Although the form and dimensions of steep vortex ripples are well studied in relation to the oscillating flow which generates them, nevertheless the accompanying fluid motion is not yet understood quantitatively. In this paper we present a method of calculation based on the assumption that the sand-water interface is fixed and that the effect of sand in suspension is, to a first approximation, negligible.
The method employs a simple conformal transformation of the fluid flow onto the exterior of a polygon, and thence onto the interior of a unit circle. The initial, irrotational flow is represented by a logarithmic vortex at the centre of the circle. Other vortices within the fluid are each represented by a symmetric system of P vortices and their images in the unit circle, P being the number of sides of the original polygon. Typically P is equal to 5. However, P is not limited to integer values but may be any rational number greater than 2 (see § 15).
To proceed with the calculation it is assumed that separation of the boundary layer takes place at the sharp crests of the ripples, and that the shed vorticity can be represented by discrete vortices, with strengths given by Prandtl's rule. (For a typical time sequence see figures 7 and 8.) After a complete cycle, a vortex pair is formed, which can escape upwards from the neighbourhood of the boundary.
The total momentum per ripple wavelength and the horizontal force on the bottom are expressible very simply in terms of the shed vortices at any instant. The force consists of two parts: an added-mass term which dissipates no energy, and a ‘vortex drag’, which extracts energy from the oscillating flow.
The calculation is at first carried out with point vortices, in a virtually inviscid theory. However, it is found appropriate to assume that each vortex has a solid core whose radius expands with time like [ε(t − tn)]½, where tn denotes the time of birth, and ε is a small parameter analogous to a viscosity. The expansion of the vortex tends to reduce the total energy (which otherwise would increase without limit) at a rate independent of ε. If the cores of two neighbouring vortices overlap they are assumed to merge, by certain simple rules.
Calculation of the effective vortex drag in an oscillating flow yields drag coefficients $\overline{C}_D$ of the order of 10−1, in good agreement with the measurements of Bagnold (1946) and of Carstens, Nielson & Altinbilek (1969). The tendency for the highest drag coefficients to occur when the ratio 2a/L of the total horizontal excursion of the particles to the ripple length is about 1·5 is confirmed. When 2a/L = 4, the drag falls to about half its value at ‘resonance’.