The Kaimal Model

The autospectral density for the longitudinal component of turbulence, according to the Kaimal model, is:

\[ \begin{equation} \frac{nS_{uu}(n)}{\sigma_u^2}=\frac{4{\widetilde{n}}_u}{(1+6.0{\widetilde{n}}_u)^{5/3}} \end{equation} \]

where \(S_{uu}\) is the auto-spectrum of wind speed variation, n is the frequency of variation, \(\sigma_u\) is the standard deviation of wind speed variation and \(\widetilde{n}_u\) is a non‑dimensional frequency parameter given by:

\[ \begin{equation} {\widetilde{n}}_u=\frac{n\ L_1}{U} \end{equation} \]

Here \(L_1 = 2.329 {^x}L_u\) where \({^x}L_u\) is the length scale of longitudinal turbulence, and \(U\) is the mean wind speed as before.

The coherence of the turbulent wind fluctuations at points separated by a distance \(\Delta r\) is defined as:

\[ \begin{equation} C(\Delta r,n)=\exp\left(-8.8\Delta r\sqrt{\left(\frac{n}{U}\right)^2+\left(\frac{0.12}{L(\Delta r,n)}\right)^2}\right) \end{equation} \]

A three-component Kaimal model is available for compatibility with the IEC standard 61400-1. The scale parameter \(\Lambda_1\) defines the characteristics of the turbulence, through the following relationships:

\[ \begin{equation} \begin{aligned} & {^x}L_u = 8.1\Lambda_1, \\ & {^x}L_v = 2.7\Lambda_1, \\ & {^x}L_w = 0.66\Lambda_1 \\ \end{aligned} \end{equation} \]

\[ \begin{equation} {\widetilde{n}}_i=\frac{n{{^x}L}_i}{U}\ \ \ i\ =\ u,\ v,\ w \end{equation} \]

\[ \begin{equation} \frac{nS_{ii}(n)}{\sigma_i^2}=\frac{4{\widetilde{n}}_i}{(1+6.0{\widetilde{n}}_i)^{5/3}} \end{equation} \]

For the longitudinal component,

\[ \begin{equation} C(\Delta r,n)=\exp\left(-H\Delta r\sqrt{\left(\frac{n}{U}\right)^2+\left(\frac{0.12}{L_c}\right)^2}\right) \end{equation} \]

where the coherence decay constant \(H = 8.8\) and the coherence scale factor \(L_c = 3.5\Lambda_1\). The standard does not define the coherence for the other two components, so the following expression is used:

\[ \begin{equation} C(\Delta r,n)=\exp\left(-H\Delta r\frac{n}{U}\right) \end{equation} \]

A more general formulation for the Kaimal model has also been introduced in which the parameters \({^x}L_u\), \({^x}L_v\), \({^x}L_w\) and \(L_c\) can be specified separately instead of specifying \(\Lambda_1\), and the parameter \(H\) and can also be specified. This can be used for compatibility with the third edition of the IEC standard 61400-1, which gives \(H=12\) and \(L_c=8.1\Lambda_1\).

Last updated 30-08-2024