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OCR for page 491
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
A Flow Model for a Displacement-Type Fast Ship
with Shallow Draft in Regular Waves
M. Kashiwagi (RIAM, Kyushu University, Japan)
ABSTRACT
A linearized 2-D boundary-value problem is
studied for the flow around a displacement-type
shallow-draft ship advancing and oscillating in
waves. In this problem, a homogeneous solution
of the wave elevation exists satisfying zero vertical
velocity on the free surface, which is introduced as
a homogeneous component in the expression of the
wave elevation. With this equation used as an in-
tegral equation for the pressure distribution associ-
ated with the disturbance of a ship, we can satisfy
the Kutta condition requiring a smooth flow at the
stern in addition to the kinematic condition of wa-
ter surface being equal to the vertical position of a
ship. An accurate numerical solution method is also
presented, using-Chebyshev polynomials for the un-
known pressure and employing a Galerkin scheme.
Excellent performance of this method is confirmed
by checking numerically Hanaoka's reciprocity theo-
rem and the energy conservation principle. Through
comparison of the results between satisfying and not
satisfying the Kutta condition, it is confirmed that
computed results based on the present flow model
are reasonable.
INTRODUCTION
Hydrodynamic problems of a shallow-draft ship
oscillating in waves with forward speed are of im-
portance not only as actual engineering problems
for developing high-speed ships but also as a mathe-
matical boundary-value problem concerning the ex-
istence of a unique solution.
In the unsteady problem for a displacement-
type ship, there is a lingering question associated
with the so-called line-integral term at the intersec-
tion of the body and free surfaces, which has made
it difficult to get a reliable solution. When a ship is
of shallow draft and thus can be represented by the
pressure distribution on the water surface, no line-
integral problem arises, thereby making the mathe-
matical treatment relatively easy.
However, there exists another problem peculiar
to a planing boat (Bessho, 1977~; not only the nor-
mal velocity but also the elevation of water surface
must be specified on the bottom of a planing boat,
and in addition the Kutta condition of smooth flow
at the stern must be satisfied as in the airfoil theory.
Theoretical investigations on this boundary-value
problem have been made by Bossho (1992) in two
dimensions. Bessho has pointed out that a solution
is unique if the elevation of water surface is speci-
fied as the body boundary condition; thus there is
no room to impose the Kutta condition, unless the
wetted length changes as in a gliding plank (Bessho
& Komatsu, 19844. Therefore, for a displacement-
type ship which is nearly vertical at the bow, it
is impossible to satisfy both the kinematic body-
boundary condition and Kutta condition.
In order to surmount this difficulty, Bessho pro-
posed a flow model (Bessho, 1992), in which a source
singularity is added at the bow intersection, corre-
sponding physically to a flow dammed in front of
the bow and then streaming out along the bottom
of a ship. However, computed results (Kashiwagi
& Bessho, 1993) based on Bessho's flow model ap-
peared eccentric in the magnitude of obtained phys-
ical quantities. Therefore, we need to reconsider
the flow model and propose a method enabling us
to impose both of the kinematic condition of water
surface being equal to the vertical position of ship's
bottom and the Kutta condition of smooth flow at
the stern.
A new flow model proposed in this paper is
physically similar to Bessho's (1992) or another flow
model proposed by Bessho & Suzuki (1986), but dif-
ferent in mathematical expressions. Since the ho-
mogeneous free-surface condition must be satisfied,
no additional terms can be allowed in the expres-
sions of the velocity potential and the vertical ve-
locity on the free surface. However, a homogeneous
wave elevation satisfying zero vertical velocity may
OCR for page 492
t
k4 k3 k. , ~ k2
In,_ ~ X
~=21 , TV
Figure 1 Coordinate system and schematic representation of generated waves on the free surface
be introduced in the expression of the wave eleva-
tion, which is used as an integral equation for the
unknown pressure distribution. The coefficient of
this homogeneous wave elevation is determined by
imposing the Kutta condition at the stern.
The integral equation is solved by representing
the unknown pressure distribution with Chebyshev
polynomials and employing a Galerkin method to
enhance the numerical accuracy. Excellent numeri-
cal accuracy is confirmed through the check of theo-
retical relations derived from Hanaoka's reciprocity
theorem (Hanaoka, 1959) and from the energy con-
servation principle associated with the damping co-
efficients in heave and pitch modes. Computed re-
sults are shown for the pressure, wave profile, hydro-
dynamic forces, and wave-induced motions, which
seems reasonable both in magnitude and variation
tendency, implying the validity of the present flow
model.
