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OCR for page 119
The Influence of a Slowly Oscillating Movement
on the Velocity Potential
C. van der Stoep, A.J. Hermans
(Delft University of Technology, The Netherlands)
Abstract
The central problem problem in this paper will be the deter-
mination and calculation of the influence of a slowly oscillat-
ing movement on the velocity potential. This is for instance
important when you want to calculate the dynamic swell-up
and the added resistance of a sailing ship. Two boundary
integral equations for the velocity potential are derived to
solve the steady and the insteady problem. It will be shown
that the potential can be represented as a area source sin-
gularity distribution over the ship hull and a source line
distribution on the ship waterline. The equations that have
been obtained in this way can be solved using an iterative
scheme. First and second order solutions of the equations
have been obtained.
Nomenclature.
B
U
g
p
71(x)
710
~1
L
H
x
z
n
Fn
G
F.S.
c'
a
Tn
Jn7 Kn, Yn
Beam at midship
Wetted surface area at rest
Ship speed
Ship block coefficient
Gravitational acceleration
Pressure
Wave elevation along hull
steady wave elevation
unsteady wave elevation
Oscillator amplitude
Length at water line
Draft at midship
coordinate system moving with velocity U
in forward direction
vertically upward
unit vector normal to ~ in outward direction
Froude number(U/~)
Green function
Free Surface
steady wave potential
unsteady wave potential
total potential
motion vector
frequency of motion
dipole strength
source strength
Chebychev polynomial of order n
Bessel functions
119
1 Introduction
The estimation of a ship speed and power was usually based
on still water performance. Assuming a potential flow the
stationary problem of calculating the ship wave restistance
is described by the Laplace equation and the conditions on
the ship hull and the free surface. Numerous computer pro-
grams have been written to tackle this problem.
In order to be able to predict ship performance in seaway
it is also desirable to be able to calculate the instationary
problem of a sailing ship. For example a ship sailing in waves
or an oscillating vessel. In preliminary design studies the use
of a fast computer algorithm could help to assist the tradi-
tional model testing methods. The central problem in this
paper will be the calculation of the added resistance and the
dynamic swell-up of a ship when it is slowly oscillating. Dy-
namic swell-up is the effect of water being pushed up around
a bow higher than can be accounted for by considering heav-
ing, pitching and incident wave alone (see also figure 1.1).
The swell-up is given by the quotient of (amplitude of rel-
ative motion) and za(oscillator amplitude).
~1
.
/ ,ho
it' ~
Id /
)
~ Ha
Figure 1.1: The dynamic 'swell-up'.
Some work in this area has been done by Blok ([1]) and
Tasaki ([2]). Tasaki was the first to introduce the term swell-
up and on the basis of experiments obtained some sort of
emperical formula for the calculation of the dynamic swell-
up, when the wave frequency, ship speed and ship block
coefficient CB are between some give limits.
The problem will be divided into several subproblems:
· The treatment of the free surface condition.
· The treatment of the body boundary condition.
OCR for page 120
The numerical treatment of the associated Green func-
tion.
· The numerical solution of the resultant integral equa-
tion.
· The calculation of the added resistance.
Every subproblem will be dealt with in this paper. The
research is carried out in cooperation with MARIN.
2 Mathematical formulation
2.1 Introduction
We consider an object moving horizontally with forward
speed U in an infinitely extended fluid. Viewed from an
inertial frame~x,y,z) attached to the ship, there is an inci-
dent stream of velocity U in the positive direction. The
formulation will be done in a Cartesian coordinate system
moving with the object. With the x coordinate in forward
direction and the z coordinate vertically upward.
The free surface elevation ~ will be given by:
Z = 71(X, y) (2.1)
The total velocity potential ~tota' can be split in a steady
and an unsteady part:
total(X' y, Z. to) = US + ¢(X, y, Z) + O(X, y, Z. t) (2.2)
steady unsteady
We are especially interested in the influence of the steady
part ¢' on the unsteady part ¢.
Subtotal has to satisfy the following three conditions:
Subtotal = 0 outside the object, z < Mix, y) (2.3)
At the free surface y~x,y) we have the dynamic and the
kinematic boundary conditions:
~ ~3., ~ `~?2 + ~ 2 + t3~2' + gz = 0 at z =
l at ~ 8c By At) = 0 at z =
and a boundary condition on the ship hull:
~ = Vn, atE (2.5)
The main difficulty in this problem is to find a solution of the
Laplace equation using boundary conditions at a free sur-
face which is still unknown. Various attempts to solve this
problem have been done by Brandsma ([3~+ [4]), Baba [5],
Sakamoto [6i, Hermans [7] and many others. Most of them
are using some sort of expansion in a small parameter usu-
ally the small ship velocity or some sort of ship slenderness
parameter.
