The university of Edinburgh^{1}

**Introduction**

Regarding the propagation of water waves under gravity which is governed by the incompressible inviscid fluid theory, there are numerous difficulties due to the nonlinear inertia terms and the nonlinear boundary condition over an unknown surface. In view of these difficulties , and with respect to the fact that propagation of water waves of especial interest are inherently two-dimensional in character, various methods are developed to replace the (nonlinear) three-dimensional theory of water waves by a two dimensional theory. Due to the nonlinearity of waves, no general analytical solution exists for solving them. Therefore, a technique of numerical solution such as finite difference, finite element and panel schemes is required to solve the flow problems. Green-Naghdi (GN) sheet theory is in the middle of a spectrum between classic perturbation methods at one end and purely numerical techniques at the other end. Through this theory the prediction of unsteady, non-periodic, free-surface flow problems becomes possible. GN fluid sheet theory reduces the three dimensions to simplify the flow-problem solution and proposes equations that can be solved numerically.

**Methods/Materials**

The one-dimensional GN continuity and momentum equations for shallow flows over variable bed topography are derived. The fully nonlinear GN equations are discretized using finite differences and are solved for a uniform Cartesian grid of specific initial and boundary conditions. GN momentum equation contains two terms that are cross-derivatives involving space and time. Since an explicit predictor-corrector scheme is incapable of solving this kind of equation, an implicit finite difference scheme such as tri- and penta-diagonal matrix solver using second-order and fourth-order differences is developed to overcome this problem. Thus, a program had to be produced from scratch. To solve the tri- and penta-diagonal matrix, it is necessary to impose boundary conditions. For instance, solid wall boundaries are located at the ends of the domain when simulating sloshing of waves in an open rectangular channel. The surface elevation at the boundary is obtained by cubic Lagrange interpol

**Result and Discussion **

Verification standard benchmark tests such as sloshing in a rectangular basin and solitary wave in open channel were performed. Sloshing in a rectangular channel was the first test case. This test reveals that (using fine and rough grids) tri-diagonal and penta-diagonal matrix solvers have grid independence. Moreover, altering the time step (*âˆ†t*) on the converged grid does not have significant effect on the results, regardless of whether a tri-diagonal or penta-diagonal scheme is used and when small initial steepness is assumed. The models developed for one-dimensional GN equations are capable of simulating the long term behavior of waves in open channels. The second verification case concerns the propagation of a solitary wave in a one-dimensional channel over a horizontal, frictionless bed. In tri-diagonal simulations, on the coarser grids (*i max* = 1001), the solitary wave loses amplitude as it dissipates energy into the production of trailing waves. The trailing waves almost disappear and the solitary wave retains its amplitude on the finer grids with

**Conclusion **

GN equations are one of the powerful equations governing deep and shallow water in the ocean. As a result of the complexity and challenging nature of GN equations, small number of researchers and scientists had attempted to employ these equations. In the present study, one-dimensional GN equations solved using two different numerical schemes (tri- and penta-diagonal matrix solvers) and very accurate simulations were produced. The results indicate that both numerical schemes have enough capability to simulate fully non-linear one-dimensional GN equations. *1. Demirbilek Z. and Webster W.C., 1992, Application of the Green-Naghdi Theory of Fluid Sheets to Shallow-Water Wave Problems, Report 1. Model Development. US ArmyEngineers Waterways Experiment Station, Coastal Engineering Research Center.Technical Report CERC-92-11. *

*2. Ertekin R.C. et al., 1986, Waves Caused By a Moving Disturbance in a Shallow Channel of Finite Width, Journal of Fluid Mechanics. 169, 275-292.*

*3. Green A.E., Laws N. and Naghdi P.M., 1974, On the Theory of Water Waves, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 338, 43-55.*

*4. Haniffah M.R. (2013) Wave Evolution on Gentle Slopes-Statistical Analysis and Green-Naghdi Modelling. PhD. thesis, St. Edmund Hall, Univ. of Oxford, UK.*

*5. Métayer O.Le, Gavrilyuk S. and Hank S., 2010, A Numerical Scheme for the Green-Naghdi Model, Journal of Computational Physics. 229, 2034-2045.*

*6. Green A.E., Laws N. and Naghdi P.M., 1986, A Nonlinear Theory of Water Waves for Finite and Infinite Depths, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 320, 37-70.*

*7. Webster W.C, Duan W.Y. and Zhao B.B., 2011, Green-Naghdi Theory, Part A: Green-Naghdi (GN) Equations for Shallow Water Waves. Journal of Marine Sci. 10, 253-258.*

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