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Hydrodynamic Resistance Assessment of Non-Monohedric Planing Hull

Forms based on Savitskys Methodology

Carlo Bertorello, University of Naples Federico II, Naples/Italy, bertorel@unina.it

Luciano Oliviero, N.A., Aerospace dr. eng. Naples/Italy, luciano.oliviero@gmail.com

Abstract

Planing hard chine hull forms are widely used for high speed small craft. While monohedric* deepvee forms have been commonly used for more than 25 years the recent trends are toward nonmonohedric** forms in which the most of hydrodynamic lift is produced by low deadrise area in theafter part of the hull. Then the deadrise gradually increases in the center and in the forebody allowinghigher angles of incidence that are beneficial to reduce wave impact forces and ship motions. Also

wetted surface is reduced in comparison to the standard monohedric form resulting in smallerfrictional resistance. The origin of this trend is in the search of better seakeeping performances and

has been possible through a general reduction in ship weight due to main engines higherpower/weight ratio and lower structural weight.

Resistance predictions based on Savitskys method are commonly used although this procedure hadbeen developed on the hypothesis of monohedric geometry for the wetted part of the hull.The results are consequently affected by errors due to wrong assessment of both hydrodynamic lift

and center of pressure.

In this paper two procedures for the application of Savitsky method to non monohedric hulls areproposed. The results relative to a recently developed non monohedric hullform are compared tothose ones obtained by standard Savitsky method and by towing tank tests. Furthermore a study forthe analysis of error propagation in Savitskys long form procedure is proposed.

*monohedric is a V bottom hull form with deadrise angle constant at least from transom to midsection** non monohedric is a V bottom hull form with deadrise angle variable along the hull length

1. Introduction

Hard chine V bottom hull form has been widely used for small and medium size HSC since early

sixties. The hydrodynamic lift achievable through such hull form is the main factor for the reduction

of the wave resistance component at Fn higher than 0.8.

The semi-empirical method for resistance assessment of such hull form proposed by Savitsky (1964)although based on a simplified geometry, has proven effective and has been widely used till

nowadays. Lacks of the method as the neglect of the spray resistance component and of some parts of

the wetted surface have proven of small relative importance (6-10%) through the comparison with

experimental results. Differences became smaller as Fn increases.

The hypothesis of monohedricity has proven not too much restrictive for a long period when deep Vee

hulls with strictly monohedric afterbody were commonly used.In the last years the search for better

seakeeping performances for planing hulls combined with the availability of higher power/weight

ratios of main engines and with lower structural weight has led to prefer non-monohedric hull forms.

These ones concentrate the hydrodynamic lift in the very after part of the hull where the bottom has

low deadrise angles generally ranging from 9 to 12 degrees. Then the deadrise gradually increases in

the center part and in the forebody allowing higher angles of incidence beneficial to reduce wave

impact forces and ship motions.

Although this geometry is noticeably different from the V plate on which has been developed the

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Savitskys method, this one is still widely used, sometimes referring to a conventional value of the

deadrise angle as suggested by Savitsky (2006). The application of Savitsky method to more realistichull forms with deadrise angles varying according to the hull length and to hull forms with multi Vee

transversal sections is considered in this paper. Furthermore the error propagation is analyzed and

assessed.

2. Application of Savitskys method to non monohedric hull forms

Non monohedric hull forms for planing craft are generally hard chine and V bottom. Several aspects

can characterize their differences from the monohedric v plate used as geometric model for the

Savitskys procedure. In this paper we will consider only two of them that are the most common. They

are the presence of double or multiple Vee in transversal sections of the bottom and the progressive

variation of the deadrise angle from stern to the bow.

2.1. Multi Vee hull forms

Double or multi Vee bottom has been tested and in case of very high relative speed successfully

applied. The part of the bottom closer to the center line that is always immersed has a lower inrespect to the outer part. This last with higher deadrise angle, has a better behaviour when water

impacts.

