Numerical Simulations of Self-Sustained Pitch-HeaveOscillations of a NACA0012 Airfoil

Simon Lapointe and Guy Dumas1

Laboratoire de Mécanique des Fluides Numérique, Département de Génie Mécanique, Université Laval,Québec, G1V 0A6, Canada

Email: simon.lapointe.5@ulaval.ca

ABSTRACT

The phenomenon of self-sustained pitch-heave oscil-lations of an airfoil at transitional Reynolds numbersis studied by two-dimensional numerical simulationsusing the Spalart-Allmaras and γ−Reθ models. Pre-dicted pitching and heaving amplitudes and frequen-cies of the oscillations are compared with previous ex-perimental results. The numerical simulations repro-duce the same trends as in the experiments. Whenlarge translational stiffnesses are used, large heavingand pitching amplitudes are obtained. These oscilla-tions can be labelled as coalescence flutter. The flowdynamics and the impact of the different turbulencemodels are also discussed.

1 INTRODUCTION

Over the course of the last decades, there has beenan increasing interest in the development of unmannedaerial vehicles (UAVs) and micro-air vehicles (MAVs).These vehicles can be used in a large variety of ap-plications and can be either remotely piloted or au-tonomous. As typical commercial aircrafts usuallyfly at high Reynolds number O(107), the small lengthscale and low velocities of UAVs and MAVs result ina flight regime with low-to-moderate chord Reynoldsnumbers (15,000 to 500,000) [6].

The flow at these Re numbers is highly nonlinear andcomplex viscous phenomena are present. Indeed, onecan typically observe an extensive region of laminarflow in the boundary layer up to the separation point. Atransition of the laminar shear layer then follows witha possible re-attachment of the turbulent shear layerthus forming a laminar separation bubble (LSB) on thesurface of the airfoil [1, 12].

Poirel’s group [8] at Kingston has worked on a se-

ries of experimental studies on an elastically mountedNACA0012 wing at Reynolds numbers in the range5.0× 104 < Rec < 1.2× 105. Wind tunnel experi-ments have shown that the rigid airfoil in this transi-tional flow regime undergoes self-sustained or limit-cycle oscillations (LCO). Typical results are shown inFig. 1. The pitch-only oscillations are characterizedby a simple harmonic motion of small amplitudes (lessthan 5.5 deg) and nondimensionnal frequencies of theorder f c/U∞ ≈ 0.05∼ 0.07. These have also been ob-served numerically in different studies [4, 9] led at theCFD lab LMFN at Laval University. These oscilla-tions have been labelled as laminar separation fluttersince laminar boundary-layer separation is key to trig-gering the phenomenon and turbulent boundary layersinhibit the appearance of oscillations.

0

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90

6.0×104 8.0×104 1.0×105 1.2×105

Pitc

ham

plitu

de(d

eg)

Rec

Small LCO - Fixed Heave (Infinite stiffness)Small LCO - large StiffnessLarge LCO - large StiffnessSmall LCO - small StiffnessLarge LCO - small Stiffness

Figure 1: Comparison of self-sustained 2 DOF oscil-lations characteristics from experiments [5].

In their recent work, Mendes et al. [5] conducted windtunnel experiments of a NACA0012 allowing two-degree-of-freedom motion in both the pitching andheave directions. Their experimental setup is illus-trated in Fig. 2. The wing has a chord of 0.156m and a

span of 0.61m (AR = 3.9). The motion is permitted bythe use of springs wrapped around two sets of pulleys.One pair of pulleys is attached to a rod at 0.186c fromthe leading edge of the airfoil to allow for the pitchingmotion. These springs have a stiffness of 58 N/m thatis kept constant in all experiments. The heaving mo-tion is constrained by another pulley system. Differentstiffnesses were tested for the plunging springs: a lowstiffness of 75 N/m and a high stiffness of 371 N/m.

Figure 2: Schematic of experimental apparatus, repro-duced from [5].

