**3. Database**

The database is generated from a full-scale experimental study of wake turbulence by Mofiadakis et al. [13]. Specifically, data is collected at several points along a complex windy terrain at an altitude of 320–330 m with seven Vestas (27–225 kW) installed in a row. The thirty cases of autospectra that were generated from the database represent free stream to fully wake affected conditions. The autospectral decay was found to be in the range of −1.36 to −1.75 for free stream conditions (all wind turbines at stand-still condition). Moreover, for notational simplicity, *βiu*, *βiv* and *βiw* are simply referred to as *β* coefficients in this section.

The database typically comprises the temporal flow velocity points. In this case, however, the database [13] has already been transformed from the temporal to the frequency domain and normalized to dimensionless form *f Sii*(*f*)/*σ*2*i* . Moreover, the temporal autocorrelation is not available, and the dimensionless autospectra are presented on a log-log scale and in turn *Sii*(*f* = 0) is not available. This limitation is overcome by assuming that *Sii*(0) = *Sii*(*f*) as *f* approaches zero. It is emphasized that the framework is designed to develop autospectral models from a database; thus the lack of a temporal database ceases to be a major issue. Having extracted *σ*2*i* from Equation (5), key information to be extracted from the database is *Ti*, the time scale. As for *Ti*, it is calculated using *Sii*(*f*) at the lowest frequency in a log-log plot; see Equation (4). Finally, the autospectral decay constant *Ai* is graphically evaluated from Equation (6).

Having thus generated time scale *Ti* and autospectral asymptotic limit *Ai*, the numerical scheme now focuses on computing the series expansion *β* coefficients. It is emphasized that these *β* coefficients determine the scaling parameter *αi*; see Equation (9). As an iterative procedure, the scheme involves selecting the *β* coefficients, beginning with the von Karman model (e.g., *β*<sup>1</sup>*v* = 1 for the lateral component) and strictly enforcing the constraint as typified by Equation (8). For completeness, the gist of the iterative procedure is included; for details see [14].

The numerical scheme minimizes the sum of two errors in a least squares sense: model's deviation from the autospectral data and the *Ai* constraint error. That is, a selected set of *β* coefficients gives a model with a value of *Ai*; stated otherwise, these *β* coefficients carry a least squares error with respect to the measured autospectral data and an error with respect to the graphically measured *Ai* value. The resulting *Ai* error is expected to be within acceptable limits for wind turbine applications (<<10%). This error is perhaps acceptable, after all, *Ai* is not rigorously defined with respect to the data sets, nor is there a standard method of determining when a computed autospectrum has reached its point of asymptotic decay. This lack of precision also means *Ai* will vary somewhat from user to user. The *Ai* constraint typified by Equation (10) also merits one final comment. For some isolated cases of data

sets, the high frequency limit does not exhibit accurately the −5/3 spectral decay [13]. For these cases, it is sensible to exclude this constraint. The next section elaborates this scenario.

Computationally, the above numerical scheme is found to be inexpensive. For example, a gradient based search algorithm (MATLAB SQP) starting with the von Karman model does not require large iteration counts for a converged model. The reason is that the computational cost, now, depends only on the length of the discrete data array, as an error between the model and this data array, and this error has to be calculated at each iteration of the search algorithm. Given the thoughtful selection of search parameters such as step size in *β*-space and convergence criteria, this numerical scheme should prove inexpensive computationally.

#### **4. Result of Autospectra**

For illustration, just one example of the vertical component *w*(*t*) is selected. Modeling is presented based on both first-order (two-term series) and second-order (three-term series) correction. And in each case, modeling covers two approaches. In the first approach, the Kolmogorov −5/3 law is not enforced; that is, by satisfying only the first two constraints, typified by Equations (8) and (9). In the second approach, all three constraints are satisfied; that is, in addition to satisfying these two constraints, the model also satisfies the Kolmogorov −5/3 law (see Equation (10)) by a minimized error. This enforcement is identified in the respective figures by "*A error* = *xx*%", where *xx*% indicates the percentage error in satisfying the −5/3 law. Typically, an "*A error*" of less than 10% is considered satisfactory. Throughout, the dimensionless autospectrum *f <sup>S</sup>ii*(*f*)/*σ*<sup>2</sup> *i* is presented against frequency *f* (Hz). Furthermore, in each figure, the corresponding von Karman model (e.g., *β*<sup>1</sup>*v* = 1 for the lateral) is also included; this helps assess how far the developed model is an improvement over the von Karman, a widely used model for the free-stream case [13]. For additional results, see Schau [14].

