Measured and predicted random wave forces near the free surface

M I C H A E L I S A A C S O N A N D J O H N B A L D W I N

Department o f Civil Engineering, University o f British Cohunbia, Vancouver, B.C. V6T llV5, Canada

Laboratory tests have been carried out to study random wave forces acting on a vertical circular cylinder in the vicinity of the water surface. For sections located near the free surface, submergence in the wave field is intermittent and this drastically alters the statistics of the wave forces in comparison to those for fully submerged sections. Expressions for the spectral density of the force and the probability density of the force maxima which account for this intermittency are summarized. The former is based on a linearization of the Morison equation, while the latter is based on the assumption of a narrow-band wave spectrum. Comparisons are presented between the experimental data and numerical predictions and are found to be quite reasonable.

Key Words: Cylinders; hydrodynamics; ocean waves; offshore structures; probability distributions; wave forces.

INTRODUCTION

Wave forces acting on structures comprised of slender structural members are generally predicted on the basis of the Morison equation in which the wave force on any section of a member is expressed directly in terms of the fluid kinematics which would occur at that section's location. For random waves, the Morison equation is usually applied in conjunction with linear wave theory to provide expressions for the spectral density and probability distribution of the forces on submerged portions of a structure. These techniques are well estab- lished and have been widely reviewed (e.g. Borgmant , Sarpkaya and Isaacson 2 and Ochi3). A linearization of the Morison equation is generally carried out in order to estimate the spectral density of the force. However, this represents an unrealistic approximation for esti- mating the probability distribution of the force, and the full Morison equation is then retained. The alternative assumption of a narrow-band spectrum is made instead in order to estimate the probability distribution of force maxima.

For a section of a member located near the free sur- face, the intermittent submergence of the section gives rise to complications in estimating wave force statistics. The intermittency o f the flow field in this region results in a significant modification to the statistical properties of the water particle kinematics. Such effects have been examined both theoretically (e.g. Tung 4'5, Pajouhi and Tung 6) and experimentally (e.g. Satyanarayana and Elang07). Anastasiou et aL s have considered the importance of second order effects on the kinematics at

Accepted October 1989. Discussion closes April 1991.

such points. These modifications to the kinematics in turn give rise to changes to the force acting on intermit- tently submerged sections. The effect of intermittency on wave forces has been considered theoretically by Tung 4 and more recently by Isaacson and Baldwin 9. Tung 4 provided expressions for the mean and standard deviation of the wave force in deep water, while Isaacson and Baldwin 9 provided expressions for the spectral density of the wave force and for the prob- ability density of the force maxima. As in the case of fully submerged sections, the Morison equation is linearized in order to estimate the spectral density, while its nonlinearity is retained in order to estimate the probability distribution of force maxima and instead, an assumption of a narrow-band wave spectrum is made.

Little experimental work has been carried out to investigate wave forces in the vicinity of the free surface and the present paper is intended primarily to provide such results. Corresponding theoretical predictions are first summarized. This is followed by a description of experiments used to measure wave forces acting on a segmented vertical circular cylinder in the vicinity of the water surface, and subjected to both uni-directional and multi-directional random waves. A comparison of the experimental results for uni-directional waves with the theoretical predictions is made.

THEORY

The Morison equation Wave forces acting on structures comprised of

slender members are generally predicted on the basis of the Morison equation. The horizontal force per unit

188 Applied Ocean Research, 1990, Vol. 12, No. 4 �9 1990 Computational Mechanics Publications

Random wave forces near the free surface: M. lsaacson and J. Baldwin

z A |

D

, / e / / / / , / i / / / / , - / / / / / / / / / / / / / / ,

Fig. 1. Definition sketch

length acting on a vertical slender structural member is expressed as:

F= Kaul u I + K,,,a (1)

where Ka=(I/2)pCaD, K,,,=t~(TrD2/4)C,,,, D is the diameter of the member, p is the mass density of water, tt and a are the horizontal water part icle velocity and acceleration respectively which would occur at the section's location in its absence, and Ca and Cm are empirical drag and inertia coefficients respectively. Extensive experimental research has been carried out to investigate the behaviour of these force coefficients under a variety of flow conditions. Nevertheless, in applications to random waves known constant values of the force coefficients are generally assumed.

