American Institute of Aeronautics and Astronautics

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Simulations of Airfoil Limit-cycle Oscillations at Transitional Reynolds Numbers

Weixing Yuan1 National Research Council (NRC) Canada, Ottawa, Ontario, K1A 0R6, Canada

Dominique Poirel2 and Baoyuan Wang3

Royal Military College of Canada (RMC), Kingston, Ontario, K7K 7B4, Canada

Mahmood Khalid4 National Research Council (NRC) Canada, Ottawa, Ontario, K1A 0R6, Canada

Structural response models addressing one-degree-of-freedom (1DOF) and two-degree-of-freedom (2DOF) aeroelastic oscillations were coupled with an in-house incompressible CFD code to perform large-eddy-based simulations (LES) for flows past rigid airfoils in free-to-rotate and free-to-rotate-and-heave conditions at low Reynolds numbers. As observed in experiments, the numerical simulations confirmed the presence of the self-sustained low-amplitude limit-cycle oscillations (LCOs) of the airfoils. It is understood that this behavior in the transitional Reynolds number regime results from the unsteadiness of the laminar boundary layer separation and its delayed recovery when compared to the corresponding static conditions.

Nomenclature b = wing span (m) c = airfoil chord length (m)

DC = drag coefficient, D / (½ρU 2bc)

LC = lift coefficient, L / (½ρU 2bc)

MC = aerodynamic moment coefficient, MEA / (½ρU 2bc2) D = drag force (N)

hD = structural damping coefficient in heave (N·s/m) Dθ = structural damping coefficient in pitch (N·m·s) f = frequency (Hz)

spf = natural structural frequency in pitch (Hz), / 2k Iθ θ π h = heave displacement of the elastic axis (m), positive upwards Iθ = mass moment of inertia about elastic axis (kg·m2)

hK = structural stiffness coefficient in heave (N/m) Kθ = structural stiffness coefficient in pitch (N·m)

1 Senior Research Officer, Institute for Aerospace Research, 1200 Montreal Road, weixing.yuan@nrc-cnrc.gc.ca, AIAA Senior Member.

2 Associate Professor, Department of Mechanical and Aerospace Engineering, poirel-d@rmc.ca, AIAA Senior Member.

3 Research Associate of RMC, Visiting Fellow at NRC, baoyuan.wang@nrc-cnrc.gc.ca. 4 Principal Research Officer, Institute for Aerospace Research, 1200 Montreal Road, mahmood.khalid@nrc-cnrc.gc.ca, AIAA Associate Fellow .

50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition09 - 12 January 2012, Nashville, Tennessee

AIAA 2012-0041

Copyright © 2012 by NRC and RMC. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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L = lift force (N), positive upwards EAM = aerodynamic moment about elastic axis (N·m), positive pitch up (clockwise)

hM = mass of all moving parts (kg) Mθ = mass of pitching parts (kg) Rec = Reynolds number based on airfoil chord length and freestream velocity, U∞c/ν U∞ = freestream velocity (m/s) St = Strouhal number based on airfoil thickness, fT/U∞ T = measured pitching oscillation period (s); airfoil thickness (m)

spT = natural structural period in pitch, Tsp=1/fsp t = time (s)

, ,u v w = Cartesian velocity components (m/s) , ,x y z = Cartesian coordinates (m)

xθ = static imbalance, non-dimensional distance (by half-chord) from EA to CG ,α θ = pitch angle (deg. or rad.), positive pitch up (clockwise)

zω = non-dimensional spanwise vorticity, zv ux y

ω ∂ ∂= −

∂ ∂

I. Introduction NTEREST in low-Reynolds-number flight vehicles has recently increased for both military and civil applications with emphasis on providing better aerodynamic performance, e.g., for emerging unmanned aerial vehicles

(UAVs). Low-Reynolds-number flows are not restricted to only small sized vehicles, but may also exist on surveillance vehicles operating at high altitude where a rarefied atmosphere restricts the flight to low Reynolds numbers. In the range of Reynolds numbers 104 ≤ Re ≤ 106, complex viscous flow phenomena can occur, such as laminar boundary layer separation, transition of the laminar shear layer, and subsequent turbulent reattachment, leading to the formation of a laminar separation bubble (LSB).1, 2 At higher angles of attack, this may be followed by an ultimate turbulent separation at the trailing edge. Previous work at the National Research Council Canada – Institute for Aerospace Research (NRC-IAR) had shown the capability to capture numerically these complex phenomena in the low-Reynolds-number regime.3

In parallel with the low-Re CFD work at the NRC-IAR, experimental aeroelastic investigations carried out at the Royal Military College of Canada (RMC) on a NACA 0012 airfoil showed self-sustained small-amplitude oscillations of the airfoil in the case where the airfoil was mounted on a support equipped to permit a free rotation.4 The investigations concluded that the trailing-edge laminar separation played a significant role in the airfoil oscillations.4, 5 It was reasoned that these aeroelastic oscillations were related to the feedback coupling mechanism that existed between the LSB behavior and the structural response, in a manner reminiscent of the classical stall flutter problem occurring at high angles of attack. The unsteady nature of the transition process made the prediction of the reattachment location difficult. In order to get a deeper insight into the physics of low-Re limit-cycle oscillations (LCOs) and their probable impact on UAVs, NRC-IAR and RMC are performing large-eddy simulation (LES)-like aeroelastic calculations for flows past airfoils oscillating at a free-to-rotate or a free-to-rotate-and-heave motion condition. This work is supported by the Department of National Defence (DND) Technology Investment Fund (TIF).