For the vertical component in Figure 2, the corresponding first-order-correction (a two-term series) models represent appreciable improvement over the von Karman, particularly for *f* > 10−<sup>1</sup> Hz. Overall, modeling still merits further improvements for *f* > 10−<sup>1</sup> Hz. For the vertical component (Figure 2b), the enforcement of the Kolmogorov −5/3 law in a least squares sense involves "*A error* = 23.56%", well above the stipulated "*A error*" of 10%. These two features, the feasibility of improving the correlation and reducing the "*A error*", is explored in the next figure based on the second-order-correction (a three-term series). The results of Figure 3 are extremely instructive in two respects. First, a comparison of the respective figures (Figure 2a compared to Figure 3a, and Figure 2b compared to Figure 3b) shows that the three-term series model improves the correlation throughout, particularly for *f* > 10−<sup>1</sup> Hz. Second, the "*A error*", which is 23.56% for the two-term series model comes down to 7.64%. Thus, this comparison shows that the three-term series model is a noteworthy improvement over the two-term series model, without or with the enforcement of the −5/3 law. To sum up: modeling based on first-order correction (a two-term series) is generally adequate, and further improvement in correlation and further reduction in "*A error* = *x*" can be achieved through modeling based on second-order correction (a three-term series).

Figure 4 shows how the autospectrum from the white-noise-driven filter for the developed vertical model compares with the one from the database and the developed model itself (specifically, Figure 4 refers to Figure 3a). As seen from this Figure 4, the developed model and simulation are almost indistinguishable. (The filter represents a single-input, single-output system driven by white noise; the design is routine and thus the details are omitted [14]).

**Figure 2.** Two-term series modeling for the vertical component (**a**) without the 'A' constraint, and (**b**) with the 'A' constraint.

**Figure 3.** Three-term series modeling for the vertical component (**a**) without the 'A' constraint, and (**b**) with the 'A' constraint.

**Figure 4.** Autospectra from white-noise-driven filter simulation, measurements and a three-term series.

#### **5. Methodology of Coherence**

As done for autospectra, for coherences also, a mathematical framework is developed for extracting interpretive coherence models from a database of flow velocity points from experimental and CFD investigations. Here as well, each velocity component is considered statistically independent of the other two. For each velocity component, the framework begins with a perturbation series expansion of the coherence; therein, the basis function or the first term of the series is represented by the corresponding coherence for HIT. The perturbation coefficients are evaluated by satisfying the theoretical constraints and fitting a curve on a set of numerically generated coherence points from a database.

In the literature, the development of the cross-spectra and coherences for the longitudinal, vertical and lateral components is scattered and piecemeal; what is more, the expressions for these cross-spectra and coherences show difference among these studies. Accordingly, this section first presents the cross-spectrum, after all, coherence is cross-spectrum that is normalized by the corresponding autospectrum (details to follow). Then it presents the coherences and finally a perturbation theory scheme for the wind-farm wake-turbulence coherence.

#### *5.1. Construction of the Vertical Cross Spectrum*

For illustration, vertical turbulence *w*(*t*) is considered under headwind conditions. Given *V*, the mean wind velocity, *τ*, the elapsed time (*<sup>t</sup>*2 − *t*1) and the correlation distance *x* = *Vτ*, the von Karman correlation function *Rww*(*x*) for vertical turbulence *w*(*t*) is given by Equation (11) [15]:

$$R\_{\rm HW}(\mathbf{x}) = \sigma\_w^2 \frac{2^{2/3}}{\Gamma(1/3)} \left[ (u)^{1/3} K\_{1/3}(u) - \frac{1}{2} (u)^{4/3} K\_{2/3}(u) \right] \tag{11}$$

where *u* = *<sup>x</sup>*/1.339*L*, *L* is the scale length and *Kn* is the modified Bessel function of the second kind. Now consider the cross-correlation between vertical turbulence *<sup>w</sup>*1(*t*) at Point 1 and *<sup>w</sup>*2(*t*) at Point 2, where these two points are separated by the across-wind distance *S*, as typified by Figure 5. For this scenario, Figure 5 shows that the correlation distance changes to the expression given in Equation (12a):