To derive an approximating solution of the problem we con-
tinue by linearizing the free surface condition (2.4) (as will
be done in the next section) and the body boundary condi-
tion (2.5~.
2.2 Linearizing the free surface condition
The free surface condition consists of two parts:
The dynamic boundary condition:
.9' + EVE V4} +gz = 2U2 at z =~(x,y) (2.6)
and the kinematic boundary condition:
~ -~9~1 _ ~ ~971 _ ~ · ~971 = 0 at z = ~7(x,y) (2~7)
One way of dealing with this problem could be neglecting
the higher order terms in 4~. Then equation (2.7) leads to:
B~ = i971 (2.8)
and the linearized free surface height will be given by:
~ 041, (2.9)
combining equations (2.8) and (2.9) leads to the well known
free surface condition:
~f2 + 90 = 0 at z = 0 (2.10)
In this paper we will obtain an asymptotic solution for the
free surface condition. Let ~ << 1 and expand the free
surface elevation around a 'known' solution ?1 = ho:
71 = ho + 6773 + ·~. (2.11)
Denote the total velocity potential by ~tota', and split this
total potential in a steady and an unsteady part as follows:
..
~'Ota`(x' y, Z' t) = UP ~ ¢(x, y, z) + c¢(x, y, z, t) (2.12)
All the terms in the free surface condition (2.6) and (2.7)
have to be expanded. This leads to the following free sur-
face condition:
For the first order problem (the steady state solution):
~ OZ + 2} V(uX + 4)) V[v~uX + O) V(uX + (/)~] = 0
| atz=yO |
(2.13)
and for the second order problem (the unsteady state solu
tio~ ):
- - - _ - -
g(¢z - Lyon - ~yHoy - Bust + vex + ¢) vat+
2V(Ux+o).Vot+V(Ux+o) ~V[V(Ux+o) EVE]+
~
Ott = 0 at z = yO
120
(2.14,
OCR for page 121
2.3 The body boundary condition
The body boundary condition for the stationary potential
reals as:
or equivalently:
.9 = Vn at the body ~(2.15)
n.V(Ux+o) = 0 atE (2.16)
And now the calculation for the instationary part (see also
for instance Timman t8] et al.~. The displacement of the
ship hull is given by:
x - x' = ~ or~x,t) (~.17)
where x denotes the coordinate system moving with velocity
U. and x' denotes the coordinate system fixed to the ship.
The body boundary condition states that: at the momentary
position of the hull, the normal velocity of the fluid is equal
to the normal velocity of the hull. Expansion of ~ round the
rest position with a small parameter ca yields:
Momentary = Total + ~ (t Subtotal (2.18)
so for the body boundary condition this leads to:
n (total + ~ A` ~total) = ~ n it,` (2.19)
combining equation (2.16) and (2.19) leads to:
~ = n ~ ~ At, - V(a ~ V(Ux + ¢~) (2.20)
now using the assumption of or~x,t) and ¢(x,t) to be oscil- and
latory:
onyx, t) = o~¢x' .-i~
¢>(x,t) = ¢,~x).-iwt (2.21) then the following is valid:
4' = -n · (i`.Ja + V(or ~ V(Ux + f)~) (2.22)
with the function G written as:
Figure 3.1: The mathematical domain.
G = (- - + - - ~(x, (; U)) (3.2)
In the same way as Brard t9] we consider the following inte-
gral formulations:
IE = /! (me () _ G Me ~ d5
EFe' + Ee + r
I] = I! (Oi {) _ G {) i) dS
EFi + Ei
(3.3)
(3.4)
this finally leads to the following body boundary condition: I { 41rSt)e 27 C De (3 5)
I = { 4~Vi x C D' (3.6)
and free surface condition (2.14):
20 - 2ic~Uo~ + US + 9¢z = ((U;~){~}
at z = 710
(2.23)
where [(U;~) denotes a linear differential operator acting
on a.
3 The leading equations
3.1 Calculation of the singularity distributions
Introduction of a Green's function G and application of
Green's theorem to the domain as can be seen in figure 3.1.
