WAVE RESISTANCE FOR HIGH SPEED CATAMARAN
H.B.Moraes*
J.M.Vasconcellos**
R.G.Latorre ***
*
Assistant Professor at Pará University / Brazil
** Associate Professor at COPPE / UFRJ – Brazil
*** Professor at University of New Orleans - USA
1. Abstract
The object of this work is investigate the wave resistance component for high speed catamarans.
Two methods were applied: the theory of the slender body proposed by Mitchel and a 3D method
used by ShipflowTM software.
The results were used for different types of twin hulls and special attention was given for the
effects of the catamaran hull’s approximation. The results comparison founded by Millward are
shown.
The investigation included also the effects of shallow water on the wave resistance component.
Special attention is given to the wave’s height generated by the craft with the objective to find out
the consequences in the river margin.
2. Introduction
The demand for high-speed craft for passenger transportation, mainly the catamarans, is
increasing significantly in the past years. Looking for being capable to attend the increasingly
demand, the international market of high-speed crafts is passing for deep changes.
Different types of crafts are being used for passenger transportation. However, the catamarans
and monohulls have been the main ship owner’s choice not only for passenger transportation but
also as a ferryboat. The investigation for more efficient hydrodynamic shapes, the engine
improvement, and lighter hulls with aluminum and composite materials application made
possible enlarge the speed.
The improvement of the craft quality, high speed, larger deck area compared with monohull craft
justifies in certain way the demand for catamarans.
The market for high speed crafts is demanding a project of catamarans with different types and
dimensions with the focus on lower resistance and power for high speed. Therefore, the
optimization of the hull resistance is fundamental for the success of a high speed catamaran hull.
The theoretical calculation of the hull resistance is complex, however its offer lower costs than
the model test evaluation. This reason justifies the effort to study many different theoretical
methods capable to show an appropriated evaluation of the hull resistance.
This research focus on two methods for catamaran wave resistance calculation in different
positions of hull separation and water depth. The 3D method based on the potential theory, used
in the SHIPFLOWTM software and the method that uses the theory of the slender body, with an
easier application were applied.
The SHIPFLOWTM program application allowed many analyses for catamaran’s hull wave
resistance. In special the interference effect between hulls and the effect of shallow water can be
investigated.
This work also look at the maximum wave’s height generated by the craft, in different water
depth. The objective is to evaluate and minimize the possible effects against the river shore and/
or nearby small boats.
2.1 – High Speed Craft
The high speed crafts have been always in the forefront naval engineering and the hydrodynamic
researching.
At the end of the 19th century and in the beginning of the 20th, there were many solutions for
alternatives types of high speed crafts, with many concepts being patented.
The first hydrofoil were build by Forlanini in 1905, achiving 61 knots. However, it was the Baron
Schertei between 1920 and 1930 that improved the hydrofoil which was created to work in calm
waters for a craft capable to operate in sea condition.
In 1716, appeared the first hovercraft, sailing at 40 knots. However, there were problems with the
air cushion not solved until the work of Rockerill and Latimrer Needham during 1950. This work
leaded to the development of a vehicle of air cushion capable of being used in ocean waters.
The high speed crafts can be classified as shown in table I.
Table I – Classification of the types of high speed crafts
Category
Types of craft
Crafts supported by air
Air Cushion Vehicles – ACV and Surface Effect Ship –
SES
Crafts supported by foils
Surface Piercing Foil (Hydrofoil) and Jetfoil
Displacement, planing and Standard monohull, catamaran, Small Waterline Area
craft for semi-displacement
Twin Hull –SWATH, Air lubricated hulls and Wave
Piercing
According to International Maritime Organization (IMO) rules and recommendations through its
special code called IMO-HSC Code (High Speed Craft) approved in the 63rd section of May 94
trough MSC.36 (63), that specifies as a high speed craft the one that has its maximum speed is
equal or the same or superior as:
V  3.7  Vol 0.1667 (m / s)
Where:
“Vol” is the displacement volume at design water line (m3).
Figure 1 shows the speed limit curve for a craft to be considered as a high speed craft.
2
(1)
60.00
Speed (knots)
40.00
High Speed Craft
20.00
0.00
0.00
200.00
400.00
600.00
800.00
1000.00
Vol - Displacement Volume (m3)
Figure 1 – Speed Limit for High Speed Craft
Regarding catamarans, the hydrodynamic performance depends on the wetted geometrical forms,
the hulls separation and the water depth where the craft is going to sail.