THEORETICAL FORMULATION
With the coordinate system shown in Fig. 1, we
consider a 2-D shallow-draft ship which is advancing
at constant velocity V while oscillating with circu-
lar frequency of encounter we. In the analysis to fol-
low, all physical quantities are nondimensionalized
in terms of the velocity V, the half length of a ship
= L/2, the gravitational acceleration g, and the
fluid density p. Consequently the nondimensional
parameters
K = V2 ~ ~ = V ~ ~ = K = e (1)
will be used, where ~ is known as the reduced fre-
quency and ~ is Hanaoka's parameter.
The fluid is assumed to be inviscid with ir-
rotational motion, introducing the velocity po-
tential, and the boundary conditions are lin-
earized. All first-order quantities are assumed to
be time-harmonic, with time dependence written
by exp~i~et). Then the relations to be satisfied on
y = 0 among the velocity potential ¢(x,y), the wa-
ter surface elevation Next, and the pressure pox) are
expressed as
(it—~ )~)(x'O) +Kr1(x) =—pax), (2)
,3y¢~(X,O)= (it- i3~)71(X). (3)
Here the pressure pox) on the free surface is associ-
ated with the disturbance of a ship.
Eliminating Next from (2) and (3) gives
( 0~ ) BY ( 0~ ) ~
on y= 0 (4)
A solution satisfying (4) and the radiation con-
dition at infinity is given by
~1
onyx ye = J ply, F(x—A, y) did, (5)
-1
where `3 \
F(x, y) = iw—~, J Gas, y), (6)
1 t°° elkly—it
Gfx,y) = Hi ~mO) Ok+—ip)2—K~k~ ~ ~
Gfx,y) is called the Green function, physically
the velocity potential due to the source singularity
of unit strength, which satisfies
(it—~ ) F(x, O_ ~ + KGy~x' O_ ~ =—5;`xy <8'
in the limit of y ~ 0_, where (6) has been used and
6(x) denotes Dirac's delta function.
From (2), (3), (5) and (8) the following relations
may be obtained:
~ 0(x,O) = | p(~)Fy~x—(,O_)d(, (9)
~1
r/(x) = ~ p(~)Gy~x—(,O_)d(. (10)
-1
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Representative terms from entire chapter:
flow model
Equation (9) is an integral equation for the un-
known pressure distribution, with the normal ve-
locity 0/~ ~ 0, it is easy to show that Gyps' y)
has the singularity of—(1/7r~lnr. On the other
hand, at a large distance from the origin, E~(z)
decays rapidly and only the sinusoidal exponential
terms remain. Thus substituting the asymptotic
form of Gyps' y) into (13) gives the asymptotic form
of the water-surface elevation as follows:
9(x) ~ k ~ kink H(k2) e-ik2X U(1 - fir)
ik,
ash >+x, (19)
r1(X) ~ k ilk H(`k~ ~ e-ikix Uf 1 - 4~)
—k 3k H(k3) eik3x
+ Eke H(
where H(kn), n = 1 ~ 4, is referred to as the Kochin
function, defined as
rl
H(kn)= J p(x) eiik7iX do, (21)
-1
where the plus sign in (21) is to be taken for n = 1, 2
and conversely the minus sign is for rz = 3, 4.
According to (19) and (20), there exist four dif-
ferent wave systems on the free surface in addition
to the homogeneous component expressed by the
last term in (20); these characteristics are schemat-
ically shown in Fig. 1.
BASIC SOLUTIONS
A shallow-draft ship is considered, heaving and
pitching in a regular incident wave, but for simplic-
ity only the case of head wave will be described here.
The basic solutions necessary in this problem
may be obtained by considering the following body
boundary conditions:
a) Heave mode (j = 3):
b) Pitch mode (j = 5):
03 (X) = 1, ¢3y (X, O_ ) = it (22)
95 (X) = X, ¢5y (X, O_ ) = iWX—1 (23)
c) Diffraction mode (j = 7):
07(X) = - 710(X) = _ eikox l 24)
¢)7y (x, 0_ ) = -into eik°X, `~; = ale + ko J (
where A, we and ko in (24) denote the (nondimen-
sionalized) surface elevation, circular frequency, and
wave number respectively of the incident wave. In
head wave, ko = k4.
