Hydrodynamic forces on floating bodies moving at low froude number

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HYDRONAUTICS, Incorporated

TEC}B\IICAL REPORT 117-15

HYDRODYNAMIC F.ORCES ON FLOATING

BODIES MOVING AT LQW FROUDE

NMBR

By G. Dagan

April 1970

Thi document has been approved for public

release and sale;. its, distribution is unlimited

Prepared for

Office of NaVal Research Department of the Navy

Qontract Nonr-33k9(OO)

NR 062-266

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-1-TABLE OF CONTENTS

Page

.A.BS TR.0 T...0 0 0 0 0

0 O00 0

0 0 0 0 0 0 o a a , 0 0 a V INTRODCJC'I'IOI\J

000.000.00.00...

_{000000000000000000 000} 0

SMALL PERTURBATION EXPANSION...00000000 3

THE PRESSURE DISTRIBUTION AND THE FORCES

ACTING ON THE BODY...,

_{000000 Oo ...000o0 000.,o}

₇

FIRST ORDER APPROXIMATION (EXAIVLE OF rfl4O

DIMENSIONALFLOW)

_{00000 00000 0000000000000, 000000000}

₁₁

FIRST ORDER APPROXIMATION (EXALE OF

TI-mEE-DIMENSIONAL FLOW) ₁₃

DISCUSSION OF RESULTS AND CONCLUSIONS...00..000000000 17

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LIST OF FIGURFS

Figure 1 - The Dependence.:of the Form Factor on the Freüde Number f or Three Geosims of a Tanker (Reproduced

from Baba,

₁₉₆₉₎

Figure 2 - Steady Flow Past a Floating Body

Figure

3 -

Two-dimensional Flow. Past a Floating Body

Figure -1 - Flow Past a Rankine Body in Shallow: Water (First

Order Approximation)

Figure ₅ -FlowPast a Slender Body of Revolution (First

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-iii-NOTATION

Primed variables are dimensional; unprimed variables are

dimensionless

a parameter

A the area enclosed by the waterline

A the area in thehorizontal plane exterior to A

C the waterline curve

Dl depth of shallow water

f complex potential

F? force vector ( =

/yL)

Frm= Ul/(gTl ) draft Froude number

Fr= Ul/(gLi)2 length Froude number

g acceleration of gravity

h(x,z) the hull function

vertical unit vector

L1 body length

= L/2T' dimensionless body half length in the example

of two-dimensional flow

n unit vector hormal to the body surface

pT pressure (p = p'/yLr)

Q. singularity strength

r radial coordinate

S body surface

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-iv-x!, y', z' cartesian coordinates (yT is vertical, z' is a complex Vari,able in the two-dimensiOnal case)

UT the velocity at infinity

y fluid specific: weight

- sinkage

= Fr small parameter

1 1

cpT velocity potential (cp = pTLT2/g2)

- free surface elevation

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v

-AES TRACT

The equations of steady free-surface flow with gravity past a ship are expanded in a small Froude number perturbation Yseries. Expressions for thevertical force nd the moment (and

the sinkage and trim, correspondingly) are derived up to second

order. Two examples of flow are solved to first ordei: a

two-dimensional flow past a ankine half-body in shallow water and the three-dimensional flowpast a slender body of revolution.

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-1-I,

INTRODUCTION

Weconsider herein floating bodies moving steadily in an

ideal heavy fluid at low speed, such that FrL = U?/(gLT)2 j

small.

More precisely, FrL is smaller than the value at which

'free-surface waves appear first behind the body.

This regitne

has not been Studied in detail, in contrast with the cases of

moderate or large F'L (wave making and planing': regimes,

res-pectively).

_{The reason for the lack of inter'est ihsmall FrL}

flow is understandable:

in the absence of wavesor spray the

reslstance is purely frictional.

The low FrL flow is still irteresting from .a theoretical

point of view, since it permits the linearization of the

free-surface cohdition while keeping the body shape unchange'd.

In

the wave making r'ange the usual solutiofls of the flow problem

are based on both linearizations of the free-surface and body

boundary conditions,

their relatve contribution to the overall

aproimation being thus obcured.

But there arealso practical reasons which justify amore

attentive invetiSationof the low FrL flow.

It is well known

that ship resistance depends on FrL even at small

_FrL.

As an

illustratiOn, we reproduce in Figure 1 the ±'esults of model

tests of Baba (Baba,

1969).

