COMPARISON BETWEEN NUMERICAL COMPUTATIONS AND EXPERIMENTS FOR by hkksew3563rd

VIEWS: 42 PAGES: 36

									           COMPARISON BETWEEN NUMERICAL COMPUTATIONS AND
       EXPERIMENTS FOR SEAKEEPING ON SHIP'S MODELS WITH
                                      FORWARD SPEED


                  C.Maury*, G.Delhommeau*, M.Ba**, J.P.Boin*** and M.Guilbaud***
*
    L.M.F. (UMR CNRS n°6598), Ecole Centrale de Nantes, BP 92101, 44321 Nantes Cedex 3,
                      France;   Tel : 33-2-40-37-25-95 Fax : 33-2-40-74-74-06
      **
           L.E.A. (UMR CNRS n°6609), ENSMA, 1 rue C. Ader, BP 40109, 86960 Futuroscope
             Chasseneuil Cedex, France; Tel : 33-5-49-49-80-87 Fax : 33-5-49-49-80-88
***
      L.E.A.-CEAT (UMR CNRS n°6609), Université de Poitiers, 43 rue de l'Aérodrome, 86036
                Poitiers Cedex, France; Tel : 33-5-49-53-70-27 Fax : 33-5-49-53-70-01


Corresponding author: Guilbaud Michel, LEA-CEAT, 43 rue de l’Aérodrome - 86036 Poitiers

Cedex, France ; tel. 33-5-49-53-70-27;fax 33-5-49-53-70-01; e-mail:guilbaud@univ-

poitiers.fr

                                              24/10/01
                                              Abstract
The recent progress of computers and numerical algorithms enables today to use the
translating and pulsating Green function in panel methods for seakeeping calculations. Two
different methods of calculations of this function, its derivatives and their integrations over
panels or segments are briefly presented and have been introduced into the seakeeping codes
Aquaplus developed at Ecole Centrale de Nantes and Poseidon at CEAT-LEA. Both codes
interchange the Fourier and boundary integrals on panels or waterline segments, the last part
being performed analytically. These methods have been used to compute flows around Wigley
or series 60 model ships. To check the numerical results, an experimental set-up has been
developed at the CEAT, which measure forces and moments on model in forced harmonic
oscillations of pitch or heave. Tests have been performed in the recirculating water channel of
Ecole Centrale de Nantes on two L=1.2m series 60 models of CB=0.6 and 0.8 block
coefficients. Unsteady wave patterns have been recorded using a resistive wave probe. The
experimental results are compared with the numerical ones.




                                                 1
                                          Introduction
The numerical difficulties to solve numerically the seakeeping problem with forward speed
lead to the necessity to check numerical results with tests. Few experimental data concerning
this problem are available and in some cases, not appropriate for accurate comparison.
Moreover, the comparison of measured and calculated global forces or motions of ship
running in waves is not an efficient check for the quality of numerical results because they
involve hydrodynamics and mechanics, (Okhusu & Wen 1996, Okhusu 1998). So comparison
with calculations does not enhanced the progress of the numerical methods and it becomes
necessary to perform new tests with local measurements (pressure distribution, wave pattern)
to improve the knowledge about the accuracy of the calculation methods. Some test results
can be found in (Okhusu 1998), but also in (Iwashita et al 1993) concerning OHS (a bulk
carrier's hull form of CB=0.8 and L/B=5.48 where L is the length and B the width), Series 60
or VLCC (very large crude carrier) ships. Two different techniques can be used in tests, free
models in waves where forces and moments but also hull motions have to be measured or
forced motion technique where the motion amplitude is known accurately and only the phase
lag reference can be subject to errors.
       Panel methods are very efficient for seakeeping calculations. For the problem without
forward speed, Green function is easy to compute. For the problem with forward speed, a lot
of numerical and theoretical difficulties appear. So, for a long time, only frequency encounter
approximation limited to thin ships (with the zero speed diffraction-radiation Green function)
or Rankine methods available for high values of Brard's parameter τ=ωeU/g (ωe encounter
circular frequency and U forward speed) have been used. Recently, using either steepest
descent method (Iwashita & Okhusu 1989), (Brument & Delhommeau 1997), (Brument
1998), (Maury 2000), Super Green function method (Chen & Noblesse 1998) or Simpson
adaptative method (Ba & Guilbaud 1995), (Nontakaew et al 1997), progress in the
calculations of the Green's function have been performed. It is now possible to develop
seakeeping codes using this Green function on small workstation with moderate
computational time and good accuracy, see (Du et al 1999,2000), (Boin et al 2000). In spite of
the numerical difficulties of its computation, the advantages of this Green function are an
automatic fulfilment of the linearized free-surface and radiation conditions, the absence of
mesh on the free-surface avoiding reflection on the boundary and the filtering of smaller wave
lengths and the use of method, it is necessary to compute accurately the integrals of the Green
function and its derivatives on elementary panel or waterline segment. Numerical quadrature


                                               2
is not able to give correct results for a panel and a collocation point close to the free surface,
(Guilbaud et al 2000). So it is needed to integrate analytically on panels or segments following
formulas given by (Iwashita 1992) for the steepest descent method or (Bougis 1981) with the
formulation of (Guevel & Bougis 1982).
        We present here two numerical methods to perform seakeeping calculations computing
accurately this Green function and its derivatives and their summation on panels or segments.
The aim is to obtain, in all cases and especially for high forward speed, the Green function and
its derivatives with a given accuracy. The first method, Aquaplus, developed at Ecole Centrale
de Nantes is founded on Bessho's formulation (Bessho 1977) and uses the steepest descent
method. The second method, developed by the CEAT, uses the formulation of (Guevel and
Bougis 1982) with computation of single integrals by Simpson adaptative method.
        We present also an experimental study using the forced motion technique in order to
compare the results with those of the two numerical methods already mentioned. Forces,
moments and also wave patterns have been measured in circulating water channel on Series 60
ship's models with length L=1.2m in forced oscillations of heave or pitch, using a
development of the experimental set-up used in (Guyot 1995) or (Guyot & Guilbaud 1995)
where a L=0.6m model have been used. The experimental set-up is a development of the one
used in (Delhommeau et al 1992).

                             Calculations of the Green function

Computation of the Green function with steepest descent method
Point source :
Let us consider a source point at P ' (x ', y ', z ') and a field point at P (x , y, z ) . The source is

moving at constant speed U and pulsating with an encounter pulsation ωe . The Green
function satisfies the linearized free surface condition and is expressed under the form (1)
derived in (Bessho 1977), which presents the main advantage to be written as a single integral
over the complex variable θ, allowing to integrate over any available paths in the complex
plane. Using this expression, one can evaluate at the same time the Green function and its
derivatives of first or higher order. Development of this method, initially proposed in
(Iwashita 1992), has been done at Ecole Centrale de Nantes (Brument & Delhommeau 1997,
Brument 1998), or more recently (Maury 2000). g is the Froude dependent (or Kelvin) part of
the Green function.



                                                   3
               1 1     
                        
                                                 
                  − 1  + g (P, P ') = 1  1 − 1  − iK0
                                                  
                                                                             θ2 (P ,P ')   k2ek2ξ − sgnc.k1ek1ξ
 G (P, P ') =    
              4π  R R1 
                 
                 
                        
                        
                                           
                                           
                                           
                                                  
                                                  
                                        4π  R R1  2π                   ∫ θ1                  1 + 4τ cos θ
                                                                                                                  .d θ (1)


            R1 
               
                           2           2           2
where           = (x − x ') + (y − y ') + (z ± z ')
            R 
               
            X = K 0 (x − x ')               g                        k1 1 + 2τ cos θ ± 1 + 4τ cos θ
                                                                     
                                       K0 =                           =
                                           U2                        
            Y = K0 y − y '                                   and     k2
                                                                                    2 cos2 θ
                                          U ωe                       
                                       τ=
            Z = K 0 (z + z ')              g                         ξ = Z + i (X cos θ + Y sin θ )
                                                                                                       π
                                                                     θ2 (P, P ') = ϕ (P, P ') −           − iε (P, P ')
                 
                  π + arccos 1                   1                                                     2
                 −
                                          if τ >
                               4τ                                                                      X
            θ1 = 
                                                  4          and
                                                                    ϕ (P, P ') = arccos
                 
                                1                1                                                 X 2 +Y 2
                 −π − i arg ch
                 
                                           if τ <
                               4τ                4                                                  Z
                                                                     ε (P, P ') = arg sh
                                                                                                    X 2 +Y 2

                                                                            −1
                                                                                              if t < 0
                sgn c = sign [ Re (cos θ )]                                
                                                                           
                                                                           
                                                ,     with     sign (t ) =  0
                                                                                              if   t =0
                sgn s = sign [ Re (sin θ )]                                
                                                                           
                                                                            1
                                                                                              if t > 0
                                                                           
                                                                           
Due to their behaviours, terms in k1 and k2 are computed differently. Terms in k2 are
computed directly in the θ -space, whereas the steepest descent method is needed for terms in
k1 due to strong oscillations in the vicinity of ±π / 2 . To apply the steepest descent method

we separate the integral over θ in different parts following the sign of terms sgnc and sgn s :
                                                 −π
                                                 2
                                      π
                                                
                                                ∫θ f1 (θ).dθ                                                 if X > 0
     θ2 (P,P ')                     −            2(P,P ')
I =∫            −sgnc.f1 (θ).dθ = ∫ f1 (θ).dθ + 
                                      2
                                                
    θ1                             θ1           
                                                     −
                                                          π          π              π
                                                 ∫ 2 f1 (θ).dθ − ∫ 2 f1 (θ).dθ + ∫ 2 f1 (θ).dθ
                                                 0                                                        if X < 0
                                                
                                                                  0               θ2 (P,P ')


                      k1e k1ξ
with f1(θ) =                      . The steepest descent method is applied in a space M defined
                  1 + 4τ cos θ
by : M = sgn c.k1.cos θ = sgn c.m . Rewriting integrals in respect to variable M , we obtain

(2) for the function and its derivatives:
                       I M
                       
                               ( )                                                          if X > 0
             ( )           θ2
      I = I Mθ        +                                                                                          (2)
                  1    
                        I (M 0 )
                       
