Dynamicinteractionsofasupercriticalaerofoilinthepres- ence

Dynamic interactions of a supercritical aerofoil in the pres-ence of transonic shock buffet

N.F. Giannelis 1, G.A. Vio 1 & G. Dimitriadis 2

1 School of Aerospace, Mechanical and Mechatronic Engineering,The University of Sydney, NSW, 2006, Australiae-mail: [email protected]

2 Department of Aerospace and Mechanical Engineering,University of Liege, 1, Chemin des Chevreuils, Liege, B-4000, Belgium

AbstractWithin a narrow transonic flight region, shock-wave/boundary-layer interactions yield large amplitude, self-sustained shock oscillations that are detrimental to both platform handling quality and structural integrity.In this study, the aeroelastic interactions between this transonic buffet instability and a spring-suspendedsupercritical aerofoil are investigated by means of Reynolds-Averaged Navier-Stokes simulations. Singledegree-of-freedom pitching simulations are performed for a range of structural to aerodynamic frequencyratios, sectional mass ratios and levels of structural damping. The results show that for a range of pitcheigenfrequencies above the fundamental buffet frequency, sychronisation of the aerodynamic and structuralmodes occurs. This so called lock-in phenomenon acts as a mechanism for large amplitude Limit CycleOscillation in aircraft structures within the transonic flow regime. The sectional mass and the addition ofstructural damping are both found to have a pronounced effect on the nature of the limit cycles.

1 Introduction

For a small envelope of Mach number and incidence combinations in the transonic flight regime, the interac-tions between shock waves and thin, separated shear-layers give rise to large amplitude, autonomous shockoscillations. This transonic buffet instability acts as a limiting factor in aircraft performance. The reducedfrequency of shock oscillation is typically on the order of the low-frequency structural modes, resulting inan aircraft that is susceptible to Limit Cycle Oscillation (LCO), and as a consequence, diminished handlingquality and fatigue life.

Hilton & Fowler [1] first observed transonic shock-induced oscillations over six decades ago, yet the physicsgoverning this complex phenomenon remains elusive. Lee [2] proposed an underlying mechanism based onacoustic wave propagation feedback. In Lee’s model, the motion of the shock wave generates downstreampropagating pressure waves, with the instability growing as it travels from the separation point through theshear layer. The separated shear layer induces a de-cambering effect, interacting with the separated flow atthe trailing edge and producing pressure waves that travel upstream along the upper aerofoil surface. Inter-action between these upstream propagating disturbances and the shock completes a feedback loop, yieldingsustained shock oscillation. An extension to this theory was posited by Jacquin et. al. [3], where acousticwaves convect not only along the upper surface, but also across the lower surface and around the leadingedge, propagating downstream towards the shock. This modified interpretation of the buffet mechanism bet-ter supports the author’s experimental findings and has also been observed computationally by Garnier &Deck [4].

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The complex shock-wave/boundary layer interactions and intermittently separated flow field inherent to thetransonic buffet phenomenon pose significant challenges to numerical simulation. Further, the fundamentalrole of the separated flow region in Lee’s [2] wave propagation feedback mechanism implies the necessityof computationally taxing scale-resolving simulations to model the instability. Nonetheless, a plethora ofcomputational investigations have successfully captured the inherent flow features of shock-induced oscil-lations through Unsteady Reynolds-Averaged Navier-Stokes (URANS) methods, albeit with an appreciablesensitivity to various simulation parameters [5–11]. In particular, the selection of a suitable turbulence clo-sure [8, 12–14], sufficient grid refinement in the shock region [9, 15, 16] and the use of Dual Time Stepping(DTS) with an acoustic temporal resolution [15, 17] have been shown to be critical in URANS modelling oftransonic shock buffet. Ultimately, the efficacy of URANS simulations in the prediction of transonic buffet isattributed to the low frequencies characteristic of shock motion, which exhibit significantly longer timescalesthan those of the shear layer eddies [10]. Such a success for a computationally efficient means of simulatingintricate aerodynamic phenomena holds promise for the numerical investigation of the complex interactionmechanisms between a buffeting flow field and a deforming structure.