The form factor, defined as the

ratio between the surplus of the actual frictional resistance

relative to the resistance corputed by the Hughes formula a1I

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Hughes formula correlates the resistance of a thin friction

plate. The form factor seems to be growing with FrL even

below the value FrL 0.18 corresponding to wave inception, This relative':increase of resistance is generally attributed to sinkage and trim related to. ree-surface effects,. In

sub-sequent sections it is shown that in the inviscid fluid

approximation there are non-zero forces and moments acting on the body, which may be computed for mall FrL,thus.. a quan-titative basis for the evaluation of the drag increase is

provided. As a matter of fact, the presence of a free-surface may influence the resistance in a subtle way by changing the

flow in the boundary layer and wake. This effect is not con-sidered at present.

The evaluation of sinkage and trim is important also in

the case of motion in shallow water. Since the speed is

gen-erally low in this case; the small FrL approxthation is no

only convenient,.but realistic too. The effect of the term of Bernoulli's equation on the suctiOn force on ships in shallow water has been recognized (see, for

instance, Saunders 1957) but no systematic quantitative

analysis has been carried:O.ut.

0u interest in the overall FrL regime has stemmed from

studies of the bow breaking wave, In our previous report

(9)

-3

front of a blunt two-dimensional body has been investigated

on the basis of a small FrT expansion. In the present report

a similar type of expansion is explored in orderto evaluate the forces acting on atwo- or three-dimensional body moving

in deep or shallow water.

II. SMALL PERTURBATION EXPANSION

With the cartesian system of Figure 2, let us consider a

steady flow of speed U' (in the -x' direction) at infinity.

The variables aremade dimensionless by referring the coordi-1

nates to the body length L'and the velocity to (gL')2 (see

Notation).

The exact flow equatidns, expressed in dimensionless variables, are a follows

vp = 0 (in the flow domain)

Li]

(on the

free-surfacey = (x,z))

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cp = -x

_FrL

(at infinity)

_£5]

We seek now the simplification of the nonlinear problem by a perturbation expansion, with FrL = as a mall parameter. Obviously, the limit c = 0 represents rest. Our xpan5ion is a limit process in which. the body

dimensions

are kept fixed, While the flow velocity tends to zero. At rest 0, T = 0. Hence., the ppropriate perturbation expansion, suggested by

Equation

£2],

is

cp

ecp1 + cp2 +

C3 +

...

£6]

£7]

Mong the free-surface cp has the following expansion

cp(x,,z) =

cp1 +

e (c

+ 1ii) +,

(c + rI2cP1y + ll1CP2y

+*n121yy)

+ ..

t8]

whereihäfl the termS on the right hand side Of Equation

_£8]

= 0.

Substituting the expansions

_{£6J, £7]}

and

[8]

into

Equations.l] -

[5]

we obt'ain the following set of equations

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(a) First order

V2cp1 = 0 cp1, 0 2 -

cp1,2

2)

-

cpih

- cp1,h, = 0 (b) Second order V2cp2 = 0 = + = C]

x

-

'z

-

cp2,h, - cp2,h, 0

(y<0)

_[9]

[10] (y = 0)

=0)

(y

= h(x,z))

£12] (y < 0) [14]

£15]

[i]

(y

= h(x,z))

[17] Vcp2 = 0

(x2+y2+z2 -

₎

£18]

cpI , 1 (x2+y2+z2 -. ) [13]

Equations

[9J -

[l3 for cp1 represent the uniform potential

flow at infinity past the body of equation y = h(x,z) (y

and confined beneath a rigid wall at y = 0, which replaces the free-surfaáe.

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IftT0NAUTICS, Incorporated

ff2dxdz

=

I

A

-6-The second approximation represents a flow' geer.tedby

a distribution of sources (Equation L153)) on y = 0. The, total

flux of the sources is given, by

I

+

(i

Jdxdz [19]

the integration being carried out on y = .0 in the exterior

of the., curve C (the waterline).

Since vanishes at infinity (Equations [l1and.3J:).., th

Gauss theorem may be ap.lied to Equation[l9in order to

trans-form the integral as fpllows . .

ff2y.dxdz =

ut, Equatibns [10] and £12] show that the integrand on the, right side of Equation [20] is zero. Hence, the total flux

of the induced source is zero arid cp2 behaves at. infinity as

o (.).