Each basic solution must be obtained so that the
Kutta condition is satisfied at the stern (x = - 1);
this condition can be imposed in the form
xl~,m1 4[p; (X) ~) = 0 (25)
To satisfy these condition, (13) may be decom-
posed as follows:
where j = 3, 5, 7. pj*(x) for each mode can be de-
termined by specifying 71i (x) according to (22)-(24),
and pH(X) associated with the homogeneous surface
elevation can be determined from (28) irrespective
of the mode. The coefficient of the homogeneous
component, Ill in (26), may be determined by sub-
stituting (26) into (25).
FORCES AND MOTIONS
Radiation problem
It is clear from (2) that integrating the pressure
over the ship's bottom gives not only hydrodynamic
but also hydrostatic forces. Therefore the results
are expressed as
L3 _ ~ p3(x)dx = w2(A33—iB33) - 2K
-1
~1
L5 _ J Is (x) do = ~2 (A3s—i B3s)
-1
~1
M3 _ J pa (x)x do = w2 (As3—i B53)
-1
Ms——J p5(x)x do = ~2 (Ass—Bs5) - 3 K
(29)
Here Aij and Bij denote the added-mass and
damping coefficients in the i-th direction due to the
j-th mode of motion; these are defined as
Aij—iBij = e2ij _ i to ij ,
where 63 = 1 and 65 = e.
We note that the hydrostatic restoring moment
included in Me is obtained with the center of gravity
assumed to be equal to that of buoyancy.
Diffraction problem
As is clear from (2), the diffraction pressure on
the free surface, p7(X), includes the contribution
from the incident wave. Thus the resulting force
is equal to the hydrodynamic wave-exciting force,
which can be written in the form
L7 _ ~ p7 (x) do = KE
-1 J (31)
Pj(~)=Pj(~)+FjP (I)' (26) ]/~7 _ J. p7(~;)~5c=KE5
t1 -l
7Ij(x) = ~ pj(~)Gy(x—(,0_) dig (27)
~1
eden = J pH (a) Gy (x - I, 0_ ) d: (28)
-1
where the nondimensional value is defined as
E' Ei (32)
Heave and pitch motions
The complex amplitude of heave (X3) and pitch
(X5) may be determined by solving the coupled mo-
tion equations of heave and pitch, given by
(m'~2 + L3) 3 - L5—= L7
—Ma_ _ (I'm + M5)—= M7 J
~1
J5 -
m,n—
-1
Tm(X) d
.~
is Tn (~)
J 1 x/~=y~x—() d(, (38)
77m = | ,iF~ it(x) do.
(39)
Eq. (37) can be analytically evaluated, and the
result can be written as
where m' = m/pt2 = 0.2 and I' = I/pi4 = 0.25m' ~r
are used, with m and I being ship's mass and mo- Lm~n = Am 2 6;m,n
ment of inertia, respectively.
NUMERICAL SOLUTION METHOD
We must solve (27) and (28~. As noted before,
the kernel function Gy~x,y) includes a logarithmic
singularity—(1/~r) in r. In addition, since (27) and
(28) are of the same form, we write these in the form
——J p(~)ln~x—(~d
al
1 ~
+ I path Gy (x—() df = it(x) ~ (34)
J -1
~
where Gy denotes the regular part of the kernel
function Gy, and the right-hand side, R(x), is to
be Ajax) or eider.
Let the pressure distribution be expressed in
terms of the first-kind Chebyshev function Tn(X)' or
equivalently the Fourier series similar to that used
in the airfoil theory:
N T ( ) N ~
where x = cost and Pn (n = 0, 1, , N) are the
unknown coefficients to be determined.
Substituting (35) into (34) and employing
a Galerkin method with weight functions of
Tm(X)/~ (m = 0, 1, , N)' we have a linear
system of simultaneous equations:
N
At, Pn { (m~n+5m,n } = firm
n=0
where
olo=21n2, cam=—(m>1) |
(40)
where dm,n denotes Kronecker's delta, equal to 1 for
m = rip and zero otherwise.
Next, the double integral in (38) must be eval-
uated with an efficient and accurate method. The
method employed here is as follows: 1) Values of
Gym) at M points of X < 2 were computed in
advance and saved, 2) on each segment Gym) was
approximated by a linear variation, 3) then the in-
tegrals with respect to ~ given by
Bn
An = | Van df l
=~ Trig) Add J
were analytically evaluated, and 4) finally the in-
tegrals with respect to x in (38) were numerically
evaluated using Clenshaw-Curtis quadrature with
variable transformation of x = cos d.
