Rigorous open boundary condition for Boussinesq-type models, applicable to complex wave fields

To perform numerical simulations of tide and tidal current in a semi-closed water area, sea levels or a certain component of tide at open boundaries, which is the sum of incoming waves and outgoing waves through the boundaries, is usually imposed as boundary conditions. It is, however, important to separate incoming and outgoing waves in the research of tidal simulations.Adopting the non-reflecting boundary scheme proposed by Hino, we show that incoming waves into a water area can really be separated from the outgoing waves. Using the scheme, simulations such as the propagation of incident waves from open boundaries in a semi-closed water area are possible. In this way wave propagation characteristics of the area can be identified. Conversely, using the obtained relations between incident waves and resultant surface elevations in the water area, waves incident to the area through the open boundaries can be estimated from the surface displacements observed in the area. This is especially useful if more than one open boundaries exist.This paper reports the process of applying the non-reflection boundary scheme to the tidal calculations and how to estimate incident waves at each open boundary. This process is applied to the tidal calculutions of Osaka Bay. First, incident component of M2 tide is estimated from the observed tidal displacements in the bay. Using the estimated incident component, tidal calculations of Osaka Bay is conducted and the results are compared with the observed data. They are also compared with those obtained by a conventional method. Through these comparisons the efficaciousness of the present method is demonstrated.

Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations

The wave-absorbing control is a control concept to absorb waves traveling in a flexible structure at actuator positions. This paper presents an approach to design a broadband compensator by applying the //«, control theory to the wave-absorbing control method. This approach aims to minimize effects of the incoming waves on the outgoing waves at the actuator positions in the sense that the //«, norm of the closed-loop scattering matrix is minimum. Vibration suppression control for a flexible beam is studied analytically and demonstrated experimentally to exemplify the controller design approach. Compensators are designed for a collocated torque actuator and angle sensor and also for a noncollocated torque actuator and bending moment sensor. Performance of the compensators is analyzed in the frequency domain, and measured open- and closed-loop transfer functions are obtained from random excitation tests. The designed compensators are shown to attain good broadband damping, and results of the experiments are shown to agree well for the range of frequency below 50 Hz with those of the numerical simulations. I. Introduction A CTIVE control of vibrations in large flexible structures has received considerable attention in recent years. The modal model is a powerful technique both for the dynamic analysis and for the control design. However, limitations on the applicability of the structural modal analysis exist1 when the requirements for vibration suppression and pointing accuracy for flexible structures become stringent. The flexible mode frequencies and shapes are extremely sensitive to inevitable modeling errors, and modal analysis cannot provide a sufficiently accurate design model over a modally rich frequency range. One alternative is the traveling wave approach. This approach is based on the property that the response of a flexible structure to a typical locally applied force can be viewed in terms of traveling elastic disturbances. Mathematically, traveling waves belong to homogeneous solutions of partial differential equations describing the vibration of continua. At controller positions, relations between incoming and outgoing wave vectors and control inputs are derived in a matrix form by representing boundary conditions in terms of the traveling wave vectors. Outgoing waves are produced by the reflection of the incoming waves and are generated by control inputs. Transfer functions from the incoming wave and control input vectors to the outgoing wave vector are called scattering and generating matrices, respectively. Control inputs are set to be in the output-feedback form. This leads to the closed-loop relations between outgoing and incoming waves. Compensators are selected so that the effects of the incoming waves on the outgoing waves are reduced in some sense by adequately selecting elements of the closed-loop scattering matrix. Characteristic elements of the wave-propagation model, such as a scattering matrix, are smooth functions with respect to frequency and are more insensitive to model uncertainties than mode frequencies and shapes. The approach can provide a sufficiently accurate model for a controller design over a modally dense frequency region, and considerable research has been done on the wave control methods.18 However these methods also have drawbacks, such as 1) the designed compensator is not guaranteed to be a causal and real function with

An explicit finite difference model for simulating weakly nonlinear and weakly dispersive waves over slowly varying water depth

An operator-splitting approach with a hybrid finite volume/finite difference scheme for extended Boussinesq models

Most water wave simulations use fixed boundaries. This simplification of course change the wave height prediction near the shore. In this paper, we will find a relation between reflected wave and incoming wave on a sloping beach. This result is important for later use in water wave simulations with moving boundaries. In this paper, we solve the 1-D nonlinear shallow water equation, using the conservative finite volume method. The scheme plus wet and dry procedure can simulate wave running up and down on a sloping beach. Comparison with Carrier and Greenspan analytical run up and down waves shows a good agreement. Then, we use this scheme to find a relation between reflected and incident wave on a sloping beach, incorporating the moving shoreline. And we obtain amplitude ratio and phase difference depending on wave frequency for moderate bottom slopes.

Run-up of non-breaking double solitary waves with equal wave heights on a plane beach

In the neighborhood of a boundary point, the solution of a first-order symmetric homogenous hyperbolic system is conveniently decomposed into fundamental waves solutions that are readily classified as outgoing, incoming, and stationary, or tangential. Under a broad hypothesis, we show that the spans of the sets of outgoing and incoming waves have nontrivial intersection. Under these conditions, local, linear, perfectly nonreflecting local boundary conditions are shown to be an impossibility.

