A review of recent developments in the understanding of transonic shock buffet

Progress in Aerospace Sciences xxx (2017) 1–46

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A review of recent developments in the understanding of transonic shock buffet Nicholas F. Giannelis a, *, Gareth A. Vio a, Oleg Levinski b a b

School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Sydney, Australia Aerodynamics and Aeroelasticity, Aerospace Division, Defence Science and Technology Group, Melbourne, VIC 3207, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Transonic shock buffet Unsteady aerodynamics Shock-wave/boundary layer interaction Nonlinear aeroelasticity Shock control

Within a narrow band of flight conditions in the transonic regime, interactions between shock-waves and intermittently separated shear layers result in large amplitude, self-sustained shock oscillations. This phenomenon, known as transonic shock buffet, limits the flight envelope and is detrimental to both platform handling quality and structural integrity. The severity of this instability has incited a plethora of research to ascertain an underlying physical mechanism, and yet, with over six decades of investigation, aspects of this complex phenomenon remain inexplicable. To promote continual progress in the understanding of transonic shock buffet, this review presents a consolidation of recent investigations in the field. The paper begins with a conspectus of the seminal literature on shock-induced separation and modes of shock oscillation. The currently prevailing theories for the governing physics of transonic shock buffet are then detailed. This is followed by an overview of computational studies exploring the phenomenon, where the results of simulation are shown to be highly sensitive to the specific numerical methods employed. Wind tunnel investigations on two-dimensional aerofoils at shock buffet conditions are then outlined and the importance of these experiments for the development of physical models stressed. Research considering dynamic structural interactions in the presence of shock buffet is also highlighted, with a particular emphasis on the emergence of a frequency synchronisation phenomenon. An overview of three-dimensional buffet is provided next, where investigations suggest the governing mechanism may differ significantly from that of two-dimensional sections. Subsequently, a number of buffet suppression technologies are described and their efficacy in mitigating shock oscillations is assessed. To conclude, recommendations for the direction of future research efforts are given.

1. Introduction Within a narrow region of the transonic flight regime, the interactions between shock-waves and thin, separated shear layers give rise to large amplitude, autonomous shock oscillations. This instability, commonly known as transonic shock buffet, acts as a limiting factor in aircraft performance. The reduced frequency of shock oscillation is typically on the order of the low-frequency structural modes, resulting in an aircraft that is susceptible to limit cycle oscillations (LCOs), and as a consequence, diminished handling quality and fatigue life. Hilton & Fowler [1] first observed transonic shock-induced oscillations over six decades ago, yet the physics governing aspects of this complex phenomenon remains elusive. Various numerical and experimental investigations have identified two distinct types of shock buffet on aerofoils. Type I buffet typically occurs at zero incidence on biconvex sections and encompasses shock oscillations on both the pressure and

suction surfaces of an aerofoil. Through the investigations of Mabey [2] and Gibb [3], a working model of Type I buffet was developed, whereby shock-wave/boundary layer interactions on both surfaces initiate phaselocked shock oscillations in opposing directions. As the shock on the upper surface moves upstream, it weakens. This permits reattachment of the separated zone and propels the shock downstream. The shock motion on the lower surfaces occurs in an identical manner, with a 180
phase shift, yielding self-sustained shock buffet cycle. As Type I buffet is critically dependent on the shock having sufficient strength to produce separation, several authors have proposed the prediction of buffet onset by the Mach number immediately ahead of the shock [2–4]. Type II shock buffet is characteristic of modern supercritical aerofoils and involves upper surface shock oscillations at non-zero angles of attack. A working model of this second type that is unequivocally accepted by the research community has yet to be determined. Early work by Pearcey [5,6], Pearcey & Holder [7] and Pearcey et al. [8] was instrumental in

* Corresponding author. E-mail address: [email protected] (N.F. Giannelis). http://dx.doi.org/10.1016/j.paerosci.2017.05.004 Received 1 March 2017; Received in revised form 22 May 2017; Accepted 25 May 2017 Available online xxxx 0376-0421/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: N.F. Giannelis, et al., A review of recent developments in the understanding of transonic shock buffet, Progress in Aerospace Sciences (2017), http://dx.doi.org/10.1016/j.paerosci.2017.05.004

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~ d fd xþ zþ τp τu τu;u τu;l fsb fα fh fα0 fh0 ω ζ V Λ δ δ A P β Cl0 λ

Nomenclature M α Re xs ap au c b τ ϕ a Mc R Ms ρ q u v T t ω S CL f yþ ~ν d ~ S Δ CDES

freestream Mach number freestream angle of attack/vortex generator pitch angle chord-based Reynolds number mean shock location downstream pressure perturbation convection velocity upstream pressure perturbation convection velocity chord span buffet period/time delay phase speed of sound local Mach number constant upper surface Mach number freestream density two-dimensional flow state vector fρ; u; v; T; ~νg streamwise velocity transverse velocity temperature time frequency blending function lift coefficient frequency nondimensional wall-normal distance eddy viscosity wall distance (length scale) local deformation rate grid size (¼ maxðΔx ; Δy ; Δz Þ) constant

h l d Cμ θramp ltail

modified length scale delaying function nondimensional streamwise distance nondimensional spanwise distance period downstream propagation period upstream propagation period upstream propagation above upper surface period upstream propagation below lower surface shock buffet frequency pitch natural frequency heave natural frequency wind-off pitch natural frequency wind-off heave natural frequency reduced frequency structural damping reduced velocity sweep angle TED deflection mean TED deflection TED amplitude pressure trailing edge flap deflection/vortex generator skew angle balanced lift coefficient dimensionless controller gain/spanwise vortex generator spacing vortex generator height vortex generator length fluidic vortex generator orifice diameter momentum coefficient shock control bump ramp angle shock control bump tail length

downstream excursion. Type C motion is qualitatively distinct from the preceding modes. The shock travels upstream, initially strengthening and then weakening, but continuing to move forward, eventually propagating forward into the oncoming flow as a free shock-wave. Although these shock motions were originally identified with oscillating aerofoils, each has subsequently been observed in rigid wing sections at certain flight conditions [12]. Considering Tijdeman Type A [11] shock motions, Lee [13] proposed an acoustic wave-propagation feedback model as the underlying mechanism governing the autonomous shock oscillations. In this model, the motion of the shock-wave generates downstream propagating pressure waves, with the instability growing as it travels from the separation point through the shear layer. The separated flow induces a de-cambering effect, and interactions with the flow at the trailing edge produce pressure waves that travel upstream in the subsonic flow above the boundary layer. Interaction between these upstream propagating disturbances and the shock completes a feedback loop, yielding sustained shock motion. Analogous to the bubble bursting mechanism of Pearcey [6], conflicting evidence has been presented in literature regarding the validity of Lee's [13] model. A mechanism underlying Tijdeman Type B [11] shock oscillations on the NACA 0012 aerofoil based on an unstable shock-wave/separation bubble interaction has also been proposed by Raghunathan et al. [14]. The authors highlight that the shock strength must be sufficient to induce a separation bubble. The appearance of this separation bubble initiates periodic motion of the shock, which is sustained through the alternating expansion and collapse of the bubble on the upper aerofoil surface. Throughout the cycle, the varying extent of the separated region acts to change the effective camber of the aerofoil, with the trailing edge playing an integral role in communicating flow states between the suction and

characterising the various forms of upper surface separation, particularly shock-induced separation bubbles, experienced by conventional aerofoils at transonic conditions. Two distinct modes of separation were identified; Model A consisting only of a shock-induced separation bubble and Model B for which trailing edge separation is either additionally present or incipient. Three variants of Model B were also identified; rear separation provoked by the formation of a bubble, rear separation provoked by the shock and a third in which rear separation is present from the outset. The investigations by Pearcey and his co-authors culminated in the first model for the prediction of buffet onset in Type II shock oscillations; a relationship between trailing edge pressure divergence and large-scale unsteadiness. For aerofoils in which separation bubbles are present, Pearcey [6] and Pearcey & Holder [7] related the onset of buffet to the Mach number or angle of attack for which the separation bubble extends to the trailing edge and bursts. This bubble bursting mechanism governing buffet onset is easily identified through the divergence of trailing edge pressure. Although bubble bursting as the cause of onset was initially supported by experimental and computational findings, recent investigations have produced conflicting evidence [9,10] and bubble bursting is now widely discounted as a potential mechanism governing shock buffet. In the seminal work of Tijdeman [11], three distinct modes of shock motion were characterised experimentally by observing the effects of sinusoidal flap deflections on the NACA 64A006 aerofoil. Type A shock motion is represented by near sinusoidal shock oscillations across the upper surface of the aerofoil, for which the shock is present throughout the entire buffet cycle but varies in strength, with maximum shock strength achieved during the upstream excursion. Type B motion resembles Type A; however, the magnitude of shock strength variation is considerably larger, resulting in a disappearance of the shock during the 2

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from the shock foot. The motion of the shock-wave generates downstream propagating pressure waves of velocity ap , with the instability growing as it travels from the separation point through the separated shear layer. As the disturbances reach the trailing edge, upstream propagating pressure waves are produced such that the unsteady Kutta condition is satisfied. These Kutta waves travel towards the shock in the subsonic region above the separated flow at a velocity au . The interaction between the upstream propagating pressure waves and the shock results in an energy exchange, completing the feedback loop and sustaining the periodic shock oscillation. The appeal of Lee's [13] model stems widely from the ability to predict the shock oscillation frequency with a simple relationship directly related to observable variables. As the shock cycle is dependent on the time taken for disturbances to propagate downstream to the trailing edge and then again upstream to the shock, the complete shock period is proposed as the sum of these two propagation times:

pressure surfaces. The understanding of transonic shock buffet described thus far reflects a classical perspective, encompassing much of the early experimental and numerical investigations that sought to identify the underlying flow physics. A comprehensive review of this early work is provided by Lee [12]. The purpose of the present paper is to provide an overview of recent developments in the understanding of transonic shock buffet; research that has followed the review of Lee [12]. Where necessary for context, an overview of the classical work will be presented, however, it is assumed the reader is broadly familiar with the various aspects of transonic shock oscillations outlined by Lee [12]. Additionally, this review is limited to Type II shock oscillations on aerofoils and wings. The justification of this limited scope is twofold. Firstly, as discussed, a working model of Type I shock buffet has been developed by Mabey [2] and Gibb [3]. Secondly, the majority of recent literature pertains to issues encountered by civil transport aircraft, namely, aircraft with supercritical aerofoil sections that are susceptible to upper surface shock oscillations. The remainder of this paper is organised as follows: Section 2 provides an overview of research considering the governing physics underlying the transonic shock buffet phenomenon, including a description of Lee's [13] wave-propagation feedback model. In Section 3, numerical studies of transonic shock oscillations are reviewed, with a particular emphasis on the sensitivity of simulations to turbulence modelling, spatial and temporal discretisation and numerical schemes in Unsteady Reynolds-Averaged Navier-Stokes (URANS) computations and the applicability of scale-resolving methods. Recent experimental investigations of transonic shock buffet on aerofoils are then discussed in Section 4. In Section 5, an area which has received particular attention over the past decade, dynamic interactions in the presence of shock buffet, is examined. Explicit focus is given to the relationship between shock buffet as an aerodynamic resonance phenomenon and the large amplitude structural oscillations that follow from this resonance. Progress towards understanding the physics governing three-dimensional shock buffet is outlined in Section 6, followed by a description of buffet suppression technologies in Section 7. Some concluding remarks and the author's perspective on critical aspects of the transonic buffet phenomenon that have yet to be addressed in open literature are then provided in Section 8.

c
x τ ¼ ∫ xs 1 ap dx  ∫ cs 1=au dx

(1)

where τ is the period of the buffet cycle, xs is the mean shock location and c is the chord. To validate the model, data from transonic wind tunnel experiments conducted by Lee [15] on the BGK No. 1 supercritical aerofoil at Re ¼ 20  106 are employed. In Fig. 2, the magnitude and phase diagrams of the pressure signals (with respect to the shock motion) from this experiment at M ¼ 0:746 and α ¼ 6:066
are shown. The contributions to magnitude and phase from the fundamental and first harmonic frequency are decomposed, and as the magnitude of the first harmonic is comparatively small, the model is developed based on the behaviour of the fundamental frequency. Evident in Fig. 2(b), the phase angle of the fundamental frequency varies approximately linearly behind the shock. Nonetheless, this is not representative of all conditions considered, with the slope of the phase dϕ=dx not typically constant for the BGK No. 1 aerofoil. It is this phase relationship that is applied to determine the velocity ap of the downstream propagating pressure waves. The upstream propagation velocity au is computed by:

au ¼ ð1  Mc Þa

(2)

where a is the local speed of sound and Mc is the local Mach number of the flow behind the shock, computed in accordance with Tijdeman [11] by:

2. Governing physics 2.1. Wave-propagation feedback

Mc ≈RðMs  MÞ þ M

Lee [13] proposed a model that enabled the prediction of shock oscillation frequency for Tijdeman [11] Type A instabilities. In Lee's [13] model, the periodic shock motions are a consequence of an acoustic wave-propagation feedback mechanism, which is shown graphically in Fig. 1 for a symmetric aerofoil with shock-induced separation emanating

(3)

where M and Ms are the freestream and upper surface aerofoil Mach numbers, respectively and R acts as a relaxation factor (0.7 to achieve good correlations with the BGK No. 1 experiments). It is important to note that calculation of ap and au is not limited to the approach described by Lee, with alternative methods provided by Erickson & Stephenson [16], Mabey [2] and Mabey et al. [4]. The predictions made by Lee's [13] model for the shock oscillation frequencies of the BGK No. 1 aerofoil are in fair agreement with the values computed experimentally through force balance spectra, particularly considering the uncertainties related to shock location and Mc . In his original work, Lee [13] found the model yields the most accurate predictions at higher Mach numbers and incidence. Although Lee's [13] model demonstrated fair agreement to experiments of the BGK No. 1 aerofoil, subsequent literature has been somewhat conflicting regarding the applicability of the original formulation of the wave-propagation feedback model. In a URANS analysis of the BGK No. 1 aerofoil at shock buffet conditions, Xiao et al. [17] reported excellent agreement between reduced shock frequencies computed through fast Fourier transform of the lift signal and that provided by Lee's [13] model. The improved predictions relative to Lee's [13] original work may be attributed to the direct availability of the wave speeds, circumventing the need for empirical correlations. With the entire unsteady

Fig. 1. Model of self-sustained shock oscillation (adapted from Lee [13]).

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2.2. Global aerodynamic mode instability A competing theory for the mechanism governing transonic shock buffet, related to the instability in a global aerodynamic mode, has been posited by Crouch et al. [10,21,22]. Global mode decomposition of the steady transonic flowfield at pre-buffet conditions reveals a marginally stable eigenvalue related to the streamwise velocity component. With increasing incidence, a Hopf bifurcation results in this eigenvalue crossing the stability boundary, yielding phase locked motion of the shock and separated boundary layer. In this section, the method employed by Crouch et al. [21] is detailed. 2.2.1. Formulation of the linearised system The method posed by Crouch et al. [21] considers the twodimensional, viscous, compressible RANS equations; encompassing continuity, streamwise momentum, transverse momentum, energy and eddy viscosity, expressed in terms of the state vector:

q ¼ fρ; u; v; T; ~νg

(4)

where ρ represents density, u and v are the streamwise and transverse velocities respectively, T is temperature and ~ν the modified viscosity. The state vector q is then decomposed into mean (q) and fluctuating (q0 ) components, such that:

q ¼ q þ q0

(5)

For conditions sufficiently close to a steady state, linearisation of the governing RANS equations proceeds by assuming the fluctuating component q0 behaves as a small perturbation to the mean flow q. The complete set of linearised equations are omitted here for brevity, however they can be expressed in the simplified operator form:

∂ A½q0  þ Bq ½q0  ¼ 0 ∂t

(6)

where A is a linear operator comprised of the time derivative terms of the RANS equations and Bq is a linear operator including the linear terms of the RANS equations and the terms resulting from the nonlinear coupling between q and q0 . The perturbations to the mean flowfield q may then be described by time-harmonic aerodynamic modes:

q0 ðx; y; tÞ ¼ b q ðx; yÞ⋅eiωt

(7)

where b q is an eigenfunction reflecting the mode shape and ω is the frequency. Substitution of Equation (7) into the system of Equation (6) and multiplication by a conditioning matrix produces the final system of equations:

Fig. 2. Magnitude and phase of pressure waves propagating downstream in separated flow region (Lee [115]).

flowfield known, the wave speeds and direction are computed directly through two-point cross-correlations of pressure fluctuations on the aerofoil surface and within the separated flow region. Similarly, Deck [18] saw excellent agreement in a Zonal Detached-Eddy Simulation of transonic flow over the OAT15A aerofoil, with differences in the computed buffet frequencies on the order of 5%. Nonetheless, in a subsequent study by Garnier & Deck [19], LES simulation of flow around the OAT15A indicated discrepancies between Lee's [13] model and experiment on the order of 60%. Jacquin et al. [20] came to analogous findings, with Lee's [13] model performing poorly at computing the buffet frequency of the OAT15A aerofoil. A modified model was suggested by the authors (detailed in Section 4), whereby the upstream propagating pressure waves also travel along the lower surface and around the leading edge, impinging the shock from both upstream and downstream directions. Although the modified model improved the frequency predictions, a difference of 36% remained, an indication that the model of Lee [13] is not robust to changes in aerofoil geometry.

iωb q þ LðqÞ⋅b q¼0

(8)

where L is a second-order differential operator. Similarly, Equation (5) is passed through the appropriate RANS boundary conditions and Riemann invariants, resulting in an eigenvalue problem to be solved for the complex frequency ω and aerodynamic mode shape b q. 2.2.2. Method of solution Crouch et al. [21] employed a finite difference approximation for the solution of Equation (8) and the associated boundary conditions and invariants. Discretisation of the steady RANS equations is performed with Roe's scheme [23], using the third-order κ scheme [24] for the inviscid fluxes and second-order central differencing for the viscous and thermal fluxes. A first-order upwind scheme is used for the turbulent convective quantities. Interpolation of the state vector and cell coefficient matrix is performed through a blended third-order upwind/fourth-order central difference scheme, to reduce the numerical dissipation induced by the upwind differencing. 4

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The authors give particular attention to the treatment of shocks. Due to the linearisation of the RANS equations, small unsteady perturbations to the flowfield may yield fictitiously large oscillations of the shock. As such, shock smoothing is employed to better resolve the flow discontinuity. This is performed in two steps. The original steady RANS flowfield is first smoothed across the entirety of the domain, producing the smoothed field qsmooth . The smoothed and original fields are then blended through:

qfinal ¼ ð1  SÞqoriginal þ Sqfinal

(9)

where S is a blending function. The discretised form of the eigenvalue problem posed by Equation (8) is then solved through the implicitly restarted Arnoldi method [25]. Calculations are concentrated about the least stable eigenvalue, which is tracked as incidence is increased. As the growth rate becomes positive, instability and flow unsteadiness ensue. 2.2.3. Validation of predictions The use of global stability for the prediction of flow unsteadiness due to transonic shock oscillation was first applied by Crouch et al. [21] to the NACA 0012 aerofoil, for which earlier experiments were conducted by McDevitt and Okuno [26]. The authors considered the high Reynolds (Re ¼ 10  106 ) data set with M ¼ 0:76. At these experimental conditions, shock buffet onset occurred at α≈3
. As evident in Fig. 3, an increase in the angle of incidence results in the least stable eigenvalue crossing the real axis at α ¼ 3:03∘ , indicative of instability due to a Hopf bifurcation. Additionally, as the incidence increases further, so too does the instability growth rate. In Fig. 4, the unstable u-velocity mode shape computed by Crouch et al. [21] is given. As evident in the magnitude plot of Fig. 4(a), unsteadiness is concentrated at the shock location and in the shear layer downstream of the shock. The phase plot provided in Fig. 4(b) indicates phase-locked motion of the shock and separated boundary layer - the shear layer thins as the shock travels downstream and thickens during upstream shock excursions. This qualitative behaviour is in good agreement with the observations made by McDevitt & Okuno [26]. The authors further investigated the influence of shock resolution and the number of shock smoothing cycles performed. The onset of instability was found to be insensitive to both parameters, with similar predictions of the critical incidence across each level of grid refinement and number

Fig. 4. u-velocity magnitude and phase for the unsteady-mode eigenfunction. NACA 0012 aerofoil results at the conditions: Re ¼ 10  106 , M ¼ 0:76 & α ¼ 3:2
(Crouch et al. [21]).

of smoothing cycles. However, an increase in shock thickness (resulting from either coarser grid resolution or an increased number of smoothing cycles) did have a pronounced effect on the critical frequency. In a further study, Crouch et al. [10] explored the origins of transonic shock buffet on the NACA 0012 through global stability analysis. In Fig. 5, the buffet onset boundary computed by the authors through URANS simulations and global stability theory are compared to the experimental results by McDevitt & Okuno [26]. The results are in excellent agreement up to M ¼ 0:8, supporting the global mode instability as the underlying mechanism for transonic buffet. Some discrepancies do appear at M ¼ 0:8, however as both the global stability and URANS results predict steady solutions at this flow condition, the poor performance here is likely due to deficiencies in the calculations of the steady flowfield. The analysis provided by Crouch et al. [10] also gives evidence against two classical indicators of buffet onset. The Mach number just ahead of the shock at the various conditions considered does not appear to yield a reliable indicator of buffet onset as postulated by Mabey [2].

Fig. 3. Least stable eigenvalues of the NACA 0012 at Re ¼ 10  106 & M ¼ 0:76 for various angles of attack (Crouch et al. [21]). 5

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The unstable global mode mechanism is also supported by the eigenvalue decomposition of the global Jacobian matrix performed by Sartor et al. [29]. The authors find that the majority of system eigenvalues are independent of the angle of incidence, excluding the eigenvalue that lies in the vicinity of the buffet frequency. As evident in Fig. 7, the least stable eigenvalue migrates towards the stability boundary with increasing incidence (indicated by the dashed line), with instability resulting from a Hopf bifurcation. A further increase in angle of attack causes this eigenvalue to exhibit an inflexion, returning to the stable region once more at buffet offset. The authors also identify that in addition to the low-frequency shock oscillations associated with shock buffet and the globally unstable mode, a medium frequency scale unsteadiness may be present in the separated shear layer. While this is not linked to the onset of buffet, it is indicative of the additional presence of a Kelvin-Helmholtz type instability that is broadband in nature. 2.3. Transonic pre-stall instability Fig. 5. Buffet onset boundary for an NACA 0012 aerofoil, with URANS simulation results and experimental data of McDevitt & Okuno [26]; Re ¼ 10  106 (Crouch et al. [10]).

