Dynamic Stall Simulations on a Pitching Finite Wing

Dynamic Stall Simulations on a Pitching Finite Wing Kurt Kaufmann, Christoph B. Merz, and Anthony D. Gardner German Aerospace Center (DLR), G¨ottingen, Germany URANS computations were carried out on a pitching finite wing model using the finite volume solver DLR-TAU. The comparison with the experimental data reveals a good agreement, especially in the region of the first occurrence of stall. Discrepancies are observed in the blade tip region, where the flow in the numerical data shows a stronger separation along with larger hysteresis effects in contrast to the experiment. An investigation with different pitching frequencies revealed a change in the sequence of dynamic stall vortex formation in the spanwise direction. For the lowest frequency the propagation of the stall vortex starts at the wing tip and then spreads rootwards, whereas for the higher frequencies the first evolution of the dynamic stall vortex starts further inboard, subsequently propagating in tipward and rootward directions. An investigation using a γ-Reθt approach demonstrated that small differences can be attributed to transition, nevertheless these can give further insight in the physics of dynamic stall and improve the comparability with the experiment. A comparison with two-dimensional simulations shows strong similarities in the sections where the vortex starts to evolve and large differences in the surrounding areas.

Nomenclature b c Cl CL Cm CM Cp f m k Re S t U∞ u x y z

y intercept Chord length, m Sectional lift coefficient Global lift coefficient Sectional pitching moment Global pitching moment Local pressure coefficient Frequency, s−1 gradient Reduced frequency Reynolds number Span, m Time, s Freestream velocity, m/s Velocity component along x, m/s Cartesian coordinate in streamwise direction, m Cartesian coordinate in spanwise direction, m Cartesian coordinate perpendicular to x and y, m

Greek and other symbols α Angle of attack, deg ∆ Difference (between two values) Subscripts 2D two-dimensional 3D three-dimensional max maximum value min minimum value r value at the wing’s root

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I.

Introduction

The aerodynamic effects of dynamic stall on pitching wings and airfoils differ significantly from the static case, as soon as the static stall angle is exceeded. The large drag and pitching moment peaks associated with this phenomenon prevent operation of helicopters in flight conditions which trigger dynamic stall over large areas of the rotor blade. To simplify the problem, most of the research on dynamic stall focuses on twodimensional pitching airfoils.1, 12, 15 While this has led to an understanding of the process and mechanism of dynamic stall, it provides only limited insight into the three-dimensional nature of dynamic stall. Several experimental and numerical studies on full helicopter configurations have been carried out in the past. These revealed the complex flow field around the rotor blades, but the combination of downwash, blade-wake-vortex interactions and the elasticity of the rotor blades leads to a limited comparability with numerical computations. An intermediate approach is to use stiff pitching finite wings with defined boundary conditions. In these configurations the blade tip vortex reduces and delays the occurrence of dynamic stall in the surrounding region, resulting in a strongly three-dimensional flow, see e.g. Le Pape et al.7 and Lorber.13 In addition, even two-dimensional experiments showed strong three-dimensional effects in the past, which may be either due to sidewall interferences, the flow through the gap between model and sidewall or a combination of both.3 To capture the three-dimensional nature of dynamic stall it is important to minimize the influence of the wind tunnel walls on the evolution of the dynamic stall vortex and the blade tip vortex in an experimental setup. Numerical investigations on three-dimensional configurations are still rare on account of the high computational costs and the lack in experimental reference material. The investigations of Spentzos et al.,17 Costes et al.,2 and Kaufmann et al.6 have demonstrated that upon the onset of dynamic stall at a certain spanwise position, the stalled region spreads rapidly in the spanwise direction resulting in the formation of an Ω-shaped vortex. A qualitative agreement between the experimental and numerical data was achieved for each of those investigations. As part of the DLR STELAR project, this work comprises three-dimensional computations using the DLR-TAU finite volume solver. The simulations were set to match the experimental setup carried out in the Side Wind Simulation Facility (SWG) in G¨ottingen.10, 11, 18

II.

Experimental and numerical setup

Figure 1: The blade tip model installed in the Side Wind Simulation Facility G¨ottingen (SWG) (left) from Merz et al.11 and the CAD model (right) including pressure taps (enlarged, blue dots) and PIV planes (green surfaces). A stiff blade tip model was built and experimentally investigated in the SWG by Merz et al.10, 11 , Fig. 1. The blade tip model has an aspect ratio of 6.2 with a chord length of 0.27 m, a span of 1.62 m and a parabolic blade tip shape often used in helicopter rotor blade configurations. The DSA-9A helicopter airfoil is used across the entire span, aside from the innermost part where due to the force transmission a thicker airfoil is used. To shift the onset of stall away from the wind tunnel walls, the wing has a positive linear twist of 5.5◦ , resulting in a larger angle of attack at the wing tip compared to the wing root. The model contains three sections equipped with 22 pressure sensors each at constant span to integrate the aerodynamic coefficients at these locations with 20 kHz. Additionally, 34 pressure transducers are spread over the upper surface to capture the propagation of the dynamic stall vortex. PIV measurements were performed with an acquisition

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rate of 1 kHz at several sections of constant span to capture the flow field above the wing’s upper side. A series of pitching motions was performed resulting in a set of different mean angles of attack, amplitudes and reduced frequencies. This study focuses on a single pitching motion with three different frequencies: αr =11◦ ±6◦ with M =0.16, Re=9×106 , k=πf c/U∞ =0.025, 0.049 and 0.076, respectively.

