Simulations of modulated plane waves using weakly compressible smoothed particle hydrodynamics

Abstract

Extreme waves, also known as ‘rogue waves’, have posed considerable challenges to maritime traffic over some time. Efforts have been directed at investigating the mechanisms governing these extreme energy localizations in oceanic environments. Modulational instability, also known as sideband instability, is one such mechanism that has been proposed to explain the occurrence of such phenomena in the framework of non-linear theory. The current work is aimed at better understanding the effects of sideband modulations on the propagation of unidirectional waves. To achieve this, a numerical wave tank (NWT) has been constructed using Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) to investigate the different parameters associated with the generation and propagation of plane, modulated waves. General Process Graphics Computing Unit (GPGPU) computing has been utilized to accelerate the computational process and improve the computational efficiency. The chosen numerical scheme has been validated by carrying out irregular waves focusing simulations to compare with available experimental data. Additionally, a Peregrine-type breather experiment has also been performed as part of the validation studies to look at energy localization within the NWT. The effects of the different parameters associated with the modulations to a plane propagating wave have been investigated using a blend of surface elevation data, eigenvalue, and frequency spectra. The effect of water depth on the perturbations to plane waves has been also investigated. The observations from these experiments can help shed light into the effects of modulations in the propagation of plane waves and help in the study of oceanic energy localization studies in future.

Appendix

Evolution of Peregrine-type breather in NWT using WCSPH

The Non-linear Schrödinger equation is the simplest theoretical model to explain the evolution of a unidirectional and narrow-banded modulated wave group in deep water, which can result in the formation of steep ocean waves through modulational instability. This equation admits a number of breather type soliton solutions. The Peregrine breather is one such soliton which is localized in both space and time, and the soliton breathes only once during its motion. The modulation evolution of the Peregrine breather solution has been investigated through wave tank simulations here.

Using the notations described in Shemer and Alperovich [34], the water surface elevation $\upzeta \left(x,t\right)$ for a wave group with carrier frequency ${\omega }_{0}$ and wavenumber ${k}_{0}$ satisfying the frequency dispersion relationship is given by:

$$\upzeta \left(x,t\right)=Re\left[a(x,t)\ {e}^{i\left({k}_{0}x-{\omega }_{0}t\right)}\right]$$

(29)

where $a(x,t)$ is the slowly varying complex wave group envelope. The characteristic wave amplitude ${a}_{0}$, the wave steepness $\varepsilon ={a}_{0}{k}_{0}$ and the wave group velocity ${c}_{g}$ are used to give the spatial NLS equation as

$$-i\frac{dA}{d\eta }+\frac{{\partial }^{2}A}{\partial {\xi }^{2}}+{\left|A\right|}^{2}A=0$$

(30)

where $\xi =$ $\varepsilon {\omega }_{0}\left(\frac{x}{{c}_{g}}-t\right); \eta = {\varepsilon }^{2}{k}_{0}x; A\left(\xi ,\eta \right)=a/ {a}_{0}$

The Peregrine soliton written in terms of these dimensionless variables is given by

$$A\left(\xi ,\eta \right)=-\sqrt{2}\left[1-\frac{4\left(1-4i\eta \right)}{1+4{\xi }^{2}+16{\eta }^{2}}\right]{e}^{-2i\eta }$$

(31)

The experiments detailed out in Shemer and Alperovich [34] were carried out in a wave tank 18 m long, 1.2 m wide and 0.9 m deep with a programmable flap type wavemaker for wave generation. The simulations in the present study are carried out in a 2D numerical wave tank which is 18.5 m long and 0.9 m deep, with a slope at the end starting from 14 m for passive wave absorption to reduce wave reflection. A flap type wavemaker at $x=0m$ is used for the wave generation. Surface elevation measurements were obtained at 13 different locations along the wave tank, as depicted in the simulation setup is provided in Fig. 21. 

Fig. 21

Numerical setup for Peregrine-type wave experiments. All dimensions are provided in meters. 13 wave gauges have been used for these simulations and located as shown in the figure. These are depicted using WGs

Fig. 22

The motion imparted by the flap wavemaker at x = 0 m. The maximum amplitude of the Peregrine breather is set at ${X}_{0}$ = 9 m. The proportionality factor is taken to be $\alpha$ = 1. The flap motion is kept at 0 for 65 ${T}_{0}$ after the tapered motion at the end to capture the observations at farther locations from the wavemaker

Simulations have been carried out using a water depth of $d=$ 0.6 m with the parameters: period ${T}_{0}=0.587s,$ corresponding to ${k}_{0}=0.587s, {\lambda }_{0}=0.538m$, $\varepsilon =0.0825$ and ${\zeta }_{0}=0.01m.$ The flap wavemaker motion is calculated using the Eqs. 29, 30 and 31 with a proportionality factor to achieve the wavemaker motion as

$$X\left(t\right)= \alpha A\left(x,t\right)$$

(32)

where $\alpha$ is the proportionality factor utilized to achieve a certain surface elevation during the wave tank simulations. The $A(x,t)$ in Eq. 32 [34] is computed to achieve the maximum of the Peregrine soliton at ${X}_{0}=9m.$ The total duration of the wave group itself has been taken to be $115 {T}_{0}$ with tapering windows of $10{T}_{0}$ over the two end periods. The wavemaker motion for $\alpha =1$ is given in Fig. 22.

