Three-Dimensional Features of the Stalled Flow field of a Rotor Blade in Forward Flight

Transcription

1 Three-Dimensional Features of the Stalled Flow field of a Rotor Blade in Forward Flight Vrishank Raghav, Phillip Richards, Narayanan Komerath, Marilyn Smith Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Tel : ; komerath@gatech.edu ABSTRACT This paper quantifies findings about breakup of radial flow in the stalled region above a retreating rotor blade, into discrete co-rotating vortical structures. It uses re-analysis of previous experimental data and new computational work using an unsteady Reynolds Averaged Navier-Stokes solver with a hybrid RANS-LES turbulence methodology. The vorticity observed in these structures is of the same order of strength as that in the shear layer over the blade between the surface, and the radial jet developing near the surface. The radial jet is weakened by the departure of these structures, so that the peak radial velocity decays towards the blade tip. These experimental results are compared with computational results for the recirculating flow over a fixed wing with and without yaw. The reported phenomena are established to be due to blade rotation, and jet profiles of similar magnitudes are captured in the computation. 1. INTRODUCTION Downstream of the stall line, the flow over a retreating rotor blade is highly three-dimensional. The stalled flow moves at low speed relative to the rotating blade, and hence the radial stresses at the surface should be transported to a substantial height above the surface into this flow. The occurrence of a large radial flow should in turn have a strong influence on the stalling process itself, thus determining the shape of the stall line. The near-surface flow field must form a powerful jet both above and below the blade surface. This jet layer must itself be highly vortical, given the no-slip condition at the surface and the strong radial acceleration of flow immediately above the surface. The magnitude, extent and profile of the radial velocity field are thus of strong interest. The effect of these near-surface flow features on the surface pressure distribution may hold the key to an unresolved question of why observed centrifugal pumping velocitiy is much less than that expected based on their predicted effects on blade lift and pitching moment through the stall process. 2. PRIOR WORK Yang [1] and DiOttavio [2] describe prior work in this area and the motivation for the experiments used in this paper. They argue why it is not necessary to simulate either the compressible flow over the leading edge or the high aspect ratio typical of rotor blades, since the interest here is in the inboard incompressible flow behind the stall line. The highly 3-dimensional flow field downstream of the stall line on a rotor blade was studied using Particle Image Velocimetry (PIV). Reference 1 used an Inflow Obstructor to generate transient stall over a single-bladed rotor in a hover facility. Reference 2 describes experiments performed in the High Advance Ratio facility at the Georgia Tech John Harper Wind Tunnel (Figure 1). Table 1 gives the rotor properties. The 2-bladed teetering rotor included collective and cyclic controls and was operated at an advance ratio of 0.3, which is in the range for dynamic stall to occur. Table 1: Rotor System Specifications Description Value Total Blade Mass (kg) Blade Span (m) Blade Chord (m) Disc Radius (m) Solidity Precone (degrees) 1.6 Max Collective (degrees) 10 Max Cyclic (degrees) 6.5 Max Tip Path Plane Tilt (degrees) 16 Blade Aspect Ratio 3.49 Motor Power (KW) 3.73 Laser sheet flow imaging confirmed that stall was occurring at the 10 degrees collective, 5 degrees cyclic condition chosen. PIV was used to measure the velocity components in the radial - axial plane nominally at the 270 o rotor azimuth, repeating as the blade passed through the measurement plane. The first set of measurements was taken at a phase corresponding to the instant when the blade had just passed through the plane, and the trailing edge was approximately 1 mm beyond, shown schematically in the inset in Figure 1. This plane enabled capture of the velocity field, very similar to what must exist above and below the trailing edge, without the blade surface obstructing or scattering the laser illumination. In cross flow planes such as the one shown in the inset of Figure 1, PIV showed the formation of several discrete vortical structures in the recirculating flow. An example is shown in Figure 2 on a schematic representation of the overall velocity field, the structures made visible first using color-coding of speed (vector magnitude) and then using vorticity contours. Numerous such instances showed that the number of discrete zones in the viewing plane was approximately 4 on average, i.e., the spacing between the structures was roughly constant, and approximated the nominal height of the separated flow region at the trailing edge of the blade, suggesting discrete vortical structures occupying the stalled region.

