In wind energy, the effect of turbulence upon turbines is typically
simulated using wind “input” time series based on turbulence spectra. The
velocity components' spectra are characterized by the amplitude of turbulent
fluctuations, as well as the length scale corresponding to the dominant
eddies. Following the IEC standard, turbine load calculations commonly
involve use of the Mann spectral-tensor model to generate time series of the
turbulent three-dimensional velocity field. In practice, this spectral-tensor
model is employed by adjusting its three parameters: the dominant turbulence
length scale

Previously, site-specific

The new form is tested across several different conditions and sites, and it
is found to be more robust and accurate than estimates relying on friction
velocity observations. Assumptions behind the derivations are also tested,
giving new insight into rapid-distortion theory and eddy-lifetime modeling –
and application – within the atmospheric boundary layer. The work herein
further shows that distributions of turbulence length scale, obtained using
the new form with typical measurements, compare well with distributions

Of the atmospheric parameters which are generally input into (or required by)
wind turbine load calculation codes, several stand out due to their
prominence in load contributions: the “mean” wind speed

To a lesser extent, some sensitivity to the
Mann-model anisotropy parameter

Within the context of obtaining site-dependent statistics of the most crucial
load-driving parameters (

Because of its widespread use in the wind industry and its inclusion in the

After deriving the eddy lifetime and giving subsequent expressions for the
turbulence length scale, this article proceeds to validation of the
underlying assumptions. Constraints implied by fitting the Mann model to
measured spectra in non-neutral conditions, given eddy lifetime and
mixing-length relations, are also tested. This includes dependence of
predicted velocity variance on model anisotropy parameter (

Relation of the turbulence length (spectral “peak”) scale to measurable
statistics is possible through the eddy-lifetime form of

A number of forms exist to estimate eddy lifetime

The

The reciprocal of eddy-damping rate,

The hypergeometric function

The
peak of the von Kármán isotropic TKE spectrum

Note

Since Eqs. (

The parameters

The rapid-distortion equations do not explicitly solve for production of
normal stresses (which sum to twice the turbulent kinetic energy) or shear
stress, though they do include terms that perturb the
stresses

Assuming a constant mean shear

A stationary equilibrium result is achieved via the eddy-lifetime
prescription together with rapid distortion of the isotropic spectral tensor
– with

Noting that the spectrum of a variable integrates to the variance of said
variable, then invoking Eq. (

The spectral Mann model (“MM”) distorts the isotropic von Kármán
spectral tensor (

In addition to the approximate expression (

As spectra fitted to Mann-model outputs correspond to distorted

Within the atmospheric surface layer (ASL), in the homogeneous stationary
limit under neutral conditions,

Since the choice of eddy lifetime form (

For the assumption testing in this section, the spectra used are measured via
three-dimensional sonic anemometers on the primary meteorological mast
located at the Danish National Test Centre for Large Wind
Turbines (Høvsøre), 1.75 km from the western coast of
Denmark

The sonic anemometers actually give a temperature very close to the virtual temperature, i.e., the temperature including buoyant effects of water vapor.

at heights of 10, 20, 40, 60, 80, and 100 m. This allows calculation of mean speeds, directions, and vertical shear of mean speed over individual 10 min records; in particular we focus on heights ofThe implications of Eqs. (

Joint distribution of isotropic (un-distorted) variance

Considering wind speeds in the typical turbine operating range of
4–25 m s

The data also show that

Ratio of observed streamwise to isotropic fluctuation magnitude
versus

The efficacy of using Eq. (

The
spectral fits were done using spectral-tensor model output over the parameter
ranges of

Joint probability density function of predicted and diagnosed
(observed) turbulent length scale, from measurements at Høvsøre over
the homogeneous eastern land sectors.

To demonstrate the statistical character of Eq. (

For the homogeneous land case in Fig.

Probability density function of turbulent length scale from
observations at Høvsøre from the homogeneous eastern land sectors.
Black: Mann-model scale from fits to spectra; dotted/blue: “mixing-length”
formulation (

While Eq. (

To demonstrate the (probabilistic) use of Eq. (

Figure

Probability density of turbulence length scale

The

To further show the behavior of

The “western lidar” at
Østerild is located

Just as Fig.

Probability density function of turbulent length scale from
observations at Østerild from the western mast/lidar. Black: Mann-model
scale from fits to spectra; dotted-blue: “mixing-length” formulation
(

As in the cases above (homogeneous land and inhomogeneous coastal), the new
form (

Towards concluding, we first revisit the motivation for (and thus context of)
this work: (1) to “close” the

A previously suggested form (

Since Eq. (

One interesting implication of the testing of assumptions then follows from
the finding that

The peak length scale also grows with
boundary-layer depth

It is also notable that Fig.

Ongoing work includes wind-speed-dependent prediction of

The eddy lifetime of

Theory and measurements support the assumption that

In terms of the classic mixing-length form

The data are within an SQL database at DTU and are not publicly available.

The author declares that he has no conflict of interest.

The author thanks the reviewers for their time and effort towards constructive criticism of the present article, and thanks are owed to Nikolay Dimitrov for discussions around probabilistic loads. This work was partly supported by the DTU Wind Energy internally funded cross-sectional project “Wind to Loads”. Edited by: Horia Hangan Reviewed by: two anonymous referees