underlying theory. It turned out that most methods produced acceptable results for the major part of the boundary layer along the hull, while all failed completely near the stern. This prompted several groups to start research on less approximate methods based on the Reynolds-Averaged Navier-Stokes (RANS) equations. By the end of the eighties a number of RANS methods for ship flows had been developed and in a new Gothenburg workshop (Larsson et al (8)), a considerable improvement was noted in the capability to predict the stern flow. This improvement was verified at a new workshop organized by Kodama et al (9) in Tokyo in 1994, where, however, the most important breakthrough was the introduction of the free surface in the RANS methods. No less than 10 methods now featured this capability; a result of the research in several groups since the mid-eighties, as will be further discussed below.

There is no doubt that the most important developments to be expected in the future will be in the viscous flow area. With the increasing power of computers more advanced methods will become available, based either on the Large Eddy Simulation (LES) technique, where the large turbulent eddies are computed and only the smaller ones modelled, or on the DNS technique, where all eddies and scales are computed.

In the following, we will first give a brief theoretical introduction to potential flow panel methods and methods of the RANS type. This will be in Section 2. Thereafter, in Section 3 an assessment will be made of current CFD capabilities. Ongoing research and short term prospects for improvements will be addressed in Section 4, and in Section 5 a forecast will be made of the exciting long term possibilities. Finally, a number of conclusions will be drawn. The emphasis will be on resistance and powering, but some reference will be made to research in other areas of ship hydrodynamics as well.

2. Theory

In this Section a short theoretical introduction will be given, covering potential flow panel methods and Reynolds-Averaged Navier-Stokes methods. For obvious reasons the description has to be brief. This holds particularly for the RANS part, which will be further discussed in later Sections.

2.1 Potential Flow Panel Methods

The flow is assumed steady, irrotational and incompressible. A Cartesian coordinate system is employed, where the origin is at the bow and at the undisturbed free surface level, x downstream, y to starboard and z upwards. U and q represent the undisturbed and disturbed velocities, respectively. Defining a velocity potential by


this potential will satisfy the Laplace equation


On the hull boundary the normal velocity is zero


and on the free surface boundary a similar relation holds. This kinematic condition may be written as


where h=h(x,y) is the equation for the wavy surface.

A dynamic free surface condition may be obtained from the continuity of stresses across the interface. In a potential flow, this condition degenerates to the simple statement that the pressure must be atmospheric at the surface, and without loss of generality this pressure may be set to zero. Neglecting surface tension and applying the Bernoulli equation the dynamic free surface boundary condition may be written


At infinity the velocity is undisturbed


g is the acceleration of gravity and R is the distance from the origin.

Finally, the radiation condition states that no below, this condition is enforced numerically. waves may be transmitted upstream. As explained

The free surface boundary conditions are nonlinear and they have to be applied at an initially unknown surface. This may be accomplished as follows. Assume that an approximate solution is known. Introduce small disturbances and δh and expand equations (4) and (5) in these quantities around the known solution. Delete terms of higher order. This yields, (see Janson (4))


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