The two-degrees-of-freedom experimental results arealso included in Fig. 1. Two types of oscillating am-plitudes were observed: small and large. Althoughboth small and large oscillations were measured usinga high plunge stiffness, only small oscillations wereobtained when a low plunge stiffness was used. In thiscase, the addition of a trip wire close to the leadingedge did not inhibit the appearance of the large oscil-lations meaning that these large oscillations are not as-sociated with laminar separation flutter but rather withcoalescence flutter. Coalescence flutter is known as adynamic instability leading to exponentially growingoscillations until aerodynamic or structural forces im-pede this growth creating the LCO behavior [5]. Theinstability responsible for the flutter occurs through thecoupling effect between the pitching and heaving mo-tions [10].

This paper aims to demonstrate the capability of mod-ern CFD to capture and reproduce the observed com-plex coupling between the flow and the elastic struc-ture.

2 COMPUTATIONAL METHODOLOGY

The fluid problem is solved with the finite volume,open source, CFD code OpenFOAM [7]. The fluidflow is governed by the mass and momentum conser-vation equations which take the following forms for anincompressible turbulent flow:

∂Ui

∂xi= 0 , (1)

∂Ui

∂t+U j

∂Ui

∂x j=−1

ρ

∂p∂xi

+(ν+νt)∂2Ui

∂x2j, (2)

where U is the ensemble averaged mean fluid ve-locity vector, ρ is the fluid density, p is the pres-sure, ν is the kinematic viscosity and νt is the tur-bulent kinematic viscosity. The fluid solver imple-ments Eqs. (1) and (2) using the finite-volume method.The transient term is discretized with a second or-der backward implicit scheme. The convection termis discretized with a second order scheme based ona linear upwind interpolation. The diffusion termis discretized with a second order accurate schemebased on linear interpolation and using an explicit non-orthogonal-corrected surface normal gradient scheme.The pressure-velocity coupling is done by means ofthe PISO segregated algorithm. The linear solver is apreconditioned bi-conjugate gradient (PBiCG) methodfor the momentum and turbulence equations, and ageneralised geometric-algebraic multi-grid (GAMG)method is used for the pressure equation.

The aeroelastic system corresponding to the experi-ments [5] is represented in the sketch of Fig. 3 alongwith the nominal parameters used in both the experi-ments and the numerical simulations. The freestreamvelocity U∞ varies from 5 to 14 m/s.

A timestep of ∆t = 2× 10−5 sec has been used in allsimulations. This allows for about 15,000 timestepsper aeroelastic oscillation and 200 timesteps per vor-tex shedding period which has been verified to bequite sufficient. An RMS convergence criterion of10−6 on all quantities is requested at each timestep.Timestep and convergence level independance studieshave shown that using a finer timestep of ∆t = 1×10−5

sec or reaching a convergence level of 10−7 did notchange the results.

Fairly periodic oscillation cycles are observed and sev-eral such periodic cycles are computed to allow for theproduction of good, stationary statistics and spectralanalysis. The typical run time for a whole simulationrequires about one week on four Intel Xeon X5560 2.8Ghz processors.

x

Equilibriumposition

CM

Figure 3: Two degrees of freedom aeroelastic model-ing.

2.1 Aeroelastic modeling

We consider an elastically-mounted rigid body withtwo-degrees-of-freedom respectively in pitch andheave. The problem is sketched in Fig. 3. A Carte-sian coordinate system is used with the y axis definedpositive downward and the x axis positive towards theright, in order to respect the aerodynamic conventionconcerning the sign of the pitching angle θ.

The dynamics of the airfoil is thus governed by:

Maxis = Iaxis θ + Dθ θ + Kθ θ + myxθ cos(θ) (3)

−L = m [y+ xθ cos(θ) θ− xθ sin(θ) θ2] (4)

+ Dyy + kyy.

where Iaxis is the moment of inertia about the elasticaxis, Dθ is the rotational damping coefficient, Kθ isthe torsional stiffness coefficient, Maxis is the momentabout the axis exerted by the fluid, m is the mass, Dyis the plunging damping coefficient, Ky is the plung-ing stiffness coefficent and L is the lift. The structuralparameters indicated in Fig. 3 were obtained in the lab-oratory by direct physical measurements and from no-flow free decay tests [5, 8]. These equations are cou-pled in the general case where the pitching axis is notcoincident with the center of mass (xθ 6= 0). In thispaper xθ = 0.1c.

