**1. Introduction**

During the past thirty years, wake turbulence and its effects on wind turbines and wind farms have been extensively investigated, primarily analytically and to some extent experimentally. The extensive literature up to 2010 has been well covered in the widely used text of Manwell et al. [1]. As for the extensive analytical investigations since, suffice it to mention, as representative samples, Keck et al. [2] for wake-turbulence modeling from the low-fidelity CFD treatment of the Navier-Stokes (NS) equations, and Carrion et al. [3] for wake-turbulence modeling from the high-fidelity treatment of NS equations. The work of Carrion et al. [3] for example, includes a concise account of the state of the art of modeling wind-farm wake turbulence.

An overview of these investigations [1–3] is included here; although extremely brief, this should help appreciate how the present work serves as a desirable adjunct of experimental and high-fidelity CFD based investigations, and why it represents a new and promising avenue of wake-turbulence modeling. Wake-turbulence modeling falls into three categories:


While major strides have been made in generating databases and in providing a much-improved understanding of wake turbulence, the current capability for modeling the one-point statistics of autospectrum, much more so, for modeling the two-point statistics of coherence, merits significant improvement. In fact, empirical exponential coherence functions are still being used [1]. Accordingly, the present study explores the feasibility of extracting the one-point statistics of autospectrum and the two-point statistics of coherence from a database.

The autospectral model extraction from a database of the present study is based on the framework due to Schau, Gaonkar and Polsky [5]. This framework guarantees that the extracted model and the autospectral data points have the same mean square value (a measure of turbulence energy), time scale and the Kolmogorov −5/3 spectral decay. For completeness, an earlier study by Gaonkar [6] should be mentioned as well; therein, the autospectral model extraction from a database is based on the framework of reference [7], in which the −5/3 spectral law is bypassed. Now it is expedient to address the development of a framework for extracting the two-point statistics models from a database. This can be approached either through cross-spectrum, which is a complex quantity involving the magnitude and phase or through coherence, which, as a spectral correlation coefficient, is a real quantity. The first approach generally leads to modeling the magnitude and not the phase, as was the case in Ref. [7]. The second approach through coherence is relatively more convenient and provides a means of capturing the two-point statistics from a database completely. The recent study due to Krishnan and Gaonkar [8] follows this second approach; although not tested against a database, the framework is formulated with a mathematical basis and the present study adopts this framework [8].

By design, these autospectral and coherence models are in closed form and they have a simple analytical structure to facilitate interrogation and interpretation of voluminous data points on autospectra and coherences. And they lend themselves well to routine use as a predictive tool. Compared to a description through such voluminous, numerically generated, autospectral and coherence data points, they describe wake turbulence analytically with better transparency and bring better understanding. Thus, these interpretive models broaden the scope and utility base of the database that invariably involves enormous resources. While the extracted models are database-specific (thus they are not predictive by themselves), the framework can be applied to any database and the model extraction is a routine exercise.

In the treatment of coherence for homogeneous isotropic turbulence for which the frozen turbulence hypothesis is applicable (HIT), the present study is motivated by and built on the earlier studies of Burton et al. [9], Houbolt and Sen [10], Frost et al. [11] and Irwin [12]. This treatment of coherence for HIT is found to show differences among these studies [9–12], and the present study, after an in-depth examination, follows Frost et al. [10] for cross-spectra and Irwin [12] for coherences. Given this background, the present study seems to provide a unified account of coherence for HIT in the treatment of wake turbulence.

To sum up: These interpretive models complement the experimental and CFD-based investigations as surrogate analytical models for both the one-point statistics of autospectrum and the two-point statistics of coherence. Moreover, this paper also demonstrates the feasibility of fruitfully exploiting the methodologies from other fields to the treatment of wind-turbine wake turbulence. And these methodologies offer promise towards providing a foothold on a formidably complex flow field inside a windfarm for engineering analysis.

## *Basic of Modeling*

A comparison of the measured autospectra of ambient atmospheric boundary layer turbulence (ABL) and wake turbulence shows that the ABL autospectrum has gone through changes in energy distribution with respect to frequency. Figure 1 [13] should help bring a better understanding of this comparison; specifically, it shows measured dimensionless longitudinal autospectrum *f S* ˇ *uu*/*σ*2*u* versus dimensionless frequency *f <sup>z</sup>*/*U*, where *z* is the mast height and *U* is the mean wind speed. These autospectra were experimentally generated at the same location in a wind farm over a complex terrain. Figure 1a refers to ABL with a turbulence intensity of 0.103, when the turbines were under stand-still conditions. Furthermore, Figure 1b refers to wake turbulence with a turbulence intensity of 0.204, when the turbines were fully operational. This change in the shape of the wake turbulence autospectrum cannot be realized through a linear superposition of a series of independently occurring changes at different frequencies on the ABL autospectrum. Thus, the autospectral morphing must be due to a nonlinear transformation of ABL. Stated otherwise, wind-farm wake turbulence could be idealized as nonlinearly transformed ABL and in turn, an earlier-developed mathematical framework for autospectral modeling of airwake-downwash turbulence could be adapted to modeling wind-farm wake turbulence as well [5,14]. (Airwake-downwash turbulence refers to the coupled flow-field of ship's airwake shed from the superstructure and the helicopter downwash. Therein [5,14], the mathematical framework "posits" that airwake-downwash turbulence is nonlinearly transformed ABL. Regarding the coherence, the framework of Ref. 8 is adopted with the same justification that is used for the autospectrum.)

