Effects of Airfoil's Polar Data in the Stall Region on The

David Marten Chair of Fluid Dynamics, Hermann-F€ottinger-Institut, Technische Universit€at Berlin, M€uller-Breslau-Str. 8, Berlin 10623, Germany e-mail: [email protected] Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 Alessandro Bianchini Department of Industrial Engineering, University of Florence, Via di Santa Marta 3, Firenze 50139, Italy Effects of Airfoil’s Polar Data e-mail: [email protected]fi.it Georgios Pechlivanoglou in the Stall Region on the Chair of Fluid Dynamics, Hermann-F€ottinger-Institut, Estimation of Darrieus Wind Technische Universit€at Berlin, M€uller-Breslau-Str. 8, Berlin 10623, Germany Turbine Performance e-mail: [email protected] Interest in vertical-axis wind turbines (VAWTs) is experiencing a renaissance after most Francesco Balduzzi major research projects came to a standstill in the mid 1990s, in favor of conventional Department of Industrial Engineering, horizontal-axis turbines (HAWTs). Nowadays, the inherent advantages of the VAWT con- University of Florence, cept, especially in the Darrieus configuration, may outweigh their disadvantages in spe- Via di Santa Marta 3, cific applications, like the urban context or floating platforms. To enable these concepts Firenze 50139, Italy further, efficient, accurate, and robust aerodynamic prediction tools and design guide- e-mail: [email protected]fi.it lines are needed for VAWTs, for which low-order simulation methods have not reached yet a maturity comparable to that of the blade element momentum theory for HAWTs’ Christian Navid Nayeri applications. The two computationally efficient methods that are presently capable of Chair of Fluid Dynamics, capturing the unsteady aerodynamics of Darrieus turbines are the double multiple Hermann-F€ottinger-Institut, streamtubes (DMS) theory, based on momentum balances, and the lifting line theory Technische Universit€at Berlin, (LLT) coupled to a free vortex wake model. Both methods make use of tabulated lift and M€uller-Breslau-Str. 8, drag coefficients to compute the blade forces. Since the incidence angles range experi- Berlin 10623, Germany enced by a VAWT blade is much wider than that of a HAWT blade, the accuracy of polars e-mail: [email protected] in describing the stall region and the transition toward the “thin plate like” behavior has a large effect on simulation results. This paper will demonstrate the importance of stall Giovanni Ferrara and poststall data handling in the performance estimation of Darrieus VAWTs. Using Department of Industrial Engineering, validated CFD simulations as a baseline, comparisons are provided for a blade in University of Florence, VAWT-like motion based on a DMS and a LLT code employing three sets of poststall Via di Santa Marta 3, data obtained from a wind tunnel campaign, XFoil predictions extrapolated with the Firenze 50139, Italy Viterna–Corrigan model and a combination of them. The polar extrapolation influence e-mail: giovanni.ferrara@unifi.it on quasi-steady operating conditions is shown and azimuthal variations of thrust and torque are compared for exemplary tip-speed ratios (TSRs). In addition, the major relevance Christian Oliver Paschereit of a proper dynamic stall model into both the simulation methods is highlighted and Chair of Fluid Dynamics, discussed. [DOI: 10.1115/1.4034326] Hermann-F€ottinger-Institut, Technische Universit€at Berlin, M€uller-Breslau-Str. 8, Berlin 10623, Germany e-mail: [email protected] Lorenzo Ferrari CNR-ICCOM, National Research Council of Italy, Via Madonna del Piano 10, Sesto Fiorentino 50019, Italy e-mail: [email protected]

Introduction Contributed by the Turbomachinery Committee of ASME for publication in the Increasing interest is presently being paid by the researchers JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received June 28, 2016; ﬁnal manuscript received July 4, 2016; published online September 13, 2016. and industrial manufacturers in rediscovering vertical-axis wind Editor: David Wisler. turbines (VAWTs), after most major research projects came to a

