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Energy 35 (2010) 3349e3363

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Energy

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Unsteady Aerodynamics of a Savonius wind rotor: a new computational approachfor the simulation of energy performance

V. D’Alessandro*, S. Montelpare, R. Ricci, A. SecchiaroliUniversità Politecnica delle Marche, Dipartimento di Energetica, Via Brecce Bianche 1, 60131 Ancona, Italy

a r t i c l e i n f o

Article history:Received 27 October 2009Received in revised form8 March 2010Accepted 10 April 2010Available online 15 May 2010

Keywords:SavoniusVertical axis wind turbineComputational fluid-dynamicsWind tunnel testing

* Corresponding author. Tel.: þ39 071 220 4359; faE-mail address: [email protected] (V. D’Ale

0360-5442/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.energy.2010.04.021

a b s t r a c t

When compared with of other wind turbine the Savonius wind rotor offers lower performance in termsof power coefficient, on the other hand it offers a number of advantages as it is extremely simple to built,it is self-starting and it has no need to be oriented in the wind direction. Although it is well suited to beintegrated in urban environment as mini or micro wind turbine it is inappropriate when high power isrequested. For this reason several studies have been carried-out in recent years in order to improve itsaerodynamic performance. The aim of this research is to gain an insight into the complex flow fielddeveloping around a Savonius wind rotor and to evaluate its performance. A mathematical model of theinteraction between the flow field and the rotor blades was developed and validated by comparing itsresults with data obtained at Environmental Wind Tunnel (EWT) laboratory of the “PolytechnicUniversity of Marche”.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Wind turbines are usually classified in two main classes: liftdriven wind turbines and drag driven wind turbines. In the formercase the aerodynamic lift is the force responsible for the rotationwhereas in the latter case the aerodynamic drag is the force whichmakes the turbine to spin.

The Savonius wind turbine is a vertical axis wind turbine(VAWT) created for the first time by the Finnish engineer SigurdSavonius in 1925. As this rotor is classified as a drag driven device,aerodynamic theories developed in order to analyze wind turbinesdriven by lift force, cannot be applied.

In the field of horizontal axis wind turbine as well as lift drivenvertical axis turbine such as the Darrieus rotor, BEM (Blade ElementMomentum) theory [1] finds its best application.

From what above observed, follows that wind tunnel tests andcomputational techniques are the only tools available for studyingSavonius wind rotors.

The aim of this work is to gain an insight into the complex flowfield developing around a Savonius (split-type) wind rotor and toevaluate its performance.

Several numerical analyses have been performed on Savoniusrotor aerodynamic using both DVM (Discrete Vortex Method) andCFD methods. Fujisawa [2] and Fernando and Modi [3] used DVM

x: þ39 071 220 4770.ssandro).

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in the prediction of the rotor performance. It is well known thatthis method does not yield a quantitatively accurate prediction ofthe rotor performance (in comparison with measured data) but itreproduces the main features of the performance curves andflow field as reported in [2,3]. CFD analyses were conducted byShinohara and Ishimatsu [4] and Redchyts and Prykhodko [5]. Inparticular an unstructured finite volume method was used in [5].In this analysis Reynolds Averaged NaviereStokes (RANS) equa-tions were solved for the computation of the turbulent flow usinga third order TVD (Total Variation Diminishing) and a fifth orderRoe flux scheme for the discretization of the convective terms,while diffusive terms were discretized by a central differencescheme; SpalarteAllmaras turbulence model [6] was used. Othernumerical studies about Savonius rotor aerodynamic perfor-mance are available in literature: these simulations were con-ducted in static conditions, varying the rotor angular positionrelative to the wind direction as in [7].

A crucial issue in the analysis of the flow field around theSavonius rotor is the treatment of the fluidesolid coupling and itsmodelling. Modelling the fluid-structure interaction (FSI) isa problem inmany industrial applications. Generally a distinction ismade between three categories of fluidesolid coupling [8]: a one-way solid to fluid reaction (solid motion influences fluid pattern butthe fluid field does not affects the solid), one-way fluid to solidreaction (fluid field moves the solid but the latter does not modifythe fluid pattern) and the two-way coupled interaction wherereciprocal influences are modelled. Obviously, in these simulationsthe accuracy of the model and its computational cost is heavily

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Nomenclature

u velocity vector [m/s]x position vector [m]xG rotation center [m]P Power [W]C dimensionless coefficientf generic time-averaged variablehfi generic angular averaged variableR rotor radius [m]Rmg radius of the rotating domain [m]N number of cells sliding at each time-stepS stage number of RungeeKutta methodH polynomial order of resistant torqueZ Number of time-steps for every roundek unit vector normal to the fluid flow panelM torque [Nm]I moment of inertia [kgm2]p pressure [Pa]k turbulent kinetic energy [m2/s2]T¼ stress tensor [Pa]I¼ identity tensorCp Power coefficientCm Torque coefficientuN wind velocity [m/s]f elliptic relaxation factor [1/s]v2 Reynolds stress normal to the wall [m2/s2]R2 coefficient of determinationDt time-step size

