Page 447 - Practical Design Ships and Floating Structures

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Good hydrodynamic characteristics in severe sea environments, adequate storage capability and
possibly the low prices of tanker hulls in the ship market are the main reasons to justify the increasing
popularity of tanker hull based production systems (Floating Storage and OMoading - FSO and
Floating Production Storage and OMoading - FPSO) among offshore oil producer companies.
The complete assessment of the dynamic behavior of moored tankers depends very much on the
accuracy of the hydrodynamic loading and response evaluation that need be performed. Potential and
viscous effects on the FSORPSO come into play equally important role on the acting flow around the
ship hull. Furthermore, translational and rotational motions of the hull have to be incorporated all
together into the analysis to produce a realistic picture of the physical problem.

Recently, Computational Fluid Dynamics (CFD) has been experiencing rapid advances due to both
computer technology progress and efficient algorithms that have been developed to solve the
Reynolds-averaged Navier-Stokes (RANS) equations used in the flow anaIysis around ship hulls,
Ratcliffe (1998). The present work is a contribution to the numerical solution of the viscous flow
around sfowly rotating ship-like bodies in the presence of currents.
To tackle such a complex, robust numerical problem one has to search for a correct balance between
accuracy and efficiency of the solution. The strategy is therefore to find a fair compromise between
accuracy and computing time. Here, the authors adopted an algorithm specially tailored, and therefore
unique, to perform an efficient solution in the usually time consuming three-dimensional
incompressible viscous flow around arbitrary shapes.

In the present solution, the complete incompressible Navier-Stokes equations will be solved through a
Finite Difference based-solver using generalized coordinates defined on a moving grid. The algorithm
used to solve the set of incompressible flow equations was recently developed by Wanderley (2001). It
assumes that water behaves as a slightly compressible fluid enough to provide the convenient coupling
of the Continuity equation. The numerical solution benefits then from the coupling but avoids the need
of any extra energy equation. Such a feature enables saving a lot of processing time, at the same time
that does not imply in any sensitive burden on accuracy or stability of the solution. The constitutive
equations will be discretised in space by second order central differences. Euler Explicit method
performs the time-marching and the Successive Over Relaxation method solves the Poisson Equation
at each iteration to calculate pressure distribution.

2 MATEHEMATICAL FORMULATION
2.1 Governing Equations

The complete discussion of the mathematical formulation of the slightly compressible approach to
solve incompressible flow are presented in detail in Wanderley (2001). The basic idea behind the
method was to include the time derivative term in the Continuity equation of the incompressible flow.
But now, by introducing a new parameter into the flow equations based on the proper compressibility
factor of the fluid, it was possible to avoid the need of any extra energy equation. The main benefits
from that was to speed up enormously the convergence rate of the numerical solution without any loss
of accuracy or numerical stability.
The 3-D slightly compressible N-S equations are written below in conservative form, in general
curvilinear coordinates (5. q, <) and in the dimensionless form.

Q, + E4 + F, + G, -+ S = 0 (1)