Experimental and Numerical Investigation on the Interactions between the Weakly Three-Dimensional Waves

Abstract

The results of laboratory experiments and numerical simulations were performed to investigate the interactions between the weakly three-dimensional waves in an ‘X’ configuration, which has a 16-degree approaching angle. In addition, another oblique two-dimensional experiment was also conducted for comparison with the ‘X’ configuration but in one single channel by removing a dummy wall in the interaction region. Our experimental results show that as the wave trains propagate into the interaction region, it is obvious that there is an increase in the wave height which reaches a maximum height of about 1.37H0 for different initial wave steepness at the center of the interaction region, and then decreases thereafter, where H0 is the input wave height. Then wave elevations at different positions downstream of the interaction region were also studied, indicating that the frequency and initial wave steepness were highly correlated with the wave-wave interaction between the weakly three-dimensional waves. For the wave with low frequency (f = 0.8 Hz), a crescent wave surface formed at the beginning of the interaction and then separated into two two-dimensional waves after the interaction, which illustrates that the waves can still keep their initial characteristic and propagate as their initial directions downstream of the interaction region. While the frequency increased (f = 1.2 Hz), three-dimensional effects appeared to dominate the interaction of weakly three-dimensional waves, especially for the large initial steepness, and the wave surfaces were also three-dimensional after interactions. Finally, numerical simulations with larger approaching angles were conducted to further understand the influence of propagation direction on the interactions between the weakly three-dimensional waves. The results suggest that intense interactions and strong three-dimensional characteristics of the wave trains downstream interactions can result from larger approaching angles.

1. Introduction

It is well known that wave-wave interactions are ubiquitous on the ocean surface and have become a topic of intensive study in these years. The wave-wave interactions, which are complex and nonlinear process, can result in a significant increase in wave height [1] that bring serious threat to structures and vessels in the ocean [2]. It is widely accepted that wave-wave interactions are also strongly correlated to the generation of freaking waves [3,4] as well as breaking [5,6,7]. Therefore, further study of wave-wave interactions is seriously necessary.

For the past decades, lots of investigations have been made to examine wave-wave interactions [8,9,10]. Pioneered by Hasselmann [11,12,13] and Zakharov [14,15], the wave turbulence theory explains the transfer of energy between different scales on the water surface. An important assumption of this theory is an asymptotic closure of the hierarchy of cumulants leading to a set of evolution equations for the spectrum variables, that is, the quadratic energy of the spatial Fourier transforms of the field variables. Then the turbulence spectrum is very well confirmed by both experimental [16] and numerical [17,18] methods. Furthermore, a new turbulence spectrum characteristic of magnetohydrodynamic turbulence can be realized on the water surface in the presence of an external magnetic field [19]. Although the wave turbulence theory has been successful in developing a statistical theory and accurately predicting the wave spectrum in a non-equilibrium stationary state, it does not account for all phenomena observed in nature or in laboratory settings, and single-point measurements limited the wave field analysis for a long time. In this connection, there is an increasing interest in this particular research that wave components with different frequencies and crests are superimposed at some position in space and time. Baldock et al. [20] found that high-order wave-wave interactions play an important function in the development of a rogue wave where wave steepness and the bandwidth of frequency have a great effect on the nonlinearity of wave groups. Analogously, Ma et al. [21] also reported that the influence of frequency bandwidth on focusing wave packets with a small steepness is negligible by examining the unidirectional focusing waves in intermediate-depth water based on JONSWAP spectra in the laboratory. In contrast, the nonlinearity increases with the increase of frequency bandwidth for cases with large steepness. The spectral evolution of wave packets was examined by Tian et al. [22] with FFT that nonlinear energy transfers between the above-peak and high-frequency band in the focusing process. In addition, Wu and Yao [23] investigated the focusing waves with a method of HHT in detail, which vividly illustrated the evolution of the time-frequency-amplitude spectrum of focusing waves. The result showed that there was a strong relationship between the focusing wave nonlinearity and the magnitude of instantaneous frequency modulation.

