This study describes the impact of postprocessing methods on the calculated parameters of tip vortices of a wind turbine model when tested using particle image velocimetry (PIV). Several vortex identification methods and differentiation schemes are compared. The chosen methods are based on two components of the velocity field and their derivatives. They are applied to each instantaneous velocity field from the dataset and also to the calculated average velocity field. The methodologies are compared through the vortex center location, vortex core radius and jittering zone.

Results show that the tip vortex center locations and radius have good comparability and can vary only a few grid spacings between methods. Conversely, the convection velocity and the jittering surface, defined as the area where the instantaneous vortex centers are located, vary between identification methods.

Overall, the examined parameters depend significantly on the postprocessing method and selected vortex identification criteria. Therefore, this study proves that the selection of the most suitable postprocessing methods of PIV data is pivotal to ensure robust results.

The wake of a wind turbine is characterized by a massive presence of vortex structures. Two main types of concentrated vortices can be identified, which are shed from the root and the tip region, respectively. The latter form strong helical structures that influence the wake of the wind turbine.

The tip vortices are generated by the pressure difference between the top and lower sides of the blade tip, which lead to a flow from the pressure side to the suction side of the blade

Since the first introduction of particle image velocimetry (PIV) applied to wind turbine aerodynamics by

Studies have also been carried out in water channel facilities.

Wind turbine tip vortices studies employing the PIV technique and VIM details.

WT: wind tunnel; WC: water channel; n.s: not specified;

It is worth remarking that, once comparing the methods, the inherent error introduced by the PIV technique must be accounted for. Table

Several vortex identification methods (VIMs) have been employed so far. However, consensus on the most suitable methodology for the study of vortices in the wake of a wind turbine has not been found yet, as shown in Table

This paper aims at comparing different vortex identification methods to evaluate their suitability to specifically study the tip vortices of a wind turbine. The methods are applied to velocity field planes that were obtained through PIV in the near wake of a wind turbine model located in a wind tunnel facility. Compared to previous investigations, the present study offers an in-depth comparison of commonly used VIMs on the same wind turbine tip vortex measurement dataset. The main goal is to identify similarities and differences of the methodologies, i.e., providing a direct insight into their application. Furthermore, a rigorous comparison of VIM application is provided, with the simultaneous study of six tip vortex parameters, namely (1) streamwise location, (2) lateral location, (3) streamwise velocity, (4) lateral velocity, (5) core radius and (6) jittering.

Thanks to the large number of analyzed samples, a statistical analysis is also included in order to give more insights into the challenges of each methodology. Three different VIMs are compared: vorticity,

The following section, Sect.

Many vortex identification methods have been proposed in the literature

Graftieaux's method

vorticity magnitude

Additional methods can be derived from the eigenvalue analysis, such as

A full description of the selected methods is given below. For a more extensive review of VIMs, the interested reader is directed to

This method identifies the vortex through a global quantity,

Figure

Sketch of Graftieaux parameters in

Depending on the grid size, the value of

The vorticity is defined as the curl of the velocity field as shown in Eq. (

In this way, the vorticity quantifies how the velocity vector changes when it moves in a direction perpendicular to it and therefore is a natural candidate for vortex identification. Indeed, this method has been used for a long time

The most common vortex identification methods are based on the analysis of the velocity gradient

In particular, the second invariant

As shown in the previous section, several vortex identification methods are based on the gradient of the velocity field. Then inherently the evaluation of flow field derivatives is necessary. In this way, the differentiation of the velocity fields from either computational data or experimental techniques (as PIV) is needed. Both normally come in a discrete format.

In the case of PIV data, the choice of the differentiation scheme becomes more relevant due to the presence of noise affecting the measurements. Noise sources include optical distortion, light sheet non-homogeneity, transfer function of charge-coupled device (CCD) cameras and particle characteristics, among others

Many methods have been developed to calculate spatial derivatives from discrete data. The most frequently used methods are based on discrete differential operators applied to the surrounding points of the position to evaluate

The backward, central and forward differencing schemes provide the simplest implementation. Nevertheless, additional schemes have been studied with the purpose of either increasing the accuracy or reducing the uncertainty of the results.

