The actuator-cylinder model was implemented in OpenFOAM by virtue of source terms in the Navier–Stokes equations. Since the stand-alone actuator cylinder is not able to properly model the wake of a vertical-axis wind turbine, the steady incompressible flow solver simpleFoam provided by OpenFOAM was used to resolve the entire flow and wakes of the turbines. The source terms are only applied inside a certain region of the computational domain, namely a finite-thickness cylinder which represents the flight path of the blades. One of the major advantages of this approach is its implicitness – that is, the velocities inside the hollow cylinder region feed the stand-alone actuator-cylinder model (AC); this in turn computes the volumetric forces and passes them to the OpenFOAM solver in order to be applied inside the hollow cylinder region. The process is repeated in each iteration of the solver until convergence is achieved. The model was compared against experimental works; wake deficits and power coefficients are used in order to assess the validity of the model. Overall, there is a good agreement of the pattern of the power coefficients according to the positions of the turbines in the array. The actual accuracy of the power coefficient depends strongly on the solidity of the turbine (actuator cylinder related) and both the inlet boundary turbulence intensity and turbulence length scale (RANS simulation related).

The modeling of vertical-axis wind-turbine (VAWT) farms has lacked researched in the last years compared to horizontal-axis wind turbines (HAWTs). The complexity of the models ranges from simple momentum models to full-rotor RANS (Reynolds-averaged Navier–Stokes equations) or LES (large-eddy simulations) simulations. While simple models are computationally inexpensive, they lack accuracy and rely on various semi-empirical corrections which may not be valid for all cases; high-fidelity simulations are simply out of the scope of many researchers and scientists due to the tremendous computational requirements.

This work proposes an actuator model integrated within an OpenFOAM solver that relies on replacing the turbine by volumetric forces exerted on the fluid; this approach eliminates the need of highly resolved meshes around the blades, thereby reducing the mesh size considerably. The forces are modeled using the steady-state actuator-cylinder model

The first category belongs to models using the momentum theory or potential flow theory. One of the simplest wake model is the Jensen model

The next category belongs to actuator models. As it was said before, these models rely on replacing the turbine by volumetric forces.

The last category of models employs full-rotor RANS simulations. Works on multiple VAWTs can be found in

The current RANS-AC has the potential of modeling entire wind farms without relying on empirical corrections for the wake or without the need of HPC (high-performance computing). Moreover, only simple input data must be entered, namely the geometrical and operational parameters and inlet boundary conditions for the simulation.

This section first presents the theory behind the AC model, then the justification of the linear solution is described in detail and a validation for the stand-alone AC is included. The last part deals with the details of the RANS-AC implementation in OpenFOAM.

The VAWT can be modeled as a hollow cylinder upon which radial volume forces

Radial forces acting on fluid.

The Euler equations are applied to the entire field, and the volume forces are represented by the forces exerted by the blades. The final form of the solution is shown in Eqs. (

The normalization of the loads can be found in

The rotational speed of the rotor normalized by

The lift and drag coefficients can be obtained from a lookup table as a function of

The turbine

The perturbation velocities can be determined if the forces are known, while the forces also depend on the perturbation velocities. The solution is iterative: first, the perturbation velocities are set to zero, then the aerodynamic coefficients are computed as well as

The aforementioned equations are only valid for low-loaded rotors (thrust coefficient), with the model including only the linear part stops being accurate at high loads; however, a relation between the induction factor

The Windspire was chosen for validation because it will be used throughout this work; therefore the results of the AC model will be compared against experimental data. Ideally, it would be better to validate against a low-solidity turbine since it meets the requirements of the AC model; nevertheless, the Windspire turbine will be used despite it having a solidity of

A particular challenge was to find polars for the DU06W200.

Lift polar for the DU06W200 airfoil at different

Drag polar for the DU06W200 airfoil at different

No attempt was made to introduce dynamic stall or flow curvature effects. Dynamic stall models can conflict with the AC model according to

The results from the AC model were compared against data from AC model results provided by

1.2 kW Windspire turbine validation and comparison.

