Traditionally, laboratory experiments and field observations have been used to study this turbulent oceanographic phenomenon and empirical formulae have been derived from these but severe limitations existed since parameters to which these derivations fitted were local (Shao, 2006). Thus, in recent time, universal derivations that can fit across a wide range of parametric requirements of structure geometry, water conditions and wave dynamics are considered essential and desirable. To this end, fluid dynamics proves a somewhat reliable model generator but traditional Eulerian approaches that discretise governing equations over a computational field divided into a grid system based on local parameters develop problems of numerical diffusion that transcends localised grid patterns and tend to encompass the entire grid so that the discretised development of the equation into an unified whole is seriously affected (Shao, 2006). More recently, to solve this diffusion effect for traditional dynamics, a particle method has been developed wherewith the discretised equation utlises individual particles in the flow as centres of development. The diffusion effect is effectively smoothed by a functional kernel that identifies and utilises the combined functions of the angular and linear momentums of each particle (Shao, 2006). The smoothing out of the diffusion effect generated at each particle location within the flow thus allows the fluid, in this case seawater in wave form, to be accounted for as an incompressible one (Shao, 2006). One such method that utilises this unique strategy is the moving particle semi-implicit method (MPS) applied somewhat successfully by Japanese scientists to wave flow patterns (Shao, 2006). The model that this paper will demonstrate is the smoothed particle hydrodynamic (SPH) method as developed and tested by Shao, 2006.
The paper shall now study a little of how this manner of computational strategy developed.
Smoothed Particle Hydrodynamics (SPH):
The smoothed particle hydrodynamic method was one of the earliest meshfree methods applying Langrangian description of motion. It was primarily proposed by Lucy (1977) and Gingold and Monaghan (1977) (source: Zhang and Batra, 2004) for problems in astrophysics in three-dimensional space (Zhang and Batra, 2004). In the conventional smoothed particle hydrodynamic (SPH) method, for a function f at a point x within a domain , the approximate value of is given as below:
= (Eqn. 2.1, p. 137, Zhang and Batra, 2004)
In this equation, is the kernel or smoothing function. The approximate value of of f depends upon two parameters - the kernel W and the dilation h, the last providing support for W. It is essential that the kernel W should have the following properties -
I) = 0, for ,
III) , here is the Dirac delta function,
IV) , and
V) =. (Zhang and Batra, 2004)
This conventional SPH method is not even zero-order consistent at the boundaries (Zhang and Batra, 2004). This forced Liu et al, 1995a,b, to introduce a corrective function that is a polynomial of the spatial coordinates, making the method order consistent (Zhang and Batra, 2004). Chen et al, (1999a,b) and Zhang and Batra, 2004, also sought to improve the conventional SPH method consistency in some manner.
It is notable that the smoothed particle hydrodynamics method is a macroscopic model but it can be considered both as a continuum and particle method (Meakin et al, 2007). This is in particular context to the fact that the computational efficiency of purely particle methods is low in comparison to purely continuum ones (Meakin