NOTE ON THE EFFICIENCY POTENTIAL OF SENI-SUW ED SNIPS. REV--ETC 1W

UNLSII JUL 76 P R PAYNE 00600-76-C-1761

NLASIED WORKING PAPER1962REV-1

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READ XNST U tION

W.P. No. 196-2 Rev 1L ~~* r(~* ., .~.S. TYPE OF REPORT & PERIOD COVERED

A Note on the Efficiency Potgntia1of Semi-Subterged Ships' e.es L

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"'Petrei-/ 7i~i 614S. PERFORMING ORGANIZATION NAME ANDADDRESS 'D-~O~AE'~rT-rttC.TS

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This report used by QP96V in their study: Advanced NavalVehicle Concepts Evaluation.

19. KEY WORDS (Continue an reverse olds, if neoseary and Identffy by block number)

Advanced Naval Vehicle Concepts Lift/brag RatioEvaluation SWAT

ANVCETechnology AssessffentSemti-Subwerged Ship

20. ABSTRACT (Continue an reverse side Itn m D.and midr dy by block number)

> - iis- -a e19smnerned miianly with~ the "lift/drag" ratio (14/D) which couldbe achieved with semi-sukcierged ships (S 3 )1 Given that the mot~ion in aseaway can be very smll, then 14/I is by Itar the =~st inportant factor inany ccxrparison with other form of vehicle. That notions can be less than

'L~a.J for conventional ships has been clearly shamn by a nuntber of eminent workers-- 1 over the last thirty years. we suggest here, in fact, that no~tions could be

LA. essentially zero, if one is prepared to accept a form which has negligible

DD oam~P 1473 EDITION OF INOItiIS OBSOLET7vt~~si'~

/ f., SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

SECURITY CLASSIFICATION OF THIS PAGEOlihen Dts Znmt*.,

20. (cont.)

-ydrostatic pitch and roll stiffness when underway, relying on dynamic pressure

forces oq stabilizer foils for stability. We conclude that in nedium and largesizes, SJ offers the possibility of very high L/D ratios in the interwediatespeed range around 60 knots, and large S3 may be cuTpetitive at 100 knots. Tb-poor showing to date seens to be principally due to designs which have thesubmerged hull too close to the surface, struts which are very thick and/orhave large wetted area, and insufficient developnent work on reducing inter-ference drag..

SECURITY CLASSIFICATION OF THIS PAGE(tWhe Date Enrtot.E)

1910 Forest Drive * Annapolis, Md. 21401 * (301) 268-6150

NONE-~

Working Paper No. 196-2July 1976Revision 1

A NOTE ON THE EFFICIENCY POTENTIAL

OF SEMI-SUBMERGED SHIPS

by

jPeter R. Payne

1910 Forest Drive Annapolis, Md. 21401

TABLE OF CONTENTS

SUMMARY AND CONCLUSIONS I

A VERY BRIEF HISTORY OF SEMI-SUBMERGED SHIPS 11

THE PERFORMANCE CALCULATIONS 18

REFERENCES 24

APPENDIX I: Calculation of Small Water Plane Hull Performance 26

APPENDIX II: Wave Resistance of Submerged Ellipsoids 37

APPENDIX III: Wedge Strut Resistance Estimate 43

SUMARY AND CONCLUSIONS

This paper is concerned mainly with the "lift/drag" ratio (LID) which could beachieved with semi-submerged ships (S0). Given that the motion in a seaway canbe very small, then L/D is by far the most important factor in any comparisonwith other forms of vehicle. That motions can be less than for conventionalships has been clearly shown by a number of eminent workers over the lastthirty years. We suggest here, in fact, that motions could be essentiallyzero, if one is prepared to accept a form which has negligible hydrostaticpitch and roll stiffness when underway, relying on dynamic pressure forces onstabilizer foils for stability. The struts which support the above-water por-tion of the ship can then be sized by structural considerations only. Ofcourse, this implies that the upper hull will descend to the water surface forlow speed operation, as indicated in Figure 1. In many ways, operation of theFigure I configuration would be similar to operation of a hydrofoil, and there

are some interesting and potentially rewarding trade-offs to be made betweenI buoyant and dynamic lift. Note also that the ability to employ a multiplicityof struts means that the upper size limitations of a conventional hydrofoilare evaded.

The SWATH configuration in Figure 1 has /2-times more wetted area than theminimum wetted area possible, so we show an alternative configuration in Figure2. This was first suggested by R.W. Priest in the fifties,* although withstruts large enough to provide hydrostatic stability.

The hydrodynamic efficiency (L/D) of such a single body is shown in Figures3 - 5, based on calculations described later in this paper. If we select60 knots as the speed of interest, we see that L/D increases markedly withsize, but also with depth of immersion in the larger sizes. For A =2000 tons,

LID 10 with the body top at the surface

20 with the body top 48 feet below the surface.I For A = 20,000 tons, there's not much difference when the draft is shallow,but a total draft of 161 feet gives L/D = 39. L/D = 50 is theoreticallyattainable, if the body is deep enough. These figures are very attractive bycomparison with other types of advanced vehicle.

So far, we have only considered conventional underwater bodies. Suppose nowwe halve the skin friction coefficient in some way; either by polymer injec-tion**or by appropriately shaping the body. The ultra-low drag underwaterbodiest are an example of such drag reduction by shaping, by virtue of extensive

*Reported by Boericke1

**See Van Mater 2 for example.

t Payne3 describes the antecedants of this technology.

