For the main challenges free surface modeling is

 For example, Akbari and Namin 24
used this approach in an SPH numerical method in wave interaction with
structures. Their developed model solves porous and pure fluid (water) flows
simultaneously by means of an equation that is equivalent to the unsteady 2D
Navier–Stokes (N-S) equations for the flows outside the porous media, while the
extended Forchheimer equation has been used for the flows inside the porous
media. In their model, the interface boundary between pure fluid (water) and
porous media is effectively taken into account by the SPH integration technique24. 

One of the main challenges free surface modeling is the tracking of free
surface evolutions. Hydrostatic assumption has been used by many authors in
previous researches 30, 31. However, the solutions based on the simple
hydrostatic assumption are not as accurate in most of engineering problems
where the vertical velocities and accelelerations are not ignorable. For this
reason, detailed investigation based on the hydrodynamic treatment of the
phenomenon is required where the vertical plane of the model area has to be
devided into a finite number of elements. Two different approaches have been
followed based on hydrodynamic assumption, which are updated mesh and fixed
mesh approaches. Studies with updated mesh mostly use an Arbitrary Lagrangian
Eulerian (ALE) technique 32, 33. When using the ALE
technique in simulations the mesh for the domain has to be regenerated and the
nodes on the boundaries of the domain have to move along with the materials to
preciesely track the boundaries, accordingly. In fixed mesh (Eulerian) approach, on the other hand, level set method usually is used by which
the free surface is represented by a signed distance function that takes the
value 0 on the free surface, takes negative values in the fluid domain and
positive values outside the fluid domain. 29, 34, 35. In most of the applications the above two
mentioned methods can track surface evolution accurately, however, the main
disadvantages of these methods are their high cost and complexities. The
improved accuracies due to the mentioned methods do not justify to suffer the
high computation expensis and model complexities in the most of the real
applications.

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