This paper presents the results obtained during the first half period of the three-year NRC-RMC collaboration. The structural equations modeling the free-to-rotate-and-heave airfoils were discretized and coupled with an NRC in-house incompressible code INSflow7. Note that the study does not account for the chordwise deformation of the airfoil itself. The current calculations confirmed the limit-cycle oscillations of the rigid but flexibly mounted airfoils.

II. Experimental Observations Poirel et al. have conducted experimental investigations of low-amplitude and self-sustained aeroelastic

oscillations of airfoils at low Reynolds numbers, which were reported in Refs. 4 and 5. The wind tunnel used for these experiments is a closed circuit facility and can achieve flow speeds ranging from 5 to 60 m/s. The wind tunnel has a test section of 0.76 m × 1.08 m and has a turbulence intensity level of less than 0.2% for the range of airspeeds considered in this work. The airspeed was measured with a Pitot-static tube located at the inlet of the test section and

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linked to a manometer. The pitch-heave apparatus was a two-degree-of-freedom (2DOF) system, composed of a rigid NACA 0012 wing moving in translation and in rotation, see Figure 1. The support structure, located outside the tunnel, had two similar top and bottom translating sub-systems, on which the rotation mechanisms were installed. Both rotation and translation mechanisms comprised two sets of pulley-springs to provide the elastic restoring force. The wing span was 0.61 m and its chord length was c = 0.156 m. End plates were installed to minimize 3D effects. The wing and the end plates resulted in a solid blockage ratio of 5%. The axis of rotation (elastic axis, EA) of the wing was located 0.01 m forward of the quarter-chord point, i.e., 18.6% chord aft of the leading edge. The mass ratio (the relative mass of the rotating wing to the surrounding air) was μ = Mθ/πρb(c/2)2 = 55.6. The airfoil motion was measured with potentiometers.

The results illustrated in this section are for a rectangular wing confined to a pure rotation, i.e., the heave DOF was held fixed. The time response revealed the loss of stability of the equilibrium point after a small perturbation, and re-stabilization on a limit-cycle oscillation (LCO) in pitch. The behavior is exemplified in Figure 2. These oscillations were self-sustained, consisting of a limit cycle, since there was no low-frequency periodicity in the upstream flow accounting for this behavior, nor in the wake or around the airfoil when it was held at a fixed angle.

Figure 1. Schematic (top view) of the 2DOF experimental setup at RMC.

Figure 2. Typical pitch angle response of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

The oscillations exhibited essentially simple harmonic motions for freestream airspeeds varying between

approximately 5 m/s and 11 m/s (5.0×104 ≤ Re ≤ 1.2×105). Outside of this range, the oscillations could not be sustained. There was a single dominant frequency that gave the spectral content of the LCO motion for the airspeed range. The spectrum displayed a main peak at the fundamental frequency, f, as well as much weaker super-harmonics at 2f, 3f, 4f and 5f, the higher super-harmonics gaining relative strength with increasing airspeed. No distinct dominating features were noticed at higher frequencies. The even harmonics accounted for a slight asymmetry in the experimental setup, whereas the odd harmonics were attributed to an intrinsic nonlinearity in the airloads.

The filtered pitch response and corresponding aerodynamic moment coefficients of the nominal configuration are shown in Figure 3. The filter cut-off frequency used for Figure 3 was 4 Hz such that only the fundamental LCO harmonic was captured. Figure 4 shows the aerodynamic moment coefficient as a function of pitch angle during one oscillation cycle. Different from Figure 3, the filter cut-off frequency was 25 Hz, and therefore, the higher harmonics are shown. The clockwise direction of the loops, as indicated by the arrow, physically means that the work done by the airflow on the airfoil was positive. In other words, the aerodynamic damping was negative and the flow transferred energy to the structure and sustained the oscillations. Hence, they were self-sustained from an aeroelastic point of view. The work per cycle, averaged over 18 LCO cycles, was calculated to be Waero = 0.0009 Nm for the nominal configuration at U∞ = 7.5 m/s. Details can be found in Refs. 4 and 5.

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Figure 3. Filtered (fc = 4 Hz) 1DOF pitch response and aerodynamic moment coefficient as a function of time. Nominal configuration (Ks = 0.30, EA at 0.18c), U = 7.5 m/s (Rec = 77,000).5 Time scale was modified from Figure 2.

Figure 4. Aerodynamic moment coefficients as a function of pitch angle during one cycle of 1DOF LCO for: i. 3rd order empirical model of experimental results, ii. Span- and phase-averaged filtered (fc = 25 Hz) forced-motion LES, iii. Unsteady linear thin-airfoil aerodynamics theory and iv. Unsteady linear aerodynamics theory modified to account for airfoil thickness. Nominal configuration: Rec = 77,000.5

It is worthwhile to add that the RMC’s 1DOF aeroelastic experiment was reproduced in a larger low-speed wind

tunnel at NRC. It used the same wing, pulleys and springs, but was mounted on a modified rig. The nominal free-stream turbulence intensity is the same in both tunnels. The self-sustained oscillations were observed but within a slightly narrower range of Re, with a smaller amplitude by ~ 0.5° - 1.0° and a larger frequency by ~ 0.5 Hz. These differences are accounted for by the increased rotational friction that was noticed with the modified rig. The reasoning is explained in Ref. 5.