In a computational investigation considering a number of aerofoils, Iovnovich & Raveh [30] identified a number of characteristics common to the buffet cycles of various wing sections. The authors classify shock buffet as a transonic pre-stall instability consistent with the unstable shock-wave/separation bubble interaction described by Raghunathan [14]. Iovnovich & Raveh employed the finite difference Riemann Elastic Zonal Navier-Stokes Solver (EZNSS) [31], using the Spalart-Allmaras [32] turbulence model with the Edwards and Chandra [33] correction to study shock buffet cycles of the NACA 0012, RA16SC1 and NACA 64A204 aerofoils. The authors found that the range of incidence for which the buffet instability is observed is narrower for the thin NACA 64A204 aerofoil than the thicker sections. Nonetheless, a common feature noted across each aerofoil examined was that the onset of global unsteadiness occurred once the shock location moved aft of the upper surface position of maximum curvature. Examining the Mach number contours and flow streamlines for each of the aerofoils, they conveyed the common qualitative flow features of the unstable shock/separation bubble interactions driving the shock motion. Typical contours taken from the NACA 0012 are reproduced in Fig. 8. The cycle begins with the shock at its most downstream location, where it interacts with the separation bubble at the shock foot in Fig. 8(a). This interaction is unstable, with the high pressure bubble pushing the shock upstream in Fig. 8(b). The upstream excursion of the shock is initially accompanied by an increase in shock strength, and hence, thickening of the separated shear layer. This is contrary to expectations from steady flow results, where an upstream shock position is typified by a reduced shock strength. The authors identify three factors that contribute to this shock strengthening:

Further, comparisons of upper surface skin-friction coefficient at various flow conditions in the vicinity of the buffet boundary are in contradiction with Pearcey's [6] bubble bursting hypothesis. At M ¼ 0:72, the separation bubble induced by the shock is present at both pre- and post-buffet flow conditions, whereas at M ¼ 0:80, the flow is completely separated in both instances, and yet, steady flow is predicted. Such findings show no correlation between the bursting of separation bubbles and the onset of flow unsteadiness. The shock motion detailed by Crouch et al. [10] provides a conflicting description of the transonic buffet phenomenon to the model posed by Lee [13]. While both models exhibit phase-locked modulation of the separated shear layer and shock location, the model posed by Crouch et al. [21] indicates pressure perturbations originating from the shock foot travel in the wall normal direction along the shock, and with lesser intensity, through the boundary layer, rather than simply through the shear layer. As this perturbation strikes the top of the shock it propagates forward and dissipates into the oncoming flow. This behaviour is evident in the pressure fluctuations given at various steps across a shock cycle in Fig. 6. The pressure perturbations also move aft as they travel along the shock, intensifying during the downstream excursion. The acoustic waves then travel around the trailing edge and propagate upstream along the lower surface. Analogous findings are also presented by the same authors for the OAT15A aerofoil [22], where comparisons between transverse velocity components from the global mode computations and Laser Doppler Velocimetry (LDV) measurements from experiment are in excellent agreement. 2.2.4. Subsequent studies Following the global mode instability proposed by Crouch et al. [21], a number of authors have reported findings that are in support of this interpretation. As discussed in Section 2.1, the modified wavepropagation model proposed by Jacquin et al. [20] encompasses precisely the qualitative flow features described by Crouch et al. [10]. Studies by Kuzmin & Shilkin [27] and Kuzmin [28] also draw links between buffet onset and a global mode instability, finding unsteadiness to be a consequence of a supercritical Hopf bifurcation of the global flowfield. The authors also note non-uniqueness of the global flowfield, where realisation of a definite solution is dependent on the time histories of the flow conditions. Studying the fixed-point stability of the BAC 3-11/RES/ 30/21 aerofoil at pre-buffet conditions, Nitzsche [9] found that, in a linear sense, shock oscillations may be attributed to a natural resonance in the steady transonic flowfield. With incidence increasing towards buffet onset, small perturbations in pitch, flap and longitudinal translation show a reduction in damping, indicative of a global mode approaching instability.

1. Wedge Effects - Pressure rise due to the shock results in flow separation emanating from the shock foot. This separated region behaves similar to a geometric wedge, strengthening the shock, as evident in the increase in the oblique shock inclination angle between Fig. 8(a) and (b). 2. Dynamic Effects - As the shock moves upstream the velocity of the shock increases the relative Mach number of the upstream flow. With shock strength related to M 2 , a strengthening of the shock is to be expected. 3. Aerofoil Curvature Effects - The expansion of flow through the sonic zone is dependent on the local surface curvature. The greater the local curvature of the aerofoil, the more pronounced the reduction in shock strength. As the downstream location of the shock is near the location of maximum upper surface curvature, the curvature effects are expected to be small relative to the wedge and dynamic effects, resulting in a net shock strengthening. As the shock achieves its most upstream excursion 6

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Fig. 6. Contours of the pressure fluctuation at eight steps during the oscillation cycle (Crouch et al. [10]).

the cycle repeating as it reaches its most downstream position. Analogous to Crouch et al. [10], Iovnovich & Raveh [30] assessed the influence of separation bubble bursting on the onset of shock buffet. In Fig. 9, the resultant steady skin friction coefficients for each of the aerofoils at buffet onset and 0:1
below the onset incidence are shown. Noticeably, both the RA16SC1 and the NACA 64A204 aerofoils show a

in Fig. 8(c), curvature effects are more pronounced, weakening the shock strength and allowing reattachment at the shock foot. It is noted that the time lag between shock weakening and complete reattachment of the separated flow is likely a result of pressure wave propagation times between the shock and trailing edge. As the shear layer reattaches, the shock strengthens as it makes a downstream excursion in Fig. 8(d), with

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Fig. 7. Change in least stable eigenvalue with angle of attack at M ¼ 0:73 (adapted from Sartor et al. [29]).

burst separation bubble both at, and prior to, onset. Conversely, the NACA 0012 steady flowfield exhibits a separation bubble at pre- and post-buffet conditions. With no apparent change in the nature of the skin friction coefficient as unsteady shock motion develops, this investigation presents further evidence against the bubble bursting theory of Pearcey [6]. 3. Numerical investigations of transonic shock oscillations on aerofoils 3.1. Reynolds-averaged Navier-Stokes simulations The intricate flow features associated with the transonic buffet phenomenon suggests the need for computationally taxing scale-resolving simulations to model the instability. Nonetheless, a plethora of numerical studies have been devoted to assessing the efficacy of URANS methods in capturing shock buffet [34–36]. Additionally, many such authors have reported good correlations to experiment of the bulk flow features under a Reynolds-averaged formulation [29,30,37,38]. The lowfrequency shock motion that is characteristic of the transonic buffet phenomenon provides an explanation for the success of the URANS approach. A number of authors [10,29] have noted that the global flow unsteadiness exists on timescales orders of magnitude longer than those of the shear layer eddies. As such, while the inherent averaging process of URANS simulation is not able to resolve turbulence of varying scales, fundamental buffeting flow features can be predicted with a fair degree of accuracy. Nonetheless, such URANS simulations do exhibit a high sensitivity to various simulation parameters, particularly the turbulence model, spatial and temporal discretisation and the numerical scheme used. The following section is devoted to highlighting the methodologies that have been most successful in reproducing transonic shock oscillations.

Fig. 8. Shock buffet cycle Mach number and flow streamline snapshots for the NACA 0012 aerofoil at developed buffet conditions, M ¼ 0:72, α ¼ 6
(Iovnovich & Raveh [30]).

viscosity model at capturing transonic shock oscillations. The authors found that only the Spalart-Allmaras and NLEVMs with functional eddy viscosity coefficients developed shock unsteadiness comparable to experiment, albeit, at a higher incidence and Mach number. In two broad spanning studies, Goncalves et al. [47] and Goncalves & Houdeville [37] employed the implicit, cell-centred finite volume CANARI code [48] to assess the influence of various numerical parameters on buffet predictions for the RA16SC1 aerofoil. The authors considered an array of two-equation turbulence closures, including the Smith k l [49,50], Wilcox k ω [51], Menter SST k ω [52], Kok k ω [53] and the high Reynolds variant of the Jones-Launder k ε [54] models, in addition to the one-equation Spalart-Allmaras [32] model. All turbulent

3.1.1. Effect of turbulence model URANS simulation of shock buffet phenomena has been shown to be particularly sensitive to the choice of turbulence model. Barakos & Drikakis [39] explored the effectiveness of various linear and nonlinear eddy viscosity models (NLEVM) using an implicit unfactored Riemann solver of third-order spatial and second-order temporal accuracy [40,41]. The authors considered transonic flow over the NACA 0012 aerofoil, taken from the experiments of McDevitt & Okuno [26] to evaluate the effectiveness of the Baldwin-Lomax [42], Spalart-Allmaras [32], LaunderSharma [43] and Nagano-Kim [44] linear k ε models and the Sofialidis-Prinos [45] k ω version of the Craft et al. [46] nonlinear eddy8

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Fig. 9. Static friction-coefficient variation along the upper surface at pre-onset and onset conditions for three aerofoils (Iovnovich & Raveh [30]).

convective quantities are discretised through a second-order accurate upwind Roe [23] scheme with flux-limited dissipation and Harten's energy correction [55]. Additionally, for each turbulence model the authors applied an analytical velocity profile in the near wall region coupled with the no-slip condition. The resulting predictions of buffet frequency and lift differential for various angles of attack are provided in Table 1. The results indicated that Menter's SST model produces the best correlations to experiment for all conditions, including the predictions of RMS pressures provided in Fig. 10. Analogous to the findings of Barakos & Drikakis [39], the k ε and k Table 1 Frequency and amplitude of the lift coefficient with various turbulence models (Goncalves & Houdeville [37]). Model

Experiment SA k l k l corrected k l SST k ε k ε SST k ε Durbin k ω Wilcox k ω Menter k ω SST Menter k ω Kok k ω SST Kok k ω Kok Durbin

α ¼ 3


α ¼ 4


f (Hz)

ΔCL

f (Hz)

88 82 – – 79.5 Steady Steady 85.2 – – 90 Steady Steady Steady

0.11 0.0146 – – 0.0084 State State 0.012 – – 0.11 State State State

100 92 Steady Steady 97.6 95.6 95.6 93.7 Steady Steady 96.6 94.6 94.6 94.6

α ¼ 5
ΔCL 0.308 0.325 State State 0.296 0.17 0.48 0.437 State State 0.33 0.26 0.26 0.26

f (Hz)

ΔCL

Steady State 100 0.55 – – – – 101.8 0.53 97.6 0.43 101.8 0.67 101.8 0.67 – – – – Steady State 95.6 0.48 96.6 0.445 96.6 0.45

Fig. 10. RMS pressure fluctuations over the RA16SC1 aerofoil - α ¼ 4
(Goncalves & Houdeville [37]).

l models perform poorly and predict very low levels of unsteadiness at the onset condition. The authors note that incorporating a realisability correction does significantly improve their behaviour. The SpalartAllmaras model is again able to reproduce the unsteadiness observed in

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OAT15A aerofoil are used as a validation dataset for the turbulent closures (M ¼ 0:73, α ¼ 3:5
), with resulting buffet frequencies and lift amplitudes reproduced in Figs. 12 and 13. Results pertaining to the scalar dissipation formulation (not shown) generally underestimate the buffet amplitude by a significant margin. With matrix dissipation employed, the LEA model performs poorly in all instances, limiting its applicability in the prediction of shock buffet phenomena. The original form of the Spalart-Allmaras model also fails to produce unsteadiness on all grids, contrary to preceding studies [21,37,39,56]. Further, only the SST model with matrix dissipation on a hybrid grid is accurately able to capture both buffet frequency and amplitude. Subsequent analysis by the authors does reveal that this is, however, an artefact of insufficient resolution of the boundary layer. The transition from hexahedral to tetrahedral cells occurs in the near-wall region within the developing shear layer, resulting in a form of numerical buffet that does not permit a robust method of solution. Consistent buffet predictions are provided in the form of the εh -RSM model, as evident in Fig. 13. The Reynolds stress model is insensitive to grid type and the form of dissipation, producing excellent correlations in buffet frequency, amplitude and RMS pressure fluctuations in the shock interaction region and at the trailing edge. In addition to the studies dedicated to assessing the influence of turbulent closures, a number of authors have reported various degrees of success with additional models. Kourta et al. [65] found a time dependent k ε model [66], where the model coefficient Cμ is related to local deformation and strain rates, performed well in buffet computations of the OAT15A aerofoil. The model is similar to the Zhu-Shih-Lumley model [67] employed by Brunet [68], who saw similar success. Soda & Verdon [69] were able to capture shock oscillations over the NACA 0012 using the LEA [63] model, however, the Spalart-Allmaras closure with upwind differencing in this study exhibited greater consistency across a range of Mach numbers. Xiao et al. [17] found the lagged k ω model proposed by Olsen and Coakley [70] produced good predictions of the mean pressure distributions of the BGK No. 1 aerofoil at buffet conditions, however, the magnitude of oscillations was significantly overestimated through the shock region. Hasan & Alam [71] and Rokoni & Hasan [72] saw excellent results in mean and RMS pressure distributions for the SC(2)-0714 aerofoil using Menter's SST model. Carresse et al. [73] and Giannelis & Vio [74] also found Menter's SST model to perform best with the OAT15A aerofoil. These two studies further explored the efficacy of the RSM-stress omega model (derived by coupling of the ω-equations with the Launder-

experiments; however, the buffet frequencies and amplitude are understated. The Kok k ω yields unsteady flow, although onset in delayed to higher incidence and no noticeable effect of the SST correction or realisability condition is observed. The Wilcox and Menter k ω models are unable to reproduce shock oscillation in the absence of the SST correction. To further examine the influence of the wall law approach on transonic buffet computations, Thiery & Coustols [56] used the explicit cellcentred, finite volume code elsA [57] to simulate transonic flow over the RA16SC1 and OAT15A aerofoils. A four-step Runge-Kutta scheme was employed to achieve second-order accurate temporal resolution, in addition to the Jameson scheme [58] with artificial dissipation for the inviscid fluxes and the Roe scheme [23] for the turbulent convective quantities, both of which achieved second-order spatial accuracy. Thiery & Coustols [56] also assessed the Spalart-Allmaras and Menter's SST models, along with the k ϕ model [59] developed at ONERA. The authors further performed simulations for the Spalart-Allmaras and Menter's SST model using two grids. The first grid is refined through the boundary layer region to achieve an average y þ ¼ 1:1 in the shock region (denoted RG). The second grid was constructed by removal of the first 20 nodes in the wall normal direction of the refined grid, such that an analytical velocity profile may be applied in this near wall region (denoted WL). The influence of the wall law approach is considered on the RA16SC1 aerofoil at M ¼ 0:732 and Re ¼ 4:2  106 . RMS pressure fluctuations are well predicted for the majority of the models, excluding the RG SST formulation for which a steady solution was produced. The RG variant of the Spalart-Allmaras model does, however, appear to produce the most consistent results, best approximating the RMS pressure fluctuations and the lift amplitudes across a range of angles of attack, as shown in Fig. 11. This indicates that while the WL approaches considered by Goncalves et al. [47] and Goncalves & Houdeville [37] may be able to reproduce the buffet phenomenon, better accuracy can be achieved with refinement through the boundary layer. Several common turbulence models were also assessed by Illi et al. [60] using the unstructured finite volume code DLR-TAU [61]. The authors investigated the influence of matrix and scalar dissipation, combined with the baseline Spalart-Allmaras model and the Strain Adaptive variant (SALSA) [62], Menter's SST closure, the Linear Explicit Algebraic k ω model (LEA) [63] and the less commonly employed εh Reynolds stress model (εh -RSM) [64]. The experiments of Jacquin et al. [20] for the

Fig. 12. Frequency and amplitude of shock motion for various turbulence models and grids with matrix dissipation (S ¼ structured, H ¼ hybrid, G ¼ grid refinement level) (adapted from Illi et al. [60]).

Fig. 11. Amplitude of lift coefficient for different turbulence models and the two RG and WL strategies (Thiery & Coustols [56]). 10

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been achieved with the Roe [28,38,73,74], Jameson [56,69,78,79] and variants of the AUSM [18,37,69] schemes. 3.1.3. Effect of spatial & temporal discretisation Several investigations have further considered the dependence of buffet calculations on the temporal formulation employed [80,81]. Rouzaud et al. [80] investigated the effectiveness of an implicit Dual Time Stepping (DTS) method in buffet computations for both an 18% thick circular arc aerofoil and the RA16SC1 supercritical aerofoil. The DTS method was found to produce results in fair agreement with experiment and with substantial improvements in efficiency relative to an explicit temporal formulation. These findings are in agreement with the previous study of Rumsey et al. [81] for the 18% thick circular arc aerofoil, where the use of subiterative DTS formulations saw improvements in both temporal accuracy and convergence rates. Due to these benefits offered by DTS, this implicit temporal formulation has been employed extensively in recent URANS investigations on transonic buffet [28,37,56,65,72,79]. The study by Rumsey et al. [81] further explored the influence of temporal and spatial resolutions on predicted buffet characteristics. For solutions performed with a subiterative DTS method, the authors found that with a temporal discretisation of approximately 170 steps per period a 1% difference in computed buffet frequency was seen when the resolution in time was increased by a factor of four. Conversely, significant variations in the predicted buffet frequency and lift amplitude were observed when doubling the grid density, with differences of 4.5% and 6.5% respectively. Variation in the computed buffet characteristics of the OAT15A aerofoil of similar magnitude with double the grid resolution were also observed by Illi et al. [60]. Iovnovich & Raveh [30] further highlighted the significance of sufficient shock resolution when performing grid convergence studies for buffeting flows, with 10% and 12% differences in buffet frequency and lift differential respectively as the shock resolution was increased from 1.5% chord to 0.3% chord. The use of a highly resolved grid in the shock region becomes particularly significant when using the linearised global stability method of Crouch et al. [21] for buffet predictions. Shock resolutions on the order of 0.15% chord were deemed necessary to accurately capture the flow physics.

Fig. 13. Frequency and amplitude of shock motion for the εh -RSM model (S ¼ structured, H ¼ hybrid, G ¼ grid refinement level, s ¼ scalar dissipation, m ¼ matrix dissipation, δ ¼ height of hexahedral cells in original hybrid grid) (adapted from Illi et al. [60]).

Reece-Rodi [75] model), which was capable of reproducing the buffet phenomenon, albeit with overestimated levels of unsteadiness in each case. 3.1.2. Effect of numerical discretisation scheme In addition to the influence of turbulence modelling, certain studies have evaluated the significance of the numerical discretisation scheme for the convective fluxes [37,69]. Goncalves & Houdeville [37] explicitly assessed the effects of the chosen numerical scheme for the mean flowfield on buffet predictions. The authors considered the Jameson [58], upwind Roe [23] with Monotone Upstream-Centred Scheme for Conservation Laws (MUSCL) [76] extrapolation and the Advection Upstream Splitting Method (AUSMþ) [77] with MUSCL extrapolation schemes. Of the considered formulations, the Jameson scheme was found to be most effective in capturing onset, however the Roe scheme with MUSCL extrapolation produced the best correlations to experiment regarding shock frequency and lift differential. The resulting buffet frequencies and amplitudes are reproduced in Table 2. The authors note that the influence of the convective flux formulation on the predicted buffet characteristics is secondary to the choice of turbulence model, with the various schemes yielding a 3% difference in buffet frequency and 10% difference in lift differential. Soda & Verdon [69] also made a comparison of transonic flow predictions made by upwind (AUSMþ) [77] and central differencing (Jameson) schemes of the inviscid fluxes for the NACA 64A010. The upwind scheme was found to improve the accuracy of the shock location, whereas the central scheme was more effective at capturing trailing edge pressures. From the plethora of URANS investigations on transonic shock buffet, it is evident that the conclusions of Goncalves & Houdeville [37], regarding numerical discretisation as a secondary consideration relative to turbulence modelling, is well founded. With appropriate choice of turbulence closure, successful reproduction of the buffet instability has

3.1.4. Effect of wind tunnel geometry With the majority of URANS studies investigating the transonic buffet phenomenon employing farfield boundary conditions, a select few authors have explored the influence of test section geometry on the computational results. Furlano et al. [82] performed an initial assessment of wind tunnel wall effects in steady conditions, using the ONERA [83] code to simulate transonic flow over the OALT25 and RA16SC1 aerofoils. Two-dimensional steady results with various turbulence models offered consistent results, with fair agreement to experiment for the OALT25 aerofoil. However, differences were observed in the pressure gradient upstream of the shock and the local Mach number on the pressure side of the aerofoil. The authors attribute these discrepancies to threedimensional effects. Performing analogous simulations with the inclusion of the entire three-dimensional tunnel geometry produced better predictions of the upstream pressure gradient and shock location. Garbaruk et al. [84] drew similar conclusions for steady transonic flow over the RAE 2822 aerofoil, stressing the need for inclusion of appropriate

Table 2 Frequency and amplitude of the lift coefficient with various discretisation schemes (Goncalves & Houdeville [37]). Model

Experiment Jameson Roe MUSCL AUSM þ MUSCL Jameson corrected

α ¼ 3


α ¼ 4


α ¼ 5


f (Hz)

ΔCL

f (Hz)

ΔCL

f (Hz)

88 90 90 90 91

0.11 0.11 0.014 0.018 0.097

100 96.6 99.7 98.6 97.6

0.308 0.33 0.3 0.307 0.327

Steady Steady Steady Steady 99.7

11

ΔCL State State State State 0.46

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computational predictions of transonic shock buffet.

wind tunnel conditions during simulation. Thiery & Coustols [56] explicitly studied the influence of test section geometry on the buffet predictions for the OAT15A aerofoil. An unconfined farfield domain is compared with the true test section geometry, including adaptable tunnel walls, with time-averaged and RMS pressure fluctuations reproduced in Fig. 14. The inclusion of test section geometry yields a pronounced improvement in both the mean pressure distribution and RMS pressure fluctuations for the OAT15A aerofoil. The authors also find that the predicted buffet frequency improves markedly, reducing to 74 Hz (relative to the experimental value of 69 Hz). Similar improvements are noted for the non-dimensional velocity profiles at various chordwise stations. Barbut et al. [85] and Braza [86] also found the effects of wind tunnel walls to be significant in the prediction of shock oscillations on the NACA 0012. The absence of the walls resulted in lower levels of surface pressures and predicted onset at higher Mach numbers and incidence. The pronounced influence of the test section geometry on the transonic flowfield demonstrated by these preceding studies suggests that the inclusion of wind tunnel walls may yield improvements in

3.2. Scale-resolving simulations Although URANS simulation is effective in capturing the lowfrequency shock oscillations that characterise transonic buffet, there remain several facets of these complex flowfields that simply cannot be reproduced. In particular, these methods are unable to provide insight into the effects of the broadband turbulent spectrum that inherently results from the intermittently separated flow. Nonetheless, the past decade has seen substantial developments in unsteady hybrid RANS/Large Eddy Simulation (LES) methods that resolve, rather than model, these turbulent scales. In combination with the continuing growth of computational power, these scale-resolving simulations are steadily becoming a feasible tool to probe the physics governing transonic buffet. One of the first investigations of transonic buffet over an aerofoil through scale-resolving methods was performed by Deck [18]. Using the FLU3M code of ONERA, Deck [18] compared URANS, Detached-Eddy Simulation (DES) and the novel Zonal DES (ZDES) predictions of periodic shock motions over the OAT15A aerofoil. A second-order accurate upwind finite volume discretisation of the Navier-Stokes equations is employed, with the upwind AUSMþ(P) [87] scheme used to resolve the inviscid flux terms. A second-order accurate implicit temporal discretisation derived from Gear's formulation [88] is also used for the unsteady simulations. Closure of the Navier-Stokes equations in the URANS computations is achieved with the Spalart-Allmaras [32] model. The DES calculations follow the original formulation developed by Spalart et al. [89] and are again based on the one-equation Spalart-Allmaras [32] model. The reader is directed to the original work for a comprehensive derivation of the DES formulation, however, the critical aspect outlined by Deck [18] is the inclusion of an eddy viscosity destruction term that is a function of the distance to the nearest wall (d). In conjunction with the production term, the computed eddy viscosity (~ν) ~ such that: scales with the local deformation rate (S)

~ν≈~Sd2

(10)

