Ocean Engng, Voi. 17, No 1/2, pp. 125-154, 1990. 0029-8018/90 $3.00 + .00
Printed in Great Britain. Pergamon Press pic
DIFFRACTED WAVE FIELD AND DYNAMIC PRESSURES
AROUND A VERTICAL CYLINDER
V. SUNDAR, S. NEELAMANI and C. P. VENDHAN Ocean Engineering Centre, Indian Institute of Technology, Madras, India
AbstractInvestigations on the hydrodynamic pressures due to regular and random waves
exerted on a large vertical cylinder in a constant water depth are reported in this paper. In the
experimental investigation, the test cylinder embedded with diaphragm-type pressure
transducers at nine different elevations was rotated about its axis to measure the dynamic pressure around its circumference. The wave field in the neighbourhood of the cylinderwas
measured at six different locations. The results of the experiments are compared with the linear diffraction theory of MacCamy. R. C. and Fuchs, R. A. [(1954) Wave forces on piles: a diffraction theory. U.S. Army Beach Erosion Board, Technical Memorandum No. 69]. In general, the agreement between the theoretical and experimental results is found to be
satisfac-tory. A comparison between the regular and random wave test results is also presented and
discussed.
1. INTRODUCTION
THE CONTINUING search for offshore oil and gas resources has driven mankind into deeper waters of the oceans, where the design of offshore structures subjected to waves
become quite complicated due to the hostile environment. Accurate assessment of not
only the total wave loads but also the wave-induced sectional forces and dynamic pressures are essential to optimise the design of such structures. The evaluation of wave
forces on vertical cylinders has been the subject of intensive research during the last
few decades.
The first empirical formulation for the calculation of wave forces on slender members (D/L < 0.2, where D is the diameter of the cylinder and L is wave length) was proposed
by Morison er al. (1950), based on the assumption that the presence of the structure
does not affect the characteristics of the incident wave field. This formulation, however,
fails to predict the wave loads accurately, for DIL > 0.2 due to the dominant wave
diffraction.
The scattering of waves from a fixed body was first recognised by Havelock (1940),
and based on which, MacCamy and Fuchs (1954) proposed a closed form solution to evaluate the hydrodynamic pressures, forces and moments on a large vertical cylinder in regular waves. While considerable work has been reported on the total wave forces on vertical cylinders, studies on wave-induced dynamic pressures are rather limited. Notable contributions on the dynamic pressures on vertical cylinders due to regular waves are those of Laird (1955), Priest (1962), Chakrabarti and Tam (1975), Endo and Tosaka (1985) and Chakrabarti et al. (1986). The results were presented in a variety of forms as a function of flow parameters. Hellstorm and Rundgren (1954) have
attempted to correlate the measured dynamicpressures around a cylindrical model with
the theoretical pressures on a plane vertical wall. Nakamura (1976) has measured the
dynamic pressures exerted on a cooling water intake structure on the coast of the Pacific
125 TECHNISCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanica Archief Mekelweg 2, 2628 CD Detft TeL:015-786873-Fax:015-181836
V. SUNDAR et al.
Ocean and concluded that the measured pressures due to irregular waves were found to be smaller in the deeper part and larger near the still water level (S.W.L) than those
given by the MacCamy and Fuchs (1954) linear diffraction theory. Caveleri et al. (1977)
have measured pressure fluctuations due to surface waves at different elevations on a
tower in the Adriatic Sea near Venice and found that the measured pressures deviate
from the linear wave theory. The deviation was observed to be greater at lower frequencies and with the increase in the depth of measurement of the dynamic pressures. Raman and Rao (1983) have measured the dynamic pressures due to laboratory wind-generated waves at three different elevations below the S.W.L. The experimental
results were compared with the MacCamy and Fuchs linear diffraction theory. It was
noticed in their study that the measured pressures of the pressure ports facing the side
wall deviated considerably from the MacCamy and Fuchs linear diffraction theory. This
phenomenon was attributed to the proximity of the flume side wall. The correlation between the measured and theoretical pressures for the other pressure transducer
orientation with respect to the wave direction was found to be quitesatisfactory at z/d
(where z is the point of pressure measurement, measured negative downwards from
S.W.L., and d is the water depth) of 0.22 and 0.44. It
should be noted that the frequency range adopted covers mostly the shallow and intermediate water depth conditions. However, significant deviations were observed between the measured andtheoretical values with increase in the depth ofmeasurement of the dynamic pressures,
similar to the observations of Caveleri et al. (1977).
