Numerical Methods for Seakeeping Problems

Frontmatter

Chapter 1. Introduction

Abstract

This chapter shows that total losses of seagoing ships decrease worldwide at an impressive rate: from about 150 to roughly 50 per year within the last 15 years. However, the share of losses caused by adverse weather conditions rises slowly and amounts now to about 50%, thus showing the relevance of this book’s topic.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 2. Fundamental Governing Equations

Abstract

In this chapter, the conservation equations for mass and momentum are derived to describe the flow of Newtonian fluids. Subsequently, the basic equations for incompressible potential flows, namely, the continuity equation (in the form of the Laplace equation) and the Bernoulli equation, are described. Finally, the equations of motion for rigid bodies are derived.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 3. Numerical Methods to Compute Incompressible Potential Flows

Abstract

This chapter illustrates the principles of computing potential flows in an unbounded fluid (without a free surface). It applies a strongly simplified source-sink method to the non-lifting flow around a body of arbitrary shape, and the steady lifting flow around an airfoil, both in two space dimensions. In the latter example, the patch method is explained: a variant of the usual panel method, if uses aims simpler formulas and yields better accuracy of the pressure force.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 4. Water Waves

Abstract

This chapter starts with the potential of linear regular waves in deep water, proceeds to nonlinear Stokes waves and continues with periodic long-crested steep waves of arbitrary order for deep and shallow water. Then methods to simulate a natural (irregular) seaway are discussed, using both linear and second-order superposition of regular waves; a higher order method is only cited. Typical wind wave spectra are described, and diagrams and formulas are given to approximate their parameters from wind speed, fetch, and duration. Finally, an example of a scatter table (probability distribution of significant wave period and height) taken from the modern ‘wave atlas’ is given.

Bettar O. el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 5. Strip Methods

Abstract

This chapter describes the traditional method for seakeeping analyses. Seventy years after the pioneering work of Ursell to determine added mass and damping of ship sections oscillating at the water surface, this book describes a highly accurate, mathematically simple, and computationally fast method for this. For the sake of brevity, the step from ship sections in forced motion to free motions of a total ship with speed ahead is made by heuristic considerations. A special feature of this chapter is the speed-dependent hydrodynamic interaction between hulls, which is important for catamarans and trimarans.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 6. Green Function Methods

Abstract

This chapter ‘Green function methods’ and all following chapters, the three-dimensional flow around the ship is computed to determine forces and moments caused by fluid pressure, thus avoiding the approximations inherent in strip methods. In this chapter, central to computing this flow is the Green function: the potential of a pulsating source under a free surface, at which it generates waves. If the source is not fixed, but translating relative to the fluid, numerical problems in computing this Green function are large; therefore, here this flow is approximated from that of a non-moving pulsating source. The sources are continuously distributed over ‘panels’ covering the underwater ship hull. In each panel, the source density is determined such that there is no flow through the ship hull at all panel centers. The chapter describes also how to take account of the nonlinear effect of fluid pressure between the instantaneous and the average waterline.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 7. Linear Rankine Source Methods

Abstract

This chapter differs from the previous chapter by using sources in the unbounded fluid as Green functions. Thus, also the free surface (including ship waves) around the hull must be covered by source panels. The described method includes the interaction of oscillatory and stationary potentials; the latter is assumed stationary not in inertial, but in ship-fixed coordinates for higher accuracy. Also, the method to satisfy the radiation conditions (no disturbance from outside, no wave reflections) is important for accuracy: For the stationary flow, a back-shift of sources by 1 panel length is used, whereas for the oscillatory flow damping is applied everywhere on the free surface.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 8. Nonlinear Rankine Panel Methods

Abstract

This chapter describes a new, fully nonlinear simulation method for ship motions and loads based on potential theory. On the hull surface, it uses a panel mesh fixed to the body, whereas on the free surface the mesh is generated a new for each time step. Special account is taken of partly submerged hull panels. To improve the accuracy, the panel method determines not only the flow potential, but also its time derivative from derivatives of the boundary conditions. The radiation conditions for the disturbance potential are satisfied using a combination of the Dawson operator for lengthwise derivatives and wave damping, leaving the potential of the incoming waves unaffected. The chapter comprises also a comparison of motions and loads in steep head and quartering waves computed by this method with results of model experiments and RANS calculations.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 9. Viscous Field Methods

Abstract

In this chapter, we briefly describe numerical approaches, based on computational fluid dynamics (CFD), developed to compute wave-induced ship motions. We do not claim completeness; instead, this chapter describes examples of numerical techniques commonly employed in this field. The listed references may be of help to the reader wishing to obtain additional details. Further on, we briefly discuss numerical errors and, specifically, procedures available to estimate discretization errors. Finally, we present an example of applying CFD to predict ship motions and loads in extreme seas.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 10. Wave-Induced Hull Vibrations

