Description
Waves propagating against an opposing current may be blocked at a certain point if the current is sufficiently strong. Such a study has been carried out by Ma et al. (2010) in a physical wave tank. The purpose of this experiment is to reproduce some of the scenarios reported in Ma et al. (2010) and verify the robustness of the improved Enhanced Spectral Boundary Integral (ESBI) method (Wang & Ma, 2015; Wang et al., 2018) for simulating fully nonlinear wave-current interactions.
Experimental Set-up
The computational domain of the Numerical Wave Tank (NWT) covers 32 peak wave lengths and is resolved into 1024 points. A pneumatic wavemaker is employed for wave generation and it is centred at one fourth of the total tank length from the left boundary, which is considered as the origin of the X-Axis. Only the part on the right is effectively used for wave propagation in this case. The current field is steady and prescribed as indicated by ‘Fitted’ in Fig. 1. The selected wave conditions are summarised in Table 1. A group of wave probes are placed along the NWT and spaced at every 1m, which is shortened to 0.5m in the range 16m - 23m.
Fig 1. Profile of the horizontally shearing current against distance. ‘Theoretical’: calculated values based on continuity equation; ‘Measurement’: values recorded in laboratory; ‘Fitted’ values obtained by numerical interpolation.
| Case No. | Wave Period (s) | Wave Height (cm) | Wave Number (m-1) |
|---|---|---|---|
| 1 | 1.0 | 2.21 | 6.6 |
| 4 | 1.1 | 2.50 | 5.2 |
| 7 | 1.2 | 2.61 | 4.2 |
Experimental Test Program
Each simulation lasts for 400 wave periods, and the last 100 periods are extracted for further analysis. Fourier transform to the recorded free surface elevation is performed to obtain the spectrum, based on which the local frequency (ω_l) corresponding to the spectral peak at all probes can be estimated.
Numerical Benchmarks
The variation of the local frequency along the NWT is demonstrated in Fig. 2. Instant frequency downshifting is observed in case 1&4, as the wave phase celerity is insufficient to penetrate the current field so that they are blocked at certain points. However, in case 7, waves are moving sufficiently fast therefore can go through the current field. In addition, the comparison of the free surface time history between the laboratory experiment data and numerical simulation results for case 1 is shown in Fig. 3, where it reveals that the surface elevation predicted by using ESBI agrees reasonably well with the experiment data, and the blockage of the waves due to the opposing current is successfully captured by the ESBI model.
Fig 2. Local frequency against distance. ‘ ̶ ’: ESBI; ‘ • ’: Measurement by Ma et al.(2010)
Fig 3. Comparison of the free surface time histories for case 1. ‘ ̶ ’: ESBI; ‘…’: Measurement by Ma et al.(2010)
Relevant References
Ma, Y., Dong, G., Perlin, M., Ma, X., Wang, G., & Xu, J. (2010). Laboratory observations of wave evolution, modulation and blocking due to spatially varying opposing currents. Journal of Fluid Mechanics, 661, 108-129.
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.