In the meanwhile, the integral of (39) can be
given analytically as follows:
a) Heave mode:
b) Pitch mode:
c) Diffraction mode:
for m=0,1,. ,N, (36)
d) Homogeneous mode:
1 ii T (x)
or _l ~
x / \~ln~x—Bide, (37)
1Zm = ~5m,O (41a)
Arm= 2(im7~ (41b)
Him =—~—i Am ~ Jm (ko ~ (41c)
Elm = (i~m~Jm(~,) (41d)
where Junk) is the Bessel function of the first kind.
Eq. (36) can be solved by a conventional Gauss
elimination method. Once the solution of each
mode is obtained in this manner, the Kutta con-
dition of (25) can be imposed in the form
Hi = ~ + Fj ~ = 0, (42)
where N
_—lymph. (43)
n=0
From (42), the unknown coefficient Fj can be
determined. Here ~ and AH correspond to the val-
ues of (43) to be computed with the results of (27)
and (28), respectively.
With these results, it is straightforward to com-
pute hydrodynamic forces, the Kochin function and
wave-induced motions.
RECIPROCITY THEOREM
Applying Hanaoka's reciprocity theorem
(Hanaoka, 1959), we can derive several relations
to be satisfied among basic solutions defined in
(22~-~24~. These relations can be used to check
the accuracy of numerical computations. Bessho
(1992) showed some relations for flat-ship problems.
However, since the present flow model is different
from that of Bessho and we are concerned with
differences between solutions satisfying and not
satisfying the Kutta condition, we extend Bessho's
results to the present case.
Skipping the details of the derivation, the first
and second theorems proposed by Hanaoka may be
written for the present flow model as follows:
The first theorem is
I
where
where
al
| Pi ~X' vjy ¢-X' dX +—hi ~j
-1 2
= | pj (x) (iy ~—x) dx + 2 dj se, (44)
Firstly, substituting (22) and (23) for ~j(x) in
(54), it follows from (29) that
Wj= - 2iw2B~j. (55)
Secondly, utilizing (13) for lo (x) and transform-
ing the result with (14) and (22), we can show that
Wj = h k ~ ki ~Hj (ki ) ~2 + k2 ~Hj (k2) ~2
+ k3—ki ~ k3 THE (k3)~ —k4 ~Hj (k4)~ ~
+Ej Hj (I)—Fj Hj (w) . (56)
Equating (55) to (56), the following relation for
the damping coefficient in the j-th mode can be
obtained:
B:i=2 2tk k iki~Hj(ki)~2+k2~Hj(k2)~2)
—b k ~ k3 ~Hj (k3) ~2 _ k4 ~Hj (k4) ~2 ~
—2 A{ Pi Hj (I) } ~ (57)
20
O
cg
CO
~ -10
10
-20 -
20 .
10
~ O
CO
-10
-20
with Kuffacond at Eh=0.5, A/.r--2.0
-1.0 4.5 0.0
x/ (L/2)
0.5 1.0
with Kuffacond. at Eh=0.5, 1/L=2.0
.... .. ~ ~ 1 .... ....
, ,,, = ;~= ~'~~~~-'"
~ PI tch ~
1~ Witch (T--g) I
.... 1 1 ....
-1.0 -0.5 0.0
x/ (L/2)
with Kuffa coed. at Fn=O. 5
20 . . . . . . . . 1 . . . ~
10 .
b0 ~_ ~ , .
,jC -10 _ DO diffraction (}~1J _
. ~ Diffraction (T~gJ
-20 _ .... 1 .. . ~ 1 .... .
-1.0 -0.5 0.0 0.5
x/ (L/2)
0.5 1.0
1.0
Fig. 2: Pressure distribution at Frr = 0.5 and )/L =
2.0 (with Kutta condition)
Table 1: Accuracy of numerical results, errors in
Hanaoka's reciprocity theorems and the energy con-
servation principle (Frz = 0.5, A/L = 2.0 in head
wave; w = 3.343, ~ = 1.672)
Eq. No.
(48)
(49)
(50)
(51)
(52)
(53)
(57)
j =3
j =5
with Kutta coed.
Error (%)
0.1918E-04
0.1643E-02
0.3692E-04
0.4653E-03
0.7245E-04
0.1491E-02
0.2117E-O1
0.2382E-01
RESULTS AND DISCUSSION
w/o Kutta coed.
