# Directional random waves in presence of horizontally shearing current

## Description

This numerical experiment aims to demonstrate that the wave-current interactions can enhance the rogue wave probability in spreading seas through reproducing the experiment carried out in the laboratory by Toffoli et al. (2011). The Enhanced Spectral Boundary Integral (ESBI) (Wang & Ma, 2015; Wang et al., 2018) method is employed to perform the numerical simulations, and comparisons between the numerical and experimental results will be made in order to verify the robustness of the ESBI model.

## Experimental Set-up

The NWT covers 78m X 58m in space and is resolved into 2048 X 512 collocation points, while directional wavemaker is installed at one fourth of the total tank length from the left boundary. Waves are absorbed at surrounding boundaries and only part on the right of the wavemaker is effectively used. The current flows obliquely in an angle of 110 degrees with respect to the main wave direction, and the configuration of the NWT is shown in Fig. 1. The maximum current velocity is 0.2m/s, and the current profile along Y=0 is approximated by using

where c_0 and L_0 are the phase velocity and wavelength corresponding to the peak spectral component, Η is the Hermitian function. The fitted current profile by using the above formula agrees well with that measured in the laboratory as indicated in Fig. 2.

The JONSWAP spectrum is employed with peak period of 1s, significant wave height 0.08m and peak enhancement factor 6. The spreading function is chosen as

where Γ is the Gamma function and N=24.

Fig 1. Configuration of the NWT

Fig 2. Fitted current velocity profile along Y=0. ‘x’: Measured in Toffoli et al. (2011); ‘ ̶ ’: Fitted

## Experimental Test Program

The numerical simulations with and without current presence are performed and each last for 500 peak periods. The free surface elevation is recorded at the locations as marked in Fig. 1. The surface time history is then used for calculating the kurtosis, which is a quantitative indication of the rogue wave probability. Then the values based on numerical simulations will be compared with those collected in laboratory.

## Numerical Benchmarks

Snapshots of the free surface elevation in space at the 150th peak period are presented in Fig. 3, where a qualitative comparison between Fig. 3(a) and (b) reveals that the number of extreme waves in presence of current are more than that without current presence. To demonstrate that quantitatively, the kurtosis at different locations along the main wave direction is estimated and compared the experimental results, as presented in Fig. 4. It shows good agreement between the curves obtained numerically and experimentally, and it is observed that the kurtosis subjects to a deceleration of its growth throughout the propagation. Meanwhile, the kurtosis preserves slightly higher values with current presence, indicating a higher probability of rogue wave occurrences.

Fig 3. The spatial distribution of the spreading wave surface

Fig 4. Kurtosis against probe locations. ‘- - -’: 2nd order prediction; ‘-o-’: without current, Toffoli et al. (2011); ‘-•-’: wit current, Toffoli et al. (2011); ‘-△-’: without current, ESBI; ‘-▲-’: with current, ESBI

## Relevant References

Toffoli, A., Cavaleri, L., Babanin, A.V., Benoit, M., Bitner-Gregersen, E.M., Monbaliu, J., Onorato, M., Osborne, A.R. and Stansberg, C.T. (2011). Occurrence of extreme waves in three-dimensional mechanically generated wave fields propagating over an oblique current. Natural Hazards and Earth System Sciences, 11(3), 895-903.

Wang, J., & Ma, Q. W. (2015). Numerical techniques on improving computational efficiency of spectral boundary integral method. International Journal for Numerical Methods in Engineering, 102(10), 1638-1669.

Wang, J., Ma, Q. W., & Yan, S. (2018). A fully nonlinear numerical method for modeling wave–current interactions. Journal of Computational Physics, 369, 173-190.