Eigenfunction Matching Method Research Articles

Propagating of Obliquely Incident, Weakly Viscous Waves over Variable Bathymetry

ABSTRACT Tsai, C.-C.; Lin, Y.-T., and Hsu, T.-W., 2016. Propagating of obliquely incident weakly viscous waves over variable bathymetry. A linear theory for weakly viscous waves propagating obliquely over variable topographies is introduced without considering the boundary layers. Using an eigenfunction-matching method, which involves including the evanescent modes that satisfy the matching conditions, the present problem can be transformed into a system of linear equations. Furthermore, by using a weak-viscosity hypothesis, the proposed method can be degenerated to the traditional eigenfunction-matching method of the potential flow if both the molecular viscosity and the bottom friction are discarded. The applicability of this model was confirmed by comparing it with available theoretical, numerical, and experimental results for both of the normal and oblique incidences.

A coupled-mode study on weakly viscous Bragg scattering of surface gravity waves

The two-dimensional scattering of weakly viscous waves over a finite region of undulating topography linking a series of small abrupt steps is studied. The free surface is assumed to be irrotational on the stationary condition. The limitation is for the case of a weak viscous wave propagating over an arbitrary topography. The viscous eigenfunction matching method (VEMM) is proposed by including the evanescent modes. And the resulted system matrix is solved efficiently by the sparse matrix solver of the Super LU. The improvement of wave reflection is obvious for waves propagating over a strongly undulating topography or a series of bars with steep side-slopes. It is noteworthy that the shift of reflection coefficient exits mainly at primary resonance and second-order resonance because of the combined effect of molecular viscosity and bottom condition. Numerical results show that the shear stress of viscous fluid leads to the energy loss. Moreover and it becomes severe for a shorter wave propagating over an undulated bottom.

Experimental and theoretical study of a cylindrical oscillating water column device with a quadratic power take-off model

The wave power extraction by a cylindrical oscillating water column (OWC) device with a quadratic power take-off (PTO) model was studied experimentally and theoretically. In the experiment, a scaled model OWC was tested in a wave flume, with an orifice being used to simulate a quadratic PTO mechanism. In the theoretical analysis, the quadratic PTO model was linearized based on Lorenz's principle of equivalent work, which allows us to perform a frequency domain analysis using an eigen-function matching method. The effects of higher harmonic components and the spatial non-uniformity of the surface velocity inside the chamber were discussed. A semi-analytical model was proposed to understand the viscous loss affecting the measured capture length. Our treatment of the quadratic PTO model was validated by comparing quasi-linear theoretical capture length and the laboratory measurement. Our results also showed that the effects of spatial non-uniformity and viscous loss could be noticeable for shorter waves.

Comparison between Consistent Coupled-Mode System and Eigenfunction Matching Method for Solving Water Wave Scattering

Both the consistent coupled-mode system (CCMS) and the eigenfunction matching method (EMM) are well-known models for simulating the propagation of small-amplitude water waves over variable bathymetry. In this study, a thorough comparison is performed through numerical experiments. For the CCMS, a bottom-sloping mode is coupled in the mildslope equation with evanescent modes, then the CCMS are discretized by the finite-element method with high-order shape functions. For the EMM, the bottom profile is approximated in terms of successive flat shelves separated by abrupt steps, and then eigensolutions on the shelves are matched by the conservation of mass and momentum. To perform error analysis, numerical solutions are compared with Roseau's analytical solution and the semi-analytical solutions of the integral equation method. Numerical results indicate that the CCMS and EMM are accurate up to six and four decimal places, respectively. On the other hand, the EMM is more efficient for short waves because multiple waves can be approximated by few shelves. In addition, improvements in their accuracy over the mild-slope system without bottomsloping mode are shown to be significant.

Analytical solution of hurricane wave forces acting on submerged bridge decks

Hurricane waves are catastrophic disasters that cause damage to coastal bridges with insufficient clearance. This study presents an analytical method to estimate the wave forces acting on a submerged bridge superstructure during a hurricane based on the potential flow approach. The boundary value problem of the submerged bridge deck under wave action is analyzed first, and the mathematical formula for the analytical solution is derived using the eigenfunction matching method. To validate the effectiveness of the analytical model, results are compared to test data from two hydrodynamic experiments on the wave action of bridge decks. Finally, the AASHTO model is discussed, and a simple method for estimating the maximum horizontal and vertical wave forces is proposed. The analysis results indicate that the proposed analytical model can be used effectively for prediction of the maximum wave forces acting on a submerged bridge superstructure during hurricane waves.

The Wiener–Hopf and residue calculus solutions for a submerged semi-infinite elastic plate

We present a solution for the interaction of normally incident linear waves with a submerged elastic plate of semi-infinite extent, where the water has finite depth. While the problem has been solved previously by the eigenfunction-matching method, the present study shows that this problem is also amenable to the more analytical, and extremely efficient, Wiener–Hopf (WH) and residue calculus (RC) methods. We also show that the WH and RC solutions are actually equivalent for problems of this type, a result which applies to many other problems in linear wave theory. (e.g., the much-studied floating elastic plate scattering problem, or acoustic wave propagation in a duct where one wall has an abrupt change in properties.) We present numerical results and a detailed convergence study, and discuss as well the scattering by a submerged rigid dock, particularly the radiation condition beneath the dock.

Water-wave scattering by submerged elastic plates

We present a solution to the water-wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method. The eigenfunction expansion depends on the solution of a special dispersion equation for a submerged elastic plate and this is discussed in detail. We show how the solution can be calculated for the case of normal incidence on a semi-infinite plate in two spatial dimensions and then extend this solution to obliquely incident waves, to a plate of finite length and to a circular finite plate in three dimensions. Numerical calculations showing various properties of the solutions are presented and a near-orthogonality relation for the eigenfunctions is used to derive an energy-balance relation.

Theoretical Evaluation of Rigid Baffles in Suppression of Combustion Instability

An analytical technique for the prediction of the effects of rigid baffles on the stability of liquid propellant combustors is presented. This analysis employs both two and three dimensional combustor models characterized by concentrated combustion sources at the chamber injector and a constant Mach number nozzle. An eigenfunction-matching method is used to solve the linearized partial differential equations describing the unsteady flow field for both models. Boundary layer corrections to this unsteady flow are in a mechanical energy dissipation model to evaluate viscous and turbulence effects within the flow. An integral instability relationship is then employed to predict the decay rate of the oscillations. Results of this analysis agree qualitatively with experimental observations and show that sufficient dissipation exists to indicate that the proper mechanism of baffle damping is a fluid dynamic loss. The response of the dissipation model to varying baffle blade length, mean flow Mach number, oscillation amplitude, baffle configuration, and oscillation mode is examined.