A review of the existing literature reveals that only limited work has been reported
on the wave-induced dynamic pressures on vertical cylinders, of which most of the work
pertains to the action of regular waves. As for the literature on the study of dynamic
pressures on vertical cylinders due to random waves, Nakamura (1976) has carried out field experiments, whereas Raman and Rao (1983) have investigated the random pressures on model cylinders exposed to laboratory wind-generated waves. Apart from
the above, to the best of the author's knowledge, there does not seem to be any other
published work related to dynamic pressures on vertical cylinders due to random waves. The measurement of the wave-induced dynamic pressures around the circumference
of a large vertical cylinder would be needed for a more rigorous verification of the linear wave theory, rather than the measurement of the total wave force. Further, the
inevitable side wall effects on the wave diffraction due to the presence of large diameter structures in the narrow laboratory wave flumes can be inferred from the measurement
of dynamic pressures around a cylindrical structure. The study of the diffracted wave field around a large vertical cylinder would be useful in fixing the minimum air gap
between the mean sea level and the deck of the offshore structures. The above reasons
prompted the investigators to carry out a detailed investigation on the dynamic pressure around a large vertical cylinder and the diffracted wave field subjected to random waves.
2. THEORETICAL CONSIDERATIONS
2. 1. Linear diffraction theory
The hydrodynamic pressure, p(t) on a large diameter vertical cylinder and the diffracted water surface elevation, lr(t) at any radial distance r, at time t are given by MacCamy and Fuchs (1954) as
Dynamic pressures around a vertical cylinder 127
'n pgHcosh[k(d+z)}
p(t)
= 'rrkacosh(kd) L LI(I) (ka)
cos[m(irO)]E-and
lr(t) = ¡"m [J1(kr)
H (kr)]
cos [m(O)]E'' (2)
m() 'n
in which p = the mass density of water, g = the acceleration due to gravity, k = the wave number (2'rr/L), a - the angular wave frequency (27rJ), where f = the linear wave frequency, a = the cylinder radius, J,,, = the Bessel function of the first kind of order
m, = the Hankel function of first kind of order m, prime denoting the derivative
of the functions with respect to their arguments, e,,, = constants with e = I and =
2 for in 1, and O = the polar angle measured clockwise from the leading edge on
the circumference of the cylinder (thus O = 1800 refers to the trailing edge) (Fig. 1). The transfer function for dynamic pressures on a vertical cylinder. TF,,,(f) and that for the diffracted water surface elevation, TFrt(f) can be obtained from the above two
equations as
TF(f)
2pg cosh [k(d+z)] ç,, 'n cos [rnerr-O)]rrka cosh (kd) QH,Y (ka)
Wave direction
'V
2a
FIG. 1. DefInition sketch.
(1)
S,,,71(f) S(f) (8) V. SUNDAR et al. and
.ka)
i TFr1(f) =, i"'4'
H) (ka)
(kr)j cos [m(O)].
(4)rn=D L
The spectral density of the dynamic pressure, S1(f) and the diffracted water surface elevation, Srg(f) can now be expressed as
S1(f) = T,(f)2 S,(f)
(5)and
Srt(f) = TFrr(f)2 S,(f)
(6)in which S(f) = the spectra! density of the incident water surface elevation, q. 2.2. Average pressure coefficient over the frequency domain
Let the pressure spectrum be approximated as
S1(f) = (Ç pg)2 ST(f) + (7)
where C,, is a pressure coefficient uniform over f. Minimising the error over the
frequency range of interest in a least-square sense, one has
(Cpg)2 =
>: s
(f)
where S,,,,, is the measured pressure spectra! density value. The pressure coefficient Ç, as defined above may be obtained for different input parameters, namely H, and f. Then, given a wave spectrum defined by a particular H and f, one can directly choose an appropriate value for C',,. It will be of interest to compare C',, from Equation (8)
with the pressure coefficient for an appropriate regular wave.
3. EXPERIMENTAL INVESTIGATIONS
3. 1. Experimental set-up
The experimental investigations were carried out in a 4 rn wide and 45 m long wave
flume at the Ocean Engineering Centre, Indian Institute of Technology, Madras, India. The water depth throughout the experiments was maintained constant at 2.5 ni. Random
waves were generated using a twin flap wave maker located at one end of the flume and its movement is controlled by a computer. A suitable wave absorber is provided at the other end of the flume to absorb effectively the incident wave energy. The test
model used for the present study is a PVC pipe of diameter 400mm and wall thickness
of 15 mm, fixed at a distance of 20 m from the wave maker. Nine diaphragm-type pressure transducers each of ± 0.5 bar capacity were embedded at different depths of submergence. One among the nine pressure transducers was fixed at the S.W.L. itself to study closely the nonlinear effects. The spacings between the pressure transducers
close to the S.W.L. were kept lower than those close to the flume bed, consideringthe
Dynamic pressures around a vertical cylinder 129
are shown in Fig. 2. The ylinder is waterproof and one of its ends is fixed rigidly to
the flume bed. A special arrangement at the top of the cylinder is provided to rotate the cylinder about its own vertical axis. This enables any desired angle of orientation of the pressure transducer with respect to the wave direction.