SIDE ELEVATION

WATERLINE

OPTIONAL ADDITIONALSTRUTS TO REDUCEBENDING MOMENT

CONTROL SURFACES-

Figure 1. A Small IVaterplane Twin 11ull (SIVAT11) Configuration Which Becomes"Foil-Borne" for Hfigh Speed Operation.

2

L SPEED

WATERLINE

CONTROL SURFACES

Figure 2. A Single Submerged Body.

3

ISO /1

160 /U2.38,

140/

a43.34A =200 L. TONSa Ib=7.O

too

04.

06

0

01 /0- 10260 0 0

SPEE INKNT

FROUDd NUM PTR IN FEET

404

160 26.68

186.76

!40 - ________

o= 93.38IA= 2,000 L. TONS

120 Ia/ 70

0

4.7

20DEPTH IN FEET d 268

0 10 20 30 40 50 60SPEED IN KNOTS

00.2 0.4 0s 0.8 L.0 1.2 L.4FROUDE NUMBER= u//a

Figure 4. Performance of a 2000 Ton Submerged Body.

ISO -- _ _ -

160'U

/4

140-- ~ -

WAS1.9A = 20,000 L. TONS

12 - a/b =7.0

o

02.IL

80 DE T N E T

40- 570.7

OL

40 - 03 4 06

0.2 0.3 SPEED IN KNOTS

0 t 02 03 04 0.5 0.6 0.7 0.8 0.9FROUDE NUMBER= h/a

Figure 5. Performance of a 20,000 Ton Submerged Body.

6

laminar flow in their boundary layers. Perhaps a more practical possibilityfor S3 is to design for a "tired,"1 low friction turbulent boundary layer overmost of the body. Stratford4 has demonstrated such a flow experimentally, andwhile no one has yet investigated the technology exhaustively for externalflows, it would appear to offer some promise.

A third alternative is drag reduction by injected air lubrication.

If such a halving of the skin friction coefficient is achieved, then the LIDratios for the three different displacements become as shown in Figures 6 - 8.Even the 200 ton size is now competitive with other vehicles at 60 knots(LID t!17) while values as high as LID = 80 are feasible in the 20,000 ton size.

It should be emphasized that all of these values will be degraded by

" Strut drag.

" Foil drag.

* Resistance of any other appendages.

By careful design, however, we can minimize the penalties involved. But sincestrut size dominates strut drag, and strut size depends principally on struc-tural loads, we have not attempted to include estimates for these parasiticitems in this paper.

We conclude that in medium and large sizes, S3offers the possibility of veryhigh LID ratios in the intermediate speed range around 60 knots, and largeS3 may be competitive at 100 knots. The poor showing to date seems to beprincipally due to designs which have the submerged hull too close to thesurface, struts which are very thick and/or have large wetted area, andinsufficient development work on reducing interference drag.

7

160 1 -~ -12.38'

/ j-q- 86.68' p140-.--I

a 43.34A= 200 L. TONS

/ a/b7.0

120 / -

100-

43.

~80- 46' /

26.0/

60-

40-

d DEPTH IN FEET 2320-

01 _ __ _

0 10 20 30 40 50 60 '10SPEED IN KNOTS

0 0,2 0.4 0.6 0.0 1.0 1.2 1.4 1.6 1.0 2.0 2

FROUDE NUMBER =U --O

0:~p .c 01Pi~fni , J .,0 o ll Sul, 11iitlo ', rgo Ivd if' Ihoi SI. i i Fri c it'll

IS ------- 1 - - *- * '- - -- -

180 d*- - ~ * - - *- * ---

140__ ____ -186.76-

a93.38iao~ ~~~~ = __ __ _.--2,000 L. TONS

a/b 7.O

!i 1 74.793.40 so-s~o

40

d DEPTH IN FEET 26.68

20 --- - __

01I____0 10 20 30 40 50 60- 70

SPEED IN KNOTSIIII I L

0 0.2 0.4 0.6 0.8 1.~p2 1.4FROUDE NUMBERu/ 2a

Figure 7. Performance of a 2000 Ton Submerged Body if the Skin FrctionCoefficient is Halved.

9

160 -- --. 5748'

£2=20,000 L. TONS

12 ob 7.0

120

20-

0 to2 04 06SPE INKNT

~ 0 0. 0.____ ____.4 0. 06 0. 0_09

200

A VERY BRIEF ISTORY OF SEMI-SUBMERGED SHIPS

There does not seem to be any totally satisfactory name for what we are dis-cussing. The abbreviation S3 does at least have the merit of extreme brevity,and so we employ it here. But we are really talking about a ship which has lowself-making wave drag, and a reduced response to the seaway. A "small water-plane area" ship will generally have both these virtues, but the descriptionis very imprecise. Also there is at least one other way of achieving the saneresult, as we shall see.

The first patent for S3 was awarded to Reuben N. Perley (1922) for a submergedmonohull with surface-piercing struts forward and aft. A similar ship witha single strut (the "Shark form") was patented by Rudolf Engleman in 1937. Amodern SWATH configuration was patented (in England) by Frederick G. Creed in1944. Thus there is nothing particularly "new" about the basic idea or concept.The problem has been in assessing its virtues and vices.

It would seem that the earliest studies in this country were carried out by agroup in BuShips (Code 420) under Owen Oakley, in the early fifties. Boericke1

summarizes much of the work done by this unusually creative team, which includedRobert W. Priest, David Winter, James L. Mills and others. Unfortunately, theywere apparently unable to arouse any interest in supporting the broad-basedRD effort which would have been required to select the best among their variousproposed configurations and then to develop it. Figure 9 depicts some of theconfigurations studied at this time.

