Environment During dynamic simulation, wind velocity and wave elevation may be read from an ascii file. 1. Wind 1.1. 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. 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 ESDU 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. Irregular waves 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. 2.2. 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 1 examples of the wave elevation and particle velocities and accelerations are shown. Figure 1. Example of wave elevation, velocities and accelerations in a sinusoidal wave. 2.3. 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 \(\mathrm {C_3\zeta_a}\) \(\mathrm {\omega C_3\zeta_a}\) \(\mathrm {\omega ^2C_3\zeta_a}\) 2.4. 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, \(\mathrm {\tilde{r}}\). \[\begin{array}{l}\begin{split}\tilde{r}&=H_r\tilde{\zeta}\\\\r&=\mathrm {Im}\{\tilde{r}\}\\\\|H_r|&=\frac{r_a}{\zeta_a}\\\\\mathrm {Arg}\{H_r\}&=\phi _r\end{split}\end{array}\] General Assumptions and Notation Force models