Vorticity for a pitching airfoil using k- ω SST 2D model; min = 0.0 [1/s]; max = 5.0 [1/s]. 

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... calculated from the first 20 oscillatory cycles, whilst the analysis is performed from the average of the last five simulated cycles. The computational simulation results using the k- ω SST model are compared with the experiments from Berton et al. (2002) and the k- ε Chien and Spalart -Allmaras model two-dimensional simulations from Martinat et al. (2008), for a pitching axis located at the center of pressure (one quarter chord aft the leading edge). The aerodynamic coefficients of lift and drag are analyzed as functions of the incidence angle α, as shown below. The different behavior of both lift and drag coefficients prediction can be observed in Fig. 2. It is found that the k- ω SST model presents a behavior that is close to the one associated with the Spalart-Allmaras model, compared to the simulations from Martinat et al. (2008). The k- ω SST model underestimates the lift coefficient at the upstroke phase; however, it has a less critical prediction for lift and drag on the downstroke phase. In the work of Martinat et al. (2008) it is possible to evaluate the results from a three-dimensional DDES k- ω SST model. The comparison of such results with those from the RANS k- ω SST model two-dimensional simulation is presented in Fig. 3. It shows that the flow approximation is fairly close for the upstroke phase, indicating that the lift and drag coefficients are not affected by three-dimensional effects. This could be also concluded from the calculated results, since an analysis of the last ten simulated cycles has shown that the aerodynamics coefficients did not change for each incidence angle. The difference comes out during the downstroke phase, in which the flow behavior is proven to be three-dimensional dependent, as concluded in works of McAllister et al. (1978) and Martinat et al. (2008). For the downstroke, it can be seen that the present two-dimensional simulation presents a smaller deviation from the experiment, when compared with the three-dimensional simulation from Martinat et al. (2008). However, both numerical predictions display high oscillation characteristics during the downstroke phase, showing that the model is not able to capture all the circulation of an oscillatory flow with this scale of complexity. Of course, a flow of such complexity of unsteadiness is not easy to model, as turbulence modelling can render misleading predictions. Also, the two-dimensional approximation can delay separation, giving results more optimistic than observed experimentally. The flow behavior of the simulation also agrees with the studies presented by McCroskey (1982) and details of unsteady calculation and solutions can be found in McCroskey (1973). During the upstroke, the flows remains attached to the profile up to 12 degrees. At a 14.4 degrees incidence angle two small recirculation regions are formed, one close to the leading and one close to the trailing edge; from this point the separation bubble and turbulent separation point start moving towards each other due to the increase of the angle of attack. As expected, the dynamic stall for unsteady pitching airfoils is delayed by the influence of its time dependent vortices, whereas static stall would be near to occur at 14 degrees. At an incidence close to 17 degrees, the trailing edge turbulent separation and the separation bubble generates a large area of recirculation, leading to the detachment of the boundary layer at 18 degrees and consequent drastic loss of lift. The flow is kept stalled during most part of downstroke, reattaching close to 7 degrees, as can be observed in the Lift Coefficient vs. Incidence, in Fig. 3, and also in Fig. 4, where the flow is reattached already at 7.2 degrees during downstroke. In the downstroke phase, where the flow detaches from the airfoil surface, it is also necessary to consider that the lower surface of the airfoil is also acting, in the other direction, producing down force, hence reducing the lift force in the phase. This occurs due to the shape of the airfoil. Since it is symmetric, both surfaces act as predicted by the Coanda Effect. The Coanda Effect explains the bending of fluids around an object due to its viscosity, which makes a fluid thick and makes it stick to a surface (Anderson & Eberhardt, 2001). Due to the difference of speed between fluid parcels of the boundary layer and in it vicinity, shear forces are created, which help attach the flow, and also force it to bend in the direction of the slower layer, the one close to the wall, trying to wrap around the object. Hence, the Coanda Effect explains why there’s no lift on a symmetrical airfoil at zero angle of attack, but when it reaches a positive incidence attitude, the flow attaches to the upper surface and due to the shape of the airfoil, the flow is bent in the trailing edge, generating downwash, which causes the lift force over the airfoil. In the downstroke phase, the