O Bokhove University of Leeds
Friday 8th August 2014 - 10:00 to 12:30
1. Some variational mechanics will be introduced first before embarking on formulating continuum models for water waves in fluids. A forced-dissipative nonlinear oscillator will be used as an example, inspired by wave motion in a laboratory Hele-Shaw cell for (breaking) water waves and beach morphodynamics. Subsequently, variational principles for 3D water waves by Miles and Luke will be derived using constraints. 2. The simpler, depth-averaged, spatially 2D, shallow water analogue of Miles' variational principle will be discretized in space and subsequently in time, using space and time finite elements, to yield a compatible geometric algebraic variational principle. Likewise, Miles’ variational principle will be discretized in space and subsequently in time, using space and time finite elements, to yield a compatible geometric algebraic variational principle. The latter interim variational principle with continuous time will be essential to find the time discretization, ensuring that there is an underlying discrete boundary element structure and that the mesh movement is soundly integrated in the variational structure to ensure numerical stability. The simple oscillator will serve as an illustrative example for the discontinuous Galerkin finite element time discretization developed. The similarities between these discrete 2D shallow water and 3D water wave models at the fre e surface will be discussed. 3. The resulting discretizations will be validated against wave tank data from the Maritime Research Institute Netherlands (MARIN) and in-house Hele-Shaw wave tank experiments (shown live). 4. Time permitting, extensions of the above results will be discussed. These concern water wave models with a vertical component of vorticity, experiments of a bore-soliton rogue wave, modelling of breaking waves, or a wave-energy device using geometric rogue-wave focussing.