Serdar Beji

A FORMAL DERIVATION AND NUMERICAL MODELLING OF THE IMPROVED BOUSSINESQ EQUATIONS FOR VARYING DEPTH

Abstract A formal derivation of the improved Boussinesq equations of Madsen and Sorens (1992) is presented to provide the correct forms of the depth-gradient related terms. Linear shoaling characteristics of the new equations are investigated by the method of Madsen and Sorensen (1992) and by the energy flux concept separately and found to agree perfectly, whereas these approaches give conflicting results for the equations derived by Madsen and Sorensen (1992). Furthermore, Nwogus (1993) modified Boussinesq model is found to produce a linear shoaling-gradient identical with the present work. Numerical modelling of the derived equations for directional waves is carried out by three-time-level finite-difference approximations. A higher-order radiation condition is implemented for effective absorption of the outgoing waves. Several test cases are included to demonstrate the performance of the model.

A fully dispersive weakly nonlinear model for water waves

A fully dispersive weakly nonlinear water wave model is developed via a new approach named the multiterm–coupling technique, in which the velocity field is represented by a few vertical–dependence functions having different wave–numbers. This expression of velocity, which is approximately irrotational for variable depth, is used to satisfy the continuity and momentum equations. The Galerkin method is invoked to obtain a solvable set of coupled equations for the horizontal velocity components and shown to provide an optimum combination of the prescribed depth–dependence functions to represent a random wave–field with diversely varying wave–numbers. The new wave equations are valid for arbitrary ratios of depth to wavelength and therefore it is possible to recover all the well–known linear and weakly nonlinear wave models as special cases. Numerical simulations are carried out to demonstrate that a wide spectrum of waves, such as random deep water waves and solitary waves over constant depth as well as nonlinear random waves over variable depth, is well reproduced at affordable computational cost.

A FULLY-DISPERSIVE NONLINEAR WAVE MODEL AND ITS NUMERICAL SOLUTIONS

A special reflecting wall 12 m long and 2.1 m high was built off the beach at Reggio Calabria, and 30 wave gauges were assembled before the wall and were connected to an electronic station on land. It was possible to observe the reflection of wind waves generated by a very stable wind over a fetch of 10 Km. The experiment aimed to verify the general closed solution for the wave group mechanics (Boccotti, 1988, 1989), for the special case of the wave reflection.Significant features on Wadden Sea wave climate are evaluated in respect of the state of the art. Main emphasis was laid on an analysis of the governing boundary conditions of local wave climate in island sheltered Wadden Sea areas with extensions being sufficient for local wind wave growth. Explanatory for significant wave heights a reliable parametrization of local wave climate has been evaluated by using generally available data of water level and wind measurements.

A time–dependent nonlinear mild slope equation for water waves

A weakly nonlinear and dispersive water wave equation, which in linearized form yields a new version of the time–dependent mild–slope equation of Smith and Sprinks (1975), is derived. The applicable spectral width of the new wave equation for random waves is found to be more satisfactory than that of Smith and Sprinks (1975). For very shallow depths the equation reduces to the combined form of Airys nonlinear non–dispersive wave equations; if the lowest–order dispersion is retained it produces the combined form of Boussinesqs equations. In the deep–water limit the equation admits the second–order Stokes waves as analytical solutions. Furthermore, by introducing a right–moving coordinate transformation, the equation is recast into a unidirectional form, rendering the KdV equation in one limit while reproducing the second–order Stokes waves in the other.

NONLINEAR REFRACTION-DIFFRACTION OF SURFACE WAVES OVER ARBITRARY DEPTHS

A study of alternatives including a shoreline evolution numerical modelization has been carried out in order to both diagnose the erosion problem at the beaches located between Cambrils Harbour and Pixerota delta (Tarragona, Spain) and select nourishment alternatives.