flow is better attached to the lower surface of the airfoil, as shown in Fig. 4, which causes the bending of the flow upwards, also generating a ‘upwash’ at the trailing edge. Due to vortex formation at the trailing edge and stall condition, the boundary layer is detached from the upper surface, causing the abrupt loss of lift. When the flow is reattached near to 7 degrees downstroke, the lower surface return to bend the air down in the trailing edge, as expected for a positive attitude incidence. Even though the k- ω SST two-dimensional simulation overestimated the aerodynamic coefficients prediction by some extent, by comparing Fig. 3 it is possible to see that the prediction of the reattachment incidence was the closest one to the experimental reference. Considering the two-dimensional simulations of Martinat et al. (2002), the latest k- ω SST simulation matched accurately with the experiment downstroke reattachment incidence. For the analyzed model performance, it is necessary to account for the influence of the SST limiter, as seen in Eq. (8), which can create a reduction of the eddy viscosity during simulation (Martinat et al. , 2002). There are many factors that can affect the flow behavior under unsteady condition. As found in experiments by McAlister (1978), not only the reduced frequency have a major importance on determining the flow characteristics and vortex dimensions, but also, other factors as the thin airfoil shape, flow velocity and rotation axis. To study the influence of the rotation axis on the vortex formation a test was made in order to compare results from simulations in which such axis was located at the center of pressure (one quarter chord aft the leading edge), and at the middle chord. The influence of the rotation axis analysis was made in order to compare the results from the k- ω SST model. It is shown by Fig. 5 that the hysteresis loops for lift and drag coefficients presents a close behavior for both tested cases. Concerning lift coefficients during pitching cycle, when the rotation axis is at the middle chord, a thinner hysteresis loop is observed in comparison with the simulations performed with the rotation axis located at the center of pressure of the airfoil. The hysteresis loop for the drag coefficients is larger when the rotation axis lies at the middle chord, for high incidence angles. Both lift and drag coefficients increase with an increase of the velocity in the near-wall region. During the upstroke phase the flow acceleration is more accentuated for the pitching axis at the center of pressure, which is responsible for the larger lift coefficient for this phase. The scenario where the rotation axis is at the middle chord presents a lower lift coefficient curve in the upstroke; however, it is possible to see a raise in the drag coefficient during this phase, which happens due to vortex formation in the trailing edge region. This vortex is responsible for the formation of the trailing edge turbulent separation region, causing the detachment of the boundary layer in that area. This implies a decrease of the boundary layer attached area on the upper surface, which is responsible for the decrease in lift and the increase in drag generation. At higher incidence angles, the separation bubble that forms close to the leading edge reveals itself more accentuated for the center of pressure rotation axis condition. The boundary layer is attached to the upper surface of the airfoil until the incidence is greater than 17 degrees; at this point, the bubble reaches the trailing edge, causing a full detachment of the boundary layer, hence leading to an abrupt lift loss. For the middle chord case, the formation of the separation bubble close to the leading edge is delayed. Unlike the center of pressure rotation axis case, in this situation, in the end of the upstroke phase, the bubble did not reach the trailing edge; this will only occur in the downstroke phase after reaching a 17 degrees angle of attack, causing a full detachment of the boundary layer. Hence, an abrupt lift loss occurs at an incidence angle close to 17 degrees during the downstroke phase. The flow will reattach at a low incidence angle, of about 7 degrees, as also predicted by the center of pressure pitching case. Both cases were able to predict the same reattachment incidence angle, close to 7 degrees, as also occur in the experiments from Berton et al. (2002). The lack of experimental data for the pitching axis at the middle chord prevents a full analysis of the flow prediction for the case. However, the simulation for the pitching axis at the center of pressure produced results that can be accurately used for the analysis of flow behavior. Therefore, it is possible to assess first impressions on the influence of the pitching axis position by comparing results from the performed k- ω SST two dimensional simulations. Simulations were carried out on a NACA0012 airfoil using the commercial software Fluent 6.4 and using the RANS model k- ω SST in order to study the behavior of flow ...