Journal of Engineering for Gas Turbines and Power FEBRUARY 2017, Vol. 139 / 022606-1 Copyright VC 2017 by ASME standstill in the mid 1990s [1,2]. At that time, horizontal-axis Case Study. The considered geometry is summarized in Table wind turbines (HAWTs) have been identified as the superior con- 1. Based on the most recent design guidelines for Darrieus rotors cept for wind energy applications. Nowadays, however, the inher- with high chord-to-radius ratios [12], the experimental model was ent advantages of the VAWT concept (independence on wind physically manufactured applying a countercambering to the direction, generator positioned on the ground, low noise emis- blades (based on the turbine’s radius), in order to compensate the sions, and good performance in misaligned flows [3]) may out- virtual camber effect [13]. weigh their disadvantages in some specific application, like, for By doing so, the airfoils were supposed to behave during the

example, the urban context [4–6] or floating platform concepts turbine rotation exactly like NACA0018 profiles. The rotor was Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 [7]. At the present state of the art, however, the global efficiencies tested in a wind tunnel with an undisturbed wind speed of 8 m/s. of Darrieus turbines still lack from those of HAWTs [8], due to Complete details on the experimental tests can be found in Refs. their intrinsically more complex aerodynamics coming from the [12,19]. revolution of blades around an axis orthogonal to the flow direction. This generates a continuous variation of the angle of attack, CFD Simulations. Moreover, the turbine was simulated using which leads to additional phenomena, like, for example, dynamic a 2D unsteady approach developed by some of the authors, whose stall [1]. details are exhaustively reported in Ref. [19]. For completeness, To improve on efficiency and then on effective design solu- the main settings of CFD simulations are anyhow briefly tions, more accurate and robust aerodynamic prediction tools and summarized. design guidelines are needed for VAWTs, for which low-order The time-dependent unsteady Reynolds-averaged Navier– simulation methods have not reached yet a maturity comparable Stokes equations were solved with the commercial code ANSYS to that of the blade element momentum theory for HAWTs’ appli- FLUENT in a two-dimensional form by employing a finite-volume cations [9]. The two computationally efficient methods that pres- method. As discussed in Ref. [19], even if the three-dimensional ently are accredited capable of capturing these unsteady approach is only able to really describe the flow field around the characteristics are the double multiple streamtubes (DMS) theory, turbine (i.e., also the real performance), some applications do not based on the momentum balances, and the lifting line theory indeed require an exact estimation of the overall performance of coupled to a free vortex wake model (LLT) [10]. Even if based on the rotor. In particular, if properly assessed, a 2D approach could the different physical approximations, both methods make use of be successfully applied to the analysis of many relevant issues tabulated lift and drag coefficients of the airfoils to compute the connected to the functioning of the Darrieus rotors, like dynamic blade forces. In this view, it is readily arguable that the accuracy stall, flow curvature effects, and wake interaction with the down- of these models is as higher as the tabulated data actually corre- wind half of the revolution. Moreover, a reliable 2D simulation, spond to the aerodynamic behavior of the airfoils in motion coupled with the simplified models to account for the main sec- onboard Darrieus rotors. ondary effects, could anyhow provide a good estimation of the To do so, the specific submodels to correct polars for dynamic overall performance of the rotor, with an enormous reduction of stall [1], finite aspect-ratio of the blades [11] or flow-curvature the computational cost with respect to a three-dimensional effects [12,13] have been studied and often applied to the models. approach. Beyond submodels, however, the quality of base polars is pivotal The fluid was air, modeled as an ideal compressible gas with to ensure high-quality simulations in low-order methods (both for standard ambient conditions, i.e., a pressure of 1.01 105 Pa and a VAWTs and HAWTs [14,15]). In particular, since the angles of temperature of 300 K. The coupled algorithm was selected for the attack (AoA) range experienced by a VAWT blade is much wider treatment of the pressure–velocity coupling. The governing equa- than that of a HAWT blade, experiencing poststall incidences at tions were discretized by a second-order upwind scheme for spa- almost every tip-speed ratio, the common practice of polar extrap- tial discretization and the bounded second order for time olation beyond the stall point [16] has here a larger effect on simu- differencing. Turbulence closure is achieved by means of the j–x lation results. SST (shear stress transport) transition model [20] in combination Moreover, airfoils in Darrieus-like motion are subject to a con- with the enhanced wall treatment for the computation of the tinuous variation of the incidence, thus, behaving similarly to boundary layer in the near-wall region. Recent studies (e.g., see pitching airfoils, for which the recent works have shown a notable Ref. [21,22]) showed the effectiveness of this transition model in modification in lift and drag coefficient between static wind tunnel performance simulations involving unsteady aerodynamics for tests and performance while in motion [17]. VAWTs, as also confirmed by the wide use in the recent literature The present study is, therefore, aimed at analyzing the effects [19]. of pre- and post-stall data accuracy in polars used for simulations The global convergence of each simulation was monitored by using either a BEM code or an LLT code. Data for these AoAs are considering the difference between the mean values of the torque indeed crucial in Darrieus VAWTs, being the ones commonly coefficient (c ) over two subsequent revolutions normalized by experienced by the airfoils in many azimuthal positions even at T the mean value over the second period of the pair. The periodicity high tip-speed ratios. Focus is particularly given to the data qual- error threshold was set to 0.1% [19]. ity in the “transition” region between the AoAs within the linear In order to allow the rotation of the machine, the simulation zone of the lift curve (attached flow) and the “deep stall” region, domain was divided into two subdomains: an inner circular zone where the airfoils tend to behave like bluff bodies [18]. In that rotating with the turbine, rotating with the same angular velocity zone, dynamic stall has also a major impact on performance, and of the rotor and a rectangular fixed outer zone, determining the the predictiveness of dedicated models is essential to obtain accu- overall domain extent. The two regions communicate by means of rate simulations. a sliding interface. The final dimensions of both domains were

