Report lOO August 1969
DEVELOPMENT AND EVALUATION OF A SECOND-ORDER WAVE-RES I STANCE THEORY
by King Eng
Prepared for the Office of Naval Research
Department of the Navy Contract Non r 263(65) Mod. 5
This document has been approved for public release and sale; its distribution is
un-limited. Application for copies may be made to the Defense Documentation Center,
5010 Duke Street, Alexandria, Virginia 2231+.
I
\/
DAVIDSON
LABORATORY
tab.
y. Scheepsböuwkunde
R-lLOOTechnische HoeschooI
De If tDAVIDSON LABORATORY
Stevens Institute of Techno'ogy Castle Point Station
Hoboken, New Jersey 07030
Report 1400
August 1969
DEVELOPMENT AND EVALUATION OF A SECOND-ORDER
WAVE-RESISTANCE THEORY
by
King Eng
Prepared for the Office of Naval Research
Department of the Navy under
Contract Nonr 263(65) Mod. 5
(DL Project
3507/102)
This document has been approved for public release and sale; Its distribution is unlImited. Application for copies may be made to the Defense Documentation Center1 Cameron Stat in, 5010 Duke Street, Alexandria, Virginia 223l4. Re-production of the document in whole or in part is permitted for any purpose of the United Stàtes Government.
Approved
lx + 70
pages3 tables, 8 figures John P. Breslin, Director
This report is taken in part from a dissertation submitted in partial fulfillment of the require-ments for the degree of Doctor of Science in the Department of Mechanical Engineering at Stevens
R_ILIOO
ABS T RA CT
A new second-order wave-resistance theory for floating bodies is developed, and then assessed by application to a parabolic strut. The problem is treated as a potential-flow problem with a centerplane
distribution of sources to represent the body. However, the kinematical boundary condition is satisfied on the surface of the body. lt is
supposed that the beam-length ratio , t , is small and that the square of
the disturbance velocities which appeared in Bernoulli's equation on the free surface is given in terms of the components of the first-order potential. lt is also assumed that the solution of the source density,
a , Is in the form of an asymptotic series in t , and the solution for
a is obtained up to the order of t2. The wave resistance is then
computed on the basis of the improvedsource density and the correction arising from the Improved representation of the free surface.
It is found that at low Froude number the expansion scheme used by Sisov, by Maruo, by Vim, and by Eggers, cah give negative resistance if the beam-length ratio is not small enough. Therefore, a new definition is adopted; that is, the sécond-order wave resistance i.s based on the
improved disturbance potential without expansion of the amp!ltude functjon of the wave-resistance integral.. The result is a marked improvement over the result obtained from the Michefl theory, and correlates rather well with experiment. lt is also found that the second-order correction to
the kinematical boundary on the body is as important to wave resistance as the non-linear free-surface correction. This is not the case when the conclusion is based on results for submerged bodies.
KEYWORDS
1.2 Potential for a Source R-1400 CONTENTS Abstract List of Figures Nomenclature INTRODUCTION
SECTION 1 THE FREE-SURFACE BOUNDARY VALUE PROBLEM . 3
1.. 1 Statement of the Problem 3
6
VI
SECTION 2 DISTURBANOE POTENTIAL OF A PARABOLIC STRUT 14
2.1 KInematic Boundary Condition on the Body Surface 14.
2.2 Solution of the Integral Equation 16
SECTION 3 WAVE RESISTANCE ... ... . . 21
3.1
DefinitIon of Second-Order Theory 213.2
Wave Resistance of a Parabolic Strut...25
SECTION 4 EXPERIMENTS 30
4.1
Description of Experiments . . . ....30
4.2
Model Characteristics and Test Conditions 314.3
Results . . . 32SECTION 5 DISCUSSION OF RESULTS AND CONCLUSIONS
...33
APPENDIX A. Source Density Due to the Effects of Froude Number. . . 34 APPENDIX B. Amplitude Function Derived From the Correction
on the Free Srface . . .. 39
APPENDIX C. Amplitude Functions Due to the Froude-Number-Dependent
Sources 47
APPENDIX D. Second-Order Amplitude Function
(Zero-Froude-Number Correction)
...55
REFERENCES 57
TABLES (l-3) 59-61
R-lL+OO
LIST OF FIGURES
1. Ratio of Second- to First-Order Wave Resistancé ofa
Parabolic Strùt According to the Expansion Procedure 62
2. Wave Resistance of a Parabolic Strut Versus Froude Number 63
3. Parabolic-Strut Model Used in Experiment 613
14A. Mounting of Drag Balance. in Midsection òf Strut (Photo) 65 LIB. Experimental Setup (Photo) . .
. 65
5A. F Ó.259;. Draft, D = 7.5 Inches. (Photo) . . 66
5B. F.=O.259;Draft, D =12.0 Inches (Photo) . 66'
6.
Data Obtained From Drag Balances .677. Resistance. Coefficient of a. Parabolic Strut 68
8. Total Drag Coefficient of an 800-FootParabolic Strut 69
9. Effective Horsepower Requirement Per Ton of Displacenent
'b' ' (L,w) (L,W) J (x) k o R-1400 NOMEN CLATURE a k sin e C e1
D draft of top section of strut (Fig. 3)
F urn
. 2gL
g gravitational acceleration
H harmonic part of the Green's function
(Eq. [1.26))
h draft of the body
si-1
Second-order amplitude function of wave
resistance integrai due to zero Froude number, to free surface, and to finite Froude-number correct ions.
Bessel function of first kind, n' order
k radius in cylindrical coordinate
(dumy variable)
g , wave number
i 2
L finite-Froude-nuinber source density (local part,
Eq. [2.151)
L semi-length of strut
m density of a single source
P,Q amplitude function of the wave-resistance integral
u, y, w
Ui, Vi, Wi
R-1400
S/u v2+ w2 , magnitude of disturbance velocity
wave resistance coefficient normalized by
2,2
2pU
x'wave résistance
distànces (Eq
[1.251)
flrst- and second-order wave-resistance coefficient
beam-length ratio of strut
(x, y, z) components of disturbance velocity (subscript 1 denotes first approximation)
x,y,z Cartesian coordinates (subscript i denotes
x1,
, ziÇ the Fourier transformed space)Um free-stream velocity
w finite-Froude-number source density (radiation
part, Eq. [2.16])
equation, of strut offset y0
equation of the frée surface
tan' ()
tant
(1+X)-
tan [t(1-x)Jt
sec2 sin 9t
sec291 sin 8i 0.577215.7 (Euler's constant) tan1 [t('i+x)] R0, R1, R2 R1, R2U
2FX
Ksec29
X1 t sec2 8
sec 8 tsèc 6
specific gravity of fluid sóurce density (distributed) total velocity potential
w
R-11+OO
Dirac delta function small parameter
Cartesian coordinates (durmy variables) coordinates of a point source
angular variables of cylindrical coordinates (wave angles)
disturbance potential (1 , 2 denote first and second approximation)
prescribed function on the free surface (the right-hand side of Eq. (1.11])
(x-') cos 8 + (y-11) sin e
(x-')
cos 8 - (y-11) sin e(l+x)
e+
e (l+x) cos 9 - y sin e(l-x)
cose + y0
sin 8 (l-x) 8 - y0 sin 8 cp, cp1, 4r(x,y)R-lLOO
INTRODUCTI ON
The resistance experienced by a floating body is composed of a viscous and a wavé-making component. According to the commonly accepted Froude's hypothesis, the viscous drag is considered to be a function of Reynolds number only, and the wave resistance Is considered to be solely dependent on Froude number. This hypothesis assumes that the viscous effects and the wave-iñduced effects do not interact and therefore permits us to study. each component independently. Thewave-making component Is the subject of this study.
