Aerodynamic Analysis Models for Vertical-Axis Wind Turbines

This work details the progress made in the development of aerodynamic models for studying Vertical-Axis Wind Turbines (VAWT's) with particular emphasis on the prediction of aerodynamic loads and rotor performance as well as dynamic stall 
simulations. The paper describes current effort and some important findings using streamtube models, 3-D viscous model, stochastic wind model and numerical simulation of the flow around the turbine blades. Comparison of the analytical results 
with available experimental data have shown good agreement.

technique de Montr6al has conducted many research on the development of computer codes for .studyingDarrieus rotor aerodynamics (Paraschivoiu, 1988).
The objective of the computer programs is to deter- mine aerodynamic forces and power output of the vertical-axis wind turbine of any geometry at a chosen rotational speed and ambient as well as turbulent wind.Three computer code variants based on the double- multiple streamtube model, stochastic wind and viscous flow field have been developed.The 3DVF viscous flow model based on Navier-Stokes equations analyses the Darrieus rotor in a steady incompressible laminar flow field by solving the Navier-Stokes equations in a cylin- drical coordinates with the finite volume method where the conservation of mass and momentum are solved by using the primitive variables p, u, v, and w (Allet et al.,  1992).Since the ambient wind has been considered to be constant the predicted loads on the blades are identical for each rotor revolution.In order to take the fluctuating nature of the wind into account a 3-D stochastic model has been developed and incorporated first into the double-multiple streamtube model to analyse the effect of atmospheric turbulence on aerodynamic loads (Brahimi et al. 1992).Actually more work are underway to incorporate the stochastic wind into the 3DVF viscous model.For the dynamic stall simulation the computer M.T. BRAHIMI ET AL.   codes developed use empirical model.Although the empirical dynamic stall models used predict well the aerodynamic loads and rotor performance, they are limited to the type of airfoil and motion used in the experiment from which they were derived.Thus, a new code based on the Navier-Stokes equations and uses the finite element method for simulating the dynamic stall around Darrieus wind turbine has been developed (Tchon  et al., 1993).Since 3-D simulation would be very expensive a 2-D simulation has been adopted.The model uses a non-inertial stream function-vorticity formulation (q,0) of the 2-D incompressible unsteady Navier-Stokes equations.The computer code was first validated for the flow around a rotating cylinder then, it was applied to simulate the flow around a NACA 0015 airfoil in Darrieus motion.The present paper presents some devel- opment of the streamtube models, the 3-D viscous model, the stochastic wind as well as. the numerical simulation of dynamic stall.
rotor are calculated by using the principle of two actuator disks in tandem.Three categories of computer codes have been developed (Fig. 1): CARDAA which uses two constant interference factors in the induced velocities calculated by a double iteration, CARDAAV code which considers the variation of the interference factors as a function of the azimuth and CARDAAX code which takes the streamtube expansion into account.These codes have been used at IREQ, Sandia National Laboratories, DAF Indal Co., IMST Marseilles and elsewhere.
For the upstream half-cycle of the rotor, the relative velocity and the local angle of attack as a function of tip speed ration "X" are given by: W

MOMENTUM MODELS
Several aerodynamic prediction models currently exist for studying Darrieus wind turbines and a complete state of the art review including the appropriate references is given by Strickland (1986)  and Paraschivoiu (1988).Generally, the main objective of all aerodynamic models is to evaluate the induced velocity field of the turbine since knowledge of this velocity field allows all the forces on the blade and the power generated by the turbine to be determined.
The first approach to analyze the flow field around vertical-axis wind turbine was developed by Templin (1974) who considered the rotor as an actuator disk enclosed in a simple streamtube where the induced velocity through the swept volume of the turbine is assumed to be constant.An extension of this method to the multiple-streamtube model was then developed by Strickland (1975) who considered the swept volume of the turbine as a series of adjacent streamtubes.Other aerodynamic methods for modeling the wind turbine are based on the vortex theory (Strickland, et al., 1980).
Basically two types of the vortex model have been used: the fixed-wake and the free-wake models.Although these models have the advantage to predict the aerodynamic loads and performance more exactly than the momentum models, they require a considerable amount of computer time.Paraschivoiu (1981) developed an analytical model (DMS) that considers a multiple-streamtube system di- vided into two parts where the upwind and downwind components of the induced velocities at each level of the The normal and tangential forces coefficients are evalu- ated for each streamtube as a function of the blade position using the blade airfoil sectional force coeffi- cients: where the blade airfoil section lift and drag coefficients, CL and CD, are obtained by interpolating the available test data using both the local Reynolds number (Re b W c/v) and the local angle of attack.