Codes and Data Sets Table 1 Geometrical details of the tested rotor To assess the inﬂuence of poststall data, a case study was ﬁrst selected from the literature [19]. Blades’ number 3 For this test case, a very complete data set was available. The Blades’ shape Straight Blades’ airfoil (attended) NACA 0018 study turbine was tested in the wind tunnel, and the experimental Radius (R) (m) 0.850 results were also used to validate an integrated CFD approach, Chord (c) (m) 0.246 which will be taken as a reference for the comparison with the Blades’ aspect ratio 12 lower order models. Moreover, for the airfoils of interest, full Solidity (r) 0.44 360 deg experimental polars were also available.

022606-2 / Vol. 139, FEBRUARY 2017 Transactions of the ASME defined according to the sensitivity analysis reported in Ref. [19] Ref. [19], is reported in Fig. 2 and will be taken as the reference in order to allow a full and unconstrained development of the tur- for all the analyses in the present study. bine wake. The core flow region of the domain was discretized with an unstructured gird, which allows localized grid refinement and coarsening to efficiently capture the aerodynamic features. In BEM Model. As discussed, two different numerical models the boundary layer region, a structured O-grid was generated with were here considered to assess the influence of poststall polars, a a row of 50 inflated layers. The first element height was chosen in blade element momentum (BEM) model, and a lifting line free þ vortex wake (LLT) method. order to guarantee the y values not exceeding the limit of 1, Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 which is recommended for the SST model. The analyses reported The BEM model is represented by the VARDAR code of the in Refs. [19,23] showed that there is a relationship between the University of Firenze (Italy) [8,11], which has been used in the sizing of the mesh elements and the rotating regime. In particular, last few years to design several industrial models of small Dar- lower TSRs are characterized by larger variations of the incidence rieus turbines. The VARDAR code is based on an improved ver- angle and higher gradients of the flow quantities in the near-blade sion of a double multiple streamtubes approach with variable region. Therefore, the requirements in terms of spatial discretiza- interference factors originally proposed by Paraschivoiu and Del- tion are stricter, since the size of the mesh elements must be claux [23]. The Glauert’s correction for the BEM theory has been reduced as the gradients increase. Two different meshes were then taken into account with the most recent improvements based on adopted for the rotating region for both study cases: a baseline the experimental data [24], together with the corrections due to mesh suitable for TSRs in the stable part of the power coefficient the blades finite aspect ratio, using the Lanchester–Prandtl model curve and a finer mesh necessary in case of lower revolution speed [25]. of the rotor. Both meshes were realized based on the mesh sensi- In order to increase the accuracy of the aerodynamic estima- tivity analyses of Ref. [19] and compared to the previous results tions, the code is embedded with several dynamic stall models of Ref. [12]. The analyses of Refs. [19,23] also showed that the (i.e., those proposed by Berg, Strickland, and Paraschivoiu [1]) required temporal discretization is almost independent on the rev- and with the stream tube expansion model presented in Ref. [1], olution speed. In detail, a constant temporal timestep was identi- although the incidence of this latter on the simulation of small tur- fied in order to limit the Courant number in proximity of the bines like those investigated in this work is reduced. blades. As a result, the angular timestep becomes directly propor- The prediction capabilities of the VARDAR code have been validated during a several-years’ experience in the design