Greek SymbolsG domain boundaryU generic calculus domain

r fluid density [kg/m3]n kinematic viscosity [m2/s]m dynamic viscosity [Pa s]l Tip-speed ratiou angular velocity [rad/s]3 rate of dissipation of TKE [m2/s3]b blockage factorh Kolmogorov length scale [m]f generic fluid-dynamic variable3ijk Ricci-Cubastro tensor

Superscript(n) time-step number(k) generic rotor roundT transpose

SubscriptsR resistantF fluidS solida aerodynamicsin domain inletout domain outletw wallG rotation centermg moving gridi RungeeKutta method substepg grid^ unit vectork summation indexm torquep power

V. D’Alessandro et al. / Energy 35 (2010) 3349e33633350

influenced by the assumptions made on the nature of the fluid(viscous-inviscid) and on the structure (rigid-deformable). Themodel treated in this work is a two-way coupling type; the struc-ture (rotor) was treated as a rigid body while the fluid wasmodelled as incompressible and viscous. The main problem in thefluid-structure interaction modelling is the procedure used to takeinto account the motion of the solid body in the solution of thefluid-dynamics equations. The strategy to solve this problem usedin this work was a SMM (Sliding Mesh Model) approach whileNaviereStokes equations were solved using the finite volume codeFLUENT. The solid body motion was treated solving the secondcardinal equation of dynamics by means of a custom MatLabnumerical algorithm able to import CFD data, calculate the rotorangular velocity and export this variable as input to the CFD code.Time marching of the solution of the second cardinal equation hasbeen executed using an Euler method in the initial steps and a fourstage RungeeKutta (or an AdamseBashfort) scheme in thefollowing steps.

Fig. 1. The Environmental Wind Tunnel of the “Polytechnic University of Marche”.

2. Experimental analysis

Experimental measurements on a full scale Savonius windturbine were carried-out in the Environmental Wind Tunnel (EWT)of the Polytechnic University of Marche, (Fig. 1). The EWT testchamber consists of three main sections: the first is used foraerodynamic tests requiring a uniform velocity distribution anda low turbulence level; the second is used to test reciprocal inter-ference effects between slender bodies; the latter is used to test

wind effects over buildings, structures, orography models whichare subjected to fully developed environmental boundary layers. Aschematic representation of the test section is reported in Fig. 2.The wind tunnel is supplied by a fan having a constant rotationalspeed of 975 RPM, consisting of 16 blades with an adjustable pitchthat ensure a regulated wind velocity in the test section between6 m/s and 40 m/s. Constant Temperature Hot Wire Anemometer(CTA HWA) measurements showed a lack of flow uniformity less

Fig. 2. Schematic representation of the Environmental Wind Tunnel of the “UniversitàPolitecnica delle Marche”.

Fig. 4. Savonius rotor analyzed in this work.

V. D’Alessandro et al. / Energy 35 (2010) 3349e3363 3351

than 2.5% and a turbulence intensity level less than 0.3% on an arealarger than 90% of the test cross section. The wind tunnel is alsoequipped with a compact heat exchanger able to control temper-ature variations inside a range of one Celsius degree. The Savoniusrotor was made in (1:1) scale and placed in the first section of thetest chamber (Fig. 3); the model overall dimensions were 1 mheight and 0.4 m diameter, so the occupied frontal area was 0.4 m2

(Fig. 3). A cross section of the rotor placed inside the wind tunnel isreported in Fig. 4. The test section transversal area is 3.16 m2 andthe solid blockage factor defined by (1):

b ¼ 0:25$Aoccupied

Afree(1)

has a value of 3.2% ensuring the possibility to use experimental datawith a simple blockage factor correction. The experimental facilities

Fig. 3. The Savonius rotor model located in

allow to measure the Savonius mean angular velocity by means ofan incremental encoder and the model torque with a load cell;a ventilated disk brake is first used to slow down the rotor, and thento keep its angular velocity to a constant mean value. The modelwas tested in a range of incoming wind velocities ranging between6 and 12 m/s with a step of a 1 m/s. The measured quantities werecombined in order to obtain the performance parameterscommonly used in the Aerodynamics of Wind Turbines [1]: i.e. thetorque coefficient (2) and the power coefficient (3). Both wereevaluated as function of the dimensionless parameter Tip-SpeedRatio (TSR) (4)

Cm ¼ M12ru2NA$R

(2)

CP ¼ P12ru3NA

(3)

l ¼ uRuN

(4)

where A ¼ 2$R$L is the frontal rotor area, R is the rotor radius and Lis the rotor height. The equations (2), (3) and (4) are also simplylinked by the equation (5).

the EWT and the measurement system.


