This research focused on depth-integrated modeling of coastal wave and surf-zone processes in support of computational fluid dynamics (CFD) simulation of ship motions. There were two components of the project. The first was the development of a numerical dispersion relation for a family of Boussinesq-type equations commonly used in modeling of coastal wave transformation. The relation depicts numerical dissipation and dispersion in wave propagation and provides guidelines for model setup in terms of temporal and spatial discretization. The second component was an extension of existing depth-integrated wave models to describe overtopping of coastal reefs and structures along with a series of CFD and laboratory experiments for model validation. The basic approach utilizing the HLLS Riemann solver performs reasonably well and produces stable and efficient numerical results for practical application.

Seakeeping analysis has traditionally been focused on dynamic response of vessels in the open ocean. As the attention is shifted to the littoral, a capability gap becomes obvious in the naval research and ship design communities. The distinct wave processes in the coastal region result in vessel loads and motions that are significantly different from those in the open ocean. Recent advances in depth-integrated models have enabled computations of wave transformation from the open ocean to the coast to provide important information for seakeeping analysis. However, such endeavors involve appreciable numerical errors and complex near-shore flows that might present a challenge to practical application.

Most coastal wave models are based on finite difference solution of Boussinesq-type equations. The depth-integrated governing equations express the vertical flow structure in terms of high-order spatial derivatives of the horizontal velocity through the irrotational flow condition. Subsequent developments have improved the dispersion and nonlinear properties, but greatly increased the complexity of the governing equations. Since the depth-integrated formulation cannot handle an overturning free surface, these early Boussinesq-type models typically utilized an empirical approach to approximate energy dissipation due to wave breaking.

Following the exponential growth of computing resources and parallel pursuit of highly nonlinear and dispersive theories, present-day computational models based on Boussinesqtype equations are being applied over vast regions from deep to shallow water with increasing resolution. The computational scheme, however, does not explicitly solve the governing system of partial differential equations. Discretization schemes involve numerical dispersion and dissipation that distort the true character of the governing equations. The leading term in the truncation error contains a derivative of the same order as the dispersion terms in Boussinesqtype equations. Such numerical errors have been studied extensively for the shallow-water equations, which represent a leading-order approximation of the Boussinesq-type equations. A wavenumber-based discretization scheme to preserve the dispersion relation of the governing equations has been proposed, but there has been no research on the effects of numerical dispersion in Boussinesq-type equations and the resulting wave propagation characteristics.

Overtopping on coastal reefs or structures becomes an important process as modeling of wave transformation extends into the nearshore region. George (2008) included effects of a bottom step as a forcing or source term in the Riemann problem and derived an approximate solver for the augmented system. The resulting model reproduces field and laboratory measurements of dam-break flows over rugged mountain terrain (George, 2011). Murillo and Garcia-Navarro (2010, 2012) generalized the effects of the bottom step as a hydrostatic force on the flow. Implementation of the resulting solver, known as HLLS (S for step), in one and two-dimensional nonlinear shallow-water models produces good agreement with analytical solutions and laboratory measurements. Application of these Riemann solvers has so far been limited to stepwise approximation of irregular topography for conservation of the hyperbolic flow character. Although these solvers do not include vertical flows to physically describe overtopping, they can better approximate the resulting characteristic flows for modeling of coastal wave processes.

This work was done by Kwok Fai Cheung, University of Hawaii at Manoa for the Office of Naval Research. ONR-0036


This Brief includes a Technical Support Package (TSP).
Coupling of Coastal Wave Transformation and Computational Fluid Dynamics Models for Seakeeping Analysis

(reference ONR-0036) is currently available for download from the TSP library.

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