Solution of the wave agitation problem through a variational solution of the Helmoltz equation

Transcription

1 Solution of the wave agitation problem through a variational solution of the Helmoltz equation V. Bovolin, A. Fortunate & E.P. Carratelli Department of Civil Engineering Eugliese@Bridge. diima. unisa. it Introduction Wave agitation problem on constant depth is governed by the well known Helmoltz equations. There is no shortage of academic or commercially available finite elements or finite difference software to solve such an equation; there are however a number of simple geometrical configurations where the computational requirements and the data preparation effort involved in running complex codes are out of proportion in comparison with the objective. Such problems are often found when carrying out feasibility studies for small harbours or coastal protection works. A Helmoltz equation solver based on a non-discretised version of Ritz's variational method was developed and tested to provide a fast and efficient tools for rectangular water basins. After validation with standard test cases the algorithm was employed for a dock inside a commercial harbour and for an equally rectangular configuration representing the gap between a beach and a coastal defence breakwater. Solution method If the water depth on a domain Q, can be assumed to be reasonably constant, the equations to be solved for the water waves complex amplitude u^ + iu. are the well known Helmoltz equations: 22,\ uf = 0, in 0 (1)

2 440 Coastal Engineering and Marina Developments with the relevant conditions on the boundaries F. If for instance F, are the wall boundaries and F, is an opening were monochromatic wave height is imposed, we have: dn on on (2) on F, = 0, Under certain conditions the problem above may be reduced to the search for a minimum of the functional: - -2a^k [ u^u.ds + -1 a«(3) According to the Ritz method, by assuming: N k=\ (4) where u,. e w- are known values of u^. e M. on Fj, c* and c* are constants which have to be determined and the ^ are the base set of functions satisfying homogeneous conditions on F,. Eqn. 3 becomes: (5)

3 Coastal Engineering and Marina Developments 441 An approximate solution is thus given by a set of coefficients cr * and cf for the base functions which are found by minimising the functional associated to the differential problem. By introducing the zero derivative condition on (5), we get: dcf = 0, (6) Eqn. 6 constitute a linear system in the unknowns c* e cf, whose solution defines, within the solution class defined by the (4), the best approximations of the functions u, e u. A particularly simple and fast numerical solution can obtained for simple, rectangular geometry by adopting polynomials in x, and x, as abase for the 0*. The solution to problem 6 can be found out by making use of standard software. The solution values are then calculated on the a number of points over a grid so that a graphical representation can be obtained and further elaboration can be carried out. In order to validate the proposed method, following the approach by Dhemellemes et al, the computational results have been compared with classical data published in the "Shore Protection Manual" (CERC). Two cases have been considered: the first where the opening width is equal to the wave length and the second one where the width is 1.4 time the wave length. The comparison is depicted in figure 1. Fig 1: Comparison of computed results (framed lines) with existing solution (thick lines).

4 442 Coastal Engineering and Marina Developments The general pattern is well reproduced, and particularly it is clear the deeper intrusion on the wave motion in the case of the larger opening. Once the validation tests have been completed the method was then applied to real harbour problems, taking into account the spectral wave energy distribution and calculating the resulting state of agitation in enclosed or semi-enclosed docks. Fig 2 gives an overall view of a large port (Salerno) which is currently used as a both a commercial and yachting harbour and is currently object of a detailed study in order to minimise the agitation during storms and make it viable for tourist development Fig 2: Salerno harbour schematic layout. The wave agitation in the main water basin (Fig 3) was computed with a reflection/refraction model, while the docks had to be dealt with separately by making use of the method described above, in order to take into account the diffraction effects.

5 Coastal Engineering and Marina Developments 443 The wave conditions on the I\ were supplied by the output of the first model. In the following the wave height at the inlet of the basin is considered to be the input, while the results of the computation are given by the maximum or the average wave height for different time periods T n- Fig 3: Internal basin: definition sketch. Fig 4 shows the local wave height for a monochromatic input wave with T=10 s andl 152 m. Fig 4: Local wave height for a monochromatic input wave (T=10 s, L 152 m). Fig 5 gives the wavelength response curve, i.e. the average internal wave height divided by the input height, as a function of the input wavelength. Different

6 444 Coastal Engineering and Marina Developments curves show the change in the agitation as the damping behaviour of the quay walls is changed from fully reflecting (a) to partially absorbing (b) a) fully reflecting D b) partially absorbing Fig 5: Non dimensional wavelength response curve (average internal wave height divided by the input height) as a function of the input wavelength. The computational economy gained through this approach can be fully exploited in order to compute the basin response to a realistic sea state: the wave spectrum is split into components W. (Fig 6 ) and the wave height H. relative to component /, wavelength L., is calculated for each grid point.

7 Coastal Engineering and Marina Developments Fig 6: Wave spectrum The resulting average wave height Hs, again for each grid point is then obtained as: The results are displayed in fig.7 (7) Fig 7: Resulting average wave height for wave spectrum of figure 6.

8 446 Coastal Engineering and Marina Developments Conclusions A variational Helmoltz equation solver can be used to calculate wave agitation in water and presents some advantages in terms of computational speed and ease of data preparation in a number of instances. One such instance is the design work for small marinas or a docks within a large harbours, which may require the analysis of wave agitation for geometrically similar but dimensionally different configurations and with a large number of input wave spectra in order to ascertain the resonance characteristics of the basin and possibly to optimise the wave dumping characteristics of the sea walls. There is also a need for faster algorithms to be incorporated in coastal models to account for the presence of piers, groins, breakwaters and other situations where the effects of wave diffraction may be important. A similar situation is the analysis of wave agitation between breakwaters and beaches on a regional basis as required in connection with civil protection and large scale erosion problems,. In these last two cases the use of large codes in not a viable option, so a cheap solver for simple geometries could be a possible alternative to the closed formulas usually employed for diffraction effects. Acknowledgements The work described in this paper was funded by the GNDCI (National Group for the Protection from Hydrogeological Disasters) LIT Project. The cooperation of the Salerno Harbour Operators Association (Assotutela) is also gratefully acknowledged. References CERC, Shore Protection Manual, 1984 Chou C.R., Han W.Y. "Wave-Induced Oscillation in "Harbor of Arbitrary Shape with Arbitrary Reflection Coefficient in Uneven Sea Bed", Computer Modelling in Ocean Engineering 91, Arcilia et al, 1991 Balkema, Rotterdam, The Netherlands. Covas J.M.A. "Harbour Resonance Computations for Saint Quay-Portieux", Computer Modelling in Ocean Engineering 91, Arcilia et al, 1991 Balkema, Rotterdam, The Netherlands. Dhemellemes F., Kadiri M., Rabeau J, "Prise en compte des coefficients de refelection des ouvrages dan les codes de calcul d'agitation dans le s ports", Reprort Electricite De France HE-42/

9 Coastal Engineering and Marina Developments 447 Hamanaka Ken-ichiro, "Open, Partial Reflection and Incident-Absorbing Boundary Conditions in Wave Analysis with a Boundary Integral Methods", Coastal Engineering 30 (1997)