3. Comparative Results Between Hull Resistance Evaluation Methods
The SHIPFLOW TM results and the slender body theory application, were compared with the
Millward [11] works for the investigation of the catamaran’s hull separation effect as well as the
shallow water effects in a wide Froude number range.
The Millward [11] result’s can be compared with catamaran’s hull in the following range:
Length / breadth (L/b)
Breadth/draught (b/T)
Block coefficient
Froude number
 6.00 – 12.00
 1.00 – 3.00
 0.33 – 0,45
 0 < Fn < 1
3.1- Hulls Geometry
Two hulls geometry were used to compare the wave resistance calculation methods. The first one
utilized the mathematical model proposed by Wigley (Figure 2) and the second geometry is a
chine hull as indicated in figure 3.
3
Figure 2: Wigley catamaran hull

B
S
Figure 3: Catamaran’s chine hull
The parabolic hull proposed by Wigley has been already submitted to exhaustive theoretical tests
and experiments, being so well known and tested. The hull geometry in a no-dimensional form of
the Wigley hull is given by the equation:
1
z2
Y  [  b(1  4 x 2 )(1  2 )]
2
d

1
1
x ,
2
2
-d  z  0
Where “b” and “d” are constants, being b=0, 1 e d=0,0625.
4
(2)
b
L
T
d
L
b
(3)
(4)
Where,
L = length
B = breadth
T = draught
The chine hull catamaran form is defined by straight water lines forming a triangle in the fore
body, a parallel central part forming a rectangle and a after body with triangular or rectangular
form, when considered a transom. The hull section has a “V” shape geometry (Figure 3).
An algorithmic was developed in the FORTRANTM for the chine hull generation. The hull is
created following the range constraints:
- S/L => 0,2  S/L  1,0
- b/T => 10  b/T  16
- Deadrise angle (degrees) 25º
- Entrance body’s length 0,3 L
- Exit body’s length 0,3 L
Where:
S= the distance between hulls (m)
L= craft’s length (m)
B= breadth (m)
T= craft’s draught (m)
The catamaran hull geometry generated by the Fortran program output the hull data in a standard
format for the SHIPFLOWTM program and the slender body program were the wave resistance
calculation are done.
Input Data
Geometry
Standard Output
Standard Output
Slender
Body
SHIPFLOW TM
SLENDER
Figure 4 – Flow chart of the algorithmic form generator
5
3.2- Wigley’s Hull Results
The wave coefficient (Cw) were used to compare the catamaran wave resistance results given by
the programs SHIPFLOWTM and SLENDER (slender body theory). The Wigley hull was tested,
in deep waters, for three different hull’s separations (S/L=0,2, 0,4 e 1,0), were S is the distance
between the hull’s center and L is the hull length.
The results for a fixed hull (related to water surface) is presented in figures 7, 8 and 9.
Wigley hull
8.00
S/ L = 0.2
Millward
6.00
Cw x 10^ 3
Slender body
Shipflow
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 7: Wave coefficient (Cw) – Wigley case – Catamaran (S/L=0.2)
8.00
Wigley hull
S/ L = 0.4
6.00
Millward
Cw x 10 ^ 3
Slender body
Shipflow
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 8: Wave coefficient (Cw)– Wigley case – Catamaran (S/L=0.4)
6
8.00
Wigley hull
S/ L = 1.0
6.00
Millward
Cw x 10^ 3
Slender body
Shipflow
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 9: Wave coefficient (Cw)– Wigley case – Catamaran (S/L=1.0)
The methods shows similar tread off results. The 3D method used by SHIPFLOW TH program and
slender body theory presented different results in Froude numbers between 0.5 and 0.7. The
difference can be explained once the SHIPFLOWTH program applied a 3D method and can taking
into account the hull interference more strong in this Froude range while Millward and Slender
program use a similar methodology (2D).
The results highlight a substantial increase in the wave resistance coefficient when the hull’s
distance is small showing the interference hull effect.
Figures 7, 8 and 9 indicate the following conclusions:
-
small S/L indicates increase in the difference between the results given by the SHIPFLOW TM
program and the Slender program
-
for 0.2  Fn  0.4 and Fn > 0.8 the results given by the SHIPFLOWTM program and the ones
given by the Slender program, compared with the Millward [11] work are very close, what
confirms the precision of all that methods for this Fround range.
