2021

[1]Christopher Julian Schauerte

[1]Institute of Aerodynamics, RWTH Aachen University, Wüllnerstraße 5a,
Aachen, 52062, NRW, Germany

# Experimental investigation on the turbulent wake flow in fully-established
transonic buffet conditions

<EMAIL_ADDRESS>Anne-Marie Schreyer<EMAIL_ADDRESS>aachen.de *

###### Abstract

The transonic flight regime is often dominated by transonic buffet, a highly
unsteady and complex shock-wave/boundary-layer interaction involving major
parts of the flow field. The phenomenon is associated with a large-amplitude
periodic motion of the compression shock coupled with large-scale flow
separation and intermittent re-attachment. Due to the resulting large-scale
variation of the global flow topology, also the turbulent wake of the airfoil
or wing is severely affected, and so are any aerodynamic devices downstream on
which the wake impinges. To analyze and understand the turbulent structures
and dynamics of the wake, we performed a comprehensive experimental study of
the near wake of the supercritical OAT15A airfoil in transonic buffet
conditions at a chord Reynolds number of $2\text{\times}{10}^{6}$. Velocity
field measurements reveal severe global influences of the buffet mode on both
the surface-bound flow field on the suction side of the airfoil and the wake.
The flow is intermittently strongly separated, with a significant momentum
deficit that extends far into the wake. The buffet motion induces severe
disturbances and variations of the turbulent flow, as shown on the basis of
phase-averaged turbulent quantities in terms of Reynolds shear stress and RMS-
values. The spectral nature of downstream-convecting fluctuations and
turbulent structures are analyzed using high-speed focusing schlieren
sequences. Analyses of the power spectral density pertaining to the vortex
shedding in the direct vicinity of the trailing edge indicate dominant
frequencies one order of magnitude higher than those associated with shock
buffet ($St_{c}=\mathcal{O}({1})$) vs. $St_{c}=\mathcal{O}({0.1})$). It is
shown that the flapping motion of the shear layer is accompanied by the
formation of a von Kármán-type vortex street of fluctuating strength. These
wake structures and dynamics will impact any downstream aerodynamic devices
affected by the wake. Our study therefore allows conclusions regarding the
incoming flow of devices such as the tail plane.

###### keywords:

Transonic buffet, Airfoil, Supercritical, Shock wave, SWBLI,Turbulent flow,
Wake,Vortex shedding

##  1 Introduction

The transonic flight regime is often characterized by the occurrence of local
supersonic regions on the suction side of airfoils and wings, which are
terminated by a shock wave (McDevitt1985, ). For sufficiently strong
interactions, shock-induced flow separation may span the entire region from
the shock to the trailing edge (Molton2013, ). For certain combinations of
Mach number, Reynolds number, and angle of attack, the transonic flow about
the airfoil becomes unsteady, and the compression shock and separated boundary
layer carry out periodic oscillations (McDevitt1985, ). The resulting complex
shock-wave/boundary-layer interaction involves major parts of the flow field
(Jacquin2009, ) and is typically referred to as transonic buffet. The self-
sustained dynamic back-and-forth motion of the shock wave is accompanied by
strong variations of the shock strength, thus inciting a periodic thickening
and shrinking of the separated shear layer Iovnovich_Raveh2012 . This self-
excited periodic phenomenon is, inter alia, associated with a highly unsteady
flow field (Molton2013, ), which results in low-frequency fluctuation of the
integral aerodynamic quantities, i.e. the lift and drag coefficients
(Crouch2008, ; Benoit_Legrain1987, ).

For specific frequencies of this periodic change between a severely separated
boundary layer and periods of mostly reattached flow, the aerodynamic
phenomenon and the aircraft structure may interact. This mechanical coupling
with the airframe can lead to a resonance of the wing (the so-called
buffeting), inciting a potentially deleterious effect on the structural
integrity (Benoit_Legrain1987, ; Crouch2008, ).

