1. STEADY AND UNSTEADY FLOW SIMULATION FOR
A FAST SHIP FORMS IMPROVEMENT
G.Alexandru, M.Nechita
Research and Design Institute for Shipbuilding, ICEPRONAV SA
Address: Portului 19A, Galati, ROMANIA
Phone: 40 36 410100; E-mail: icepronav@icepronav.ro
ABSTRACT NOMENCLATURE
The study of the flow around the ship represents Aij - added mass coefficients;
an important step in designing an optimized hull
form concerning the low resistance criterion. A Bij - damping coefficients;
modern approach of this design process should B - breadth;
consider the flow simulation around fast ship CW - wave resistance coefficient
forms employing CFD analysis.
The paper presents the numerical analysis of d - draught;
steady and unsteady flow for a fast vessel in or- E j - wave exciting forces;
der to estimate the ship behavior (including ship g - gravity acceleration;
motions and the unsteady forces acting on the GM - transverse metacentric height;
ship hull) in sea conditions.
Numerical simulations were necessary to pro- K - wave number;
vide useful information about the pressure and KG -center of gravity above keel;
velocity fields on the wetted hull surface, the k yy - pitch radius of gyration;
ship behavior in order to make the modification
of the bodylines more efficient. LCG - longitudinal center of gravity;
For the unsteady problem is proposed a seakeep- L - length between perpendiculars;
ing computation method taking into account the p - fluid pressure;
influence of the steady wave field for estimating SH, SF - hull and free-surface area;
the hydrodynamic forces acting on the ship hull U - ship speed;
and the response functions. ρ - water density;
The proposed computation methods predicts the ∇ - volumetric displacement;
steady wave field in good accuracy, this being σ - sources density;
one important condition for the consequent un-
steady wave field computations. The steady ζ - wave elevation;
problem is solved so that the fully nonlinear χ - angle of encounter wave;
free-surface condition is satisfied and evaluating λ - wave length;
consequently the influence terms of the steady
wave field on the unsteady wave field. The un- ω 0 - wave circular frequency;
steady boundary value problem is liniarized as- ω e - encounter circular frequency;
suming the small amplitude of the incident
waves and ship motions. The boundary condi- Ψ - total velocity potential;
tions for the unsteady problem are satisfied on Φ - steady velocity potential;
the exact steady free-surface and wetted surface φ - unsteady velocity potentials;
of the body. The numerical results are carefully ξ j - ship motions;
compared with experiments.
Finally, it is emphasized that the present simula-
tion is confirmed to be effective to solve the
steady and unsteady flow for a fast ship.
1. INTRODUCTION

2. Nowadays, the necessity of designing high-speed φ 0 is the incident wave potential, φ 7 represents
vessels demands a careful analysis of their be-
scattering potential and φ j (j=1∼6) are radiation
havior in seaway. The efficiency and accuracy of
numerical simulation are essential for the im- potentials
provement and optimization of hull forms. On Combining the kinematic boundary condition
this basis, the ship designers will choose the and dynamic boundary condition, the exact free (5)
most advantageous of several design options be- surface boundary condition to be satisfied by the
fore carrying out the model tests. total velocity potential has the following form:
Applying numerical simulation methods for a
ship form design involves a thoroughly valida- ∂ 2Ψ ∂Ψ ∂Ψ 1
+ ∇Ψ ⋅ ∇ + ∇Ψ ⋅ ∇ + ∇ Ψ ⋅ ∇ (∇ Ψ ) 2
tion of the computer code used on this purpose. ∂t 2
∂t ∂t 2
In this paper, a seakeeping computation method (2)
∂Ψ
is proposed taking into account the steady wave +g = 0 on z = ζ ( x, y; t )
field influence on the unsteady pressure acting ∂z
on the ship hull. The numerical results are com- Taking into account the decomposition of the
pared with experimental data obtained for a total potential, the steady wave field must satis-
modern fast vessel of which form was subject to fy:
several bodylines modifications.