Figure 2: The hemispherical grid domain, the surface grid at the wing’s tip and the wing’s root, as well as a section of the grid at the middle of the span and at the rotation axis (from left to right). The numerical simulations in this study were carried out using the DLR-TAU code.16 The URANS equations were closed by the Menter SST turbulence model8 for all three frequencies computed, using a central method with artificial matrix dissipation to solve the viscous fluxes and the inviscid fluxes. In addition, the Menter SST turbulence model coupled with the γ-Reθt transition transport model9 was used for the highest frequency computed to investigate the effect of transition. Because of convergence problems for this approach, the viscous and inviscid fluxes were solved by a second order Roe scheme and a central method with artificial scalar dissipation, respectively. For the fully turbulent and the transition computations a dual time stepping approach with 3000 time steps per pitching period were used and between 200-600 Newton iterations were needed to converge each pseudo-time step. Therefore, Cauchy control was used assuming convergence of the pseudo-time step when the relative variation of the last 20 iterations was below a threshold of 10−7 for each of the aerodynamic coefficients (CL ,CD and CM ), as well as the maximum eddy viscosity and the total kinetic energy, resulting in a drop of the Newton residuals by at least one order of magnitude. The mesh generated was based on the two-dimensional study of Richter et al.,15 the knowledge obtained in the three-dimensional DLR-TAU computations of Kaufmann et al.6 and on a grid sensitivity study shown at the end of this paper, see Table 2 and Figure 14. A hemispherical grid domain with a radius of 500 chords, symmetry conditions at the middle line and farfield conditions at remaining boundaries was built, see Fig. 2. The wing’s surface boundary layer is discretized with 30 prisms in the normal direction and tetrahedrons outside the boundary layer. The prisms are anisotropic, being stretched by a factor of 3 in the spanwise direction in the region outside of the parabolic blade tip and the thickened root. The height of the first prismatic layer and the stretching factor in the wall normal direction were adjusted to reach y + ≤1 and the boundary layer thickness, respectively. The refined trapezoidal area with a streamwise length of 6c, a maximum height of 5c and cell sizes of l/c=3.3% was stretched in the spanwise direction from the root beyond one chord over the tip to resolve the wake and the vortices generated by the wing.

III. A.

Results

Comparison with the experimental data

The integrated aerodynamic coefficients of the DLR-TAU simulations and the experimental data at the spanwise positions y/S=0.49, 0.68 and 0.86 are compared in Figure 3. Only the pressure information at the sensor positions was used in the numerical simulations to obtain comparable aerodynamic coefficients. The forces were then integrated using a trapezoidal rule. The experimental data was recorded with 20 kHz and averaged over 160 pitching cycles. For clarity, the standard deviation is plotted for every 50th point only. Additionally, the values for 5 single cycles are plotted to receive an impression of the cycle to cycle variations of the experimental data. For the three sections (y/S=0.49, 0.68 and 0.86, see Figure 3) larger lift coefficients are reached during the upstroke in the numerical data. At y/S=0.49 (Figure 3 top), one lift peak is obtained in the numerical data, whereas two exist in the experiment. As shown by Merz et al.,11 the first peak is a result of a vortex formation further outboard than y/S=0.49 and the associated accelerated flow. The second peak originates from the vortex formation at y/S=0.49. After stall lower lift coefficients are reached in the experiment. Eventually, reattachment takes place at the same angle of attack. The lift

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y/S=0.49

1.5

Cm

Cl

y/S=0.49

0

2

−0.2

1 0.5 y/S=0.68

y/S=0.68

0

2 1.5

Cm

Cl

DLR-TAU Exp. phase averaged Exp. 5 single cycles

−0.4

−0.2

1 0.5 y/S=0.86

y/S=0.86

0

2 1.5

Cm

Cl

DLR-TAU Exp. phase averaged Exp. 5 single cycles

−0.4

−0.2

1

DLR-TAU Exp. phase averaged Exp. 5 single cycles

−0.4

0.5 4

6

8

10

12 αr , ◦

14

16

4

18

6

8

10

12 αr , ◦

14

16

18

Figure 3: Comparison of the aerodynamic coefficients between the experimental data and the DLR-TAU computations for the αr =11◦ ±6◦ , k=πf c/U∞ = 0.049 dynamic case at y/S=0.49 (top), y/S=0.68 (middle) and y/S=0.86 (bottom).

