New Perspectives in Fluid Dynamics by Chaoqun Liu

By Chaoqun Liu

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Then, given that at z = zα , uα = ∇ φα , a fully nonlinear version of a Boussinesq type model in terms of η and uα is written: . ìï ìï ìï é z 2 1 2 ù ht + Ñ íæçè h1-x + eh ö÷ø íua + m 2 í ê a - h12-x - eh h1-x + ( eh ) ú Ñ ( Ñ ua ) 2 6 ûú îï ëê îï îï üü üïïï é ù 1 + ê za + ( h1-x - eh ) ú Ñ éÑ ( h1-x ua ) ù ýýý = O m 4 ë û 2 ë û þïþïïþ ) ( . (65) ( ) . ( ) ua t + e ( ua Ñ ) ua + Ñh + m 2 R + em 2 S = O m 4 (66) where R= . ( . ( 2 é1 ù 1 2 z Ñ ( Ñ ua t ) + za Ñ éÑ h1-x ua t ù - Ñ ê ( eh ) Ñ ua t + ehÑ h1-x ua t ú ë û 2 a ë2 û .

As recommended in [5] “a suitable grid is normally crucial to the success of a finite element model. In our case, the following rules must be fulfilled for its generation. a. Element side lower than the local depth. b. Minimum of 20 to 25 elements per wave length. c. Courant number always lower than one in the whole domain”. 2. Real case study using the WACUP model The fortification of S. Lourenço da Cabeça Seca (lighthouse of Bugio) – Tagus estuary (Portugal) – has endured over the course of four centuries the continuous action of waves and currents, 49 50 New Perspectives in Fluid Dynamics as well as bathymetric modifications resulting from the movement of significant quantities of sand in the area where it is located.

2. Mathematical formulations We start from the fundamental equations of the Fluid Mechanics, written in Euler’s variables, relating to a three-dimensional and quasi-irrotational flow of a perfect fluid [Euler equations, → or Navier-Stokes equations with the assumptions of non-compressibility (dρ / dt = divv = 0), → irrotationality (rotv = 0) and perfect fluid (dynamic viscosity, μ = 0)]: ux + vy + wz = 0 ut + uux + vuy + wuz = - px r vt + uvx + vvy + wvz = - py r (1) wt + uwx + vw y + wwz = - pz r - g uz = wx ; vz = w y ; vx = uy with p =0 at z = η ( x, y, t ), w = ηt + uηx + vηy at z = η ( x, y, t ), and w = ξt + uξx + vξy at z = − h 0 + ξ ( x, y, t ).

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