To simulate transient wind turbine wake interaction problems using limited wind turbine data, two new variants of the actuator line technique are proposed in which the rotor blade forces are computed locally using generic load data. The proposed models, which are extensions of the actuator disk force models proposed by

In recent years, large eddy simulation (LES) of wind farms has become feasible due to the emergence of fast computers and the development of the actuator disk and line methods. Hence, today it is possible to capture the interaction of the wind flow between the wind turbines (WTs) and transient problems in the atmospheric boundary layer (ABL), including the variation in the mean wind and turbulence associated with the diurnal cycle

This kind of input data is normally sufficient for performing computations using the actuator disk method (ADM), as the loading in this method is spread over the surface (the disk) representing the swept rotor area

In order to develop a generic load model for the ALM, we exploit the recent achievements of the ADM to improve the calculation of forces from local information of the flow over the disk while keeping the input parameters at a minimum. Using local flow properties allows the model to have a better performance in complex inflow conditions, such as ABL profiles in flat or complex terrain, or for rotors subject to partial- or total-wake impact. The first example is the analytical ADM developed by

The novelty of the present work is the development of two new ALMs which, based on the earlier developed analytical and numerical ADM approaches, are capable of simulating transient WT wake flows without the need of blade geometry and local airfoil data. The models are verified and compared for a wide range of flow cases, covering both uniform inflow and partial-wake impact subject to non-turbulent and turbulent ambient conditions in the ABL. Furthermore, the capabilities of the ALMs to improve the time-dependent response of the turbine are studied.

Example of the node distribution strategy on lines for the ALM

This work is organized as follows. First, in Sect.

The basic idea behind the actuator disk is to replace the three-dimensional geometry of the blades by equivalent forces acting on the flow. This strategy allows us to save computational costs, avoiding the need to resolve the blade boundary layer and replacing the otherwise boundary-fitted mesh around the solid blade surface by a coarse mesh with distributed volumetric forces

Figure

It should be noted that, in this work, the only difference between the ALM and the ADM is the number of lines

Finally, the total thrust,

To avoid numerical instability, the normal and tangential forces in each node are then distributed by means of a three-dimensional Gaussian function using a regularization kernel (

WTs in wind farms are affected by non-uniform velocity fields due to the ABL flow, the influence of the topography and the upstream WT wakes. This creates the necessity of a local force calculation in the ALM and the ADM in order for the model to give a better and more realistic response to this complex velocity field. The local calculation of

Based on the two types of actuator models (ALM and ADM) and the three approaches to calculate the local forces (BE, analytical and numerical), six models in total are implemented. Table

Actuator disk and actuator line models employed in this work, describing the differences in the local force calculation and the chosen tip and root corrections.

The first ADM combining BE theory with computational fluid dynamics (CFD) is from

First, the general operating regime of the WT needs to be obtained, which is related to the reference velocity

Once the operating regime is determined, the forces are calculated for each node, as described in the following. First, the relative velocity

The local angle of attack is determined as

The analytical local force calculation was originally proposed by

In this work, the implementation of this model is done following the next steps. First, in the cases where

A local calculation of forces using a numerical approach was originally proposed by

In the formulation from the original publication, with no tip and root corrections included, the non-corrected normal

In order to apply the tip

In the cases when the upstream reference wind velocity

The numerical ADM implementation requires a calibration process where the turbine is operated to face different uniform inflow cases. This process allows us to establish a relationship between local velocities at the ADM and the corresponding unperturbed velocities: firstly, the relation between

Example of the table defining the relation between the average velocity on the disk (

Example of the calibration table to obtain the relations between the local velocity on the node (

To construct Table

Afterwards, in each time step of the wind farm simulation, the ADM implementation takes the average velocity over the disk,

The tip and root corrections adopted in each ADM and ALM are summarized in Table

In the ALM there is also the need of introducing another type of correction to address the problem of the thickness of the shed vortices in the ALM produced by smearing the blade forces into the CFD domain. This has lately been discussed and solved in the works by

The root correction

In this work the open-source software OpenFOAM (version 2.3.1) is used in conjunction with the Simulator for Wind Farm Applications (SOWFA) libraries

Both single-turbine and double-turbine interaction configurations are performed for the uniform inflow case. In the single-turbine case, two domains are used. For the mesh sensitivity analysis a cubic domain configuration with dimensions of 6

Horizontal domain size and cell size in the mesh refinement zones for the non-turbulent inflow cases:

For the ABL inflow, a neutral stratified condition is simulated, and the turbulent solution is achieved by means of a precursor simulation with periodic boundary conditions. In this stage, a horizontal domain extension of

In the farm stage, the mesh is refined in the wake zone and near the turbines. Figure

Domain size and mesh refinement zones for the ABL inflow case:

In the present study, the widely employed NREL 5 MW reference WT

The results are separated between the uniform and turbulent ABL inflow conditions. First, a single turbine is simulated with uniform inflow to compare all the variations in the ADM and the ALM proposed in this work. This comparison is done for a below-rated inflow velocity

First, a preliminary mesh sensitivity study is necessary to find the dependency of the force's distribution and the wake on the number of cells along the diameter. This pre-study can be found in Sect.