STOCHASTIC WIND MODEL
The earlier aerodynamic models for studying Vertical Axis Wind Turbine (VAWT) are based on a constant incident wind conditions and are thus capable of predict- ing only periodic variations in the loads (Veers, 1984), (Malcolm, 1987), (Marchand et al., 1987), (Strickland, 1987) and (Homicz, 1988).As a result, the predicted loads on the blade are identical for each rotor revolution and we have no information about the effect of turbulence on the rotor.Indeed the atmospheric tui'bulence as seen by rotating wind turbines has become an important factor for studying stochastic aerodynamic loads and turbulent flow effects have been identified as one of the major source of rotor blade fatigue life.Continuing the development of the DMS model, a new code (CARDAAS) has been developed to predict loads on VAWT by taking the fluctuating nature of the wind into account.The velocity field of the wind is assumed to be a linear superposition of a steady or mean component and a fluctuating component.The main objective of the wind model is to simulate the turbulent velocity fluctua- tions.It includes both the streamwise and lateral com- ponent of the turbulent velocity.The one dimensional variations of this turbulent wind are introduced by creating time series of the wind velocity at a fixed point upwind the rotor and assuming that the wind speed is constant in a plane perpendicular to the mean wind direction.
The turbulent wind speed downstream of the fixed point is obtained by calculating a time delay in the time series.The decrease in the streamwise velocity as the flow passes through the rotor is taken into account by assuming a linear variation in the streamwise direction.The method used for the 3-D wind model is to simulate wind speed time series at several points in the plane perpendicular to the mean wind direction (Fig. 2).For each point the time series is generated to represent the variation about the mean velocity in the longitudinal and vertical directions.The relative velocity and the local angle of attack are: W2= [Or (V + uf)sinO vfcosO] 2 + [(V quf)cosO vf sirtO]2cos 2 (5) arcsin [((V + uf)cosO vfsinO)cos]/W (6)   where f represents the turbine rotational speed, r the local rotor radius, V the induced velocity for each streamtube as a function of the azimuthal angle 0, uf and vf the fluctuating velocities and the meridional angle.
The fluctuation velocities due to the turbulent wind are represented by a Fourier time-series (Brahimi, 1992): + where V f represents the normalized fluctuation veloci- + ties, V f Vf/ Vi( V f =u Vy), and o "+ represents the normalized turbulence intensity, cr cr/Vi.The un- known Fourier coefficients A + and B+. are given in terms J J of the non dimensional spectral power density, , the dimensionless frequency band A'q and a random phase angle Oj by: Af (26PjA +) 1/2 sin(Oj) (8) The fluctuation velocities are performed by using a Fast Fourier Transform, then the aerodynamic loads are evaluated for each streamtube as function of the blade positions using the total flow velocities.

THREE-DIMENSIONAL VISCOUS MODEL
Since the DMS codes do not take the viscous effects into account a computer code named 3DVF (Allet, 1993) has been developed.This code analyses the Darrieus rotor (Fig. 3) in a steady incompressible laminar flow field by solving the Navier-Stokes equations in a cylindrical coordinates using the finite volume method.The conser- vation of mass and momentum are solved using the primitive variables p, u, v, and w.The effect of the spinning blades is simulated by distributing source terms in the ring of control volumes that lie in the path of the turbine blades.
By using Fig. 3 the relative velocity V ret is calculated by the following equation: Vret Vn en + V'o eo ( Vabs e .)en The only unknown parameters in the above equation (Eq.( 10)) is the absolute velocity, V abs, which is computed by using the Navier-Stokes equations.The discretization method used to solve the governing equations is based on the finite volume method (FVM) proposed by Patankar  (1980).For velocity-pressure coupled flows, a staggered grid system is known to give more realistic solutions and is adopted in the present study.The calculating method based on the control volume approach used here is the widely known "SIMPLER" algorithm.Details of the governing equations with the numerical procedure are given by Allet (1993).The motion of the Darrieus blades are time averaged and introduced through the source terms into the momentum equations.The source terms are valid for all the computational cells that lie in the path of the turbine.