From a different vantage point, Lewis' studies5 of seakeeping in the mid-fif-ties led to the first semi-submerged ship (S3) described as such, and a formwhich he patented in 1959.6 The lines are given in Figure 10. This was thefirst time, apparently, that a form had been consciously designed for super-critical operation in head seas; although supercritical operation was knownto be possible with conventional forms under certain rather special conditions.Insofar as one can tie it down to one individual, Lewis was clearly the inven-tor of the supercritical ship.

The work of this intellectually fruitful decade was summarized in a number ofimportant papers which appeared in the period 1959-1962. Mandel's 7 study, andthat of Lewis and Breslin are the best and the most complete. But Mandel'sstatement that "none of the new ship types . . . is likely to supplant moretraditional ships and aircraft . " may have discouraged some further work,because it was so easily taken out of context.

At this point, one might say that the most severe seakeeping problem - highspeed in head seas - had been solved by Lewis' invention of the supercriticalship (SCS) and that this concept could be applied to some of the configurationsalready proposed. But the monohull SCS was very tender in roll, had insuffi-cient deck space for many purposes, and experienced rather extreme motionsat resonance.

11

,".- , -I

1'r T-7------- --

_ . . l IZ J C;

]761

Figure 9. Configurations Studied by the Code 420 (Preliminary Design) Teamin the Early Fifties. (From Bocricke l)I

12

4 0 In P

Til - 7, - -

I IL an0 i a. ;: I I nL

NN -i -J -i -, i' J, NiL

39 33 9 It

C4.

131

In the period 1964-66, Payne built two small, manned, supercritical catamarans(FICAT I and II) which avoided the first two of these three problems. Thefirst of these craft is shown in Figure 11. A reduction in wave drag wasachieved partly by the reduction in individual hull beam which is permittedby the catamaran configuration, and partly by cancellation interference ofthe wave trains from each hull. In other words, interference was substitutedfor submergence of the main buoyancy volume. This avoided some of the practicalproblems associated with S3 struts, excessive draft, etc., and resulted in amore conventional ship.

The two FICAT's were operated "at sea" in the Chesapeake area for some hundredsof hours, during the years 1965-66, under widely varying conditions. Modeltests (Figure 12) were also conducted, and it seemed clear that at least a 50%wave drag caiicellation was being achieved for all Froude numbers above about0.5, with some evidence of total cancellation at F =O.7.9 The FICAT's werethe first SCS configurations to be analyzed theoretically10'11'12 so far asresistance was concerned. Band found generally good agreement between Michell'swave drag integral and experiment, including wave drag calculated from photo-graphs of the local surface elevation.

Despite the significant progress made, the Payne team was never able to findsupport for its research, which was therefore discontinued in early 1969, sothat attention could be concentrated on supercritical planing hulls. Adefinitive assessment of FICAT vis-a-vis other configurations was never made,and so this must remain a question mark.

Coincident with Payne's departure from the field, Leopold 131 re-introducedthe Creed configuration and built the first Litton TRISEC15 man-carrying model.This was the first true S3 actually reduced to practice, and its seakeepingability was very impressive. The problem of excessive motions near resonancewas solved by using inclined active foils which acted as both rudders andpitch angle control.

Litton was also unsuccessful in obtaining funding to pursue the research, butthe basic concept has since been an ongoing project in the Navy, principallyat NSRDC and NAVSEC.

Shortly after Leopold, Lang 1,7patented some very innovative improvements,and was later able to build a 190 ton "manned model." Again, because of lackof funds, this design was not "optimized," and has a much higher resistancethan the minimum possible. Lang adopted and improved the active foil stabili-zation concept, and linked it to a simple autopilot to achieve minimum motionsat all speeds.

In this present paper, we have suggested that, since active stabilization isessential while underway, one might as well dispense with hydrostatic stabilityand thus avoid the rather severe resistance penalties which it imposes. We alsosuggest that the hulls should be much deeper in the water if high efficiencyis to be achieved. Then, as a final improvement, the operational limitationsimposed by this deep draft can be ameliorated by hinging the strut assembly insuch a way that the above and underwater hulls can be brought together forlow speed, low draft operation. Although there may he some size limitationsto this last suggestion, the basic technology is not far removed from that usedin the new variable wing sweep bombers.

14

Y4*i

L.

I--

is

4J

Itr

-

LL.

iiiij iiiii l' L

160

114V00 co.ntn

-14

i~ (nO

4J -

(o

. 4J

4) 4J

4d .4

L.

T 71-Y Me

17

TilE PERFORMANCE CALCULATIONS

Oddly enough, the basic theoretical tools for an overview assessment were devel-oped by Havelock prior to the need for them. The four relevant papers are:

"The Wave Resistance of a Spheroid" 18

"The Wave Resistance of an Ellipsoid" 19

"The Moment on a Submerged Body Moving Horizontally" 20

"The Forces on a Submerged Body Moving Under Waves"121

For the present paper, we are interested in wave resistance only, the equationsfor which are given in Appendices I and II. Since the step from Havelock'sequations to numerical results is not entirely trivial, we first checked ourresults with the only known previous solution, the "slender body" numericalevaluation by Wigley.22 As Figure 14 shows, the agreement is good.

As recounted in Appendix I, we then compared the theory with tank test measure-ments of residual resistance; again with satisfactory agreement for our presentpurposes.