lt was In 1898 that Mlchell1 gave the first approximate solution of the problem of wave resistance for a thin ship-shaped body moving on the surface of a perfect fluid of Infinite extent. This formula successfully depicts the nature of the variation of wave resistance with Froude
nUmber, but the result appears to be in error when compared with
experimental evidence. Many attempts have been made to Improve Michell's formula by some rational: approximate corrections.21 However, the
success of these undertakings has been very limited. In an effort to correct the situátion, theorists in the field have developed a higher-order wave theory in terms of a systematic expansion in small parameters such as the beam-length ratio of the body.'2'13 The first-order term of the expansion is identical with MichelPs formula, and the higher-order terms enable us to improve the boundary condition on the body and on the free surface. This technique, which was introduced by Peters and
Stoker,12 can give a successive approximation up to any order. However, the second-order term becomes so complicated that numerical computations seem scarcely possible.
The treatment of wave-making theory for submerged bodies such as two-dimensional symmetrical airfoils, cylinders, spheres, and spheroids
Is relatively simpler. By using the linearized free-surface condition and an image method, Havelock, in 1928,1 gave a second approximation for a submerged cylinder and, in his later 1936 paper,15 was able to construct
RlLeOO
a complete formal solution to the problem up to any order. Again by using the linearized free-surface condition, Bessho16 presented a higher-order solution for a submerged cylinder and spheroid. Tuck, in 1965, showed the work of Havelock and Bessho to be inconsistent and proved that the most important contribution to the second order is the free-surface non-linearity which they had neglected In their analyses. Tuck's
conclusions were strengthened by Salvesen's results in the case of the submerged two-dimensional foil. 18
Sisov, in 1961,
formulated a second-order wave-resistance theory for a thin ship,19 as did Maruo, in 1966, In a more general way.2° Vim, in1966, treated the problem of higher-order wave theory in slender ships,21 and Eggers, in 1966, made another approach to the same problem.22 At
present no definite conclusion can be reached as to the merit of the works cited above, because of the difficu'ty involved in numerical evaluation.
The present study Is concerned with the assessment of second-order theory in predicting the wave resistance of a body. Specifically, the analytical work is concerned with finding a disturbance potential of a body up to the second order In beam-length ratio, and determining corresponding wave resistance. The analytical result is compared with the experimentally determlnèd wave resistance instead of with the usual
residuary. The procedure is applied to a strut-like model with a syrwnetrical parabolic cross-section. The wall-image approximation Is used to calculate the free-surface correction to wave resistance, so that the numerical work can be made more tractable. For most surface ships, the operating Froude number is small, and in these cases such an
R-1400
Section 1
THE FREE-SURFACE BOUNDARY VALUE PROBLEM
1.1 STATEMENT OF THE PROBLEM
Consider a body which is at rest in a semi-infinite fluid through which a stream of constant velocity U in the negative x-direction is flowing (see sketch below). Throughout the discussion which follows, a right-hand Cartesián coordinate system with the z-axis positive upward and the origin at the undisturbed level of the free surface at the center section of the body is used. The equation of the body surface and that of the free surface are denoted, respectively, by
and
The fluid Is assumed inviscid and incompressible and the flow is considered Irrotational.
R- 11OO
Under the assumed conditions there exists a velocity potential and a disturbance potential p related by
=
cf(,,'r)
+.:1L
(1.3)
u
which satisfies continuity in the form of Laplace equations
= o
i
and expresses theve!ocity. ata point of the fluid In the forjn
'
:
where: u,v,wareothponents o.fthe:disturbance velocity..
The boundary condítiór Ì t,hÎchthere is no'f 1owacross the surfaces
y = y(xz)
and z = Z(x,y) yieldsand
o
The Bernoulli equation in the. present cale ¡s
ifD +-[('r+
Z)+
2:
(1.8)
I
(1.9)
and the condition p = Oat the free surfce yields
= tWoi4.
we then obtain
or
Differentiating Eq. (1.9) with respect to x and noting that
(j
t (U\
+
--
t\)
Z j) - C3R-1400
Similarly differentiating Eq. (1.9) with respect to y , we get the
result
Eliminating Z from Eqs. (1.7), (1.9a), and (1.9b), we can write
the exact boundary condition at the free surface as
uo +)
.$z
+
J)
Thus, we are required to find a solution of Laplace's equation which satisfies both the linear equation (1.6) on the body surface and the hon-linear equation (1.10) on the unknown free surface.
The free-surface condition (1.10) is inconvenient, not only because
it is to be applied on a surface of unknown location, but also because
It is non-linear, so that the superposition technique is not applicable. The condition may be transferred to the plane z O by means of a Taylor". expansion, such as
('x1)
l;iL(,o) + i:;
+
( .9a) (l.9b)(,o) +
T(','',o)
Z!o)+
.
etc.R
ikoo
which, substituted into (1.10) with (1.9) applied, gives terms up to the second order In disturbance velocity:
¿1.11)
Equation (1.11) is still non-linear. However, If we employ the method of successive approximation, it is possible. to obtain a solution for the system. Let cp1 denote a solution which satisfies (1.11), with the
right-hand side equal to zero; that Is,
Then, letting the, right-hand side of (1.11) be a fuñction of cp1 , one
obtains, to the second order,
(4!L
À1Li4
\b)
- iJ ò . b'ØaI5 (4.
This procedure, which was suggested by Professor. L.. landweber, can be continued to obtain linear approximations to (1.10) of higher order.
1.2 POTENTIAL FORA SOURCE
Let (x,yz) be a right-handed coordinate system as shown in the sketch on page .3. iet a source of strength m be located at the point
P(11,C)
in the fluid(<o)
, and let the right-hand side of Eq. (1.11) be a prescribed function i(x,y) ,,on the surface z = O Suppose the coordinate system, the source point, ad ,(x,y) are moving(1.13)
as
=
V'b
-r ('ì.306where cp Is a harmonic function everywhere in the fluid region except
at P , and behaves like mIR In the neighborhood of P (R Is the
distance from P). By virtue of the properties of the Dirac delta function, the disturbance potential p satisfies the equation
V24
-
4)
Th-')
-ì
(i.c
This is to say that cp Is harmonic everywhere but Indeterminate at the
point P(0,110,C0) in the fluid. The condition at the free surface is of the form
R-l+00
(&\
-)'.)
The other conditions for p areo
- o'rrt_.4) ,w
(;-)
, -» o
(1.17)Although the solution of this problem is known,13 It seems desirable to exhibit the derivation here to clarify the specific form of the complete solution that is desired In this analysis.
The Fourier transform method Is chosen to solve the boundary value problem as posed by Eqs. (1.14) to (1.17). Reference 23 states that If F* (x1) Is the Fourier transform of F(x), then the following relation Is true:
F
< > F()=
(1.15)
If the Fourier transforms of p and i are denoted by
t4)
(2.. .**
t-')c=
R-1400=
(
,
)
e.'
&
i
ç'P
(' ')
) e..
c frL1+,)
=__
then the transforms of Eqs. (1.14) and (1.15) with respect to x and y can be written as
-zm (-L)
e00)
(1.18)=
4
(tc,,1)
(1.19)cr4=o
(1.20)R-11+0O
becomes
4
j;)
The inverse transform of with respect to z1 , cp , represents
the particular solution of Eq. (1.18):.
