This work aims to develop an analytical model for the streamwise velocity and turbulence in the wake of a wind turbine where the expansion and the meandering are taken into account independently. The velocity and turbulence breakdown equations presented in the companion paper are simplified and resolved analytically, using shape functions chosen in the moving frame of reference. This methodology allows us to propose a physically based model for the added turbulence and thus to have a better interpretation of the physical phenomena at stake, in particular when it comes to wakes in a non-neutral atmosphere. Five input parameters are used: the widths (in vertical and horizontal directions) of the non-meandering wake, the standard deviation of wake meandering (in both directions) and a modified mixing length. Two calibrations for these parameters are proposed: one if the users have access to velocity time series and the other if they do not. The results are tested on a neutral and an unstable large-eddy simulation (LES) that were both computed with Meso-NH. The model shows good results for the streamwise velocity in both directions and can accurately predict modifications due to atmospheric instability. For the axial turbulence, the model misses the maximum turbulence at the top tip in the neutral case, and the proposed calibrations lead to an overestimation in the unstable case. However, the model shows encouraging behaviour as it can predict a modification of the shape function (from bimodal to unimodal) as instability and thus meandering increases.

The CPU cost of classical computational fluid dynamic models is too high to deal with all the different cases needed to estimate and optimize the performances of a wind farm. Thus, so-called engineering models have been developed to estimate the power loss due to wakes at a low computational cost (e.g.

The stability of the atmospheric boundary layer (ABL) influences the wake recovery

For the turbulent kinetic energy (TKE), it is common to model only the maximum value of added turbulence, which can be computed with the Crespo model

The present work aims to propose a physically based model that predicts both the mean and the variance (i.e. turbulence) of the axial velocity in the wake of a wind turbine. The advantage of basing our model on physical interpretations is that it gives more room for further improvements, as we know which assumptions were made and how it degrades the results. Moreover, the proposed model is dependent on atmospheric stability, since it influences both the velocity and the turbulence fields in the wake (see companion paper). Many models, such as the I&Q2018 model, do not take atmospheric stability into account, assuming that stable and unstable cases compensate for each other and that a calibration on neutral cases is thus sufficient. This approach is valid for monthly or yearly estimations of wind farms' performances. But some applications of the future wind industry, such as digital twins, need estimations over a day, an hour or even smaller periods. In such cases, the stability must be taken into account. Since we showed in the companion paper that stability mainly affects the wake meandering, this phenomenon must be decoupled from the wake expansion to take the ABL stability into account. To do so, the breakdowns described in the companion paper are reused and briefly explained in the following.

A field in the MFOR can be written as an unsteady translation of the same field in the FFOR through Eq. (

These terms are thoroughly described and quantified in the companion paper, where they are separated into pure terms (I, III and IV) and cross terms (II, V, VI and VII).

Term (I) is the convolution of

The proposed analytical model is based on the velocity and turbulence breakdowns (Eqs.

In the second section of this work, the datasets are presented: for the calibration of the model, a dataset from the MOMENTA project is used, and for the validation the neutral and unstable cases obtained from the large-eddy simulations (LESs) from the companion paper are reused. The third section presents the derivation of the model. The fourth section shows the chosen calibration methods, and the fifth section presents the corresponding results. All these results are discussed in the sixth section, followed by the conclusion.

The analytical model developed in this work is based on LES datasets generated with the Meso-NH solver

To close the set of equations, the subgrid TKE equation is resolved, and all the subgrid quantities are written as a function of this subgrid TKE, the resolved variables and a Deardorff mixing length. A grid nesting method allows for simultaneously having a vertical and horizontal mesh size of 1.5 and 0.5 m in the wake region for the two datasets and a domain large enough to compute the largest eddies of the atmosphere. The model and numerical parameters are described in more detail in the companion paper.

Two different LES datasets are used in this work: the first one for creating and calibrating the model and the second one for testing the model's results. Inflow conditions of these datasets can be found in Table

List of LES cases.

The calibration dataset contains six simulations, with four different ABL stabilities and three different thrust values. The simulated turbine is 92 m in diameter and has a hub height of 80 m. The turbine's data were obtained in the context of the MOMENTA project

To perform such simulations, a precursor without heat flux is first simulated in a domain of 19 km

After these two steps, the coarsest computational domain (horizontal resolution of 37.5 m) is ready: two grid nestings are then applied to reach a resolution of 1.5 m in the most refined domain. Then, 10 min of dynamics is used to let the flow establish itself in the wake of the wind turbine, and the post-processing is performed on the following 50 min of dynamics. The data are sampled at 1.2 Hz, which is the approximate limit between resolved and subgrid TKEs for these simulations (equivalent to 4 times the mesh size).

