Effect of vorticity on steady water waves

Abstract

Two-dimensional, finite-depth periodic steady water waves with variable vorticity ω=γ(ψ) and large amplitude a are computed for a large number of cases. In particular, the effect of a shear layer at the top, the middle or the bottom is considered. The maximum amplitude amax varies monotonically with the vorticity function γ(ċ). It is increasing if the stagnation point is at the crest, and is decreasing if the stagnation point is in the interior of the fluid or on the bottom. Relationships between the amplitude, hydraulic head, depth and mass flux are investigated.

Analyticity of periodic traveling free surface water waves with vorticity

“…For Stokes waves one can show that the maximal value of the horizontal fluid velocity in the flow is attained at the wave crest and for regular waves the wave speed exceeds this maximal value, while for the waves of greatest height these two values are equal (and consequently the wave crest is a stagnation point since the vertical fluid velocity there is zero) [1], [20], [21]. For rotational waves the existence of waves of this type is currently at the level of conjectures supported by formal considerations and numerical simulations (see the discussions in [7], [19], [15], [16]). Homogeneity (constant density) implies the equation of mass conservation…”

Regularity of Steady Periodic Capillary Water Waves with Constant Vorticity

“…There is an extensive research literature in the area of water waves with vorticity, see [2,3,10] for existence results, [14,22] for matters of uniqueness, [6][7][8]19] for symmetry results and [9,15,18] for regularity results. We would also like to mention the important numerical simulations from [13,16].…”

Extrema of the dynamic pressure in an irrotational regular wave train

“…The approach used in the present paper relies heavily on the irrotational nature of the flow, and cannot be applied to deal with flows having non-zero vorticity. Nevertheless, provided that no flow-reversal occurs, numerical simulations 9,10,15,16,26 and some analytical investigations 14 indicate that we should expect qualitatively somewhat similar results.…”