Spalart et al. [89] suggested replacing the original length scale d with a modified scale:

d~ ¼ minðd; CDES ΔÞ

(11)

where Δ ¼ maxðΔx ; Δy ; Δz Þ represents the grid size and CDES is a model constant. The dependence of the length scale on the maximum grid extension is a natural representation, as this distance governs the wavelengths that can be resolved by the simulations. While the standard DES formulation seeks to combine the best attributes of both RANS and LES simulations, the application to transonic buffet, where intermittent thin-layer separation is present, may be problematic. In particular, premature switching to LES mode may occur within the RANS boundary layer, yielding fictitious grid-induced separation. In an attempt to remedy this, Deck [18] developed the ZDES method. In this approach, the user explicitly identifies the RANS and LES regions at the outset of the simulation. The attached shear layer region and the entire shock/boundary layer interaction zone are treated exclusively in RANS mode, while the separated flow region aft of the trailing edge is resolved in LES. Deck [18] further took advantage of an innovative two-threedimensional grid coupling technique. The method was developed by Mary & Sagaut [87,90] under the LESFOIL project to reduce the highdensity grid resolution inherent in scale-resolving simulations. In Fig. 15, a typical grid topology for this coupling method is shown. Notably, only zones 3 and 6 are treated in LES. Further, threedimensional resolution is limited to zones near the aerofoil and in the wake, with the remainder of the domain computed in two-dimensions, reducing the node count by a factor of two relative to an entirely

Fig. 14. Effect of wind tunnel walls on the unsteady pressure distributions of the OAT15A aerofoil (α ¼ 3:5
, M ¼ 0:73, Re ¼ 3  106 ) (Thiery & Coustols [56]). 12

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longitudinal velocity profiles and fluctuations, as evident in Fig. 17. Discrepancies are evident in these profiles upstream of the shock; however, the author attributes this to the calculations being performed in a fully turbulent manner, whereas the experiment employed a fixed transition trip. Power spectral analysis of the wall pressures, constructed through a parametric autoregressive technique [91] due to the limited available simulation time, indicate the dominant buffet frequency is well captured by ZDES at the experimental condition and by URANS and DES at higher incidence. Flattened spectra at higher frequencies, an indication of random white noise associated with varying scales of turbulent structures, are also best captured by the ZDES simulations. Analogous to Xiao et al. [17], Deck [18] assessed the efficacy of Lee's [13] model at predicting the buffet frequency of the OAT15A aerofoil. Using a frequency-wave number spectrum [92] to compute the downstream wave propagation velocity, the computed buffet frequency derived from the computations is in excellent agreement with the experiment, differing by approximately 6%. Although this initial study appears to support Lee's [13] model, a subsequent investigation by Garnier & Deck [19] contradicts the early findings. Again using the second-order accurate in space and time FLU3M solver, the authors employed a modified Roe scheme [23] for the convective fluxes, integrating the Ducros et al. [93] sensor to locally adapt the solvers dissipation. A zonal RANS/LES method is again employed to resolve the turbulent eddies, however, the entirety of the suction side and wake region are now treated through LES, with the pressure side resolved in two-dimensional RANS mode. For sub-grid scale modelling, the Selective Mixed Scales Model (SMSM) [94] has been used. Investigating the influence of spanwise grid extent, Garnier & Deck [19] found a two-fold increase in the domain size (from 3.65% chord to 7.30% chord) significantly reduced the computed pressure fluctuations at the trailing edge. As shown in Fig. 18, the larger transverse grid extent (B1 and B2) better captures the unsteady pressures at the trailing edge by allowing three-dimensional coherent structures to develop, limiting their intensity relative to the predominantly two-dimensional structures

Fig. 15. Two-three-dimensional grid (Deck [18]).

three-dimensional grid. The resultant calculations clearly indicate that the ZDES method is superior to both URANS and classical DES in terms of mean and RMS pressure fluctuations and general character of the oscillating flowfield (evident in the RMS of longitudinal velocity fluctuations in Fig. 16). Notably, the ZDES results overestimate the degree of pressure fluctuation both through the shock region and at the trailing edge, as well as predicting a shock location marginally upstream of the experiment. Further, while the URANS and DES simulations require a higher angle of incidence relative to the experiment before unsteadiness is observed (α ¼ 4:5
and α ¼ 4
respectively), ZDES is able to capture shock oscillation at the experimental incidence of α ¼ 3:5
. The ZDES simulations also provide good predictions of downstream

Fig. 16. Longitudinal velocity fluctuation, RMS field (Deck [18]). 13

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Fig. 18. RMS pressure distribution for the OAT15A (adapted from Garnier & Deck [19]).

the near-field flow to compute wave convection velocities, Garnier & Deck [19] also assessed the accuracy of Lee's [13] model for buffet frequency prediction. A significant discrepancy is apparent between the LES buffet frequency of 72 Hz and the analytical prediction of 110 Hz. The authors do note, however, an upstream propagating pressure disturbance along the lower surface of the aerofoil, consistent with the global mode instability of Crouch et al. [21]. Furthering Deck's [18] study, Grossi et al. [38] compared the effectiveness of URANS and Delayed DES (DDES) simulations in the prediction of transonic buffet for the OAT15A section. The authors employed the density based finite volume Navier Stokes Multi Block (NSMB) code [95], with the third-order accurate total variation diminishing (TVD) variant of the upwind Roe scheme [23] with MUSCL extrapolation for the convective fluxes and a second-order accurate backward Euler dual time stepping scheme for temporal discretisation. For the two-dimensional URANS computations a number of turbulence closures were investigated, from which, the Edwards-Chandra modified Spalart-Allmaras model with compressibility correction (EDW-CC) [33] was found to best capture the flow features observed at equivalent conditions to the experiments. The hybrid DDES formulation is developed from the underlying EDW þ CC RANS turbulence model, and employs a modified length scale relative to the standard DES formulation of Equation (11), defined by:

d~ ¼ d  fd maxð0; d  CDES ΔÞ Fig. 17. Velocity profile predictions (Deck [18]).

(12)

where fd is a delaying function. The analysis provided by Grossi et al. [38] indicates the formulation of turbulent quantities has a significant effect on the flow topology predicted. From the DDES computations, the separated flow immediately aft of the shock develops as a primarily two-dimensional Karman instability. As the shock moves upstream, a spanwise undulation develops. From this perturbation, a secondary instability emanating from the primary vortices ensues and the two-dimensional structures break down, exhibiting a strongly three-dimensional character downstream. Further, power spectral densities (PSDs) of the lift time history do not display sharply resolved peaks, but rather, dispersed bumps. Such spread in the frequency spectrum is indicative of an aperiodic and broadband buffet response. Conversely, the URANS simulations show a periodic character with no appearance of a secondary instability. The differences in flow topology predicted by the URANS and DDES approaches can be seen in Fig. 21, which provides a map of the separated flow regions over a buffet cycle. Further discrepancies between URANS and DDES simulations are

observed with the original grid (A). Evidently, the intensity of pressure fluctuations is overestimated relative to the experiment in each instance. Additionally, the averaged pressure distribution of Fig. 19 indicates a downstream mean shock location. The authors note that the level of numerical dissipation imparts an appreciable influence on the predicted shock location. The high dissipation simulation (B1) yields a mean shock location further aft, with the converse applying in the low dissipation case (B2). Relative to the earlier work by Deck [18], the LES simulations performed by Garnier & Deck [19] yield more consistent predictions of longitudinal velocity profiles and fluctuations along the entire chord. As shown in Fig. 20, regardless of the degree of numerical dissipation, the longitudinal velocity fluctuations are well captured. Spectral analysis of the wall pressure fluctuations reveals a dominant buffet frequency of 72 Hz, consistent with the findings of Deck [18] and marginally higher than the experimental value of 69 Hz. Using two-point two-time correlations of the fluctuating pressures in 14

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Fig. 19. Mean pressure distribution for the OAT15A (adapted from Garnier & Deck [19]).

Fig. 21. Spatiotemporal evolution of flow separation on the upper surface (Grossi et al. [97]).

present in the analysis of statistical flow properties, particularly in the shock/boundary layer interaction region and at the trailing edge. The DDES simulations predict with fair accuracy the shock location, range of shock travel and peak pressure RMS in the shock region, however, the pressure fluctuations at the trailing edge are significantly overestimated, as found by Deck [18]. Conversely, although the URANS simulations yield a shock location downstream relative to the experiment, mean and RMS trailing edge pressures are well captured. Mean and RMS longitudinal velocities are also best represented by the URANS approach. Such results suggest that the preferred method of solution may be dependent on the particular features of buffet to be investigated, with the DDES approach providing a richer resolution of qualitative flow features and the URANS method better predicting the statistical properties.

Fig. 20. Longitudinal velocity RMS profiles (adapted from Garnier & Deck [19]).

15

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Grossi et al. [96] extended their previous study, performing both twoand three-dimensional URANS simulations, in addition to threedimensional DDES with the SALSA [62] turbulence model and the third-order AUSMþ [77] scheme for the convective fluxes. While reformulation of the underlying RANS model in the DDES approach improves the prediction of pressure fluctuations at the trailing edge and shifts the shock location in the URANS method upstream, the findings are generally in accordance to Grossi et al. [38]. The results of both papers are presented in greater detail in Grossi's dissertation [97]. As an alternative to the ZDES method presented by Deck [18], Roidl et al. [98] developed a novel zonal hybrid RANS-LES method to compute transonic shock oscillations over the DRA 2303 aerofoil [99]. The computations employ a mixed second-order accurate centred/upwind AUSM [100] scheme for spatial discretisation of the inviscid fluxes, with secondorder central differencing for viscous terms. Temporal discretisation also achieves second-order accuracy through an explicit 5-stage Runge-Kutta method. To resolve turbulent quantities, the domain is segmented as shown in Fig. 22. The Spalart-Allmaras [32] model is used in the RANS zone and the Monotone Integrated LES (MILES) [101] approach in the LES region. The hybrid method proposed employs a synthetic turbulence generation method (STGM) at the LES inlet and localised control planes to modify turbulence production in a buffer region between the two zones. Further details of the formulation are provided by Zhang et al. [102]. Two distinct grids were developed for the computations; a pure LES grid of resolution Δxþ ≈100, Δy þ ≈1 and Δzþ ≈20 for the streamwise, wall normal and spanwise directions, and a hybrid grid of equivalent resolution in the LES zone and coarser streamwise and spanwise spacing in the RANS region. Although no direct validation of either method relative to experiment is provided, the hybrid and pure LES approaches yield consistent predictions for statistical flow properties (Fig. 22), Reynolds stresses (Fig. 23(b)) and pressure PSDs. The hybrid method is thus able to achieve an equivalent degree of accuracy relative to pure LES with a significant reduction in the grid requirements, in this instance, reducing the node count by a factor of two. 4. Experimental investigations of transonic shock oscillations on aerofoils Recent years have seen few experimental investigations in open literature that explicitly study the nature of transonic shock oscillations on aerofoils. Many such studies were conducted following the seminal work of Tijdeman [11] in classifying the various types of shock motion observed on the NACA 64A006 excited by sinusoidal trailing edge flap deflections. McDevitt et al. [103], McDevitt [104], Mabey [2] and Mabey et al. [4] each performed wind tunnel tests on biconvex aerofoils. Experiments considering shock oscillations over supercritical aerofoils were conducted by Finke [105], Stanewsky & Basler [106], Roos [107,108],

Fig. 23. Validation of the Zonal RANS-LES method (M ¼ 0:72, Re ¼ 2:6  106 , α ¼ 3
) (Roidl et al. [98]).

Hirose & Miwa [109] and Lee [15,110], while McDevitt & Okuno [26] investigated the NACA 0012 buffet boundary. Although the volume of transonic buffet experiments in recent years has diminished, a number of comprehensive studies have appeared which have been instrumental in

Fig. 22. Schematic of the zonal RANS-LES method for transonic flow around the DRA 2303 profile (Roidl et al. [98]). 16

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furthering the understanding of the underlying flow physics. An extensive computational and experimental campaign has been undertaken at ONERA over the past decade, seeking to characterise the buffet phenomenon and develop alleviation techniques. A complete overview of the programme is given by Dandois et al. [111]. In particular, experiments on the supercritical OAT15A performed in the S3Ch Continuous Research Wind Tunnel at the ONERA Chalais-Meudon Center have been detailed by Jacquin et al. [20,112]. 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 the experiment. The model ensured a fixed boundary layer transition at 7% chord through the installation of a carborundum strip on the upper and lower surfaces. The experiments conducted under this campaign sought to develop an extensive experimental database for the validation of numerical buffet simulations. The model was fitted with 68 static pressure orifices and 36 unsteady Kulite pressure transducers through the central span, mitigating the three-dimensional effects from sidewall boundary layers. Adaptable upper and lower wind tunnel walls (Fig. 24) further alleviated wall interference, allowing for a test section Mach number uncertainty of 104 . A sublimating product was applied to the models upper surface, which permitted oil flow visualisations for characterisation of turbulent regions and shock motion. The authors employed Schlieren cinematography to observe qualitative flow features and two-component Laser Doppler Velocimetry (LDV) to capture quantitatively the longitudinal and vertical velocity fluctuations during the buffet cycle. Further, steady and unsteady pressure measurements produced mean and RMS pressure data, along with spectral content for the pressure fluctuations. The test programme undertaken by Jacquin et al. [20] consisted of an angle of attack sweep at M ¼ 0:73 to obtain data for buffet onset, as well as Mach number sweeps at α ¼ 3
and α ¼ 3:5
. The chord-based Reynolds number was kept approximately constant at Re≈3  106 for each test point. The wealth of steady and time-resolved pressure and velocity field measurements produced during the programme have since seen the OAT15A aerofoil become somewhat of a benchmark transonic buffet test case, as evidenced by the multitude of numerical studies on this section in the recent years [18,22,56,73,74]. The extensive analysis performed by Jacquin et al. [20] also served to provide novel insights of the physics governing transonic shock oscillations. Spectral analyses of the unsteady pressure signals, as shown in the

spectrogram in Fig. 25(a) and sound pressure level contours in Fig. 25(b), indicate that the two-dimensional buffet phenomenon is time-invariant and essentially modal in nature. Excluding the intermittent shock region in Fig. 25(b), where frequency spreading is present due to the lowfrequency shock oscillation, the spectra are dominated by a single frequency. Such behaviour is in support of the global mode instability theory proposed by Crouch et al. [21]. In analysis of the two-dimensional character of the buffet phenomenon, Jacquin et al. [20] observed somewhat conflicting results. While spectral analysis revealed a constant spanwise distribution of sound pressure levels across the central span, oil flow visualisations in the separated region indicated the presence of three-dimensional structures. The authors note that the velocity associated with the pressure fluctuations is an order of magnitude larger than that of the three-dimensional structures observed in the wall velocity field. As such, Jacquin et al. [20] hypothesised the buffet instability may result from the superposition of strong two-dimensional and weaker three-dimensional global modes. Jacquin et al. [20] also applied a similar line of reasoning to Lee [13], that the observed buffet period for an aerofoil consists of the sum of disturbance convection time-scales, to develop the modified wavepropagation feedback model briefly detailed in Section 2.1. The constituent time-scales include the time taken for a disturbance originating at the shock foot to propagate to the trailing edge (τp ) and an acoustic time-scale which defines the time delay between perturbations emanating from the trailing edge impinging on the shock (τu ). The authors note that reduction of the global physics governing the buffet phenomenon to a simple empirical model is subject to several difficulties, particularly in the computation of appropriate convection velocities. Whereas Lee [13] postulated τu represented the time taken for acoustic waves to convect upstream above the upper aerofoil surface from the trailing edge (τu;u ), Jacquin et al. [20] proposed such disturbances could also propagate towards the shock along the lower surface, rounding the leading edge and impinging on the shock from upstream (τu;l ). In the latter, the authors found τu ¼ τu;u þ τu;l produced the most consistent results relative to experiment. Pressure fluctuations from the Kulite transducers indicated disturbance propagation along both surfaces and a predicted buffet frequency with the modified definition of τu in better agreement with the experiment. Jacquin et al. [20] also detailed difficulties in the calculation of the convection velocity ap in the computation of τp . Two-point cross correlation of pressure fluctuations are employed by the authors to compute ap , with comparable velocities obtained to those of Lee [13]. These disturbances do not lend themselves to a simplified interpretation, and different values for the convection speed would be obtained if these perturbations were viewed in light of local stability theory and characterised the propagation of a Kelvin-Helmholtz type instability. The authors conclude that while the modified propagation model better represents the flow characteristics observed in their experiments, a robust and simple model of the phenomenon remains elusive. The shock oscillations are driven by two cooperating mechanisms; propagation of perturbations emanating from the shock foot in a region of receptivity and forcing of these perturbations through the convection of acoustic waves. The complexity of the constituent flow mechanisms, as they are currently understood, does not permit the construction of a generalised model. A contrary perspective has been provided by Hartmann et al. [113], whose investigations support a second modified acoustic wavepropagation mode. Experiments were conducted on the supercritical laminar-type DRA 2303 aerofoil at Re≈2:7  106 , with a freestream Mach number between 0:67  M  0:76 and incidence range of 0
 α  4
. Steady and unsteady pressure measurements, time-resolved stereo particle-image velocimetry (TR-SPIV) and Schlieren imaging are used to characterise the unsteady flowfield across the range of flow conditions. Additionally, the experiments are repeated at equivalent conditions in the presence of an artificial acoustic source downstream of

Fig. 24. OAT15A supercritical profile in the S3Ch transonic wind tunnel (Jacquin et al. [20]). 17

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Fig. 25. Spectral analyses of OAT15A pressure fluctuations (Jacquin et al. [20]).

2303 aerofoil, extending the original study to include a mid-range driver and horn to allow for control of the acoustic driving frequency. This subsequent study details a single flow condition of M ¼ 0:73 and α ¼ 3:5
. The baseline aerofoil exhibits Tijdeman Type A [11] shock motion analogous to the cycle represented in Fig. 26, at a frequency of 129 Hz. Two-point two-time cross correlations of both pressure and velocity signals are used to compute disturbance propagation velocities. Application of Lee's [115] model resulted in fair prediction of the buffet frequency (146 Hz), however, the authors suggest a modification to the original model concerning the characteristic length for upstream acoustic propagation. A change in the characteristic length from c xs to lshock =c in Fig. 28 produced a significant improvement in the predicted frequency, which correlates well with the experiment at 133 Hz. Phase-averaged correlations of the streamwise and normal velocity fluctuations revealed highly correlated large-scale structures downstream of the shock. Noting that a mechanism for noise generation is the impinging of these vortical structures on the trailing edge [116,117], Hartmann et al. [114] computed an acoustic disturbance of approximately 1030 Hz is emitted as the vortices pass the trailing edge. A pure sine wave at this frequency was then driven through the installed speaker to amplify the effects of these high frequency perturbations. PSDs of the shock location show frequency content at both the undisturbed buffet frequency and the acoustic driving frequency, with two additional modes appearing within this frequency band, indicating the shock motion is receptive to high frequency forcing. Hartmann et al. [114] go on to provide a detailed analysis of the buffet mechanism strongly linked to noise generation and disturbance propagation. This proposed mechanism (for which a single oscillation cycle is shown in Fig. 29) is similar to that of Lee [13], consisting of a feedback loop between noise production at the trailing edge and the location of the shock. As the shock oscillates, the strength of vortices emanating from the shock foot and the extent of the separated region vary. Both of these factors influence the distribution of the Lamb vector, an indicator of trailing edge noise intensity [116,117]. Such changes in the Lamb vector result in acoustic waves of varying sound pressure levels which emanate from the trailing edge. A decrease in sound pressure level

the test section. The artificial source is achieved through removal of the upper and lower wind tunnel walls, with the freestream chamber acting to amplify acoustic disturbances at a frequency of 248 Hz. The authors find that in the absence of the acoustic source, shock buffet is observed with M ¼ 0:72 and α ¼ 3
. The time series in Fig. 26 shows the velocity field and corresponding pressure distribution for a single buffet period at this condition. The shock oscillation is characteristic of Tijdeman Type A [11] motion at a frequency of 126 Hz. At equivalent flow conditions, the addition of an acoustic source has a significant influence on the nature of the shock oscillations. PSDs of pressure fluctuations at the midchord indicate the frequency content is dominated by the acoustic forcing frequency of 248 Hz; the acoustic source has resulted in mode switching of shock oscillations to the first harmonic of the undisturbed aerofoil. Additionally, time series of the velocity fields in Fig. 27 show a dissimilar oscillation cycle relative to the Type A sinusoidal motion in Fig. 26. As the shock moves upstream from time ~t , the supersonic region contracts and the shock loses strength. At t ¼ ~t þ 5Δt the primary shock region has practically vanished, and is accompanied by the formation of a weaker secondary shock, indicating the beginning of a new cycle. Such an intermittent presence of the primary shock, weakening and disappearing during the upstream excursion, cannot be characterised by any of the classical Tijdeman types. At higher Mach numbers, the intensity of the acoustic source is comparatively smaller, imparting less influence on the shock motion and producing lower amplitude oscillations. The results presented by Hartmann et al. [113] categorically represent transonic shock oscillations as an acoustic phenomenon. In the absence of an external source, noise is produced at the trailing edge, with intensity varying dependent on the extent of the separated region. Larger separated regions, as seen at higher Mach numbers, result in reduced sound pressure levels emanating from the trailing edge, and hence, a reduction in unsteadiness. Through comparison of both forced and unforced experiments, the authors propose the noise generation at the trailing edge drives the shock motion, which in turn, varies the extent of the separated flow region and completes the feedback loop. Hartmann et al. [114] carried out further experiments on the DRA

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Fig. 26. Time sequence of synchronously measured pressure distribution and velocity field for M ¼ 0:72, α ¼ 3
- no acoustic source (Hartmann et al. [113]).

motions, and the associated shock oscillations, on aerofoils at transonic conditions [120–122]. In an investigation involving transient pitching of a NACA 64A010 with shock-induced separation, Davis & Malcolm [123] observed distinct resonance in the aerodynamic coefficients when the system was excited at a particular reduced frequency. The measured gain and phase relationships are reminiscent of a pair of complex conjugate eigenvalues in a single degree-of-freedom harmonic oscillator. Similar resonant behaviour was also found by Despre et al. [124] during sinusoidal trailing edge deflections of the OAT15A aerofoil in a buffeting flow, where the maximum amplitude of shock-wave oscillation occurred at excitation frequencies near the buffet frequency. As will be discussed, the prevalence of such an aerodynamic resonance near shock buffet onset has significant implications for the nature of shock oscillations in dynamic aerofoil systems. In examining the fixed-point stability of the BAC 3-11/RES/30/21 aerofoil at pre-buffet conditions, Nitzsche [9] also drew analogy between shock buffet and a single degree-of-freedom oscillator. The DLR-TAU [61] was employed with central differencing of the inviscid fluxes and second-order backward Euler dual time stepping for temporal discretisation. Using the LEA variant of the k ω turbulence model [63], buffet onset was captured by sweeping through incidence at M ¼ 0:75 and Re ¼ 4:5  106 . Hysteresis was observed during the incidence sweeps, with buffet persisting to lower angles of attack during the down

results in a downstream shock excursion, and vice versa. Although not explicitly considering large-scale shock motion, the complex mechanisms governing the production of the trailing edge disturbances detailed by Hartmann et al. [113,114] are investigated experimentally by Alshabu et al. [118]. For the BAC3-11 aerofoil at M ¼ 0:71, Re ¼ 3:3  106 and α ¼ 0
, pressure fluctuations at the trailing edge gain strength as they propagate upstream, before interacting with the supersonic region and weakening. Unsteady pressure measurements indicate the dominant frequency of these trailing edge disturbances is in accordance with the findings of Hartmann et al. [114] and on the order of 1 kHz. Further experiments conducted by Alshabu & Olivier [119] on the BAC3-11 aerofoil showed that interaction between the shock and upstream propagating disturbances resulted in degeneration of the shock into compression waves coupled with the production of vortices in the boundary layer. The vortices propagate toward the trailing edge, initiating a new cycle of trailing edge pressure waves in an analogous manner to the mechanism detailed by Hartmann et al. [114]. 5. Dynamic interactions in the presence of transonic shock oscillations Beginning with the original work of Tijdeman [11], a number of experimental studies have investigated the influence of forced harmonic 19