The higher order terms may be obtained in a siniilr

way. The solution of the second order approcimation 'being already extremely tedious the uefu1ness of higher terms

for computational puLrposes is doubtful.

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HYDRONAUTICS, Iricorporated

IIL,

THE PRFSURE DISTRIBUTION AND THE FORCES ACTING ON' THE BODY

The dimensionless pressure has the folldwing' expansion which resUJts frotn the ernoul1i Eqiation and Equation E6

+

2

p y +

-The force on the body is obtained by the pressure inte-gration in the form

is the buoyancy, while F1 is the dynamic force acting on

a hull submerged beneath a rigid waThi at y = 0 in a uniform stream. The D'Alembert paradox holds in this case, so that

F1 is also vertical.

By successive application of Green's theorem F1 found to be equal to -. ₁ 2 S [1 -

(vcp)2

=

ffdS

- (vcp1)2] ndS + Ve Vp2) ndS +

.. =

F

+ 2F1

+ e4F2 + 22] - Cpi, )

dxdz

[23]

-II

4Vcp1 cp2 + 211

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-8-Where. A is the area exterior to the waterline in the plane

y= 0.

F2 has properties similar to ttose of F1. App lring

Gauss' theorem to a volume bounded by the body, the plane = 0* and a large hemisphere, we find

because v(vcp1 Vcp2) P and at infinity IVCP1I = 1 + o(r.2)

and _lVcp2i

= o(r).

Hence, F2 is also vertical and the

D'Alembrt paradox hold at second, as well as at higher. order. The evaluation of forces and moments. b,y the aid of the

values of Vc21 and Vcp2 on the horionta1 surface Arather than those on the hull S may be useful in the case in which and

cp2 are determined numerically or by the aid of an electrolytic

analog.

The moments acting on the hull may be found in a similar

way.

In the case of a two-dimensional flow, in the x,y plane

past a body of shape . .

* The force component resulting from the pressure on the wetted

hull s.tri between C and thèinteiectiOnbf .thèhul ãhd the

free-surface is 0(e6 ) or less.

=

1 I (vv2)

ndS

= -

ff:

cpi vcp2) dxdz .21l.] \\

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fdx

Icl(

] dz + ic4 df1 -- dz + / dz = 'yL3(c2F1 + c4F2 +

0.0)

[27]

BCD ECD BCD

[26]

Since at infinity df1/dz = -1 + o(l/z) and

df2/dz = o(l/z), the integration niay be carried out on the

two segments AB and DE, rather the curve BCD, like in the

three dimensional case.

At rest (e = 0) the body isin equilibrium under its

weight and the buoyancy force. The dynamic components of

the vertical force F1,F2.. cause a singkage Tt, _{Since, by}

definition AT'/T' is small, it may be determined approximately by the aid of the following relationship

-9-y=h(x)

_[25]

Equations [9] - [18] remain unchanged, excepting the terms

in z and which have to be deleted,

The expressions f or the force and moment may be rendered more compact by the use of complex variables. With z a com-plex variable this time, f = cp + j the complex potential and

F F - iF the complex force (Figure

3),

we easily find

d ₁ z

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-10-where A' is the area enclosed by the waterline C at

equilib-rium. Hence

L'3 'Ut2

U'4

T' - A'T' F1 + g2L'2 F2

+ 00

In the two-dimensional case Equation [27J reduces to

U'2 U!2 U'2

-+

() () F2

+ 000

Equations [281j and [29j show that although the first order vertical force e2F1 gives, in dimensional variables,

a gravity-independent component of the total force, the

sinkage T' is dependent on FrL. Since the increase in the frictional resistance is somehow proportional to T'/T', it is FrL-dependent even at first order.

The. computation of cp2 (Equations

It has been determined analytically in the case of flow pasta two-dimensional semi-infinite box (Dagan and Tulin, 1969). In the case of bodies of more complex shape, the second order

term may be found by numerical solution or analog models. In

the following sections, in which some analytical results are

presented, only the first order term c is retained0

T'

[28

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IWDRONAUTICS, 'Incorporated

-11-IV. FIRST ORDER APPROXIMATION' (EXAMPLE OF TWO-DIMENSIONAL FLOW)

If the mapping of the complex ielocity on the z plane,

w1 df1/dz is known, the cornputation of F1 (Equation[26])

and that of the moment M1, is straightforward and reduces to

an integration in the complex plane.