Context 2

... simulation was based on the experiments from Berton et al. (2002). For the prescribed motion, the minimum and maximum incidence angles were 6 and 18 degrees , respectively, the reduced frequency for the experiment was given as = 0.188 at a Reynolds number of , for a 1-meter-chord NACA 0012 airfoil. To achieve the best possible results, static mesh test was made for an incidences of 0 and 6 degrees, leading to 10% and 5% deviation for both lift and drag aerodynamic coefficients, respectively, in comparison with experimental results from Sheldal et al. (1981). For these settings, the mesh refinement level ensures y-plus values that are lower than one in near-wall region across the whole domain. Results are calculated from the first 20 oscillatory cycles, whilst the analysis is performed from the average of the last five simulated cycles. The computational simulation results using the k- ω SST model are compared with the experiments from Berton et al. (2002) and the k- ε Chien and Spalart -Allmaras model two-dimensional simulations from Martinat et al. (2008), for a pitching axis located at the center of pressure (one quarter chord aft the leading edge). The aerodynamic coefficients of lift and drag are analyzed as functions of the incidence angle α, as shown below. The different behavior of both lift and drag coefficients prediction can be observed in Fig. 2. It is found that the k- ω SST model presents a behavior that is close to the one associated with the Spalart-Allmaras model, compared to the simulations from Martinat et al. (2008). The k- ω SST model underestimates the lift coefficient at the upstroke phase; however, it has a less critical prediction for lift and drag on the downstroke phase. In the work of Martinat et al. (2008) it is possible to evaluate the results from a three-dimensional DDES k- ω SST model. The comparison of such results with those from the RANS k- ω SST model two-dimensional simulation is presented in Fig. 3. It shows that the flow approximation is fairly close for the upstroke phase, indicating that the lift and drag coefficients are not affected by three-dimensional effects. This could be also concluded from the calculated results, since an analysis of the last ten simulated cycles has shown that the aerodynamics coefficients did not change for each incidence angle. The difference comes out during the downstroke phase, in which the flow behavior is proven to be three-dimensional dependent, as concluded in works of McAllister et al. (1978) and Martinat et al. (2008). For the downstroke, it can be seen that the present two-dimensional simulation presents a smaller deviation from the experiment, when compared with the three-dimensional simulation from Martinat et al. (2008). However, both numerical predictions display high oscillation characteristics during the downstroke phase, showing that the model is not able to capture all the circulation of an oscillatory flow with this scale of complexity. Of course, a flow of such complexity of unsteadiness is not easy to model, as turbulence modelling can render misleading predictions. Also, the two-dimensional approximation can delay separation, giving results more optimistic than observed experimentally. The flow behavior of the simulation also agrees with the studies presented by McCroskey (1982) and details of unsteady calculation and solutions can be found in McCroskey (1973). During the upstroke, the flows remains attached to the profile up to 12 degrees. At a 14.4 degrees incidence angle two small recirculation regions are formed, one close to the leading and one close to the trailing edge; from this point the separation bubble and turbulent separation point start moving towards each other due to the increase of the angle of attack. As expected, the dynamic stall for unsteady pitching airfoils is delayed by the influence of its time dependent vortices, whereas static stall would be near to occur at 14 degrees. At an incidence close to 17 degrees, the trailing edge turbulent separation and the separation bubble generates a large area of recirculation, leading to the detachment of the boundary layer at 18 degrees and consequent drastic loss of lift. The flow is kept stalled during most part of downstroke, reattaching close to 7 degrees, as can be observed in the Lift Coefficient vs. Incidence, in Fig. 3, and also in Fig. 4, where the flow is reattached already at 7.2 degrees during downstroke. In the downstroke phase, where the flow detaches from the airfoil surface, it is also necessary to consider that the lower surface of the airfoil is also acting, in the other direction, producing down force, hence reducing