Numerical Modeling of Stall and Poststall Events of a

Kuldeep Singh1 Department of Electromechanical Engineering, University of Beira Interior, R. Marqu^es D’Avila e Bolama, Covilh~a 6201-001, Portugal e-mail: [email protected]

Jose Carlos P ascoa Associate Professor Department of Electromechanical Engineering, University of Beira Interior, R. Marqu^es D’Avila e Bolama, Covilh~a 6201-001, Portugal e-mail: [email protected]

Numerical Modeling of Stall and Poststall Events of a Single Pitching Blade of a Cycloidal Rotor In the present work, a numerical study is carried out to compare the performance of seven turbulence models on a single pitching blade of cycloidal rotor operating in deep dynamic stall regime at moderate Reynolds number. The investigated turbulence models were: (i) kx-shear stress transport (SST), (ii) kx-SST with c, (iii) transition SST (c–Reh), (iv) scale adaptive simulation (SAS), (v) SAS coupled with transition SST, (vi) SAS with c, and (vii) detached eddy simulation (DES) coupled with transition kx-SST. The wake vortices evolution and shedding analysis are also carried out for the pitching blade. The performance of the investigated turbulence models is evaluated at various critical points on the hysterias loop of lift and drag coefficients. The predictions of the investigated turbulence models are in good agreement at lower angle of attack, i.e., au  20 deg. The detailed quantitative analysis at critical points showed that the predictions of SAS and transition SST-SAS turbulence models are in better agreement with the experimental results as compared to the other investigated models. The wake vortices analysis and fast Fourier transport analysis showed that the wake vortex characteristics of a pitching blade are significantly different than those for the low amplitude oscillating blade at the higher reduced frequency. [DOI: 10.1115/1.4040302] Keywords: pitching blade, deep dynamic stall, wake vortices, vorticity diffusion and stretching, FFT analysis

1

Introduction

Cyclorotor is a fluid propulsive device, which has several pitching blades. These blades turn around an axis, which is perpendicular to the direction of flow to produce thrust and lift. Figure 1 depicts cyclorotor (cycloidal rotor). The cycloidal profile is obtained by a point on circle when it rolls about the axis without slipping. In the case of cycloidal rotor, the blade mounted on the periphery of circular disk follows a cycloidal profile as depicted in Fig. 1 by the trajectory of the blade. The blades of the cycloidal rotor are also oscillating about the pitching axis while following this cycloidal path. The cycloidal rotor generates aerodynamic force by combining the rotational and oscillating motion. Each blade of the cycloidal rotor, in addition to oscillating around a fixed point, also rotates around the center of the rotor. The combined motion around these two points causes the blades to change their respective angles of inclination as described by Leger Monteiro et al. [1], Gagnon et al. [2], Xisto et al. [3]. Thus, at each rotation, the blades cyclically vary their respective angles of attack. The studies such as Yun et al. [4] and Sirohi et al. [5] showed that the direction and intensity of the generated force can be altered by changing the phase and amplitude of the control rod that deliver the cyclic movement of the blades. This remarkable feature allows the use of the cycloidal rotor for propulsion, support, and control of air and water vehicles in both military and civilian applications [6]. Thus, the aerial vehicle, whose main source of propulsion and support is the cycloidal rotor, can take off and land vertically, take-off and land on short runways, hover, and move in any direction perpendicular to the axis of rotation [7]. Moreover, the conventional aero-planes are powered by gas

1 Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 26, 2018; final manuscript received May 10, 2018; published online June 27, 2018. Assoc. Editor: Philipp Epple.

Journal of Fluids Engineering

turbine, which requires afterburner for boosting thrust for the application like take-off from small runways [8–10]. Cyclorotors have been overlooked in the literature because of their very complex kinematics and fluid flow pattern apart from being heavy as compared to conventional screw propellers. There are evidences in the literature, which proves that cyclorotor propellers are efficient as compared to helicopters in terms of thrust and power ratio [11]. These propellers have also better maneuvering capabilities as well as vertical take-off and landing. Technological advancement over the past four to five decades in the materials and computational methods enable the researchers to understand and develop technology for practical applications of cyclorotor propellers. In the present work, single pitching blade of cycloidal rotor is considered under deep dynamic stall conditions at relatively moderate Reynolds number, Re  105. The pitching motion of the blade is found in numerous practical applications apart from cycloidal rotors, viz., helicopter blade, vertical axis wind turbine, highly maneuverable fighters, etc. Obviously, it has been investigated extensively both experimentally and numerically. Wernert et al. [12] identified the main flow features and categorized them into the following four stages: (i) attached flow at low angles of attack, (ii) development of the leading edge vortex (LEV), (iii) the shedding of the LEV from the suction surface of the blade, and (iv) re-attachment of the flow. For a pitching blade, significantly higher value of lift is obtained even beyond the angle of static stall until LEV shed away in the wake region. This is an advantage used by cyclorotors. The increment in the lift is obtained because of the formation and growth of LEV on the suction side of blade. The convection of LEV on the suction surface of the blade introduces nonlinearly varying pressure field and produces transient variation in forces and moments. These forces and moments are found fundamentally different than their static counterparts, Lee and Gerontakos [13]. As LEV passes the trailing edge of the blade and goes to the wake region, fully separated flow is experimentally visualized over the

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Fig. 1 Schematic diagram of cycloidal rotor with pitching blade

suction side of the blade. This is accompanied by sharp fall of lift and increase in drag. During the down-stroke of the pitching blade, the formation of secondary vortex is identified which results in an increase in secondary lift. With the further movement of the blade, when the angle of attack becomes low enough, the flow starts to attach again. Flow remains attached during the rest of the downward and upward movement until the formation of LEV follows the same cycle. The detailed description of these flow features and dynamic stall events can be found in the reviews of McCroskey and Philippe [14] and Carr [15]. Cycloidal rotors operate at the moderate Reynolds number, i.e., Re  105. Poirel et al. [16] reported that for this range of Reynolds number, the flow in pitching airfoils is highly nonlinear. Deep dynamic stall process for such flows are associated with leading edge separation and transition of boundary layer from laminar to turbulent flows, Poels et al. [17], Rudmin et al. [18], and Ouro et al. [19]. Flow under these circumstances is highly sensitive to local adverse pressure gradients. Thus, accurate predictions of flow separation are a challenging task. Selection of adequate turbulence model for numerical analysis on deep dynamic stall process at moderate Reynolds number, which is inherently unsteady and characterized by laminar to turbulent transition and relaminarization, is essential [20–22]. Spentzos et al. [23] investigated dynamic stall over an oscillating blade at Reynolds number in the range of 104–105 using two equation unsteady Reynolds-averaged Navier–Stokes (URANS), k–x model. They reported a reasonable agreement of numerically predicted velocity profiles and pressure distributions with the corresponding experimental data. Martinat et al. [24] investigated Spalart–Allmaras (SA) model, k–e model, shear stress transport (SST) k–x model, organized eddy simulation and delayed detached eddy simulation (DES) for modeling the nonequilibrium turbulence effects on an oscillating blade. The Reynolds number investigated in their study was varied from Re ¼ 105–106. Im and Zha [25] used delayed detached eddy simulation to investigate dynamic stall over NACA0012 airfoil at 17 deg, 26 deg, 45 deg, and 60 deg angle of attack. Tadjfar and Asgari [26] used k–x SST turbulence model to study the influence of a tangential blowing jet in dynamic stall of a NACA0012 airfoil at Reynolds number of 106. Using these turbulence models, although they obtained qualitative flow topology yet none of the turbulence model predicted satisfactorily lift and drag at the peak and during the down-stroke phase. In a series of studies on oscillating blade, Wang et al. [27–29] further investigated the ability of various turbulence models to simulate the dynamic stall at relatively low Reynolds numbers. Wang et al. [27] carried out a numerical study on an NACA0012 airfoil at Re  105 on a two-dimensional (2D) computational domain. The investigated turbulence models were standard k–x and SST k–x. This study revealed that the numerical results predicted by the investigated turbulence models are in fair agreement with the corresponding experimental data at low angle of attack. The numerical results deviated from the experimental results at high angle of attack. Overall, they concluded that the turbulence model SST k–x is comparatively better. In the extension of this 011103-2 / Vol. 141, JANUARY 2019