In the simulations, the flow solver is coupled withthe dynamics of the elastically mounted rigid airfoil.The calculation algorithm, performed at each timestep,goes as follows:

1. The instantaneous fluid flow around the airfoil iscomputed;

2. The lift and moment exerted by the flow on theairfoil are calculated;

3. The equations of motion, Eq. (3) and (4), aresolved using a fifth-order Runge-Kutta scheme todetermine the pitch rate θ, angular position θ, ver-tical velocity y and position y;

4. The vertical and angular positions and veloci-ties of the airfoil are thus updated and the nexttimestep can proceed.

2.2 Turbulence modeling

In the present study, turbulence closure is achieved bythe Boussinesq eddy-viscosity approximation and theeddy viscosity νt is modeled by the Spalart-AllmarasRANS model [11] or the γ−Reθ transition model [3].The S-A model is a one-equation turbulence model forthe modified turbulent viscosity ν. This model doesnot need wall functions since it provides near-wallresolution and is based on empiricism and argumentsfrom dimensional analysis. Since simulations are per-formed at transitional Reynolds numbers, the γ−Reθ

model is also tested to investigate the impact of thetransition phenomena. It is a correlation-based transi-tion model built on transport equations using only localvariables. It adds two transport equations to the k−ω

SST model.

Previously published comparisons of URANS calcula-tions (algebraic, one-equation and two-equations tur-bulence models) with experimental measurements forpitching only (rotorcraft-type) oscillating airfoils indi-cated that the main weakness of the computational andtheoretical methods was the turbulence modeling [2].It is already known that unsteady calculations maybe strongly affected by the choice of turbulence andtransition models [13]. Although the Spalart-Allmarasmodel has been built up based on aerodynamic con-siderations, it was developed from steady flow casesand shares the same basic limitations as other RANSmodels for flows characterized by massive separation.URANS turbulence modeling is thus obviously chal-lenged here and may be criticized. However, it is im-portant at this point to recall that the current paper ob-jective is not so much to provide precise quantitativeresults for the entire parametric space, but to give ba-sic insights into the complex unsteady flow physics ofself-sustained oscillating airfoils.

Despite the uncertainty of the quantitative predictionsfor massively separated flows, we argue that the cur-rent URANS study is nonetheless of significant practi-cal interest as it provides the right physical trends andthus, useful physical insights for the first time ever asfar as the author knows.

2.3 Boundary conditions and space dis-cretization

The pitch-heave oscillating airfoil problem is heresolved in an accelerated translational frame of refer-ence which thus requires moving body and movinggrid capabilities to simulate the pitching motion.

In OpenFOAM, a non-conformal interface can be usedin order to avoid deforming mesh and remeshing in theclose proximity of the airfoil. To take into account thepitching motion, the inner part of the mesh, locatedinside a radius of 2 chords about the pitching axis,rotates rigidly with the body while the outer part re-mains stationary as shown in Fig. 4. At the interface,interpolation is calculated by a General Grid Interface(GGI) algorithm. The approach has been thoroughlyvalidated in previous studies using Fluent and Open-FOAM [2, 4].

Figure 4: Computational domain and grid details:mesh size ∼ 80,000 cells.

Varying velocity is imposed at the inlet (50c upstreamof the airfoil) and boundaries far above and below thewing to take into account the vertical motion of the air-foil while constant mean static pressure is imposed atthe outlet. Acceleration terms are added as body forcesin the momentum equation. Adequate near-body reso-lution (320 cells on both the upper and lower surfaces)is used to capture accurately the vorticity gradients andto satisfy the turbulence model requirement for the firstcell thickness, namely y+ ≤ 1 on the airfoil surfaceover the whole cycle [3]. Mesh refinements close tothe airfoil in both the streamwise and wall-normal di-rections have been shown to not influence the results.

