12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

EFFECT OF BOUNDARY-LAYER TRIPPING ON TRANSONIC BUFFETFOR A LAMINAR-FLOW WING SECTION

Pradeep Moise
Markus Zauner
Aerodynamics and Flight Mechanics Group Aerodynamics and Flight Mechanics Group
University of Southampton
University of Southampton
Southampton, Hampshire, SO17 1BJ, UK
Southampton, Hampshire, SO17 1BJ, UK
pradeep890@gmail.com
m.zauner@soton.ac.uk

Neil D. Sandham
Aerodynamics and Flight Mechanics Group
University of Southampton
Southampton, Hampshire, SO17 1BJ, UK
n.sandham@soton.ac.uk

ABSTRACT

comfort, issues in aircraft control, structural fatigue and even
failure (Lee, 2001). Thus, transonic buffet is detrimental to aircraft performance and manoeuvrability and for these reasons,it has been extensively studied with a focus on understanding
and controlling its occurrence (Giannelis et al., 2017).

Transonic buffet refers to the self-sustained periodic oscillations that can occur in transonic flows over wings and isgenerally associated with strong shock wave/boundary layerinteractions. Since it can cause large variations in lift andlimit the flight envelope of aircraft, it has been studied extensively. Some studies have classified transonic buffet as eithera laminar or a turbulent buffet based on the boundary layer
characteristics at the foot of the shock wave and suggested
that the mechanisms underlying the two are different. In a recent study (Moise et al., 2021), we have shown by performing
wall-resolved Large-Eddy Simulations (LES) of laminar buffet that the features of this type resemble those reported for
turbulent buffet, albeit under different flow conditions. In thepresent study, we present LES of both types of buffet for similar flow conditions, allowing a direct comparison. Turbulent
buffet is achieved by forcing boundary layer transition wellupstream of the shock foot. These results are compared withthose for laminar buffet obtained at the same flow conditions(without enforcing transition). It is shown that unlike the freetransition case, where multiple shock waves exist, there is onlya single shock wave when the boundary layer is tripped. Furthermore, for the flow conditions studied, buffet amplitude issignificantly reduced for turbulent buffet. However, the buffet frequency remains approximately the same suggesting thatmechanisms for both types are similar. This is further corroborated by a spectral proper orthogonal decomposition whichshows that buffet and other dynamical characteristics for thetwo types are consistent.

Based on the BL characteristics at the foot of the shock
wave, transonic buffet has been classified as either laminar or
turbulent buffet (Brion et al., 2020). Turbulent buffet is characterised by the BL developing on the aerofoil’s suction surface
transitioning to turbulence well upstream of any shock waves
present in the flow field. It has been extensively studied andhas been shown to occur due to a global instability (Crouch
et al., 2007, 2019; Timme, 2020), but the physical mechanismsthat drive the instability remain unclear. By contrast, for laminar buffet (where the BL remains laminar from the leadingedge to the shock foot) it has been proposed in Dandois et al.
(2018) that the mechanism sustaining it is different and associated with a bubble-breathing phenomenon. However, in ourrecent study (Moise et al., 2021), we have shown using numerical simulations and a modal reconstruction that laminar
buffet has characteristics that are essentially similar to thosereported in previous literature for turbulent buffet. Furthermore, we have shown that bubble breathing occurs due to ahigh-frequency phenomenon related to vortex shedding similar to those reported in the resolvent analysis performed in(Sartor et al., 2015). We have shown that this is distinct from
buffet, but can accompany it.
Simulations in Moise et al. (2021) were carried out only
for laminar buffet. Here, we further explore the buffet types
by performing Large-Eddy Simulations of turbulent buffet forconditions similar to those at which laminar buffet is observed
by forcing transition of the BL to turbulence. To achieve transition, we use a trip that is located on the surface of the aerofoil
which is modelled by a synthetic, unsteady, wall-normal jetthat perturbs the BL. The simulations results are analysed using various tools including spectral proper orthogonal decomposition (Towne et al., 2018) and a flow reconstruction basedon the same so as to make direct comparisons between laminarand turbulent types of transonic buffet.

