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Wind spectra |
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Wind spectra |
The NPD spectrum, at elevation $z$ above MWL, is defined as \begin{equation} S(f,z) = 3.2\ U_\textrm{ref}^2 \left( \frac{z}{10} \right)^{0.45} \left[1 + \tilde{f}^n \right]^{-5/3n} \end{equation} where $n$ is 0.468, $U_\textrm{ref}$ is the user specified ref. mean speed and $\tilde{f}$ is \begin{equation} \tilde{f}(z) = 172\ f\ \left( \frac{z}{10} \right)^{2/3}\ \left( \frac{U_\textrm{ref}}{10} \right)^{-3/4}\ \end{equation} The associated 1-hour mean wind speed is \begin{equation} U_\textrm{z}(z) = U_\textrm{ref} \left[ 1 + 0.0573\sqrt{1 + 0.15\ U_\textrm{ref}} \ln{\left( \frac{z}{10} \right)} \right] \end{equation}
The API RP 2A (1993) spectrum, at elevation $z$ above MWL, is defined as \begin{equation} S(f,z) = U_\textrm{z}^2\ I_\textrm{z}^2\ f_p^{-1} \left[1 + 1.5\ \left( \frac{f}{f_p} \right) \right]^{-5/3} \end{equation} where \begin{equation} f_p = 0.025\ \frac{U_\textrm{z}}{z} \end{equation} The associated 1-hour mean wind speed is \begin{equation} U_\textrm{z}(z) = U_\textrm{ref}\ \left( \frac{z}{10} \right)^{0.125} \end{equation} The turbulence intensity is \begin{equation} I_\textrm{z}(z) = \begin{cases} 0.15\ \left( \frac{z}{20} \right)^{-0.125} & \text{if $z \leq$ 20 m} \\ 0.15\ \left( \frac{z}{20} \right)^{-0.275} & \text{otherwise} \end{cases} \end{equation}
The ESDU spectrum, at elevation $z$ above MWL, is defined as \begin{equation} S(f,z) = 4\ I_\textrm{z}^2\ U_\textrm{z}\ L_\textrm{u}\ \left[1 + 70.8 \left( \frac{f\ L_\textrm{u}}{U_\textrm{z}} \right)^2 \right]^{-5/6} \end{equation} The associated 1-hour mean wind speed is \begin{equation} U_\textrm{z}(z) = \frac{u_*}{0.4}\ln{\left( \frac{z}{z_0} \right)} \end{equation} Here $u_*$ is the friction velocity, calculated from the user specified ref. mean speed as \begin{equation} u_* = \sqrt{C_{d10}}\ U_\textrm{ref} \end{equation} and $z_0$ is the boundary-layer scaling parameter given by \begin{equation} z_0 = 10\ \exp{\left[ -\frac{0.4}{\sqrt{C_{d10}}} \right]} \end{equation} where $C_{d10}$ is the drag coefficient taken as \begin{equation} C_{d10} = \begin{cases} 0.0023 & \text{if $U_\textrm{ref} \geq$ 27.85 m/s} \\ 0.001\ (0.49 + 0.065\ U_\textrm{ref}) & \text{otherwise} \end{cases} \end{equation} The integral length scale is \begin{equation} L_\textrm{u}(z) = \frac{50 z^{0.35}}{z_0^{0.063}} \end{equation} The turbulence intensity is \begin{equation} I_\textrm{z} (z) = u_*\ 7.5\ \eta\ \left[0.538 + 0.09\ \ln{\left( \frac{z}{z_0} \right)} \right]^{\eta^{16}} \left[U_\textrm{z}\ \left(1 + 0.156\ \ln{\left(\frac{u_*}{f_C\ z_0} \right)} \right) \right]^{-1} \end{equation} where \begin{equation} \eta = 1 - \frac{6\ f_C\ z}{u_*} \end{equation} and $f_C$ is the Coriolis parameter, calculated from the user specified absolute value of site latitude, $\psi$, as \begin{equation} f_C = 2\ \Omega \sin{\psi} \end{equation} where $\Omega$ is the Earth's rotation speed, taken to be 72.9e-6 rad/s.