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curves obtained at y/S=0.68 and y/S=0.49 show a qualitatively similar behavior with the exception that only one peak is present in the experimental data at y/S=0.68. Furthermore, stall begins at an earlier time point and is less pronounced in the experiment. Following stall the lift values are lower in the experiment and a similar reattachment behavior is observed. However, the lift curve at the outer section and y/S=0.86 exhibits a different behavior. This section is located directly next to the parabolic wingtip and is strongly influenced by the 3D wingtip flow. The lift gradient is reduced and the lift curve becomes nonlinear as the blade tip vortex effects start to increase. In comparison to the other sections, the maximum lift peak and the hysteresis effects are reduced. In the numerical data the lift stall is more pronounced and reattachment sets in sooner than in the experiment. During attached flow, a small offset of the pitching moment coefficients between experiment and simulation can be observed for all three sections. The drop in the pitching moment is shifted to an earlier time point in the experiment compared to the CFD for the two inner sections, whereas for the outer section a similar behavior is observed. For all three sections the peak values of the simulations exceed the phase averaged determined negative peaks in the pitching moment. The flow reattachment of the section at y/S=0.86 starts earlier in the simulation, whereas for the sections at y/S=0.49 and y/S=0.68 reattachment of the experiment and the simulation takes place at the same time. Stall is not a phase locked event and phase averaging partly cancels out the peak values, see also Ramasamy et al..14 Instead, conditional averaging can be performed by measuring the peak heights and positions in each cycle of the original data and averaging those values. Table 1 shows the peak values and their time of occurrence of the phase averaged, the conditional averaged experimental data and the numerical simulation. The lift peaks of the phase averaged data are reduced by 8-10% and the pitching moment peaks by 28-35% in comparison to the conditional averaging. When comparing the conditional averaged values with the numerical simulation, the peak values obtained are mostly at the upper limit of the standard deviation. The extreme values of the aerodynamic coefficients over the span are plotted in Figure 4, where the numerical prediction of the forces is evaluated at 12 additional sections across the wing’s span in order to get an impression of the spanwise spreading of the separation. In the numerical data, two lift peaks can be seen at y/S=0.45 and y/S=0.68 across the span which also occur later than in the other sections. The pitching moment exhibits three peaks across the span at y/S=0.31, y/S=0.49 and y/S=0.68. Except for the tip and root region, the minimal pitching moment occurs in a relatively short timeframe in the simulation. In contrast to this, the sectionwise Cmmin values of the experiment are spread over a longer period of time. The onset of separation is comparable between the experiment and the simulations, however the propagation varies, particularly in spanwise direction. In the experiment the separation area is smaller and the spanwise propagation speed of the separation is slower, whereas in the numerical data separation sets in almost simultaneously over a large spanwise area. Table 1: Comparison between the experiment and the DLR-TAU data for the maximum lift, minimal pitching moment and their time of occurrence. Section

Exp. phase averaged

Exp. conditional averaged

DLR-TAU

1

Cl,max t/T at Cl,max Cmmin t/T at Cm,min

1.75±0.12 0.416 -0.25±0.05 0.433

1.89±0.08 0.409±0.014 -0.32±0.04 0.431±0.010

1.94 0.432 -0.40 0.455

2

Cl,max t/T at Cl,max Cmmin t/T at Cm,min

1.69±0.10 0.375 -0.25±0.05 0.417

1.87±0.09 0.385±0.013 -0.32±0.05 0.422±0.023

2.09 0.443 -0.41 0.449

3

Cl,max t/T at Cl,max Cmmin t/T at Cm,min

1.54±0.06 0.415 -0.17±0.05 0.43

1.67±0.07 0.423±0.019 -0.23±0.05 0.476±0.053

1.75 0.448 -0.25 0.0454

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0 DLR-TAU Exp. phase averaged Exp. conditional averaged

