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Modeling of the Unsteady Separated Flow over Bilge Keels of FPSO Hulls under Heave or Roll Motions

Yi-Hsiang Yu09/23/04

Copies of movies/papers and today’s presentations may be downloaded fromHttp://cavity.ce.utexas.edu/kinnas/fpso/

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Introduction FPSOs are tanker like floati

ng hulls which are used for production, storage and offloading of oil.

FPSO hulls have often been found to be subject to

The main focus of this research is to model the unsteady separated viscous flow over the bilge keels of a FPSO hull subject to roll motions and to determine its effect on hull forces.

excessive roll motions, and the installation of bilge keels has been widely used as an effective and economic way of mitigating the roll motions of hulls.

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Overview of the Presentation Numerical Formulations

Governing Equations Numerical Methods Non-linear Term Treatment The Effect of the Moving Grid

Results An Oscillating Flow over a Vertical Plate Submerged Body Undergoes Heave or Roll motions FPSO Hull Subjects to Roll Motions

Conclusions and Future Work

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Numerical Formulation Governing Equation Non-Dimensional Governing Equation (Navier-Stokes Equation & Continuity Equa

tion)

where U represents the velocity; Q is the force term; and R indicates the viscous term. The definitions of the column matrices for the Navier-Stokes equation are given as

where the Reynolds number is define as Re = Umh/ν ; and the length scale, h, is a representative length in the problem being solved.

( ) , 0U

UU Q R Ut

v

uR

fy

p

fx

p

Qv

uU

y

x

2

2

Re

1,,

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Cell Based Finite Volume Method (Collocated variable, non-staggered grid arrangement) According to the integral formulation of the Navier-Stokes equation

and to the Gauss divergence theorem, a semi-discrete integral formulation of the momentum equation can be given as

where Sij is the area of the cell; and ds represents the length of each cell face. A cell center based scheme is applied where “i,j” is the center of the cell (non-staggered grid: unknown value u, v, p are located at the cell center).

when calculating the flux, the value on the cell face (at D) is needed. It can be obtained from Taylor series expansion.

, , ,

1( )

Rei j n i j i jfaces

U US UV ds QS

t n

U, V, P

Cell Based

i,j

1 4

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l

m

j

k

where 1,2,3,4 represent the node point; represent the face number of the cell.

j k l m

A B

C

D

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Crank-Nicolson Method for Time Marching

where f represents the summation of the convective terms, the viscous terms and the pressure terms at the present time step n and the next time step n+1.

Pressure-correction Method SIMPLE method (Patankar 1980)

where p’ is the pressure correction, V’face is the velocity correction term, әp’ /әn is the pressure correction derivative with respect to the normal direction of the cell face, V*face = (u*; v*) is the predicted velocity vector obtained from the momentum equation.

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, , 2

n nn ni j i j

f fU U t

*

* *

*

,

'

'

face face face face

faceface face

p p p

pV V V V t

np

t ds V dsn

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Appropriate Pressure-Correction EquationSince our 2-D Navier-Stokes solver uses non-staggered grid, the scheme has to be modified somewhat to avoid the checkerboard oscillation problem.

where aij is the coefficient of the unknown velocity in the momentum equation; "av" indicates the average value obtained from the cell center value; and "d" represents the value calculated directly at the face center.

Non-linear Terms TreatmentThe momentum equation can be rearranged as

where u* is the unknown velocity at T = n + 1; the coefficient “a” is also a function of the velocity at T = n+1 which can be obtained from the previous iteration; and dij is the coefficient of pressure in the momentum equation.