$$
\mu = \frac{\sigma}{1.339L} \sqrt{1 + \left(\frac{V\tau}{S}\right)^2} \tag{12a}
$$

where *σ* = *S*/*L*. Now, the cross-correlation *Rw*1*w*2 (*x*) can be expressed as in Equation (12b) [10]:

$$R\_{\overline{w}\_1\overline{w}\_2}(\mathbf{x}) = \sigma\_{\overline{w}}^2 \frac{2^{2/3}}{\Gamma(1/3)} \left[ (u)^{1/3} K\_{1/3}(u) - \frac{1}{2} (u)^{4/3} K\_{2/3}(u) \right] \tag{12b}$$

**Figure 5.** Correlation Distance *P*(*t*1)·*Q*(*t*2) for negligible Stream-wise Separation and across-wind separation 'S'.

The Fourier transform of *Rw*1*w*2 (*τ*) is the cross-spectrum *Sw*1*w*2 (*ν*) [10]:

$$S\_{w\_1 w\_2}(\nu) = \sigma\_w^2 \frac{2^{\frac{2}{3}}}{\Gamma\left(\frac{1}{3}\right)} \cdot \frac{1}{\sqrt{2\pi}} \left(\frac{1}{1.339}\right)^{-\frac{6}{3}} \left[\frac{8}{3} 1.339^2 \left(\frac{\sigma^{\frac{5}{3}}}{z^{\frac{5}{6}}}\right) K\_{\frac{5}{6}}(z) - \left(\frac{\sigma^{\frac{11}{3}}}{z^{\frac{11}{6}}}\right) K\_{\frac{11}{6}}(z)\right] \tag{12c}$$

where,

$$z = \frac{\sigma}{1.339} \sqrt{1 + \left(1.339\nu\right)^2} \tag{12d}$$

In Equation (12d), *ν* represents the dimensionless frequency *ν* = *ωL*/*V* and *σ* = *S*/*L*, the dimensionless distance.

#### *5.2. Coherence for HIT*

Coherence is also referred to as spectral correlation coefficient in that it quantifies the normalized cross-correlation between the turbulence velocities at two points as a function of frequency. For illustration, consider the vertical turbulence velocities at two points which are separated by a distance S, as typified by Figure 5. By definition, coherence is given by:

$$\mathbb{C}\_{w\_1 w\_2}(\nu, \nu) = \frac{|S\_{w\_1 w\_2}(\nu)|}{\sqrt{S\_{w\_1 w\_1}(\nu) S\_{w\_2 w\_2}(\nu)}} \tag{13}$$

where *Sw*1*w*2 (*ν*) is the cross-spectrum between vertical turbulence *<sup>w</sup>*1(*t*) at Point 1 and *<sup>w</sup>*2(*t*) at Point 2, and similarly *Sw*1*w*1 (*ν*) and *Sw*2*w*2 (*ν*) are the corresponding autospectra of *<sup>w</sup>*1(*t*) and *<sup>w</sup>*2(*t*). For HIT, cross-spectrum is real and *Sw*1*w*1 (*ν*) ≈ *Sw*2*w*2 (*ν*). Therefore, coherence from Equation (13) simplifies to Equation (14).

$$\mathcal{C}\_{w\_1 w\_2}(\sigma, \nu) = \frac{|\mathcal{S}\_{w\_1 w\_2}(\nu)|}{\mathcal{S}\_{w\_1 w\_1}(\nu)}\tag{14}$$

where *Sw*1*w*2 (*ν*) is given by Equation (12c) and *Sw*1*w*1 (*ν*) is the von Karman vertical spectrum as given in Equation (15) [15].

$$S\_{w\_1 w\_1}(\nu) = \frac{\sigma\_w^2}{\pi} \left[ \frac{1 + \frac{8}{3} (1.339)^2}{(1 + 1.339^2)^{11/6}} \right] \tag{15}$$

As seen from Equation (14), *Cw*1*w*2 (*<sup>σ</sup>*, *ν*) is a ratio of the cross-spectrum from Equation (12c) and the autospectrum from Equation (15). It is expedient to reiterate that this autospectrum is due to von Karman [15] and that the cross-spectrum is due to Houbolt and Sen [10], as an extended version of the von Karman spectral equations that accounts for the cross-correlation between vertical turbulence velocities at two points; also see Figure 5. After some algebra, Equation (14) simplifies to Equation (16) [8].