2.1.1. Equivalent Deadrise Angle

A method to assess the equivalent deadrise angle * can be obtained through the constance of themain section form coefficient.

BT

AC MM =

T T

Figure 1: Double Vee main section and modified equivalent section

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With reference to Figure1 if we call A* the area not comprised inside the main section, the problem is

to determine the right angle triangle with the larger cathetus equal to B/2 of area equal to A*. * willbe the angle between the hypothenusas a* an the largest cathetus B/2

With reference to a polyhedral section composed by n sides:

( )2

1 1

1

*

=

ii

ii

n i

jjji

b

tgbbbB

tg

This transformation does not modify the areas as they are constant in respect to the form coefficients

but modifies the perimeter of the sections and consequently the wetted surface.

2.1.2. Wetted surface variation

In the hypothesis of prismatic hull form the variation of hull surface is proportional to the variation of

perimeter consequent to the transformation.

If a is the wetted perimeter of the true section and a the wetted perimeter of the transformed

section, it follows:

a

aa

'1 =

[ ] ++

=+=

n

i

iiii

n

ii

n

iiv tgbtgbTaaa 21

2

11

1

[ ] 21

*2*** 122

' tgB

tgB

Taaa v ++

=+=

( )

( )

++

++

=n

iiii tgtgbT

tgtgB

T

a

1

2

12

*2

1*2

1

12

1

In the hypothesis of prismatic hull, it follows:

aLSW ='' aLSW =

''a

a

S

S

W

W ='

'

a

aSS WW =

( )a

SS WW

=1

'

WS Wetted surface of the real hull'

WS Wetted surface of the equivalent hull

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These consideration allow to evaluate the increase in frictional drag due to the effective wetted

surface. As a reference from some investigated cases of multi Vee hull forms, the wetted surface

variation is in the order of few percentages.

2.2. Variable deadrise hull forms

The Savitskys method in the long form or in the short simplified versions is applied also to non

monohedric hull form in the general professional practice. The difference of such hull form from the

constant deadrise V plate for which the method has been originally developed is considered using an

average value of the deadrise angle. This value is suggested by individual practice or considering the

deadrise variation along the hull length. Savitsky (2006) suggests to use the deadrise value measured at L from the stern.

This type of simplified assumption can be effective for hydrodynamic lift assessment, provided an

appropriate average deadrise value had been chosen, but does not result in the true value of the

center of pressure longitudinal position that is strictly connected to the deadrise trend along the ship

length. With the average value, typically, the result gives a position of the center of pressure muchmore forward than it really is. The consequent longitudinal trim is higher then that observed in the

reality. The total resistance evaluation is not correct and approximate in excess.

In this paper the possibility of a more rigorous and effective application of Savitsky method to non

monohedric hull forms is presented.

2.2.1. Physical model and base hypothesis

The considered physical model is the typical one for a planing Vee plate. While the plate presents a

rectilinear motion the streamlines below the bottom have a divergence toward the after part of the

plate. It is common practice to divide the flow field around the plate into two components: the

longitudinal aligned with the motion direction and the vertical with direction opposite to the gravityforce. With these assumptions the motion longitudinal field can be considered as relative to a plane

plate with angle of incidence different from zero. The transversal field is considered as the field

around a wedge with zero lift angle.

Gravitational effects can be considered negligible in respect to inertial ones. Furthermore the

following hypothesis have to be taken into account:

Ideal non compressile non viscous fluid. Twodimensional flow field Uniform transversal pressure distribution Longitudinal position of hydrodynamic center of pressure connected only to longitudinal

pressure distribution.

2.2.2. Procedure

a)The hull is divided into N transversal sections.

With reference to a standard system of coordinates with x axis aligned with the keel and oriented

toward the bow, the generic i-th section is defined by the transversal plans i, with abscissa xi, and

1+

i , with abscissa xi+1.

the generic i-th section will be characterized by an average abscissa

2

1* ++= iiixx

x and by a constant

deadrise angle ( )ii x = .