* * 1 1( )face av

dij ijav

P PV V t ds

a n a n

* *, , ,i j i j nb i j cnb

a u a u d p ds

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When the grid is moving,

additional terms need to

be taken into account.

where (ugrid, vgrid) is the velocity of the moving grid; and

represents the total change in the value of u with both increment in time and the corresponding change in the location of the point. When the above equation is substituted into the momentum equation

( ) ( )grid grid

u u u uu v

t t x y

/u t

( ) ( ) ( )grid grid

U U Uu v UU Q R

t x y

Moving Grid

gridu

gridv

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Problem Description The main focus of this research is to model the unsteady separated

viscous flow over the bilge keels of a FPSO hull subject to roll motions and to determine its effect on hull forces.

It can be simplified as three different problems: Oscillating flow over a vertical plate. Free surface effect (linear and non-linear). Submerged body with

or without the bilge keels. Consider the effect of the

bilge keels and the effect of

the free surface

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Results Oscillating Flow Past a 2-D

Vertical Plate

GRID FOR OSCILLATING FLOW PAST A FLAT PLATE

Top

Bottom

Inflow Outflow

Plate

Direction of oscillating flow

Grid size: 201x101Domain size: -8 to 8 in x-direction and 0 to 8 in y-direction

0, 0, 0u p

vy y

0, 0, 0u p

vy y

2

2

cos( )

0, 0

mu U t

pv

x

2

2

cos( )

0, 0

mu U t

pv

x

0, 0, 0v p

ux x

Um, T: Amplitude, and period of oscillating flow

h: height of the plate

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Drag & Inertia Coefficient for a Range of

Kc=UmT/h (0.5 < Kc < 5)

2

20

ˆ3 ( )cos

4dm

FC d Drag coefficient

hwU

2

3 20

ˆ2 ( )m

m

KC F sinC d Inertia coefficient

hwU

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X

Y

-0.5 0 0.5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The grid of the cylinder

X

Y

-0.5 0 0.5-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The grid of the hull without bilge keels

X

Y

-0.5 0 0.5-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The grid of the hull with bilge keels

Submerged Body Motions

/ / ( )( / ) ( )( / )n n n S S nP n U t U Ugrid U n U Ugrid U S

:

: ,

Invicid Free slip boundary condition

Viscous U V velocity of the moving body

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Arclength

Pre

ssu

re

1 2 3-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25FVM (Invicid)BEM (Potential)

The pressure distribution along the cylinder

Arclength along the submerged hull

Pre

ssu

re

1 2 3

-2

-1.5

-1

-0.5

0

0.5

BEMFVM

The pressure distribution along the hull

Potential Flow Results of the Hull Undergoing the Heave Motion at t/T=0.25

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The pressure distribution along the submerged hull without bilge keels

Potential Flow Results of the Hull Subject to the Roll Motion

FVM

FVM

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Viscous Flow Results of the Submerged Hull with Bilge Keels Subject to the Roll Motion

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Results of the Unsteady Separated Viscous Flow over the Bilge Keels of a FPSO Hull Subject to Roll Motions

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Previous Results

More details can be found in Kacham [2004], and Kakar [2002]

X

Y

-10 -5 0 5 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

The wave profile at t/T=3.6 for FPSO hull undergoing roll motion

XY

-10 -5 0 5 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

The wave profile at t/T=3.75 for FPSO hull undergoing roll motion

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Conclusion A numerical scheme for solving the Navier-Stokes equati

on has been developed. It is well validated with experimental results in the case o

f an oscillating flow past a vertical plate. The method is also applied to the case of submerged bo

dies which are subject to forced heave or roll motions. The numerical results have shown good agreement with the potential solver (boundary element method).

Then, the method is applied to the case of a FPSO hull undergoing roll motions. The effects of the bilge keels and of the free surface are also taken into account. The numerical results is improved after including the terms for a moving grid in a fixed inertial coordinate system.

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Future Work More convergence studies in time and space are

necessary. The capability of the solver to handle the non-orthogonal

grid geometry still needs to be improved (Some small oscillating behaviors exist around the bilge keels area).

More investigations on the nonlinear free surface effect are needed.

Extend the model in 3-D and compare with experiments and other numerical results.