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Fig. 27. Time sequence of synchronously measured pressure distribution and velocity field for M ¼ 0:72, α ¼ 3
- acoustic source (Hartmann et al. [113]).

locations in Fig. 30, where the resonant peaks align just below the onset reduced frequency of ω ¼ 0:63. From Fig. 30, the phase responses of each of the excitations are consistent in the vicinity of the resonant peak. Specifically, the shock buffet resonance is independent of the mode of excitation and is characterised by a  180
phase reversal with a characteristic slope. Considering the pitch phase response, Nitzsche [9] found that the shock buffet eigenfrequency appears where the shock motion changes from inverse to regular motion. Davis & Malcolm [123] encountered precisely this behaviour in their transient pitching experiments on the NACA 64A006. In a comprehensive numerical study on the interaction between forced harmonic heaving and shock buffet of the NACA 0012, Raveh [125] uncovered a buffet lock-in mechanism related to the aerodynamic resonance detailed by Nitzsche [9]. URANS simulations using the finite difference code EZNSS [31] were performed with the modified k ω turbulence model [126], and were validated against the static and dynamic experimental data of McDevitt & Okuno [26] and Landon [122], respectively. Considering the nominal condition (M ¼ 0:72, Re ¼ 10  106 , α ¼ 6
), at which autonomous shock oscillations are observed for the rigid aerofoil, sinusoidal excitations in heave of varying frequency and amplitude were applied to the wing section. The resultant aerodynamic responses indicated that for sufficient amplitudes of motion and driving frequencies in the vicinity of the rigid buffet frequency, the buffet flow response synchronises with the aerofoil motion. This behaviour is evident in Fig. 31, where the excitation amplitude increases, the frequency band for which lock-in occurs also broadens. This phenomenon is

Fig. 28. Sound wave propagation and geometric flow properties (Hartmann et al. [114]).

sweep. For increasing angles of attack in the pre-buffet flowfield, frequency responses of the lift signal, excited by small oscillations of a trailing edge flap, developed more lightly damped characteristics and pronounced resonant peaks at frequencies approaching the buffet frequency. Similar behaviour was observed when varying the excitation kinematics to include perturbations in pitch and longitudinal translation, as indicated by the frequency responses of the shock and separation point

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Fig. 29. Schematics of the flow characteristics for one oscillation cycle of the shock-wave motion, (a) when the shock is in its most downstream position, (b) shock-wave moves upstream at its maximum velocity, (c) turning point of the cycle with the shock being in its most upstream position, (d) shock-wave moves downstream at its maximum velocity; the thickness of the circles represents the strength of the vortices, ωI , ωII , ωIII , ωIV (Hartmann et al. [114]).

likened to flowfield of an oscillating cylinder, whereby the shedding frequency prevalent in the von Karman vortex street synchronises with the driving frequency [127]. At conditions which exhibited lock-in, Raveh [125] noted that for excitation frequencies below buffet, the lift response leads the aerofoil motion. The converse is observed at higher driving frequencies, where the lift response lags the motion. The lift responses were also seen to be symmetric about the rigid buffet mean lift and exhibited harmonic responses at frequencies above the rigid buffet fundamental. Such observations allowed the extraction of a simplified gain-phase model, shown for the lift-curve slope in Fig. 32. Analogous to Nitzsche [9], the lift-curve slope undergoes a phase reversal and exhibits a resonant peak at approximately the buffet frequency. Raveh [125] further posited as to how the lock-in phenomenon may provide a mechanism for LCO of aircraft in the transonic regime. Shock oscillations may serve to excite a lightly damped structural mode during flight conditions near the stability boundary, yielding increasing amplitude modal oscillations. With increasing amplitudes lock-in occurs, and acts to reduce the excitation loads, with this cycle then repeating. Raveh & Dowell [128] performed further numerical studies on the lock-in mechanism for the NACA 0012 aerofoil under equivalent conditions, analysing the pre-buffet response to aerofoil motion and the influence of lock-in on the flowfield topology. At pre-buffet conditions, the flow was found to behave analogous to a linear oscillator, as per the findings of Nitzsche [9]. As the mean angle of attack increases towards onset, so too does the resonant frequency of the flow, with a corresponding decrease in damping. Onset is characterised by zero damping in the flow. At conditions in the region of developed buffet, the shock motion for the rigid aerofoil resembles Tijdeman [11] Type A oscillations. From the Mach number contours in Fig. 33, strong pressure fluctuations at the trailing edge are observed during the shocks upstream excursion (Fig. 33(d)), yielding prominent vortex shedding into the wake region (Fig. 33(e)–(g)).

Simulations of forced harmonic oscillations in heave reveal qualitatively different aerodynamic responses depending on whether the excitation frequency is above or below the buffet frequency. At sufficient amplitudes for synchronisation to occur, driving frequencies above the buffet fundamental result in quasi-harmonic oscillation of the aerodynamic coefficients, with smaller shock travel and less variation in shock strength. This is illustrated by the Mach contours in Fig. 34. The severe vortex shedding prevalent during the rigid shock oscillation cycle of Fig. 33 has been quenched. This smoothing of the flowfield was also observed for small heave oscillations (equivalent to an induced incidence of 0:1
) at the buffet frequency. At excitation frequencies below the buffet, vortex shedding is again quenched, however an expansive separated region exists aft of the shock. The aerodynamic responses are no longer harmonic and exhibit large fluctuations due to large-scale shock motion. The authors note that qualitatively similar behaviour was observed for excitations in pitch. A recent investigation by Giannelis & Vio [129] confirmed analogous flow features for a harmonically driven OAT15A supercritical aerofoil. Iovnovich & Raveh [130] continued to explore the lock-in mechanism on the NACA 0012, with a particular focus on the aerodynamic response due to pitch and flap excitations. At pre-buffet conditions, pitch excitation of the aerofoil was found to yield a phase lead in the lift response for excitation frequencies below the buffet, with a phase reversal at the buffet frequency. The converse was observed for both the moment coefficient for pitch excitations and the lift coefficient for flap excitations, where the responses exhibit a phase lead at higher driving frequencies. These characteristics persist in developed buffet flow for pitch excitations, and the combination of phase lead and significant aerodynamic fluctuations support the transonic LCO mechanism posed by Raveh [125]. The aerodynamic response due to flap motions is fundamentally different, acting to attenuate the buffet lift response and eliminate the resonance lift response. These findings support the use of trailing edge deflections as an effective means of shock buffet alleviation.

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Fig. 31. Combinations of excitation frequencies and angles of attack leading to responses in both the excitation and buffet frequencies and in the excitation frequency only; α ¼ 6
, M ¼ 0:72, Re ¼ 10  106 (Raveh [125]).

excitation considered. This is illustrated by the aerofoil pitch and shock location time histories in Fig. 35, where the shock location is phaselocked to the aerofoil motion. Additionally, analysis of the timeresolved velocity fields and phase-averaged Reynolds stress distributions indicates the aerofoil motion does not significantly influence the underlying buffet mechanism, which remains consistent with the authors earlier findings [113,114]. Raveh & Dowell [133] extended the work on shock buffet lock-in to spring-suspended aeroelastic systems. In a pitching system, lock-in did not occur when the structural eigenfrequency was below the buffet. At these conditions, the moment-coefficient response remained harmonic, with frequency content concentrated at the rigid aerofoil buffet frequency. The shock oscillation acts as an external forcing to the system, driving the pitch motion at the buffet frequency. For pitch natural frequencies above the buffet, lock-in occurs following a long transient beating. Both the aerodynamic and structural responses exhibit only the pitch natural frequency and converge to large amplitude LCO (Fig. 36(a) with ζ ¼ 0). Such findings provide further support for the transonic LCO mechanism described by Iovnovich & Raveh [130], where a phase lead in the pitching moment for pitch natural frequencies above buffet drives the instability. The influence of structural damping on the aeroelastic response of the pitching aerofoil was also investigated by Raveh & Dowell [133]. As shown in Fig. 36(a), the addition of a small amount of structural damping (ζ ¼ 0:5%) is sufficient to quench the lock-in phenomenon for the particular parameter set considered. A coupled pitch-heave aerofoil has also been considered by the authors, with structural natural frequencies fα ¼ 1:1fsb and fh ¼ 0:9fsb . Synchronisation of the structural and aerodynamic responses to the pitch mode is observed, with large amplitude LCO in pitch as evident in Fig. 36(b). A linear flutter analysis of this coupled system indicates the instability is not the result of flutter. The analysis is repeated with the structural natural frequencies reversed (fα ¼ 0:9fsb and fh ¼ 1:1fsb ). The results show low amplitude oscillations exclusively at the buffet frequency, as expected from earlier studies [125,128,130] where the moment and lift responses exhibit phase lags to pitch and heave excitations, respectively. Iovnovich et al. [134] have further suggested that the lock-in phenomenon may be responsible for inflight LCO encountered by the F-16. Recent literature in the field has continued the exploration of aeroelastic systems in the presence of shock buffet, concentrating on classifying the influence of various structural parameters on the lock-in phenomenon. In particular, Quan et al. [135] investigated the influence

Fig. 30. Displacement magnitude and phase with respect to excitation of the estimated shock foot (squares) and the separation point (circles) for three different excitation kinematics at α ¼ 4
(Nitzsche [9]).

Although the series of studies presented by Raveh and her co-authors are purely numerical, the lock-in mechanism has been evidenced experimentally by Hartmann et al. [131]. Following from earlier experiments of the DRA 2303 aerofoil [113,114], the authors investigated forced and flow-induced oscillations of the section using TR-SPIV and unsteady pressure transducers. In the elastically suspended configuration, coupled pitch and heave motions enable an energy exchange from the fluid to the structure, as dictated by the mean work coefficient of Dietz et al. [132]. For the particular parameter set investigated, such an exchange attenuates the amplitude of shock oscillation. Forced pitch/ heave motion of the aerofoil revealed synchronisation of the shock with the driving frequency across the various amplitudes and frequencies of 22

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both the pitch and lift responses, a consequence of the shock buffet frequency synchronising with the pitch mode. The sensitivity of the aeroelastic response of a pitching aerofoil to changes in mass ratio and structural damping observed by Giannelis et al. [136] also support the findings of Quan et al. [135]. As shown in Fig. 38(a), the mass ratio does not influence the onset frequency of lockin, with the aerodynamic and structural modes synchronising at fα =fSB0 ≈1 in all cases. The results also indicate that sectional mass has little influence on the LCO amplitude; however, an increase in the mass ratio yields a narrowing of the lock-in region and reduces the rate of convergence to a fully developed limit cycle. From Fig. 38(b), the addition of structural damping to the aeroelastic system imparts greater influence to the emergence and nature of shock buffet lock-in. With a small degree of damping, both the LCO amplitude and the extent of the lock-in region decrease appreciably. Carrese et al. [73] also investigated the aeroelastic behaviour of the OAT15A aerofoil, assessing the influence of flight condition, structural rigidity and natural frequency on the dynamic response of a pitching aerofoil system. At the nominal flight condition, the findings are in accordance with previous studies [133,135], whereby lock-in ensues when the structural natural frequency exceeds that of the buffet. However, a dependency on the structural rigidity (as dictated by the torsional stiffness) and the flow condition is also shown. The static trim position of the aerofoil is inherently linked to both these factors, and in particular, a lower pitch frequency (corresponding to a lower torsional stiffness) may yield a static trim position that lies outside the buffet envelope for the aerofoil. While this may produce a steady aeroelastic response in a flowfield that would, for a rigid aerofoil, exhibit shock oscillation, the converse may also apply. Flight conditions that yield a steady aerodynamic response for a rigid aerofoil may be perturbed to cross the buffet boundary when elastic effects are considered. Giannelis et al. [138] identified similar behaviour, where a coupled pitch-heave OAT15A aerofoil in pre-buffet conditions is excited by a regulation gust load, resulting in the system synchronising with the pitch mode and developing large amplitude LCO. A series of recent publications by researchers at the Northwestern Polytechnical University China have also drawn links between the transonic buffet instability and other aeroelastic phenomena, including single degree-of-freedom (SDOF) nonlinear flutter [139–141], hereafter denoted SDOF flutter. Gao et al. [142] studied the relationship between SDOF flutter and buffet on the NACA 0012 aerofoil, free to oscillate in pitch, through URANS simulation. For structural natural frequencies in the vicinity of the buffet, and angles of attack up to 1
below onset, LCO can occur in the pitching system even for eigen frequencies below the buffet, as shown in Fig. 39. This occurrence is tied to the lock-in phenomenon of Raveh [125,133] as the coupled response frequency is at the structural natural frequency if the wind-off pitch frequency is above the buffet. No qualitative change in the nature of the LCO responses is observed as the angle of attack varies from pre- to post-buffet onset, suggesting a similar mechanism is responsible for SDOF flutter and buffeting. Gao et al. [143] provide further insight into the correlation between SDOF flutter and buffet on the NACA 0012 aerofoil, constructing an ARX reduced-order model for the aerodynamics from URANS simulation and analysing the resultant eigenvalue problem for the coupled aeroelastic system. Analogous to classical bending-torsion flutter, SDOF flutter is found to be the result of modal coupling. However, rather than two or more structural modes coalescing, the instability stems from the coupling of a structural and aerodynamic mode. For this instability to arise, two requisite criteria must be adhered to. Firstly, the fluid must exhibit sufficiently low damping i.e. the static aerofoil is at an angle of attack near to buffet onset. Secondly, the instability only occurs for structural natural frequencies close to the buffet, in accordance with the authors earlier findings [142]. Additionally, two distinct response characteristics are identified for a system experiencing SDOF flutter, dependent on the structural natural frequency and mass ratio. At higher structural frequencies and mass ratios, the instability is governed by the structural

Fig. 32. Lift-curve slopes vs frequency ratio; α ¼ 1:5
, α ¼ 6
, M ¼ 0:72, Re ¼ 10  106 (Raveh [125]).

of structural natural frequency, mass ratio and structural damping through URANS simulations for an elastically suspended NACA 0012 in pitch. With varying pitch frequency, lock-in first occurs when the structural and buffet frequencies are approximately equal, and seen to persist until the pitch frequency is approximately double the buffet. This lock-in region contracts with both increasing mass ratio and structural damping. While the upper threshold of the region decreases with an increase in the structural parameters, the lower bound is insensitive to these changes, remaining at approximately the buffet frequency in all cases. In an extension of the earlier work by Giannelis & Vio [74], similar findings to Quan et al. [135] have been presented by Giannelis et al. [136] for the supercritical OAT15A aerofoil. Pitching simulations using the commercial finite volume code ANSYS Fluent [137] indicate the nature of the aeroelastic response of the system is sensitive to the ratio of structural and aerodynamic frequencies. With reference to Fig. 37, four frequency regions of qualitatively distinct behaviour are identified. For fα0 =fSB0 < 0:6, highly nonlinear pitch and lift responses develop, with comparatively large amplitude oscillations. In the regions bound by 0:6 < fα0 =fSB0 < 1 and fα0 =fSB0 > 1:8, the pitch response behaves as a single degree-of-freedom oscillator, excited by the unsteady shock oscillations. Within the frequency band 0:6 < fα0 =fSB0 < 1 resonance is observed in 23

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Fig. 33. Mach contours along a buffet cycle at α ¼ 6
, M ¼ 0:72, Re ¼ 107 . (a) Beginning of buffet cycle; (b) 27% of buffet cycle; (c) 41% of buffet cycle; (d) 55% of buffet cycle; (e) 61% of buffet cycle; (f) 68% of buffet cycle; (g) 75% of buffet cycle; (h) 82% of buffet cycle; (i) 90% of buffet cycle (Raveh & Dowell [128]).

Fig. 34. Mach contours along a cycle of heave aerofoil motion of fh =fb ¼ 1:55, α ¼ 6
, M ¼ 0:72, Rec ¼ 107 . (a) Beginning of heave cycle; (b) 21% of heave cycle; (c) 43% of heave cycle; (d) 64% of heave cycle; (e) 85% of heave cycle; (f) 106% of heave cycle (Raveh & Dowell [128]).

The interactions between classical bending-torsion flutter and transonic buffet have also been investigated by Zhang et al. [144]. The authors considered the Benchmark Active Controls Technology (BACT) model [145], a rectangular pitch-heave system with the NACA 0012

mode, and the flutter frequency locks-in to the structural frequency. At lower structural frequencies and mass ratios, instability in the fluid mode dominates and the coupled frequency is the characteristic fluid frequency (the buffet frequency at post-buffet conditions). 24

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Fig. 35. Shock location and aerofoil pitch at ωf ¼ 0:6942, hf =c ¼ 0:00175, αf ¼ 0:3
and xf =c ¼  0:0515 (Hartmann et al. [131]).

Fig. 37. Effect of frequency ratio on pitch response amplitude and frequency (M ¼ 0:73, α ¼ 3:5
, Re≈3  106 , μ ¼ 50, ζ ¼ 0%) (Giannelis et al. [136]).

aerofoil section. URANS simulations are performed at M ¼ 0:71, Re ¼ 3  106 , α ¼ 0
and a reduced velocity of V  ¼ 0:65, 10% higher than the reduced flutter velocity. At these conditions, buffet onset occurs at α ¼ 4:8
. The resulting response is a form of nodal-shaped oscillations of alternating diverging and damped behaviour, as shown in Fig. 40. The authors note that as the system is in a supercritical state, the response is not likely due to a beating superposition of two modes. Rather, the

Fig. 36. Time histories of aeroelastic response in the present of buffet (Raveh & Dowell [133]).

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Fig. 39. Amplitudes of responses against the structure reduced frequency kα . Buffet reduced frequency ksb ¼ 0:18 (Gao et al. [142]).

Fig. 40. Nodal-shaped oscillations of the structural displacements (Zhang et al. [144]). Fig. 38. Effect of mass ratio and structural damping on pitch response amplitude (Giannelis et al. [136]).

with variations in the structural natural frequency reveal the presence of mode veering [147,148]. At pitch frequencies below and well in excess of the buffet frequency, the aeroelastic system undergoes forced vibration at the buffet frequency. For structural frequencies approximately equal to the buffet frequency, an exchange between the structural and fluid modes occurs, with the aeroelastic response following neither the buffet nor pitch frequencies. Within the lock-in region, coupling between aerodynamic and structural modes results in instability in the pitch degree of freedom and the response tracks the structural natural frequency. In accordance with their previous work, Gao et al. [146] assert that lock-in is a consequence of a form of SDOF flutter, where the instability results from the coupling of the structural system with an unstable fluid mode.

oscillations result from the interaction between the flutter and buffet modes. As shown in Fig. 41, when the pitch angle exceeds the buffet onset angle, the lift response begins to exhibit higher frequency components, the most dominant of which is the buffet frequency. The change in aerodynamic loading causes the structural modes to dissociate, migrating from the flutter frequencies to their respective natural frequencies. This is accompanied by an abrupt phase reversal between the two structural modes, inhibiting the energy transfer between the fluid and structure and producing the damped oscillations evident in Fig. 41 at approximately t ¼ 3 s. The emergence of nodal-shaped oscillations in the work of Zhang et al. [144] prompted further analysis of the mechanism governing buffet lock-in by Gao et al. [146]. Linear stability analysis of the coupled NACA 0012 aeroelastic system is performed, employing the reduced-order aerodynamic model detailed by Gao et al. [142]. The authors note that the asymmetry of the lock-in region about the buffet frequency is not representative of a pure resonance phenomenon, as was observed in vortex-induced vibrations of an oscillating cylinder for wind-off structural frequencies both above and below the characteristic flow frequency. Rather, analysis of the structural and least stable fluid mode eigenvalues

6. Three-dimensional transonic shock buffet 6.1. Experimental investigations The investigations of transonic shock buffet on aerofoils outlined in the preceding sections surmise that the governing physics of the phenomenon is essentially two-dimensional. As discussed by Jacquin et al. [20], even in the case of a constant cross-section, rectangular wing section, the dimensionality of the buffet instability is questionable. Early

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where incipient and massively separated flow is present. Steady and unsteady pressure transducers are positioned at various spanwise stations, with static pressure orifices along a chordwise strip at 28% span. Additionally, fast-response anodic aluminium pressure-sensitive paint (AA-PSP) is applied to the wing surface for temporal resolution of the surface pressure distributions. The aeroelastic response of the wing is recorded through videogrammetric deformation measurement at two flow conditions; condition 1 (½α; M ¼ ½0
; 0:86) in which a small degree of trailing edge separation is present and condition 2 (½α; M ¼ ½0
; 0:92) where massive shock-induced separation is observed (consistent with Pearcey Type-B3 separation [8]). The dynamics of the unsteady flowfield, and consequently, the structural response, at the two conditions considered each exhibit unique characteristics. In the weak shock-wave/boundary layer interaction condition 1, spectral analysis of the pressure fluctuations reveals several harmonic peaks. While some of these peaks represent harmonics of the test section, a characteristic shock oscillation frequency is present. The authors attribute the high level of unsteadiness at this condition to an aeroacoustic feedback loop between the shock and trailing edge pressure perturbations. Time sequences of surface pressure distributions, such as those of Fig. 49, show shock motion concentrated at the wingtip, correlating well with numerical studies [152–155]. The shock oscillations also exhibit a highly aperiodic character, with broadband structural oscillations appearing at outboard stations. The strong shock-wave/boundary layer interactions of condition 2 produce a relatively steady separation line, owing to the inertia of the shock-induced separation. While this reduces the levels of pressure fluctuations, the structural response is excited at the dominant aerodynamic frequency in a resonant response analogous to that described by Nitzsche [9]. Lawson et al. [156] have detailed an experimental programme conducted under the Buffet Control of Transonic Wings (BUCOLIC) collaborative project between the Aircraft Research Association and the University of Liverpool. The experiments investigated transonic flow over the RBC12 half wing-body model, a variant of the B60 model used to study nacelle installation effects [157] and a representative geometry of modern civil transport aircraft. The wing is equipped with a variety of measurement apparatus, including steady and unsteady pressure transducers, dynamic pressure-sensitive paint (DPSP), accelerometers and wing root strain gauges, which are employed to assess the efficacy of a number of commonly used buffet indicators for onset prediction. The buffet indicators considered are derived from recommendations put forth by the ESDU [158], and include wing root strain divergence, RMS wingtip acceleration, lift-curve slope break, pitching moment break, axial force divergence and trailing edge pressure divergence. The onset predictions of each of the indicators is summarised in Fig. 50 across a range of Mach numbers. As shown, all indicators excluding axial force divergence and pitching moment break give consistent results throughout the Mach number range considered. A further comparison between the strain gauge divergence and DPSP onset predictions is given in Fig. 51. The onset of buffeting through strain gauges is 0:5
below the onset of large-scale flow unsteadiness determined from the DPSP measurements. This is rectified by considering the frequency range for which pressure oscillations develop. While broadband fluctuations of the flowfield develop at α ¼ 3
and coincide with onset through DPSP measurements in Fig. 51, pressure oscillations in the frequency band of 75–150 Hz originate at α ¼ 2:7
. The strain gauge onset is thus a result of excitations in this lower frequency band, where the structure is more responsive to external forcing. The authors suggest that at onset, the structure is more receptive to pressure fluctuations due to incipient separation, rather than large-scale shock oscillations. Further analysis of the unsteady pressure transducer and DPSP data acquired by Lawson et al. [156] has been undertaken by Masini et al. [159]. Proper orthogonal decomposition of the DPSP snapshots has been employed to identify dominant features of the buffeting flow. In Fig. 52, the first four modes of a clean wing configuration are shown, with the highest energy mode associated with the structural response of the wing.