The general situation being considered is indicated in Fig. I. The free surface elevation corresponding to the random wave field is assumed to be a zero mean, stationary, random Gaussian process described by a specified wave spectrum S~(~). For points near the still water level, submergence in the random wave field will be intermittent due to fluctuations of the free surface elevation. The spectral densities and probability distri- butions of the water particle kinematics will conse- quently be different from those of continuously submerged points. In order to account for the effects of this intermittent submergence for sections of a member which are close to the still water level, the Morison equation may be applied in conjunction with the corres- ponding particle kinematics which take account of the intermittency. A brief summary of the pertinent equations for a statistical force model which includes intermittency effects in the manner indicated above is given here. For supporting details the reader is referred to Isaacson and Baldwin 9.

Spectral density The Morison equation may be applied to the kine-

matics of the intermittent flow field in order to develop an expression for the spectral density of the corres- ponding force acting on a section of a member. The use of the complete Morison equation proves to be difficult and a simplification may be developed by a suitable linearization of the nonlinear drag component. Using a suitable drag linearization factor based on the prob- ability density of the horizontal velocity tt in the inter- mittent flow field 5, and applying expressions provided

by Pajouhi and Tung 6 for the velocity and acceleration spectra, Isaacson and Baldwin 9 obtained the following expression for the force spectral density:

• Q2(b)H2(o~)S~(o~) (2)

where a~ and a,, are the standard deviations of r /and tt ~espectively, b = h/a~ is a dimensionless measure of the section elevation h, and:

~ 3 = E [ i u l 3 l E[U2 ] (3)

H,, = o~G(z) (4)

G(z) = cosh [k(z + d)] (5) sinh(kd)

S Q(b) = Z(X) dX (6) b

I

l I r = - - S,,~ (w) do~ ~ l t ~ r I 0

_ 1 H , , ( o ~ ) S ~ ( o ~ ) do~ (8) tTulT~ I 0

Also E[] denotes the expected value, d is the still water depth, z is distance measured vertically above the still water level, and w and k are the angular frequency and wave number respectively, which are related by the linear dispersion relation. In the above /3 is a drag linearization factor chosen such that llJn[ is approxi- mated as {3u, Q(b) corresponds to the Gaussian prob- ability function, r is the correlation coefficient between u and r/, and S,,~ is the cross spectral density between tl and r/.

Probability density o f force The derivation of the probability density of the inter-

mittent wave force is more difficult since a linearization of the Morison equation is now unrealistic and the full nonlinear drag term should be retained. Tung 4 has pro- vided expressions for the mean and standard deviation of the force:

[ 2rr' ( b ) #F = Kaa 2 - Q(b) + 2L(0, b, r) + ~ Z

O.~= z~2 4 ~,ao,,13Q(b) + Z(b)brZ(6 + r2b 2 3r2)1

+ K~,a~Q(b) - ~,~ (10)

where

S i L(a ,b , r )= Z(z) Z(y) dy dz (11) a iv

Here w = ( b - rz/r', and r ' = , f f Z T . Note that only the drag component contributes to the mean of the intermittent wave force, since the mean of the intermit- tent horizontal acceleration is zero.

Applied Ocean Research, 1990, Vol. 12, No. 4 189

Random wave forces near the free surface: M. lsaacson and J. Baldwht

Probability density o f force maxima The probability distribution of the force maxima/O is

generally of greater interest than that of the force itself. An expression for this may be derived by assuming that the wave spectrum is narrow-banded and concentrated near a single frequency w0. This assumption allows the peak force for each wave to be calculated determin- istically, and the Rayleigh distribution of wave heights may then be used to develop an expression for the prob- ability density of the force maxima 9. Alternative expressions arise for sections below and above the still water level. For a section at or below the still water level, the force maximum in each wave occurs when the section is submerged, and thus is unaffected by inter- mittency effects. The probability density of force maxima then corresponds to that for continuously submerged sections (see Borgmanl~ On the other hand, for sections above the still water level, the prob- ability density of force maxima is influenced noticeably by the intermittent submergence. The corresponding expressions are given as follows.

