Several models are available for generating high-frequency wind fluctuations within a three-dimensional (3D) space. These models can generate realistic wind fields that can be used for load estimation on structures such as bridges, wind turbines, and buildings. For wind turbine design and load calculations, the International Electrotechnical Commission (IEC) standards recommend two commonly used models: the Mann uniform shear model and Kaimal spectral and coherence model . A notable advantage of these two models is simulating realistic small-scale turbulence without exorbitant computational time and resources. In contrast, large eddy simulation (LES) or other numerical solutions of the Navier–Stokes equations have proven to be computationally expensive and unfeasible for the wind turbine design process.

While high-frequency fluctuations have more influence on the stresses and fatigue loads experienced by the blades and tower of a wind turbine, low-frequency fluctuations can significantly affect the overall energy production and capacity factor of a wind farm. In the context of floating offshore wind turbines, low-frequency wind fluctuations may be of significant importance in terms of dynamic response and loading since these structures can have very low natural frequencies . Low-frequency fluctuations are also crucial for meandering wakes behind wind farms, affecting power fluctuations and dynamic loads. The dynamic wake meandering model of uses the low-frequency turbulence to move the wake deficit, but it uses a normal turbulence spectrum that does not take into account the excess power spectral energy at low frequencies often seen offshore . Thus, we need a fast method for simulating realistic low-frequency wind fluctuations that can be easily integrated with high-frequency wind fields to get a comprehensive spectral range representation.

Here, we present a method for simulating low-frequency wind fluctuations based on the two-dimensional (2D) spectral tensor introduced in . At low frequencies, only the longitudinal (u) and lateral (v) wind components have strong fluctuations since, at least close to the ground, the presence of the land or sea blocks the vertical large-scale movements. Thus, the vertical wind (w) fluctuations at low frequencies attenuate or weaken considerably, rendering the turbulence 2D. The 2D turbulence model only describes the u and v fluctuations in the low-frequency range and assumes that these fluctuations do not vary in the vertical direction. The algorithm to generate stochastic wind fields from the 2D turbulence model is similar to the one described in . The 2D wind field is represented as a discrete Fourier series, which takes the mean squared amplitude of the Fourier coefficients from the 2D spectral model. These coefficients are then multiplied by a random Gaussian field. Subsequently, the resulting product's inverse discrete Fourier transform yields the stochastic wind field.

Section 2 of this paper describes the low-frequency, 2D turbulence model, along with model validation details. Section 3 outlines the process for simulating 2D wind fields containing 2D turbulence. Section 4 describes combining 2D and 3D wind fluctuations to create turbulence boxes that represent a wide spectral range. Finally, a discussion regarding the effect of anisotropy on the 2D turbulence and some basic guidelines to generate 2D wind fields for the wind turbine design load process is presented in Sect. 5.