Some general trends were then studied. Figure 15 gives numerical results for abody of revolution, and Figure 16 shows the effect of varying the cross-sectionalshape for a fixed submergence of the centerline. The corresponding total resis-tance ratio is given in Figure 17, and the inverse, LID in Figure 18. Changingthe cross-sectional shape clearly has a very small effect compared with achange in depth, so this effect was ignored in subsequent calculations.

The final calculations are summarized in Figures 3 - 8, which have already beendiscussed at the beginning of this note.

Some limited work on strut resistance was done during the course of this work,and is summarized in Appendix 111. This is considered to be incomplete.

18

15

011

IS

ILI

EXACT '00,176b I=--ISOLUTIOS_ ....

0 " 1 0 WIGLEY'S APPROXIMATION

0 0.2 0.4 0.6 0.8 1.0 1.2LENGTH FROUDE NUMBER F20 U W/2, /"a-

Figure 14. Wigley's "Slender Body" Numerical Approximation in Comparisonwith the Exact Solution of Hlavelockts Equation for the W av'eD~rag of a Prolate Ellipsoid.

19

SWBMERGENCE DEPTH RATIO

0.4- h =I

0.3 2 ---

Ih L3/ 7

~4I

2 .1

0.060.1 1

z

La 0.06 ± -T - 6

z 100.02 --

0.0

3 4 56 7 89LO 3 4 5 6 7 10.0 2 3

SUBMERGENCE FROUDE NUMBER, Fh: U/ AI-W

Figure 15. The Effect of Slenderness and Submergence on the Wave Dragof a Prolate Ellipsoid.

h submergence of centerlineb radius of body

20

1.0 -.._ - . ...... .

0.-0.1

SPHERE -. 1 2t4

I/ \L.

/ CI \uT.10

0.05 I ,\ .0.099

I_-0.03 0112c

I -\

O0-0.0-2b

- I \-

"_ _ _ _ _ I•0.04. ___,_

002 _______________

-j Io.- I0.01

,10: 1 ELLIPSE

-- I I \j I

0.0001 . - _ _ I _ _ _ !- . ___ I__I__I__ I

0.1 , I O 1.5 0.0 150 00.0

SUBMERGENCE FROUDE NUMBER u/gh

Figure 16. The Effect of Varying the Cross-Section of a Submerged Ellipsoid.

21

1.07 b-1O2 6 FT

/ I " 2.0" 1.01 8 0.

-t0h m45.6 FT

.2. 0

A 2000 L. TONS 1.01o 114.0 FT.°oil- .0 .°SLUOS/,T

-01 FOR ALL CASFS

h d

h a 11.4 FT

* 0.05

-II A.1

O/L d 02

10.01.01

-AU 0.5

0.005

,//' //0.001-- / /

0.0000-

/1 j " Il.l, C,.vd I iI',oidS, ,It./ I.0n1.t t1111 I"1t, dt

o.000, ... L J_LLLLLL. I. .I I_ .l ..uol 0, 10 I5 to 0

UlMFO IMIIN: F ROFtI. NUMI* R '

190 ii--

,18- I, _ooo.

IIId0JA

d=0.5

0 0.2A 200 L. TONSI Iia 14.0 FT

140 p 2.0 SLUGS/FT3

120- ___ _ -(7 7 b

--

2.0

1.0 10 10

2.

0.5

40 0.5

20 1.0.01

20 __ 0_._2.0

020 40 80 s 100 120SPEED IN KNOTS

Figure 18. Efficiency of Submerged Ellipsoids at Constant Draft.

23

REFERENCES

1. Boericke, H., Jr. "Unusual Displacement Hull Forms for Higher

Speeds," Int. Shipbuilding Progress, Vol. 6,No. 58 (June 1959).

2. Van Mater, P.R. "A Review of Viscous Friction Prediction Proce-dures and Viscous Friction Reduction Conceptsfor Advanced Naval Vehicle Missions in the1980-2000 Time Period." Payne Inc. Working

Paper No. 181-6 (March 1976).

3. Payne, P.R. "Comment on 'Shaping of Axisymmetric Bodies forMinimum Drag in Incompressible Flow"'. Journalof Hydronautics, Vol. 9, No. 3 (July 1975).

4. Stratford, B.S. "An Experimental Flow with Zero Skin FrictionThroughout its Region of Pressure Rise," Journalof Fluid Mechanics, Vol. 5, Pt. 1 (January 1959).

S. Lewis, E.V. "Ship Speeds in Irregular Waves." Trans. SNAME(1955).

6. Lewis, E.V. "Ship." U.S. Patent 2,974,624 (14 March 1961).

7. Mandel, P. "A Comparative Evaluation of Novel Ship Types."Paper presented at SNAME Spring Meeting (June1962).

8. Lewis, E.V., and "Semisubmerged Ships for High Speed Operation inBreslin, J.P. Rough Seas," Third Symposium Naval Hydrodynamics,

High Performance Ships (ACR-65) (September 1960).

9. Band, E.G.U. "Analysis of Resistance Measurements of FICATCatamaran Model in Ship Towing Tank and WindTunnel." Payne Division of Wyle LaboratoriesWorking Paper No. 1001-6 (November 1968).

10. Payne, P.R. "Application of Michell's Resistance Integralto FICAT Wave Drag." Payne Division of WyleLaboratories Working Paper No. 1001-3 (October1968).

11. Payne, P.R. "Squatting of Hulls Which are Half-Bodies ofRevolution." Payne Division of Wyle LaboratoriesWorking Paper No. 1001-7 (November 1968).