                       
                                 sgn s =−1
                                            − I (M 0 )
                                                       sgn s =1
                                                                    ( )
                                                                + I Mθ
                                                                      2
                                                                                     if X < 0




                                                        4
                                
                                                   1                
                                                                     
                                
                                                                    
                                                                     
                                
                                                 i.m                
                                                                     
                                
                               ∞
                                                                     
                                                                     
                                                                   2
                                                                             e φ(M )
with    I (β ) = sgn c.sgn s ∫                          m                             dM
                              β                                    
                                i.sgn s. (m − τ )2 1 − 
                                                                                        2
                                                        (          
                                                                           m
                                                         m − τ )2   1 −            
                                                                                       
                                
                                                                                   
                                                                                     
                                                                            (m − τ )2 
                                                                           
                                
                                             (m − τ )2              
                                                                     
                                
                                                                    
                                                                     
                                                                                                                 2
                                                              m        
                                                                        
and     φ (M ) = (m − τ )2 Z + i.m.X + i.sgnsY (m − τ )2 1 − 
                                             .                m − τ )2  .
                                                             (         
                                                                        
                                                             

It must be noticed that the term (m − τ )2 is out of the square root avoiding the existence of a
branch cut in the M -plane, which simplifies quite significantly the research of steepest
descent lines. Integration is done following a path on which the imaginary part of φ is
constant, suppressing the oscillations of integrands, in a direction where the real part
decreases. Steepest descent lines do not exist in all cases because they must be entirely
included in a θ -space area where sgnc and sgn s are constant. Moreover, the steepest descent
line is determined numerically with a given accuracy. In the case of a failure in the research of
a steepest descent line, the starting point β is moved to β ' in order to find another line. A
complementary integral between β and β ' is then introduced and computed in θ -space. β '
must be chosen carefully to avoid large oscillations of the complementary integral.
Panel and segment source :
The solution of 3-D seakeeping problem with forward speed by the boundary element method
needs to compute an integral of a source distribution over panels on the body or segments on
the waterline. Thus the computation of the Green function associated with a panel or a
segment of constant source density is developed. The analytical quadrature of Kelvin part g of

(1) over a panel or a segment (Iwashita 1992) gives us expressions I S =                                         ∫∫ gds       for a panel
                                                                                                                 ∆S


with N vertices and I L =                 ∫ gdl       for a straight line:
                                          δC

                                                    1 k2ξl     1
                                                       e − sgnc ek1ξl
            i     N        θ2 (P ,Ql )    2Sl       k2         k1                     i        1 N        S 
 IS = −
          2πK 0
                  ∑∫      θ1             ψl (θ )           1 + 4τ cos θ
                                                                             dθ +
                                                                                    πK0τ 2   ∫ ∑  ψ (θ)
                                                                                              0
                                                                                                            l
                                                                                                                                   νl (η)d η
                  l =1                                                                            l =2      l        θ=θ2 (P,Qη )

       i β (Y1 −Y2 )     N          θ2 (P ,Ql )        1       e k2ξl − sgnc.e k1ξl
IL =                     ∑∫                                                         dθ                                               (3)
           2π            l =1
                                  θ1
                                                  (ξl +1 − ξl ) 1 + 4τ cos θ
with :Ql vertices of panel (l = 1, N ) and Sl area of triangle (Ql ,Ql +1,Ql −1 )


                                                                      5
                    (
       Xl = K 0 x P − xQ
                                  l
                                      )
        Yl = K 0 yP − yQ                  (with the sign of (yP − yQ ) constant for l = 1, N
                              l                                           l



                    (
        Zl = K 0 z P + zQ
                              l
                                      )
        ψl (θ ) = (ξl +1 − ξl )(ξl −1 − ξl )            and       ξl = ξ (P,Ql )

                   X ηYl '− Xl 'Yη             Zl ' (X η + Yη2 ) − Z η (X η Xl '+ Yl 'Yη )
                                                       2

       νl (η ) =                          +I
                          2
                        X η + Yη2                 (X η2 + Yη2 ) (X η2 + Yη2 + Z η2 )
                 X η = X 1 + η (X l − X 1 )                       X l ' = Xl − X 1
       where Yη = Y1 + η (Yl −Y1 )                      and       Yl ' = Yl −Y1
                   Z η = Z 1 + η (Zl − Z1 )                       Zl ' = Zl − Z 1

Expressions of the derivatives may be found in (Maury 2000) or (Ba et al 2001a,b). Here
again, the steepest descent method is useful to evaluate efficiently terms in k1 . For this reason
and the fact that θ2 varies with Ql , computation of integrals corresponding to each vertex l ,

are done separately. Due to this decomposition of the sum, singular points θl appears in

integrands when ψl (θ ) equals to zero. The values of these singular points are known and

evaluated by the (4) below:
                                                                                Xel = XQ              − XQ
                            Ye ± Xe 2 + Ye 2 + Ze 2                                        (l +1)             l
                                                 l  with
              θl = 2 arctan  l
                            
                                    l       l
                                                                                Yel = YQ             −YQ           (4)
                                 Xel − i.Zel        
                                                     
                                                                                           (l +1)      l


                                                                                 Zel = ZQ             − ZQ
                                                                                             (l +1)         l



Paths of integration must avoid these singular points, and the contribution of poles must be
taken into account. It is necessary to evaluate this contribution in θl , referred to vertices Ql

and Ql +1 , when the paths used for both integrals ( l and l + 1 ) surround the singular point

considered.
       When the influenced point is far from the panel (or the segment), we can optimise the
evaluation of the Green function using an averaged path of integration to integrate directly the
whole sum from θ1 to θc . Only one integral has to be evaluated and there is no singular point

to take care. Nevertheless, θ2(l ) being a function of the vertices Ql , we have to complete this

expression by a sum of integrals from θc to each θ2(l ) as written in (5). Those integrals are

done either in θ -space or with steepest descent method:


                                                           6
                                   θc    N                               N        θ2 (P ,Ql )
            I S (or I L ) =   ∫   θ1
                                        ∑ − sgnc .f1,l (θ ).d θ + ∑ ∫
                                        l =1                             l =1
                                                                                 θc
                                                                                                − sgnc .f1,l (θ ).d θ (5)

                                                  1 k1ξl
                                                     e                                                                              
                       −i         2S l            k1                                               i β (Y1 − Y2 )        e k1ξl     
                                                                                                                                     
where     f1,l (θ ) =                                                 or
                                                                               f1,l (θ ) =                                          
                                                                                                                                     
                      2πK 0 ψl (θ ) 1 + 4 τ cos θ                    
                                                                     
                                                                                                   2π (ξl +1 − ξl )                 
                                                                                                                       1 + 4 τ cos θ 
                                                                                                                                     

        Concerning the terms in k2 , we can easily use a common path for each integral

associated with each vertex of the panel. Integrands have no oscillatory behaviour so we have
no constraint on the path of integration. Expression (3) is decomposed as in (5). Here again,
we have to complete the integral by a sum of integrals from θc to each θ2(l ) .

        It had been shown, (Maury 2000) that the difficulty to calculate increase with the order
of derivation but decrease from the function itself to the segment integral and to the panel one.
This is due to the fact that as the power in k1 is higher the integrands decreases more rapidly.

Computations of the Green function with a Simpson Adaptative method
Point source
The Green function used is given in (Guevel & Bougis 1982) and developed in (Ba &
Guilbaud 1995) with the dimensionless parameters ω = ωe                                         0
                                                                                                    /g , F =U / g        0
                                                                                                                             , τ = ωF

Kc = 4ω 2 where           0
                              is a reference length. It can be written as the sum of 3 functions

G = G0 + g = G 0 + G1 + G2 , defined by:

                                                                 1  1  1
                                               G 0 (P , P ') =      R R−  , (6)
                                                                  0 
                                                                         1
                                                                           
                                π 2 K g (K ξ ) + g (K ξ ') − K g (K ξ ) + g (K ξ ') 
                                                                                            
                         1           1  1                     2  1                       
                                                                                        d θ  , (7)
                               ∫
                                            1       1  1               2       1  2
        G 1 (P , P ') =
                        π 0    0
                                                       1 + 4 τ cos θ                        
                                                                                             
                               
                                                                                            
                                                                                             
                          θc'
                                 Z g (Z ξ ) + g 2 (Z 3 ξ ') − Z 4 g 2 (Z 4 ξ ) + g 2 (Z 4 ξ ')         
                                                                                                               
                         
                          ∫ −i 3  2 3                                                                        
                                                                                                      dθ     
                         0
                                                       4 τ cos θ − 1                                          
                                                                                                               
                         
                          θc −αc                                                                              
                         
                                        g (Z ξ ) + g (Z ξ ') − Z g (Z ξ ) + g (Z ξ ') 
                                     Z3  1 3                                                                  
                         + ∫ −i                       1     3           4  3      4        3     4      d θ
                                                                                                               
                         
                          θ'                                                                                  
                                                                                                               
                                                              4 τ cos θ − 1                                    (8)
                      1       c
                                                                                                               
     G 2 (P , P ') =     
                                                                                                              
                                                                                                               
                                                      (
                     π 0  2πK c (1 − i ) e Kc ξc + e Kc ξc          )                                         
                                                          '
                         
                         −                                                                                    
                                                                                                               
                         
                                                                αc                                            
                                                                                                               
                                        τ sin θc                                                              
                         
                          π2                                                                                  
                                                                                                               
                         
                                     g (K ξ ) + g (K ξ ') − K g (K ξ ) + g (K ξ ') 
                                   K3  3 3                                                                    
                         +
                          ∫                         3     3            4  1      4        1     4      dθ 
                         
                          θc + αc                          1 − 4 τ cos θ                                      
                                                                                                               
                         
                         
                                                                                                              
                                                                                                               
                                                                                                               




                                                                 7
               0
                   ξ = z + z '+ i [(x − x ') cos θ + (y − y ') sin θ ]
   where                                                                   with ξc=ξ(θ=θc), ξ'c=ξ'(θ=θc),
                 ξ ' = z + z '+ i (x − x ') cos θ − (y − y ') sin θ 
               0                                                      
The integration limits are given by:
               if τ < 1/ 4
                                  θc = θ 'c = αc = 0,
               
               
               
               if 1/ 4 < τ < 1/ 2 θ = arcos (1/4τ ), θ ' = 0, α = 10−6 ,
               
               
               
