Variational Principles for Water Waves From the Viewpoint of a Time‐dependent Moving Mesh

Abstract

The time-dependent motion of water waves with a parametrically-defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic PDEs. The aim is to study the effect of transformation on variational principles for water waves such as Luke's Lagrangian formulation, Zakharov's Hamiltonian formulation, and the Benjamin-Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian, and some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

The stability of capillary waves on fluid sheets

“…Following Benjamin & Olver (1982) and Bridges & Donaldson (2011), we assume that we may write our system (in a frame of reference moving with the base wave) in the Hamiltonian form (6.1) with Z(ξ, t) = (x, y,x,ŷ, φ,φ) T and where J is a 6 × 6 matrix whose elements are derivatives of the components of Z ξ . Importantly, in this case (6.1) is degenerate since J is singular; in particular the kernel of J contains the vector Z ξ (Bridges & Donaldson 2011). Accordingly the associated symplectic form is degenerate and the conditions of the theory of MacKay (1987) are not satisfied.…”

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

“…Finally, to transform the dynamic boundary condition into parametric form, we use (2.3) from Bridges and Donaldson (2011) and the Cauchy-Riemann equations to write φ t (x, y, t) = φ t (µ, ν, t) − 1 J (y µ y t + x µ x t ) φ µ − 1 J (y µ x t − x µ y t ) ψ µ .…”

“…Following the approach of Bridges and Donaldson (2011) (cf. equations (A-7) and (3.8) in Bridges and Donaldson (2011)), with the details confined to appendix A, the boundary conditions become…”

“…This ensures that all of the modes for x ∈ [0, L], both symmetric and anti-symmetric, are periodic for x ∈ [0, 2L] and thus we can utilize Fourier analysis in combination with the conformal mappings (Dyachenko et al, 1996;Bridges and Donaldson, 2011;. conformal modulus which needs to be found as part of the solution procedure.…”

Liquid sloshing in a horizontally forced vessel with bottom topography

“…To numerically calculate these velocites, the partial derivatives with respect to x and y are written in terms of derivatives with respect to µ and ν via chain rule and (1.1) (Bridges and Donaldson, 2011 and y , x, Φ and Ψ are evaluated on an (N + 1) × (N + 1) grid in the interior of the computational domain via (3.29)-(3.32) respectively. The resulting derivatives of these functions with respect to µ are then calculated via central finite differences, making then accurate to O(N −2 ).…”