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In this study, large eddy simulation (LES) is combined with a turbine model to investigate the wake behind a vertical-axis wind turbine (VAWT) in a three-dimensional turbulent flow. Two methods are used to model the subgrid-scale (SGS) stresses: (a) the Smagorinsky model; and (b) the modulated gradient model. To parameterize the effects of the VAWT on the flow, two VAWT models are developed: (a) the actuator swept-surface model (ASSM), in which the time-averaged turbine-induced forces are distributed on a surface swept by the turbine blades,

Revived attention towards vertical axis wind turbines (VAWTs) has been observed in recent years. VAWTs are claimed to have several advantages over the conventional horizontal axis wind turbines (HAWTs), such as: insensitivity to the wind direction and, consequently, the absence of any yaw equipment; lower noise, due to the lower tip-speed of the blades; and placement of the drive train system on the ground [

Almost 40 years after Darrieus’ patent was registered (in 1931), researchers, mainly from some North American institutes (such as the National Research Council of Canada, NASA Langley Research Center and the Sandia National Laboratories), started to study VAWTs extensively. Rigorous and exhaustive experimental data from both wind tunnel and field measurements for different sizes of VAWTs were collected by these institutes in the 1970s and 1980s [

Besides experimental investigations, analytical and numerical studies have also been employed with the aim of VAWT performance prediction. The aerodynamic models that have been proposed for VAWTs can be classified in two main groups: streamtube models [

Simulation of VAWTs by solving the Navier–Stokes equations with numerical techniques was pioneered by Rajagopalan and Fanucci [

The present work aims at simulating the turbulent wake of a VAWT in a large eddy simulation (LES) framework by solving the full filtered Navier–Stokes equations in three dimensions, using an actuator swept-surface and an actuator line method to model the turbine effects on the flow and two turbulence modeling techniques to parameterize the subgrid-scale (SGS) stresses. It is worth mentioning that it is the first time that LES has been applied and validated for the simulation of VAWT wakes by using either the actuator swept-surface model (ASSM) or the actuator line model (ALM) techniques. The LES framework and model formulations are presented in Section 2. Next, we simulate the wake behind a straight-bladed VAWT for one special case and compare the results with corresponding experimental data. The numerical and experimental setups are described in Section 3, and the results are presented in Section 4. A summary of the study is given in Section 5.

LES solves the filtered incompressible Navier–Stokes equations, which can be written in rotational form as:
_{i}_{i}_{p}_{ij}_{i}

Two closure models for parameterizing the SGS stresses are considered: the traditional Smagorinsky model and the modulated gradient model.

The traditional Smagorinsky model [_{S}

The modulated gradient model (MGM), which was proposed by Lu and Porté-Agel [

If Δ̃_{x}_{y}_{z}_{ij}

To evaluate the SGS kinetic energy, _{sgs}_{ε}_{ij}_{sgs}

Replacing the above expression for _{sgs}

To ensure numerical stability and that energy does not transfer locally from unresolved to resolved scales, we use the following clipping procedure:

With _{ij}

In this study, two methods are used to model the turbine-induced forces: (a) the actuator swept-surface model (ASSM); and (b) the actuator line model (ALM). Both models are based on blade-element theory, and therefore, they do not require resolving the boundary-layer flow around the turbine blade surfaces, which greatly reduces the computational cost requirements. In the blade-element theory, the turbine blades are divided into

_{b}_{∞} is the unperturbed wind velocity magnitude and _{local}_{local,n}_{local,s}_{rel}_{local,n}_{n}_{local,s}_{b}R_{s}_{n}_{s}

The lift and drag forces (see _{L}_{L}_{c}_{D}_{D}_{c}_{c}_{L}_{D}

In the ASSM approach, the actuator swept surface is defined as the surface swept by the blades. For example, in the case of a straight-bladed VAWT, this actuator swept surface is an actuator cylinder. We distribute the time-averaged turbine-induced forces on this surface and integrate them over time and space.

