Predictions of the dynamic wake meandering model (DWMM) were compared to flow measurements of a scanning Doppler lidar mounted on the nacelle of a utility-scale wind turbine. We observed that the wake meandering strength of the DWMM agrees better with the observation, if the incoming mean wind speed is used as advection velocity for the downstream transport, while a better temporal agreement is achieved with an advection velocity slower than the incoming mean wind speed. A subsequent investigation of the lateral wake transport revealed differences to the passive tracer assumption of the DWMM in addition to a non-passive downstream transport reported in earlier studies. We propose to include the turbulent Schmidt number in the DWMM to improve (i) the consistency of the model physics and (ii) the prediction quality. Compared to the observations, the thus modified DWMM showed a root-mean-square error reduction by 2 % for mean velocity deficit and 1 % for the turbulence intensity, relative to the unmodified DWMM, in addition to better temporal agreement of the dynamics. This is in contrast to an error increase of 35 % and 36 % if only a more accurate downstream transport velocity is used without including the turbulent Schmidt number.

Wind turbine wakes impinging on other wind turbines within a wind farm are a significant source of power losses, and they decrease the lifetime of affected wind turbines. Wake meandering is a low-frequency horizontal and vertical oscillation of the entire wake

Modeling approaches for wake meandering can be grouped into two categories. The first group are computationally expensive large-eddy simulations (LESs) that solve filtered flow equations at a high temporal and spatial resolutions

Illustration of wake meandering at an isolated wind turbine as assumed by the dynamic wake meandering model. Large-scale turbulence of the inflow displaces the wake of a wind turbine in the spanwise direction while it is transported downstream.

The DWMM has seen validation efforts in literature, which are reviewed in the following. The underlying passive scalar assumption of the DWMM has been accepted with the exception of the downstream transport velocity of wake meandering, which is slower than the mean wind speed

The above-mentioned discrepancies between the passive tracer assumption of the DWMM and observed transport behavior warrant closer examination. Specifically, assuming the wake as a passive tracer in the cross-stream directions and non-passive in the streamwise direction is physically inconsistent. Also, an investigation of the impact of the downstream advection velocity on the predictions of the DWMM has not been carried out so far. Further, previous validation efforts for the velocity deficit and the turbulence intensity predicted by the DWMM focused on validating all components of the DWMM simultaneously, with the exception of

Therefore, this paper will compare the wake dynamics modeled by the DWMM to the wake dynamics observed with field measurements. Further, we investigate how differences between modeled and observed wake meandering dynamics affect the predictions of the DWMM for the effect of wake meandering on the mean velocity deficit and the turbulence intensity. These research questions will be studied across a wide range of atmospheric conditions using field measurements of two pulsed Doppler lidars at a utility-scale wind turbine.

This section introduces first the DWMM (Sect.

The dynamic wake meandering model (DWMM) was introduced by

The quasi-steady velocity deficit is modeled with the steady-state, axisymmetric thin shear layer approximation of the Navier–Stokes equations with an eddy-viscosity turbulence closure

The system of partial differential equations given by Eq. (

The thin shear layer equations (Eqs.

The DWMM uses the hypothesis that the wake can be modeled as a passive tracer that is transported by the large-scale turbulence structures of the atmospheric boundary layer. The process can be imagined as a continuous sequence of velocity deficits emitted by the wind turbine that are passively transported by the large-scale turbulence

We will compare two assumptions for the downstream advection velocity in the results: (1) the advection velocity is the same as mean wind speed with

The vertical component of wake meandering cannot be computed directly, because measurements for the right-hand side of Eq. (

We assume that a suitable choice of

Further, Eq. (

The research site and the measurement setup are the same as reported in

The site consists of an isolated wind turbine located at Kirkwood Community College in Cedar Rapids, Iowa (Fig.

Satellite image of the measurement site with the location of the wind turbine (© Google Earth). The wind turbine coordinates are 41.9165° latitude and

We installed two pulsed Doppler lidars on the roof of the nacelle (Fig.

Photo of the front-mounted Doppler lidar on the nacelle of the wind turbine

The Doppler lidar mounted towards the rear of the nacelle was used to scan the wake. It was configured to average 3000 pulses per velocity estimate and use six points per range gate. This leads to an effective sampling frequency of approximately 3 Hz and spatial resolution of 18 m along the laser beam. A signal-to-noise ratio (SNR) threshold of

Simultaneously, the front-mounted Doppler lidar was used to measure the inflow. It was configured to average 5000 pulses per velocity estimate and uses six points per range gate. It was programmed to measure the lateral velocity component with a horizontal, fixed beam at a

The input values of the DWMM and the reference of the validation use data from separate measurement instruments to keep them independent. The model input is generated from the SCADA data and the measurements of the front-mounted Doppler lidar. The measurements of the rear-mounted Doppler lidar are used as the reference for the validation.