3 RESULTS

3.1 Aeroelastic characteristics

Simulations were first performed in order to comparewith the experimental results of Mendes et al. [5]. Thesame parameters as in the experiments were used forthe inertia, mass, rotation damping, rotational stiffnessand heaving stiffness. The heaving damping was firstset at Dh = 3.5 N·s/m (ζ = 0.1) which was establishedas an educated guess following a personal communi-cation with Poirel.

To start with, no initial perturbations are imposed. Theairfoil was initially at an angle of attack of zero degreewith no angular or heaving velocity. Using these ini-tial conditions only small LCOs developed; no largeLCOs are obtained. The pitching amplitudes recordedare very similar to the cases where only the pitchingmotion was allowed (see Fig. 1). The plunging ampli-tudes are very small, almost negligible. Changing theheaving stiffness from small (75 N/m) to large (371N/m) has no influence on the results.

The second step was to give the wing an initial pertur-bation. Providing the wing with an initial heaving ve-locity (v0 = 1m/s), the results obtained are very differ-ent. Using the higher heaving stiffness, LCOs of largepitching and heaving amplitudes are generated. How-ever, when the smaller heaving stiffness is used, nooscillations of significant amplitude occur as in the ex-periments. Figure 5 presents the pitching and heavingamplitudes as well as frequencies of the large LCOsobtained.

Although the results do not show a good quantita-tive agreement with the experimental measurementsof Mendes et al. [5], similar trends are observed.Large pitching amplitudes are obtained over the wholeReynolds number range with both turbulence modelsas well as in the laminar simulations (no turbulencemodel added). The numerical simulations also presenta decrease in pitching amplitudes at Reynolds numbershigher than 90,000, a trend also present in the experi-ments. However the simulations tend to predict highermaximal angles of attack than the experimental mea-surements.

Frequencies obtained from the computations are quiteclose to the experiments, albeit slightly higher. Dis-crepancies between the results can be caused by theturbulence modeling and the two-dimensionnality ofthe simulations since 2D URANS predictions have atendency to overestimate the lift due to the artificialvortex coherence in the spanwise direction. It is of in-terest to note that the frequencies of the two-degrees-

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Experiments [5]2D-γ−Reθ

2D-SALaminar

(a) Pitching amplitudes

0

0.2

0.4

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0.8

1

6.0×104 8.0×104 1.0×105 1.2×105

Hea

ving

ampl

itude

y/c

Rec

2D-γ−Reθ

2D-SALaminar

(b) Heaving amplitudes

0.03

0.04

0.05

0.06

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0.1

6.0×104 8.0×104 1.0×105 1.2×105

fc/U

∞

Rec

Experiments [5]2D-γ−Reθ

2D-SALaminar

(c) Frequencies

Figure 5: Comparison of self-sustained 2 DOF oscil-lations characteristics from numerical simulations andexperiments.

of-freedom oscillations are very close to those of thepitch-only oscillations despite their different charac-teristics.

Figures 6 and 7 show the airfoil’s pitching and heavingamplitudes at Rec = 75,000 over the whole run along

with the power spectral density of the responses. Bothoscillations are periodic and they have the same dom-inating frequency as expected. This is typical of allcases over the range of Reynolds numbers of interest.

time (s)

y/c

0 1 2 3­80

­60

­40

­20

0

20

40

60

80

(a) Pitch history

Frequency (hertz)

PSD

ofpitchresponse

100

101

102

10­4

10­3

10­2

10­1

100

101

102

(b) PSD of pitch response

Figure 6: Airfoil pitching amplitude and frequencyover the whole run at Rec = 75,000.

time (s)

y/c

0 1 2 3­0.4

­0.2

0

0.2

0.4

(a) Heaving history

Frequency (hertz)

PSD

ofverticaldisplacement

100

101

102

10­5

10­4

10­3

10­2

10­1

(b) PSD of heaving motion

Figure 7: Airfoil heaving amplitude and frequencyover the whole run at Rec = 75,000.