2 The discovery of discrete structures and the resolution of the radial jet layer in the stalled flow field raised curiosity about their significance. A review of the literature on the nature of the separated flow over lifting surfaces reveals very little about the dynamic stall flow field, but does include some studies on fixed wings under stalled conditions. Gregory et al [ 3 ] detected 3-dimensional flow patterns on airfoils based on surface visualization. Winkelmann and Barlow [ 4 ] focused on surface oil streaklines on a stalled rectangular wing. Weihs and Katz [5] reported cells in the post-stall flow field over straight wings. Yon [ 6, 7 ] reported coherent structures and cells in the separated flow and fluctuations attributed to these cells in the wake of a stalled rectangular wing. Boeren and Bragg[ 8 ] found low-frequency oscillations in cases of 2-D leading edge separation. Spanwise separation cells were seen to arise from trailing-edge separation, but this was mostly steady. Thus from the above, it appears that spanwise cells have been observed in the stalled flow over fixed wings, and even on nominally two-dimensional airfoil test models; however these arise from the shear at the outer edge of the recirculation zone, driven by the kinetic energy of the external flow. This paper follows up on the above findings, with quantitative analysis of the results, extension to measurements performed further along the blade chord towards the separation line, and computational prediction efforts of the radial flow problem. The measurements now reported for the first time were also obtained during the wind tunnel entry where the results in Reference 2 were acquired, but they are being presented now after detailed quantitative analyses. At the end a new result is added, confirming that the observed phenomena are unique to the rotating blade case. 3.1 Objectives 3. COMPUTATIONAL SETUP The computational portion of this effort was the first qualitative attempt to isolate the underlying cause of the features observed in the experiment for future quantitative analysis. To obtain these results, first an infinite wing was examined at varying amounts of sideslip, followed by the simulation of a rotating wing. These simulations allow the identification of the source of different flow field characteristics. Computational fluid dynamics (CFD) has been first used to estimate the overall performance parameters of the test case. The low-speed, highly three-dimensional, turbulent test case poses a substantial challenge in grid and time resolution. The OVERFLOW [ 9, 10 ] code has been applied to test the hypothesis on the character of the vortical structures. Initial deconstruction of the flow characteristics has been conducted via computations that seek to match the test conditions and are correlated with the PIV data. 3.2 Grid Description Several structured grids were used in this study. The sectional characteristics were first verified using a two-dimensional O-grid to evaluate the influence of the finite trailing edge, as observed in Figures 3 and 4. Details of grid studies were reported in Smith et al. [11]. The final grids used for the results presented herein were observed to provide integrated force and moment results comparable to finer grids when flow was attached, as well as averaged results within 2-4% of one another for separated flows. The three-dimensional computational grid (Figures 3 and 4) consisted of sectional O-grids with 120 points normal to the airfoil surface and 971 along the circumference of the airfoil, including 21 on the blunt trailing edge. The initial cell spacing normal to the surface is 5.0 x 10-6 c, (where c is the chord) which represents y^+ <= 1 for the Reynolds number in the study. The outer boundary was located at 17c from the airfoil. There were 121 points along the radius of the blade. The airfoil numerical simulations were correlated with results from existing experimental campaigns to ensure grid independence (using the RANS k -SST turbulence model), most notably the incompressible tests reported by Abbott and Von Doenhoff [12] and Critzos et al [13]. Yawed correlations were accomplished using data from Purser and Spearman [14]. Details of these correlations are provided in Ref. [ 9]. 3.3 Code Description Simulations were computed using the OVERFLOW (2.1z) code with the 4 th -order central-difference scheme along with the ARC3D diagonalized Beam-Warming scalar pentadiagonal scheme. Dissipation was calculated using the TLNS3D dissipation scheme. The 4 th -order smoothing coefficient was set to the default of 0.04, and the 2 nd -order smoothing coefficient was set to 0.0 for subsonic and 2.0 for transonic flows. RANS turbulence models have been shown by Shelton[15] and Smith[16] to have difficulty predicting the character of stall for some airfoils, while LES-based models have been able to more accurately capture the performance characteristics[8]. Therefore, as the simulations require angles of attack above stall, the turbulence simulation applies a hybrid RANS-LES method (HRLES-SGS) that applies the k -SST method for attached flows and resolves the k-equation with a subgrid-scale model away from the wing. The hybrid RANS-LES method requires that a time-accurate simulation be computed, as the characteristics