BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
CONFORMAL MAPPING IN HYDRODYNAMIC
C. CIOBANU*, M. CAŢĂ**, A. R. ANGHEL***
Abstract: Hydrodynamic problems can be solved directly by using the
coefficients of the mapping function. The ship hull forms have been described
by Lewis two-parameter conformal mapping with practical applicability for
any types of ships. We present here an algorithmic method solving directly
the problems which appear in architecture naval domain concerning the
contour of ship's cross section. The application was made in Java language
and creates both a text file and a graphical chart.
Keywords: conformal mapping, hydrodynamic, ship cross section, ship hull
form representation.
1. Introduction
During the last 20 years, in the area of naval hydrodynamics as well as in other
domains of mechanical engineering, a growing interest has occurred towards the
algorithmic methods of solving some definite problems.
The shapes of the ships have been described around a point as it is difficult to describe
them according to analytical mappings.
It is necessary to approximate the ship's shape by continuous functions, in order to get
some practical results. A method which has imposed itself during the last few years is that
of multi-parameter conformal mapping, with good results also in the case of extreme
bulbous forms.
The advantage of conformal mapping is that the velocity potential of the fluid around
an arbitrary shape of a cross section in a complex plane can be derived from the more
convenient circular section in another complex plane. In this manner, hydrodynamic
problems can be solved directly by using the coefficients of the mapping function.
The general transformation formula is given by:
n
f ( Z )   s  a 2 k 1 Z  2 k 1 ,
k 0
with f ( Z )  z , z  x  iy is the plane of the ship's cross section.
Z  ie  e  i - is the plane of the unit circle,
 s is the scale factor,
a1  1,
a2 k 1 are the conformal mapping coefficients ( k  1, ... , n ),
n is the number of parameters.
Therefore we can write in turn:
*
Department of Mathematics-Informatics, Naval Academy “Mircea cel Batran”, Constanta
**
IT Department, Naval Academy “Mircea cel Batran”, Constanta
***
Nautical Sciences Department, Naval Academy “Mircea cel Batran”, Constanta
Conformal mapping in hydrodynamic
86
n
x  iy   s  a 2 k 1 (ie  e i ) ( 2 k 1) ,
k 0
n
x  iy   s  a 2 k 1 (i ) ( 2 k 1) e ( 2 k 1) [cos(  )  i sin(  )]  2 x 1 ,
k 0
n
x  iy   s  (1) k a 2 k 1e ( 2 k 1) [i cos( 2k  1)  sin( 2k  1) ] .
k 0
From the relation between the coordinates in the z - plane (the ship's cross section) and
the variables in the Z - plane (the circular cross section), it follows:
n
x    s  (1) k a 2 k 1e ( 2 k 1) sin( 2k  1) ,
k 0
n
y    s  (1) k a 2 k 1e ( 2 k 1) cos( 2k  1) .
k 0
Now by using conformal mapping approximations, the contour of the ship's cross
section, follows from putting   0 in previous sections. We get:
n
x0    s  (1) k a 2 k 1 sin( 2k  1) ,
k 0
n
y 0   s  (1) k a 2 k 1 cos( 2k  1) .
k 0
The breadth on the waterline of the approximate ship's cross section is defined by
n
B0  2 s  , with    a 2 k 1 ,
k 0
and the draft is defined by
n
D0  2 , with    (1) k a 2 k 1 .
k 0
The breadth on the waterline is obtained for    2 , that means:
x
2
n
n
k 0
k 0
   s  (1) k a 2 k 1 sin( 2k  1)  2 , hencex 2   s  a 2 k 1 and B0  2 x / 2 .
B
The scale factor is  s  0 and the draft is obtained for   0 :
2
n
n
k 0
k 0
y 0   s  (1) k a 2 k 1 cos( 2k  1)  0, hencey 0   s  (1) k a 2 k 1 and D0  y 0 with
s 
d0