For h ~< 0:

p(g.) = f~- exp(-�89 r 2) [~-~ exp [ -~ ~'c(2~"- ~u

F o r h > 0:

for ~ < ~'~ (12) for ~ i> g'~

f r(3`)6(~') for ~ = 0 ~ 0 for 0 < ~'< 3 ̀

p(~') = ~ (~'- 7 ) e x p [ - �89 (~-- 3`)2- 7~'c1 (13)

] " for 3 ̀< ~" < ~-~+ 3 ̀~ ' c exp[ - �89 ~'c(2~'- ~'~)]

for ~" i> ~'~, + 3'

In the above b is the Dirac delta function, and:

2 , ~ p - oj2g,,,HrmsG(z ) (14)

~%- "/~" (15) KaHr~G(z) b 2

3" = 2~'c (16)

P(v) = 1 - exp( - 3`~'c) = I - exp( - b2/2) (17)

Here F'is the wave force maximum and H~m~ is the root- mean-square wave height. ~'r is a measure of the relative contributions of inertia and drag forces which may con- veniently be expressed in terms of a representative Keulegan-Carpenter number K. For a wave field with a specified significant wave height Hs and peak period Tp, such a Keulegan-Carpenter number may be defined as:

K = 7rHs 1 D tanh(kd) (18)

where k is the wave number corresponding to the peak period Tp. Taking Hrms = Hs]~-2, and taking z ~- 0 near the water surface, ~'~ may be expressed directly in terms of K as:

71"2Cm ~'~ = - - (19)

CdK

The probability density function derived above is based on the assumption that one force maximum occurs per wave, even though this may be zero for some waves due to non-submergence of a section. Thus equation (13) incorporates a finite probability P(3`) at ~" = 0, corresponding to force maxima of zero magni- tude, and I"(3,) is thus the probability that H < 2h. Although the retention of zero force maxima in equation (13) may be considered somewhat artificial, they may be excluded if required, in which case the remaining portion of the probability density function would be multiplied by 1[ [ 1 - I1(3,)].

Expressions for the mean/~r and standard deviation or of the force maxima may be obtained from equation (13) and are given by:

~,r = [l - I'(3`)1,2/2U~ + e(~-c) (2o) rc

+ 3 ` 2 ~ [ 1 - 2 Q ( ~ - c ) ] ] - # ~ " r ~ - ~ c 2 J(21)

Note that these values are based on the probability dis- tribution given by equation (13) so that as h becomes large, the zero magnitude force maxima predominate and both #r and or tend to zero.

Single largest force maxinntnl In engineering applications it is often the single

largest value F,,, of the force maxima /O which is of primary interest. The probability distribution function of the single largest value of the force maxima which occurs in a sample of N successive waves, considered independent and to occur in a specified duration, can be expressed as:

P ( s = [ P ' (Fro = F ) ] N (22)

where P' (F) may be developed from equation (13). The corresponding probability density function of F,,, and the expected value E[/~,,] of the single largest force maximum may thereby be obtained.

The number Ny of the non-zero force maxima will be less than the number of waves N due to non- submergence of the section for some waves, and is given by:

NI= [1 - F(3`)] N (23)

For a specified duration of N waves, the highest section elevation h which becomes submerged corresponds to the value of h when N f = 1. On the basis of equations (19) and (23) this corresponds to:

b =--=h [2 In(N)] 1/2 (24) 0" 7

The associated wave height is H = 2 h , so that equation(24) then corresponds to the well-known expression for the expected value of the single largest wave height occurring in a specified duration.

EXPERIMENTS

Experiments have been carried out to investigate the wave forces acting on segments of a fixed vertical

190 Applied Ocean Research, 1990, Iioi. 12, No. 4

Random wave forces near the free surface: M. Isaacson and J. Baldwhl

circular cylinder which are in the vicinity of the free surface and are described here. Corresponding results for fully submerged segments have been presented by Isaacson and Nwogu~

Table 1. Sununary o f target wave spectra characteristics

Test No. fp (Hz) Hs (m)

Description o f experiments The experiments have been carried out in the ocean

engineering basin of the Hydraulics Labora tory of the National Research Council, Canada. The basin is 30 m wide, 19.2 m long and 3 m deep, and is equipped with a 60 segment wave generator capable of producing regular and random uni-directional and multi- directional waves. Fixed walls along portions of the two sides adjacent to the generator may be used to increase, through wave reflection, the test area in the basin. Energy absorption beaches are placed along the remaining sides of the basin.

The test cylinder used has a diameter of 0.17 m and is 2.4 m high, with the upper 1.5 m divided into nine segments as indicated in Fig. 2. Each segment is fitted with two load cells oriented orthogonally so as to measure the segment forces in-line and transverse to the incident wave direction. The cylinder was wrapped in polyvinyl to make it watertight and smooth. Extensive static and dynamic calibration tests were carried out. These indicated very little coupling between the seg- ments, and that the natural frequency of the segments

V

2

1"

�9 0 . 1

0.1

' 0 . 1

1

0.15 m I

0.15 m

--1-- 0.35 m

-5 0.35 m

0 .885 m

m

m

m

ITI

m

2 m

1 !