The wind turbine rotational speed and pitch are set according to the controller's database and the calculated upstream velocity. Since all the cases are computed at a similar inflow velocity, similar values of the thrust coefficient are obtained in the simulations. To include the influence of the thrust coefficient in the model, two additional cases with a degraded thrust coefficient are also computed, with the same inflow as the neutral case. To reduce the thrust, the pitch value is increased from 0 to 3 and 4.5

The second set of simulations, hereafter called the validation dataset, is based on the neutral and unstable cases that are described in the companion paper. The simulated turbine is a modified version of the Vestas V27: it is a three-bladed rotor with a diameter

For the validation simulations, the wake centre's

Time series of the wake centre's lateral

Another method is proposed here to compute the unsteady wake centres in the calibration dataset. A passive scalar (similar to a pollutant) is emitted at the rotor disc with a concentration value of 1 at each time step. This new variable is only driven by the advection scheme, in accordance with the passive tracer of the DWM theory, and only marginally impairs the code's performance. By supposing that this variable follows the wake, the unsteady wake centre is deduced from the centre of mass of this pollutant at each downstream position. The results lead to a low-frequency behaviour similar to the constant-flux method used in the companion paper but with fewer outliers (see Fig.

Table

Figures

Inflow conditions for the calibration cases.

Inflow conditions for the validation cases.

In the left panel the mean velocity is plotted. The calibration dataset (Fig.

The middle panels of Figs.

In the calibration dataset, the amount of TKE increases as the imposed heat flux increases. In the validation dataset, this is not the case since the neutral case is at a higher velocity at hub height, but the TI of the unstable case is indeed higher than that of the neutral case.

The right panels of Figs.

However, in the unstable cases, the velocity profile becomes nearly constant above a given height, leading to low values of

In this section, we derive an analytical model for the dominating terms of Eqs. (

Description of the most-used notations in this part and the following.

Based on the literature

To model term (IV) or

The derivation of a model for

It is proposed here to assume that the turbulence in the MFOR is solely driven by wake-generated shear, as in

From the literature

Combining Eqs. (

In Eq. (

Profiles of the modified mixing length (turbulence to shear ratio) for the different simulations.

It was thus decided to neglect shear in the formulation and to add the contribution of the inflow turbulence with a maximum function. This is a strong assumption that impacts the results (see Sect.

Figure

One can see that there are two distinct values: one inside the wake and one outside the wake. Inside the wake, the value is fairly constant (except in the bottom of the wake where it increases chaotically, probably due to the effect of the ground). It only seems to vary with the streamwise distance, and it was thus assumed that

Note that in Eq. (

For the PDF of wake meandering, the central limit theorem leads to a Gaussian distribution

In the following, the dependency of the variables on coordinate

In Eq. (

An analytical form of term (I) can then be deduced from Eqs. (

Since it has been shown in the companion paper that term (II) of Eq. (

Even though the reasoning of

To fulfil the conservation of momentum as in Eq. (

For the turbulence, a model has been found for terms (III) (Eq.

With the same assumptions as for term (I), it is possible to derive an analytical formulation for term (III) of Eq. (

Combining the chosen models for the wake meandering distribution and the added turbulence in the MFOR (Eqs.

At this point, the added turbulence in the FFOR is the sum of two terms that are identical if the coordinates

From Eq. (

It can be noted that in the absence of meandering, i.e. for

The model's equations are based on five variables: the wake widths in the MFORs

As described in Sect.

The resulting wake widths in the MFOR as a function of the downstream distance are plotted with solid lines in Fig.

Wake width in the MFOR for the different cases of the calibration dataset. Solid lines: results from the LES (Eq.

For all these reasons, the chosen function for the calibration is the following:

Parameters for the wake width in the MFOR.

The modified mixing length

Normalized modified mixing length for the different cases of the calibration dataset.

In the first approach, it is thus decided to fit the mixing length with a linear function of

Parameters for the mixing length.

The widths of the wake centre's distributions,

The resulting amount of meandering in the horizontal (top figure) and vertical directions (bottom figure) for the six cases of the calibration dataset can be found in Fig.

Normalized standard deviation of the wake centre from the LES (solid lines) and results from the base calibration (dashed line) and the engineering calibration (dotted line).

To model the amount of meandering,

Therefore, we hereby propose (dotted lines in Fig.

The issue is that

Ratio of turbulence averaged over a disc to the total turbulence, for different disc sizes.

Figure

Even though a fully physical approach would require a measure of the stability and an in-depth study of the turbulence spectrum as a function of the ABL conditions, the objective here is to propose an analytical model that is easy to implement and use. It is thus proposed to model the ratios

A least-square fit is used to determine the value of the parameter

Parameters for the large-scale turbulence function.

Results of the analytical velocity model for the different calibrations (dashed blue, dotted red and dash-dotted orange lines) in the neutral case compared to Meso-NH (solid black line) and the I&Q2018 model (dotted red line). Lateral

In this section, we analyse the results of the new model described in the precedent sections. For the streamwise velocity, the model is described with Eq. (

The base calibration is defined with Eqs. (

The engineering calibration uses the same equations except for the wake meandering, where Eqs. (

Finally, we also propose the “best” version of the model. Knowing that the calibration produces errors, it was interesting to see what would be the results of the “best calibration possible”, i.e. with parameters

Additionally, in Figs.

The results for the streamwise velocity field in the FFOR can be found in Figs.