INTRODUCTION
Self-sustained periodic oscillations that occur in transonicflows over wings under certain flow conditions are associatedwith a phenomenon referred to as transonic buffet (Helmut,
1974). Transonic buffet, which will also be referred to here
as simply ‘buffet’, can lead to large variations in lift, possibly due to large-scale streamwise motion of shock waves andthe periodic separation and reattachment of the boundary layer(BL). This variation in lift, in turn, can lead to passenger dis-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
METHODOLOGY
Large-Eddy Simulations, spectral proper orthogonal decomposition and a modal reconstruction based on the latterare employed in this study. Unless otherwise mentioned, themethodology adopted here is the same as that detailed in Moiseet al. (2021). The major aspects of the latter study are highlighted here for completeness. Two types of simulations are
performed, the first being the free-transition case, where theBL transitions naturally (i.e., same as Moise et al., 2021) anda forced-transition case where a trip is introduced.
The in-house flow solver SBLI is used to perform the simulations on transonic buffet. It has been used in a numberof previous studies to examine laminar buffet (Zauner et al.,2019; Zauner & Sandham, 2020b,c; Moise et al., 2021). The
compressible Navier-Stokes equations are solved in a dimensionless form with the the required scales being the aerofoil
chord, the freestream density, velocity and temperature. For
spatial discretisation, a fourth-order finite difference scheme
is employed, while a third-order Runge-Kutta scheme is used
for time discretisation. A total variation diminishing scheme
is employed to capture shock wave features. A spectral-error
based implicit LES approach is adopted (Jacobs et al., 2018;
Zauner & Sandham, 2020a). This has been previously validated against Direct Numerical Simulations for flows where
buffet is observed (Zauner & Sandham, 2020c).
The aerofoil chosen is Dassault Aviation’s laminar, supercritical V2C profile with a blunt trailing edge of thickness
0.5% chord. Multi-block structured grids (C-H type) weregenerated for each incidence angle using an in-house, opensource code (Zauner & Sandham, 2018). The C-shaped blockis of radius 7.5 while the other blocks have a streamwise lengthof 4.5. The aerofoil is extruded in the spanwise direction witha uniform grid spacing of 5×10−4 (see grid-convergence studies) and 1 × 10−3 for the cases of tripped and free transition,
respectively. The span is chosen for both cases as 5% of theaerofoil chord based on previous studies (Zauner & Sandham,
2020c). The spanwise direction is denoted by z, while thestreamwise and the third orthogonal Cartesian direction are labelled as x and y, respectively. The curvilinear circumferential
and radial directions are ξ and η. The chordwise direction isreferred to as x′ .
Zonal characteristic boundary conditions (associated witha buffer zone) are applied at the outflow boundary, while integral characteristic boundary conditions are applied on otherouter boundaries (see Moise et al., 2021, figure 1a). The spanwise direction is chosen as periodic. The aerofoil is treated as
an isothermal and no-slip wall, except at the location of thetrip. Typical grid features in the vicinity of the aerofoil are
shown in Fig. 1 (only every 15th point shown for clarity). The
different colours used for grid lines indicate different blocks
while the green circle is centred about the tripping location.
The grid refinement downstream of the trip was determinedsuch that the turbulent BL features are appropriately captured
a posteriori, with the grid spacing on the aerofoil surface inwall units being δ ξ + ≈ 15, δ η + ≈ 1 and δ z+ ≈ 10. The dimensionless time step was chosen as δt = 3.2 × 10−5 .
Turbulent buffet can occur when the Reynolds number(based on aerofoil chord and freestream velocity) is sufficiently high and other conditions are conducive for naturaltransition to occur close to the leading edge (e.g. Lee, 1989)or when the BL is forced to transition to turbulence in experiments (e.g., using trips, Brion et al., 2020) or numerical studies(e.g. Xiao et al., 2006; Garbaruk et al., 2021). Here, due to themoderate values of Reynolds numbers studied, we adopt a synthetic jet which forces transition at a streamwise position, xt .

Figure 1. Grid features in the vicinity of the V2C aerofoil(every 15th point). The green circle shows tripping location.

Figure 2. Isosurfaces based on the Q−criterion at an arbitrary time instant for the case M = 0.735, forced transition.