of the unsteady flow field are more accurately simulated. Unsteadiness in the resultant force or moment from the simulations, indicating the presence of shed vorticity and/or separation is depicted using error bars about each computational data point. In these circumstances, the mean data point was computed by averaging the data over a minimum of three periodic cycles. The error bar limits indicate the average of the minimum and maximum values of the data cycles used to compute the mean. 4. RESULTS 4.1 Profiles of Radial Velocity The radial jet flow develops over the surface because of the rotation of the blade. However, at the blade surface, the no-slip condition must apply and hence there cannot be any radial velocity. This means that there must be a boundary layer under the radial jet flow. The radial acceleration is also experienced at the bottom surface of the blade where there is attached flow. The cross-flow 2

3 velocity map should therefore show strong radial jet layers very near both surfaces, but there must also be an underlying boundary layer adjacent to each surface. This boundary layer is very thin, and it is not expected that PIV will resolve this in the present experiments. Thus, what should be a double-peaked radial velocity profile with the blade surface between the two peaks, should appear instead with a single peak without the blade boundary layer seen. This is what occurs at the most inboard radial stations in the plane 1mm downstream of the trailing edge, as shown in Figure 5. The blade surface in each image was located again with reference to video images taken of the blade with ordinary lighting in addition to the laser sheet. Analyzing the blade dynamics suggests that the observed flapping angle implies a much smaller aerodynamic contribution than what is predicted from high Reynolds number lift coefficient tables; This is reasonable given the Reynolds number range and blade aspect ratio used here. Figure 5 shows that the radial jet is confined to the region close to the blade. The jet appears to exist above and below the blade trailing edge. This is an illusion as explained above, and in fact there should be two jets, separated by the thin boundary layers above and below the blade. Measurement resolution with PIV is inadequate to capture this double-peaked feature. It does appear in other velocity profiles as the radial station moves further outboard to the tip of the rotor. For the vorticity bounds estimated in this paper it is assumed that the peak of the jet is what is seen and occurs where it is seen, and that the radial velocity must drop from that maximum value to zero in the distance between the peak location and the surface of the rotor. This assumption is used to estimate the vorticity in the boundary layer above the blade. For these purposes, the phenomena on the bottom surface of the blade are ignored. 4.2 Discrete Structures Driven By Radial Jet Layer Figure 6 shows a sample result from Ref. 2 of the root mean square (RMS) variation of the radial velocity profile at r/r= If the radial flow were a stabilizing influence inhibiting the formation of discrete structures, then the RMS fluctuation intensity in radial velocity should be higher in the region well above the wall and lower in the region where the strong radial flow exists. If the radial flow is a driver of instability, then high fluctuations should be seen in the regions of high radial velocity nearer the wall. In most cases, the root mean square fluctuation peaked at the jet, thus showing that the vortical structures are powered by the vorticity in the jet shear layer, and not by the freestream at the other edge of the recirculating flow. Figures 7 and 8 show the averaged radial velocity profiles such as the one in Figure 6, normalized by local blade speed r. The magnitude of the peak is decreasing as one moves outboard. In these figures, unlike in Figure 6, the radial velocity component is transformed so that it is parallel to the blade surface (the blade is flapped up 4+ degrees) in order to more accurately consider the vorticity in the boundary layer. 4.3 Vorticity Contours in the Crossflow Plane Figure 9 shows the vorticity contours in the cross-flow plane at r/r=0.7. It is clear that the overall jet shear layer has broken into several discrete structures in the radial window shown. The structures also appear to be lifting off the surface at the more outboard locations a trend also seen in the measurement window centered at r/r = and beyond (not shown). 