!
Experimental Investigation of Ursell's
of Wavemaking by a Rolling Cylinder
William Curtis McLeod and Tsuying HsiehIowa Institute of Hydraulic Research, State University of Iowa, Iowa City
ntroduction
The energy expended by a rolling ship is divisible into three components: (1) viscous shear resistance in the boundary layer along the surface of the hull, (2) form resistance dueto
sepa-ration of the boundary layer near the bilges, and (3) the
energy expended in the formation of surface waves.
An experimental investigation of the surface waves pro-duced by certain cylinders of shiplike sections while rolling in otherwise quiet water is the purpose of the present study. According to F. Ursell [1], the amplitude of the surfacewaves
may be predicted under certain conditions. The conclusions of Ursell's study are that the wave amplitude is related to the _Yhull cross section (primarily the beam-draft ratio)Ate
loca-tion of the roll axis with respect to the water surface;.%e
period of roll,t!and the amplitude of roll. Ursell evaluated the wave amplitudes for a particular family of two-dimensional forms. A startling prediction was that one member of this family would generate no waves. It was of great interest, then, to attempt to verify these predictions experimentally.
No
A amplitude of surface wave
beam of cylinder
CE w coefficient of wave-damping energy
CEt coefficient of total damping energy draft of cylinder
potential energy of cylinder at an angle of heel
Et total energy loss per cycle
wave energy propagated per rolling cycle
gravitational acceleration
GM metacentric height of model
vertical distance of the roll axis above the water
surface 4n,/gT.,
natural frequency of rolling
number of complete model oscillations
p angularity parameter
T period of one complete oscillation
W weight (or displacement) of model
x distance in direction of travel of wave X abscissa of hull cross section
y wave height, measured from undisturbed level
of water surface
Y ordinate of hull cross section
roll amplitude of model wave length
mass density of water
The Ursell Cylinders
Ursell considered the family of cylinders represented by the following set of parametric equations:
X = 9/16 (B/2 + D) sin p + 25/48 (B/2- D) sin p + 1/16 (B/2 + D) sin 3p- 1/48 (B/2 - D) sin 5p
Y = 9/16 (B/2 + D) cos p- 25/48 (B/2 - D) cos p (1)
-1/16 (B/2 + D) cos 3p + 1/48 (B/2- D) cos 5p
where X and Y are the horizontal and vertical coordinates of the cross section with beam B und draft D, and p is an angu-larity parameter. The origin of coordinates is located at the intersection of the centerline of the hull cross section and the water plane.
The wave amplitude to be expected is given by Ursell in
the following equation:
A = 0.63 K20 (B/2 + D) [(B/2 - 1.26 D) (B/2 + 1.05D)
+ 0.015h (B/2 + 26D)] (2)
where K = 47E2/ gT2, A is the amplitude of the surfacewave, g is the acceleration of gravity, T is the period of one cycle, 0 is the amplitude of roll, and h is the vertical distance of the roll axis above the water surface. When the roll axis is located at the water plane, equation (2) reduces to
A =- 0.63 K26 (B/2 + D) (B/2- 1.26 D) (B/2 + 1.05 D) . (3) It is seen that, when the hull cross section has a beam-draft ratio of 2.52, the wave amplitude is zero.
oo
2
4
5
(Mode/ c, B/D = 3.5
Fig. 1 Transverse Model Sections
Three cylinders, constructed according to the parametric equations (1), were obtained from the David Taylor Model Basin primarily for this study. The cross-sectional shapes of these models are given in Figure 1. The models each had a
length of 9.0 feet with a constant section, and a displacement of 149 pounds. Those designated as a, b, and c had
beam-draft ratios of 2.00, 2.52, and 3.50,
respectively. Theircharacteristics are given in Table 1.
f'
r r
0111'1
.) The period of rotting was chosen so that the dimensionless
natural frequency of rolling was the same for each model, I. e.,
ne D 112' 02
Or
T1 DI
T2 =
D,where n is the natural frequency of rolling and T = 1/n is the roll-ing period.
The predicted values of A/D, the ratio of wave amplitude to draft, for these cylinders, obtained from (3) for the case when the rolling axis is at the undisturbed level of the free surface, are as follows:
Model A/D
a 0.000179
0 0.000871
Table 1.
