Solitary Wave Propagation over a Complex Shelf
This benchmark consists in a single solitary wave propagating up a triangular shaped shelf with an island feature located at the offshore point of the shelf. As computational domain the extended region [-9,44.6]x[-13,13] was considered. The initial condition for the free surface elevation consisted in a soliton, whose approximate expression taken from Tonelli and Petti (2009) is given by:
η(x,0)=A sech [(x-x0)√(3A/4H3)]
besides an initial velocity correction is imposed as:
u(x,0)= η(x,0) √(gH)/H
with x0=-3.3 m; A=0.39 m; H=0.78 m; g=9.81 m/s. Thus, the initial conditions imposed were, h(x,y,0)= η(x,0)+0.78; qx(x,y,0)= h(x,y,0) u(x,0) and qy=0. Two NLSW models were used, one with and the other without dispersion. The non-dispersive NLSW model considers two numerical schemes (second and third order), with three mesh resolutions (2.5, 5 and 10 cm) and varying friction ranging from 0.005 to 0.035. The model including dispersion implements a third order numerical scheme for the NLSW terms and second order for dispersive terms, uses a resolution of 10 cm (490 by 260 cells) and a varying friction ranging from 0.005 to 0.035. The breaking criteria used, although is not the same, is based on the hybrid criteria from Kazolea et al. (2014). The dispersion coefficient is set to 1/21.
Some conclusions can be extracted from the numerical experiments performed. First, it should be mentioned that for this problem configuration A/H = 0.5 in the left-hand side of the domain where propagation of the initial perturbation takes place. This makes dispersion mandatory in this part of the domain. But at the same time nonlinearity becomes predominant when approaching the shelf and in the breaking zone. Therefore, a hybrid model is needed to suitably represent both stages of the flow: the dispersive-dominated flow and the non-linear shallow water dominated flow. For numerical comparison, we used first a NLSW model without dispersion. In this case, increasing numerical resolution from 10 cm to 5 cm and 2.5 cm has a minor effect, producing similar (bad) results. Varying friction in the NLSW non dispersive model has negligible effects on points located in front of the obstacle and the effect of varying friction it is mostly felt at points behind the obstacle, more while “more behind” with smaller friction producing results closer to measured data. When the model including dispersion is used much better results, closer to measured data are obtained, mainly for free surface elevation, with larger discrepancies in the v component of the velocity. Figure 8 shows the comparison between numerical results and measurements for the free surface elevation at the nine locations considered. For locations in front of the obstacle (x=7.5 and x=13m) the shape and amplitude of the signal is well captured and friction has some minor effect. For points located behind (x=21 and x=25m) friction plays an important role and a better fit to observations is obtained for lower values of friction. The breaking criteria starts acting at the shelf, where the flow becomes shallower, in particular at y=0m this happens at x=13m. In Figure 8, in the graphic for location x=13m - y=0m it can be observed how the breaking criteria starts acting and the soliton starts to dissipate, while at location x=7.5m - y=0m it can be observed how it is fully present. Figure 9 depicts the comparison between model results and measurements for the two components of the velocity at x=13m y=0m location. The structure of u component is captured but model fails to reproduce the v component at this location. Nevertheless, some doubts about the accuracy of the velocity component in the y-direction at ADV1 and ADV2 locations has previously been mentioned (Shi et al., 2012), arguing that too small values were recorded.
Figure 8. Comparison of free surface elevation measurements with NLSW dispersive model (varying friction) at the 9 given locations.