$$C\_{w\_1 w\_2}(\sigma, \nu) = \frac{0.597}{23869(z/\sigma)^2 - 1} \left[ 4.781(z/\sigma)^2 z^{5/6} \mathcal{K}\_{5/6}(z) - \frac{1}{2} z^{11/6} \mathcal{K}\_{11/6}(z) \right] \tag{16}$$

As for the longitudinal and lateral velocity components, the cross-spectra are given by Equations (17a) and (18a) and the coherences are given by Equations (17b) and (18b). In the literature (e.g., [9–12]), the expressions for cross-spectra and coherences from one set of study do not completely agree from another set.

$$S\_{\mathfrak{u}\_1\mathfrak{u}\_2}(\nu) = 0.1946 \sigma\_\mu^2 \frac{z^{5/6}}{\left(1 + (1.339\nu)^2\right)^{11/6}} \left[\mathcal{K}\_{5/6}(z) - \frac{z}{2}\mathcal{K}\_{1/6}(z)\right] \tag{17a}$$

$$\mathcal{K}\_{\mathfrak{u}\_{1}\mathfrak{u}\_{2}}(\sigma,\nu) = 0.9944 z^{5/6} \left[ \mathcal{K}\_{5/6}(z) - \frac{z}{2} \mathcal{K}\_{1/6}(z) \right] \tag{17b}$$

$$S\_{\overline{v}\_1 v\_2}(\nu) = 0.0727 \sigma\_v^2 \left(\frac{\sigma^{5/3}}{z^{5/6}}\right) \left[\frac{8}{3} K\_{5/6}(z) - \frac{\sigma^2}{1.339^2 z} K\_{11/6}(z) + \frac{z}{2} K\_{1/6}(z)\right] \tag{18a}$$

$$\mathcal{L}\_{\mathbb{P}1\mathbb{P}2}(\boldsymbol{\sigma},\boldsymbol{\nu}) = \frac{0.597}{2.8687(z/\sigma)^2 - 1} \left[ 4.781(z/\sigma)^2 z^{5/6} \mathcal{K}\_{5/6}(z) - \frac{1}{2} z^{11/6} \mathcal{K}\_{11/6}(z) \right] \tag{18b}$$

Given this background, it is emphasized that in the present study, the expressions of cross-spectra as typified by Equations (12c), (17a) and (18a) for the vertical, longitudinal and lateral components agree with those of Frost et al. [11]. As for coherence, the corresponding expressions given by Equations (16), (17b) and (18b) agree with those of Irwin [12]. Figures 6–8, respectively, show vertical, longitudinal and lateral coherence between Points 1 and 2 as a function of dimensionless frequency *ν* = *ω*L/*V* for *σ* = *S*/*L* = 0, 0.1, 0.2, ... 1. For *σ* = 0, Point 2 merges into Point 1 and in turn the cross-spectra become the respective autospectra and thus the coherence represents the perfect coherence. For example, as seen from Figure 6 for the vertical coherence, *Cw*1*w*2 (*<sup>σ</sup>*, *ν*) → 1. Similarly, as seen from Figures 7 and 8, *Cu*1*u*2 (*<sup>σ</sup>*, *ν*) → 1 and *Cv*1*v*2 (*<sup>σ</sup>*, *ν*) → 1. Exactly the opposite happens with increasing *σ* = S/L. That is, with increasing *σ*, the correlation between these two points decreases and so does the corresponding coherence. For example, as seen from Figures 6–8, *Cw*1*w*2 (*<sup>σ</sup>*, *ν*) → 0, *Cu*1*u*2 (*<sup>σ</sup>*, *ν*) → 0 and *Cv*1*v*2 (*<sup>σ</sup>*, *ν*) → 0 as *σ* → ∞. Moreover, as seen from these figures, the coherence decreases rapidly for *ν* > 1 or so.