The 2D, incompressible, and isotropic turbulence has the spectral tensor form of 1ϕij(k1,k2)=E(k)πkδij-kikjk2, where E(k) is the energy spectrum, k is the magnitude of the horizontal wave vector k=|k|=k12+k22, and δij is the Kronecker delta. The assumption of incompressibility is an approximation. observe some divergence in a horizontal plane at wind-turbine-relevant heights. We assume that the energy spectrum is given by 2E(k)=ck3L2D-2+k27/3, where c is a constant and a scaling parameter, and L2D is the corresponding length scale of low-frequency fluctuations. This particular shape of Eq. () is inspired by the spectra. The variance of any horizontal velocity component can be found by 3σ2=σ112=σ222=∫0∞E(k)=98cL2D2/3. Due to isotropy, the variance is the same for both wind components. Now, let us introduce scale-independent anisotropy in the energy spectrum. We replace the horizontal wave vector k=|k|=k12+k22 with κ, where 4κ2=2(k12cos⁡2ψ+k22sin⁡2ψ), and 0<ψ<π/2 is the anisotropy parameter. Now, the energy spectrum with anisotropy parameter takes the form of 5E(κ)=cκ3L2D-2+κ27/3. When ψ=π/4, k=κ and Eq. () takes the form in Eq. (). By inserting E(κ) into Eq. () we can obtain two-point cross-spectra χij2D and one-point spectra Fij2D using 6χij2D(k1,Δy)=∫-∞∞ϕij(k1,k2)exp⁡(-ik2Δy)dk2, where Fij2D(k1)=χij2D(k1,0) is the one-point cross- or auto-spectrum depending on whether the component indices i and j are different or equal. The anisotropy parameter ψ determines the spectral distortion in the wavenumber domain and the spectrum magnitudes of longitudinal and transverse wind components. When the 2D turbulence is isotropic (ψ=π/4), F112D=35F222D in the k1-5/3 range. For the anisotropic cases, the ratio can be found using 7F112DF222D=35cot⁡2ψ. The anisotropy parameter can be obtained from measured spectra at frequencies below ∼10-3 Hz. As observed from the analysis of real offshore measurements in , the subrange below f<10-3 Hz follows a S(f)∝f-5/3 relation, where S(f) is the velocity spectrum in terms of frequency. Thus, ψ can be evaluated as 8ψ=arctan⁡35Sv(f)Su(f), for f<10-3 Hz, corresponding to fluctuations with a period longer than approximately 16 min. The energy spectrum must be attenuated at the wavenumbers corresponding to small-scale 3D turbulence. This is necessary because we assume low-frequency fluctuations are independent of high-frequency fluctuations, and at very high wavenumbers, only small-scale 3D turbulence is present. This high wavenumbers range is referred to as the inertial subrange. The turbulence is isotropic in this range and follows a power law . For practical reasons, we attenuate the low-frequency turbulence at wavenumbers higher than 1/zi, where zi is the boundary-layer height. This implies that any eddy with a length scale smaller than the boundary-layer height would be considered 3D turbulence. The attenuated E(κ) can be defined as 9E(κ)=cκ3L2D-2+κ27/311+κ2zi2. Here, the attenuation factor 1/(1+κ2zi2) is an activation function that ensures the energy spectrum smoothly drops to zero for wavenumbers greater than 1/zi. This drop is accelerated due to an increased negative slope of the spectrum for κ>1/zi, i.e., E(κ)∝κ-11/3. Sigmoid functions such as a hyperbolic tangent or a logistic function can also be used as an attenuation factor. From Eq. () we can obtain Fij2D as follows: 10F112D(k1)=cΓ116L2D113{-p2F156,1;12;p-72F156,1;12;p+22F1-16,1;12;p}102πΓ73L2D2-zi2sin⁡3(ψ)a56+L2D143b22d73zi3sin⁡3(ψ), and 11F222D(k1)=ck12(-zi4a16L2D113Γ176552πL2D2-zi22bΓ73sin⁡(ψ)-9-262F1-16,1;12;p+p2{15-302F1-16,1;12;p-592F156,1;12;p}+352F156,1;12;p+15p23F156,1;12;p+p{-54+882F1-16,1;12;p+92F156,1;12;p})-L2D1432bd73zisin⁡(ψ), where a=1+2k12L2D2cos⁡2(ψ),b=1+2k12zi2cos⁡2(ψ),p=L2D2bzi2a, Γ is the Gamma function, and 2F1 is the hypergeometric function. The two-point cross-spectra χ112D(k1,Δy) and χ222D(k1,Δy) for the attenuated energy spectrum in Eq. () to our knowledge do not have any analytical solution but can be obtained through numerical integration techniques. An example of F11 with and without attenuation at high wavenumbers is shown in Fig. .

It is important to note that although this model utilizes the wavenumber information to generate a spatial field containing large-scale fluctuations, Taylor's frozen turbulence hypothesis can be used to sweep the spatial field into a frequency domain. More intricate models, such as those presented by and , characterize spatiotemporal turbulence structures as a function of both wavenumber and frequency. However, for the sake of simplicity, the model presented here disregards the temporal variation or distortion of eddies.

The 2D turbulence model combined with the Mann uniform shear model for 3D turbulence was validated against measurements from two offshore sites: 10 Hz ultrasonic measurements from the FINO1 research platform in the North Sea and line-of-sight (LOS) wind measurements from a forward-looking nacelle lidar in the Hywind Scotland offshore wind farm. The corresponding model parameters that fit the measurements at these two sites can be found in . A good agreement was recorded between observed and predicted auto-spectra, cross-spectra, and co-coherences. The measured data were classified into different atmospheric stability classes, and it was found that for a 1 h time series, low-frequency fluctuations existed in all stability classes. However, the relative strength of 2D turbulence, compared to 3D turbulence, was more dominant during stable stratification. For the 1 h time series, the mesoscale turbulence peak corresponding to L2D was also not observed. At both sites, the low-frequency turbulence was in the F(k)∝k1-5/3 range. For the FINO1 site, the measured value of ψ was close to 45° in the low-frequency range, representing isotropic 2D turbulence. At Hywind Scotland, we observed ψ<40° reflecting the anisotropy in the 2D turbulence.

