Investigation of depth effects on the wave exciting low frequency drift forces by different approximation methods

• Slow drift forces were identified by model tests in bichromatic and irregular waves.
• Second order potentials can be neglected for surge slow drift force in deep water.
• Second order potentials are important for numerical predictions in shallow water.
• In practice, second order potential effects are dominated by incident wave potential.
• Scattering second order potentials are important only for larger difference frequencies.

The paper presents an experimental and numerical investigation on the slowly varying wave exciting drift forces acting on a body of simple geometry subjected to bichromatic waves and long crested irregular seas. The body is axis-symmetric about the vertical axis, like a vertical cylinder with a rounded bottom and it is restrained from moving. Three different water depths representing shallow, intermediate and deep waters are considered.The objective is, firstly, to investigate the relation between the water depth, the incident wavelength and the resulting slowly varying drift force. The other objective is to assess the accuracy of different approximations to the solution of the second order problem. With this aim in mind, the quadratic transfer functions are calculated with a boundary element method using several different levels of approximation for the second order forces, as described in the following sentences. The most complete approximation solves the boundary value problem completely up to the second order. The first-order approximation is similar to the former; however, the second order boundary value problem is simplified by neglecting the free surface forcing. The simpler method is of zeroth-order with respect to the difference frequency and it is commonly known as Newman's approximation. A fourth approximation is evaluated, which combines Newman's approximation with a contribution from the second order incident wave potential.A fifth approximation is applied to the vertical second order forces only and it consists on, first, calculate the steady vertical second order force in monochromatic waves corrected by an additional setdown, and second, apply Newman's approximation together with the corrected second order steady vertical forces. The additional setdown is derived from the second order incident bichromatic wave potential as the difference frequency tends to zero.Second order WAMIT code is used for the frequency domain hydrodynamic calculations. The numerical results are compared with experimental data in bichromatic waves and in irregular waves.This study shows that the contribution from the second order velocity potential must be considered for shallow waters calculations to achieve accurate results. For small difference frequencies, second order scattering potential effects are small; therefore, in this case, a good practical approximation consists on considering the second order potential is contributed by the incident waves only.