We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable $w$ into the fluid’s area, with the real line of $w$ mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov ( J. Appl. Mech. Tech. Phys. , vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to the Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identify powerful reductions that allow one to find general classes of particular solutions.
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Numerical algorithms for water waves with background flow over obstacles and topography
We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).
more » « less- NSF-PAR ID:
- 10369687
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Advances in Computational Mathematics
- Volume:
- 48
- Issue:
- 4
- ISSN:
- 1019-7168
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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