Error (~o)
0.4755E-05
0.3412E+OO
0.8947E-05
0.3424E+OO
0.2063E-04
0.3423E+OO
0.1334E+O1
0.1822E+O1
An example of numerical errors in six equations
shown by (48~-~53) and the energy-conservation re-
~ without Kuttacond. at Eh=0.5, 4/~-2.0
L ~ II=
- I ~ fl ev (jag) I -
0.0 0.5 1.0
x/ (L/2)
-0.5
without Kuffacond at En=0.5 4/~2.0
100, , .... , '
.
| ~ patch (jug) |
0.5 1.0
100 without Kuffacond. at Eh=0.5, 1/L=2.0
O ~
-50 ~ ~ Di~frac"~ (my) ~
. ~ L Diction (Tag) |
-100 .... 1 .... ~ .... ~ ....
-1.0 -0.5 0.0 0.5
x/ (L/2)
50
~ O
-50
-1.0 -0.5 0.0
x/ (L/2)
Fig. 3: Pressure distribution at Frz = 0.5 and A/L =
2.0 (without Kutta condition)
,.o
4
2
g
O
-2
6
4
2
~ O
5
-2
~ '
with Kuffacond. at Fn=0.5, 2/~2.0
~ ..
~ :~_~
-3 -2 -1 0 1 2 3
x/ (L/2)
at Fn=O. S. 4/0—2. 0
0 1 2 3
with Kuffa coed.
' 1 ' 1
.-
t~,f ix~;~
~
l
-5 ~ -3 -2 -1
x/ (L/2)
~ with Kuffa coed.
, ~ ~ 1-
1 ~
1 L'
~ W.
, amp---? ,xX^~
~
. ,,, ,
-5 ~ -3 -'
6
4
2
o
-2
with Kuffa coed.
.,,
2 I
I0 ~
-5 ~ -3
^~ ph=n ~ 2/L=2. 0
~ Diffraction |
1 1
. 1 . ~ ~ _
-1 0 1 2 3
x/ (L/2)
at Fn=O. 5, 2/~—2. 0
~ ' . ~ 1 ' 1
~ ~ 1
mix 1
=
_ . 1
incident breve + | I
.— Diffraction brave | j
, . 1 . ~ . 1 . i,
_ -1 0 1 2 3
x/ (L/2)
Fig. 4: Wave profile on the free surface (with Kutta
condition): Solid and dotted lines are by the
exact expression, symbols to and x) are by the
asymptotic expression.
ration for Bjj shown by (57) is presented in Ta-
ble 1 for both cases of satisfying and not satisfying
the Kutta condition, for En (= V//;' = 0.5 and
A/L = 2.0 in head wave.
We can see that the accuracy with Kutta con-
dition is virtually perfect. For the case of not sat-
isfying the Kutta condition, the accuracy is a lit-
tle worse but still within allowable tolerance. The
accuracy for other parameters of En and A/L was
confirmed to be of the same order as Table 1, im-
plying that the present boundary-value problem is
without Kuffa coed.
En
50 without Kuffa coed.
25
~ O
-25
-50
50
25
~ O
-25
on
eln
25
~ O
-25
-50 -
-5
at Fn=O. S. 1/I^2. 0
. ~ ~ ' 1 '
~ i__
I ~ ~ -tin
~ , I ~
i 1 1 :
I | Heave ~
., . 1 ., .
-1 0 1 2 3
x/ (L/2)
at Fn=O. 5, A/~—2. 0
, ~ 1
1 1 1 1 1
=
L_0 -- I hi
I ~ I lPitchl i
! '. I .'i
1 2 ^
-5 ~ -3 -2 -1 0
x/ (L/2)
without Kuffacond. at Fn=0.5, A/~-2.0
~ ., ., . ~ .,
1 ~ 1 1
—
1 1 ~ ~
I ~ I Di ffraction ~
1 . 1 ., . 1 .':
-3
without Kuffa coed.
-2 -1 0 1 2 3
x/ (L/2)
at Fn=O. 5, 2/ -—2. 0
' 1 ' ' ' 1 '
1 1 1 1 ~
A- -t------r~ ~----~
!~
r| I ! ~ i
| | Incident drive + Li
| | Diffrection - diva | i
1 .'1 . ~ . i .'1
-1 0 1 2 3
x/ (L/2)
Fig. 5: Wave profile on the free surface (without Kutta
condition): Solid and dotted lines are by the
exact expression, symbols (o and x ) are by the
asymptotic expression.
successfully solved.