Cm ¼ CPl

(5)

Fig. 6. The Savonius rotor torque curve.

The reported experimental results refer to a Savonius rotorhaving no twist (0�) and having end plates. Twist angle is defined asthe characteristic angle of the helix. The end plates are devices aptto limit the three-dimensional flow behaviour around the modeltips. This allows the comparison of the experimental results to thenumerical computational data from a 2D domain. The goodagreement between numerical and experimental data reported inthe next sections, confirms as negligible the effects of the residual3D vortices due to the presence of the end plates in the experi-mental model.

In Figs. 5 and 6 are represented the power curve and the torquecurve in dimensionless form; data are collected for differentincoming wind velocities; the blue arrow show the direction ofincreasing angular velocities. The trend exhibited by the curves is ina good agreement with previous authors data [2e4]. The Cp versusTSR curvewas interpolated bya cubic polynomial curveobtaining anR2 value greater than 0.96. The Cm versus TSR curve (Fig. 6) showsinstead a linear behaviour down to TSR values of 0.6 and a curvebending for lower TSR values corresponding, for the experimentalconditions of fixed wind velocities, to low rotational speed veloci-ties. The dimensionless performance curves reported in Figs. 5 and 6show the absence of a meaningful dependence on the Reynoldsnumber (6) referred to the investigated incident wind velocities.

Re ¼ uN$2Rn

(6)

3. Computational model description

The Savonius rotor analyzed in this work is a split-type as clearlyshown in Fig. 4. The computational 2D domain U (Fig. 7), delimitedby the boundary G, is occupied by the fluid in a portion named UF

and by the solid in a portion named US such that:

UFWUS ¼ U (7)

and

UFXUS ¼ B (8)

In UF a sub-domain K is defined by the following expression

K ¼nðx; yÞ˛UF : ðx� xGÞ2þðy� yGÞ2� R2mg

o(9)

Fig. 5. The Savonius rotor power curve.

K represents a rotating frame around a fixed axis normal to therepresented plane and passing for the point ðxG; yGÞ:

Inside the K domain an unstructured triangular mesh havinga curvilinear orthogonal refinement near the walls (Fig. 8) wasadopted, in {UF\K} (the white area inside the rectangle) bothcurvilinear and orthogonal structured elements were used.

4. Flow field mathematical model

The flow field around a Savonius wind rotor at the Reynoldsnumber tested (ranging from 221,000 to 294,000) exhibitsa turbulent behaviour. In selecting the numerical approach to thesimulation of a turbulent flow themain features of turbulencemustbe considered: highly unsteadiness, three-dimensionality, presenceof different size eddies and coherent structures. Moreovera turbulent flow also exhibits random fluctuations of the fluid-dynamic variables on a broad range of scales (in space and time).This feature makes the Direct Numerical Simulation (DNS) unre-alizable for flows of engineering interest. In fact a DNS is based onthe numerical solution of NaviereStokes equations by a computa-tional grid fine enough to allow for all the significant structures ofthe turbulence (in space and in time) to be captured. This impliesusing a grid size determined by the turbulent smallest scale h thatcan be estimated using the Kolmogorov theory [11]. A DNSapproach also requires a computational domain dimension as largeas the largest turbulent eddy.

For this reason in a DNS the number of grid points in eachdirection must be at least L/h (where L is computational domainsize), but this ratio is proportional to Re3/4 [11].

In the three-dimensional computational domain the totalnumber of grid points should therefore scaled as Re9/4. The time-

Fig. 7. Schematic representation of the Computational Domain.

Fig. 8. Mesh adopted near the rotor blades.

Fig. 10. Flow chart of fluid-rigid body coupling algorithm.


step size, on the other hand, must be at least T/th (where T is thetemporal scale of the largest turbulent scales and th is the Kolmo-gorov time-scale). Kolmogorov theory shows that the ratio T/th isproportional to Re1/2 [11] hence the total operations required toperform a DNS is proportional to Re11/4 which becomes easilyprohibitive due to the limitations on computer speed and memoryfor industrial interest problems.

RANS approach has been developed on the idea of decomposingthe velocity components, the pressure and the density (if the flow iscompressible) into two parts: a time-averaged part and a fluctu-ating part (10).

f ¼ fþ f0 (10)

RANS equations have been obtained introducing (10) in theNaviereStokes equations and assuming that the operator used forthe averaging satisfies the following properties: linearity, deriva-tives and average commutation, double average (the averaginghave not effect on the averaged variables) [11]. Using this approachthe CPU e time can be heavily reduced and numerical simulationsof flows with engineering interest can be performed. A RANSapproach implies a reduction of the number details in thecomputation of the flow field details as the turbulent kinetic energyspectra are fully modelled.

