Trajectories of fluid particles in a periodic water wave

Abstract

We compute trajectories of fluid particles in a water wave that propagates with a constant shape at a constant speed. The Stokes drift, which asserts that fluid particles are pushed forward by a wave, is proved using a new method. Numerical examples with various gravity and surface tension coefficients are presented.

New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

“…One should then use instead a parametric representation of the free surface, such as the arc length coordinate. The corresponding integral relations and equations can be easily obtained from (25), (30) and (31). These elementary derivations are left to the reader.…”

“…one obtains two conjugate nonlinear singular integro-differential equations for η. Either equation (30) or equation (31) can be used to compute the solution, the other one can be used to check the accuracy of the computed approximation. For pure gravity waves (τ = 0), the imaginary part (31) of (28) becomes…”

“…Here, c is Stokes' first definition of wave celerity [23,25] and comparisons with another phase velocity can be found in [26]. Note also that this frame issue also appears when investigating, for example, fluid particle trajectories [27][28][29] because the latter are different in different frames (i.e. trajectories are not Galilean invariant).…”

Capturing the flow beneath water waves

“…The flow beneath water waves can be captured by different numerical strategies, as outlined above. The following articles [7][8][9][10][11][12][13][14][15][16] are of interest to the problems discussed here. As will be presented herein, we adopt a boundary integral formulation (through Green's third identity) as well as conformal mappings, in order to compute our features of interest with great precision.…”

Nonlinear water waves

“…On the other hand, Clamond [16] formulates on the basis of numerical simulations some conjectures that refine the present understanding of the velocity field beneath a travelling water wave in irrotational flow. Chen et al [17] and Okamoto & Shōji [18] also contribute to the study of particle paths, the first being concerned with numerical and experimental studies, and the second with analytical and numerical approaches.…”