As the buffet onset constrains the usable range of the flight envelope
(Benoit_Legrain1987, ; Molton2013, ), its understanding is one of the major
challenges in the design of modern civil aircraft (Crouch2019, ; Jacquin2016,
). The number of applications of supercritical airfoils increases with the
increasing demand for fast air travel. Therefore, the overall efficiency of
modern civil aircraft becomes more and more relevant and the significance of a
thorough understanding of the buffet phenomenon increases, as it is a
prerequisite to identify and potentially shift the limits of the flight
envelope.

Several possible explanations for the underlying mechanisms of this
oscillatory shock motion have been suggested. The well-established model
proposed by Lee1990  and refined by Hartmann2013  describes this motion as a
feedback loop. According to the model, two sequential phases of downstream and
upstream propagation of disturbances form the recurring buffet cycle. The
shock motion initiates pressure waves that subsequently propagate downstream
to the trailing edge. The interaction with the latter emits disturbances that
travel upstream along the airfoil surface. These disturbances provide the
energy required to maintain the back and forth motion of the shock wave and
thus close the feedback loop. Crouch et al. Crouch2008  linked the buffet
onset with a global instability of the flow, and thus contributed
significantly to the perception of the phenomenon and enhanced the prediction
of the buffet onset.

The majority of studies concerned with transonic buffet aim at contributing to
a profound understanding of the underlying mechanisms to mitigate buffet and
finally prevent the structural response. Substantially less attention has been
dedicated to the evolution of the buffet-induced turbulent wake. Regarding the
practical implementation in civil aircraft, however, also the influence of the
wake on aerodynamic devices downstream of the wing, such as tail plane control
surfaces, is crucial, as it may affect their performance.

The formation of wakes downstream of airfoils has been studied for either low-
Reynolds number flows Gunasekaran2018 , non-buffet conditions Epstein1988 ;
AlshabuOlivier2008 , or mostly laminar flow configurations Zauner2020 . Only
Szubert et al. Szubert2015  addressed fully-established shock buffet in
turbulent flow at high-Reynolds number and replicated the conditions
identified relevant for pronounced buffet by Jacquin et al. Jacquin2009  in
terms of Mach number, angle of attack, and chord-based Reynolds number.

The suction-side-bound oscillatory flow mode of the airfoil and the complex
dynamics inherent to the turbulent wake appear to be coupled Szubert2015 ;
Zauner2020 . A similarly complex interplay between the two domains was also
observed for other transonic flows that comprise pronounced vortex streets,
e.g. double-wedge profiles with strong separation Franke1989 .

For flow configurations with large wake fluctuations in the context of
unsteady shock motion, Lee et al. Lee1994  studied the formation of upstream
traveling waves. These waves were advanced as an essential element to predict
the periodic shock oscillation Lee1990 . Tijdeman Tijdeman1977  interpreted
these waves as an immediate consequence to changes in the flow conditions at
the trailing edge of the airfoil and denoted them as Kutta waves.

For laminar buffet conditions, Zauner and Sandham Zauner2020  identified a
complex interplay between Kelvin-Helmholtz instabilities in the surface-bound
flow towards the rear part of the airfoil, the formation of a von Karman
vortex street aft of the trailing edge, and subsequently shed pressure waves
traveling upstream towards to the shock wave. Based on a modal analysis of an
established buffet flow using proper orthogonal decomposition, Szubert et al.
Szubert2015  showed an interaction between the vortex shedding mode and the
buffet mode. Events of massive flow separation – not necessarily driven by
developed buffet – may thus give rise to von Kármán-like vortex shedding in
the vicinity of the trailing edge and beyond in the near wake Szubert2015 .
Pressure sensors placed along the airfoil upper surface and in the near wake
captured strong secondary fluctuations (in addition to the primary buffet
oscillation) over a broad range of chordwise stations. These fluctuations were
attributed to a von Kármán mode and persisted throughout the entire near-wake
domain downstream of the trailing edge Szubert2015 . Power spectra of the
wall-pressure signals showed the characteristic low-frequency ($St_{c}=0.075$)
footprint of shock buffet.

For a more global, integral perspective of an entire aircraft, it is necessary
to also quantify the dynamics that the buffet phenomenon imposes on the
aircraft structure and control surfaces via the wake of the wing or airfoil.
The quantitative importance of the wake unsteadiness to the aerodynamic
performance of a civil transonic aircraft has yet to be determined. As a first
step, the present work assesses the region around the airfoil rear section and
the near wake in detail for fully-established buffet conditions, since fully-
established buffet represents the most severe dynamic conditions.