Fig. 9. Domain boundaries.

RANS equations for an incompressible flow are here reported(11), (12).

V$u ¼ 0 (11)

vtuþ ðu$VÞu ¼ �Vðp=rÞ þ nV2uþ V$ R¼ (12)

The system (11), (12) must be closed introducing a suitableexpression for the Reynolds stress tensor R¼ ¼ �u05u0 bymeans ofa turbulence model.

A widely used approach uses the “extra-quantities”: turbulentkinetic energy (TKE) k, its dissipation rate 3 and the Boussinesqapproximation (13).

R¼ ¼ nT

�VuDVuT

�� 23k I¼ (13)

with k ¼ ð1=2Þu0iu0i:The unknown variables k and 3 are obtained solving the two

transport equations (14) and (15).

vtkþ ðu$VÞk ¼ Pk � 3þ V$½ðnþ nT ÞVk� (14)

where Pk ¼ nTS2 and S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

qwith Sij ¼ ð1=2Þðvjui þ viujÞ:

vt3þ ðu$VÞk ¼ c31Pk � c323T

þ V$

��nþ nT

s3

�V3

�(15)

with T time-scale

T ¼ min

"max

"k3;6

ffiffiffik3

r #;

1cm

ffiffiffi3

p ak

Sffiffiffiffiffiy2

p #(16)

Fig. 11. Aerodynamic torque treatment in the RungeeKutta sub-steps.

Fig. 12. Tip-speed ratio versus time (Experimental value: l¼ 0.735).


The model used in this work also includes a transport equationfor y2 (Reynolds stress normal to the wall) and for f (elliptic relax-ation factor) representing non-local effects [13]. The turbulentviscosity nT is defined as:

vT ¼ cmy2T (17)

A transport equations was also solved for y2 and f.

vty2 þ ðu$VÞy2 ¼ kf � y2

k3þ V$

hðnþ nT ÞVy2

i(18)

f � L2V2f ¼�cf1 � 1

�2=3� y2=kT

þ cf2Pkk

(19)

L is the length scale

L ¼ CLmax

"min

"k

32

3;

1cm

ffiffiffi3

p k32

Sffiffiffiffiffiy2

p #; ch

v34

314

#(20)

The following values for the calibration constants included inthe turbulence model [13] were used

Fig. 13. Plot 3d of Moment coefficient, tip-speed r

cm ¼ 0:22 c31 ¼ 1:4 c32 ¼ 1:9c ¼ 1:4 c ¼ 0:3 s3 ¼ 1:3
f1 f2CL ¼ 0:23 Ch ¼ 0:23 a ¼ 0:6
The y2 � f model was used for the CFD computations of the flowfield around a Savonius wind rotor because it was designed in orderto represent the tendency of the wall to suppress transport innormal direction without requiring a full second moment closurehence limiting the CPU e time [13]. Moreover this model showedgood performance in the prediction of the flow fields with strongseparations [14] that affects Savonius rotors too.

In order to take in account the rotor rotation in the computa-tional domain the fluid flow equations can be solved following twodifferent approaches: locating the rotor in a non-inertial frame or inan inertial one.

In the first case the solution obtained is strictly connected withthe angle of attack of the rotor (angle between the cord of therotor and the wind direction). Moreover if a constant angularvelocity is imposed the Coriolis force effect on the turbulencescalar quantities is completely avoided. Equation (10), in tensorialnotation, for a two-dimensional flow field in a non-inertial frameis expressed by (21)

atio and time (Experimental value: l¼ 0.735).

Fig. 14. Irregularity degree as function of the mean tip-speed ratio.


vuivt

þ v

vxjuiuj ¼ �1

r

vPvxi

þ nV2ui �v

vxju’iu

’j � 23i3kuuk (21)

� � � �in which P is the pressure term modified in order to include thecentrifugal potential as in (22)

P ¼ p� 12ru2r2 (22)

It’s well know from turbulent flows theory [11,12] that multi-plying (21) by u’i and averaging, a transport equation for TKE can beobtained. Starting from (21) an equation for TKE in non-inertialframes can be derived; in this case it’s easy to note as the termrelated to the Coriolis force in (21) disappeared (23).

23i3ku’iuuk ¼ 23i3ku’iuuk ¼ 0 (23)

hence Coriolis force effect is completely neglected in Eddy-Viscosity Models (EVM) using the transport equation for TKE. Thisis directly related to the scalar nature of the TKE. Second Moment

Fig. 15. Cm� q trend polar plot (the wi

Closure (SMC) can solve this problem producing an increasing inCPU e time respect every EVM.