The function G has to fulfill the following condition:
-HUG + 2i~UG~ + U2G<~: + gG: at ~ =0 (3.1)
At EFi and Life the following equations (2.23) and (3.1) are
valid:
~0 = _{~(~' + ~2¢> + 2icoU~ _ u2 ~ 'it} (3.7)
;,G ~ {HUG - 2i~U ~ - U {~2 } (3.8,
Combining equations (3.7) and (3.8) leads for equation (3.3)
to:
dG 4~ e
(me .~ - G ~' ~ =
121
-OeG - -Use-
g g 0*
U 4)e~ G _ -{(o)G
OCR for page 122
- g ¢)eG - -U ~ eG +
+ _ `9 Ve G (3 9)
regrouping the expressions at the right hand side of equa-
tion (3.9) leads to the following formulation:
2i~ut3~(~)eG) + U 0( ~c G (e 8= ~ _ - C`oe'G
9 dx 9 dx g
(3.10)
and almost the same for equation (3.43. The ~F, part of
equation (3.3) leads to:
|/ (¢e ~'n - G ~) ~dS = - / -U 4>e G dy
~Fe'
J 16e J y
C°° -Cf
and for the ~Fi part:
g |'-at c,~ - aT ~T + ~na(~)Gfx,()dy
Cf
- - J| 13~)Gf x~ ()dSt = 47r<~(x) (3.15)
F.S.
with:
CYt = COS(OX, t)
OT = COS(Ox, T)
Ctn = COS(OX, n)
(3.16)
with a choice of y(~)-O the following expression will be
obtained:
4,ro~x) =- /. [cJ(~)Gtx,()dst- + - /c~nat()Gtx,()d
/* u2 d~e dG ~ .1 -. 9 C
_ / ~ G - °e~ ~dy 1 1 1 ^
c~J c, g {J ~0~ + g J~ {(o)G(z,()dS~ (3.~7'
- - || {(ve)G dS (3.11)
, using the body boundary condition (2.22) we may obtain a
description of the potential funtion ~ by means of a source
distribution of the following form:
||~¢i~ - G~°i~dS = |-~ ,,~iG - ¢>i ,, ~dy
EFi CF
| 2i~U Gd (3.12)
adding equations (3.11) and (3.12) results in (with ~8i =
~ _ oN.
_ ~ _ ~ J.
Ie + Ii = IJ~ ~(~)e _ ~¢'iGdS +-U /`Oe-oi~Gdy
~ Cf
U2 /~¢li _ 0¢le'Gdy + U /'¢,e-¢'i) ;3~ dy
Cf C'
-|| {~)e ~Gd S - - || l:~¢'i ~Gd S
rFe ~Fi
+ | j`<;6i-°e) ~3 dS
The source and vortex distribution are defined as:
= ¢>e-`/)i
B¢ii 0¢e
a = ~ _ _
equation (3.13) will now transform into:
| 7~) d G(z, f ) d St - | | a(~)Gf x, t!~)dS~
g | ~y(~)Gtx, (,d71 + _ | ?~) aG(~ . f)
(3.14)
122
- 41rn · (i~or + V(~. V(Ux + ¢~) = - 2lrafx)
| | ~) 6 ( 'f ) dS~: + -| ~ c7~) 6 f( ' f) d
- ~ || ~C(~) f ( ' f ) d St
F.S.
Remark:
We consider the following potential ~:
(3.18)
~ = ||-dS (3.19)
the following is valid (Kellogg [10]~: if the density a of the
distribution on ~ is continuous at x, the normal derivative
of the potential ~ approaches limits as X approaches x along
the normal to ~ at x from either side.
(3.13) These limits are:
( ~'n )+ = -2~a~x) + || a(~) ,9/ dSt (3.20)
( ~n )- = +2~a~x) + || a(~) ~/ dSt (3.21)
So when using the R sources the resulting equations looks
like:
+ 21ra(x' - ||a ~ '<;)dSf + ... = Vn (3.22,
and when the -R sources are used (as in Brard [9]) the
resulting equation transforms into:
OCR for page 123
-2,ra(x) - ||a ~ '()dSf t
= Via (3.23)
3.2 Calculation of source strength and potential
function
We will use an expansion of a, ~ and G in the small param-
eter w:
a(x) = aO(x) + wai(x) +
¢(x) = fO(X) + woi(x) ~
G(x,() = Go(x,() + h7Gi(x,<) + (3.24)
substitution of these expansions in equations (3.17) and (3.18)
leads for the first order problem to:
- 4lrn ~ V(cx · V(Ux + ¢)) = - 21raO(x)
aO(~)0 ~( '()dSt + ~ Now) t~( ~ ()dr
~Cf
- ~ / / ~Co(~)~ ~( ' () dSt (3.25)
F.S.
and Jo is given by:
- 41r~o(X) =
-| | ao(~)Go(x, ()dS~ + -| ('naO(~)GO(X' ()d?1
~ of
+~ || t~o(O)Go(X, ()dSf (3.26)
F.S.
