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Boussinesq–Green–Naghdi rotational water wave theory
Abstract Using Boussinesq scaling for water waves while imposing no constraints on rotationality, we derive and test model equations for nonlinear water wave transformation over varying depth. These use polynomial basis functions to create velocity profiles which are inserted into the basic equations of motion keeping terms up to the desired Boussinesq scaling order, and solved in a weighted residual sense. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Explicit coefficients are given at O(μ 2) and O(μ 4), while more generalized basis functions are given at higher order. Nonlinear performance is somewhat more limited as, for reasons of complexity, we only provide explicitly lower order nonlinear terms. Still, second order harmonics may remain good to kh ≈10 for O(μ 4) equations. Numerical tests for wave transformation over a shoal show good agreement with experiments. Future work will harness the full rotational performance of these systems by incorporating turbulent and viscous stresses into the equations, making them into surf zone models.
Highlights ► We derive and test Boussinesq–Green–Naghdi type water wave equations without any irrotationality assumptions using polynomial basis functions ► Excellent convergence is demonstrated for linear properties ► Asymptotic rearrangement may be used to improve properties for a given order of approximation by redefinition of basis function coefficients ► Numerical solutions show good agreement with experiments.
Boussinesq–Green–Naghdi rotational water wave theory
Abstract Using Boussinesq scaling for water waves while imposing no constraints on rotationality, we derive and test model equations for nonlinear water wave transformation over varying depth. These use polynomial basis functions to create velocity profiles which are inserted into the basic equations of motion keeping terms up to the desired Boussinesq scaling order, and solved in a weighted residual sense. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Explicit coefficients are given at O(μ 2) and O(μ 4), while more generalized basis functions are given at higher order. Nonlinear performance is somewhat more limited as, for reasons of complexity, we only provide explicitly lower order nonlinear terms. Still, second order harmonics may remain good to kh ≈10 for O(μ 4) equations. Numerical tests for wave transformation over a shoal show good agreement with experiments. Future work will harness the full rotational performance of these systems by incorporating turbulent and viscous stresses into the equations, making them into surf zone models.
Highlights ► We derive and test Boussinesq–Green–Naghdi type water wave equations without any irrotationality assumptions using polynomial basis functions ► Excellent convergence is demonstrated for linear properties ► Asymptotic rearrangement may be used to improve properties for a given order of approximation by redefinition of basis function coefficients ► Numerical solutions show good agreement with experiments.
Boussinesq–Green–Naghdi rotational water wave theory
Zhang, Yao (author) / Kennedy, Andrew B. (author) / Panda, Nishant (author) / Dawson, Clint (author) / Westerink, Joannes J. (author)
Coastal Engineering ; 73 ; 13-27
2012-09-17
15 pages
Article (Journal)
Electronic Resource
English
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