We characterize the temporal and spatial structure of the separated boundary
layer and near wake with particular focus on the variation over the buffet
cycle. A careful assessment of the wake structure and involved dynamics is the
essential groundwork for understanding the complex interaction of a fully-
established buffet configuration with a tail plane. To the authors’ knowledge,
this is the first experimental investigation to focus explicitly to the
evolution of the buffet-dominated periodic fluctuation of the near wake field
by means of optical flow measurement techniques.

In Sec. 2, we introduce the experimental methods and setup, followed by a
detailed discussion of the turbulent flow field in Sec. 3. We present our
conclusions in Sec. 4.

##  2 Experimental methods and setup

We briefly introduce the experimental facility and investigated airfoil
models, before presenting an overview of the applied measurements arrangement.
The section is complemented with fundamental considerations of vector field
post-processing that form the basis for subsequent sections.

###  2.1 Wind-tunnel facility and model

Our experiments were conducted in the closed-circuit Trisonic Wind Tunnel at
the Institute of Aerodynamics, RWTH Aachen University. The Trisonic Wind
Tunnel is an intermittently-operated vacuum-type indraft facility. Based on
interchangeable test sections and adjustable nozzle geometries, it can provide
Mach numbers between $0.3<\mathrm{M}_{\infty}<4.0$. Continuously run screw-
type compressors evacuate a vacuum chamber with a total volume of $380\,m^{3}$
downstream of the test section and transport the air through a dryer bed
before it enters a large $180\,m^{3}$ air reservoir at ambient conditions.
Turnaround times of approximately 7 minutes are achieved. The relative
humidity of the air is kept below 4 % to preclude condensation effects and
falsifications of the measured shock location (Binion1988, ). A measurement
cycle is initialized by the actuation of a fast-acting valve that allows the
high and low pressure sections to retrieve equilibrium conditions, thus
drafting the measurement medium from the balloon through the test section into
the vacuum tanks. In each measurement cycle, stable flow conditions are
obtained for 2 to 3 seconds, depending on the configured Mach number. The
obtainable Reynolds number is a function of the selected Mach number and the
ambient conditions of the resting, pre-dried air in the reservoir. The
turbulence intensity based on the freestream velocity fluctuation of the wind
tunnel is below 1 %.

For the present study, measurements were conducted at
$\mathrm{M}_{\infty}=0.72$, which is equivalent to a free-stream velocity of
$u_{\infty}=235$ $\mathrm{m}\mathrm{/}\mathrm{s}$. For the investigated
airfoil, a chord-based Reynolds number of $Re_{c}=2\cdot 10^{6}$ is obtained.
These conditions require the transonic operation mode. The transonic test
section features a rectangular cross section of 400 mm x 400 mm and a total
length of 1,410 mm.

Figure 1: Illustration of the transonic test section with adjustable walls

It is equipped with flexible upper and lower walls to compensate the influence
of solid walls confining the flow and to emulate conditions in terms of
streamline contouring observed in the equivalent free, unconfined flow around
the same airfoil geometry (McDevitt1982, ).

We studied buffet on the supercritical OAT15A airfoil. The two-dimensional
wind tunnel model has a relative thickness of 12.3 % and features a total span
of 399 mm, which is equivalent to the width of the test section. A chord
length of $c=150\,\text{mm}$ was chosen, giving an aspect ratio of approx.
2.66. The upper and lower half of the model were manufactured from stainless
steel, along with a trailing edge made of metal particle-filled epoxy resin.
The center of rotation for the angle of attack is located at about mid-chord
of the airfoil.

Fig. 1 shows a schematic representation of the airfoil installed in the
optically accessible test section.

###  2.2 Measurement arrangement

Flow visualizations are obtained with a focusing schlieren setup, where
extended grids illuminated by correspondingly large illuminated surfaces
replace the point-shaped light sources used in classical schlieren
arrangements Schardin1942 ; Settles2001 . A focusing schlieren system inspired
by Weinstein’s approach Weinstein1993  has been developed by the authors
(Schauerte_Schreyer2018, ) to explore flow configurations in the transonic and
supersonic flow regime.

