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Wave definition
The regular wave is defined by its period, its amplitude, its phase relative to the origin and its direction. The direction is given by the angle \(\theta\) from the X-axis. For an Airy wave, the wave elevation writes at the origine of the reference frame :
where:
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\(A_{wave}\) is the wave amplitude
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\(\omega_{wave}\) is the wave circular frequency
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\(t\)is the time
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\(f_{wave}\) is the wave phase
The irregular wave is defined as a sum of regular waves having randomly chosen phases between 0 and 360.
The instantaneous wave elevation at a dedicated point is :
The parameters to be defined are :
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The number of waves component \(N_h\). This number of components must be at least 100 (default value) to ensure a correct representation of the wave frequency energy
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The wave heading (0 degrees is a wave propagating along the \(O_x\) axis from the negative value to the positive value of X)
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The trigger number for the randomly phase algorithm in order to ensure a non repetitiveness . This number is a 5 digits odd integer.
Maximum wave height :
Some procedure are automatically done to ensure that during the duration of the simulation defined by the user, the maximum wave elevation encountered at the FPF location is reached. The user has to define the ratio Hmax/Hs (a value of 1.86 is a common value). A specific pre simulation of the wave height is performed and a dephasing is at the following time : \(t = 2.*T_p + 3*(T_{FIN}-2.*T_p)/4\) with Tp the wave spectrum peak period.
Embedded wave :
Another feature is the possibility of using an embedded stream function to replace the irregular wave when reaching maximum Hs. The wave signal is modified over one and half period, centred on the time of maximum height. For one period (again centred on the time of maximum height), the stream function is used to define wave elevation, wave velocity and acceleration. A transition of a quarter is used to go from the irregular wave to spectra and after one period of the stream function back to the irregular wave. In the transition region, the following coefficient is defined:
with tref the reference time for maximum wave height and tper the period.The elevation is then a blend of the irregular wave elevation (multiplied by 1-coef ) plus the stram function height (multiplied by coef ). Note that the period is by default the period of the irregular wave around the maximum wave height (See Rainey and Camp, "Constrained non-linear waves for offshore wind turbines," The Science of Making Torque for Wind - Journal of Physics: Conference Series 75 (2007) ).
The associated amplitude Ai at the elementary wave at the pulsation wi is deduced from the wave spectrum defined by the user or using an analytical wave spectrum . The different formulas defining the wave spectra available in DeepLines are presented below :
JONSWAP wave spectrum
The formulation used for the JONSWAP spectrum is given by :
where :
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s = 0.07 if f < \(f_o\)(\(f_o\)=1/TP)
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s = 0.09 if f > \(f_o\)
is a constant determined in order to ensure that the spectrum energy is equal to those with a Pierson Moskovitz.
Ochi-Hubble wave spectrum
The Ochi-Hubble wave spectrum is defined by :
where
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\(Ai = \frac{Hsi^2}{4} \frac{Bi^\lambda}{\Gamma(\lambda i)}\)
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\(Bi = \frac{ai}{4}\omega pi^4\)
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\(ai = 4\lambda i + 1\)
Triangle type wave spectrum
| Component | Spectral model |
|---|---|
| Swell peak | \(S(f) = 0.08474\frac{H^2_5}{f_p}(26.2\frac{f}{f_p} - 22)\) if \(\frac{22}{26.2} f_p < f < f_p\) \(S(f) = 0.08474\frac{H^2_5}{f_p}(26.2 - 22\frac{f}{f_p})\) if \(f_p < f < \frac{26.2}{22} f_p\) \(S(f) = 0\) else |
| Last swell peak in the upper frequencies | \(S(f) = 0.08474 \frac{H^2_5}{f_p}(26.2\frac{f}{f_p} - 22)\) if \(\frac{22}{26.2} f_p < f \le f_p\) \(S(f) = 0.08474 \frac{H^2_5}{f_p}(26.2 - 22\frac{f}{f_p})\) if \(\frac{22}{26.2} f_p < f < f_0\) \(S(f) = 0.17\frac{H^2_5}{f_p}(\frac{f}{f_0})^-6\) if \(f_0 \le f (f_0 = 1.1 f_p)\) |
Wallops wave spectrum
| Component | Spectral model |
|---|---|
| Wind sea peak (WALLOPS type spectrum) | \(S(f) = 0.206625\frac{H^2_5}{f_p}(\frac{f}{f_p}) e^{\frac{-1}{4}(\frac{f}{f_n})^-20}\), \(0.85f_p < f\) |
Gaussian wave spectrum
The spectrum is characterized through its significant wave height (Hs), peak period (Tp) and dimensionless shape parameter s coefficient.