-
It can be seen for the Froude’s number higher than 0,8 in all observed relations of S/L (0.2 ,
0.4 and 1.0) exists a converging value of Cw around 0,0002.
-
It also can be percept as smaller the space between the twin hulls, higher is the Cw curves
elevation for Froude’s number around 0,5.
Regarding the previous results been taken with fixed hull related to the water surface, research
was done with the utilization of SHIPFLOWTM program in a free condition to find out the Cw
curve trade off, the hull under water volume’s variation and the wetted surface for different
Froude numbers.
7
Figures 10 and 11 show and compare the given results for the Wigley monohull in fixed and free
conditions, and in shallow and deep water (L/h=5, L = length, h = depth)
Wigley monohull - free
14.00
12.00
Shalow water ( L/ h=5 )
Cw x 10^ 3
10.00
Deep water
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 10: Cw values for monohull in a free condition
Wigley monohull - fixed
14.00
12.00
Cw x 10^ 3
10.00
Shallow water ( L/ h = 5 )
Deep water
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 11: Cw values for monohull in a fixed condition
Results indicate relevant difference in the Cw values for Froude’s number between 0.3 and 0.6
for the free and fixed monohull condition related to the water surface. Those differences are even
higher when the depth effect is included. When the sea or river bottom is close to the hull the
craft draft is increased and as a consequence displacement and wetted area is increased.
The same test were done for twin hull craft. Figures 12, 13 and 14 are shown the results given for
the Wigley catamaran hull for different twin hull space (S/L) and depths in a free condition.
8
Wigley catamaran - free
S/ L = 0,2
14.00
12.00
Shallow water ( L/ h = 5 )
Cw x 10^ 3
10.00
Deep water
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 12: Cw value’s for Wigley Catamaran in a free condition and S/L = 0.2
Wigley catamaran - free
S/ L = 0,4
14.00
12.00
Cw x 10^ 3
10.00
Shallow water ( L/ h = 5 )
Deep water
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 13: Cw value’s for Wigley Catamaran in a free condition and S/L = 0.4
9
Wigley catamaran - free
S/ L = 1,0
14.00
12.00
Cw x 10^ 3
10.00
Shallow water ( L/ h = 5 )
8.00
Deep water
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 14: Cw values for Wigley Catamaran in a free condition and S/L = 1.0
It can be understood from the Figure 12, 13 and 14, when compared to the Figures 7, 8 e 9 that
there is a tendency to higher values of Cw for the free condition for Froude numbers between 0.3
and 0.6 for monohulls. Regarding the Catamaran, this effect is even higher when the hull’s
distance is small, being that effect increased when the shallow water effect occurs. For Froude
numbers greater than 0.7 results shown that the Cw values are basically the same for monohull as
well as catamaran it does no matter the hull’s distance, what leads us to conclude. For Froude
numbers greater than 0,7 the hull interference practically does not exist and the catamaran wave
resistance can be calculated by two times the monohull resistance.
In the previous Figures, Cw values for Froude numbers between 0.3 and 0.6 are greater in the
free case. That happens because of the sinking of the hull.
Figures 15, 16, 17 and 18 show an increased percentage of the under water volume as well as the
wetted hull surface. The monohull volume increases approximately 12% related to the fixed hull
condition in deep water and approximately 20% in shallow waters (L/h = 5).
In the Catamaran’s case this effect is powered, as a consequence of the hull’s interference, as can
be seen in the graphics of figure 16, 17 and 18. Figure 17 shows an increasing of 33% in the
volume when the twin hull distance is 0.2 (S/L) in shallow waters and Froude numbers between
0.3 and 0.5. It is also shown that the Cw curve reaches a pick, for deep water around 0.5, and
shift the peak for Froude number around 0.4 when the shallow water effect occur.
