The added resistance of ships

1. Computation ofAdded Resistance of Ships
by Using a Frequency-domain
Rankine Panel Method
Beom-Soo Kim, Jae-Hoon Lee, Yonghwan Kim
Department of NavalArchitecture and Ocean Engineering
Seoul National University
InternationalAcademic Research Exchange Workshop on
Ship and Ocean Engineering (SOE Workshop)
January 20-21, 2018
Osaka University

2. Research Backgrounds
IMO recently made the EEDI regulation
regarding the restriction of greenhouse gas
emissions from ships
Added resistance in waves, which can be up to
30% of calm-water resistance, should be
predicted accurately in terms of fuel consumption
Several theoretical approaches have been
introduced with varying complexity and
accuracy in the estimation of added resistance
Accurate and efficient prediction of added
resistance is necessary both for the ship design
and operation
< Calm water condition >
- 1 -
< Under wave condition >

3. State-of-the-Art Review
• Frequency-domain Rankine panel method
• Evaluation of the added resistance using the near-field method
Faltinsen et al. (1980)
Computation of the added resistance in a seaway along with 2-D strip method to
analyze the motion of the ship
Kim and Kim (2011)
Formulation of the near-field method for the 3-D time-domain Rankine panel
method (WISH)
H. Söding and V. Shigunov
(2015)
Computations of added resistances of various ship models by using the 3-D
frequency-domain Rankine panel method (GL Rankine)
- A frequency-domain 3-D Rankine panel method is used to solve the basic
seakeeping problem and to estimate the added resistance in present study
- The implemented method is verified by comparing results of a practical ship
- 2 -
P.D. Sclavounos, D.E.
Nakos (1990)
Development of 3-D frequency-domain Rankine panel method (SWAN) for the
solution of both steady and unsteady problems
H. Iwashita et al. (1993)
Development of 3-D frequency-domain Rankine panel method applying
different infinity (radiation) conditions in all forward-speed and frequencies
H. Iwashita (2016)

4. Boundary Value Problem
• Boundary value problem of the velocity potential (Ψ)
Governing equation 2
  0 in the fluid domain
Combined free surface
boundary condition
  2  
1
   g  0 on z 
tt t
2 z
Body boundary condition

U n 
n on S
n t
B
Basis potential,
Steady wave potential,
 Decomposition of total velocity potential
 x,t Ux   x
+x
+x t
, 
O(1)
 
Unsteady wave potential,  
 Perturbation of boundary conditions about z  0 or mean body
• Linearized boundary value problem
Governing equation 2
  0 in the fluid domain
Combined free surface
boundary condition
(Steady wave potential)
U 2
  2U      2U 
1
  g  U   
xx x x
2 z zz x
 U2
  2U  
1
  U  
1
  on z  0
xx x
2 zz x
2 zz
Combined free surface
boundary condition
(Unsteady wave potential)
  2U  2 U 2
 U


U   
U 
tt tx t xx
x x x

1
  g   U    0 on z  0
2 z zz t x
Body boundary condition
 6    m1,m2 ,m3   n U 
 
j
n m  on S
n j1  t  m4 ,m5,m6   nx U 
j j j B
< Definition of the system >
- 3 -

5. Evaluation of the Basis Potential
ii) Second derivatives of basis potential on each panel:
iii) M-terms over the hull surface:
are also used on the free surface boundary condition and the evaluation of added resistance
x  x'2
 y  y'2
 z  z '2
,r '  x  x'2
 y  y '2
 z  z '2
 r r '
 
S
B
  1

1 dS where r 
 
3 zx x zy y zz z
 m1  xxnx  xyny  xznz 

m2  yxnx  yyny  yznz 

 m    n   n   n
 6 2 1 x 2 y y 1 x
 m4  ym3  zm2  ny U3 z  nz U2 y 

m5  zm1  xm3  nz U1 x  nx U3 z 
 m  xm  ym  n U    n U   


 
  1

1 dS
 
xj xj  r r '
S
B
j
SB
G*dS
 
2

xjxk

xk

j
j
 r  r
r ' r '
x    
S
B
G *dS 

where G*   1

1  for j 1,2, G*   1

1  for j  3
B
2
   1

 1 dS 

U n
 
 n r n r ' n
S
j j k
- 4 -
 2

iv)
x
,
x x
• Double-body linearization
-Regards the basis potential as the double-body flow (z  0 on z  0)
- Dirichlet-type integral equation (source-method) (Wu, 1991, Chen and Malenica, 1998)
i) Basis potential and its derivatives on each panel:

6. Frequency-domain Formulation
• Radiation problem (j=1,…,6)
𝑔
-Assume weak upstream influence (𝑟 =
𝑈𝜔
> 1/4)
0
0
• Radiation potentials and diffraction potential
g
e g
eit

• Obtaining the steady-state limit of the unsteady response is the principal objective
• Incident wave potential (Unit amplitude)

0
zixcosiysin
x,t i
6
j1

it   j 1,...,6: radiation potentials
j  7 : diffraction potential
Diffraction Problem (j=7)


 
 Ree A
0 
7  
j
j 
j
j
j
j j

g  F , on z  0
 
z
2
  0, in the fluid domain





 j
 i
n  m on B
 n
Radiation Condition

• Radiation condition (Nakos, 1990)
 
7
7
7 0
g

 G , on z  0
7
z
  on B
2
  0, in the fluid domain




 

 n n
Radiation Condition

𝑥 = 𝑥0 𝑈
𝑧
❑2𝜙 =
0
𝑥
Rigid lid condition
Upstream edge
< Upstream boundary condition >
 
- 5 -
0
  
iU x  x0  0: zero wave elevation condition
 x 
  
2
iU  x  x  0: zero wave velocity condition
 
x
 

7. Numerical method
• Green’s second identity
• Higher-order B-spline panel method (potential-method)
- Basis function: second-order polynomial variation over the neighbor panels
- Efficient and robust evaluation of the differentiations of velocity potential
1 x : position vector of a field point
x ': position vector of boundary points (sources)
where, Gx; x '
1
2 x  x '
FS
B
Gx; x 'dx ',
FSB
x '
Gx; x 'dx ' x '
Gx; x 'dx '
z ' n
x '
x FS B
n
x  
=
7 8 9
4 5 6
1 2 3
2
2
2
2
2
4
1
 1  3 
2
x  , 
3
 x  

2 2
 
2  


32
  
2
 1 
  x 
2
bx  2 x  ,

  

 x 
3  ,

 x 
3
2 2
 2 
 2  

Free surface boundary condition
Body boundary condition
(FS)or(B)
  ,,,
x
n
 
 
9
0,0
1
k
k
B k a
64 k 1
k 1
 

x 64



1

9
B1,0
ka
1 6 1
6 36 6
1 6 1
-4 0 4
-24 0 24
4 0 4
B0,0
:
1,0
B :
< Panel index >
- 6 -

8. Equation of Motion
• The pressure on the hull about the mean position
• Wave induced unsteady force
 
     
 
 
6
1 2
added mass coefficients
i a i ij
B
ij j j i
j j i
it


j1
 
 
 
F t  p  p n ds Re e X   a i
b  c
  i ij ij 


 


 a  
Re

 i    n ds 

B 


 

bij  Im i    n ds
 
B 

 j

2 
 damping coefficients
ij
c   
a  1
   gz
n ds restoring coefficients
 2  i
B  

 


B
• 6-DoF equation of motion



Xi  Rei

0 
7   
0 
7 nids exciting force

6
2
ij ij ij ij j i ij
 i 1,...,6, m : mass matrix, Newman(1977)
 m  a  i
b  c 
  X ,

j1
2 2
a t
 1
p  p        gz  a  
1
   gz
  t 



 
  xB
- 7 -

9.   
 
  
(0)
2
1
(1)*
3 4 5 1
* (0)
1
2
2
1
2
I d 3 4 5 I d 3 4 5
1
SB 2
1
SB 2
g 1
dl
Re i
 *
 n
F  Re    ( y x)      
( y x)
WL 2   sin
 
  U    g(  y x) n dS
 
 
 Re  n dS
 
 *

 Re  i

1
 
1

1


(0)
 1
(1)* (2)
2 2 2
B B
SB
S S
 U    n dS

1 
1
Re
     1  
 U     n dS    U    n dS
  1      1
  
  
 