Computed pressure distributions for Fn = 0.5
and A/L = 2.0 are shown in Fig. 2 for the case of
satisfying the Kutta condition and in Fig. 3 for the
case of not satisfying. As expected a big difference
can be seen between Fig. 2 and Fig. 3 near the stern
(~ = -1), and the pressure distribution with Kutta
condition looks reasonable.
Figures 4 and 5 show the wave profiles for the
same parameters as that in Figs. 2 and 3. The wave
amplitude is normalized in terms of the maximum of
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Gap in Wave Amplitude at Bow at Eh=O. 5
~ . , . . ~
- —— fioeve | r | /X3
- Pi I:` oh ~ /X! (L/2J ~ ~ 1 l
" ~ | Diffraction | r | /a _ ~
—~
_ -~;'~;
,_. ......... , ,.,., , ,,.,.,.,.,., , ,,.,.,,, ,, ,.,,, ,,.,.,.,.,.,,,. 1
Motion Free I r I /a
, I I
0 1 2/L
Fig. 6: Amplitude of homogeneous component of the
wave elevation, corresponding to the gap in
wave amplitude at bow, at Fn= 0.5.
10.0
an
an
2.0
on
-2.0 —
1 n
ns
on
.5
-1.0
with Kuffa coed. at Fn=O. 5
- 1 ~ 1 ~ ~ ! ~ .
T T 1
A33/p (L/2) -2~/a'
- --- B33/p (L/2J a>.
: . 1 _ ,
Ll
0 1 2
Reduced frequency:
with Kuffa coed. ^~ Fn=0 5
___
. i it,:
__
1
an
an
l l l _
~~
_ . 1 . _
3 4 0
a. (L/2) /V
.
10 .
o
-10
-20
B55/p (L/2) -2/3a
----- B55/p (~/2) a).
! 1 1 1 1 _
1 2 3 4
Reduced frequency: w.(L/2)/V
Fig. 7: Added-mass and damping diagonal coefficients
in heave and pitch (with Kutta condition), at
Fn = 0.5.
each mode in the radiation problem and in terms of
the incident-wave amplitude in the diffraction prob-
lem. In the case of not satisfying the Kutta condi-
tion (Fig. 5), the amplitude is of order of 25, which
seems quite unrealistic. In contrast, the results with
Kutta condition (Fig. 4) seems reasonable, judging
from the order of wave amplitude. The open circles
and crosses in Figs. 4 and 5 are the results computed
by asymptotic expressions shown as (19) and (20~.
Slight difference from the exact results based on (13)
can be seen only in close proximity to the stern and
bow, which means that the evanescent wave term is
relatively very small.
As shown by (13), the flow model in the present
paper gives necessarily a gap in the wave ampli-
tude at the bow, corresponding mathematically to
the amplitude of a homogeneous component of the
wave elevation, Ace. The nondimensional value of
this amplitude is shown in Fig. 6 for various modes
at Fn = 0.5. For the realistic case of a ship freely
-10 _
without Kuffa coed. at Fn=O. 5
-em 1 _ 03/p(~/2,2-2X/~2 1 ~
, ~: ,~
1 2 3 4
Reduced £=equency: a'. (L/2) /V
without Kuffa coed. at Fn=O. 5
1 1 1 , ~ :
B55/p (L/2J -2K/3Q)
B55/p (L/2) Jane
~ 1 ~ 1 1 1
0 1 2 3 4
Reduced frequency: ~ (L/2J /V
Fig. 8: Added-mass and damping diagonal coefficients
in heave and pitch (without Kutta condition),
at Fn = 0.5.
with Kuffacond. at Fn=0.5, head rave without Kuffacond. at Fn=0.5, head wave
40 ~ _ lE3l/pUa(L/2) | | I ~~ 100 l ~ —
ga.o~ ~KSl/2 And 2 °~Isal/2ga~
i' ~~ :=T- ~2 20$
0 1 2 3 4 0 1 2 3 4
4/L 2/L
1.0
0.8
0.0 -
withKuffacond. at Eh=0.5, head wave
1 ~ 1 1 ~ _
Heave ( I X3 I /a)
_ In Pitch ( I X5 1 /kOa) _
i0.6 . _ __
0.4 __ _ _
0.2 ~
/,
/,
al'
. .~ _
_ I L
0 1 2
1/L
_ _ ~
3 4
Fig. 9: Upper: Wave exciting forces in heave and
pitch, Lower: Motion amplitudes of heave and
pitch (with Kutta condition), at En = 0.5 in
head wave
oscillating in waves (denoted as motion free), the
gap in amplitude at the bow decreases as the wave-
length increases. We note that the value of Is in
(13) corresponds physically to the amount of fluid
periodically dammed at the bow and then streaming
down along the bottom of a ship.