The input variables of the DWMM and the measurements from which they are taken are listed below:

The instantaneous wake center position, the turbulence intensity added by wake meandering, and the reduction of the mean velocity deficit due to wake meandering are extracted from the measurements of the rear-mounted Doppler lidar. The processing steps listed below and illustrated in Fig.

The data processing steps to quantify the effect of wake meandering on the mean velocity deficit and the turbulence intensity are illustrated for an example at a downstream distance of

The line-of-sight velocities are gridded on a polar coordinate system

The transformation to a Cartesian coordinate system is made using

An instantaneous velocity deficit is computed with

The instantaneous position of the wake center is detected with the centroid of the velocity deficit with

The instantaneous velocity deficit in the meandering frame of reference (

The temporal mean and the standard deviation of

The reduction of the mean velocity deficit due to wake meandering is then quantified by the amplitude difference of two Gaussian functions fitted to the mean velocity deficit in the NFOR and the MFOR, respectively. The difference will be denoted as

The turbulence added by wake meandering is quantified by the laterally averaged difference in turbulence intensity between the NFOR and the MFOR. It will be denoted as

Lastly, a temporal mean is also removed

The above processing steps are applied for downstream distances between

The measurement uncertainties are propagated to the model predictions and the reference data with a Monte Carlo method. We created 100 resamples of the measurement data with random fluctuations added that were drawn from a normal distribution with a standard deviation equal to the measurement uncertainty (except for the azimuth for which a uniform distribution was used). The model predictions and the wake quantities were computed for each of the 100 resamples, and the propagated measurement uncertainty was quantified as the root-mean-square error. If results are normalized in Sect.

In the case of the DWMM, the Monte Carlo approach was implemented in two stages. First, we estimated the uncertainty of

The first part of the results will focus on the modeling of wake meandering itself, while the second part will focus on the validation of the predicted effects that wake meandering has on the mean velocity deficit and the turbulence intensity. A set of 43 cases, each covering an approximately 14 min period, was selected from the measurement data of the campaign. The selection criteria were a mean wind speed above 5 m s

First, the predictions of the instantaneous wake center positions will be validated. The root-mean-square error (RMSE) and the correlation coefficient between the predicted wake center position (Eq.

The correlation coefficient

Next, we will investigate the effect of the downstream advection velocity on the predicted wake-center positions. We will compare predictions using the reduced downstream advection velocity given by Eq. (

The effect of the downstream advection velocity on the correlation coefficient

Despite the increase in correlation coefficient, using

It was previously observed that using

The time series of the observed and the predicted wake center positions at

Error between observed and predicted wake meandering strength with

Better temporal agreement of the DWMM when using a downstream advection velocity slower than the mean wind speed was observed in several studies

Based on our own findings and the literature review, we believe that the discrepancy between temporal agreement and wake meandering strength points towards a short-coming of the passive-tracer assumption of the DWMM. In the following section, we will provide a hypothesis to address this problem. Other possible explanations for the overestimation that were tested on the data and rejected are listed in Appendix B.

We hypothesize that the transport of the wake with large-scale turbulence is more akin to the transport of momentum than the transport of a passive tracer and that, subsequently, the turbulent Schmidt number should be considered in the modeling of wake meandering. The turbulent Schmidt number characterizes the ratio between the turbulent transport of momentum and the turbulent transport of passive scalars. Previous experiments indicated that momentum is transported less efficiently than scalars in turbulent wakes

We use the diffusion theory of

The lateral transport velocity of the wake center (

The turbulent Schmidt number was determined with Eq. (

Turbulent Schmidt numbers of wake meandering (Eq.

Following our hypothesis, we modified the DWMM to account for a reduced momentum transport efficiency by including

Observed wake meandering strength and predicted wake meandering strength of the DWMM. Panel

The second part of the results will validate the predictions of the DWMM for the effect of wake meandering on the mean velocity deficit and the added turbulence intensity. The DWMM will be compared to the observations in Sect.

First, we validate the predicted reduction of the mean velocity deficit due to wake meandering. The modified DWMM treating the wake as non-passive (i.e., using

The reduction of the mean velocity deficit as a function of the wake meandering strength at a downstream distance of

Next, the model predictions of the added turbulence intensity due to wake meandering are validated. We observe that both the modified DWMM and the observations show an increase of the added turbulence intensity with the wake meandering strength (Fig.

The added turbulence intensity as a function of the wake meandering strength at a downstream distance of

To investigate the effect of the proposed modification to the DWMM on the comparison with the observations, we compare three versions of the DWMM:

The original DWMM uses the passive tracer assumption (

A semi-modified DWMM only accounts for the slower downstream advection velocity without modifying the lateral transport (

The fully modified DWMM as proposed here treats the wake as non-passive (

Table

The similar results for the original DWMM and the fully modified DWMM are explained by the temporally averaged validation approach used here, where the errors of a downstream transport that is too fast and a lateral transport that is too efficient mostly cancel out, which might also explain why the issue was not noticed in previous validations. Only when including dynamics into the validation as in Sect.