As mentioned above, most of the studies are conducted in the condition of unidirectional focusing of wave components. However, it is important to keep in mind that waves in real sea states are always multidirectional, and consequently, it is essential to investigate how directionality affects wave-wave interactions. Recently, it attracted more attention in the research of weakly three-dimensional wave interactions which occur when two wave trains approach at an angle and is quite common in the real sea state since the crossing state has been identified as one of the factors contributing to freaking waves in recent years [24,25,26]. Significant work has been done by Onorato et al. [27] with Nonlinear Schrodinger (NLS) equations to explain the formation of freaking waves in weakly three-dimensional sea state and proposed that the modulational instability is of great importance for the occurrence of freaking waves. Meanwhile, Toffoli et al. [28] also showed a result that there is an increase in the probability of freaking waves in weakly three-dimensional seas. Gramstad et al. [29] considered the effects of approaching angles, differences in peak frequency as well as the spectral shape of two wave trains and provided that a positive correlation exists between kurtosis and the maximum unstable growth rate of two Stokes waves. Luxmoore et al. [30] also examined carefully how the kurtosis can work as an indicator of nonlinearity in the state of waves interacting with an approaching angle, showing that variations in the directional propagation of the components affected the kurtosis as well as wave crest height exceedance probabilities.

In this study, an ‘X’ configuration experiment was conducted firstly with an approaching angle of 16° to investigate the influence of directional propagation of wave trains with different frequencies and initial steepness. Then a numerical simulation by a reliable and accurate wave model is implemented for further investigations of the influence of approaching angles on wave-wave interaction since the space limitations where only a few experiments with relatively small approaching angles were conducted. The paper is arranged as follows: the experimental setup is described in detail in Section 2. Experimental results and discussion are presented in Section 3. Numerical results are shown in Section 4. Lastly, conclusions are drawn in Section 5.

2. Experimental Setup

2.1. Experimental Facility

The experiments were conducted in the wave basin, which is 50 m long, 24 m wide, and with a water depth of 0.6 m at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China. A servo-controlled serpentine wavemaker that has 70 individual 0.34 m wide paddles is installed along one side of the basin. While towards the end of the basin, a wave-absorbing beach is equipped to mitigate reflections. In the wave basin, an ‘X’ configuration which consisted of two wave channels of 0.8 m width, was installed to isolate two identical wave trains before superposition (see Figure 1). In the following, the two wave channels are called the A flume and the B flume, respectively, for convenience, as shown in Figure 1. Two coordinate systems are used here, as is seen in Figure 1: one is a local coordinate system (x,y) with the x direction following the A/B flume and the y direction perpendicular to the A/B flume, and the other is a global coordinate system (x_G,y_G). Since the bottom of the two flumes varies over a distance, the beginning of the two flumes is located at x = 3.5 m. A glass sidewall was constructed between intersecting parts of the channels to observe the spatial variations of surface elevation profiles. In addition, the ‘X’ configuration was protected with wave absorbers to reduce wave reflections. It is evident from Figure 1 that all of these features exist.

During the experiment, the time series of water surface elevations were measured by a series of well-calibrated capacitance wave probes of the order ±1 mm, along the wave channels, which are denoted in Figure 1 by dots. Each experiment was repeated three times to ensure repeatability. To find out whether waves retain a two-dimensional structure before superposition in the ‘X’ configuration, we measured the wave surface elevations by two in-line probes, as seen in Figure 1, which also examined the features of the wave surface after the interaction. Upstream and downstream of the interaction region, the wave probes were arranged with a distance of 1.17 m, while adjustments were made to the distance in the interaction region.