A resistance-type wave probe fixed at a distance of 14 m from the paddle was used
to register the time history of the incident wave elevation. Another wave probe fixed
around the test model at six different locations registered the time history of the scattered waves. The leads from the pressure transducers and the wave probes were
connected to multi-channel carrier frquency amplifiers and wave monitors, respectively, and the signals from this system were acquired continuously using a personal computer.
The general instrumentation set-up used for the present study is shown in Fig. 3. 3.2. Experimental procedure
The wave probes and the pressure transducers were calibrated prior to and after each
set of experiments. The test model was exposed to random waves with
PiersonMoskowitz spectra having a spectral width of = 0.6 and two different energy
levels. In addition to this, experiments were also conducted for random waves with a
narrow band spectrum having . = 0.3. The characteristics of these generated spectra are displayed in Table 1.
The experimental data were acquired at a sampling rate of 20 samples/sec from each channel for a time span of 51.2 sec. About 45-50 waves were collected at this time
span. The test cylinder was rotated about its own vertical axis in steps of 30° from O = O to 180°. In order to carry out a comparative study between regular and random wave tests, the cylinder was also subjected to regular waves of different heights and periods. The wave heights ranged from 3 cm to 25 cm and wave periods ranged from 0.6 to 2.0 sec.
4. RESULTS AND DISCUSSION
4.1. General
The time history of the simulated random waves and the corresponding dynamic
pressures were analysed and tested for their normality. The instantaneous water surface
elevation and the dynamic pressures were found to follow the Gaussian distribution reasonably well. The frequency domain analysis of the measured time series was performed using the FF1 technique. The raw spectral estimates were smoothed using the nine point moving average method as proposed by Daemrich et al. (1980). The frequency range of the generated
wave spectra covered mostly the deep water
conditions.TABLE 1. CHARACTERSTtCS of TItE LABORATORY-StMULATED WAVE SPECTRA
Type of Significant wave Peak Spectral width S. No. spectrum height H (cm) frequency f) (Hz) parameter ,
i Narrow band 6.6 0.74 0.3
2 PM1 4.46 0.76 0.6
FLUME SIDE WALL
/ //// / / / ////////////////////////
Wave direction Wove probe for Incident Wove\\\\\\\\
\\\ \\\\\\\\\\\\\\\\N
FLUME SIDE WALL
PLAN
Flange screwed to tighten SWL o -v-Q V. SUNDAR et al. o 90 120 30 1503210
270Wave probe (to br fixed any where
T around the test cylinder) for diffracted
wove measurement 180
G I Coupling for pipe extension Mild steel channel Scn-rx7.Scm
Pressure Transducers V
Pipe 40cm dia 15cm waLl thickness
Golvanished iron shaft
Coupling /Rubber gasket
Mild steel circular plate
/
1 c ni thickI
/ Flume bed Stainless steel Brass washer Mild steel channel
bolt + nut
ALL DIMENSIONS ARE IN cm
FIG. 2. Experimental set-up.
'JI Q Q Ç,-' e -t
Dynamic pressures around a vertical cylinder 131
FIG. 3. General instrumentation set-up.
4.2. Theoretical and measured dynamic pressure spectra
Though the dynamic pressures exerted on the test cylinder were measured for O =
o to 180° in steps of 30°, in order to make the presentation simple, onlytypical plots
showing the comparison between the theoretical and measured pressure spectra for O = 0°, 90° and 180° for the cases of narrow band and PM2 spectra are shown in Figs 4ac and Figs 5ac. respectively. The corresponding incident wave spectrum is also
superposed. In general, the correlation between the theory and experiments is found
to he satisfactory. It is seen that the agreement between the theory and experiment is
good even at S.W.L. for the case of narrow band spectrum, whereas deviations are
observed at S.W.L. for the broad band cases, showing that the effect of scattering and
nonlinearity is more predominant for a broad band spectrum than for the case of a
narrow band spectrum. It is also noticed in these plots that the spectral peaks of both theoretical and measured pressure spectra shift towards the lower frequency zone as
the depth of submergence of the pressure transducer increases. This obviously is due
to faster decay of the high frequency components with respect
to depth. It
is also observed that at zld = - 0.36, the measured peak pressure spectral value is less than1% of that measured at S.W.L. Therefore, further results and discussions will only be
confined to pressure transducer locations with z/d from 0.0
to - 0.26.