10
Variatoin of the submerged volume ( % )
Wigley monohull - free
35.00
30.00
Shallow water ( L/ h = 5 )
25.00
Deep water
20.00
15.00
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Variation of the submerged volume ( % )
Figure 15: Volume variation for monohull in a free condition
Wigley catamaran - free
S/ L = 0,2
35.00
30.00
Shallow water ( L/ h = 5 )
25.00
Deep water
20.00
15.00
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 16: Volume variation for Catamaran in a free condition and S/L = 0.2
11
Variation of the submerged volume ( % )
Wigley catamaran - free
S/ L = 0,4
35.00
30.00
Shallow water ( L/ h = 5 )
25.00
Deep water
20.00
15.00
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Variation of the submerged volume ( % )
Figure 17: Volume variation for Catamaran in a free condition and S/L = 0.4
Wigley catamaran - free
S/ L = 1,0
35.00
30.00
Shallow water ( L/ h = 5 )
25.00
Deep water
20.00
15.00
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 18: Volume variation for Catamaran in a free condition and S/L = 1.0
12
Variation of the wetted surface area ( % )
Wigley monohull - free
20.00
15.00
Shallow water ( L/ h = 5 )
Deep water
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Variation of the wetted surface area ( % )
Figure 19: Wetted surface variation for monohull in a free condition
Wigley catamaran - free
S/ L = 0,2
20.00
Shallow water ( L/ h = 5 )
15.00
Deep water
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 20: Wetted surface variation for Catamaran in a free condition and S/L = 0.2
13
Wigley catamaran - free
S/ L = 0,4
Variation of the wetted surface area ( % )
20.00
Shallow water ( L/ h = 5 )
15.00
deep water
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Variation of the wetted surface area ( % )
Figure 21: Wetted surface variation for Catamaran in a free condition and S/L = 0.4
Wigley catamaran - free
S/ L = 1,0
20.00
Shallow water ( L/ h = 5 )
15.00
Deep water
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 22: Wetted surface variation for Catamaran in a free condition and S/L = 1.0
The same effect that happens in the volume also happens in the wetted surface, however with an
increased percentage smaller, around 19% in the most critical condition (shallow waters and
relationship S/L = 0.2).
14
3.3 - CATAMARAN CHINE HULL RESULTS
Chengyi [12] work shows that the hull shape of symmetrical catamaran hull has little influence in
the results of catamaran wave resistance. To investigate a different catamaran chine hull
geometry (hull separation and water deep effect) with SHIPFLOWTM program an algorithm was
used that quickly modifies the geometric forms of the hull without a lot of effort.
The hull generated for analysis has the following characteristics:
Length
Breadth
Draught
Deadrise Angle
(L): 40 m
(b): 3,5 m
(T): 1.5 m
(A): 25º
The results of wave resistance were obtained using the program SHIPFLOW TM using the same
twin hull separation and water depths analyzed in the Millward [11] work.s.
The nomenclature used in the variables it was defined according to outlines below:
S = space between hulls
h=depth
b=breadth
Figure 23: 40 meter catamaran hull section
L = length
Figure 24: Waterline shape
The results are analyzed accordingly to Froude´s numbers (Fn) and speeds (V) presented in table
II.
15
Table II - Froude number and speed for a 40 m chine hull catamaran
Fn
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
V ( m/s )
3,96
5,94
7,92
9,90
11,90
13,86
15,84
17,82
19,80
V ( knots )
7,70
11,54
15,39
19,24
23,12
26,93
30,78
34,62
38,48
Figures 25, 26 and 27 show the main results obtained by the program SHIPFLOW TM for chine
hull catamaran in the free condition.
Catamaran 40m - free
Catamarã 40m - livre
S/ L = 0,2
16.00
14.00
águas
rasas ( L/ hwater
=6)
Shallow
(L/h=6)
(L/h=5)
Shallow
águas
rasas ( L/ hwater
= 4 ) (L/h=4)
Deep
water
águas
profundas
12.00
Cw x 10^ 3
águas
rasas ( L/ hwater
=5)
Shallow
10.00
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 25: Comparison of CW for S/L = 0,2 in several water depths.
16
Catamaran 40m - free
Catamarã 40m - livre
S/ L = 0,4
16.00
14.00
Cw x 10^ 3
12.00
Shallow
águas
rasas ( L/ h water
=6)
(L/h=6)
Shallow
águas
rasas ( L/ h water
=5)
(L/h=5)
(L/h=4)
águas
rasas ( L/ h water
=4)
Shallow
10.00
águas
profundas
Deep
water
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 26: Comparison of CW for S/L = 0,4 in several water depths.
