be neither generalized nor predicted on the basis of relatively idealized

situations. There may exist one or more critical spacings in an array

for which the force coefficients, Strouhal number, etc., may exhibit

dual values. The blockage effect on the flow though a tube bundle is

far more complex than that on the flow about a single rigid cylinder. The results obtained in a confined flow are not, therefore, appropriate

for the determination of the interference effects in unconfined flow about tube bundles.

Experiments with two particular riser configurations have shown that the drag and inertia coefficients are independent of the Reynolds number and reach their terminal values at relatively small values of K. The inertia coefficient is considerably larger than that predicted by the potential theory and shows that some fluid mass is entrapped within the bundle as a consequence of the solidification of the tube configuration.

It appears that experiments will continue to play a major role in the determination of the appropriate force-transfer coefficients for the type of tube bundles discussed herein.

12 outer tubes central tube Configuration - I Do/Dc = 0.3256 DP/DC = 4.3950 D /D = 0.0740 o p Configuration - II Do/Dc = 0.4186 D /D = 4.3950 p c Do/Dp = 0.0952

6.0 5.0 4.0 3.0 2.0 1.5

1.0-0.9 0.8 10 6.0 5.0 s 4.0 3.0 2.0 1.5 Cm Cd

Fig. 2a Drag and Inertia Coefficients versus K for Configuration - I.

y_ Cm Cd 15 20 30 40 50 A.

AO A

AO At Re/K = 1250 Re/K = 700 Oi 11146. Ike 70 90 100

IV

% 150 200

1.0_

0.9 0.8 1 I 1 I I I 1 10 15 20 30 40 50 70 90 100 150 200

References

Arita, Y., Fujita, H. and Tagaya, K., (1973), A Study of the Force of Current Acting on a Multitubular Column Structure, OTC Paper 1815. Bushnell, M. J., (1977), Forces on Cylinder Arrays in Oscillating Flow,

OTC Paper No. 2903.

Chakrabarti, S. K., (1978), Wave Forces on Multiple Vertical Cylinders, Jour. Waterways etc. Div., ASCE, WW2, pp. 147-161.

Chakrabarti, S. K., (1978b), Discussion of 'Added Mass of Pile Group' (R. J. Gibson and H. Wang), Jour. Waterway etc. Div., ASCE, WW2,

pp. 256-258.

Chen, Y. N., (1972), Fluctuating Lift Forces of the Karman Vortex Streets on Single Circular Cylinders and in Tube Bundles, Part-3: Lift Forces

in Tube Bundles, Jour. Eng. for Industry, Trans. ASME, Vol. 89, 623. Crua, A., (1967), Druckverlustmessunger an Glattrohrbundeln, Sulger

Brothers, Report No. 1387.

Ball, D. J. and Cox, N. J., (1978), Hydrodynamic Drag Forces on Groups of Flat Plates, Jour. Waterway etc. Div., ASCE, WW2, pp. 163-173.

Dalton, C. and Helfinstine, R. A., (1971), Potential Flow Past a Group of Circular Cylinders, Jour. of Basic Eng., Trans. ASME, pp. 636-642. Dalton, C. and Szabo, J. M., (1976), Drag on a Group of Cylinders, ASME

Paper No. 76-Pte-42.

Engineering Sciences Data Unit, ESDU, (1975), Fluid Forces on Lattice Structures, Data Sheet 75011.

Gibson, R. J. and Wang, H., (1977), Added Mass of Pile Group, Jour. Waterway etc. Div., ASCE, WW2, pp. 215-223.

Hammeke, R., Heinecke, E. and Scholz, F., (1967), Warmeubergangs und Drucklustmessungen an Queranstromten Glattrohrbundeln, Insbesondere bei Hohen Reynoldsgablen, International J. Heat Mass Transfer, Vol.

10, pp. 427-435.

Laird, A. D. K. and Warren, R. P., (1963), Groups of Vertical Cylinders Oscillating in Water, Jour. Eng. Mechs. Div., ASCE, Vol. 89, EM1, pp. 25-35.

Ross, C. W., (1959), Large Scale Tests of Wave Forces on Piling, U. S. Corps of Engineers Beach Erosion Board, Tech. Memo. No. 111. Sachs, P., (1972), Wind Forces in Engineering, Pergamon Press, Oxford. Sarpkaya, T., (1976), In-Line and Transverse Forces on Smooth and

Sand-Roughened Cylinders in Oscillatory Flow at High Reynolds Numbers,

Naval Postgraduate School Tech. Report No. NPS-69SL76062, Monterey, CA. Sarpkaya, T. and Isaacson, M. de St. Q., (1978), Wave Forces on Offshore

Structures - Theory and Application, (in press).

Spring, B. H. and Monkmeyer, P. L., (1974), Interaction of Plane Waves with Vertical Cylinders, Proc. 14th Coastal Eng. Conf., ASCE, Vol.

III, pp. 1828-1847.

Tanida, Y., Okajima, A., and Watanabe, Y., (1973), Stability of a Circular Cylinder Oscillating in Uniform Flow or in a Wake, Jour. of Fluid Mech., Vol. 61, pp. 769-784.