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b) Nnni = ,1...,,1 Long Form Savitskys Method is applied in the hypothesis of

monohedric hull with constant deadrise angle i .

The sections with index*

11, ++>

nnwsxL are discharged as they do not contribute to

hydrodynamic lift .

c) Nnni = ,1...,,1 the following quantities are known:

*

i

i

x

from hull geometric characteristics

iC p

iw s

i

LL

,

,

by Savitskys Method

d) nni ,1...,,1 = we consider a planing plate with incidence angle i .

Applying the conformal mapping based on Schwarz-Christoffel differential equations we can

determine the function:

=

iws

iiL

xCpCp

,

that is:

( ) ( )

[ ]

+

+=

+

+

=

=

+

==

1,1

cos1cos2cos1lg

cos1cos1

arccos12

1lgcos1cos1

cos1

1

1cos1

cos1

2

,,,

2

22

21

i

i

i

i

i

i

i

m

iiiiiii

imimiws

ii

ii

senx

sensensenx

x

x

x

x

L

x

senV

pCp

where is a motion field characteristic dimension and is an arbitrary variable in conformalmapping.

e) nni ,1...,,1 = , when the functions

=

iws

i

ii

L

xCpCp

,

*

and their maximum values max,iCp ,have been calculated

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their non dimensional values in correspondence of*

ix , can be determined and the function:

max,

,

*

*

i

iws

ii

iCp

L

xCp

Cp

=can be obtained.

The value of this function in the way of i-th section represents the weight of the contribute of this

section to the hydrodynamic lift for a hull with constant deadrise i .

f) The valuews

CP

L

Lof the examined hull has been obtained as averaged weight of the determined

values*

iCp ni ,...,1= as shown in the following formula:

=

n

ii

n

i

iws

CPi

ws

CP

Cp

CpL

L

L

L

1

*

1

*

g) When determinedws

CP

L

Lthrough Bc (Projected beam at chine that is considered constant), the

factor e and the wetted length Lws can be calculated according to Savitskys method:

39.221.5

175.0

2

+

=

CvL

L

ws

CP

and

Lws = Bc

from that:

c

ws

CPCP B

L

LL =

and

2

5,25.0

1.1055.0012.0

v

LO

C

C

+=

h) Savitskys procedure is applied again. A first tentative value for the longitudinal trim angle is

fixed and CLO is obtained by LOC /1.1

.

When CLO e CL are known the deadrise angle of the equivalent hull can be determined and by theSottorf formula the bare hull total resistance RH can be evaluated. In factwhen and are known it is

possible to evaluate , and then F, Rn, SWL, DF and RH.

The equilibrium condition on which Savitskys procedure is based can be verified. If the values of the

determined forces do not verify the equilibrium a new value for is fixed and the process iscontinued to convergence.

i) The first tentative value for can be obtained by the position: LOC =1.1 LC and then by 1.1

LOC

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2.2.3. Wetted Surface

It is necessary to consider the wetted surface of non monohedric hull form. In the Savitskys procedure

for a monohedric hull form the following formula is used

cos

2

Cws

BS =

In the case of non monohedric hull form we can write:

( )dx

x

BS

wsL

Cws =

0cos

integrating, we obtain:

++

=

1

0

0

1

01 cos

cos

1

1lg

sen

senLBS wsCws

Where:

( )( )wsLx

x

====

1

0 0

3. Comparison with experimental results

An experimental test to validate the proposed procedure has been prepared. The differences in the bare

hull resistance values for monohedric hull forms obtained by Savitskys method and by experimental

tests had been already investigated at Naples University towing tank. Bare hull resistance of strictly

monohedric (80% of the length) hull forms with different deadrise angles had been experimentally

assessed. The difference from the Savitskys procedure result can be divided into two parts. The first is

due to known components that it is possible to asses, essentially the model aerodynamic resistance.

The second part is due to known and unknown factors that anyway are almost impossible to assess

directly. The most important are the spray and the vortex resistance components.

