Harry B. Bingham

A new Boussinesq method for fully nonlinear waves from shallow to deep water

A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z -level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Pade approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis.

Boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity field as well as the linear and nonlinear properties. First, an exact solution to the Laplace equation is given in terms of infinite–series expansions from an arbitrary z–level which is a constant fraction of the still–water depth. This defines the fully dispersive and fully nonlinear water–wave problem in terms of five variables: the free–surface elevation and the horizontal and vertical velocities evaluated at the free surface and at the arbitrary z–level. Next, the infinite series operators are replaced by finite–series (Boussinesq–type) approximations. Three different approximations are introduced, each involving up to fifth–derivative operators, and these formulations are analysed with respect to the linear–velocity profile, linear dispersion and linear shoaling. Nonlinear characteristics are investigated by a perturbation analysis to third order for regular waves and to second order for bichromatic waves. Finally, a numerical spectral solution is made for highly nonlinear steady waves in deep and shallow water. It can be concluded that the best of the new formulations (method III) allows an accurate description of dispersive nonlinear waves for kh (wavenumber times water depth) as high as 40, while accurate velocity profiles are restricted to kh < 10. These results represent a major improvement over existing Boussinesq formulations from the literature.

An efficient flexible-order model for 3D nonlinear water waves

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211-228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss-Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature.

A numerical study of crescent waves

In this paper, a high-order Boussinesq model is used to conduct a systematic numerical study of crescent (or horseshoe) water wave patterns in a tank, arising from the instability of steep deep-water waves to three-dimensional disturbances. The most unstable phase-locked (L2) crescent patterns are investigated, and comparisons with experimental measurements confirm the quantitative accuracy of the model. The unstable growth rate is also investigated, as are the effects of variable nonlinearity. The dominant physical mechanism is clearly demonstrated (through time and space series analysis) to be the established quintet resonant interaction, involving the primary wave with a pair of symmetric satellites. A numerical investigation into oscillating crescent patterns is also included, and a detailed account of the complicated oscillation cycle is presented. These patterns are shown to arise from quintet resonant interactions involving the primary wave with two unsymmetric satellite pairs. Pre-existing methods for analysing the stability of steep deep-water plane waves subject to three-dimensional perturbations are extended to provide accurate quantitative estimates for the oscillation period. A possible explanation for their selection in experiments is also provided. Finally, we use the model to conduct a series of experiments involving competition between various unstable modes. The results generally show that multiple instabilities can grow simultaneously, provided that they are of roughly equivalent strength. Results using random perturbations also match observations in physical experiments both in the form (i.e. two- or three-dimensional) and the location of the initial instability. The computational results are the first examples of highly nonlinear (to the breaking point) deep-water wave modeling in two horizontal dimensions with a Boussinesq model. The efficiency of the model has allowed for a quantitative study of these phenomena at significantly larger spatial and temporal scales than have been demonstrated previously, providing new insight into the complicated physical processes involved.

A hybrid Boussinesq-panel method for predicting the motion of a moored ship

Abstract This paper describes a new computational technique for predicting the wave-induced motion of a restrained floating body in restricted water. A combination of established methods is used in an attempt to account for the most important physical processes involved in this complicated problem, while keeping the computational burden modest. Potential theory is invoked to describe both the wave transformation over the bathymetry of the harbour, and the hydrodynamic interaction between the waves and the floating structure. Modified Boussinesq theory is used to predict the transformation of the waves as they propagate from deep water into to the harbour or bay where the body is moored. This model includes the effects of shoaling, refraction and non-linear wave–wave interaction, and most importantly sub-harmonic generation. This flow is then linearised locally, at the structure, to provide the incident wave forcing term in the equation of motion which is solved in the time-domain. Linear wave radiation and diffraction forces are computed using a constant-strength panel method, while the instantaneous, non-linear, point mooring forces are included exactly. The model is validated for the linear problem, and non-linear calculations are compared with experimental measurements for a ship moored in an L-shaped harbour. Qualitatively, the non-linear features of the dynamical system are successfully captured by the model. Some tuning, in the form of empirically obtained damping coefficients, is, however, necessary in order to get a reasonable prediction of the response amplitudes near resonance, when the linear hydrodynamic damping is very small.

