Buoy Design

A Spar Buoy Design for Oceanographic Data Telemetry Seaconsult

Donald 0. Hodgins Marine Research Ltd, Vancouver, British Columbia and

Dobrocky

Brian N. Lea Seatech (Nfid) Limited, St John’s, Newfoundland

[Original manuscript received 10 February 1981; in revised form, 3 July 19811

ABSTFLWT The design of a spar buoy, employed as one component of a current monitoring system deployed during 1980in Davis Strait, and some of its response characteristics to ocean wavesare presented. The buoy was36.5ft (Il. 13m) in length with Upperand lower masts0.5ft (15.2 cm) in diameter and buoyancy hulls, centrally located, 2ft (61.Ocm) in diameter. Powerfor the system wassuppliedfrom lead-acid batteries in a ballast tank on the lower end. Telemetty electronics werelocatedin a l-ft (30.5-cm)diameter caseon the Upperend, below the antenna. The buoy weighedabout 1,635lb (746kg) and could be man-handled at sea with relatively light lifting equipment. It had a dampedperiod of 16sand a signijicant heave responseof about 62%of the significant wave height of theforcing spectrum. Scalemode1testsindicated that the buoy would be operational in heavy seaswell up to about 20ft in height with pitching anglesof lessthan 10” off vertical. Observations at seabave shown that the buoyfollowed swellwaves better than the 62%heave responsefigurewould imply and had negligiblepitching motions except during severestorms. Thedesignwasjudged to be successfulinproviding a stablebasefor VHF transmissionand is recommendedfor usein other applications.

RÉSUMÉ On présente dans cet article les caractéristiques et le comportement dans les vagues du large, d’une bouée Spar qui forme une composante intégrale d’un systèmede surveillance du courant dansle détroit de Davis, institué en 1980.La bouée a 36.5 pieds (11.13m) de long et possèdeun mât supérieur et inférieur de 0.5 pied (15.2cm) de diamètre chacun, et une coque de jlotabilité de Ipieds (61.Ocm) de diamètre situé au centre de la bouée.L’alimentation est assuréepar despiles à plomb et à acide situéesdans un caissonde lest logé dans la partie inférieure. Les circuits électroniques de télémétrie se retrouvent dans une boîte d’un pied (30.5cm) de diamètre logéedansla partie supérieureau dessousde l’antenne. La bouéepèse1,635 livres (746kg) environ et peut être manipuléeen mer à l’aide de grues légères.Elle a une période amortie de 16s et une réponseégale à 62% de la hauteur de la vague signihcative du spectreforçant. Les épreuvesmenéessur les modèlesrévèlent que la ATMOSPHERE-OCEAN 19 (2) 1981, 158-171 0705-5900/81/0000-0158$01.25/0 @Canadian Meterological and Oceanographic Society

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bouée demeure opérationnelle dans des houles sévères atteignant jusqu’à 20pieds de hauteur, et l’angle du tangage est inférieur à 10” de la verticale. Les observations en mer révèlent que la bouée suivait les houles de merplus$dèlement qu’une réponse de 62% aurait suggéré et avait un tangage négligeable excepté au cours des tempêtes violentes. On juge que la conception de cette bouée est un succès qui offre une base stable pour la transmission à B.m et l’on recommande son application à d’autres usages.

1 Introduction

The telemetry of ocean current and water property data gathered on deep water moorings with conventional instruments is an important method of acquiring information at sea. It allows these data to be used in a real-time manner, for example, by offshore drilling operators for detecting interna1 waves or monitoring currents for oil spill and ice motion prediction, and it offers scientific investigators the ability to greatly increase sampling rates, since data cari be logged and processed at manned stations with sophisticated equipment. A critical element of any system based on VHF transmission is the surface-piercing buoy containing at a minimum the transmitter and its antenna. In this paper we report on the design and behaviour of a spar buoy, deployed during August and September 1980 in Davis Strait, as the surface component on three separate current-meter moorings. When faced with the problem of designing the buoy we discovered a dearth of practical experience to draw upon for the behaviour of a spar light enough to be managed at sea by a small boat and crew, and yet able to withstand open ocean wave conditions. Thus, we approached its design from first principles, guided by the Woods Hole Oceanographic Institution’s 53-ft spar buoy (see Berteaux, 1976, pp. 89 and 215). Our observations of its behaviour in the field have shown the new design to be a success; on September 18, for example, it continued to transmit throughout a severe storm with 45kt winds and a 20-ft sea. ( We felt, therefore, that the buoy design and our observations of its behaviour would be of interest to other oceanographers who may be contemplating similar projects. In this paper we have concentrated on the spar buoy itself - the put-pose and composition of the entire system are topics covered in a separate communication (Hodgins and Westergard, 1981). TO provide some background for the project, however, the system configuration is shown in Fig. 1. Two of the moorings contained two InterOcean 195RX electromagnetic current meters (15- and 30-m depths) while the third mooring held three such meters at 15,30 and 120 m. Al1 subsurface moorings were of conventional design, taking the drag force of the spar buoy into account. The main subsurface buoyancy was provided with a 58-in. (1.47-m) steel sphere on each mooring. Reserve buoyancy was obtained with 37-in. (0.94-m) steel spheres. Each current meter was hardwired to the spar buoy via the tether connecting the spar to the uppermost subsurface buoy.