4.3. Measured pressure spectral density around the circumference of the cylinder Typical plots showing the variation of the measured pressure spectral density around
the circumference of the cylinder for two different z/d values (z/d
= 0.0 and - 0.08)
for the narrow band, PMI and PM2 cases are shown in Figs 6ac. It isseen in general
E E P7.. 6 Channel car rier frequency -Apple Computer for Data Dota Storage for further analysis L. amplifier Acquisition P, Pi
Power from main
p? 6 Channel car rier frequency Voltage Stabilizer P4 - PS amplifier P6 = o - _lncident Wve H1 'p 0. Wave Diffracted Wave Monitor Amplifier
Wa
En'
E u E ci 20 lo 6 o O t i E E o 12 Q- Q- C-if) if) 10 o
ti 2
o -E o o in E Z ¡do-0.26 -12 C-- C.- (J)6
I 1 I 1.!. 2.0 0.2 1.4 2.0 0.2 0.0 Frequenicy (Hz) Legend: Exp (Sf ) ) ...Theory (S (f))- Wcive (S(f))
FIG. 4a. Comparison of theoretical and experimental pressure spectrum for
0 00. 2
k
Z /do- 0. 16 O I I I i Z /d 0.0 _.1__... I z/d o-000 126
t I Ii
o in 12 6jt "\ 0-2 0.8 0 Frequency (Hz) Legend: Exp (S (f)) .Theory (Sf)) - - Wcive(S(f))
FIG. 4b. Comparison of theoretical and experimental
pressure spectrum for O = 900.
L) th r. E L) j:- j:- In 0
6
3
E o a a II) 0-90 0.60 o . 0.00 z /d-O.16 I Z Id-0.2612
9
6
3
I i 1-4 2.0 02 O-8 1-4 2.0 z/d 0.0 ud s-0.08 12 12. 12-9
9 63
o i I I o6 U Q, U) e-,--.. o E U E a a 12 11)2 O 02 Legend: Exp (S(f)) i / d r-0.16
6
12 6I.
I I I 0.8 1.4 2.0 Frequency (Hz) ,Theory (S(f))
-z/d - 0.08 .126
FIG.4c. Comparison of theoretical and experimental pressure spectrum for (0
= 180°. t U Q, U) E o C-
C-
f) -11) 0.80 0.40 0.00-,
WOVe I6
0.2 0.6 1.6 (S 2.0 11(f)) u 'J) o U Q, I I q U) E U E u E 12 C-a C-a ud - 0.26 12u o, 'n o E u E cy' 20 t 10 Q. Q. 2 0 0.2 Legend: Exp.(5 (f)) 10 Frequency (Hz) ,Theory (S (f)) - - Wave (Sq(fl)
FIG. 5a. Comparison of theoretical and experimental
pressure spectrum for 0
00. 15 10 5 o E C- C- '-n o 15 10 O u E C C k z /d o-0.16 V. -i I ".... z /d 0.26 Ï Z /d° 0.0 15 z /d 0-0.08 Is 0.8 1.4 2.0 02 0.8
i.'
2 0 10 10 s 5 t12 E u E Q-2- Q- (t) o z íd - 0.16 15 10 s 0 0.00 02 7.
(
Frequency (Hz).---Legend: Exp. (S (f)) ...Theory (S(f) ) Wove (S (f)) FIG.5b. Comparison of theoretical arid experimental
pressure spectrum br O = 90°. 'J Il, r-d E o o o E o E o E 15 z Id 0.26 Q- C-Q- 0.60 C-LI) (J) 10 C-Iii -I O . 4Q s 15 10 6 2 \ z Id i-0.08 15 10 5 0.2 0.6 1.4 20 o 2 06 16
10 5 o 2 0 E Q. Q- U) 15 10 5 0 u I) E C C- U) E o E a a 080 0.40 00 -.1
6" \
Z Id - 0.16 L 12 15 10 5 o 6 o o z Id 0.0 02 0.8 14 2.0 02 0.8 1.6 Frequency (Hz) Legend: Exp.(S(f» Theory (S(f))
-WQVe (Sqq(f))FIG. 5c. Comparison of theoretical and experimental
pressure spectrum for O = 180°.
z Id0.26 -15 15 z/ds-0.08 :1' r 10 s
)
.1
O20 0 E o E alo a V) a 4 o 0.2
L
o 9Q° zId-0.08 OoO 0 30° 0 600 r 90°6
t
0 0 o E o 12 Frequency (Hz) 6 o 4 2 o 0.2 lid oDO I -12 6 tFIG. óa. Measured
pressure spectral density around the
circumference of a vertical
cylinder (narrow band).
E o ZIdo-0.08 9r180° 12 ° 150° C 012 0° C-If) 6 'J
k
12
Sijii(f) z/do0.0 O o 0° o 30° o 60° 2.0 0.8 I I o 08 1.4 200 3 2 t o In E Frequency (Hz) 1, e... O E o E cn a a u) 0 0.2 IlIl z /d0.O_ O -8O 1s0 120° FIG.
6h. Measured pressure spectral density around the circunilerence of a vertical cylinder (PM 1).
