Short branch cut approximation in twodimensional hydrodynamics with free surface
Abstract
A potential motion of ideal incompressible fluid with a free surface and infinite depth is considered in twodimensional geometry. A timedependent conformal mapping of the lower complex halfplane of the auxiliary complex variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid's surface. The fluid dynamics can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider the short branch cut approximation of the dynamics with the small parameter being the ratio of the length of the branch cut to the distance between its centre and the real line of w. We found that the fluid dynamics in that approximation is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including moving square root branch points and poles. These solutions involve practical initial conditions resulting in jets and overturning waves. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 May 2021
 DOI:
 10.1098/rspa.2020.0811
 arXiv:
 arXiv:2003.05085
 Bibcode:
 2021RSPSA.47700811D
 Keywords:

 Physics  Fluid Dynamics;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 24 pages, 5 figures