160 / Donald 0. Hodgins and Brian N. Lea spar buoys

6

-

telemetry

,a

Aanderaa RCM- 4 (selfrecording)

200m

I

I

I

4 n.m.

2 n.m.

1 n.m.

South west

Fig. 1 Configuration of the current-metering

0 North East

system deployed in Davis Strait.

Each spar buoy had three functions: it powered the current meters and transmitters, preprocessed the current data for transmission, and served as a stable base for the antenna. As shown schematically in Fig.’ 1, the drillship Ben Ocean Lancer served as the reception base for the data. 2 The buoy design Our design of the spar buoy was conditioned

by the following five objectives:

1) To minimize the total weight and projected area so as to minimize deployment and recovery problems at sea and water and wind drag forces, without sacrificing structural rigidity. 2) To optimize the distribution of buoyancy and ballast elements to give the maximum righting moment while minimizing the heave and pitch response to swell waves. 3) To optimize the tether point attachment so as to minimize tilt in large currents, without worsening the buoy response to waves. 4) To make the buoy large enough to withstand the wave climate, and strong

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Tripod with liqht 8 radar rrtlrctor Transmitter housing 1’ 0 Al

Upper foam

6”6 Al

mart tilled

Main buoyancy hull - ‘Opm tilled 6 Al

2’

Lower mast water filled with drainage ports d Al pipe

6”

2I*.,o

Ballast tank containing batteries Al

2’ I

MODEL DIMENSIONS IN INCHES OVER PROTOTYPE DIMENSIONS IN FEET

PROTOTYPE

(a)

: MODEL = 30 : I

(b) Fig. 2 Spar buoy a - Prototype design b - Scale model.

enough to survive some level of ice or supply boat impact sufficiently well to permit field repair. 5) To utilize readily available materials to meet our fabrication schedule (8 weeks). The final design is shown in Fig. 2a*. The buoyancy

hulls - both the

*Both S.I. and British units are used throughout this paper. Because many manufacturers’ specifications are presently in British units it was necessary to do the design in these units, and we decided to retain them here in discussing the buoy itself. Wave and mooring data are presented in S.I. units where practical. Where British units appear first they are followed by their S.I. equivalent in parentheses.