.
2. Lewis conformal mapping
A simple transformation of the cross sectional hull form will be obtained with n = 2 in
the Lewis transformation. An extended description of the representation of ship hull
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
87
forms by Lewis two-parameter conformal mapping is given by Kerczek and Tuck [5] in
1969.
The two-parameter Lewis transformation of a cross section is defined by:
z  f ( Z )   s a 1 Z   s a1 Z 1   s a3 Z 3 ,
where a1  1,  s is the scale factor and the conformal mapping coefficients a1 and a 3
are called Lewis coefficients.
Then, for z  x  iy and Z  ie  e  i , that is Z  ie  cos    i sin    , we
have:
x    s [e sin(  )  a1e  sin   a3 e 3 sin 3 ], hence
x   s (e sin   a1e  sin   a3 e 3 sin 3 )
and
y   s [e cos(  )  a1e  cos   a3 e 3 cos 3 ], hence
y   s (e cos   a1e  cos   a3 e 3 cos 3 ) .
For   0 we obtain the contour of the so- called Lewis form expressed as:
x0   s (sin   a1 sin   a3 sin 3 ) ,
y0   s (cos   a1 cos   a3 cos 3 )
or as:
x0   s [(1  a1 ) sin   a3 sin 3 ] ,
y0   s ((1  a1 ) cos   a3 cos 3 ) ,
where the scale factor  s is:
Bs
Ds
or  s 
,
2(1  a1  a3 )
2(1  a1  a3 )
in which Bs is the sectional breadth on the load waterline and Ds is the sectional draft.
The half breadth to draft ratio H 0 is given by:
B
1  a1  a3
.
H0  s 
2 Ds 1  a1  a3
An integration of the Lewis form delivers the sectional area coefficient  s :
s 
As
 1  a12  3a32
 
Bs Ds 4 (1  a3 ) 2  a12
in which As is the area of the cross section,
d

As  2  xdy   s2 (1  a12  3a32 )
0
2
s 
and
Bs  Ds  2[(1  a3 ) 2  a12 ] .
Now the coefficients a1 , a 3 and the scale factor  s will be determined in such a
manner that the sectional breadth, the draft and the area of the approximate cross section
and of the actual cross section are identical.
Conformal mapping in hydrodynamic
88
( H 0  1)(1  a3 )
and, by putting that into the
1 H0
expression of  s , we get a quadratic equation in a 3 :
From H 0 expression, we obtain a1 
2
2
 H 1
   H 1
3 2
 s (1  a3 )   0  (1  a3 ) 2  s    0  (1  a3 ) 2 
a3 ,
4 4  H 0  1
4
 H 0  1
2
therefore