($(,. ;L
))
=
¿(+ Q;;)
d
yru
(1vL).... t_+'?)
(1.21)
The homogeneous solution of Eq. (1.18) is of the form
=
Equation (1.20) requires that B(x1,y1) = O , because z ¡s always
negative. Hence, the general solution of Eq. (1.18) is
7
e'
I.ç0\j:)
1.22)
where A(x1,y1) ¡sto be determined fromthe boundary condition (1.19). Since Ç<0, then in the neighborhood of z = O ,
-= -o>o
R_1LOO
q4
+
L**i .\
.4
S where=
The Inverse Fourier transform of xi,yi,z) with respect to x1 and yj. will yie1d cp(x,y,z) of the system of Eqs. (i.l) tó (1.16).
1
-T'
111) d
'4L
Changing the variables
x1,y
by letting1L
,(-L))}
dj
and noting that thé integral representation of
hR
Is of the form/
--if
#(Xcose4Yne))
(-'i
dd
we can write R-1400
=
(-&
+
+
--
;\
o+
+
f
=
R
1 (-- ('e-1
+ (
whe re(1.25)
Ç, k:)
=
-i-J
Íek{)_
(%-(oSO #(1ne)
&ce
.tÓ
When integrating with respect to k from k = O to k = in the function H , we must go around the singular point of k = k0 sec2 8
Since H
is a harmonic function, it is necessary to interpret the24
integral properly. Havelock suggested a method of artificial viscosity which interprets H as follows:
I.
Ltj7Ll
i
.dkde
()O
To fix the singular point, we can fix the sign of the 8 integration by
TI 2
reducing the range of integration from
8=0
to 8=jd
-\
L
(e..4-a
.'+
o(k
ei)
G (1.26)k-
- (sce))j
W= (-)(Qs- ('-)'ne
= (- (oG (-) 6:flQ
>'
0ec
It can be shown that H decays exponentially in x far upstream
(x
> > o)
of the source point, and behaves ltke/
-
_j
-\)cbsO)
os-i4.e.
Q
far downstream (x < < O) . Thus, the asymptotic behavior of q far
upstream is
and far downstream it is
-im1)2..
>(hSQ'
b(-cose)
c4-t0' n)e.
î-t
1'
0(i-) C.os)
COS )¿0
(1.27)
(1.28)
With H defined by Eq. (1.26), the disturbance potential Cp given by Eq. (1.25) satIsfies (1.17) automatically, and therefore it is the solution of the boundary value problem which is posed by Eqs. (1.111) to
(1.17). An interesting feature of this solution is that it shows hi the non-linear free-surface correction (x,y) enters the problem. It
Is as though a distribution of sources with an associated strength were distributed on the surface z = O . Note that t(x,y)
is an approximation of the right-hand side of Eq. (1.11), and that no additional constraint is placed on this function in the derivation of cp
R-lkOO
The term mIR1 In Eq. (1.25) is a potential solution ofa point source In infinite fluid. The quantity (mIR1 + mIR2) Is the solution of the problem If the free surface Is replaced by a wall (I.e., cp2 = at z = O); thus the term m/R2 is comonly known as the wall Image. Equation (1.27) indicates that no gravity wave Is present far ahead of the disturbance, whereas Eq. (1.28) shows that gravity waves are present behind the disturbance. Equation (1.28) is usually identified as the free-wave part of the disturbance potential.
R_1LOO
SectIon 2
DISTURBANCE POTENTIAL OF A PARABOLIC STRUT
The main concern of this section is to determiné a centerplane distribution of sources as a disturbance potential for a parabolic strut. However, the kinematical boundary condition is satisfied on the surface of the body, rather than at the centerplane as is true In usual thin-ship theory. It is supposed that the thickness-to-length ratio, t , is
small, and.that the right-hand side of Eq. (1.11) is given by the first-order disturbance potential. It is also assumed that the solution of
the source density , , ¡s In the form of an asymptotic series in t,2°
and the aim of the present analysis is to obtain up to the order t2.
Throughout the discussion, all the length dimensions are normalized by the semi-length ,
. ; the source density and all the velocity dimensions
are normalized by the free-stream velocity, U
2.1 KINEMATIC BOUNDARY CONDITION ON ThE BODY SURFACE
From Section 1.2, the disturbance potential for a distribution of sources on the plané y = 0, -1 < x < , - < z < O , moving at a
constant speed below the surface z = O on which the right-hand side of Eq. (1.11) ,
4i(x,y)
, is prescribed, is given by the expression=
-k
;+
where where
R0
Jj ('-+
+(-=
j (-+
(#?
k (;
7c--,°
It has been shown in Section 1.2 that cp satisfies the conditions
c+Iktó
=*Ç1)
oFor the solution of the complete boundary value problem as posed in Section 1.1, one condition remains to be satisfied. This ¡s the "no-flow
through" boundary condition, which is given by Eq. (1.6). The specialized form of (1.6) for a parabolic strut is
[_
=
R-lkOOKe
(2.3) ..4J ok +
se.e)
L.dO(4') k
'czl:
(2 2a) .(2.2b) (2.2c) (2. k) (2.5) (2.6) (2.7)where
= t-t (,ç
O M.)EquatIon (2.8) Is a Fredholm Integral equation of the first kind in which the source density , a , is the unknown. The homogeneous
solution is usually taken to be Identically zero for thin bodies and this solution Is adopted In the present analysis.
2.2 SOLUTION OF THE INTEGRAL EQUATION
The function
4(,11)
Is of the order t2 , because it is composedof the square of the disturbance velocities, and for bodies with a y-plane of symetry, it Is an even function of 11 ; that ¡s,
R-11+00.
Introduction of the disturbance potential p , given by Eq. (2.1), to the kinematical boundary condition, given by Eq. (2.6), yields the expression
lì.
=
4-
iç%stad
The terms of order t2 in the last integral of Eq. (2.8) are
(2.9)
+
?J6
t4(+(i1
R-l'400
4*3+
(3))
(2. 10)But the. quantity
l.#V-'3
)=Q
is an odd function of 11 ; hence, Eq. (2.10) ls identically zero. The
remaining terms of the last integral in Eq. (2.8) are of the order t3,
and will be omitted since we are only Interested in obtaining the solution to the order t2 . Equation (2.8) reduces to
a- = (2. 12)
Equation (2.12) is the well-k.nown.thin-body approximation.
O(t3)
(2.11)It should be noted that the kinematical boundary condition on the body surface is independent of the free-surface correction up to the second order.
Equation (2.7) shows that both y0(x) and
y(x)
are of order tUsing an iteration technique, we find that the solution of a up to the order t (i.e., a= ta1) is
R-l#OO
Replace a in Eq. (2.li) by [_2y(x) + t2a2(x,z;1t)] it in the following manner:
-J4
t'-od It--
:
'(t)
dçd)
Ic4ijjI%
I)
Ici
tt can be shown that the last integral of (2.13) is of the order t3
Keeping terms up to the order t2 we can write Eq. (2.13) as
=
and write /I{O1b(
()-()\
-
I
ShÇ0 -o1u
Thus, the solution of Eq. (2.8) for the source density a of a thin strut up to the order t Is
'3_,= tG7+ taj
(2.14)where
=
(2.15)Çç;t»
0(t3)
R-lkOO
___
+
):)
..Jto
/
VÇi:c.