**Figure 1.** (**a**) Measured longitudinal velocity autospectrum of ambient atmospheric boundary layer turbulence, and (**b**) Measured longitudinal velocity autospectrum of wake turbulence.

#### **2. Methodology of AutoSpectra**

The Lateral component *v*(*t*) is selected for providing details of the autospectral model extraction methodology [5]. The methodology remains the same for the vertical component with minor changes in the parameters used in the constraint equations. However, for the Longitudinal component, additional changes in the constraint equations are also required [14]. One-sided autospectrum is used throughout.

Statistical independence of velocity components is assumed [9]. The mean square value, time scale and autospectral asymptotic limit law are different for each component of non-homogeneous turbulence according to Kolmogorov's −5/3. The framework combines four elements: (1) A mathematical formulation based on a perturbation series expansion of the autocorrelation/autospectrum functions; (2) Extraction of time scale and autospectral asymptotic limit from the database; (3) Development of constraint equations in closed form to ensure that the developed model satisfies the requirements related to normalization, time scale, and autospectral asymptotic limit; and (4) Evaluation of the constants in the series expansion subject to satisfying the constraint equations and fitting a curve on a set of selected autospectral data points in a least squares sense.

#### *2.1. Lateral Wake Turbulence*

The perturbation series for the autocorrelation of lateral wake turbulence velocity *v*(*t*) can be expressed as in Equation (1).

$$\check{R}\_{vv}(\tau) = \beta\_{1v} R\_{vv}(\tau) + \beta\_{2v} R\_{vv}^2(\tau) + \beta\_{3v} R\_{vv}^3(\tau) + \dots + \beta\_{nv} R\_{vv}^n(\tau) \tag{1}$$

The calculated autocorrelation as well as series expansion autocorrelation follow the properties of normalized autocorrelations, that is, *<sup>R</sup>vv*(0) = *Rvv*(0) = 1. The Fourier transform of Equation (1) gives the series expansion for the autospectrum *<sup>S</sup>vv*(*f*):

$$\widetilde{S}\_{\text{vv}\_{\text{vv}}}(f) = \beta\_{1\text{v}} S\_{\text{vv1}}(f) + \beta\_{2\text{v}} S\_{\text{vv2}}(f) + \beta\_{3\text{v}} S\_{\text{vv3}}(f) + \dots + \beta\_{n\text{v}} S\_{\text{vvn}}(f) \tag{2}$$

where *Svvn*(*f*) is the Fourier transform of *Rnvv*(*τ*).

$$S\_{vvu}(f) = 4 \int\_0^\infty R\_{vv}^u(\tau) \cos(2\pi f \tau) d\tau \tag{3}$$

The autospectrum is typically normalized with respect to dimensional time scale *Tv*, which is traditionally defined as *Tv* = ∞ 0 *<sup>R</sup>vv*(*τ*)*d<sup>τ</sup>*. With *σ*<sup>2</sup> *v* , the mean square value, Equations (4) and (5) typify the normalization:

$$\tilde{S}\_{vv}(0) = 4\sigma\_v^2 \int\_0^\infty \tilde{R}\_{vv}(\tau)d\tau = 4\sigma\_v^2 T\_v \tag{4}$$

$$\frac{1}{\sigma\_{\upsilon}^{2}} \int\_{0}^{\infty} \tilde{S}\_{\upsilon\upsilon}(f) df = \tilde{R}\_{\upsilon\upsilon}(0) = 1 \tag{5}$$

According to Kolmogorov's spectral law, the autospectrum model should decay as given in Equation (6), where *Av* is a scaling parameter determined from the data, and *fh f* represents high frequencies.

$$\frac{f\mathbb{S}\_{vv}\left(f\_{\rm hf}\right)}{\sigma\_v^2} = A\_v \left(fT\_v\right)^{-2/3} \tag{6}$$

The basis function *Rvv*(*τ*) on the right-hand side of Equation (1) is the von Karman lateral correlation function as given in Equation (7).

$$R\_{vv}(\mathbf{x}) = \frac{2^{2/3}}{\Gamma(1/3)} \left(\frac{a\_{\upsilon}\tau}{T\_{\upsilon}}\right)^{1/3} \left[K\_{1/3}\left(\frac{a\_{\upsilon}\tau}{T\_{\upsilon}}\right) - \frac{1}{2}\left(\frac{a\_{\upsilon}\tau}{T\_{\upsilon}}\right)K\_{2/3}\left(\frac{a\_{\upsilon}\tau}{T\_{\upsilon}}\right)\right] \tag{7}$$

The scaling parameter *αv* in Equation (7) ensures that the relation in Equation (4) is satisfied.

## *2.2. Constraint Equations*

The autospectrum model is constrained by Equations (4)–(6). The extracted model as given in Equation (1) is substituted to obtain the constraint Equations (8)–(10) for the expansion series co-efficients [5].

Because *R vv*(0) = *Rvv*(0) = 1, Equation (1) gives:

$$1 = \beta\_{1v} + \beta\_{2v} + \dots + \beta\_{nv} \tag{8}$$

Satisfying Equation (4), and integrating both sides by Equation (1) gives [5,14]

$$
\mu\_v = 0.373417 \beta\_{1v} + 0.199591 \beta\_{2v} + 0.12236 \beta\_{3v} + \dotsb \tag{9}
$$

Similarly, satisfying Equation (6) leads to [5,14]

$$\frac{A\_v}{a\_v^{2/3}} = 0.186176(\beta\_{1v} + 2\beta\_{2v} + 3\beta\_{3v} + \cdots) \tag{10}$$
