Environment During dynamic simulation, wind velocity and wave elevation may be read from an ascii file. 1. Wind The wind field is assumed to be 2-dimensional, i.e. propagating parallel to the horizontal plane. The model includes gust spectra both in the mean direction and normal to the mean wind direction. The wind gust (the varying part of the wind velocity) is assumed to be a Gaussian stochastic process. The varying part of the wind velocity in the mean direction is described by either the Harris, the Davenport, the Wills, the Sletringen, the ISO 19901-1 (NPD) or the API wind gust spectrum. 1.1. Wind simulated in the time domain If a Davenport, Harris or ISO 19901-1 (NPD) wave spectrum is used, the wind velocity may be simulated in the time domain by use of a state-space model. See State-space model driven by white noise and (Kaasen 1999). If this method is used, one will only get wind in the main direction (no transverse gust) and no admittance function will be used. 1.2. Wind profile The wind profile used for all wind spectra is described by \[\bar{u}(z)=\bar{u_r}(\frac{z}{z_r})^\alpha\] where \(z\) height above water plane \(z_r\) reference height, normally \(10 ~ \mathrm{m}\) \(\bar{u_r}\) average velocity at the reference height \(z_r\) above the calm water level \(\alpha\) height coefficient \((0.10 - 0.14)\) \(\bar{u}\) average velocity at height \(z\) For the ISO 19901-1 (NPD) and API wind spectra, \(\mathrm {z_r}\) is always \(10~\mathrm {m}\). \(\mathrm {\alpha }\) is \(\mathrm {0.11}\) for the ISO 19901-1 spectrum. The average wind velocity is calculated for each body at the height for which wind force coefficients have been calculated. 1.3. Davenport wind spectrum \[S_u^+(z,\omega )=\frac{4\kappa\bar{u}^2r^2}{\omega (1+r^2)^{4/3}}\] where \(S_u^+\) spectral density in the mean wind direction \(\omega\) angular frequency \(r\) \(\displaystyle = \omega \frac{L_r}{2 \pi \bar{u} (z)}\) \(L_r\) reference length of wind turbulence (\(1200 ~ \mathrm{m} - 1800 ~ \mathrm{m}\) suggested) \(\kappa\) surface drag coefficient (\(0.001 - 0.003\) suggested) Recommended values of \(\mathrm {L_r}\) and \(\mathrm {\kappa}\) can be found in (Sigbjørnson, 1977). Note that \(\mathrm {\bar{u}}\) is a function of \(\mathrm {z}\). Suitable scaling factors for the state-space model in State-space model driven by white noise are \[\lambda_f=\frac{\bar{u}(z)}{L_r},\quad \lambda_s=4\kappa L_r\bar{u}(z)\] The non-dimensional coefficients of the fitted transfer function are \[\begin{array}{lll}a'_0=0.63611957&a'_1=2.04293375&a'_2=1.00000000\\\\b'_0=0.17350619&b'_1=1.29021181\end{array}\] 1.4. Harris wind spectrum \[\begin{array}{l}\displaystyle S_u^+(z,\omega )=A\frac{4\kappa\bar{u}^2r}{\omega (2+r^2)^{5/6}}\\\\A=1\end{array}\] The same parameters and scaling factors as for the Davenport spectrum apply for the state-space model in State-space model driven by white noise. The non-dimensional coefficients of the fitted transfer function are \[\begin{array}{lll}a'_0=7.79434308&a'_1=6.67987775&a'_2=1.00000000\\\\b'_0=5.83768758&b'_1=1.51253903\end{array}\] 1.5. Wills spectrum (modified Harris) This spectrum is identical to the Harris spectrum except for the factor \(\mathrm {A}\). \[A=0.51\frac{(2+r^2)^{5/6}}{(r^{0.15}+\frac{9}{8}r)^{5/3}}\] 1.6. Sletringen wind spectrum \[S(f)=u_{r}^p\{\frac{a_1(\displaystyle \frac{z}{z_r})^{-q}}{(B_1+f)^{5/3}}+\frac{a_2}{(B_2^n+f^n)^{5/(3n)}}\}\] where \(S(f)\) is the power spectral density \(\displaystyle \mathrm{\left[\frac{\left(\displaystyle m/s\right)^2} {Hz}\right]}\) \(u_r\) is the 1-hour mean wind speed at \(10 ~ \mathrm{m} ~ \displaystyle \mathrm{ \left[m/s \right]}\) \(z\) is the height \(z_r\) is the reference height \((10 ~ \mathrm{m})\) \(f\) is the frequency \(\gamma\) is the temperature stability parameter, recommended values are in the range \(\displaystyle \mathrm{ 10 ~ \frac{K}{km} - 20 ~ \frac{K}{km} }\) \[\begin{array}{l}\displaystyle n=n_0+n_1e^{-c_n\rho }\\\\\displaystyle B_1=b_1\frac{u_r}{z}\\\\\displaystyle B_2=b_2u_re^{c_{b2}\rho }\\\\\displaystyle \rho =\frac{\gamma }{(\displaystyle \frac{u_r}{10~\mathrm {\displaystyle m/s}})^\varepsilon }\end{array}\] The values for the parameters have been derived from the Sletringen measurements as shown below. 1.6.1. Table 3.1 Sletringen measurements parameter values Parameter Value \(\mathrm {a_1}\) \(\mathrm {2.03\cdot 10^{-5}}\) \(\mathrm {a_2}\) \(\mathrm {1.18\cdot 10^{-5}}\) \(\mathrm {\varepsilon }\) \(\mathrm {1.40}\) \(\mathrm {p}\) \(\mathrm {3.07}\) \(\mathrm {q}\) \(\mathrm {1.50}\) \(\mathrm {b_1}\) \(\mathrm {1.82\cdot 10^{-2}}\) \(\mathrm {b_2}\) \(\mathrm {3.56\cdot 10^{-4}}\) \(\mathrm {c_{b2}}\) \(\mathrm {0.293}\) \(\mathrm {n_0}\) \(\mathrm {0.281}\) \(\mathrm {n_1}\) \(\mathrm {0.428}\) \(\mathrm {c_n}\) \(\mathrm {0.183}\) 1.7. ISO 19901-1 wind spectrum The ISO spectrum was previously referred to as the NPD spectrum. For strong wind conditions the design wind speed, \(\displaystyle u(z,t)~[\mathrm {m/s}]\) at height \(z~[\mathrm {m}]\) above sea level and corresponding to an averaging time period \(t\leq t_0=3600~\mathrm {s}\) is given by \[u(z,t)=U(z)[1-0.41\cdot I_u(z)\cdot \ln(\frac{t}{t_0})]\] where the 1 hour mean wind speed \(\displaystyle U(z)~[\mathrm {m/s}]\) is given by \[\begin{array}{l}\displaystyle U(z)=U_0[1+C\cdot \ln(\frac{z}{10})]\\\\C=5.73\cdot 10^{-2}(1+0.15\cdot U_0)^{0.5}\end{array}\] and where the turbulence intensity factor \(\mathrm {I_u(z)}\) is given by \[I_u(z)=0.061+0.043\cdot U_0(\frac{z}{10})^{-0.22}\] where \(\displaystyle U_0~\mathrm {[m/s]}\) is the 1 hour mean wind speed at \(10~\mathrm {m}\). For structures and structural elements for which the wind fluctuations are of importance, the following 1 point wind spectrum shall be used for the longitudinal wind speed fluctuations: \[\begin{array}{l}\displaystyle S(f)=\frac{\displaystyle 320\cdot (\frac{U_0}{10})^2\cdot (\frac{z}{10})^{0.45}}{\displaystyle (1+f_m^n)^{5/(3n)}}\\\\\displaystyle f_m=172\cdot f\cdot (\frac{z}{10})^{2/3}\cdot (\frac{U_0}{10})^{-0.75}\end{array}\] where \(\mathrm {n=0.468}\) and where \(S(f)\) is the spectral density at frequency \(f, ~ \left[ \mathrm{m^2/s} \right]\) \(z\) is the height above sea level \(\mathrm{[m]}\) \(U_0\) is the 1 hour mean wind speed at \(10 ~ \mathrm{m}\) above sea level \(~ \mathrm{\left[ m/s \right]}\) \(f\) is the frequency, \(1/600 ~ \mathrm{Hz} \leq f \leq 0.5 ~ \mathrm{Hz}\) According to the ISO 19901-1 document, the spectrum formula is defined on the domain \([1/600,~0.5]~\mathrm {Hz}\). In the SIMO implementation, the spectrum is set to zero above \(0.5~\mathrm {Hz}\) and limited in