2.4 m

r

Fig. 2. Sketch o'f test cylinder showing segment dimen- sions

1 0.32 0.43

2 0.41 0.39

3 0.51 0.31

in air was about 20 Hz, well above the wave frequencies of interest.

A water depth of 2 m was used for all the tests, and the cylinder was located 9 m away from the wave generator. The cylinder was subjected to random long-crested and short-crested waves described by a two-parameter Pierson-Moscowitz spectrum and a fre- quency independent cosine power directional spreading function with various values of the spreading index. An array of nine capacitance-type wave probes located in front of the cylinder was used to provide measurements of the water surface elevation. Directional spectra were obtained from these using the maximum entropy method. A summary of the target significant wave heights and peak frequencies for the three uni- directional wave conditions described here is given in Table 1. These correspond to Keulegan-Carpenter numbers in the range K = 6-10, where K is defined in equation(18), and to Reynolds numbers of about 100,000.

The wave trains were each synthesized for a duration of 204.8 s and the time series of the wave force on each segment was recorded. Data was sampled at a rate of 20 Hz using the G E D A P data acquisition system Of the National Research Council and stored on an on-line HP1000 computer .

RESULTS AND DISCUSSION

In order to illustrate the effect o f the intermittent flow near the free surface, and to assess the accuracy of the formulations developed, the numerical predictions are compared here to experimental results. The three target wave spectra together with the corresponding measured wave spectra are shown in Fig. 3. The figure illustrates how the measured wave spectra do not reproduce the smooth shape of the target spectra exactly, and so indicates one reason for discrepancies between measured and predicted force results.

Force spectral density Figure 4 shows a comparison between the measured

and predicted wave force spectra for segment 6 for the three wave conditions indicated in Table 1. As indicated in Fig. 2, segment 6 is located just above the still water level. The predicted wave force spectra were calculated with force coefficients of Cd = 0.7 and Cm = 1.7 (see Isaacson, Nwogu and Cornett~2). The broken curve corresponds to predictions based on a neglect of inter- mittency effects (see, for example, Borgman~), whereas the solid curve corresponds to the inclusion of intermit-

Applied Ocean Research, 1990, 1Iol. 12, No. 4 191

L

Random wave forces near tile free surface: M. Isaacson and J. BaMwin

it . itS

0 . 0 ~ 2 5

i t . e l l

0 . 0 I l l

it.it 0 , i t $ . 5 I . i t

F r e q u e n e 7 ( H X )

I

I . $ t . $ 0.5 1.0

Frequency (Hz)

_-.r -,.. E

t o

0.0500-

0 . 0 3 7 5 -

0 . 0 2 5 0 -

0 ,012.5 -

0.0000 O0 i.s

(a)

!

2 . 0

E

o i, B

l . t l i t

I . I I

I . l l t I I

J 0 . i t - -

0 . i t O . J I . i t

f r e q u e l r 7 ( l g )

",r" -...

to

0 . 0 4 5 -

0 . 0 3 0 -

0 . 0 1 5 -

m 0 . 0 0 0 t_t t . 0 0 . 0 0.5 1.0 .5

Frequency (Hz)

1 ~ 0

it

N o

v

t . O t l

I . I I I I

I1.111

Q.801

I.| I . I

F r e q ~ e n r ( R I )

0 . 0 2 4 -

O . O l S -

r r

0 . 0 1 2 -

E t.o

, i -

o

g v

C

0 . 0 0 6 -

* .e ~ 0 0 0 0 e.e m.$ 1.1 0 .0

/ 0.5 1.0 I.,5

Frequency (Hz)

(c)

Fig. 3. Comparison o f measured and target wave spectra for the three tests. (a) test 1, (b) test 2, (c) test 3

i 2 . 0

tency effects (2). Although there is a notable discrep- ancy between either of these and the measured spectra, those which take account of intermittency agree more closely with the measured spectra. Nevertheless, the theory underpredicts the peak spectral density and over- predicts the tail of the distribution at higher frequen- cies. As already indicated by Fig. 3, part of the

irregularity in the measured spectra arises because the wave spectra as measured do not reproduce the smooth shape of the target spectra exactly.