A number of experimental studies have considered the influence of forced harmonic motions on an aero-foil at transonic conditions [18–21]. Additionally, for harmonic excitation in the presence of shock-inducedseparation, a number of authors have reported aerodynamic resonance for driving frequencies near to thefundamental frequency of shock-oscillation [22–24]. The nature of this resonance has been formalised byRaveh [25–27] as a frequency lock-in phenomenon, whereby for sufficient amplitudes of motion at excitationfrequencies in the vicinity of the buffet frequency, the buffet flow response synchronises with the aerofoilmotion. Raveh & Dowell [28] extended the work on shock buffet lock-in to spring-suspended aeroelasticsystems, finding synchronisation of the aerodynamic and structural eigenfrequencies in both pitch, heaveand coupled simulations. As a significant implication of these findings, the authors propose shock buffetlock-in as a possible mechanism governing transonic LCO instabilities. More recent literature in the fieldhas continued the exploration of aeroelastic systems in the presence of shock buffet, concentrating on classi-fying the influence of various structural parameters, particularly the ratio of structural and shock oscillationeigenfrequencies, on the lock-in phenomenon [29–31].

Following the work of Giannelis & Vio [29], this study aims to further the understanding of the shock buffetlock-in phenomenon in aeroelastic systems and corroborate the trends observed by Quan et. al. [31] forthe NACA0012 on the supercritical OAT15A aerofoil section. Particular emphasis is given to classifyingthe effects of frequency ratio, mass ratio and structural damping on frequency synchronisation. The articleproceeds in Section 2 with a description of the computational approach employed in the static aerofoil sim-ulations. Section 3 then presents the main results of the static case, including validation of the turbulencemodels against experimental data and the nature of the buffeting flow field at the experimental condition.In Section 4, the methodology used in the aeroelastic simulations is briefly presented. This is followed byboth time and frequency domain analysis of the dynamic aerofoil systems, with a discussion of the relation-ship between the respective structural parameters and the presence of shock buffet lock-in. Section 5 thenconcludes the paper with a summary of the major findings.

2 Numerical method

2.1 Test case

This study investigates the flow field around the OAT15A supercritical aerofoil at transonic buffet conditions.Experiments on this section have been performed in the S3Ch Continuous Research Wind Tunnel at theONERA Chalais-Meudon Center and are detailed by Jacquin et. al. [3, 32]. A wind tunnel model of 12.3%relative thickness, 230 mm chord, 780 mm span and a 1.15 mm thick trailing edge was constructed for theexperiment. The model ensured a fixed boundary layer transition at 7% chord through the installation of acarborundum strip on the upper and lower surfaces.

458 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

The experiments conducted at ONERA sought to develop an extensive experimental database for the valida-tion of numerical buffet simulations. The model was fitted with 68 static pressure orifices and 36 unsteadyKulite pressure transducers through the central span to mitigate 3D effects from sidewall boundary layers.Adaptable upper and lower wind tunnel walls further alleviated wall interference, allowing for a test sectionMach number uncertainty of 10−4. The investigation applied a sublimating product to the model surface,permitting oil flow visualizations for characterisation of turbulent regions and shock motion. The authorsemployed Schlieren cinematography and Laser Doppler Velocimetry (LDV) to observe qualitative flow fea-tures during the buffet cycle. Further, steady and unsteady pressure measurements yield mean and RMSpressure data, along with spectral content for the pressure fluctuations.

The test programme undertaken by Jacquin et. al. [3] consisted of an angle of attack sweep at M∞ = 0.73to obtain data for buffet onset, as well as Mach number sweeps at α∞ = 3◦ and α∞ = 3.5◦. In this study,the data at M∞ = 0.73 and α∞ = 3.5◦ are employed to validate the numerical method. This case is used asa baseline from which dynamic aeroelastic simulation of a spring-suspended OAT15A aerofoil is performed.