Recently, RMC has extended the experimental investigation from one to two degrees of freedom by enabling the heave motion of the airfoil. The aeroelastic dynamics was studied in terms of the pitch-heave kinematics, as well as related to the work done by the flow, for a variety of structural stiffness coefficients and position of elastic (rotation) axis. In-depth analysis is provided in Ref. 6 and some results of the nominal 2DOF test case will be illustrated later in this paper for the comparison purpose.

III. Numerical Simulations The in-house block-structured code INSflow,7 developed for computing three-dimensional (3D) unsteady

incompressible flows, was applied in this study. INSflow has been used for a number of LES and unsteady Reynolds-averaged Navier-Stokes (URANS) calculations for various flows in incompressible regimes. Recent numerical investigations of low-Reynolds-number and flapping-wing aerodynamics can be found in Refs. 3, 5 and 8.

References 3 and 5 present large-eddy simulations of the laminar separation and laminar-turbulent transition of flows past stationary and pitching airfoils/wings at Reynolds numbers in the transitional regime. The research work in Ref. 8 attempted simulations of flapping wings in large-amplitude flapping motions.

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The integral form of the conservation laws of mass and momentum was used in INSflow. A fully implicit second-order temporal differencing scheme was used in the discretization, which made the algorithm stable for large timesteps. The discretization of the convective and diffusive fluxes was carried out in a co-located variable arrangement using a finite-volume approach that was second-order accurate in space. The coupling of the pressure and velocity was handled using the SIMPLE algorithm.9 The continuity equation was transformed into a pressure-correction equation that had the same general form as the discretized momentum equations. The use of the co-located variable arrangement on non-orthogonal grids required that the SIMPLE algorithm be slightly modified to dampen numerical oscillations. A pressure-velocity coupling method for complex geometries, used by Ferziger and Perić,10 was implemented, where an additional pressure gradient term as described by Rhie and Chow11 was subtracted from the velocity value at the surface of the control volume to prevent non-physical oscillations. A number of two-equation turbulence models and two sub-grid-scale (SGS) models were implemented for Reynolds-averaged Navier-Stokes (RANS) and large-eddy simulations.

The calculations were performed on moving grid configurations. The velocity of the grid movement was included in the governing equations in an inertial frame of reference. In order to avoid artificial mass sources generated by the grid velocity, a space conservation law was introduced to ensure a fully conservative property in the computations, as applied by Demirdžić and Perić.12

A linear 2DOF structural model was assumed for the rig, leading to the following equation of motion of the aeroelastic system:

0.5 ( ),EA EAI M cx h D K M tθ θ θ θθ θ θ− + + = (1)

0.5 ( ).h h hM h M cx D h K h L tθ θθ− + + = (2)

In the nominal (two-spring) configuration, the masses of all moving parts and that of the pitching part only were Mh = 2.5 kg and Mθ = 0.77 kg, respectively. The mass moment of inertia about the elastic axis located at 18.6% of the chord length from the leading edge was Iθ = 0.00135 kg·m2 and the structural stiffnesses as dictated by the springs were Kh = 1484 N/m and Kθ = 0.30 N·m for the two types of motion, respectively. The structural damping properties were represented by the classical linear viscous model, Dh = 10 N·s/m and Dθ = 0.002 N·m·s. The static imbalance, non-dimensional distance (by half-chord) from the elastic axis to the center of gravity was xθ = 0.19. The right hand sides of Eqs. 1 and 2 represent the aerodynamic moment and lift, respectively. When the heave DOF was held fixed, the above system of equations was reduced to a 1DOF system and in effect only a simplified version of Eq. 1 needed to be resolved:

( ).EA EAI D K M tθ θθ θ θ+ + = (3)

The structural equations were discretized in a fully implicit manner with second-order accuracy in time. The first derivative was calculated using a three-point backward differencing method:

1 1

1 23 4 ( ),2

n n nn O t

tθ θ θθ

+ −+ − +

= + ΔΔ

(4)

where the superscript (n+1) represents the new time level. When the use of the above equation was repeated, the second derivative was discretized using a five-point backward stencil:

1 1 1 1 2 3

1 2 22

3 4 9 24 22 8( ) ( ).2 4

n n n n n n n nn O t O t

t tθ θ θ θ θ θ θ θθ

+ − + − − −+ − + − + − +

= + Δ = + ΔΔ Δ

(5)

IV. Computed Results and Discussion

A. Nominal Test Case of NACA 0012 Airfoil in Free Rotation As in the experiments described earlier in this paper, a NACA 0012 airfoil was chosen as the first test case. The

airfoil had a chord length of c = 0.156 m. The freestream velocity was specified as U∞ = 7.5 m/s, resulting in Re = 77,000. Three-dimensional (3D) LES-based calculations were carried out on O-H type meshes. The farfield boundaries were located at least 10 chords away from the surface of the airfoil. The span was first assumed to be