This system overcomes some of the limitations of classical schlieren
configurations, in particular the close to infinite depth of focus that
results in strong line-of-sight integration. We require the ability to focus
on narrow slices of the flow to be able to analyze our highly unsteady and
partially three-dimensional flow of interest.

The flow was illuminated by a 3x3 array of 15W high-power LEDs together with a
constant-current source with an electric output adjustable between 50 W and
120 W. The source grid consists of CNC-machined alternating clear and opaque
lines, the cutoff grid is an exact photographic negative of the source grid. A
large 470 mm Fresnel lens, placed slightly ahead of the source grid, captures
major portions of the light intensity, illuminates the source grid evenly
bright by reshaping the light cone, and redirects the light beam through the
test section past the model and onto the schlieren lens. The illuminated
measurement area around the spanwise centerplane of the airfoil has a diameter
of 280 mm. However, the outer edges are subject to vignetting due to the light
beam passing through two consecutive circular windows. An effective field of
view (FOV) without vignetting of $d_{FS}\approx 220\textrm{mm}$ was achieved,
using a magnification $M_{FS}$ of 0.2 between the two grids in conjunction
with the relay optics and a 35 mm camera lens.

To temporally resolve the oscillatory motion of the shock, the schlieren
visualizations were recorded with a Photron SA-5 CMOS high-speed camera at a
frame rate of 20 kHz and with a short exposure of
$1.9\text{\,}\mathrm{\SIUnitSymbolMicro s}$. For the high-speed recording, a
FOV of $180\text{\,}\mathrm{mm}$ width and $130\text{\,}\mathrm{mm}$ height is
projected on the camera sensor, yielding an effective resolution of 4.05
px/mm.

The focusing schlieren setup was operated with a horizontally aligned cutoff
grid to capture vertical density gradients. From these measurements, we
extracted the temporal development of the SWBLI and the near wake, the shock
position and amplitude, as well as the shock oscillation frequencies.

To acquire detailed velocity data, we applied particle image velocimetry
(PIV). Planar measurements are performed in a streamwise/wall-normal plane
along the centerline of the airfoil model.

Figure 2: Overview of the measurement location captured by the PIV setup

A Litron Nano PIV double pulsed Nd:YLF laser, operating at a wavelength of 527
nm, produces a light sheet of 1.5 mm thickness. The large laser-pulse energy
of 200 mJ/pulse with a pulse duration of 4 ns ensures a high signal-to-noise
ratio. The flow is seeded with di-ethyl-hexyl-sebacate (DEHS) tracer droplets
with a mean diameter of 1 $\mu m$. Particle images are captured using two
identical FlowSense EO 11M CCD cameras with a resolution of 11 megapixels,
which corresponds to a sensor resolution of 4,008 pixels x 2,672 pixels at a
pixel size of $9\,\mu m$. Both cameras are focused on the same field of view
(FoV) to increase the acquisition rate to 10 Hz. The FoV covers both the
vicinity of the airfoil trailing edge and the near wake. It is
$145\text{\,}\mathrm{mm}$ wide and $80\text{\,}\mathrm{mm}$ high, thus
spanning a streamwise domain from 0.6 c to 1.7 c. Based on the magnification
of 4.04, a spatial resolution of
$27.56\text{\,}\mathrm{p}\mathrm{x}\mathrm{/}\mathrm{m}\mathrm{m}$ is
obtained. With a final interrogation-window size of 16 px x 16 px at an
overlap of 50 %, the resulting velocity maps feature a vector spacing of 0.29
mm. Relevant parameters of the focusing schlieren and PIV tests are summarized
in table 1.