2.2. Generation of Weakly Three-Dimensional Waves

To investigate the interaction between the weakly three-dimensional wave trains, two identical regular waves of opposite approaching directions were generated first using a serpentine wave maker. In this study, the method for the simulation of multi-directional irregular waves was adopted to generate two identical wave trains with ±8° from x_G respectively (see Figure 1). With the linear theory, it is possible to express the elevation of the surface at an arbitrary point (x_G, y_G) as a linear superposition of regular waves with different frequencies and directions:

$$\eta (x_G,y_G,t)={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^M{a}_{ij}\cos (k_{i}x\cos {\theta }_j+k_{i}y\sin {\theta }_j-{\omega }_{i}t-{\epsilon }_{ij})}}$$

(1)

where N represents the number of wave components, M represents the number of wave directions; θ_j represents the jth wave direction in the global coordinate system, herein θ_j = ±8°; a_ij represents the corresponding wave amplitude; ω_i represents the radian frequency of the ith wave component; ε_ij represents phase shift; and k_i represents wave number, which can be obtained from the linear dispersion relation:

$${({\omega }_i)}^2=k_{i}g\tanh (k_{i}d)$$

(2)

where d represents the depth of water and g represents the gravitational acceleration. It is important to note that k_i indicates the magnitude of the wave number in the global coordinate system.

Figure 2 illustrates the sketch of the wave-maker in the global coordinate system, in which the y_G direction is just parallel to the mean position of the wave paddle with y_G = 0 m defined at the center of the first wave paddle, while the x_G direction is perpendicular to y_G corresponding to Cartesian coordinates. In this study, the identical wave trains with opposite approaching angles can be generated by managing phase differences between adjacent paddles of a serpent-type wave maker:

$$\sin \theta =L/nD$$

(3)

$$n\mathsf{\Delta }\phi =2\pi$$

(4)

where L represents the wavelength; n represents the number of the paddle in the range of wavelength; D represents the width of a paddle, and Δφ represents the phase difference of adjacent paddles. Thus, the phase difference Δφ can be obtained by substituting Equation (3) into (4):

$$\mathsf{\Delta }\phi =kD\sin \theta$$

(5)

Therefore, the wave-maker driving signal of the nth paddle for generating the oblique regular waves can be written by (get M = N = 1, ε_ij = 0, x = 0 for regular wave):

$$X(n,t)=E\sin (knD\sin \theta -\omega t)$$

(6)

Here E_i = a_i/K(f_i, θ), and K(f_i, θ) represents the wavemaker amplitude transfer function:

$$K(f_i,\theta )=\frac{2(\cosh 2k_{i}d-1)}{(\sinh 2k_{i}d+2k_{i}d)\cos \theta }$$

(7)

2.3. Wave Conditions and Wave Trains

In this study, two monochromatic waves with different initial wave steepness were investigated, as is seen in

Table 1. Wave parameters. (In this table, f represents the frequency; H₀ represents the initial wave height at x = 3.5 m; IMS = H₀k/2 represents the initial wave steepness).

Case	f(Hz)	k(m−1)	H0(cm)	IMS	Comments
A1	0.8	2.77	8.07	0.11	non-breaking
A2	0.8	2.77	14.84	0.21	non-breaking
A3	0.8	2.77	17.11	0.24	non-breaking
A4	0.8	2.77	18.77	0.26	breaking
B1	1.2	5.81	3.78	0.11	non-breaking
B2	1.2	5.81	4.79	0.14	non-breaking
B3	1.2	5.81	7.07	0.21	non-breaking
B4	1.2	5.81	8.43	0.24	non-breaking
B5	1.2	5.81	12.50	0.36	breaking

. During each experiment, three duplicated measurements were conducted with a record of 30 s as well as a sample interval of 0.005 s. The parameters offered in Table 1 are estimated from the surface elevations recorded at the position x = 3.5 m as the preliminary wave parameters.