In summary, the low-frequency turbulence model has four input parameters:

σ2D2 the variance exhibited by low-frequency fluctuations (excluding the attenuation);

Here, we follow the procedure of to simulate low-frequency, anisotropic wind fields. The 2D turbulence is assumed to be statistically homogeneous in horizontal directions and constant in the vertical direction. Taylor's frozen turbulence hypothesis is also employed to convert the wavenumber domain into the frequency domain. The wind field will be simulated on a horizontal grid with N1 and N2 grid points in the longitudinal and transverse directions, respectively. The length of the grid in two directions would be L1=N1⋅dx and L2=N2⋅dy. Following , the incompressible, homogeneous, 2D velocity field can be written as a sum of discrete Fourier modes: 12ui(x)=∑kexp⁡(ιk⋅x)Cij(k)nj(k), where ∑k denotes the sum over all wave vectors k, where ki=m2π/Li for m=-Ni/2,…,Ni/2. Cij(k) are the Fourier coefficients, and nj(k) are independent Gaussian stochastic variables. Here, the summation over repeated indices is assumed. The solution to Eq. () is approximately 13Cij(k)nj(k)=1L1L2∫Aui(x)exp⁡(-ιk⋅x)dx, where ∫Adx is integration over the area L1×L2. The process of obtaining Cij involves multiplying Eq. () with its complex conjugate, which gives 14Cij*(k)Cij(k)=∫ϕij(k′)∏(l=1)2sinc2(kl-kl′)Ll2dk′, where sincx≡(sin⁡x)/x. In the case if Ll≫L2D, where l=1,2, Eq. () can be simplified to 15Cij*(k)Cij(k)=(2π)2L1L2ϕij(k). The length scale L2D corresponding to the mesoscale turbulence peak is quite large, usually on the order of 105 to 106 m. Simulating a high-resolution wind field containing the wavenumbers corresponding to L2D would be costly in terms of computation time. Usually, L2≪L2D when simulating wind fields for single wind turbine load calculations. So, the simplified relation in Eq. () no longer holds true. We have observed that if we simplify the sinc2 function for L1 and replace it with 2π/L1 but integrate the sinc2 function for L2, we would get simulated spectra much closer to the target spectra. 16Cij*(k)Cij(k)=2πL1∫ϕij(k1,k2′)sinc2(k2-k2′)L22dk2′ To speed up the numerical integration, the limits of integration are k2-2π/L2 to k2+2π/L2. A correction factor is multiplied, compensating for the loss in variance due to the limited integration interval. This problem with discretization has been discussed in detail by . The Fourier coefficients obtained from Eq. () or Eq. () after taking a matrix square root are then multiplied by a random Gaussian field. The resulting product's inverse discrete Fourier transform would yield the wind field.

Figure  illustrates the effect of discretization on the simulated spectra. In Fig. b when L2≪L2D, Cij(k) obtained via Eq. () underestimate F11(k1) and overestimate F22(k1) at very low k1 values. In such cases, Cij(k) must be evaluated using Eq. ().

Figure  shows the simulated u and v low-frequency fluctuations, where the input parameters are L2D=15 km, σ2=0.6 m2 s-2, zi=500 m, and ψ=43°. These parameters, with the exception of L2D, are representative of typical neutral conditions for 8<U‾<10 ms-1 at the FINO1 offshore site. Here, large-scale coherent structures can be identified along the longitudinal axis for the u component. We can also observe the almost equal variance in the u and v fluctuations due to ψ being close to the isotropic value of 45°. The one-dimensional (1D) spectra of this simulated 2D wind field are illustrated in Fig. a. The spectra derived from the simulated wind field are in excellent agreement with the theoretical spectra mentioned in Eqs. () and (). Normalized cross-spectra (co-coherence, the real part of the cross-spectrum divided by the auto-spectrum) for the simulated 2D wind field components are also compared with the theoretical expression in Eq. (). In Fig. , co-coherence of u and v is plotted as a function of k1 for separations ranging from 750 to 7500 m. Once the lateral separation distance Δy approaches L2D (in this case 15 km), the normalized cross-spectra decrease significantly.