For illustration reasons and for the sake of clarity, the size of the grid cells with respect to the diameter of the rotor is shown as being much bigger than in the simulations. The turbine forces are distributed among grid points, whose corresponding control volumes intersect the actuator swept surface. For instance, arc
_{rel}_{L}_{D}_{b}_{b}_{i}

In the ALM, the turbine forces are distributed vertically along lines that represent the blades of the wind turbine. In this model, which was introduced by Shen and Sørensen [

First, each grid point whose corresponding control volume encompasses the blade line is identified, and then, the airfoil data and subsequent loading are determined by computing local angles of attack from the movement of the blades and the local flow field. For instance, in _{i}

By explicitly representing the individual blade motions, the ALM enables us to capture in more detail the complex and unsteady vortical structure of the near-wake flow. Consequently, it is likely to improve the near-wake predictions with respect to the ASSM. The application of the ALM seems particularly crucial for VAWTs, since, in contrast to HAWTs, a blade of a VAWT passes through the wakes generated by other blades, as well as its own wake; this fact makes the ALM approach attractive, because it allows us to study the blade-wake interaction phenomenon more realistically.

In this section, the details of the numerical simulations, as well as the experimental setup, with which our results are compared, are presented. We have chosen the experimental study by Brochier

In 1986, Brochier

The water channel was vertical, and the driving force of the flow was gravity. The length of the channel in the streamwise direction (vertical direction in the experiment) was 1.5 m, and the cross-section of the channel (normal to the streamwise direction) was a square of a side length of 20 cm (see

The LES code is a modified version of the code described by Albertson and Parlange [_{x}_{y}_{z}

To compute the spatial derivatives, a Fourier-based pseudospectral scheme is used in the horizontal directions, and a second-order finite difference method is used in the vertical direction. The governing equations for conservation of momentum are integrated in time with the second-order Adams–Bashforth scheme.

The boundary conditions in the horizontal directions are mathematically periodic. For the bottom and top boundary conditions, the instantaneous surface shear stress is calculated using the Monin–Obukhov similarity theory [

In the simulations for this case, the same dimensions as in the experiment are used in the _{x}_{y}_{z}_{x}_{y}_{z}

Because of the periodic boundary conditions in the

In the _{xy}

As discussed in Subsection 2.2, for both VAWT models (ASSM and ALM), we need tabulated data for lift and drag coefficients. The conventional source for aerodynamic characteristics of symmetrical airfoils, used in VAWTs, is the tabulated data of Sheldahl and Klimas [_{L}_{D}

The turbine center mast is also modeled as a cylindrical body, which applies a drag force on the fluid. The drag coefficient for the mast is considered to be equal to 1.0. In our simulations, the tip speed ratio (defined as the ratio between the velocity of the blades and the incoming flow velocity) of the turbine is taken to be 3.85 to match that of the experiment. The inflow of the simulation is a uniform flow with a velocity of 15 cm/s and a turbulent intensity of 2.5%, as in the experiment.

In this section, the results of our simulations for the above-described case are presented for five different combinations of the turbine and SGS models. Three simulations are performed for the ALM (with the MGM and the traditional Smagorinsky model with _{S}_{S}

_{0}.

As can be observed in the figures, the wake of a VAWT is not symmetric in the spanwise direction (_{b}_{b}

To obtain a more quantitative understanding of the turbine wake, mean velocity profiles in the spanwise direction and at the equator height of the rotor are plotted in _{s}

In

At this point, we are able to draw conclusions regarding the links between the mean velocity field and the turbulence predictions of different models. First of all, as discussed earlier, the fact that the ASSM predicts lower levels of TI in the near wake leads to slower wake recovery. This can be seen clearly in _{S}

By observing the presented results, one can conclude that the ALM, coupled with the MGM or the Smagorinsky model with _{S}_{S}

To show the grid resolution sensitivity of the computational results, we performed simulations for the ALM-MGM and ASSM-MGM model combinations with three different grid resolutions (the resolution is varied in the horizontal directions, which is critical in capturing the blade motions). It was observed that, for the range of resolutions considered here, resolution had only a small effect on the results.