of three tional to the revolution speed of the rotor. In the present case 2 study, angular timestep in the range between 0.135 deg and H-Darrieus rotors, having swept areas of 1, 2.5, and 5 m , respec- 0.27 deg were used. tively, and two or three blades, either straight or helix-shaped Figure 1 shows the comparison between experiments and CFD [11,19,26]. The 1:1 models of all the rotors were tested in differ- calculations in terms of power coefficient versus TSR. Very good ent wind tunnels (both with closed and open-jet). In all cases, the agreement is readily noticeable with an only partial discrepancy in code was able to predict correctly both the power curves at differ- correspondence of the curve peak, where CFD overestimated the ent wind speeds and the starting ramps of rotor and is then consid- power output. This phenomenon is probably due to a different ered predictive for the turbine typology investigated in this study. modeling of such a difficult condition, in which the experienced incidence angles range becomes narrow enough to modify the LLT Model. The LLT model used in this study was recently dynamic stall characteristics, or even suppress its onset, with a implemented in the HAWT and VAWT wind turbine simulation consequent variation of torque extraction in the second quadrant. code QBlade [27] by some of the authors. A slight overestimation can be noticed also at low TSRs (where It is based on the nonlinear lifting line theory [28] for the calcu- separations are larger and more difficult to predict). Notwithstand- lation of blade forces and on a free vortex wake method represent- ing this, CFD analyses can be definitely considered representative ing the flow field [29]. Detailed descriptions of the of the turbine behavior. implementation and different contributing models can be found in On these bases, the same numerical setup was applied to the Refs. [30–33]. While the free wake approach has a computational simulation of a single blade of the same rotor. This solution has cost significantly higher than the momentum balance, used in been already exploited in the recent studies [13,15] whenever BEM methods to model the flow field, its inherent advantage lies detailed analyses of the aerodynamics on the airfoils were needed. in a more accurate physical representation of the transient rotor In particular, by doing so, interferences between the blades are wake development. As a result, the LLT model can not only pre- avoided and flow conditions on the airfoils are only due to the dict rotor performance and loads for steady state operation, as the flow–blade interactions. The power coefficient curve of the hypo- BEM, but also for highly transient events. Transient behavior thetic single-blade rotor, simulated with the CFD approach from occurs, for example, in many of the design load cases (DLCs’) in

Fig. 1 Comparison between wind tunnel experiments and CFD Fig. 2 Power curve of a single blade as a function of the TSR simulations for the study turbine (simulated with CFD)

Journal of Engineering for Gas Turbines and Power FEBRUARY 2017, Vol. 139 / 022606-3 the IEC 64100-1 guidelines, during turbine startup, or rotor dis- polars at Re ¼ 150,000 in the range between AoA ¼ 0 deg and placement (as for example, found in ﬂoating platform applica- AoA ¼ 90 deg, which is of particular interest for VAWTs func- tions). However, as transient operating conditions are not tioning. Upon examination of the results, it is apparent that the simulated in this study, both methods can be easily compared and deep-stall region (after approximately AoA ¼ 40 deg) is quite sim- ideally the results between the BEM and the LLT method should ilar between the two data sets, except for a slight underestimation perfectly match, as all calculations are based on the same lift and of XFoil predictions. As discussed by Bianchini et al. [16], data drag polar data. sets are even more similar at very high AoAs, not reported here

