1. Modeling of Oscillating Flow Past a Vertical Plate Spyros A. Kinnas, Yi-Hsiang Yu, Hanseong Lee, Karan Kakar Ocean Engineering Group, Department of Civil Engineering The University of Texas at Austin, Austin, TX, USA The 13th International Offshore and Polar Engineering Conference ISOPE-2003, Hawaii, May25-30, 2003

2. Overview Introduction Governing Equations, Boundary Conditions and Numerical Formulation. Comparison of results from the 2D unsteady Euler, Navier-Stokes Solvers with measurements. Visualization of separated flow field. Summary and future work.

3. Introduction Motivation & Objectives  Develop and validate a method for the prediction of the effects of bilge keels on FPSO Hull motion.  Understand the physics of the unsteady separated flow about the bilge keels of a hull subject to roll motions, and predict the corresponding hydrodynamic coefficients.

4. Motivation & Objectives In this work, we apply and validate our model to a bilge keel (vertical plate) subject to horizontal sinusoidal inflow.

5. Literature Review Numerical Work: - - Gentaz et al. 1997 - - Korpus and Falzarano 1997 - - Vassalos et al. 2000 - - Yeung et al. 1992, 1996, 1998, 2000, 2002 Experiments on Vertical Plate: -Keulegan and Carpenter 1958 -Sarpkaya and O’Keefe 1995

6. Numerical Formulation Governing Equation Non-Dimensional Navier-Stokes equations KC: Keulegan-Carpenter number, Re: Reynolds number,

8. Numerical Method Finite Volume Method Ni’s Lax-Wendroff Method for time Artificial dissipation (viscosity) ~ only in the case of Euler solver SIMPLE Method for pressure correction

9. Euler and Navier-Stokes Solver Choi & Kinnas 2000, Choi PhD thesis 2000 -Development of Euler Solver and Navier-Stokes solver. Kakar MS thesis 2002 - FPSO hull motion - Application of unsteady 2D Euler solver of flow over bilge keels for FPSO hull motion. artificial viscosity  Euler Solver ~ Applied artificial viscosity  Navier-Stokes Solver ~ Viscous terms

10. Force Calculation on the plate Morison’s equation as follows [Sarpkaya and O’Keefe (1995)]

11. Results Convergence Study At KC=1

12. Convergence Study At KC=1

13. Force History comparison

14. Drag coefficient for a range of KC (0.5 < KC < 10)

15. Inertia coefficient for a range of KC (0.5 < KC < 10)

16. Streamline & vorticity contour at 0*T/4 for KC=1 Streamline & vorticity contour at 1*T/4 for KC=1 ESNS ESNS

17. Streamline & vorticity contour at 2*T/4 for KC=1 Streamline & vorticity contour at 3*T/4 for KC=1 ES NS ESNS

18. KC=1

19. KC=10

20. Summary  A finite volume method was applied to the solution of oscillating flow around a vertical flat plate. The same plate and tunnel dimensions as in the experiment of Sarpkaya and O'Keefe have been used.  It was found, that either the Euler or the Navier- Stokes solver produced force coefficients and separated flow fields which were close to each other and also close to the measurements, especially for lower values of the Keulegan- Carpenter number.

21. Summary  For higher KC numbers, the viscous terms start to be more important, and the effects of turbulence may need to be taken into account.  The Euler solver has already been extended in the case of FPSO hull sections, with and without bilge keels, and some preliminary results have been presented in Kinnas, et al., 12th Offshore Symposium, Texas section of the SNAME, 2003.

22. Future Work Further investigations with different Re numbers and KC numbers. Continue with the extension in the case of FPSO hull sections to study the effects of bilge keels.

23. The End