This paper investigates the application of the roller approach for breaking waves in a 1D hybrid finite-volume finite-difference weakly-nonlinear Boussinesq-type model. The vorticity transport equation is employed to model the movement of vorticity through the fluid. This allows vertical profiles of horizontal velocity and undertow to be computed. Previous implementations of this method caused numerical dissipation that influenced the physical behaviour of the breaking process. The use of a hybrid scheme overcomes this issue as the need to filter flow variables in the surf-zone is removed. Greater numerical stability increases the flexibility of the calibration parameters, allowing finer control over the breaking process and a more detailed investigation of the underlying physics. The mechanism used to dissipate energy during breaking is derived from physical principles and the Boussinesq equations are retained throughout the breaking procedure, providing a realistic description of the hydrodynamics throughout the surf-zone. The dissipative performance of the proposed model is discussed and compared with other state-of-the-art approaches, proving the feasibility and value of using a rotational roller model with a finite-volume finite-difference scheme to model surf-zone hydrodynamics with a Boussinesq-type model. Tests involving waves breaking on a sloping beach are performed to validate against results from physical experiments, demonstrating the model to be capable of accurately resolving profiles of the free surface, velocity and undertow. The resulting new model overcomes many of the issues encountered by previous Boussinesq solvers based on the same approach and provides significant improvements in the accuracy of predictions of breaking wave processes. The proposed approach is very flexible and can be used in any hybrid finite-volume finite-difference weakly-nonlinear Boussinesq-type model.

A Three-Dimensional Numerical Model of Surface Waves in the Surf Zone and Longshore Current Generation over a Plane Beach

This study considers the pre- and post-breaking behavior of Weakly-nonlinear (WNL) and Fully-nonlinear (FNL) Boussinesq-type (BTE) models. The main inquiries are: 1. Can the FNL model accurately reproduce shoaling behavior and the estimation of parameters crucial for describing breaking initiation? 2. How can the depiction of breaking be enhanced to accurately reproduce CFD calculations? Previous findings indicate that linear, KdV, or WNL models fail to reproduce shoaling for waves near breaking. While FNL models have shown accurate results in shoaling solitary waves (for example, Wei et al., 1995), it is uncertain if this robustness extends to shorter wavelengths. To address this, we analyze wave crest and velocity field properties across various wave conditions, comparing them to CFD calculations based on an LES/VOF model (Derakhti and Kirby, 2014) or a boundary element method potential flow solver (Grilli and Subramanya, 1996). Our tests reveal shortcomings in both the FUNWAVE 1-centered grid scheme (Wei and Kirby, 1995) and FUNWAVE-TVD (Shi et al., 2012) for accurately reproducing wave shoaling up to the breaking point in steep regular waves and solitary waves, respectively. We resolve this by employing a finite difference scheme with a staggered grid, enhancing the FNL Boussinesq model and ensuring precise numerical calculations of dispersion effects and shoaling. BTE models, assuming irrotational flow, utilize breaking closure models to identify breaking onset and calculate dissipation. Breaking onset is typically determined using geometric criteria, while post-breaking dissipation employs 0-equation models for eddy viscosity based on empirically derived geometric criteria. Our study demonstrates that these representations overpredict the immediate wave height decay after breaking. To address these issues, we (i) implement the B-criterion for a more robust kinematic threshold for breaking onset, where B = us/c, us is the surface velocity and c is the crest translation speed. This criterion was initially proposed by Barthélemy et al. (2018) for intermediate and deep water and validated by Derakhti et al. (2020) for all depths. (ii) For energy dissipation due to breaking, we adopt a revised model based on a 1-equation closure for turbulent kinetic energy (TKE), as suggested by Nwogu (1996). Finally, we derive a parameterization of TKE for determining eddy viscosity without relying on the closure model.

A Boussinesq-type model is applied herein to study wave propagation and wave breaking over complex bathymetries reproducing common coastal features, namely plane and barred beaches, submerged bars and rip channels. A hybrid finite volume–finite difference numerical scheme solves a set of, in the horizontal plane, two-dimensional extended Boussinesq equations where both nonlinear and dispersive effects are relevant and nonlinear shallow water equations where nonlinearity prevails over dispersion. The shock-capturing features of the finite volume method enable an intrinsic representation of spilling wave breaking and runup. Comparisons with experimental data indicate that the numerical model adequately simulates wave transformation over submerged bars, correctly capturing wave breaking onset and termination including the related energy dissipation. The development of breaking-induced currents and their interaction with wave propagation are also well represented within the applicability range of the governing equations.

Surf zone dynamics simulated by a Boussinesq type model. III. Wave-induced horizontal nearshore circulations

Open boundaries (OBs) are usually unavoidable in numerical coastal circulation simulations. At OBs, appropriate open boundary conditions (OBCs) are required and a good OBC should be able to let outgoing waves freely pass to the exterior of a computational domain without creating reflections at the OBs. In the present study, a methodology has been developed to predict two parameters, phase speed c_r and decay time T_f, in a standard OBC formulation, so that the OBC is significantly improved compared to commonly used existing OBCs with specified c_r and T_f. For the conditions where wave period is unknown, the OBC with approximated c_r and T_f may be applied and a test reveals that this OBC is able to yield good results in typical coastal flow conditions. In addition, a Swing-Door Boundary Condition (SDBC) is proposed and tested for application at an offshore open boundary where both incoming and outgoing waves exist.

The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.