Model Characteristics
Characteristic Model a Model b Model c
Area sq. in. 38.03 38.03 38.03
Beam in. 9.00 10.00 11.75
Displacement lb. 149.00 149.00 149.50
Draft in. 4.50 3.97 3.36
Metacentric height in. 0.457 0.548 1.150
Length ft. 9.00 9.00 9.00
Period" sec. 1.93 1.73 1.54
Location of roll axis W.S. W.S. W.S.
- 17 -
Schiffstechnik Bd. 10 - 1963 - Heft 50x, in inches
Equipment and Procedure
The experimental work was conducted in the IIHR towing channel [2]. The channel is rectangular in cross section with
a depth of 9.5 feet, a width of 10.0 feet, and a length of
approximately 300 feet. Parabolic wave absorbers, connected to an overflow weir, are located at each end of the channel.
Figure 2 shows an overall view of one of the models mounted in the channel with its longitudinal axis placed perpendicular to the axis of the channel at approximately the one-third point from the south end. The water level was maintained at the elevation of the top of the weir to reduce the amplitude of the reflected waves. It was attempted to consider in each run only the data recorded before the reflected waves returned to the
model. 4 VU Schiffstechnik Bd. 10 1963 Heft 50 18
-teir8
- -7-Fig. 3 Equipment in Model
cessary clearance in the center of the model for the oscillator,
gyro, and other equipment. It also provided a convenient
means of connection between the model roll axis and rigid support. Two small ball bearings spaced 5 feet apart were used to connect the lower legs of the aluminum frame to the model. The upper horizontal member of the frame was
con-Did Gage Supporting Bar
Note, Drawing not to scale Fig. 2 Model Mounted in Channel
Insulators
1
A mechanical oscillator was used for all forced rolling ex-periments and also for bringing the model up to a maximum
roll angle for declining-angle experiment. The oscillator Hinge
installed in one of the models is shown in Figure 3. It
consisted essentially
for two pairs
of eccentric weightsmounted on each of two counter-rotating vertical shafts.
Since the two shafts are located on the longitudinal axis Icr Surface, of the model, a pure rolling moment is set up about the
center of gravity of the model. No pitching or yawing mo- Fig. 4 Schematic of Vibrating Needle
ment is present. The rolling moment may be varied from
zero to the maximum by changing the amount of eccentric weight, the eccentricity of the weights, or the vertical distance between the pairs of eccentric weights. By means of a worm drive and a speed reducer, the two vertical shafts are con-nected to a 1/25-horsepower direct-current motor. Power to the drive motor is supplied by a variable-speed direct-current
generator. By varying the generator speed, the eccentric
weights can be driven at any desired angular velocity. For the, forced rolling experiments the frequency of the oscillator was held constant at the resonance frequency of the model. De-clining-angle results were obtained by bringing the model up
to a maximum roll angle by means of the oscillator at
resonance frequency, then rapidly stopping the oscillator by interrupting the power supply.
Roll angles and model periods were obtained by means of a Giannini vertical gyro transmitter mounted nearthe
mecha-nical oscillator in the model as shown in Figure 3. The data permitted roll-angle measurements to be made to the nearest 0.1 degree.
Fulfillment of the conditions of the Ursell theory neces-sitated the provision of a fixed roll axis. An inverted U-shaped aluminum frame was chosen for this purpose, as shown in
Figures 2 and 3. The shape of the frame provided the
ne-Springs
Mounting Plate
Fig. 5 Vibrating-Needle Apparatus
TkIE
nected to the towing carriage in the channel. The aluminum material gave sufficient rigidity to the roll axis, yet was easy to handle.
A vibrating-needle apparatus, used for measuring the ampli-tude of the small surface waves generated by a rolling model, may be seen schematically in Figure 4 and photographically in Figure 5. Connected to the mounting plate are the electrical coils, stop blocks, springs, and a vibrating needle. The coils are similar to those in a door bell, which produce an alter-nating field at 60 cycles per second. The alteralter-nating field causes the horizontal stem, to which the needle is connected, to vibrate vertically at the above rate. The stop blocks limit the vertical travel to 3/16 inch. The springs tend to stabilize
the motion into a more regular pattern of vibration. The
needle, which is connected to the horizontal stem, travels alter-nately in and out of the water surface in a vertical pattern. Surface-tension effects, which are present with some other types of pickups such as a resistance wire, can be significantly reduced or possibly eliminated in this manner.