12. Band, E.G.U. "Evaluation of Michell's Resistance Integral as

Adapted to FICAT Wave Drag." Payne Division ofWyle Laboratories Working Paper No. 1001-4(October 1968).

24

REFERENCES

13. Leopold, R. "A New Hull Form for High-Speed Volume LimitedDisplacement-Type Ships." Presented at the SNAMESpring Meeting (May 1969).

14. Leopold, R. "Marine Vessel." U.S. Patent 3,447,502 (June 1969).

15. Marbury, F., Jr. "Small Prototypes of Ships - Theory and Practical

Example." Naval Engineers Journal (October 1973).

16. Lang, T.G. "High-Speed Ship with Submerged Hulls." U.S.Patent 3,623,444 (November 1971).

17. Lang, T.G. "Hydrodynamic Design of an S3 Semi-Submerged Ship."9th Symposium Naval Hydrodynamics, Vol. 1, Uncon-

ventional Ships Ocean Engineering (ACR-203)(August 1972).

18. Havelock, T.H. "The Wave Resistance of a Spheroid." Proc. ofthe Royal Society A, Vol. 131 (1931).

19. Havelock, T.H. "The Wave Resistance of an Ellipsoid." Proc. ofthe Royal Society A, Vol. 132 (1931).

20. Havelock, T.I. "The Moment on a Submerged Solid of RevolutionMoving Horizontally." Quart. Journal Mech. andApplied Math., Vol. V, Pt. 2 (1952).

21. Havelock, T.H. "The Forces on a Submerged Body Moving UnderWaves." Quart Trans. of the Inst. of NavalArchitects (1954).

22. IWigley, IW.C.S. "Water Forces on Submerged Bodies in Motion."Transactions of the Royal Inst. of Naval Archi-

tects, Vol. 95 (1953).

25

APPENDIX I

CALCULATION OF SMALL WATER PLANE HULL PERFORMANCE

26

In considering the feasibility of a new ship concept, an early question is"what is its transport efficiency?" Ie typically evaluate the calm water"lift/drag ratio" (the inverse of the more conventional R/D) in order toanswer this, and it's clear that approximate figures for "L/D" are quiteadequate for establishing feasibility so long as they are realistic. In thecase of S3, it is rather simple to compute resistance, and it's also clearthat, due to small motions in a seaway, performance will not degrade muchwith increasing sea state; except possibly near resonance.

The significant resistance components are

RW Hull wave making drag

Hull friction drag (considered together in this analysis)BS Hull pressure drag

DSW Strut wave making and spray drag

D IStrut skin friction dragSS IStrut pressure drag

Di Strut/hull interference drag

DA Wind resistance

The purpose of this Appendix is to present equations for the first three resis-tance components.

Hull Wavemaking Drag

This is perhaps the most critical term, because it will dominate the optimizationof strut length.* On the other hand, it would be needlessly complicated todetermine the wave drag of different hull shapes at this stage in the analysis,when we are looking for overall trends. It's sufficient to determine a"standard" variation of wave drag with slenderness ratio and immersion depth,recognizing that subsequent variations of hull volume distribution may enable usto improve on this result.

Such a "tndard" variation is provided by Havelock's analysis of a prolatespheroid , where he obtains

RW = 1282ga3c3A2e-2/f2fe-2t2/f2 [J,/ 2 (z)] 2 dt (I.1)

0

* A trade-off which does not seem to have been addressed in the literature.

27

where

C = (b/a)2 the eccentricity

a,b are the major and minor semi-axis lengths

f = u//g, the submergence Froude number

h = submergence of the spheroid centerline

J3/2 (z) is the Bessel Function of the first kind, of order 3/2

rz /

A = 4c 2ogl+1- C 1

f2K

t is a dummy variable.

The displacement of a prolate ellipsoid is

Pg 4ab 2 (1.2)

Thus, from equations 1.1 and 1.2

- 961r A [sin z - cos z]2 dt (1.3)

0

The term in the Ibracket is a function only of b/a and is platted in Figure1.1, and tabulated in Table 1.1.

Equation 1.3 is easy to integrate numerically as it stands, so that there islittle point in seeking further simplification.* In Figure 1.2, we have com-pared it with some experimental measurements of net residual resistance fromAppendix 7 of Reference 1.2. Since the experimental data is for "streamline"bodies, the aft portions of which are quite unlike an ellipsoid, the agreementseems excellent, and quite sufficient for our present purposes.

* Because of the exponential term, it converges well as t increases.

28

4-J

co

14-4

.0

Ii4J

292

Table 1.1. Functions of the Fineness Ratio b/a

(b/a) C A If (b/a) }

IE-03 .9999995 2. 5000211E-07 1.884984613-05

IE-02 .99995 2.5014505E-05 l.8868605E-03

SE-02 .99874922 6.3161389E-04 4.7946157E-02

1E-01 .99498744 2.5905254E-03 .19936531

.2 .9797959 1.1260011E-02 .89917777

.3 .9539392 2.8651639E-02 2.3880292

.4 .91651514 6.0075642E-02 5.2374138

.5 .8660254 .11643375 10.622599

.6.8 ..22249714 21.234275

.7 .71414284 .44496157 44.383883

.8 .6 1.0231108 106.54675

.9 .43588989 3.5216047 382.42625

.95 .31.22499 10.994751 1207.5799

.98 .19899749 45.337269 5086.5397

.99 .14106736 130.40137 14689.081

30

0.1 MODEL 4157

a SUBMERGENCE DEPTH RATIO0 h/b: 2.544

'0h/b = 2.84

6 - -,- -h/b = 3.164- ----- - ,hv=3.488

4- THEORY EXPERIMENT

0

CrC

W

4-

z

a -\

0.001-a 4 5 'ID 2 3 4 5 6 7 10.0 2 3

SUBMERGENCE FROUDE NUMBER Fh:u/,f-h-

Figure 1.2. Comparison Between the Theoretical Wave Drag of a 7:1 ProlateEllipsoid and Some Experimental Measurements.