Yamamoto, T., (1976), Hydrodynamic Forces on Multiple Circular Cylinders, Jour. Hydraulic Div., ASCE, Vol. 102, HY9, pp. 1193-1210.

Zdravkovich, M. M., (1977), Review of Flow Interference Between Two

Circular Cylinders in Various Arrangements, Jour. of Fluids Eng., Trans. ASME, Vol. 99, pp. 618-633.

2. RESULTS OF RISER TESTS

AT DANISH INSTITUTIONS

Laboratory tests with a bundle-type riser model having the same

diameter ratios as Sarpkaya's Configuration - I

have been conducted

by the Danish Hydraulic Institute (DHI) and the Institute

of

Hydro-dynamics and Hydraulic Engineering (ISVA) at the Technical

Univer-sity of Denmark.

The model has been tested under 5 different flow

conditions:

Oscillating flow,

Oscillating flow superposed on a constant in-line flow,

Oscillating flow superposed on a constant cross flow,

Constant acceleration,

Steady current.

Only a brief review of the results for the total bundle will be

given here.

For the steady current tests the relative force

distri-bution on the outer pipes will be presented for three different

Rey-nolds numbers (forces were measured on

individual pipes).

A least squares fit of the measured data to the Morison formula

was used to determine the hydrodynamic

force coefficients

Cc

and

Cm

for test conditions (1) - (3).

However, for test conditions (2) and

(3) the inertia coefficient is not well defined, and a

comparison

be-tween test conditions should preferably be made on the basis of a

to-tal force and the toto-tal velocity.

The results for test condition (1) compare fairly well

with the

results presented by Sarpkaya for corresponding KC-values

(K-values

in Sarpkaya's paper).

The Cd-values found in the DHI/ISVA-tests are

somewhat lower than Sarpkaya's.

This might be due to the following

two reasons:

(i) For the considered range of KC-numbers, tests were

conducted at higher Reynolds number at DHI/ISVA.

(ii) The blockage

ratio was smaller in the DHI/ISVA test facility.

The transverse force coefficient, CL,

(based on maximum force and

maximum in-line velocity) was about one third of the Cd-values.

For test condition (2) the Cd-values decreased 10-25%, while

CI,

remained within the same range as for test condition (1).

Test condition (3) gave Cd-values of the same magnitude as (1).

The transverse force coefficient, however, increased significantly

(by a factor of 2 to 4)

as compared to (1).

The constant acceleration tests resulted in a C -value of 2.08

for the total bundle, which is very close to the calculated

poten-tial flow value of 2.17 (Sarpkaya).

Finally, the steady current test resulted in Cd-values of

approxi-mately the same magnitude as for test condition (2).

CL was here

25 -50% higher than for (1).

As mentioned above the relative force distribution on the outer

pipes will be presented for the model exposed to a steady current.

The average in-line force coefficient Cd

(= T/1/2

pd 172) and the

maximum total force coefficient C for the outer pipes, both divided

by the average in-line force coefficient for the central pipe are

shown in Figs. 3 b and 4, respectively.

The numbering of pipes

ap-pears from Fig. 3a.

The in-line drag on the whole bundle is taken

mainly by the upstream pipes, whereas the total (vectorial) force

ACKNOWLEDGMENT

Project manager for the tests mentioned above was N.-E. Ottesen

Hansen, Senior Engineer, DHI, in collaboration with Vagner Jacobsen,

Research Engineer, ISVA.

2.0 1.0 Cd outer pipes Cd central pipe

z

1

Fig. 3a

Numbering of pipes

FLOW

DI RECTION

PPE

4 3

1 2 3 4 5 6 NUMBER

Fig. 3b

Relative average in-line force distribution

on outer pipes

). PIPE

2 3 1. 5 6 NUMBER

Fig. 4

Relative maximum total force distribution

on outer pipes

(Re is based on the pitch-diameter of the model)

4.0 3.0 2.0 1.0 CMQX outer pipes

/

/0\

\\

_\

\ \

\

\x

\ \

\

0

A

\

\\

X.

,,Z

/

/

b

/

Cd, central pipe

x/

/

/

.,

/

N

/

'0

Re 3 10-5

--\

_\

A : = x Re = 6 x 10-5

_

x:

---

0: Re

= 10x 10-5

H. LUNDGREN

Professor of Marine Civil Engineering

B. MATHIESEN

M.Sc.-Student

H. GRAVESEN

Research Engineer

WAVE LOADS ON PIPELINES

ON THE SEAFLOOR

(From

BOSS '76, Vol. I, pp. 236-247)

Institute of Hydrodynamics and Hydraulic Engineering Denmark

Technical University of Denmark

Consultant, Danish Hydraulic Institute

Institute of Hydrodynamics and Hydraulic Engineering Denmark

Technical University of Denmark

Danish Hydraulic Institute Denmark

ABSTRACT

Pressure distributions around a pipe (Fig.

2) and total forces

upon it (Figs. 4- 5) have been determined in model tests by

oscillat-ing a pipe, as well as the sea bed attached to it, horizontally in

still water.

Smooth and rough pipes of 150 mm diameter have been

studied in the Reynolds number range from 105 to 2.5

105.