162 / Donald 0. Hodgins and Brian N. Lea cylindrical and tapered sections - were formed from l/4-in. (6.35mm) 6061-T6 aluminum plate. During fabrication the tanks were filled with expanding polyurethane foam. As shown in Fig. 2 the buoy could be assembled by bolting the flange plates together between the buoyancy tanks, making shipping and handling easier. The lower mast extended through the cylindrical tank and was welded to its upper flange plate increasing the structural rigidity. Both upper and lower masts were made of 6-in. (15.3-cm) diameter schedule 40 6351-T6 aluminum pipe. The lower mast filled with water when submerged through ports adjacent to each end plate thus effectively increasing the buoy’s mass under dynamic forcing. Importantly though, the lower mast was also foamed where it passed through the buoyancy hull so that only the lower 8 ft were water filled. Similarly the upper mast was foamed to the transmitter housing and welded to the lower flange plate. The signal processor and transmitter electronics were contained in the upper housing (l-ft (30.5cm) diameter welded aluminum, l/4-in. (6.35-mm) thick). Above this we placed a 5-ft (1.52-m) aluminum tube tripod which held the antenna. Power was supplied to the upper housing with Brantner sealed cables (through-hull fittings top and bottom) secured to the exterior of the buoy. Power was supplied by two lead-acid batteries mounted in the ballast tank. Additional lead ballast was added to an accessible compartment, separate from the sealed power supply, to trim the buoys to leave 7 ft of mast above the still water level (SWL). The tether point, giving a near-zero overturning moment in the anticipated 3-kt currents, was located on the lower flange of the cylindrical buoyancy hull. In trials, however, we found that this attachment point could be varied by about + 12 in. (30.5 cm) without producing any appreciable overturning tendency. All flange plates were cut from l/2-in. (1.27-cm) 6061 -T6 aluminum plate. The half-wave dipole antenna was fastened in a socket to the upper plate of the tripod. This part of the design proved to be a weak point since we found that the antenna had broken just above the socket a day or two after a severe storm on September 25. This failure presumably resulted from a type of pitch-induced fatigue. In future deployments a lighter and stronger whip antenna will be used to eliminate this problem. 3 Some response characteristics a Theoretical Calculations One of the most important parameters to be determined during the design process was the heave response to wave forcing. The simplest equation governing this single degree of freedom motion is (Berteaux, 1976, p. 56): i + 2n.t + p*x = (F&n,)

cos(m

+ 0)

(1)

where X

= the vertical displacement water level,

of the buoy water line with respect to the still

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Fig. 3 A dimensional sketch showing the observed free heave response of the prototype spar buoy.

n = b = m, = P = FCI = 0 = CJ =

b/2 m,,

the linear damping coefficient (to be found), the virtual mass (actual plus added mass), natural heave frequency, the exciting force (see below), the exciting wave frequency, the phase angle between the force and the wave (see below).

One of the main difficulties in evaluating solutions to (1) lies in determining numerical values for b and m,. From the solution for damped free oscillations the damped period, Td, is related to p and n by Td = 27&m.

(2)

By measuring (or estimating) Td and the amplitude of free oscillations over successive cycles, IZ can be calculated from the logarithmic decrement of amplitudes according to n Td = In (A,&+,).

(3)

Figure 3 shows the damped response of our buoy, obtained by submerging the upper mast nearly to the transmitter housing and allowing it to move freely. The design proved to be so highly damped that the 16-s period was difficult to determine accurately, as was the approximately 1-ft amplitude on the second cycle. The experiment was, in fact, repeated three times and the period and amplitude values averaged. Periods were obtained by timing the up-crossing points of the SWL (painted on the buoy) with the sweep second hand on a watch. We estimate the accuracy here to be about I!I 1 s. One-foot increments were also marked on the upper mast and used to estimate the

164 / Donald 0. Hodgins and Brian N. Lea amplitudes to within kO.25 ft (7.6cm). The effects of these errors on the calculated heave response are discussed later. Solving (3) for n gives 0.112 s-i. Hence, from (2) the natural heave frequency p = 0.408 s-l, corresponding to a period of 15.4 s. The restoring constant, C, for the buoy is pg&, where A, is the upper mast cross-sectional area; C = 12.53 lb ft-’ (182 N m-i) in this design. By definition p2 = C/m,, hence m, = 75.4 slugs (1100 kg). Since the actual weight is about 51.1 slugs (746 kg), the added mass, m’, is approximately 24.3 slugs (355 kg). The submerged volume (to the SWL) is 20.5 ft3 (0.58 m3); thus, with p = 1.99 slugs fte3 (1024 kg mP3) the added mass coefficient, C,,,, is approximately 0.66. Finally, b = 2 nm, or 16.9 slugs SK’ (247 kg s-l). The forced wave solution of (l), written normalized with respect to the wave amplitude, r, is (Berteaux, 1976, p. 58): x-=r

e-kD

mu

J

(C - m’02)2 + (bo)2 cp2 -"2)2 + 4n202 cos(“t+++o)

