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Numerical treatment of breaking

A disadvantage of numerical models for wave propagation considered in this chapter is their inability to model spilling breakers. The discontinuity of the free surface that develops at the spilling crest leads to a singularity in the numerical solution, resulting in a breakdown of the calculations. Models based on Lagrangian representation of the free surface (MEL and fully Lagrangian) can simulate overturning waves and, therefore, with sufficient spatial and temporal resolution they can resolve micro-plungers originating at wave crests during the initial stages of spilling breaking. However, they are unable to continue the calculations after self-contact of the free surface occurs and the resulting solution becomes non-physical. This makes impossible applying such models to steep travelling waves and severe sea states.

Removing a singularity at the breaking crest can help in continuing calculations with only a minor effect on the overall wave behaviour. This can be achieved by implementing artificial local dissipation in the vicinity of a wave crest prior to breaking. With this approach all small-scale local features would disappear from the solution, but the overall behaviour of the wave would still be represented with good accuracy. Practical implementation of the method includes using of a breaking criterion to initiate dissipation right before breaking occurs. The dissipation is usually enforced by including damping terms in the free-surface boundary conditions (Haussling and Coleman, 1979; Subramani et ah, 1998; Guignard and Grilli, 2001). Recently, a breaking model based on an advanced breaking criterion and an eddy viscosity dissipation model was developed by Tian et al. (2012) and implemented in a spectral model of wave evolution (Tian et ah, 2012; Seiffert and Ducrozet, 2018). The method demonstrates a good comparison with the experiments and allows to apply spectral models to simulate the evolution of severe sea states with breaking waves.

In this chapter we use a method of treatment of spilling breaking which uses the same basic concept but differs in the details of realisation. The method includes dissipative suppression of the breaker and correction of crest shape to provide accurate post-breaking behaviour of the wave. There are several conditions such a method should satisfy; (i) to act locally in the close vicinity of a developing singularity without affecting the rest of the flow; (ii) to simulate energy dissipation caused by breaking; (iii) to be mesh-independent, that is, the change of the effect with changing mesh resolution should be within the accuracy of the overall numerical approximation and (iv) to be naturally included into a problem formulation representing an actual or artificial physical phenomenon. The development of a spilling breaker is associated with a rapid growth of surface curvature. Therefore, the local dissipation effect satisfying these conditions can be created by adding a term —o d/da (дк/dt) to the right hand side of the free surface dynamic condition (8, 9). This term with a small coefficient к which acts locally at the region of fast curvature changes and suppresses breaker development without affecting the rest of the wave. To minimise the undesirable effect of dissipation, the action of the damping term is limited both in time and in space. Breaking dissipation is triggered when the maximal acceleration of fluid particles at the crest exceeds a specified threshold aon and is turned off when the maximum acceleration falls below a second lower value a0fj. Spatially, the action of the breaking model is limited by the half-wavelength between the ascending and descending zero-crossing points delimiting a breaking wave crest.

The activation of the damping term makes it possible to continue the calculation beyond the breaking event. However, the resulting shape of the wave crest is different from the actual crest after the breaking. Since local dissipation suppresses breaking, the local behaviour of the wave crest is different from the real one. Overturning of the crest does not occur, and for a sufficiently intense breaking, it can significantly affect the shape of the entire wave around the crest. To account for this difference, we increase the surface tension around the crest. Numerical tests show that large surface tension produces an effect similar to that of the peak overturning, changing the shape of the crest and reducing the error in the profile of the post-breaking wave. This effect is achieved by the surface tension term in the right side of the free surface boundary condition (8,9) with an appropriately selected coefficient 7br added to the natural value of 7. It should be emphasised that the desired effect is only possible for values of the qbr much larger than the actual ones and it is used only in the regions and during the periods when the breaking model operates.

To summarise, the intensity of the dissipation (er), the acceleration thresholds to activate and deactivate the dissipation (aon and aDff) and the strength of the surface tension for the correction of the shape of the crest (7br) constitute the four parameters of the breaking model. The functions of the parameters of the model are as follows. Parameter aon defines the beginning of the breaking process, a regulates the rate of energy dissipation, a0ff controls the duration of the breaking and the total amount of energy dissipated and 71,r corrects the shape of the breaking crest. It should be noted that being a heuristic model, the breaking model requires calibration of parameters to achieve optimal performance for each particular case. For this chapter, the parameters were selected by running a small number of numerical cases for different parameters values. The following values of the parameters are selected for calculations presented below: aon = g, «off = <7/2, cr = 0.0002 m3/s and 71,r = 0.01 m3/s2. Additional work is required for systematic study of the effects of the parameters on model performance and to establish a rational procedure to select the parameters for different breaking conditions.

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