and for order h':
-4~i (x)-~ J a~(~) aft (x,() dS~ + U2 J ol(g) ~G~o(x,() dr1 =
-47rn · in + ~ [aO(~3(='() dS~ _ rug Jo c~aO(~) (inn +
+ 9 ~ r (em ( ~ ,0 ~ + , O (< ,~ ~ ~ (A <) + ~0 (SO ~ ) of, (A <) dS,:
F.S.
and ¢~ is given by:
3.3 The incorporation of the body boundary
condition
The right hand side of equation (2.22) contains a vector ork
or (but) = { °rk(t) (ik X Xj k - 49596 (3.29)
here C>k(t) is the deflection in translational motion for k =
1,2,3 and for k = 4,5,6, Skit) represents rotation angles
about the Xk_3 -axis. So for example for k=l(surge) this
looks like:
4) = -icons-nl¢==-n2O=y-naves (3.30)
An
when ~ << 1 then the expansion (3.24) will start with
leading order ~ (a = Day + h'2~2 + · · ) and then equa-
tion (3.25) will transform into:
- 21ra~ - ,/ JP(a] ~ ° + ~ on ) + · · ·
~=o
=-no (3.31)
when o = (~(1) then the expansion will start with leading
order zero and equation (3.25) will look like:
- 27raO - ,/ / aO i, o + = -n! O== - n2¢=y - n3O=z (3.32)
it should also be noted that for the calculation of the steady
wave potential ~ the same matrix equation (3.25) as for the
unsteady potential ~ can be used! But of course the right
hand side for in now looks like:
~ = -no U (3.33)
(fin
3.4 Solution of the integral equation.
(3.27)
_ 47ri1(X)=
(ao(~)~l(x~ A) + al(~)GO(x, ())dS~
+-| ~n(aO(~)~l(X, () + al(~)GO(x' ())dr'
9
C'
-- || (Jo + L:l)Go(X,() + {o(¢o)~l(x,<)dS~
F.S.
(3.28)
The solutions obtained by the singularity distributions are
two coupled integral equations. Equations (3.25) and (3.26)
will be solved using an iterative scheme. In this scheme use
has been made of the numerical evaluation of the wave re-
sistance Green function as done by Newman (~11~+ t12~).
The numerical solution of equation (3.25) and (3.26) is car-
ried out using a finite element method. The wetted body ~
is divided into N triangular panels and integration is done
using a piecewise constant variation of the source strength
a(~).
| | = ~( | | a(~) ~ ~ d So (3 .34)
In this way a set of N linear equations for the N source
strengths aj is obtained. The Green function contains a
1/A-term. The integration of these 1/A-terms is carried out
by a subroutine developed at MARIN t13] using analytical
expressions of those integrals in order to avoid large errors
from numerical integration for points close to the panel.
123
OCR for page 124
4 The Green's function
4.1 Behavior of the Green's function.
In Wehausen and Laitone t14] the Green's funtion of an os-
cillating source is given:
G(x,(; U) = -- + - - ~(x,(; U) (4.1)
with the function ~ given by:
2 7r
~(x,(;U) = 9 JlF(8,k)d~dk
O L1
7r
+ 9 ||F(8,k)d~dk (4.2)
2 7r L2
where F(8,k) is given by:
F(8, k) = k e ~ cos(k(y-A) sin 8)
the paths Lo and L2 are given in figure 4.1.
with Kit - k4 the poles of F(8,k):
1 - j1 -4T cos
/; ~2r cos ~
-W 6' ~ (O' 72r) (4.4)
~ = 1 + AL 4rcos8~ ~ ~ (0, 72r) (4.5)
Why = 1 - ;1 4rcos8~ ~ ~ (72r~1r) (4.6)
~ = _ `9 ~ 6' ~ (12r,~) (4.
with ~ = ugly, so T ~ 1 this leads to:
~j: + ~(~2) (4.8)
/;: it- Ucosp + + (~( ) ( )
when for instance U ~ O then k2 and k4 will go to oo and
the paths Lo and L2 will coincide with there poles located
at Kit = k3 = w2/9. see also figure 4.2.
and when ~ ~ O the poles Kit and k2 will move to the origin
and the paths Lo and L2 will look like as can be seen in
figure 4.3.
However this is not correct, when ~ ~ O a factor k can be
removed from the function F(8,k) and looks like this:
K1 K2
O ~
K4
Figure 4.1: The contours L1 and L2.
K1 ~ K3
_ L ~
I \ ~ , L1 ~ L2
O
Figure 4.2: The location of the poles when U ~ 0.