Fig. 41. Pitching displacements and lift coefficient in one cycle of nodal-shaped oscillation (Zhang et al. [144]).

experiments of buffeting flows on three-dimensional wings, although finding some degree of commonality in the underlying fluid-dynamic phenomena, highlighted distinct behaviours that are unique from the two-dimensional buffet characteristics observed on aerofoils. Roos [149], for instance, examined unsteady pressure fluctuations over a high aspect ratio, transport-type swept wing configuration in the NASA-ARC 14-foot transonic wind tunnel. From the experiments, chordwise pressure perturbation convection velocities were found to be consistent with earlier studies on two-dimensional aerofoils [108]. However, deep within the buffet region, large-scale unsteadiness is found to be most severe at the wingtip. This is contrary to the predominantly constant spanwise distribution of pressure fluctuations observed in experiments over full-span aerofoil models. Additionally, the characteristic flow frequencies were found to be approximately an order of magnitude higher than those of a two-dimensional aerofoil, with pressure spectra exhibiting broadband bumps for Strouhal numbers in the vicinity of 0:2 < St < 0:4. These broadband spectral characteristics have also been noted by Benoit & Legrain [150] in transonic experiments of shock buffet over a transport-type wing in the ONERA Chalais Meudon S3Ch wind tunnel. The authors further investigated the buffeting flowfield of a constant RA16SC1 cross-section wing of moderate aspect ratio with zero and moderate sweep. In the rectangular wing case, shock oscillations were seen to originate at the root and extended outboard with increases in incidence. Although three-dimensional effects contributed to less organised flow dynamics, narrowband pressure spectra were obtained with peaks corresponding to the frequencies observed in experiments of a two-dimensional RA16SC1 aerofoil. Overall, the nature of the threedimensional buffet instability for the unswept wing was found to correlate well with the experiments on two-dimensional sections. Conversely, results of the swept wing configuration are indicative of distinct flow phenomena. As for transport-type wings, shock unsteadiness first emerges at the wingtip and progresses inboard with incidence. At developed buffet conditions, pressure fluctuations are again of highest intensity at the wingtip. These fluctuations are broadband in nature, and present a qualitative distinction to the organised and periodic buffet cycles observed on aerofoils. More recent experiments of three-dimensional shock buffet have verified the early findings of Roos [149] and Benoit & Legrain [150]. Steimle et al. [151], for instance, studied unsteady transonic flow over a flexible transport-type swept wing with BAC 3-11/RES/30/21 sectional geometry, assessing the fluid-structure coupling mechanism at conditions 27

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12 unsteady pressure transducers along the suction surface of the wing at 50% and 60% span. Steady pressure readings were acquired at an additional six spanwise stations and aerodynamic forces measured by the tunnel load balance [164]. Additionally, aeroelastic deformations of the wing are provided through videogrammetric recording, such that wing twist and bending along the span is available at each incidence. Trip dots are installed at 10% chord on the suction surface and data is presented for incidence sweeps at M ¼ 0:85 with Re ¼ 0:949  106 and Re ¼ 1:515  106 . As the distribution of sensors is insufficient to completely capture shock motion at the lower Reynolds number condition, the authors direct their discussion to the results recorded at Re ¼ 1:515  106 . The qualitative flow features identified by Koike et al. [163] are again consistent with earlier experiments of three-dimensional shock buffet. In particular, the pressure spectra reveal that with increasing incidence, three distinct characteristic responses emerge:

Higher frequency modes are also identified, which correspond to shockinduced unsteadiness concentrated at various spanwise bands. In particular, modes 2, 4 and 5 exhibit high spatial amplitudes at inboard, central and outboard spanwise stations, respectively. A dependence of the characteristic frequencies of each mode on the local chord is also identified, with the frequency content of shock unsteadiness migrating to higher frequencies at outboard stations. Additionally, by reconstructing the pressure field through combination of the aerodynamic modes, the authors have provided insight into the nature of spanwise shock dynamics. At angles of attack in the vicinity of onset, inboard propagating pressure perturbations are observed from approximately 87% span, running along the shock from tip to root. These spanwise perturbations are, however, dependent on incidence. At developed buffet conditions an additional outboard propagating perturbation also appears, which is associated with the higher-frequency bumps in the pressure spectra. Dandois [160] has presented the results of two wind tunnel tests conducted at ONERA for the investigation of closed-loop buffet control under the Aerodynamic Validation of Emission Reducing Technology (AVERT) project. The AVERT model is a swept half wing-body configuration of OAT15A aerofoil section and with sufficient rigidity to produce negligible oscillations in the bending and torsion modes. Test are conducted for Mach numbers between 0.78 and 0.86, with Reynolds number (based on mean aerodynamic chord) ranging between 2:83  106 and 8:49  106 . The model was instrumented with 86 static pressure tappings at four spanwise stations, 57 unsteady pressure transducers at seven spanwise sections and three stations equipped with two accelerometers each. A sublimating oil film was also applied to the upper wing surface to track separation characteristics. The investigations explicitly considered the effects of both Mach number and Reynolds number on buffet onset. The effect of an increase in Reynolds number is to delay buffet marginally throughout the regime considered in the experiments. This delay is independent of incidence and is equal to 0:1
at Re ¼ 4:72  106 and 0:4
at Re ¼ 8:49  106 , as compared to the baseline results at Re ¼ 2:83  106 . Mach number effects on onset are more pronounced. A higher Mach number is accompanied by a downstream shock displacement, with onset incidence decreasing with an increase in Mach number. Consistent with the findings of Roos [149] and Benoit & Legrain [150], the experiments of Dandois [160] have questioned whether buffet in a three-dimensional sense can be explained by the two-dimensional models described in Section 2. From the PSDs of pressure fluctuations shown in Fig. 53, the buffet instability appears as a broadband bump at frequencies between 0:2  St  0:6. In the experiments of an equivalent two-dimensional wing section performed by Jacquin et al. [20], the instability appears as a well-resolved peak with St ¼ 0:06. A physical explanation for this disparity is offered by the spanwise propagation of so-called buffet cells described by Iovnovich & Raveh [161] (see Section 6.2). The PSDs in Fig. 53 further highlight the presence of a secondary Kelvin-Helmholtz type instability, manifesting as a broadband bump between 1  St  4. This secondary instability has also been noted in the global stability analysis performed by Sartor et al. [29], and represents a shear layer interaction downstream of the shock. For investigation of the three-dimensional buffet mechanism, the research community has shown significant interest in the NASA Common Research Model (CRM) geometry [162]. The CRM is a generic transporttype aircraft developed by NASA for the validation of numerical routines, and has been the subject of extensive study in the American Institute of Aeronautics and Astronautics Drag Prediction Workshops. Buffet experiments have been explicitly conducted on an 80% scaled variant of the CRM by Koike et al. [163] in the 2 m  2 m transonic wind tunnel of the Japanese Aerospace Exploration Agency (JAXA). In a similar light to the experiments conducted by ONERA on the OAT15A aerofoil, the wind tunnel tests performed by JAXA look to establish a comprehensive database for the validation of three-dimensional buffet simulations. In the experiments of Koike et al. [163], the model was equipped with

1. At low incidence (α < 3
), the shock is essentially stationary and low intensity pressure fluctuations are evident, with resolved spectral peaks corresponding to the tunnel natural frequencies. 2. At moderate incidence (3
< α < 5:5
), small amplitude shock oscillations are present with a spectral bump at St≈0:3. This spectral bump corroborates the findings of Roos [149] and Dandois [160]. 3. At high incidence (α > 5:5
), large amplitude, aperiodic shock oscillations are observed, which exhibit low-frequency, broadband spectra. Pressure spectra of this form are in accordance with the experiments of Benoit & Legrain [150]. Cross-correlation and analysis of coherence between unsteady pressure measurements at the two spanwise stations by the authors has provided further insight into the nature of spanwise pressure fluctuations on three-dimensional wings. For both moderate and high incidence conditions, spanwise undulations originated near the wing root and propagated outboard towards the tip. The convection speeds of these spanwise perturbations decreases with an increase in angle of attack. These findings have been complemented by analysis of fast-response PSP data for buffeting flow over the CRM detailed by Sugioka et al. [165].

6.2. Numerical investigations Exploration of the three-dimensional buffet mechanism through wind tunnel experiments have been supplemented by a number of extensive computational investigations of the phenomenon. These studies have further indicated that three-dimensional effects may significantly alter the nature of shock oscillations, which are particularly sensitive to planform geometry [20,161]. Owing to advances in computer processing power, numerical simulation through URANS or scale-resolving methods has become an indispensable tool in probing the three-dimensional nature of the buffet phenomenon. Brunet & Deck [152] have provided one of the earliest studies on three-dimensional shock buffet simulation. The Zonal-DES method of Deck [18] described in Section 3 is employed in ONERA's elsA code [57] to simulate shock buffet of the CAT3D half wing-body model studied experimentally by Caruana et al. [166]. Computations are performed at M ¼ 0:82, Re ¼ 2:8  106 and α ¼ 4:2
, a condition for which established buffet was observed in the experiments. DES computations are limited to regions of separated flow, encompassing the entire wing-body wake and the outboard portion of the upper wing surface. The simulations successfully capture the massively separated and unstable regions on the outboard 50% of the wing. As seen in Fig. 42(a), Kelvin-Helmholtz type instabilities become prominent downstream of the shock, with eddy rollup observed in the separated region and a distortion of the shock towards the tip. Fig. 42(b) further highlights these effects, where the maximum pressure fluctuations are due to shock motion. The curved shock pattern is a distortion of the steady flow resulting from the massively separated zone that produces an upstream shock displacement. 28

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Fig. 42. Visualisation of pressure waves (Brunet [152]).

Results of the DES calculations correlate well with the experiment in terms of mean and RMS pressure distributions along the span. Comparisons between RANS simulations including a variety of turbulence models and grid densities showed that in all instances the RANS computations underestimated the levels of separation. The authors suggest that accurately resolving turbulent scales may be a necessity to model correctly the three-dimensional buffet instability. Analysis of the instantaneous dilation field ∇ðρuÞ in Fig. 43 reveals apparent pressure wave propagation from the trailing edge. This radiation process is inherently three-dimensional; large-scale structures impinge on the trailing edge and generate acoustic waves which propagate upstream (Fig. 43(a)) and along the span (Fig. 43(b)), a dispersion which affects the dynamics of the separation line. Sartor & Timme [153] have performed RANS and URANS computations using a number of turbulent closures with the DLR-TAU code [61], confirming the shock motions on the outboard portions of the wing as a characteristic of three-dimensional shock buffet. The RBC12 half wingbody model is investigated, corresponding to the geometry studied by Lawson et al. [156]. A number of turbulence models were considered, including the negative Spalart-Allmaras [32], Menter's k ω SST [52], a k ω LEA [63] and an explicit algebraic Reynolds stress model in the form of a Realisable Quadratic Eddy Viscosity Model (RQEVM) [63]. At small incidence, each of the models produced consistent predictions of shock-induced separation. At higher angles of attack, the SpalartAllmaras and RQEVM models yield similar lift predictions, while the

Fig. 43. Visualisation of pressure waves (Brunet [152]).

SST model predicts the largest extent of separation. Although the various closures result in different estimates for the critical angle at which largescale separation (and implicitly, buffet onset) occurs, each produces similar flow features. Nonetheless, comparisons between steady and unsteady simulations show that both the extension of shock-induced separation to the trailing edge and the appearance of a kink in the lift curve slope, two commonly applied indicators of buffet onset, are unreliable measures for the emergence of large-scale unsteadiness. Unsteady simulations found that both the Spalart-Allmaras and Menter's SST models provide similar predictions for onset (α≈3:1
), although the severity of buffet is highly dependent on the particular model chosen. The LEA and RQEVM models produced similar results to one another, but predict onset at a higher incidence relative to the previous closures (α≈3:8
). PSDs of the lift signal also showed that onset is dominated by narrowband frequency content consistent with periodic motion. At higher angles of incidence, the instability adopts a broadband 29

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character characterised by aperiodic motions. Whereas Brunet & Deck [152] suggested that scale-resolving simulations may be necessary to capture three-dimensional buffet, Sartor & Timme [153] have shown that the URANS approach is capable of reproducing the dominant flow features. With increasing incidence, the unsteady shock motions first become apparent at the wingtip and grow in spanwise extent towards the fuselage. The increase in incidence is also accompanied by the development of broadband frequency content, with the dominant frequency reducing as the shock unsteadiness spreads to spanwise stations of larger chord. This highlights a fundamental distinction between two- and threedimensional buffet characteristics, where at developed buffet conditions for a three-dimensional geometry, shock oscillations typically exhibit aperiodic motions. Subsequent studies by Sartor & Timme [154,155] have compared the prior URANS results to DDES computations. Both methods show excellent agreement regarding shock position inboard of the crank, however, the outboard mean shock foot trace of the DDES results exhibit a straighter character, more consistent with the experimental results of Lawson et al. [156]. Standard deviations of the surface pressure distributions derived from DDES computations are also in better agreement with the experiments, supporting Brunet & Deck's [152] notion that scale-resolving simulations may be necessary to obtain a comprehensive description of the fundamental flow physics. Nonetheless, both methods are able to reproduce self-sustained shock oscillations outboard of the crank. From Fig. 44, turbulent structures are seen extending from the shock foot in both cases, with the separated zones shifting the shock foot toward the leading edge. Due to the resolution of turbulent eddies, the DDES computations show a rich turbulent spectrum, which also manifests in the time histories of the aerodynamic coefficients. Consistent with the experiments of Koike et al. [163], the separated regions migrate from inboard to outboard sections, producing regular yet aperiodic oscillations of the integral forces. PSDs of the force coefficients further demonstrate this aperiodic behaviour, with low-frequency broadband peaks between 150 and 300 Hz in both approaches. In analysing instantaneous velocity divergence fields, the authors came to similar conclusions as Brunet & Deck [152], finding that acoustic wave propagation is an inherently three-dimensional process where both streamwise and spanwise radiation of disturbances influence shock and separation dynamics. Extending their earlier work, Sartor & Timme [167] studied the effect of Mach number on the emergence of unsteady phenomenon on the RBC12 geometry. RANS simulations found that at smaller Mach numbers, the appearance of a kink in the drag polar provided a fair indicator of the onset of shock buffet. This break in the polar is, however, an unreliable metric, a consequence of unconverged steady simulations that does not appear at higher Mach numbers. Unsteady simulations across a range of conditions showed that onset incidence decreases with Mach number. Additionally, higher Mach number simulations exhibit lower magnitude pressure fluctuations, with unsteadiness concentrated in the separated flow regions and the standard deviation of lift oscillations plateauing soon after onset. Conversely, although the extent of separated flow at lower Mach numbers is reduced, unsteadiness develops across the entirety of the suction surface and is accompanied by large amplitude pressure fluctuations. With confidence in the efficacy of URANS methods to produce the main flow features associated with three-dimensional shock buffet, Iovnovich & Raveh [161] conducted a numerical parametric study to assess the effects of sweep angle and span on the buffet instability mechanism. Using the finite difference RANS solver EZNSS [31], three baseline configurations were investigated: infinite-straight, infinite-swept and finite-swept wing models, each with a constant RA16SC1 section. For the infinite-straight wing model, nonuniform lateral structures develop during upstream shock excursions and the shock travels farther forward than in the two-dimensional case due to the increased separation associated with three-dimensional effects. Nonetheless, excluding the localised three-dimensional character during parts of the oscillation cycle, the

Fig. 44. Instantaneous values of eddy-viscosity ratio at α ¼ 3:8
, M ¼ 0:80 and Re ¼ 3:75  106 (Sartor & Timme [154]).

buffet amplitudes and frequencies are consistent between two- and threedimensional simulations, and are in fair agreement with the predominantly two-dimensional mechanism for infinite-straight wings described by Jacquin et al. [20]. Analysis of the buffet behaviour of infinitely-swept wings at variable sweep angles revealed ranges of sweep for which qualitatively distinct flow mechanisms are observed. At low sweep angles (Λ < 20
), the buffet mechanism is similar to the infinite-straight and two-dimensional cases, with outboard shock oscillations primarily along the chord and minimal lateral pressure disturbance propagation. The surface pressure RMS contours in Fig. 45 show that with increasing sweep in this regime, the 30

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Fig. 45. Surface pressure RMS levels for various sweep angles: infinite-swept configuration at nominal buffet conditions (Iovnovich & Raveh [161]).

two time instances. The authors describe the physical mechanism governing the propagation of these buffet cells as follows:

shock travel becomes increasingly nonuniform along the span. For moderate sweep angles (20
 Λ < 40
), a lateral flow mode begins to dominate, which eliminates the two-dimensional buffet instability through pressure perturbations propagating outboard from the wing root. The unsteady shock motions observed at moderate sweep are distinct from the two-dimensional buffet mechanism and are characterised by the periodic spanwise convection of so-called buffet cells, as evident in the surface pressure fluctuation maps of Fig. 46 for Λ ¼ 30
at

1. Pressure fluctuations are generated at the aft λ shock of the wing root and emanate outboard. 2. As a positive pressure fluctuation propagates to the spanwise station of the λ shock tip, unstable shock-wave/boundary layer interaction ensues due to an increase in local shock strength, resulting in an

Fig. 46. Surface pressure coefficient and fluctuation maps at two time snapshots during shock-buffet oscillations: Λ ¼ 30
infinite-swept configuration and nominal buffet conditions (Iovnovich & Raveh [161]). 31

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frequency-domain analysis has been performed on the RBC12 half wingbody configuration at conditions approaching buffet onset using the DLRTAU code [61]. The authors note that the high dimensionality of the fluid Jacobian prohibits the use of direct methods for eigenvalue calculations due to inordinate memory requirements. Additionally, the stiffness of the linearised aerodynamics near the onset of instability is problematic; conventional iterative methods, such as the incomplete lower-upper (ILU) preconditioned generalised minimal residual (GMRES) approach [170], are particularly susceptible to stall. These issues are circumvented by use of an advanced iterative solution scheme, the generalised conjugate residual solver with deflated restarting (GCRO-DR). A complete description of the theory and implementation of the method is provided by Xu et al. [171]. Timme & Thormann [169] performed frequency sweeps at four angles of attack below buffet onset (α ¼ 3:03
), using a synthetic torsional mode for excitation. As shown in the lift coefficient frequency response of Fig. 48, as the incidence approaches onset, a distinct resonant peak emerges at St ¼ 0:11. Within this low-frequency range, the aerodynamic response leads the structural excitation, a potential indicator of a destabilising aerodynamic mode that was also observed in two-dimensional stability analysis of aerofoils. Contrary to the two-dimensional case, however, interesting response characteristics emerge for Strouhal numbers between 0.3 and 0.7. Marginal increments in incidence result in the appearance and amplification of secondary peaks, which coincide with the typical frequency band for which broadband spectra have been reported in buffeting flows on three-dimensional wings and which may reflect the presence of additional unstable aerodynamic modes. The complex-valued unsteady pressure distribution has also been examined, in particular, for excitation frequencies corresponding to the low-frequency peak (St ¼ 0:11) and the dominant secondary peak (St ¼ 0:51). At St ¼ 0:11, unsteadiness is concentrated at the outboard shock foot. The pressure fluctuations grow in spanwise extent and intensity with an increase in incidence, consistent with prior experimental and computational observations [150,153]. At the higher-frequency excitation, the unsteady flow exhibits a distinct character. Minimal pressure fluctuation was observed at α ¼ 2:9
; however, upon closer approach to onset, high intensity unsteadiness is evident at the wingtip shock foot and downstream of the shock. Relative to the low-frequency excitation, the shock centred unsteadiness is concentrated at the wingtip. Considering explicitly the eigenspectrum across the various angles of attack, an increase in incidence is accompanied by the migration of several eigenvalues in the frequency range 0:3 < St < 0:7 towards the imaginary axis. Again, the distinct behaviour in this higher-frequency band suggests a

upstream displacement of the shock and an increase in shock-induced separation. 3. The corresponding negative pressure fluctuation also propagates from the wing root along the aft λ shock, but does not yield an increase in shock strength. As such, as the perturbation passes the λ shock tip, it propagates outboard along a constant (aft) chordwise position, with flow remaining attached aft of the shock. 4. With both positive and negative pressure fluctuations emanating periodically from the wing root, outboard propagating buffet cells result, which at an individual spanwise stations exhibit the chordwise shock motion characteristic of two-dimensional buffet, but which stem from an inherently three-dimensional phenomenon. This spanwise propagation of buffet cells appears to be tied to a higher-frequency (relative to two-dimensional buffet) aerodynamic mode, as indicated by the incremental mode decomposition conducted by Ohmichi et al. [168] on the CRM configuration. In addition to a change in the nature of unsteadiness, Iovnovich & Raveh [161] found that an increase in sweep is also accompanied by a reduction in buffet amplitude (as shown in the RMS contours of Fig. 45) and an increase in frequency. A further increase in sweep (Λ  40
) yields a stationary shock position due to stall. From these results, the authors relate the effects of sweep angle to incidence. An increase in sweep, as with an increase in incidence, results in an increase in buffet frequency, up to the point at which the wing stalls and shock oscillation ceases. Regarding the finite-swept wing configurations, tip-vortex interactions were found to contribute to the buffet mechanism. The mean upper surface pressure coefficients for variable span length at a nominal sweep of Λ ¼ 20
are shown in Fig. 47. For b=c ¼ 2, pressure fluctuations were limited to the tip, where tip-vortices dominate and shock buffet does not develop. At longer span extents, the lateral propagation mechanism found in infinite-swept wings is also observed; however, the mechanism is somewhat more complex due to the interaction between the outboard pressure fluctuations due to shock oscillation and the wingtip vortices. The main oscillation character along the span is consistent with the infinite-swept wing results, with irregular shock motion emerging at outboard stations. While Iovnovich & Raveh [161] have provided a physical description of the three-dimensional buffet mechanism, the recent work of Timme & Thormann [169] has shown that the origins of the instability may lie in globally unstable aerodynamic modes; an analogue to the twodimensional instability detailed by Crouch et al. [21]. Linearised

Fig. 47. Surface pressure coefficient for various span length cases: Λ ¼ 20
finite wing configurations at nominal buffet conditions (Iovnovich & Raveh [161]). 32

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Koike et al. [163]. In a similar light, integration of the Zonal-DES method of Deck [18] into the JAXA code FaSTAR [177] by Ishida et al. [178] has also produced favourable correlations to the CRM experimental data, albeit with shock locations displaced upstream. The CRM configuration has also been subject to aerodynamic shape optimisation with buffet constraints by Kenway & Martins [179], in which the optimisation successfully drove the buffet boundary of the optimal configuration away from the design-point. 7. Control of transonic shock oscillations 7.1. Trailing edge deflections A common feature among the various models governing the transonic buffet phenomenon presented in Sections 1 and 2 is a coupling between the large-scale excursions of the shock and pressure fluctuations at the trailing edge. It follows that control of this instability can be achieved by modulating the flowfield at either the shock-wave/boundary layer interaction zone or at the trailing edge. For the latter, the efficacy of trailing edge flap deflections for the control of shock oscillations have been investigated in the classical literature [180,181]. While static trailing edge flap deflections may effectively alter the flow state to a stable region, they are often accompanied by detrimental aerodynamic effects, particularly appreciable drag penalties, rendering these methods undesirable for the control of shock oscillations. Under the BUFET’N Co [182] project at ONERA, a Trailing Edge Deflector (TED) was developed to take advantage of the shock stabilisation achieved by modulating flow at the trailing edge while minimising the effects on the global flow characteristics. Despre et al. [124] detail the design of the TED (as shown in Fig. 54), where a servo-motor is connected to the tab (spanning 1–3% chord) on the pressure surface of the model. Various control strategies employing the TED were investigated both experimentally and computationally using the viscous-inviscid interaction code VIS15 [183,184] on the two-dimensional OAT15A section. Static TED deflections were shown to reduce the incidence of buffet onset, however, the lift at onset increased. Open-loop sinusoidal deflections at developed buffet conditions with particular phase shifts were briefly able to stabilise the shock oscillations. It was observed that an increase in the deflector angle resulted in downstream shock excursions, and vice versa. Nonetheless, closed-loop control was deemed necessary for a robust control strategy. Using unsteady pressure signals at the rear of the shock-wave/boundary layer interaction zone, the control law involved positive TED deflections during upstream shock excursions and negative TED deflections during downstream shock travel - actuation that opposes the natural motion of the shock. For a pressure signal P, the TED deflection δðtÞ is governed by:

  δðtÞ ¼ δ þ A Pðt  τÞ  P

Fig. 48. Frequency response of lift coefficient due to flexible torsion mode (Timme & Thormann [169]).