Figure 5 shows a comparison of the measured and predicted values of the standard deviation aF of the intermittent wave force for those segments above the still water level. The broken curves correspond to the

192 Applied Ocean Research, 1990, Vol. 12, No. 4

Randonl wave forces near the free surface: M. Isaacson and J. BaMwin

BO

I

O5

6G

4 0

20J

OOI O0

MEASURED

BO

(a)

t !

I 0 !,5

Frequency (Hz)

:2:

"z

0r

8 0

6 0

PREDICTED

,40

2 0 -

0.0 O0

/ \ \ \

(a)

0.5 I '0 1'5 2 0

Frequency (Hz)

"r-

v

ur

2"-

ii ,o Frequency (ttz)

v

B O -

60'

4.0.

2 0 -

0.0 0.0

i \ / \

/ \ co)

, i

0.5 1.0 1.5

Frequency (Hz) 2 0

6 0

4 0

2 0

O0 ! I I

0 5 I 0 15

Frequency (Hz)

6 0 -

4 0 -

2 0 -

0 0 "

o o o

/ \ / \ ~ (c)

,

0 5 1-0 1.5

Frequency (Hz) 2 0

Fig. 4. Comparison o f measured and predicted force spectra for segment 6. - - , inclusion o f intermittency, (2); . . . . , neglect o f ihtermittency. (a) test 1, (b) test 2, (c) test 3

Applied Ocean Research, 1990, Vol. 12, No. 4 193

Random wave forces near the f ree surface: M. lsaacson and J. BaMwin

1.0

h

Hs

0.5

0 I ~ 2.0 4.0 6.0 %

1.0 - -

.H s Oa)

o I I I ~_

3.0 6.0 9.0 ~o O'F

1.O

h (c) Hs

05

0 I ! ! 4.0 8.0 12.0 %

Fig. 5. Comparison o f the measured and predicted standard deviation o f the force as a funct ion o f segment elevation h. ----, standard deviation based on (2), - - , standard deviation based on (10). (a) test 1, (b) test 2, (c) test 3

standard deviation given by the square root of the area under the spectral density curve, whereas the solid curves correspond to the closed-form expression (10) provided by Tung 4. The former predictions are some- what lower than those based on the closed-form expres- sion since they are based on a linearization of the Morison equation, whereas expression (10) is based on the complete Morison equation. This is similar to a corresponding difference for fully submerged sections. The theoretical values agree reasonably well with the measured values except near the still water level where they significantly overpredict aF.

Force maxima .

Figure 6 shows representative comparisons for test 1 of the measured probability density distributions of the

wave force maxima for segments 6, 7 and 8 located above the still water level with the corresponding numerical predictions based on equation (13). Sections below the still water level are not included in these results since intermittency effects are then predicted to be absent. The effects of intermittency become more pronounced the farther the segment is above the still water level. The observed irregularities in the measured probability distributions arise in part because they each correspond to a single wave record, whereas the smooth theoretical curves correspond to the expected distri- butions obtained from many records. In spite of the scatter in the measured probability distributions, the general shifts in the distributions that occur with height above the still water level are the same for both the measured and predicted probability distributions. The most notable difference between the predicted and measured probability distributions is the large spike observed in the measured distributions at small values of the force maxima, particularly for segments 7 and 8. The theoretical probability density of the force maxima as given by equation (19) predicts a spike at zero, corre- sponding to a probability of P. For test 1 the values of P for segments 6, 7 and 8 are 0.1, 0.62 and 0.93 respect- ively. It is possible that this spike corresponds to the measured spikes at small non-zero values of force maxima, on account of errors in force calibration.

Of greater interest than the probability density distri- butions of the peak forces are the statistics deriving from these distributions, such as the mean and standard deviation of the force maxima and the single largest maximum force occurring in a specified duration. Figure 7 shows the variation of the mean tq of the force maxima with segment elevation h above the still water level. Below the still water level intermittency has no effect on the peak forces and the distribution with depth follows the expected profile associated with the hyper- bolic functions appearing in the expressions for the par- ticle kinematics. However, above the still water level the mean peak force is not properly defined unless inter- mittency effects are considered. The decay of the mean of the force maxima with elevation above the still water level is shown. Note that the mean force maxima obtained above takes account of zero force maxima as discussed earlier so that ~- decreases with increasing h. (On the other hand, the mean of the non-zero force maxima would increase with h, although the probability of encountering such maxima diminishes with increasing h.) The force maxima are noticeable at elev- ations up to h---Hs. Agreement between predictions taking account of intermittency effects and the experi- mental results is generally good. The predicted values of /q- are somewhat higher than the measured peak forces at elevations nearer the still water level and somewhat lower at elevations above about 0.5Hs.