K1
1 it..
o
K2
K3 Ka
I ~ J > L2
o
Figure 4.3: The location of the poles when ~ ~ 0.
F (8 k) ek(Z+~+i(~-~)C°s8)cos(k(y-77)sin8) (410
with only one pole, located at k2 = k4 = U2 C9os2 8
124
OCR for page 125
4.2 Calculation of the Green function.
Newman (~114+ t124) has written two papers in the Journal
of Ship research about the evaluation of the wave resistance
Green function: one of of the calculation of the double inte-
gral and one for the calculation of the single integral on the
centerplane. The Green function is written in the following
form (see Wehausen and Laitone t14~:
1 1
Tn(X) = costn arccosx) -1 < x < +1 (4.14)
so the function 95x; is approximated by:
N
f(x) = ~ cmTm~x) (4.15)
m=0
The coefficients cm can be found according to:
N
1 = ~ 4~ s(k~cos8)cos(ky sin y~dkd ~Cm = N-~ f(~)Tm(~n) (4.16)
0 0
1 7r
- 4 | e-Z sec2 ~ sings sec 8) costs sec2 ~ sin 8) sec2 add
o
(4.11)
The quantities are defined as can be seen in figure 4.4.
> X
.1
Ir .
S(
rage
Ye 1
-
\G?
urce ~f ield
Figure 4.4: Location of field-, source- and image point.
with R = ~x2 + y2 + z2
As has been done by Newman this integral can be split in a
Double and a Single integral as follows:
2 + 2 ~ °° e-kz+iklxl see ¢+ky cam ~
Double = -i ~ / CoS ~ 2 . dEd;
or J J k - COS ~ + tE
_~x O
(4.12)
+ 2
Single = 4iH~-x) ~ sec2 Be-zsec28+ixsec8+ilylsec28sin~db
2
(4.13)
4.2.1 Double integral
The double integral as given by equation (4.12) will be ap-
proximated by Chebychev polynomials as done by Newman.
In order to approximate a function f of one variable x in the
normalized range i-1,+1] the Chebychev polynomial of order
n is defined by:
with c0 = 1,em _ 2, the double prime indicates that the
first and the last terms in this summation are multiplied by
2. The coordinates on are given by:
(see Fox and Parker [15~)
an = cost N ~(4.17)
For the 3D-case the situation is completely equivalent. In
equation (4.12) logarithmic singularities are present when
R = 0. These singularities must be subtracted and approx-
imated first in order to improve convergence of the approx-
imation. The final approximation is given by:
16 16 8
D ~ S + ~CijkTiff(R)]Tj(-1 +-8)T2k(7ra)
i=o j=o k=0
(4.18)
The function f(R) is defined so as to transform the interval
(O,oo) into (-1,+1) see figure 4.5.
and ~ are defined as:
x = Using
z + iy = R cos heir = pew (4.19)
Ti, Tj and T2k are Chebychev polynomials. S is the loga-
rithmic part of the double integral. The Chebychev coeffi-
cients Cijk are calculated and tabulated by Newman. Also
the differentiated Green function has to be evaluated, which
contains terms like dGfx,(~/dox (see equation (3.25) and
f (A)
25
/1~1 /1~
/ ~ / 4 / 10/ ~
Figure 4.5: The transformation function f(R).
OCR for page 126
(3.27~. Each part of the expansion (4.18) has to be differed- where Ye (x) denotes the Bessel function of the second kind,
tiated and evaluated analytically. The following terms have ~ equals - ~ and F(~) Dawson's integral:
to be evaluated:
~, = A · ~ (4.20)
with the transformation matrix A given by:
OR
OR
By BY
OR
LIZ LIZ
(4.21)
and also for the singular part S. For example the next fig-
ure 4.6 is obtained. The singular character is well shown
here.
4.2.2 Single integral
The single integral as given by equation (4.13) is evaluated
by Newman at the centerplane, i.e. a special case where the
source and field point are in the same longitudinal plane.
This is especially important when analyzing thin ships. The
Figure 4.6: Function plot of the Double integral.
case y 7t 0 will be dealt with later. For instance with the
use of Fade approximations. The centerplane integral looks
like this:
2 ~
Sax, Y. Z)lY=0 = -8H(-x) l sec2 ee-Z see ~ sinks sec 8)db
o
(4.22)
In each of the different x-z regions (see figure 4.7), the inte
gral will be approximated differently.
region A: (small z) an expansion involving differentiated
Bessel functions of the second kind:
S -F( )- d2 4 23
n on! dz277 t2 i~ )+ ~: ~ · ~
126
o
1
2
3
4
5
6
O 1 2 3 4 5 6 7 8 9 x
~,> (C)' ' ' ' '
- (B) "~` (B) ~
Figure 4.7: Domains for the approximation of the single
integral.