Table 1: Overview of experimental PIV and FS parameters. Particle image
velocimetry setup Focusing Schlieren setup
---
Sensor resolution | 4008 x 2672 | px x px |  | Sensor resolution | 704 x 520 | px x px
FOV width | 150 | mm |  | FOV width | 180 | mm
FOV height | 80 | mm |  | FOV height | 150 | mm
Acquisition rate | 10 | Hz |  | Acquisition rate | 20,000 | Hz
Exposure | 11 | $\text{\,}\mathrm{\SIUnitSymbolMicro s}$ |  | Shutter speed | 1/525,000 | s
Spatial resolution | 27.56 | px/mm |  | Spatial resolution | 4.05 | px/mm

###  2.3 Identification of buffet phases

Preceding analyses of the buffet flow around the OAT15A airfoil (see
Schauerte_Schreyer2022a ) with a field of view covering the entire suction
side from 0.1 c to 1.1 c. revealed large-scale global modifications of the
flow topology throughout the buffet cycle (see also Fig. 3).

Since the shock motion covers almost $20\,\%$ of the chord and is coupled with
intermittent flow separation and reattachment, statistics for discrete phases
of the buffet cycle are more meaningful than a global mean. The contour maps
in Fig. 3 show the shock’s most upstream location (a), an intermittent chord
wise postion during downstream (b), and upstream (d) motion, as well as the
shock most downstream location (c).

Figure 3: Phase-averaged streamwise velocity maps and profiles for
characteristic phases of the buffet cycle.

Individual buffet phases can be unambiguously assessed based on the
instantaneous shock location, the shape of the shock front, and the shock
inclination near the airfoil contour for each uncorrelated snapshot
Schauerte_Schreyer2022a . Based on a sufficiently large overlap between the
fields of view in the present and the earlier Schauerte_Schreyer2022a
measurement campaign, the latter quantities were applied to the rear portion
of the airfoil and the near wake, yet excluding the actual shock wave. The
procedure and evaluated parameters are clarified in Fig. 4.

First, the integral velocity magnitude within the rectangular analysis window
in Fig. 4 (4) is evaluated in terms of the percentage of vectors that exceed
or stay below a certain threshold ($u\leq 0.25\,u_{\infty}$). Threshold values
were previously identified to uniquely describe the chordwise location of the
shock and thus allow the definition of characteristic phases within the buffet
cycle. This procedure allows to pre-assign the snapshots to the respective
buffet phases. This criterion was complemented by considering the height of
the low-momentum bubble (see Fig. 4 (5)). However, due to similar velocity
defects for phases of final upstream motion and incipient downstream motion,
some ambiguity remains in this first classification.

Figure 4: Phase assignment for instantaneous velocity fields in the wake

Second, the streamwise velocity magnitude at three chordwise stations
(location (1), (2), and (3); Fig. 4) were compared with reference values from
the predefined phase assignment (see Fig. 3) for each snapshot.

An analysis of the thus finally obtained phase assignment in the field of view
covering the entire airfoil suction side (studied in Schauerte_Schreyer2022a )
revealed that each buffet phase induces characteristic velocity defects. These
deficits are modulated by the direction of motion and strength of the shock
wave. This third criterion is independent of the velocity deficit in the
strongly separated region and can eliminate the last ambiguity in the phase
assignment.

Finally, as a consistency check, the velocity profiles at the chordwise
stations between $x/c=0.8$ to $x/c=1.1$ of the present case and the case
studied in Schauerte_Schreyer2022a  were examined for congruence. The results
showed excellent agreement (less than $0.6\%$ deviation), which proofs that we
can successfully identify the respective buffet phase of a snapshot without
directly knowing the shock position.

##  3 Results and discussion

Since the flow varies considerably throughout a buffet cycle, and these
variations have a strong impact on the wake of the airfoil, we begin this
section with a short discussion of the flow topology on the airfoil suction
side for fully-estasblished buffet conditions. We then briefly discuss the
rear section of the airfoil and near-wake region ($0.6\leq x/c\leq 1.7$) on
the basis of ensemble-averaged velocity maps. Subsequently, we discuss the
wake structure in detail, using phase-distinguished representations of
instantaneous and averaged velocity fields in the streamwise and vertical
directions. We also provide a discussion of turbulent quantities and the
vorticity, as well as a spectral analysis of the vortex shedding at the
trailing edge.