The AC model can be incorporated into one of the OpenFOAM solvers by taking advantage of the source terms in the Reynolds-averaged Navier–Stokes equations. The solver used is called

Notice that

Boundary conditions at time 0 for the computational domain.

In order to prove that the RANS-AC has been implemented correctly, the power coefficient of the RANS-AC will be compared against that of the stand-alone AC model. A sensitivity analysis concerning the thickness of the cylinder and the turbulence intensity

It must be clear that the simpleFoam solver interprets

RANS-AC results from two different meshes verified against the stand-alone AC.

In summary, all variables have to be initialized at time 0 in the domain. In order to initialize the values of all variables, a set of equations is needed. Equations (

The distance from the turbine to the outlet does not seem to affect the result. In this case, the outlet was placed 10

Since a volumetric field is created initially, at the end of the simulation it is possible to visualize these forces using Paraview. Figure

Volumetric forces acting on the cylinder. Flow goes from left to right. Free stream speed is 8 m s

This section is meant to test the capabilities of the RANS-AC in a multi-turbine environment. Experimental data from a small wind farm of VAWTs were found in

Four turbines in a row. The distance from turbine to turbine is 11.31

Fish schooling configuration. Turbines are placed in a counterclockwise-rotating fashion. The flow is also from left to right.

Since the numerical simulation must be provided with the turbulence length scale in order to compute the turbulent kinetic energy dissipation rate at the inlet, a study on the width of the wake was conducted in order to find the appropriate turbulence length scale. The results in Table

In order to obtain a value for

Wake width at different downwind stations. The width begins to reach a stable value at 7

Once the new value of

Wake widths obtained by following an iterative procedure. All the plots are located at 7

The power coefficients from this array were obtained from data published by

Power coefficients of the four-turbine array. Panel

Another plot concerning the velocity and turbulence intensity along the center line is included in Fig.

Wake across the turbine array. The axis for

Power coefficients of the 18-turbine array.

A plot similar to Fig.

It was observed that a portion of the angles of attack of the turbines that were free of blockage were above stall according to Fig.

It must be clear that this fault is due to the wrong predictions of

As seen in Sect.

Comparison of

Next, a comparison of the wake of the turbine is presented. The LES wake is taken at the Equator. The width of the wake

Comparison of the LES wake at different downwind distances with the current RANS-AC results.

Wind farm array of 18 turbines using the 25 m radius turbine from

Finally, the same 18-turbine wind farm simulation is done, except this time the turbine from

The RANS-AC was successfully implemented in OpenFOAM. This was confirmed by the fact that it could achieve a power coefficient very similar to the stand-alone AC. Guidelines for selecting the mesh size and the thickness of the actuator were also given along with inlet boundary conditions for the RANS simulation. The model was validated against multi-turbine experiments, and good agreement was found concerning the trend of the power coefficients in a row of four VAWTs. Unfortunately these multi-turbine experiments were done using small turbines which had a high solidity; this caused the model to wrongly predict the angles of attack, namely overestimating the angles of attack and thus getting the wrong coefficients of lift and tangential force. Thus results from the array of 18 turbines could not match the experimental data. An additional verification against a large VAWT LES simulation

Codes are available at

Data sets are available at

This video explains the results of the 18-large-turbine wind farm. Pressure and velocity fields are shown (

The supplement related to this article is available online at:

EMO was the main author of this work; he developed the stand-alone actuator-cylinder code and the actuatorCylinderSimpleFoam solver for OpenFOAM. MS contributed to the guidance, suggestions and some of the writing in LaTeX as well as some figures. FJSO contributed to the guidance, suggestions and revision of this work.

The authors declare that they have no conflict of interest.

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

The main author of this work would like to thank CONACYT (Consejo Nacional de Ciencia y Tecnología) as well as the OpenFOAM community.

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