4.4 Quantitative Estimates Figure 10 compares the average vorticity in these discrete structures to its RMS variation at various radial stations. The structures grow stronger on average as one moves outboard, and the variance in their strength also increases. Figure 11 summarizes data from measurement planes moved forward along the chord towards the separation line from the trailing edge. It shows that the peak of the radial velocity profile drops in magnitude as radial distance increases, contrary to what might be expected from the increasing radial acceleration at the surface at outer locations on a spinning disk. The explanation for this is in Figure 10, where increasing portions of the vorticity generated from the surface are entrained in the discrete structures, and this weakens the radial jet layer. It also shows that this weakening cannot be attributed to the chordwise variation of the radial velocity profile and therefore the angularity of the flow velocity near the surface. The structures clearly weaken if one goes close to the separation line and beyond. Figure 12 makes a quantitative comparison of the average vorticity in the two shear layers bounding the radial jet to the average structure strength. The lower shear layer is the boundary layer between the solid blade surface and the peak of the radial jet profile. This is by far the stronger shear layer and is produced by the shear between the radially accelerated flow and the no-slip condition at the surface. This layer starts out being almost an order of magnitude (factors of 7 or 8 have been observed) stronger than the discrete structures at inboard locations. It then weakens, even as the structures themselves get stronger, so that it is only 3 to 4 times as strong as the structures at the most outboard locations evaluated. Thus at these outboard locations, the discrete structures carry away 25 to 30% of the jet. The sense of the discrete structures is the same as that of the upper shear layer where the jet velocity decays into the largely stagnant zone above it. However, since the fastest-moving fluid is entrained into these structures and carried away, the jet strength decays. The vorticity estimated from the radial velocity profile for the upper shear layer is much less than 50% of the strength of individual structures. The jet decay thus occurs mostly through the departure of discrete structures. A final point from new experiments (figure not shown) is that when the blade is held fixed at the same location as that used in the rotating blade PIV measurements, with an angle of attack and mid-radius velocity matched to the conditions of the rotating case (including the inflow velocity correction), the strongest vortical structures seen are irregular, and weaker by more than an order of magnitude than the discrete structures in the rotating case. 3

4 4.5 Computational Correlations As previously noted, experimental PIV observations of the crossflow plane showed the formation of several discrete vortical structures in the recirculating flow. The experimental hypothesis of the character of the flow field, shown in Figure 2 as a schematic representation of vorticity contours, suggested that discrete vortical structures occupy the stalled region, and that these arise due to the radial flow near the surface, caused by rotation. This hypothesis was examined using the CFD analysis. Examination of a NACA 0012 semi-infinite wing at angles of attack just above stall, yielded weak discrete vortical structures, but they could not be directly correlated to the discrete structures observed in the rotor experiment. The influence of cross flow on the wing in stall was next examined. As the yaw angle is increased for angles of attack above stall (12 o and 16 o angles of attack were studied), the addition of cross flow results in the formation of periodic vortex structures emanating from the separation point. An example of the character of the structures predicted by CFD simulation is provided in Figure 13. Prior researchers (References 1 and 2) have argued that these structures will appear in both incompressible and compressible Mach regimes, so that experimental evaluation at incompressible speeds is sufficient. To examine that hypothesis, the computational simulations were performed at Mach numbers of 0.2, 0.3, 0.5 and 0.6. These periodic structures were observed at all Mach numbers for yawed flow, supporting the experimental observations reported in the fixed-wing literature. It should be noted that this periodic vortex distribution is not observed with the same fidelity using RANS turbulence models on the same grid. Since RANS turbulence models tend to shift the location of the stall angle of attack to higher values, this behavior will not occur at the same angles of attack. In addition, it has been observed that the statistical averaging in the RANS turbulence models tends to smear the characteristics of the vorticity unless highly refined grids, beyond the scope of most engineering level applications, are utilized[17]. As an example, Figure 14 illustrates the character of the vorticity in stall using the same grid and numerical options for the results in Figure 13, this time with the k -SST turbulence model. Further analyses of these flows are warranted to investigate the behavior of these structures as Mach numbers and angles of attack are modified. While it was previously discussed that these vortex structures exist in a qualitative sense, quantitative correlations can be made with the experimental data. To obtain these correlations, the computation of the rotor blade was performed at 270 o in an inertial frame. The simulation was performed using the angle of attack equivalent to the collective angle plus cyclic angle, along with the rotation rate of the rotor. The coning angle of the experiment was not included. In addition to correlation with experimental data, these results were also compared with the infinite wing in a yawed flow that estimates the amount of radial flow as observed in experiments at the r/r=0.586 radial location. This