In the case of non monohedric hull form, the difference between experiment and theory is larger due to

the further error consequent to the inappropriate Savitskys procedure application.

The comparison of Savitskys and experimental results can be synthesized in the following:

for monohedric hull forms

R(experimental)= R(Savitskys) + R (aerod) + R(unknown)

for non monohedric hull forms:

R(experimental)= R(Savitskys) + R* + R (aerod) + R(unknown)

where R* is the component due to inappropriate Savitskys application.

3.1. Experimental set up

The resistance tests were conducted in the Naples University towing tank (130 m x 9 m x 4.20 m).

The towing carriage is able of speed up to 7 m/s. The models have been tested without turbulence

stimulators. The considered not-monohedric hull form has the following main characteristics as shown

in Table 1 below:

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Table 1: Non-monohedric Hull Form Parameters

Loa Lwl Bmax Bwl T Displacement Static Wetted Surf.

(m) (m) (m) (m) (m) (kg) (m2)

2.36 2.22 0.614 0.513 0.104 103.7 1.11

This model refers to a 100 Loa high speed motor-yacht. The investigated Fn values range from 0.46

to 1.23.

Figure 2 Towing tests of non monohedric (left) and monohedric (right) hull forms.

3.2. Comparison

In the Table 2 and in the diagrams reported in Figure 3 the assessment of the R* component relative to

the model described in the previous paragraph is presented.

In Table 2 the RH values obtained by experimental tests and by three different application ofSavitskys method are reported:

betaTransom values are relative to Savitskys long form application when using the deadrise

angle measured at transom;

betaLpp/4 values are relative to Savitskys long form application when using the deadrise

angle measured at 1/4 L from the stern;

NON MONOHEDRIC values are relative to the application of the presented procedure. In

this case effective is the deadrise angle of a fictitious monohedric hull equivalent to the real

one according to Savitskys procedure.

The difference of the results obtained in the first two cases is negligible due to marginal variation of

the deadrise angle in the after part of the hull. The result obtained through the application of the

proposed procedure is remarkably different. It highlights the R* component previously identified and

allows its evaluation.

From the diagrams it is appreciable at first sight how the results of Savitsky NON MONOHEDRIC

proposed procedure have a better and closer fit to the experimental ones. The residual gap between the

two curves is given by aerodynamic and sprays components.

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Table 2: RH values obtained by experimental tests and by three different application of Savitskys

method

EXPERIMENTALSavitsky

betaTransom

Savitsky

betaLpp/4

Savitsky NON

MONOHEDRIC

RH RH RH RH effective[N] [deg] [N] [deg] [deg] [N] [deg] [deg] [N] [deg] [deg]

V

[m/sec

]

4 73 3.15 61 2.26 8.40 61 2.28 8.70 68 2.93 18.5

5 92 4.22 77 2.55 8.40 77 2.57 8.70 86 3.42 19.4

6 103 5.31 92 2.68 8.40 92 2.70 8.70 98 3.33 16.3

RH = RH ( V )

60

70

80

90

100

110

4 5 6

V [m/sec]

RH[N]

EXPERIMENTAL

Savitsky betaTransom

Savitsky betaLpp/4

Savitsky NOT MONOHEDRIC

Figure 3 Diagrams of RH curves obtained by experimental tests and by three different application of

Savitskys method.

4. Analysis of error propagation in Savitskys long form procedure

The aim of this analysis is to evaluate the sensitivity of Savitskys procedure in regard to the unitaryvariation of input data meaningful figures. The sensitivity is assessed as relative error on the output

data that is bare hull resistanceHR

.