presented the uniform shear model for small-scale turbulence in the neutral atmosphere. We combine the two models, assuming that the large-scale and small-scale fluctuations are independent. Combining the mesoscale and microscale turbulence in the frequency or wavenumber domain requires an assumption of weak or no correlation between the two scales. provided a qualitative framework for combining the spectra at large and small scales in the atmosphere by a simple superposition method. On a similar pattern, noted that the 1D spectrum of streamwise velocity in a turbulent pipe flow has a bimodal representation of high- and low-wavenumber modes. These two modes are associated with small- and large-scale turbulent motions, respectively. Superposing these two modes gives a good general representation of the measured spectra over the whole wavenumber domain since the two modes are uncorrelated.

Figure  displays a 2D + 3D turbulence wind field of u and v wind components for relatively smaller dimensions. The 3D wind field is generated by the Mann uniform shear turbulence model, which has three input parameters: the dissipation parameter αϵ2/3=0.01 m4/3 s-2, the turbulence length scale L3D=50 m, and the anisotropy parameter Γ=2.5. The values of the parameters selected here are typical of offshore wind conditions at the FINO1 site for neutral conditions . The 3D turbulent wind field is also generated by the procedure presented in . Since the wind fields are assumed statistically independent, they can be added to get the combined fluctuations. In this case, the 2D wind field components are directly added to all the vertical planes of the corresponding 3D turbulence box. We can observe the increased variance in the combined 2D and 3D wind fluctuations. The large-scale coherent structures are still dominant, but we now also observe smaller structures. A smaller vertical slice of the same wind field is illustrated in Fig. . Here, one can observe the large shear in the v component compared to the u component. This implies that the phase difference between v fluctuations at different heights is higher (φv>φu), as observed in atmospheric turbulence measurements at multiple sites . The 1D two-sided spectra of the 3D turbulence wind field by itself and combined with the low-frequency fluctuations are shown in Fig. b and c, respectively. The resulting spectra add individual 2D and 3D wind field spectra over the wavenumber domain.

The u and v co-coherences of simulated combined 2D + 3D wind field at different lateral and vertical separations are illustrated in Fig. . The co-coherences are plotted for lateral and vertical separations ranging from 150 to 450 m. At lower k values, the low-frequency fluctuations are fully coherent for all vertical Δz separations, and we obtain co-coherence values close to 1. This is because the low-frequency fluctuations are assumed to be constant in the vertical direction at any instant. The same can not be said about the lateral Δy separations, as we have observed a decrease in the u co-coherence of low-frequency fluctuations for increasing lateral separations in Fig. a.

A method to generate the low-frequency wind fluctuations is introduced. This method utilizes the spectral tensor presented by to generate 2D stochastic wind fields for the longitudinal u and lateral v wind components. The generated wind fields contain large-scale and low-frequency wind fluctuations called 2D turbulence. The model employs four input parameters: (i) the variance characterizing low-frequency fluctuations σ2D2, (ii) a length scale corresponding to large-scale flow structures L2D, (iii) an anisotropy parameter ψ, and (iv) a cutoff or attenuation length zi. The simulation method uses the wind field presented as a discrete Fourier series, where Fourier coefficients are derived from the 2D spectral model. The coefficients are then multiplied by a random Gaussian field. Subsequently, the product's inverse discrete Fourier transform yields a 2D wind field featuring low-frequency, anisotropic wind fluctuations. Issues arising from the discretization, such as underestimation of the spectral density at very low wavenumbers and periodicity, are also addressed in this study. Some guidelines to simulate the wind fields containing 2D turbulence are also provided in the context of wind energy applications.

The 2D turbulence wind field can be added to a 3D turbulence field to get the spectral representation over a wide frequency range. We combined the 2D turbulence wind field with a 3D turbulence field generated using the Mann uniform shear turbulence model. The spectra and co-coherences from the combined simulated 2D + 3D turbulence wind are compared with the theoretical expressions, and an excellent agreement was observed. The 2D turbulence simulation program is open-source and can be accessed via the link in the “Code availability” section.