As is seen in Figure 3, the comparison of wave elevations measured upstream of the interaction region is provided, where the solid lines represent the surface elevations measured on the right-hand side in the A flume; the dash lines represent that measured on the left-hand side in the A flume; the dash-dot lines represent that measured on the left-hand side in the B flume. It is obvious that the wave-maker drive signals defined aforementioned were verified to generate an appropriate agreement of the surface elevations, which were recorded from the probes in-line in the A flume as the wave train propagated to the interaction region, as is seen in Figure 3a. Additionally, representative wave surface elevations in the B flume are also provided in Figure 3b. The outcomes show that an excellent agreement has been achieved between the surface elevations in the two channels. Therefore, by making use of this facility, two virtually similar wave trains can be produced; more important, the interaction of these weakly three-dimensional wave trains might be explored further. Herein, it is noted that only the first few waves were concerned on account of the influence of reflections from the interaction region contributing to contaminating initial waves.

3. Experimental Results

3.1. Evolution of the Wave Height

In order to investigate the variations in wave height, the evolution of the non-dimensional wave height along the centerlines of two flumes for the regular wave cases with different frequencies, as well as preliminary steepness, are presented in Figure 4. In these panels, the circles represent the wave heights measured on the extension line of the A flume, and the squares represent that on the extension line of the B flume (i.e., the red points in Figure 1).

As shown in Figure 4, the variation of wave heights for the corresponding positions of two wave trains at the entrance of the interaction region (x = 11.64 m) is asymmetrical: the wave height corresponding to the A flume is larger, while it is smaller corresponding to the B flume. This asymmetrical change is not observed from the images, implying that the diffractions and reflections from the side-wall during the active interactions may have a non-negligent influence on the change of wave profiles. As the waves propagate further, it is interesting that the increasing rate of wave height at the two sides is not consistent, but the wave heights reach their maximums in the center of the interaction region. At the same time, it should be noted that the effect of initial wave steepness on the increasing rate of wave height is not apparent during the interaction process, i.e., the ratio between measured and initial wave height at the same position is almost the same (Figure 4a–c), which means that the IMS seems to have little influence on the interactions of two wave trains for the low frequency (f = 0.8 Hz) absent breaking. However, downstream of the center of the interaction region, the wave height decreases at the same rate, which is close to linear fitting. For cases with the same preliminary wave steepness (Figure 4b,d), violent changes occur in the wave height during the interactions for the large frequency, resulting in complicated changes of wave profiles as observed. Following the report of [31], when the wave steepness is bigger than 0.3, the three-dimensional instability would dominate. While for the case with large frequency, it is much easier to reach its threshold, suggesting that the frequency plays an important role in the interactions between weakly three-dimensional wave trains, especially for waves with high frequency.

Additionally, variations in the frequency and initial input steepness do not appear to strongly affect the maximum wave height, and the linear least-squares fit is H_max = 1.37 H₀, as can be seen in Figure 5. Two reasons may explain this result. First is the diffraction and reflection during the interactions between weakly three-dimensional wave trains that redistribute the energy. Second is the approaching angle of two wave trains which may be an important factor in changing the maximum wave height. However, we need another experiment to study it further.

3.2. Variation of Surface Elevations Downstream Interaction Region

In the case of examining the influence of the weakly three-dimensional interactions, the surface elevations downstream of the interaction region are presented in Figure 6 and Figure 7. In the figures, the solid lines represent the surface elevations recorded at the right side of the B flume, and the dash lines represent the surface elevations recorded at the left side of the B flume. It is observed that the surface elevations measured by two in-line probes are nearly the same, and more importantly, the wave can keep its initial characteristic at the position x = 17.4 m for the case with a relatively small wave steepness (IMS = 0.11, Figure 6a), where the first two probes installed downstream of the interaction region. As the waves propagated further, the waves were still two-dimensional. However, there is evidence of a three-dimensional attribute on the lateral wave surface after nonlinear interactions indicating that the larger wave steepness can impact the weakly three-dimensional interactions even with lower frequency from Figure 6b. Compared to large frequency (Figure 7), it is noted clearly that three-dimensional impacts seem to dominate the wave surface despite the fact that the initial steepness is small. This result confirms further that the high frequency plays an essential function in the weakly three-dimensional interactions, as discussed above.