                                     c                     c     c

               if τ > 1/ 2
                                  θc = arcos (1/4τ ), θ c = arcos (1/2τ ), αc = 10−6
                                                         '
               
               
The functions Kj(θ) and Zj(θ) are given by :

       1 + 2τεj cos θ + (−1)j 1 + 4τ cos θ
Kj =                                               for j = 1 to 4, εj = +1 if j = 1,2; εj = −1 if j = 3, 4
                        2F 2 cos2 θ

                            τ            1 − 2τ cos θ + (−1)j 4τ cos θ − 1
              K5 =                ; Zj =                                   for j = 3, 4
                        F 2 cos θ                    2F 2 cos2 θ
The functions gi(ξ) are defined by :

            g1 (ξ ) = e ξ ε1 (ξ ) if 0<arg (ξ ) < 2π, g 2 (ξ ) = e ξ E1 (ξ ) if − π<arg (ξ ) < π;
            g 3 (ξ ) = e ξ  ε1 (ξ ) + 2i π  if 0<arg (ξ ) < 2π,
            with: ε1 (ξ ) = E1 (ξ ), if ℑ (ξ ) ≥ 0, or ε1 (ξ ) = E1 (ξ ) − 2i π, if ℑ (ξ ) < 0;

where ℑ corresponds to the imaginary part and E1 is the complex-valued exponential integral
function:
                             ∞                                             ∞
                                e−t                                        e −ξ t
                    E1 (ξ ) = ∫     dt if − π < arg (ξ ) < π ; E1 (ξ ) = ∫        dt if ℜ (ξ ) > 0
                              ξ t                                        1
                                                                             t

The main difficulties concerns the Fourier integrals (G1 and G2) and the singular behaviour
close to τ=1/4; furthermore, the integrands become very oscillating when z and z' are both
close to 0.
       For the numerical calculations, we have replaced the 4th order Runge-Kutta method
used in (Ba & Guilbaud 1995) with a variable step by an adaptative quadrature method,
(Nontakaew et al 1997). The integration is done using a Simpson 1/3 method with five points
to avoid to calculate more that once each value of the integrand needed. (Guttman 1983) has
proposed an expression to relate the absolute error ε for an interval with length ln to the global
error εtot for the total path with length L, ε=(15εtotln)/L. This method presents various advan-
tages as the time reduction for a prescribed error, the possibility of computing simultaneously
the real and imaginary parts. The behaviour of the integrands close to π/2 must be also taken




                                                          8
into account. Asymptotic developments have been made close to this value (Nontakaew et al
1997).
Panel or segment source
The boundary integrations are performed by an analytical method which has been shown to
more accurate and efficient that the combined method mixing Gauss point method and
analytical one of (Boin et al 2000). This analytical method is based on a Stokes theorem
transforming the surface integral into a contour one (Bougis 1981):

                                                 d2          m    f (ζk +1 ) − f (ζk )
                               Is =         ∫∫      2
                                                      fds = ∑ C k                      (10).
                                            S
                                                 dζ         k =1      ζk +1 − ζk

In this equation, use of the properties of the modified complex exponential integral functions

have            been        done,           concerning       integration          ∫g   j
                                                                                           (ξ ) ξ = g j (ξ ) + ln ξ
                                                                                              d                       and


∫ (∫ g    j            )
              (ξ ) ξ d ξ = g j (ξ ) + (ξ + 1) ln j ξ − ξ with ln ξ = ln ξ + i arg(ξ ) , and 0<arg(ξ)<2π,
                 d

for    j=1,3           or   -π<arg(ξ)<π             if    j=2.       Concerning    the         derivatives,     we    have
              d                       1              d2                      1  1
g j (ξ ) =       g j (ξ ) = g j (ξ ) − and g j (ξ ) = 2 g j (ξ ) = g j (ξ ) − + 2 . The integration of
              dξ                      ξ              dξ                      ξ ξ
the term G0 has been extensively studied in the Rankine methods. For G1, the integration on a
panel with N vertices gives:
   1
            
                   1
                          
                                  
                                        D − D             D ' − D '                 
                        π 2    
                                   N     1k                1k                       
                                              2k                     2k
            
                   
   ∂ / ∂x Gds = 1 A/ L 
                          
                               dθ 
                                    C K D − K D  +C ' (−1)i+1(K D ' − K D ' ) (11)
                                                                                          
∫∫  i  1 π  0  ∫ 1 + 4τ cos θ ∑ k  1 1k 2 2k  k 
                                  k=1                                   1   1k   2 2k 
                                                                                          
 S  2
   ∂ / ∂x 2 
            
             
                     2 2 0
                    A / L 
                          
                                   
                                   
                                   
                                         2
                                                   2     
                                          K1 D1k − K2 D2k 
                                                              2
                                                                          2
                                                               K1 D '1k − K2 D '2k
                                                                                        
                                                                                        
                                                                                          
   
         i       
                         0
                                  
                                   
                                                                                     
                                                                                          
                                                                                          

                 
                Ck 
for i=1,2,3 and   = (q ∓ ir sin θ )(xk+1 − xk ) − (p − ir sin θ)(yk+1 − yk ) . M k (x k , y k , z k ) is
                 
                C'k 
                 
                 
the node k of a panel with MN+1=M1; the outer unit normal to the panel is next = (p, q, r ) and:
                                    χk +1
         
                  (g1(ξ)d ξ )d ξ  χk
                                      j

          ∫   ∫
Djk =               k +1     k
                                      j
                                        j=1,2. ( )' are deduced from the ( ) ones by replacing χk by χj′k
                  χj − χj                                                                       j




χk  K
 j 
 
       j 
 k =      z + z k + i (x − x k ) cos θ ± (y − y k ) sin θ   , j=1,2                 (12)
χ ' 
 j   lo                                                   
 
 
                                                                           .
         Concerning the first derivatives, coefficients ( D jk ) are deduced from the previous ones



                                                                 9
by replacing the two integrations by a unique one. For the second derivatives, the coefficients
  ..
( D jk ) involved only the functions g1. Furthermore, we have the following relations: x1=x,

A=icosθ if i=1;x2=y, A=isinθ if i=2 and x3=z, A= 1 if i=3. Integrations for G2 involve terms
very similar to the one of equation (11) and are given in appendix 1.
       For the integration on a segment, a similar expression to equation (10) can be obtained:
                                            df        f (ξ2 ) − f (ξ1 )
                                IL =   ∫ d ξ dl =        ξ2 − ξ1
                                       Cl


Details can be found in (Boin et al. 2000).
                                       Numerical results
Integration on an isolated panel
Computations of the Green function and its first and second derivatives by the two methods
developed in the previous paragraph have shown a very good agreement, (Brument &
Delhommeau 1997), (Brument 1998). To test the new methods of integration, we have
performed calculations of the various integrals involving the Green function and its first
derivatives on an isolated panel with an unit side and with its centre of mass located at z=-.51;
the unit outer normal is directed along the y-axis. Only cases where differences can be
observed are presented. The calculations have been performed for a field point M located on
the three different axes. Examples of calculations of the integrals for the function are plotted
on Fig. 1 with the location of M on the y-axis. The two analytical integration methods give the
same results ensuring the accuracy of the computations, and the results can be considered as a
reference to check the results of the numerical integration methods. Far from the panel, the
simple Gauss method induces very low errors with a very short computational time. Similar
results are obtained for the first derivative with respect to x as shown on Fig. 2, also along the
y-axis. Nevertheless, close to the peaks of the curves, numerical integration with a unique
Gauss point leads to some errors and even the method with four Gauss points is unable to give
accurate results close to y=0 for the integration of the imaginary part of the x–derivative. The
results concerning the y-derivatives will not be presented as leading to null values. Results of
the integration for M on the y-axis are presented on Fig. 3 and 4, respectively for the function
and the x-derivative; they show greater difficulties when the field point is close to the free
surface, the Gauss method requiring more and more points when the absolute value of zM
decreases. This effect is more sensitive for the derivative than for the function. For example,
four points are needed for zM<-.0.5, 16 points for zM<-.3, 36 are required for zM>-.1. Closer to


                                                 10
the free surface even 36 points are not enough to obtain a good accuracy and only analytical
methods can give correct results. This last fact is very important mainly for free surface
elevation calculations. The derivatives imply more difficulties to be integrated than the
function, as observed in (Maury 2000).
        Figure 5 plots the real parts of the integrations on a waterline segment ( -
0.433≤x≤0.433; -0.25≤y≤0.25) of the function g (left graph) and of its derivative ∂ g ⁄∂x (right
one) for a field point parallel to the axis x with z=-0.01 and y=2 for F=0.32 and ω = 2 .
Downstream the source point, only numerical methods using more than 8 Gauss points can
give accurate results. Figure 6 presents similar results for a field point describing the z axis.
Even numerical integration with 16 Gauss points are unable to calculate accurately the
integrals close to the free surface and only the analytical integration can be used. The
agreement is the same as for a panel but integrations are more difficult to perform for a
segment than for a panel.

Computations in the seakeeping codes

The use of the third Green's formula applied to a domain closed by the body surface, the free
surface and a surface located at infinity leads to an integral equation to compute the velocity
potential. Due to the radiation condition the contribution of the surface at infinity vanishes and
                                        N
                                                      ∂G                  M
                                                                                 ∂G
                         Vn (x , y, z ) = ∑ σk ∫∫          ds(M ') + F 2 ∑ σl ∫      dl                (13)
                                       k =1     ∆Sk
                                                      ∂n M               l =1 δC
                                                                                 ∂nM
                                                                               l


the integral on the free surface can be transformed into a line integral on the waterline. After
discretisation on the hull surface in N panels with M waterline segments, for a symmetric flow
(only sources σ are used), the body condition leads to the following equation after derivation:
        This previous integration technique has been used to compute the boundary integrals
of (13). The first application presented is for a Wigley hull (with relative thickness B/L=0.1
and relative draft T/L=.0625) defined by :

                                            (                     )
                                                                                                              4
y / B = 1 − (z /T )  1 − (2x / L )  1 + 0.2 (2x / L ) + (z /T ) 1 − (z /T )  1 − (2x / L ) 
                     2                  2                    2         2             8                  2

                                                                                                   
in forced heave and pitch motion. Added mass and damping coefficients (Aij and Mij being
respectively the damping coefficient and the added mass for force or moment i and motion j, V
being the immersed volume), are plotted Fig. 7 versus the non dimensional frequency (with
l0 = L , at Froude number F=0.2. Results of both codes for all coefficients are in good

agreement in spite of the different number of panels, showing that the convergence has been
reached : 522 for Poseidon and 750 for Aquaplus. Some differences are visible close for


                                                          11
values of τ close to the singular value of τ=ϖF=1/4. Theses results have been also compared
with the experimental measurements of (Gerritsma 1988), and the computations of (Lin &
Yue 1990) using 3D singularity method with a Green function satisfying the unsteady free
surface boundary condition and (Wang et al 1997), using the Chapman approach with a two
and half dimensional method, both in the time domain. Here also the agreement is reasonably
fair; results are very good for CM33 and CM55, and the graphs show similar variation with the
frequency but on nearly parallel curves for CA33 and CA55. For the cross coupling coefficients,
the results are in relatively good agreement but in some cases, some discrepancy either with
numerical methods or measurements can be observed.