2.2 Flow solver

Simulations are performed using the commercial, cell-centered finite volume code ANSYS Fluent. The2D density-based implicit solver is used to formulate the coupled set of continuity, momentum and energyequations. The inviscid fluxes are resolved by an upwind Roe flux difference splitting scheme with theblended central difference/second-order upwind MUSCL scheme for extrapolation of the convective quan-tities. All diffusive fluxes are treated with a second-order accurate central-difference scheme. Gradients forthe convection and diffusion terms are constructed through a cell based Least Squares method and solved byGram-Schmidt decomposition of the cells coefficient matrix.

Three turbulence models are considered for closure of the Navier-Stokes equations; the Spalart-Allmarasmodel (SA) [33], Menter’s k − ω SST [34] and the Stress-Omega Reynolds Stress Model (SORSM), astress-transport model derived from the omega equations and the Launder-Reece-Rodi (LRR) model [35].The SA and SST models have been used extensively in URANS buffeting simulations, successfully captur-ing the bulk flow buffeting features. Reynolds Stress Models (RSM) have been less widely applied in theavailable shock buffet literature, however Illi et. al. [16] observed excellent correlations to the OAT15Aexperimental data set using a εh RSM model. The SORSM is derived by taking the second-moments ofthe exact momentum equations, yielding an additional four transport equations for the Reynolds stresses,together with an equation for the dissipation rate. All turbulent transport equations are solved segregatedfrom the coupled set of continuity, momentum and energy equations, with second-order accurate upwinddiscretisation of the turbulent quantities.

2.3 Spatial & temporal discretisation

Calculations in this study are performed on a two-dimensional CH-type structured grid (Figure 1(b)) withfar-field boundaries located 80 chord lengths from the profile, as shown in Figure 1(a). The domain is dividedinto two zones; a laminar region upstream and along 7% of the aerofoil chord forward section and a turbulentregion in the remainder of the domain to represent the experimentally imposed boundary-layer transition.

Three grids have been generated to assess mesh independence, with the grid parameters provided in Table1. Refinement levels are primarily dictated by shock resolution across the aerofoil surface, with minorrefinement adopted in the wall normal direction. A wall y+ ≈ 1 is achieved at each level of refinement.Grid convergence is assessed based upon steady flow pressure distributions at M∞ = 0.73 and α∞ = 2.5◦

using the SA turbulence model. Mesh independent solutions are achieved with Grid B and thus, this gridis employed for all subsequent simulations. Grid B is comprised of 285 nodes along each surface of theaerofoil profile, 96 nodes in the wake and 92 nodes in the wall normal direction.

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(a) Far-field Grid Topology (b) Near Wall Grid Topology - Grid C

Figure 1: Computational Grid

Grid Size (i× j) Shock Resolution (c)A 288× 86 0.005B 381× 92 0.0035C 472× 98 0.0025

Table 1: Computational Grid Properties

For time-accurate solutions, an implicit second-order accurate backward Euler DTS scheme is used. Amaximum Courant-Friedrichs-Lewy (CFL) number of five is imposed for the pseudo-time stepping at eachphysical time step. During the developed buffet cycle, a minimum of 10 Newton sub-iterations are requiredto reduce the L1 norm of the residuals to 10−5. In order to resolve the propagating pressure waves inher-ent to the shock oscillation phenomenon, near acoustic temporal resolution is required. As such, a fixednondimensional time step of ∆t = 0.01 is used in all time-accurate solutions.

3 Static aerofoil simulations

3.1 Turbulence modeling

The choice of the turbulence model used to explore the buffeting flow of the OAT15A aerofoil is madebased on correlations to the experimental mean and RMS pressure data, in addition to the predicted buffetfrequency and lift differential. In Figure 2 the mean and RMS pressure coefficients for each of the turbulencemodels are given. Each of the closures results in shock unsteadiness at the experimental conditions, albeitwith varying degrees of accuracy. The SA model severely under-predicts the degree of pressure fluctuationdue to shock oscillation, as evident in the abrupt pressure recovery in Figure 2(a) and small amplitude RMSpressure in Figure 2(b). Conversely, the RSM model overestimates the magnitude of pressure fluctuationsdue to shock motion, along with the range of shock travel, which covers approximately 30% of the chord.The mean pressure distribution and trailing edge pressure fluctuations are, however, in fair agreement withexperiment. Nonetheless, of the three closures considered the SST model exhibits the best correlation to theexperiment. The mean pressure distribution is in excellent agreement, and whilst the amplitude of pressure


fluctuations due to shock motion and trailing edge separation are slightly underestimated, the range of shocktravel and mean shock location is well captured.