:r
F(x) = e-2 |et2dt 0 ~ x < oo (4.24)
o
region B: an expansion in Neumann series, products of
Bessel functions of the first kind and modified Bessel func-
tions of the second kind:
S = 2e 2 ~(-1) J2n+1~) [K-n(2Z)+Kn+l(2Z)]
(4.25)
where Jew) denotes the Bessel function of the first kind and
Kn~z) the modified Bessel function of the second kind.
Region C: large distances form the origin, steepest-descent
expansion. The final expansion looks like this:
S ~ -2 ieh(°)+it am, (2n + 1 ~Bn(` _ in+ 2 (p + i`~'nr~n + 2 ~
(4.26)
where only the coefficient Bn has to be evaluated.
For all the expansions in the different domains (A)-(C) the
differentiated Green's function has to be evaluated. An ex-
ample for the Green function can be seen in figure 4.8 . The
wave character is well observed here.
OCR for page 127
OCR for page 128
OCR for page 129
OCR for page 130
OCR for page 131
OCR for page 132
Representative terms from entire chapter:
free surface
Figure 4.8: The Single integral at y _ 0.
5 Pade Approximations.
5.1 Introduction
The basis for the Pade approximation technique is the formal
Taylor series expansion. From this basis a Pade approxima-
tion can be found. It is also possible for a Taylor expansion
to be divergent and the Pade expansion to be convergent
and also vice versa. For instance the Taylor expansion of
the exponential function:
oo rat 1 1
en = at, a! = 1 + x + 2x2 + 6x3 + · · · rC = oo (5.1)
and for instance the Euler function:
oO
E(x) = 1 1 + tdt = 1-x + 2x2 - 6x3 + ~ ~ rc = 0(5.2)
o
The idea of Pade expansion is to approximate the function
by a rational funtion of the following form:
Definition: (see Baker t16~+ [17~) We denote the L,M Pade
approximant of I(X) by:
[LL/M] = Q (( )) (5 3)
where P~,(x) is a polynomial of degree at most L and QM(X)
is a polynomial of degree at most M. The formal power series
of f(X) reads as:
When we require:
oO
f (X) = ~ aixi (5 4)
i=o
f(X) - [L/M] = O(XL+M+1) (5 5)
Then the coefficients of PL and QM can be found according wich will be denoted by:
to the following scheme (Baker):
127
+ aOql
+ at + aOq2
al + al-! at + · · · aoq' =
al+ + at + a'-m+~qm =
al+m + ai+m-~ at + alum =
an - O n
n no
Q ~ ~
~A
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 n ~n-R
0 nls
o
-0.005
-0.01
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0.02
0.015
0.01
0 - non
~ . ~ ~
- 1 -0.8 -0.6 -0.4 -0.2
0 025
~ 0.02
Q 0.015
0.01
0.005
o
-0.005
-0.01
-0.015
_ . ~ . ~v . ~ . ~u u . c u . ~U . ~U . ~1
0.025
0.02
Q 0.015
0.01
0.005
o
-0.005
-0.01
-n nip; t ;
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0 no
~o no
Q
-0.01
-0.02 1 , . . . . . . .
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 6.3: Hull side wave profiles of WIGLEY.
128
changing of variable to s = see ~ leads to:
oo 2
se-ZS Ids (5~13)
1 ~
Taylor expansion of f~x,y,z) leads to:
f(x,y,z) = f(x,O,z)ly=0 +Y (~'yY ~ IY=0
+ ~ y2 02f(~, Y'Z) 1 + . (5.14)
applying this to (5.13) leads to:
oo 92
f (x, y, z) = | ,~ sin(sx)ds
so
1 r
_ 1 y2 J S3 >~e-zs sin(sx)ds + O(y4) (5 15)
1
This is the formal Taylor series expansion. Numerical in-
vestigation reveals the fact that this series is not converging
very well. Perhaps Pade approximation could lead to valu-
able results. Numerical experiments give raise to the follow-
ing table (see table 1):
(x, y, I)
(-1,.1,-1)
(.1,.5,.1)
(~1,.9,.1)
(~1,.9,.9)
(~5,.1,.9)
(~5,.9,.5)
f (x, y, z)
.30124
.0053
.0029
.00252
.2132
.1608
~-
. Taylor
.5179
2.7 102
1.1 105
.0044
.2132
l 2.11
Pade[2/2]
.3019
.0073
.006
.00254
.2132
.168
.