For the special case of a parabolic strut , ci and 2 are of the form
(2.17)
c.oste 51(&)+
Zco s (ta)
+ -
(.k)
6-Q
+
k)
w
=
:p
(tf)) O(COS -)-
Z%+
Cos where Q..te
¡JA
cete
>.k.ec3G..
2.16)
(2.18)
(2. 19) (2.20)(2.21)
whereJ= -+
(i.3(11
))+L
+
w
L+W=
R-1400
For the purpose of numerical analysis, the function .' sin Bt(l-x2)
Is approximated as follows:
+ (2.22)
where J0 and J1 are Bessel functions of zero and first order, respectively. The right-hand side of Eq. (2.22) behaves like (l-x2) where t Is small, and it becomes a bounded oscillation when t
becomes large. This substitution particularly suits the need of Eq. (2.21), because the main contribution to W is at the range of 8 where (2.22) gives the best approximation (i.e. , t is small); and It has
the advantage of uncoupling the 8 and x variables. Inserting (2.22) into (2.21),.we obtain W in the form
:i: a[;ft
(4) '' (.)
4 z os ('-)
(4)
(_z4s
-Z%
J --.CO5
*))}
(R-lkOO
Sectioñ 3 WAVE RESISTANCE
3.1
DEFINITION OF SECOND-ORDER THEORYThe wave resistance derived from an analysis of momenta at control surfaces taken at large distances from the body is given by the
equa t on13
where Z is defined by the pressure condition on the free surface
(Eq. 1.9), and Cp is the disturbance potentiaL The asymptotic
behavior of cp Is given by Eq. (1.28), which can be written as
(O
LjJQ
(*-) c.us(4(ii)
6tG ¿:;)
do
--
*1
JL
-))
CbS
e
¿a
(3.2)At large distance downstream (i.e. , x
« o)
, only the square terms ofthe disturbance velocities will give a non-zero contribution to the wave resistance.2° Hence, Eq. (3.1) is reduced to
RJf(
(-
()Z]
ralo -.
Í((5#
7..-
t'
-R-1LOO11-.
L
11cr-n-)
+')
4-A.IÌ
));
0jJi
rt(-s
dch
For symmetrical bodies with centerplane distribution, 2 and P1
(3.3)
Introducing Eq. (3.2) to the above integral and making use of the Fourier double-integral theorem, we have the following final wave-resistance. express Ion :15
(++#i)
sec
de
(3.4) where cÓ(-ç
(t
ds
c4 -4(.1l) coS(..)
e
dclrk
where
The definition of second-order wave resistance used In References 19-22 is obtained by making use of the expansion
o
"P4. t.(=
1i(cr,';k)#))
e
j1
R_1L#OOa by Um , and all lengths by the semi-length L , R in. Its
non-dimensional form can be written as
R =
(?+
)
G
0
(3.5)JJ)
os
(ß1)od
. . . (3.6) a = ta1 + t3a2 + (3.7a) ta,2 + (3. 7b) ptp1 + tp +
(3.7c) Q = tQ1 + t2Q +. (3.7d) RtR1+ t3%
+ (3.7e) where (3.8R-11400
The first-order wave resistance , R1 , Is positive definite, but the
second order , R2 , can be negative. Unless t can be made arbitrarily
small, then at some Froude numbers the.second-order correction (I.e., R = t2R1 + t3R2) will make R negative (see Fig. I and Table l).
If we examine the procedure more closely, we find there ¡s no assurance that the third, or higher, order can eliminate the possibility of R's becoming negative. Although the procedure is mathematically consistent,
It Is unrealistic. Let us state the definition of second-order wave resistance adopted ¡n the present analysis.
j+-t+ (Q-tQ) cose1çj
(3.10)o
where=
k)
¿
(3.11) Lo s(4L.)1«
14 L
=
G(Ç
;k) e.
¿. ¿7
+
JE
e.
dd
04
This, however, retains terms up to t4, but it is. correct, since wave drag Is a second-order force.
R-1400
3.2
WAVE RESISTANCE OF A PARABOLIC STRUTOn this section, the aim Is to.developthe necessary formulas for computing the second-order wave resistance of a parabolic strut according to Eq. (3.10).
The source densities a. and a2 are given by Eqs.
(2.17)
and (2.18), respectively. The quantity In needof further discussion Is the free-surface correction , i(x,y). From Section 1 ; $(x,y) Is defined as__
-((+
(4+
(t)
(3.13)
where p. Is the first-order disturbance potential and is given by the
expression . .
-I-ø
t ,ù=
d1
kk
ö;ç (3.1's).To compute the wave resistance, we have to take the Fourier transform of the function x,y) . Because of the large amount of computer time
required, for such a calculation, the wall-image approximation Is used to estimate this contribution. For most surface ships,. the operatIng Froude number is small, and in these cases such an approximation is reasonably valid. For the present case, the wall-image velocity potential is defined by the expression
,,
' (3.15)
=
R11+OO
Substituting Eqs. (3.16) through (3.18) into (3.13), we can write $(x,y) as
I
1f_
7t
4
Applying the expansion given in Eq. (3.7b) , 4' (x,y) Is defined as
tç') =
j
-ii.
r'
It should be noted that $2 (x,y) is defined only when (x,y) is
outside the body.
The first-order amplitude function P1 is zero because of symmetry, and Qi is given by
=
(3.21) (3.16) (3.17) (3.18) (3.19) (3.20)Jw
Ji--flÇ
fl(b)-b))ir )U
(P-û R- I '400Let (P2+ ¡Q2) of Eq. (3.12) be of the form
ç.QL;)= (ib+1w*IL+Ir)#443)
(3.22)
From Appendices B to D, the quantities
1b 1L and are identically
zero, and the remaining quantities are given as follows:
b=
3:. +
c1O7L)ICOS())
(z5
35
44q4.
golz\
,(\
___
-
L
+
V=
11j
{ t
+
A .4
(A+
b)--:ì
F1(-)
(+)
J+
.ces
(bJ= Cos() {
where(__(
\
(L_Cz) s-
C4 (i+
1g-î)
-
({-+
L +
#b3(!-e-)
/
(z-T)c
4
4
C + 1J
-
+
i Z t*cN
+
/
ZC C\
_z(:
(-eJ
4y)
et-F C (i- c) 1))
+
_-
(i-2.e).
-R-lkOO -C;(3z)
- .ft
+ sn (
s:r () +')
+
¡it
b -
+
4ì()
+
- o.ozzz
b3=
= o.c4tzt-j
The term
(Ib+IJb)
can be identified as the contribution to the amplitude function from the zero-Froude-number correction to the kinematical boundary on the body. Correspondingly,[(IL+Iw)
+is from the finite Froude number and
(I+iJ)
is from the free-surface correction. The wave resistance versus Froude numbers for threedifferent sets of theoretical results is shown in Fig. 2 and tabulated in Table 2 (the sets are the first order, the second order, and the second order without free-surface correction Eiie. , I +iJ is omitted in
p p R_lLOO
A=
-
ze.os(4)
ti -l-'--\zii\) t
F(') =n.() - (o()
---Cos() . the calculation]). z 1).T
-
J_J4 '-4;::c
tft43
u,=
V-
-ossde
owhich can be written as
°°
-where K and K1 are the zero and first-order modified Bessel functions
of the second kind, and D is the draft of the top section. Hence at
sufficient depth of submergence and low enough Froude number, the drag of the center portion is predominantly due to the effect of viscosity (i e
R_lLOO
Section L EX PERI MENTS
DESCRIPTION OF EXPERIMENTS
The model used in the experiment was a parabolic strut whose principal dimensIons are shown in Fig. 3. The strut consists of three separate sections assembled so that the drag of the surface-piercing portion and thé center portion can be measured independently. This setup is necessary becausé the méjor contribution to the wave resistance comes from the portion of the strut that is closest to the free surface. For the purpose òf model design, a rough estimate of. the wave resistance of the center ection, in percentage of the total, is
-ft
DI
-;t O
R-11400
of the upper part, and establishing the viscous drag coefficient from the center section, we can determine the wave resistance of the upper section under the assumption that the viscous effects and the wave-induced effects do not interact.