The added-mass and damping coefficients (only
for diagonal coefficients) versus the reduced fre-
quency (w = we{/V) are shown in Fig. 7 for the case
of satisfying the Kutta condition and in Fig. 8 for
the case of not satisfying. All results change dras-
tically around ~ = 0.5, which is equal to ~ = 1/4
because ~ is given by we(/V = 2wFrz2. Since no
experimental results are available, it is difficult to
make a definitive judgement. However, at least we
can say that the results satisfying the Kutta condi-
tion seem much better than the results without the
Kutta condition.
1.4
1.2
1.0
0.8
~ 0.6
.~
no
no
0.0
.
without Kuffacond. at Fn=O.S, head cave
~T
, 1 !
~ it'
--it 1~ -~---1
~ 1 ~ , 1 1 . ,,
I ~' I | Heave (lX3l/a)
;- | I ~ -- Pitch ( I X5 1 /kOa) n
I I 1 ' I I r I
0 1 2 3 4
2/L
Fig. 10: Upper: Wave exciting forces in heave and
pitch, Lower: Motion amplitudes of heave and
pitch (without Kutta condition), at Fn = 0.5
in head wave
Computed results of the wave-exciting forces
and wave-induced motions are shown in Fig. 9 for
the case with Kutta condition and in Fig. 10 for
the case without Kutta condition. We note that
the results without Kutta condition are unrealistic
judging from other past computations for a surface-
piercing body and related experiments. On the
other hand, the results for the case of satisfying the
Kutta condition seem to be reasonable in magnitude
and variation tendency, which supports the validity
of the proposed flow model.
CONCLUDING REMARKS
In the hydrodynamic problem of a displacement
type ship with shallow draft, a solution must satisfy
the kinematic condition of water surface being equal
to ship's vertical position and the Kutta condition of
flowing out smoothly at the stern. However there
has been no convincing theory accomplishing this
requirement, and in fact it is known that a solution
satisfying only the kinematic condition gives very
large pressure and wave elevation near the stern.
To overcome this problem, a new flow model was
proposed in this paper, which introduces a homoge-
neous wave elevation satisfying loo/by = 0 into the
integral equation for the pressure distribution, spec-
ifying the surface elevation as the boundary con-
dition. The coefficient of this homogeneous com-
ponent can be determined by imposing the Kutta
condition at the stern.
Numerical results were confirmed to be very ac-
curate through checking several relations derived
from Hanaoka's reciprocity theorem and the energy-
conservation principle. It was also confirmed that
computed results satisfying the Kutta condition
were reasonable judging from the magnitude and
variation tendency of the pressure, wave profile, hy-
drodynamic forces, and wave-induced motions. Fi-
nally we should note that conventional results with-
out Kutta condition, which were also shown in this
paper, were much different from the results by the
present flow model and thus not acceptable even as
an approximate solution.
REFERENCES
Bessho, M., "On Hydrodynamic Forces Acting on
an Oscillating Gliding Plank," Journal of Kansai
Soc. of Nav. Arch., Vol. 165, (1977), pp. 49-57.
Bessho, M. and Suzuki, K., "Porpoising Instability
of Two-Dimensional Oscillating Planning Plate,"
Mem. of National Defence Academy, Vol. 25, No. 1,
(1986), pp. 25-43.
Bessho, M., "Hydrodynamic Forces Acting on an
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a Constant Speed," Mem. of National Defence
Academy, Vol. 30, No. 1, (1992), pp. 41-49.
Bessho, M. and Komatsu, M., "Two-Dimensional
Unsteady Planing Surface," Journal of Ship Re-
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Hanaoka, T., "On the Reverse Flow Theorem Con-
cerning Wave Theory," Proceedings of 9th Japan
National Congress for A pplied Mechanics, ( 1959~.
Kashiwagi, M. and Bessho, M., "A New Flow Model
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