Excitation of trapped water waves by the forced motion of structures

Abstract

A numerical and analytical investigation is made into the response of a fluid when a two-dimensional structure is forced to move in a prescribed fashion. The structure is constructed in such a way that it supports a trapped mode at one particular frequency. The fluid motion is assumed to be small and the time-domain equations for linear water-wave theory are solved numerically. In addition, the asymptotic behaviour of the resulting velocity potential is determined analytically from the relationship between the time- and frequency-domain solutions. The trapping structure has two distinct surface-piercing elements and the trapped mode exhibits a vertical ‘pumping’ motion of the fluid between the elements. When the structure is forced to oscillate at the trapped-mode frequency an oscillation which grows in time but decays in space is observed. An oscillatory forcing at a frequency different from that of the trapped mode produces bounded oscillations at both the forcing and the trapped-mode frequency. A transient forcing also gives rise to a localized oscillation at the trapped-mode frequency which does not decay with time. Where possible, comparisons are made between the numerical and asymptotic solutions and good agreement is observed. The calculations described above are contrasted with the results from a similar forcing of a pair of semicircular cylinders which intersect the free surface at the same points as the trapping structure. For this second geometry no localized or unbounded oscillations are observed. The trapping structure is then given a sequence of perturbations which transform it into the two semicircular cylinders and the time-domain equations solved for a transient forcing of each structural geometry in the sequence. For small perturbations of the trapping structure, localized oscillations are produced which have a frequency close to that of the trapped mode but with amplitude that decays slowly with time. Estimates of the frequency and the rate of decay of the oscillation are made from the time-domain calculations. These values correspond to the real and imaginary parts of a pole in the complex force coefficient associated with a frequency-domain potential. An estimate of the position of this pole is obtained from calculations of the added mass and damping for the structure and shows good agreement with the time-domain results. Further time-domain calculations for a different trapping structure with more widely spaced elements show a number of interesting features. In particular, a transient forcing leads to persistent oscillations at two distinct frequencies, suggesting that there is either a second trapped mode, or a very lightly damped near-trapped mode. In addition a highly damped pumping mode is identified.