                      4. Experimental set-up and test conditions
An experimental device to measure forces and moments on ship model in forced oscillations
of heave and/or pitch has been designed and developed at LEA-CEAT (Guyot 1995, Guyot &
Guilbaud 1995), Fig. 8. This experimental set-up enables also to measure the unsteady wave
pattern around this model. The system is driven by an electric motor (1) with a variable speed
of rotation. The model (2) is moved through a set of counter gears and cranked belts. Pure
heave is obtained through the heave rod (5) (after uncoupling the pitch rod (8) or inversely).
With the 2 eccentrics and rods, a combined heave-pitch motion is possible. The model is fixed
to the oscillator through a 3-component dynamometer (3), which can be replaced by a rigid
modulus in order to avoid deformation of the set-up during the wave pattern measurements.
The motion amplitudes are from 0 to 20 mm for heave, 0 to 6 ° in pitch, with frequencies
ranging from 1 to 6 Hz. To keep a good accuracy of the added mass and damping coefficient
measurements, we need to reduce the inertia forces and so an effort has been done to build
models as light as possible by use of composite materials (carbon fibber); the masses are
M=0.8 and 1 kg for the two L=1.2 m long series 60 models for respectively the block
coefficient CB=0.6 and 0.8.
       The three-component dynamometer is composed of three force transducers. These
transducers being sensible to forces normal to their axis, a system for uncoupling forces has
been designed, so transducers are only in contact with compression or traction forces. Two
±25 daN and one ±10 daN force transducers are used. One displacement transducer records
the model motion : (6) for heave and (7) for pitch.
       Free surface elevations are recorded using a resistive probe (4) consisting of 2 parallel
chromel wires (0.2 mm diameter and 150 mm high, separated from 10 mm) held by a plexiglas



                                              12
frame to avoid electric perturbations. The electric resistance between the two wires is function
of the water elevation, the 2 wires forming a branch of a Wheatstone bridge supplied with
alternative current (3.0 kHz) to prevent water electrolysis, so the output signal has to be
demodulated. This probe is moved by a traversing mechanism driven by a stepper motor in the
streamwise direction (x), and can be moved transversally (y) by a manual system.
Measurements have been performed every 40 mm in both directions (80 mm in the y direction
for y>240 mm), on only one side of the model, due to the symmetry of the flow, on a total
length of 1.88 m long and 0.40 m wide. The bow of the model is located at y=0 and x=240
mm and the rear part at x=1440 mm. Close to the model, due to the size of the probe, the
number of wave measurements has been reduced (for y≤80 mm for the CB=0.6 model and
y≤120 mm for CB=0.8 one). The number of measurement locations is about 350 for each
configuration, following Fig. 9, which plots the relative locations of the model and of the
probe measurements.
       Force and free surface elevation measurements have been performed with a PC com-
puter with a 12 bits analog to digital converter using a four channel sample-hold box; 6000
samples for global measurements and 1024 samples are recorded for each channel at data
acquisition frequency of 100 Hz. Four channels are used for force measurements (motion and
dynamometer), but only two (motion and wave probe) for the wave pattern. For these latter
measurements, the data acquisition time is about 10s for each point. For each signal, a Fourier
analysis first achieves a rough determination of the motion frequency by looking for the
maximal energy spectrum. Around this first value, 100 points are distributed for ±0.1 Hz with
a frequency step decreasing closer to the first determination of the frequency, and the
fundamental response is obtained from the maximum amplitude of the signal with respect to
the frequency. This method is more accurate than the FFT method used in (Guyot 1995), but
the computational time is somewhat larger. This determination is applied to the various
signals to determine the transfer function of each signal with respect to the motion, enabling to
calculate the motion frequency, the signal spectra and more particularly the response at same
frequency that the input. So amplitude and phase with an arbitrary origin of signals and by
using a calibration table, force, moment or wave elevation amplitudes and phase lags with
respect to the forced motion are computed. For added mass and damping measurements, tests
are performed two times at the same frequency, once in air to record inertia forces and once in
water to measure total forces, inertia plus hydrodynamic forces, the last ones being obtained
by subtracting inertia forces from total forces taken into account the phase lags; then the added


                                               13
mass and damping coefficients are calculated. For the wave height measurements, the
calibration is achieved by static measurements, but takes into account the water velocity U, so
3 different calibrations (fifth order polynomial) have been determined.
         After a development on a small model (L=0.3 m) in the recirculating water channel of
the CEAT with a 0.3 m wide test section, the experiments have been performed in the 2*1 m2
test section of the recirculating water channel of Ecole Centrale de Nantes, where the
maximum water velocity is 1.7 m/s. Only pure motions have been studied with amplitudes of
a3=10.8 mm (a/L=0.09) and a5=1.8 ° for U=0.13-0.7 and 1 m/s, corresponding to Froude
numbers based on the model length 0.04≤F≤0.29 and frequencies of 2.5 to 6 Hz for the global
measurements (Brard parameter 0.21≤τ≤3.84). For the wave measurements the lower velocity
was increased till 0.4 m/s to reduce the wave reflection by the walls of the test-section. So, the
frequencies were 0.87≤f(Hz)≤3.92, corresponding to Brard parameter between 0.22 and 2.5.
The Reynolds numbers of the tests are between 0.2.106 and 1.2.106.

Comparison of the numerical and experimental results on the series 60 models
The first results ("initial calculation"), for the CB=0.6 hull at Froude number F=0.2 only for
the heaving motion, are presented Fig. 10 versus the non dimensional frequency fL/U. They
are obtained by neglecting the waterline integral and show large oscillations of the
coefficients, probably due to the existence of irregular frequencies, as already mentioned in
(Du et al 2000). By adding the waterline integral ("wi"), it is clear that the amplitude of
oscillations have strongly decreased. We have then introduced a classical technique to
suppress these frequencies ("tif") by adding an horizontal flat surface divided into panels
inside the hull. On each of these inner panels, a condition of zero normal velocity have been
written. The oscillations of the curve are nearly entirely suppressed but are still present with
weak amplitudes if the waterline integral is neglected. It shows the relation between this
integral and the existence of irregular frequencies. These results have been observed for both
codes.
         The results of computations for the two CB=0.6 and 0.8 models have been compared
with the measurements described in the previous paragraph. Both numerical results are for a
grid of 245 panels on the half hull. Figures 11 and 12 plot the added mass and damping
coefficients CMjj=Mjj/ρLn and CAjj =Ajj/ρωLn for j=3 (n=5) and 5 (with n=3) versus the non-
dimensional frequency fL/U, respectively for heave and pitching forced motions at the Froude
number F=0.2. On these figures, the dashed lines are for Poseidon code, the full ones for


                                               14
Aquaplus code and the symbols correspond to the present test measurements. The two
methods show very similar results with oscillations of the result for some values of the
frequency. The curves are similar for both ship models (CB=0.6 or 0.8), with higher values of
the coefficients as the block coefficient increases, except for CA55. Nevertheless, in some
cases, some differences on the absolute values of maximum or minimum (for example, CM33
for the CB=0.8, figure 12) or on their locations can be observed (for CA55 for the CB=0.6, Fig.
11). The test results are generally in relatively good agreement with the results of the
computations, except for CM55, where the test results are quite weak, may be due to problems
in the measurements.
       Figures 13 present the different measured wave-elevation patterns for both CB=0.6
(upper graphs) and 0.8 (lower ones) models for the same test configuration (F=0.2; f=3Hz;
τ=1.35); graph a is for heave and graph b for pitch. The recorded wave patterns have the same
characteristics whatever are the test configurations and the model motion : two zones in V-
shape with the tip in the upstream direction are visible with strong amplitude values, at the
bow and at the stern (these amplitude values being stronger at the stern). The whole wave field
is contained in this V-shape pattern with an opening angle and amplitudes which increase with
the block coefficient. For instance, for the CB=0.8 model, just after the bow wave, a small area
with low wave-amplitude appears, and three zones with strong amplitude are clearly separated
along the hull. For the pitch, the back V-shape pattern presents globally higher wave
amplitudes in comparison with heave, but the areas of strong wave amplitudes are reduced. It
must be noticed than for heave fore and aft waves have the same phase but there is a phase lag
of about 180 ° for the pitch motion.
       A more accurate approach of the wave-elevation contours is furnished by the Fig. 14
and 15 where respectively axial and transverse cuts of the unsteady wave pattern amplitudes
for the CB=0.8 model at F=0.2 and f=4 Hz (τ=1.79), far from an irregular frequency, both for
heave and pitch motions. Figure 14 plots the wave amplitude for y/L=0.2; upper graph is for
heave and the lower one is for pitch. The agreement between the 2 codes is quite good, but it
is necessary to have a larger number of panels (about 1000 while about 500 seems to be
enough for the force calculations) than for computation of global forces. It can be observed on
this figure than the amplitudes behind the stern (x/L>1.1) are greatly overestimated, effect
which has not been yet explained. If values of the maxima and minima seems to be correctly
predicted, their locations are not very accurate with regard to the tests. The transverse cuts,
Fig. 15, are plotted for x/L=0.8 and 0.9 (close to the model stern). Upper graphs are for


                                              15
x/L=0.9 and the lower ones, for x/L=0.8. Left ones are for heave motion and right ones for
pitch. Here also, the 2 codes are in good agreement except some phase lags which may be
probably due to a different number of panels. It must be noticed that the values of both
components of wave elevation are weak and that the step between two measurement points are
larger than some oscillations observed on the calculations. The results of computations have
also been compared to present measurements. The general trend is predicted but with some
discrepancies concerning the location of the maxima and minima of the curves.

