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When the Froude number F of a free-surface flow ranges between 0.3 and 3, the flow is unstable and frequently characterised by free surface undulations, with the undular hydraulic jump being a seminal flow in hydro-environmental mechanics. The presence of the free surface undulations significantly affects the flow field, with major velocity and pressure field redistributions between successive crests and troughs. All the current theoretical models to simulate undular hydraulic jumps are limited to two-dimensional flow conditions, ignoring all the relevant three-dimensional flow effects, namely shock-wave drag, turbulent breaking and turbulent stresses. These aspects are critically accounted for in this review article, where a depth-averaged Boussinesq model which approximately accounted for 3D flow effects was presented, constituting the first attempt in this line. The model predictions were compared with experimental results from different sources for F1 < 1.5, with F1 the inflow Froude unnumber, resulting a reasonable agreement with observations. The curvature distribution parameter was found to controlling the wave length, and an approximate value was obtained based on ideal fluid flow computations. The new depth-averaged model did not include the effect of flow concentration in the centerline, observed physically, but this 3D feature is analysed at the first wave crest based on an improved treatment of flow curvature, highlighting the impact of the ratio qCL/q on the velocity profile features with qCL the centerline discharge. The main limitation of the new model, presented in this critical review, originated from approximating a complex 3D flow by a depth-averaged model. However, the predictions with the new approximate treatment of 3D effects produced better results than those previously reported. [ABSTRACT FROM AUTHOR]

An extension of the Boussinesq-type models to weakly compressible flows is derived in the fully nonlinear case. The dispersive properties are consistent with the linear theory of compressible fluids at the long-wave limit. The particular case of a vanishing Mach number gives a quasi-incompressible model, intended for coastal wave simulations, which is a hyperbolic version of the Serre–Green–Naghdi equations, with a new treatment of the bathymetric terms. Both the compressible and quasi-incompressible models are hyperbolic four-equation models on an arbitrary bathymetry, with an exact equation of energy conservation. In addition, these models are extended to hyperbolic and fully nonlinear five-equation versions with improved dispersive properties. A remarkable property of the quasi-incompressible model with improved dispersive properties is that it is possible to decrease significantly the sound velocity, and thus the computational time, with the same accuracy or even slightly better. The numerical results show good agreement of the quasi-incompressible model with experimental data and the capability of the compressible model to calculate the decrease of tsunami velocity due to compressible effects. • Derivation of a Boussinesq-type model with compressibility effects. • The model is hyperbolic and fully nonlinear. • Model with improved dispersive properties. • New treatment of the bathymetry. • The model predicts the decrease of tsunami velocity due to compressibility. [ABSTRACT FROM AUTHOR]

Early Warning Systems (EWS) are nowadays becoming fairly standard in river flood forecasting or in large scale hydro-meteorological predictions. For complex coastal morphodynamic problems or in the vicinity of complex coastal structures, such as harbours, EWS are much less used because they are both technically and computationally still very challenging. To advance beyond the state-of-the-art, the EU FP7 project RISC-KIT ( www.risc-kit.eu ) is developing prototype EWS which address specifically these topics. This paper describes the prototype EWS which has been developed for the case study site of the harbour of Zeebrugge, situated in Flanders along the Belgian coast, allowing the validation of the newly developed tools. The challenge for this EWS and DSS (Decision Support System) is selecting the right number, type, and detail of the models in order to get a sufficiently detailed and trustable results, while keeping calculation time limited in order to allow fast and frequent predictions. In general, waves inside harbours are a combination of locally generated wind waves and offshore wave penetration at the port entrance. Outside a prediction environment, the conditions inside the harbour could be assessed by superimposing processes. The assessment can be carried out by using a combination of a spectral wave model (i.e. SWAN) for the wind generated waves and a Boussinesq type wave model (i.e. Mike 21 BW) for the offshore wave penetration. Finally, a 2D hydrodynamic model (i.e. TELEMAC) can be used to simulate the overland flooding inside the port facilities. To reproduce these processes under a EWS environment, an additional challenge is to cope with the limitations of the calculation times. This is especially true with the Boussinesq model. A model train that integrates process-based modelling, for wind generated waves, with a smart simplification of the Boussinesq model for the wave penetration effects, is proposed. These wave conditions together with the extreme water levels (including storm surge) can then be used to simulate the overtopping/overflow behaviour for the quays. Finally, the hydrodynamic model TELEMAC is run for the inundations inside the port facilities. The complete model train was integrated into the Deltares Delft-FEWS software for scenario simulating to showcase the potential for real time operations. [ABSTRACT FROM AUTHOR]

Intermediate-depth, Boussinesq-type modelling is used to generalize previously known results for surface water waves propagating over arbitrarily shaped topographies. The improved reduced wave model is obtained after studying how small changes in the linear dispersion relation (over a flat bottom) can become dramatically important in the presence of a highly fluctuating topography. Numerical validation of the dispersive properties, regarding several possible truncations for the reduced models, are compared with the complete (non-truncated) linear potential theory model. Moreover, linear L2-estimates are extended from the analysis of KdV-type models to include the improved Boussinesq systems in contrast with potential theory. Discrepancies observed among the different possible reduced models become even more important in the wave-form inversion problem. The time reversal technique is used for recompressing a long fluctuating signal, representing a highly scattered wave that has propagated for very long distances. When properly back-propagated (through a numerical model), the scattered signal refocuses into a smooth profile representing the onset of the ocean's surface disturbance. Previous Boussinesq models underestimate the original disturbance's amplitude. The improved Boussinesq system agrees very well with the full potential theory predictions. [ABSTRACT FROM PUBLISHER]

Abstract: A non-linear wave propagation model, based on the higher order depth-integrated Boussinesq-type equations for breaking and non-breaking waves, was applied to predict irregular wave transformation in two horizontal dimensions. A new source function, adapted for the proposed equations, is introduced inside the computational domain, to generate the desired short-crested waves. The dissipation due to the roller is introduced in the momentum equation in order to simulate wave breaking. Bottom friction and sub-grid turbulent processes are also introduced in the model. At the open boundaries a damping layer is applied together with a radiation boundary condition. Model results are compared with experimental measurements, containing tests with normal or oblique to the shore long- and short-crested irregular waves. The comparisons show that the model is able to simulate successfully the non-linear evolution of a unidirectional or a multidirectional wave filed in the nearshore zone, under the effects of refraction, shoaling, and breaking. [Copyright &y& Elsevier]

We briefly analyze some characteristics of the behavior in very shallow waters i.e. near the shoreline of high-order (dispersive-nonlinear) Boussinesq-type equations. By using the solution as test flow conditions we illustrate the behavior of both purely dispersive and dispersive-nonlinear contributions near the shoreline. It is also shown that Boussinesq-type equations can be more usefully handled in the swash zone if written in terms of the total water depth. [Copyright &y& Elsevier]