Cm,min

Cl,max

2 1.8

−0.4

0.45

t/T at Cm,min

t/T at Cl,max

1.6

−0.2

0.4 0.35 0.3

0.5

0.45

0.4 0.2

0.4

0.6 y/S

0.8

1

0.2

0.4

0.6 y/S

0.8

1

Figure 4: Comparison of maximum lift and minimum pitching moment (top) and their occurrence (bottom) over the span. The pressure distributions of the three sections indicated in Figure 1 (right) are plotted in Figure 5 for three time instants to capture the separation process. The experimental data was averaged over the 160 pitching cycles. The first row shows the phase averaged pressure distributions for αr =14.0◦ ↑, where the standard deviations are relatively small and only minor cycle to cycle variations exist. The second and third row show the conditional averaged data of the maximum lift and minimal pitching moments so as to capture the flow physics of every single cycle. In the first row, the pressure distributions match well, except for the suction peak which is more pronounced in the numerical data. The reason for the differences in the pitching moment of Figure 3 during attached flow is a variation in pressure observed at the rear part of the airfoil’s suction side, which can only be seen in the zoom in Figure 6. At Cl,max the suction peak of all three sections collapsed, except for the section at y/S=0.86 of the experiment where only a small reduction of the suction peak is visible in comparison to αr =14.0◦ ↑. In the numerical pressure distribution two pronounced suction areas at y/S=0.49 between x/c=0.1-0.3 and at y/S=0.68 between x/c=0.25-0.8 can be seen. In contrast the experimental data do not show these areas, as for the section at y/S=0.49 after the remaining suction peak at the nose an almost linear increase of the pressure values on the suction side is visible. Nevertheless, beside the suction peak and the suction area comparable pressure values are reached. At y/S=0.68 the experiment revealed an area of low pressure directly behind the remaining suction peak up to the pressure sensor at x/c=0.24, thus further upstream in comparison with the numerical data. At Cm,min the numerical and experimental data display a close correlation. Nonetheless, the numerically determined pressure values are reduced at the rear part of the two inner sections, while the experiment mostly showed a flat pressure distribution. For the outer section small differences can be observed in the front part of the suction side where the experiment exhibits a remaining suction peak, whereas the numerical simulations predict a small pressure buckle between x/c=0.16 and 0.24. In Figure 7 the streamwise velocity components of the conditional averaged PIV measurements (left) and the numerical data (right) at Cmmin (compare Figure 5 bottom) are shown for the three sections as indicated in Figure 1 (right). The PIV acquisition was carried out for 10 cycles and the data depicted here are averaged over the 5 nearest neighbors of the sectionwise Cmmin of each cycle. DLR-TAU indicates higher maximum speeds in comparison with the experiment. This applies to the accelerated flow above the vortex as well as the backflow close to the surface. For the two inner sections the 6 of 17 American Institute of Aeronautics and Astronautics

Cp

−10

DLR-TAU αr =14.0◦ ↑ Exp. αr =14.0◦ ↑

DLR-TAU αr =14.0◦ ↑ Exp. αr =14.0◦ ↑

DLR-TAU αr =14.0◦ ↑ Exp. αr =14.0◦ ↑

DLR-TAU Cl,max Exp. Cl,max

DLR-TAU Cl,max Exp. Cl,max

DLR-TAU Cl,max Exp. Cl,max

DLR-TAU Cmmin Exp. Cmmin

DLR-TAU Cmmin Exp. Cmmin

DLR-TAU Cmmin Exp. Cmmin

−5 0

Cp

−10 −5 0

Cp

−10 −5 0 0

0.5 x/c

1

0

0.5 x/c

1

0

0.5 x/c

1

Figure 5: Comparison between the experimental and numerical data of the pressure distributions at y/c=0.49 (left), y/c=0.68 (middle) and y/c=0.86 (right) for different time instants.

−0.4 Cp

−0.2 0

DLR-TAU Exp.

0.2 0.4 0.6

0.8 x/c

1

Figure 6: Zoom of the pressure distribution’s rear part of the section at y/S=0.49 and αr =14.0◦ ↑.

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(a) y/S=0.49 PIV

(b) y/S=0.49 DLR-TAU

(c) y/S=0.68 PIV

(d) y/S=0.68 DLR-TAU

(e) y/S=0.86 PIV

(f) y/S=0.86 DLR-TAU

Figure 7: Streamwise velocity components of the averaged PIV measurements (left) and the numerical data (right) at the sectionwise Cmmin for the 3 sections shown in Figure 1 (right).

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experiment and the simulations correspond well with each other. The separation areas in the images are of the same size, nevertheless higher backflow velocities are observed in the numerical data, especially in the rear part of the airfoil. Differences occur in the outer section, where in the experimental data no backflow region can be identified, whereas in the simulation a leading edge vortex is created. The vortex positions of the numerical data correlate well with the additional suction seen in Figure 5. B.

Global dynamic stall evolution

(a) αr =8.0◦ on the upstroke

(b) αr =15.0◦ on the upstroke

(c) αr =16.3◦ on the upstroke

(d) αr =16.7◦ on the upstroke

(e) αr,max =17.0◦

(f) αr =11.6◦ on the downstroke

Figure 8: Evolution around the DSA-9A wing for the dynamic stall case of k=πf c/U∞ =0.049: Visualized by means of isosurfaces of the λ2 criterion5 and Cp contour plots. The evolution of the separation of the DLR-TAU computations around the finite wing are illustrated in Figure 8. The vortices were visualized with isosurfaces of the λ2 criterion5 and color coded with Cp for several time instants. In all snapshots the blade tip vortex and the wake are clearly visible. At αr =8◦ on the upstroke the flow is fully attached, the blade tip vortex is directly generated at the blade tip and is relatively small in comparison to the other time instants. Separation starts to occur at the leading edge of the parabolic blade tip and the vortex wraps around the blade tip, Figure 8b. In parallel the blade tip vortex moves further inboard and trailing edge separation sets in along the entire span. At αr =16.3◦ on the upstroke the dynamic stall vortex starts to evolve at the leading edge between y/S≈0.49 and y/S≈0.71. Simultaneously, the separation at the parabolic blade tip spreads further inboard. High spanwise separation propagation speeds are reached and at αr =16.7◦ on the upstroke the dynamic stall vortex and the parabolic 9 of 17 American Institute of Aeronautics and Astronautics