(13)

where δ and P are the mean TED deflection and pressure measurement, respectively. The amplitude A and time delay τ were parameters that were varied throughout the investigation. With an optimal set of parameters, Despre et al. [124] saw significant reductions in the severity of shock buffet through active TED control, a decrease in shock oscillation amplitude of approximately 66% in the experiments. A subsequent study by Caruana et al. [185] employing the TED for buffet control identified similar static behaviour, and with small modifications in the control law achieved comparative control effectiveness to Despre et al. [124]. As shown in Fig. 55(a), static TED deflections delay the onset of buffet to higher lift coefficients, however, in the developed buffet region this control strategy is ineffective at reducing the magnitude of pressure fluctuations. The efficacy of closed-loop TED deflections is clearly evident in Fig. 55(b), where the amplitude of shock travel is substantially lower with the control law active. Caruana et al. [186] further investigated the effectiveness of TED

different physical mechanism underlying three-dimensional buffet. The preceding discussion illustrates the fundamental advances in the understanding of three-dimensional shock buffet attributed to computational efforts. However, these investigations are supplemented by an array of additional numerical studies, which act to corroborate the underlying fluid-dynamic phenomena. Xiong & Liu [172] have employed DES to successfully capture three-dimensional shock buffet on a swept wing with a supercritical aerofoil, providing further evidence to the three-dimensional nature of pressure wave propagation. Illi et al. [173,174] employed both URANS and DDES computations on the CRM geometry to show that the large vortices shed from the main wing at buffet conditions have a significant influence on the flowfield at the horizontal tailplane. Ribeiro et al. [175] have detailed modifications to the Lattice-Boltzmann based commercial solver PowerFLOW [176], which allow excellent buffet predictions relative to the experiments of 33

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Fig. 49. Representative pressure distribution time sequence on the upper wing surface measured by AA-PSP, condition 1 flow ½α; M ¼ ½0
; 0:86, Re ¼ 1:05  106 , Δt ¼ 0:667 ms. (Steimle et al. [151]).

Fig. 50. Comparison of buffet onset indicators (Reproduced by kind permission of Aircraft Research Association Ltd Lawson et al. [156]).

Fig. 51. Comparison of buffet strain gauge response with DPSP and CFD unsteadiness (Reproduced by kind permission of Aircraft Research Association Ltd Lawson et al. [156]).

control on a three-dimensional wing-fuselage model experimentally at ONERA's S2Ma wind tunnel. Static TED deflections provide an increase in aerodynamic performance analogous to the two-dimensional effects, delaying onset to higher values of lift. While drag penalties are incurred at lower lift coefficients, the authors note that multiple TED actuators along the span may be employed to impart artificial wing twist, allowing optimal configurations to minimise lift-induced drag at various flight conditions.

Sinusoidal TED deflections in open-loop on a three-dimensional wing are qualitatively different from those of the two-dimensional section previously investigated. Caruana et al. [186] found that the presence of three-dimensional separated flow structures must be considered, with open-loop deflections in many cases exacerbating the shock motions and increasing levels of structural vibration, particularly in torsion. The inherently three-dimensional flowfield and non-uniform spanwise shock

34

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Fig. 52. Spatial component of dominant modes related to the structural response and spanwise shock unsteadiness. The spatial amplitudes are coloured from blue to red, representing opposite signs. The numbers in parentheses represent the POD energy of the mode and the cumulative POD energy from the first mode until the respective mode (Masini et al. [159]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

an effective means of buffet alleviation. Numerical simulations of the NACA 0012 aerofoil at developed buffet conditions and with a trailing edge flap axis of rotation at 80% chord are performed. The lift coefficient Cl is employed to determine the present state of the system due to the relative smoothness compared to pressure fluctuations or shock location. The flap deflection βðtÞ is governed by the control law:

location also poses issues for closed-loop TED control, particularly as the principle flow direction does not coincide with the axis of the deflectors and transducers. By low-pass filtering of the measured pressure signals, the authors demonstrated computationally an effective means of controlling shock oscillations on the outboard sections of the wing by TED deflections through the central span. The resultant shock displacements and commanded TED deflection are shown in Fig. 56. Caruana et al. [186] do note, however, that such a control strategy may be detrimental to the flow over the central span and further investigations are required with more complex control laws. Although the use of trailing edge flap deflections may be accompanied by various detrimental aerodynamic effects, Gao et al. [187] have demonstrated that with a closed-loop control law this form of actuation is

βðtÞ ¼ λ½Cl ðt  τÞ  Cl0 

(14)

where Cl0 is the balanced lift coefficient, λ is the dimensionless gain and τ is the time delay in nondimensional time. Exploring the influence of the various parameters, the optimal balanced lift coefficient is found to be that pertaining to the unstable steady solution and is obtained by

35

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Fig. 53. PSDs for different chordwise locations at 60% span; M ¼ 0:82, Re ¼ 2:5  106 (Dandois [160]).

Fig. 55. Effects of static and dynamic TED deflections of OAT15A aerodynamic characteristics (Caruana et al. [185]).

the three control parameters, shock oscillations are completely quenched with reasonable control effort, as shown in Fig. 58. Although not strictly related to trailing edge deflections, Liu & Yang [188] have shown, computationally, that modification of the suction surface flowfield through installation of a wall-normal protrusion aft of the shock may provide an effective means of buffet load alleviation. The NASA SC(2)-0714 aerofoil, studied experimentally by Bartels & Edwards [189], is employed as a baseline from which subsequent simulations investigating the effect of chordwise location and protrusion height of the microtab were developed. The installation of such a passive control device imparts a geometric effect on the flowfield (as discussed in Section 2.3). The microtab alters the dynamics of the main vortices shed from the shock-wave/boundary layer interaction by introducing a secondary vortex. The height difference between vortices fore and aft of the microtab results in a tendency of velocity variations, which (for certain positions of the protrusion) weaken interactions aft of the shock. The authors found a microtab positioned at 80% chord with a protrusion height of 0.75% chord was the most effective configuration for buffet load alleviation, reducing buffet amplitudes by 93% while shifting frequency content to a low-frequency mode associated with the secondary vortices.

Fig. 54. Trailing edge deflector (TED) (Caruana et al. [185]).

iteratively varying the flap deflections until Cl0 equals the unstable steady lift coefficient. The authors note that this value differs from the timeaveraged lift coefficient and that the variation in drag incurred from these control deflections is negligible. Changes in the gain and time delay have a significant influence on the controller's effectiveness. As seen in Fig. 57, there are distinct regions in the λ τ parameter space for which positive control effectiveness is obtained. With optimal combinations of 36

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Fig. 58. Closed loop control with λ ¼ 0:3, τ ¼ 30=36T0 and Cl0 ¼ 0:357 (Gao et al. [187]).

mechanical (passive) and fluidic (active) vortex generators in the suppression of transonic shock oscillations. Through the production of corotating vortices, vortex generators act to energise and stabilise the boundary layer, promoting attached flow and inhibiting the shockinduced separation that is instrumental to the buffet phenomenon [190,191]. Caruana et al. [185,186] demonstrated the usefulness of mechanical vortex generators experimentally on a rectangular OAT15A wing section. A single configuration was considered by the authors, consisting of rectangular vortex generators at a 30
incidence, with a height equivalent to the local boundary layer height (h ¼ 1 mm) and length l ¼ 5h. The vortex generators were positioned at 33% chord, 17% chord upstream of the mean shock location, with a spanwise spacing of λ ¼ 10h. With the vortex generators installed an improvement in aerodynamic performance is achieved at incidence close to buffet onset, however, drag penalties are incurred at lower angles of attack. As shown in Fig. 59(a), the additional energy imparted to the boundary layer by the vortex generators acts to reduce the width of the wake, and correspondingly, the drag coefficient at higher angles of attack. At conditions that exhibited buffet in the clean wing configuration, the mechanical vortex generators provide an effective means of suppression. From Fig. 59(b), the authors show the low-frequency pressure fluctuations associated with large-scale shock motions do not develop in the incidence range considered in the experiments. Higher frequency fluctuations related to turbulent eddies in separated flow regions are also attenuated at all test points. Furthering the BUFET’N Co project, Dandois et al. [192] have conducted a parametric study on the effects of various design parameters in both mechanical and fluidic vortex generators. Numerical simulation of transonic flow over a full-span constant cross-section OAT15A-CA aerofoil at conditions with significant shock-induced separation was employed to probe the parameter space. For the mechanical vortex generators, the influence of skew angle and spanwise spacing (β and λ in Fig. 60, respectively) has been assessed, in addition to the chordwise positioning. The height and length of the vortex generators remains fixed in the study, with h ¼ 1:3 mm and l ¼ 5h. Considering the lift and wall pressure distributions as indicators of the parameters effectiveness, a skew angle of β ¼ 0
is optimal, producing higher lift and a steady, downstream shock location. In accordance with the previous work of Godard & Stanislas [193], the optimal spanwise spacing is found to be

Fig. 56. Control of three-dimensional buffet by closed-loop TED deflections (Caruana et al. [186]).

Fig. 57. Effective control regions for different λ and τ at M ¼ 0:70, α ¼ 5:5
and Re ¼ 3  106 (Gao et al. [187]).

7.2. Vortex generators Under the guise of the BUFET’N Co project [182], a number of researchers at ONERA have further assessed the effectiveness of both 37

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Fig. 61. Design parameters for fluidic vortex generators (Dandois et al. [192]).

1:3 mm, a spanwise spacing λ ¼ 10d and jet static pressure equal to the local static pressure, yielding a momentum coefficient Cμ ¼ 2  103 . The optimum skew angle exhibits a dependence on the pitch angle, with higher pitch angles yielding a lower optimal skew angle. For the conditions considered, the authors determine an optimal set of α ¼ 30
and β ¼ 60
. With these optimal parameters, inserts were designed for a threedimensional half wing-body model to examine the behaviour of the mechanical and fluidic vortex generators experimentally. Both methods were effective in suppressing the large-scale separation present in the clean wing configuration. Molton et al. [194] and Dandois et al. [195] continued the work of Dandois et al. [192] and detailed a comprehensive experimental investigation into the efficacy of both mechanical and fluidic vortex generators on a swept half wing-body model. The reference configuration exhibits large-scale separation at outboard sections, with established buffet at M ¼ 0:82 and α ¼ 3:5
. In the clean configuration, the oil flow visualisations in Fig. 62 show the flow is fully separated downstream of the shock past 50% span. Further, trailing edge pressure divergence, Kulite pressure signals, PIV 2C and LDV 3C measurements all indicate sustained shock oscillation at these conditions. For the controlled test cases, four configurations have been investigated and are summarised in Table 3. The test space consists of one mechanical configuration (VGm), two continuous fluidic vortex generators with varying skew (VGF4 and VGF5) and one of pulsed fluidic actuation (VGFp). The oil traces in Fig. 62 show the mechanical vortex generators suppress flow separation along the majority of the span, excluding the region between 50 and 60% span

Fig. 59. Effects of mechanical vortex generators on rectangular OAT15A wing section (Caruana et al. [186]).

Fig. 60. Design parameters for mechanical vortex generators (Dandois et al. [192]).

λ ¼ 6h. Smaller spacing results in destructive interference between adjacent vortices, reducing lift. Additionally, as long as the vortex generators are positioned upstream of the shock, the chordwise location has a negligible influence on the aerodynamic performance. Regarding the fluidic vortex generators, Dandois et al. [192] investigated the effects of varying pitch and skew angles (α and β in Fig. 61, respectively). The orifice is positioned at 21% chord, with diameter d ¼

Fig. 62. Oil flow visualisations for the uncontrolled and controlled configurations: M ¼ 0:82 and α ¼ 3:5
(Dandois et al. [195]). 38

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Table 3 Parameters for various vortex generator configurations [194]. Configuration

Control type

Dimensions (mm)

λ

βð
Þ/LE

Y=b (%)

No. of VGs

1 2 3 4

Mechanical (VGm) Fluidic (VGF4) Fluidic (VGF5) Fluidic pulsed (VGFp)

h ¼ 1:3, l ¼ 5h d¼1 d¼1 d¼1

12h 6d 6d 11:5d

0 60 30 60

51–89 53–82 53–82 50–84

27 40 40 25

mass flow rate for the continuous fluidic vortex generators shows that flow rate effects saturate for Cμ  5  104 . Regarding pulsed fluidic vortex generators, the forcing frequency has a profound influence on their effectiveness. As shown in Fig. 63(b), low-frequency pulsing results in shock smearing, a characteristic of large-scale shock motion. At higher frequencies (Strouhal number St  0:24), the shock is stabilised and located marginally upstream relative to the continuous fluidic blowing. Molton et al. [194] demonstrated that mechanical, continuous and pulsed fluidic vortex generators in open loop are each effective in delaying buffet onset by suppressing large-scale separation. Further, fluidic blowing may be the preferable control strategy, providing similar control actuation to mechanical vortex generators but with the absence of a drag penalty at lower incidence. Further evidence supporting the effectiveness of mechanical vortex generators for buffet suppression has been provided by Timme & Sartor [196]. Both steady and unsteady URANS simulations with the DLR-TAU [61] code are performed on the half wing-body model RBC12, with geometry derived from their earlier investigations [153,154]. The simulations are performed for various angles of attack at M ¼ 0:80 and Re ¼ 3:75  106 . Co-rotating mechanical vortex generators are integrated at 32% chord between 64% and 91% span with spanwise spacing λ ¼ 7:7h, taper ratio of 0.6 and aspect ratio of 1.3. From the steady simulations, unconverged solutions are found at α ¼ 3
in the clean configuration and α ¼ 3:6
for the controlled case. Unsteady simulations confirm that these unconverged solutions are, in fact, due to the development of large-scale unsteadiness and provide buffet onset conditions for the respective configurations. The topological effects on the flow evidenced through surface pressure distributions are in agreement with earlier studies [111,194]. The vortex generators produce complex shock/vortex interactions that shift the mean shock location downstream and inhibit the development of massively separated flow. The integration of vortex generators also significantly affects the frequency content of the aerodynamic coefficients. For the clean wing shown in Fig. 64(a), a distinct peak is observed between 200 and 300 Hz near onset, which develops a more broadband character with increasing incidence. For the controlled wing in Fig. 64(b) however, low-amplitude, high-frequency content dominates at onset. The buffet instability manifests at onset as a lowamplitude broadband peak, which becomes dominant as the incidence is increased. Accompanying the recent surge in research of the three-dimensional buffet mechanism, a number of publications from JAXA have further explored the physics governing buffet suppression through mechanical vortex generators. Kouchi et al. [197,198] have conducted wind tunnel experiments to investigate shock motions on a two-dimensional NASA SC(2)-0518 aerofoil with vortex generators. Focusing-Schlieren imaging [199] and wavelet analysis revealed that although vortex generators successfully suppressed onset to higher incidence, they significantly increased the three-dimensionality of the flowfield and incited aperiodicity of the pressure fluctuations. Wavelet spectrograms of the timespace flowfield maps further revealed that even in a suppressed buffet state, frequency content relating to the low-frequency buffet mode and its harmonics appears intermittently, with the rate of appearance diminishing with lower spanwise vortex generator spacing. Koike et al. [200] have also investigated the differences between installation of vortex generators on two- and three-dimensional wings experimentally on the CRM, with the two-dimensional geometry derived from the 65% span aerofoil section. Although not explicitly considering buffet conditions,

where a recirculation zone exists. A complex but near periodic threedimensional shock/vortices interaction pattern is also present in the shock foot region, with the vortices acting to deform the shock along the span. The flow topology of the continuous fluidic vortex generators (not shown) is strikingly similar, suppressing separation in the outboard regions and producing analogous shock/vortex interaction tracks. Comparison of pressure distributions between the controlled and uncontrolled cases (Fig. 63(a)) show that both mechanical and fluidic control strategies shift the mean shock location downstream by approximately 20% chord. The consistency of trailing edge pressures between the controlled cases, in addition to the RMS pressure distributions and pressure spectra (not shown) confirm both control strategies effectively damp out unsteadiness downstream of the shock. An analysis of varying

Fig. 63. Effects of mechanical and fluidic vortex generators on wall pressure distributions (Dandois et al. [195]). 39

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Raghunathan et al. [206] have provided a recent review on the applicability of SCBs to the suppression of shock buffet. The findings reported therein are not repeated here and the reader is directed to the work of Raghunathan et al. [206] to gain context for the following discussion. Eastwood & Jarrett [207] have performed a numerical investigation using the commercial finite volume solver ANSYS Fluent [137], optimising two- and three-dimensional SCBs for drag reduction and buffet alleviation. The authors investigated the effects of spanwise width, spacing and SCB edge design for three-dimensional bumps, in addition to ramp angle, which is applicable in both the two- and three-dimensional designs. The various design parameters are shown in Fig. 65. For designpoint aerodynamic improvements, two-dimensional SCBs outperform the corresponding three-dimensional arrays. The usefulness of threedimensional SCBs arises in off-design conditions, where streamwise vorticity is produced in a manner similar to vortex generators. By reducing the length ltail or increasing the ramp angle θramp , a stronger spanwise pressure gradient is achieved at the SCB tail, producing stronger vortices which are more effective at suppressing the separated flow downstream of the shock. Using a number of buffet onset metrics based on wall shear stress from steady simulations, Eastwood & Jarrett [207] ascertained that three-dimensional SCBs successfully reduced the largescale separation associated with buffet. Further, the buffet suppression criteria exhibited a strong correlation to the design-point vortex strength, permitting an assessment of both design-point aerodynamic performance and improvements in buffet margin. These objectives are shown to be in direct conflict, leaving the applicability of buffet alleviation through SCBs subject to a trade-off with design-point performance. A following study by Bogdanksi et al. [208] arrived at contrary conclusions on the usefulness of three-dimensional SCBs for buffet suppression. Both steady and unsteady RANS simulations have been performed with the DLR-TAU code [61] to investigate shock buffet control on the aerofoil of the Pathfinder transonic laminar wing [209]. Three different SCB geometries are considered and shown in Fig. 66, with a downhill simplex algorithm employed to optimise the height and chordwise positioning for minimum drag at two design-points. At both conditions, each of the SCBs degrades the buffet behaviour of the aerofoil. As shown in Fig. 67, buffet onset occurs at an incidence approximately 0:5
below the baseline in each of the controlled cases, with higher buffet amplitudes observed within the buffet region. The extended bump does provide an improvement in the separation behaviour of the aerofoil (Fig. 68), where the Hill-Shaped Control Bump

Fig. 64. Power spectral density of lift coefficient (Timme & Sartor [196]).

comparisons between the tests indicated the three-dimensional wing exhibits greater sensitivity to installation effects. The subsequent paper by Ito et al. [201], performing RANS computations on infinite-span threedimensional wings for various sweep angles, confirmed this sensitivity is a consequence of cross-flow effects, which become more pronounced as sweep increases.

7.3. Shock control bumps In a similar manner to vortex generators, contoured shock control bumps (SCBs) provide a passive means of shock control by modulating a flowfield near the shock-wave/boundary layer interaction region to reduce separation. The use of such devices for drag reduction is well documented [202,203], with extensive studies into the technology conducted under the EUROSHOCK and EUROSHOCK II projects [204,205].

Fig. 65. Three-dimensional SCB geometry (Eastwood & Jarrett [207]). 40

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Fig. 66. SCB geometries investigated by Bogdanski et al. (Bogdanski et al. [208]).

Fig. 67. Lift polar for the three SCBs, M ¼ 0:76 (Bogdanski et al. [208]). Fig. 68. Separated area over angle of attack for the three SCBs, M ¼ 0:76 (Bogdanski et al. [208]).

