Stokes Waves in a Constant Vorticity Flow

Abstract

The Stokes wave problem in a constant vorticity flow is formulated via a conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newtonconjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks. (2000). Primary 76B15; Secondary 76B07, 30C30, 65T50. Mathematics Subject Classification

Stokes waves with constant vorticity: I. Numerical computation

“…Moreover, we predict that the limiting wave at the large vorticity limit is rigid body rotation of a fluid disk. Dyachenko & Hur (2018) will confirm it. 4.4.…”

“…We note that wave A closely resembles (Figure 4(b) Vanden-Broeck 1996, for instance) at the zero gravity limit. Dyachenko & Hur (2018) will study in detail the limiting wave at the end points of the zero-to-touching branches of the c = c(s) curves as the strength of positive vorticity increases unboundedly or, equivalently, as gravitational acceleration vanishes.…”

Shallow water models with constant vorticity

“…Moreover, Constantin, Strauss, and Varvaruca conjectured that either a limiting wave would exhibit stagnation with a 120 • corner, or its surface profile would overturn and intersect itself at the trough line. This will be numerically studied in [DH17].…”

“…Of course, at the most prominent stage of breaking, an element of the fluid surface becomes vertical; a portion of the surface overturns, projects forward, and forms a jet of water; see [Per83], for instance. Overturning Stokes waves with constant vorticity will be numerically studied in [DH17]. By the way, the profile of a periodic traveling wave is necessarily the graph of a single valued function in the irrotational setting.…”

Exact free surfaces in constant vorticity flows

“…Recently, Dyachenko & Hur (2019b,c) (see also Dyachenko & Hur 2019a) found numerically a family of gravity waves with constant vorticity, in the absence of the effects of surface tension, whose free surface approaches the limiting Crapper wave in an irrotational flow as gravitational acceleration vanishes. Hur & Vanden-Broeck (2020) took matters further and provided numerical and asymptotic evidence of a new exact solution.…”