Results of the analytical velocity model for the different calibrations (dashed blue, dotted red and dash-dotted orange lines) in the unstable case compared to Meso-NH (solid black line) and the I&Q2018 model (dotted red line). Lateral

Our model (with any calibration) behaves very similarly to the I&Q2018 model in the neutral case (Fig.

In the unstable case (Fig.

The proposed model is better in that regard, showing a larger wake expansion due to the higher predicted meandering compared to the neutral case. It shows that the determination of the velocity deficit in non-neutral cases necessitates more than only the total streamwise turbulence. In this case, one can note a discrepancy between the best version and the two calibrations of our model. It is due to an overestimation of the wake width in the MFOR for the unstable case (not shown here). Indeed, the neutral and unstable cases have similar wake widths in the MFOR, while they have different total turbulence intensities

With the same plotting convention as in Figs.

Results of the analytical streamwise turbulence model for the different calibrations (dashed blue, dotted red and dash-dotted orange lines) in the neutral case compared to Meso-NH (solid black line) and the I&Q2018 model (dotted red line). Lateral

Results of the analytical streamwise turbulence model for the different calibrations (dashed blue, dotted red and dash-dotted orange lines) in the unstable case compared to Meso-NH (solid black line) and the I&Q2018 model (dotted red line). Lateral

In the neutral case (Fig.

The unstable case shows the main shortcomings of the I&Q2018 model and the added value of our model. As shown previously, the I&Q2018 model gives similar results between the unstable and neutral SWiFT cases because they have similar inflow

Except for the upstream turbulence profiles, the inflow conditions used in the I&Q2018 are very similar between the neutral and unstable cases. Consequently, the purple profiles are alike in Figs.

However, the calibration of our model leads to an overestimation of the streamwise turbulence, in particular in the near wake. Since there are not many differences between the basic and engineering calibrations, it is not attributed to the meandering calibration (these two calibrations only differ by the meandering modelling) but rather to the overestimated

The best version of the model gives interesting results, showing that if a better calibration was achieved, in particular for the modified mixing length, the results of the model would be better. This question is further detailed in the next section.

The previous section shows the results of the model developed in this paper. It is quite good for the streamwise velocity field but can be improved for the turbulence, where the fully empirical model of

The authors want to emphasize that the presented work is a first step toward a fully physically based model for turbulence profiles that depend on atmospheric stability. In the companion paper, it was shown that the turbulence in the wake of a wind turbine is the sum of several terms, and here we present a methodology to analytically model the most important of these terms. Even though a fully usable calibration is proposed for anyone who would like to test the model, the main purpose of this work is to demonstrate how the rotor-added turbulence and meandering turbulence can be modelled from simple functions.

Results of the axial turbulence analytical velocity model in the neutral case, for different values of parameter

In Figs.

On one hand, all of these assumptions make the measure of

Besides a better calibration, the model could benefit from conceptual improvements. Indeed, the best version of the model (orange curve in Figs.

At several points of the reasoning, the atmospheric shear, i.e. the dependence of

Calibration parameters of the model.

A second improvement that could be done concerns the near wake. As mentioned in Sect.

For both of these improvements, some solutions were tried: not neglecting the

Finally, modelling the additional terms of Eq. (

This work is the second part of a two-step study that aims to model the turbulence in the wake of a wind turbine based on the meandering phenomenon. In the companion paper, the velocity and turbulence in the FFOR were broken down into different terms, some of which were shown to be negligible. In the present work, an analytical model is proposed for the dominating terms of the velocity and turbulence breakdowns, i.e. the meandering turbulence and the rotor-added turbulence. The originality of this work is that it allows for the independent modelling of the effects of meandering (and thus of the ABL stability) and the wake expansion and that it gives the whole turbulence profile rather than only the maximum value. For the velocity, it writes

The model has been tested on two LESs of a single wind turbine wake in a neutral and unstable atmosphere. For the velocity, the results are satisfactory, either in the vertical or in the lateral direction. The model performs better than the model from

This is the first step toward a fully analytical, physically based model for turbulence and velocity profiles in the wake of a wind turbine that takes into account atmospheric stability. For future work, the treatment of shear must be improved to model vertical turbulence profiles more realistically. The MFOR velocity deficit function could be replaced by a more accurate function in the near wake to improve the model's results in this region. It would also be interesting to derive an analytical model for the other terms of the turbulence breakdown.

Finally, this model can currently only be used for one turbine, as it predicts only the streamwise velocity and turbulence but necessitates the upstream lateral and vertical turbulence. For the model to be usable for multi-turbines, an expression for every term of the Reynolds stress tensor (or at least the diagonal terms to get the total TKE) would be needed, which implies a model for the lateral and vertical velocities

The Meso-NH code is open source and can be downloaded on the dedicated website. The authors can provide the source code of the modified version 5-4-3 that was used in this work. The data used for the plot presented here and in Part 1

EJ wrote the analytical model with FB. All the authors worked on the interpretation of the results. The paper was written by EJ with feedback from FB and VM.

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 made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors would like to thank the different stakeholders of the MOMENTA project

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