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0.064c and was appropriately discretized in the spanwise dimension with 17 grid points. This scheme was indeed successfully used in the previous work with the NACA 0012 airfoil, in which a prescribed motion schedule was used in concordance with the experimental observations.5 In other configurations, the span was extended to 0.2c where finer grids were implemented for accuracy. The maximum value of y+ for the first control cell center at the wall was assured to be less than 1. The inner part of the O-type mesh pitched rigidly, in unison with the airfoil, while the outer part remained stationary. The mesh layer between the inner and outer parts was dynamic, rather than sliding, and deformed during the airfoil pitching motion. This functioned satisfactorily as long as the deformations remained small. The use of the stationary mesh for the outer part eased the implementation of the farfield boundary conditions. The timestep used in the current computations was Δt = T/3840, where T is the period of one experimental pitching cycle with f = 1/T = 2.9 Hz for the nominal 1DOF case measured in the experiments.

Initial conditions, for both fluid and elastic equations, are needed to perform the calculations. The 3D calculations were started or interpolated from a previous prescribed-motion airfoil study (cf. Ref. 5).

The 3D LES revealed self-sustained limit-cycle oscillations in pitch. The overall characteristics of the pitching oscillations are summarized in Table 1. Figure 5 to Figure 8 illustrate the computed results from the current calculations. Although the computations on the fine grid are not yet conclusive, it seems that the medium mesh delivered results slightly better than those from the coarse grid when compared with the experimental data, in particular, the peak values of the pitch angle and the oscillation frequency. Therefore, the results from the medium grid (O481×097×033) were chosen for the following analysis.

Table 1 Summary of overall characteristics of the pitching oscillations

Cases Far field Span y+ Cycles Peaks (°) Frequency (Hz) Work (Nm) Experiment 4.80-5.10 2.9 0.0009

O481×065×017 10c 0.064c < ∼0.2 15 4.08-5.26 2.82 0.00081 O481×097×033 10c 0.20c < ∼0.6 13 4.77-5.29 2.85 0.00081 O961×129×033 25c 0.20c < ∼0.4 7 4.22-5.22 2.81 0.00064

Figure 5. Computed pitch angle response of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000, on mesh O481×097×033.

Figure 5 shows the computed pitch angle response of the 1DOF test case. The time is normalized by the

experimental LCO period. The predicted pitch amplitudes in general were slightly smaller than those observed in the experiment, Figure 2. As shown in Figure 6, the power spectra of the pitch angle and the pitching moment coefficients displayed a first harmonic frequency at 2.85 Hz, which was in good agreement with the measurement (2.9 Hz). Smaller peaks at 3f, 5f, 7f, and 9f were also observed, confirming the existence of the odd super-harmonics and the nonlinearity in the airloads. Figure 7 plots the computed aerodynamic moment coefficient responses during three pitching cycles, starting from about zero pitch angle when the airfoil was pitching up. As shown in the figure, the predicted pitching moment was positive (nose-up) at the start near zero degrees when pitching up, indicating the fluid provided energy to the airfoil sustaining the limit-cycle oscillations. The phase of the aerodynamic pitching moment coefficient lagged that of the pitch angle at zero degrees, which is coincident with the experimental

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observations shown in Figure 3. The plot of the computed aerodynamic moment coefficients as a function of pitch angle during one quasi cycle shown in Figure 8 confirms the clockwise loop as observed in the experiments (Figure 4). The work integrated along the loop was calculated to be 0.00081 Nm, which was somewhat smaller than the one measured in the experiment (0.0009 Nm), resulting in slightly lower pitching amplitudes in the computations than in the experiment. It is noted that the pitching moment coefficients shown in Figure 7 and Figure 8, as opposed to Figure 3 and Figure 4, are instantaneous; hence the strong, high frequency, unsteadiness being observed. However, the high frequency physics has no impact on the LCO.4, 5

Figure 6. Power spectra of the computed pitch angle response (top) and aerodynamic moment coefficient (bottom) of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

Figure 7. Computed aerodynamic moment coefficient response of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

Figure 8. Computed aerodynamic pitching moment coefficients of the 1DOF NACA 0012 airfoil as a function of pitch angle during one quasi cycle at U∞ = 7.5 m/s, Rec = 77,000.

It is worthwhile to mention that the flow around the static airfoil at zero degrees is nearly symmetrical, as shown

in Figure 9. Quasi symmetric open-separation zones at the trailing edge – laminar separations without transition to turbulence and the consequent turbulent reattachment – were observed on both surfaces of the airfoil. The symmetry could be disturbed due to disturbances from the inflow, boundary layer or any imperfections on the airfoil surface. If the airfoil were disturbed from the original centered position (0°), the flow would tend to recover the symmetrical condition. To confirm and elucidate the flow physics, the flowfields of the 1DOF pitching airfoil are analyzed in Figure 10.

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Figure 9. 2D LES computed spanwise vorticity ( zω ) distribution of a static NACA 0012 airfoil at α = 0°, U∞ = 7.5 m/s, Rec = 77,000.

