Wind Spectra

1. Definitions

\(z\)

height above calm water level

\(z_r\)

wind reference height, normally 10 m

\(\bar{u}_r\)

average velocity at the reference height \(z_r\) above the calm water level

\(\theta\)

average wind propagation direction

When using parametric spectra to describe the wind field the wind is assumed to propagate in the horizontal plane. Further, any spatial variation is neglected except in the vertical direction. The time varying part of the wind is assumed to be a Gaussian stochastic process.

For a given wind propagation direction, \(\theta\), the wind speed is divided into a mean and a fluctuating component which varies with time:

\[u_\theta(z, t) = \bar{u}(z) + u_{f,\theta}(z, t)\]

Optionally, a fluctuating component transverse to the the mean wind direction may be included:

\[u_t(z, t) = u_{f,t}(z, t)\]

2. Mean wind profile

The mean wind profile, \(\bar{u}(z)\) is given by

\[\bar{u}(z)=\bar{u}_r(\frac{z}{z_r})^\alpha\]

where

\(\alpha\)

height coefficient (0.10 - 0.14)

3. 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

\[x(\omega ,z)=(1+(\frac{\omega \sqrt{A}}{\pi \bar{u}_r})^{4/3})^{-1}\]

where \(\mathrm {A}\) is the characteristic area of the structure.

For bodies with Quadratic Wind Coefficients, the admittance function is applied to the wind spectrum both in the mean direction and, if included, for the transverse wind spectrum:

\[\begin{array}{l}S_u^{+'}(\omega )=x^2(\omega )S_u^+(\omega )\\\\S_v^{+'}(\omega )=x^2(\omega )S_v^+(\omega )\end{array}\]

4. Wind spectra for the main direction

The fluctuating component in the mean direction is realized from a spectrum function. Multiple spectrum functions are available as described in the following sections.

4.1. Davenport wind spectrum

The Davenport wind spectrum is given by

\[S(f, z)=\frac{4\kappa\bar{u}_r^2r^2}{\omega (1+r^2)^{4/3}}\]

where

\(\omega\)

angular frequency, \(\omega = 2\pi f\)

\(r\)

\(\displaystyle = \omega \frac{L_r}{2 \pi \bar{u} (z)}\)

\(L_r\)

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).

4.2. Harris wind spectrum

The Harris wind spectrum is given by

\[S(f, z)=A\frac{4\kappa\bar{u}_r^2r}{\omega (2+r^2)^{5/6}}\]

4.3. Wills spectrum

This spectrum is identical to the Harris spectrum except for the factor \(\mathrm {A}\).

\[A=0.51\frac{(2+r^2)^{5/6}}{(r^{0.15}+\frac{9}{8}r)^{5/3}}\]

4.4. Sletringen wind spectrum

The Slentringen wind spectrum is given by

\[S(f,z) = \bar{u}_{r}^p\{\frac{a_1(\displaystyle \frac{z}{z_r})^{-q}}{(B_1+f)^{5/3}}+\frac{a_2}{(B_2^n+f^n)^{5/(3n)}}\}\]

where

\[\begin{array}{l} \displaystyle n=n_0+n_1e^{-c_n\rho }\\\\ \displaystyle B_1=b_1\frac{\bar{u}_r}{z}\\\\ \displaystyle B_2=b_2\bar{u}_re^{c_{b2}\rho }\\\\ \displaystyle \rho =\frac{\gamma }{(\displaystyle \frac{\bar{u}_r}{10~\mathrm {\displaystyle m/s}})^\varepsilon } \end{array}\]

and

\(\gamma\)

is the temperature stability parameter, recommended values are in the range \(\displaystyle \mathrm{ 10 ~ \frac{K}{km} - 20 ~ \frac{K}{km} }\)

The values for the parameters have been derived from the Sletringen measurements as shown below.