Figure 8 shows both the pitching and heaving responsefor one LCO cycle at two Reynolds numbers. Thisshows that there is phase-shift between the pitchingand heaving motions as the wing does not reach itshigher angle of attack when it is at the highest verticalposition. To quantify this phenomenon, φ is definedas the phase-shift between the pitching and heavingmotions. It is presented in degrees to compare withprevious studies even though, in the present case, themotions are not necessarily purely sinusoidal. Hence,for example, φ = 90 degrees corresponds to a shift ofa quarter cycle between pitching and heaving.

At Rec = 75,000 the phase-shift is of 40 degrees (∼1/10 cycle) while at Rec = 117,500 it is of 97 degrees(∼ 1/4 cycle). Looking over the range of Reynoldsnumbers, the phase-shift increases with the Reynolds,from 24 degrees at Rec = 64,000 to 117 degrees atRec = 139,000. These results are presented in Table 1.

The evolution of the vertical force coefficient Cy andthe aerodynamic moment coefficient Cm in one LCO

t/T

pitch(deg)

y/c

0 0.2 0.4 0.6 0.8 1­100

­50

0

50

100

­1

­0.5

0

0.5

1

y/c

θ(t)

(a) Rec = 75,000

t/T

pitch(deg)

y/c

0 0.2 0.4 0.6 0.8 1­100

­50

0

50

100

­1

­0.5

0

0.5

1

y/c

θ(t)

(b) Rec = 117,500

Figure 8: Pitching and heaving responses for one cycleof LCO at two Reynolds numbers.

Re θmax (y/c)max f c/U φ

64000 48◦ 0.25 0.089 26◦

75000 55◦ 0.36 0.076 40◦

85500 59◦ 0.43 0.071 52◦

96000 60◦ 0.45 0.066 72◦

107000 58◦ 0.45 0.059 83◦

117500 54◦ 0.45 0.059 97◦

128000 49◦ 0.44 0.055 108◦

139000 45◦ 0.42 0.055 117◦

Table 1: Two DOF self-sustained oscillations charac-teristics.

cycle is shown in Fig. 9 at the same two Reynolds num-bers.

t/T

Cy

CM

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1­4

­3

­2

­1

0

1

2

3

4

­1

­0.5

0

0.5

1

Cy

CM

(a) Rec = 75,000

t/T

Cy

CM

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1­4

­3

­2

­1

0

1

2

3

4

­1

­0.5

0

0.5

1

Cy

CM

(b) Rec = 117,500

Figure 9: Vertical force and moment coefficients forone cycle of LCO at two Reynolds numbers.

Looking first at Cy, it increases rapidly from t/T = 0to t = T/8. Then follows a sharp decrease (up tot = 3T/8 at Rec = 75,000 and up to t = T/4 at Rec =117,500) followed by a small plateau slightly above 0around t = 3T/8 and then another decrease to a nega-tive value. The moment coefficient has a somewhat op-posite behavior. It is negative and decreasing at the be-ginning of the cycle and reaches its lower value shortlyafter Cy reaches its peak. It then increases and goes

through a small plateau similar to Cy before increasingto a positive value. For both coefficients the oppositebehavior is observed in the second half of the cycle.

3.2 Flow dynamics

The instantaneous spanwise vorticity and pressurecontours at eight times in a half-cycle are shownin Figs. 10 and 11 at Rec = 75,000 and Rec =117,500. The flow dynamics is somewhat similar atboth Reynolds numbers. The wing begins its cycleby pitching up, which causes the increase in Cy anddecrease of Cm shown in Fig. 9. The pivot point be-ing relatively close to the leading edge, the high angleof attack creates an important negative moment. Cyreaches its peak value around t = T/8, when a lead-ing edge vortex starts to appear. This separation ofthe flow causes deep stall and the rapid decrease oflift from t = T/8 to t = 3T/8 or t = T/4 dependingon the Reynolds. Cm reaches its lowest value aroundt = T/4 when the wing is at its highest angle of at-tack and then increases as the wing starts to pitchdown. The small plateau observed in both lift andmoment coefficients is caused by a trailing edge vor-tex that forms around t = 3T/8 at Rec = 75,000 andt = T/4 at Rec = 117,500. This vortex is bigger atRec = 117,500 and thus has a sligthly larger impacton the force and moment. At all Reynolds number, a2P vortex shedding mode [14] is observed as two pairsof vortices are shed in each oscillation cycle.