0.8
2 0
s
i)
-f)lj zId0.O 8 18O -1S0 -12 6 o 12 o E 12 12 6 O o 4, O n E E L) 2' 12 -. Q. C-Q. Ii) V) 4 6 2 0 0 O? 01 1 .4 20 02 0.8 2. 0 Frequency (Hz)FIG. 6c. Measured pressure spectral density
around the circumference of
that the dynamic pressure around the circumference of the test cylinder at any depth
of submergence decreases with increase in the value of O from O to 1200, beyond which
there is an increase in the pressure spectral density. It is to be noted from Figs 6b and c that the measured pressure spectra at S.W.L. for the PM1 and PM2 cases show secondary peaks for O = 0-90° at frequencies beyond 1 Hz, which are absent for the case of narrow band spectra. It reveals that the pressure transducers at O = 0-90° are mostly exposed to the direct impact of the incident waves and the effect of diffraction
is more pronounced beyond 1 Hz, whereas at O = 90-180°, (being the shadow region), the pressure transducers in this region are expected to sense the dynamic pressures after the waves get scattered by the cylinder, especially at higher frequencies. This
explains the reason for the presence of secondary peaks at S.W.L. for O = 0-90° beyond
1 Hz and its absence for O = 120-180° even beyond 1 Hz. It is also noticed from these
plots that the circumferential variation of the spectral density of the dynamic pressure
is quite large from O = 30-90°, when compared to that from O = 120-180°.
4.4. Theoretical and experimental peak pressure spectral values
A plot showing the theoretical and measured spectral peak values of the dynamic pressures around the circumference of the test cylinder at different elevations (z/d =
0.0 to - 0.26) for all the incident wave spectra referred to in Table i are shown in Fig.
7. The correlation between the theoretical and measured peak pressure spectral values
Dynamic pressures around a vertical cylinder 141
Narrow nd E 03 z /d - 0-08 - O-16 -026 0 60 120 180 0 60 120 180 0 60 120 180
Pressure tronsducer orientation w.r.t. wove direction in degrees
-FIG. 7. Comparison of theoretical and measured spectral peaks around a vertical cylinder.
E '-' E D, ci > o 10 5
tI
s D * u . . u u00
A A A A o A o 0.
o s o.
o A L D V s o A A u D V.
o u D.
S O o1A
A u u D D.
o u u D D o-" 1.0 V V V V y D V D V D V u A A A A A L A A y y V V V V V V V V D y V 05 V y y u D u o D u D V V Vy VV V V 01 J i t I I i t i t I t J t tTheory Prese rit
E xpt. PM 1 PM 2 s o O6 E . zOG £ A E1 u D V V loo 50
V. SUNDAR et al.
is found to be good even at S.W.L. for the case of narrow band spectra, whereas
deviations are observed in the case of PM1 and PM2 at S.W.L. This is due to the
dominant diffraction and nonlinear effects existing for the case of broad band spectra
as discussed in Section 4.2.
4.5. Theoretical and measured zeroth pressure spectral moment
In order to visualise clearly the total energy of the dynamic pressure induced at
different elevations due to the three differentincident wave spectra mentioned in Table
1, a comparative plot showing the theoretical and measured zeroth pressure spectral moment is given in Fig. 8. The zeroth moment at any elevation is a maximum at O =
00 and decreases as O increases up to 120°, beyond which it increases slightly. The
correlation between the theory and experiments, in general, is found to be good. Here
again it is seen that at zid - 0.36, the total measured pressure energy observed is
less than 1% of that observed at S.W.L. as mentioned earlier. The deviation between
the measured and theoretical zeroth moment of the dynamic pressure at z/d
= - 0.36
is greater. Such a phenomenon has also been observed by Raman and Rao (1983) and
Caveleri et aI. (1977). X X X X 4 A A u D V y u D V y e e o S O o O A 8 u D V y e
.
A A A u D A A u u D O e.
S A 4 A £ e e X XJI
i A A 8 A u D D U D V V y y e X e G A A A A u u u D D O y A A e e A A u D e X * X X Narrow band 0-3 z/d o -0.0 e -0.1k -l-24 -E u 9 E E 1.0 £ 8 u a. 05¿u
D . D . u u u Q--C y ç, y V D y 7 o y D y V D y V 0.1 e V 3 E 0.05 e e S S -C o NJ X X Tb.ory Pr,i.nt Expt.E0-6
PM 1 o o AA o 8 E PM2 =0-6 Q.
o £ 8 u D y V X Ö 0 60 120 180 0 60 120 180 0 60 120 180Pressure transducer orientation w.r.t. wave direction in degrees
-FIG. 8. Comparison of theoretical and measured zeroth moment of the pressure spectrum around a vertical cylinder.