magnitude below \(1/600~\mathrm {Hz}\). Suitable scaling factors for the state-space model in State-space model driven by white noise are \[\lambda_f=\frac{(\displaystyle \frac{\bar{u}(z)}{10})^{3/4}}{172},\quad \lambda_s=320(\frac{\bar{u}(z)}{10})^2\] The non-dimensional coefficients of the fitted transfer function are \[\begin{array}{llll}a'_0=0.60398949&a'_1=4.64963688&a'_2=5.65818257&a'_3=1.00000000\\\\b'_0=0.39434210&b'_1=1.74889195&b'_2=0.87403007\end{array}\] ISO 19901-1 wind spectrum for a mean wind speed of 20 m/s. In the SIMO implementation, the magnitude is limited for frequencies between 0 and 1/600 Hz as shown in Figure 1. Figure 1. ISO 19901-1 wind spectrum for a mean wind speed of 20 m/s 1.8. API wind spectrum The API spectrum representing typhoon wind conditions may be written \[S(f)=\frac{\sigma ^2(z)}{f}\frac{F}{(1+1.5F)^{5/3}}\] where the following definitions shall be utilised. \(S(f)\) spectral wind energy density distribution \(f\) frequency \([ \mathrm{Hz}]\) \(F\) \(= \displaystyle \frac{f}{f_p}\) \(f_p\) \(= \displaystyle \frac{\beta U(z)}{z}\) \(\beta\) frequency parameter (default value is \(0.025\)) \(\sigma(z)\) \(= \begin{cases} 0.15 U(z) \left( \displaystyle \frac{z}{z_s} \right)^{-0.125} , & z \leq z_s \\ \\ 0.15 U(z) \left( \displaystyle \frac{z}{z_s} \right)^{-0.275}, & z > z_s \end{cases}\) \(U(z)\) 1-hour mean wind speed at \(z\) meters above mean water line \(z_s\) thickness of surface layer 1.9. Normal wind spectrum The wind gust spectrum normal to the mean wind direction may be combined with any of the above spectra and is described according to (Simiu, 1978) by \[S_v^+(z,\omega )=\frac{\kappa\bar{u}^2Kx'}{\omega (1+9.5x')^{5/3}}\] where \(S_v^+\) spectral density normal to mean wind direction \(K\) scale factor (\(K = 17\) is used in the program) \(x'\) \(= \displaystyle \frac{\omega z}{2 \pi \bar{u}}\) The normal wind spectrum is not included for the state-space model (time domain generation). 1.10. Admittance function Due to the fact that high frequency wind fluctuation has low spatial correlation, an admittance function is introduced. The admittance function is structure-dependent, and serves mainly as a low-pass filter for the gust spectra. The admittance function, proposed by (Davenport, 1977), is given by \[x(\omega ,z)=(1+(\frac{\omega \sqrt{A}}{\pi \bar{u}})^{4/3})^{-1}\] where \(\mathrm {A}\) is the characteristic area of the structure. The admittance function is applied both for the gust spectrum in mean direction and for the gust spectrum normal to the mean wind direction: \[\begin{array}{l}S_u^{+'}(\omega )=x^2(\omega )S_u^+(\omega )\\\\S_v^{+'}(\omega )=x^2(\omega )S_v^+(\omega )\end{array}\] The effect of the admittance function is included in the generated wind time series for each body, but not for the state-space model (time domain generation). 