Contrary to the pitch-only oscillations, laminar sep-aration of the boundary layer is not the triggeringphenomenon of the self-sustained pitch-heave oscil-lations [4, 9]. The large pitch-heave oscillationshave been obtained numerically with a fully turbulentmodel (Spalart-Allmaras) and experimentally evenwhen placing a trip wire close to the leading edge, re-vealing the secondary role of the boundary layer state.The large oscillations reported here are thus not associ-ated with laminar separation flutter as is the case withthe pitch-only oscillations. The LCOs observed hereare rather described as coalescence flutter. The insta-bility causing coalescence flutter is due to the couplingbetween the vertical and torsional degrees of freedomof the airfoil [10]. If the center of gravity of the wingis placed at the pivot point (xθ = 0 in Eqs. (3) and(4)) then the equations of motion are not coupled and,indeed, simulations performed with this value did notproduce any oscillations.

Figures 12 and 13 compare the contours of vorticityand turbulent viscosity ratio (νt/ν) between simula-tions with the Spalart-Allmaras model and simulations

(a) Vorticity contours

(b) Pressure contours

Figure 10: Instantaneous vorticity and pressure contours at Rec = 75,000.

with the transition model at the same instant in a cycleat Rec = 75,000. There are some significant differ-ences in the turbulent viscosity levels between thesetwo simulations as the results obtained with the transi-tion model present lower levels of turbulent viscosityclose to the airfoil and in the wake. The flow dynamicsobserved are somewhat in between those of a laminarsimulation and those of the fully-turbulent S-A model.This is reflected in the vorticity contours which showmore unsteadiness in the solution computed with theγ−Reθ model. Despite this difference in the instanta-neous flow fields, the global results are very similar asboth models predict similar pitching and heaving am-plitudes and frequencies.

4 CONCLUSION

Self-sustained pitch-heave oscillations of an airfoilhave been studied to gain some understanding of thephenomenon. Two-dimensional URANS simulationswere performed using both the Spalart-Allmaras andthe γ− Reθ models. Predicted oscillation frequen-cies were quite close to the experimental values whilepitching amplitudes did not exhibit such a good agree-ment with the measurements from wind tunnel exper-iments. However, they still showed comparable trendsand similarities. Discrepancies between the numericaland experimental values could be caused by the 2D as-sumption of the flow which has important impacts in

(a) Vorticity contours

(b) Pressure contours

Figure 11: Instantaneous vorticity and pressure contours at Rec = 117,500.

the case of massively separated flows.

As opposed to the pitch-only simulations which arecaused by laminar separation of the boundary layer,these large oscillations are caused by an instability oc-curing through the coupling between the pitching andheaving motions and are thus labelled as coalescenceflutter. The analysis of the flow fields showed impor-tant boundary layer separation at both the leading andtrailing edges, causing large vortex shedding.

ACKNOWLEDGEMENTS

Financial support from NSERC Canada and FQRNTQuébec is gratefully acknowledged by the authors.Computations were performed on the Colosse super-computer at the CLUMEQ HPC Consortium and theGPC supercomputer at the SciNet HPC Consortium,both under the auspices of Compute Canada. The au-thors would also like to thank Prof. D. Poirel for accessto his experimental data.

(a) Vorticity contours (b) Viscosity ratio contours

Figure 12: Instantaneous vorticity and turbulentviscosity ratio contours at Rec = 75,000, Spalart-Allmaras.

(a) Vorticity contours (b) Viscosity ratio contours

Figure 13: Instantaneous vorticity and turbulent vis-cosity ratio contours at Rec = 75,000, γ−Reθ.

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