An electrical circuit is closed during the time the needle is in contact with the water and open when the needle is above the water surface. A low-pass filter between the needle appa-ratus and the recorder allows passage of signals with frequen-cies of the order of the wave frequenfrequen-cies, but blocks signals with frequencies of the order of the needle vibration. There-fore, with a constant frequency of needle vibration a relation-ship was obtained and recorded between the time of submer-gence and the time of nonsubmersubmer-gence of each cycle. For proper operation the frequency of the needle vibration must be much greater than the frequency of the water-surface fluc-tuations. The relative resistance of the total system must also be large in comparison with the contact resistance between the actual needle and the water. A dial gage is provided for cali-bration in quiet water by adjusting the entire needle appara-tus to various elevations relative to the water surface. The error in wave measurements was determined to be less than
0.0001 foot by comparing the recorded data with the dial
gage.
The vibrating-needle apparatus was located on the center-line of the channel and two wave lengths from the model. The computed wave length, assuming deep-water conditions, was approximately 15 feet for Model b.
The data for the roll angles and the water-surface profiles were electrically recorded using a Sanborn twin-channel re-corder. These data were recorded simultaneously. A sample record of a wave profile is shown in Figure 6.
In a recent study on the effect of roughness applied to a model ship hull [3], it was concluded that, by the addition of
Fig. 6 Sample Data Record
suitable roughness material, the form drag of the model may more closely resemble the form drag of the corresponding prototype. Consequently the majority of the experiments were conducted with plastic cylinders, 1/8 inch in diameter and 1/16 inch in height, cemented to the centerline of each bilge at a spacing of 1 inch for the entire 9-foot length of the model. As a basis for comparison, a few measurements were also made on Model b without roughness.
Data Analysis
It was of interest to measure not only the wave damping but also the total damping in order to evaluate the importance of discrepancies between theory and experiment. For these purposes two different experiments were conducted.
From declining-angle measurements obtained by allowing
a model to oscillate freely, a plot of a 0vs--N may be
ob-tained, in which N is the number of complete rolls. The energy loss per cycle in which the maximum angle changes from et
to 09 is the change in potential energy. But the potential
energy E, at an angle of roll ID is given by
Et, W GM (1 cos ID) (4)
where W is the displacement of the model and GM its meta-centric height. The energy loss per cycle is then A Et) / AN when AN = 1. This suggests that the slope of the curve of
E N may be interpreted as an instantaneous rate of energy damping. From (4) we have
dEP = W GM sin 0
de
dN dN
It was also necessary to correct the data for the damping due to the parasite drag of the plastic cylinders. By compar-ing the energy dampcompar-ing at large angles with and without these roughness elements, it was determined that the results should be modified by a coefficient of 0.873. Hence, the total energy damping per roll, Et, is given by
Et = 0.873W GM sin 0
de
. dNIt remains to evaluate the part of the energy damping due to wave generation. The potential energy in one cycle of wave length X is
P.E. = 1/2 Qg f y2 dx (6) where p is the mass density of the fluid, x is distance in the direction of travel of the wave, and y is the ordinate of the wave, measured from the undisturbed level of the free
sur-face.
The profiles of the measured surface waves were found not to have a pure sinusoidal shape, as is shown in Figure 6. For a sinusoidal wave it is well known that the kinetic energy associated with the wave is equal to its potential energy. Less familiar is the result that the same equality applies, for small values of An., even when the wave is not sinusoiCal. This may be shown by expressing the wave form as a Fourier series, whence it is readily found that the equality of potential and kinetic energies applies separately to each Fourier wave com-ponent. Thus the total energy per wave is twice that given by (6).