1.11. Computation of wind at multiple locations In some situations it is necessary to compute wind velocity at multiple locations during time domain simulation. A typical example of this is when computing wind force on slender elements. For wind spectra where the spectral density does not depend on the vertical position this is simply achieved by scaling the pre-generated wind time series with the wind profile, Equation (1). This applies for the following wind spectra: Davenport Harris Wills The spectral density of the following wind spectra depends on the vertical position, which means that the approach described above will not give correct results: Sletringen ISO 19901-1 (NPD) API Simiu normal gust To account for the vertical variation in spectral density, wind velocity is pre-generated at multiple levels in an evenly distributed grid from \(z=z_{min}\) to \(z=z_{max}\): \[u_{gen,i}(t,z=z_i),i=[1,nz]\] During timedomain simulation, the wind velocity at an arbitrary vertical position \(\mathrm {z}\) is found using linear interpolation between the pre-generated time series before the wind profile is applied: \[u(t,z)=u_{gen}(t,z)\cdot (\frac{z}{z_r})^\alpha\] Computation of wind velocity outside the specified domain can be enabled by the user. If so, the nearest pre-generated wind velocity is used when computing wind velocity outside the domain: \[u_{gen}(t,z)= \begin{cases} u_{gen}(t,z_{min}), & \text{if} z < z_{min} \\ u_{gen}(t,z_{max}), & \text{if} z > z_{max} \\ \end{cases}\] 2. Waves 2.1. Linear potential model Linear wave potential theory is used throughout the study. The incoming undisturbed wave field is determined by the wave potential \(\mathrm {\Phi_0}\). \(\mathrm {\Phi_0}\) defines a long-crested sinusoidal wave. Unidirectional wave spectra are thought of as a sum of a large number of regular waves at different frequencies. Short-crested waves are constructed by introducing a directional distribution in addition to the frequency distribution. Further details of the method used are outlined below. The wave potential \(\mathrm {\Phi_0}\) for a regular wave is, according to Airy’s theory, expressed by \[\Phi_0=\frac{\zeta_ag}{\omega }C_1\cos\big(\omega t-kx\cos(\beta )-ky\sin(\beta )+\phi _\zeta\big)\] where \(\zeta_a\) wave amplitude \(g\) acceleration due to gravity \(k\) wave number, \(~ \omega^2 = gk \tanh(k d)\) \(\beta\) direction of wave propagation. (\(\beta = 0\) corresponds to wave propagation along the positive x-axis.) \(\phi_\zeta\) wave component phase angle \(\mathrm {C_1}\) is given by \[C_1=\frac{\cosh(k(z+d))}{\cosh(kd)}\] where \(\mathrm {d}\) is the water depth. In deep water \(\mathrm {C_1}\) tends to \[C_1\approx \exp(kz)\] We then obtain the following relations for the particle velocities and accelerations in the undisturbed wave field \[\begin{array}{l}v_x=\zeta_a\omega \cos(\beta )C_2\sin(\alpha )\\\\v_y=\zeta_a\omega \sin(\beta )C_2\sin(\alpha )\\\\v_z=\zeta_a\omega C_3\cos(\alpha )\\\\a_x=\zeta_a\omega ^2\cos(\beta )C_2\cos(\alpha )\\\\a_y=\zeta_a\omega ^2\sin(\beta )C_2\cos(\alpha )\\\\a_z=-\zeta_a\omega ^2C_3\sin(\alpha )\end{array}\] where \(\mathrm {\alpha =\omega t-kx\cos(\beta )-ky\sin(\beta )+\phi _\zeta}\). Using the deep water approximation we have \[C_1=C_2=C_3=\exp(kz)\] Taking into account finite water depth we have \[\begin{array}{l}\displaystyle C_1=\frac{\cosh(k(z+d))}{\cosh(kd)}\\\\\displaystyle C_2=\frac{\cosh(k(z+d))}{\sinh(kd)}\\\\\displaystyle C_3=\frac{\sinh(k(z+d))}{\sinh(kd)}\\\\\end{array}\] The surface elevation is given by \[\zeta=\zeta_a\sin(\alpha )\] Similarly, the linearized dynamic pressure is given by \[p_d=-\rho g\zeta_aC_1\sin(\alpha )\] In Figure 2 examples of the wave elevation and particle velocities and accelerations are shown. Figure 2. Example of wave