Catamaran 40m - free
Catamarã 40m - livre
S/ L = 1,0
16.00
14.00
Shallow
águas
rasas ( L/ hwater
=6)
(L/h=6)
(L/h=5)
Shallow
water
(L/h=4)
águas rasas ( L/ h = 4 )
Deep
water
águas profundas
Cw x 10^ 3
12.00
Shallow
águas
rasas ( L/ hwater
=5)
10.00
8.00
6.00
4.00
2.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 27: Comparison of Cw for S/L = 1,0 in several water depths
Figures 25, 26 and 27 allowed the following conclusions:
-
Figure 25 shows that critical Cw (the peak of Cw curve) tends to increase when the water
depth decreases. It is also observed that the hulls interference is practically the same for
Froude numbers higher than 0,7 in any water deep.
-
For shallow water, the peak of Cw curve is close to 0,4 Froud number.
-
Between 0,6 and 0,7 Froud number the wave resistance decrease when the water deep
decrease for all S/L range.
17
-
For Froude numbers lower than 0,3 the results are practically the same for all S/L and L/h
range.
-
Figures 26 and 27 show that when increasing the separation between twin hulls, Cw
coefficient achieves the lower value. However, Cw coefficient maintain the critical value for
Froude number 0,4 considering L/h <5.
-
Figure 27 shows the same tendency of the Figures 25 and 26, however, the interference
effects between hulls is small.
-
As shown in Millward [11] works it is observed that there is a tendency of null interference
between catamarans hulls for numbers of Froude smaller than 0,3 and negative interference
for Froude numbers larger than 0,7 considering any water depth.
-
Results indicate the tendency of Cw peak occurs for lowers Froud number in shallow water.
For deep water, critical Cw occurs for Froude number 0,5. When water depth decreases, this
critical Cw occurs around Froude number 0,4.
-
The effect of the hull separation, added to the shallow waters cause significant increase in
critical Cw.
According to Chengyi [12] works, when the distance between hulls is small and the craft speed is
high, the flow between hulls tends to suffer the blockage phenomenon. In this case, the surface of
the water is lifted in a violent splash. According to tests results with catamaran round bottom
hulls (arc bilge) and chine bilge hulls, the following expression is used to calculate the Froude
number (Frb):
Frb 
10
L
S
[(
) 2  1]
b S  Cp.b
(5)
Where,
L = Craft length (m)
S = Distance between two hulls center
b = Breadth
Cp = Prismatic hull coefficient
The tests [Chengyi [12]] indicate that the interference between catamaran hulls, that cause a
reduction in the resistance, occurs for Froude number higher than 0,5 (Fn>0,5).
For different S/b, Froude number where begins the negative interference - Fro (reduction in the
resistance) can be calculated by the following expressions:
18
  
S
Fro(  1,6)  0.55  0.057 
3
b
 (0,1L) 
Fro(
2
(A)
  
S
 2,0)  0.55  0.050
3 
b
 (0,1L) 
2
(B)
2
2
   (C)
S
Fro(  2,6)  0.55  0.046
3
b
 (0,1L) 
(6)
Fro(
   (D)
S
 3,2)  0.55  0.044 
3
b
 (0,1L) 
2
   (E)
S
Fro(  6,0)  0.55  0.042 
3
b
 (0,1L) 
-
Using the equations proposed by Chengyi [12], for blockage Froude (Frb) and the Froude
number where the null interference (Fro) begins, the following values were calculated for the
catamaran analyzed in Figures 25, 26 and 27 respectively.
For Figure 25 (S/L=0,2 - k/b = 2,66) => Fro = 0,640 and Frb = 1,00
For Figure 26 (S/L=0,4 - k/b = 5,33) => Fro = 0,634 and Frb = 1,59
For Figure 27 (S/L=1,0 - k/b =13,33) = >Fro = 0,632 and Frb = 2,67
-
It is observed that smaller distance between hulls, larger the value of Fro and the beginning of
the negative interference. In the catamaran case (chine hull), the negative interference begins,
according to the equation of Chengyi [10], in numbers of Froude higher than 0,63.
The submerged volume and wetted area variation had the following results:
Catamaran 40 m - free
S/ L=0.2
35.00
Variation of the submerged volume (%)
30.00
Shallow water (L/ h=5)
25.00
Deep water
20.00
15.00
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 28: 40 m catamaran submerged volume variation in the free condition and S/L = 0,2
19
35.00
Catamaran 40 m - free
S/ L=1.0
Variation of the submerged volume (%)
30.00
Shallow water (L/ h=5)
25.00
Deep water
20.00
15.00
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 29 - 40 m catamaran submerged volume variation in the free condition and S/L = 1,0
The 40 m catamaran submerged volume suffers significant increase in shallow water and when
the interference effect is present. The percentage of sinking is 33%. The results show that when
both effects are present the resistance increases to very high value.