2
2
2
8 s 
 4 s  H 0  1   2    H 0  1  8  H 0  1 
3 


a 
 1 
a

2




3
s
4
 
  H 0  1  
4   H 0  1    H 0  1 
  3




2
 H  1   4 s  
 4
 1 
   s   0
  1  0
4 
H

1


 
 0


4 s
Now, putting
 

m  3  4  s  1  4 s
 

2
 H 0  1 
 ,

 H 0  1 
2
 

n  8 s  2  8 s
 

 H 0  1 
  2m  6 ,

 H 0  1 
 

p  4  s  1  4 s
 

 H 0  1 
  1  m  4 ,

 H 0  1 
2
we obtain the equation
ma32  na3  p  0,
that means:
a3 
 m  3  9  2m
.
m
The Lewis form with the other solution of a 3 in the quadratic equation, namely:
a3 
 m  3  9  2m
m
are looped. As ships are „better behaving” thus solution is not considered.
It is obvious that a transformation of a half immersed circle with radius r will result
for  s  r , a1  0 and a3  0 .
A typical and realistic form is presented below. More precisely we have considered a
dry bulk carrier of 55.000 tone deadweight capacity.
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
89
3. Java application
The following application was made in Java language and creates both a text file and a
graphical chart.
Fig. 1. The interface of Java application
The following code is associated with “Input datum generation” option of the menu
and generates a file named “input.txt” that contains the coordinates of points that will be
used for plotting the graph.
if (command.equals("Input datum generation"))
{
for(int i=0;i<x.length;i++)
{bs[i]=2*x[i];
ds[i]=y[i];
if(bs[i]==0 || ds[i]==0) ss[i]=0;
else ss[i]=as[i]/(bs[i]*ds[i]);
if(ds[i]==0) h0[i]=0;
else h0[i]=bs[i]/(2*ds[i]);
c1[i]=3+4*ss[i]/Math.PI+(1-4*ss[i]/Math.PI)*((h0[i]-1)/(h0[i]+1))*((h0[i]1)/(h0[i]+1));
a3[i]=(-c1[i]+3+Math.sqrt(9-2*c1[i]))/c1[i];
a1[i]=(h0[i]-1)*(1+a3[i])/(1+h0[i]);
if((1+a1[i]+a3[i])==0) ms[i]=0;
else ms[i]=bs[i]/(2*(1+a1[i]+a3[i]));
for(int j=0;j<=18;j++)
{
x0[i][j]=ms[i]*((1+a1[i])*Math.sin(Math.PI*j/36)a3[i]*Math.sin(3*Math.PI*j/36));
y0[i][j]=ms[i]*((1a1[i])*Math.cos(Math.PI*j/36)+a3[i]*Math.cos(3*Math.PI*j/36));
}
Conformal mapping in hydrodynamic
90
}
try{
BufferedWriter bw=new BufferedWriter(new FileWriter("input.txt"));
for(int ii=0;ii<x.length;ii++)
for(int jj=0;jj<=18;jj++)
{
bw.write(Double.toString(x0[ii][jj]));
bw.write(" ");
bw.write(Double.toString(y0[ii][jj]));
bw.newLine();
}
bw.close();
} catch (IOException ioe) {}
The pairs of coordinates are obtained from the following initial values of the
parameters: the half breadth (x), the draft (y), and the area of the cross section (as), for a
dry bulk carrier. The half breadth (x), and the draft (y) have been measured from the
midship cross section, and the area of the cross section (as) has been calculated with
Trapezium Method.
x={0, 7.2, 11.6, 14.4, 16, 16.4, 16.4, 16.4, 16.4, 16.4, 16.4, 16.4, 16.4, 16.4, 16.4, 16.4,
16.4, 16, 13.6, 8, 0};
y={0, 4.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4, 12.4,
12.4, 12.4, 12.4, 12.4, 0};
as={0, 35.092, 147.86, 243.61, 318.06, 368.77, 386.73, 395.16, 397.3, 397.3, 397.3,
397.3, 397.3, 397.3, 397.3, 396.74, 391.72, 368.86, 297, 184.04, 26.66};
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 4.4
0.37959227939554996 - 4.361565688761223
0.7675237112740548 - 4.248033467995359
1.1713048071087337 - 4.064601263156071
1.5968472482251417 - 3.819559306313815
2.0478066019290666 - 3.5237572139950872
2.525084671943123 - 3.1899091226418377
3.026527288600797 - 2.8317838967454927
3.5468399661753223 - 2.463335164450289
4.077728934315969 - 2.0978299469936363
4.608259605183426 - 1.7470346590530128
5.125409620754668 - 1.420513266817868
5.614780253238265 - 1.1250846719431238
6.061419017697706 - 0.864475469055108
6.4506986445143575 - 0.6391908469132694
6.769193576765462 - 0.4466114808216764
7.005495176429541 - 0.28130881380664985
7.15091084341811 - 0.13555619970210706
7.2 - 9.390341369784692E-17
0.0 - 12.399999999999999
0.38792599765378244 - 12.298301765780753
0.8011173857334878 - 11.997695984237794
1.2627242773367489 - 11.511362088556229
1.791813378200004 - 10.860327473165992