Numerical simulation of lowest-order short-crested wave instabilities

A numerical study of doubly periodic deep-water short-crested wave instabilities, arising from various quartet resonant interactions, is conducted using a high-order Boussinesq-type model. The model is first verified through a series of simulations involving classical class I plane wave instabilities. These correctly lead to well-known (nearly symmetric) recurrence cycles below a previously established breaking threshold steepness, and to an asymmetric evolution (characterized by a permanent transfer of energy to the lower side-band) above this threshold, with dissipation from a smoothing filter promoting this behaviour in these cases. A series of class Ia short-crested wave instabilities, near the plane wave limit, are then considered, covering a wide range of incident wave steepness. A close match with theoretical growth rates is demonstrated near the inception. It is shown that the unstable evolution of these initially three-dimensional waves leads to an asymmetric evolution, even for weakly nonlinear cases presumably well below breaking. This is characterized by an energy transfer to the lower side-band, which is also accompanied by a similar transfer to more distant upper side-bands. At larger steepness, the evolution leads to a permanent downshift of both the mean and peak frequencies, driven in part by dissipation, effectively breaking the quasi-recurrence cycle. A single case involving a class Ib short-crested wave instability at relatively large steepness is also considered, which demonstrates a reasonably similar evolution. These simulations consider the simplest physical situations involving three-dimensional instabilities of genuinely three-dimensional progressive waves, revealing qualitative differences from classical two-dimensional descriptions. This study is therefore of fundamental importance in understanding the development of three-dimensional wave spectra.

A non-periodic spectral method with application to nonlinear water waves

Spectral methods are very efficient and powerful for solving periodic problems. A new spectral method is developed for problems with no spatial periodicity, and demonstrated for water waves. The method splits the potential into the sum of a prescribed non-periodic component and an unknown periodic component. Computed results are compared with experiments by Shemer et al (1998).

A non-linear wave decomposition model for efficient wave-structure interaction. Part A

This paper deals with the development of an enhanced model for solving wave-wave and wave-structure interaction problems. We describe the application of a non-linear splitting method originally suggested by Di Mascio et al. 1], to the high-order finite difference model developed by Bingham et al. 2] and extended by Engsig-Karup et al. 3,4]. The enhanced strategy is based on splitting all solution variables into incident and scattered fields, where the incident field is assumed to be known and only the scattered field needs to be computed by the numerical model. Although this splitting technique has been applied to both potential flow and Navier-Stokes solvers in the past, it has not been thoroughly described and analyzed, nor has it been presented in widely read journals. Here we describe the method in detail and carefully analyze its performance using several 2D linear and non-linear test cases. In particular, we consider the extreme case of non-linear waves up to the point of breaking reflecting from a vertical wall; and conclude that no limitations are imposed by adopting this splitting. The advantages of this strategy in terms of robustness, accuracy and efficiency are also demonstrated by comparison with the more common strategy of solving the incident and scattered fields together.

The Ultimate Boussinesq Formulation for Highly Dispersive and Highly Nonlinear Water Waves

A new set of highly accurate Boussinesq-type equations are presented. The equations are formulated in six variables, counting vectors as one: The surface elevation, the horizontal gradient of the velocity potential at the free surface, the horizontal and vertical velocities at the free surface and the horizontal and vertical velocities at an arbitrary z-level. The arbitrary z-level is determined by optimizing the vertical distribution of the horizontal and vertical velocity components compared to linear fully dispersive wave theory. A Stokes type analysis is performed and the new formulation is found to retain high accuracy in the linear and nonlinear characteristics for wave numbers as high as kh=30. A numerical model is developed in one horizontal dimension and results are presented for the case of a highly nonlinear solitary wave propagating over a constant depth. The numerical results are shown to be in excellent agreement with exact solutions.

An overset grid approach to linear wave-structure interaction

A finite-difference based approach to wave-structure interaction is reported that employs the overset approach to grid generation. A two-dimensional code that utilizes the Overture C++ library has been developed to solve the linear radiation problem for a floating body of arbitrary form. This software implementation has been validated by performing time-domain simulations to evaluate the dynamic forces applied to a half-submerged cylinder and a rectangular barge in response to a prescribed motion. A Gaussian displacement is used to introduce a range of wave frequencies, thereby allowing the measurement of the body response over the frequency range of interest. The radiation added-mass and damping coefficients of both bodies have been evaluated and compared to exact analytical solutions. The numerical and analytical results show good agreement when the modes of excitation and response are the same. The cross-coupled results are in qualitative agreement, but show some quantitative variations that may be related to slight differences in the fluid domain geometry. For both the cylinder and the barge, the effects of bottom slope on the coefficients are found to be minimal.Copyright