work, Wang et al. [28] investigated RNG k–e, transition SST (cReh) and DES coupled with SST k–x turbulence models. The authors found that the numerical predictions of DES coupled with SST k–x turbulence model are in good agreement with the corresponding experimental data. Further investigations in this series by the same authors [29] on the same configuration under similar operating conditions, considered transitional turbulence model, transition SST (c-Reh), DES coupled with SST and RNG k–e model. The authors arrived on the same conclusion, which was reported in their previous study [28]. Despite being large body of literature on oscillating blade, till late none of the turbulence model predicts stall and poststall event satisfactorily. Although, direct numerical simulation (DNS) [30] and large eddy simulation (LES) [31] can resolve the issue of turbulence modeling. But the computational power demand and time required for these simulations is very high, which is not economical from industrial point of view. URANS-based turbulence model is still the work horse for industrial requirements. Recently, Egorov and Menter [32] developed the scale adaptive simulation (SAS)-based turbulence model that is found to be a good compromise between LES and RANS turbulence models. The computational demand of SAS based models is equivalent to RANS models, whereas the predicted flow feature capabilities are equivalent to LES simulations, Menter and Egorov [33]. Recently, Zheng et al. [34] compared the performance of SAS and DES turbulence models on a static NACA0021 airfoil at the angle of attack 60 deg. In this study, they reported that the mean flow properties were in good agreement for the SAS simulation as compared to DES studies. They also found that the second-order turbulent statistics in the near wake region was just reasonably in agreement with the experimental results. Hence, in the present work, the recently introduced SAS-based turbulence models, with three variants, along with well explored SST k–x turbulence model are studied. To the authors’ knowledge, this is the first attempt to investigate SAS-based turbulence models for pitching blade operating at moderate Reynolds number. A numerical study is carried out to compare the performance of seven turbulence models. The investigated turbulence models are: (i) kx-SST, (ii) kx-SST with c, (iii) transition SST (c–Reh), (iv) SAS, (v) SAS coupled with transition SST, (vi) SAS with c, and (vii) DES coupled with transition kx-SST. The dynamic stall event of a single pitching NACA0012 blade is captured using the above-mentioned turbulence models and their comparative analysis is presented in detail. In all the previous studies, qualitative trend and flow features were presented. In the present work, a detailed quantitative analysis of prestall, stall, and poststall events is presented. The deviation of numerical results from the corresponding experimental results is quantified and discussed. The numerical analysis is extended to the wake region. It is important to understand the vortex shedding frequencies in the wake region, as in the case of cycloidal rotors, the wake region of the succeeding blade may affect the performance of the preceding blade [35]. The vortex shedding phenomenon for an airfoil undergoing deep dynamic stall conditions is limited in the known literature. The available studies on oscillating airfoils are mainly Transactions of the ASME

focused on high frequency, low amplitude oscillation of airfoil at low Reynolds number (1000–15,000). The practical application of such studies is in micro- and nano-air vehicles (NAVs), which is entirely different than the macro-objective of the current study. The pitching reduced frequency for cycloidal rotors is in the range of 0.05–0.4, whereas micro- and nano-air vehicles are operated at reduced frequency 3, [36]. The operating Reynold number is almost one order smaller for these applications as compared to the present case. Because of the nonavailability of the vortex shedding phenomenon analysis in the literature, a qualitative comparison of the present study was also done with the available studies on low amplitude oscillating airfoil or static airfoils.

2

Problem Description

In the present study, a single pitching blade of a cycloidal rotor is investigated under deep dynamic stall conditions at moderate Reynolds number, see Fig. 2. The transient behavior of the blade surface and unsteady boundary layer and various events associated with flow, i.e., laminar to turbulent transition and relaminarization are presented on a pitching NACA0012 blade. The numerical results obtained from seven turbulence models are presented qualitatively and quantitatively. The computational domain considered in the present study is shown in Fig. 2, following the work of Lee and Gerontakos [13]. A two-dimensional computational domain was investigated for numerical analysis as shown in Fig. 2(a). In the experimental study, Lee and Gerontakos [13] found that the flow is twodimensional and uniform. The nonuniformity in the flow measured by hot wire probe was with-in 64% of the freestream value. Hence, the two-dimensional computational model is sufficient to predict the flow mechanism of the present problem. The length and width of the computational domain were 45C  18C, where C ¼ 0.15 m is the chord length of the blade/airfoil. The NACA0012 airfoil was kept at a distance 15C downward in the direction of flow and at the widthwise center of the computational domain. The origin of the coordinate was located at the leading edge of the airfoil with x, and y in the streamwise and normal directions, respectively. The dimensions of the channel enclosing the airfoil to simulate the mainstream flow were sufficiently large so that the formation of flow vortices near the airfoil is not affected by the walls of the channel. The whole computational domain is divided into two zones, viz., fixed mesh zone and dynamic mesh zone. These two zones

are separated by a circular interface, shown in Figs. 2(a) and 2(b). The dynamic zone oscillates with the sinusoidal mode: a ¼ 10 deg þ 15 deg sin (18.67t). The center of oscillation was at 25% of the chord length from leading edge of the airfoil as shown in Fig. 2(c). 2.1 Mathematical Model. In the present study, the flow is assumed to be two-dimensional, unsteady, incompressible, and turbulent with transition. The equations of continuity and conservation of momentum with the above assumptions are given as @ ðui Þ ¼0 @xi

(1)