only because they are thought to have a scarce influence on the Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 Polars. To assess the impact of polars’ poststall data, a typical performance of the rotors since those AoAs are experienced only design condition was figured out. in the very first motion, during the startup phase of the machines. When designing a new machine, two scenarios can in fact be Major differences in the data sets selected for this study, how- attended. In the first scenario, the designer owns the experimental ever, arise in the near-stall region (13 deg < AoA < 40 deg), where full-360 deg polars of the airfoil he wants to use (after virtual cam- experiments revealed an abrupt drop of performance just after the ber correction [12]): this is indeed the most favorable situation, static stall angle of AoA ¼ 13 deg. but not always possible due to the cost of airfoils’ wind tunnel This drop was not found in the results of the XFoil simulation, tests. In some cases, however, the full-360 deg polars of similar which smoothly connects the maximum lift point to the extrapo- airfoils are available in the literature and can be taken as a refer- lated data in the deep-stall region, where the airfoil behaves simi- ence for similar studies. On the other hand, very often designers larly to a thin plate [37]. For the sake of brevity only the lift make use of tabulated data or panel-method simulations (e.g., polars are plotted here; drag polars show, however, a similar with a software like XFoil [34]) to obtain the characteristic lift behavior. Instead of a sudden drop, the experimental drag data and drag coefficients within the linear regime (i.e., before deep obviously show a sudden increase around the static stall angle stall, where potential flow theory is valid) and then extrapolate while the drag in the XFoil data shows a much smoother distribu- beyond that point using different poststall methods [16]. tion between the pre- and post-stall region (see also Refs. [15,16]. Under these preconditions, in the present study, the same two A similar behavior can be observed at all Reynolds numbers, scenarios were first reproduced: experimental full-360 deg polars with a progressively lower discrepancy between experiment and of the NACA 0018 airfoil were in fact taken from a recent work XFoil, when the Reynold number and the maximum lift increases. [13], whereas “extrapolated data” were recreated using XFoil (Ncrit ¼ 9, free transition on the blade) before the static stall angle (AoA ¼ 13 deg), coupled with the ubiquitous postextrapolation Results: Polar Data Sensitivity method by Viterna and Corrigan [35]. It has to be noticed, however, that, prior to the study, a sensitiv- Based on the two polar sets described above, the power curve ity analysis was made on the poststall extrapolation model of the virtual single-blade rotor was first simulated using both accounting also for the Montgomerie model [36], which was codes and compared to CFD predictions. In these first simulations, available in both the codes. For the purposes on the present analy- dynamic stall was neglected in order to focus the attention solely ses, the two models indeed gave very similar results if applied on on the effect of two different tabulated polar data sets. The com- the same XFoil calculations: for brevity reasons, only the parison is shown in Fig. 4. Viterna–Corrigan model was then considered here. Upon examination of the results, some first outcomes become In both scenarios, polars were available at different Reynolds apparent: 4 5 numbers, ranging from 5.0 10 to 3.5 10 , i.e., for all the Both theories, using the same tabulated polars, give substan- attended aerodynamic regimes on the studied blades at a wind tially coherent results. This is indeed an important outcome speed of 8 m/s. In particular, it is worth remembering that, during as testifies the consistency of the two modeling approaches the turbine functioning, the airfoils undergo a variety of Reynolds over a large range of tip-speed ratios; numbers, in dependence of both the TSR and the azimuthal posi- In case of simulations using numerical data, obtained from tion. For the selected wind speed, however, the available polars XFoil and the Viterna–Corrigan extrapolation, the shape of were fully adequate to cover the entire range of relative speeds on the power curve is more similar to that predicted by CFD, the blades. Both the BEM and the LLT models moreover have even if the optimal TSR is only anticipated, and the power internal interpolation tools to define the proper data in each func- coefficient is overestimated throughout the whole curve. This tioning conditions attended on the rotor (AoA and Reynolds num- behavior suggests that the real performance of the airfoils ber). As an example, Fig. 3 compares the lift curves of the two has been somehow over predicted, as will be shown later on;

Fig. 3 Comparison of lift curves at Re 5 150,000: experiments Fig. 4 Power coefﬁcient versus TSR for the analyzed single- [13] versus XFoil simulations plus Viterna–Corrigan blade rotor: CFD versus predictions with BEM and LLT models extrapolation using either experimental polars or XFoil 1 Viterna

022606-4 / Vol. 139, FEBRUARY 2017 Transactions of the ASME If experimental coefﬁcients are used, the optimal TSR and the peak efﬁciency are more accurately estimated but the shape of the curve is very poorly described in the whole unstable part (i.e., lower TSRs). In particular, it is apparent that both theories now under predict the airfoils’ performance at lower tip-speed ratios, when larger oscillations of the AoA are attended.

