3.Linear Wave

WAVE EXCITATION FORCES AND MOTIONS INTRODUCTION In this lecture, three different approaches used to predict the linear wave forces and resulting motion responses and loads on floating offshore platforms will be discussed. The results of various international co-operative studies to investigate the correlation between the different methods will be summarised. Finally the recent advance in various techniques will be highlighted. MOTION RESPONSE PREDICTIONS USING THE MORISON APPROACH Since the early 1970s various investigators have adopted the Morison approach to predict wave exciting forces and the resulting responses and loads on various types of floating and compliant offshore platforms. The features common in the linear analysis tools developed and described in References 1, 2 and 3 can be summarised as follows: a) Airy wave theory is adopted. Amplitudes of wave and platform motions are assumed to be small. This assumption permits the linear superposition of the wave forces acting on the restrained structure due to the wave particle motions and hydrodynamic forces acting on the structure due to rigid-body oscillations of the platform in calm water. b) The platform can be divided into several volume elements. If the sectional dimensions of these elements are less than about 1/5th of the wave length, the wave and motion induced forces can be assumed to be concentrated in the centre of these volume elements. If one of the dimensions of these volume elements is large compared to the wave length, the two or three dimensional source distribution methods which are discussed in the next sections should be adopted. c) The wave and motion induced forces are calculated on each volume element, assuming that the rest of the structure is no present. In other words, the interference between the elements of the structure is not taken into account. The total force acting on the structure is obtained by summing the forces on each volume element. d) Hydrodynamic forces due to rigid body velocity are linearised and the wave forces due to wave particle velocities are neglected. In the following a brief summary of the Morison approach and the application of this method to predict wave exciting forces and response will be discussed. WAVE INDUCED FORCES The wave induced force on each member of an offshore structure is assumed to consist of the following force components: WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 2 a) Dynamic Pressure Force: This force is due to the hydrodynamic pressure change below the surface of a wave train while the wave is proceeding. It is assumed that the presence of the member does not interfere with the flow field. The pressure forces can be calculated in the following form: ds n t x k e g H ds n p FkyWS) cos( 5 . 0 = = (1) where n : Unit normal vector ds : Area element of surface S When the sectional dimensions of the volume element (say height, H, and beam B) in the wave field are small compared to the wave length, i.e. H / 50 Slowly varying second- >50 >50 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 14 order forces The 20th ITTC Conference Ocean Engineering Committee undertook another comparative study by correlating the first-order wave exciting force and resulting response values as well as wave drift force and moment coefficients predicted for a semisubmersible by about twenty different organisations using their computer programs. The predictions will also be correlated with the measurements. Although the comparative studies undertaken to-date show generally poor correlations when the results are closely examined one may that the spread in results from different computer programs based on identical hydrodynamic theory was just as large as that from programs based on different theories. This may lead us to conclude that the problem of disagreement in the first-order motion response predictions reduces to a problem of validation of computer software and accurate modelling rather than inaccuracy of the underlying hydrodynamic theory. ADVANCES IN FIRST ORDER WAVE FORCE AND RESPONSE CALCULATIONS ZERO SPEED UNSTEADY FLOW PROBLEM: This problem has almost reached the most advanced state in its reduction to a computationally efficient and reliable form. Therefore it is of interest for direct application to vessels without forward speed, but also as a model to provide guidance for the more difficult cases involving steady and unsteady forward motion. A substantial number of computer programs have been developed to solve this problem for arbitrary three dimensional floating or submerged bodies, in water which is infinitely deep or of constant depth. These programs were widely used in the design of offshore platforms. However the limitations of the first generation programs were apparent when various comparative studies were carried out (Refs. 10,11) In order to improve this unsatisfactory situation the second generation panel programs have been developed with various numerical refinements. These include fast algorithms for evaluating the free surface Green function with controlled accuracy, and the use of an iterative solver for the linear system of algebraic equations leading to the value of the unknown potential on each panel. The use of these programs have made it possible to increase the maximum number of panels from a few hundred to several thousand, and to establish convergence of the solution as this number is increased. Comparisons of the first-order wave and rigid-body induced forces and motion responses obtained for a deep water floater and a ship using the second-generation panel programs show significant improvements. Full details of the comparative study are given in Ref.12 THE FORWARD-SPEED STEADY PROBLEM: One of the most difficult problems in ship hydrodynamics involve steady or unsteady motion with forward speed. The difficulties are associated with the analytical formulation and the numerical problems in developing solutions. The uncertainties regarding the analytical formulation are associated with the issue of WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 15 linearization of the free-surface. Well known integral expressions exist for the steady free-surface Green function, with forward velocity included. However these functions, and presumably also the exact solution of the linearised problem, contain subtle and very complicated mathematical singularities. An