!     @ ðui Þ @ ðuj ui Þ @p @ @ui @uj þ  qu0i u0j q þ l þ ¼ @t @xj @xj @xj @xj @xi (2) where u and u0 are the mean and fluctuating velocity components, p is pressure, q is density, t is time of working fluid, l is the fluid dynamic viscosity, x is coordinate, and subscripts i and j denote the directions of the Cartesian coordinates. The term qu0i u0j is the Reynolds stress tensor. In order to solve Eq. (2), the Reynolds stresses term has to be modeled. Boussinesq approximation is used to calculate Reynolds stresses. Seven turbulent models, viz: (i) kx-SST, (ii) kx-SST with c, (iii) transition SST (c–Reh), (iv) SAS, (v) SAS coupled with transition SST, (vi) SAS with c, and (vii) DES coupled with transition kx-SST, are used for the closure of the governing equations. Detailed description of these models can be found in Ref. [37]. 2.2 Boundary Conditions. The boundary conditions imposed on the computational domain are shown in Figs. 2(a) and 2(b). At the inlet, velocity components and turbulence intensity are specified. A turbulence intensity of 0.08% is considered in this study, based on the experimental measurements of Lee and Gerontakos [13]. The blade surface and the walls of the rectangular channel are given nonslip walls. The circular section in Figs. 2(a) and 2(b) is the interface, which separates dynamic zone, i.e., the zone inside the circle containing the blade and the static zone, i.e., the domain outside of the circle. A sinusoidal oscillating motion,

Fig. 2 (a) Computational domain, (b) dynamic mesh zone, and (c) blade pitching arrangement

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at ¼ ao þ amsin(xt), was given to the dynamic zone by writing a user-defined function. The center of oscillation was 25% of the chord length from the leading edge of the blade. 2.3 Numerical Method and Solution Procedure. The continuity and momentum equations (1) and (2) are solved numerically using FLUENT, a commercial CFD solver based on the finite volume method. A second-order upwind interpolation scheme is used to spatially discretize all the governing equations. A second-order implicit scheme is used for temporal discretization. SIMPLE algorithm [38] is used for pressure–velocity coupling. The solutions were considered converged when the residual of the governing equations is lower than 105. 2.4 Detailed Studies on Time and Grid Dependency. In the present study, a nonuniform structured mesh was used to discretize the numerical domain. Grids were generated using the commercial grid generation package ICEM/CFD. Grids were refined near the wall and at entry of the computational domain so that the viscous sublayer is resolved accurately. It was ensured that wall yþ is less than unity in the above region. A typical grid used for the numerical analysis and wall yþ on the surface of blade is shown in Fig. 3. In order to select the optimum grid size for the computational domain, grid-dependent studies were carried. Three grids with cell density of 143,740, 313,530, and 452,480 cells were investigated. O-grids were used in the dynamic mesh zone and in the region surrounding the dynamic zone up to a distance of 5C. The number of nodes on each suction and pressure side of the blade were 100, 150 and 500, respectively, for the three investigated grids. The number of nodes on leading and trailing edge were 60, 80 and 100 for these three grids. Thus, total number of nodes on the periphery of the blade was 320, 460, and 1200, respectively, for the investigated grids. The number of nodes on the leading edge and trailing edge were placed such that Dx varied from 0.25 mm to 0.16 mm for coarse to refined grid. The grid spacing (Dx) along suction side and pressure side varied from 1.45 mm to 0.3 mm for coarse to refined grid.

Fig. 3 (a) A typical grid close to blade surface and (b) wall y1 on the blade

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The turbulence model kx-SST was chosen for the griddependent studies with the assumption that all the grid selected for this model would give grid independent solution for all other turbulence models investigated in this study. All the numerical simulations were carried out at a reduced frequency of k ¼ 0.1 and mainstream Reynolds number of Re ¼ 135,000. Results obtained from the grid dependent studies are shown in Figs. 4(a) and 4(b) for lift and drag coefficient. It can be seen from the figures that the lift coefficient predicted by first grid having 143,740 cells is higher as compared to the other two grids during the upward motion of the blade. Although, the peak value of lift coefficient achieved from first grid before stall is lower than that of the other two grids. Moreover, the variation of lift coefficient during the down-stroke of the blade is entirely different than the other two grids. Similar observations were made for the drag coefficient. It can also be observed from the figure that the lift coefficient obtained from Grid2 (313,530 cells) and Grid3 (452,480 cells) is almost overlapping for the entire cycle of pitching motion. The lift coefficient predicted by these two grids deviates slightly in between 19 deg and 23 deg during upstroke and 20–17 deg during down-stroke. However, the maximum deviation was limited to 3.88%. Hence, the grid with 313,530 cells was selected for further studies. It is not only the grid, which affects the accuracy of the numerical simulation, but also the time-step selected for the simulation. Hence, it is mandatory to carry out time-independent studies. In the present study, three time steps were investigated with step size t ¼ 5  105, 1  105, and 5  106. The predicted values of lift coefficient and drag coefficient for eighth pitching cycle are shown in Fig. 5. It can be observed from the figure that the time steps with stepping size, t ¼ 1  105 and t ¼ 5  106, give

Fig. 4 Grid dependence study using SST-kx turbulence model at Re 5 1.35 3 105: (a) lift coefficient and (b) drag coefficient

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Fig. 5 Time dependence study using SST-kx turbulence model at Re 5 1.35 3 105

almost identical results. Lower time-step would be computationally extensive. Hence, the time-step size of t ¼ 1  105 is selected finally for the further investigation. The uncertainty in the numerical modeling was estimated by the method described by Celik and Li [39] for five grids. The maximum estimated uncertainty in the numerical calculation of lift and drag coefficient was found to be 8.01% and 12.23%, respectively.