In deep waters, the effects that take to an abrupt increase of the submerged volume increase to
13% (Fr = 0.5) and than low down to 11% for Fr > 0.6.
For 0,6 < Fn < 0,8 the submerged volume decrease in shallow waters compared with deep waters.
Following, the variation of the wetted surface will be shown for catamaran separation hull (S/L)
of 0,2 and 1,0 in shallow waters (L/h=5) and deep water.
Variation of the wetted surface area (%)
20.00
Catamaran 40 m - free
S/ L=0.2
16.00
Shallow water (L/ h=5)
Deep water
12.00
8.00
4.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 30 - 40 m catamaran wetted surface variation in the free condition and S/L = 0,2
20
Variation of the wetted surface area (%)
20.00
Catamaran 40 m - free
S/ L=1.0
15.00
Shallow water (L/ h=5)
Deep water
10.00
5.00
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Fn
Figure 31 - 40 m catamaran wetted surface variation in the free condition and S/L = 1,0
The wetted surface variation for low Froude number (to approximately 0,3) is similar in all the
cases, however when Froude number reaches 0,4 a substantial increase in the wetted surface is
verified, mainly when the interference between hulls and the shallow water effects happen.
Millward [11] works was used to compare the results generated by the program SHIPFLOWTM
although his works did not contemplate the free craft condition.
The results were expressed in the form of non-dimensional (C *) wave resistance coefficient,
adopted Millward [11] works.
R* 
R
8
b 2T 2
g

L
C* 
R*
Fn 2
Where,
R = wave resistance
L = catamaran length (m)
b = breadth (m)
T = draft (m)
 = Fluid density (kg/m3)
g = acceleration of gravity (m/s2)
Fn = Froude number
21
(7)
(8)
Figure 32: Catamaran C* coefficient for S/L = 0,2 in shallow and deep waters (Millward[11])
Catamaran 40m - fixed
S/ L = 0,2
18
16
Shallow water ( L/ h = 5 )
14
Deep water
12
C*
10
8
6
4
2
0
0.20
0.40
0.60
0.80
1.00
Fn
Figure 33: Catamaran C* coefficient for S/L = 0,2 in shallow and deep waters (SHIPFLOWTM)
Figures 32 and 37 show that the water depth as well as the distance between catamaran hulls has
low influence in the C* values for Froude numbers smaller than 0,3 and larger than 0,7. For
Froude numbers higher than 0,7 there is a convergence of C* values.
22
Figure 34: Catamaran C* coefficient for S/L = 0,4 in shallow and deep water (Millward[11]).
Fn
0.20
0.40
0.60
0.80
1.00
0
2
4
*C
6
8
Deep water
10
Shallow water ( L/ h = 5 )
12
14
S/ L = 0,4
Catamaran 40m - fixed
Figure 35: Catamaran C* coefficient for S/L = 0,4 in shallow and deep water (SHIPFLOWTM).
23
Figure 36: Catamaran C* coefficient for S/L = 1,0 in shallow and deep water (Millward[11]).
Catamaran 40m - fixed
S/ L = 1,0
10
8
Shallow water ( L/ h = 5 )
Deep water
C*
6
4
2
0
0.20
0.40
0.60
0.80
1.00
Fn
Figure 37: Catamaran C* for S/L = 1,0 in shallow and deep water (SHIPFLOWTM)
The C* results obtained by SHIPFLOWTM program for the 40 m catamaran when compared with
Millward [11] results, shown that the C* curve has the same aspect and it presents very close
results.
2.4 - CATAMARAN TRIM INVESTIGATION
Study for trim verification in high speed was conduct. Computational tests were conducted for
hull separation (S/L) 0,2 and 1,0. Shallow water (L/h=5) and deep waters were investigated.
Figures 38 and 39 present the results in a graphic form.
24
Draught ( m )
3.00
2.90
2.80
2.70
2.60
2.50
2.40
2.30
2.20
2.10
2.00
1.90
1.80
1.70
1.60
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Catamaran 40m - free
S/ L = 0,2
Deep water
Fn=0,4 Fn=0,5 Fn=0,6 Fn=0,7
Fn=0,2
Fn=0,3
Fn=0
Fn=0,8
0.00
10.00
20.00
30.00
Fn=0,9
40.00
Length (m)
Figure 38: Trim variation for different Froud numbers (S/L=0.2 and deep water).