Figure 10 shows the instantaneous flowfields of the 1DOF NACA 0012 airfoil obtained from the computations

on the medium grid. Laminar boundary layer separation and laminar-turbulent transition can be clearly seen, e.g. in plots a and b. The flattened pressure plateaus indicate the flow separation owing to the adverse pressure gradient. At θ ≈ 0° when the airfoil was pitching up (Figure 10a), an open-separation zone was observed on the upper surface near the trailing edge since the laminar separation occurred too close to the trailing edge to let the flow transit to turbulence. This happens usually at low angles of attack according to the low-Re aerodynamics at static conditions (cf. Ref. 2). Otherwise, laminar-turbulent transition or a laminar separation bubble (if phase averaged) was observed. The transition process was evidenced by the vortex shedding from the primary separated shear layer and rolling up on the lower surface at θ ≈ 0° when the airfoil was pitching up and on the upper surface at θ ≈ 5°, see spanwise vorticity distributions in Figure 10a (red color) and Figure 10b (blue color), respectively. In this case, the pressure plateau terminated when the transition of the separated shear layer caused a rapid increase in the surface pressure as shown by the dashed line in Figure 10a for the lower surface at θ ≈ 0° when pitching up and the solid line in Figure 10b for the upper surface at θ ≈ 5°.

The snapshots shown in Figure 10 were extracted from a quasi pitching cycle of the 3D LES. They represent flowfields at critical time instants of a limit-cycle oscillation. Starting from plot a, where the airfoil was pitching up through the center position at zero degrees, the flow separation (bubble, if phase averaged) on the lower surface caused noticeable suction, resulting in a nose-up pitching moment, and thus contributed to the pitching-up rotation. During the pitching-up motion, the angle of attack of the airfoil increased and the separation on the upper surface moved upstream according to the low-Re aerodynamic behavior at static conditions. Subsequently, the nose-down pitching moment, mainly caused by the pressure suction on the airfoil upper surface, reached its maximum near the maximum pitch angle (plot b), followed by a pitch-down rotation (plots c-h). Because of the time delay, the upstream movement of the separation on the upper surface continued to about 3° (plot c). At this time instant, this separation on the upper surface started moving downstream (plot d), which was consistent with the low-Re aerodynamics at static conditions. The downstream movement of the separation increased the pitching arm and thus retained a nose-down pitching moment. This separation movement continued from 3° (plot c) to about -1.5° (plot f). Corresponding to the low-Re aerodynamic behavior at static conditions, the separation point on the low surface moved upstream when the nose-down angle increased (plots e-h). Owing to the upstream movement of the separation point on the lower surface and the downstream movement of the separation point on the upper surface, the separation zones were comparable on both sides and the pitching moment changed sign from nose down to nose up at about -2° (plot g), resulting in a pitching moment close to zero. The pitch-down rotation continued until the positive (nose-up) pitching moment reached its maximum at about -5° (plot i), leading to a subsequent pitch-up rotation (plots i-j).

The pressure fluctuations near the trailing edge (e.g. in plots b and i) corresponded to the vortices shedding and rolling up from the primary separated shear layer. A closer look indicates that these fluctuations did not directly attribute the fluctuations of the aerodynamic pitching moment coefficients in Figure 7 and Figure 8. A rough estimate of the vortex shedding frequency based on Figure 10a and Figure 10b showed frequencies of ∼1200 Hz (St = 3) and ∼600 Hz (St = 1.5) at θ ≈ 0° when pitching up and θ ≈ 5°, respectively, while the fluctuations of the aerodynamic pitching moment coefficients at these two time instants were ∼276 Hz (St = 0.69) and ∼145 Hz (St = 0.36). The local convective velocities near the shedding vortices at the trailing edge, which were close to the freestream velocity, were measured and used for the estimate. Since the vortex shedding on the right plot of Figure 10a is not clear, the (zoomed) pressure fluctuations shown on the left plot of Figure 10a was used for the estimate of the vortex travelling distance at θ ≈ 0° when pitching up. According to this rough estimate, it is believed that the latter (fluctuations of aerodynamic pitching moment) corresponded to the classic von-Karman vortex shedding, as indicated in the measurements of the airfoil wake velocities (250 Hz).

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a) θ = 0.21° pitch up

b) θ = 4.94° near ultimate

c) θ = 2.91° pitch down

d) θ = 1.26° pitch down

e) θ = 0.04° pitch down

f) θ = -1.21° pitch down

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g) θ = -1.83° pitch down

h) θ = -2.99° pitch down

i) θ = -5.06° near ultimate

j) θ = -0.01° pitch up

Figure 10. 3D LES computed instantaneous surface pressure and spanwise vorticity ( zω ) distributions on the mid-span of the 1DOF NACA 0012 wing during a quasi pitching cycle at U∞ = 7.5 m/s, Rec = 77,000.

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The actual 3D flow structure is presented in Figure 11, which shows iso-surfaces of spanwise, streamwise, and cross-stream-direction vorticities over the top surface of the rectangular wing at two time instants (corresponding to phases b and e in Figure 10). The spanwise vorticity in the top plots clearly show the 2D shear layer and its distortion to a 3D structure near the trailing edge. In addition, the streamwise and cross-stream-direction vorticities confirm non-uniform flow structure along the wing span, implying probable inaccuracy of 2D assumptions used in the 2D calculations, as discussed later.

a) ωz = -100 and -20, θ = 4.94° b) ωz = -100 and -20, θ = 0.04° (pitch down)

c) ωx = -10 and -1, θ = 4.94° d) ωx = -10 and -1, θ = 0.04° (pitch down)

e) ωy = -10 and 1, θ = 4.94° f) ωy = -10 and 1, θ = 0.04° (pitch down)

Figure 11. Computed iso-surfaces of non-dimensional instantaneous spanwise (top), streamwise (middle), and cross-stream-direction (bottom) vorticities over the 1DOF NACA 0012 rectangular planform wing at U∞ = 7.5 m/s, Rec = 77,000.