Improved Boussinesq-type equations for spatially and temporally varying bottom

ABSTRACT Boussinesq-type equations with improved linear dispersion characteristics are derived for spatially and temporally varying bottom. Starting from the first principles, spatial variations and temporal movements of seabed due to underwater earthquakes, landslides and alike are incorporated into the Boussinesq-type equations. The momentum equation is then manipulated by the partial replacement technique so that a generalized Boussinesq set of equations with improved dispersion characteristics is obtained. For an impulsive bed motion-simulated wave profiles are compared with experimental measurements. Waves generated by an ellipsoidal slump moving down on an inclined plane are also numerically simulated to disclose the effect of a newly derived term. Overall, the new set of equations is expected to provide more accurate representation of wave motions due to bottom movements by correctly modeling accelerative bed effects and propagation of relatively shorter waves.

Kadomtsev–Petviashvili type equation for entire range of relative water depths

ABSTRACT A Kadomtsev–Petviashvili type equation valid for the entire range of relative water depths; namely, shallow, intermediate, and deep, is derived. The new equation is capable of simulating the shallow water cnoidal waves and the deep water second-order Stokes waves equally well besides accounting for wave shoaling due to varying water depths. Linear shoaling properties of the equation for unidirectional sinusoidal waves are in complete agreement with the energy flux concept. A finite-difference scheme is adopted for numerical solution of the equation to demonstrate its performance against test cases and its wide range of possible applications.

VARIATIONS IN NONLINEARLY EVOLVED NEARSHORE SPECTRA AND THEIR SIGNIFICANCE IN THE ESTIMATION OF WAVE FORCES

Detailed studies have been undertaken to assist in the design of major extensions to the port of Haifa. Both numerical and physical model studies were done to optimise the mooring conditions vis a vis the harbour approach and entrance layout. The adopted layout deviates from the normal straight approach to the harbour entrance. This layout, together with suitable aids to navigation, was found to be nautically acceptable, and generally better with regard to mooring conditions, on the basis of extensive nautical design studies.Hwa-Lian Harbour is located at the north-eastern coast of Taiwan, where is relatively exposed to the threat of typhoon waves from the Pacific Ocean. In the summer season, harbour resonance caused by typhoon waves which generated at the eastern ocean of the Philippine. In order to obtain a better understanding of the existing problem and find out a feasible solution to improve harbour instability. Typhoon waves measurement, wave characteristics analysis, down-time evaluation for harbour operation, hydraulic model tests are carried out in this program. Under the action of typhoon waves, the wave spectra show that inside the harbors short period energy component has been damped by breakwater, but the long period energy increased by resonance hundred times. The hydraulic model test can reproduce the prototype phenomena successfully. The result of model tests indicate that by constructing a jetty at the harbour entrance or building a short groin at the corner of terminal #25, the long period wave height amplification agitated by typhoon waves can be eliminated about 50%. The width of harbour basin 800m is about one half of wave length in the basin for period 140sec which occurs the maximum wave amplification.Two-stage methodology of shoreline prediction for long coastal segments is presented in the study. About 30-km stretch of seaward coast of the Hel Peninsula was selected for the analysis. In 1st stage the shoreline evolution was assessed ignoring local effects of man-made structures. Those calculations allowed the identification of potentially eroding spots and the explanation of causes of erosion. In 2nd stage a 2-km eroding sub-segment of the Peninsula in the vicinity of existing harbour was thoroughly examined including local man-induced effects. The computations properly reproduced the shoreline evolution along this sub-segment over a long period between 1934 and 1997.In connection with the dredging and reclamation works at the Oresund Link Project between Denmark and Sweden carried out by the Contractor, Oresund Marine Joint Venture (OMJV), an intensive spill monitoring campaign has been performed in order to fulfil the environmental requirements set by the Danish and Swedish Authorities. Spill in this context is defined as the overall amount of suspended sediment originating from dredging and reclamation activities leaving the working zone. The maximum spill limit is set to 5% of the dredged material, which has to be monitored, analysed and calculated within 25% accuracy. Velocity data are measured by means of a broad band ADCP and turbidity data by four OBS probes (output in FTU). The FTUs are converted into sediment content in mg/1 by water samples. The analyses carried out, results in high acceptance levels for the conversion to be implemented as a linear relation which can be forced through the origin. Furthermore analyses verifies that the applied setup with a 4-point turbidity profile is a reasonable approximation to the true turbidity profile. Finally the maximum turbidity is on average located at a distance 30-40% from the seabed.