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# Development Focus: New Wave Generators

This article highlights developments to be released in FLOW-3D version 10.0.

Currently, FLOW-3D can only simulate linear and Stokes waves. The linear wave has a sinusoidal surface profile and small wave steepness, while the Stokes wave is nonlinear and has larger wave steepness, sharper crests and flatter troughs. The waves are generated at the mesh boundary using Airy's linear wave theory (Airy, 1849) and Fenton's fifth-order Stokes wave theory (Fenton, 1985), respectively. Although the two theories can deal with many wave problems in practice, limitations exist.

The linear wave theory works only for small amplitude waves, which has quite limited application because coastal and ocean engineers are mainly interested in large waves that cause the greatest damage to structures. Although the Stokes wave theory works for larger amplitude waves in deep water, it fails for long waves in shallow water, i.e., cnoidal waves. A cnoidal wave is a nonlinear wave and has even sharper crests and flatter troughs than a Stokes wave. The differences between linear, Stokes and cnoidal waves can be found in Figures 1 and 2.

Figure 1. Comparison of profiles of the different progressive waves.

## Fourier Series Wave Generation Method

Figure 2. Applicability ranges of various waves (after Le Méhauté, 1976,
Sorensen, 2005 and USACE, 2008). d: mean water depth;
H: wave height; T: wave period; g: gravitational acceleration

In FLOW-3D version 10.0, a new wave generator has been added using the Fenton's Fourier series method (Fenton, 1999). Different from the linear, Stokes and other wave theories that have certain application ranges, this method works for all kinds of oscillatory waves in deep and shallow water, including linear, Stokes and Cnoidal waves (see Figure 2 for details). More than that, it possesses higher order accuracy than other theories (USACE, 2008). Fenton's Fourier series method is thus the recommended wave generator for any linear or nonlinear oscillatory wave simulation. The existing linear and Stokes wave generators are still retained in version 10.0 to serve special needs of users. The animations below show linear, Stokes and Cnoidal waves generated by Fenton's Fourier series method.

## Solitary Wave Generator

A solitary wave is a nonlinear, non-oscillatory wave. It has a single crest, no trough and is completely above the undisturbed water level. It is a good approximation of a shoaling cnoidal wave as its crests become shorter and the troughs longer. It is often used to describe Tsunami waves caused by earthquakes and large-scale landslides. In FLOW-3D version 10.0, a solitary wave generator is available using McCowan's theory (McCowan, 1891). This theory is more accurate than Boussinesq's theory (1871) and is highly recommended by Munk (1949). The simulation below is the result of a solitary wave striking a structure.

## Random Wave Generator

In coastal and ocean engineering, a regular wave like a Stokes or Cnoidal wave is often used to represent a design wave in analysis of wave interactions with offshore structures. However, when wind acts on the sea surface, what we observe are many waves with different wavelengths, periods and amplitudes moving in different directions, which are referred to as irregular or random waves.

In FLOW-3D version 10.0, random waves can be generated at a mesh boundary as a superposition of many linear component waves of different amplitude and period. These waves propagate into the computational domain to form a random sea. For each of these component waves, its amplitude and frequency are calculated using the wave energy spectrum. Initial wave phases, however, are random. The Pierson-Moskowitz (P-M) spectrum for fully developed sea (Pierson-Moskowitz, 1964) and the JONSWAP spectrum for fetch-limited sea (Hasselmann, 1973) are implemented. Users can choose either of them or use their own wave spectrum defined in a data file. For now, all the component waves are assumed to travel in the same direction at the wave boundary and directional wave spectrums are not considered. The animation below is an example of random waves generated using the P-M spectrum.

## Initial Wave Condition

Previously in FLOW-3D, a wave could not be defined as an initial condition. The solver must run for sufficiently long time to allow the oscillation from the wave boundary to reach everywhere in the computational domain and evolve into steady wave motion. To shorten computation time, a new development has been made to initialize a wave at the beginning of a simulation. As an initial condition, wave elevation and water velocity can be defined throughout the computational domain using the same wave generator and the same wave parameters as at the wave boundary. The animation below shows the initial condition and the simulation result of a Stokes wave.

## References

• Airy, G. B., 1845, Tides and Waves, Encyc. Metrop. Article 102.
• Boussinesq, J. 1871, Theorie de L’intumescence Liquide Appelee Onde Solitaire ou de Translation se Propageant dans un Canal Rectangulaire, Comptes Rendus Acad. Sci. Paris, Vol 72, 755-759.
• Fenton, J. D., 1985, A Fifth-Order Stokes Theory for Steady Waves, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 111, No. 2, 216-234.
• Fenton, J.D., 1999, Numerical Methods for Nonlinear Waves, in Advances in Coastal and Ocean Engineering, Vol. 5, ed. P.L.-F. Liu, 241-324, World Scientific: Singapore.
• Hasselmann, K., Barnet, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Muller, P., Olbers, D.J., Richter, K., Sell, W., and Walden, H., 1973, Measurement of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP), Report, German Hydrographic Institute, Amburg.
• Le Méhauté, B.,1976, An Introduction to Hydrodynamics and Water Waves, Springer-Verlag.
• McCowan, J., 1981, On the solitary wave, Philosophical Magazine, Vol. 32, 45-58.
• Munk, W. H. 1949, The Solitary Wave Theory and Its Application to Surf Problems, Annals New York Acad. Sci., Vol 51, 376-423.
• Pierson, W. J., and Moskowitz, L., 1964, A proposed spectral form for fully developed wind seas based on the similarity theory of S.A. Kitiagordskii, J. Geophys. Res., vol9, 5181-5190.
• Sorensen, R. M., 2005, Basic Coastal Engineering, Springer, 3rd edition.
• USACE (U.S. Army Corps of Engineers), 2008, Coastal Engineering Manual EM 1110-2-1100, Part II, Washington, DC.

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