For illustration, we perform several numerical simulations with the nonlinear Variational Boussinesq Model ( Adytia and van Groesen, 2012), to test and validate the method. The simulations aim to generate harmonic waves with period 55 s in a numerical basin with a depth of 2 m and length 15L, where L is the wavelength. The waves are
generated at x=0 with the (bidirectional) elevation influxing. At both ends of the basin, sponge-layers are placed to damp the waves. To test the adjustment-scheme, and the required length of the adjustment interval, various values of the amplitude are considered, corresponding to wave steepness in between ka=0.0075 and ka=0.12. Target Selective Inhibitor Library screening In Fig. 4 simulations with the linear model are shown in the first row, and simulations with the nonlinear model without and with adjustment in the second and third row respectively. The appearance of spurious free waves is clearly pronounced when the nonlinear simulation is performed without the adjustment scheme. By using
the length of the adjustment interval according to the information in Table 1, the results with the fully nonlinear VBM give good agreement with the 5th order Stokes waves ( Fenton, 1985) as illustrated in Fig. 5. A relative error of 2% compared to the 5th order Stokes wave has been used to determine the minimal length of the adjustment interval. To analyze the resulting www.selleckchem.com/products/ABT-263.html harmonic evolution in more detail, a Fast Fourier Transform (FFT) analysis of the
time series at each computational grid point has been performed. Fig. 6 shows the first-order (solid line) and the second-order (dotted line) amplitudes for various simulation methods: with the linear code (upper left plot), with the nonlinear code without adjustment (upper right plot), and with an adjustment interval of 2L (lower Selleck Dolutegravir left plot) and 5L (lower right plot). Since a linear influxing method misses the bound (second and higher) harmonics, a direct influx in the nonlinear model shows the release of spurious waves that compensate the missing bound waves. These spurious waves travel as free wave components, with opposite phase compared to the missing bound harmonic components in the linear influx signal (see also Fuhrman and Madsen, 2006). By applying an adjustment interval of sufficient length, shown in the lower right plot of Fig. 6, the second harmonic grows slowly to nearly steady in the adjustment zone, taking some energy from the first harmonic. If the length of the adjustment zone is not sufficiently long, for instance 2L2L as in the lower left of Fig. 6, the model is still releasing spurious waves. Since the performance depends on a nontrivial relation between the strength of the nonlinear waves to be generated and the length of the adjustment zone, as shown in Table 1, the method is still somewhat ad hoc and further investigations are desired.