31

L .. . . . . . . . . . . . . _ . .. .. . . . . .

0.1

*! MODEL A 4164SUBMERGENCE DEPTH RATIO

p /b =3-119h/b a 2.738

4- -- -- /b =2.47

THEORY EXPERIMENT

- \ THEORY FOR 7:1 ELLIPSOID!SOII;E

4-z in

El

0.001 Ia 4 5 6 7 8 91l 2 3 4 5 6 7 8 910.0 3

SUBMERGENCE FROUDE NUMBER Fh =u/hgh

Figure 1.2. Continued.

32

0.1,- ..

MODEL 4165SUBMERGENCE DEPTH RATIO

-0 -0 z3.178B =2.852

T-- 4 hb a2.52THEORY EXPERIMENT

4I -%

4 .

O.

" 0

'hi 4

0 o ..1. .I I-1

SUBMERGENCE FROUDE NUMBER Fhzu/0/h"

Figure 1.2. Continued.

33

zib 0... .,... ... . . ..., I..... i........ ... ,,. . ... .. . . . , .. ..."...

Ifull Friction and Pressure Drag

Hoerner gives a relationship for this which has been widely accepted. De-fining

D = C 1 -u S (1.4)BS Dwet 2 wet

+3 (b 3/23

= wetB + (] (.5)

where C is the usual flat plate skin friction coefficient, which may be con-veniently expressed as

Cf = 0.427 (1.6)fB (log 1 0 R2 a - 0.407)2.64

R2a = 2au (Reynolds number based on length) (1.7)

v 1.25 x 10 - 5 (ft2/sec) for normal water

There is no simple relationship between the wetted area S and b/a, becauseother factors are involved, principally the prismatic coeffcient,

= VolumeP Ta 22nrab 2

A typical variation of

C = SwetS 47rab

with b/a and C as given in Figure 1.3. C S is obviously less at the lowerprismatics; bu then, so is the displacement, so this tends to cancel out.For the purpose of preliminary performance calculations, therefore, we mightas well use the spheroid relationship

Swet 2na2 (b b+ s l(1.10)

where

£ = 1- (b/a)2

Since we are generally working to a given displacement, the hull volume V isknown, and since

V =4 7 a3 (b) 2 a 2/3 . 1/3 (1.11)

34

Fl 0.600 065 DATA FOR SERIES 58 FORMS FROM REF. 1.2

C)0. 7PRISMATIC COEFFICIENT Cp 2VOL/ 2 7r ab2

1.0~ ~ CCLIDER WTH SPHERICAL ENDS HR

PROLATE SPNERIOD

I0.8

0.7-

vSLENDER QUADRATIC SPINDLE

FALSEZERO

0 0.2 0.4 0.6 0.8 1.0

DIAMETER kLENGTH a

Figure 1.3. Wetted Area Coefficient, as a Function of P.inelless Ratioand Prismatic Coefficient.

35

APPiENlD IX I RITIIliNCIS

I H. I lvelock, T.II. "The Wave R, sist:incte of" a Splheroid, ' ' Proc. of'tlhe tRoal Soc ielv A, Vol. 131 (1931).

I..2 (~ert I*, M. "Rsi st ainck Expe rilli'.t s on :i ys' culnat I Series ofSt ream il ined B~odi es ofr Revolu ili on - for Appl icat ionto the 00sin i' of Iillgh Speed Sibma i.rine s."ivid IV. T'layIlor Model itasin Report C--:297,ATI 8I.176 (Apri 1 1950.

1 .3 ll ivrn r. , .F. I: lii d -l, niiii c Iiae. PubihI i shed 1)by th, Alit hIoi-.(1.18 Biu;t od rive, M id I land 1'i rk, N 07,132)( 196!;).

I I I I I I I

APPENDIX 11

WAVE RESISTANCE OF SUBMERGED ELLIPSOIDS

37

The problem of the wave resistance of generalized ellipsoids ha. been solved byHavelockll 1. The purpose of this Appendix is to recast his equations into aform suitable for numerical evaluation.

Case of a > b > c

From Hlavelock11 1

322gpa2b2c2 e-2Koh[Int (1) + Int (2)] (11.1)

(2-ao) 2 (a 2_b 2) 3/2

4

The volume of the ellipsoid is - abc. Thus

R 247rabc -2Koh [ l( + mt (2)] (11.2)

A (2-a ) 2 (a2_b 2)3/2 e [hnt ( I

In Ilavelock's notation K 0 g/u 2 . We will employ the submergence Froude numberf = u/rgh, so that

K - g 2K h = 2/f2 (11.3)o fgho

For numerical integration, b f . .. ... d . .... 2 -53/2

a = (2A)(K (cA + 2 abcX1 -/ (11.4)

where X > ,b ,c . (This form of the well known ntegral is due to and 2In nondimensional form, if 1 b/a c c/a gduX/e B

b .. . d ... - 2 fi i -3/2 (11.5)

Rw 242 fc- -2/f1

2 .2... e 2 [Int(1) + lnt(2)] (11.6)

