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OCR for page 225
Calculation of Nonlinear Water Waves around a 2-Dimensional
Bully in Uniform Flow by Means of Boundary Element Method
K. Suzuki
Yokohama National University
Yokohama, Japan
Abstract
In this paper, nonlinear wave making phenomena a-
round a two dimensional body is studied. The analysis
uses the boundary element method based on Cauchy's in-
tegral theorem and the wave profile is calculated by time
marching integration based on the semi-Lagrangian ap-
proach. Steepening or Breaking waves can be simulated
by this scheme.
Two problems are discussed; a semi-circular mound
in shallow water, and a floating body with semi-infinite
length. In cases including a uniform flow component, nu-
merical treatments have some difficulties. In the semi-
circular mound problem, nodal points on the free surface
move out of the calculation region, and in the floating
body problem, nodal points concentrate in front of the
bow. In the present work, numerical difficulties caused
by these problems are settled and numerical examples are
given for several cases.
1. Introduction
Many numerical schemes hare beets presented fur the
purpose of the analysis of free surface flu: jar receipt ~ C 1' S
Some of the schemes, however, steed bigly perf~r,~, e
computers with very large memory storage. T~ tl~:>se cases
some difficulties remain in practical applications. In Flee
problems of nonlinear water waves with steepening or
breaking, many complicated numerical procedures and
considerable CPU time are needed. If we neglect viscos-
ity and assume irrotational motion, a numerical scheme
based on the less complex boundary integral equation cart
be employed, which does not need high performance com-
puters.
For the problem of two dimensional free surface flow,
Longuet-Higgins and Cokelet t1] introduced the boundary
element method ~ abbreviated as BEM ~ based on Green's
integral theorem, in which they used the mixed Eulerian-
LagraIIgiall method in order to follow nodal points on
the free surface by means of time marching integration.
This scheme was modified by Vine and Greenhow t2],
who employed BEM based on Caucl~y's integral theorem
and highly accurate scl~e~e of time marching integra-
1 1;
225
tion. Similar schen~es were developped in last decade and
iIIvestigatioIls of two dimensional nonlinear water wav
including steepening or breaking have been carried out
t3] t4] t5] t6] . In most cases, however, wave-making pheno~ll-
ena around a body ir1 uniform flow A] have slot been tried.
The main difficulty is that nodal points on the free surface
can move according to the uniform flow compoIlerlt.
In the present study, two problems are described; a
semi-circular mound in shallow water, and a rectangu-
lar floating body with semi-i~lfinite length. In tile for-
mer problems, nodal points on the free surface will pass
through the downstream boundary, and in the latter prob-
lem, nodal points will concentrate at the stagnation region
in front of the bow, if the suitable way cannot be found.
In this paper, numerical treatments for these problems
alla related ones are discussed, alla several numerical ex-
amples are given. Some checks about the validity of the
present numerical method are also given by comparing
the results to the other theories and experilnellts, a',`:l ail\
varying the number of nodal points, tile size otcalcuJ ,ti -all
region and tile time interval.
2. Basic Equations
Two examples of nonlinear wave making pllenolllella
around a two dimensional body in uniform flow are dis-
cussed in this paper.
I) Semi-circular mound in shallow water as in Fig. 1.
2) Rectangular floating body with sen~i-infinite length as
in Fig. 2.
IY
N ~ 1 2 3 Free Surface |
_ Ct
C=C+uC~
Bottom so .
h Cal
, ~ Cal ~ J
Fig. 1 Sell~i-circular mound ill shallow water.
x
OCR for page 226
The downstream boundary is open in tile first problem,
and a stagnation region exists in front of the bow in the
second problem. When using the numerical method ex-
plained below, various numerical difficulties are arisen be-
cause of these features of the problem. Numerical treat-
ments for each of these problems are explained in later
sections.
~ (y
N ~ .2 3 ~ C ~ ,NF Ax
_, '/'''`~//C//~"~/"' .
Ci I I
/////l/////f/Cp/i////////~`ti/////~/
Fig. 2 Rectangular floating body with semi-infinite
length.
The coordinate system is taken as shown in Fig. 1 or
Fig. 2. For the sake of convenience, all equations in this
paper are normalized by a characteristic length 1, and a
uniform flow U. For example, the normalised time t is
expressed as U/l x real time. In the first problem, the
radius a of semi-circular mound can be chosen as the char-
acteristic length, and in the second problem, the draught
d can be chosen as the characteristic length. If we neglect
the fluid viscosity and also assume irrotational motion,
the governing equation of the flow field is J~aplace's e.~.~-
tion. The complex velocity potential call be i,~trod''~d
as follows,
win; t) = Aid, y; t) + idle, y; t) (,1)
z = ~ + is (2)
where ~ is the velocity potential and ~ is the stream func-
tion. If the contour C of clockwise direction is chosen and
the singular point zo is considered on C as shown in Fig.