tip vortex have fused, building a separation line along the leading edge from y/S≈0.31 up to the blade tip. Additionally, several Ω-shaped vortex structures begin to evolve, the largest of which is formed at y/S≈0.59. At αr,max =17.0◦ the flow over the entire wing is separated and the first Ω-shaped structures are transformed into ring shaped structures as they are shed from the wing. Moreover, another large Ω structure is formed further inboard at y/S≈0.37. At this time instant, the blade tip vortex bends directly behind the trailing edge and expands afterwards. Eventually, at αr =11.6◦ on the downstroke reattachment sets in, a suction peak along the span is visible and the remaining small vortex structures are propagating downstream. In addition, the blade tip vortex assumes a spirally twisted form. C.

Frequency variation

Figure 9 shows the vortices shed from the wing for the frequencies k=πf c/U∞ =0.025 (left) and k=0.076 (right). With increasing frequencies stall is shifted to higher angles of attack and the aerodynamic forces become stronger. For the lowest frequency the chronology of vortex formation differs significantly from the others. In this case, separation starts from the blade tip as for higher frequencies, but at lower angles of attack (αr =14.0◦ ), see Figure 9a. Moreover, trailing edge separation across the entire span is visible. Large discrepancies are observed in the subsequent time instants. Originating from the blade tip the separation only spreads in the rootward direction and a dynamic stall like vortex is formed at y/S≈0.85 near the blade tip at αr =15.0◦ accompanied by CLmax of the entire wing. Afterwards, at αr =15.9◦ the separation continues to spread rootwards up to y/S≈0.35 and the first large vortex structure is shed from the wing. Its shape is not as round as for the other frequencies and it is stretched in the vertical direction. As a consequence of the downstream moving vortex structure close to the blade tip, strong disturbances are present in the blade tip vortex. Finally, at αr =16.5◦ as the overall CMmin is reached, a vortex is wrapped around the entire wing building a smooth coherent vortex structure. The separation behavior of the case with k=0.076 is similar to the case with k=0.049, although as a result of the higher pitching frequency the separation starts at higher angles of attack and the accompanying forces are higher. At αr =15.9◦ separation at the blade tip and trailing edge separation along the span are visible. In Figure 9d, the first dynamic stall vortex is formed at the leading edge between y/S≈0.37 and 0.71. This is slightly further inboard than for the k=0.049 dynamic stall case. Afterwards, the separated areas join and extend from the blade tip up to y/S≈0.28 at the maximum angle of attack, where also the overall CMmin of the wing is reached. Eventually, the entire wing is separated and multiple Ω structures are formed. The rapid loss in lift is accompanied by an expansion and a weakening of the blade tip vortex. D.

Transition effects

A possible reason for the differences observed is that the experiment was performed without tripping and the numerical investigations were carried out fully turbulent. To investigate the transition effects the Menter SST computations were additionally carried out using a γ-Reθt transition transport model for the highest frequency of k=0.076. The approach of Menter et al.9 was implemented in DLR-TAU by Grabe et al.4 In a first step, the farfield values of the turbulent intensity were calibrated to match the turbulence level in front of the wing with the turbulent intensity measured in the SWG. In Figure 10 the aerodynamic coefficients of lift (left) and pitching moment (right) are plotted for the sections at y/S=0.49, y/S=0.68 and y/S=0.86. At y/S=0.49 (Figure 10 top, left) the lift values during the upstroke for both numerical approaches are comparable and higher than in the experiment. The two detail views clarify the kinks obtained in the experimental data (left) as well as in the γ-Reθt simulations (right). Both are accompanied by the stagnating transition movement on the suction side near the airfoil’s nose. The lift overshoot at y/S=0.49 occurs at the same angle of attack for the experimental data and the γ-Reθt model and is delayed in the Menter SST simulations. Another similarity between the experiment and the transition model is the existence of two lift peaks for which the first is at the same time instant and the second is earlier in the simulation. The reason for the earlier stall between the two numerical approaches is that in the γ-Reθt computations trailing edge separation sets in earlier compared to the fully turbulent computations. This is in better agreement with the experiment as shown in the rear part of the pressure distribution at αr =14.0◦ ↑ in Figure 11 (left). Nevertheless, the lift peak values of both simulations are higher. During downstroke the lift values observed are in good accordance between the experiment and the transition computations, whereas in the fully turbulent simulations higher lift values are observed. Lift coefficient fluctuations are observed in the second half of the downstroke. These are a result of the integration process in the γ-Reθt

