A new theoretical framework, based on an analysis in the moving and fixed frames of reference (MFOR and FFOR), is proposed to break down the velocity and turbulence fields in the wake of a wind turbine. This approach adds theoretical support to models based on the dynamic wake meandering (DWM) and opens the way for a fully analytical and physically based model of the wake that takes meandering and atmospheric stability into account, which is developed in the companion paper. The mean velocity and turbulence in the FFOR are broken down into different terms, which are functions of the velocity and turbulence in the MFOR. These terms can be regrouped as pure terms and cross terms. In the DWM, the former group is modelled, and the latter is implicitly neglected. The shape and relative importance of the different terms are estimated with the large-eddy-simulation solver Meso-NH coupled with an actuator line method. A single wind turbine wake is simulated on flat terrain, under three cases of stability: neutral, unstable and stable. In the velocity breakdown, the cross term is found to be relatively low. It is not the case for the turbulence breakdown equation where even though the cross terms are overall of lesser magnitude than the pure terms, they redistribute the turbulence and induce a non-negligible asymmetry. These findings underline the limitations of models that assume a steady velocity in the MFOR, such as the DWM or the model developed in the companion paper. It is also found that as atmospheric stability increases, the pure turbulence contribution becomes relatively larger and pure meandering relatively smaller.

The wake behind a wind turbine is characterized by a decrease of wind velocity and an increased level of turbulence compared to the inflow properties, leading respectively to a decreased generated power and increased loads for downstream turbines. The stability of the atmospheric boundary layer (ABL) influences the wake recovery

Schematic of the wake meandering phenomenon. The mean (red) and instantaneous (blue) wake outlines are plotted at two different time steps:

The evolution of the time-averaged wake may thus be considered the combination of two phenomena: on one hand the wake expansion and dissipation due to the turbulent diffusion and on the other hand the wake meandering due to large-scale forcing of the ABL. Most analytical models are calibrated directly in the frame of reference linked to the ground (called hereafter fixed frame of reference or FFOR) against reference data averaged over the meandering time period, and the wake widths are written as a function of the turbulence intensity (TI) upstream the turbine

This can be achieved with the use of the moving frame of reference (MFOR), which is moving with the wake centre at each time step. The unsteady velocity field in the MFOR is thus equivalent to the velocity field that would be observed if there was no meandering. Due to the spreading caused by the meandering, the mean velocity deficit in the FFOR is weaker and wider compared to the mean velocity deficit in the MFOR (dashed red and blue profiles in Fig.

In the DWM, the total turbulence (defined as the temporal variance of the velocity field) in the FFOR in the wake can be computed as the sum of two components:

The objective of this work is to write the velocity and turbulence in the FFOR as a function of the velocity and turbulence in the MFOR and show the underlying DWM assumptions that neglect some terms. The importance of these missing terms for both velocity and turbulence is evaluated. The reference results come from large-eddy simulations (LESs) of an isolated wind turbine wake over flat terrain. Three cases of stability, approximately corresponding to the SWiFT benchmark

This work is separated into two articles. In the present one, the breakdown of the velocity and turbulence is presented and applied to the LESs datasets. In the companion paper, the results are used to build a new analytical model for velocity and turbulence in the wake of a wind turbine. The first part of the present article is dedicated to the development of the velocity and turbulence breakdowns, i.e. the expression of the velocity and turbulence fields in the FFOR as a function of their counterparts in the MFOR. In the second part, the numerical framework is detailed: it describes the SWiFT cases, the LES code Meso-NH, the numerical set-up, the wake tracking algorithm and the limitations of these tools. In the third part, the LES datasets are used to quantify the error induced by the approximations necessary to write Eqs.

To lighten the mathematical formulations, the notation

For the turbulence equation, one can write from Eqs. (

Through Eq. (

Similarly to the velocity field, Eq. (

It has thus been shown in this section that the mean velocity turbulence fields in the FFOR can be broken down into two types of terms: pure terms – (I), (III) and (IV) – and cross terms – (II), (V), (VI) and (VII). In models where the wake is considered steady in the MFOR and advected as a passive tracer (such as the DWM or the model developed in the companion paper), the pure terms are modelled but the cross terms are implicitly neglected. The error induced by this assumption is verified in this work with LESs.

The breakdowns of the mean velocity and turbulence fields in the FFOR described in Sect.

MESOscale Non-Hydrostatic (Meso-NH) is a finite volume and finite difference open-source research code for ABL simulations developed by the Centre National de Recherches Météorologiques and the Laboratoire d'Aérologie. The model is described in detail in

The turbulence closure is of the order 1.5: an additional equation is introduced for the subgrid kinetic energy

To model the wind turbine, the ALM is used, following

The numerical parameters used for the three simulations are presented in Table

Numerical parameters used in Meso-NH.

The flow field is initialized with a constant-velocity profile equal to the geostrophic wind. A constant-temperature profile is set up to an arbitrarily defined ABL height, capped by an inversion region of 5 K over a depth of 50 m. The geostrophic wind, ABL height, surface roughness

In the first domain

Schematic of the simulation set-up with Meso-NH.