The wall-normal momentum component of the jet is given byn(x − xt )2sin(ki2 z) sin(ωi t + φi )ρuη η=0 = ∑ A exp −
2σ 2
i=1
(1)
where ρ is the density and uη , the wall-normal velocity. The
variation in the streamwise direction is that of a Gaussianwith amplitude, A = 0.05, centred about xt = 0.2, the streamwise trip position on the aerofoil surface, with a standard deviation, σ = 0.00833. The variation in time and spanwise
directions are sinusoidal, with three modes superposed (i.e.,n = 3), with spanwise wavenumbers, k = {120π, 160π, 160π}
and angular frequency, ω = {26, 88, 200} and temporal phase
φ = {0, π, −π/2}. The spanwise wavenumbers and amplitude
were chosen based on results from the free-transition case suchthat the former is approximately 10 times the BL thickness at
xt and the latter is a third of the velocity at the edge of the BL,
while other parameters were found by trial-and-error.
The effect of forcing transition is shown in figure 2 using
isosurfaces of Qinc = ||Ω||22 − ||S||22 = 100. Here, Qinc is theincompressible version of the second invariant of the velocity gradient tensor, while S and Ω are the rotation and strainrate tensors, respectively, and ||.||2 denotes the second norm.
The isosurfaces essentially highlight the regions where rotation dominates over shear strain. This Q−criterion (Hunt et al.,1988) implies the presence of vortices and indicates that fluid
elements are locally revolving/swirling about some axis. The
presence of vortices of varying length scales formed downstream of the trip location (xt = 0.2) suggests that by x ≈ 0.3
the BL is fully turbulent. The isosurfaces in the figure arecoloured by the local axial velocity, and it can be inferred fromthe sudden reduction in velocity at x ≈ 0.5 that a shock wave
exists well-downstream of the transition location, as shown in
the following sections (e.g., see Fig. 5). This indicates that2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
turbulent buffet occurs for the forced-transition case.
The Reynolds number (based on chord and freestream
velocity) and the incidence angle are fixed at Re = 5 × 105
and α = 5◦ , respectively. The parameters that are varied are
the freestream Mach number, M, and the transition type, with
M = 0.7 and M = 0.735 examined for both free- and forcedtransition cases. Additionally, the fluid is assumed to be a perfect gas with a specific heat ratio, γ = 1.4 satisfying Fourier’s
law of heat conduction with Prandtl number, Pr = 0.72. It isalso assumed to be Newtonian and satisfying the Sutherland’slaw, with the Sutherland coefficient, CSuth = 110.4/268.67 ≈0.41.
Spatio-temporally coherent structures in the flow field canbe extracted using a spectral proper orthogonal decomposition
or SPOD (Lumley, 1970; Towne et al., 2018). Here, we use
the streaming algorithm and the numerical code provided in
Schmidt & Towne (2019) to perform SPOD. This approachsolves the eigenvalue problem

(a)
-1

-0.5

0

0.5

1
0.1

(b)

10

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-2

10 -3
10 -4

Z

S(x1 , x2 , St0 ) W(x2 )ψi (x2 , St0 )dx2 = λi ψi (x1 , St0 )

10 -5

(2)

Ω
10 -6

where ψi (x, St0 ) and λi are the i−th eigenfunction (or SPOD
mode) and eigenvalue, respectively, of the cross-spectral density tensor, S(x, x′ , St0 ), at a given Strouhal number (dimensionless frequency based on chord and freestream velocity),
St0 . Here, S was computed based on spatio-temporal data collected on a two-dimensional plane, z = 0 (with x and x′ being
any two points on the plane), while W is the weight matrix associated with the quadrature on the grid (a weighted 2-norm).
The eigenvalues are indexed such that λ1 > λ2 > λ3 ..., implying that most-energetic SPOD mode is ψ1 with higher indicesrepresenting modes of lower energies. The energy content ofmodes other than ψ1 were negligible (not shown) and thus,
these modes were not considered. The temporal data at various grid points was sampled at Fs = 12.5 (based on chord andfreestream velocity and about two orders of magnitude higher
than observed buffet frequencies) for a duration of 70 dimensionless time units (≈ 10 buffet cycles). These were then divided into blocks of total time, Tζ = 44 and S was computed
using the Welch’s method with 50% overlap.
The flow field data, q(x,t), was decomposed using SPOD
and buffet features were isolated by finding ψ1 (x, Stb ), where
Stb is the buffet frequency. Note that this SPOD mode is associated with the most dominant coherent motion present in
the flow field. The reconstructed flow field based on only the
buffet mode is then obtained by combining the spatio-temporalfeatures of this SPOD mode with the mean flow field, q (seeMoise et al., 2021, for further details) and is given byq̃(x,t) = q(x,t) + Renpoλi ψi (x, Stb ) exp (2πiStb t)

10 -7
10 -8

10 -1

10 0

10 1

Figure 3. Comparisons of (a) time- and span-averaged pressure coefficient variation on the aerofoil surface (dashedcurves: pressure surface) and (b) power spectral density of lift
fluctuations for various choices of spanwise grid spacing.

downstream and buffet amplitude (circles) reduced, but for theother two cases these features and buffet frequency match adequately. Thus, δ z = 5 × 104 was used in all other simulations.