Locating the rotor in a inertial reference frame with constantangular velocity a not high-fidelity physical modelling is performedbecause the Savonius during its rotation exhibits a strong variationin angular velocity; for this reason in this work an approach basedon an inertial frame with variable rotor angular velocity was used.In particular a SMM approach was adopted in order to take intoaccount the rotation of the rigid body around a fixed axis. Thecomputational grid is decomposed into two sub-grids, where theinner K domain is able to rigidly rotate with respect to the outerone. The rotational sliding grid set-up, used in this work is dis-played in Fig. 7. The commonly used Fluid-Dynamics conservationlaws (i.e. momentum, mass and turbulence quantities) must besuitably modified in order to take into account the grid motion. Atthis purpose a mesh motion flux term, related to mesh motionvelocity ug, was introduced. Hence the transport equation fora generic fluid flow variable f becomes (24) [9]:

nd direction is parallel to q¼ 90�).


vtðrfÞdUþ ru� ug

f$ndG� ðGVfÞ$ndG ¼ 0 (24)

ZUF

IvUF

IvUF

the main problem due to the relative motion between the meshelements of the K and {UF\K} domains is related to the fluxes at theboundaries. The adopted solution procedure provides an appropriatechoice of time-step size (25) obtained by fixing the number of nodesthat slide in each time-step and knowing the distance between twoconsecutivenodeson thegrids interfaceDs, theangularvelocityat theprevious time-step solvedu(n � 1) and the radius of the rotating frameR. In this way there are not overlapped cells at the sliding boundaryand a conformal grid is assured in every time-step solved.

DtðnÞ ¼ Ds$Nuðn�1Þ$Rmg

(25)

RANS equations closed with y2 � f model must be, obviously,completed with a set of boundary conditions. The imposedboundary conditions are:

� a fixed inflow velocity at Gin (26), (see Fig. 9);

uðxÞ ¼ uin cx˛Gin (26)

Fig. 17. Torque release comparison between the approach developed in this work andthe approach with a constant tip-speed ratio e Polar frame (Experimental value:l¼ 0.735).

� a zero normal pressure gradient at Gin;� no-slip at the rotor blades (27);

uðxÞ ¼ uek^ðx � xGÞ cx˛vUs (27)

� Gw is treated as wall in order to reproduce the test chamberof the Environmental Wind Tunnel Laboratory of the Poly-technic University of Marche and to create a virtual windtunnel.

� an outflow physical boundary condition was used at Gout [10].� the turbulent intensity and the hydraulic diameter were fixedon Gin (the same values used in the experimental tests werereproduced).

� At Gin, Gw and on the rotor blades a zero normal gradientcondition for the turbulent quantities was fixed

� For the fluid domain the following initial condition was set:

uðx; t ¼ 0Þ ¼ 0 cx˛UFyK (28)

Fig. 16. Torque release comparison between the approach developed in this work and thl¼ 0.735).

5. Solid body mathematical model

The fluid-structure coupling algorithm presented in this paperprovides, for each time-step, the solution of the fluid-dynamic field(by means of the finite volume code FLUENT) and then evaluatesthe angular velocity of the rotating system. These analyses wereperformed by a numerical algorithmwritten in MatLab language. Aflow chart of the algorithm is shown in Fig. 10.

The fluid flow induces the motion of the solid body by means ofaerodynamic interactions. The structural deformations wereneglected and only the rigid body kinematics was considered;hence the motion of the solid is evaluated integrating the secondcardinal equation of dynamics for 1-DOF (Degree of Freedom)rotating system (29)

e approach with a constant tip-speed ratio e Cartesian Frame (Experimental value:

Fig. 18. Mean tip-speed ratio versus rotation number (Experimental value: l¼ 0.735).


I _u ¼X

MextG;k (29)

k

which is referred to the fixed rotation axis of the system. Torquesacting on the rigid body are the aerodynamic moment Ma andresistant torqueMr. The first is expressed per unit of rotor length by(30), inwhich n is the outward normal unit vector while the secondone can be expressed in a general way by means of a polynomialfunction of the angular velocity with coefficients ak (32).

Ma ¼IvUS

ðx � xGÞ^½ T¼ ðu; pÞ � n�dG (30)

with

T¼ ðu; pÞ ¼ m�Vuþ VuT

�� p I¼ (31)

Mr ¼ XH

k¼0

akuk

! bMr (32)

Fig. 19. Mean power Coefficient versus rotation

In this work a linear function of the angular velocity (posingH¼ 1 in (32)) was used as the resistant torque in order to reproducethe experimental conditions previously described.

As a consequence the equation (29) become:

I _u ¼ LjMaj � jMrj (33)

where L is the rotor height. The initial condition adopted in solving(33) is given by:

uð0Þ ¼ u0 (34)

u0 was assumed to be 16 rad/s in order to achieve a suitable time-step size in the first time-step solved for flow governing equationsnumerical solution. The rotor moment of inertia resulted froma CAD evaluation of the model geometry.