1 ∂Φ
∇Φ ⋅ ∇( ∇Φ ) 2 + K 0 = 0 on z = ζ S (3)
2. FORMULATION 2 ∂z
∂Φ
The coordinate system used for the formulations = 0 on S H (4)
∂n
has the origin located on the undisturbed free-
surface, at the midship, having the x axis orient- where S H represents the wetted hull surface and
ed towards the fore part, y axis towards the port
g
side, and z upwards. The ship is advancing with K0 = 2 .
constant speed U in oblique regular waves en- U
countered at angle χ (180° heading means head For solving this problem, an iterative numerical
waves). Incident wave amplitude A and ship procedure developed by Jensen et al. (1986) was
iω t implicated.
motions ξ j e e ( j = 1 ~ 6) are assumed to be
Radiation condition is satisfied by shifting the
small. For the considered irrotational ideal fluid, collocation points of the free surface one panel
the total velocity potential Ψ fulfills Laplace’s upward. Source panels of the free surface are
equation: placed one panel length above still water plane.
In our case the ship has transom stern and there-
Ψ ( x , y , z ; t ) = UΦ ( x , y , z ) + Φ t ( x , y , z ; t ) fore it was assumed that the flow separates and
(1) the transom stern is dry (atmospheric pressure at
iω e t
= UΦ ( x , y , z ) + ℜ[φ ( x , y , z ) e ] the edge of the transom).
After some iteration steps, the numerical proce-
where: Φ - steady wave field, φ - unsteady wave dure is stabilized around a certain solution for
field the steady potential Φ and free surface elevation
and ω e = ω 0 − KU cos χ is encounter circular ζS.
Consequently we can proceed with calculus for
frequency where ω 0 and K represents circular
the unsteady velocity potentials. Assuming small
frequency and wave number of incident wave amplitude of the incident wave and ship motions,
6
gA the free surface boundary condition (2) can be
φ= (φ 0 + φ 7 ) + iω e ∑ ξ j φ j
ω0 j =1
liniarized around previously calculated steady
φ 0 = ie Kz −iK ( x cos χ + y sin χ ) free surface z = ζ S . The final form of the free
surface boundary condition for the unsteady po-
tentials can be found in Nechita, Iwashita et al.
(2000).

3. Body boundary conditions for the unsteady po-
tentials have the following forms: The ship motions ξ j can be determined by solv-
ing six simultaneous equations, Newman (1978):
∂φ j U
= nj + mj ( j = 1 ~ 6) on S H
∂n iω e 6
∑ [−ω ( M ij + Aij ) + iω e Bij + C ij ]ξ j = E i
2
(5) (10)
∂φ7 ∂φ e
=− 0 on S H j =1
∂n ∂n where M ij is body mass matrix and C ij is the
matrix of total static restoring forces.
Obtaining the velocity potential on the ship sur-
face, the steady pressure and the wave elevation iω t
can be easily calculate by: The diffraction wave ℜ[ζ 7 e e ] is given by:
ζ7 − iτ /ν
pS =
ρU 2
2
[
1 − ( ∇Φ )
2
on S H ] =
A 1 + 1 ∇Φ ⋅ ∇ ∂Φ
(1 +
1
iK 0τ
∇Φ ⋅ ∇ )φ7
(11)
(6)
ζS =
1
1 /(2 K 0 )
1 − ( ∇Φ )
2
[ on z = ς s ] K0 ∂z
where ρ is fluid density. 3. COMPUTATIONAL METHOD
The unsteady pressure ℜ[ pe iω et ] is given as fol- The numerical method used is a desingularized
lows, Timman & Newman (1962): panel method proposed by Jensen et al. (1986)
and developed for the unsteady flow calculations
p = − ρ (iω e + UV ⋅ ∇)φ by Bertram (1990). Steady wave potential Φ
ρU 2 6
(7) and unsteady wave potentials φ j are represented
−
2
∑ξ j =1
j ( β j ⋅ ∇)(V ⋅ V ),
with constant source distributions on body sur-
face and free surface panels.
e j ( j = 1,2,3)
where V = ∇Φ and β j = 
e j −3 × r ( j = 4,5,6) Φ ( P)  σ S (Q) 
Second term of the unsteady pressure is due to φ j ( P) 

=−
S
∫∫ +SF
 G ( P, Q)dS
σ j (Q) 
(12)
dynamical restoring force of the unsteady motion H
within steady flow field.