The latter schemes are defined for a single-variable function and applied in one dimension at a time. Conversely, the velocity field can be influenced by the complete spatial coordinates. Therefore, the velocity gradient should depend on the surrounding flow.

Table

Summary of differentiation schemes and implementation

The analysis shown in the rest of this paper relies on the stereo-PIV dataset from previous work by

The experiments were carried out in the closed-loop wind tunnel at the Technische Universität Berlin. The wind turbine, Berlin Research Turbine (BeRT)

The wind turbine model produces a

Front view sketch of Berlin Research Turbine (BeRT) (1), cameras (2) and laser sheet (3), left. Camera system, middle. BeRT and calibration target in the test section, right.

The stereo-PIV system consisted of a Quantel dual-Nd:YAG double laser with energy of 171 mJ, a mirror arm, laser sheet optics and two cameras (CCD-chip). Additionally, an ILA synchronizer receives information of a reference blade azimuthal angle from a light sensor located in the nacelle. In this way, the phase-locked measurement is achieved by coupling the laser and blade position.

The measurement plane was horizontal and was centered on the tip location when the blade was in the horizontal position. In this study, only one vortex age is analyzed,

Operation and image acquisition details.

A two-dimensional analysis is carried out on the dataset. To conduct a three-dimensional analysis of the vortex structures, additional parallel planes are needed. Therefore, only the two in-plane velocity components are used (

The application to obtain the vortex properties follows, while the statistical analysis of the available data is described at the end of this section.

The velocity fields are analyzed through the application of the VIMs described in Sect.

Figure

Flowchart of the implementation of the vortex identification methods and schemes.

Results are presented in a normalized form and bounded by unity. In the case of

Field of view of PIV measurements and area of interest of the current study. Axes are normalized by the grid spacing.

The tip vortex, after being shed, is both translating and rotating at the same time. Considering this, the convection velocity (downstream,

The core radius is calculated using the following steps.

The induced velocity field

The swirling velocity is analyzed through vertical and horizontal lines, Eq. (

A spline line is fitted to the swirling velocity curves. The radius,

The procedure is repeated for each VIM and scheme and applied to both the average and the instantaneous velocity fields. Figure

Both vortex characteristics, convection velocity and core radius are normally used to describe the evolution of the vortex at different ages

A statistical analysis is made over the complete dataset of the instantaneous velocity fields. In this way, the vortex center, convection velocity and core radius are analyzed in terms of their location and magnitude variations.

An ellipse is used to define the jittering characteristic zone, similar to the work of

The results are presented as follows. First, an overview of the VIMs and schemes applied to the average velocity field are shown using the vortex center locations and convection velocities based on the three identification parameters

Second, a statistical analysis of the complete set of instantaneous velocity fields is performed. In this way, the vortex center locations are studied in order to define the shape, distribution and surface of the jittering zone described by each VIM and scheme. Additionally, these results are used to compare the scattering of the convection velocities and core radii.

Figure

Graftieaux's method has been developed for stationary vortices, while indeed, the test case is a superposition of the vortex-induced velocities and the streamwise flow, which convects the vortex downstream. The latter difficulty is overcome by subtracting the background velocity.

Figures

Vorticity and

Normalized vorticity magnitude on the area of interest.

Normalized

The presence of multiple maxima and the ring-like distribution of the

On the other hand, the presence of multiple maxima might also be due to small-scale structures within the vortex, as suggested by

Vortex center locations for different differentiation schemes. The vorticity magnitude contour based on the least-squares scheme is shown.

Average convection velocities.

Among these hypotheses, the first one seems the most suitable. It is possible that artifacts are produced on some of the schemes applied, where the concentration of seeding is diminished. These artificial peaks are not present in results using Graftieaux's method because the methodology includes information from a larger number of grid points.

In fact, 8 and 24 points are employed to estimate the parameter

Jittering evaluation using Graftieaux's VIM with

Jittering zones over the area of interest with Graftieaux and vorticity VIMs.

Regarding the position of the vortex centers, the locations are shown in Fig.