(a) Mean Pressure Coefficient (b) RMS Pressure Coefficient

Figure 2: Pressure Correlations

In addition to the pressure data, the various turbulent closures are also assessed for accuracy of the predictedbuffet frequency and lift differential. The resultant buffet characteristics are given in Table 2. Each of themodels yields a higher buffet frequency relative to the experiment; however, with a maximum deviation ofapproximately 10%, the buffet frequency is generally predicted well. The best agreement to the experimentalfrequency is achieved with the SST closure, exhibiting a deviation of 5%. The SST model also producesfair agreement in the peak-to-peak lift differential during the buffet cycle. This model is deemed to bestrepresent the experimental buffet conditions, capturing both the shock oscillation and alternating trailingedge separation inherent to transonic buffet. As such, the remainder of this study proceeds with the SSTclosure.

f (Hz) ∆CLSA 74.18 0.01SST 72.55 0.15SORSM 74.07 0.33Exp 69 0.11

Table 2: Computed Buffet Characteristics

3.2 Buffet response at experimental conditions

The lift coefficient time and frequency responses at M∞ = 0.73 and α∞ = 3.5◦ are provided in Figure 3.The time history in Figure 3(a) shows the lift oscillations of the fully developed buffeting flow. Lift is seento oscillate in a Period-1 LCO of constant amplitude, with no pronounced nonlinear effects. This is furtherevident in Figure 3(b), where the frequency content is concentrated at the buffet frequency for the fullydeveloped buffeting flow. Minor peaks are also observed at the second and third harmonic of the buffetfrequency; however, they exhibit insufficient power to influence the time history. The periodic nature ofthe buffet observed at these conditions is characteristic of Type A shock buffet as described by Tijdeman &Seebass [36].

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(a) CL Time Response (b) CL Frequency Response

Figure 3: Lift Coefficient Time & Frequency Responses at M∞ = 0.73 & α∞ = 3.5◦

4 Aeroelastic simulations

To explore the aeroelastic interactions between a buffeting flow and the OAT15A aerofoil constrained tosingle degree-of-freedom motion in pitch, Fluent’s Six-DOF Rigid Body solver is employed. For the fluid,all simulations are performed at the experimental condition, with M∞ = 0.73 and a mean α∞ = 3.5◦. Atthis condition, the static aerofoil buffet reduced frequency of fSB0 = 0.43 is obtained, as calculated through:

f =2πfc

U∞(1)

where f is the reduced frequency, f is the circular frequency, c is the aerofoil chord and U∞ is the freestreamvelocity at the experimental condition. The single degree-of-freedom aeroelastic simulations are initialisedwith the steady-state LCO solution of the static aerofoil. In the present computations, the aeroelastic casesare modelled as spring-mass-damper systems. The equation of motion for the pitching system is thus:

Iα(α+ 2ζωαα+ ω2αα) = M1/4c (2)

where α, α and α are the pitch displacement, velocity and acceleration respectively, ωα is the pitch naturalfrequency, ζ is the structural damping and M1/4c is the pitching moment about the quarter-chord point. Theelastic axis and centre of gravity are enforced at the quarter-chord point such that all moments in Equation 2are taken about this point. The pitch moment of inertia Iα is computed by:

Iα = µπρ∞b3r2α (3)

where ρ∞ is the freestream density, b is the aerofoil semi-chord and r2α = 0.75 is the radius of gyration,which is held constant across all simulations. The parameter µ is the sectional mass ratio, given by:

µ =m

πρ∞b2(4)

where m is the aerofoil mass. In the aeroelastic simulations, nominal conditions of µ = 50 and ζ = 0%are chosen. The analysis proceeds by sweeping through a range of wind-off pitch eigenfrequencies (fα0),spanning through 0.5 < fα/fSB < 2.0. Simulations are run until convergence to a steady state LCOresponse in the pitch displacement is observed. A minimum nondimensional time for simulation of 2500 isalso imposed, as defined by:


t =ta∞c

(5)

where t is the nondimensional time, t is the physical time and a∞ is the freestream speed of sound. From thetime domain responses, the maximum amplitude lift coefficient and pitch displacements are extracted. Thefrequency content of the time histories is then determined through a Fast Fourier Transform (FFT).

In order to characterise the effect of mass ratio on the aeroelastic response, the preceding analysis is repeatedat µ = 100 and µ = 200. Spectrograms are also constructed by Short-Time Fourier Transform (STFT) toobserve the variations in frequency content during lock-in. The process is repeated at µ = 50 with ζ = 1%and ζ = 2% to assess the effects of structural damping on shock buffet synchronisation.

4.1 Effect of frequency ratio

In Figure 4, the variation of pitch displacement amplitude and dominant response frequency are graphedagainst frequency ratio. The parameter fα represents the dominant pitch response frequency once a steadystate LCO is achieved. The plots are divided into four distinct regions, where qualitatively dissimilar be-haviour is observed.

(a) Pitch Amplitude (b) Pitch Frequency Ratio

Figure 4: Effect of Frequency Ratio on Pitch Response Amplitude and Frequency

From Figure 4(b) it is evident that at frequency ratios below 0.6, the pitch response is concentrated at the firstsubharmonic of the buffet frequency. The LCO amplitude in Figure 4(a) is also seen to be approximatelytwice the magnitude observed in the adjacent region. This abrupt change in frequency and the accompanyingincrease in LCO amplitude is attributed to the low spring stiffness required to maintain a constant mass ratioin this range of frequencies. In the neighbouring region (0.6 < fα0/fSB0 < 1.0), the oscillating shock actsas an external forcing to the aeroelastic system. The pitching motion is driven at the buffet frequency, and agradual increase in LCO amplitude is evident as the frequency ratio approaches unity.

From approximately fα0/fSB0 = 1, lock-in of the shock oscillation frequency to the pitching mode isobserved. In Figure 4(b), the structural response is shown to be concentrated at the modal frequency throughto fα0 ≈ 1.8fSB0 . Throughout this region a distinct resonance is present in the pitch amplitude, withthe maximum amplitude LCO response occurring at fα ≈ 1.5fSB0 . This shift in the maximum amplitudeaeroelastic response to higher frequencies relative to the dominant aerodynamic mode is supported by similarfindings for forced harmonic motions [22, 25, 26]. At fα0 ≈ 1.8fSB0 offset of the shock buffet lock-inphenomenon occurs. The pitch response settles to a low amplitude LCO driven at the frequency of shockoscillation, essentially behaving as a single degree-of-freedom harmonic oscillator.

Similar behaviour is also observed in the lift coefficient response of Figure 5. At low frequencies, the reducedpitch stiffness incites a larger amplitude aerodynamic response. Although not apparent in Figure 5(b), the

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subharmonic of the pitch response of Figure 4(b) at low frequency ratios is also present in the aerodynamicresponse, albeit less prominent than the fundamental buffet frequency. The lock-in region is again apparentfor 1 < fα0/fSB0 < 1.8, with a gradual onset at approximately the static buffet frequency and abrupt offset atthe high frequency bound. In Figure 5(b) the buffet frequency sychronises with the pitch mode throughout thelock-in region. Somewhat dissimilar to the pitch response however, the frequency of maximum aerodynamicresonance occurs near the centre of the lock-in region, with fα0 = 1.4fSB0 . The magnitude of aerodynamicresonance is significant. The four-fold increase in lift amplitude (relative to the static aerofoil) highlights thesevere detriment shock buffet, and the associated lock-in phenomenon, poses to aircraft controllability.