Li.2 MODEL CHARACTERISTIcS AND TEST CONDITIONS
The model consists of three fiberglass parabolic cells with
aluminum ribs Inside each cell, at about one-foot intervals, to prevent the cell from warping and the water from sloshing (see Fig.
M).
A drag balance was installed on the top section and in the center section of the strut (Fig. kA shows the balance mounted in the center section; Fig. 3 shows the 0.12-inch gap between sections). The cells were flooded with water to provide the weight needed to keep the strut submerged in water. A Hama-type boundary-layer stimulator, which measures O.O'42-inch In height, was placed k Inches from the leadin.g edge to stimulate turbulence. Figure 4B shows the experimental setup, with the modelrestrained in trim and heave but free to roll.. The free-to-roll provisi:on Is designed to prevent the model from developing any side forces and roll
moments due to model misalignment, which could cause damage to the top drag balanceañd the model itself. The roll axis is inclined 15 degrees forward. This makes the model stable in roll wheñ it is towed in the
forward direction (see Figs. 5A and
5B).
For all runs, the roll angle measured by the accelerometer wasless than0.5
degrees, which was considered an acceptable model misalignment for the experiment. Thestrut was tested at two different drafts (of the top sectIon), 7.5 and 12 inches. Figures 5A and 5B show the strut béing towed In Tank 3 at the Davidson Laboratory.
The span of the center section is 1,5 inches and the measured resist-ance of the section includes the resistance of two gaps. For the case of
the 7.5-inch draft, the top section has a span of 7.5 inches and the measured resistance includes only one gap. Therefore, if we normalize.
the measured resistances of the two sect ions by, their respective wetted areas, and assume that all the gap résIstances are equal, we fiAd that.,
R-1400
when we take the difference ¡n drag coefficient of the two sections, the gap contributlonis automatically eliminated.
4.3 RESULTS
The resistance measurements from the two balances for two different drafts of the. strut are shown in Fig. 6 and tabulated h, Table 3. The corresponding drag coefficients normalized by their respective wetted areas, (the area between the gaps Is excluded) are shown in FIg. 7. Each balance Is accurate to within ±0.005 lb, and all measurements can be repeated to within the same range. Figure 7 shows that the free-surface effect begins to reach the midsection at Froude number above 0.26 for
7.5
inches of midsection submergence. At 12 Inches of submergence, this effect decreases; considerably, but it persists. However, with the aid of the ITTC and Schoenherr friction curves, data from the center balance are sufficient to establis.h the viscous-drag coefficient for the strut..When this newly established viscous drag is eliminated from the top balance, the remainder is the uncoupled wave-making component, under Froude's hypothesis. The wave résistance which appears above Froude number 0.26 , in the center balance at 7.5 inches of submergence, can be
obtained in a similar way. The sum of the wave-making components of the two sections is taken to be the wave resistance of the infinite strut; and the correspon:ding coefficients based on the wetted area of the top section are. shown at the bottom of Fig. 7.
FIgure 8 shows the total drag coefficient for an 800-foot parabolic ship at standard condition, and Fig. 9 gives the corresponding effective horsepower, requirement. The form drag appearing In Fig. 8 is obtainèd by eliminating the contributions from the gap, the turbulent stimulator, and the Schoenherr friction at test condition from the viscous drag in Fig. 7. The resistance of the Hama stimulator Is given in Reference 26, and the contri bution from the. gap is assumed equal to the difference between the Schoenherr and the ITTC friction.
R- 11+00
Section 5
DISCUSSION OF RESULTS AND coNcLusioNs
According to the expansion procedure that was used In References 19-22 (see p.23), the second-order theory can give negative wave
resistance. This is revealed by the result of the parabolic strut shown In FIg. 1. Taking a strut with a beam-length ratio of 0.1 as an example.,
the wave resistance , R =t2R1 + t3R2 , becomes negative at Froude
numbers 0.205 < F < 0.215 . This is physically unrealistic. Therefore
a new definition Is adopted in the present analysis; that is, the second-order wave resistance. Is based on the improved disturbance potential without expansion of the amplitude function ¡n the wave-resistance integrai (see p. 21+). The results are shown in Fig. 2
together with the results from application of the Michell theory and the data from tank testing. A remarkable différence between first- and
second-order theoretical resistance is revealed by the figure. Depending on the Froude number at which thè comparison is taken, the ratio of the
first to the second order can be as high as 3.63 at Froude number 0.21. The intermediate result of the second order (without free-surface
correction) indicates that the correction to the kinematical boundary on the body is as important as the non-linear free-surface correction. This conclusion is contrary to the one which was based on the results for submerged. bodies)'18
Although the second-order resistance in genéral is still higher,it corresponds rather well tothé data obtained from tank,testiñg. Such correlation is not surprising, because the effect of the boundary layer near the body surface dampéns the waves generated by the body. This Is particularly true for the stern wave system. In the Froude number range 0.27 F 0.33 , the free-surface correction becomes
insignificant and the agreement with experiment lessens. This may be due to the wall-image approximation used in computing the disturbance velocity on the free surface. . . .
R-1400
Appendix A
SOURCE DENSITY DUE. TO.THE.EFFEcTS OF FROUDE NUMBER
The portion of the source density which depends. on Froude nùmber according to Eq. (2.19) Is
where H is defined ¡n Section 2 ár,d can be written as
1h1r4+
î:
¿+
k ()-tsecQ)
The limiting process is to be taken after the reva1 .of the singularities. The singularities which may lie on the first or the fourth quadrant
depending on the sign of w. and i will consequently require a
different contour fór evá.luation of their corresponding residues. Since the procedure Is straightforward, the detall is omitted in this appendix.
Ça I
. (x-i) > w > O
and a> O
N=
do
¿ de +
e
e)
¿i.
(A .2).
LW
=
j:
and
R-1OO
Case 2:
(x-i) < O .-'-
< O and < O.L, ) e
d'dG
A..UQe) e
¿
e)
c K
de
where ,
J s4
#ç'}
os(kCs+j
(-)
C.oS +'ii, se
('c.oe
-Note that A(k,e) = A(k,-e) , and that iii and 2 can be written in
a common form Ì which is the same expression as Eq. (A.2) In another form. That is,
=
tJÍJA()
de
+JA(k1e)
kdG
-j
de
H
(A .3)
Denoting
and recognizing that
Jcos()ds
=
tk(k)
i
R-lkOO
;;L
3:Ç7
d
p
d
TL) (J kcoso
Jkose
_____
&+
{
c.cs(k(#>)
dk
-
e.