 
Estimation of Added Resistance
• Added resistance in the near-field method is evaluated by direct integration of second-
order pressure on a body surface
• Perturbations of motions, pressure, wave elevation, and the normal vector of the surface
are used to obtain the second-order pressure
• Formulation of added resistance based on the near-field method (K.H. Kim et al., 2010)
(I) Waterline-integral component
st
(II) 1 -order-normal-related component
(III) Kinetic-energy component
(IV) Motion & potential coupled component
(V) Basis-potential-related component
 

- 8 -

 
 
2 0 2
2 2   * * 0
1 1 1 5 3 6 2 1 5 6 1 1 5 5 6 6 1
2 2
5 6
2 2
4 5 4 6
2 2
4 6 5 6 4 5
1 1
2 4
0 0
0
2 2
 
where, n 0
 n , n 1
n n , n     n , n   Re   n ,
 
  


H 
1  2   
2 




 
  
 

10. Frequency-domain vs. Time-domain
• Comparison of Rankine panel methods on different domains
- For the comparison of results, the calculation conditions are the same in:
linear problem, double-body basis flow, near-field method
Elements
Frequency-domain method
(Present)
Time-domain method
(Kim et al., 2011, WISH)
Common
conditions
Higher-order B-spline Rankine panel method (potential method)
Extension to
nonlinear problems
Restrictive Unconstrained
Formula of free surface
boundary condition
Combined
Separated
(Kinematic & Dynamic)
Radiation condition
Upstream rigid lid condition
(Rectangular-type domain)
Rigid lid
Rayleigh’s artificial damping
at the free surface end (O-type domain)
Damping zone
Additional filtering None
K
Removing saw tooth waves, j  ck jk jk 
k0
Simulation time per case 1-5 minute 1-3 hours
- 9 -

11. Solution Grids - 1
• Computation grids
- Rectangular-type free surface domain
- Number of panels: 2,000 (body), 8,000 (free surface)
- The disturbed waves should be captured in the domain
i) Upstream truncation distance: 0.5λ
ii) Downstream truncation distance: 0.8λ
iii) Transverse truncation distance: 2.0λ
• Additional wake patch for transom-stern ships
- Reed et al., 1990, D.E. Nakos and P.D. Sclavounos, 1994
- Physically, the free surface separates from the transom which is shallow
- Additional Smooth detachment conditions on the transom is applied
 xT , y hxT , y
  h
xT , y xT , y

 x
 x
x
x=xT
h
𝜍
< Example of a computation grid >
< Grids around a transom (KVLCC2 )>
- 10 -
< Definition of geometry around a transom >

12. Solution Grids - 2
uu uv vv u v
• Body-fitting grid generation (A. Hilgenstock et al., 1986)
- Improvement of skewness of free surface panels around the blunt hull forms
- Solving a Poisson equation generates smooth grids
x, y: Physical domain
Transformation
u,v: Rectangluar computation domain

xuu xuv xvv  J Pxu  Qxv 
y y y  J Py  Qy 

< Before fitting > <After fitting >
< Before fitting > < After fitting >
• Improvement on the grid around the bow of blunt ship
- The solutions of B-spline panel method highly depend on geometrical properties of neighboring panels
- Discordance of local axes and high skewness of panels deteriorate the solutions
<Magnitude of disturbed potential>
- 11 -

13. Results: Wigley III
• Test conditions & Main dimensions
• Motion responses &Added resistance
β (deg.) Fn L (m) B (m) T (m) CB
180 0.20 3.0 0.3 0.1875 0.462
<Hull geometry>
- 12 -
< Heave motion > < Pitch motion > <Added resistance >
- Due to its unconventional (slender) section shape:
large pitch motion → large added resistance in the resonance waves
- The present results show good agreements with those obtained by experiments and the
time-domain Rankine panel method (WISH)

14. Results: KVLCC2 - 1
• Test conditions & Main dimensions
• Motion responses &Added resistance <Hull geometry>
and the long-wave region
- 13 -
< Heave motion > < Pitch motion > <Added resistance >
- Motion responses and added resistance in short-wave region are almost identical to
those obtained by time-domain Rankine panel method and the experiments
- The present results underestimate the added resistance in both the peak value region
β (deg.) Fn L (m) B (m) T (m) CB
180 0.142 3.2 0.58 0.208 0.8098