(4)

where r = forcing wave amplitude k = 2n/L, the wavenumber, L = wavelength,

D = the buoy draft, 4 = tan-’ [-2no/(p2 - 02)], and o = tan-’ [-bo/(C - m ‘02)] In this solution the wave forces are considered to act on the ballast tank at draft D; hence, the eekD term in (4). When cos(ot + 4 + o) = 0 the expression in (4) gives the maximum heave for a wave component of amplitude r and frequency o. This curve, called the Response Amplitude Operator (RAO) and denoted by Y(f), evaluated using the parameters derived above, is shown in Fig. 4a for a range of wave frequencies covering open-ocean conditions. A typical energy spectrum, S(f), for offshore Labrador and Davis Strait is also shown in Fig. 4a, calculated by the Marine Environmental Data Service from Datawell Waverider buoy data. The peak period is 11.4 s and the spectrum is characteristic of a sea state where swell predominates. The corresponding significant wave height, H,, is 6.4 m. Taking this as representative of a design sea state for the spar buoy, the mean square heave height, p, was calculated as the statistical response to this spectrum, i.e. p

=s yY.fwJwf =sm RWd! 0

0

(5)

where S(f) and Ycf) are defined in Fig. 4a. The heave response spectrum R(f) = Pcf>.S(f) is shown in Fig. 4b. The shaded portions of Ycf) and R(f) show the bounds corresponding to the errors in the ,measured values of Td and AiIAi+1-

A Spar Buoy Design for Oceanographic Denoting

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the significant heave by h, we found that h,=&p = 4.0 m,

(6)

analogous to the definition of H,. This value is approximately 62% of the significant wave height, H, (6.4 m) and results mainly from the separation of the peaks of Ycf) and S(f). Observations of the spar behaviour during its deployment indicated that it followed swell waves rather better than this percentage would imply, suggesting that the buoy’s damped period is slightly lower than the 16 s indicated in Fig. 3. It is clear, however, that statistically the buoy tends to heave somewhat less than the wave heights. This is desirable, in that resonance with swell is not likely to occur, but at the cost of having the wave crests run up the buoy mast. This feature is discussed further below in light of the scale-model test results. That resonance can indeed occur in buoys with low damping is shown in the model results of Adee and Bai (1970). The estimate of Y(f) is very sensitive to errors in the estimate of p and hence in Td and Ai/Ai+l , since asp + o the value of Y(p) becomes large. Thus ifin Fig. 4, p is overestimated, the peaks in Y(j) and S(f) are separated and such parameters as h, are underestimated. For example, solving for the significant heave height with Td = 15 s (AJA,,, = 6) gives h, = 5.02 m or 78% of H,. Since this value differs so much from the value found previously, one must be very cautious in basing the design solely on this type of calculation. b Scale Model Tests Therefore, in order to lend support to the results calculated above, and to examine the buoy’s pitch response, a number of scale-model tests were conducted. A drawing of the 30: 1 (prototype:model) scale model is shown in Fig. 2b. It was made from the same or similar materials as the prototype buoy, but some compromises were necessary to achieve the correct weight-tobuoyancy relationship. Specifically the antenna, tripod and’ instrument housing were omitted and the buoy hull and cone were slightly increased in size from the true scale. Five tests were designed to look at the response to swell waves of varying amplitude. The forcing wave was scaled in time to preserve the ratio of the prototype wave period (12 s) to the damped heave period (16 s), i.e. T, (model) = Td (model) x T, (prototype)/T, = 3.1 x 12/16 = 2.3 s,

(prototype)

(7)

where the damped model period (3.1 s) was measured in the same way as the prototype. To preserve dynamic similitude between the model and prototype wave and heave motions it remained to establish the correct scaling for the length scale, C. For modelling the heave motion of axisymmetric floating bodies the scaling of heave to wave amplitude is a function only of C/h, where h is the

166 / Donald 0. Hodgins and Brian N. Lea

60

2

Heave Response Spectrum

40

“E 2 E

20 (b)

1 .! 5

f(Hd

Fig. 4 a - A typical design energy spectrum for waves in the Labrador Sea Scf) and the Response Amplitude Operator YCf) for the spar buoy b - The heave response spectrum Rcf)

wavelength of deep water waves (Newman, c!,:&, is simply related to the wave periods,

1977, p. 42). Since h = 2ng/oz,

a value very close to the model scale. (We note that this scaling is equivalent to Froude scaling in oscillatory flows. Ordinary Froude scaling is not applicable since there is no time-steady velocity scale established in the experiment; in this case, the dimensionless parameter is oe’l\/ge, where u