. Since the wave energy is propagated at the group velocity
which, in the present case, may be considered to be that for
deep water, i. e., half the phase velocity, the energy
pro-pagateci, per cycle is half the energy per wave length, and consequently equal to (6). However, since waves are pro-pagated from each side of the model during one cycle of
rol-(5)
- 19 - Schiffstechilik Bd. 10 1963 Heft 50
s c(A.,/10,4
ling; We-obtain finally, for the loss--of energy due to wave-. Taking:during .one cycle of rolling;
E = eg S y2 dx
:Plots of dimensionless a e amplitude against roll angle for the three models are sl-h-.)wii; in Figure. 7: In contrast with the less successful attempts ,by Moreira [4] and Stefun. [5], who
. also conducted rolling 'experiments on Ursell cylinders it is
-believed that the present data were taken ,Under
corresponding more nearly to Ursell's assuMptions. The nici'St 'important of thege probably, is that the cylinders were re-strained to roll about a.fixed axis at the undiSturbed:levellof the water surface. ,Another ;condition, which may be a conse. quence of the first, is the constancy of the amplitude of during the measurernents. In neither of the previous., studies was a fixed roll axis provided.
It is seen from Figure 7 that the measured wave ainpli-2. :hides are about 34 percent greater than the theoretical
valuesfor
Model a, but about:16 Pereent smaller than the theorelidal'
4.5 4.0 3.5 3.0 2.5 2.0 1-5 05 00 /.8 /.6
Results and :Discussions
Mode/ a 2 4
6 8 /0
12 9, in degrees Model b 4 6 8 .10 /2/4 -16 18 9, in degrees Schiffstechriik Bd. 10 .-1963 Heft 50Fig. 7b Wave Amplitude
Fig. 7a Wave Amplitude
1.0
00 1,12 3 4
67 h
9Fig.7c. Wave Amplitude
prediction for /Model c.-.For both?.Mbdel.s. a and :-C at Snia.11
angles of example; fOr.zing6.of roll smaller. than 7 - degrees thcraré sOMe`
overlap the theoretically .wa.i!e:amplitude -.curves. Since Cisell'S.the-O-ri`AsSuines rna1l angles of roll, agreement
- these points ;iS'-'..Veri:Sighificant.:
. .
For Model b tile theory prediet:-. zero Wave amplitude; but -measurable values ,:wefe..Obtained exPerithentally: However, these we-re tiboUt one-half the' amplitudes : for Model. a and
about one-sewrithAif those fOr Model e: asr-shoWn in .1."igiffe fl.
The.iheashred wave amplitude'. for Mndel I gave tWOpuryes Of :data, one fOr1)._sniallerjhan 16.degfees, the other:af...0 greater than 1,..2"degr.e6s-.:.Bfwp-pri 10. and l3 degreeino, data
were obtained - - associated with
.the:bcpurrence...of-latnina:r Separationzat the:Smaller ,angles and
turbulent separatiOn_at.thel.4rieF..-;If this were theZelse;'''ohe
- would .expeet eOnsiderablY.C.' redueed -wave ampliti;des Model h at theSe-rshiall :anglet..,at-=iliehigheY Reynolds.-
mini-hers attainable . with ,..Stiificient1y.`;...1.arger,:,,cylinder.S.,-; Model b is a form of minimum Wave: making = one'WoUld:a3.,,
,pect that its Wave g(;r 'would
ges in flow pattern due viscous.6fIectS than would -the (idler models. 20 -4 .3 6C 60 50 30 2.0 oc.j Wade' a
1111111=11111
11111=M111111111_-IMIlififf
MEE
111111111WW11111
IMEMENINEEM
WillifFAME
MANIPMEgir
MIES111111111
2 4 .E.vevrnent ' - . . Fig.:,13,1Comparisan of Wave Amplitudes -6 8 /2 .14 .16-,78 14 16 18/6 /4 /2 10 8 6 4 2 /60 140 120 100 80 41 60 40 20 00 Model b 00 2 4 6 8 /0 12 14 16 /8 6, m degrees Model c 0 Fig. 9b Wave-Damping Energy Fig. 9c Wave-Damping Energy
Curves of wave-damping energy against angle of roll, ob-tained by applying equation (7), are plotted in Figure 9. It is apparent that the wave damping energy for Model b is much smaller than that for Models a and c. For instance, at 0 =
10 degrees, E, is 11.6 times greater for Model a than for
Model b, and 45.0 times greater for Model c than for Model b; at 0 = 14 degrees, E. is 43.3 times greater for Model a than for Model b. The ratios for 10 degrees would probably be even greater at higher Reynolds numbers, as was indicated
above.
