Nomenclature
Chapter 2 Chapter 2 Literature Review
2.3 Numerical Simulations
down. A predictor-corrector-smoothing procedure was presented to predict whether the grid points needed to be adjusted. This numerical treatment was not robust.
as pointed out b~T Titov and Synolakis (1995). Nevertheless. Hibbert and Peregrine (1979) gave the first quantitative and realistic solution of the uniform bore behavior during the run-up process.
Titov and Synolakis (1995) solved the characteristic form of the shallow water equations using finite difference methods and used it to model the propagation and run-up of solitary waves. The characteristic equation was solved using the Godunov scheme to avoid the numerical instabilities problem associated with wave breaking.
The moving shoreline was treated the same as that of Hibbert and Peregrine (1979) by adding and subtracting grid points according to the shoreline position, except the boundary conditions imposed on the shoreline were modified as the following to avoid stability problems:
d:rs
=
0dt
at (2.11)where :1"8 is the location of the shoreline. 'Tl is the wave amplitude measured from the initial water level, and h is the water depth. The wave amplitude evolution and maximum run-up for non-breaking and breaking solitary waves were computed and compared with experimental results. However. small oscillations can still be found around the breaking point in their simulations. and tlw second boundary conditions in Eq.2.11 was wrong (see Zhang (1996)) and need to be corrected to provide good prediction of run-up.
Zhang (1996) developed a finite-difference scheme for the shallow water equations using the Lax-""endroff scheme to investigate non-breaking solitary wave run-up. The run-up was modeled by remapping the grid points at the surface according to the
instant shoreline position. Based on his numerical simulations, Zhang (1996) found that '·the maximllm. run-up of a solitary wave Vf'cdicted by the shallow water' equations was dependent on the initial location of the solitary wave and its value was not llniqlle becallse the waUe became incn;asely steepened given long time to tmvel in the absence of the dispersive effects. " Zhang (1996) also investigated the frequency dispersion and three-dimensional wave run-up upon a vertical wall using his numerical scheme. The computing domain mapping technique proposed by Zhang (1996) apparently treats the shoreline movement well and will be used in the numerical scheme developed in the present study.
Dodd (1998) investigated wave run-up. overtopping, and regeneration by solving the non-linear shallow water equations using a Roe-type Riemann solver, which was developed in gas dynamics to track shock waves. An energy dissipative term repre- senting bottom friction was included in the model. In the scheme. a minimum local depth dmin was defined to treat the moving shoreline. When the water depth in the cell is less than dmin , the cell was considered "dry", otherwise, the cell was occupied by water ('·wet·'). The shoreline was defined as the separation line between the "dry"
cell and the "wet" cell. Dodd (1998) conducted simulations of wave propagation and overtopping including random waves and compared them with experimental results, good agreements were found from his investigation.
In summary, the models utilizing non-linear shallow water wave equations, al- though having the limitation of failing to provide depthwise variations in velocity and omitting frequency dispersive effects, appear to have the ability to model aspects of the wave breaking process and the corresponding run-up for solitary waves. "The well-documented but unel;plained ability of the shallow water eqllations to provide qllantdati71dy C07'Tect TlI.nup T'e:mlts even in parameter- mnges where the underlying assu.mptions of the governing equations aTe violated" (Titov and Synolakis (1995))
need further investigation and will bc given attention in this thesis.
Boussincsq type models have also been used widely to simulate wave breaking and run-up. They can represent the nOll-linear effects and dispersive effects theoretically to any degree of accuracy and can descrihe most wave phenomena. However. a special breaking term has to be included in the momentum conservation equation to model the dissipation associated with wave breaking. The term must incorporate coefficients that need to be calibrated by field or experimental data. This drawback limits the application of the Boussinesq models. Pedersen and Gjevik (1983) developed a finite- difference scheme for the Boussinesq eqnations using a Lagrangian description. which can predict the non-breaking run-up process and also the possibility of wave breaking during run-down. The maximum run-up predicted using this numerical model was larger than the experimental data of Hall and Watts (1953). Peterson and Gjevik (1983) suggested that this difference was due to surface tension and friction effects that were neglected in the numerical model. It was also found that the friction effects became less important as the depth in the channel increases.
ZeIt and Raichlen (1990) developed a Lagrangian representation of the Boussinesq equations and used a finite-element mociel to investigate non-breaking solitary wave run-up on two-dimensional and three-dimensional bathymetry. ZeIt (1991a) applied this model to the case of the run-up of both non-breaking and breaking waves on a plane beach. "Vave breaking was parameterized with an artificial viscosity term in the momentuIll equation. and the bottom friction was also modeled as a term quadratic in the horizontal water velocity. ZeIt (1991a) found that non-hydrostatic effects associated with the frequency dispersion term in the Boussinesq equations reduced the tendency of waves to break and improved the agreement of the numerical results with the laboratory run-up data. \Vhen calibrated with laboratory data. the model of ZeIt (1991a) could provide reasonable predictions of the wave run-up process.
In additioll, ZeIt (1991b) studied the landward inundation of non-breaking solitary waves that propagate up a non-planar slope.
1\mnerical solutions of the Laplace equations and the N avier-Stokes equations also have been used in wave run-up investigations as the comput.er power has increased and the algorithms used to solve complex systems have beell developed. Grilli, Svendsen, and Subramanya (1997) solved a fully non-linear potential flow model (the Laplace equation) using the boundary element t.echniques (BE1\l) , and used it to calculate various characterist.ics of breaking solitary propagation and run-up. In contrast to t.he depth-averaged models like the shallow water equations and the Boussinesq models, the vertical structure of the water particle velocities could be treated by the numerical model. The detailed wave breaking information including the shape of the plunging jet generated by the wave breaking, the celerity, and water particle velocity as well as the wave shoaling and overall wave profile were reported. However, this numerical model cannot predict maximum run-up since the computation stops when the plunging jet impinges the free surface. In Chapter 5, the numerical results of Grilli et al. (1997) will be compared to experimental results obtained from the present investigation.
Lin, Chang and Liu (1999) developed a numerical model solving the Reynolds equations for the mean fiow field and the k - f equations for the turbulent kinetic energy, k, and the turbulence dissipation rate, f. and applied the model to wave breaking and run-up problems. The free-surface locations and movement were tracked by the volume-of-fiuid (VOF) method proposed by Rirt and Nichols (1981). Their numerical results agreed with the experirnental results in terms of the wave profile and velocities. but fail to provide the jet and splash-up information. which may be due to the inaccuracy of the free surface tracking techniques used.