6. Numerical methods

RANS equations coupled with the turbulence model, weresolved by means of a second order Finite Volume Method (more

number (Experimental value: l¼ 0.735).

Table 1Comparison between numerical and experimental results.

Numerical data Experimental data D[%]

l 0.513 l 0.53 �4.04Cp 0.200 Cp 0.20 �0.5

l 0.617 l 0.64 �3.66Cp 0.221 Cp 0.22 �0.48

l 0.735 l 0.76 �3.27Cp 0.240 Cp 0.23 2.5

l 0.847 l 0.90 �5.77Cp 0.234 Cp 0.24 �0.67

l 1.009 l 1.02 �1.09Cp 0.237 Cp 0.24 �2.95

l 1.090 l 1.10 �0.92Cp 0.216 Cp 0.22 �3.7


details can be found in [16]). The discretization of the convectiveterms was obtained bymeans of MUSCL [10] approach. All diffusiveterms were discretized with a central difference scheme and non-orthogonal correction was treated using a Green-Gauss cell basedapproach [10]. Time-integration was performed with a secondorder accurate implicit scheme in order to cancel numerical diffu-sion due to temporal discretization [16]. Pressureevelocitycoupling was solved using a SIMPLEC [17] approach. The fluid-structure coupling algorithm, presented in this paper, implies thatthe aerodynamic torque is known only from previous time-step.This requires an explicit scheme for the time-integration of thesecond cardinal equation of dynamics with a time-step size fixedfor mesh conformity reasons, see (25). An a priori stability analysiscould not be performed hence strong stable time-integrationmethod was needed. For this reason a four stage RungeeKuttatime-integrationmethodwas used for the second cardinal equationof dynamics. The fourth order accuracy guaranteed by the Run-geeKutta method is a surplus, because of the time-integrationscheme used in fluid-dynamic code is second order accurate;anyway this does not heavily increase the computational load. Atwo point and a tree point explicit AdamseBashfort scheme weretested too.

In the first time-steps an Euler method [15] was used, obtainingfrom (33) u(n þ 1).

uðnþ1Þ ¼ uðnÞ þ DtI

�LjM’

ajðnÞ � jMrjðnÞ�

(35)

Fig. 20. Comparison between numerical and experimental power curve.

after solution initialization the RungeeKutta method wasemployed, hence

uðnþ1Þ ¼ uðnÞ þ DtXsi¼1

biKi (36)

with

K ¼ 1IðLjM’

aj � jMrjÞ (37)

and

Ki ¼ f

0@tðnÞ þ ciDt;uðnÞ þ Dt

Xsj¼1

aijKj

1A (38)

the coefficients {aij}, {ci} and {bi} completely characterize thegeneric RungeeKutta method and are collected in the well knownButcher matrix (39).

used in the form given by (40)

The condition (41) ensures the scheme consistency [15] andbeing the RungeeKutta method a one step scheme then theconsistency implies stability and hence convergence [15].

Xsi¼1

bi ¼ 1 (41)

The aerodynamic moment is known only in the time-step (n)and it is unknown in the sub-steps between (n) and (nþ 1): so, inorder to limit the computational load, these values were calculatedby means of an interpolating fourth order polynomial function

Fig. 21. Comparison between numerical and experimental torque curve.

Fig. 22. Comparison between results obtained with time-integration schemes (Experimental value: l¼ 0.735).


using previous time-steps data and the coefficients reported inButcher matrix (40) (see Fig. 11).

The fourth order polynomial coefficients were calculated ondata reported in (42).

X ¼n�

Mði�4Þa ; tði�4Þ

�;�Mði�3Þ

a ; tði�3Þ�;�Mði�2Þ

a ; tði�2Þ�;�

Mði�1Þa ; tði�1Þ

�;�MðiÞ

a ; tðiÞ�o

ð42Þ

Hence they were used for calculating the requested aero-dynamics torque at the time tðiþ1=2Þ (43) (according to the Butchermatrix).�Mðiþ1=2Þ

a ; tðiþ1=2Þ�

(43)

The second cardinal equation of dynamics was also solved usingboth explicit 2-points and 3-points AdamseBashforth schemes [15]expressed in (44) and (45) respectively.

uðnþ1Þ ¼ uðnÞ þ DtðnÞ

2

�3KðnÞ � Kðn�1Þ

�(44)

Fig. 23. Comparison between results obtained with time-

uðnþ1Þ ¼ uðnÞ þ DtðnÞ

12

�23KðnÞ � 16Kðn�1Þ þ 5Kðn�2Þ

�(45)

As shown in the following paragraphs, results are quite inde-pendent from the time marching scheme. A comparison betweenthe approach just described, based on the fluid-rigid body model-ling, and the widely used approach in the aerodynamics of windturbines based on a constant tip-speed ratio is also presented in thefollowing.