1 or Fig. 2, the following equation can be introduced by
Cauchy's integral theorem,
icxw~zo; t) + / (; Adz = 0 (3)
c Z—To
where cat = or, if the contour is smooth at zO. If ~ is given
at z0 ~ C¢, part of C ), the real part of eq. (3) can be
taken. On the contrary, if ~ is given at zo ~ C,~, part of C
), the imaginary part of eq. (3) call be taken. Taking the
real and imaginary parts of eq.~3), following equations are
obtained.
-Ct?,~(Zo; t) + Re,/ ~ Adz = 0 on Cal, (4)
(matzo; t) + Im,| ~ ; Adz = 0 on Cal (5)
c Z—To
These formulae are integral equations of Fredholm type of
else second kind, in which ~ is unknown on C¢, (eq.~4~) and
is unknown on Cat (eq.~5~) respectively. To solve these
integral equations, nodal points are distributed along the
contour C as shown in Fig. 1 or Fig. 2. We 1lave NF
nodal points on the free surface which are movable and
a total of N1D nodal points on the contour C. On each
boundary element divided by these nodal points, it is as-
sumed that the complex velocity potential w varies lin-
early in z. Using the well known procedures by means
of these linear boundary elements, the following discrete
expressions are obtained with respect to eq. (4) and (5),
where
-NP ~
Re ~ Ek,nWn = 0 on ~
n=!
-NP
Im ~ Ek,nWn =
n= ~1
'` (Gil
_ O (~t C'7/~ ~ ~ ~
Zk— Zn—1 Zn —Zk Zk — Zn+1 Zn+-l
[k,n = In + In
Zn — Zn—~ Zn—~ — Zk Zn — Zn+ -1 Zn
Zk—Zk—2 Zk—1 —Zk
[k,k—~ = ID
Zk—~ —Zk - 2 Zk - 2 —Zk
1 Zk+i — Zk
[k,k = no
Zk—1 —Zk
rk,k+1 = k Zk+2 n1 Zk+2 —Zk
Zk+1 —Zk+2 Zk+1 —Zk
Zk
Zk
(8)
(9)
(10)
(11)
Since terms including known ~ and ~ remain in deft hand
side of eq. (6) and (7), these terns must be transposed
to the right hand side. A set of sin~ultaneous equations is
thus obtained. Since the coefficient matrix of this equa-
tion system has a property of diagonal superiority, Gauss-
Seidel iterative method can be used to solve the sin~ulta-
neous equation.
In order to follow nodal points and velocity potential
values on the free surface, the niixed Eulerian-Lagrallgian
formulation is employed. Namely the dynamical free sur-
face boundary condition is expressed as follows by using
the material derivative
dig = 1 {A + (~1~) }-COY+ 2' (12)
where TO = gl/U2. In eq. (12), a uniform flow component
is taken into consideration. The kinematical free suface
boundary conditions are also expressed as follows.
dx = 0+ = Re d- (13)
Y = -¢ =-Im— (14)
dt By dz
Integrating the ordinary differential equaticlls (12)-~( l1)
numerically with time increment, the `~-~-e profile CR.11
be obtained at each time step. Hanll~illg s predict.,r-
corrector method is used as the time marching integral loll
technique. This method needs the values at the first three
time steps, which are calculated by means of Runge-Kutta
method.
Wave making drag acting on the mound or the rect-
angular floating body can be estimated in each time step.
Since the pressure coefficient is expressed as
226
OCR for page 227
C P - Po = 1 - (ok) - (my) - 2^yoY - 2 bf (15)
by Berno~lli's theorem, the coefficient of wave making
drag is obtained by the following pressure integral,
Cw= --lu2 =-2,iCpn~ds, (~16)
where s is the girth of the mound or the wetted bow part of
the floating body, and no is the direction cosine of outward
normal on s to ~-axis.
3._Semi-circular Mound in S_allow Water
3.1 Treatments for Numerical Computations
General descriptions of the numerical method used in
this work are given in the preceding section. In practi-
cal free surface computations, however, some difficulties
caused by the uniform flow component must be settled.
First the initial condition locust be given. In general,
the initial condition is given by U = 0 at t = 0 and the
flow is increased gradually to uniform to maintain the
stability of the numerical computations. In the present
study, however, the uniform flow is give at t = 0 as
~ = :e on C+, ~ = 0 on Cat,, (17)
in order to eliminate the accelerations effect on to Fly--.
This initial condition gives no influence Ott the stability- of
the numerical computations. This procedure is effective
also for saving CPU time.