(a) Lift Magnitude Ratio (b) Lift Frequency Ratio

Figure 5: Effect of Frequency Ratio on Lift Response Amplitude and Frequency

4.2 Effect of mass ratio

The maximum LCO amplitude and dominant response frequency of the pitch angle for varying mass ratioand a range of structural eigenfrequencies is presented in Figure 6. From Figure 6(b), the mass ratio has littlebearing on the lock-in onset frequency, with synchronisation occurring at approximately the static aerofoilbuffet frequency. Additionally, the mass ratio imparts little influence on the LCO amplitude throughout thelock-in region, with peak amplitudes differing by a marginal 3% between µ = 50 and µ = 200. Nonetheless,an apparent effect of mass ratio on shock buffet synchronisation is reflected in the range of the lock-in region.As mass ratio increases the lock-in region contracts, reducing the frequency range of synchronisation by 17%between µ = 200 and the nominal condition. Analogous findings also persist in the lift coefficient response.


Figure 6: Effect of Mass Ratio on Pitch Response Amplitude and Frequency


In order to discern the effects of mass ratio at a given frequency ratio, Figure 7 presents the time historiesof the lift coefficient and pitch displacement at fα = 1.25fSB0 for three distinct mass ratios. As evident inFigure 6(a), once a converged LCO is achieved the amplitude of oscillation in both lift and pitch is scarcelyaffected by the sectional mass. Even with an eight-fold increase in mass, a reduction in amplitude of 9%and 15% is observed in the lift and pitch responses, respectively. An increase in mass ratio however, doesinfluence the rate of convergence to a fully developed LCO, lengthening the time required to achieve a steadystate response. Further, a pronounced beating in the time histories is evident in the lift response during shockbuffet synchronisation. This beating response becomes more pronounced as the mass ratio is increased.

(a) Lift Coefficient (b) Pitch Displacement

Figure 7: Lift and Pitch Response Time Histories with Variable Mass Ratio - fα = 1.25fSB0

Figure 8 shows the progression to shock buffet lock-in in the lift and pitch responses explicitly. In Figure 8(a),the lift response exhibits frequency content concentrated at fSB0 and it’s second and third harmonics. Fromthe outset of the simulations, the pitch eigenfrequency is also apparent in the aerodynamic response. Asthe simulation progresses, the shock buffet frequency fades and the superharmonics are drawn towards thesecond and third harmonics of the pitch mode. The response in pitch is somewhat simpler. Following theinitial beating involving the fundamental shock oscillation and pitch frequencies, the steady state responseevolves to a simple period-1 LCO, oscillating at the structural eigenfrequency.


Figure 8: Short-Time Fourier Transforms of Lift Coefficient and Pitch Displacement - µ = 400

4.3 Effect of structural damping

The effects of structural damping on the appearance of shock buffet lock-in are more pronounced than thoseof mass ratio. In Figure 9 the pitch LCO amplitude and dominant frequency are given for a range of frequency

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ratios and various degrees of structural damping. From Figure 9(b), it is evident that the onset frequency forlock-in is invariant to damping in the pitch mode. However, as shown in Figure 9(a), the addition of dampingdoes significantly reduce the amplitude of the pitch response, with a 57% reduction in peak magnitudebetween ζ = 0% and ζ = 2%. The lock-in region also shows an appreciable contraction between theseconditions, narrowing by 50%.


Figure 9: Effect of Structural Damping on Pitch Response Amplitude and Frequency

The trends observed in Figure 9 are qualitatively consistent with Quan et. al. [31], however, Raveh & Dowell[28] have reported that a small addition of structural damping (ζ = 0.5%) is sufficient to quench the lock-inphenomenon at fα = 1.13fSB0 . Such discrepancies may be a consequence of the differing sections (whereasthe current study is performed on the supercritical OAT15A aerofoil, the preceding analyses investigated thesymmetric NACA0012), whereby the range of the lock-in region for a symmetric aerofoil is more sensitiveto the addition of structural damping. Nonetheless, further analysis is required to establish the nature of thesedifferences. Again, the findings for the pitch displacement are indicative of the trends observed in the liftcoefficient.