non-rotating simulation will aid in deconstructing the source of the flow field phenomena. Figure 15 shows the computed radial velocity profile at the r/r=0.586 radial location for the rotating frame. The strength and location of the jet peak are well predicted by both the rotating and, while not shown, also the yawed flow simulations. This indicates that the jet is primarily influenced by the amount of the cross flow on the blade, whether generated by a rotating blade or a fixed wing in cross flow. Above the jet, the wing in cross flow yields a velocity distribution very similar to the experimental quantities, while the rotating blade does not. (Note however that the experiment is for a blade at 270 degrees, where there is no yaw). This difference between the other data and the rotating blade can be explained by the appearance of vortices through which the velocity profile was obtained. Additional simulation time to permit averaging over a longer period will be necessary to determine a comparable time-averaged correlation for this correlation. The computational results at the larger scales indicate that the jet lies totally on the upper surface, with no secondary jet located on the lower surface, as hypothesized from theory and supported from experimental results. However, when the normal scales are expanded, the secondary jet located at the lower surface is clearly observed, as noted in Figure 15, plotted at 1mm behind the trailing edge. The influence of coning angle will be further studied. The coning angle as well as other aspects of the experimental configuration could more accurately be modeled using overset grid techniques. In this simulation, a high-resolution grid could be used to model the aerodynamics over the blade while a less refined background grid would allow simulation of the wind tunnel free stream conditions and transient effects caused by the passing of each rotor blade. These overset grid techniques might also yield insight into how transient effects from one blade alter the stall characteristics of the next. Another potential numerical influence on the flow in this region that should be studied is the boundary condition applied along the wake cut. Finally, while the normal grid spacing was set to a y+ of slightly less than 1, it may be that additional grid points are necessary in this region close to the airfoil lower surface. The influence of y+ and the number of grid points within this critical area will be further analyzed. Figure 16 shows the vorticity in the trailing edge cross-flow plane (compare with experimental Figure 2 whose radial extent is approximately z/r=0.01 and below). A similar contour plot was also made for the non-rotating case, but there were no discrete vortical features similar to those in the experiment to be seen. Therefore the identified radial flow structures above the radial jet are, as expected, clearly due to the rotation. Discrete structures are seen, however at this stage of computation development, resolving the shear layer breakup is left to future efforts as the qualitative efforts here progress to quantitative investigations. 5. CONCLUSIONS 1. In the recirculating flow over a retreating rotor blade under stalled conditions, the radial velocity along the blade develops a sharp jet profile pointing outward. 2. Discrete vortical structures occur in the flow field above the radial jet, similar to those breaking off a shear layer. These are not phase-locked to the rotor azimuth. 3. The spanwise spacing of structures is approximately the height of the separated flow region. 4. The strongest root-mean square velocity fluctuation in the flowfield is near the peak of the radial velocity profile, indicating that the source of vorticity of the discrete structures is the jet layer, rather than the external flow. 4

5 5. The discrete vortical structures carry on the order of 30% of the vorticity in the shear layer between the radial jet and the blade surface. 6. Averaged radial velocity profile data show that the averaged peak radial velocity decreases with increasing radial distance at a given chordwise location, contrary to what would be expected in the absence of the formation and breakup of discrete vortical structures. 7. From the above, it is concluded that the breaking away of the discrete structures is the mechanism responsible for keeping the radial jet from increasing as radius increases. 8. Discrete quasi-periodic but spatially repeatable vortical structures are also observed in computational simulations under cross-flow conditions. There is a substantial difference in the radial velocity between the yawed fixed wing and the rotating wing cases. 9. The magnitude comparisons show that the radial jet development, the limiting of its strength by the breakup into discrete structures, and the discrete structures themselves, are first-order phenomena in the flow field of the rotor blade in retreating blade stall and may have significant effects on the lift and pitching moment of the blade through stall region. 10. These experimental results, compounded with initial CFD results described herein, tentatively indicate that computational predictions that ignore the radial acceleration and resulting cross flow may not represent the true fluid dynamics of the rotating blade undergoing dynamic stall. 