followed by the predicted lift and drag coefficients is identical to the experimental results. The comparison of present numerical results with the numerical results of Wang et al. [27] and Rahman et al. [40] indicates that the present numerical results are in agreement with the numerical results available in the literature. It can also be observed that the numerical results of Wang et al. [27] are fluctuating in nature, whereas insignificant fluctuation can be observed in the present numerical simulation despite of same turbulence model. The present numerical results are closer to the experimental results compared to those reported by Wang et al. [27]. The possible reason for this variation is grids and time-step. In the present numerical simulation, structured grids are used whereas in the study of Wang et al. [27], unstructured grids were used. Time step considered in the present study is 105, whereas time-step in the study of Wang et al. [27] was 7  103, which is almost three orders higher. The maximum deviation is found in the numerical results of Rahman et al. [40] obtained from Realizable-ke model, which might be due to the difference in turbulence model, grid size and time used in the study of Rahman et al. [40]. In order to confirm the predictions of computational model, another test case was considered for validation following the work of Wernert et al. [12]. The analysis was carried out for the experimental conditions of Wernert et al. [12]. For this test case, blade oscillates with the sinusoidal mode: a ¼ 15 þ 10 sin(41.89t). The center of oscillation was at 25% of the chord length from leading edge of the blade. The mainstream Reynolds number and reduced frequency for this case were 3.73  105 and 0.15, respectively. The nondimensional velocity is plotted for upstroke angles, au ¼ 22 deg and au ¼ 24 deg in the normal direction to the blade axis at specified angles at x/C ¼ 0.5 in Figs. 6(c) and 6(d). In the study of Wernert et al. [12], the results were presented in the format of contours and streamlines. However, the same group utilized the experimental results of Wernert et al. [12] for the validation of numerical study in subsequent work of Spentzos et al. [23]. Hence, the experimental and numerical data used for validation in the present study were extracted from the work of Spentzos et al. [23]. It can be observed from Figs. 6(c) and 6(d) that the present numerical results using transition SST-SAS turbulence model are in good agreement with the experimental results of Wernert et al. [12] and numerical results of Spentzos et al. [23] obtained from URANS, k–x model. Based on the analysis, it can be concluded that the present numerical methodology is capable to reproduce the results reported in the literature and analysis can further be used for comparative assessment of various turbulence models.

3 2.5 Validation of the Computational Model. Although throughout this paper numerical results are compared and evaluated against the experimental results of Lee and Gerontakos [13] yet in this section, numerical results with SST-kx turbulence model are compared with the corresponding experimental results of Lee and Gerontakos [13], numerical results (SST-kx model) of Wang et al. [27], and numerical results (Realizable-ke model) of Rahman et al. [40]. The predicted lift and drag coefficients are shown in Fig. 6. It can be seen from the figure that the predicted lift coefficient is in fair agreement with the experimental results when the angle of attack, au < 21 deg, whereas predicted drag coefficient is in good agreement at angles, au < 15 deg during the up-stroke. Significant deviation in the predicted lift-coefficient can be observed from au ¼ 21 deg to ad ¼ 20 deg down-stroke. Similarly, predicted drag coefficient is not in agreement with the experimental results from au ¼ 15 deg to ad ¼ 18 deg. This particular region belongs to stall and poststall events. The flow is highly separated over this range of angle of attack. It indicates about the inability of SST-kx turbulence model to predict such flow. These observations are consistent with most of the numerical studies. It can also be observed from Fig. 6 that the qualitative trend Journal of Fluids Engineering

Results and Discussion

The results obtained from numerical study using seven turbulence models for aerodynamic coefficients and nondimensional wake velocity are compared with the experimental results of Lee and Gerontakos [13]. A qualitative as well as quantitative comparison of lift and drag coefficients is presented at few critical points on the hysteresis loop. A quantitative comparison of nondimensional wake velocity is documented at eight selected angle of attacks for which experimental results were available. The main flow features, e.g., laminar to turbulent transition and relaminarization of flow on suction surface, are presented. A detailed analysis is carried out on the vortex stretching and shedding downstream of airfoil. Although the experimental results are not available for wake vortices yet the qualitative discussion is presented in the comparison with the low amplitude pitching airfoils at higher reduced frequencies and lower mainstream Reynolds number. The frequency of wake vortices is accessed using fast Fourier transformation (FFT) and documented in Sec. 3.6. 3.1 Flow and Aerodynamic Coefficients in Hysteresis Loop. Lift and drag coefficients for eighth pitching cycle for SAS and transition SST-SAS turbulence models are presented along with the experimental data of Lee and Gerontakos [13] in Fig. 7. JANUARY 2019, Vol. 141 / 011103-5

Fig. 6 Comparison of ((a) and (b)) lift and drag coefficient obtained from (SST-kx model) with corresponding experimental results of Lee and Gerontakos [13], numerical results (SST-kx model) of Wang et al. [27], Comparison of ((c) and (d)) nondimensional velocity in the normal direction to the axis of blade for upstroke angles 22 deg and 24 deg with the experimental results of Wernert et al. [12] and numerical results of Spentzos et al. [23]

For quantitative analysis of lift coefficient, flow results are compared at eight critical points with the experimental results. These points are marked in the lift and drag hysteresis loop and shown in Figs. 7(a) and 7(c). For lift coefficient, these points are corresponding to ‹ formation of LEV (au  12 deg), › spread of flow reversal (au  21 deg), fi turbulent break-down (au  22 deg), fl stall (au  24 deg),  end of stall (ad  24.5 deg), – formation of secondary vortex (ad  24.9 deg), † beginning of flow reattachment (ad  14 deg), and ‡ fully re-attached flow (ad  1 deg). 3.1.1 Assessment of Numerical Modeling for Prestall and Stall Event. The most significant characteristics of a pitching blade is, delay in the inception of stall as compared to static counterpart as reported by Raffel et al. [41] and Lee and Gerontakos [13]. All the turbulence model investigated in the present study captured this delay very well. It can be observed from Fig. 7(b) that the angle of attack for dynamic stall conditions varies from 23 deg to 25 deg for the investigated turbulence models whereas static stall angle observed in the experimental study of Lee and Gerontakos [13] was 13 deg. It can also be seen from Fig. 7(b), during upward stroke, numerical predictions of lift coefficients collapse to a single line up to au  21 deg, which indicates that the predicted slope of angle of attack-lift curve is captured well by the investigated turbulence models. Thereafter, distinct trend in the prediction of lift coefficient is observed. The maximum value of lift coefficient predicted numerically by kx-SST, kx-SST with c, SAS, transition 011103-6 / Vol. 141, JANUARY 2019

SST-SAS, transition SST-DES, SST-c-Reh, and SAS-c models, is: 2.25, 2.26, 2.26, 2.25, 2.28, 1.86, 2.06, respectively, corresponding to an angle of attack (au): 23.36 deg, 25 deg, 23.5 deg, 23 deg, 25 deg, 22.3 deg, and 23.2 deg, respectively. Whereas, the experimental values reported by Lee and Gerontakos [13] for maximum lift coefficient was 2.41 at an angle of attack of au ¼ 24.7 deg. These values are corresponding to point Clfl in Table 1. It can be seen that all the turbulence models underpredict the lift coefficient. The maximum deviation is observed in the numerical results obtained from SST-c-Reh and SAS-c models. All other turbulence models predicted almost a similar value for lift coefficient. The predicted value of angle of attack corresponding to the onset of stall is lower as compared to the experimental value for all the turbulence models except transition SST (c-Reh) and SST-DES. For these two turbulence models, the numerically predicted angle of attack is slightly higher than the experimentally measured value. Similar to the lift coefficients, quantitative analysis of the drag coefficient is done at five critical points as shown in Fig. 7(c). The value of drag coefficient corresponding to these five points is given in Table 2. It can be observed from the tabulated data that the drag coefficient prediction of SAS and transition SST-SAS turbulence models are compared well with those of the experimental values. Although, the hysteresis loop of drag coefficient exhibits some fluctuation around the experimental values of drag coefficient for all the turbulence models when angle of attack varies from au  18 deg to ad  17 deg. The angle of attack corresponding to drag-stall compares well with the experimental data. The lowest angle of attack corresponding to stall, au ¼ 24.4 deg, Transactions of the ASME