Once the mesh resolution is chosen, the next step is the comparison of the six models for a basic case with a single turbine facing a uniform inflow velocity of

Model comparison for the single-turbine case under uniform velocity inflow conditions of

In order to analyze how the differences in the force's distribution affect the wake, in Fig.

Velocity magnitude comparison of the wake using different models for the single-turbine case under uniform velocity inflow conditions of

Model comparison for two turbines under uniform velocity inflow conditions of

The next step is to analyze how the models perform under non-uniform inflow conditions, in this case when WT1 generates a wake which afterwards affects WT2. Particularly for this work, the ADM-airfoil is always used to simulate WT1 in order to create the same wake inflow condition for WT2. Also, in WT2 the reference velocity

Figure

ALM comparison of the time-varying power output

Model comparison for two turbines under uniform velocity inflow conditions of

In order to study how the models distribute the forces in this uneven spatial inflow, in Fig.

The last step is to compare the model performance in a turbulent neutral ABL inflow condition, which is related to the problem of simulating wind farms in real field conditions. For this inflow condition, the same layout of two turbines as the uniform inflow case is simulated. At hub height a time-averaged velocity of

As in the previous case, the ADM-airfoil is used to simulate WT1 in order to create the same wake inflow condition for WT2. Also, in WT2 the reference velocity

Model comparison for two turbines in an ABL under horizontally averaged velocity inflow conditions at hub height of

Model comparison of the power output time variation normalized with its time-averaged value (

In Fig.

Model comparison for two turbines in an ABL under average velocity inflow conditions of

In order to analyze how all the models distribute the forces in this turbulent wake inflow, in Fig.

Model comparison for the time-averaged normal force distribution (N m

Model comparison for the time-averaged tangential force distribution (N m

In this work the previous developments for an ADM with local force calculations based on analytical

When a single turbine facing a uniform free-stream inflow is simulated, a close normal and tangential force distribution along the blade is found between all the models. The difficulties in capturing the root and tip force distributions obtained with the airfoil data are the major source of differences with the simpler models. The two new ALMs have the same distribution as the corresponding ADMs, with the only difference being the particular tip correction needed in the ALMs.

The models are tested in the challenging case of a WT partially affected by an up-steam wake, for both uniform non-turbulent inflow and turbulent neutral ABL. The analytical and numerical approaches manage to correctly capture the different force distributions at the different regions of the rotor, with a consistent overestimation of the normal force on the free side and a sub-estimation on the waked side. Looking at the tangential force distribution, the numerical approach tends to overestimate the values on the free side. In general, the analytical approach shows a slightly better performance in wake impact cases compared to the numerical one. The three ALMs show the additional capacity of capturing higher frequencies in the power output variation in time. In the uniform inflow case, the clear sign on how the power is reduced when one of the blades passes through the wake region is captured in an equivalent way by the three models. This added feature of the ALM is also visible in the ABL inflow case. The numerical approach has shown higher power fluctuations in both the ADM and the ALM. Finally, the extension of both the analytical and numerical approaches from the ADM to the ALM has shown promising results, opening the possibility to simulate commercial wind farms in transient inflows with the ALM without the restriction of private manufacturer blade data.

For this study the well-known ADM-airfoil and the ALM-airfoil are chosen. The inflow velocity is fixed at

This pair of models is tested for a wide range of mesh resolutions, which are defined by the cell size in relation to the rotor diameter

In Fig.

In Fig.

In order to see how the force distributions converge when the mesh is refined, the L2 error for the normal force

From the results obtained in this mesh sensitivity study, the

Mesh sensitivity study for the ADM-airfoil and the ALM-airfoil: instantaneous vorticity on a plane at hub height. Part of the extension of the force distributions is marked with a rectangle, assigning a total rectangle width of

Mesh sensitivity study for the ADM-airfoil and the ALM-airfoil: normal

Mesh sensitivity study for the ADM-airfoil and the ALM-airfoil:

The SOWFA framework on which this work is based is made available by NREL (

All data presented in this study can be made available upon request.

GPND developed the model code, performed the simulations, plotted the results and wrote the manuscript. ADO, HA, JNS and SI contributed with ideas, corrections and modifications both in the choice of models and in the results and text.

At least one of the (co-)authors is a member of the editorial board of

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

The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC).

This paper was edited by Alessandro Bianchini and reviewed by two anonymous referees.