38

Defining

2 b2-c2 2_2al a2-b 2 1-b2(I7

Havelock gives a 2 2 b2)(l+t2 )(1a2t21/ 2]2 e2t 2 /f 2 dt

Int(1) f (1 - a 22) 3/2

0 1 )

where the Bessel Function J3/2 is defined in Appendix I. Thus the integral canbe written as3/

1/a1win1 a 2/ e-2t2/f2 [(sin q)/q - cos q]2

Int(l) =irJ q (1-a 2 t 2 ) 2 312 dt

0

where q= (1-b 2 )(1+t 2 22(j_2)(~t)(l-ai1 t 2

qf V4 2 (11.9)

fhY

( =h/a)

For the second integral, lavelock givesInt() =f [3/2(p)2 e-2t 2 /f 2

Int(2) (12e2 3/2 dt (II. 10)

(a12 t 2-03)

where

p = (1-b 2 )(1+t 2 )(a 1 2t 2-1)

p =4 2 (11.11)

and 13/2 is the modified Bessel Function of the first kind, of order 3/2.

Since

I1/2 (z) = vr2/lrz sinh z, I-1 1 2(z) = /2/z cosh z

and I 2n I (a standard form)In~ = In- - n

n+l - z n

1-1 1

3/2 = 1/2 - E 1/2

39

I ,(p) v2-/rp [cosh (P) - inh(p) 1 (.12p

S= -2t2/f2 [cos(osh(p) - sinh(p.Int(2) P (II.13)

Wt 2 2 P 22_ )3/2

Equation 11.6 can now be evaluated, using equations 11.5, 11.8 and 11.13.

Case of a > c > b

From Havelock, after some manipulation

R W 24 bc e 2 f2A-= 2( 232Int(3) (11.14)A (2-a ) 2 -_2)/

whereInt(3)

-2t2 /f2dtS (1 + 2 t 2 t

2 22

r - r 22t2)1 d (11.15)

where 2 2 _62a2 .1-l2

and

Thus 11.14 can be evaluated using 11.15 and 11.16.

40

Wave Drag of a Sphere

From Htavelock, after some manipulation

R 3 -2/f2 23 -2t 2/f22/ (l+t) et dt (11.17)

0

where f = u/g, the submergence Froude number.

This solution is a useful check case for the general equations when programmedfor numerical solution.

41

APPENDIX II REFERENCES

11.1 Havelock, T.H. "The Wave Resistance of an Ellipsoid." Proc.of the Royal Society, A., Vol. 132 (1931).

11.2 Band, E.G.U., and "The Pressure Distribution on the Surface ofPayne, P.R., an Ellipsoid in Inviscid Flow." Payne Inc.

Working Paper No. 101-1S (July 1972).

42

APPENDIX III

WEDGE STRUT RESISTANCE ESTIMATE

I

43

During this study, it was felt that a base-ventilated wedge section strut mighthave some attractions. Accordingly, the following calculation was carried out.

Surface Pressure Drag

The equations for the force per unit length, P/Z, on a wedge are given byKorvin-Kroukovsky and Chabrow I I .1 ; viz

P/ 22pkbv cos a f(Y) (1II.1)

where W0 and e is the wedge included angle

w/2

f(y) =f + sin y n (1 y )-n] c y sin y dy

00

/ 2

4 cos a (1 + sin y) n(cos y)l-n sin y dy

n w- 20 On ir2

b is the total strut width

Figure III.1 and Table III.1 give the variation of P with 0 obtained fromnumerical integration of these equations.

Some curve fits to these results are of value.

For 0 < 0 < 300

C 8o

P (89.9543 + 0.517358 5°)

For 0 < 0 < 1800

Cp 1.1107 x 10-2(0) - 6.1332 x 10-5(00" 2

+ 2.31799 x 10-7(6) 3 - 6.02678 x 10-10(00) 4

+ 7.85803 x 10-13 (0)5

44

E-- >-- b

1.0,

UO "

orim

L-

0.6 rd "0

CO

0

IL

I,.

0.2

U

44

a

0

0.2"

0 20 ° 40° 60 ° 80 ° I0WEDGE HALF ANGLE 8/2 IN DEGREES

Figurc 1II.1. The Drag Force Acting on the Forward Faces of a W~edgein Tw¢o Dimensional Inviscid Flow.

4S

Table 111.1

0 (deg.) cp

2 0.021978

4 0.043473

6 0.064498

8 0.085062

10 0.105178

12 0.124855

14 0.144104

16 0.162935

18 0.181357

20 0.199379

22 0.217011

24 0.234262

26 0.251140

28 0.267656

30 0.283816

40 0.359552

50 0.427527

60 0.488569

70 0.543405

80 0.592685

90 0.636976

100 0.676800

110 0.712603

120 0.744788

130 0.773720

140 0.799723

150 0.823082

160 0.844058

170 0.862884

180 0.879762

46

Surface Skin Friction

Korvin-Kroukovsky and Chabrow give the surface velocity asx

s _ Cos y(I.2

u (1+sny) (111.2)

where x is related to the dummy variable y by

x = 2kb Cos + sin y)n(cos y)l-n sin y dy (111.3)

In principle, then, we can compute the local skin friction force and integrateit. For the present study, because of time limitations, we employan approxi-mation which is simpler, although less accurate. We assume that the surfacepressure P is a constant so that the pressure drag P is given by

P

UT = Ps - P., (III.)