These conjectures are supported by the analysis of torque coef- Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 ficient distributions over a single revolution. For example, Figs. 5 and 6 compare CFD predictions to those of the low-order methods using the analyzed polars at two relevant tip-speed ratios of TSR ¼ 2.2 (unstable part of the curve) and TSR ¼ 3.9 (stable part of the curve), respectively. Figure 5 clearly shows that the XFoil þ Viterna polars were only able to provide a torque coefficient trend somehow comparable to CFD upwind (at least in terms of position and magnitude of Fig. 7 AoA and lift coefficient versus azimuthal angle at the peak), even though they were not adequate to model the abrupt TSR 5 2.2 for the analyzed single-blade rotor: BEM model using performance drop after the blade had reached the optimal inci- either experimental polars or XFoil 1 Viterna dence (see Fig. 7). As a result, the torque between # ¼ 90 deg and # ¼ 180 deg is remarkably overestimated as well as the downwind extraction. On the other hand, experimental polars gave a very poor estimation in the largest parts of the revolution. To explain this, Fig. 7 shows the attended incidence angles over the revolution for the BEM code (similar considerations apply to the LLT model but are not reported here for clarity reasons) in combination with the lift

Fig. 8 AoA and lift coefﬁcient versus azimuthal angle at TSR 5 3.9 for the analyzed single-blade rotor: BEM model using either experimental polars OR XFoil 1 Viterna

coefficient that is taken from tabulated polars. As attended, the differences in the torque coefficient trends arise from the abrupt decrease of lift just after stall (also coupled with an increment of Fig. 5 Torque coefficient versus azimuthal angle at TSR 5 2.2 drag) that is present in experimental polars but not found in the for the analyzed single-blade rotor: CFD versus predictions XFoil data. with BEM and LLT models using either experimental polars or Conversely, at TSR ¼ 3.9, AoA variation (Fig. 8) is small XFoil 1 Viterna enough not to exceed the static stall angle, then both data sets actually predict the same performance. This phenomenon has been recently discussed in case of VAWTs in Ref. [15], where the “actual” polars produced by the airfoils in motion on a Darrieus turbine were reconstructed directly from the computed (with CFD) flow fields. The analysis in fact demonstrated that the lift coefficient did not reveal an abrupt stall just after the static stall angle. Tests in Ref. [15] were repeated for a NACA 0018 airfoil and two cambered airfoils at two different chord-to-radius ratios, obtaining a constant agreement on the described trends. This behavior is also consistent with many experimental campaigns on pitching airfoils (e.g., see Ref. [17]), which revealed the lack of abrupt performance drop in airfoils subject to continuous variations of the incidence angle. In particular, these studies showed that, whenever the airfoil just passes through the static stall angle without remaining in that condition for a sufficiently long time, the large performance drop measured in the time-averaged tests in the wind tunnel is bypassed. Moving from these considerations, the possibility of Fig. 6 Torque coefficient versus azimuthal angle at TSR 5 3.9 “smoothing” the experimental lift and drag experimental polars for the analyzed single-blade rotor: CFD versus predictions was explored. This technique has been proposed with very good with BEM and LLT models using either experimental polars or results also in case of HAWTs [38] and more recently used in XFoil 1 Viterna VAWTs analyses [11].

Journal of Engineering for Gas Turbines and Power FEBRUARY 2017, Vol. 139 / 022606-5 More speciﬁcally, it was here thought to replace the data in the transition region after stall with the Viterna–Corrigan extrapolation just after the experimental static stall angle. This procedure, followed in the past in the BEM code whenever small aspect ratios force to recompute the ﬁnite-AR data with the Leicester–Brandt model [11], has shown indeed a good matching between theoretical predictions and experiments.