elevation, velocities and accelerations in a sinusoidal wave. 2.2. Phase angles of wave particle motions The surface elevation, \(\mathrm {\zeta}\), is selected as the reference when describing waves and wave-induced responses. \[\zeta=\zeta_a\sin(\alpha )\] where \[\begin{array}{l}\alpha =\omega t+\phi _p+\phi +\phi _\zeta\\\\\phi _p=-kx\cos(\beta )-ky\sin(\beta )\end{array}\] \(\mathrm {\phi _p}\) is a position-dependent phase angle. Table 1 and Table 2 present summaries of phase angles and amplitudes of wave particle velocities and accelerations in x, y and z directions: \(\mathrm {C_2}\) and \(\mathrm {C_3}\) are depth- and frequency-dependent functions, compare with Linear potential model. Table 1. Summary of phase angles referenced to wave elevation Direction Velocity Acceleration X 0 \(\mathrm {\pi /2}\) Y 0 \(\mathrm {\pi /2}\) Z \(\mathrm {\pi /2}\) \(\mathrm {\pi }\) Table 2. Table 3.3 Summary of motion "amplitudes" Direction Displacement Velocity Acceleration X \(\mathrm {\cos(\beta )C_2\zeta_a}\) \(\mathrm {\omega \cos(\beta )C_2\zeta_a}\) \(\mathrm {\omega ^2\cos(\beta )C_2\zeta_a}\) Y \(\mathrm {\sin(\beta )C_2\zeta_a}\) \(\mathrm {\omega \sin(\beta )C_2\zeta_a}\) \(\mathrm {\omega ^2\sin(\beta )C_2\zeta_a}\) Z \(\mathrm {C_3\zeta_a}\) \(\mathrm {\omega C_3\zeta_a}\) \(\mathrm {\omega ^2C_3\zeta_a}\) 2.3. Complex notation The complex harmonic wave component is defined by \[\tilde{\zeta}(\omega _k)=\zeta_a\exp\big(i(\omega _kt+\phi _\zeta+\phi _p)\big)\] \[|\tilde{\zeta}|=\zeta_a\] \[\mathrm{Arg} \{ \tilde{\zeta} \} = \omega t + \phi_\zeta + \phi_p\] The surface elevation is \[\zeta=\mathrm {Im}\{\tilde{\zeta}\}=\zeta_a\sin(\omega t+\phi _\zeta+\phi _p)\] All other responses, \(\mathrm {r}\), are related to the surface elevation by complex transfer functions, \(\mathrm {H_r}\), and can be derived from a complex harmonic function, \(\mathrm {\tilde{r}}\). \[\begin{array}{l}\begin{split}\tilde{r}&=H_r\tilde{\zeta}\\\\r&=\mathrm {Im}\{\tilde{r}\}\\\\|H_r|&=\frac{r_a}{\zeta_a}\\\\\mathrm {Arg}\{H_r\}&=\phi _r\end{split}\end{array}\] 2.4. JONSWAP spectrum \[S_\zeta^+(\omega )=\frac{\alpha g^2}{\omega ^5}\exp(-\beta (\frac{\omega _p}{\omega })^4)\gamma ^{\exp(\displaystyle \frac{(\frac{\omega }{\omega _p}-1)^2}{2\sigma ^2})}\] The wave spectrum contains the following parameters \(\alpha\) spectral parameter \(\omega_p\) peak frequency, \(\displaystyle \omega_p = \frac{2 \pi}{T_p}\) \(\gamma\) peakedness parameter \(\beta\) form parameter, default value \(\beta = 1.25\) \(\sigma\) spectral parameter with default values \(\) \(\sigma_a = 0.07 \quad\) for \(\quad \omega < \omega_p\) \(\) \(\sigma_b = 0.09 \quad\) for \(\quad \omega > \omega_p\) Significant wave height, \(\mathrm {H_s}\), is often used instead of \(\mathrm {\alpha }\) to parameterize the spectrum. \[\alpha =(\frac{H_s\omega _p^2}{4g})^2\frac{1}{0.065\gamma ^{0.803}+0.135}\] Another formula giving the same \(\mathrm {\alpha }\) is \[\alpha =5.061\frac{H_s^2}{T_p^4}(1-0.287\ln(\gamma ))\] Equation (39) and Equation (40) have been implemented in the program as alternatives to specifying \(\mathrm {\alpha }\). \(\mathrm {\gamma }\) may normally be taken as \[\begin{array}{l}\displaystyle \gamma =\exp\left[3.484(1-0.1975~\delta \frac{T_p^4}{H_s^2})\right] \\\\ \displaystyle \delta =0.036-0.0056\frac{T_p}{\sqrt{H_s}}\end{array}\] However, for a two parameter JONSWAP spectrum, the following limits on \(\mathrm {\gamma }\) are valid \[\begin{array}{lll}T_p\geq 5\sqrt{H_s}&&\gamma =1.0\\\\T_p\leq 3.6\sqrt{H_s}&&\gamma =5.0\end{array}\] The following alternatives for specifying the spectrum are available. \(\mathrm {\alpha ,\omega _p,\gamma ,\beta ,\sigma _a,\sigma _b}\) \(\mathrm {H_s}\), \(\mathrm {T_p}\), \(\mathrm {\gamma }\), assuming \(\mathrm {\beta =1.25}\), \(\mathrm {\sigma _a=0.07}\) and \(\mathrm {\sigma _b=0.09}\). \(\mathrm {\alpha }\) is calculated from Equation (39) \(\mathrm {H_s}\), \(\mathrm {T_p}\), assuming \(\mathrm {\beta =1.25}\), \(\mathrm {\sigma _a=0.07}\) and \(\mathrm {\sigma _b=0.09}\). \(\mathrm {\gamma }\) is calculated from Equation (41) and \(\mathrm {\alpha }\) is calculated from Equation (39). 2.5. Pierson-Moskowitz spectrum \[S_\zeta^+(\omega )=\frac{a}{\omega ^5}\exp(-\frac{b}{\omega ^4})\] where the constants \(\mathrm {a}\) and \(\mathrm {b}\) are given for the two-parameter spectrum as \(\mathrm {\displaystyle b=\displaystyle \frac{1}{\pi }(\frac{2\pi }{T_z})^4=\frac{496.1}{T_z^4}}\), \(\mathrm {\displaystyle a=\frac{bH_s^2}{4}=\frac{124H_s^2}{T_z^4}}\), and for the one-parameter spectrum (the so-called ITTC-spectrum) as \(\mathrm {a=0.0081g^2}\), \(\mathrm {\displaystyle b=\frac{3.11}{H_s^2}}\). And where \(H_s\) significant wave height \(T_z\) zero-crossing wave period \(g\) acceleration of gravity 2.6. Two-peaked spectrum A two-peaked spectrum according to (Torsethaugen, 1996) has been implemented. 2.7. Numerically defined spectrum This is a wave spectrum specified by the user. The spectrum is defined by spectral values and frequencies. Spectral values outside the given frequency range are assumed to be zero. Linear interpolation is used within the frequency range. 2.8. Combination of wave spectra The sea state is described as a sum of two spectra: a wind sea contribution and a swell contribution. \[S_{\zeta, \mathrm{TOT}}^{+} (\beta, \omega) = S_{\zeta,1}^{+} (\omega) \theta_1 (\beta - \beta_1) + S_{\zeta,2}^{+} (\omega) \theta_2 (\beta - \beta_2) \tag{3.37}\] in which \(S_{\zeta,1}^+, S_{\zeta,2}^+\) describe the one-sided wave spectra (power spectra of wave elevation). \(\theta_1, \theta_2\) describe the directionality of the waves. Unidirectional waves and cosine spreading functions are included. \(\beta\) direction of wave propagation. For more information on the power spectra, see Spectral analysis. The spreading function satisfies the relations \[\begin{array}{llll}\displaystyle \int_{-\pi /2}^{\pi /2}\theta _j(\beta )\mathrm {d}\beta &=1.0&&\displaystyle -\frac{\pi }{2}<\beta <\frac{\pi }{2}\\\\\theta _j(\beta )&=0&&\displaystyle \frac{\pi }{2}<\beta <\frac{3\pi }{2}\end{array}\] \[\int_0^\infty S_{\zeta,1}^+(\omega )\mathrm {d}\omega +\int_0^\infty S_{\zeta,2}^+(\omega )\mathrm {d}\omega =\sigma_\zeta^2\] in which \(\mathrm {\sigma _{\zeta}^2}\) is the variance of the surface elevation. 3. Current The current is described by a profile with specified directions and speeds at different levels. Linear interpolation is used to determine current velocity between points defined in the profile. If the profile does not cover the complete water column the current is assumed to be constant outside the tabulated range.