A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed

A new multi-layer irrotational Boussinesq-type model is proposed for both linear and nonlinear surface water waves over mildly sloping seabeds. The model is formulated in terms of computational horizontal and vertical velocity components within each layer and satisfies exact kinematic and dynamic fr...

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Bibliographic Details
Published in:Journal of fluid mechanics 2018-05, Vol.842, p.323-353
Main Authors: Liu, Z. B., Fang, K. Z., Cheng, Y. Z.
Format: Article
Language:English
Subjects:
Accuracy
Amplitude
Boussinesq approximation
Boussinesq equations
Coasts
Coefficients
Components
Computer applications
Computer simulation
Dispersion
Engineering
Errors
Free surfaces
Gravitational waves
Harmonic functions
JFM Papers
Laboratories
Mathematical models
Model accuracy
Multilayers
Ocean floor
Propagation
Random waves
Shallow water
Shoaling
Surface water
Surface water waves
Surface waves
Theoretical analysis
Time integration
Transfer functions
Velocity
Velocity distribution
Velocity profiles
Water depth
Water waves
Wave dispersion
Wave phase
Wave propagation
Wave velocity
Wavelengths
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Summary:A new multi-layer irrotational Boussinesq-type model is proposed for both linear and nonlinear surface water waves over mildly sloping seabeds. The model is formulated in terms of computational horizontal and vertical velocity components within each layer and satisfies exact kinematic and dynamic free-surface conditions as well as kinematic seabed conditions. Using a Stokes-type expansion, a theoretical analysis of the new multi-layer model is carried out to examine both linear and nonlinear properties, including wave celerity, velocity profiles, shoaling amplitude, second- and third-order transfer functions and amplitude dispersion. The dispersive coefficients in the governing equations are determined by optimizing the linear celerity or linear velocity profiles. For example, the four-layer model shows extremely high accuracy and is applicable up to $kh=667$ –800 (where $k$ is the wavenumber and $h$ is a typical water depth) with a 1 % error in wave phase celerity, and up to $kh=352$ –423 with a 1 % error in the linear velocity components. The super- and subharmonic transfer functions are extremely accurate up to $kh=300$ (1 % error), the third-order harmonics and amplitude dispersion are accurate up to $kh=477$ (1 % error), and the shoaling property is optimized to cover the range of $0
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2018.99