                                         Conclusion
We have presented a numerical and experimental study of the radiation flow about ships
advancing in waves. The numerical codes have been developed with the assumption of linear
theory. Both are first order panel methods using the diffraction-radiation with forward speed
Green function in the frequency domain. By satisfying the body condition on the hull, we
obtain a linear system of equations. Using a velocity based formulation with only a source
distribution, this distribution can be calculated and then the whole flow. The main feature of
these methods is an accurate computation of the coefficients of the linear system. These
coefficients involve boundary integrals of the Green function and its first derivatives over
panels on the hull or segments on the waterline. The boundary integrals are calculated
accurately after permutation of the boundary and single Fourier integrals. The first code uses
the Bessho formulation and integration of complex oscillating integrands are performed by
steepest descent method. The second code uses the formulation of Guével and Bougis with the
single integrals involving the complex exponential integral function calculated by a Simpson
adaptative method where the local step varies with the oscillations of integrand. The accuracy
of these boundary integrations have been shown, even close to the free-surface, by comparison
of results of the two codes on elementary panels or waterline segments. This comparison
shows that the derivatives are more difficult to integrate than the function and that for a field
point close to the free-surface, it is impossible to use a numerical method of integration like
the Gauss one with a reasonable accuracy. Integrations on a segment are more difficult to
perform that on a panel but easier than the calculation of the function or of its derivatives.
Finally these integrations have been introduced in seakeeping codes which have been applied
to a Wigley hull and on two Series 60 ones with CB=0.6 and 0.8.
       In order to check the validity of the calculations, we have performed an experimental
study in the recirculating water channel where not only global measurements as added-mass



                                               16
and damping have been measured but also local ones. The wave pattern has been measured
and amplitude and phase-lag with respect to the motion have been recorded on about 350
points on half the free surface for heave and pitch motions. The measurements have been
performed for various values of the Froude number and of the motion frequency.
       The comparison of the numerical results with available numerical method or with the
present test measurements have been done. The agreement between the two codes is excellent.
For the Wigley hull, the agreement is also good with other results, particularly for the added-
mass of the pure coefficients. Some discrepancies appear for the damping, but the curves
versus the frequency are nearly parallel. A good prediction of the cross coupling coefficients
has been obtained in spite of the low values of these coefficients and of the difficulty to
measure them. Concerning the Series 60 hulls, results have shown the existence of irregular
frequencies, but less abrupt than these observed at zero forward speed. Calculations performed
by neglecting the waterline integral show an increase of the amplitudes of the oscillations
showing that this integral has a strong damping effect on the irregular frequencies. So a
technique to remove these frequencies has been used. The hulls are closed by a flat horizontal
surface, slightly immersed, where a zero velocity condition has been written. These
frequencies have been almost completely suppressed, at least the lower of them. Results
concerning added-mass, damping and cross-coupling coefficients calculated by the two codes
are in good agreement between them and with the other results available. Finally some
comparison have been done concerning the unsteady wave field. Some difficulties appear for
the comparison of the wave pattern amplitudes. The general shape is predicted but with some
differences of maxima and minima along the hull. Downstream of the stern, the calculations
overpredict the amplitude of the wave field, phenomena which has not be yet explained and
will need further experiments.

                                         Appendix 1
By use of equation (10), the integration of G2 on a panel gives:

     1
                
                     1
                            
                             
     
                
                     
                            
                             
                   1
     ∂ / ∂ x G ds = 1/ L  I + I + I + I  with:
                             
∫∫   
            i    2
                 
                      
                     π    0  1
                                   2   3   4
 S    2
     ∂ / ∂ x 2 
                    1/ L2 
                            
     
              i 
                 
                     
                      
                          0
                             




                                               17
            1 
                                         
                                                             E − E                    E ' − E '                      
                                         
                                           N                  3k                       3k                           
              θc'                                                                                                     
                                                                          4k                            4k
                                −i       
                                            C                                                                          
      I 1 = A  ∫
                                         ∑                 Z E − Z E  + C ' (− 1)i +1 (Z E ' − Z E ' )  d θ
                                                               3 3k         4 4k     k                            4k  
              0
             2             4 τ cos θ − 1  k =1 k
                                           
                                                                                                        3    3k    4
                                                                                                                           
            A                                               2                        2                            
             
                                         
                                           
                                                                             2
                                                              Z 3 E 3k − Z 4 E 4k                      2
                                                                                         Z 3 E ' 3k − Z 4 E '4 k       
                                                                                                                           
                                                                                                                   

            1                            
                                                              F − F                                  F ' − F '                    
              θc −αc                     
                                           N                   1k                                     1k                         
                                                                                                                                     
                                                                          2k                                          2k
                                −i                                                                                                
      I 2 = A  ∫                          C
                                           ∑                  Z F − Z F  + C '                       (−1) (Z F ' − Z F ' ) d θ
                                                                                                                i +1
              '                                                           4 2k                                               2k  
                              4τ cos θ − 1  k =1 k
                                                               2
                                                                   3 1k
                                                                                    
                                                                                      k
                                                                                                         2
                                                                                                                      3    1k   4
                                                                                                                                     
                                                                                                                                        
             2  θc                       
            A                          
                                           
                                                                             2
                                                               Z 3 F1k − Z 4 F2k                                     2
                                                                                                        Z 3 F '1k − Z 4 F '2k       
                                                                                                                                        
                                           
                                                                                                                                  
                                                                                                                                        

                       1                                                          1                              
                                                                                                                 
                                                                       χc +1                          χ 'c +1   χ 'c 
                                                                                                                      
                                −2π (1 − i ) αc
                                                                          k        k                      k         k
                                                                N
                                                                     e
                                                                             −e χc
                                                                                                   e         −e 
                 I 3 = Ac Kc 
                                    τ sin θc
                                                               ∑ C k χk +1 − χk + (−1)i +1  C 'k χ 'k +1− χk 
                                                                    
                                                                    
                                                                                                                      
                                                                                                                      
                                                                                                                      
                                                                                                                 c 
                         2 2                                 k =1
                                                                    
                                                                         c        c
                                                                                      1                  c
                                                                                                                      
                          Ac Kc                                                                                      
                                                                 
                                                                                                                 

       1                       
                                                            H − H                                    H ' − H '                   
         π2                    
                                 N                           1k                                       1k                         
                                                                                                                                     
                                                                         2k                                           2k
                       1       
                                                                                                                                   
                                                                                                                                       
I 4 = A  ∫                     ∑ C k                      K H − K H  + C '                         (−1) (K H ' − K H ' ) d θ ,
                                                                                                               i +1
                  1 − 4τ cos θ  k =1                       3 1k         4 2k     k                              3    1k   4 2k 
                                                                                                                                       
        2  θc +αc              
                                                             2                                        2                          
      A 
                                                                            2                                            2
                                 
                                                            K 3 H 1k − K 4 H 2k                    K 3 H '1k − K 4 H '2k        
                                                                                                                                     
                                 
                                                                                                                                  
                                                                                                                                       

Coefficients Ejk, Fjk, F2k, H1k and H2k are given by:

                                       χk +1                                                        χk +1                                              χk +1
          
                     (g2(ξ)d ξ )d ξ  χk                          
                                                                                (g1(ξ)d ξ )d ξ  χk                 
                                                                                                                                     (g 3 (ξ)d ξ )d ξ  χk
                                         j

           ∫   ∫                                                   ∫    ∫
                                                                                                    3

                                                                                                                       ∫   ∫
                                                                                                                                                           4



E jk =                 k +1
                                         j
                                                 j=3,4, F1k =                     k +1     k
                                                                                                    3
                                                                                                            , F2k =                    k +1
                                                                                                                                                           4
                                                                                                                                                             ;
                     χj − χj    k
                                                                                χ3 − χ3                                              χ4 − χ4      k




                                                                    χ5 +1
                                                                     k
                                                                                                                       χ6 +1
                                                                                                                        k
                                                                                                                 
                                        ∫    ∫ (g 3 (ξ)d ξ )d ξ  χ
                                                                     k                      ∫   ∫ (g1(ξ)d ξ )d ξ  χ  k

                            H 1k =                                   5
                                                                            , H 2k =                                     6
                                                                                                                                 .
                                                  χ5 +1 − χ5
                                                   k       k
                                                                                                     χ6 +1 − χ6
                                                                                                      k       k




In the expressions (12) for χk and χj′k , Kj must be replaced by Zj if j=3,4 and by Kj-2 if j=5,6;
                             j



            χck 
              K
             
furthermore  k  = c z + z k + i (x − x k ) cos θc ± (y − y k ) sin θc   and Ac=icosθc if i=1;
             
            χ 'c  Lo                                                     
             
             
             

Ac=isinθc if i=2 and Ac=1 if i=3.