In this study, large eddy simulation (LES), coupled with a turbine model, is used to investigate the 3D unsteady turbulent wake flow behind a vertical-axis wind turbine (VAWT). To model the SGS momentum fluxes, two SGS models are used, namely, the traditional Smagorinsky model and the modulated gradient model (MGM) [

Different combinations of VAWT and SGS models are tested and compared against each other, and a fairly good agreement with the experimental data is observed. It is found out that, in general, the results obtained with the ALM agree better with the measurements and also with the physics of the flow than the ones from the ASSM. The inherent advantage of the ALM in resolving the periodic location of the blade forces allows one to capture the unsteady-periodic nature of the wake and, thus, the blade-wake interaction, which is a characteristic of the aerodynamics of a VAWT rotor. The slight superiority of the ALM over the ASSM was shown quantitatively in the better prediction of the turbulence intensity and, consequently, of the wake recovery, which is very important in designing and optimizing potential VAWT arrays. However, it is worth bearing in mind that the ASSM could potentially be the preferable turbine model in simulations of VAWT wind farms, due to its lower grid resolution requirement. Regarding the SGS models, although both tested models showed reasonable turbulence modeling capabilities, it should be noted that the traditional Smagorinsky model needs its model coefficient to be optimized, depending on the local flow characteristics, while the MGM proved that it can, with its theoretically-determined model coefficient, capture the mean flow and turbulent characteristics of a VAWT wake. This advantage of the MGM is expected to be even more decisive in the case of VAWTs placed in turbulent boundary layer flows, where the MGM has previously been shown to have a superior prediction ability (without turbines) compared with the standard Smagorinsky model [

Potential sources of the observed discrepancies between simulation results and measurement data can be summarized as: (a) inherent limitations of the proposed numerical model; (b) inherent measurement errors in the experiment; and (c) inaccuracies in the tabulated airfoil data. It is worth noting that, in effect, the final results are considerably dependent on the lift and drag coefficient curves as a function of the angle of attack.

As mentioned by Paraschivoiu [

This research was supported by EOS (Energie Ouest Suisse) Holding, the Swiss National Science Foundation (grant 200021-132122 and grant IZERZ0-142236) and the Swiss Innovation and Technology Committee (CTI) within the context of the Swiss Competence Center for Energy Research “FURIES: Future Swiss Electrical Infrastructure”. Computing resources were provided by the Swiss National Supercomputing Centre (CSCS) under project ID s306. Special thanks go to Charles Meneveau for his sage and insightful suggestions. We thank the two anonymous reviewers whose constructive comments helped improve this work. We also thank Jacques Boppe and Fabien Mandrillon from Bogga Ltd. (Lausanne, Switzerland) for the useful discussions during the early stages of this project.

The authors declare no conflict of interest.

Schematic of a cross-section of a vertical-axis wind turbine (VAWT) blade rotating about the turbine axis [

Diagram of forces acting on the blade.

Schematic of a cross-sectional view of the actuator swept-surface in the computational mesh.

Schematic of a cross-sectional view of the actuator line in the computational mesh.

Schematic of y-z view (normal to streamwise direction) of the experimental setup.

Schematic of x-z view of the experimental setup.

x-y view (normal to the blades) of the computational domain.

Contours of the normalized mean streamwise velocity, _{0}, in a horizontal plane at the equator height of the rotor for the actuator swept-surface model (ASSM): (

Contours of the normalized mean streamwise velocity, _{0}, in a horizontal plane at the equator height of the rotor for the actuator line model (ALM): (

(

Comparison of the spanwise profiles of the normalized mean streamwise velocity, _{0}, at different downstream locations: (_{S}_{S}_{S}

Contours of the normalized mean streamwise turbulence intensity (resolved part), _{u}_{0}, in a horizontal plane at the equator height of the rotor for the ASSM: (

Contours of the normalized mean streamwise turbulence intensity (resolved part), _{u}_{0}, in a horizontal plane at the equator height of the rotor for the ALM: (

Contours of the normalized instantaneous streamwise velocity, _{0}, in a horizontal plane at the equator height of the rotor for the two turbine models: (

Comparison of spanwise profiles of the normalized mean streamwise turbulence intensity (resolved part), _{u}_{0}, at five downstream locations: (_{S}_{S}_{S}

The effect of the grid resolution on the spanwise profiles of the normalized mean streamwise velocity and mean streamwise turbulence intensity (resolved part) for the ASSM (