Since the nodal point NF shown in Fig. 1 can Gove
according to the uniform flow component, the treatment
of the downstream boundary is more difficult. If this
boundary is fixed, nodal points on the downstream free
surface will move out of the calculation region. In order
to avoid this difficulty, a new nodal point is introduced
on the upstream surface, and a nodal point which passed
through the downstream boundary is deleted. The de-
tailed numerical process can be described as follows:
1) As shown in Fig. 3, a fixed upstream boundary and
an initial downstream boundary are given. On the free
surface, NF nodal points which are movable with time
marching are given.
2) The downstream boundary can Gove corresponding to
the nodal point NF. Unless the nodal point NF- 1 pass
through the initial downstream boundary, the computa-
tion is continued without changing numerical procedures
as shown in Fig.3 (1~.
3) If the nodal point NF—1 passed through the initial
boundary, the downstream boundary is changed to new
position of the nodal point NF- 1 at next time step. At
the same time, as in Fig. 3, the nodal point NF is deleted,
the nodal point slumbers are replaced, and the new nodal
point 1 is added between nodal point NP and 2.
The position and the velocity potential of the new nodal
point 1 is given simply like
x~ = 2(\XNP+X2), Y! = 2(YNP+Y2), ¢1 = 2(¢NP+¢2~.
(18)
This replacement technique is convenient, because the cal-
culation region can be kept almost the sense size.
NP 1 2 3 1xF-3 I\F-2 1xF-1 NF
( 1 ) a=
,.-_ I I
A ,: B C
NP 1 2 3 NF-3 NF-2 NF-l NF del.
It ~ t t t ~ t t
NP 1 2 Nf-4 FF-3 FF-2 NF-l NF
(2)
1' 1
A BD C
A: Upstream Boundary ( f i x )
B: I ni t ial Downstream Boundary
C: Downstream Boundary ( movable )
D: New Downstream Boundary
Fig. 3 Treatment of nodal points on the free surface.
For the movable downstream boundary-, the ~onditi.:'n
for ~ must be given, because this boundary- is C¢. as in Fig.
1. In the finite difference method, the zero-extrapolation
technique is ordinally used for an open boundary of this
kind. In this analysis, however, BIDM is employed, and an-
other way must be found. When the disturbance velocity
potential ~ is introduced as
~ = X + A, IF = XNF + (PNF, (~19)
the downstream conditions for ~ are given below, which
utilize the solution form based on the linearized free sur-
face condition.
1) If YNF > 0,
for y ~ o,
for y < 0,
2) andifyNF < 0,
where
by sinh kh + cosh kh
IF kYNF sinh kh + cosh kh
Oslo key + h)
IF kYNF Sigh kh + cash kh
(20)
(21)
= IF hl~k~y + h) (22)
k - so tanh kh = 0. (23)
These equations satisfy the condition of ~ = IF at y =
YNF and the continuity conditioll, and eq. (20) is the
linear extrapolation of eq. (21~.
As described in the previous section, in order to follow
the wave profile in each tinge step, the numerical integra-
tions of differential equations (13), (14) are needed. For
227
OCR for page 228
this purpose, dw/dz On tile free surface must be evalu- 2.D
ated. Several methods are known, however, the following
two simplest methods for approximations of dw/dz were
used for the present problem.
1) Upstream difference
2) Downstream difference
3.2 Numerical Examples
_ . .. . _
dw wn—Wn-1 (124)
dz Zn—Zn-l
dw we'—Wn+1 (25,
dz Zn—Zn+l
Numerical examples are shown for the semi-circ~lar
mound of radius a = 0.1m in the uniform flow of U =
0.5m/sec and the water depth ht = depth/a) = 0.25.
Some examples are also given for the other speeds of urti-
form flow or the other water depths.
As the first example, else schemes for dw/dz on the free
surface have to be examined numerically. The computed
wave profiles based on the schemes of upstream difference
arid downstream difference are shown in Fig. 4. Ilk this
example, numerical conditions are chosen as; the position
of downstream boundary Amid,, = -7.5' the position of up-
stream boundary :Uma~ = 10.0' the number of nodal points
on the free surface NF = 100 ~ length of all elements are
equivalent ), and the time interval At = 0.05. Fig. 5
shows Glee wave drag coe~cier~ts for the sense cases. In
Fig. 4 and 5, the scheme of downstream difference seems
to be a suitable one for this problems. However, if ex-
tending the wave height to vertical direction as in Fig.6,
the reflected wave front the downstream boundary can be
observed. For this reason, the upstream difference which
can simulate the steep wave as in Fig. 4 is employed as
the scheme for dw/dz.
1=3.50 ( 0.70 see ) DOWN-STREAM DIFFERENCE
t=2.55 [ 0.51 see ) UP-STREAM DIFFERENCE
~ ~ 4 ,~
Fig. 4 Wave profiles based On tile scllellle of upstream
difference and downstream difference.