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(a) 1.6Hz αr =14.0◦ on the upstroke

(b) 4.9Hz αr =15.9◦ on the upstroke

(c) 1.6Hz αr =15.0◦ on the upstroke

(d) 4.9Hz αr =16.9◦ on the upstroke

(e) 1.6Hz αr =15.9◦ on the upstroke

(f) 4.9Hz αr,max =17.0◦

(g) 1.6Hz αr =16.5◦ on the upstroke

(h) 4.9Hz αr =16.5◦ on the downstroke

Figure 9: Dynamic stall evolution around the DSA-9A wing for the frequencies k=πf c/U∞ =0.025 (left) and 0.076 (right): Visualized by means of isosurfaces of the λ2 criterion5 and Cp contour plots.

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y/S=0.49 y/S=0.49

.5 1.5

.5 0.5 2 2

Cm

−0.2

y/S=0.68

y/S=0.68

0

2

Cl

Cm

y/S=0.68 1.5 y/S=0.68

0 0

−0.2

1 0.5 y/S=0.86

0

2 1.5

Cm

Cl

CmCm

−0.4

−0.2

1 −0.4

0.5 4

6

8

10

12 αr , ◦

14

16

4

18

−0.2y/S=0.86 −0.2 −0.4 −0.4

DLR-TAU fully turbulent DLR-TAU γ-Reθt Experiment

6

8

10

12 αr , ◦

14

16

18

y/S=0.86 Figure 10: Comparison of the aerodynamic coefficients between the experimental data and the DLR0 TAU computations for the α =11 ±6 , k= 0.076 dynamic case at y/S=0.49 (top), y/S=0.68 (middle) and y/S=0.86 y/S=0.86 (bottom). 0 r

◦

◦

CmCm

.5 1.5 1 1

.5

−0.2y/S=0.49 −0.2 −0.4 −0.4

−0.4

0.5

.5 1.5 1 1

1.5 1

.5 0.5 2 2

0

2

Cl

1 1

y/S=0.49

0 0 CmCm

2 2

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−0.2 −0.2 −0.4 −0.4

DLR-TA DLR-TA DLR-T Experim DLR-T Experim

model. As the transition location is moving downstream on the upper side small pressure waves are created and as they pass the sensor position additional lift is produced, see Figure 11 (right). −0.4

−10 Cp

−0.2 Cp

DLR-TAU fully turbulent DLR-TAU γ-Reθt Experiment

0

−5

0.2 0

0.4 0.5

0.6

0.7

0.8 x/c

0.9

0

1

0.2

0.4

0.6

0.8

1

x/c

Figure 11: Pressure distribution of the dynamic stall case with k=0.076 at y/S=0.49 for αr =14.0◦ ↑ (left) and for αr =10.0◦ ↓ (right). As for the frequency of k=0.049, the pitching moment of the Menter SST simulations differs from the experimental data during the upstroke, see Figure 10 (top, right). For the γ-Reθt computations a better agreement with the experiment is obtained. As the pressure sensors of the experiment are mainly distributed over the suction side, small pressure differences at the pressure side can have a large impact on the integrated forces. In this case, for low angles of attack slightly higher pressure values at a pressure sensor at x/c=0.95 are obtained in the γ-Reθt in comparison to the fully turbulent computations (not shown here) leading to the differences obtained in the pitching moment. The pitching moment stall at y/S=0.49 first occurs in the γ-Reθt approach, then in the experiment and ever later in the fully turbulent simulations, see Figure 10 (top, right). This is also the case for the occurrence of the pitching moment peak, which is clearly higher in both simulations compared to the experiment. At y/S=0.68 during the first half of the upstroke the lift curves are similar for the numerical simulations and lower in the experiment, see Figure 10 (middle, left). In the second half the experimental data approaches the lift values obtained in the simulations. As for y/S=0.49, two lift peaks are present in the transition simulations, however the first peak is much smaller than the second. In the experiment a broad range of lift overshoot can be observed. The second lift peak of the γ-Reθt computations and the peak in the fully turbulent simulations are significantly higher than in the experiment and also than in any other sections. The remaining process is similar to the process at y/S=0.49. At low angles of attack the pitching moment of the y/S=0.68 section (Figure 10 middle, right) is similar to the pitching moment at y/S=0.49. At the beginning comparable values between the transition model and the experiment are obtained and the pitching moment in the fully turbulent simulations is lower. For this section a good agreement of the starting pitching moment stall can be observed between experiment and transition simulations, whereas again stall is delayed in the fully turbulent approach. High pitching moment peaks are observed at this section, the highest for the γ-Reθt model and again reduced values for the experiment. As a consequence of the blade tip effects, the gradient of the lift curve for the outermost section is reduced compared to the other sections, see Figure 10 (bottom, left). At the end of the upstroke a nonlinear behavior is present in all the curves. The lift overshoot again starts first for the γ-Reθt model, afterwards in the experiment and the fully turbulent simulations. The peak values are comparable between the two numerical simulations and lower for the experimental data. Noticeable differences are obtained in the hysteresis effects, where for the experiment these are relatively low and the highest can be recognized for the γ-Reθt model. At first, the pitching moment at this section has a similar behavior as for the other sections, Figure 10 (bottom, right). At low angles of attack the fully turbulent computations reveal the lowest values, the experiment the highest and the values of the transition model are in between. Pitching moment stall occurs first for the γ-Reθt simulations, followed by the experiment and the fully turbulent simulations. The pitching moment peak at this section is significantly reduced in comparison to the other sections. However, for this section the largest peak is reached in the fully turbulent computations.