The size of the domain of interest (

The ALM is activated once the flow is established in the most refined domain, and after a 10 min spin-up to let the wake flow establish, the instantaneous velocity is extracted at one plane upwind of the turbine and several planes downwind, according to Fig.

The rotational velocity of the wind turbine

The wake meandering is characterized by the time series of the wake centre coordinates

To determine the wake centre at each time step, an algorithm based on the conservation of momentum in the wake is used

Result of the wake tracking at an arbitrary time step at

To compute terms (I) to (VII) of Eqs. (

Given that the ground is located around

The wake tracking and the computation of each term of Eqs. (

The error induced by the choice of the 1 Hz sampling has been estimated with a 95 % confidence interval and discussed in Appendix

Due to numerical limitations, the duration of the simulations was constrained to 80, 40 and 10 min (see Table

Finally, the streamwise component of the velocity is computed in the following, in both MFOR and FFOR. In all the following, the mean streamwise component of the velocity will be noted

Once the Meso-NH simulations are performed,

The objective of this section is to quantify the importance of each term and to estimate the error induced by neglecting the cross terms in the velocity and turbulence breakdowns, for instance in the DWM model or in the model developed in the companion paper. The focus is on the neutral case to keep a concise section. The normalized root mean square error (RMSE) indicator (Eq.

In Eq. (

Contribution of terms (I) and (II) from Eq. (

From this first observation, it seems acceptable to neglect term (II). The effect of this assumption can also be measured with a global variable. It has been chosen to investigate the error induced by neglecting term (II) on the available power, since predicting the power output of a farm is a direct application of analytical models. The available power is here defined as

From Fig.

Available power predicted by (I) (blue) and (I)

The same study is performed for the turbulence field in the wind turbine wake. The vertical turbulence profiles for the neutral case are plotted for different levels of approximation, at different positions downstream in Fig.

Streamwise turbulence profiles in the wake of the wind turbine for different levels of approximation. Neutral case at three positions downstream (1, 5 and 8

Adding the covariance term (V) along with terms (III) and (IV) (purple curve in Fig.

To quantify more clearly these differences, the maximum axial turbulence

Normalized maximum turbulence in the wake for different levels of approximation as a function of the downstream position for the three simulation cases. The RMSE of

In the neutral case, neglecting the cross terms leads to an underestimation of about 6 % to 12 % of the maximum turbulence in the wake. In the far wake (beyond

For the unstable case, the same orders of magnitude are observed for the different

It has been shown in this section that neglecting term (II) as in the DWM model or in the companion paper leads to a rather accurate velocity deficit in the wake and a reasonable estimation of the available power (less than 2 % overestimation) for a wind turbine inside the wake, as long as it is positioned beyond

In this section, the turbulence fields in the wake of the wind turbine are compared for the three cases of stability. The influence of atmospheric stability on each term of Eq. (

2D maps of the different terms in Eq. (

The values of each term of Eq. (

For the neutral case of stability (Fig.

The same as Fig.

Figure

The same as Fig.

In the stable case (Fig.

The reference turbulence in the FFOR

For all cases, the non-zero values of each term in the near wake (first column of every figure) are mostly distributed around the tip of the blades. For pure terms (III) and (IV), they are spatially smoothly distributed at

Term (III) or

At a fixed

Evolution of the maximum value of terms (III) and

Term

2D map of the added turbulence in the MFOR, normalized by the velocity at hub height.

Despite strongly different values of

The value of the cross terms (V), (VI) and (VII) is zero either if there is no meandering (i.e.

Term (VI) can be viewed as the varying part of turbulence: before being moved by the meandering and averaged, this term is the varying part of the square of the deviation from the mean (in opposition to

Term (VII) is always negative from its mathematical formulation: similarly to the viscous dissipation in the Navier–Stokes equations, it is a sink of energy. It has negligible values in all the stability cases. This last result should be taken with care: if the analogy with the viscous dissipation holds for this term, it means that it concerns small-scale eddies, i.e. variations of the wind velocity at high frequency. Yet, as explained in Sect.

It is important to note that all these results are sensitive to the wake tracking method: despite that the method used here being among the most reliable available in the literature, there are always frames where the tracking failed, plus the limitations described in Sect.

In models predicting wake meandering such as the DWM, it is assumed that the turbulence in the wake can be separated into two parts: the turbulence generated by the rotor and the turbulence generated by meandering. In this work, the turbulence in the FFOR has been developed as a function of the two terms aforementioned, and it appears that three cross terms are missing, thus implicitly neglected in DWM-type models. A similar conclusion is drawn for the velocity, with one missing term.