The surface elevation readings at 4 different locations given in Shemer and Alperovich [34] are used to look at the evolution of the modulation along the wave tank. Considering the numerical dissipation present in WCSPH models, a proportionality factor of $\alpha =1.5$ was used during the wave generation to obtain a background wave amplitude at ${X}_{0}=9m$, similar to that observed in the experimental results detailed in Shemer and Alperovich [34]. The simulation observations along with the experimental results given in [34] are depicted in Fig 

Fig. 23

The evolution of a modulated wave group in a wave tank studied through surface elevation readings at different locations. The left plot shows the results from experiments carried out by Shemer et al. [34] The right plot shows the results from the WCSPH simulations in the present study

23.

From the surface elevation plots in Fig. 23, it can be observed that there are significant differences between the experimental results in Shemer and Alperovich [34] and the observations using the WCSPH simulations. This can be attributed to the numerical dissipation present in WCSPH. This results in gradual decrease of the background wave amplitude with increasing distance from the wavemaker at $x=0m.$ The evolution of the modulation can be observed clearly in the experimental results from the left plot with the maximum modulation amplitude observed at $x=11.6m.$ To get a better picture of the modulation evolution in our simulations, the surface elevation observations at more locations along the wave tank are depicted in Fig. 

Fig. 24

The evolution of the modulated wave group in the numerical wave tank using a flap type wavemaker with $\alpha$ = 1.5. Surface elevation readings at six different locations are depicted with respect to the wave group velocity. The center of the modulated wave group is at $\left(t-x/c_g\right)T_0= 0$

24. The surface elevation observations for the WCSPH simulations at six different locations along the numerical tank are shown in Fig. 24. It can be seen from these plots that the background wave amplitude gradually starts to decrease with increasing distance from the wavemaker due to numerical dissipative effects.

However, the growth of the modulation can be observed clearly by looking at the ratio of the peak modulation amplitude to the average background wave amplitude. For the sake of subsequent break-down of the observations, this will be referred to as the amplitude ratio $\chi$ in further discussions. At $x=1.85 m$, the ratio $\chi$ is observed to be 1.31. The growth of the modulation down the numerical wave tank can be observed from the plots at $x=2.25m$ and $x=5.65m$. The amplitude ratio $\chi$ is found to increase to 1.49 at $x=2.25m$ and 1.92 at $x=5.65m$, where it is found to reach the maximum during the simulation duration. The modulation amplitude starts diminishing after this position, with the amplitude ratio between the modulation and the background going down to $1.8$ and $1.5$ at $x=7.65m$ and $x=11.6m$ respectively. These observations depict the evolution of a modulated wave group for a Peregrine-type breather solution for NLSE in our developed NWT. However, due to the numerical dissipation in the WCSPH model, the amplitude, and the location of the maximum evolution amplitude in the simulation results differ from theoretical and experimental results given in Shemer and Alperovich [34].

A second set of simulations was performed using a larger proportionality factor of $\alpha =3.0$ using the same set of parameters. Wave breaking is not observed during the course of this simulation. The surface elevation readings at six separate locations for these simulations are given in Fig. 

Fig. 25

The evolution of the modulated wave group in the numerical wave tank using a flap type wavemaker with $\alpha$ = 3.0. Surface elevation readings at six different locations are depicted with respect to the wave group velocity. The center of the modulated wave group is at $\left(t-x/c_g\right)T_0= 0$

25. Similar to the case of $\alpha =1.5$, it is observed from the plots in Fig. 25 that the background wave amplitude decreases with increasing distance from the wavemaker. However, the ratio of the modulation amplitude to the background amplitude shows a growth from $x=0m$ to $x=6.25m$. As observed from the elevation readings in Fig. 25, the amplitude ratio $\chi$ shows an increase from the initial ratio of 1.3 at the wavemaker to $\chi =1.35$ at $x=2.25m.$ The modulation gets enhanced further down the numerical wave tank as observed from the plots at $x=4.25m,$ $x=5.65m$ and $x=6.25m$. The amplitude ratio $\chi$ evolves through 1.47 and 2.125 at $x=4.25m$ and $x=5.65m$ respectively to a maximum of $\chi =2.8$ at $x=6.25m$. Similar to the case of $\alpha =1.5$, the amplitude ratio is observed to reduce after this location, with the values observed being $1.8$ and $1.66$ at $x=9m$ and $x=9.25m$ respectively. The numerical experiments carried out for the Peregrine-type breather extend our validation efforts for the numerical WCSPH scheme utilized in our studies of plane modulated waves. It can be ascertained from these simulation results that the developed numerical scheme is able to capture the modulation evolution process that results in the formation of large Peregrine type breather solutions to the NLSE, which is very similar to observed extreme waves in oceans. However, there are discernible differences when compared to previous experimental studies on these waves. Such disagreements can be attributed to the numerical dissipation which is necessary for the stability of the weakly compressible scheme in our studies. This mainly gets manifested for waves with smaller amplitudes and larger distances from the wavemaker. Thus, for the larger proportionality factor that we have used in our study, the modulation growth was observed to be more prominent. The phenomenon of numerical dissipation in our model and methods to reduce its effects will be investigated in future studies to be carried out by the authors.