According to the numerical conditions, soluble examples
are computed. Fig. 7 shows the cvu~p~ted wave profiles
for U = 0.3, 0.4, 0.5m/sec, where h = 2.5 at Teal time =
228
Cw=Rw/[poU2)
_
~.6
'.2
o. 8
n ~
UP-STREAM D I FFERENCE
/ DO`N-STRERM D I FFEREN~
u.u- I.0 2.0 3.0 4.0
Fig. 5 Wave drag coefficients based on the schemes of
upstream difference and downstream difference.
1=3.5 ~ 0.7 see )
Fig. 6 Wave profile based on tlte scheme of downstream
difference.
Go
4.O
0.5 m/s
0.4 m/s
0.3 m/s
- ~ . O
Fig. 7 Wave profiles for U = 0.3, 0.4, 0.5m/sec.
( h = 2.5, real time = 0.51sec ~
0.51sec in all cases. The difference of wave steepening
points can be simulated well ill this example. Fig. 8
shows the computed wave profiles for h = 1.25,2.5,5.0,
where U = 0.5m/sec and t = 0.5. Fig. 9 shows flee
wave drag coefficients for the same cases. The case of
h = 1.25 is the limit of what call be computed by the
present technique. As shown in Fig. 9, wave breaking
occurs immediately after the wave profile ill Fig. 8. As is
well known, numerical techniques based on BEM cannot
simulate the wave after breaking.
OCR for page 229
~ n
-°.51
-1. 0_
Fig. 8 Wave profiles for h = 1.25, 2.5, 5.0.
~ U = 0.5m/sec, t = 0.5 ~
Cw=Rw/ [po.U2 )
z . c
1.6
1 ~
D. 3
DEPTHS: I .25 1 0. 125 r )
DEPTHS: 2.50 ~ 0. 250 m )
f
DEPTH/a: 5.00 1 0.-500 m )
1 . 0 2. 0 3. 0 4. 0
Fig. 9 Wave drag coefficients for h = 1.25, 2.5, 5.0.
( U = 0.5m/sec )
2.
2.0
1.61
1.2 .
n R ~
_ . _
0.4 ./
0.0 _
Cw=Rw/ [poU2 )
or
DECO. 05 /
I\\
Dl=O. 07
~ \~
Fig. 10 Wave drag coefficients for the cases of
At = 0.03, 0.05, 0.07.
37t h st op t=2. 59 1 O. 518 see ) Dt=O. 07
1 ~~ 1
DEPTH/e : 5. 00
DEPTH/e : 2. SO
_.._.._.._. DEPTH/e : 1. 25
Fig. 11 Wave profile for the case of At = 0.07.
For the first example, ejects of the time interval are
exemplified. The wave drag coefficients for the cases of
lit = 0.03, 0.05, 0.07 do not show serious differences caused
by the time interval ~ Fig. 10 ). In Fig. 11, however, the
wave breaks unusual in case of lit = 0.07. Since the wave
profiles for At = 0.03 and 0.05 are almost saline in this
problem, At = 0.07 is considered as a rough time step for
the present problem.
0.5~
n n
-0.5
t.0_
0.5
- 9. 0
-7.0
XMIN : -lO.0
XMIN : -7.S
Fig. 12 Upstream free surfaces for the cases of
xmin = -7.5 and-10.0.
2;~,
T-0.40
T=t.20
r=2. Do
T=2. 80
T=3. 1 2
Fig. 13 Time history of wave profile. ~ U = 0.5m/sec,
Cumin = - 10.0, NF = 157, At = 0.04 ~
In some of tile examples, small undesirable ocillatioIIs
can be seen on the upstream surface. Whell the numerical
conditions are changed to Cumin = -10.0 alla NF = 115,
those undesirable ocillations are suppressed ~ Fig. 12 ).
In the final three examples, Cumin = -10.0, NF = 157,
alla /\t = 0.04 are used ~ Fig. 13, 14 and 15 ). The com-
puted time history of the wave evolution is given as in Fig.
13. At the final time step, the wave becomes very steep.
In Fig. 14, this steep wave profile just before breaking
is conspired with Else result based Ott the finite difference
method by Miyata et. al.~84~9] Both results show a fairy
good agreen~ellt. The wave profile before steepness pre-
dicted by the present method is also compared with the
experi~nelltal result by Miyata et.al.~8~9] in I?ig. 15. Since
the experimental wave profile is replotted front the pub-
lished photograph, small errors are probably included, anal
229
OCR for page 230
the time step is not equivalent ifs both cases, because ini-
tial conditions are different with each other. Though clear
conclusions cannot be described for the above reasons, the
present method can be regarded as one of the powerful
simulation tools for real phenomena of steep wave.