Abstract

In this chapter, we describe modern numerical methods used to compute global hull girder vibrations. We also discuss the causes and effects of such vibrations and the importance of the associated damping. In addition, we present sample applications.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 11. Additional Forces and Moments

Abstract

Whereas previous chapters dealt with the water flow and the corresponding pressure force and moment on the hull, this chapter describes approximations of other influences. Especially the small moments in excitation, restoring, and damping of roll motions require taking account of additional influences, e.g., fins, of the lift effect of hull and rudder in oblique flow, of the roll damping by bilge keels, and of the control forces required to keep the ship’s intended speed and course both in reality and during simulations. Also, recommendations are derived for the required control forces to attain stable motions not overly influencing the free motions. Often neglected effects are included, e.g., the influence of the stern wave on metacentric height, which may amount to about 1 m at 15 knots speed.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 12. Special Topics

Abstract

This chapter deals with the influence of sails and suspended loads on roll motions and how to take account of forces and moments generated by roll damping tanks. Also, the dangerous wave-riding condition is discussed, in which fast ships may be accelerated by steep following waves up to the wave celerity. The chapter explains also how motion restraints, for instance, in case of contact of a ship with a fixed obstacle, can be handled in computations.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 13. Further Transfer Functions

Abstract

This chapter explains how transfer functions (i.e., linear responses to regular waves) for hull pressure oscillations can be determined in strip methods more accurately than using the standard procedure, i.e., by eliminating second derivatives (‘m terms’) of the steady potential. For multi-body vessels (catamarans, etc.), a pressure formula is given that accounts also for the interaction of waves between the hulls. Furthermore, the chapter deals with computing relative motions between water (especially, the water surface) and ship-fixed points, including the disturbance of incoming waves by the ship; with forces and moments in cross sections of the hull girder; and with the wave- and motion-induced vertical water motion in a moonpool (a vertical channel in certain marine structures).

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 14. Drift Force and Added Resistance

Abstract

This chapter deals with the stationary second-order force exerted on the body by regular waves. Of most interest is the longitudinal force component, the added resistance. The chapter gives a rigorous derivation of the second-order force, including effects which are usually neglected in the relevant literature. Results of the added resistance determined by a Rankine source method using these formulas are compared with results of model experiments for a sharp and a full ship.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 15. Comparison Study

Abstract

A more comprehensive comparison of responses to regular waves is given in this chapter for a 6500TEU containership. It shows the results of model experiments, of a linear strip method, a Green function method including nonlinear corrections, a linear and a fully nonlinear Rankine source method, and a RANS method. Compared are ship motions as well as linear and nonlinear section forces and moments in three transverse sections. The ship speed is varied between 0 and 15 knots. Only longitudinal (head and following) waves are covered; the maximum wave height (in full scale) is 10 m. Results show that motions can be well predicted by linear three-dimensional methods, but accurate section loads in steep waves require nonlinear computing methods. Apparently, both experiments and nonlinear computational methods have difficulties with loads in steep waves. Nonetheless, carefully computed results are of sufficient accuracy for practical application.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 16. Ships in Natural Seaways

Abstract

This chapter starts with describing statistics of Gauß processes, which result from linear responses to a linearized stationary seaway. Then computing the distribution of nonlinear responses to one or several correlated Gauß processes is illustrated as an example. For more general nonlinear cases, simulation cannot be avoided, but this may require excessive time to directly count the rate with which seldomly occurring events like capsizing or extreme loads occur. A technique for reducing the effort of such computations by orders of magnitude is based on the fact that similar wave events occur in seaways of different significant height with rates differing by known factors. Finally, the superposition of probability distributions for stationary seaways to long-term distributions and the concept of design waves is described.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

Chapter 17. Miscellaneous Topics

Abstract

In this chapter ‘Miscellaneous topics’, the following is shortly dealt with

• A frequently-used method for simulating roll motions of ships, which treats surge, roll, and the coupling of roll with the other four kinds of motion nonlinearly, while these other motions themselves are linearized, using pre-calculated transfer functions. The method uses the extremely fast method by Grim to determine the heel restoring moment in waves from previously computed results.
• The above method is extended to damaged ships by simulating the in- and outflow of water through openings and the water motion within ship compartments and on deck, using special methods for (a) extremely shallow and (b) deeper flooded compartments.
• Nonlinear motions of planing boats in waves are simulated by extensions of the classical Wagner method developed already about 1930 for starting and landing sea planes.
• Mathematical quantities (‘perturbators’) and their operations are constructed to simplify both the derivation and the programming of second-order formulas, e.g., drift forces and springing excitation.
• For passenger transport, often the occurrence of sea sickness must be reduced as much as possible. An ISO norm for estimating the frequency of occurrence is sketched and evaluated for normal catamarans and those with longitudinally staggered hulls (‘weinblums’). The weinblums have not only less resistance, but are also less prone to generate sea sickness.

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding

18. Correction to: Water Waves

Bettar Ould el Moctar, Thomas E. Schellin, Heinrich Söding