15. Results: KVLCC2 - 2
• Components analysis for added resistance
< (I) and (III) components of added resistance > < (II) and (IV) components of added resistance >
- Waterline-integral component (I) is the most dominant component and the present
method underestimates it over the short-wave and the peak value region
- Kinetic-energy component (III) is the second dominant component and the present
 
 

 
 

(0)
* 1
3 4 5 1
* (0)
1
*
1 1
2 2
1
2
1
I d I d
WL
SB 2
1
SB 2
n
dl
dS

(I) : 
g Re      
( 3 4 y 
5 x)     
( 3 4 y 
5 x)
  sin
 (1)* 
 U    g(  y x) n
 
 
Re  n dS
 
  (0)
 U    n dS



1

(II ) :  Re i
III : 
1


SB 2
(IV ) :  Re  i
method underestimates it over the long-wave region
- 14 -

16. Results: KVLCC2 - 3
• Magnitude of 1st-order disturbed wave profiles adjacent to the body
θ
<Top view, definition of θ> < Short-wave region, λ/𝐿 = 0.30 > < Peak value region, λ/𝐿 = 1.20 >
- Waterline-integral component (I) 𝖺 (relative wave height )2
- The present method underestimates the magnitude of 1st-order disturbed wave around
the bow than the results obtained by time-domain Rankine panel method
- These discrepancies lead to the underestimated value of component (I) over the short-
wave and peak value region
- Different free surface panel distributions adjacent the body on each method may cause
1 1
2
I d 3 4 5 I d 3 4 5 dl
1
sin

*
 n(0)
  y x)   ( y x)
WL 2  
(I) : 
g Re    (
these discrepancies
- 15 -

17. Results: KVLCC2 - 4
• 2nd-order pressure distribution of added resistance on hull surface
(λ/𝐿 = 1.20, peak value region)
- Distribution of Kinetic-energy component (III) on the upper area of the bow is
different from the distribution of time-domain Rankine panel method
- At the bulbous bow of KVLCC2, the ascending/descending flow along the bow is
severe
- The present method exaggerates the vertical flow which leads to larger suction
pressure in kinetic-energy component (III)
< 1st-order-normal-related component (II) > < Kinetic-energy component (III) > < Motion & potential coupled component (IV) >
  (1)*
 3 4 5 1
1
SB 2
 
 U    g(   y x) n dS
 
 Re i  
* (0)
1
1
2 SB 2
 

1
 Re  n dS  
*
1
  SB 2
  (0)
 U    n dS

 
 1
 Re  i
- 16 -

18. Results: KVLCC2 - 5
- Discrepancy of kinetic-energy component (III) on the upper area of the bow are
intensified in the long wave region
- The differences extends downward because the longer waves affect in deeper depths
- Differences in applied techniques and differences in free surface panel distribution
may occur the discrepancies on 2nd-order pressure distributions
• 2nd-order pressure distribution of added resistance on hull surface
(λ/𝐿 = 2.00, long-wave region)
< 1st-order-normal-related component (II) > < Kinetic-energy component (III) > < Motion & potential coupled component (IV) >
  (1)*
 3 4 5 1
1
SB 2
 
 U    g(   y x) n dS
 
 Re i  
* (0)
1
1
2 SB 2
 
 

1
 Re  n dS  
*
1
SB 2
  (0)
 U    n dS

 
 1
 Re  i
- 17 -

19. Concluding Remarks
- 18 -
• Afrequency-domain Rankine panel method is proposed
- The method is based on the higher-order B-spline potential method
- Smoothing of free surface panels is achieved by using the body-fitting grid generation technique
• Demonstration of proposed method
- Overall seakeeping results show good agreements with the results from references
- The present method slightly underestimates the added resistance of KVLCC2 in both the peak value region and
the long-wave region
• Discrepancies in added resistance of KVLCC2
- Waterline-integral component (I) and Kinetic-energy component (III) are the most dominant and the present
method underestimates these components
- Underestimation of disturbed wave in the bow area leads to the discrepancy of waterline-integral component (I)
- Overestimation of the velocity of vertical flow on the upper bow area leads to the discrepancy of
kinetic-energy component (III)
- The present method needs improvement in prediction of velocity potential around the bow of blunt ships

20. Thank you for listening
Q &A
InternationalAcademic Research Exchange Workshop on
Ship and Ocean Engineering (SOE Workshop)
January 20-21, 2018
Osaka University