For the constants and for the input data the relative error will be the ratio between the unit variation of

the smallest significant figure and the value of the considered characteristic. As an example is reported

the relative error for the speed value of 25.7 m/s that will be used in the following:

sec7.25m

V =

Smallest significant value: 0.7

Unitary variation: 0.1 (25.65 < V < 25.75)

Relative error:

31089.3

7.25

1.0

==V

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4.1. Procedure Development

1-C

VgB

VC = ( ) ( ) ( ) ( ) ( )( )

+= CCCC

C

V dBgdgBgBV

dVgBgB

dC 21

21

2

1

+=

C

C

V

V

BB

gg

VV

CC

21 ( )CV BgVC += 21

( )CV BgVC

++=2

1

2- 22

2

1C

L

BV

C

=

( ) ( ) ( ) ( )[ ]2

22

222222

2

1

22

2

1

2

1

++

=

C

CCCCC

L

BV

dBBVBdVVBVdBVd

dC

C

C

L

L

B

B

V

V

C

C

=

22

( ( )CL BVC

= 22

CL BVC 22 +++=

3-6.0

00 0065.0 LLL CCC =

( ) ( ) ( ) ( ) dCdCCdCdC LLLLL = 6.0

00

4.0

00 0065.00039.0

( ) ( )( ) ( ) dCdCCdC LLLL = 6.0

00

4.0

0 0065.00039.01

( )( ) ( )

4.0

0

6.0

0

00039.01

0065.0

+=

L

LL

LC

dCdCdC

+

=

4.0

0

4.0

0

6.0

000

0

0039.01

0065.0

0039.0 L

L

L

L

LL

L

L

L

C

C

C

C

CC

C

C

C

( ) ( )

+

=

4.0

0

4.0

0

6.0

00 0039.01

0065.0

0039.00 L

L

L

LL

L

CC

C

CC

C

L

4.0

0

4.0

0

6.0

00 0039.01

0065.0

0039.00

+

=

L

LC

LL

L

CC

C

CC

C

LL

4-

+= 2

2

1.10 0095.00120.0

21

V

LC

C

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( )

( ) ( ) ( )( )

++

+

+=

VVV

V

V

L

dCCdCC

d

CddC

22

4

1.1

2

21.0

0

220095.0

006.0

0095.0012.01.1

21

21

+

=

V

V

VVLL

L

C

C

CCCC

C2

2

2

21.1

00

0 0190.00190.0006.01

1.1 21

2

2

2

2

1.1

0

0

0

0190.0006.0

0190.01.1

21

V

V

V

V

L

L

L

C

C

C

C

C

C

C

+

=

( )

2

2

2

2

1.1

0

0190.006.0

0190.01.1

21

0

V

C

V

L

C

C

C

CVL

++=

5- +=F += ddd F

+

=

FFF

F

+=

F

F

FF

6- LWS = F BC ( )CFCF dBB += ddLWS CFWS BL +=

7- VV

VV mm

= ( )dVV

V

V

VdVdV mmm +

= V

V

VV mm +=

8-

FCmn

BVR =

( ) ( ) ( )[ ] ( ){ }

dBVdBVdBVdVBdRFCmFCmCFmmFCn++=

2

1

FCmn BVR +++=

9- ( )FnF

CRLogC

=2

1

242.0

F

F

n

n

F

F

FC

C

R

R

C

C

C

+

=

10lg

1

10lg

1

2

242.0

21

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nF R

F

F

C

C

C

1 0lg12 1.021

21

+=

10- FFF CCC +=' FFF CF

FC

F

FC C

CCC += '''

11-'

22

cos2

1F

CFmF C

BVD

=

( ) +

+

+

+

+

=

'

''22

222cos2

cos4

1

F

F

C

C

m

m

F

F

FCmFFC

dC

B

dB

V

dVddCBVdD

( ) ( )[ ]

dtgCBV FCmF +'22

2

cos2

cos4

1

( ) tgFCmFF C

BVD +++++= '22

12-

tgD

R FH +=cos

( )( ) ( )

dtgd

dsenDdDdR FFH

++

+=

22 coscos

cos

+

+

+

=

2cos

costg

Rtg

D

D

R

D

R

R

HF

F

H

F

H

H

( )

+

++= t

H

D

H

F

R tgR

tgR

D

FH

2

cos

cos

4.2. Application Example

To give an example the previous formulas have been applied to the resistance assessment for the hull

considered in paragraph 2 at a speed of 50 kn.