G ( P, Q ) is the Green function with
Wave exciting forces acting in j-th direction are
computed by integrating the unsteady pressure 1
G ( P, Q ) = ( x − x' ) 2 + ( y − y' ) 2 + ( z − z ' )2
given by incident wave potential and scattered 4π
potential over the ship hull: (P is the field point and Q is source point).
σ S (Q) and σ j (Q) are the constant source
Ej τ 1 strength over the panels surface for steady and
ρgA
=i
ν ∫∫ (1 + iK τ V ⋅ ∇)(φ
SH 0
0 + φ 7 )n j dS (8)
unsteady velocity potentials, respectively.
The body surface and the free surface are dis-
where ν = Uω 0 / g . cretized into a finite number of constant source
Added mass and damping coefficients Aij and panels. The source panels on S F are shifted one
Bij acting in i-th direction due to j-th motion are panel above the free surface, the collocation
given by: points on S H coincide with the geometric center
of each panel and those on S F are shifted one
Aij Bij 1 panel upward in order to force numerically the
− +i = ∫∫ (1 + V ⋅ ∇)φ j ni dS
ρ ρω e iK 0τ radiation condition.
SH
(9)
1
−
2( K 0τ ) 2 ∫∫ ( β
SH
j ⋅ ∇)(V ⋅ V )ni dS 4. EXPERIMENTS

4. The experiments for the fast ship model were Z
carried out at ICEPRONAV SA having the prin- Y
X
cipal particulars listed in Table 1.
Table 1. Principal particulars of the model
L [m] 4.80 KG [m] 0.20
B [m] 0.72 LCG [m] 2.02
d [m] 0.10 GM [m] 0.33
∇ [m ]
3
0.105 kyy/L 0.25 Fig.2 Computational grids of the fast ship
(NH=637, NF=1091)
The dimensions of the towing tank are
The steady wave profile along ship side for lin-
L×B×H=300×12×6 [m].
ear and nonlinear computation is presented in
The final hull form has been improved taking
Fig. 3. The linear computation represents in fact
into consideration the numerical results and ex-
the first iteration result and the nonlinear itera-
perimental tests. The principal hull form im-
tive process converges after 11 steps. In dashed
provement consists of bodylines modification at
lines, the wave profiles behind transom stern are
the aft part of the ship, in order to assure a low
presented.
resistance coefficient and a proper sinkage/trim
The significant difference between linear and
in the ship speed range.
nonlinear wave resistance coefficient is due to
the high Froude number. Although the computed
5. RESULTS
wave elevations along ship-side have compara-
5.1 STEADY WAVE FIELD ble values, except for the aft half, the differences
The perspective view of the final hull form is become evident for the steady wave contour
presented in Fig 1. In order to carry out the fully plots on the free surface and pressure distribu-
nonlinear computation, the hull was defined con- tion on the hull (see Fig. 7-8, Appendix A).
sidering the freeboard. After some numerical
0.030
tests, it was adopted 637 panels for the hull grid h/L Linear C W =0.698 10
. -3
and 1091 panels for the free surface (see Fig. 2). 0.020 Nonlinear C W =1.027 10
. -3
The panels on the hull surface are regenerated in Experiment C W =1.251 10
. -3
each iteration step considering the steady wave 0.010
A.P .
profile along ship side obtained in the previous 0.000
iteration. F.P .
-0.010
Linear ship-side
Z -0.020 Linear center-line
X Y Nonlinear ship-side
Nonlinear center-line x/L
-0.030
1 0.5 0 -0.5 -1 -1.5
Fig.3 Steady wave along ship-side at Fn=0.6
5.2 UNSTEADY WAVE FIELD
For the unsteady problem, the body and free sur-
face discretizations are made accordingly with
Fig.1 Hull discretization for the fast ship steady wave elevation previously obtained. The
discretized hull surface will have a wavy shape
on the upper boundary and the collocation points
of the free surface will be placed on the fully
nonlinear steady wave surface. For satisfying the
radiation condition was used the same shifting
technique as in the steady flow case.