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E.

Comparison between 2D and 3D stall

Historically, dynamic stall simulations were mostly carried out on two-dimensional configurations. This study aims to compare three-dimensional and two-dimensional computations for the dynamic stall case with k=0.049. In a first step, static computations were carried out to investigate the reduction of the effective angle of attack at each of the three sections that is initiated by the blade tip vortex, see Figure 12. The gradient of the 2D slope (m2D =0.115) is higher than for the 3D sections, as no downwash is present in the 2D computations. In addition, the y intercept b of the 2D computations is lower than for the 3D sections, because of the finite wing’s linear twist. The 3D slopes of the inner two sections (my/S=0.49 =0.098 and my/S=0.68 =0.091) are comparable, whereas the gradient of the outer section is strongly reduced (my/S=0.86 =0.080). As a result of the linear twist of the finite wing, at low angles of attack the outermost section shows the highest lift values, followed by the middle section and the innermost section. The gradients m and the y intercepts b of the static Cl curves in Figure 12 of the three 3D sections and of the 2D computations are used for a linear extrapolation to observe the pitching motions for the 2D dynamic stall simulations in the following way: α2D =(m3D ∗ α3D + b3D − b2D )/m2D . For the 2D simulations the corresponding pitching motion for the three sections are: α2D =11.52◦ ±5.12◦ for y/S=0.49, α2D =11.27◦ ±4.75◦ for y/S=0.68 and α2D =9.23◦ ±3.75◦ for y/S=0.86 all using the same frequency of k=0.049. 1.5

Cl

1 0.5 2D 3D y/S=0.49 3D y/S=0.68 3D y/S=0.86

0 −0.5 −5

0

5 α2D ,

◦

10 and αr ,

15

20

◦

Figure 12: Comparison of lift for static DLR-TAU computations using Menter SST between the twodimensional case and the three-dimensional case at the three sections. The lift and pitching moment curves for the three 2D simulations and the three sections of the 3D dynamic stall case with αr =11◦ ±6◦ and k=0.049 were compared and are shown in Figure 13. For this purpose, all angles of attack for the 2D case were converted to the angle of attack at the wing’s root in the following way: αr = (m2D ∗ α2D − b3D + b2D )/m3D . In Figure 13 (top, left), the section of the 3D case at y/S=0.49 is compared with the 2D simulation using α=11.52◦ ±5.12◦ . The lift values observed during the upstroke in 3D agree well with the 2D simulations, however the gradient of the 2D case is slightly reduced in comparison the 3D case. The occurrence of the lift peak is shifted to higher angles of attack for the 2D case and to a later phase of the pitching motion. After stall strong fluctuations are present in the 2D case and the lift values are higher than in the 3D case. Finally, reattachment sets in earlier in the 2D simulations. In the pitching moment (Figure 13 top, right) similar values are obtained during the upstroke. The earlier occurrence of stall for the 3D case is also visible in the pitching moment. Nevertheless, higher peak values are obtained in the 2D simulations, followed by an earlier start of reattachment. Figure 13 (middle) shows the aerodynamic coefficients of the y/S=0.68 section and the 2D simulations using α=11.27◦ ±4.75◦ . The lift values of the finite wing are again comparable with the 2D simulations during the upstroke. Although a larger offset in lift is visible, the gradient of the two curves are in better agreement compared to the y/S=0.49 section. At y/S=0.68, the lift peak values of the 2D case are lower than in 3D. In contrast, the pitching moment peak of the 2D case is higher than for the corresponding finite wing’s section, see Figure 13 (middle, right). Nevertheless, the 2D stall occurs later and the hysteresis is much less pronounced than in the 3D case and also in comparison with the 2D simulation using α=11.52◦ ±5.12◦ . In 14 of 17 American Institute of Aeronautics and Astronautics

addition, the reattachment takes place at significantly higher angles of attack for the 2D case. In the bottom row of Figure 13, the outermost section is compared with the 2D computation using α=9.23◦ ±3.75◦ . In this case, only the first phase of the upstroke shows similarities with the 2D case, afterwards the 3D curve becomes nonlinear because of the blade tip vortex effects. In the 2D case no stall is present, whereas in the 3D case separation does occur, albeit to a lesser degree than at the other sections. Altogether, the analysis leads us to assume that although the 3D case shows similarities in the magnitude of stall in areas where it is triggered, this does not hold true for locations away from the stall triggering point, as they are influenced by the stall event. This leads to an earlier onset of stall with more pronounced hysteresis in 3D and the maximum stall peaks are weakened as they propagate in spanwise direction. 0