Since

Poo 21 1 2P2 PS + y PUS

1 2 1 2Pu s P". - Ps + f pu

1 2 P= -PU - b-

1 2 P= P u [1 - Cp] where CP =1 pu2bk (111.5)

therefore total friction drag 2

Df (2ct) Cf -p 2 [1 - Cp] (111.6)

where c is the chord and k the wetted length.

Based on frontal area, since

b

c 2 tan (0/2)

Df _ CfCD f 1 Pu2 b. tan (0/2) p(1I.7)

47

Base Drag

SPRAY ,-LINE ,f

-X CAVITY T--x~ ~ -A , a__P g

Uo (V-z)

Figure 111.2. Side View of Strut.

We can compute the base pressure Ap. using Iloerner's anproximation, or directlyfrom experimental observations. From Hoerner's approximation (p. 3-21)

B c C CfCf cos (0/2) f 2 tan (/2) cos (O/2) (111.8)

Then12 tan (0/2) cos (0/2))1/ 1 2

APB = 0.135 C / Pu (III.9)f

As Figure 111.3 shows, this agrees well with experimental observations.

48

1.2 -170.6

E) EXPERIMENT

1.0 I APPROXIMATE 0.

c'J

Q 0.8- -0.4

CL C~i0 0

z -0.3

0

U

01

0 200 400 600 800 1000WEDGE HALF ANGLE 81p IN DEGREES

Figure 111.3. Comparison Between Iloerner's Approximation for Base DragPressure, and CP = CD - CP. where C P is determined

from Figure 111.1, and CD from Iloerner's Summary of ExperimentalData. (Cf =.004) D

49

The depth z of the ventilation is then given by

AB= pgz

i.e.

z9~~~ = .6 2 tan (0/2) cos (0/2) 1/3 (1.0u2 Cf

This is plotted as an equivalent Froude number in Figure 111.

u /Irg- is a kind of Froude number, therefore, dependent only upon 0 and C f.Typical values are:

For Cf = .004, 0/2 = 50 and

U = 10 20 30 40 s0 knots

Z= 2.1 8.4 18.9 33.6 52.5 ft

The total base drag of the strut will be

DB (.1- zpg)bz + b(2.-z) ApB

ipgbz2 + b(t-::) 1 Pu2 zg= 2Y 2u2

pgb 2 T pgbz(2k-z)

I pgb(z 2+ 9Z) (1.1

where

z is given by equation 111.10.

so

3.5-

3.0

2.5-

2.0-

0.000.004

0.00

0.5

10-

0 50 100 15* 200WEDGE ANGLE 8 IN DEGREES

Figure 111.4. Ventilation Froudc Number as a Function of Wedpe Angle.

SprayDr.

From Ogilvie111.2 we take the spray surface elevation to be (see Figure 111.2)

Co (O1xXe/2] )~xl dA (1.2

where K g/u2

The wetted area associated with this is

AS=J idx (III. 13)AS =o .d

In the numerical integration of equation 111.12, a singularity occurs at A = 0.This difficulty can be avoided by integrating to infinity from a small valueAV and determining the residue as follows. Since X1 << 1, cos (exA/2) 1.0,except right at the bow (x 0). Also sin (/K- x) = A~3"x.

.'. 0 = e - x dX

0

ex f I l ) 2 (iA) 3 (HA) 4 dX~ J "1! 2! 3! 4! + A

0

OxHX (fiA 1) 2 (fi I) 3 (fix 1) 4 IO - 1 2.2! + 3.3 4.4! + ( .14)

Equation 111.12 is plotted in Figure 111.5 for some arbitrary values of If and x.

Total Strut Resistance

If z < X (or gz/u < g,/u 2 , or u2/gz > 11/gk)1 2 1 2 be

-Pu bWp + -i Pu Cf tan (-02) (1 CP

pgz2 b + b(,E-z) - pu 2 (zg/2u2) + (spray drag)(II.15)

52

From Ogilvie11.

Z(X,O) = 1 - -i X Y- dui

X = XK/1H

K = g/u 2

4.0 -- A

3.0 -- X 1. -25. -- - -...-. .

I--

o 1 2 3 460 s 70

DISTANCE ALONG WEDGE IN FEET- X

Figure 111.5. Spray Sheet Surface Flevition from Equation 111.12.

t = Draift ill feetThis case is for ui 25 ftlsec, 0 =7.50

If u2/gz < u2/gL the terms in the curly bracket become

Y Pgz 2 b (III.lSa)

1 2If we base CD on Pu bt

C = C + C f (1 Cp) + 2 t)) g--2 (1116)

or + (spray drag)

S(for u2/gz < u2/gX)

Note that

gz =0.0675(2 tan (0/2) cos (8/2))1/3u 2 0.67 Cf (III.lO)

u 2f )

Some typical results are given in Figure 111.6. Spray drag has been omittedfrom the calculation because of time limitations.

54

0 100 000t 0

Ci 4 0 0 c;jN 00 0 0 e4.

. 7-

cQ+

0 4-

U 44

...... ._ ...~- . _ 0 )

Cd

W -(0 10

U) w 41

40 2 0

03~ I-11140 H

APPENDIX III REFERENCES

III.1 Korvin-Kroukovsky, B.V., "The Discontinuous Fluid Flow Past anand Chabrow, F.R. Immersed Wedge." Sherman M. Fairchild

Publication Fund Paper Preprint No. 169(October 1948).

111.2 Ogilvie, T.F. "The Wave Generated by a Fine Ship Bow."Proc. Ninth Symposium on Naval Hydrodynamics(1972). Also: Report No. 127, DNAME, Univ.of Michigan (1973).

56