                                                               Nomenclature

Aij                                              : damping



                                                                            18
A,Ac                                                : coefficients used in the integration over panels

B                                                   : ship width

Ck 
 
  = (q ∓ ir sin θ ) xk+1 − xk − (p − ir sin θ)(yk+1 − yk ) : coefficient
 
C'k 
                     (         )
 
 

CAij, CMij                                          : damping and added-mass coefficients

CB                                                  : block coefficient

                                            χk +1
           
                         (g1(ξ)d ξ )d ξ  χk
                                             j

            ∫   ∫
D jk =                     k +1     k
                                             j
                                                           : coefficients j=1,2
                         χj − χj

Ei                                                  : complex-valued exponential integral function

                                            χk +1
           
                         (g2 (ξ)d ξ )d ξ  χk
                                              j

            ∫   ∫
E jk =                     k +1      k
                                              j
                                                           : coefficients j=3,4
                         χj − χj

                                           χ 3 +1
                                             k

       ( g m (ξ )dξ ) dξ 
      ∫ ∫                  χ3k
Flk =                                               : coefficients with m=1 if j=1 and m=3 if j=2
            χ 3k +1 − χ 3k

                         k1e k1ξ
f1(θ) =
                  1 + 4τ cos θ

                                             1 k1ξl
                                                e                                                            
                   −i 2Sl                    k1                                i β (Y1 −Y2 )       e k1ξl    
                                                                                                              
f1,l (θ ) =                                                        or f (θ ) =
                                                                                                             
                                                                                                              
                  2πK 0 ψl (θ ) 1 + 4τ cos θ                                   2π (ξl +1 − ξl )              
                                                                                                 1 + 4τ cos θ 
                                                                        1,l
                                                                   
                                                                                                             

    F =U / g               0
                                                    : Froude number

g                                                   : gravity acceleration

g                                                   : Froude number dependant part of the Green function

gi (i=1,3)                                          : modified complex-valued exponential integral function

G=G0+G1+G2                                          : Green function

           θ2 (P ,P ')
I =    ∫ θ1
                         − sgn c.f1 (θ ).d θ : integral in the steepest descent method, θ-space



                                                                       19
IS =    ∫∫ gds , I      L
                            =   ∫ gdl          : integrals of the Green function or derivatives on a panel or a
         ∆S                     δC


segment

                                       χk +1
         
                    (g3 (ξ)d ξ )d ξ  χk
                                         5

          ∫   ∫
H 1k =                k +1      k
                                         5
                                                      : coefficients
                    χ5 − χ5

k1,k2                                          : poles of the Green function in the steepest descent method

 Kc = 4ω 2


K0= g /U 2                                     : wave number

Kj                                             : poles of the Green function in the formulation of Guevel and

Bougis, j=1,5 (k1=K1 and k2=K2)

l0                                             : reference length

ln                                             : length of a single interval

L                                              : ship length

M=k1cosθ

M = sgn c.k1.cos θ = sgn c.m :space to calculate the steepest descent method

Mij                                            :added-mass

N                                              : number of vortices for a panel

P(x,y,z),P'(x',y',z')                          : field and source points

Ql, l=1,N                                      : vortices of a panel

R
          ( x − x ' ) + ( y − y ') + ( z ∓ z ')
                       2                2              2
   =                                                      : distances to the source point and its symmetrical with
R1 

respect to the free surface

sgnc=sign[Re(cosθ)]

sgns=sign[Re(sinθ)]




                                                                  20
sign(t)=-1 if t<0; 0 if t=0;1 if t>0

Sl                                           : area of triangle (Ql,Ql+1,Ql-1)

T                                            : ship draft

U                                            : forward speed of the ship

V                                            : immersed volume of the hull

Vn                                           : normal velocity

x,y,z                                        : reference frame fixed to the body

X = K 0 (x − x ');Y = K 0 y − y ' ; Z = K 0 (z + z ')


             (            l
                              )
Xl = K 0 x P − xQ ;Yl = K 0 yP − yQ , Zl = K 0 z P + zQ
                                                          l
                                                                       (           l
                                                                                       )
X η = X1 + η (Xl − X1 ); Yη = Y1 + η (Yl −Y1 ); Z η = Z 1 + η (Zl − Z1 )

Xl ' = Xl − X1; Yl ' = Yl −Y1; Zl ' = Zl − Z1

Xel = XQ              − XQ ; Yel = YQ                 −YQ ; Zel = ZQ        − ZQ
             (l +1)           l              (l +1)       l        (l +1)      l




          1 − 2τ cos θ + (−1)j 4τ cos θ − 1
Zj =                                        for j = 3, 4 : complementary poles
                      2F 2 cos2 θ

αc=10-6                                      : coefficient used to calculate the Green function

β,β'                                         : starting points for the integration in the M-space

                                    Z
ε (P, P ') = arg sh
                                  X 2 +Y 2

εtot,,ε                                      : absolute error on total interval L or elementary interval with

length ln

ζ,,ζ'

θ                                            : complex variable

θc,θ'c                                       : limits of integration for the Fourier integrals




                                                                 21
     
      π + arccos 1                        1
     −
                                   if τ >
θ1 =               4τ                     4
     
     
                     1                    1
     −π − i arg ch
     
                                    if τ <
                    4τ                    4

                                π
θ2 (P , P ') = ϕ (P , P ') −      − i ε (P, P ')
                                2

              Ye ± Xe 2 + Ye 2 + Ze 2 
                                   l 
θl = 2 arctan  l
              
                      l       l
                                       
                                       
              
                   Xel − i.Zel        
                                       

            X ηYl '− Xl 'Yη         Zl ' (X η + Yη2 ) − Z η (X η Xl '+ Yl 'Yη )
                                            2

νl (η ) =                      +I
                   2
                 X η + Yη2               (X η2 + Yη2 ) (X η2 + Yη2 + Z η2 )

ξ = Z + i (X cos θ + Y sin θ ) or ξ (ξ ') = z + z '+ i [(x − x ') sin θ ∓ (y − y ') cos θ ] / l 0

ξl = ξ (P,Ql )

σ                                      : source density

τ=ωeU/g                                : Brard parameter

                               X
ϕ (P, P ') = arccos
                             X 2 +Y 2

                                                                                  2
                                                                       m        
                                                                                 
φ (M ) = (m − τ Z + i.m.X + i.sgnsY (m − τ
                      )2          .                            )2   1−
                                                                      (         
                                                                                 
                                                                       m − τ )2 
                                                                      

χck 
  K
 
  = c z + z k + i (x − x k ) cos θ ± (y − y k ) sin θ  
 k                                                   c 
χ 'c  Lo 
 
                                      c                    
 
 

ψl (θ ) = (ξl +1 − ξl )(ξl −1 − ξl )

ωe                                     : encounter frequency

ω = ωe       0   /g




                                                          22
                                           References

Ba, M. and Guilbaud, M. 1995 A fast method of evaluation for the translating and pulsating

Green's function. SHIP TECHNOLOGY RESEARCH, 42, 2, 68-80.

Ba M., Boin J.P., Delhommeau G., Guilbaud M. et Maury C. 2001a Calcul de tenue à la mer

avec la fonction de Green de diffraction-radiation avec vitesse d'avance. Proceedings: 8èmes

Journées de l'Hydrodynamique, 125-140.

Ba M., Boin J.P., Delhommeau G., Guilbaud M. et Maury C. 2001b Sur l'intégrale de ligne et

les fréquences irrégulières dans les calculs de tenue à la mer avec la fonction de Green de

diffraction-radiation avec vitesse d'avance. C.R. ACAD. SCI. PARIS, 329, IIb, 141-181.

Bougis, J. 1981 Bessho, M. 1977 On the Fundamental Singularity in the Theory of Ship

Motions in a Seaway. MEMOIRS OF THE DEFENCE ACADEMY JAPAN, XVII, 3, 95-

105.

Boin, J.P., Guilbaud, M. and Ba, M. 2000 Sea-keeping computations using the ship motion

Green's function. Proceedings: ISOPE2000 Conference, Seattle, 398-405.

Bougis J. 1981 Etude de la diffraction-radiation dans le cas d'un flotteur indéformable animé

d'une vitesse moyenne constante et sollicité par une houle sinusoïdale de faible amplitude.

Thèse de doctorat, Université de Nantes.

Brument, A. et Delhommeau, G. 1997 Evaluation numérique de la fonction de Green de la

tenue à la mer avec vitesse d'avance. Proceedings: 6èmes Journées de l'Hydrodynamique,

Nantes, 147-160.

Brument, A. 1998 Evaluation numérique de la fonction de Green de la tenue à la mer. Thèse

de Doctorat, École Centrale de Nantes.

Chen, X.B. and Noblesse, F. 1998 Super Green function. Proceedings: 22nd Symposium on

Naval Hydrodynamics, Washington, 860-874.




                                               23
Du, S.X., Hudson, D.A., Price, W.G. and Temarel, P. 1999        Comparison of numerical

evaluation techniques for the hydrodynamic analysis of a ship travelling in waves. Trans.

RINA, 141, 236-258.

Du, S.X., Hudson, D.A., Price, W.G. and Temarel, P. 2000           A validation study on

mathematical models of speed and frequency dependence in seakeeping. Trans. RINA, 214,

181-202.

Delhommeau, G., Ferrant, P. and Guilbaud, M. 1992, Calculation and measurement of forces

on a high speed vehicle in forced pitch and heave. APPLIED OCEAN RESEARCH, 14, 119-

126.

Gerritsma, J. 1988 Motions, wave loads and added resistance in waves of two Wigley hull

forms. Delft University of Technology, Ship Hydromechanics laboratory, Report n° 804.

Guével, P. and Bougis, J. 1982 Ship-Motions with Forward Speed in Infinite Depth. INT.

SHIPBUILDING PROG., 29, 103-117.

Guttman, C. 1983 Etude théorique et numérique du problème de Neumann-Kelvin pour un

corps totalement immergé. Rapport de Recherche n°177, ENSTA.

Guilbaud, M. Boin, J.P., and Ba, M. 2000 Frequency domain numerical and experimental

investigation of forward speed radiation by ships. Proceedings: 23rd Symposium on Naval

Hydrodynamics, Val de Reuil, Tuesday session, 110-125.

Guyot, F. 1995     Étude expérimentale de la résistance ajoutée d'une maquette de navire

soumise à des oscillations forcées harmoniques: étude du champ de vagues instationnaires

associé. Thèse de Doctorat, Université de Poitiers.

Guyot, F. and Guilbaud, M. 1995 Force and free surface elevation measurements on a series
                                                          th
60 CB=0.6 ship model in forced oscillations. Proceedings: 5 ISOPE Conference, The Hague,

507-514.




                                              24
Iwashita, H. and Okhusu, M. 1989 Hydrodynamic Forces on a Ship Moving at Forward Speed

in Waves. J.S.N.A. Japan, 166, 87-109.

Iwashita, H. 1992 Evaluation of the Added-Wave-Resistance Green Function Distributing on

a Panel. Mem. Fac. Eng. Hiroshima Univ., 11, 2, 21-39.

Ishawita, H., Ito, A., Okada, T., Okhusu, M., Takaki, M. and Mizoguchi, S. 1993 Waves

forces acting on a blunt ship with forward speed in oblique sea (2). T. Soc. Naval Arch. Japan,

173, 195-208.

Lin, W.-M., and Yue, D. 1990 Numerical solutions for Large-Amplitude ship motion in the

time domain. Proceedings: 18th Symp. on Naval Hydrodynamics, Ann Arbour, 41-66.

