The normal way to calculate wave-induced motions and loads on ships and large volume structures is to use potential theory Free surface effects are included, but vortex shedding is usually neglected in theoretical procedures This is generally an appropriate assumption. An exception is resonance roll motion of ships and barges Eddymaking damping is then of equal importance as the wave radiation damping. In practical calculations viscous roll damping is accounted for by empirical formulae.

This chapter presents a rational theoretical approach that includes both the effect of free surface waves and vortex shedding Forced harmonic roll oscillation of an infinitely long horizontal cylinder will be studied The method can in principle be applied to any forced body mode The body has to have sharp corners so that the separation points are well defined No boundary layer calculation is then needed A fundamental assumption of the method is that vorticity is concentrated in thin free shear layers. The theory is based on a time-step integration method where we in each time-step have to solve a potential flow boundary value problem outside the thin free shear layers.

Two different cases are treated

• Rigid free surface

• Free surface waves generated by the body are allowed to propagate

When the free surface is rigid, an "image-body" moving with opposite phase and with the same amplitude as the real body has to be introduced At each tune-step we have to solve a Fredholm's integral equation of the second kind By dividing both the body surface and the free shear layers into line-elements with dipole and source distribution over the body and dipole distribution over the free shear layers, the integral equation may be represented by a linear equation system where the unknowns are the fluid velocity potentials at the body element midpoints.

In the case of a moving free surface we also have to represent the part of the free surface close to the body by linear elements In the far field the influence of the body is represented by the sum of a source and a horizontal dipole satisfying the linearized time-dependent free surface condition. The problem is treated as an initial value problem with the velocity potential on the free surface given at each time-step, and the fluid particle velocity on the free surface as the additional unknown variable in our problem

The equivalence to the Kutta condition is that we require the potential lump at the separation points to be continuous and that vorticity is shed parallel to the body surface on one of the sides of the free shear layers at the separation points. To start up the time simulation a discrete vortex with a given position and circulation is introduced into the fluid in the first time-step. The position and strength of this vortex are calculated on the basis of a pure potential theory calculation as described by Rott (1956) and Pullin (1978).

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