Pade[O/4]
.3031
-.0018
-.00012
.00255
.2132
.177
Table 1: comparison between different approximation tech-
niques.
So in some cases Pade approximation leads to better results.
6 Computational results.
6.1 Wave profiles of Wigley Hull.
Some examples for a parabolic Wigley Hull (Shearer t194)
will be calculated. The Wigley Hull is a mathematically
defined form (see figure 6.1~:
y = By _ `227~2~1 _ ~ Z >~2~
with -2
0 . 02
0.015
0.01
0.005
O
-0 .005
-O . 0 1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.06
n 0 04
0.02
o
-0 .02
-0 .04
-n
n
n
n
0.02
-0 .04
O . u
0.04
n 0.02
o
-0 .02
-0 .04
-O . VIE
-1
. ~ ~
-1 -0.8 -0.6 -0 .4 -0.2
0.04
-n no
4
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
. ~ _A
6
2
o
-2
4 ~
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
. v 0 ~ ~
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 6.6: Wave heights for the unsteady motion (En = 0.31).
130
A ~ ~ ~
Figure 6.2: Wave profile on a 48 x 8 grid (Fn = 0.20~.
It can also be observed that only near the ship bow the wave
height changes substantially when decreasing the grid size.
So local grid refinement near the ship bow will lead to more
accurate results.
For a specific value for the Froude number (Fn = 0.348),
the calculation can be compared with values calculated by
Tsutsumi t21] and Dawson t22] as can be seen in figure 6.4.
In all our calculations the wetted body ~ is divided into
triangular panels. This has been done such that to obtain a
symmetrical grid with respect to the y-0 plane. In order to
test the grid-independence of the algorithm, the symmetry
with respect to this y-0 plane has been removed. Results
of this can be observed in figure 6.5. The algorithm seem to
be relatively grid-independent.
For the unsteady motion 0 the water heigths for the six dif-
ferent types of motion have been calculated using the same
matrix equation (3.25) as for the unsteady motion o but now
with different right hand sides (for instance equation (3.30~.
see figure 6.6.
For this figure the following right hand sides for the unsteady
~
wave motion ~ have been used:
surge:-And TOil: w~zn2-yn3)
sway:-con2 pitch:-Un3
heave:-~n3 yaw: Un2 (6.4)
Of course as mentioned is the paragraph dealing with the
incorporation of the body boundary condition, these right
hand sides are not complete. For instance when ~ = (~1)
the right hand side for surge looks like:
_
-no ¢~=-name-n3¢\2Z (6.5)
But already some interesting features of the unsteady ware
character or the dynamic swell-up factor can be observed
here. In these figure there is a sway-yaw and a pitch-heaste
correspondence. This can also be seen from the right hand
side of equation (6.4~. But a different order for the magni-
tude: sway O(~) and yaw (ECUS is evident.
6.2 Added Resistance.
Once all the charactertic quantities are known, the pressure
can be determined from Bernoulli's equations. (added) Re-
sistance can be calculated directly by integration of first and
second order pressure, or by means of conservation of mo-
mentum, derived in a similar way as done by Huijsmans t23~.
-
Figure 6.4: Wigley Hull - Wave Profiles for Fn = 0.348
0.02
0.015
0.01
0.005
O
-0 .005
-0.01
Figure 6.5: Grid independence of the algorithm (Fn=0.2663.
7 Concluciing remarks.
We presented an asymptotic method for the calculation to
the influence of a slowly oscillating movement on the velocity
potential. Especially the steady~unsteady wave interaction
is important. The free surface condition for the unsteady
wave component was derived and the body boundary con-
dition. The potential function, source strength and Green
function are expanded in the small parameter w. A first
order approximation for the unsteady wave have been ob-
tained using calculation that only involves the evolution of
the steady wave Green function. This Green function must
also be used when calculating the steady state character-
istics. Future plans involves the calculation of (added) re-
sistance in the computer program as well. Also at the Ship
Hydromechanics Laboratory at the Delft University of Tech-
nology some model tests will be performed. So the calcu-
lated results can be compared with these test.
Acknowledgments
The author wishes to thanl: R.H.M. Huijsmans and
H.C. Raven of M GRIN for their valuable comments.
131
References
t1] Blok, J.J. and Huisman, J.,'Relative motions and swell-
up for a frigate bow.' The Royal Institution of Naval
Architects (1983~.
t2] Tasaki, R.,'On shipment of water in head waves', 10th
ITTC. London, (1963~.
Brandsma, F.J., Hermans, A.J.,'A quasi linear free sur-
face condition in slow ship theory,' Shiffstechnik 32 Heft
2 pp. 25-41.