It could be observed that for Froude numbers up to 0,3 dynamic trim is significant and the craft
sinks parallel.
When the catamaran achieve Froude number around 0,4 the catamaran acquires trim by stern
with significant draught variation. Above Froude number 0,5 the sinking happens practically with
the same trim angle with small variations.
Figure 39 presents the same test but in shallow water (L/h=5).
Figure 39: Trim variation for different Foude numbers (S/L = 0,2 L/h=5).
The bottom effect increases the dynamic trim as show in figure 39. Comparing with the sinking
in deep waters is observed that this sinking is larger in 29%.
Above Froude number 0,6 the stern reduces the sinking reaching draught of approximately 2,3m,
that is practically equal the value achieve in deep waters (2,4m). Therefore it can be concluded
that the effect of shallow waters only provokes significant sinking variation for the Froude range
between 0,4 and 0,6.
25
2.5 - INVESTIGATION OF THE WAVES GENERATED BY THE HULL
The objective of the analysis of waves generated by the passage of the craft is the verification of
the waves heights spread starting from the catamaran hull to a certain distance. Here the distance
was considered as 60% of the hull length.
Figure 40 shows the positioning of a catamaran hull in x-y plane. The points in the graph
represent the center of each panel used to represent the free surface.
Catamaran 40 m S/L = 0.2
Catamarã
40m S/ L of
= 0,2
Position of
the Centers
the panels
Posição dos centróides dos painéis
0.00
-2.00
-4.00
Eixo X - Extensão transversal
-6.00
-8.00
-10.00
-12.00
-14.00
-16.00
-18.00
-20.00
-22.00
-24.00
-26.00
-28.00
-20.00
0.00
20.00
40.00
60.00
80.00
Eixo X - Extensão longitudinal
Figure 40: Position of a 40 m catamaran hull and the free surface.
Figures 41 to 44 present a traverse cut plane showing the elevation of the points of the free
surface that define the wave height or elevation of each panel center of the free surface. These
points are defined starting from the lateral of the hull of the catamaran in deep water and shallow
waters (L/h=5) condition and hull spacing S/L = 0,2, that is considered to be the most restrictive.
26
Altura das ondas geradas ( m )
Height of the waves (m)
Catamarã
m - free
Catamarã
40m40
- livre
S/ L = S/L
0,2 - águas
( L/ h =water
5)
= 0.2rasas
shallow
Fn = 0,3
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
-1.10
-1.20
-1.30
-1.40
-1.50
Fn = 0.3
-30.00
-20.00
-10.00
0.00
Eixo Y - distância transversal do casco ( m )
Height of the waves (m)
Altura das ondas geradas ( m )
Figure 41: Wave height generated in shallow waters (L/h = 5) for Fn = 0,3
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
-1.10
-1.20
-1.30
-1.40
-1.50
Catamarã 40m - livre
Catamarã 40 m - free
S/ L = 0,2 - águas profundas
S/L
= 0.2 deep water
Fn
= 0,3
Fn = 0.3
-30.00
-20.00
-10.00
0.00
Eixo Y - distância transversal ao casco ( m )
Figure 42: Wave height in deep water for Fn=0,3
27
Altura das ondas geradas ( m )
Height of the waves (m)
Height
Height
of of
thethe
waves
waves
(m)
(m)
Height of the waves (m)
Catamarã 40 m - free
S/L = 0.2 shallow water Fn = 0.4
Catamarã 40m - livre
S/ L = 0,2 - águas rasas (L/ h = 5 )
Fn = 0,4
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
-1.10
-1.20
-1.30
-1.40
-1.50
-30.00
-20.00
-10.00
0.00
Eixo Y - dista6ncia transversal ao casco ( m )
Altura das ondas geradas ( m )
Figure 43: Wave height in shallow water (L/h = 5) for Fn = 0,4
Height of the waves (m)
Catamarã
40m40
- livre
Catamarã
m - free
S/ L = 0,2 - águas profundas
S/L = 0.2 deep water
Fn = 0.4
Fn = 0,4
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
-1.10
-1.20
-1.30
-1.40
-1.50
-30.00
-20.00
-10.00
0.00
Eixo Y - distância transversal ao casco ( m )
Figure 44: Wave height in deep water for Fn = 0,4
Figures 41 to 44 indicate:
-
The SHIPFLOWTM program allowed to calculate the wave high generates by the catamaran
hull and also investigate this effect in the river shore.