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B. Nominal Test Case of NACA 0012 Airfoil in Pitch and Heave As mentioned earlier, the 2DOF LCOs of the NACA 0012 airfoil were observed recently at RMC when the

heave DOF was not held fixed.6 To obtain more details of the flow physics, numerical simulations of the 2DOF LCOs were performed in parallel. The calculations were started from the aforementioned 1DOF pitching oscillations. Different from that in the 1DOF simulations, the outer part of the O meshes remained in pure heave motion. The preliminary results are shown in Figure 12 and Figure 13.

Figure 12 shows the computed pitch angle and heave response of the 2DOF NACA 0012 airfoil compared with experimental data. As shown in the plots, the predicted heave amplitudes (1.2-1.3 mm) were almost identical to those observed in the experiments (1.1-1.3 mm). The pitch angle predicted on the medium mesh (∼5.5°) was slightly better than the one predicted on the coarse grid (∼5.3°), when compared with the measured data (∼5.5°). The predicted LCO frequency was about 2.9 Hz, comparable to the measurement (3.1 Hz). Also, the predicted phase shift was in good agreement with the experiments.

a) Experiment b) CFD O481×065×017

c) CFD O481×097×033 d) CFD O961×129×033

Figure 12. Pitch angle and heave response of the 2DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

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Figure 13 shows the computed aerodynamic coefficients of the 2DOF NACA 0012 airfoil as a function of pitch angle and heave displacement, respectively. The lift coefficient as a function of pitch angle reflected nonlinear and unsteady effects. Same as in the 1DOF LCO, the aerodynamic pitching moment coefficient as a function of pitch angle during one quasi pitch cycle confirmed the clockwise loop, indicating positive work done by the flow at the centre position near 0° pitch angles. Moreover, the loop of the lift coefficient as a function of the heave displacement was clockwise and denoted positive work done by the lift force as well. This resulted from the positive lift coefficient at zero heave displacement when the airfoil heaved upwards, meaning that the lift force fed energy to the airfoil sustaining the limit cycle oscillations in heave. Further investigations will be carried out by completing the computations on the fine grids.

Figure 13. Computed instantaneous aerodynamic coefficients of the 2DOF NACA 0012 airfoil as a function of pitch angle and heave displacement during one quasi LCO cycle at U∞ = 7.5 m/s, Rec = 77,000 on mesh O481×097×033.

C. Other Lessons Learned Some lessons learned during the course of the project are briefly described in this section, which might be

helpful to the readers when new numerical models are developed or similar applications are initiated.

1. 2D vs. 3D It is well known that the laminar-turbulent transition process and turbulent flows are three-dimensional. 3D

simulations are necessary to elucidate accurately the detailed flow physics. However, for demonstration purposes during algorithm development and/or quick solutions of engineering applications, representative 2D simulations are worthwhile to test. Although 2D simulations cannot capture the complete 3D effects, it is believed 2D simulations may capture some major features of the flow. Towards this assessment, 2D simulations were performed for both 1DOF and 2DOF LCOs of the NACA 0012 airfoil at transitional Reynolds numbers.

The 2D calculations were performed on O-type meshes. The setups were similar to the aforementioned 3D simulations. However, the 2D 1DOF calculations were started from a flowfield of the airfoil at a static condition,

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followed by 2DOF computations. The 2D LES calculations predicted 1DOF LCOs as listed in Table 2. Some representative 1DOF and 2DOF results are demonstrated in Figure 14 - Figure 17.

Although the 2D LES predicted both 1DOF and 2DOF LCOs, the computed pitch angles and heave displacements were noticeably smaller than those from the 3D LES and the experiments. It is believed that this was attributed to the over-prediction of the vortex shedding from the primary separated shear layer when the airfoil transversed the neutral center position (0°). Figure 17 shows the 2D LES computed instantaneous surface pressure and spanwise vorticity ( zω ) distributions of the 1DOF NACA 0012 airfoil as if traversed the neutral center position (0°) during the pitch-down phase. Compared with Figure 10e, the pressure fluctuations on the upper surface near the trailing edge and the corresponding vortex rolling up predicted by the 2D LES (Figure 17e) were noticeably more intensive than those from the 3D LES. This over-prediction caused earlier laminar-turbulent transition or shortened the transition process, and thus resulted in an earlier flow reattachment if phase averaged. This numerically earlier transition shortened the pressure suction on the upper surface or the resulting pitching arm, and thus reduced the nose-down pitching moment coefficient and lowered the maximum pitch angle.

Same as in the 1DOF calculations, the inaccuracy appeared in the 2D 2DOF simulations. As shown in Figure 16, both pitch angle and heave displacement predicted by the 2D LES were slightly smaller than those observed in the experiments or 3D LES, as shown in Figure 12. The flow physics will be further analyzed as soon as the 3D computations are complete.