In order to explain the influence of the interactions of weakly three-dimensional wave trains better, an extra experiment with just one direction (i.e., oblique two-dimensional experiment) was likewise taken into consideration as a comparison, as is shown in Figure 8. The oblique two-dimensional experiment was also conducted in this ‘X’ configuration with a wall in place, i.e., sealed the A flume in the interaction region so that there was only one wave propagating along the B flume. In the A flume, wave absorbers were set up aimed to dissipate the wave energy as well as minimize the effect of wave representation. As a result, two wave trains were still generated, just like the weakly three-dimensional experiment, so the initial conditions for the two experiments were kept the same. Nevertheless, it was only possible for the B flume that the waves may propagate to the end in this way. Details of wave elevations were measured just like that in the weakly three-dimensional wave experiments by two wave probes in-line.

According to Figure 8a, comparisons of the wave elevations between weakly three-dimensional and oblique two-dimensional experiments after interactions for the case with a low frequency (f = 0.8 Hz). The surface profiles are almost the same in amplitude and phase between the experiments as IMS = 0.11. Furthermore, the increase of the steepness could not increase the deviation of the surface elevations apparently until the breaking occurred (IMS = 0.26), whose wave height decreased due to the breaking. However, a large deviation occurred between the weakly three-dimensional and oblique two-dimensional experiments for the largest frequency (f = 1.2 Hz) case, even though the steepness is low (IMS = 0.11). With the increasing steepness, there is an interesting phenomenon that a large deviation appeared in phase (Figure 8b), resulting in the wave profiles measured in a weakly three-dimensional experiment lagging behind that in a two-dimensional wave at the same location. This result may be attributed to the directionality superposition that the propagating of energy could be blocked during the interaction, resulting in increasing of energy densities and even breaking occurring.

4. Numerical Simulation of Weakly Three-Dimensional Interaction

4.1. Numerical Method

A simulation of nonlinear wave interactions in the ‘X’ configuration was conducted using a non-hydrostatic model during this investigation [32] by combining the finite element and the finite volume method to solve the three-dimensional Euler equations. In this model, a method of vertical boundary-fitted was employed to simplify the discretization of the momentum and achieve the Poisson equation that is symmetric as well as positive definite by defining the pressure at the edge of the layer and the vertical velocity at the center of the layer. Then the governing equations for free-surface wave flow according to the conservation of mass and momentum can be expressed as follow:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0.$$

(8)

$$\frac{\partial u}{\partial t}+\frac{\partial u^2}{\partial x}+\frac{\partial uv}{\partial y}+\frac{\partial uw}{\partial z}=-\mathrm{g}\frac{\partial \mathsf{\eta }}{\partial x}-\frac{\partial q}{\partial x}.$$

(9)

$$\frac{\partial v}{\partial t}+\frac{\partial uv}{\partial x}+\frac{\partial v^2}{\partial y}+\frac{\partial vw}{\partial z}=-\mathrm{g}\frac{\partial \mathsf{\eta }}{\partial y}-\frac{\partial q}{\partial y}.$$

(10)

$$\frac{\partial w}{\partial t}+\frac{\partial uw}{\partial x}+\frac{\partial vw}{\partial y}+\frac{\partial w^2}{\partial z}=-\frac{\partial q}{\partial z}.$$

(11)

where u, v, and w represent the velocity components in the x, y, and z directions, respectively, and η represents the free surface elevation. Herein, in order to reduce rounding errors, the pressure can be divided into non-hydrostatic pressure q as well as hydrostatic pressure g(η-z). Boundary conditions are required for obtaining a unique solution at the free surface η(x, y, t); the kinematic boundary is given by

$$\frac{\partial \eta }{\partial t}+u\frac{\partial \eta }{\partial x}+v\frac{\partial \eta }{\partial y}=w{|}_{z=\eta }.$$

