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of a mean component and a smaller wave induced component varying in time, and thus the strengths of the shed vortices vary with time, resulting in a (weak) periodic spatial variation of vortex strengths away from the body, (b) that the vortices in the far wake are subject to a ‘concertina-like' effect in which the wave-induced velocities cause a periodic dilation and extension of the vortex spacing in a sense parallel to the wave direction. The former mechanism (a) is dependent on the wave amplitude and frequency and is approximately independent of the wavelength of the wave, whilst the wavelength is important in the latter mechanism (b). The waves considered in this work have wavelengths many times that of the body size (as is generally the case of towed bodies in the open sea), and thus are not subject to significant diffraction effects around the body.

It is a well known that the idealisation of a vortex street to an infinite system of point vortices is only neutrally stable—that is to say, an infinitesimal disturbance introduced into the system will grow infinitely slowly (Lamb (1932)), and moreover there is only a single configuration of horizontal to vertical vortex spacing for which this neutrally stable configuration holds; all other spacing ratios are unstable. For an assumed two-dimensional flow and an infinite periodic point vortex model, the imposition of an additional two-dimensional approximation to simulate the effect of the wave motion on the point vortices as described by mechanisms (a) and (b) leads to a degeneration or breaking-up of the wake structure (Graham and Arkell (1992)), characterised by a process of vortex pair dissolution, in which vortex pairs are seen to eject from the central wake.

In reality a surface gravity wave induces orbital velocities which decay exponentially with depth in deep water, but are in phase in the plane normal to the wave propagation direction. Therefore a towed vertical cylinder in waves experiences a spanwise variation in velocity which consists of both horizontal and vertical components and leads to three-dimensional vortex shedding, vortex stretching, and inevitably a much more complicated velocity field.

Computational Method

Two flow simulation codes were used in the present work. A two-dimensional viscous vortex-in-cell code was developed using a Finite Volume approach. The method uses a streamfunction-vorticity formulation, in which vorticity is convected on moving particles (vortices) and diffusion is carried out on the mesh (see Graham and Arkell (1992)). A three-dimensional vortex code (FEMVOR) has also been developed in order specifically to investigate some of the three-dimensional effects. FEMVOR, as in the two-dimensional code, employs a particle representation of the vorticity field, in this case using vortons:


where κα is the strength of the αth vor ton, and δ the Dirac delta function. In the present method the velocity field is calculated on a fixed mesh by projecting the volume-integrated vorticity (vortonicity) carried by the moving vortons onto that mesh. Integration of the field equations is performed over fully unstructured tetrahedral-element meshes using a piecewise linear Galerkin Finite Element method; this allows for compact coding and efficient spatial representation. The mesh generator was the FELISA system (Peraire, Morgan and Peiro (1990)), which is an Advancing Front type and is highly efficient for complex geometries and allows the user to modify mesh densities through application of point, line and triangular source distributions.

Vortonicity is interpolated from the particles onto the mesh nodes, and division of this quantity by the node control volume yields values of vorticity on the mesh. Changes to the vorticity field which arise through diffusion and stretching are re-interpolated back onto the particles at the end of a time step. New particles are created at mesh nodes if the vorticity magnitude there is finite at the end of a time step and no particles originally contributed to that node. New particles therefore materialise at a diffusion front. The method developed here is a hybrid one—similar to the two-dimensional code —in which only the convection is done by tracking particles, all other processes being implemented

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