Fig. 7 Comparison of numerical results for ((a) and (b)) lift coefficient at ‹ formation of LEV (au  12 deg) › spread of flow reversal (au  21 deg) fi turbulent break-down (au  22 deg) fl stall (au  24 deg)  end of stall (ad  24.5 deg) – formation of secondary vortex (ad  24.9 deg) † beginning of flow re-attachment (ad  14 deg) ‡ fully re-attached flow (ad  1 deg) and ((c) and (d)) drag coefficient obtained from SAS and transition SST-SAS turbulence models at ‹ formation of LEV (au  12 deg) › drag coefficient cross-over (au  17 deg) fi turbulent break-down (au  22 deg) fl stall (au  24 deg)  end of stall (ad  24.5 deg) Table 1 Comparison of numerically predicted lift coefficient with experimental results of Lee and Gerontakos [13] at critical points

Cl‹ Cl › Cl fi Cl fl Cl  Cl – Cl † Cl ‡

Table 2 points

Cd‹ Cd › Cd fi Cd fl Cd 

Lee and Gerontakos [13]

kx-SST

Transition-SST

SAS

SST-SAS

SST-DES

kx-SST-c

SAS-c

1.15 1.97 2.04 2.41 1.45 1.41 0.77 0.04

1.11 2.09 2.1 2.25 1.18 1.35 0.52 0.06

1.09 1.7 1.79 2.26 1.29 1.34 1.1 0.08

1.13 1.89 1.9 2.26 1.39 1.8 0.8 0.02

1.17 2.05 2.1 2.25 1.28 1.24 0.86 0.03

1.1 1.8 1.82 2.28 1.28 1.35 1.09 0.03

1.11 1.81 1.84 1.86 1.16 2.06 0.62 0.18

1.11 1.98 2.03 2.06 1.6 2.01 0.94 0.02

Comparison of numerically predicted drag coefficient with experimental results of Lee and Gerontakos [13] at critical

Lee and Gerontakos [13]

kx-SST

Transition-SST

SAS

SST-SAS

SST-DES

kx-SST-c

SAS-c

0.15 0.21 0.46 0.93 0.5

0.13 0.26 0.8 1.09 0.61

0.12 0.186 0.36 1.13 0.39

0.15 0.3 0.47 1.04 0.45

0.13 0.22 0.45 0.94 0.31

0.11 0.19 0.38 1.08 0.41

0.11 0.26 0.5 1.02 0.56

0.12 0.17 0.51 1.02 0.42

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was observed for kx-SST turbulence model and the highest, au ¼ 25 deg, for transition kx-SST. The experimentally measured value of angle of attack corresponding to stall was 24.6 deg, which is predicted reasonably well by all other turbulence models investigated in the present study. 3.1.2 Assessment of Numerical Modeling for Poststall Event. The poststall flow features are even more complicated because of formation of secondary vortices. The flow mechanism is discussed in Sec. 3.3. Due to the formation of secondary vortices, the lift coefficient increases after the stall condition, i.e., from point  to – and afterward lift coefficient continuously decreases as pitching angle decreases down-stroke. All the turbulence models captured secondary rise in the lift coefficient. But after secondary fall in the lift coefficient, all the turbulence models exhibited oscillating lift coefficient in the region, 24.9 deg  ad  12 deg. Although the lift coefficient oscillates around the experimentally measured value of lift coefficient yet amplitude of oscillation is different for different turbulence model. As the angle of attack decreases further, because of flow relaminarization, oscillation in lift coefficients decreases and the difference among predicted lift coefficient by various turbulence models also decreases. During the down-stroke movement of the blade for numerical prediction of SAS, turbulence model is the closest to the experimentally measured value among all the investigated turbulence models.

Fig. 8

It is well documented in the literature that the prediction of stall and poststall event is not very well captured by reported RANS and URANS turbulence models [19–21]. In the present numerical studies, SAS, transition SST-SAS, and SAS-c turbulence models were investigated in addition to the others, which are not reported yet for pitching blade operating in deep-dynamic stall conditions. Although, the turbulence models such as kx-SST, transition kxSST, SST-cReh, and SST coupled with DES were investigated and the deviation from the experimental results was reported. But the quantification of deviation from the experimental results is essential for rotor designer, which might provide a fair idea to the designers of cycloidal rotor, vertical axis wind turbine, helicopter blades, etc., about the uncertainty aroused because of the turbulence modeling. Based on which, one might take a decision whether to go for computationally extensive LES/DNS studies or URANS results are within the acceptable limits. A summary of the predicted lift coefficient corresponding to the selected eight critical points are given in Table 1. It can be analyzed from the data presented in Table 1 that the numerical results obtained from SAS and SST-SAS turbulence model are in better agreement with the experimental results. Similar to the lift coefficients, the secondary rise and fall of the drag coefficient was also observed in the numerical predictions which was not witnessed in the experimental results. This is due to the formation and shedding of secondary vortices, which is

Intermittency-contour chronology using transition SST and transition SST-SAS turbulence models

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discussed in detail in Sec. 3.3. These vortices are also responsible for poststall fluctuating drag coefficient. The drag coefficient predicted by kx-SST-c-Reh model overshoots after experiencing poststall lowest value of 0.56 at ad ¼ 23.8 deg and attains a peak value of 0.72 at ad ¼ 17.9 deg. This behavior significantly deviated from the experimental results as well as from the numerical results obtained from all other turbulence models.

3.2 Laminar to Turbulent Transition and Relaminarization on the Suction Surface. A pitching blade in deep dynamic stall conditions at moderate Reynolds number undergoes laminar to turbulent transition with the increase in the angle of attack during upstroke movement and poststall flow relaminarization during down-stroke movement of the blade. Accurate prediction of onset of laminar to turbulent transition, transition length, and relaminarization can improve the accuracy of the numerical predictions. Vizinho et al. [42] through their novel phenomenological transition model showed that the numerical predictions of the newly developed transition model are better than that using SA and SAc-Reh turbulence models. Despite the accurate prediction of transition onset by SA-c-Reh turbulence model, the transition length was not in the agreement with the experimental results for the numerical analysis carried over 6:1 prolate-spheroid. The pitching

blade considered in this study also undergoes a complex flow phenomenon with laminar to turbulent transition and relaminarization. Hence, in this section, analysis is focused on laminar to turbulent transition and relaminarization. The transition SST (c-Reh) and transition SST-SAS models are designed to model flow transitions with the intermittency (c) and momentum thickness Reynolds number (Reh) being solved as transport quantities. The specified value of intermittency at inlet of the computational domain is unity and a zero flux condition at the blade surface. Therefore, the flow is fully turbulent for a unity value of the intermittency and fully laminar for zero value of the intermittency. Thus, the flow transition can be identified from the contours of intermittency as the value between zero to unity shows the occurrence of flow transition. Figure 8 shows the contours of the intermittency for two transition turbulence models, viz., transition SST (c-Reh) and transition SST-SAS. The onset of transition is identified at an angle 12 deg as can be seen in Figs. 8(c) and 8(o). Prior to this, flow remains laminar close to the airfoil surface as shown in Figs. 8(a), 8(b), 8(m), and 8(n). These values are close to the experimental value (au ¼ 12.9 deg) of Lee and Gerontakos [13] corresponding to the transition. As the angle of attack increases to au ¼ 14 deg, the predicted transition by SST turbulence model restricted to location 0.2  x/C  0.6 as can be seen in Fig. 8(d). On the contrary to this, transitional flow can be