The stroke

of the oscillation has been varied between 6 and 16 pipe diameters.

The direction of oscillation has been either normal to the pipe or

at an angle of 450 .

Since the two roughnesses, k, applied have no significant

influ-ence on the stability of the pipe, as compared with the smooth pipe,

it is tentatively concluded that there is 'enough turbulence' in the

water from the preceding stroke as to make the boundary layer

com-pletely turbulent at the phase that is critical for the stability.

The suggested consequence of this is that the Reynolds numbers of

the tests are sufficiently large for the application of the

(prelim-inary) results presented here to the much larger Reynolds numbers in

the field.

The flow has been visualized in Fig.

3 for the range of phases

that are critical for the stability.

It is recommended to discontinue the use of the Morison formula

because the drag, lift and inertia coefficients

_{are far from}

con-stant.

_{Instead, a stability coefficient, Cst, has been defined}

_by

Eqs. 12 and 13 (Figs. 6-9).

Under the assumption that the Reynolds number has little

influ-ence, the values of Cst have been plotted in Fig. 10

as a function

of the Keulegan-Carpenter number, KC, which represents the stroke to

diameter ratio.

Presumably, the large scatter for small KC-values

is due to the deviations of the accelerations of the oscillating pipe

from those of a purely sinusoidal motion (Fig. 1).

While the mean design curves in Fig. 10 probably correspond to

sinusoidal oscillations, it is concluded that the forces in

irregu-lar waves may differ essentially from those in reguirregu-lar waves.

A few tests with a pipe at an angle of a = 45° with the wave

mo-tion indicate that it is somewhat conservative to multiply the forces

found for a pipe at right angles to the flow by sin a

(Fig. 11).

The geotechnical aspects of the pressure difference between the

two sides of the pipe (Fig. 12)

is mentioned in Sec. 5.

The preliminary test results are on the safe side because they

have not been corrected for the blockage (limited depth of water

under the pipe).

1. INTRODUCTION

Most literature dealing with wave forces on pipelines (see

Refer-ences) express the maximum forces in terms of Morison's formula,

which was an outstanding contribution to hydraulic engineering at

a time when little was known of the physics of the flow.

According to Morison, the horizontal force, Fh, is the sum of two

components:

the drag force and the inertia (or mass) force, while

the vertical (or lift) force, Fv, is analogous to the drag force.

Unfortunately, however, it has been found that the drag, mass and

lift coefficients are far from independent of the phase in the

oscil-latory wave.

Physically, this is easy to understand:

The flow

pat-tern does not depend only upon the instantaneous velocity of the

un-disturbed flow but also upon 'the history of the velocity'.

In

ad-dition, the influence of a pressure gradient depends upon the flow

pattern that exists when the gradient occurs.

Mathematically

speak-ing, the weakness of the Morison formula could be said to be the

superposition of drag and lift terms, which are quadratic in the

undisturbed velocity, upon the inertia term, which is linear in the

pressure gradient.

In recent years it has been pointed out that, for sinusoidal

mo-tions, there is no need for Morison's formula if the forces

are

ex-pressed just in terms of the dimensionless numbers

Reynolds number

_{Re = vmax Div}

₍₁₎

Relative pipe roughness

k/D

( 3 )

Relative bed roughness

_kb/D

(4)

where

vtnax

= maximum velocity of the

sinusoidal motion

( )

- pipe diameter

(6)

= kinematic viscosity

( 7 )

2 A = full stroke of the sinusoidal motion

(8)

= roughness of pipe

( 9 )

= roughness of bed

(10)

This point of view has been applied in the present paper.

2.

DESCRIPTION OF EQUIPMENT AND TESTS

Larger Reynolds numbers cannot be obtained in a normal wave

flume,

but only in an oscillating water tunnel or in a basin where the pipe

is oscillated instead of the water.

Though a 15 m long oscillating

water tunnel was built 20 years ago at the Technical University of

Denmark, it was decided to develop equipment for oscillating the

pipe, partly because of the small height of the oscillating water

tunnel, partly because the enormous inertia of the water would make

it impossible to impose any but a nearly sinusoidal motion.

In a 600 mm wide flume a pipe of 150 mm diameter was mounted on

a 6 m long smooth 'sea bed' supported on a carriage oscillating

along the flume with a maximum stroke of 2.4 m.

In order to avoid

generation of surface waves, the pipe was placed on the underside

of the sea bed, which had a distance of 550 mm from the bottom of

the flume.

With a total water depth of 650 mm in the flume it was

found that the surface waves generated by the displacement of the

sea bed structure were negligible.

v [-ngs]

_{Water movement :}

1 0

90°

900 1800 3600 2700 180°

N8O°

360°

2';0°

/./11

Fig. 1

Oscillatory velocity of water relative to pipe

9 = 23°

= 34°

9 = 45

9 = 68°

9 = 90°

Fig. 2

_{Pressure distributions in Tests No.}

₁₂

16

-Fig. 3

"'_*.ENOU4110

Flow patterns relative to pipe in Test No. 33

at phases (0 = 214°, 225°, 248°, 2700