RESULTS
The temporal variation of the span-averaged lift coefficient, CL , for time past transients (t > t0 ) for the free- andforced- transition cases are compared in figure 4a for different
M. The power spectral density (PSD) of the fluctuating component of CL is compared in figure 4b. Strong lift oscillations,characteristic of buffet, can be seen for the free-transition caseswith a peak observed in the PSD at Stb = 0.09 and 0.16 for
M = 0.7 and M = 0.735, respectively (highlighted by circles).
The oscillations are significantly subdued when transition isforced. At M = 0.7, no clear lift oscillations or a peak in the
PSD are discernible, while at M = 0.735, the PSD(CL′ ) associated with buffet drops by almost two orders of magnitude, with
Stb = 0.13. Thus, in the parameter range studied, it appears
that forcing BL transition leads to the suppression or reduction
in strength of buffet when compared to free transition. Thissuggests two possibilities: (i) buffet amplitude at any flow condition (i.e., M, α and Re) might be reduced or (ii) buffet onset
might be shifted to higher α for forced-transition cases. The
latter implies that at higher α, there might be relatively higher
buffet amplitudes for the forced-transition cases. This is notfurther explored here. Another important aspect seen here isthat the buffet frequency remains approximately the same forboth free- and forced-transition cases, which corroborates withthe proposition in Moise et al. (2021) that laminar and turbu-(3)

Grid-convergence studies
Studies examining the effect of the grid were performed
for the forced-transition case (free-transition cases have been
examined in Zauner et al., 2019). The grid spacing in different directions (x, y and z) was varied independently. It wasobserved that the solutions were most sensitive to the spatial
resolution in the z−direction. A comparison of the mean pressure coefficient for M = 0.735, Re = 5 × 105 and α = 5◦ when
the grid spacing in the spanwise direction is varied is shown in
Fig. 3. The solutions for the lowest resolution is found to beinadequate, with the mean shock wave position being further

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

(a) 1
0.9

0.8

0.7

0.6

0.5
0

(b)

10

20

30

40

50

60

70

10 0

10 -2

10 -4

10 -6

10 -8

10 -1

10 0

10 1

Figure 4. (a) Temporal variation of the lift coefficient, CL ,past transients and (b) the power spectral density of its fluctuating component for different M.

lent buffet have similar mechanisms. This should be contrasted
with the bubble-breathing phenomenon reported in Dandois
et al. (2018) which occurs at a frequency an order of magnitude higher (St ≈ 1). From Fig. 4a, it can also be inferredfrom the temporal variations that the time-averaged CL is approximately the same at a given M for both free- and forcedtransition cases. Another interesting feature is the presence ofbumps in the PSD at an intermediate frequency of St ≈ 0.5 anda high frequency St ≈ 1 for both cases at M = 0.735. The latter
will be shown to occur due to wake modes reported in Moiseet al. (2021).
Instantaneous contours of the streamwise density gradientare plotted in Fig. 5 for both free- and forced-transition casesfor M = 0.735 at approximate times when the lift achieves
a local maximum or minimum. Multiple shock waves, characteristic of laminar buffet at low Re (Ackeret et al., 1947;
Zauner et al., 2021), are observed for the free-transition case(a,b), while the envelope of the supersonic region delineatedby the sonic line (grey curve), varies significantly with time.
By contrast, only a single shock wave is present for the forcedtransition case (as is commonly reported in literature) and thestreamwise excursions of the shock are relatively low. Notethat the shock position, xs ≈ 0.5, which is well downstream of
xt implying turbulent buffet. Thus, buffet amplitude at moderate Re is shown to be sensitive to the BL characteristics upstream of the shock wave. These findings align with the global
linear stability analysis results in Garbaruk et al. (2021), wherea buffet mode becomes unstable at lower α when the BL istripped further downstream. This suggests that for a constantFigure 5. Density-gradient contours at high- (a,c) and lowlift (b,d) phases for the free- (a,b) and forced-transition (c,d)
cases for M = 0.735.