Figure 8 shows the variation of the standard devi- ation as of the force maxima with segment elevation h above the still water level. Above the still water level, the standard deviation is predicted to increase initially with elevation h and then to decrease. Agreement between predictions taking account of intermittency effects and the experimental results is generally reason- ably good, except for test 3 shown in Fig. 8(c), for which the standard deviation appears to be over- predicted at lower values of h.

194 Appl ied Ocean Research, 1990, Vol. 12, No. 4

Random wave forces near the f ree surface: M. Isaacson and J. BaMwin

hi,,. v L

O.3i

I . I I

I ' l . t l . I i . I

SEGWIVNT I a

e.o i . t i i . i

l i / r 1 6 2 I N )

HFIHH l i . l l l a . i

0 . 3 6 -

0 . 3 0 -

0 . 2 4 -

fa. O.I8-

r~

0 . 1 2 -

0 0 6 -

0.00

0.0

(a)

2.0 4 0 s o 6 o , o o 1 2 o J4o F o r c e (N)

b.

1.7$

I . I I

i - l $

i . 1 $

e . t O.I

~ | G M I N T 1 7

I . O 4 . I ql.gl II.10 I g l . I I

F o r c e (N)

0.'75 -

0 . 6 0 -

0 . 4 5 -

0 3 0 -

0 . 1 5 -

0 . 0 0 nl .o 0 . 0

Co)

2.o 4'0 ~o ~o ,~o ,~o Force (N)

L.

i . i

S . l l

i.g

t . t l !

.'.i ILo e'~.l I.i

$ 1 G W E N T 1 8

c,

(c)

1 .15-

! . 5 0 -

1 . 2 5 -

1 . 0 0 -

0 . 7 5 -

0 . 6 0 -

0 . 2 5

0 O0

0 . 0

, ~ fin fl, n ~, , n , , n a.o O.* t.o" no.o u#.o l*.* uO.o 2 . 0 .0 .0 B.O ! 0 . 0 | 0 1 4 0 ! 0

' . . . . ~") F o r c e (N)

Fig. 6. Comparison o f the measured and predicted probabifity density o f the force maxinla f o r test 1. (a) segment 6, (b) segment 7, (c) segment 8

Appl ied Ocean Research, 1990, Vol. 12, No. 4 1 9 5

Random wave forces near the free surface: M. Isaacson and J. BaMwin

h

1.0

0.5

0 z5 rt; 5.o

1.0

h

Hs

0.5

0 2.o % 4.o

h

I ts

1 , 0 B

0.5

0 ! D. 2.5 5.0

1.0

h . �9

Hs

0.5 )

0 I I 5.0 10.0 ~

Fig. 7. Comparison o f measured and predicted mean o f force maxima. (a) test 1, (b) test 2, (c) test 3

1.0

h

H s

0.5

0 ! ~. 2.5 5.0

Q

1.0 - ~ h (c)

Hs 0.5

0 I 2.0 4.0 6.0

Fig. 8. Comparison o f measured and predicted stan- dard deviation or force maxima. (a) test 1, (b) test 2, (c) test 3

There are a number of effects which have not been incorporated into the theoretical predictions of the probabili ty distribution of the force maxima which might be used to explain some of the observed discrep- ancies between theory and experiment. The first is due to differences between the target and generated wave spectra as indicated in Fig. 3. The theoretical results are based on the use of the target wave spectra rather than the measured wave spectra.

The second is associated with the assumption of a narrow-banded spectrum. The measured wave spectra are not narrow-based as the theory assumes, and this will give rise to a greater number of force maxima than is assumed, depending on the spectral width parameter, so that the corresponding probabili ty distribution will

also be affected. The measured force maxima were determined by scanning the force time series and the number of maxima obtained from the experimental data is significantly higher than that based simply on the duration of record and the peak period.

A third effect which has not been included is due to the measured forces relating to segments of finite height, whereas the theoretical predictions relate to forces on segments of infinitesimal height. The segment heights were 0.1 m, which is not small in relation to the significant wave height. A finite segment height gives rise to a number o f force maxima corresponding to the water surface rising only a short distance up a segment. This gives rise to relatively small forces which would not be reflected in the theoretical predictions.