DYNAMIC STALL SIMULATION
Dynamic stall is an unsteady flow phenomenon which refers to the stalling behavior of an airfoil when the angle FIGURE 3 Angles, velocity vectors and forces for Sandia 17-m VAWT.
of attack is changing rapidly with time.It is characterized by dynamic delay of stall to angles significantly beyond the static stall angle and by massive recirculating regions moving downstream over the airfoil surface.
In the case of Darrieus wind turbine, when the operational speed approaches its maximum, all the blade sections exceed the static stall angle, the angle of attack changes rapidly and the whole blade operates under dynamic stall conditions.This increases the unsteady blade loads and structural fatigue (Ham, 1967, Philippe  et al., 1973, Gormont, 1973, McCroskey et al., 1976 and  McCrosky, 1981).Semi-empirical dynamic stall models have already been included in our computer codes based on DMS models as well as on the 3-D viscous model.These are namely the Gormont model (Gormont, 1973), MIT model and Indicial model (Paraschivoiu et al.,  1988, Proulx et al., 1989).Although the dynamic stall models predict well the aerodynamic loads and performance on Darrieus wind turbine, they are limited to the type of airfoil and motion used in the experiment from which they were derived.
A new code for simulating the dynamic stall around Darrieus wind turbine has been developed, it is called "TKFLOW" (Tchon et al., 1993).This code is based on the Navier-Stokes equations and uses the finite element method.Since 3-D simulation would be very expensive a 2-D simulation has been adopted.The model uses a non-inertial stream function-vorticity formulation (q*,o) of the 2-D incompressible unsteady Navier-Stokes equa- tions.The computer code was first validated for the flow around a rotating cylinder (Tchon et al., 1990).Then, it was applied to simulate the flow around a NACA 0015 airfoil in Darrieus motion.
The vorticity transport equation and the stream func- tion compatibility are given by: ---" [(.0 g-V])e.O + 2S.,] (14 The stream function compatibility equation is: The effective viscosity is given by v v + v where v represents the eddy viscosity.The computational mesh used in the TKFLOW is an hybrid one composed of a structured region of highly stretched quadrilateral ele- ments in the vicinity of solid boundaries and an unstruc- tured region of triangular elements elsewhere (Fig. 4).

RESULTS AND DISCUSSION
The prediction of the performance coefficient vs tip speed ratio (TSR) for Sandia 17-m wind turbine using the DMS model is given by Fig. 5. Comparison with vortex model (VDART3) (Strickland et al., 1980) and experimental data (Paraschivoiu, 1988) shows that the prediction by CARDAA code is well improved using CARDAAV and CARDAAX codes.Figs. 6 and 7 show the effect of atmospheric turbulence on the angle of attack and on the aerodynamic torque distribution.Unlike the periodic distribution predicted by CARDAAV, when turbulence is included the angle of attack distribu- tion varies from one revolution to another.Furthermore, the ensemble-averaged aerodynamic torque distributions (Fig. 7) do not coincide with the periodic distribution.
The simulation of the dynamic stall hysteresis loop using 3DVF code with indicial model is given by Fig. 8. Compared to CARDAAV results, predictions in the upwind and downwind regions of the turbine are well compared to experimental data (Akins, 1989).The per- formance predictions at different tip speed ratio (Fig. 9) is also well predicted using 3DVF code.16 14 -90 -75 ,i.-"" '""'..,, RPM=508 ,.";;"': ';"! ":".: i/' ,-" ".,, being used in many places and are judged to be satisfac- tory because of its inexpensive approaches it is very important to include atmospheric turbulence into account especially for large wind turbines.In the case of wind turbine interferences such as wind farms the 3DVF seems to be more suitable since it can compute the flow velocity everywhere in the rotational plane as well as in its vicinity.The dynamic stall simulation using Navier- Stokes equations presents a good tool to predict the unsteady flow for Darrieus motion and more effort are underway to introduce a turbulence model in the present code.

Acknowledgements
In the above aerodynamic codes the model used for dynamic stall is based on semi-empirical methods.. The simulation of the flow around an airfoil in Darrieus motion using Navier-Stokes solver is given by Fig. 10 in term of streamline evolution.Results using TKFLOW code can predict the region where the dynamic stall may occur.

CONCLUSION
Aerodynamic loads and performance of Darrieus wind turbine depend on the flowfield of the wind through the surface swept by the blades.The computer codes devel- oped in this study represent a good tool for calculating the aerodynamic loads and performance of the vertical- axis wind turbines.Although the DMS models are still FIGUREDMS model; CARDAA, CARDAAV, and CARDAAX computer codes.

FIGURE 2
FIGURE 2 Schematic of 3-D wind simulation.