A model of the flow around bilge keels of FPSO hull sections subject to

roll motions

The University of Texas at AustinOcean 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

• Conservation form

t x y

U F G Q

,uv

U 2 2 /1 ,

/ /Reu xu

u y v xuv

F

2

/ /1 ,2 /Re

uv u y v xv v y

G x

y

p x fp y f

Q

x

y

u

v q

)(ˆˆ)ˆˆ(ˆˆ

)Euler(ˆ)ˆˆ(ˆˆ

0ˆ

2 StokesNaviervfpvvtv

fpvvtv

v

• Ni’s Lax Wendroff method for time

11 1 1 2 4

, , ,

2 2 1 1 4 4 1 1

( ) ( )

where

( ), and ( )

n n nc

c A B C D

n n n nii jj iiii jjjj

t

U U U

U U U U

2 21

2

,, , ,

( )2

( )

nnn ni j i j

i j i j

n ni j i j c

c A B C D

ttt t

U UU U

U U

• Artificial Viscosity

• SIMPLE method for Pressure Correction

B C

DA

Node 1

to satisfy the continuity equation: 0u vx y

Oscillating Flow past a Flat Plate• Fluid domain and Boundary Conditions

• Definition of Keulegan-Carpenter Number

• Horizontal force on Plate (Morison’s equation)/mKC U T h

2

2

2d m

m

F C cos cos C sinKhwU

2

20

3 ( )cos4d

m

FC dhwU

2

3 20

2 ( )m

m

KC F sinC dhwU

where 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)

Euler Solver Navier-Stokes Solver

• Vorticity & Streamlines at t=T/4 (KC=1)

• Vorticity & Streamlines at t=T/2 (KC=1)

Euler Solver Navier-Stokes Solver

• Vorticity & Streamlines at t=3T/4 (KC=1)

Movie (KC=1)

Movie(KC=10

)

• Comparison of force over one period (KC=1)

FPSO Hull Section Motion• Fluid domain and Boundary Conditions

2

* (for u),u uy g

(for p)p gvt

2

(for v), andv vy g

• Body motion (Roll motion)( ) ( )ot sin t

• Added mass and damping coefficients66

66 2ˆ (Added Mass)

4aa

b

6666 2

ˆ(Damping)

4bbbgb

2b

• Boundary Condition on Free-surface

• 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.