Wave Modeling
Theory: period and depth set the wave
Overview
When generating waves in CFD, what should be decided first?
The wavelength. Period alone doesn't fix a wave: length and speed change with depth, linked by the dispersion relation. Once wavelength is known, domain size, grid fineness and generation/absorption zone lengths follow. Crest speed and energy speed also differ; when waves reach a structure and how much energy they carry are set by group speed. Let's calculate for several periods and depths.
Dispersion relation
Deep/shallow limits and energy
In deep water (depth > half wavelength) wavelength scales with period squared; in shallow water (depth < a twentieth) speed depends only on depth. P is the energy carried per metre of crest by a wave of height H.
Airy's wave theory
The linear theory of water waves was compiled in 1845 by British astronomer George Airy and is still called Airy wave theory. Stokes later gave nonlinear theory for steeper waves, and solitary and cnoidal wave theories were developed for shallow water. Modern CFD still generates waves at the inlet by imposing these theoretical velocities and surface shapes, and the theories remain the benchmark for checking that a simulation propagates waves correctly.
Worked examples
Example 1: wavelength and speed by period and depth
| Period | Depth | Wavelength | Phase speed | Group speed | Wave power (H = 2 m) |
|---|---|---|---|---|---|
| 6 s | 100 m | 56 m | 9.4 m/s | 4.7 m/s | 23 kW/m |
| 10 s | 100 m | 156 m | 15.6 m/s | 7.8 m/s | 39 kW/m |
| 10 s | 20 m | 121 m | 12.1 m/s | 9.3 m/s | 47 kW/m |
| 10 s | 5 m | 68 m | 6.8 m/s | 6.3 m/s | 32 kW/m |
| 14 s | 100 m | 297 m | 21.2 m/s | 11.9 m/s | 60 kW/m |
The shallow-water limit √(gh) is 7.0 m/s at 5 m depth; the 10 s wave approaches it at 6.8 m/s.
Example 2: CFD grid (10 s, 2 m, deep water)
| Item | Value |
|---|---|
| Wavelength | 156 m |
| Steepness H/L | 0.013 |
| Horizontal cell (80 per wavelength) | 1.95 m |
| Vertical near surface (15 per height) | 0.13 m |
| 3-wavelength 2D domain | about 240 × 45 cells near the surface |
A 10 s wave shrinks to less than half its wavelength coming from deep water to 5 m depth.
The period is unchanged while speed drops, so wavelength shortens. In deep water waves don't feel the bottom; in shallower water the seabed slows them. Group speed falls and energy bunches up, so height grows until the wave breaks — why waves steepen near shore. In CFD, shallower regions with shorter waves need finer grids, so cell size should vary with depth. And deep-water group speed being half the phase speed helps estimate how long a wave group takes to cross the domain.
Handling waves in CFD
- Find wavelength from period and depth, then size grid and domain.
- Use generation and absorption (damping) zones of 1–2 wavelengths.
- Use nonlinear theory (e.g. Stokes) for steep waves.
- Use time steps of 1/100–1/500 of the period and check Courant number.
- Compare wave height after several wavelengths with theory to check numerical damping.
“The waves faded away”
In an analysis of a floating body in waves, generated waves had decayed to about 60% of theoretical height by the time they reached the body. The horizontal grid had only about 25 cells per wavelength, and numerical viscosity damped the waves as they travelled. Refining to 80 cells per wavelength with finer cells near the surface cut decay to a few percent. Since forces on the body scale roughly with wave height, grid coarseness had directly underestimated loads.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Deep-water formula in shallow water | Wavelength overestimated | Solve dispersion relation |
| Coarse grid | Waves decay | 60–100 cells/wavelength |
| No or short absorption zone | Reflections contaminate | 1–2 wavelengths of damping |
| Energy at phase speed | Wrong arrival time | Use group speed |
| Linear theory for steep waves | Wrong shape and forces | Nonlinear theory |
I'd like to learn related topics.
Related pages include VOF waves, ship resistance prediction, mooring analysis, sloshing and propeller CFD.
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