-
The effect of shallow water is very important when consider the wave high generate by the
catamaran hull.
28
4 – CONCLUSION
The SHIPFLOWTM and SLENDER computer software present good results to analyze the
catamaran in a preliminary design phase.
Some conclusions were extracted from catamaran simulation:
-
For S / L > 0,6 are not verified significant effects of interference between the hulls and the
wave resistance can be calculated as a twin hulls infinitely separated.
-
The type of the hull section is not a factor that modifies the wave resistance significantly.
-
The maximum positive interference happens around Froude number = 0,5 for deep waters and
0,4 for shallow waters.
-
Difference between SHIPFLOWTM and SLENDER program is high for smaller ration S / L
-
For 0.2 ≤ Fn ≤ 0.4 and Fn > 0.8, the results obtained by SHIPFLOWTM and SLENDER
program, compared with Millward [11] works are very close.
-
It is observed that for Froude numbers above than 0,8, in all S/L range (0.2, 0.4 and 1.0) a
convergence exists for values of Cw around 0,0002.
-
It is also observed that as smaller the spacing between hulls greater is the elevation of the
curve of Cw, for Froude number around 0,5.
-
When the effect of shallow waters is present, the interference is negative for Froude number
starting from 0,7.
-
The effect of the hull separation added to the shallow water effect cause a significant increase
in critical Cw value.
-
The wave high generate by the hull can be studied by SHIPFLOWTM. The results can be used
to study and minimize the effect in river shore.
29
5 – REFERENCES
1. Harward, SV. AA. "Resistance and Propulsion of Ships" , New York
2. J.L. Hess & A.M.O. Smith,1964, "Calculation Of Non-lifting
Arbitrary Three- Dimensional Bodies".
Potential
Flow
About
3. Nishimoto, K., 1998, "Semi-deslocamento Aplicando Mecânica dos Fluidos Computacional".
4. Dawson, C.W., 1977, "A practical Computer Method for Solving Ship Wave Problems"2º
International Conf. On Numerical Ship Hydrodynamics, Berkeley.
5. Picanço, Hamilton P.,1999, "Resistência ao Avanço: Uma Aplicação de Dinâmica dos
Fluidos Computacional" MSc, Thesis, COPPE/UFRJ, Rio de Janeiro.
6. Williams, M. A., "Otimização da Forma do Casco de Embarcações Tipo Swath", MSc,
Thesis, COPPE/UFRJ, 1994, Rio de Janeiro.
7. Havelock, T. H., "The Calculation of Wave Resistance". Proc. Of the RS, series A, vol. 144,
1934.
8. Newman, J. N. , Marine Hydrodynamics. The MIT press, Cambridge, Massachusetts and
London, England, 1977.
9. Lunde, J. K. "On the Linearized Theory of Wave Resistance for Displacement Ship in Steady
and Accelerated Motions". Trans SNAME, vol. 59, pp. 25-85
10. FLOWTECH, 1998, "SHIPFLOWTM 2.4, User manual".
11. Millward, A., 1992, "The Effect of Hull Separation and Restricted Water Depth on
Catamaran Resistance" The Royal Institution of Naval Architects, 1992
12. Chengyi, Wang "Resistance Characteristic of High-Speed Catamaran and its Application"
Marine Design and Research Institute of China, 1994.
13. Everest J. T. "Some research and hydrodynamics of catamarans and multi-hulled vessels in
calm water" Trans. N.E.C.I.E.S., vol. 84, pp. 129-148, 1968.
14. Kostjukov A. "Wave resistance of a ship near walls", J. Gydromech, vol. 35, p 9-13, 1977
15. Insel M. and Molland A. F. "An investigation into the resistance components of high speed
displacement catamaran" Royal Institution of Naval Architects, Spring Meeting, paper no.
11,1991.
16. Jane's "High-Speed Marine Transportation, 1996-97 Edition
30
17. Fry ED & Graul T. "Design and application of modern high-speed catamarans, Marine
Technology, 1972.
18. Michell, J. H., "The Wave Resistance of a Ship". Phil. Mag., Series 5, vol. 45, pp.106-123,
1898.
31