2.401684100620313 - 10.072126717503046
3.0985866018268724 - 9.179053254299323
3.880931969129568 - 8.216121653366487
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
4.739051145744875 - 7.2188781471258325
5.655521684915399 - 6.221207109864638
6.606042592630648 - 5.253281233494108
7.560800017935294 - 4.339793118210406
8.48623293127177 - 3.4985866018268745
9.347080487873725 - 2.7397787100128728
10.108573376537262 - 2.065429492860541
10.738621427524972 - 1.4697795134187672
11.209849781828026 - 0.9400359278670318
11.501346007307355 0.45765059185190893
11.6 - 3.1817737930515983E-16
0.0 - 12.4
0.9695927227771434 - 12.327840619714813
1.944733565044777 - 12.113613567379202
2.930047447243021 - 11.763938993915474
3.9283799547705125 - 11.289415685833687
4.940070726302039 - 10.703999471024305
5.962409937890854 - 10.024192051421918
6.98931888289643 - 9.268094477535561
8.011280252117743 - 8.454388414013625
9.015526548757673 - 7.6013129863845785
9.986477300251579 - 6.725705032744664
10.906398566113543 - 5.842166010194338
11.756242858990795 - 4.962409937890855
12.516615045097605 - 4.094834202820641
13.168800927405503 - 3.2443396681191756
13.69579064649361 - 2.412409357037982
14.08322907340362 - 1.5974372097109169
14.320230015898307 - 0.7952812372818264
14.4 - 5.571995877088869E-16
0.0 - 12.4
1.4447494468819484 - 12.357211223349532
2.876227440502829 - 12.22887089578949
4.281418639575981 - 12.01507724698215
5.647807407229181 - 11.7160598558048
6.963597180010219 - 11.332273141786308
8.217895485528187 - 10.864517024151944
9.400856710276546 - 10.31407537178904
10.503777451078967 - 9.682861340431112
11.51914233286065 - 8.973557920589078
12.440621335659433 - 8.189742056207427
13.263022731229409 - 7.3359815394127805
13.98220847777592 - 6.417895485528191
14.594981175118248 - 5.442171437743701
15.09895329063407 - 4.4165348912567755
15.492410221622796 - 3.3496700772069086
15.774178806633437 - 2.251094000901881
15.943512136479818 - 1.1309887729903787
16.0 - 7.948603499914596E-16
0.0 - 12.399999999999995
1.840581732952125 - 12.388792005257582
3.648532074302054 - 12.352801496505045
5.392737134602704 - 12.285113998609923
91
7.045019988619084 - 12.174809585696966
8.581370697551405 - 12.007825269405775
9.982908320195035 - 11.768077605532065
11.236514484837253 - 11.438767765320721
12.33510031574988 - 11.00377854225767
13.277493296051148 - 10.449066171304956
14.067956314733586 - 9.763949877003723
14.715375942476687 - 8.942208739433068
15.232179220669819 - 7.982908320195039
15.633056417552375 - 6.890897650588609
15.9335800688406 - 5.6769394153164106
16.148817303766194 - 4.357460954192624
16.292032508553877 - 2.9539393636343343
16.373570797624566 - 1.491958761961492
16.4 - 1.0504054111380305E-15
0.0 - 12.4
2.012171372783688 - 12.403804153521342
3.982634593440278 - 12.411712784951785
5.8717976628765705 - 12.413477880286536
7.644162202468813 - 12.392879517630627
9.270032959434054 - 12.328953756004
10.726848251039943 - 12.197591524866288
12.000045956039962 - 11.973398257117212
13.083411148349544 - 11.631685885898017
13.978886589245663 - 11.150459464499473
14.695863658373927 - 10.51226070960339
15.250006434273175 - 9.705740210635781
15.66169314000404 - 8.726848251039948
15.954184904150598 - 7.579559912471257
16.15165000077426 - 6.276081629166139
16.277181185442807 - 4.836521482466492
16.350943797000614 - 3.288041882772558
16.388582945888324 - 1.6635484017930549
16.4 - 1.1718805651469799E-15
0.0 - 12.4
2.0982753089079567 - 12.411337271817176
4.150287733775987 - 12.441274556993472
6.112190947498201 - 12.477891066766766
7.944812655266027 - 12.502307333367467
9.615604705295548 - 12.490096507541207
11.100158413288915 - 12.413122247531955
12.38318718233919 - 12.241676631970842
13.458914635591018 - 11.946770723548118
14.330846785528461 - 11.502419660782271
15.01094849602403 - 10.887764196844861
15.518284809126804 - 10.088881436935006
15.877223862669707 - 9.100158413288918
16.115327655687807 - 7.925131658332751