Maury, C. 2001, Etude du problème de tenue à la mer avec vitesse d'avance quelconque par

une méthode de singularités de Kelvin. Thèse de Doctorat, École Centrale de Nantes.

Nontakaew, U., Ba, M. and Guilbaud, M. 1997 Solving a radiation problem with forward

speed using a lifting surface method with a Green's function. AEROSPACE SCIENCE AND

TECHNOLOGY, 8, 533-543.

Okhusu, M. and Wen, G. 1996 Radiation and diffraction waves of a ship at forward speed.

Proceedings 21th Symposium on Naval Hydrodynamics, Trondheim, 29-44.

Okhusu, M. 1998 Validation of theoretical methods for ship motions by means of experiment.

Proceedings: 22nd Symposium on Naval Hydrodynamics, Washington, 341-358.

Wang, C.-T., Horng, S-.J. and Chiu, F.-C. 1997 Hydrodynamic forces on the advancing

slender body with speed effects. INT. SHIPBUILDING PROG., 44, 438, 105-126.



List of figures

Figure 1 Integrals of the Froude dependant part of the Green function for a field point

describing the y-axis




                                              25
Figure 2 Integrals of the first x-derivative of Froude dependant part of the Green function for a

field point describing the y-axis



Figure 3 Integrals of the Froude dependant part of the Green function for a field point

describing the z-axis



Figure 4 Integrals of the x-first derivative of the Froude dependant part of the Green function

for a field point describing the z-axis



Figure 5 Integrals of the Froude dependent part of the Green function and x-derivative for a

field point describing a parallel to the x-axis (z=-0.01;y=2)



Figure 6 Integrals of the Froude dependent part of the Green function and x-derivative for a

field point describing the z-axis



Figure 7 Added mass and damping coefficients on the Wigley hull (F=0.2)



Figure 8 Experimental set-up and motion oscillator



Figure 9 Locations of the wave measurements



Figure 10 Influence of the waterline integral on the added mass and damping coefficients on

the Series 60 CB=0.8 (F=0.2)



Figure 11 Added mass and damping coefficients on the Series 60 CB=0.6 (F=0.2)


                                               26
Figure 12 Added mass and damping coefficients on the Series 60 CB=0.8 (F=0.2)



                        a) heave motion (CB=0.6: top; CB=0.8: low)

                         b) pitch motion (CB=0.6: top; CB=0.8: low)

Figure 13 Measured waves amplitudes (F=0.3, f=3Hz, τ=1.92)



Figure 14 Comparison of longitudinal cut of wave elevation y/L=0.2

(Series 60 CB=0.8 hull; F=0.2; τ=1.79)



Figure 15 Comparison of transverse cut of wave elevation

(Series 60 CB=0.8 hull; F=0.2; τ=1.79)




                                            27
                                                                       z
                                               -0.5                         0.5                                                       x
                                                                                                               -0.01

                                                                                                               -1.01




ℜ ∫∫ gds                                                                                 ℑ ∫∫ gds
   S                                                                                        S
                                                                                                      Aquaplus
                           Aquaplus                                                                   Poseidon
  0.06                     Poseidon                                              0.15                 Numerical integration 1 Gauss point
                           Numerical integration 1 Gauss point                                        Numerical integration 4 Gauss points
  0.05
                           Numerical integration 4 Gauss points                                       Numerical integration 16 Gauss points
  0.04                     Numerical integration 16 Gauss points
  0.03
  0.02                                                                             0.1

  0.01
       0
  -0.01
                                                                                 0.05
  -0.02
  -0.03
  -0.04
  -0.05                                                                             0
  -0.06
  -0.07
  -0.08
                                                                                 -0.05
       -6   -5   -4   -3    -2   -1   0    1   2    3    4    5    6                  -6   -5   -4   -3   -2    -1   0   1   2    3       4   5   6
                                      y                                                                              y


       Figure 1 Integrals of the Froude dependant part of the Green function for a field point
                                        describing the y-axis

           ∂g                                                                       ∂g
ℜ ∫∫          ds                                                           ℑ ∫∫        ds
   S
           ∂x                                                                S
                                                                                    ∂x

                                                                                                      Aquaplus
                      Aquaplus                                                                        Poseidon
  0.14                Poseidon                                                   0.12                 Numerical integration 1 Gauss point
                      Numerical integration 1 Gauss point                                             Numerical integration 4 Gauss points
  0.12                Numerical integration 4 Gauss points                         0.1
                                                                                                      Numerical integration 16 Gauss points
                      Numerical integration 16 Gauss points                      0.08
    0.1
                                                                                 0.06
  0.08
                                                                                 0.04
  0.06
                                                                                 0.02
  0.04                                                                              0

  0.02                                                                           -0.02
                                                                                 -0.04
       0
                                                                                 -0.06
  -0.02
                                                                                 -0.08
  -0.04
                                                                                  -0.1
  -0.06                                                                          -0.12
  -0.08                                                                          -0.14
       -6   -5   -4   -3    -2   -1   0    1   2    3    4    5    6                       -5                        0                        5
                                      y                                                                              y




                                                                            28
Figure 2 Integrals of the first x-derivative of Froude dependant part of the Green function for a
field point describing the y-axis

ℜ ∫∫ gds                                                                                   ℑ ∫∫ gds
    S                                                                                          S


  0.075                                                                              0.4
                                                                                                     Aquaplus
                     Aquaplus                                                                        P oseidon
                     P oseidon                                                                       N um erical   integration    1 G auss point
   0.05
                     N um erical   inte gration    1 G auss point                                    N um erical   integration    4 G auss points
                     N um erical   inte gration    4 G auss points                                   N um erical   integration    1 6 G auss points
                                                                                     0.3             N um erical   integration    3 6 G auss points
  0.025              N um erical   inte gration    1 6 G auss points
                     N um erical   inte gration    3 6 G auss points

        0
                                                                                     0.2
  -0.025


   -0.05
                                                                                     0.1

  -0.075


    -0.1                                                                              0
        -1           -0.75             -0.5               -0.25         0              -1          -0.75             -0.5               -0.25          0
                                        z                                                                              z


        Figure 3 Integrals of the Froude dependant part of the Green function for a field point
                                         describing the z-axis

             ∂g                                                                                    ∂g
ℜ ∫∫            ds                                                                         ℑ ∫∫       ds
    S
             ∂x                                                                                S
                                                                                                   ∂x

        0.3                                                                            0.3            Aquaplus
                                                                                                      P oseidon
                                                                                                      N um erical   inte gration   1 G auss point
        0.2                                                                            0.2
                                                                                                      N um erical   inte gration   4 G auss points
                                                                                                      N um erical   inte gration   1 6 G auss points
                                                                                       0.1            N um erical   inte gration   3 6 G auss points
        0.1

                                                                                           0
            0
                                                                                      -0.1
        -0.1
                                                                                      -0.2
                       Aquaplus
        -0.2           P oseidon
                       N um erical   integration    1 G auss point                    -0.3
                       N um erical   integration    4 G auss points
        -0.3
                       N um erical   integration    1 6 G auss points                 -0.4
                       N um erical   integration    3 6 G auss points
        -0.4                                                                          -0.5

        -0.5                                                                          -0.6
            -1         -0.75                -0.5             -0.25          0             -1         -0.75                 -0.5            -0.25           0
                                              z                                                                             z


Figure 4 Integrals of the x-first derivative of the Froude dependant part of the Green function
for a field point describing the z-axis




                                                                                29
                                                                                                           ∂g
ℜ ∫ gdc                                                                                          ℜ∫           dc
  δCl                                                                                             δCl
                                                                                                           ∂x

    0.5                                                                                 5

    0.4                                                                                 4

    0.3                                                                                 3

    0.2                                                                                 2

                                                                                        1
    0.1
                                                                                        0
        0
                                                                                    -1
    -0.1
                                                Aquaplus                                                                         Aquaplus
                                                Poseidon                            -2
                                                                                                                                 Poseidon
    -0.2                                        Numerical method 1 Gauss point                                                   Numerical method    1 Gauss point
                                                Numerical method 4 Gauss points     -3
                                                                                                                                 Numerical method    4 Gauss points
    -0.3                                        Numerical method 8 Gauss points                                                  Numerical method    8 Gauss points
                                                Numerical method 16 Gauss points    -4                                           Numerical method    16 G auss points
    -0.4                                                                            -5
            -8      -6       -4           -2           0           2                        -8        -6            -4           -2           0           2
                                   x                                                                                         x


Figure 5 Integrals of the Froude dependent part of the Green function and x-derivative for a
field point describing a parallel to the x-axis (z=-0.01;y=2)

                                                                                                           ∂g
ℜ ∫ gdc                                                                                          ℜ∫           dc
  δCl                                                                                             δCl
                                                                                                           ∂x

    0.3                                                                                 7
                         Aquaplus
                         Poseidon                                                                            Aquaplus
                         Numerical integration 1 Gauss point                            6
                                                                                                             Poseidon
                         Numerical integration 4 Gauss points                                                Numerical integration 1 Gauss point
                         Numerical integration 8 Gauss points                                                Numerical integration 4 Gauss points
                                                                                        5
                         Numerical integration 16 Gauss points                                               Numerical integration 8 Gauss points
    0.2                                                                                                      Numerical integration 16 Gauss points
                                                                                        4


                                                                                        3


                                                                                        2
    0.1

                                                                                        1


                                                                                        0


        0                                                                           -1
                 -0.2                    -0.1                          0                              -0.2                            -0.1                     0
                                   z                                                                                         z


Figure 6 Integrals of the Froude dependent part of the Green function and x-derivative for a
field point describing the z-axis




                                                                                   30
        2.25                                                                                      2.25
                                         Poseidon 522 panels
                                         Lin Yue (1990)
              2                                                                                          2
                                         Tests Gerritsma ex (1988)
                                         Chun-Tsung (1997)
        1.75                             Aquaplus 750 panels                                      1.75

            1.5                                                                                        1.5




                                                                                  A33(L/g)0.5/ρV
M 33/ρV




        1.25                                                                                      1.25

              1                                                                                          1

        0.75                                                                                      0.75

            0.5                                                                                        0.5

        0.25                                                                                      0.25