Figure 10 shows the dimensionless coefficients of wave-damping energy CE:
E,
CE
. (8)W GM 82
Evidently, CE for Model b tends to zero when the angle of roll decreases, while the other curves do not have this ten-dency. Thus, although Model b generated some waves, con-trary to the predictions of the theory, nevertheless the trend agrees with the theory.
25 20 9, M degrees 20
1111111111111
111111111111111111
1111111111111111111
,11111115111111111111111
IIHNIIENEI1111
1111111BORIIII1
111111171aMill
111111111111111MMIII
11111111111111111111M
11111111111.1111
6 8 /0 12 14 16 18 20 /V, Number of RollsFig 11 Declining-Angle Curves
Declining-angle curves for the three models are plotted in Figure 11. Figure 12 shows the dimensionless coefficient of total-damping energy,
t CEt
W GM derived from the declining-angle data.
(9) - 21 - Schiffstechriik Bd. 10 1963 Heft 50 /.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00 Fig. 10 Coefficients of Wave-Damping Energy Mode/ c Model o b
24 6 8 10
12 14 16 /8 2 4 6 8 M degrees 10 12 /60 /40 120 100 Mode/ a Z. 80 60 40 20 Fig. 9a Wave-Damping Energy 00 2 4 6 5 10 12 /4 /6 6, M degrees8, in degrees
Fig.12 Coefficients of Total Damping Energy
Curves of E,/Et against 0 are shown in Figure 13. It is
seen that, for Model b, the wave-damping energy is always smaller than 2.0 percent of the total-damping energy, and, as indicated above, would probably be even less at higher Reynolds numbers. On the other /land, the wave damping ener-gies for Modles a and c are of the other of 22 and 42 percent of the tots/ damping energy respectively. Moreover, for
Mo-del b the (E/E)vs-0 curve tends to zero at small angles of
roll, while for Models a and c this does not occur. This again is evidence of agreement between the -theory and the experi-mental results.
Conclusions
Ursell's predictions of the amplitudes of surface waves generated by a particular family of rolling cylinders are in fairly good agreement with experiment, for relatively small
angles of roll. For larger angles of
roll, the agreement worsens, but the theory may serve as an aid to finding the approximate value of the wave amplitude.The fact that the ratio of the wave-damping energy to the total-damping energy for the model of theoretically zero wave
Schiffstechnik Bd. 10 -- 1963 Heft 50 22
-0,5
0.1
0 2 4
6 8 10 12 14 16 /8 206, in degrees
Fig. 13 Wave-to-Total Damping Energy Ratios
damping is so small in comparison with the others strongly 1
emphasizes the applicability of Ursell's theory for predicting i approximately the actual wave damping of cylindrical sections.
Acknowledgements
This project was conducted at the Iowa Institute of Hydrau-lic Research, with financial support from the Office of Naval Research under Contract Nonr-1611(01). The authors wish
to thank Professor Philip Hubbard for his assistance with
electrical instrumentation and so thank Professor Louis Land-weber for his guidance in the experimental work and in the preparation of this paper.
(Eingegangen am 7. November 1962) References
Ur sel 1, F.: "On the Rolling Motion of Cylinders in the Surface of a Fluid." Quart. Jour. Mech. and Applied Math.,
Vol. II, Pt. 3, 1949.
Ma r ti n, M.: "The Iowa Towing Tank" Iowa Institute of Hydraulic Research Report to ONR, Nov. 1953.
Martin, M., McLeod, C.,
and Landweber, L.:
"Effect of Roughness on Ship-Model Rolling." Schiffstech-nik, Bd. 7,Heft 36, 1960.
M or ei r a, D. R.: "The Effect of Beam-Draft Ratio on Roll Damping." Massachusetts Institute of Technology Thesis, Department of Naval Architecture, 1955.
St ef u n, G. P.: "Experimental and Theoretical Investi-gations of Roll Damping." David Taylor Model Basin
Hydromechanics Technical Note No. 4, 1955.
4.0 3.5 3.0 2.5 1.3 2.0 (.3 /,5 /.0 0.5 00 Model a Model b Mode
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