16.261077816511097 - 6.576732081963353
16.34159437192304 - 5.0769147670881205
16.3805055690423 - 3.455695023108268
16.39611606418416 - 1.7496523379173234
16.4 - 1.232836977851964E-15
0.0 - 12.4
92
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.4
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.4
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
Conformal mapping in hydrodynamic
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.4
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.4
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.4
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.4
2.120788786475164 - 12.413306945880853
4.1941237694314575 - 12.44900403280683
6.175046248477997 - 12.494733093904317
8.023423315002322 - 12.530919273607507
9.7059608736487 - 12.532230280805823
11.197767307164252 - 12.469476768686182
12.48336659098911 - 12.311823009080905
13.557097015130761 - 12.029155522008024
14.422873333212952 - 11.594446208466763
15.093333294483939 - 10.985946576384604
15.588431186236871 - 10.189060845584926
15.933578383823935 - 9.197767307164254
16.157461428952423 - 8.0154878266859
16.28968975675114 - 6.6553427416996485
16.358436399060594 - 5.139770068067918
16.388235044855662 - 3.4995310587637385
16.398085738247836 - 1.7721658154845303
16.4 - 1.2487751671123301E-15
0.0 - 12.399999999999999
2.1148702899013063 - 12.412789144525522
4.182599853321034 - 12.44697205547314
6.158522418146268 - 12.490305546911063
8.002757608474797 - 12.523397571561407
9.682207427686002 - 12.521153867045847
11.172107215236887 - 12.454661907704486
12.457030744601287 - 12.29338245091502
13.531286161379516 - 12.007497644144983
14.398680766536005 - 11.570253641789815
15.071675416620893 - 10.96013572263336
15.569990628070986 - 10.162724999197106
15.918763522842237 - 9.172107215236892
16.146385015192443 - 7.991734380723207
16.28216805470504 - 6.634677035172124
16.35400885206733 - 5.12324623773619
16.386203067521972 - 3.488007142653315
16.397567936892504 - 1.766247318910673
16.4 - 1.2445852271863736E-15
0.0 - 12.399999999999997
2.0626571276889383 - 12.408221084745097
4.080935504328584 - 12.429045887769608
6.012748669236603 - 12.451245588613059
7.82044409799292 - 12.457040880441424
9.472653783162292 - 12.423437397810147
10.945733080274003 - 12.323964739946094
12.224695082615582 - 12.130699268997684
13.303582000882692 - 11.816431167054565
14.185253185312678 - 11.356826060566489
14.880608939530477 - 10.732431562136535
93
15.407307446153649 - 9.930389337211398
15.78806635508385 - 8.945733080274005
16.048668545956748 - 7.782180736199495
16.215811363585054 - 6.452363524690247
16.31494889376933 - 4.977472488826525
16.36827689981844 - 3.3863427936608645
16.392999877112082 - 1.7140341566983053
16.4 - 1.2076214458807672E-15
0.0 - 12.399999999999997
1.8877318322787247 - 12.395967160213422
3.738759298721103 - 12.380958534113976
5.518179645499846 - 12.346466359749751
7.194575891291457 - 12.279037543304527
8.741474048339656 - 12.161310740744948
10.138479300967678 - 11.97336660699717
11.372018796582596 - 11.694297923653716
12.435645353876863 - 11.30389098515205
13.329886119173175 - 10.784301706901605
14.06165098038037 - 10.12160995900532
14.643245283094082 - 9.30714362571883
15.091058060621148 - 8.338479300967677
15.424018774076892 - 7.220048306073139
15.661930978133796 - 5.963303375319051
15.823799334390396 - 4.586431083130775
15.926266444957927 - 3.1136258591201558
15.982268073343706 - 1.5739711583871545
16.0 - 1.1084652596010834E-15
0.0 - 12.400000000000002
1.4760480599151795 - 12.378249831857874
2.9276960800966614 - 12.311431466906733
4.331626989797154 - 12.19497138376765
5.666620266729935 - 12.021671540740519
6.914431432713245 - 11.782315093924838
8.060481841842734 - 11.466454537623578
9.094315966063338 - 11.063327338656455
10.009799095881057 - 10.562835115686907
10.805045901376387 - 9.956517763952531
11.48208844742968 - 9.23845396425721
12.04630979180324 - 8.406024242442086