)--I
(A.k)
We have a closed solution for the last integral of (A.k), and we now integrate the remaining integral by parts, obtaining the final expression for in the form
f
+
(&+kcos)
)
oI (t4koso)
2kd+[(icos)_Fe
)
-it-
4Cos()
+
¿G -4.jos
(x
6(ì
áe
where
F
- --
kc.os (k) +
(q))
;
o = J. S?SZ0(
4
-_( «-
(A .5)whe re
:i5=
_t)ose_Lse)
,.
:::À.
((ose)-áde
t
. JL z_Q&#ìcos
V?&O;,)OEec1G
-Iii
dg
-+.jjj4kos) v:A:
R-1OO .j
(.f4f)
+ (°s
+
,1-\
4+f ic)
£&10
¿ULd)dK
ed
(A. 6) (A. 7)j(4Qe
de)
k
jci_k:e
ZóSO
dojdk
ed 4
¿de
R_lLOO
For small t , Eq. (A.8) can be written
1T
-JÍ(i-)
os6 s
(&) z%se
(&&°
(-ke'
dkcSe
4OCtl)
{srre
c..os#-
rn.
)O6G
+ ii'-)
w
=
4{te
(tos
-))--Z)
-
- QOSL+V=
(*)
tt
\-(A.8)
u yo=
j(z
-
(-) Cos
-o
Appendix B
AMPLITUDE FUNCTION DERIVED FROM THE CORRECTIONON THE FREE SURFACE
Definition of Terms In Appendix B
=
-
j
(i-fl
;4 (z4))
R-JOO
X3
= J+2
d
X4
( Cos1(t
39(._\)
Qos(»1(_)')
K3
4Z
DeinItion of Terms in Appendix B (Cont'd)
(z
4(
(z
s (
i:
R-1400
t)
s() ¿d%
R-l00
14
tr)
B. )where $2(x,y) is defined by Eq. (3.20), whIch, when introduced in the above equation, yields the expression
_T'.1.j
''
\ßs(.it)
¿)(
d?(kT
j J1_1
Ç)14)_i.4)
+
4.>'L
c.os(e'c1«
JO& Ò)1j
'SU-t(8.2)
the last Integral of Eq. (B.2) is omitted because it is of the order t
integrating the reñaining integral by parts with respect to x , and.
then interchanging the order of integration to evaluate the y Integral,
gives the result . .
_____
____
_ß1bC-t
¿1;h1)
The real part of Eq. (B.3)
,
I , Is identically zero, and the
imaginary part , , is given by the expression
whe re o b? R_lLÖo
tI
k-'c.os
Có51(-
(B 6)
Replace x by Ex +
(l-I-i)]
, then by (l-2) , in Equation (B.5);and replace x by
Ex -
(l-)]
, then by (2g-l) , in (B.6). Thenq(1)
and q(2) assume the following form:
cÇ
pI
J(+z
ncos(K+1
o
Consider the integral
jj(+) i& (
os (s+o)
dd
û It can be written as A1 jI (8.5) (B. 7 ))
(zO (zj- )
n jcsj1(-))
(B.8)'I=
(K1-çT(
È'1=
Ê¡'t
k\t
(t4Zb)\ ('h)
4)e"
chc
o R-lkOOrA2
= >
() co-)
(#-)
&d
K5<5')- A1
(t
A=
5n
(
5(ß4&
¿
+ ,
(Ç
(
os(1(-
dc
u1(Ì1- 4)-A(Z4
'Q +
(ç+ tç-L +
-143g;)
*
42)
)i
(:.) C.G5(J(#) e
+
(Ç+ KçZ4)
A
= Li1 (1+
-K) +
..1 R- 1k 00 (' (*4
(4
(;\-iP;) + (A-z,Ç)
=
where- -'i-
(K4# K
'B= ti1(K-(3') +A(K-K + .t)1p1(K3+\Ç)
):'.1% ¡B=
tJ1(LK
+
(2
- u
(i+
RI
-
- ,
(-))i
155= Z
( K-zi)
--.
(tJ) K,
=
4s
cos())+
-J% cos(1')
oi
(?-p) si.(') .k.)
ßÇd'
- z cos
- z(31 s
= -.
5i{{-z(i\O)+()}
+
+(i-.)(«1)+ï)}
where
y = 0.5772157
(Euler's constant)
'J
Jeos(.i)
R-11400
co s (i)) -
sì n.(ì)(i-_))
¿os (i.ij (z- i)
()
(i-44+
os(P)
-
(-i))
cos()
-+
o(4('
.)
Sur\.(t.)D3
-
(i4)1s()
-134=
u(i+
'. sn(Lç
-
--Qos(j. t-2n ed
(z.-t) S
(-I) QQS
For integration, the function xn(x) may be fitted by another curve with equal area ànd the same end conditions; that Is,
-xLn(x)
2x-3x2+xIntegrating by parts to get B4 and 85 in terms of -xLn(x) ,wè then replace the latter by (2x-3x2+x) , to get the final expressions
of B4 and 85
B4,=
os(IiI{ii.z
+
{
Jr
i
Jfl4.&\
Ñ1
)Ç R-140ÒThe sum of the Bn's yields the quantity [q:t1)+q2)J whIch appeared in Eq. (B.4). Thus the amplitude function can be written
os({z(i_fr)(()+Y)_
(z_j))
.4
(tJ(iI«%) i.
ï)
+
4
4
(tX
* I
>t))
j
{1i4 .t1+ t))(4 (X) +)-3
.(a+
.+.) +
,$ß4
R-1400
Appendix C
AMPLITUDE FUNCTIONS DUE TO THE FROUDE-NUMBER-DEPENDENT SOURCES
The contribution to the amplitude functions from the Froude-number-dependent sources is defined as
C)
Iw+.I.:Jw=
1k4
:i;.
);..
t'4
(c.2)where W
and L are given by Eqs. (2.23) and (2.20), respectively.Ìntroducing W and L to the above expressions, and then Interchanging hè order of integration, we get this result:
4
p" 4 qj:
ÇI4)
j).(3-...(U)
(44)_)
u ()-ti)
(#A)
)
(c.4)+
(P.t4#
-çe
(C.3).where.
+
=
A
(iV
=
0s ()
=
-
Sn(
R-14004.= *+.+y
r[-tc.
+
')
(4C)
kcos
s13 ¿'d
e
(C.9a)
(c.9b)
=
i.
L2' 4_3ZCF"yi=
;no 3ck
(c.9c), t.u4c
=
( f(i+z+
G$$
13de
L
5Lc
\ so
24.2()
e
s&the
(&!*)j
(a3+
')(4
(+
sO)
When the integrand is rearranged takes the form
ko
ì
()6
(Ci4))
(cA.ø;L)'
(i#&e_
Z6'¿ce
(C.9h)+)
j . (c.9d) (C. 9e) (C.9f) (C.9g)2se'ckce
()
R- lkOO (i')
c3.
kY1L3G (s;r-
c') ke
Js
-
j
(t# t) (ie +
+
6(
.-ÇQ ( HQ¼s 5 o-6
;floò-.
cQc:E
)
(tt) z)(+ (j')
I
e+
i
l#'
'j
Ik6no4-c
Q+C3 430All the inner integrals of Eq. (C.lO) can be evaluated by the residue theory, and r5C1) reduces to
-
j
-i-
+
G¿e
43Clí
___c.
CJ+4d
c.zs. JM+&
C.(#i) jkcik
J +e>
2C.J i+c.)