[4] Brandsma, F.J., Low froze number expansions for the
wave pattern and the wave resistance of general ship
forms. Phd. Thesis. Delft (1987~.
t5] Baba, E., 'Wave resistance of ships in low speed.' Mit-
subishi Technical Bulletin 109 (1976~.
t6] Sakamoto, T. and Baba, E.,'Minimization of resistance
of slowly moving full hull forms in short waves.' (1985~.
t7] Hermans, A.J. and Huijsmans, R.H.M., 'The effect
of moderate speed on the motion of floating bodies.',
Schiffstechnik, 34 (1987~.
t8] Timman, R., Hermans, A.J. and Hsiao, G.C., Water
Waves and Ship Hydrodynamics. Delft University Press
(1985).
[9] Brard, R., 'The reprensentation of a given ship form
by singularity distributions when the boundary condi-
tion on the free surface is linearized.' Journal of Ship
Research, 16 (1972) 79-92.
t10] Kellogg, O.D., Foundations of Potential Theory,
Springer Verlag, Berlin (1929~.
t11] Newman, J.N., 'Evaluation of the wave-resistance
Green function: Part 1 - The double integral.' Jour-
nal of Ship Research, 31~1987) 79-90.
t12] Newman, J.N., 'Evaluation of the wave-resistance
Green function: Part 2 - The single integral on the cen-
terplane.' Journal of Ship Research, 31 (1987)145-150.
t13] Dercksen, A., 'Pa,~el integration of 1/A-function for
nearbylocations.' IvIARINreportno.50831-1-RF, Wa-
geningen (1988)
t14] Wehausen, J.V. and La~tone, E.V.,'Surface waves',
Hand~ok of Physics ( 1960~.
[15] Fox, L. and Parker, I.B., Chebychev polynomials in nu-
merical analysis. O~ford University Press U.K. (1970)
[16] Baker, G.A. jr.,Essentials of Pade appro~cimants, Aca-
demic Press, New York (1975~.
t17] Baker, G.A. jr. and Graves-Morris, P., Pade approxi-
mants, part I 1~ II, Encycl. of Mathematics (1981~.
t18] van Gemert, P.H., A linearized surface condition in low
speed hydrodynamics. Delft University of Technology
(1988)
t19] Shearer, J.R. and Cross, J.J., 'The experimental de-
termination of the components of ship resistance for a
mathematical model', Transactions of the Royal Insti-
tute of Naval Architects, vol.107 London, pp 459-473
(1965~.
t20] Kitazawa, T. and Kajitani, H., 'Computations of wave
resistance by the low speed theory imposing accurate
hull surface condition', Proc. Workshop on Ship Wave-
Resistance Comp., Bethesda, Marylalld (1979) 288-305.
t21] Tsutsumi, T., 'Calculation of the wave resistance of
ships by the numerical solution of Neumann-Kelvin
problem',P~c. Workshop on Ship Wave-Resistance
Comp., Bethesda, Maryland (1979) 162-201.
t22] Dawson, C.W., 'Calculations with the XYZ free surface
program for five ship models', Proc. Workshop on Ship
Wave-Reststance Comp., Bethesda, Maryland (1979)
232-255.
t23] Huijsmans, R.H.M. and Hermans, A.J., 'The effect of
the steady pertubation potential on the motions of a
ship sailing in random seas', 5th Int. Conf. on Numer-
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t24] Jensen, G., Mi, Z.X. and Soding, H., ' Rankine source
methods for numerical solutions of the steady wave re-
sistance problem,' Proc. 16th Symp. on Naval Hydrody-
namics., Berkeley (1986~.
DISCUSSION
. Nicholas Newman
Massachusetts Institute of Technology, USA
It is very interesting to see this expansion about the w=0 solution
complementing the studies at Delft where n = 0 is the basis. But here
the applications are less obvious. Added resistance is mentioned by
the authors, but it is generally negligible for small w. Perhaps
seakeeping problems in following or quartering seas are more
appropriate applications?
AUTHORS' REPLY
We would like to thank Professor Newman for this discussion; it will
give us the opportunity to clarify this. The problem is not only the
calculation of the added resistance, which can indeed be called small
for small w (but will be calculated just as well), but also the
determination of the dynamic swell-up. This swell-up is shown to be
an important factor-for instance, when looking at the phenomenon
of shipping of W green waterW. About your last remark, seakeeping
problems in, for instance, following seas, is also an interesting thing
to investigate. But when the incoming waves catch up with the ship,
this ship is actually ~ridingW on a wave. A highly unstable situation
occurs which cannot be considered by us thus far.
132