6. ACKNOWLEDGMENTS This work was funded by the US Army Research Office. The Technical Monitor is Dr. Tom Doligalski. The authors would also like to acknowledge the HPC support from the NASA Ames Research Center, and the support/inputs of Mr. Ben Koukol and Mr. Nicholas Liggett in the wing-in-cross-flow computations. Extensive discussions with Mr. James DiOttavio regarding the experiments are gratefully acknowledged. 7. REFERENCES [1] Yang, J., Ganesh, B., and Komerath, N., Radial Flow Measurements Downstream of Forced Dynamic Separation on a Rotor Blade," Paper , AIAA, June [2] DiOttavio, J., Watson, K., Komerath, N., and Kondor, S., Discrete Structures in the Radial Flow Over a Rotor Blade in Dynamic Stall," AIAA Paper , August [3] Gregory, N., Quincy, V. G., O'Reilly, C. L., and Hall, D. J., Progress Report on Observations of Three Dimensional Flow Patterns Obtained During Stall Development on Aerofoils, and on the Problem of Measuring Two-Dimensional Characteristics". NPL Aero Report 1309, Aeronautical Research Council, Fluid Motion Sub-Committee, Vol. 18, No. 8, 1980, pp [5] Weihs, D. and Katz, J., Cellular Patterns in Poststall Flow over Unswept Wings," AIAA Journal, Vol. 21, No. 12, 1983, pp [6] Yon, S., Coherent Structures in the Wake of a Stalled Rectangular Wing, Ph.D. thesis, University of California, San Diego, and San Diego State University, [7] Yon, S. and Katz, J., Cellular Structures in the Flow Over the Flap of a Two-Element Wing," NASA CR , [8] Broeren, A. and Bragg, M., Spanwise Variation in the Unsteady Stalling Flow field of Two-Dimensional Airfoils," Paper , AIAA, [9] Chan, W.M. and Meakin, R.L. and Potsdam, M.A., CHSSI Software for Geometrically Complex Unsteady Aerodynamic Applications, AIAA , [10] Jespersen, DC and Pulliam, TH and Buning, PG, Recent Enhancements to OVERFLOW, AIAA , [11] Smith, M.J., Koukol, B. C.G., Quackenbush, T., and Wachspress, D., "Reverse- and Cross-Flow Aerodynamics for High-Advance-Ratio Flight," Proceedings of the 35th European Rotorcraft Forum, Hamburg, Germany, Sept 22-25, [12] Abbott, I. H. and von Doenhoff, A. E., Theory of Wing Sections, Dover, [13] Critzos, C.C., Heyson, H.H. and Boswinkle Jr, R.W, Aerodynamic characteristics of NACA 0012 airfoil section at angles of attack from 0 to 180, NACA TN-3361, [14] Purser, P. E., and Spearman, M. L., "Wind-Tunnel Tests at Low Speed of Swept and Yawed Wings Having Various Plan Forms", NACA TN 2445", December [15] Shelton, A. B., Braman, K., Smith, M. J. and Menon, S.. Improved Turbulence Modeling for Rotorcraft, presented AHS Annual Forum, Grapevine, TX, Also to appear Journal of the American Helicopter Society. [16] Smith, M.J., Wong, T.C., Potsdam, M, Baeder, J. and Phanse, S., Evaluation of CFD to Determine Two-Dimensional Airfoil Characteristics for Rotorcraft Applications, American Helicopter Society 60 th Annual Forum, Baltimore, MD, [17] Szydlowski, J. and Costes, M., Simulation of flow around NACA 0015 airfoil for static and dynamic stall configurations using RANS and DES, Office National d'etudes et de Recherches Aerospatiales. [4] Winkelmann, A. E. and Barlow, J. B., \Flow eld Model for a Rectangular Planform Wing Beyond Stall," AIAA Journal, 5

6 Figure 1: 2-bladed teetering rotor. Inset shows trailing edge measurement plane image superposed on blade image at 270 degrees azimuth Figure 3: Two-dimensional grid section and closeup of airfoil O-grid Figure 2: Discrete structures in the spanwise cross flow. Top: schematic of vector field at trailing edge. Middle: Vorticity contours. Bottom: Postulated orientation of structures. Figure 4: Closeup of the grid at the leading and trailing edges 6

7 Figure 5: Profile of radial velocity at r/r=0.58, averaged over 100 instants. Figure 8: Profile of the average variation of radial velocity at r/r=0.814, averaged over 100 instants, normalized with respect to local blade rotational speed Figure 6: Profile of the root-mean square variation of radial velocity at r/r=0.814, averaged over 100 instants. Figure 9: Contours of vorticity in units of s -1 above the blade, showing the discrete structures, centered at r/r=0.7. Figure 7: Profile of the average variation of radial velocity at r/r=0.7, averaged over 100 instants, normalized with respect to local blade rotational speed Figure 10: Radial variation of the average vorticity of discrete structures, and their RMS variation, in the trailing edge plane. 7

8 Figure 11: Radial variation of the peak of the radial velocity profile, in crossflow planes at different chordwise stations. Figure 14: Iso-vorticity surfaces computed for a cross-flow of 30o yaw angle for a NACA0012 infinite wing at 12o angle of attack using a RANS k -SST turbulence model. Figure 12: Radial variation of the average strength of the lower and the upper shear layers with respect to the average strength of the structures Figure 15: Computed radial velocity profile at r/r=0.586 for a rotating blade, 1mm behind the trailing edge. Figure 13: Isovorticity surfaces (letft) and vorticity contours (right) computed for a cross-flow of 30o yaw angle for a NACA0012 infinite wing at 12 o angle of attack using the HRLES-SGS turbulence method.. Figure 16: Velocity profile of the component of velocity along the surface of the blade, at r/r=0.586, computed vs. experimental. 8