In order to liustrate the procedure and to determine the influence of the shallowness on sinkage, we consider a sym-metrical Rahkine body (Figure .i-) generated by a sou'ce and a

sink at a distande 2a. For convenience, we normalize the

georrietrical variables.with respect to Tt rather than L'. The complex velocity is (Kennard,

₁₉₆₇₎

W1(z) =

i +

[co.tanh' - cotanh (z+a)]

where Q,is the dimensionless strength.of the sources. The first order vertical force is (Equation26])

-.t -.,

F1 =

-

(l-w12

dz=-

_I

- 2cotanh (z-a) cotanh 7r(z+a)

fl (z -a)

cotanh

2D

ir(z+a) 2:'lr(z-a) .;rr(z+a)

- cotanh - cotanh + cotanh

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cotanh

-12-where _{= L'/2T' and D = D/T'O}

The integral of Equation

[3l

may be .carried out. in a

closed form, the result being

sinh ₂

= 2n

sinh (+a)

+ [cotan

(+a)

+ cotan

wa .'ia

1 cosh -s.- sinh - tanh

I. - cosh + sinh

'ira

tanh

The..quantities Q and a are.related to the geometrical parameters and D through the following relationships

2 'ir 'ira

1 - - arctan (tan . cotanh

cosh ₌ sinh + cosh.

D

F1, which is a suction forcë has been computed for different values of and D by the aid of Equations t32j,

33]

and

_C3+].

_{The results are.represented graphically in}

Figure k. It is worthwhile to note that for a given ratio

L'/T' the shape .of the body changes somehow when Dt/Tt varies.

Disregarding this variation, we can draw from Figure _k a

quantitative estimate of the shallowness influence.

2]J

E32'i

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l3

-For instance for a value of FrL =0.05 we. find a

relative sinkage ATt/T = 0.05 for L?/2Tt _{= 7 and} Dt/T!. _{= 2.}

The sinkage increases rapidly with thelength andthe shal

lowness.

V. FIRST ORDER APPROXIMATION (EXA1VLE OF ThREE-DIMSIONAL FLOW)

The analytical integrationof 1 -

(v)2

iong the bodr

surface (Equation [22]) or on the plane.y _{= o} (Equation 23J). is

cumbe±'ome even in the simple case of a Rankine body iac.

deep water,

For this reason we shall stüdonly the cas of a

slender. body of revolution in deep water, sufficient in order

to get orders of magnitude of sinkage and trim.

Withthe notations of Figure 5, the équatio of the body surface:is

r = ö

₃₅

where Is the.slenderness raIIL parameter. Weare. going

to expand.cp1 i a small ô perturbation series. Sitice _Pi

represents uniform flow past the body beneath arigid wall we may extend the flow by reflectiOn across y = 0. This way

(20)

slender body. The results are readily found in different books (for instance, Cole

_1968).

In the case of a smooth body, whose nose and tail are not too blunt, the pressure onthe:body is given by (Cole,

₁₉₆₈₎

Pi =*(1 -

p1,

2'.L i2)

=- 222n

2 +

O(nö)

where

d.

(x) =.'2wh

2

B1(x) =

I-j1sgn

!) _{n Ix-1d}

_E37j

Even in the frame of theslender body appoximation,

.the computation of the 2

Qrder term i tedious. For this ±'.eason we truncate the expansion of _{Pi (Equation36]) after}

the first. term, the result being of practical valüè only if

n5 >>1.

At this approximation the veitica1 force and the moment acting on the body are

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F1=-k632

L

dS1 dx Idx + O(5)

[381

where 0 < a 1.

After substituting in Equation [38 and carrying out

the integration we have

OQftlO

a(1-a)

In particular, for a symmetrical body (a =

C 1J

F1

= 53.6 on

5

M, =-k32nö

As an example we

Figure 5 with a shape

= 1 -h(x) = 1 1 dS hxdx 0(o3)

I

+ [kp] dx 1

--consider the nonsymmetrical body of

given by 2 < < (0 x 1-a) (a< x < 0) -1 1 2

(22)

The center of gravity of the area A (enclosed by the

waterline) is located at

XG = (l-2a)

The moment of the pressure component _p1 with respect to XG is therefore

52n b (l-2a)

1'

_G

-5

_a(l-a)

The moment of inertia of the area A with respect to

XG

is found from quation [1+0]

[(la)3 + a31 -' (l2a)2 _[1+6]

