2023/01/10 14:00 Distinguished Prof. Philip Li-Fan Liu(National University of Singapore)

Poster:Post date:2023-01-07
NCU IHOS Seminar Announcement

Title:Recent development of 2DH models with applications to 3D water waves


Speaker:Distinguished Prof. Philip Li-Fan Liu

National University of Singapore
Chair Professor @ NCUIHOS


Place:S-325, Science Building 1

  In the last forty years, many depth-integrated 2DH models have been developed for simulating water wave propagation from deep-water to shallow-water. These models can be divided into two general categories: Boussinesq-like models and Green-Naghdi-like models. The former models adopt the Boussinesq approximation and flow irrotationality or weak rotationality. The latter models approximate the vertical profile of the horizontal and vertical velocity components in a polynomial form in terms of the vertical coordinate. The corresponding 2DH equations for the horizontal and vertical velocity components are obtained by invoking the Weighted Residual (Galerkin) method. The pressure field is related to the velocity field through the vertical momentum equations and the dynamic free surface boundary conditions. However, to close the problem, the horizontal velocity components associated with the highest degree of polynomial term are required to be zero in an ad-hoc way. On the other hand, the analytical expression for the approximated pressure field is also missing.
  Yang and Liu (2020, 2022, hereinafter referred as YL-models) presented two sets of depth-integrated wave--current models. The major difference between YL-models and other existing models is that the YL-models are based on the depth-integrated continuity equation and momentum (Euler’s) equations in terms of horizontal velocity components and free surface elevation. These equations are exact and the free surface and bottom boundary conditions are also satisfied. More importantly, the vertical velocity component and the pressure field can be expressed analytically in terms of the horizontal velocity and the free surface elevation. Approximating the vertical profiles of the horizontal velocity components by polynomials, and using the Galerkin and subdomain methods, GK and SK models were presented. The SK models demonstrate better accuracy than GK models and Green-Naghdi-like models in terms of various wave properties by a theoretical linear analysis, which were also verified by several numerical validations (Yang & Liu 2020). The SK models were also extended to simulate waves interacting with arbitrarily sheared currents (Yang & Liu 2022), which cannot be treated by other existing models.
  In this talk a new model will be presented. In this model, the horizontal velocity components are discretised by finite elements. Employing the Galerkin method, a set of depth-integrated model equations for horizontal velocities at the finite element nodes and the free surface elevation are derived. The model equations are general and can be applied not only to water wave propagation and scattering problems, but also transient and steady hydraulic flow problems. In this talk we will focus on the former. A Stokes wave-type Fourier analysis is conducted on the new models to examine models' linear wave properties, including linear wave phase velocity, group velocity, shoaling gradient, vertical profiles of horizontal and vertical velocities and vertical profiles of non-hydrostatic pressure field. In terms of linear wave phase velocity, using only two linear elements in the vertical direction, the maximum relative error is 2% at kd ~ 14.7, which is already in deep water condition. When four linear elements are used in the model the phase velocity is within 2% relative error for kd ~ 127.9, which is equivalent to an infinite depth. The nonlinear wave property is only scrutinized for the model with two linear elements for the second-order wave amplitude, which is still reasonably accurate up to kd ~ 6.0. The performance of the new models are much better than the existing models mentioned above. The embedded Doppler-shift effects of wave--current interactions are also examined analytically. These 2DH model equations are solved by a finite difference method (FDM). Numerical validations are conducted using the two-linear-element model to study regular/irregular wave transformation over a submerged shoal and newly-conducted experiments of waves interacting with linearly sheared currents; very good agreements are obtained between the numerical results and laboratory experiments.
Last modification time:2023-01-07 PM 3:20

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