The considerably larger error of the semi-modified DWMM compared to the other two implementations is explained by the overestimation of the wake meandering strength observed in Sect.

The root-mean-square error (RMSE) between the observations and three versions of the DWMM. The left column shows the error percentage for the reduction of the normalized mean velocity deficit due to wake meandering (

A test of the existing formulation of the DWMM and a new formulation that incorporated additional physics was presented. The test site was an isolated wind turbine in Cedar Rapids, Iowa. A Doppler lidar deployed on the nacelle of the wind turbine scanning the velocity field of the wake at hub height was used as reference to which the models were compared. A second Doppler lidar and the SCADA data of the wind turbine were used to initialize the wake meandering models.

The results for the instantaneous wake center position exposed an issue with the passive tracer assumption of the existing formulation of the DWMM. The wake meandering strength had better agreement with the observation, if the mean wind speed were used for the downstream transport; at the same time, a better temporal agreement is reached if the downstream transport used a special wake velocity to more accurately represent the advective transport. Analyzing the transport behavior of the wake, we found that both the downstream transport of wake meandering as well as the lateral wake displacement showed differences compared with the DWMM, assuming a passive scalar transport. Therefore, we propose to include the turbulent Schmidt number in the DWMM to account for the less efficient turbulent transport of momentum compared to a passive scalar in addition to the a slower downstream transport velocity. This will also make the DWMM physically more consistent, because the wake is considered fully non-passive with this modification, while previously it has been treated as non-passive in the downstream direction and passive in radial direction.

A comparison of the thus modified DWMM with measurements showed that it reconciles the previously noted discrepancy of statistics and dynamics. The DWMM using only the more accurate downstream transport velocity had an error increase of 35 % for the mean velocity deficit reduction and 36 % for the added turbulence intensity compared to the original DWMM using the passive tracer assumption. The DWMM that included the Schmidt number in addition to the more accurate downstream transport velocity had an error reduction for those statistics by 2 % and 1%, respectively (and better temporal agreement for the dynamics of wake meandering).

In future work, we propose a validation of our findings with a different experimental approach or through simulations to exclude site factors or methodological biases. It would also be interesting to investigate if the variability in the MFOR can be fully explained with the small-scale turbulence part of the DWMM. However, the latter requires wake measurements with a higher temporal and spatial resolution.

The normalized mean velocity deficit of the

Thrust coefficient curves of six wind turbines from manufacturer data (first compiled by Abdulrahman, 2017) and the ensemble average, which is assumed as the

The following hypotheses for the overestimation of the wake meandering strength observed in Sect.

Temporal variations of the downstream advection velocity during a 14 min period would lead to a reduced (increased) amplitude of the wake meandering during times with faster (slower) than average advection velocity. Utilizing the outside points of PPI of the wake scanning lidar to gain a time series of the wind speed, we found that the effect on the predicted wake center position is too small to explain the overestimation.

A misalignment of the wind turbine could contaminate the lateral velocity measured by the front-mounted Doppler lidar with contributions from the longitudinal velocity. We used the yaw angle reported in the SCADA data and the mean wake center position within the wake scanning lidar's field of view to quantify the yaw misalignment of the wind turbine. The overestimation did not show any relationship to the average or trend.

The overestimation persists if the mean instead of a linear trend is removed from

In case any remaining flow distortion of the wind turbine affecting

We had the hypothesis that the onset of wake meandering is delayed due to a sheltering effect within the near wake until entrainment has reached the wake center. However, this assumption seems unrealistic based on the fact that the wake-scanning lidar shows wake meandering within the near wake. Testing the hypotheses on the data led to a increased of the RMSE for

The data to replicate the figures are available on Zenodo (

PB contributed to the data curation, formal analysis, conceptualization, methodology, software, validation, visualization, and writing (original draft). FPA contributed to the conceptualization, funding acquisition, project administration, supervision, and review and editing. CDM contributed to the funding acquisition, resources, data curation, investigation, and review and editing.

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 Kirkwood Community College for their cooperation and allowing access to their wind turbine. We also extend our appreciation to Clipper Windpower for granting access to technical data on the Liberty Wind Turbine.

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant nos. 200021_172538 and 200021_215288), the Bundesamt für Energie (grant no. SI/502135-01), the National Science Foundation (grant no. 1101284), and the Center for Global and Regional Environmental Research (CGRER), University of Iowa.

This paper was edited by Rebecca Barthelmie and reviewed by two anonymous referees.