Parameter Value

\(\mathrm {a_1}\)

\(\mathrm {2.03\cdot 10^{-5}}\)

\(\mathrm {a_2}\)

\(\mathrm {1.18\cdot 10^{-5}}\)

\(\mathrm {\varepsilon }\)

\(\mathrm {1.40}\)

\(\mathrm {p}\)

\(\mathrm {3.07}\)

\(\mathrm {q}\)

\(\mathrm {1.50}\)

\(\mathrm {b_1}\)

\(\mathrm {1.82\cdot 10^{-2}}\)

\(\mathrm {b_2}\)

\(\mathrm {3.56\cdot 10^{-4}}\)

\(\mathrm {c_{b2}}\)

\(\mathrm {0.293}\)

\(\mathrm {n_0}\)

\(\mathrm {0.281}\)

\(\mathrm {n_1}\)

\(\mathrm {0.428}\)

\(\mathrm {c_n}\)

\(\mathrm {0.183}\)

4.5. ISO 19901-1 wind spectrum

The ISO spectrum is also referred to as the NPD spectrum and is defined as follows:

\[S(f, z)=\frac{\displaystyle 320\cdot (\frac{\bar{u}_r}{10})^2\cdot (\frac{z}{10})^{0.45}}{\displaystyle (1+f_m^n)^{5/(3n)}}\\\\\]

where

\[f_m=172\cdot f\cdot (\frac{z}{10})^{2/3}\cdot (\frac{\bar{u}_r}{10})^{-0.75}\]
\[n = 0.468\]

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 \(1/600~\mathrm {Hz}\).

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.

image101
Figure 1. ISO 19901-1 wind spectrum for a mean wind speed of 20 m/s

4.6. API wind spectrum

The API spectrum representing typhoon wind conditions may be written

\[S(f, z) = \frac{\sigma(z)^2}{f_p(1+1.5\frac{f}{f_p})^{5/3}}\]

where

\[f_p = \frac{\beta \bar{u}(z)}{z} \label{eq-api-2}\]
\[\sigma(z) = \begin{cases} 0.15 \bar{u}(z) \left( \displaystyle \frac{z}{z_s} \right)^{-0.125}, & z \leq z_s \\ \\ 0.15 \bar{u}(z) \left( \displaystyle \frac{z}{z_s} \right)^{-0.275}, & z > z_s \end{cases} \label{eq-api-3}\]

and

\(\beta\)

frequency parameter (default value is \(0.025\))

\(z_s\)

thickness of surface layer

4.6.1. ESDU Wind Spectrum

The ESDU wind spectrum given in (API RP 2MET, 2014) is intended to represent tropical storms. The spectrum is parameterized by the function

\[S(f, z) = \frac{4 I_u(z)^2 U_{w,1h}(z) L_{u,x}(z)}{\left[ 1 + 70.8 \cdot \left( f \frac{L_{u,x}(z)}{U_{w,1h}(z)} \right)^2 \right]^{5/6}}\]

where

\[L_{u,x}(z) = \frac{50 \cdot z^{0.35}}{z_0^{0.063}}\]
\[U_{w,1h}(z) = \frac{u_*}{0.4} \ln \left( \frac{z}{z_0} \right)\]
\[I_u(z) = \frac{u_* \cdot 7.5 \cdot \eta \left[ 0.538 + 0.09 \cdot \ln \left(\frac{z}{z_0} \right) \right]^{\eta^{16}}}{1 + 0.156 \cdot \ln\left(\frac{u_*}{\widehat{f_C} \cdot z_0}\right)} \cdot \frac{1}{U_{w,1h}(z)}\]
\[u_* = \sqrt{C_{d_{10}}} \bar{u}_r\]
\[z_0 = 10 \cdot e^{\frac{-0.4}{\sqrt{C_{d_{10}}}}}\]
\[C_{d_{10}} = \begin{cases} \left( 0.49 + 0.065 \bar{u}_r \right) \cdot 10^{-3} &, \bar{u}_r < 27.85 \mathrm{\ m/s} \\ 0.023 &, \bar{u}_r \geq 27.85 \mathrm{\ m/s} \end{cases}\]
\[\eta = 1 - 6 \widehat{f_C} \frac{z}{u_*}\]
\[\widehat{f_C} = 2 \cdot 7.29 \cdot 10^{-5} \sin|\Psi|\]

and

\(\Psi\)

Site latitude in decimal degrees \(\mathrm{\left[ deg \right]}\)

5. Transverse wind spectrum

The wind spectrum normal to the mean wind direction may be combined with any of the above spectra and is described according to (Simiu, 1978) by

\[S_v^+(z,\omega )=\frac{\kappa\bar{u}_r^2Kx'}{\omega (1+9.5x')^{5/3}}\]

where

\[x' = \frac{\omega z}{2 \pi \bar{u}_r} \label{eq-simiu-2}\]
\[K = 17 \label{eq-simiu-3}\]