Vorticity effects on steady nonlinear periodic gravity-capillary waves in finite depth

Periodic gravity-capillary waves on a fluid of finite depth with constant vorticity are studied theoretically and numerically. The classical Stokes expansion method is applied to obtain the wave profile and the interior flow up to the fourth order of approximation, which thereby extends the works of Barakat & Houston (1968) J. Geophys. Res. 73 (20), 6545–6554 and Hsu et al. (2016) Proc. R. Soc. Lond. A 472 , 20160363. The classical perturbation scheme possesses singularities for certain wavenumbers, whose variations with depth are shown to be affected by the vorticity. This analysis also reveals that for any given value of the physical depth, there exists a threshold value of the vorticity above which there are no singularities in the theoretical solution. The validity of the third- and fourth-order solutions is examined by comparison with exact numerical results, which are obtained with a method based on conformal mapping and Fourier series expansions of the wave surface. The outcomes of this comparison are surprising as they report important differences in the internal flow structure, when compared with the third-order predictions, even though both approximations predict almost perfectly the phase velocity and the surface profiles. Usually, this occurs when the wavenumber is far enough from a critical value and the steepness is not too large. In these non-resonant cases, it is found that the fourth-order theory is more consistent with the exact numerical results. With negative vorticity the improvement is noticeable both beneath the crest and the trough, whereas with positive vorticity the fourth-order theory does a better job either beneath the crest or beneath the trough, depending of the type of the wave.