Transcript A model of the flow around bilge keels of FPSO hull

A model of the flow around bilge keels
of FPSO hull sections subject to roll
motions
The University of Texas at Austin
Ocean Engineering Group
S. A. Kinnas, Y.-H. Yu,
B. Kacham, and H. S. Lee
Supported by the Offshore Technology Research Center (OTRC)
Introduction
• Motivation
-
FPSO hulls have often been subject to amplitudes
of roll motion which seem to be larger than the
predicted using potential flow solvers.
Introduction
• Objectives
Develop efficient computational model to predict
the hydrodynamic coefficients of FPSO hulls
in roll motions.
- Investigate effectiveness of bilge keels on roll
mitigation
-
Numerical Formulation
• 2D Continuity and Momentum Equations

  vˆ  0



vˆ

   ( vˆ vˆ )  p  fˆ
tˆ




vˆ
2

   ( vˆ vˆ )  p  fˆ   vˆ
tˆ
( Euler)
( Navier  Stokes)
• Conservation form
U F G


Q
t x y
u 
U   ,
v 
u  1  2u / x 
F   
u / y  v / x  ,
Re
uv


 


 
 p x  f x 
uv

u
/

y


v
/

x


1
 

Q  
G   2 
,

p y  f y 
 v  Re

 2v / y 
 
v
q
y
u
2
x
• Ni’s Lax Wendroff method for time
U inj1
n
2



U

U
(

t
)


Uin j  
 t   2 
 t  i  j
 t  i  j 2
n
U in j 

c  A, B ,C , D
2
( Ui , j )cn
• Artificial Viscosity
U1n 1
U1n 

c  A, B , C , D
B
C
A
D
( U1 )cn  t ( 2  4 )
Node 1
where
2   2 ( ii U1n   jj U1n ), and 4   4 ( iiii U1n   jjjj U1n )
• SIMPLE method for Pressure Correction
to satisfy the continuity equation:
u v

0
x y
Oscillating Flow past a Flat Plate
• Fluid domain and Boundary Conditions
• Definition of Keulegan-Carpenter Number
KC  U mT / h
• Horizontal force on Plate (Morison’s equation)
2F
2
 Cd  cos  cos  Cm
sin
2
K
 hwU m
3 2 F ( ) cos 
Cd   
d
2
4 0
 hwU m
Cm  
2 KC
3

where
2
0
F ( ) sin
d
2
 hwU m
  t
• Background
- Sarpkaya & O’Keefe’s Experiment (1995)
• Comparison of results from Euler and Navier-Stokes
solvers, and Sarpkaya’s measurements
Inertia Coefficients (Cm)
Drag Coefficients (Cd)
• Vorticity & Streamlines at t=0xT (KC=1)
• Vorticity & Streamlines at t=T/4 (KC=1)
Euler Solver
Navier-Stokes Solver
• Vorticity & Streamlines at t=T/2 (KC=1)
• Vorticity & Streamlines at t=3T/4 (KC=1)
Euler Solver
Navier-Stokes Solver
Movie
(KC=1)
Movie
(KC=10)
• Comparison of force over one period (KC=1)
FPSO Hull Section Motion
• Fluid domain and Boundary Conditions
• Boundary Condition on Free-surface
u  2
*

u (for u),
y
g
v  2

v (for v), and
y
g
p
  gv (for p)
t
• Body motion (Roll motion)
 (t )   o sin(t )
• Added mass and damping coefficients
a66 
aˆ 66
(Added Mass)
2
4b
ˆ 66
b
b66 
4 b 2
b
(Damping)
g

2b
• Grid and geometry details
• Comparison of Added mass and Damping coefficients
for Heave motion (B/D=2, No bilge Keel)
• Vorticity contour and Streamlines for roll motion
(6% bilge keel, roll angle=0.05(rad), and Fn = 0.8)
t = 0xT
t = T/4
t = T/2
t = 3T/4
• Comparison of added mass and damping coefficients
for various sizes of bilge keel
Added mass Coefficients
Damping Coefficients
Conclusions
• Developed CFD model to solve the Euler equations around a
flat plate, and an FPSO hull section subject to roll motions.
• Validated the present method :
- Flat plate subject to an oscillating flow:
Euler results comparable to those from Navier-Stokes
and experimental data
- Hull in Heave motion:
Present method predicted hydrodynamic coefficients well.
- Hull in Roll motion:
* Euler solver is capable of capturing the separated flow
from bilge keels.
* Present solver can predict expected increase in added mass,
and damping coefficients with increasing bilge keel size.
Future Work
• Implement viscous flow terms in the case of
FPSO hulls with and without bilge keels
• Use the modified hydrodynamic coefficients at
each FPSO hull section, to correct the results
from potential flow solvers.
• Validate the method with existing and future
experiments on FPSO hulls to be carried out at
OTRC’s Wave Basin.