Table 2 Summary of 2D LES results of the pitching oscillations

Cases Far field y+ Cycles Peaks (°) Frequency (Hz) Work (Nm) Experiment 4.80-5.10 2.9 0.0009 O481×065 10c < ∼0.2 13 4.16-4.94 2.76 0.00048 O481×097 10c < ∼0.66 16 3.82-4.93 2.67 0.00064 O481×129 25c < ∼0.3 15 3.36-4.69 2.74 0.00052 O961×129 25c < ∼0.37 11 3.84-4.77 2.72 0.00058

Figure 14. 2D LES computed aerodynamic moment coefficient response of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000, on mesh O481×065.

Figure 15. 2D LES computed aerodynamic moment coefficients of the 1DOF NACA 0012 airfoil as a function of pitch angle during one quasi cycle at U∞ = 7.5 m/s, Rec = 77,000, on mesh O481×065.

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a) CFD O481×065 b) CFD O481×097

Figure 16. 2D LES computed pitch angle and heave response of the 2DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

c) CFD O961×129

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d) θ = 1.96° pitch down

e) θ = -0.08° pitch down

f) θ = -1.75° pitch down

Figure 17. 2D LES computed instantaneous surface pressure and spanwise vorticity ( zω ) distributions of the 1DOF NACA 0012 airfoil when traversed the neutral center position (0°) during the pitch-down phase at U∞ = 7.5 m/s, Rec = 77,000, on mesh O481×065.

2. Influence of the Flow Model

In the LES computations, the Smagorinsky sub-grid scale (SGS) model was used. To verify the SGS model effect, 2D laminar simulations were performed for the 1DOF LCOs. The laminar results are compared in Figure 18 and Figure 19. Without damping effects of the eddy viscosity from the SGS model, the laminar solution reached the steady state of the limit-cycle oscillations faster than the 2D LES did, meaning the SGS model numerically smeared the flow unsteadiness resulting from the laminar-turbulent transition process to some extent and delayed the LCO development. Although the laminar simulations predicted self-sustained oscillations, as the 2D LES did, the laminar calculations did not show a clear clockwise positive-work loop around zero degrees in the pitching moment – pitch angle chart either, see Figure 19.

Another test case using a first-order upwind scheme delivered a completely wrong solution. After a nine-cycle time period of the 2nd-order laminar computations (Figure 18), the calculations were switched to a first-order scheme when the airfoil was pitching up. The results of the 1st-order scheme are shown in Figure 20. After several pitching cycles, the airfoil was re-stabilized at about 2°. In the computations, the upwind scheme introduced numerical dissipation, mimicking the eddy viscosity effect. However, the numerical dissipation introduced by the 1st-order scheme was so strong that it smeared the major physical unsteadiness, including transition and turbulence processes, resulting in a non-realistic situation. The resulting unreal pitching moment was numerically balanced by the structural stiffness.

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Figure 18. 2D LES computed pitch angle response of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000, on mesh O481×065. Left: laminar; right: 2D SGS LES.

Figure 19. Computed (laminar solution) pitching moment coefficients of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

Figure 20. Computed (1st-order laminar solution, solid line, preceded by the 2nd-order laminar solution, dashed line) pitching moment coefficients of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

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3. Influence of Numerical Scheme for the Structural Model Alternatively, a common four-point stencil can be found in mathematics books:

1 1 2

1 22

2 5 4 ( ).n n n n

n O tt

θ θ θ θθ+ − −

+ − + −= + Δ

Δ (6)

A Taylor series analysis confirmed that the truncation errors of the discretized five-point and four-point schemes

(Eqs. 5 and 6, respectively) for the second derivative of the structural equation were of the same order. The coefficient of the truncated error of the five-point format was smaller than that of the four-point format. In order to check the impact of this discrepancy on the solutions, two laminar test cases were performed for the 1DOF LCOs. The solution using the five-point format is shown in the left plot of Figure 18. The solution from the four-point format was performed after about nine pitching cycles using the five-point format. The results are given in Figure 21. The calculations using the four-point format predicted similar results as the five-point format, but with slightly higher fluctuations of the pitching amplitude.

Figure 21. Computed (laminar solution using the four-point format, solid line, preceded by the five-point format, dashed line) pitch angle response of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000.

4. Influence of Boundary Conditions and Initial Solutions

Since the effects of boundary conditions and initial solutions may be “remembered” by the flow for a considerable time, they can have a significant effect on the results. In this study, two different boundary conditions were tested in the laminar simulations for the 1DOF LCOs: one with a freestream velocity specified on all farfield boundaries and a reference point of pressure selected at the upstream farfield; the other with a freestream velocity specified for the inflow and the pressure specified for the outflow. The former enforced the exact mass conservation at the farfield boundaries while the latter was more powerful in its ability to avoid reflection of errors introduced at the outlet boundary. The discrepancy between the results was marginal and the second boundary conditions provided a slightly clearer clockwise positive-work loop in the pitching moment – pitch angle chart (not shown here).