α, buffet intensity would increase with an increase in streamwise extent of the laminar BL. Comparing the wake of theaerofoil at the low-lift phase (b,d) with that at the high-lift
phase (a,c), we see that for both free and forced transition,
we have a stronger vortical organisation resembling the vonKármán vortex street.

SPECTRAL PROPER ORTHOGONAL DECOMPOSITION
The eigenvalue spectra based on the dominant eigenvalue,
λ1 , for both forced- and free-transition cases are shown in Fig.
6. The peaks at low-frequency (highlighted by circles) matchthe buffet frequency seen for the lift coefficient. Since the intermediate frequency occurs close to the harmonic of the buffetmode and since no clear peak is present in the SPOD spectra it4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Buffet mode
10 -1

Wake modes

10 -2

10 -1

Figure 8. Spatio-temporal contours of C̃ p for the cases of (a)free and (b) forced transition. The isolines for M̃loc = 1 (solid
black curve) and C̃ f = 0 (solid green curve) are also overlaid.

10 0

Figure 6. The eigenvalue spectra of the dominant eigenvalue
from SPOD for free and forced transition at M = 0.735.

value (red contours), implying that these two regions are outof phase in a buffet cycle. Similarly, the maximum reduction
occurs approximately at the mean shock wave position (black
curve). Other minor differences between the free- and forcedtransition cases can be accounted for by the presence of multiple shock waves and a larger buffet amplitude for the formercase. Note that multiple shock waves were observed for freetransition cases even in the absence of buffet (see M = 0.69
case in Moise et al., 2021) implying that these are features of
low-Reynolds number flows and are not related to buffet (Zauner et al., 2021). Also, the unstable mode associated withbuffet obtained through global linear stability analysis of solutions of steady Reynolds-Averaged Navier Stokes equationsin other studies have a striking resemblance to the buffet mode
for the forced-transition case seen here (e.g., compare Fig. 7
bottom left in present study with density contours shown in
Fig. 12, Sartor et al., 2015). Together, these results indicatethat the origins of laminar and turbulent buffet are the same.
To understand the associations between shock wave andBL separation in buffet, a flow reconstruction based only on
the buffet mode and the mean flow field (see Eq. 3) is examined. A spatio-temporal diagram showing contours of thepressure coefficient on the aerofoil surface, C̃ p , along with thecurves delineating separated-flow regions (i.e., skin-friction
coefficient, C̃ f = 0) and the sonic line (i.e., M̃loc = 1, based on
the local flow field at a distance 0.5 from the aerofoil surface)are compared in Fig. 8. Here, ‘∼’ denotes a reconstructed
quantity, while φ denotes the phase based on a single buffet
cycle, with φ = 0◦ and 180◦ denoting the low-lift and high-lift
phases, respectively. The variations of C̃ p are more gradualfor the free-transition case, which is expected due to the multiple unsteady shock waves present in the LES field. By contrast, the pressure jump is abrupt for the forced-transition case.
Also, for the former case, the upstream motion of the shock
wave is characterised by large flow separation (approximately,0◦ ≤ φ ≤ 150◦ ), while the flow remains fully attached when
the shock moves downstream. By contrast, for the latter case,the flow is mostly separated downstream of the shock. However, the motion of the separation point in different phases of
the cycle is similar for both cases and the differences seem tobe due to stronger shock motion for the free-transition case ascompared to the forced-transition case. This is especially seen
when considering the qualitative variation of the isoline C̃ f = 0(green curves).