5. In Fig. 4, 5 are presented the comparisons be- 6. CONCLUSIONS
tween computed values of heave and pitch mo-
tions and experimental data in regular head Through the present study, the following conclu-
waves. sions can be reached:
1 .5 0 1. The proposed method is efficient to investi-
ξ 3 /A Pr ese nt metho d
Exp er imen t gate steady and unsteady flows for hull form
1 .2 5 analysis of fast ships;
2. Successive hull form modifications based on
1 .0 0
numerical simulations determined a de-
0 .7 5 crease of the wave resistance coefficient
confirmed by experiments from the initial
0 .5 0 value of 1.667×10-3 to the final one of
1.251×10-3 for Fn=0.6;
0 .2 5
3. The present desingularized Rankine panel
λ /L
0 .0 0 method is confirmed to be effective in solv-
0.00 1.00 2 .0 0 3 .0 0 4.00 5.00 ing the unsteady problem, taking into ac-
Fig.4 Heave motion in head sea at Fn=0.6 count the nonlinear steady wave field.
The agreement between numerical results and 7. REFERENCES
experiments seems satisfactory except for the
pitch motion in the range of longer wavelength. JENSEN, G., MI, Z-X., SOEDING, H. – Rank-
1 .5 0
ine Source Methods for Numerical Solutions of
ξ 5 /KA
Pre se nt metho d the Steady Wave Resistance Problems – 16th
Exp er iment
1 .2 5 Symposium on Naval Hydrodynamics, Berkeley,
1986, pp. 575-582.
1 .0 0
NECHITA, M., IWASHITA, H., IWASA, H.,
0 .7 5
HIDAKA, Y., OHARA, H. – Influence of the
0 .5 0
Fully Nonlinear Steady Wave Field on the Un-
steady Wave Field of a Blunt Ship Advancing in
0 .2 5 Waves – Transaction of the West-Japan Society
of Naval Architects, No. 101, 2000, pp. 37-48.
λ /L
0 .0 0
0.00 1.00 2 .0 0 3.00 4.00 5.00
TIMMAN, R., NEWMAN, J. N. – The Coupled
Fig.5 Pitch motion in head sea at Fn=0.6 Damping Coefficients of a Symmetric Ship –
Journal of Ship Research, 5/4, 1962, pp. 1-6.
Fig. 6 presents the contour wave plot of the com-
puted diffraction wave for λ/L=0.5. Due to the NEWMAN, J. N. – The Theory of Ship Motions
slenderness of the fore part of the ship, the sig- – Advances in Applied Mechanics, 18, 1978, pp.
nificant crests of the diffraction appear after the 221-283.
midship. This fact is better observed in Fig. 9,
Appendix A. BERTRAM, V. – Ship Motions by a Rankine
1
source Method – Ship Technology Research, 37,
0
.
9
0
.
8
- -4 -2
0 . -8 -6 0 . 0 . 0
. 0 . 0 . 00 2 0 .
3 0. 0
2 11 0 . 0.
618
1423 1990, pp. 143-152.
L
0
.
7
y
0
.
6
/
0
.
5
0
.
4
0
.
3
0
.
2
0
.
1
0
-
1 -
0
.5 0 0
.
5 1
x
/
L
Fig.6 Computed contour plot of diffraction wave
at Fn=0.6, λ/L=0.5, χ=180° and t= 0.

6. APPENDIX A
0
-0
-0
Linear computation -0
-0.5
0.5
0
-0.5 0 0.5 1 1.5 2
x/L
Nonlinear computation
ζ /L
-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005
Fig.7 Steady wave contour plots, Fn=0.6
Y
Linear computation
X Z
Nonlinear computation
cp: -0.06 -0.05 -0.04 -0.02 -0.01 0.00 0.01 0.02 0.04 0.05 0.06
Fig.8 Steady pressure distribution on the hull
Z
X
Y
Fig.9 Perspective view of diffraction wave at Fn=0.6, λ/L=0.5, χ=180° and t= 0.