approximate linearization of the free-surface condition, without restricting the geometry or Froude Number, is known as the Kelvin Neumann approach in wave resistance theory. Several computer programs have been written utilising the Kelvin Neumann approach and excluding the short-wave component of the spectrum in evaluating the Green function. However, due to numerical convergence problems the routine use of these programs is not as yet practicable. Of course this situation could change if fast algorithms are used for the Green function, and if effective numerical filtering of the short wave length singularity can be implemented in a rational manner. THE FORWARD-SPEED UNSTEADY PROBLEM: The same techniques used for the steady problem can be extended to unsteady ship motions in waves. Having developed and validated computer programs based on 3-D source distribution technique for zero forward speed Chan (Ref. 9) extended the formulation of Greens function to determine the wave loading on a marine vessel travelling with a forward speed in deep and shallow water. The Green function formulation developed by Chan represents a translating and oscillating source in infinite and finite water depths. Newman (Refs.13 and 14) and Telste and Noblesse (Ref.15) have developed algorithms for the computation of a Green function representing an oscillating source or translating source for infinite water depth. However, the oscillating source potential or translating source potential cannot satisfy the undisturbed free surface condition for the unsteady forward motion problem. Inglis and Price (Ref. 16) and Wu and Eatock Taylor (Ref. 17) have independently modified the expression for Greens function representing a translating and pulsating source which satisfies the undisturbed free-surface condition for the unsteady forward motion problem. However, these modified expressions of the Green function are only applicable to deep water. The formulations derived in Ref. 9 are also applicable to finite and infinite water depths and allow for the interaction between steady waves (Kelvin Waves) and the unsteady waves (Radiation Waves) and satisfy the undisturbed free surface conditions. WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 16 REFERENCES 1. HOOFT, J.P., A mathematical Model of Determining Hydrodynamically Induced Forces on a Semisubmersible, Trans. Of SNAME, Vol. 79, 1971. 2. PAULLING, J.R., Elastic Response of Stable Platform Structures to Wave Loading, Proc. Of the Intl. Symposium on the Dynamics of Marine Vehicles and Structures, London, 1974. 3. OO, K.M. and MILLER, N.S. Semisubmersible Design: The effect of different geometries on Heaving Response and Stability, Trans. R.I.N.A., 1977. 4. FRANK, W., Oscillation of Cylinders in or below the Free Surface of Deep Fluids, NSRDC Report 2375, 1967. 5. SALVESEN, T., TUCK, E.O. and FALTINSEN, O., Ship Motions and Sea Loads Trans. SNAME, Vol. 78, 1970. 6. KIM, C.H., CHOU, F-S and TIEN, D., Motions and Hydrodynamic Loads of a Ship Advancing in Oblique Waves, Trans. SNAME, Vol. 88, 1980. 7. GARRISON, C.J. Hydrodynamic Loading of Large Offshore Structures: Three-Dimensional Source Distribution Methods in Numerical Methods in Offshore Engineering, Ed. Zienkiewicz, Lewis, and Stagg, Wiley Series in Numerical Methods in Engineering, 1978. 8. HOGBEN, N and STANDING, R.G., Wave Loads on Large Bodies, Proc. Of Intl. Symposium on Dynamics of marine Vehicles and Structures in Waves, London, 1974. 9. CHAN, H.S., A Three Dimensional Technique for Predicting First and Second Order Hydrodynamic Forces on a Marine Vehicle Advancing in Waves, Ph.D. Thesis, University of Glasgow, 1990. 10. TAKAGI, M., ARAI, S.I., TAKEZA, S., TANAKA. K. and TAKARADA, N., A Comparison of Methods for Calculating the motion of a semisubmersible, Ocean Engineering, Vol. 12, No. 1, 1985. 11. EATOCK TAYLOR, R. and JEFFERYS, E.R., Variability of Hydrodynamic Load Predictions for a Tension Leg Platform, Ocean Engineering, Vol. 13, No. 5, 1986. 12. NIELSEN, I.G., HERFJORD, K. and LOKEN, A., Floating Production Systems in Waves: Results from a Comparative Study on Hydrodynamic Coefficients, Wave Forces and Motion Responses, Report of the Workshop on FPS200, Bergen, Norway, 1989. WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 17 13. NEWMAN, J.N., Evaluation of the Wave Resistance Green Function: Part 1 The Double Integral, Journal of Ship Research, Vol. 13, No. 2, 1987a. 14. NEWMAN, J.N., Evaluation of the Wave Resistance Green Function: Part 2 The Single Integral on the Centreplane, Journal of Ship Engineering, Vol. 13, No. 3, 1987b. 15. TELSTE, J.G. and NOBLESSE, F., Numerical Evaluation of the Green Function of Water-Wave Radiation and Diffraction, Journal of Ship Research, Vol. 30, No. 2, 1986. 16. INGLIS, R.B. and PRICE, W.G., Calculation of the Velocity Potential of a Translating, Pulsating Source, Trans. RINA, Vol. 123, pp 163-175, 1981. 17. WU, E.X. and EATOCK TAYLOR R., A Greens Function Form for Ship Motions at Forward Speed, International Shipbuilding Progress, Vol. 34, 1987. 18. SARPKAYA, T. and ISAACSON, M., Mechanics of Wave Forces on Offshore Structures, Van Nostran Reinhold Company, New York, 1981. 19. HALLAM, M.G., HEAF, N.J and WOOTTON, L.R., Dynamics of Marine Structures: Methods of calculating the dynamic response of fixed structures subject to wave and current action, CIRIA, Underwater Engineering Group, London, 1978. WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 18 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 19 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 20 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 21 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 22 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 23 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 24 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 25 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 26 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 27 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 28 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 29 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 30 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 31 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 32 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 33 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 34 WAVE EXCITATION FORCES AND MOTIONS - A. INCECIK 35