Virtual Wave Tank - Generating Gravity Waves in PreonLab

Before going deeper into the setup in PreonLab we need to introduce some basics about waves. Free-surface gravity waves, as they occur in bodies of water, can be described by the free-surface elevation at a certain space and time. A regular sinusoidal wave is described by:

$$
\eta(x,t)=A_w\cos(kx-\omega t),\quad
k=\frac{2\pi}{\lambda},\quad
\omega=\frac{2\pi}{T}.
$$

For gravity waves, the wave number and frequency are related via the dispersion relation given by: $$ \omega^2 = gk \tanh(kh), $$

Wavemaker:

Most modern wave tanks use one of two types of wavemaker, i.e., a flap or piston-type (see Figure 2). These systems consist of a moving wall placed in a tank onto which an oscillatory motion is applied. So-called Biésel transfer functions (BTF), which are derived using linear wave theory, describe the relation between the wavemaker actuation and the corresponding wave it induces. For each wavemaker type, there is a corresponding BTF. 

Irregular waves:

Using the above linear wave theory and BTFs, arbitrary irregular waves can be composed through the superposition of multiple waves with varying amplitudes and frequencies. Simple examples are biochromatic waves which consist of two waves with different frequencies as shown in Figure 3.

The concept of superposition can be further extended to obtain statistically realistic waves. E.g., the Joint North Sea Wave Project (JONSWAP) model provides the amplitudes for a set of frequency modes given a set of imput varialbes such as the significant wave height and peak period and frequency. For more details on JONSWAP wave spectra see, e.g., [2].

Using the basic wavemaker theory, we model a sine-shaped regular wave in a numerical wave tank setup. For this, we will set up a tank according to the dimensions in [3] which for completeness are given in Figure 4. The only difference in our setup is the shift of the probe distance to the wavemaker from 24.6 to 15.0 meters. The choice for probing at closer to the wavemaker, is to reduce the time needed for the wave to reach the probe and therefore minimize the required computational time.

For 2D wave tank simulations, the walls and wavemaker depicted above can be represented by solid planes. For a 3D wave tank, the tank can be modeled as a box with fixed width, or with the use of period boundaries. A general overview of the tank as used withing PreonLab is shown in Video 1.

As can be observed in Video 1, the total dimensions of the complete wave tank are quite large, in particular the length. Consequently, a large number of particles would be needed to resolve the full fluid volume. To minimize the total number of particles, and the required computational cost, we employ a non-uniform particle resolution. The Continuous Particle Size (CPS) feature of PreonLab allows the user to specify regions with a different particle size. In case of free-surface waves the tank is filled with coarse particle and refinement is applied only in the region close to the free-surface. This choice is based on the fact that most of the wave energy in deep water waves is transported close to the free surface. In Video 2, the effect of using CPS is clearly visible.

Wavemaker Actuation For Regular Waves:

Generating Irregular Waves:

As already stated, with the use of superposition, an irregular wave can be constructed simply by adding together regular sine waves with varying period, amplitude, and phase angle. We demonstrate this concept using the wave tank setup from [3] which we used in the previous case. We replace the regular sine wave with a bichromatic wave comprised of two regular sine waves of different frequencies which we added together.

Although not demonstrated here, keyframing of wavemakers using BTFs in PreonLab can be used for more realistic scenarios such as ocean waves too. Ocean waves are composed of a wide range of frequencies and amplitudes, rather than a single uniform wave. Wave spectra describe how the total energy of the sea surface is distributed across these different wave frequencies. It provides a statistical representation of the sea state, capturing both the dominant waves that carry most energy and the smaller waves that are less dominant. Spectra are fundamental for modeling wave dynamics, predicting loads on structures, and understanding energy transfer within the ocean surface. The demonstrated concept of superposition for the bichromatic wave can simply be extended with a larger number of frequency components that are based on a realistic, statistically backed wave spectrum.  

One widely used model for representing a wave spectrum is the JONSWAP spectrum, developed from extensive measurements during the Joint North Sea Wave Project [2]. It provides a practical way to reconstruct the energy distribution across individual wave frequencies using only a small set of parameters, such as the peak frequency, significant wave height, and a peak enhancement factor. This makes it particularly useful in engineering and coastal applications, where a realistic, yet computationally efficient representation of a sea state is needed. By capturing the concentration of energy around the dominant waves, the JONSWAP spectrum offers a more accurate description of developing seas than simpler, fully developed models.

The resulting realistic ocean waves can be explored in PreonLab to analyze all sorts of marine applications such as floating structures in waves. A nice example is that of parametric rolling of a ship due to head waves as shown in Video 4. Parametric roll is an unintuitive phenomenon in which a side-to-side roll can occur due to alignment of the ships natural roll period and the encounter frequency of the head waves. It is an unintuitive effect which needs to be constantly monitored by ship masters to mitigate capsizing of the vessel either by changing course or ship speed. As the video progresses, it can be clearly seen how the resonant roll motion can intensify to dangerous levels.