Smoothed lift polars are reported for completeness in Fig. 9 for Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 some relevant Reynolds numbers within 0 deg and 90 deg to focus the attention on the near-stall region. Using those data, simulations were reproduced using both codes (Fig. 10). Upon examination of the results, it is apparent that the accuracy of simulation was increased notably, both in terms of absolute values and of shape of the curve. In particular, the simulations now benefit from a correct estimation of the stall angle and its associ- ated performance, coupled with a more realistic transition to deep-stall conditions. Fig. 11 Torque coefficient versus azimuthal angle at TSR 5 2.8 This is testified, for example, by the torque profile trend at for the analyzed single-blade rotor: CFD versus predictions TSR ¼ 2.8 showed in Fig. 11, which compares the torque coeffi- with LLT using experimental, XFoil, and smoothed experimental cient, computed with the three different polar data sets. For brev- polars ity reasons, only results of LLT simulations are now presented. Especially, in the downwind half of the rotation, the prediction is notably improved. In the upwind half, the torque is under pre- From the results shown in Fig. 11, it can be concluded that the sudden performance drop in polars after static stall measured in dicted. As the AoAs for TSR ¼ 2.8 lay, however, in the unstable part of the polar curve, dynamic stall is thought to heavily impact the wind tunnel should be smoothed when using that polars in on the upwind rotor performance (not accounted for in any of the simulation codes in order to reproduce the observed behavior of simulations in this first part of the study). rotating airfoils operating under a continuous variation of the incidence angle.

Results: Dynamic Stall Models Previous results clearly showed that the accurate static polars in the stall region are needed to obtain reliable simulations using low-order methods like the BEM or the LLT. To assess the impact of this data, however, it is mandatory also to analyze the effects of dynamic stall [1]. Several experimental and numerical studies have in fact put in evidence that dynamic stall is a phenomenon that is largely present on Darrieus VAWTs blades [39,40]. More speciﬁcally, since the angle of attack changes rapidly and can also exceed the static-stall angle, a whole Darrieus turbine blade can work in dynamic-stall conditions [1]. In these conditions, when the angle of attack becomes too large, the wind ﬂow separates from the leading edge, causing the airfoil to stall or simply lose lift; dynamic stall overdrives aerodynamic lift when the AoA decreases rapidly. On these bases, many efforts have been made to reproduce the effects of dynamic stall in low-order simulation models using either semi-empirical or theoretical approaches Fig. 9 Smoothed lift curves used in the analysis for some rele- [1,26]. vant Reynolds numbers

Fig. 10 Power coefﬁcient versus TSR for the analyzed single- Fig. 12 Power coefﬁcient versus TSR for the analyzed single- blade rotor: CFD versus predictions with BEM and LLT models blade rotor: CFD versus predictions with BEM and LLT models using experimental polars smoothed with the Viterna–Corrigan using smoothed experimental polars (with dynamic stall sub- model after stall models enabled)

022606-6 / Vol. 139, FEBRUARY 2017 Transactions of the ASME Both codes that are used in this study have integrated speciﬁ- cally adapted dynamic stall models. For the purposes of this analysis, in particular, the BEM code made use of the Berg model [39] with an AM of 30, which recently showed a good accuracy for Darrieus rotors [1]. The LLT method of QBlade instead used an unsteady aerodynamics model that integrates a Beddoes–Leishman type dynamic stall model [41] in indicial for-

mulation. Due to the different formulations and the requirement Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 of several empirical time constants in the Beddoes–Leishman formulation, both models are not directly comparable and can yield slightly different results for different operating conditions. How- ever, when these two models were enabled in the simulations, the modifications of the predictions from both codes followed the same trends: for low TSR’s the cP values were slightly increased and for higher TSR’s, around the optimum TSR, cP values were decreased, as shown in Fig. 12. In both cases, using the dynamic stall models lead to an overall Fig. 13 Torque coefficient versus azimuthal angle at TSR 5 2.8 (single-blade rotor): CFD versus BEM and LLT models using much better prediction of both the curve shape and the peak value smoothed experimental polars of the power coefficient. To highlight the effects of the two different dynamic stall models on the torque profiles, Figs. 13 and 14 show, respectively, the azimuthal torque coefficient distributions without and with dynamic stall models activated at TSR ¼ 2.8. If one analyzes and compares in further details these torque profiles, it is worth remarking that the addition of the dynamic stall model was able to give a much more accurate estimation of the performance in proximity of maximum incidence in the upwind region. This better description of stall also led to a more accurate prediction of the torque profile trend between 90 deg and 180 deg, i.e., where the dynamic stall model allows accounting for differences in the lift and drag coefficients when entering or leaving stall conditions. Likewise, the performance predictions in the downwind part, i.e., between 180 deg and 360 deg, also benefit from the inclusion of dynamic stall models into the simulations. It is finally concluded that both theories, if provided with accurate polars and proper dynamic stall submodels, revealed a very good accuracy. This is testified by Table 2, which directly com- Fig. 14 Torque coefficient versus azimuthal angle at TSR 5 2.8 pares some relevant simulations with the different models enabled (single-blade rotor): CFD versus BEM and LLT models (with in terms of variation of the power coefficient values with respect dynamic stall) using smoothed experimental polars to the CFD benchmark, and by Table 3, which instead gives the coefficient of determination R2 (Eq. (1)) of the torque coefficient