              0                                                                                          0
                   0   1    2    3          4           5            6   7                                    0   1   2   3       4   5   6   7
                                     ϖ                                                                                        ϖ



        0.16                                                                                      0.09


        0.14                                                                                      0.08

                                                                                                  0.07
        0.12

                                                                                                  0.06


                                                                                  A55(L/g) 0.5 /ρVL2
            0.1
2
M 55 /ρVL




                                                                                                  0.05
        0.08
                                                                                                  0.04
        0.06
                                                                                                  0.03

        0.04
                                                                                                  0.02

        0.02                                                                                      0.01

              0                                                                                          0
                   0   1    2    3          4           5            6   7                                    0   1   2   3       4   5   6   7
                                     ϖ                                                                                        ϖ



            0.2                                                                                   0.05


                                                                                                         0
        0.15

                                                                                              -0.05
                                                                                  A53(L/g)0.5/ρVL




            0.1
M 53 /ρVL




                                                                                                       -0.1


                                                                                              -0.15
        0.05

                                                                                                       -0.2

              0
                                                                                              -0.25


     -0.05                                                                                             -0.3
                   0   1    2    3          4           5            6   7                                    0   1   2   3       4   5   6   7
                                     ϖ                                                                                        ϖ



        0.05                                                                                           0.6


              0


     -0.05
                                                                                                       0.4
                                                                                  A35/(L/g) 0.5 /ρVL
M 35/ρVL




            -0.1


     -0.15

                                                                                                       0.2
            -0.2


     -0.25


            -0.3                                                                                         0
                   0   1    2    3          4           5            6   7                                    0   1   2   3       4   5   6   7
                                     ϖ                                                                                        ϖ


                       Figure 7 Added mass and damping coefficients on the Wigley hull (F=0.2)




                                                                             31
                   1 Motor – 2 Ship model
               3 Dynamometer or rigid modulus
                  4 Wave probe 5 Heave rod
                     6 Heave transducer
                      7 Pitch transducer
                         8 Pitch rod



      Figure 8 Experimental set-up and motion oscillator




Figure 9 Locations of the wave measurements




                    32
         0.012                          initial calculation (without wi and tfi)                    0.008
                                        calculation with wi without tif                                                              initial calculation (without wi and tif)
         0.011                          calculation with wi and tif                                 0.007                            calculation with wi without tif
                                        calculation without wi with tif                                                              calculation with wi and tif
          0.01                                                                                      0.006                            calculation without wi wit tif

         0.009
                                                                                                    0.005
         0.008
                                                                                                    0.004
         0.007
                                                                                                    0.003
 CM 33




                                                                                             CA33
         0.006
                                                                                                    0.002
         0.005
                                                                                                    0.001
         0.004
                                                                                                        0
         0.003

         0.002                                                                                      -0.001

         0.001                                                                                      -0.002

            0
                 0          2             4                  6                 8       10                    0   1   2       3         4        5         6       7        8    9   10
                                                  fL/U                                                                                         fL/U


Figure 10 Influence of the waterline integral on the added mass and damping coefficients on
                               the Series 60 CB=0.8 (F=0.2)


                                                                                                     0.008
         0.012
                                                                                                     0.007
                                                      Aquaplus
                                                      Tests
          0.01                                                                                       0.006
                                                      Poseidon
                                                                                                     0.005

         0.008                                                                                       0.004

                                                                                                     0.003
CM 33




                                                                                             CA33




         0.006
                                                                                                     0.002

                                                                                                     0.001
         0.004
                                                                                                        0

                                                                                                    -0.001
         0.002
                                                                                                    -0.002

            0                                                                                       -0.003
                 0      1       2   3         4          5       6         7       8   9                     0   1       2       3         4          5       6        7        8   9
                                                  fL/U                                                                                         fL/U



        0.0007                                                                                      0.0006


        0.0006                                                                                      0.0005


        0.0005                                                                                      0.0004


        0.0004                                                                                      0.0003
CM 55




                                                                                             CA55




        0.0003                                                                                      0.0002


        0.0002                                                                                      0.0001


        0.0001                                                                                          0


            0                                                                                    -0.0001


    -0.0001                                                                                      -0.0002
                 0      1       2   3         4          5       6         7       8   9                     0   1       2       3         4          5       6        7        8   9
                                                  fL/U                                                                                         fL/U


                     Figure 11 Added mass and damping coefficients on the Series 60 CB=0.6 (F=0.2)




                                                                                            33
  0.025                                                                                                                                    0.025

                                                                          Aquaplus
                                                                          Tests                                                             0.02
    0.02                                                                  Poseidon


                                                                                                                                           0.015
  0.015
CM33




                                                                                                                                         CA33
                                                                                                                                            0.01
    0.01

                                                                                                                                           0.005

  0.005

                                                                                                                                                0


        0
            0                     1               2   3                      4          5   6   7               8               9                   0   1   2   3   4          5   6                7      8     9
                                                                                 fL/U                                                                                   fL/U



0.0015                                                                                                                                     0.001




  0.001                                                                                                                                  0.0005
CM 55




                                                                                                                                         CA55




0.0005                                                                                                                                          0




        0                                                                                                                                -0.0005
            0                     1               2   3                      4          5   6   7               8               9                   0   1   2   3   4          5   6                7      8     9
                                                                                 fL/U                                                                                   fL/U


                Figure 12 Added mass and damping coefficients on the Series 60 CB=0.8 (F=0.2)
                                                                                                                    C   =   6
                                                                                                                            ,
                                                                                                                            0




                                                                                                    =
                                                                                                    C   0   6
                                                                                                            ,




                  C   =   ,
                          0   6

                      B




                                      C   =
                                          0   ,
                                              6

                                      B




                C B = 0 .6                                                                                                                                                             C   0
                                                                                                                                                                                           =   6
                                                                                                                                                                                               ,




                                                          C   B   0
                                                                  =   ,
                                                                      6




                                                                                                                                                                                                   h /L

                                                                                                                                                                                                        0 .0 0 7 0
                                                                                                                                                                                                        0 .0 0 6 3
                                                                                                                                                                                                        0 .0 0 5 6
                                                                                                                                                                                                        0 .0 0 4 9
                                                                                                                                                                                                        0 .0 0 4 2
                                                                                                                                                                                                        0 .0 0 3 5
                                                                                                                                                                                                        0 .0 0 2 8
                                                                                                                                                                                                        0 .0 0 2 1
                                                                                                                                                                                                        0 .0 0 1 4
                                                                                                                                                                                                        0 .0 0 0 7




                                  C B= 0 .8




                                                                                        a) heave motion (CB=0.6: top; CB=0.8: low)




                                                                                                                                    34
                                                                                                                 C   =   ,
                                                                                                                         6
                                                                                                                         0




                                                      6
                                                  C =0.
                                                   B




                        C =0. 6
                          B



                                                                                                     C
                                                                                                     =   0   6
                                                                                                             ,




        C   =   0
                ,   6

            B




                                      C   =
                                          0   ,
                                              6

                                      B




      C B = 0 .6                                                                                                                                   C   0
                                                                                                                                                       =   6
                                                                                                                                                           ,




                                                            C   B   =
                                                                    0   ,
                                                                        6




                                                                                             C =0.
                                                                                              B  6
                                                                                                                                                               h /L
                                                                                                                                               L
                                                                                                                                             h /




                                                                                                                                                                 0 .0 0 7 0
                                                                                                                                                                 0 .0 0 6 3
                                                                                                                                                                 0 .0 0 5 6
                                                                                                                                                                 0 .0 0 4 9
                                                                                                                                                                 0 .0 0 4 2
                                                                                                                                                                 0 .0 0 3 5
                                                                                                                                                                 0 .0 0 2 8
                                                                                                                                                                 0 .0 0 2 1
                                                                                                                                                                 0 .0 0 1 4
                                                                                                                                                                 0 .0 0 0 7




       C B = 0 .8




                                                                                     b) pitch motion (CB=0.6: top; CB=0.8: low)

                                                          Figure 13 Measured waves amplitudes (F=0.3, f=3Hz, τ=1.92)


       0.015


      0.0125
                                                                            C B = 0 ; a 3 /L= 0 0 9 ; F = 0 .2 ; f= 4 H z ; τ= 1 .7 9

                                                                            y/L= 0 .2
                          0.01


      0.0075
A/L




       0.005


      0.0025


                                  0


      -0.0025
                                                                            0                                                0.5         1                       1.5
                                                                                                                                   x/L




       0.015
                                                            C B = 0 .8 ; a 5 = 1 .8 °; F = 0 .2 ; f= 4 H z ; τ= 1 .7 9

                                                            y/L= 0 .2
                        0.01

                                                                                      Aquaplus
                                                                                      P oseidon
                                                                                      T e sts
A/L




       0.005




                                  0




       -0.005
                                                                            0                                                0.5         1                      1.5
                                                                                                                                   x/L


                                      Figure 14 Comparison of longitudinal cut of wave elevation y/L=0.2


                                                                                                                                   35
                                        (Series 60 CB=0.8 hull; F=0.2; τ=1.79)
                  X/L=0.9 Heave                                                    X/L=0.9 Pitch
      0.014                                                            0.018

                                                                       0.016
      0.012

                                                                       0.014
       0.01
                                                                       0.012

      0.008                                                             0.01
A/L




                                                                 A/L
                                           Aquaplus
      0.006                                                            0.008
                                           Poseidon
                                           Tests
                                                                       0.006
      0.004
                                                                       0.004

      0.002
                                                                       0.002

         0                                                                0
              0        0.1        0.2    0.3          0.4                      0         0.1       0.2   0.3   0.4
                                  y/L                                                              y/L



                  X/L=0.8 Heave                                                    X/L=0.8 Pitch
      0.012                                                            0.016


                                                                       0.014
       0.01

                                                                       0.012

      0.008
                                                                        0.01
A/L




                                                                 A/L
      0.006                                                            0.008


                                                                       0.006
      0.004

                                                                       0.004

      0.002
                                                                       0.002


         0                                                                0
              0        0.1        0.2     0.3         0.4                      0         0.1       0.2   0.3   0.4
                                  y/L                                                              y/L


                             Figure 15 Comparison of transverse cut of wave elevation
                                      (Series 60 CB=0.8 hull; F=0.2; τ=1.79)




                                                            36

								
To top