(12)

While at the bottom of the impermeable layer z = −h(x, y), the kinematic boundary condition can be written as:

$$-u\frac{\partial h}{\partial x}-v\frac{\partial h}{\partial y}=w{|}_{z=-h}.$$

(13)

In the end, the free surface equation can be derived by applying kinematic boundary conditions (12) and (13) to the continuity Equation (8) by the integral form along with water depth:

$$\frac{\partial \eta }{\partial t}+\frac{\partial }{\partial x}{\displaystyle {\int }_{-h}^{\eta }udz}+\frac{\partial }{\partial y}{\displaystyle {\int }_{-h}^{\eta }vdz}=0.$$

(14)

4.2. Numerical Simulation

Even though the precision and capability of the non-hydrostatic model have already been exhibited by comparing it with other experimental and numerical data in different events [33,34,35], it is still necessary to demonstrate the accuracy of the numerical model to simulate the process of weakly three-dimensional wave interactions. Following the measurements of the first two wave probes (i.e., at x = 3.5 m), the initial conditions of the inflow for subsequent numerical simulation were determined based on the parameters in Table 1. Comparisons of wave elevations simulated by the numerical method (solid lines) and measured in experiments (dash lines) are shown in Figure 9a,b upstream and Figure 9c,d downstream about the interaction region. It is apparent that a good agreement can be seen between the numerical and experimental results both in wave amplitude as well as phase, indicating that the non-hydrostatic model performs amazingly well to simulate the weakly three-dimensional wave interactions.

Figure 10 shows the position of maximum wave occurrence and its wave height for different approaching angles. As is shown in the figure, the non-dimensional position Δs/s increases with the increase of the approaching angle, in which the s represents half of the horizontal length of the interaction region and Δs represents the distance up-shift from the center of the interaction region, implying that the position shift is more significant for the condition with a larger approaching angle. Meanwhile, Figure 10b indicates that the maximum wave height is affected by the approaching angle a lot. As the approaching angle increases, the maximum wave height decreases at a certain angle range and then increases monotonically, reaching about 1.65 times when the approaching angle got to a degree of 90. This result is in accordance with what [36] published that with the increasing of the wave approaching angle, the wave height would also increase by an analysis of short-crested waves.

5. Conclusions

This research investigates the evolution of weakly three-dimensional waves with a relatively small approaching angle of 16°. Experimental observations show that the interaction of weakly three-dimensional wave trains plays a significant role in the development of the extreme wave. There is a clear increase in the wave height, which the maximum wave height of about 1.37H₀ achieved even for different initial steepness and frequencies when reaching the center of the interaction region. Then lateral wave elevations measured from the downstream of the interaction region were examined. It implies that the frequency and initial wave steepness are highly correlated with the wave-wave interactions between the weakly three-dimensional waves. For cases with low frequency, the wave-wave interactions seem to be reversible in our experimental configuration in that the waves can still keep their initial feature and propagate in their initial directions downstream of the interaction region. However, for cases with large frequencies, the deviation of lateral wave elevations increases with the increase of initial wave steepness. In addition, the two-dimensional experiment also confirms the studies mentioned above. At the same time, it is noted that the apparent change of phase occurred during the weakly three-dimensional interactions.

In addition, a non-hydrostatic model is also employed to investigate the weakly three-dimensional interactions, which shows that the model can simulate the interaction of two wave trains accurately. Thereafter, more cases with different approaching angles from 16° to 90° were conducted for further study of the influence of approaching angles on the interaction process. The research indicates that a larger approaching angle can result in more intense interactions and strong three-dimensional characteristics after interactions. More importantly, the maximum wave height that occurred due to interactions were observed to up-shift with increasing approaching angle, and wave height also increases as the wave approaching angle increases.