12.505685022164903 - 7.460481841842736
12.869884438368818 - 6.407289518624407
13.149302685683606 - 5.256196094739135
13.35408237531453 - 4.021044135674133
13.49320077052138 - 2.7193182668963467
13.573683469567966 - 1.3714611686179892
13.6 - 9.651003558708342E-16
0.0 - 12.399999999999999
1.0340420729621906 - 12.382280099717866
2.044961818483845 - 12.327247209480433
3.0108523264049074 - 12.229432862405888
3.912157434784615 - 12.080216142250054
4.732652325037954 - 11.868527526397983
5.460205235215455 - 11.581764892550783
6.087270999830458 - 11.206858034610427
Conformal mapping in hydrodynamic
94
6.611085318313334 - 10.731407577247174
7.033548947818976 - 10.144818785039785
7.36081202752367 - 9.439350800934108
7.602589039738864 - 8.61100731977506
7.771253115899253 - 7.660205235215456
7.880773263436724 - 6.5921726766970306
7.9455686107920585 - 5.4170460654175585
7.979359226733989 - 4.149656124856002
7.994093096226718 - 2.80901380021834
7.999023428114188 - 1.417527341051886
8.0 - 9.977124377920012E-16
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0 - 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
The graphical representation of the points shows the contour of the ship’s cross section
of the dry bulk carrier.
Fig. 2. The graphical representation
4. Conclusions
This is a mathematical solution in order to obtain the contour of the ship’s cross section
of different types of ships, using conformal mapping approximations.
The advantage of conformal mapping is that the velocity potential of the fluid around
an arbitrary shape of a cross section in a complex plane can be derived from the more
convenient circular section in another complex plane.
References
1. Athanassoulis, G.A., and Loukakis, T.A.: An Extended-Lewis Form Family of Ship
Sections and its Applications to Seakeeping Calculations, International Shipbuilding
Progress, 1985, Vol. 32, No. 366.
2. Ikeda, Y., Himeno, Y., and Tanaka, N.: A Prediction Method for Ship Rolling.
Technical Report 00405, Department of Naval Architecture, University of Osaka
Prefecture, Japan, 1978.
Bulletin of the Transilvania University of Braşov • Vol. 13(48) - 2006
95
3. Journee, J.M.J.: Verification and Validation of Ship Motions Program SEAWAY,
Tehnical Report 1213a, 2001, Delft University of Technology, Ship Hydromechanics
Laboratory, The Netherlands, Internet: www.shipmotions.nl.
4. Journee, J.M.J., and Massie, W.W.: Offshore Hydromechanics, Lecture Notes, First
Edition, January 2001, Delft University of Technology, Ship Hydromechanics
Laboratory, The Netherlands, Internet: www.shipmotions.nl.
5. Kerczeck, C.V. and Tuck, E.O.: The Representation of Ship Hulls by Conformal
Mapping Functions. Journal of Ship Research, 13(4), 1969.
6. Lamb, H.: Hydrodynamics. Dover Publications, Inc., New York, 6 edition, 1932.
7. Tasai, F.: Improvements in the Theory of Ship Motions in Longitudinal Waves. In
Proceedings 12th I.T.T.C, 1969.
8. Ursell, F.: On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid.
Quaterly Journal of Mechanics and Applied Mathematics, II, 1949.
9. Vantorre, M. and Journee, J.M.J.: Validation of the Strip Theory Code SEAWAY by
Model Tests in Very Shallow Water, Colloquium on Numerical Modelling, 23-24
October 2003, Antwerp, Belgium, Internet: www.shipmotions.nl
Transformarea conformă în hidrodinamică
Rezumat: Problemele de hidrodinamică pot fi rezolvate direct folosind
coeficienţii transformării. Formele navei au fost descrise de transformarea
conformă cu doi parametri a lui Lewis cu aplicaţii practice pentru orice tip de
navă. Aici prezentăm un algoritm ce rezolvă direct problemele care apar în
domeniul arhitecturii navale privind conturul secţiunii transversale a navei.
Aplicaţia informatică a fost realizată în limbajul Java şi creează atât un fişier
cu date de ieşire cât şi o reprezentare grafică.
cheie: transformarea conformă, hidrodinamică,
transversală a navei, reprezentarea corpului navei.
Cuvinte
secţiune