In the case of vorticity and

Contour distribution of the vortex center locations over the area of interest from vorticity VIM.

Probability distribution of the vortex center locations using the vorticity VIM and CD schemes. Additionally, red lines show fitting curves.

Vorticity and

Therefore, the estimation of the convection velocity is recommended with the smoother VIMs and schemes, i.e., Graftieaux's method or vorticity and

Based on selected operational parameters (see Table

Probability distribution of the vortex center locations while the vorticity VIM is applied using BD, CD and LS schemes.

The vortex center locations are identified on each instantaneous velocity field, which constitutes the complete PIV dataset. For readability, only the Graftieaux

Figure

Normalized convection velocity in the streamwise direction.

Instantaneous vorticity magnitude with the central differentiation scheme and quiver lines of the velocity field.

From Fig.

Normalized core radius.

Figure

Even though the area swept by the scattering of the estimated vortex center locations is similar in magnitude, in the case of vorticity and

Figure

Figure

Similarly, in the lateral direction, Fig.

Overall, the estimation of the vortex center location is influenced by the VIM and scheme. In the same way, the convection velocity and the core radius are affected by the implemented methodology.

Convection velocity magnitudes from the average velocity field and the statistics after instantaneous velocity field analyses.

To see the effect of VIMs and schemes on the instantaneous convection velocity, Fig.

Graftieaux's VIM, Fig.

Several estimations fail, such as

However, in the case of Graftieaux, vorticity and

The visible ring-like concentration in Fig.

Table

The average convection velocity is also estimated from the average flow field (red line). However, It should be noted that average results must not be overstated because of the tip vortex jittering

In the case of the vortex core radius, even when all the scattering of the convection velocity influences its calculation (see Sect.

In this way, the magnitude of the core radius is

Several vortex identification methods (VIMs) and implementation schemes have been applied to the two components of the velocity field data in the near wake of a wind turbine model, obtained through PIV measurements. The methodology was applied to the average velocity field as well as the instantaneous velocities resulting in a statistical analysis of the PIV dataset.

In the case of the average flow field, the chosen VIMs and schemes provide different magnitude distributions of the identification parameters. Nevertheless, the vorticity and

Through the statistical analysis, it is concluded that different methodologies lead to different interpretations of the tip vortex behavior. Even though the jittering zone is found to be ellipsoidal for all the VIMs and schemes, the probability density function of the vortex center locations varies in the streamwise direction from one single peak with the Graftieaux, vorticity and

The multiple peaks, found in some identification parameters, are determined to be an artifact produced by certain schemes. The latter can be avoided using either Graftieaux's VIM or vorticity and

It is concluded that the vortex center locations are within a small variation range, and their comparability is viable independently of the VIM or scheme. Nevertheless, first-order schemes, such as backward and forward differences, should be avoided.

The convection velocity presented a higher dependency on the VIM and scheme applied. Therefore, and keeping in mind that the results have shown good comparability regarding the vortex center locations, it is recommended to use the information of several vortex ages instead of the swirling velocity approach to estimate the convection velocity. Conversely, the vortex core radius only showed a grid step variation between VIMs and schemes. Further studies might include analytical approaches which predict the tangential velocity profiles of a vortex from which the vortex core is estimated to also check their applicability.

Overall, Graftieaux's method is the recommended VIM to track the tip vortex. Indeed, the method does not use differentiation and has shown to be independent of the number of neighboring points used. Moreover, it presents the lowest standard deviation between all the methodologies applied here.

Jittering zone of each VIM and scheme.

Graftieaux probability contours.

Vorticity probability contours.

Velocity field data and results can be provided by contacting the corresponding author.

RSV carried out the measurement campaign, worked on the methodology, performed calculations and analysis, and wrote the paper. SC implemented and tested the methodology with assistance and supervision from RSV. SC, SB, MM, CNN, AB and COP contributed with comments and discussions about each section in the paper.

Some authors are members of the editorial board of

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

This research has been supported by the ANID PFCHA/Becas Chile-DAAD/

This paper was edited by Horia Hangan and reviewed by two anonymous referees.