To
NF= 157 DT=~. 04
CRL. BY H. ~ I YRTR
.ol
Fig. 14 Wave profiles by the present method and the
finite difference method. ~ U = 0.5m/sec ~
~ n
NF= 157 DT=O. 04
EXP. BY H. MITRTR
1.01
Fig. 15 Wave profiles by Else present method and else
experiment. ~ U = 0.5m/sec ~
4. Rectangular Floating Body with Semi-infinite Length
4.1 Treatment for Numerical Computations
As described in section 3.1, tile means of solving some
numerical difficulties have to be given also for this prob-
lem. If the deep water deftly is assumed, the boundary
conditions of upstream and downstream are simply writ-
ten as
= x on the upstream boundary, (26)
~ = kit on the downstream boundary (27)
respectively, because there is no free surface at tile down-
stream boundary in this case. However tile condition (27)
is not applied to the top and bottom nodal point at the
downstream boundary, wl~ich are regarded as points of Cal
on the body and the bottom in the present calculation. If
flee water depth become shallower, the problem becomes
more complicated and other considerations will be needed
for both boundary conditions.
In order to start the computation, an initial condition
is needed. Though the uniform flow condition is employed
on all C¢, region as in eq. (17), it is not able to be applied
to the present problem, because of existence of stagna-
tion point on the bow. For this type of tile flow field, the
numerical solution of the double model flow can be intro-
duced as the initial condition. The double model solution
can be obtained by tile boundary conditions of
= 0 on the body and tile free surface at rest, (28)
~ = -ah on the bottom (29)
with eq. (26) and (27~. The obtained values of ~ on the
free surface are employed as tl~e initial condition with the
other boundary conditions.
The most serious problem is caused by the existence of
the stagnation region. Because of the uniform flow com-
ponent, the nodal points will concentrate in front of the
bow and the distance between nodal points NP and 1
will become larger and larger. This causes unusual wave
profile around Else nodal point 1, which is shown in the
subsequent section by a numerical example. In this case,
the replacement technique of nodal points as explained
in section 3.1 cannot be introduced, because the verti-
cal boundary wills the nodal point NF is not movable
and nodal points in front of the bow cannot be deleted
in order to simulate the wave breaking. To counter this
difficulty, long elements are introduced on y = 0 before
the nodal point 1 as shown in Fig. 16. Following bound-
ary conditions are imposed on nodal points on these long
elements.
fib = ;z on the long element region (30)
These long elements act as wave suppression plates.
ALONG ELEMENTS:
Fig. 16 Long elements on tile upstream surface.
In order to evaluate dw/dz on tile free surface, the
following three methods are used.
1) Upstream difference ~ same as eq. (24)
2) Downstream difference ~ same as eq. (25)
3) Centered difference
dw z7~_zW76_~} ~Zn—An- + Wz,,_zW7,+ - In—an+
= R]
dz ~Zn—An-. + ~Zn—An+ -.
Eq. (31) corresponds to tile weighted mean of the up-
stream difference and the downstream difference. Nodal
points on the free suface can be followed by the above
numerical procedures. For the nodal point NF, how-
ever, the horizontal velocity component Re~dw/dz) is ne-
glected, because this nodal point must be restricted to
move along tile bow.
4.2 Numerical Examples
Before allowing several numerical examples, tile nu-
merical accuracy of tile double model solution lllUSt be
230
OCR for page 231
studied, because it is employed as the initial condition for
the present problem. Two numerical solutions of Oman =
20.0 and 80.0, where Cumin = -10.07 are compared with the
analytical solution based on Schwarz-Christvffel transfor-
mation t10] in Fig. 17. The results are slightly different,
though the same tendency of ~ is obtained. The extension
of the upstream boundary does not improve the numeri-
cal solution. Since the numerical solution is employed as
the initial condition, it cannot be avoided that ~ includes
small errors initially. However, initial flow velocities on
the free surface are accurate, because the velocities are
obtained from derivatives of ¢.
''\
"amp
"amp
. -to
As
me,
"
"A
'% —~
-- — — — NUMERICaL SOLUTION: SHOD so.n
-- aNaLrtlC SOLUTION
~ —
it SOLUT l ON : XHqX 20.0 ~
Fig. 17 Double model solutions.
As discussed in section 3.2, several numerical treat-
ments and conditions are studied. These are carried out
as d = 0.1m and U = l.Om/sec ~ Fig. 18 ~ Fig. 24 ).
First, the electiveness of long elements on the upstream
free surface is verified. Long elements are arranged before
:r = -10.0, :e = -10.0 ~ -8.0 is divided by 10 normal el-
-8 -7 -6
WITH LONG ELEMENTS
DO,'N-STRERM DIFFERENCE I ~ t18
-5
- 4
UP-STREAM D I FFERENCE t ~ 1.6
-2
_ 7
/ ~ r
I I I I I I I I ~-l
-9 -8 -7 -6 -5 -4 -3 -2 - ~ x
CENTERED D I FFERENCE I = Z6 / \ r
I 1 4 1 1 ~ I I me HI >
-9 -8 -7 -6 -5 -4 -3 -2 - ~ x
Fig. 19 Wave profiles based on tile schemes of upstream,
downstream and centered difference.