Figure 7. Contours of density of the buffet (left) and wake(right) modes for the free-transition (top) and forced-transition(bottom) cases for M = 0.735. The sonic line based on timeaveraged flow is shown using the black curve.

is not studied here. Bumps in the spectra in the high-frequency
range of St ∼ O(1) (largest amplitude in the range highlighted
by a diamond) are discernible and are related to vortex shedding, as shown next.
The contours based on the real part of the density fieldof the buffet mode and the most-dominant wake mode obtained in SPOD are shown in Fig. 7 for the two cases. The
wake modes for both cases show a clear von-Kármán vortex
street pattern, with the vortical length scale being larger for
the free-transition case. The buffet modes for the two cases
also have a similar flow structure. For example, the contours
indicate that when a reduction in density from the mean value
occurs on most parts of the suction surface (blue regions), thedensity in the wake of the aerofoil increases above the mean

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
CONCLUSION

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D. J & Sandham, N. D 2018 An error indicator for finite
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Lee, B. H. K 1989 Investigation of flow separation on a supercritical airfoil. J. Aircr. 26 (11), 1032–1037.
Lee, B. H. K 2001 Self-sustained shock oscillations on airfoils
at transonic speeds. Prog. Aerosp. Sci. 37 (2), 147–196.
Lumley, J. L 1970 Stochastic tools in turbulence, 1st edn. Academic Press.
Moise, P, Zauner, M & Sandham, N 2021 Large eddy simulations and modal reconstruction of laminar transonic buffet. arXiv:2110.13237 [physics.flu-dyn], accepted, J. Fluid
Mech.
Sartor, F, Mettot, C & Sipp, D 2015 Stability, receptivity, andsensitivity analyses of buffeting transonic flow over a profile. AIAA J. 53 (7), 1980–1993.
Schmidt, O. T & Towne, A 2019 An efficient streaming algorithm for spectral proper orthogonal decomposition. Comput. Phys. Comm. 237, 98–109.
Timme, S 2020 Global instability of wing shock-buffet onset.
J. Fluid Mech. 885, 1–32.
Towne, A, Schmidt, O. T & Colonius, T 2018 Spectral proper
orthogonal decomposition and its relationship to dynamic
mode decomposition and resolvent analysis. J. Fluid Mech.
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Xiao, Q, Tsai, H. M & Liu, F 2006 Numerical study of transonic buffet on a supercritical airfoil. AIAA J. 44 (3), 620–
628.
Zauner, M, De Tullio, N & Sandham, N. D 2019 Direct numerical simulations of transonic flow around an airfoil at
moderate reynolds numbers. AIAA J. 57 (2), 597–607.
Zauner, M, Renou, A, Dandois, J, Brion, V, Moise, P &Sandham, N 2021 Numerical and experimental analysis oftransonic buffet for different airfoil geometries at moderate
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Wall-resolved Large-Eddy Simulations of laminar andturbulent transonic buffet are performed. The latter is achievedby forcing transition using a wall-normal jet on the aerofoil
surface well-upstream of the shock wave. Simulations forthis forced-transition case were performed for the same
flow conditions under which laminar buffet occurs whenboundary layer transition is instead allowed to occur naturally(free-transition case). The flow conditions chosen are a
Reynolds number (based on freestream velocity and aerofoil
chord) of Re = 500, 000, an incidence angle, α = 5◦ , for
two different freestream Mach numbers of M = 0.7 and0.735. It was observed that when transition is forced, buffet
is eliminated (M = 0.7) or reduced in amplitude (M = 0.735)
as compared to the corresponding free-transition case. Also,a single shock wave was present when the transition isforced, while multiple shock waves were observed for thefree-transition cases, which agrees with results from Ackeret
et al. (1947). However, the dynamics of the system seemto be similar for both free- and forced-transition cases. For
example, the frequency for the two are the same when buffet
occurs. Similarly, a spectral proper orthogonal decomposition
showed that the flow is dominated by modes at two different
frequencies. The lowest-frequency mode was identified as due
to buffet and comparisons between free- and forced-transition
cases showed that, when the variations in buffet amplitudeis accounted for, the features of the modes were essentiallysimilar. This study provides further evidence that laminar andturbulent buffet have the same origins. Importantly, this study
links the turbulent buffet features at high Reynolds numbers(Re > 106 ) reported in literature (e.g. Sartor et al., 2015)
and laminar buffet features reported at moderate Reynolds
numbers of Re ∼ O(105 ) (e.g., Zauner & Sandham, 2020c;
Moise et al., 2021).

Acknowledgement This work was carried out
through computational time provided by UK Turbulence Consortium through the EPSRC grant EP/R029326/1 and funding
from the EPSRC grant EP/R037167/1. We also acknowledge
the use of ARCHER2 (UK national supercomputing facility)
and the IRIDIS High Performance Computing Facility (University of Southampton), and associated support services, inthe completion of this study. The V2C airofoil geometry was
kindly provided by Dassault Aviation.

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