The trim angle e has the following value HYDRONAUTICS, Incorporated

C2M

IxTT

-16-According to Equation [28] the relative sinkage, in the latter case, would be

T' - kOFrLbn 5. 52n 5 (l-2a) gLt a (1-a) [(l_)3 ± a3] -[13] [1+5 [1+7]

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-17-VI. DISCUSSION OF RFSULTS AID CONCLUSIONS

The small Froude number perturbation expansion provides

a, convenient way to compute the, vertical force and momen,t acting on a floating body. At first order, in the rigid wal.l

approximation, the.relative sinkage and the trim are

pro-portional to FrL2. Excepting very simple shapes (like those

considered, in the preceding sections), the force has to be determined numerically. If the hull is represented, in a slender body approximation, by a.linear source distribution, the numerical effort is moderate in both.cases of deep and

shallow. water, . If a more accurate estimate is sought, the

first or'der flow problem may be easier solved by using an electrolytic tank and a small model of the actual hull. The

second order approximation may also be determined by a tom-bined analog and numerical procedure. But, for the small FrL

number encdunte,red in applications, the second order effect is

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REFERENCF

Baba, E., Study on Separation of ship Resistance Components,

Mitsubishi Tech. Bul., No.

59, p. 16, 1969.

Cole, D. J., Perturbation Methods in Applied. Mathematics, Blaisdell Pub. Co., _p.

260, 1968.

Dagan, G. and Tulin, M. P., A Study of Gravity Free-Surface

Flow Past Blunt Shi.p Bows, HYDRONATJTICS, Incorporated Technical Report

117-lI, p. 17, 1969.

Kennard, E. H., Irrotational Flow of Frictionless Fluids,

]J.T.M.B, Rep.

2299, p. kl2, 1967.

Saunders, H. E., Hydrdynamics in Ship Desigr-i, SNAME, _{Vol. 1,}

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0 AO

A

0

0 o

4.2mMODEL

A

7mMODEL

o

10 mMODEL

FIGURE 1 - THE DEPENDENCE OF THE FORM FACTOR ON THE FROUDE NUMBER FOR

THREE GEOSIMS OF A TANKER (REPRODUCED FROM BABA, 1969)

0.12 0.14 0.16 0.18 0.20 0.22 0.245

FL

FRICTIONAL (EXPT.) - FRICTIONAL (HUGHES)

K-FRICTIONAL (HUGHES) 0i7 0.6 0.5 0.4 K 0.3 0.2 0.1

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FIGURE 2 - STEADY FLOW PAST A FLOATING BODY

I I

z = x + iy

FIGURE 3 - TWO-DIMENSIONAL FLOW PAST A FLOATING BODY

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HYDRONAUTICS, INCORPORATED F1 200 100 50 10 2 4 6 8 10 12 14 16 18 20 22 24 L'/2T'

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FIGURE 5 - FLOW PAST A SLENDER BODY OF REVOLUTION (FIRST ORDER APPROXIMATION)

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UNCLASS IFIED

Security Classification

FORM

NOV6S (PAGE 1) _UNCLASSIFIED

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3. REPORT TITLE

HYDRODYNAMIC FORCF. ON FLOATING BODIES MOVING AT LOW FROUDE NUTVIB

4. DESCRIPTIVE NOTES (Type ofreport andinctosire dates)

-

- -Technical Report

5-AU THORI5I -(First name, middle initial, last name)

Dagan, G.

6. REPORT DATE

April 1970

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28

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TFCHNICAL REPORT Tb. NO. CF NEFS -5 117-15 Sâ. CONTRACT OP GRANT NO Nonr-33)49(OO) NR

062-266

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II ABSTRACT

The equations of steady free-surface flow with gravity past a

ship are expanded in a small Froude number perturbation series.

Ex-pressions for the vertical force and the moment (and the sinkage and trim, correspondingly) are derived up to second order. Two

ex-amples of flow are solved to first order: _{a two-dimensional flow} past a Rankine half-body in shallow water and the three-dimensional

flow past a slender body of revolution.

(43)

UNCLASSIFIED Security Classification

'4. _{KEY WORDS}

HydrodyramiC force Sinkage and trim Small Froude Number Perturbation Solution LINK A ROLE WT ROLE LINK B LINK C ROLE FORM 1473 (BACK) 1 NOV 6 S/N 0102-014-6800 UNCLASSIFIED.