The variation of the theoretical and experimental ,. around the circumference of the cylinder for different z/d, for PM1 and narrow band spectra, is shown in Fig. 9. Since
Ep for both PM1 and PM2 are nearly the same, only PMI has been considered.
The general correlation between the experimental and theoretical ,, is found to be satisfactory for z/d up to - 0.26, beyond which large deviations of experimental values from the theory are observed. This discrepancy may be attributed to the fact that at
lower levels in deep water, as seen earlier, the magnitude of dynamic pressure isvery
small, and a slight error in their measurement would get magnified considerably in the calculation of the fourth spectral moment leading to the possible increase in . It is seen
that the theoretical and experimental e,, are almost constant around the circumference of 4.6. Theoretical and measured spectral width parameters of dynamic pressures
The spectral width parameter, e,, for the dynamic pressure is given as /
= 11
-\ mm4
where m denotes the nth spectral moment and is given by
nz=J
S(f)df.
0.4 0.2 oc zldr 00 o C D C D u . D . a u o O o o O O J I IDynamic pressures around a vertical cylinder 143
z Id r -008 D D u o o Theor1 D a a D D D a U u o o o o o o s o a U O o D D U o
- .
.
s D . o . s O s t I I o s . o s o u o u o u o s o s o . u u- s
s L I I o o . u s s e . s u u . 0 60 120 180 0 60 120 180 0 60 120 180 eFIG. 9. The variation of spectral width parameter around the circumference of a vertical cylinder.
ZIdr-016 LEGENO NQr row bn For PM
. D . u u Q D O z/d r-0.36 o o o o D z Id o-0. 26 z/do-0.48 O C o O o o f opt. o 08 06 a-w 0.4 E Q 0.2 Q. 00 Q u 0.6
the cylinder for z/d up to - 0.26, beyond which dominant variation in 1, is observed
especially for O > 600.
A typical plot showing the correlation of theoretical and experimental
,,
along thedepth for three different O values (0 00, 90° and 180°) for both narrow band and PM1 spectra is depicted in Fig. 10. It is seen from this plot that the correlation between the theoretical and experimental e,. values are better for the narrow band spectrum than for the PM1 spectrum up to zid = - 0.26. The reason for the deviations between the theory and the experiments for z/d beyond - 0.26 has been discussed earlier.
00 -0.2 -0.4 -0.6 0.0 -02 -04 -0.6 00 -02 -0.4 -0.6 -0.8 0.0 02 0.4 06 08 00 02 0.4
SpectraL width parometeç Ep
Fio. IO. Variation of spectral width parameter along the depth of a vertical cylinder. 06 0-8 o O o .
io
IFor Narrow band
0.3
e=o
o o o.
o I - Theory o - Experiments . s 1 o O.01
For PM1 E Q.6eo
o o o C -I s o0
o-
.
o i O 90 O I o o -I 50 . o . . s s I o S I 9:9Q o o a O so 50o
-o I G180 o S I O-es Io
o s s S 5 o s I Ol80 o o c I V. SUNDAR e! al.15 13 1.1 09 .4 2
Dynamic pressures around a vertical cylinder 145
0 30 60 90 120 150 180
e
FIG. 11. Comparison of regular and random wave pressure results.
0 30 60 90 120 iSO 180 X o o X o z/d 00 11 C o o z
.9
X Q X X o X 0 o 7 + X V -f .5 -+ + + o o o z/d-016 z /d-026 X-
.4 o o Q f 2 I i + ( i * I i J4.7. Comparison of regular and random wave results
The comparison of normalised pressures C,, (see Section 2.2.) as a function of O for
different z/d values based on regular and random wave tests is shown in Fig. 11. Tn the
case of random waves, the scattering parameter ka was computed using peak spectral
frequency. The wave frequency of the regular wave could not be matched with this
exactly due to the experimental limitations. The normalised pressures C,, from the
random wave tests with PM1, PM2 and narrow band spectra were obtained as discussed in Section 2.2. The theoretical plots of Ç obtained using the MacCamy and Fuchs linear diffraction theory (1954) are also superposed on these plots. It is seen that the
experimental results, from both regular and random wave tests, deviate from the theory for the transducer location at S.W.L. This deviation is observed to be more pronounced
LEGEND
Theory (ka 0.411(
ReguLar wove tests (ka 0.411
Rondom wave tests (PM 1) (ko 046(
Random wove tests (PM2flka0.46(
Random wave t,st (Narrowbarid)(ka0 ''L
(j 6
146 V. SUNDAR et al.
for O = 900 for the case of regular waves, whereas this is not so for the random wave pressure tests. This may be due to the fact that the wave reflection from the side wall for the case of a monochromatic wave establishes its presence, whereas for random
waves, since it is a combination of many frequency components, the effect of the
reflected waves from the side wall gets nullified by the scattering waves from the cylinder. It is seen that the regular wave tests yielded consistently higher values of C than the theory, whereas the opposite trend is noticed in the case of random wave pressure tests. This can partly be attributed to the fact that the average pressure coefficient C,, is assumed to be constant over the whole range of frequencies and also the small difference in the ka value considered for comparison (ka = 0.411, 0.44 and 0.46 for regular wave, narrow band and for the two PM spectra, respectively). It is interesting to note that the agreement between the MacCamy and Fuchs theory and the experimental results for C,, obtained for the narrow band spectrum is found to be good even at S.W.L., unlike the results from the PM1 and PM2 spectra. It is observed
that the C',, values for the case of a narrow band spectrum for zid below - 0.08 almost coincide with the results for the PM1 spectrum.