the lift force in the phase. This occurs due to the shape of the airfoil. Since it is symmetric, both surfaces act as predicted by the Coanda Effect. The Coanda Effect explains the bending of fluids around an object due to its viscosity, which makes a fluid thick and makes it stick to a surface (Anderson & Eberhardt, 2001). Due to the difference of speed between fluid parcels of the boundary layer and in it vicinity, shear forces are created, which help attach the flow, and also force it to bend in the direction of the slower layer, the one close to the wall, trying to wrap around the object. Hence, the Coanda Effect explains why there’s no lift on a symmetrical airfoil at zero angle of attack, but when it reaches a positive incidence attitude, the flow attaches to the upper surface and due to the shape of the airfoil, the flow is bent in the trailing edge, generating downwash, which causes the lift force over the airfoil. In the downstroke phase, the flow is better attached to the lower surface of the airfoil, as shown in Fig. 4, which causes the bending of the flow upwards, also generating a ‘upwash’ at the trailing edge. Due to vortex formation at the trailing edge and stall condition, the boundary layer is detached from the upper surface, causing the abrupt loss of lift. When the flow is reattached near to 7 degrees downstroke, the lower surface return to bend the air down in the trailing edge, as expected for a positive attitude incidence. Even though the k- ω SST two-dimensional simulation overestimated the aerodynamic coefficients prediction by some extent, by comparing Fig. 3 it is possible to see that the prediction of the reattachment incidence was the closest one to the experimental reference. Considering the two-dimensional simulations of Martinat et al. (2002), the latest k- ω SST simulation matched accurately with the experiment downstroke reattachment incidence. For the analyzed model performance, it is necessary to account for the influence of the SST limiter, as seen in Eq. (8), which can create a reduction of the eddy viscosity during simulation (Martinat et al. , 2002). There are many factors that can affect the flow behavior under unsteady condition. As found in experiments by McAlister (1978), not only the reduced frequency have a major importance on determining the flow characteristics and vortex dimensions, but also, other factors as the thin airfoil shape, flow velocity and rotation axis. To study the influence of the rotation axis on the vortex formation a test was made in order to compare results from simulations in which such axis was located at the center of pressure (one quarter chord aft the leading edge), and at the middle chord. The influence of the rotation axis analysis was made in order to compare the results from the k- ω SST model. It is shown by Fig. 5 that the hysteresis loops for lift and drag coefficients presents a close behavior for both tested cases. Concerning lift coefficients during pitching cycle, when the rotation axis is at the middle chord, a thinner hysteresis loop is observed in comparison with the simulations performed with the rotation axis located at the center of pressure of the airfoil. The hysteresis loop for the drag coefficients is larger when the rotation axis lies at the middle chord, for high incidence angles. Both lift and drag coefficients increase with an increase of the velocity in the near-wall region. During the upstroke phase the flow acceleration is more accentuated for the pitching axis at the center of pressure, which is responsible for the larger lift coefficient for this phase. The scenario where the rotation axis is at the middle chord presents a lower lift coefficient curve in the upstroke; however, it is possible to see a raise in the drag coefficient during this phase, which happens due to vortex formation in the trailing edge region. This vortex is responsible for the formation of the trailing edge turbulent separation region, causing the detachment of the boundary layer in that area. This implies a decrease of the boundary layer attached area on the upper surface, which is responsible for the decrease in lift and the increase in drag generation. At higher incidence angles, the separation bubble that forms close to the leading edge reveals itself more accentuated for the center of pressure rotation axis condition. The boundary layer is attached to the upper surface of the airfoil until the incidence is greater than 17 degrees; at this point, the bubble reaches the trailing edge, causing a full detachment of the boundary layer, hence leading to an abrupt lift loss. For the middle chord case, the formation of the separation ...