ements, and x = -8.0 ~ 0.0 is divided also by 120 normal
elements. By employment of these long elements, unnat-
ural waves induced on the upstream surface, shown in the
upper example of Fig. 18, are suppressed as in the lower
example. In this example, the centered difference scheme
is used to calculate dw/dz oft the free surface. The pre-
dicted wave profiles at final time step by three schemes of
downstream difference, upstream difference and centered
difference are shown in Fig. 19. Iior the present prob-
le~n, both the downstream difference and the upstream
difference are not suitable, because computations failed
1' r
WITHOUT LONG ELEMENTS
i'
-4 -3
-2
-9 -8 -7 -6 -5 -4 -3 -2
~_-
Fig. 18 Wave profiles for the cases without and with long
elements on the upstream surface. ( t = 2.4, At = 0.04 )
231
t
OCR for page 232
without sufficient wave overturning. In the case of the
centered difference, however, the plunging breaker can be
simulated. The centered difference includes both informa-
tion from upstream and downstream. In order to express
the flow field in the stagnation region, both are needed.
Differences in tlte wave drag coefficient are also observed
with respect to the upstream difference and the centered
difference as in Fig. 20. In all following examples, the
scheme of centered difference is employed.
Cw=Rw/ (Ed UZ )
1.8
1.5
1.2 .
0.9
0-~::
0.3;
it/
UP-STREAM D I FFERENCE
/ ~ CENTERED D I FFERENCE
Fig. 20 Wave drag coefficients based on the schemes of
upstream and centered difference.
Effects of the size of calculation region and the tilde
interval are examined. Fig. 21 shows the wave profiles at
t = 2.4 for the cases of Oman = 10, 20, 30, and Fig. 22
slows the wave drag coefficients for tl~e stance cases. The
wave overturning point is closer to the bow of floating
1
,.~
1~
-2 - 1
XM9X- 3G. O
- x ~ q x - 2 ~ . 0
Xerox- 1 a.
. .
1. O
n ~
x
Fig. 21 Wave profiles for the cases of Oman = 10, 20, 30.
1VF = 130, t = 2.4, i\t = 0.04 ~
232
body as ~ma: increases. On the contrary, the extension
of the upstream boundary has no effect for the numerical
solution.
Cw=Rw/ Spa Us ~
. . _
1.0 .
0.~- . //
0.6. /
~ 4 .
0.2 .
0.0-
~ — XMqX- I 0. 0
XM4X=20. 0
—-— xmQx- 30. 0
, , , , , t
1.0 2.0 3.0 4.0
Fig. 22 Wave drag coefficients for the cases of
xmac = 10, 20, 30.
Fig. 23 shows the wave drag coefficients for the cases
of At = 0.02, 0.04, 0.08, Slid 0.12, where Oman = 20. At
At = 0.02 and 0.04, almost the sense results are obtained.
As in Fig. 24, the detailed simulation of wave breaking is
Flown, where t = 0~ 2.6 and /\t = ().02.
Cw=Rw/ (pU U2 ~
,~,f-#
- DT=O. 02
i' ~ DT=O. 04
—- DT=O. 08
DT=O. 12
0.2
Fig. 23 Wave drag coefficients for the cases of
/\t = 0.02, 0.04, 0.08, 0.12.
OCR for page 233
/ \ r
1
x
-1
Fig. 24 Wave breaking simulation.
U = l.Om/sec,xma~ = 20, t = 0 ~ 2.6, /` t = 0.02 ~
According to the above studies about numerical treat-
ments and conditions, simulations of mave making phe-
nomena are carried out for Fa = U/~/~d = 0.5, 0.8 and
1.0 as in Fig. 25, 26 and 27 respectively. In the case of
Fa = 0.5 a spilling breaker is obtained, though the other
cases show plunging breakers. Since reliable experimental
data is not available, these solutions cannot be compared
with experiments. However, under the nurr~erical treat-
ment that the numerical solution of double model flow
is employed as the initial condition, these solutions must
be considered as accurate ones, if the above mentioned
studies are acceptable.
\ r
1
_ ~
x
..
-1
Fig. 25 Wave breaking simulation for F.' = 0.5.
~ t = 0 ~ 0.9, fit = 0.02 ~
I r
1
A/
-2
x
. -1
Fig. 26 Wave breaking simulation for F,l = 0.8.
~ t = 0 ~ 1.92, At = 0.032 ~
~~ Y
1
/,'\/:
y
-1
Fig. 27 Wave breaking simulation for Ed = 1.0.