(4
t)2
(C. 10) (C. 11)We then change the variable on the first two terms of the last integral of (C.11) by letting
(d
15\
1dk
+
de
j
+
4%)4Ñ)J (4
jt?j
(j o+
3Z$C
Ç(zf
-\).Z.)
d'-- àL:.7- (,A_ zth-ç I t QL'Y7L I C. t''° 1
)
kjcr
I)_ iç_(_
c .1- .yM)L
J'ItJtP)
R- 11400 c fi-
j)%_+
4-'
We can obtain, r (2) , r (i) , and r.
s c C
fol lowing results:
- ZCn (z
4
t()
öl
¿'3.
+ì4-(.)4
-..riE)
4&[.4+
:c;
in a similar way, with the
(4#zt.')
j(.4;'Ç)
s -
.T'-.)
{-c
(C.12)
(C.13) The final result of s of the form
+
3z eR_lLOO C.t
ik
A(ZC)
ja
(c4#
1jj:'
ItJ4L+
n (zc.) - ___
e1)
+
n(c+ji')--f 4-L
Q -
.4. "+
+
(i 4c+zc4
(1114- iiE"\
4\3
3ce1
(c.k) (C. 1 5)In Eq. (,C.6) the major contributions of f5 and are
(r5(1) + r5(2j
) and (rc(i) + rc(2) -
J)
respectively; the contributionsfrom ÇF (»
+ ) and (r + F(2) ) are comparatively small. To
avoid the lengthy computation of small quantities, we can closely approximate these integrals by using the technique described below.
Consider an Integral A , which Is a typical form of the integrals
that appeared in Eqs. (C.9e) to (C.9h).
t
I '
o#(cz._6;o')sw.o e=
>I
j S
(fl
y
+
)
dû
Since O c 1 and C2X1 =
cv1 =
> i , the main contribution of (C.16)o_e R-1400
=
F I 6)c-
£flO
TtO (4.);) ( 4.t)
( b4)
A (Q
= -->;
(s;nOfcc-
-zke
Jô'()tM*
de
k
(C.17)
(C..18)
'
:'ddk
Equation (C.20) should béa good approximation to (C.l6), because the
-2k sin e -2k8
error introduced by replacing e by e us compensated for by replacing [k2 sin2 e + y12) (k2
e+2)]
by (v1t)2 Using the above-described technique to evaluate Eqs. (C.9e) to (C.9h), we have the following:(c.2o) where 81 and 62 are constants between zero and one. A parametric study of Eqs. (C.16), (C.17),and (C.18) reveals the relationship
AL(0)
Au()
A J9)Equation (C.19) is true for a range of values of t and c , espécially
._.(t) .._.(z
Ç
Ti+ J
=
3(ZC-i)4 4b1(i-c3)-
4b)
+
({-ze)# O(i)
(c.21)-
iz&
zC
-(C.22) where=
o.riszi4-ir
t,
= -o.o3zzz3.
3=
The final expression of (C.7) and (C.8) is
13+_
1j.. _Çj
5Cv'
-f= z-cT-cV)Wzc)4(j-)
[-
-')
c + i4civ-
r)
+
¿'eI '
4(C(zQ)+13
ji:J
+
*c._Iz(i+cW+.
Cz(ZT
(t4c)T)
where zC(t2-+)+
cL(zc.)
+_ZC..ti
i'\
U,
Q3ê1+
c(1-cU)
+
-
(-.z)
4-Appendix D
SECOND-ORDER AMPLITUDE FUNCTION (ZERO-FROUDE-NUMBER CORRECTION)
The contribution to the amplitude function from the zero-Froude-number correction to the kinematical boundary is given by
4(
(i-3
(1#In Eq. (D.l), the real part
1b
is identically zero, and the Imaginary part
b can, when integrated by parts twice,
be written as
cs1)
D .2)The non-oscillatory part of the Integrand In Eq. (D.2) is fitted by a polynomial. That is,
(
t5+5+3'')
which, when substituted in (D.2), yields the final expression for J1,
-
-
3Zf (
3qoz' os(Ò
R-1400
REFERENCES
MICHELL, J. Hi,, ,"The.Wave Resistance of a Ship," Phil. Maq., Vol. 45, 1898, p. 106.
HAVELOCK, T. H., "Calculation Illustrating the Effect of Boundary Layêr on Wave Resistance,":Trans. Inst. of Naval Arch., March 948. WU, T. Y. T., "Interaction Between Ship Waves and Boundary Layer,"
International Seminar.ón Theoretical Wave Resistance, Vol. III, August 1963.
L,. WIGLEY, C., "Effects of Viscosity on Wave Resistance," International. Seminar on Theorètical Wave-Resistance, Vol. III, August 1963. WIGLEY, C., "The Effect of Fluid Viscosity on Wave Resistance Using
a Modification of Laurentieff's Method - A More Accurate Calculation," C. Wigley, Flat 103, 609 Charterhouse Square, London EC1, England,, May 1966.
INUI, T., "Study on Wave Making Resistance of Ships," Soc. Naval Arch., Japan, 60th Anniversary Serles, Vol. 2, 1957.
BRESLIN, J. P. and ENG, K., "Calculation of the Wave Resistance of a Ship Represented by Sources Distributed Over the Hull." Paper presented at the International Seminar on Theoretical Wave
Resistance, University of Michigan, Ann Arbor, Mich.., August 1963 (DL Report 972, July 1963).
EGGERS, K., "On the Determination of the Wave Resistance of a Ship Model by an Analysis of its Wave System," Proc. International Seminar on Theoretical Wave Resistance, Ann Arbor, Mich.., August 1963, pp. 1313-1352.
KOBUS, H, E., "Examination óf Eggers' Relationship Between Transverse Wave Profiles and Wave Resistance," J. Ship Research, Vol. 11,
No. L1, December 1967.
SHARMA, S. D., "A Comparison of the Calculated and Measured Free-Wave Spectrum of an muid in Steady Motion," Proc. International Seminar on Theoretical Wave Resistance, University of Michigan, Ann Arbor, Mich., August 1963, pp. 201-257.
IKEHATA, M., "The Second Order Theory of Wave-Making Resistance," J. Soc. of Naval Arch., Japan, Vol. 117, 1965.
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PETERS, A.S. and STOKER, J. J., "The Motion of a Ship as a Floating Rigid Body ina Seaway." Communication in Pure Appl. Math, 10, 399-490 (1957).
WEHAUSEN, J. V. and LAITONE, E. V., "Surface Wave," Encyclopedia of Physics, Vol. IX, Fluid Dynamics III, Springer-Verlag,
Berfln, 1960.
1k. HAVELOCK, T., "The Vertical Force on a Cylinder," Proc. Royal Soc., A, Vol. 122, p. 387 (1928).
HAVELOCK, Ï. "The Forces on a Circular Cylinder Submerged in a Uniform Stream," Proc. Royal Soc., A,.Vol. 157, p. 526 (1936). BESSHO, M., "On the Wave Resistance Theory of a Submerged Body,"
Soc. Naval Arch., Japan, 60th Anniversary Series, Vol. 2, 1957. TUCK, E. O., "The Effect of Non-Linearity at the Free Surface on
Flow Past a Submerged Cylinder," J. Fluid Mech., Vol. 22, Part 2, June 1965.
SALVESEN, N., "Ön Second-Order Wave Theory for Submerged Two DimensionalBodies," Proc. 6th Nava.1 Hydrodynamics Symposium, Vol. II, Washington, D. C., 1965.
SISOV, V. G., "Contribution to the Theoryof Wave Resistance of Ships in Calm Water," Izv. Akad., Nank, SSSR, Dept. of Tech. Sci., Mechanics. and Machine Constructions, No. 1, pp. 75-85, 1961
(Bureau of Ships Translation No. 887).