Table 3 reports the uncertainties due to the various factors involved and the global error assessment. It

can be noticed that the total uncertainty value is about 12%. Similar tables have been developed for the

errors due to the uncertainty margin of deadrise angle, speed value, displacement , longitudinal trim , projected chine beam BC It can be noticed that speed values and longitudinal trim are the mostinfluencing factors on the global error. For space reasons the tables relative to deadrise angle and

speed values are reported only.

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Table 3: Global error assessment

DATA Relative error

Symbol Value Unit Value Unit Symbol Value Formula

Inputdata

BC 2.600 m

BC 3.85E-04 0.04% = 0.001 / BCg 9.81 m/sec2 g 1.02E-03 0.10% = 0.01 / g

V 25.7 m/sec2 V 3.89E-03 0.39% = 0.1 / V 46598 N 2.15E-05 0.00% = 1 / 2.99 deg 0.052 rad 3.34E-04 0.03% = 0.001 / 20.0 deg 0.349 rad 5.00E-03 0.50% = 0.1 / 1.1883E-06 m2/s

8.42E-05 0.01% = 1e-10 /

1026 Kg/m3 9.75E-04 0.10% = 1 /

F 0.501 F 2.00E-03 0.20% = 0.001 / F(Vm/V) 0.99 (Vm/V) 1.02E-02 1.02% = 0.01 / (Vm/V)

CF 0.0004 CF 2.50E-01 25.00% = 0.0001 / CF

Outputdata

CV 5.09

CV 4.59E-03 0.46%

CL 0.020 CL 9.55E-03 0.95%

CL0 0.039 CL0 5.06E-03 0.51% 0.908 9.62E-02 9.62%

F 1.409 F 6.27E-02 6.27%

LWS 3.66 m LWS 6.31E-02 6.31%

Vm 25.4 m/s Vm 1.41E-02 1.41%Rn 7.84E+07

Rn 7.72E-02 7.72%

CF 2.1E-03 CF 1.10E-02 1.10%

CF' 2.5E-03 CF' 4.86E-02 4.86%

DF 8533 N DF 1.42E-01 14.19%

RH 10979 N RH 1.18E-01 11.80% RH 1295 NRHmin 9684 N

RHmax 12274 N

Table 4: Error due to the uncertainty margin of deadrise angle

DATA Relative error

Formul

a Unit Symbol Value

Formul

a Symbol Value Formula

Inputdata

BC 2.600 m

BC 0.00E+00 0.00%

g 9.81 m/sec2 g 0.00E+00 0.00%

V 25.7 m/sec2 V 0.00E+00 0.00% 46598 N 0.00E+00 0.00% 2.99 deg 0.052 rad 0.00E+00 0.00%