Table 2 Relative deviation of cp values between different tested conﬁgurations and CFD benchmark (1-blade rotor)

Exp. polars Exp. polars smoothed Exp. polars smoothed þ dynamic stall

cP Relative deviation (%)

TSR CFD BEM LLT BEM LLT BEM LLT

1.67 0.077 106.5 111.7 28.6 37.7 154.5 48.1 2.23 0.175 81.7 99.4 49.1 25.1 18.9 31.4 2.78 0.347 31.1 59.7 11.0 5.2 1.2 3.5 3.34 0.405 18.5 21.0 12.6 5.4 0.5 2.0 3.89 0.407 4.7 6.1 6.4 6.4 3.7 12.3 4.45 0.373 14.5 7.8 2.1 11.3 8.8 16.1

Table 3 Coefﬁcient of determination of ct curves at TSR 5 2.8 for different tested conﬁgurations

Exp. polars Exp. polars smoothed Exp. polars smoothed þ dynamic stall

TSR CFD BEM LLT BEM LLT BEM LLT

2.80 — 0.19 1.09 0.39 0.67 0.69 0.94

Journal of Engineering for Gas Turbines and Power FEBRUARY 2017, Vol. 139 / 022606-7 2 curves at TSR ¼ 2.8. The R indeed allows calculating an index of R2 ¼ coefficient of determination similarity between the two curves [42] RANS ¼ Reynolds-averaged Navier–Stokes Re ¼ Reynolds number SST ¼ shear stress transport 360P deg TSR ¼ tip-speed ratio c c 2 ðÞT# TCFD;# U absolute wind speed (m/s) #¼0deg ¼ R2 ¼ 1 (1) VAWTs ¼ vertical axis wind turbines 360P deg þ 2 y ¼ dimensionless wall distance Downloaded from https://asmedigitalcollection.asme.org/gasturbinespower/article-pdf/139/2/022606/6175472/gtp_139_02_022606.pdf by Tu Berlin Universitaetsbibl.im Volkswagen-haus user on 20 December 2019 ðÞcT # cT ave #¼0deg Greek Symbols The improved accuracy of this codes, coupled with the well- # ¼ azimuthal angle (deg) known computational efficiency (both models run an entire power 3 curve simulation in only few minutes on a standard personal com- q ¼ air density (kg/Nm ) puter), suggests that they can still represent a standard in the r ¼ solidity industrial design activity, especially for a first selection of the tur- x ¼ turbines’ revolution speed (rad/s) bine configuration. Subscript Conclusions ave ¼ average value In the study, the impact of polar data in the stall region on the References accuracy of Darrieus VAWTs’ simulation using either a BEM or a [1] Paraschivoiu, I., 2002, Wind Turbine Design With Emphasis on Darrieus Con- LLT model was assessed. These two approaches are indeed the cept, Polytechnic International Press, Montreal, Canada. two most used low-order methods commonly applied in the first [2] Aslam Bhutta, M. M., Hayat, N., Farooq, A. U., Ali, Z., Jamil, Sh. R., and Hus- design of new rotors. sain, Z., 2012, “Vertical Axis Wind Turbine—A Review of Various Configura- tions and Design Techniques,” Renewable Sustainable Energy Rev., 16(4), It was shown that the accuracy of data in the stall region of the pp. 1926–1939. polars is crucial to obtain accurate results, which are largely [3] Bianchini, A., Ferrara, G., Ferrari, L., and Magnani, S., 2012, “An Improved dependent on the input coefficients. 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