4.8. Incident and diffracted wave spectra
4.8.1. Comparison between theory and experimental values. In order to study the
pattern of the diffraction by the test cylinder, six different locations around the cylinder (viz., D1D6 as shown in Fig. 12) were chosen to measure the time history of the
N NN N N 'NNNNN.NNNN
Flume sude wall
0
0W 06
drecton
Flume sude wall
600cm B 80cm 05 B 50cm Test cylinder 40cm dia.
FIG. 12. Location details of the diffracted wave probes.
E 'J
o
CN
Dynamic pressures around a vertical cylinder 147
diffracted wave elevation. B/D values of 1.25 and 2 have been used, where B is the
distance of the wave gauge from the centre of the test cylinder and D is the diameter of the cylinder. Though experiments were conducted for all the three spectra given in Table 1, only selected cases are taken up for discussion.
The comparison of the theoretical and the measured diffracted wave spectra for the narrow band case, for locations Dl. D2, D3 and D5, are shown in Fig. 13a. The first two locations are considered in preference to D4 and D6 in order to examine the possible side wall effects. The corresponding incident wave spectrum is also superposed
on these plots. It is seen that for all these four locations the diffracted wave energy is greater than that of the incident wave energy. The agreement between the theory [Equation (6)] and the experiments are found to be satisfactory. However, the deviations
observed especially near the spectral peak may be due to the nonlinear effects, which are not accounted for in the theory.
In order to demonstrate the effect of wave steepness, wave elevation spectra at locations D1D6 for PM1 and PM2 cases are shown in Figs 13bd, which reveals that
the diffracted wave energy is greater than the corresponding incident wave energy as
observed in the narrow band case. It is surprising to note from these figures that the deviation between the measured and theoretical diffracted wave energy is smaller for the case of PM2 (larger energy level) when compared to the case of PM1 (smaller energy level), consistently for all the locations considered. Based on the foregoing discussions, it is inferred that the correlation between the theoretical and measured diffracted wave spectra for the case of the narrow band spectrum is better than in the case of the two broad band spectra considered in this study.
4.8.2. Diffracted wave spectrum around the cylinder. The trends in the variation of the diffracted wave energy around the cylinder at the locations discussed in the
earlier section for the three incident wave spectra were similar. Hence, only
arepresentative plot showing the variation of the measured spectral density of the diffracted waves corresponding to the narrow band case is shown in Fig. 14 for BID of 1.25 and 2.0. As seen earlier the diffracted wave energy is significantly higher compared to the incident wave energy. The diffracted wave energy is found to be maximum at location D5, apparently due to the build-up of scattered waves in front of the cylinder around this location.
4.8.3. Theoretical and measured zeroth moment and peak values of the diffracted wave spectra. A comparison of the theoretical and the measured zeroth moment and peak values of the diffracted wave spectrum at selected locations around the cylinder is shown in Fig. 15. As discussed earlier, the zeroth moment and the spectral peak of the three incident wave spectra are smaller than the respective values of the diffracted wave. It is seen consistently that the zeroth moment and the spectral peak are a maximum at location D5. It is also noticed that the measured zeroth moment and the spectral peak is a minimum at location D6. A study of the above two parameters show
that the BID value does not significantly influence the diffracted wave energy behind
the cylinder when compared to the locations in front of the cylinder. Similarly the above two parameters are almost identical for locations Dl and D2 near the side wall, except that an increase in their values is noticed at D2 for the incident wave spectrum
Frequency (Hz)
Legend. Exp.(Srr
(f)) ...Theory (5rr (f))---,Wove
Fio. 13a. Comparison of theoretical and experimental diffracted
wave spectrum (narrow band).
20 10 20 lo o I I D1 I .12
6
o -115 12-6
o o C- (J) 20 I D2 I12
6
O12
6
L) U) E o C- C- (J-) -r. Iv I D) ii I\\ I I D5 i 0.2 0 1.4 20 02 01 1-l. 206 2 20 10 3 2 3 2 D2 PHI 4 ). /\\. t...