To quantify the importance of each of these terms and estimate the error induced by the assumptions of such models, LESs with an actuator line are performed to model the wake of an isolated wind turbine inside an ABL. The modelled turbine is the modified Vestas V27 used in the SWiFT campaign of measurements, and three cases of atmospheric stability are investigated: near-neutral, unstable and strongly stable. The instantaneous wake centre is detected at different planes downstream of the turbine (from 1 to 8

Neglecting the cross term of the mean velocity equation leads to small differences in the computation of the mean velocity profile in the FFOR. For the neutral case, the corresponding error leads to a less than 2 % overestimation of the available power in the wake of the wind turbine for a turbine located further than 3

Neglecting cross terms in the computation of turbulence in the FFOR leads to vertical profiles where the imbalance between the turbulence at the bottom tip and the top tip is underestimated. Adding the three missing cross term allows us to correct this error and reduce the overall RMSE.

In the unstable case, the meandering term is dominating the total streamwise turbulence whereas in the stable case, it is the turbulence added by the rotor which is dominant. In the neutral case, these two terms are of similar magnitude and overall larger than the cross terms. These cross terms, especially the so-called covariance term, however show local values sufficiently strong to significantly correct the maximum axial turbulence in the wake.

The statistical convergence of the data has been assessed and showed that increasing the sampling frequency would most likely improve the reliability of the stable case but would have little effect on the two other cases. On the other hand, increasing the simulation time would probably change the unstable results but has a low impact on the other cases. The uncertainty is the highest on the cross terms of the turbulence breakdown equation, but the pure terms are subject to only small uncertainty. For a better interpretation of these terms, it may be important to perform ensemble simulations to get more reliable fields.

It must be noted that these conclusions are drawn on the results of three particular cases of atmospheric stability and one model of turbine that can be regarded as rather small compared to modern rotors. The orders of magnitude given in this work should not be considered universal but are a good indication that for an accurate version of DWM-type models, the cross terms (or at least the covariance term) must be taken into account. In the companion paper, an analytical model for the dominant terms is developed on the neutral and unstable cases presented herein.

In this Appendix, an analysis of the statistical convergence of the three simulations is proposed to give a glimpse to the reader of the uncertainty of the presented data. Indeed, the quality of our data is limited in two ways:

First, a sampling frequency of 1 Hz has been set for the three simulations. This value has been chosen accordingly to the SWiFT benchmark

Moreover, a length of 80 min was initially the target for the three simulations. However, the unstable simulation is computationally expensive, and the stable simulation diverges to unrealistic results if it runs for too long due to the strong negative heat flux. Consequently, the segment length was reduced to 40 and 10 min for the unstable and stable cases respectively. Even for the 80 min though, it is not sure a priori that the simulation has run for long enough to have statistically converged results.

In the most refined domain, the Meso-NH code runs at a time step close to the millisecond (see Table

For term (VII), it is assumed that the confidence interval is equal to the square of the confidence interval of term (II). This may be a strong approximation, but this term is anyway found to be negligible, and thus plays only a small role in the total turbulence budget.

Vertical profiles at

Vertical profiles at

Vertical profiles at

Vertical profiles at

The confidence interval is plotted in the shaded area in Figs.

This confidence interval quantifies the uncertainty towards the small-scale turbulence. It is thus not surprising that it gets higher values in the stable case (where turbulence is mostly of small scale) and in particular for the turbulence in the MFOR and for term (VI), which is attributed to the spatial non-homogeneity, and thus to the small-scale variation. A higher sampling frequency for the stable case would seemingly improve the reliability of the simulation, but for the two other cases, the sampling at 1 Hz seems suitable.

The length of the simulations (80, 40 and 10 min for the neutral, unstable and stable cases) has been arbitrarily chosen depending on numerical constraints. Even though some works in the literature are based on 10 min long simulations, it should be ensured that this choice is sufficient to take into account all the large-scale variations of the ABL.

To do so, the velocity time series are separated into two equal sub-segments (thus of length 40, 20 and 5 min for the neutral, unstable and stable cases). The whole post-process is reproduced on these sub-segments, and the resulting profiles are plotted in dashed black lines in Figs.

This concept of convergence should however be taken with care here, because the ABL is in permanent evolution due to the constant heat flux at the ground that changes progressively its characteristics. This is particularly true for the unstable case where the mean wind direction is not equal among the two sub-segments and where the atmospheric conditions will likely never reach a quasi-steady state.

Conversely to the sampling frequency, this method measures the uncertainty of the large-scale turbulence. It is thus not surprising to see that the uncertainty is higher in the unstable case (Fig.

As a conclusion, a higher sampling frequency could give more reliable results in the stable case, and a longer simulation time may have been needed for the unstable case. This is particularly true for term (VI) which has the largest level of uncertainty. Nevertheless, the uncertainty on the other terms and the streamwise turbulence in the MFOR seem sufficiently low to maintain the conclusions drawn at the core of this article.

The code Meso-NH is open-source and can be downloaded on the dedicated website (

EJ developed the equations and performed the simulations with VM. All the authors worked on the interpretation of the results. The manuscript has been written by EJ with the feedback of FB and VM. The data used for the plot presented here and in Part 2 are available under this online deposit:

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper was edited by Raúl Bayoán Cal and reviewed by two anonymous referees.