7. Numerical results

Savonius wind rotor performance, obtained from numericalcomputations, was evaluated for every angular position occupiedby the rotor using the parameters (2), (3) as function of (4).

Performance parameters were also averaged on the single rotorrotation obtaining (46) and (47)

hli ¼ 12p

Z2p0

lðqÞdq (46)

integration schemes (Experimental value: l¼ 0.735).

Fig. 24. Comparison between tip-speed ratio results obtained with two different grid refinements (Experimental value: l¼ 0.833).


1Z2p

hCPi ¼2p

0

CPðqÞdq (47)

the integral averages were numerically calculated by means ofa second order accurate method (48) and (49).

hli ¼ RuN

(1

ðZ � 1ÞXZ�1

k¼1

12½uðqkÞ þ uðqkþ1Þ�

)(48)

hCPi ¼ 1ð1=2Þru3NA

(1

ðZ � 1ÞXZ�1

k¼1

12½MðqkÞuðqkÞ

þMðqkþ1Þuðqkþ1Þ�)

ð49Þ

In these equations qk is the angular position of the rotor ata generic time-step (a zero value corresponds to the rotor alignedwith flow direction as showed in Fig. 15) and Z is the number oftime-steps in each rotor rotation. Z is fixed at 68 for mean TSRslower than one and at 157 for TSRs greater or equal to one. This is

Fig. 25. Comparison between torque coefficient results obtained with

due to a variation in N caused by the necessity of diminishing time-step size values at TSRs greater or equal than one.

A stable periodic operating condition was reached by theSavonius rotor model as clearly visible in Fig. 12, where the tran-sient behaviour ended in about 3 s. The periodic value of the TSR isdirectly related to the variable torque that a drag based windturbine shows in relation with the angular rotor position.

The unstable aerodynamic torque experienced by the rotor wascalculated and reported in Fig. 13. It is evident that there is a strongvariability during rotor operation that relates to time and TSR. Thisis in agreement with the strong irregularity degree of this kind ofwind turbine that was also calculated.

It is worthy introducing EðkÞ as the vector that contains theangular velocities for the generic (k)-th rotation.

EðkÞ ¼nuðkÞ1 ;u

ðkÞ2 ;.;u

ðkÞZ

o(50)

k ¼ 1;2;.;Nsp with Nsp defined as the number of the rotor rota-tions in the periodic steady state.

Maximum and minimum angular velocity for the (k)-th roundare respectively uðkÞ

max and uðkÞmin:

two different grid refinements (Experimental value: l¼ 0.833).

Fig. 26. Contour of turbulence intensity (Experimental value: l¼ 0.735).


uðkÞmax˛E

��cuðkÞ˛E; uðkÞ � uðkÞmax (51)

�

uðkÞmin˛E

��cuðkÞ˛E; uðkÞ � uðkÞmin (52)

The mean angular velocity for the (k)-th round is defined as:

uðkÞm ¼ 1

2

�uðkÞmax þ uðkÞ

min

�(53)

hence the irregularity degree for the (k)-th rotation is defined as:

iðkÞ ¼ uðkÞmax � uðkÞ

min

uðkÞm

(54)

The irregularity degree for a given functioning point isdefined as

I ¼ 1Ns

XNs

k¼1

iðkÞ (55)

where Ns is the number of complete rotor rotations in which themean performance parameters, previously, defined, exhibita steady behaviour.

The parameter reported in (55) was calculated for all mean tip-speed ratios computed in this work. The trend is reported in Fig. 14.The irregularity degree (55) shows a decreasewith the increasing ofthe mean tip-speed ratio.

Aerodynamic torque coefficient data averaged in time (56)(averaging procedure was performed only in the periodic steadystate) makes it possible to evaluate Savonius rotor torque release

Fig. 27. Contour of velocity magnitude

(see Fig. 9). The integral (56) was calculated with the sameapproach adopted for (46) and (47).

CmðqÞ ¼ 1t2 � t1

Zt2t1

Cmðq; tÞdt (56)

The polar plot reported in Fig. 15 shows a clockwise rotationof the Cm� q trend by increasing the mean tip-speed ratio. Thiseffect is evident for tip-speed ratio values less than one andseems to disappear, or even revert, at tip-speed ratios greaterthan one. This also allows to compare the modelling approachused in this work with the more classical approach based ona constant tip-speed ratio during all the numerical simulation.Figs. 16 and 17 clearly show a strong difference in the torqueprediction during the rotor rotation and moreover the overlap jeteffect is less important in the approach based on a constant tip-speed ratio.