(HSCB) and wedge designs perform poorly at off-design conditions. Additionally, while the levels of separation increase with incidence, there is no direct correlation to the lift variation, suggesting the amount of 41

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these various simulation parameters. The work of Goncalves & Houdeville [37] made significant progress in mapping the combinations of parameters that successfully modelled the instability in a single test case. An extension of this work should be sought in future research efforts, comparing the effective combinations of simulation parameters between various codes. The growing applicability of hybrid RANS/LES methods to transonic buffet simulation has also been discussed. For capturing spectral data and the effects of secondary instabilities, such scale-resolving methods are the only way forward from a computational perspective. Continued developments in the efficiency of such approaches is a necessity. Nonetheless, the current requirements of high fidelity simulations to model the buffet instability pose difficulties for the integration of buffet considerations early in the aircraft design process. The modal nature of buffet suggested by global stability theory may remedy this issue. Modal-based reduced order models, derived from methods such as proper orthogonal decomposition or dynamic mode decomposition of the flowfield, could provide a computationally efficient means of integrating buffet constraints into aerofoil and wing optimisation. A number of comprehensive experimental investigations have been detailed. The extensive database provided by the experimental buffet campaign at ONERA, particularly for the OAT15A aerofoil, has been indispensable for the validation of numerical simulations. These experiments have also provided a wealth of data for the development of a promising theory of the mechanism governing buffet onset. To classify definitively the mechanism of buffet onset, comprehensive experimental studies must be performed on aerofoils of various geometry; encompassing thin sections, classical NACA profiles and the inclusion of flap and slat deflections. Such investigations will allow to research community to ascertain whether the mode of buffet exhibits any geometrical dependency or whether the phenomenon is intrinsic to all aerofoils. The understanding of dynamic interactions in the presence of shock buffet has progressed significantly in recent years. The exploration of the aerodynamic resonance corresponding to buffet onset and the associated lock-in phenomenon determined by Raveh [125] offer a potential mechanism for limit cycle oscillations in aircraft at transonic conditions. Additionally, research efforts conducted at Northwestern Polytechnical University indicate lock-in is not representative of a classical aerodynamic resonance, but rather, an unstable coupling between structural and fluid modes. Although the phenomenon has been explicitly observed by Hartmann et al. [131] experimentally, the majority of investigations in this field have been primarily concerned with two-dimensional geometries. The work by Iovnovich et al. [134] provides a stepping stone for future research efforts in looking to confirm whether lock-in is a realisable phenomenon in practical aircraft structures. Although costs will be high, experimental investigations will be imperative in this sense, supplemented by three-dimensional coupled Computational Fluid Dynamics/Computational Structural Mechanics simulation with highfidelity time-resolved or modal structural models. Research investigating the nature of three-dimensional shock buffet has indicated that the predominantly two-dimensional phenomenon observed on aerofoils and straight wings does not extend to swept-wing geometries. For planforms consistent with modern civil transport aircraft, unsteady shock motions are first apparent at outboard sections and progress inboard with increasing incidence. Frequency spectra of pressure fluctuations on three-dimensional wings indicate the presence of two broadband modes: the low-frequency shock oscillation that characterises buffet and a higher frequency bump representing a secondary Kelvin-Helmholtz type instability. The reduced buffet frequencies pertaining to these three-dimensional geometries are significantly higher than those for equivalent two-dimensional aerofoils. While the threedimensional buffet instability is yet to be completely understood, initial studies seem to indicate the presence of additional unstable aerodynamic modes, with the appearance of spanwise propagating pressure disturbances linked to the lateral convection of buffet cells. Exploration of the nature of autonomous shock oscillations on threedimensional geometries will be a significant topic in years to follow.

separated flow present is a poor indicator of the performance of a buffet suppression technology. A similar conclusion was reached in evaluating the influence of vortex strength on the buffet response. The three designs exhibit unique vortex structures, yet similar onset behaviour follows, again indicative of a poor buffet onset metric. The subsequent study by Bogdanski et al. [210] further assessed the viability of various metrics as buffet indicators, in addition to evaluating the performance of two innovative SCB geometries. A link is drawn between the flow effects incurred through SCBs and classical vortex generators, with SCBs generally able to produce half the streamwise vorticity as relative to mechanical vortex generators. Correlations are also developed between the levels of flow unsteadiness and the maximum lift coefficient, as well as a proposed buffet indicator: α

high BI ¼ ∫ αlow ⋅ClRMS dα

(15)

where αlow is the highest incidence where all bumps exhibit steady flow and αhigh is the lowest angle for which all bumps show lift oscillations. Minimum coefficients of determination for the correlation of the maximum lift and the buffet indicator to flow unsteadiness are 0.70 and 0.66, respectively, indicating both these quantities serve as fair metrics for evaluating the efficacy of a buffet control technology. The authors also investigated two forked bumps, and while the buffet response was marginally improved relative to previous designs, each of the SCBs exhibited degraded buffet behaviour relative to the baseline. 8. Concluding remarks The phenomenon of self-sustained shock oscillation in the transonic regime is one that inhibits the performance of modern aircraft. Oscillatory airloads degrade the handling characteristics of aircraft and the inherent structural vibrations associated with the unsteady flowfield have the potential to induce large amplitude limit cycle oscillations, degrading the structural integrity of a platform through cyclic loading. Although over six decades has passed since the transonic buffet instability was identified as a significant problem in aeronautics, a cohesive explanation of the underlying mechanisms governing this phenomenon and robust buffet alleviation technologies continue to elude the research community. This review has outlined the progress of recent research efforts in furthering the understanding of the mechanism underlying buffet onset. Acoustic wave-propagation feedback models have resulted in excellent predictions of the buffet frequency in certain cases, however, evidence to the contrary has also been presented by a number of authors. The transonic pre-stall instability model has linked the onset of buffet to geometric features of aerofoils, with an unstable shock-wave/separation bubble interaction driving the shock motion. A relationship between buffet onset and a globally unstable aerodynamic mode has been established, with support for this model from both experimental and numerical studies steadily accumulating. To ascertain which, if any, of the currently prominent physical models of shock buffet correctly identifies physics governing the instability, future researchers should look to a cohesive comparative study of predictions made by each of the models, developed under a consistent framework and validated against a comprehensive experimental database. The substantial body of research devoted to numerical simulation of the transonic shock oscillations illustrates that URANS methods are capable of capturing the low-frequency shock motions and intermittent boundary layer separation inherent to shock buffet. In many instances, the buffet frequency, pressure fluctuations and velocity distributions predicted through simulation correlate well with experiment. A caveat of these positive results is that the simulation of shock buffet through URANS becomes more so an art than a science. A demonstrable sensitivity to the selected turbulence model, numerical scheme and spatial and temporal discretisation exists in the literature. The successful reproduction of the transonic buffet phenomenon involves a complex interaction between 42

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While the effects of sweep angle on the flow topology have been characterised numerically, the influence of taper, twist and other geometric parameters remain unknown. For future research efforts to gain a comprehensive understanding of geometric effects, any numerical studies will need to be supplemented by experimental validation, for which, the NASA Common Research Model offers an attractive test case. Finally, control of shock oscillations for two-dimensional sections has been successfully achieved through closed-loop actuation of trailing edge deflectors. Complexities have been observed when attempting to extrapolate this technology to three-dimensional wings. These difficulties may inherently be a result of the lack of understanding of the three-dimensional buffet phenomenon and exploration of the physics underlying this instability will aid the development of more effective control laws. Mechanical and fluidic vortex generators have also been shown to be effective buffet suppression technologies. Parametric studies of the design variables have led to a thorough understanding of vortex generator configurations that are most effective at dispersing regions of separated flow. Additionally, continual development in pulsed fluidic vortex generators for buffet alleviation is expected. These configurations do not suffer the drag penalties incurred by mechanical vortex generators at cruise conditions and can be employed in closed-loop to yield more effective control strategies. Relative to trailing edge deflections and vortex generators, the suitability of shock control bumps to mitigate shock oscillations has not been demonstrated. For this technology to become fruitful, optimisation of the bump shapes and positions must explicitly consider a reliable indicator of buffet onset.

[17] Q. Xiao, H.M. Tsai, F. Liu, Numerical study of transonic buffet on a supercritical airfoil, AIAA J. 44 (3) (2006) 620–628. [18] S. Deck, Numerical simulation of transonic buffet over a supercritical airfoil, AIAA J. 43 (7) (2005) 1556–1566. [19] E. Garnier, S. Deck, Large-eddy simulation of transonic buffet over a supercritical airfoil, in: Direct and Large-Eddy Simulation VII, ERCOFTAC Series, vol. 13, Springer, Heidelberg, 2010, pp. 549–554. [20] L. Jacquin, P. Molton, S. Deck, B. Maury, D. Soulevant, Experimental study of shock oscillation over a transonic supercritical profile, AIAA J. 47 (9) (2009) 1985–1994. [21] J.D. Crouch, A. Garbaruk, D. Magidov, Predicting the onset of flow unsteadiness based on global instability, J. Comput. Phys. 224 (2) (2007) 924–940. [22] J.D. Crouch, A. Garbaruk, D. Magidov, L. Jacquin, Global structure of buffeting flow on transonic airfoils, in: IUTAM Symposium on Unsteady Separated Flows and Their Control, Springer, 2009, pp. 297–306. [23] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (2) (1981) 357–372. [24] B. Van Leer, Upwind-difference methods for aerodynamic problems governed by the Euler equations, Lect. Appl. Math. 22 (Part 2) (1985) 327–336. [25] R.B. Lehoucq, D.C. Sorensen, C. Yang, ARPACK Users' Guide: Solution of Largescale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, vol. 6, Siam, 1998. [26] J.B. McDevitt, A.F. Okuno, Static and Dynamic Pressure Measurements on a NACA 0012 Airfoil in the Ames High Reynolds Number Facility, NASA TP-2485, National Aeronautics and Space Administration, 1985. [27] A. Kuz’min, A. Shilkin, Transonic buffet over symmetric airfoils, in: Computational Fluid Dynamics 2006, Springer, 2009, pp. 849–854. [28] A. Kuz’min, Bifurcations and buffet of transonic flow past flattened surfaces, Comput. Fluids 38 (7) (2009) 1369–1374. [29] F. Sartor, C. Mettot, D. Sipp, Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile, AIAA J. 53 (7) (2014) 1980–1993. [30] M. Iovnovich, D.E. Raveh, Reynolds-averaged Navier-Stokes study of the shockbuffet instability mechanism, AIAA J. 50 (4) (2012) 880–890. [31] M. Adar, Y. Levy, A. Gany, Numerical simulation of flare safe separation, J. Aircr. 43 (4) (2006) 1129–1137. [32] P.R. Spalart, S.R. Allmaras, A one-equation turbulence model for aerodynamic flows, La Rech. Aerosp. 1 (5) (1994). [33] J.R. Edwards, S. Chandra, Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock-separated flowfields, AIAA J. 34 (4) (1996) 756–763. [34] L.L. Levy Jr., Experimental and computational steady and unsteady transonic flows about a thick airfoil, AIAA J. 16 (6) (1978) 564–572. [35] J.W. Edwards, J.L. Thomas, Computational methods for unsteady transonic flows, Unsteady Transonic Aerodyn. 120 (1989) 211–261. [36] C.M. Maksymiuk, T.H. Pulliam, Viscous transonic airfoil workshop results using ARC2D, in: Proceedings of the 25th AIAA Aerospace Sciences Meeting, Reno, NV, 1987. [37] E. Goncalves, R. Houdeville, Turbulence model and numerical scheme assessment for buffet computations, Int. J. Numer. Methods Fluids 46 (11) (2004) 1127–1152. [38] F. Grossi, M. Braza, Y. Hoarau, Prediction of transonic buffet by delayed detachededdy simulation, AIAA J. 52 (10) (2014) 2300–2312. [39] G. Barakos, D. Drikakis, Numerical simulation of transonic buffet flows using various turbulence closures, Int. J. Heat Fluid Flow 21 (5) (2000) 620–626. [40] G. Barakos, D. Drikakis, Implicit unfactored implementation of two-equation turbulence models in compressible Navier-Stokes methods, Int. J. Numer. Methods Fluids 28 (1) (1998) 73–94. [41] G. Barakos, D. Drikakis, An implicit unfactored method for unsteady turbulent compressible flows with moving boundaries, Comput. Fluids 28 (8) (1999) 899–922. [42] B.S. Baldwin, H. Lomax, Thin layer approximation and algebraic model for separated turbulent flows, in: Proceedings of the 16th AIAA Aerospace Sciences Meeting, Hunstville, AL, 1978. [43] B.E. Launder, B.I. Sharma, Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc, Lett. Heat Mass Transf. 1 (2) (1974) 131–137. [44] Y. Nagano, C. Kim, A two-equation model for heat transport in wall turbulent shear flows, J. Heat Transf. 110 (3) (1988) 583–589. [45] D. Sofialidis, P. Prinos, Development of a non-linear, strain-sensitive k-ω turbulence model, in: Proceedings of the 11th Symposium on Turbulent Shear Flows, Grenoble, France, 1997, p. 2, 89–94. [46] T.J. Craft, B.E. Launder, K. Suga, Development and application of a cubic eddyviscosity model of turbulence, Int. J. Heat Fluid Flow 17 (2) (1996) 108–115. [47] E. Goncalves, J.C. Robinet, R. Houdeville, Numerical simulation of transonic buffet over an airfoil, in: Proceedings of the 3rd International Symposium on Turbulence and Shear Flow Phenomena, Senda, Japan, 2003. [48] V. Couaillier, Numerical simulation of separated turbulent flows based on the solution of RANS/low Reynolds two-equation model, in: Proceedings of the 37th Aerospace Sciences Meeting and Exhibit, Reno, NV, 1999. [49] B. Smith, The k-kl turbulence model and wall layer model for compressible flows, in: Proceedings of the 21st Fluid Dynamics, Plasma Dynamics and Lasers Conference, Seattle, WA, 1990. [50] B. Smith, A near wall model for the k-l two equation turbulence model, in: Proceedings of the 24th AIAA Fluid Dynamics Conference, Colorado Springs, CO, 1994. [51] D.C. Wilcox, Reassessment of the scale-determining equation for advanced turbulence models, AIAA J. 26 (11) (1988) 1299–1310.

Acknowledgements This research is partially funded by the Defence Science and Technology Group (MyIP: 6786), Australia. References [1] W.F. Hilton, R.G. Fowler, Photographs of Shock Wave Movement, NPL R&M No. 2692, National Physical Laboratories, 1947. [2] D.G. Mabey, Oscillatory flows from shock induced separations on biconvex aerofoils of varying thickness in ventilated wind tunnels, in: AGARD CP-296, Boundary layer effects on unsteady airloads, Aix-en-Provence, France, 14–19 September, 1981, p. 11, 1–14. [3] J. Gibb, The Cause and Cure of Periodic Flows at Transonic Speed (Ph.D. thesis), Cranfield Institute of Tech., 1988. [4] D.G. Mabey, B.L. Welsh, B.E. Cripps, Periodic Flows on a Rigid 14% Thick Biconvex Wing at Transonic Speeds, RAE-TR-81059, British Royal Aircraft Establishment, 1981. [5] H.H. Pearcey, Some Effects of Shock-induced Separation of Turbulent Boundary Layers in Transonic Flow Past Aerofoils, ARC R&M-3108, Aeronautical Research Council, 1955. [6] H.H. Pearcey, A Method for the Prediction of the Onset of Buffeting and Other Separation Effects from Wind Tunnel Tests on Rigid Models, AGARD TR 223, National Physics Laboratory, 1958. [7] H.H. Pearcey, D.W. Holder, Simple Methods for the Prediction of Wing Buffeting Resulting from Bubble Type Separation, NPL AERO-REP-1024, National Physics Laboratory, 1962. [8] H.H. Pearcey, J. Osborne, A.B. Haines, The interaction between local effects at the shock and rear separation-a source of significant scale effects in wind-tunnel tests on aerofoils and wings, in: AGARD CP-35, Transonic Aerodynamics, Paris, France, 1968, p. 11, 1–23. [9] J. Nitzsche, A numerical study on aerodynamic resonance in transonic separated flow, in: Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Seattle, WA, 2009. [10] J.D. Crouch, A. Garbaruk, D. Magidov, A. Travin, Origin of transonic buffet on aerofoils, J. Fluid Mech. 628 (2009) 357–369. [11] H. Tijdeman, Investigations of the Transonic Flow Around Oscillating Airfoils (PhD Thesis), Delft University of Technology, TU Delft, 1977. [12] B.H.K. Lee, Self-sustained shock oscillations on airfoils at transonic speeds, Prog. Aerosp. Sci. 37 (2) (2001) 147–196. [13] B.H.K. Lee, Oscillatory shock motion caused by transonic shock boundary-layer interaction, AIAA J. 28 (5) (1990) 942–944. [14] S. Raghunathan, R.D. Mitchell, M.A. Gillan, Transonic shock oscillations on NACA0012 aerofoil, Shock Waves 8 (4) (1998) 191–202. [15] B.H.K. Lee, Investigation of flow separation on a supercritical airfoil, J. Aircr. 26 (11) (1989) 1032–1037. [16] A.L. Erickson, J.D. Stephenson, A Suggested Method of Analyzing Transonic Flutter of Control Surfaces Based on Available Experimental Evidence, NACA RM A7F30, National Advisory Committee for Aeronautics, 1947. 43

N.F. Giannelis et al.

Progress in Aerospace Sciences xxx (2017) 1–46 [85] G. Barbut, M. Braza, Y. Hoarau, G. Barakos, A. Sevrain, J.B. Vos, Prediction of transonic buffet around a wing with flap, in: Progress in Hybrid RANS-LES Modelling, Springer, 2010, pp. 191–204. [86] M. Braza, NACA0012 with aileron, in: P. Doerffer, C. Hirsch, J.P. Dussauge, H. Babinsky, G.N. Barakos (Eds.), Unsteady Effects of Shock Wave Induced Separation, Springer, 2010, pp. 101–131. [87] I. Mary, P. Sagaut, Large eddy simulation of flow around an airfoil near stall, AIAA J. 40 (6) (2002) 1139–1145. [88] M. Pechier, Previsions numeriques de l’effet magnus pour des configurations de munitions (Ph.D. thesis), 1999. [89] P. Spalart, W. Jou, M. Strelets, S. Allmaras, et al., Comments on the feasibility of les for wings, and on a hybrid rans/les approach, Adv. DNS/LES 1 (1997) 4–8. [90] B. Raverdy, I. Mary, P. Sagaut, N. Liamis, High-resolution large-eddy simulation of flow around low-pressure turbine blade, AIAA J. 41 (3) (2003) 390–397. [91] S.M. Kay, S.L. Marple, Spectrum analysis - a modern perspective, Proc. IEEE 69 (11) (1981) 1380–1419. [92] L. Larchev^eque, P. Sagaut, T.H. Le, P. Comte, Large-eddy simulation of a compressible flow in a three-dimensional open cavity at high Reynolds number, J. Fluid Mech. 516 (2004) 265–301. [93] F. Ducros, V. Ferrand, F. Nicoud, C. Weber, D. Darracq, C. Gacherieu, et al., Largeeddy simulation of the shock/turbulence interaction, J. Comput. Phys. 152 (2) (1999) 517–549. [94] E. Lenormand, P. Sagaut, L.T. Phuoc, P. Comte, Subgrid-scale models for largeeddy simulations of compressible wall bounded flows, AIAA J. 38 (8) (2000) 1340–1350. [95] J.B. Vos, A. Rizzi, A. Corjon, E. Chaput, E. Soinne, Recent advances in aerodynamics inside the NSMB (Navier-Stokes Multi-Block) consortium, in: Proceedings of the 36th AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, 1998. [96] F. Grossi, M. Braza, Y. Hoarau, Delayed Detached-Eddy Simulation of the transonic flow around a supercritical airfoil in the buffet regime, in: Progress in Hybrid RANS-LES Modelling, Springer, 2012, pp. 369–378. [97] F. Grossi, Physics and Modeling of Unsteady Shock Wave/boundary Layer Interactions over Transonic Airfoils by Numerical Simulation (Ph.D. thesis), National Polytechnic Institute of Toulouse, 2014. [98] B. Roidl, M. Meinke, W. Schr€ oder, Zonal RANS-LES computation of transonic airfoil flow, in: Proceedings of the 29th AIAA Applied Aerodynamics Conference, Honolulu, Hawaii, 2011. [99] J.L. Fulker, M.J. Simmons, An experimental investigation of passive shock/ boundary-layer control on an aerofoil, in: EUROSHOCK-Drag Reduction by Passive Shock Control, Springer, 1997, pp. 379–400. [100] M.S. Liou, C.J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107 (1) (1993) 23–39. [101] J.P. Boris, F.F. Grinstein, E.S. Oran, R.L. Kolbe, New insights into large-eddy simulation, Fluid Dyn. Res. 10 (4–6) (1992) 199–228. [102] Q. Zhang, W. Schr€ oder, M. Meinke, A zonal RANS-LES method to determine the flow over a high-lift configuration, Comput. Fluids 39 (7) (2010) 1241–1253. [103] J.B. McDevitt, L.L. Levy Jr., G.S. Deiwert, Transonic flow about a thick circular-arc airfoil, AIAA J. 14 (5) (1976) 606–613. [104] J.B. McDevitt, Supercritical Flow about a Thick Circular-arc Airfoil, NASA-TM78549, National Aeronautics and Space Administration, 1979. [105] K. Finke, Unsteady shock wave-boundary layer interaction on profiles in transonic flow, in: AGARD CP-183, Flow Separation, Gottingen, Germany, 27–30 May, 1975, p. 28, 1–11. [106] E. Stanewsky, D. Basler, Experimental investigation of buffet onset and penetration on a supercritical airfoil at transonic speeds, in: AGARD CP-483, Aircraft Dynamic Loads Due to Flow Separation, Sorrento, Italy, 1990, p. 4, 1–11. [107] F.W. Roos, Surface pressure and wake flow fluctuations in a supercritical airfoil flow field, in: Proceedings of the 13th AIAA Aerospace Sciences Meeting, Pasadena, CA, 1975. [108] F.W. Roos, Some features of the unsteady pressure field in transonic airfoil buffeting, J. Aircr. 17 (11) (1980) 781–788. [109] N. Hirose, H. Miwa, Computational and experimental research on buffet phenomena of transonic airfoils, in: Symposium Transsonicum III, Springer, 1989, pp. 489–499. [110] B.H.K. Lee, F.A. Ellis, J. Bureau, Investigation of the buffet characteristics of two supercritical airfoils, J. Aircr. 26 (8) (1989) 731–736. [111] J. Dandois, P. Molton, A. Lepage, A. Geeraert, V. Brunet, J.B. Dor, et al., Buffet characterization and control for turbulent wings, AerospaceLab 6 (2013) 1–17. [112] L. Jacquin, P. Molton, S. Deck, B. Maury, D. Soulevant, An experimental study of shock oscillation over a transonic supercritical profile, in: Proceedings of the 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Ont, 2005. [113] A. Hartmann, M. Klaas, W. Schr€ oder, Time-resolved stereo PIV measurements of shock-boundary layer interaction on a supercritical airfoil, Exp. Fluids 52 (3) (2012) 591–604. [114] A. Hartmann, A. Feldhusen, W. Schr€ oder, On the interaction of shock waves and sound waves in transonic buffet flow, Phys. Fluids (1994–present) 25 (2) (2013) 026101. [115] B.H.K. Lee, Transonic buffet on a supercritical aerofoil, Aeronaut. J. 94 (935) (1990) 143–152. [116] M.S. Howe, Trailing edge noise at low Mach numbers, J. Sound Vib. 225 (2) (1999) 211–238. [117] R. Ewert, W. Schr€ oder, On the simulation of trailing edge noise with a hybrid LES/ APE method, J. Sound Vib. 270 (3) (2004) 509–524.