Fig. 9 Variation of skin friction coefficient for kx-SST-c-Reh turbulence model ((a) and (b)) and transition SST-SAS turbulence model ((c) and (d)) at various angles of attack

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seen over the entire airfoil in the numerical prediction of SSTSAS turbulence model for this angle of attack, Fig. 8(p). The fully turbulent flow predicted by numerical simulations was earlier as compared to the experimental observation. The experimental measurement showed fully turbulent flow at a  21 deg, whereas numerically predicted value is au 16 deg, Figs. 8(e) and 8(q). During the down-stroke movement of the airfoil, flow remains fully turbulent for ad  2 deg, Figs. 8(g)–8(k), 8(s), and 8(v). For SST-SAS turbulence model, flow relaminarization takes place at ad ¼ 2 deg, shown in Fig. 8(w), which is close to the experimentally observed value ad ¼ 1.1 deg. Flow relaminarization predicted by the transition SST (c-Reh) turbulence model was at ad ¼ 1.7, Fig. 8(l). The numerical prediction of flow transition of SST-SAS turbulence model is in better agreement with the experimental results as compared to that of the transition SST (c-Reh) turbulence model. The transition is also captured in the plots of skin friction coefficient as shown in Fig. 9, plotted for transition SST (c-Reh) and transition SST-SAS turbulence model. During the upward movement of the airfoil, laminar to turbulent transition identified by fluctuating skin friction coefficient for angle of attack, 12 deg  au  15 deg as shown by circle in Figs. 9(a) and 9(c). The relaminarization is captured in Figs. 9(b) and 9(d) during the downstroke movement of the airfoil at ad  2 deg. The similar conclusions were arrived from Fig. 8 also. 3.3 Separation and Re-Attachment of the Flow Over the Suction Surface of the Pitching Blade. The main flow features for a pitching blade in deep dynamic stall conditions at moderate Reynolds number are: (i) formation of thin flow reversal layer near the trailing edge and upward spread of flow reversal, (ii)

formation of laminar separation bubble (LSB) near the leading edge and eventually formation of LEV, and (iii) triggering of stall by turbulent separation of LEV at a short distance downstream of the leading edge as reported by Lee and Gerontakos [13] and McCroskey et al. [43]. In order to analyze the flow features over suction surface of pitching blade, the Q-criterion proposed by Hunt et al. [44] is used. This method is based on the second invariant of the velocity gradient tensor which defines an eddy structure as a region of positive second invariant of Q. The expression for Q is given in the below equation: Q¼

 1 kXk2  kSk2 2

(3)

where kk denotes the trace of tensor, X is vorticity tensor, and S is strain rate tensor. Thus, according to Q-criterion, a region with positive Q indicates about the region where rotation has overcome the strain. Q-contours are presented for turbulence models SAS and SAS coupled with SST as the performance of these turbulence models are not investigated earlier for an airfoil pitching in deep-dynamic stall regime. A chronology of Q-contours for SAS and SST-SAS turbulence models at various angles of attacks is shown in Figs. 10 and 11. It can be seen from Fig. 10 that during the upstroke, flow remains attached at angle of attack au  13 deg except the trialing edge where flow reversal is identified even at lower angle of attack. As the angle of attack increases beyond 13 deg, a very thin layer of reverse flow is identified at the bottom of thick turbulent layer on the suction surface of pitching blade. In Fig. 10(a), velocity vector and x-velocity (U) contours are shown in the offset along with Q-contours. Flow reversal is identified up to downstream location X/C ¼ 0.82. With increase in the angle of

Fig. 10 Q-contour chronology using SAS turbulence model

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attack during the upstroke movement of pitching blade, the thickness and span toward the leading edge of the flow-reversal zone increase. The flow reversal identified by SST-SAS appears at an angle of attack au  10 deg. This layer of reversed flow near the suction surface is highly unstable and easily broken down into small vortices, which can be seen in Figs. 11(a) and 11(b). In the experimental study, Lee and Gerontakos [13] reported occurrence of flow reversal at au ¼ 12.9 deg at a location S/C ¼ 0.84. The numerical predictions of SAS turbulence model are in better agreement with the experimental findings as compared to SSTSAS turbulence model for this part. It should also be noted down that Q-contours presented in Figs. 10 and 11 are not for the same angle of attacks. The plots are extracted at an angle of attack where significant change in flow mechanism is noticed. Upstream of the flow reversal, boundary layer remains attached. With further increase in angle of attack, formation of LSB is identified at au  18.2 deg as can be seen in Figs. 10 and 11(c). The flow downstream of LSB is found to remain attached. As the blade continued to pitch up, appearance of LEV is identified at angle of attack, au  20 deg at the downstream location X/C ¼ 0.2. It covers 25% of the suction side of the blade. With further increase in the angle, LEV rapidly grows up and convects downstream. The increase in the lift coefficient observed in Fig. 7 is due to formation and convection of LEV. The detachment of LEV from the suction surface of the blade is identified in Figs. 10(f) and 11(h) for SAS and SST-SAS turbulence models, respectively. These are the stall conditions predicted by these turbulence models. Both, the SAS and SST-SAS turbulence models predicted early detachment of LEV compared to the experimental counterpart, which resulted in lower lift coefficient corresponding to the stall

condition. Further upstroke movement of blade resulted in complete detachment of LEV and its shedding into wake region. During the stall, flow remains fully separated and characterized by presence and rearward convection of secondary vortices. Poststall, formation of trailing edge vortex is identified at an angle of attack, au ¼ 23.9 deg as can be seen in Figs. 10(g) and 11(h). Formation and growth of small secondary vortices can also be observed in Figs. 10(i), 10(j), and 11(i)–11(k), which yielded oscillating lift and drag coefficient witnessed in Fig. 7. Poststall, a complex flow mechanism was observed. Formation of secondary vortices and their shedding continued as observed from Figs. 10(k)–10(t) and 11(l)–11(v). As angle of attack (ad) decreases to  6 deg, a gradual front-to-rear reattachment of the flow is observed. Fully attached flow was observed at ad  1 deg. The experimentally observed angle of attack for flow reattachment was 1.1 deg. From the above discussion, it can be concluded that major flow features are identified by SAS and SST-SAS turbulence models. The numerical results for the critical points are in fair agreement with the experimental observation. However, the accurate prediction of stall condition and near poststall flow features was not captured by both the turbulence models. 3.4 Assessment of Numerical Modeling in the Wake Region. The nondimensional velocity in the wake region is shown in Fig. 12. The velocity was extracted at a point one chord downstream of the trailing edge and compared with the corresponding experimental results of Lee and Gerontakos [13]. The numerically predicted velocity by SST-c-Reh and SAS-c turbulence model almost overlapped with the prediction of transition SST (c-Reh) and SAS turbulence model, respectively. Hence, the numerical