196 Applied Ocean Research, 1990, Vol. 12, No. 4

Random wave forces near the free surface: M. Isaacson and J. BaMwhl

Additional effects relate to nonlinearities in the wave flow, and to uncertainties and variations in the values of the force coefficients in this region of the cyl- inder. Thus the fluid velocity and acceleration are not constant above the still water level and differ from the predictions of linear wave theory. Likewise, the force coefficients may exhibit significant variability in this region.

Extreme forces Figure 9 shows a comparison between theoretical

expected values E[F,,,] and measured values of the single largest force maximum recorded on each segment corresponding to the duration of the total wave record for each test. In all cases the total duration was 204.8 seconds and on the basis o f the narrow-band assump- tion this corresponds to the passage of 65, 84 and 111 waves for tests 1,2 and 3 respectively. Notable features

h

Hs

1.0

0.5

!

(a)

7.5 ^ 15.0 ElF m} (ND

h

H s

1.0 B

0.5

I I 10.0 ^ 2 0 . 0

EiFm] (I'4)

(b)

1.0

h

Hs

0.5

"<___ ~ (c)

I I 10.0 ^ 20.0

EiFml (N)

Fig. 9. Comparison of theoretical and measured vahte of the shlgle largest force maxinntnl. (a) test 1, (b) test 2, (c) test 3

of these results are that this force value remains approximately constant with elevation above the still water level and that the force may be significant at elev- ations h approaching Hs. This flattening of the curve is confirmed by the experimental results although there is a sizeable amount of scatter. Apar t f rom the factors mentioned earlier, this scatter is also due in part to the experimental results representing a single set o f wave records, whereas the theoretical predictions correspond to the expected value of the extreme force obtained from many sets of wave records.

Multi-directional waves Finally, it is of interest to present selected experi-

mental results o f forces due to multi-directional waves, although the theoretical predictions have not been made for such a case. Because a random multi-directional wave train corresponds to components propagating over a range of directions, there will now be a signifi- cant force transverse to the principal wave direction. Figure 10 shows the in-line and transverse force spectra for a wave condition corresponding approximately to test 2, but with a spreading index s ~ 1. The figure pre- sents results for segments 6, 7 and 8. The figure exhibits the trend of Fig. 4 which shows the magnitude of the spectral density to decrease for the higher segments which are submerged more infrequently. As expected, there is now a significant transverse force component associated with the directional spreading of the waves.

CONCLUSIONS

Experiments have been carried out to measure random wave forces in the vicinity of the free surface which act on a slender vertical circular cylinder. Recent theoret- ical predictions for this case are summarized and a comparison between the measurements and the theoretical predictions is presented. The intermittent submergence of sections near the water surface mark- edly influences the statistics o f the wave forces.

The predicted force spectral density accounting for the intermittency of submergence is based on a linearization of the Morison equation. Above the still water level, intermittency effects give rise to a signifi- cant reduction in the magnitude of the force spectral density. This is observed in the experiments.

The predicted probabili ty density of the force maxima accounting for the intermittency of submergence is based on the assumption of a narrow- band wave spectrum. F o r sections of a member which are below the still water level, the intermittency of submergence does not affect the characteristics of the force maxima, whereas for sections of a member which are above the still water level, the intermittency of submergence should be taken into account in predicting the probabilistic properties of the wave force.

Comparisons of the experimental results with theoretical predictions of mean of the force maxima, and the expected value of the single largest force maximum which occurs within a specified duration are presented. The latter remains approximately constant with elevation above the still water level, particularly for longer durations, and may be significant at elev- ations above the still water level which approach Hs.

Applied Ocean Research, 1990, Vol. 12, No. 4 197

Random wave forces near the f ree surface: M. lsaacson and J. Baldw#t

pl

N

T. I

$+I

IP.l

I . I I . I

IN-LINE FORCE

S I G M E H T 18

i . I I . i 1.5 1.1

r r e q u e n � 9 7 ( R z )

N

z

w]

$.1

11.71

M

z

I . l |

I . I t , I

TRANSVERSE FORCE

S E G M E N T 18

I , l I . l I .5

I r r , q u l n r ( H z )

| .1

I . !

I , I

I I . I

! . I I I . I

! ! t l . l l , l I . I I . I

F r e q m e l e ~ ( H z )

i l G M I N T 1 7

A u it,,,

.'2'

"t 1.4

l . l

I . I t . I I

S I G M I N T 17

I ! !

l . I I , l I .S l , l

r r e q ~ e l e 7 ( X z )

N mE %.