The longitudinal cross-spectrum *Su*1*u*2 (*ν*) and coherence *Cu*1*u*2 (*<sup>σ</sup>*, *ν*) are typified by Equations (17a) and (17b), respectively, and Figure 7 shows coherence *Cu*1*u*2 (*<sup>σ</sup>*, *ν*) as a function of dimensionless frequency *ν* = *ω*L/*V*; all of this merits revisiting. The reason is that *Su*1*u*2 (*ν*) and in turn the corresponding coherence can become negative at high frequencies. As seen from Equations (17a) and (17b), respectively, *Su*1*u*2 (*ν*) and *Cu*1*u*2 (*<sup>σ</sup>*, *ν*) can take on negative values for *<sup>K</sup>*5/6(*z*) ≤ *<sup>z</sup>*/2*K*1/6(*z*). See Figure 9, which is a recasting of Figure 7 for a much expanded vertical scaling (1 to 10−<sup>6</sup> in Figure 9 in comparison to 1 to 10−<sup>2</sup> in Figure 7). The crosses (\*) in Figure 9 indicate the termination of the curve to avoid generating negative coherence values. Given the state of the art and one's initiation into cross-spectra and coherence, it is difficult to come up with a basis for these negative values of cross-spectrum and coherence for HIT; a resolution of this difficulty would require further research [11].

**Figure 6.** Vertical Coherence *Cw*1*w*2(*<sup>σ</sup>*, *<sup>ν</sup>*).

**Figure 8.** Lateral Coherence *Cv*1*v*2 (*<sup>σ</sup>*, *<sup>ν</sup>*).

**Figure 9.** Longitudinal Coherence with an expanded scale.

#### *5.3. Coherence Modeling for Wind-Farm Wake Turbulence*

Wind farm wake turbulence deviates from HIT. Accordingly, the framework for coherence modeling from a database accounts for this deviation based on perturbation theory. Here as well, the framework assumes the same topology that was assumed in the development of the basis functions; for illustration vertical coherence *Cw*1*w*2(*<sup>σ</sup>*, *ν*) is selected.

Let *Cw*1*w*2 (*<sup>σ</sup>*, *ν*) represent the vertical coherence of wake turbulence. The framework begins with a perturbation series expansion of *C w*1*w*2 (*<sup>σ</sup>*, *ν*) (essentially the same procedure applies to the other two components):

$$\check{\mathbb{C}}\_{w\_1 w\_2}(\sigma, \,\nu) = \mathbb{C}\_{1w} \mathbb{C}\_{w\_1 w\_2}(\sigma, \,\nu) + \mathbb{C}\_{2w} \mathbb{C}\_{w\_1 w\_2}^2(\sigma, \,\nu) + \cdots + \mathbb{C}\_{nw} \mathbb{C}\_{w\_1 w\_2}^n(\sigma, \,\nu) \tag{19}$$

The basis function or the first term of the series is given by Equation (16). Since *Cw*1*w*2 (*<sup>σ</sup>*, *ν*) = 1 for *σ* = 0, Equation (17) is subject to the constraint:

$$\mathcal{C}\_{1w} + \mathcal{C}\_{2w} + \dots + \mathcal{C}\_{nm} = 1\tag{20}$$

The second condition that *C w*1*w*2 (*<sup>σ</sup>*, *ν*) = 0 for *σ* = ∞ is automatically satisfied since *Cw*1*w*2 (*<sup>σ</sup>*, *ν*) = 0 for *σ* = ∞. The coefficients in the series *Ciw* are evaluated by satisfying the theoretical constraint of Equation (20) and fitting a curve on a set of selected numerically generated coherence points in a least squares sense.

For illustrations, longitudinal coherence of wake turbulence *C u*1*u*2 (*<sup>σ</sup>*, *ν*) is selected with a twoterm perturbation series (also see Equation (19)):

$$\dot{\mathbb{C}}\_{u\_1 u\_2}(\sigma, \nu) = \mathbb{C}\_{1u} \mathbb{C}\_{u\_1 u\_2}(\sigma, \nu) + \mathbb{C}\_{2u} \mathbb{C}\_{u\_1 u\_2}^2(\sigma, \nu) \tag{21}$$

Specifically, consider *C*1*u* = 0.7 and *C*2*u* = 0.3 (also see constraint Equation (20)). Descriptively stated, this case represents wake turbulence, which deviates weakly from HIT. It is plausible that this case belongs to wake turbulence at locations that are downwind of the first two rows or so. Therein, wake turbulence is expected to deviate only weakly from HIT as depicted in Figure 10.

**Figure 10.** Longitudinal Coherence for HIT and for wake turbulence weakly deviating from HIT.