The Savonius rotor mean performance, at the operating point,was evaluated averaging the parameters reported in Figs. 18 and 19once the asymptotic behaviour was reached. The average wasperformed on the samples number corresponding to the steadycondition. In Fig. 18 the effect of the average on the instantaneoustip-speed ratio is also shown. These data were used for numerical/experimental data comparison.

Several operating points were simulated in order to obtaincomplete numerical performance curves. Numerical and exper-imental data are reported in Table 1 and graphed in Figs. 20 and21. In all the simulations reported in this paper the inflow windvelocity has been fixed to 9 m/s and the turbulent intensity

(Experimental value: l¼ 0.735).

Fig. 28. Flow pattern near rotor blades (Experimental value: l¼ 0.735).


at the inflow section was fixed to 0.2% as in the experimentalset-up.

Aerodynamic performance, obtained by the numerical simula-tions, shows a very good agreement with experimental data. Thisallows to use the computational model for a fluid-dynamic analysisof a Savonius wind rotor.

Time marching of the second cardinal equation was performedusing an Euler method in the initial step and a four stage Run-geeKutta or the AdamseBashfort schemes (two points and threepoints) in the following steps. Figs. 22 and 23 show small differ-ences in rotor performance calculation using these three differenttime-integration schemes; the main differences are evident in therun up phase. RungeeKutta scheme offers a better estimation of thepower coefficient of the rotor while a notmeaningful difference canbe noted in the tip-speed ratio calculations between the schemesadopted.

A sensitivity analysis on the results was performed about theinfluence of the yþ (57) value.

Fig. 29. Velocity magnitude profiles in the wake re

yþ ¼ usyn

(57)

where us is the friction velocity defined as us ¼ ffiffiffiffiffiffiffiffiffiffiffisw=r

pand y is the

distance of fist computational cell from the closest wall.A coarser grid near the rotor blades was built in order to

guarantee yþ values ranging from 0.1 to 90 on the blade suctionside and from 0.1 to 40 on the pressure side. The results obtainedon the coarse grid configuration was compared with experi-mental results and with computational results calculated ona finer grid with low yþ values (ranging from 0.1 to 4 on thesuction side and from 0.1 to 2 on the blade pressure side).Comparisons show a little influence of the grid refinement on theprediction of the Savonius rotor performance as enlightened inthe Figs. 24 and 25.

Moreover the computational model developed in this workallow to highlight the following flow features:

1. The wake induced by the Savonius rotor is showed in Fig. 26where a turbulent intensity map is reported. The flow patternenlightens a cyclic behaviour. A velocity magnitude map of thewake is presented in Fig. 27.

2. A quantitatively description of the wake is presented in Figs. 29and 30 where the velocity magnitude profiles in the wakeregion are reported. It’s quite easy to understand as the recir-culation zones produce a velocity decrease.

3. An analysis of sequential frames allows to distinguish that thecentral vortex is continuously supplied by the vortices devel-oping on the advancing blade tip and behind the returningblade.

Fig. 28, in which the maximum value of the velocity magnitudefield was fixed at 12 m/s, shows a zoom of the flow pattern previ-ously highlighted; it shows the presence of some recirculationzones in the suction side of the advancing blade and of the overlapjet. This jet starts from the concave side of the advancing blade anddevelops toward the returning blade. The increment of the pressureon the impinging side reduces the negative contribution of the

gion e rotor aligned with the wind direction.

Fig. 30. Velocity magnitude profiles in the wake region e rotor orthogonal to the wind direction.


returning blade to the overall torque. This is confirmed by theexperimental tests. Moreover counter-rotating vortices are presentbehind the returning blade. Both these structures evolve in thebigger vortex located downstream.

8. Conclusions

An extended wind tunnel testing programwas conducted in theEnvironmental Wind Tunnel of the “Polytechnic University ofMarche” on a split-type Savonius wind rotor. The experimentalfacilities allowed to evaluate rotor performance expressed bymeans of the dimensionless parameters usually used in the aero-dynamics of wind turbines. The use of dimensionless quantitiesallows to verify the Reynolds number effect on the Savoniusperformance. Graph reported in Figs. 5 and 6 underlines theabsence of a meaningful Reynolds number dependence on theinvestigated wind velocities. A computational methodology able tocalculate the flow field around the rotor was also developed. RANSequations (closed using y2 � f turbulence model) were solved inorder to obtain accurate information about the flow field. The rotorblades were treated as rigid bodies and their behaviour weremodelled by means of the Second Cardinal Equation of Dynamic.The experimental data were used to validate the developedcomputational methodology. The comparison of performance dataobtained by numerical simulations and experimental measure-ments shows very good agreement. This suggests the use of thisnumerical method for studying new blades shapes in order toproduce better rotor performance. The results exhibited bynumerical simulations allowed to gain an insight into the flow fieldmean features.

References

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