~ t = 0 ~ 2.4, At = 0.04 ~
The present results are compared with the other the-
oretical and numerical results. Dagan and Tulin t11] ob-
tained the wave profile ill front of the rectangular body
by a perturbation method based on small Froude number
expansion. In Fig. 28, wave height at the bow ~7 based
on the present method is plotted for If, with their re-
sult. The present result is not equivalent to ~7 = 0.5F,' by
Dagan alla Tulin. Wave steepening is related to their sec-
ond order solution, but wave breaking phenomena cannot
be explained by their analytical approaches. Finally as
shown in Fig. 29, wave profile for F`' = 1.0 is compared
with the result based on the similar method by (;rosen-
baugl~ and Yeung t73. Fairy good agreement is observed
except the sharpness of overturning waves.
233
OCR for page 234
1.0~
-
0.5
· PRESENT CAL.
PAGAN &TULIN
.
/
/
/
, ,
0 0.5 Ed 1.0
Fig. 28 Wave height at the bow by the present method
and by Dagan and l'ulin.
A\ r
- 1
/. -
·:
PRSENT CAL.
GROSENBAUGH & YEUNG
1
Fig. 29 Wave profiles for F`` = 1.0 by the present method t4]
and by Grosenbaugl~ and Yeung.
5.Conclusiot1
In the present study, flee simulation method of two
dimensional nonlinear water waves based on BEM and
the mixed Eulerian-Lagrangian approach is applied to two
wave making problems around a body ifs uniform flow; the
sen~i-circular mound in shallow water and the rectangular
floating body with semi-infinite length.
For numerical difficulties caused by respective prob-
lems, some treatments are given and those effectiveness
are confirmed numerically by several examples. The pre-
sent method can simulate the nonlinear wave snaking phe-
nomena including steepening or breaking, but cannot sim-
ulate the wave after breaking. In the case of the floating
body problem, overturnig waves ~ plunging breaker ~ in
front of the bow can be simulated. Numerical validations
234
of the present method are shown by examples for several
cases with numbers of nodal points, sizes of calculation
region, and time intervals. For the detailed experimental
verifications, reliable data is needed. The present method,
however, can be regarded as one of the powerful simula-
tion tools for the nonlinear wave making phenomena. The
present method can be extended to general problems of
two dimensional wave making phenomena.
Acknowledgement
The author wishes to express his deep appreciation to
Prof. M. Ikehata and Emeritus Prof. H. Maruo of Yoko-
han~a National University for their useful suggestions and
encouragements. Allis paper was written flails at flee Uni-
versity of British Columbia under the financial support of
the Government of Canada Award. The author would also
like to express his deep gratitude to Prof. SKI. Calisal of
U.B.C. and the World University Service of Canada. He
thanks also Mr. D. McGreer and Dr. J.L.K. Chan of
U.B.C., and Mr. D. Jimbo, Mr. N. Nakajilua, Mr. S.
Masuda and Mr. K. Furusawa of Y.N.U. for their kind
cooperations.
References
[1] Longuet-Higgins, M.S. and Cokelet, E.D.: "The de-
forn~ation of Steep Surface Waves on Water, I. A Nu-
r~lerical Method of Computation", Proc. R. Soc. (A),
350 (1976)
[2] Vinje, T. and Brevig, P.: "Nonlinear Ship Motions",
Proc. 3rd Illt. Conf. Numerical Ship Hydrodynan~cs
(1981).
[3] Greenhow, M. and villje, T.: " Extreme Wave
Forces on Submerged Wave Energy Devices", Ap-
plied Ocean Res., Vol. 4, No. 4 (1982).
Lin, W.-N., Newman, J.N. and Yue, D.K.: "Nonlin-
ear Forced Motions of Floating Bodies", 15th Syrup.
On Naval Hydrodynamics ( 1984).
t5] Takagi, K., Naito, S. and Nakal~'ura, S.: "Co~npu-
tatiorl of Nonlinear Hydrodynamic Forces on Two-
Dimensional Body by Boundary Element Method",
Journal of Kansai Soc. of Naval Archtects of Japan,
Vol. 197 (1985). ~ in Japanese )
t6] Schultz, W.W., Ra~berg, S.E. and Gri~n, O.M.
: "Steep and Breaking Deep Water Waves", 16th
Symp. on Naval Hydrodynamics ( 1986 ).
A] Grosenbough, M.A. and Yeung, R.W.: "Nonlinear
Bow Flows - An Experimental and Theoretical I~-
vestigation", 17th Sync. on Naval Hydrodyrlarnics
(1988).
t8] Miyata, M., Matsukawa, C. and Kajitalli, H.: "A
Separating Flow near the Free Surface", Osaka Int.
Colloquim on Ship Viscous Flow ( 1985).
OCR for page 235
[9] Miyata, M., Matsukawa, C. and Kajitani, H. : 6th Ed., p287.
"Shallow Water f low with Separation and Breaking
Wave", Jounal of Soc. of Naval Archtects of Japan,
Vol. 158 ~ 1985 j.
t10] Milrte-Thotnso~n
" Theoretical Hydrodynamics",
t11] Pagan, G. and Tulin, M.P.: "Nonlinear Free-Surface
Effects in the Vicinity of BluIlt Ship Bows", 8th
Snap. on Naval Hydrodynamics (1970~.