20.0 deg 0.349 rad 2.50E-02 2.50%= 0.5 /

1.1883E-

06 m2/s

0.00E+00 0.00%

1026 Kg/m3 0.00E+00 0.00%

F 0.5011889 F 0.00E+00 0.00%(Vm/V) 0.98905 (Vm/

V)0.00E+00 0.00%

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CF 0.0004 CF 0.00E+00 0.00%

Outputdata

CV 5.09

CV 0.00E+00 0.00%

CL 0.020 CL 0.00E+00 0.00%

CL0 0.039 CL0 1.59E-02 1.59%

0.908

3.28E-03 0.33%

F 1.409 F 2.11E-03 0.21%

LWS 3.66 m LWS 2.11E-03 0.21%

Vm 25.4 m/s Vm 0.00E+00 0.00%Rn 7.84E+07

Rn 2.11E-03 0.21%

CF 2.1E-03 CF 3.01E-04 0.03%

CF' 2.5E-03 CF' 2.53E-04 0.03%

DF 8533 N DF 5.54E-03 0.55%

RH 10979 N RH 7.21E-030.72

%

RH 79 NRHmin 10900 NRHmax 11058 N

Table 5: Error due to the uncertainty margin of speed value

DATA Relative error

Formula Unit Symbol Value

Formula Symbol Value Formula

Inputdata

BC 2.600 m

BC 0.00E+00 0.00%

g 9.81 m/sec2 g 0.00E+00 0.00%

V 25.7 m/sec2 V 3.89E-03 0.39% = 0.1 / V 46598 N 0.00E+00 0.00%

2.99 deg 0.052 rad 0.00E+00 0.00% 20.0 deg 0.349 rad 0.00E+00 0.00%

1.1883E-

06 m2/s

0.00E+00 0.00%

1026 Kg/m3 0.00E+00 0.00%

F 0.5011889

F 0.00E+00 0.00%

(Vm/V) 0.98905 (Vm/

V) 0.00E+00 0.00%

CF 0.0004 CF 0.00E+00 0.00%

Outputdata

CV 5.09

CV 3.89E-03 0.39%

CL 0.020 CL 7.78E-03 0.78%

CL0 0.039 CL0 4.09E-03 0.41% 0.908 7.25E-02 7.25%

F 1.409 F 4.67E-02 4.67%

LWS 3.66 m LWS 4.67E-02 4.67%

Vm 25.4 m/s Vm 3.89E-03 0.39%Rn 7.84E+07

Rn 5.06E-02 5.06%

CF 2.1E-03 CF 7.21E-03 0.72%

CF' 2.5E-03 CF' 6.07E-03 0.61%

DF 8533 N DF 6.06E-02 6.06%

RH 10979 N RH 5.01E-025.01

% RH 550 N

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15/15

RHmin 10429 N

RHmax 11529 N

5. Conclusions

The paper presents a mathematical model to treat non monohedric hull geometries in the frame of

Savitskys procedure for planning hull resistance assessment. To consider multi Vee transversal

sections suitable formulas for the evaluation of an equivalent deadrise angle and for the assessment of

the relative wetted surface have been proposed. Equivalent deadrise angle can be used instead of the

standard average angle generally adopted in professional practice.

The proposed model for the variable deadrise hull form has been validated through experimental tests.

The results of the proposed procedure have a better and closer fit to the experimental values than those

obtained by Savitskys standard long form method.

The analysis of the propagation of the errors highlights the relative importance of the different factors

in relation to the uncertainty of the final result and gives the designer a clear picture of the matter.

This work is one step in the development of a design modulus relative to the powering performancesof high speed small craft to be implemented within a Multi Attribute Decision Making procedure.

6. Acnowledgements

This work has been financially supported by University of Naples Federico II within the frame of

2005-2006 research program.

7. References

Savitsky, D., DeLorme, M.F. Datla, R., (2006): Inclusion of Whisker Spray Drag in Performance

Prediction Method for High Speed Planing Hulls, The Society of Naval Architects and Marine

Engineers, TECHNICAL REPORT SIT-DL-06-9-2845 March 2006

Keller, J.B., Ting, L., (1977): Optimal Shape of Planing Surface at high Froude NumberJournal ofShipResearch, Mar. 1977

Milne-Thomson, L.M., (1968): Theoretical Hydrodynamics Chapter X, London Mc Millan & Co.

Savitsky, D., (1964): Hydrodynamic Design of Planing Hull, Marine Technology vol.1 , No 1,1964

Korvin-Kroukovsky . B.V., Savitsky, D., Lehman, W.F., (1949):Wetted area and center of pressure

of planning surfacesReport n. 360 August 1949 Experimental Towing Tank Stevens Institute of

Technology,

Pierson. and Leshnover,. (1948): An analysis of the fluid flow in the spray root and wake regions of

flat planing surfaces, SIT-DL-48-335 , Stevens Institute of Technology

Sottorf W., (1934): Experiments with planing SurfacesNACA Technical Memorandums739 March 1934