Fig. 11 Q-contour chronology using turbulence model SST coupled with SAS

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Fig. 12 Velocity in wake region

results are presented for rest of the investigated turbulence models. The wake region can easily be identified by crust and trough in the nondimensional velocity plots in Fig. 12. The momentum deficit wake region is identified in Figs. 12(a)–12(e). It can be seen from this figure that the thickness of the wake region and velocity deficit varies significantly with the varying angle of attack. It can also be observed that the numerical results for prestall and near stall (i.e., au ¼ 11–24 deg) are in better agreement with the experimental results as compared to those corresponding to immediate poststall conditions (i.e., a ¼ 25–23 deg downstroke). The maximum deviation is observed for SST-SAS turbulence model at a ¼ 25 deg. The momentum deficit and momentum surfeit region were identified in Figs. 12(f) and 12(g) for SSTSAS and SAS turbulence models. The stall angle in the experimental studies was au ¼ 24.7 deg, whereas numerically predicted stall angle by SST-SAS turbulence model was au ¼ 23 deg. The poststall flow is characterized by formation and shedding of the secondary vortices and a rapid change in flow pattern was 011103-12 / Vol. 141, JANUARY 2019

observed in Figs. 10 and 11 as angle of attack varies from the poststall occurrence to ad  20 deg. This mismatch in the prediction of stall and poststall event leads to a huge difference in the wake velocity. As the blade continues to move downward, wake velocity predicted by numerical study comes closer to the experimental studies. Figure 12(h) shows that the numerical prediction of SAS turbulence model is closer to experimental results compared to other turbulence model when angle of attack decreases to ad ¼ 9 deg. 3.5 Vorticity Diffusion and Stretching in the Wake Region. The nondimensional z-vorticity (xz  C/U) contours of the wake region are shown in Fig. 13 at various angles of attack for a pitching blade. It can be seen from Figs. 13(a)–13(t) that the wake vortices do not exhibit any particular pattern for the whole pitching cycle. The shape, size, strength, and trajectory of the vortices are influenced strongly by angle of attack, which is continuously Transactions of the ASME

Fig. 13 Nondimensionalized z-vorticity contours of wake region at various angles of attack for SAS turbulence model

varying for the pitching blade. The wake vortices observed in the experimental studies of Bohl and Koochesfahani [45] and Ashraf et al. [46] on NACA0012 and NACA0015 airfoil were found to be diminished just after two cords downstream of the trailing edge. Bohl and Koochesfahani [45] reported that the reduction in vorticity was due to both the vorticity diffusion and vorticity stretching with a contribution factor of 0.7 and 0.3, respectively. However, in the present study, the wake vorticity was not diminished in the vicinity of trailing edge. This indicates that the wake vortices for the pitching airfoil in deep dynamic stall regime at moderate Reynolds number are entirely different than the case of low amplitude pitching of airfoil at comparatively lower Reynolds number. The studies of Bohl and Koochesfahani [45] and Ashraf et al. [46] were conducted at low Reynolds number, high reduced frequencies, and for low amplitude oscillation, i.e., a ¼ 65 deg and a ¼ 62 deg, respectively. For the similar configuration and operating conditions, Schnipper et al. [47] identified the occurrence of 16 vortices per oscillation including Karman vortex, reverse Karman vortex, 2P wake, 2P þ 2S wake, and wakes ranging from 4P to 8P. Here, S stands for single vortex and P for vortex pair. On the contrary to the studies of Bohl and Koochesfahani [45] and Ashraf et al. [46], the studies of Koochesfahani [48] and Schnipper et al. [47] highlighted the presence of wake vortices even in the far downstream region. These observations are in consensus with the finding of the current study. Moreover, the number of wake vortices observed in the present study was 4–5 times higher compared to that reported by Schnipper et al. [47]. It can also be observed from Figs. 13(a)–13(t) that at low angle of attack, vortices with anti-clockwise circulation are shed from the pressure side of the blade, whereas those with clockwise circulation are shed from the suction side of the blade. These vortices Journal of Fluids Engineering

are released from the tip of trailing edge. The wake vortices associated with the pitching movement with angle, 5 deg  a  5 deg are the vortices typically found in low amplitude oscillating airfoil. The size of the vortices corresponding to these angles is small as compared to high angle of attack and frequency of the vortices is high. The experimental study of Koochesfahani [48] highlighted the formation of a smooth undulating wake, which carried Karman vortices shed by natural wake for a NACA0012 blade oscillating at an amplitude of am ¼ 2 deg and 4 deg with a frequency of 0.5 Hz. For the same amplitude but at higher frequency (1.85 Hz), wake vortex analysis revealed formation of double-wake structure. At higher frequencies (4–6 Hz), the formation of alternating vortices is such that the vortex with the positive (counter-clockwise) circulation at top and negative circulation at the bottom was also found. However, in the present study, the vortices identified at most of the angle of attacks are not similar to Karman or the reverse Karman vortex. For instance, at au ¼ 15 deg and ad ¼ 0 deg some resemblance of Karman vortices can be seen in Figs. 13(e) and 13(r). As the blade continues to pitch up, the vortices with the positive circulation are released from the suction side of the trailing edge, shown in Figs. 13(f)–13(q). It appears that the counter-clockwise vortex has shifted clockwise vortex upward, which eventually affects the wake vortices. Near the stall conditions, i.e., au > 23 deg only positive circulation vortices appeared in the wake region, shown in Fig. 13(i). For the maximum angle of attack, the vortex with negative circulation is released from the midsection of suction side of blade and its size is bigger than the positive vortices. During the down-stroke movement of the blade, the sizes of both þve and ve circulation vortices are bigger and their frequency is low. The experimental study of Hu et al. [36] on JANUARY 2019, Vol. 141 / 011103-13

airfoil under deep dynamic stall condition at moderate Reynolds number. This was also concluded from the discussion of Fig. 13. It can also be observed that the peak amplitude is obtained at lower Strouhal number, i.e., at lower frequencies. The Strouhal number based on the pitching frequency was 0.013 (marked as A, B, C, and D in Fig. 14) in the present study and first seven peaks were observed at StA ¼ 0.099, 0.152, 0.193, 0.246, 0.323, 0.381, and 0.441 for SST-SAS turbulence model. The same data for SAS turbulence model was observed at StA ¼ 0.0804, 0.187, 0.215, 0.292, 0.439, 0.524, and 0.61. The prediction of these two turbulence models is deviating from each other. However, the deviation in these results cannot be quantified because of the nonavailability of experimental data. The low frequency (low Strouhal number) is corresponding to the vortices released near the stall conditions, which is bigger in size. Moreover, the shedding frequency of the smaller vortices is higher. These vortices are released at low amplitude oscillation. This finding is in consensus with the study of Onoue and Breuer [51] on passively pitching plate.