\(S(\omega) = \frac{1}{2\pi} \frac{m_0}{\sigma^2(2\pi)^2}exp[-\frac{(\omega - \omega _0)^2}{2\omega^2(2\pi)^2}]\)
where:
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S (w) Spectral density at wave frequency w, [\(m^2/s\)]
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w Wave frequency, [\(rad/sec\)]
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w0 Spectrum peak frequency, [\(rad/sec\)]
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s Dimensionless shape parameter, [-]
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m0 \((Hs/4)^2\)
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Hs Significant wave height, [\(m\)]
Torsethaugen wave spectrum
The spectrum is characterized through its significant wave height (Hs), peak period (Tp) and factor (\(a_f\)).
\(T_f=afHs^{1/3}\)
\(T_p<Tf :\)wind dominated range
\(T_p>Tf :\)swell dominated range
The spectrum is the sum of a wind sea and a swell contribution:
\(S(f) = E_1S_{n1}(F_{n1}) + E_2S_{n2}(f_{n2})\)
With:
\(f_{nj} = f.T_{Pj}\)
\(E_j = 1/16*Hs_{Sj}^2T_{Pj}\)
\(S_{nj}(f) = G_0A_{\gamma}\Gamma {_Sj\gamma{Fj}}\)
For the simplified model
\(\Gamma{_Sj} = fnj^{-4}exp(-f_{nj}^{-4}); j=1.2\)
\(G_0 = 3.26\)
\(\gamma_{F1} = \gamma^{\exp[\frac{(f_{n1} - 1)^2}{2\sigma^2}]}\) , \(\sigma = 0.07 for f_{nj}<1 and \sigma =0.09 for f_{nj}>=1\)
\(\gamma_{F2} = 1\)
\(A_{\gamma 1}=(1+1.1[\ln (\gamma)]^{1.19}) \gamma\)
\(A_{\gamma 2}=1\)
| Range | Spectral model |
|---|---|
| Wind dominated | Primary Peak \(H_{S1}=H_{Sw}=r_{pw}H_s\) \(T_{P1}=T_{Pw}=T_{p}\) \(\gamma =35[(2 \pi H_{SW})/(gT_P^2)]^{0.857}\) Secondary Peak \(H_{S2}=H_{Ssw}=(1-Rpw^2)^{1/2}H_s\) \(T_{P2}=T_{Psw}=T_f +2.0\) \(\gamma=1\) And \(r_{pw}=0.7+0.3 \exp[-(2(T_f-T_p)/(T_f-2H_s^{1/2}))^2]\) |
| Swell dominated | Primary Peak \(H_{S1}=H_{Ssw}=r_{ps}H_s\) \(T_{P1}=T_{Psw}=T_p\) \(\gamma =35[(2p H_S)/(\gamma T_f^2)]^{0.857}[1+6(T_p-T_f)/(25-T_f)]\) Secondary Peak \(H_{S2}=H_{Sw}=(1-R_{ps}^2)^{1/2}H_s\) \(T_{P2}=T_{Pw}=6.6H_{SW}^{1/3}\) \(\gamma =1\) And \(r_{ps}=0.6+0.4 \exp[-((T_p-T_f)/0.3/(25-T_f))^2]\) |
Lognormal spectrum:
The three main parameters are: * The significant wave height \(H_s (m)\), * The peak period \(T_p (s)\), * The standard deviation \(\sigma (Hz)\).
The other parameters are used to reconstruct the wave time elevation and the same as other spectra. It is possible to use directional spreading.
Definition of an irregular wave based on a lognormal spectrum
The spectrum is defined as
With \(S_d=\sqrt{ln(\sigma^2{T_p}^2)-1}\) and \(\mu=-ln(T_p)+{S_d}^2\)
The lognormal spectrum can be defined as individual wave object as well as in a wave set or environment set.