1.5

Cm

Cl

2

−0.2

1 0.5

0

2 1.5

Cm

Cl

3D y/S=0.49 2D α=11.52◦ ±5.12◦

−0.4

−0.2

1 0.5

0

2 1.5

Cm

Cl

3D y/S=0.68 2D α=11.27◦ ±4.75◦

−0.4

−0.2

1 3D y/S=0.86 2D α=9.23◦ ±3.75◦

−0.4

0.5 4

6

8

10

12 αr , ◦

14

16

18

4

6

8

10

12 αr , ◦

14

16

18

Figure 13: Comparison between the DLR-TAU computations for the 2D and 3D dynamic case with k=0.049.

F.

Grid sensitivity study

Four different grids were generated to investigate their sensitivity on the following dynamic stall test case: αr =11◦ ±6◦ M =0.16, Re=9×105 , k=πf c/U∞ =0.1. Step by step, the grids were refined in the crucial regions of the flow field. For the four grids the minimum and maximum cells size on the surface, the points in the boundary layer and the size of cells in the refined areas close to the wing, see Figure 2 (right), were gradually refined. A description of the grid resolutions can be found in Table 2. Figure 14 shows the lift (left) and pitching moment curve (right) integrated across the entire wing for the four grids used. For the coarse mesh the largest discrepancies can be observed. During the upstroke the

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Table 2: Resolution of the investigated unstructured grids of the DLR-TAU grid sensitivity study. Grid name

coarse

medium

fine

very fine

minimum cell size at leading and trailing edge maximum cell size on the surface tetrahedron size in stall area points in boundary layer grid nodes

0.1% chord 4% chord 6.7% chord 20 5,252,267

0.067% chord 1.67% chord 5% chord 25 11,108,788

0.033% chord 1% chord 3.3% chord 30 21,142,546

0.025% chord 0.75% chord 3.3% chord 35 27,223,196

lift values are lower than for the other grids used. Also the width of the pitching moment is broader than for the other computations. For the other three grids used the coefficients at attached flow are similar. The onset of dynamic stall is shifted to higher angles of attack as the mesh fineness increases, while the changes in the onset angle of dynamic stall decrease. Differences especially in the stalled region are still visible for all four cases. Besides these, minor changes occur between the fine and the very fine grid and a qualitative grid convergence can be noticed. The position of the stall onset and all main features of dynamic stall are captured by both grids. Therefore, all further investigations were carried out on the fine grid. coarse medium fine very fine

CL

CM

1.5

0

−0.1 coarse medium fine very fine

1 −0.2 4

6

8

10 12 ◦ αw ,

14

16

18

4

6

8

10 12 ◦ αw ,

14

16

18

Figure 14: Grid sensitivity study of the αr =11◦ ±6◦ , M =0.16, Re=9×106 , k=πf c/U∞ =0.049 dynamic stall case. Forces integrated across the entire wing.

IV.

Conclusions

A pitching finite wing model was numerically investigated with the finite volume solver DLR-TAU. The dynamic stall case with k=πf c/U∞ =0.049 showed a good agreement with the experimental data, nevertheless stall occurred later than in the experiment and the spanwise spreading of the stall was faster in the numerical simulations. It can be concluded that the onset of separation is comparable between the experiment and the simulations, but the propagation especially in the spanwise direction differs. In the experiment the separation area is smaller, whereas separation in the numerical data sets in almost simultaneously over a large spanwise area. A dynamic stall case with a lower frequency of k=πf c/U∞ =0.025 showed strong differences in the spanwise spreading of separation. In contrast to the higher frequencies, the separation starts at the blade tip and afterwards spreads only in rootward direction, whereas for the higher frequencies a leading edge vortex evolves at a further inboard section and then spreads in both directions. In addition to the fully turbulent computations, a γ-Reθt approach was tested for the k=πf c/U∞ =0.076 case. The results obtained showed an earlier start of trailing edge separation, in better accordance with the experimental data. A comparison with two-dimensional simulations shows strong similarities in the sections where the vortex

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starts to evolve and large differences in the surrounding areas. The 3D and 2D cases are comparable in the magnitude of stall in areas where it is triggered, but this does not hold true for locations away from the stall triggering point, as they are influenced by the stall event. This leads to an earlier onset of stall with a more pronounced hysteresis in 3D and the maximum stall peaks are weakened as they propagate in the spanwise direction.

Acknowledgments The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at the Leibniz Supercomputing Centre (LRZ, www.lrz.de).

References 1 Carr,L.

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