4

Fig. 14 Fast Fourier transformation analysis of wake vortices at various locations for (a) SAS turbulence model and (b) SSTSAS turbulence model

a root fixed flat plate undergoing oscillation at 60 Hz revealed formation of clockwise and counter clockwise vortices in the wake region with multiple trajectories. The present study revealed the formation and shedding of vortices in the wake region with a single trajectory. 3.6 Vortex Frequency-Fast Fourier Transformation Analysis. In order to analyzed the frequency of the shaded vortex, FFT analysis was carried out on the velocity component in the direction of the flow (Ux). The velocity data were extracted at eleven discrete locations in vertical direction at one chord downstream of the trailing edge. The results obtained from this study for two turbulence models, viz., SAS and SST-SAS are presented in Fig. 14 in terms of Strouhal number, StA ¼ ðA  fs Þ=U1 . Strouhal number was calculated based on the peak-to-peak amplitude of pitching (A) and vortex shedding frequency, fs. A distinct feature of pitching airfoil is the presence of multiple peak at various Strouhal number. The studies on static and low amplitude oscillating airfoil revealed the presence of single or two peaks in the energy spectra versus frequency plots, Rodriguez et al. [30], Goyaniuk et al. [49], Yarusevych and Boutilier [50], etc. This indicates multiple vortices are shed for the pitching 011103-14 / Vol. 141, JANUARY 2019

Conclusions

In the present work, a numerical study was carried out to compare the performance of seven turbulence models on a single pitching blade of cycloidal rotor operating in deep dynamic stall regime at moderate Reynolds number. The investigated turbulence models were: (i) kx-SST, (ii) kx-SST with c, (iii) transition SST (c-Reh), (iv) SAS, (v) SAS coupled with transition SST, (vi) SAS with c, and (vii) DES coupled with kx-SST. The wake vortices evolution and shedding analysis was also carried out for the pitching blade. Based on the present study, the following conclusions can be arrived at: Prestall flow characteristics and aerodynamic coefficients predicted by all the turbulence models are in the agreement with the experimental results. With the increase in the angle of attack, performance of the investigated turbulence models was deviated near the stall and poststall angle of attack. The maximum value of lift coefficient predicted numerically by kx-SST, kx-SST-c-Reh, SAS, transition SST-SAS, transition SST-DES, kx-SST-c, and SAS-c-models, was: 2.25, 2.26, 2.26, 2.25, 2.28, 1.86, 2.06, respectively, corresponding to an angle of attack (au): 23.36 deg, 25 deg, 23.5 deg, 23 deg, 25 deg, 22.3 deg, and 23.2 deg, respectively. Whereas, the experimental value of maximum lift coefficient was 2.41 at an angle of attack of au ¼ 24.7 deg. All the investigated turbulence models deviated significantly from the experimental results for the range of angle of attack, a  20 deg during upstroke movement and a  17 deg during down-stroke movement of blade. Poststall predictions of SAS turbulence model were better as compared to other investigated turbulence models. The quantitative analysis at critical points showed that the predictions of SAS and SST-SAS turbulence model were in reasonably agreement with the experimental results. The angle of attack has significant effect on the wake vortices. The vortices observed in the wake region at lower angle of attack were smaller in size. The vortices with positive and negative circulation were released from the pressure and suction side of the blade at trailing edge for lower angle of attack. With the increase in the angle of the attack, vortices with positive circulation were released from the suction side of the blade at trailing edge. These vortices migrated the location of negative circulation vortices upward on the suction side of the blade. Near the stall condition, the size of wake vortices was bigger than that observed at lower angles. The wake vortices corresponding to the same angle of attack during up-stroke and down-stroke were entirely different. The trajectory of the vortices was also changed with the movement of the blade. Moreover, vortices were released with the single trajectory irrespective of the upward or downward movement of the blade. The wake vortices released during one pitching cycle were 4–5 times higher for the present case as compared to the oscillating airfoil at low amplitude. Transactions of the ASME

Multiple nonharmonic peaks observed in the plot of Strouhal number at eleven discrete locations in the direction normal to the flow revealed the presence of wake vortices with distinct features. The vortices released at lower angle of attacks were smaller in size but their frequency was higher. The reverse was true for the vortices shed at angle of attack close to the stall event. Overall, it can be concluded that the performance of SAS and SST-SAS turbulence models to predict the aerodynamic coefficients for a pitching blade in deep dynamic stall regime at moderate Reynolds number is better compared to other five turbulence models investigated in this study. The wake vortices analysis and FFT analysis showed that the vortex characteristics of a blade pitching in deep dynamic stall regime at moderate Reynolds number is significantly different than that for the low amplitude oscillating blade at the higher reduced frequency as available in the literature.

Acknowledgment This work has been supported by the project Centro-01-0145FEDER-000017 - EMaDeS - Energy, Materials and Sustainable Development, co-financed by the Portugal 2020 Program (PT 2020), within the Regional Operational Program of the Center (CENTRO 2020) and the European Union through the European Regional Development Fund (ERDF). The authors wish to thank the opportunity and financial support that permitted to carry on this project.

Funding Data European Regional Development Fund (Centro-01-0145FEDER-000017). Fundacao para a Ciencia e a Tecnologia (PT2020).

Nomenclature C¼ Cd ¼ Cl ¼ f¼ fs ¼ k¼ P¼ Re ¼ S¼ St ¼ t¼ U¼

chord length, m drag coefficient lift coefficient pitching frequency, Hz vortex shedding frequency, Hz reduced frequency, xC=2U1 vortex pair mainstream Reynolds number based on chord length, qU1 C=l single vortex Strouhal number ðA  fs Þ=U1 time, s velocity, m/s

Greek Symbols a ¼ angle of attack c ¼ intermittency h ¼ momentum thickness l ¼ dynamic viscosity q ¼ density x ¼ angular velocity (pitching) xz ¼ z-vorticity Subscripts d ¼ down-stroke u ¼ upstroke x ¼ crossflow 1 ¼ mainstream

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