*

z

I . I

I , I

I I I . I I . I

I I G M I N T | !

I ! I

t , S I . ! I . $ I . I

r r e q u e n e 7 ( M z )

I ' l l r

%.

m I . I $

t . I I . I

$ 1 G M K N T f 8

I+$ I+l I .S | . 1

r r e q u e n r ( H s )

Fig. 10. hi-line and transverse force spectra on segments above the still water level f o r nndti-directional waves corresponding to test 1, with s = 1

198 Appl ied Ocean Research, 1990, Vol. 12, No. 4

Random wave forces near the free surface: M. lsaacson and J. BaMwin

Differences between the measured and theoretical forces are attributed to differences between the target and generated wave spectra, the assumption of a narrow-band spectrum, the finite width of segments used in the experiments, nonlinearities in the wave flow, and uncertainties in the force coefficients.

Measurements of force spectra due to multi- directional random waves are also presented. These exhibit a considerable transverse force component.

NOMENCLATURE

a

b Ca Cm D d F r G(z)

g H I-I,, h K

1G Km k L (a ,b , r ) N N; P P Q

r s~ S~,,1

II

horizontal component of fluid acceleration h/o~ drag coefficient inertia coefficient cylinder diameter still water depth wave force wave force maximum function from linear wave theory, see equation (5) gravitational constant wave height velocity transfer function, see equation (4) section elevation above the still water level representative Keulegan-Carpenter number, see equation (18) drag parameter, �89 pDCa inertia parameter, P 0rD2/4) Cm wave number see equation (11) number of waves number of non-zero force maxima cumulative probability probability density function Gaussian probability function, see equation (6) correlation coefficient, see equation (8) spectral density o f variable x cross spectral density between tt and r/ peak period horizontal component of fluid velocity

Z Z

~( .), F

# p o"

L

vertical coordinate, see Fig. 1 Gaussian probability density, see equation (7) Dirac deIta function see equation (16) see equation (17) free surface elevation mean fluid density standard deviation wave angular frequency dimensionless force maxima, see equation (14) dimensionless constant related to K, see equations (15) and (19)

REFERENCES

1 Borgman, L. E. Statistical models for ocean waves and wave forces, Advances in Hydroscience, 1972, 8, 139

2 Sarpkaya, T. and lsaacson, M. Mechanics o f Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 1981

3 Ochi, M. K. Stochastic analysis and piobabilistic prediction of random seas, Advances hl Hydroscience, 1982, 13, 217

4 Tung, C. C. Statistical properties of wave force, Journal o f Enghwering 2~lechanics Div., ASCE, 1975, 101, 1

5 Tung, C. C. Statistical properties of the kinematics and dynamics of random gravity wave field, Journal o f Fhdd AIechanics, 1975, 70, 251

6 Pajouhi, K. and Tung, C. C. Statistics of random wave fields, J. Waterway, Harbours, and Coastal Div., ASCE, 1975, 101,435

7 Satyanarayana, M. and Elango, K. Effect of free surface fluctua- tions on the probability density functions o f wave kinematics, Proc. Second hldian Conference b~ Ocean Engineering, 1983, 31

8 Anastasiou, K., Tickell, R. G. and Chaplin, J. R. The nonlinear properties of random wave kinematics. Proc. 3rd. Int. Conf. on the Behaviour o f Offshore Structures, BOSS "82, 1982, Vol. 1, 493

9 Isaacson, M. and Baldwin, J. Random wave forces near the free surface, Journal o f IVaterway, Port, Coastal and Ocean Div- ision, ASCE, 1990, 116, 232

10 Borgman, L. E. Wave forces on piling for narrow-band spectra, Journal o f IVaterways and Harbors Div., ASCE, 1965, 91, 65

11 lsaacson, M. and Nwogu, O. Short-crested wave forces on a ver- tical pile, Proc. 7th Int. Conf. on Offshore ]Lfeehanics and Attic Engineering, Houston, Texas, 1988, 47

12 lsaacson, M., Nwogu, O. and Cornett, A. Short crested wave forces on a cylinder, Proc. Ocean Structural D)'namics Syrup., OSDS '88, Corvallis, Oregon, 1988, 376

Applied Ocean Research, 1990, Vol. 12, No. 4 199