2 /1
\..
o 1I o o O o V) E u E o E o D2 12 C-30 PM? C-(I) U) 12 ti) 20 It- J..6
10 I t 6 Q OIL- ...L
0.2 0.8 2 0 02 0.8 1.). 2.0 Frequency (Hz) --Legend: Exp. (5rr (f)) ...Theory (Sr r (i)) Wove (Sql) U)) FIG.I 3h. Comparison of theoretical and experimental diffracted
2 .; o t" E o 4 o -0.2 o I L... 08 1.4 2.0 o t" E o 0.2 0.8 Frequency (Hz)--Legend Exp. (r r () ) ...Theory (5rr()) -- Wuve(Sq
(f))
-FIG. 13c. Comparison of thcoretical and expciiinciìtal diffracted
wave spectrum. D4 PHi -.2 1.4 2.0 o E L)
i
D3 04 P112 - C-PlI 2 12 C-20 C-20 12 ti) U) U-) Jj . 10 J 6 io / D3 Pill 3 2i
8 32 16 02 / II
\.
ç 0.8 D5 PM 2 2 3 o 2 0 Frequency (Hz) Legend Exp.( 5rr () ...Theory (Sr r (f))-- Wove (S qq(fl)FIG. 13d. Comparison of theoretical and experimental diffracted wave spectrum.
-2 r. D6 PHi 06
-PH 2UtS
12 3 2 n 1 n n L) 4) In E o 15 C- C-L!) 10 o 4) (n E o n O o -, 0 2 0.8 l. 20L) E O u Narrow bond S.111(flot D? 20 2 0 Ql U at 06 15 10 -5 I I O 0.23 0.59 V. SUNDAR er al. (f 0.94 1.29 165 2.00 Frequency
(Hz).--FIG. 14. Comparison of measured incident and diffracted wave elevation spectrum.
PM2 (higher energy level), since the reflections from the side wall are expected to be greater for steeper waves. Based on the percentage difference of the measured values of the zeroth moment and the peak spectral value from their respective theoretical values, it is found that the agreement between the theory and experiments for both these parameters is better for the case of a narrow band spectrum as compared to the broad band incident wave spectra.
5. CONCLUSIONS
The hydrodynamic pressures exerted around a vertical cylinder due to regular and random waves were measured in a wave flume. The diffracted wave field around the cylinder at six different locations was measured. The measured spectral density of the
Norrow band r 1.25 D S (flot OS 30 at Dl atO3 20 S1 (fI 10
Dynamic pressures around a vertical cylinder 153
Locaflon of the wove probe
Fio. 15. Variation of spectral peak and zeroth moment of incident and diffracted wave field.
dynamic pressure and the diffracted wave elevation around the cylinder is compared with a theoretical formulation based on the MacCamy and Fuchs linear diffraction theory. The following conclusions have been drawn based on the present study:
The agreement between the measured and theoretical pressure spectrum exerted on a vertical cylinder is found to be better for a narrow band incident wave spectrum when compared to a broad band case.
The spectral peaks of the dynamic pressure of both experiments and theory shift towards the lower frequency region as the absolute of the relative depth of submergence of the pressure transducer zId increases.
E . 'J q' 32 28 A A
.
A E L) G' a-C A A 2 CA-
o o A .2 24 -A a' A.
C > 0) A Q) > A G) s C A A A °' > 20 A S S a' C a' o 16 A s A o O A A o 5 o .0 a E4-
s o S O a, 12 o 3-o o.
o.
C L) G) u G' u u I u C 4 u u u a o u a a u o2-1-
. a o a o o o o I I I I I I I Do Di Dz D3 D D5 D6 Do Di Dz D3 D D5 Db 44 ATheory E opt. Name of the spectrum o a s Narrow band PM i 40 A A PM? 10 -A
t36
Af9
A A L' 0) E A L)V. SUNDAR et al.
For any zid, the spectral peak of the dynamic pressure around the circumference of the cylinder reduces with the increase in O from O to 120°, beyond which it
increases slightly.
The circumferential variation of the peak of the dynamic pressure is quite large
from O = 30-90° when compared to the same from O = 120-180°.
The agreement between the theoretical and experimental spectral width parameter of the dynamic pressure spectrum is found to be good up to z/d = - 0.26, beyond which deviations are observed.
The correlation between the regular wave test and the random wave test based
on the average pressure coefficient applied over the frequency domain is found
to be encouraging.
The side wall reflection is more dominant for the regular waves when compared
to the random waves.
The diffracted wave energy around the cylinder for the selected locations is found
to be more than the incident wave energy.
The experimental diffracted wave spectrum overestimates the corresponding theoretical value by around 15%.
The spacing parameter BID has a significant influence on the measured spectral
peak and zeroth moment of the diffracted wave spectrum in front of the cylinder,
whereas its influence on the said parameters for the wave field behind as well as
in phase with the cylinder is insignificant.
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