[52] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (8) (1994) 1598–1605. [53] J.C. Kok, Resolving the dependence on freestream values for the k-turbulence model, AIAA J. 38 (7) (2000) 1292–1295. [54] W.P. Jones, B.E. Launder, The prediction of laminarization with a two-equation model of turbulence, Int. J. Heat Mass Transf. 15 (2) (1972) 301–314. [55] A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys. 71 (2) (1987) 231–303. [56] M. Thiery, E. Coustols, URANS computations of shock-induced oscillations over 2D rigid airfoils: influence of test section geometry, Flow, Turbul. Combust. 74 (4) (2005) 331–354. [57] L. Cambier, M. Gazaix, elsA: an efficient object-oriented solution to CFD complexity, in: Proceedings of the 40th AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, 2002. [58] A. Jameson, W. Schmidt, E. Turkel, Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes, in: Proceedings of the 14th AIAA Fluid and Plasma Dynamics Conference, Palo Alto, CA, 1981. [59] J. Cousteix, V. Saint-Martin, R. Messing, H. Bezard, B. Aupoix, Development of the k-ϕ turbulence model, in: 11th Symposium on Turbulent Shear Flows, Institut National Polytechnique, Universite Joseph Fourier, Grenoble, France, 1997. [60] S. Illi, T. Lutz, E. Kr€ amer, On the capability of unsteady RANS to predict transonic buffet, in: Proceedings of the Third Symposium Simulation of Wing and Nacelle Stall; Braunschweig, Germany, 2012. [61] D. Schwamborn, T. Gerhold, R. Heinrich, The DLR TAU-Code: recent applications in research and industry, in: Proceedings of the European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006; Egmond Aan Zee, The Netherlands, 2006. [62] T. Rung, U. Bunge, M. Schatz, F. Thiele, Restatement of the Spalart-Allmaras eddyviscosity model in strain-adaptive formulation, AIAA J. 41 (7) (2003) 1396–1399. [63] T. Rung, H. Lübcke, M. Franke, L. Xue, F. Thiele, S. Fu, Assessment of explicit algebraic stress models in transonic flows, Eng. Turbul. Model. Exp. 4 (1999) 659–668. [64] S. Jakirlic, K. Hanjalic, A new approach to modelling near-wall turbulence energy and stress dissipation, J. Fluid Mech. 459 (2002) 139–166. [65] A. Kourta, G. Petit, J.C. Courty, J.P. Rosenblum, Buffeting in transonic flow prediction using time-dependent turbulence model, Int. J. Numer. Methods Fluids 49 (2) (2005) 171–182. [66] A. Kourta, Computation of vortex shedding in solid rocket motors using timedependent turbulence model, J. Propuls. Power 15 (3) (1999) 390–400. [67] T.H. Shih, J. Zhu, J.L. Lumley, A new Reynolds stress algebraic equation model, Comput. Methods Appl. Mech. Eng. 125 (1–4) (1995) 287–302. [68] V. Brunet, Computational study of buffet phenomenon with unsteady RANS equations, in: Proceedings of the 21st AIAA Applied Aerodynamics Conference, Orlando, FL, 2003. [69] A. Soda, N. Verdon, Investigation of influence of different modelling parameters on calculation of transonic buffet phenomena, in: New Results in Numerical and Experimental Fluid Mechanics V, Springer, 2006, pp. 487–495. [70] M.E. Olsen, T.J. Coakley, The lag model, a turbulence model for non equilibrium flows, in: Proceedings of the 15th AIAA Computational Fluid Dynamics Conference, Anaheim, CA, 2001. [71] A.B.M.T. Hasan, M.M. Alam, RANS computation of transonic buffet over a supercritical airfoil, Procedia Eng. 56 (2013) 303–309. [72] A.A. Rokoni, A.B.M.T. Hasan, Prediction of transonic buffet onset for flow over a supercritical airfoil-a numerical investigation, J. Mech. Eng. 43 (1) (2013) 48–53. [73] R. Carrese, P. Marzocca, O. Levinski, N. Joseph, Investigation of supercritical airfoil dynamic response due to transonic buffet, in: Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, 2016. [74] N.F. Giannelis, G.A. Vio, Aeroelastic interactions of a supercritical aerofoil in the presence of transonic shock buffet, in: Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, 2016. [75] B.E. Launder, G. Reece Jr., W. Rodi, Progress in the development of a Reynoldsstress turbulence closure, J. Fluid Mech. 68 (03) (1975) 537–566. [76] B. Van Leer, Towards the ultimate conservative difference scheme. V. A secondorder sequel to Godunov's method, J. Comput. Phys. 32 (1) (1979) 101–136. [77] M.S. Liou, A sequel to AUSM: AUSMþ, J. Comput. Phys. 129 (2) (1996) 364–382. [78] J.T. Xiong, Y. Liu, F. Liu, S. Luo, Z. Zhao, X. Ren, et al., Multiple solutions and buffet of transonic flow over NACA0012 airfoil, in: 31st AIAA Applied Aerodynamics Conference, 2013. [79] V. Brunet, S. Deck, P. Molton, M. Thiery, A complete experimental and numerical study of the buffet phenomenon over the OAT15A airfoil, ONERA Tire Part 35 (2005) 1–9. [80] O. Rouzaud, S. Plot, V. Couaillier, Numerical simulation of buffeting over airfoil using dual time stepping method, in: Proceedings of the European Conference on Computational Fluid Dynamics ECCOMAS CFD 2000; Barcelona, Spain, 2000. [81] C.L. Rumsey, M.D. Sanetrik, R.T. Biedron, N.D. Melson, E.B. Parlette, Efficiency and accuracy of time-accurate turbulent Navier-Stokes computations, Comput. Fluids 25 (2) (1996) 217–236. [82] F. Furlano, E. Coustols, S. Plot, O. Rouzaud, Steady and unsteady computations of flows close to airfoil buffeting: validation of turbulence models, in: Proceedings of the Turbulence and Shear Flow Phenomena Conference, Stockholm, Sweden, 2001, pp. 211–216. [83] A.M. Vuillot, V. Couaillier, N. Liamis, 3D turbomachinery Euler and Navier-Stokes calculations with a multidomain cell centered approach, in: Proceedings of the 29th Joint Propulsion Conference and Exhibit, Monterey, CA, 1993. [84] A. Garbaruk, M. Shur, M. Strelets, P.R. Spalart, Numerical study of wind-tunnel walls effects on transonic airfoil flow, AIAA J. 41 (6) (2003) 1046–1054.

44

N.F. Giannelis et al.

Progress in Aerospace Sciences xxx (2017) 1–46 [150] B. Benoit, I. Legrain, Buffeting prediction for transport aircraft applications based on unsteady pressure measurements, in: Proceedings of the 5th AIAA Applied Aerodynamics Conference, Monterey, CA, 1987, pp. 225–235. [151] P.C. Steimle, D.C. Karhoff, W. Schr€ oder, Unsteady transonic flow over a transporttype swept wing, AIAA J. 50 (2) (2012) 399–415. [152] V. Brunet, S. Deck, Zonal-detached eddy simulation of transonic buffet on a civil aircraft type configuration, in: Advances in Hybrid RANS-LES Modelling, Springer, 2008, pp. 182–191. [153] F. Sartor, S. Timme, Reynolds-Averaged Navier-Stokes simulations of shock buffet on half wing-body configuration, in: Proceedings of the 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, 2015. [154] F. Sartor, S. Timme, Delayed detached-eddy simulation of shock buffet on half wing-body configuration, in: Proceedings of the 22nd AIAA Computational Fluid Dynamics Conference, Dallas, TX, 2015. [155] F. Sartor, S. Timme, Delayed detached-eddy simulation of shock buffet on half wing-body configuration, AIAA J. 55 (4) (2017) 1230–1240. [156] S.G. Lawson, D. Greenwell, M. Quinn, Characterisation of buffet on a civil aircraft wing, in: Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, 2016. [157] J. Peiro, J. Peraire, K. Morgan, O. Hassan, N. Birch, The numerical simulation of flow about installed aero engine nacelle using a finite element Euler solver on unstructured meshes, Aeronaut. J. 96 (956) (1992) 224–232. [158] ESDU, An Introduction to Aircraft Buffet and Buffeting, ESDU 87012, ESDU, July 1987. [159] L. Masini, S. Timme, A. Ciarella, A. Peace, Influence of vane vortex generators on transonic wing buffet: further analysis of the BUCOLIC experimental dataset, in: Proceedings of the 52nd 3AF International Conference on Applied Aerodynamics, Lyon, France, 2017. [160] J. Dandois, Experimental study of transonic buffet phenomenon on a 3D swept wing, Phys. Fluids 28 (1) (2016) 016101. [161] M. Iovnovich, D.E. Raveh, Numerical study of shock buffet on three-dimensional wings, AIAA J. 53 (2) (2014) 449–463. [162] J. Vassberg, M. Dehaan, M. Rivers, R. Wahls, Development of a common research model for applied CFD validation studies, in: Proceedings of the 26th AIAA Applied Aerodynamics Conference, Honolulu, Hawaii, 2008. [163] S. Koike, M. Ueno, K. Nakakita, A. Hashimoto, Unsteady pressure measurement of transonic buffet on NASA Common Research Model, in: Proceedings of the 34th AIAA Applied Aerodynamics Conference, Washington, D.C, 2016. [164] M. Ueno, M. Kohzai, S. Koga, H. Kato, K. Nakakita, N. Sudani, 80% scaled NASA Common Research Model wind tunnel test of JAXA at relatively low Reynolds number, in: Proceedings of the 51st AIAA Aerospace Sciences Meeting, Grapevine, TX, 2013. [165] Y. Sugioka, D. Numata, K. Asai, S. Koike, K. Nakakita, S. Koga, Unsteady PSP measurement of transonic buffet on a wing, in: Proceedings of the 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, 2015. [166] D. Caruana, A. Mignosi, M. Correge, A. Le Pourhiet, Buffeting active control in transonic flow, in: Proceedings of the 21st AIAA Applied Aerodynamics Conference, Orlando, FL, 2003. [167] F. Sartor, S. Timme, Mach number effects on buffeting flow on a half wing-body configuration, Int. J. Numer. Methods Heat Fluid Flow 26 (7) (2016) 2066–2080. [168] Y. Ohmichi, A. Hashimoto, Numerical investigation of transonic buffet on a threedimensional wing using incremental mode decomposition, in: Proceedings of the 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, 2017. [169] S. Timme, R. Thormann, Towards three-dimensional global stability analysis of transonic shock buffet, in: Proceedings of the AIAA Atmospheric Flight Mechanics Conference, Washington, D.C, 2016. [170] Y. Saad, M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (3) (1986) 856–869. [171] S. Xu, S. Timme, K.J. Badcock, Krylov subspace recycling for linearised aerodynamics analysis using DLR-TAU, in: Proceedings of the 16th International Forum on Aeroelasticity and Structural Dynamics, St Petersburg, Russia, 2015. [172] J. Xiong, F. Liu, Numerical simulation of transonic buffet on swept wing of supercritical airfoils, in: Proceedings of the 543rd AIAA Fluid Dynamics Conference, San Diego, CA, 2013. [173] S.A. Illi, C. Fingskes, T. Lutz, E. Kr€amer, Transonic tail buffet simulations for the common research model, in: Proceedings of the 31st AIAA Applied Aerodynamics Conference, San Diego, CA, 2013. [174] S.A. Illi, T. Lutz, E. Kr€amer, Transonic tail buffet simulations on the ATRA research aircraft, in: Computational Flight Testing, Springer, 2013, pp. 273–287. [175] A.F. Ribeiro, B. Konig, D. Singh, E. Fares, R. Zhang, P. Gopalakrishnan, et al., Buffet simulations with a Lattice-Boltzmann based transonic solver, in: Proceedings of the 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, 2017. [176] D.P. Lockard, L.S. Luo, S.D. Milder, B.A. Singer, Evaluation of PowerFLOW for aerodynamic applications, J. Stat. Phys. 107 (1–2) (2002) 423–478. [177] A. Hashimoto, K. Murakami, T. Aoyama, K. Ishiko, M. Hishida, M. Sakashita, et al., Toward the fastest unstructured CFD code ‘FaSTAR’, in: Proceedings of the 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Nashville, TN, 2012. [178] T. Ishida, A. Hashimoto, Y. Ohmichi, T. Aoyama, K. Takekawa, Transonic buffet simulation over NASA-CRM by Unsteady-FaSTAR code, in: Proceedings of the 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, 2017. [179] G.K. Kenway, J. Martins, Aerodynamic shape optimization of the CRM configuration including buffet-onset conditions, in: Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, 2016.

[118] A. Alshabu, H. Olivier, I. Klioutchnikov, Investigation of upstream moving pressure waves on a supercritical airfoil, Aerosp. Sci. Technol. 10 (6) (2006) 465–473. [119] A. Alshabu, H. Olivier, Unsteady wave phenomena on a supercritical airfoil, AIAA J. 46 (8) (2008) 2066–2073. [120] R.J. Zwaan, NLR 7301 Supercritical Airfoil Oscillatory Pitching and Oscillating Flap. Compendium of Unsteady Aerodynamics Measurements AGARD-R-702, Advisory Group for Aerospace Research and Development, 1982. [121] S.S. Davis, NACA 64A010 Oscillatory Pitching. Compendium of Unsteady Aerodynamics Measurements AGARD-R-702, Advisory Group for Aerospace Research and Development, 1982. [122] R.H. Landon, NACA 0012 Oscillatory and Transient Pitching. Compendium of Unsteady Aerodynamics Measurements AGARD-R-702, Advisory Group for Aerospace Research and Development, 1982. [123] S.S. Davis, G.N. Malcolm, Transonic shock-wave/boundary-layer interactions on an oscillating airfoil, AIAA J. 18 (11) (1980) 1306–1312. [124] C. Despere, D. Caruanal, A. Mignosi, O. Reberga, M. Correge, H. Gassot, et al., Buffet active control-experimental and numerical results, in: Proceedings of RTO AVT Symposium on Active Control Technology of Enhanced Performance Operational Capabilities of Military Aircraft, Land Vehicles, and Sea Vehicles; Braunschweig, Germany, 2001. [125] D.E. Raveh, Numerical study of an oscillating airfoil in transonic buffeting flows, AIAA J. 47 (3) (2009) 505–515. [126] D.C. Wilcox, Formulation of the kw turbulence model revisited, AIAA J. 46 (11) (2008) 2823–2838. [127] E.H. Dowell, K.C. Hall, J.P. Thomas, R.E. Kielb, M.A. Spiker, C.M. Denegri Jr., A new solution method for unsteady flows around oscillating bluff bodies, in: IUTAM Symposium on Fluid-structure Interaction in Ocean Engineering, Springer, 2008, pp. 37–44. [128] D.E. Raveh, E.H. Dowell, Frequency lock-in phenomenon for oscillating airfoils in buffeting flows, J. Fluids Struct. 27 (1) (2011) 89–104. [129] N.F. Giannelis, G.A. Vio, Investigation of frequency lock-in phenomena on a supercritical aerofoil in the presence of transonic shock oscillations, in: Proceedings of the 17th International Forum on Aeroelasticity and Structural Dynamics, Como, Italy, 2017. [130] M. Iovnovich, D.E. Raveh, Transonic unsteady aerodynamics in the vicinity of shock-buffet instability, J. Fluids Struct. 29 (2012) 131–142. [131] A. Hartmann, M. Klaas, W. Schr€ oder, Coupled airfoil heave/pitch oscillations at buffet flow, AIAA J. 51 (7) (2013) 1542–1552. [132] G. Dietz, G. Schewe, H. Mai, Amplification and amplitude limitation of heave/ pitch limit-cycle oscillations close to the transonic dip, J. Fluids Struct. 22 (4) (2006) 505–527. [133] D.E. Raveh, E.H. Dowell, Aeroelastic responses of elastically suspended airfoil systems in transonic buffeting flows, AIAA J. 52 (5) (2014) 926–934. [134] M. Iovnovich, D. Michaels, M. Adar, D.E. Raveh, Numerical study of the transonic limit cycle oscillation phenomenon on the F-16 fighter aircraft, in: Proceedings of the 16th International Forum on Aeroelasticity and Structural Dynamics, St Petersburg, Russia, 2015. [135] J. Quan, W. Zhang, C. Gao, Z. Ye, Characteristic analysis of lock-in for an elastically suspended airfoil in transonic buffet flow, Chin. J. Aeronaut. 29 (1) (2016) 129–143. [136] N.F. Giannelis, G.A. Vio, G. Dimitriadis, Dynamic interactions of a supercritical aerofoil in the presence of transonic shock buffet, in: Proceedings of the 27th International Conference on Noise and Vibration Engineering, Leuven, Belgium, 2016. [137] ANSYS, Fluent 12. 1 User Manual, ANSYS Inc, 2011. [138] N.F. Giannelis, J.A. Geoghegan, G.A. Vio, Gust response of a supercritical aerofoil in the vicinity of transonic shock buffet, in: Proceedings of the 20th Australasian Fluid Mechanics Conference, Perth, Western Australia, 2016. [139] E.H. Dowell, J. Edwards, T. Strganac, Nonlinear aeroelasticity, J. Aircr. 40 (5) (2003) 857–874. [140] M.J.d.C. Henshaw, K.J. Badcock, G.A. Vio, C.B. Allen, J. Chamberlain, I. Kaynes, et al., Non-linear aeroelastic prediction for aircraft applications, Prog. Aerosp. Sci. 43 (4) (2007) 65–137. [141] O.O. Bendiksen, Review of unsteady transonic aerodynamics: theory and applications, Prog. Aerosp. Sci. 47 (2) (2011) 135–167. [142] C. Gao, W. Zhang, Y. Liu, Z. Ye, Y. Jiang, Numerical study on the correlation of transonic single-degree-of-freedom flutter and buffet, Sci. China Phys. Mech. Astron. 58 (8) (2015) 1–12. [143] C. Gao, W. Zhang, Z. Ye, A new viewpoint on the mechanism of transonic singledegree-of-freedom flutter, Aerosp. Sci. Technol. 52 (2016) 144–156. [144] W. Zhang, C. Gao, Y. Liu, Z. Ye, Y. Jiang, The interaction between flutter and buffet in transonic flow, Nonlinear Dyn. 82 (4) (2015) 1851–1865. [145] J.A. Rivera, B.E. Dansberry, R.M. Bennett, M.H. Durham, W.A. Silva, NACA0012 Benchmark Model Experimental Flutter Results with Unsteady Pressure Distributions, National Aeronautics and Space Administration, 1992. [146] C. Gao, W. Zhang, X. Li, Y. Liu, J. Quan, Z. Ye, et al., Mechanism of frequency lockin in transonic buffeting flow, J. Fluid Mech. 818 (2017) 528–561. [147] N.C. Perkins, C.D. Mote, Comments on curve veering in eigenvalue problems, J. Sound Vib. 106 (3) (1986) 451–463. [148] P. Meliga, J.M. Chomaz, An asymptotic expansion for the vortex-induced vibrations of a circular cylinder, J. Fluid Mech. 671 (2011) 137–167. [149] F. Roos, The buffeting pressure field of a high-aspect-ratio swept wing, in: Proceedings of the 18th Fluid Dynamics and Plasmadynamics and Lasers Conference, Cincinatti, OH, 1985.

45

N.F. Giannelis et al.

Progress in Aerospace Sciences xxx (2017) 1–46

[180] B.H.K. Lee, Effects of trailing-edge flap on buffet characteristics of a supercritical airfoil, J. Aircr. 29 (1) (1992) 93–100. [181] B.H.K. Lee, F.C. Tang, Transonic buffet of a supercritical airfoil with trailing-edge flap, J. Aircr. 26 (5) (1989) 459–464. [182] E. Coustols, V. Brunet, R. Bur, D. Caruana, D. Sipp, BUFETN Co: a joint ONERA research project devoted to buffet control on a transonic 3D wing using a closedloop approach, in: Proceedings of the CEAS/KATNET II Conference; Bremen, Germany, 2009. [183] J.C. Le Balleur, P. Girodroux-Lavigne, A semi-implicit and unsteady numerical method of viscous-inviscid interaction for transonic separated flows, La Rech. Aerosp. 1984 (1) (1984) 15–37. [184] J.C. Le Balleur, P. Girodroux-Lavigne, A viscousinviscid interaction method for computing unsteady transonic separation, in: Numerical and Physical Aspects of Aerodynamic Flows III, Springer, 1986, pp. 252–271. [185] D. Caruana, A. Mignosi, C. Robitaillie, M. Correge, Separated flow and buffeting control, Flow, Turbul. Combust. 71 (1–4) (2003) 221–245. [186] D. Caruana, A. Mignosi, M. Correge, A. Le Pourhiet, A. Rodde, Buffet and buffeting control in transonic flow, Aerosp. Sci. Technol. 9 (7) (2005) 605–616. [187] C. Gao, W. Zhang, Z. Ye, Numerical study on closed-loop control of transonic buffet suppression by trailing edge flap, Comput. Fluids 132 (2016) 32–45. [188] J. Liu, Z. Yang, Numerical study on transonic shock oscillation suppression and buffet load alleviation for a supercritical airfoil using a microtab, Eng. Appl. Comput. Fluid Mech. 10 (1) (2016) 529–544. [189] R.E. Bartels, J.W. Edwards, Cryogenic Tunnel Pressure Measurements on a Supercritical Airfoil for Several Shock Buffet Conditions, NASA-TM-110272, National Aeronautics and Space Administration, 1997. [190] T. Barber, J. Mounts, D. McCormick, Boundary layer energization by means of optimized vortex generators, in: Proceedings of the 31st AIAA Aerospace Sciences Meeting, Reno, NV, 1993. [191] J. Lin, Control of turbulent boundary-layer separation using micro-vortex generators, in: Proceedings of the 30th AIAA Fluid Dynamics Conference, Norfolk, VA, 1999. [192] J. Dandois, V. Brunet, P. Molton, J.C. Abart, A. Lepage, Buffet control by means of mechanical and fluidic vortex generators, in: Proceedings of the 5th Flow Control Conference, Chicago, IL, 2010. [193] G. Godard, M. Stanislas, Control of a decelerating boundary layer. Part 1: optimization of passive vortex generators, Aerosp. Sci. Technol. 10 (3) (2006) 181–191. [194] P. Molton, J. Dandois, A. Lepage, V. Brunet, R. Bur, Control of buffet phenomenon on a transonic swept wing, AIAA J. 51 (4) (2013) 761–772. [195] J. Dandois, A. Lepage, J.B. Dor, P. Molton, F. Ternoy, A. Geeraert, et al., Open and closed-loop control of transonic buffet on 3D turbulent wings using fluidic devices, Comptes Rendus Mec. 342 (6) (2014) 425–436. [196] S. Timme, F. Sartor, Passive control of transonic buffet onset on a half wing-body configuration, in: Proceedings of the 16th International Forum on Aeroelasticity and Structural Dynamics, St Petersburg, Russia, 2015.

[197] T. Kouchi, S. Yamaguchi, S. Koike, T. Nakajima, M. Sato, H. Kanda, et al., Wavelet analysis of transonic buffet on a two-dimensional airfoil with vortex generators, Exp. Fluids 57 (11) (2016) 166. [198] T. Kouchi, S. Yamaguchi, S. Koike, S. Yanase, T. Nakajima, M. Sato, et al., Wavelet analysis of unsteady shock-wave motion on two-dimensional airfoil with vortex generators, in: Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, 2016. [199] T. Kouchi, C. Goyne, R. Rockwell, R. Reynolds, R. Krauss, J. McDaniel, FocusingSchlieren visualization in direct-connect dual-mode scramjet, in: Proceedings of the 18th AIAA/3AF International Space Planes and Hypersonic Systems and Technologies Conference, Tours, France, 2012. [200] S. Koike, K. Nakakita, T. Nakajima, S. Koga, M. Sato, H. Kanda, et al., Experimental investigation of vortex generator effect on two- and three-dimensional NASA common research models, in: Proceedings of the 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, 2015. [201] Y. Ito, K. Yamamoto, K. Kusunose, S. Koike, K. Nakakita, M. Murayama, et al., Effect of vortex generators on transonic swept wings, J. Aircr. 53 (6) (2016) 1890–1904. [202] J. Birkemeyer, H. Rosemann, E. Stanewsky, Shock control on a swept wing, Aerosp. Sci. Technol. 4 (3) (2000) 147–156. [203] B. K€ onig, M. P€atzold, T. Lutz, E. Kr€amer, H. Rosemann, K. Richter, et al., Numerical and experimental validation of three-dimensional shock control bumps, J. Aircr. 46 (2) (2009) 675–682. [204] E. Stanewsky, J. Delery, J. Fulker, W. Geißler, EUROSHOCK: Drag Reduction by Passive Shock Control; Results of the Project EUROSHOCK, Tech. Rep.; AER 2CT92-0049 supported by the European Union 1993–1995, Vieweg, Braunschweig, 1997. [205] E. Stanewsky, J. Delery, J. Fulker, P. de Matteis, Synopsis of the project EUROSHOCK II, in: Drag Reduction by Shock and Boundary Layer Control, Springer, 2002, pp. 1–124. [206] S. Raghunathan, J.M. Early, C. Tulita, E. Benard, J. Quest, Periodic transonic flow and control, Aeronaut. J. 112 (1127) (2008) 1–16. [207] J.P. Eastwood, J.P. Jarrett, Toward designing with three-dimensional bumps for lift/drag improvement and buffet alleviation, AIAA J. 50 (12) (2012) 2882–2898. [208] S. Bogdanski, K. Nübler, T. Lutz, E. Kr€amer, Numerical investigation of the influence of shock control bumps on the buffet characteristics of a transonic airfoil, in: New Results in Numerical and Experimental Fluid Mechanics IX, Springer, 2014, pp. 23–32. [209] T. Streit, K. Horstmann, G. Schrauf, S. Hein, U. Fey, Y. Egami, et al., Complementary numerical and experimental data analysis of the ETW Telfona Pathfinder wing transition tests, in: Proceedings of the 49th AIAA Aerospace Sciences Meeting, Orlando, FL, 2011. [210] S. Bogdanski, P. Gansel, T. Lutz, E. Kr€amer, Impact of 3D shock control bumps on transonic buffet, in: High Performance Computing in Science and Engineering, vol. 14, Springer, 2015, pp. 447–461.

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