As a first approach to modelling waves, a numerical simulation was prepared to generate regular waves by replicating the setup of a tow-tank. The initial study was in two dimensions, as shown in Figure 6-1.
Figure 6-1. Full size tow-tank model for the wave study.
The tank was 520m long and 40m deep. It employed a rigid floor-hinged paddle wave maker at one end and a gently sloping beach at the other to reduce reflections. A wavelength of 60m was chosen to ensure deep water conditions for the depth of tank, yet retain a reasonable number of particles per wavelength. In this case, particles of 2m diameter were used. As noted in the Chapter 4, the location of the free surface can be determined to a location smaller than the diameter of the particles.
Wave-maker paddle
hinged at the floor
B K Cartwright, 2012. p.78 The paddle was oscillated about the tank floor to generate the waves. The paddle amplitude was chosen to give a wave height of 2m close to the paddle. The simulation was run for 120 seconds, which was about 20 periods of the 60m wavelength.
The numerical simulation successfully generated waves, however, the wave height reduced significantly along the length of the tank. This is shown visually in Figure 6-2, where the upper 10m of depth has been magnified about 100 times to show the diminishing 2m wave height over the 520m length of the tank. Note the SPH distributions are not representative of anything very much due to the vertical magnification (they are in fact still largely hexagonal in distribution). Normalised wave height as function of distance from the paddle is shown in the curve of Figure 6-3.
Figure 6-2. Wave height from the full-size wave tank decreased with distance from the wave-maker. (Wave height has been exaggerated in this image. The maximum wave height was about 2m.)
Figure 6-3. Normalised wave height with distance from the paddle for a 60m wave and 2m particles. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7
Distance from Wavemaker, wavelengths
No rm a li s e d W a v e He ig h t 2m SPH 60m wave length Beach Wave-maker paddle
B K Cartwright, 2012. p.79 It is understood that the loss of wave height exhibited in Figure 6-2 is due to small amounts of energy being lost at each of the many numerical smoothing calculations performed (McCue et al 2004) as the wave propagates along the tank.
The parameter set used provided a reasonable compromise between the numerical stability and damping and was developed over many cases in the course of this work. The parameter set used is shown in Table 6-1. It is slightly less-damped (lower smoothing length to radius ratio, and lower artificial viscosity parameters),than those arrived at by Kalis (2007) for his work on free fall lifeboats using the same software.
Smaller SPH particles were reported by McCue et al (2006) to provide less dissipation. The use of smaller particles requires more computational effort, and so a compromise is required.
Table 6-1 SPH Parameter Sets
Parameter Set
Smoothing length to radius ratio
Smoothing length limits Anti-crossing parameter
Artificial Viscosity coefficients
min max Alpha Beta
Cartwright 1.8 0.0 4 x spacing 0.02 0.02 0.02
For the regular wave setup of Figure 6-1, different size particles were trialled to observe the effect of particle size. Figure 6-4 shows the wave height with distance from the paddle for 1m, 2m and 4m particles for a 60m wave length. Less energy dissipation is observed for the small particles, agreeing with the findings of McCue et al (2004), but is still unacceptable for ship motion studies.
Figure 6-4. Normalised wave height with distance from the paddle for 1m, 2m and 4m particles.
The effect of SPH particle size on computation effort is shown in Figure 6-5, where the number of SPH particles and CPU time (in seconds) is shown for the 1m, 2m and 4m
0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7
Distance from Wavemaker, wavelengths
No rm a li s e d W a v e He ig h t 1m SPH 2m SPH 4m SPH
B K Cartwright, 2012. p.80 particles. The CPU effort is linearly related to the number of SPH particles for these
simulations. The CPU time for the 1 metre SPH simulation was 4300 CPU-seconds, or about 72 minutes on one CPU (Intel Xeon 2.4 GHz), or at best 18 minutes on a 4-CPU (reduction in CPU-time is less than linear with number of CPUs for the code used here).
Extrapolating the trend exhibited in Figure 6-4 to a particle size that would limit wave height loss to about 10% over the 6 wavelengths modelled here would require an SPH particle size of about 0.1m. Extrapolating the trend for CPU effort shown in Figure 6-5 to a simulation with 0.1m SPH would require 7,200 minutes (5 days) of CPU effort for a similar simulation in 2D. If the model was a 3D wave tank, of dimensions 100m wide, this would be 1000 particles wider than the present 2D model, requiring an estimated 7.2 million minutes of CPU time. This is impractical with the computing resources available to this project.
Figure 6-5. Computation time for 2D simulations using 1m, 2m and 4m particles.
More recent work (Jones and Belton 2006, Guilcher et al 2007) suggested an improved time- stepping algorithm as a possible solution to reducing the wave decay in the propagation of free surface waves, however this was not demonstrated. A still more recent work (De Padova et al 2009) also showed dissipation in the waves modelled by their custom code which presumably incorporated many of the improvements of recent publications. The simulated waves of De Padova et al (2009) included both regular and irregular waves, but were in a tank with a sloping floor, which ultimately led to breaking on a shore, so even those waves were not directly comparable to a constant depth wave tank for ship motion predictions. DePadova et al (2009) concluded that a small value of the empirical alpha coefficient of the artificial viscosity was required for numerical stability, but the result was still too dissipative for accurate wave reproduction.
Consequently, the generation of waves by replicating the wave tank physics with current SPH formulation and currently affordable computer power is not a viable approach to predict the response of a ship, as the waves developed are not sufficiently consistent in height along the tank. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 1 2 3 4 5 SPH particle diameter, m Number of SPH CPU Time, seconds
B K Cartwright, 2012. p.81