235
OCR for page 236
DISCUSSION
by R.C. Ertekin
I think your paper lacks quite important
references on the upstream waves that can be
seen in your figures. It is well known by now
(see the three papers by Bai et al.; Choi and
Mei; and Ertekin & Qian) that when a
disturbance moves in finite depth, then
upstream waves (solitons) will be generated
if the blockage coefficient is significantly
high (like yours) and the depth Froude number
is not very small (>0.2). So the upstream
waves that you obtain are not necessarily
"undesirable oscillations" but a gift of
nature. By the way, you are solving Laplace's
equation and there is no difference between
the body moving (steady) in an otherwise calm
water and the fixed body placed in an uniform
oncoming flow.
With regard to the "open-boundary"
conditions you can very well calculate the
phase speed at these boundaries and use
Orlanski's scheme coupled with the
Sommerfeld's radiation condition. Your results
show that your "open-boundary" conditions are
reflective.
Author's Reply
In the case of solitons, the waves
propagate from the body to the upstream. In my
case, however, the upstream waves appear
around the nodal point 1 and propagate to the
downstream direction. It is caused by the
numerical technique of the addition of new
nodal point 1 and can be avoided by the
extension of the calculation region. In the
present case, the wave breaking occurs before
the generation of solitons. In near future, I
would like to simulate the soliton by the
present technique.
Exactly speaking, your opinion about the
radiation condition is right. For the
practical use, however, we usually need the
simple and numerical radiation condition. For
example, in the research field of Rankine
source method, several numerical radiation
conditions are employed. In the present
method, the combined technique of the upstream
difference approximation of dw/dz, the
replacement of nodal points on the free
surface and the employment of the linear
solution form at the downstream boundary can
be expected as the numerical radiation
condition.
DISCUSSION
by C.G. Kang
Usually there is singular behavior at the
intersection point between the body and the
free surface. Even if the potential and the
stream function are not singular at the point,
the velocity is singular when the intersection
angle is not 90 degrees. Could you show us how
to remove the singular behavior? Greenhow
showed that the solution using fine grids is
poorer than that using coarse grids. Did you
check the convergence of the velocity at the
intersection point?
Author's Reply
As described in my paper, the intersection
point NF is treated as the free surface nodal
point, and its horizontal velocity component
calculated by Re(dw/dz) is ignored. Along the
bow, only this intersection point is movable,
that is, the other nodal points on the bow
under the free surface are fixed. In this
approximation, the velocity at the
intersection can be obtained without
difficulty.
DISCUSSION
by J.H. Hwang
I congratulate on your fine presentation.
Your calculation is seemed to be basically
based on Vinje-Brevig method. Could you give
some comments on major advantages of your
calculation in the numerical scheme including
the treatment of the intersection point
between the free surface and the body.
Author's Reply
As described in my paper, the intersection
point NF is treated as the free surface nodal
point, and its horizontal velocity component
calculated by Re(dw/dz) is ignored. Along the
bow, only this intersection point is movable,
that is, the other nodal points on the bow
under the free surface are fixed. In this
approximation, the velocity at the
intersection can be obtained without
difficulty.
Discussion
by J.W. Kim
We would like to comment on your treatment
of the downstream condition and your finite
difference schemes.
236
OCR for page 237
The downstream condition given in
Eqs.(20)-(22) is based on the steady linear
solution. But your calculation is made on an
unsteady problem. In a transient stage many
components of waves with different wave
lengths are evolved and eventually hit the
downstream boundary. The wave components which
do not satisfy the dispersion relation in (23)
will be reflected back to the computational
domain. Even for the wave components
satisfying the equation (23), this equation
cannot distinguish incoming or out-going waves
with respect to the computation domain.
You have tried various difference schemes
in your paper and the final choice was made
from computational results. We do not
understand how one can choose a specific
finite difference scheme if we don't know the
correct result in advance. We strongly believe
that one should decide a certain numerical
scheme for a given problem based on rational
mathematical analysis, not after comparing
with the known result.
Author's Reply
Strictly speaking, your comments are
true. For numerical treatments in my paper,
however, it is not suitable to discuss
separately the downstream condition and the
finite differnce schemes of dw/dz. These
treatments connect with each other through
eqs.(12)-(14), that is, ~NF in eqs.(20)-(22),
which is time dependent variable, is
determined from egs.(12) and (19). In this
treatment, the position of down stream
boundary is not fixed. If we find a suitable
way to estimate the wave number as a time
dependent variable, these numerical treatments
will be improved more precisely. We should
not pursue an ideal, but find more convenient
way.
237
OCR for page 238
Representative terms from entire chapter:
downstream boundary