A Spar Buoy Design for Oceanographic

Data Telemetry / 167

in the Froude number has been replaced by CM. Equation (8) is readily derived from this expression.) Three criteria were used to judge the model response: how well it followed the waves - i.e. did it plunge in the waves and threaten to submerge -, the wash level, and the tilt angle, both recorded as maxima and defined in Fig. 2b. The “wash level” is a term coined to describe how far water levels produced by waves rise above the SWL. The results for the five swell tests are shown in Table 1 (Nos 1 to 5). An average wave period of 2.5 s was, in fact, obtained, or about 12.9 s in the prototype. We found that for swell conditions the tilt of the model always remained small (less than 10”) and maximum wash levels were the equivalent of about 4.8ft, or 2.2ft below the instrument box, for 15ft wave heights. These values were scaled from the model tests. As anticipated from the difference between the damped heave and forcing wave periods, the buoy lagged the wave crest, allowing the wave to run up the upper mast and produce such a high wash level. Had the buoy’s damped period been more closely matched to the swell period it presumably would have followed the waves better, but may also have been closer to a resonant condition. Thus for swell waves up to about 15 ft in height we did not anticipate any serious problems due to overtopping or excessive pitching. In fact, the model buoy was found to be very stable during these tests, In the remaining three tests the wave periods were reduced and amplitudes kept large (Nos 6 and 7) and moderate (No. 8). Test No. 7 corresponded to the largest waves that could be generated in the flume with this wave period and, in fact, swamped the buoy. The greatest pitching motion at low wash levels (i.e. where it was not swamped) corresponded to the shortest wave periods (5.2-s prototype) in test No. 8. These three “short” wave tests were much more severe than prototype conditions because of flume limitations. In each case the wave speed was measured together with the period, and the wavelength calculated. These were compared with the equivalent prototype wavelengths for deep water waves. When scaled up, the test waves were found to be only about one-half the length of deep water waves with the equivalent period and amplitude, and hence much steeper than ocean waves. Thus, because the buoy performed adequately (taking test No. 6 as its limit) under these wave conditions we were much more confident that it could do its job in Davis Strait. The prototype buoy is shown in Figs 5a and b and the model is seen in Fig. 5c during one of the short-wave tests. Figure 5b is a photograph taken during towing tests to determine whether this buoy would tilt seriously when tethered in high currents and to determine how much drag force the buoy would exert on its subsurface mooring. This is clearly an important parameter for the mooring design, and our results are shown in Table 2. The prototype buoy was towed behind a small boat and the tension in the tow line was measured with a dynamometer in the line. The boat’s speed was obtained from successive radar fixes on a shoreline building. The measurement

TABLE

1.

Results from the model tests Model

Conditions

Prototype

Conditions

Wave Heighta f&R (in.)

Wave Period T7## (9

Wave Heighta HP (ft)

Wave Period

1

2.4

2.5

5.4

12.9

2 3 4

2.6 3.8 5.7

2.6 2.5 2.5

5.9 8.6 12.8

13.4 12.9 12.9

5 6.5 Untethered Buoy 6 6.0 7 6.4

2.6

14.6

13.4

1.4 1.2

13.5 14.4

8.1 7.0

1.0

6.8

5.2

Test No. Tethered Buoy

8

3.0

aCrest-to-trough height. bHighest water level above the SWL in prototype units (ft).

Model Buoy Response Wave Effect buoy follows wave well 7, buoy follows wave moderately well 7, 1,

wave interacts with buoy wave-buoyinteraction fairly strong

Wash Heightb (ft)

Tilt From Vertical (deg.)

-0.7

-0

-1.2 -1.7 -2.6

-0 -0
-4.8


-3 -7

-15” < 25”

<3

i; 17

A Spar Buoy Design for Oceanographic

Data Telemetry

(b) Fig. 5 a - The prototype spar buoy b - The prototype spar buoy under tow c - The model buoy during flume tests

/ 169

170 / Donald 0. Hodgins TABLE

and Brian N. Lea

2. Summary of towing test results Speed

Tension

Test No.