The best initial conditions are invariably those which have been preserved from previous simulations. This methodology, as shown earlier, was used in the 3D LES calculations. In order to test simulation behaviors, three other initial solutions were tested in the 2D LES calculations of the 1DOF LCOs. The first one was a flowfield obtained on the same grid at a static condition with a stationary airfoil; the second one was a stationary fluid with a stationary airfoil; and the third one was a stationary fluid with an airfoil in a prescribed harmonic pitching motion. The results of the first test case were depicted in the right plot of Figure 18. The other two are shown in Figure 22. All three calculations reproduced 1DOF LCOs. In the first case, lower-amplitude oscillations remained for a while. The other two cases showed irregular behavior in the beginning phase reflecting the fluid-structure interaction. The airfoil oscillations of all three test cases converged to a steady state of the LCOs after about ten shaking cycles, as shown in the figures.

As a side note, it is believed that the low-amplitude LCOs observed in the beginning phase of the 2D LES (right plot in Figure 18) are different from those observed at the first stage of the experiments (cf. Figure 2). It is speculated that the former was purely numerical whereas the latter was related to the small, but nevertheless present, dry friction of the support structure.

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Figure 22. 2D LES computed pitch angle responses of the 1DOF NACA 0012 airfoil at U∞ = 7.5 m/s, Rec = 77,000. Left: started from stationary fluid and stationary airfoil; right: started from stationary fluid and prescribed airfoil motion.

5. Effects of the Airfoil Geometry

It is well known that symmetric airfoils are commonly used as tail empennage of aircraft, including emerging UAVs. The tail is the most important part to the longitudinal stability and control of aircraft. On the other hand, symmetric airfoils are hardly ever used as the aircraft main wing. To demonstrate the potential occurrence of the LCOs on the cambered airfoils, mimicking a real-world UAV main wing, an artificial 1DOF pitching case was designed. For this purpose, a low-Re SD7003 airfoil was chosen. The chord length (c = 202.6 mm) and the Reynolds number (Re = 60,000) were selected corresponding to Ref. 3, resulting in a freestream velocity of 4.5 m/s. Similar to the static configuration in Ref. 3, a 2D C-type mesh with 737×65 grid points was used in this study. The structural parameters used for the study of the NACA 0012 airfoil were simply adopted for the cambered SD7003 airfoil. The elastic axis (xEA) was also located at a distance of 18.6% of the chord length from the leading edge. In addition, in order to make the model more realistic, a mean pitching moment (CM, External = 0.02) was added to account for a non-zero angle of incidence typical of cruise conditions, as shown in the following equation:

( ) .EA EA EA

Aero External

I D K M t Mθ θθ θ θ+ + = + (7)

The 1DOF 2D LES calculations were started from a static-configuration solution at α = 4° as indeed

investigated in the previous studies. The computed pitch angle response of the SD7003 airfoil from the preliminary 2D LES calculations is shown in Figure 23. Note that the time was normalized by the structural frequency in pitch (fsp = 2.37 Hz), which is different from that of the NACA 0012 airfoil, where a more relevant experimental period was applied. The preliminary results verified the presence of potential oscillations on the main wings of UAVs at cruise conditions. Similar to the simulations for the NACA 0012 airfoil, the aerodynamic stiffening effect can be clearly seen, resulting in higher oscillation frequencies than the natural structural frequency. Since the applied residual mean pitching moment coefficient CM = -0.02 was in fact not corrected to xEA/c = 0.186, the mean angle of attack for this 1DOF pitching case was understandably different from 4°. It should be noted that for this cambered airfoil, the laminar separation occurred only on the upper surface of the airfoil. Since there was no triggering mechanism invoking a switch of separation events between the upper and lower surfaces, the airfoil oscillations of the cambered airfoil were less intensive than in the symmetric case. It is believed that the Reynolds number, the location of the elastic axis, and the residual mean pitching moment (or mean angle of attack) play important roles in promoting this kind of oscillations. An experimental test case is under design and the refined computations will be undertaken for further investigations.

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Figure 23. 2D LES computed pitch angle response of the 1DOF SD7003 airfoil at U∞ = 4.5 m/s, Rec = 60,000.

V. Conclusions A numerical exercise was carried out for free-to-rotate and/or free-to-rotate-and-heave aeroelastic airfoils

(NACA 0012 and SD7003) at low Reynolds numbers. The numerical results confirmed the presence of the limit-cycle oscillations for the NACA 0012 airfoil. It was observed that during the time delay that occurs in restoring the airfoil to its statically stable position, the laminar separation and its related transition (bubble) moved forwards and rearwards, causing a self-sustained fluctuation or switching of the pitching moment, and thus the negative aerodynamic damping. The predicted LCO frequency was in good agreement with the experiments while the pitching amplitude showed the same amplitude compared to the measured data. For the cambered airfoils, further investigations are needed to conclude if the numerically observed oscillations can converge to a steady state LCO.

Acknowledgments The authors gratefully acknowledge the support of the Department of National Defence of Canada (DND)

through the Technology Investment Fund (TIF) program.

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NACA0012 Airfoil at Transitional Reynolds Numbers”, 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, Tennessee, 9 - 12 Jan 2012.

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AIAA Journal, Vol. 21, No. 11, pp. 1525-1532, 1983. 12 Demirdžić, I. and Perić, M., “Finite Volume Method for Prediction of Fluid Flow in Arbitrarily Shaped Domains with

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