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TABLE 1. RATIO OF SECOND- TO FIRST-ORDER WAVE RESISTANCE OF A PARABOLIC STRUT ACCORDING TO THE EXPANSION PROCEDURE
R = t2R1+t3R2+... R2 = R2(0)+R2(p)+R2(F)
R2 (o) , ZERO-FROUDE-NUMBER CORRECTION
R2 (F) , FINITE-FROUDE-NUMBER CORRECTION R2(p) , FREE-SURFACE cORRECTION F R2(0)/R1 R2(p)/R1 R2(F)/R1 R2' .20 -0.792 -4.054 -4.808 -9.654 .21 -1.986 -4.562 -5.214 -11.762 .22 -0.362 -3.931 -4.70,9 -9.002 .23 -0.030 -2.611 -4.224 -6.865 .24 -0.997 -2.750 -4.281 -8.028 .25 -1.848 -3.323 -4.667 -9.838 .26 -0.477 -2.787
-4576
-7.840 .27 1.451 -0.756 -3.716 -3.021 .28 1.239 -0.087 -3.488 -2.336 .29 0.655 0.102 -3.370 -2.613 .30 .047 0.183 -3.290 -3.060 .31 -0.578 0.192 -3.329 -2.715 .32 -1.175 0.055 -3.525 -4.645 .33 -1.405 -0.410 -3.816 -5.631 .34 -0.371 -1.307 -3.901 -5.579 .35 1.947 -1.976 -3.490 -3.519 .36 3.542 -1.835. -2.991 -1.284 .37 3.977 -1.375 -2.713 -0.111 .38 3.914 -0.951 -2.569 0.391+ .39 3.720 -0.620 -2.461 0.639 .40 3.515 -Ò.364 -2.353 0.798R-1400
TABLE 2. WAVE-RESISTANCE COEFFICIENT x 1O OF A PARABOLIC STRUT
Beam/Length = 0.1
F First Order Second Order
Second Order Without Free-Surface Correct ¡on .20 0.2794 0.1028 0.1734 .21 0.2639 0.0726 0.1400 .22 0.2022 0.11714 0.1616 .23 0.14805 0.21496 0.33714 .214 0.6239 0.2547 0.3729 .25 0.141458 0.1621 0.2516 .26 0.3261 0.2169 0.2578 .27 0.5567 0.5047 0.5129 .28 1.0413 0.8904 0.8818 .29 1.14878 1.1828 1.1627 .30 1.6751 1.2512 1.2246 .31 1.5635 1.0882 1.0681 .32 1.2615 0.8050 0.8115 .33 0.9401 0.5631 0.6166 .34 0.7585 0.5026 0.6159 .35 0.8246 0.7062 0.8809 .36 1.1857 1.2013 1.4270 .37 1.8371 1.9757 2.2324 .38 2.7384 2.9951 3.2560 .39 3.8298 4.2138 4.41483 .140 5.01456 5.5810 5.7577
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TABLE 3. DATA OBTAINED FROM DRAG BALANCES
Draft = 7.5 in. Draft = 12.0 ¡n.
Velocity ¡n fps Drag in lb Top Mid.
3.822
.975 .9893.717
.9283.620
.876 .911.3.526
.810
.8703.417
.714
.8203.320
.643.770
3.217
.601 .7313.140
.566.704
3.052
.533 .669Velocity
in fps
Drag in ¡b Top Mid3.852
.702 1.0433.720
.665 .9943.617
.607 .9243.523
.573 .8903.419
.507 .8363.313
.442 .7613.217
.405 .7303.134
.374 .7043.040
.361 .659r
2.908
.347 .6162.808
.308
.5762.673
.288
.535 -2.593 .268 .4952.502
.243 .459w
z
4
I-(n (n w o: w >4
o: wo
o:o
(I) o: ILo
I;.o
z
o
u
w U)u-o
o
-8
o:R:t2 R1+13R2
R2: R2 (O )+ R2 (F) R2 (pl R2(0),ZERO-FROUDE-NUMBER CORRECTION R2 (F)1 FINITE-FROUDE--NUMBER CORRECTION R2 (p), FREE-SURFACE CORRECTION 4-0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 FROUDE NUMBER2.4 2.0
o
X i-I.6 zw
C.) IL.u-w
o
o
1.2 "J a' .j:-I (I, U)J
"J > 4 0.4 es e
EXPErIMENT4L RESULTS SECOND-ORDER THEORYSECOND ORDER (WITHOUT FREE-SURFACF CÖlPFCTIAr
I
- - - --
FIRST- ORDER THEORY-..',
j BEAM/ LENGTH: O. IO/1
/
li
/
Ryj /r-
2 2 a2pUP
I't
/
t
/
I
/
/
0.20 0.22 0.24 026 028 0.30FROUDE NUMERF
¡
1,
/ 1/ /£
/
,
0.32 0.34 0.36 0.38R-11+O0 60.00 W.. L. 0.12 GAP-1
I
t
0.12 GAPI
HALF BODY 0F REVOLUTION
I
-ALL DIMENSIONS ARE INCHES D:DRAFT OF TOP SECTION
WETTED AREAs 0.8388 FT2/IN OF SUBMERGENCE
DISPLACEMENT: 8.68 LB/IN
FIGURE 3. PARABOLICSTRUT MODEL USED IN EXPERIMENT
t
6.00i
t
15.00 48.24 f 15.00FIGURE. Li.A., MOUNTING OF DRAG BAINCE IN MIDSECTION OF STRUT
.':
---/
I, L - _,; ) R-1h00.2 ¡.0 0.8 0.6 0.4 Q2 O R-1400 DRAFT 0F TOP SECTION 7.5 ¡N. 12.0 IN. BALANCE 'ro P
o
MIDs
A 2.4 2.832
3.6 4.0MODEL SPEED ¡N FEET PER SECOND
w
08
tu t-wo
C 1.0< 0.2
o
D DRAFT OF TOP SECTION
ST: TOPI
1-SECTION WETTED AREA
SM:MIDJ (EXCLUDED GAP AREA)
BEAM! LENGTH: 0.10
SM:12.58 FT2
0.__-.----_._
0.22\ITTC FRICTION
SCHOENHERR FRICTION
WAVE RESISTANCE DERIVED FROM
EXPERIMENTAL DATA (BASED ON S1:6.29 FT2)
VISCOUS DRAG 0F STRUT
INFINITE STRUT
-7.5 IN. SECTIONFROUDE NUMBER
0.26 028 0.30 0.32 034
D(IN.)
ST(FT2) TOP
MID7.50
6.29
0
S
12.00 10.07 A I .2 1.4 1.6 1.8 2.0 REYNOLDS NUMBER X IOz
o
0.6 w (n NI.0
4
w4
0.8I-u
z
o
o
0.64
w wo
XI-z
0.4u
u. u. wo
u
o
4
Q -J4
I-o
I-C O EXPERIMENTAL RESULTS SECOND-ORDER THEORYFIRST- ORDER THEORY BEAM / LENGTH: 0.10
WETTED AREA:I.61x105 FT2(BOTTOM AREA EXCLUDED)
0.20 0.22 0.24 0.26 FROUDE NUMBER REYNOLDS NUMBER X
I
/
/
/
/
/
/
/
/
SCHOENHERR FRICTION PLUS FORM DRAG
SCHOENI-IERR FRICTION (59° F SEA WATER)
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