In Figure 10, the time histories of the lift coefficient and pitch displacement at fα = 1.25fSB0 at three levelsof structural damping are given. A significant reduction in LCO amplitude with the addition of dampingis evident in both the lift and pitch responses of Figure 10(a) and 10(b) respectively. The longer transitionto steady state LCO observed in Figure 7, a consequence of the increasing mass ratio, is, however, not aspronounced as the degree of structural damping is increased.


Figure 10: Lift and Pitch Response Time Histories with Variable Structural Damping - fα = 1.25fSB0


Similar to the STFT results presented in Figure 8, the dominant lift response frequency at fα = 1.25fSB0

and ζ = 5% in Figure 11(a) transitions from the static buffet frequency to the pitch natural frequency, withthe first three harmonics again prevalent. Regarding the pitch response frequency content in Figure 11(b),the shock buffet frequency is seen to swiftly vanish and the structure oscillates primarily at the pitch eigen-frequency.


Figure 11: Short-Time Fourier Transforms of Lift Coefficient and Pitch Displacement - ζ = 5%

5 Conclusion

In this paper, the transonic buffeting flow over the OAT15A supercritical aerofoil has been analysed throughURANS simulation. Grid independence is assessed relative to a steady flow pre-buffet condition and isachieved with a medium density grid. Following spatial convergence, various turbulence models are con-sidered for the time-accurate solutions. It is found that Menter’s SST model yields unsteady solutions thatcorrelate well with the experimental results. The buffeting flow field at the experimental condition has beenanalysed, and is shown to be characteristic of Type A shock motion, with periodic shock excursions alongthe upper surface.

Aeroelastic simulations at the experimental flow conditions have also been performed for a single degree-of-freedom pitching aerofoil. A sensitivity analysis on the aeroelastic response with a variable structuralfrequency has been conducted and has identified four frequency regions of qualitatively distinct behaviour.With fα0/fSB0 < 0.6, highly nonlinear pitch and lift responses develop, with comparatively large amplitudeoscillations. In the regions bound by 0.6 < fα0/fSB0 < 1 and fα0/fSB0 > 1.8, the pitch response behavesas a single degree-of-freedom oscillator, excited by the unsteady shock oscillations. Within the frequency re-gion bound by 0.6 < fα0/fSB0 < 1 resonance is observed in both the pitch and lift responses, a consequenceof the shock buffet frequency synchronising with the pitch mode.

The effects of mass ratio on the shock buffet lock-in phenomenon have been characterised by sweeps acrosspitch natural frequency for a range of sectional masses. The mass ratio is shown to have no effect on the onsetof lock-in, with the aerodynamic and structural modes sychronising at fα/fSB0 ≈ 1 in all cases. The resultsalso indicate that sectional mass has little influence on the LCO amplitude; however, an increase in the massratio yields a narrowing of the lock-in region and reduces the rate of convergence to a fully developed limitcycle. The addition of structural damping to the aeroelastic system is of greater influence to the emergenceand nature of shock buffet lock-in. With a small degree of damping, both the LCO amplitude and the extentof the lock-in region decrease appreciably.

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The trends identified in the aeroelastic simulations of the pitching system are in accordance with previousstudies. Lock-in is observed for a region of frequency ratios where the pitch natural frequency exceeds thefrequency of shock oscillation, and an increase in both mass ratio and structural damping act to contract thisregion. Ultimately though, each study in the available literature investigating the aeroelastic response of anaerofoil section in the presence of transonic shock oscillations has been purely computational. Althoughthere are significant technical challenges in conducting aeroelastic experiments in the transonic regime, itis imperative that the effects observed in these investigations are confirmed and validated by wind tunneltesting.

Acknowledgements

The authors would like to thank Dr Robert Carrese from RMIT University and Dr Oleg Levinksi fromthe Defence Science and Technology Group Australia for their comprehensive insight into transonic buffetphenomenon, and for providing the preliminary test cases from which the present work was developed. Thisresearch was partially funded by the Defence Science and Technology Group.

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