Buoy Reynolds number UDlv

Drag Coefficient 2TIpU’A

Water Conditions

2.1 :::

250

0.93 x iv

0.96

dead

300 475

1.11 x 10s 1.47 x 105

0.81 0.74

calm

2.5 3.0

400* 500*

1.11 x 1w 1.08 1.33 x 105 0.94 Average: 0.91

2-ft wind wave

*Peak values v = 1.9 x 10-S ft* s-1 D = mast diameter = 0.5 ft p = 1.987 slugs ft+ A = projected area, 20.9 ft2

accuracies would be about +5% for tension and kO.2 kt for speed. The drag coefficient averaged 0.91 and was used in our mooring loading calculations. As shown, the maximum drag force was approximately 5001b at 3 kt in choppy waves. We note that the buoy Reynolds numbers shown in Table 2 are below the range where appreciable drag reduction can occur. This is due to the choice of the mast diameter for the length scale. If, however, the Reynolds numbers are calculated using the hull diameter they do fall in the range where drag reduction may be anticipated at higher flow speeds. This, indeed, seems to be borne out by the overall drag coefficients shown in Table 2, which decrease with increasing speed. It is also interesting to compare the values of line tension that can be calculated using published drag coefficients (e.g. see Schlichting, 1968, p. 17) applied to each element of the buoy below the SWL with the measured tensions. This calculation is shown for two cases, U = 2.1 and 3.3 kt, in Table 3. The agreement with observed values of T is within the measurement accuracy. The reduction in the overall buoy CD is also predicted and is produced by the reduction of drag force on the buoyancy hulls and the ballast tank corresponding to a reduction in drag coefficient for these elements due to increasing Reynolds number. 4 Conclusions

Our observations of the buoy’s performance in the field showed it to be good, exceeding, in fact, our expectations from the tests described in Section 3. Some general conclusions are: 1) The design is workable with no modification to the overall configuration. It functioned well in heavy sea states, could be manhandled with light equipment and was durable enough to withstand deployment and recovery over the stern roller of a supply boat. 2) The analyses outlined above, together with scale-model tests, were

A Spar Buoy Design for Oceanographic TABLE

Data Telemetry

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3. Predicted drag force and drag coefficient for the spar buoy 7J = 2.1 kt Buoy Element

U = 3.3 kt

Area w

10-SRe

GJa

C, Area W)

10mSRe

CDs

C,. Area W)

3.75

0.93

1.2

4.50

1.47

1.2

4.50

3.13

2.34b

1.1

3.44”

3.67b

0.7

2.19”

6.00 4.00 4.00 20.88

3.74 0.93 3.74

0.7 1.2 0.7

4.2Oe 4.80 2.80c 20.78

5.87 1.47 5.87

0.3 1.2 0.3

1.8oC 4.80 1.20” 15.01

Upper mast Tapered buoy hull Cylindrical buoy hull Lower mast Ballast tank Totals Predicted values

T CD

262 lb 1.00

464 lb 0.72

Measured values, Table 2

T CD

250 lb 0.96

475 lb 0.74

aFrom Schlichting (1968) bBased on average diameter ‘Add 10% for end effects p = 1.987 slugs ft-’

essential parts of the design process and have proved to be reasonably accurate and to give answers on the safe side. We recommend this buoy design for use in similar applications. Acknowledgements

The development of this buoy and mooring system was part of the 1980 drilling operation at Hekja O-71 in Davis Strait carried out by Aquitaine Company of Canada Ltd as operator for the Baffin-Labrador Group of companies comprising Aquitaine Company of Canada Limited, PetroCanada Exploration Inc., Home Oil Company Ltd, Hudson’s Bay Oil and Gas Company Ltd, Murphy Oil Ltd, Pancanadian Petroleum Ltd ang SOQUIP. We thank Aquitaine, and Mr Howard Westergard of that company, for encouragement and permission to publish this material. The calculation shown in Table 3 was pointed out to us by Mr W.H. Bell of the Institute of Ocean Sciences, Patricia Bay. References B.H. and K.J. BAI. 1970. Experimental studies of the behaviour of spar type stable platforms in waves. Report NA-70-4. College of Engineering, University of California, Berkeley. BERTEAUX, H.O. 1976. Buoy Engineering. John Wiley and Sons, New York, 3 14 pp. HODGINS, D.O. and H.G. WESTERGARD. 1981. Internal waves in Davis Strait and their ADEE,

measurement with a real-time system. Proc. Port and Ocean Engineering Under Arctic Conditions, POAC 81, Qu6bec City, Qub., July 27-3 1. NEWMAN, J.N. 1977. Marine Hydrodynamics. MIT Press, Cambridge, Mass. 402 pp. SCHLICHTING, H. 1968. Boundary Layer Theory. 6th Ed., McGraw-Hill, New York.