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

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

• Yi-Hsiang Yu
• 09/23/04
• Copies of movies/papers and todays presentations
  may be downloaded from
• Http//cavity.ce.utexas.edu/kinnas/fpso/

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Introduction

• FPSOs are tanker like floating hulls which are
  used for production, storage and offloading of
  oil.

• FPSO hulls have often been found to be subject to

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.

• 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.

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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 Equation)
• 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/? and the length scale, h, is a
  representative length in the problem being
  solved.

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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.

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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 n1.
• Pressure-correction Method
• SIMPLE method (Patankar 1980)
• where p is the pressure correction, Vface is
  the velocity correction term, ?p /?n is the
  pressure correction derivative with respect to
  the normal direction of the cell face, Vface
  (u v) is the predicted velocity vector
  obtained from the momentum equation.

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• Appropriate Pressure-Correction Equation
• Since 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 Treatment
• The 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 n1 which can be obtained from
  the previous iteration and dij is the
  coefficient of pressure in the momentum equation.

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• Moving Grid

• 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

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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

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• Drag Inertia Coefficient for a Range of
• KcUmT/h (0.5 lt Kc lt 5)

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• Submerged Body Motions

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• Potential Flow Results of the Hull Undergoing the
  Heave Motion at t/T0.25

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• Potential Flow Results of the Hull Subject to the
  Roll Motion

The pressure distribution along the submerged
hull without bilge keels
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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

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Conclusion

• A numerical scheme for solving the Navier-Stokes
  equation has been developed.
• It is well validated with experimental results in
  the case of an oscillating flow past a vertical
  plate.
• The method is also applied to the case of
  submerged bodies 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.