###  3.1 Shock-buffet induced variation of the flow topology

In our previous study Schauerte_Schreyer2022a ; Schauerte_Schreyer2022b , we
acquired high-speed focusing schlieren sequences capturing the temporal
evolution of the flow topology, in particular the low-frequency shock
oscillation. An overview of selected phases of the buffet cycle is given in
Fig. 5, which emphasizes the huge periodic variation of the flow, synchronized
with a back and forth motion of the shock wave that covers approx. $20\,\%$ of
the airfoil chord. The images indicate a strongly turbulent wake with numerous
turbulent eddies of varying size.

Figure 5: Focusing schlieren snapshots capturing the global variation of the
flow topology over the full buffet cycle.

We identified eight characteristic phases throughout a buffet cycle, primarily
determined by the respective shock location and direction of shock motion
(either upstream or downstream). Phase 1 (Fig. 5 (a)) corresponds to the most
upstream shock location close to the reversal point, for which the separated
boundary layer thickens strongly and the wake is of large surface-normal
extent. The downstream motion of the shock wave is accompanied by a continuous
straightening of the shock and a successive reduction in separation
Schauerte_Schreyer2022a ; Schauerte_Schreyer2022b , which can be seen in phase
3 according to Fig. 5 (b). The minimum extent of the separation region is
reached in phase 5 (see Fig. 5 (c)), around the shock’s most downstream
location. While the shock wave is oblique in phase 1 (shock is at an angle of
about $70^{\circ}$ to the horizontal line), it is almost normal to the surface
in phase 5 (Fig. 5 (c)). At the onset of the shock’s upstream motion in phase
7 (see Fig. 5 (d)), the shock starts to tilt back and the separation region
increases. The distinct lambda shock pattern is characteristic of this type of
flow and is discernible in all snapshots, however the vertical triple point
location changes considerably with the current chordwise location of the shock
wave and its direction of motion.

To account for the large-scale variation of the global flow topology, we
carried out a phase-averaged description of the velocity fields from PIV
according to the phases discussed in the previous paragraph and following the
procedure outlined in Sec. 2.3. Fig. 3 shows the resulting phase-averaged
streamwise velocity contours and profiles for the phases corresponding to the
Schlieren visualizations shown in Fig. 5. We present a brief summary of the
strongly varying flow topology and behavior throughout a buffet cycle here,
since it lays the basis the subsequent exploration of the wake properties. An
in-depth discussion of this data is provided in Schauerte_Schreyer2022a  and
Schauerte_Schreyer2022b .

The first phase includes all instantaneous flow representations with far
upstream shock locations. It is characterized by massive flow separation (see
Fig. 3 (a)). The severity of the flow separation is further substantiated in
the great velocity deficit in the near-wake velocity profile at $x/c=1.1$.
Phase 2 (not shown) represents the phase of incipient downstream motion of the
shock wave. Compared with phase 1, the shock line is less oblique. This trend
is seen to continue in phase 3 (Fig. 3 (b)), in which the vertical extent of
the wake and its related momentum deficit have reduced by almost $75\,\%$
compared with phase 1. The continued downstream displacement of the shock wave
from $0.35\,c$ to $0.40\,c$ between phases 3 and 4 is accompanied by a further
contraction of the turbulent wake, most apparent in fuller velocity profiles
close to the wall (not shown here; see Schauerte_Schreyer2022a ;
Schauerte_Schreyer2022b ). Phase 5 (see Fig. 3 (c)) denotes the most
downstream shock location $x/c\approx 0.48$. The shape of the shock line is
now exactly normal, and the $\lambda$-region has reached a minimal extent. The
flow is completely attached all along the suction side, and the wake influence
is only noticeable in the immediate vicinity of the airfoil contour. Phases 6
(not shown) and 7 (Fig. 3 (d)) capture the incipient upstream motion of the
shock. The maximum upstream shock velocity is not yet reached, however, both
wake size and the velocity deficit close to the wall are greater than in the
corresponding equivalent shock location in phases 3 and 4 during the shock
downstream motion. The velocity profiles of phase 7 are significantly less
full than for phase 5, and the wake increases. The $\overline{u}$ velocities
therefore contain negative values all the way until the TE. The distinct
obliqueness of the shock line and the large $\lambda$-structure are both
finely reproduced in the PIV data. The final phase of the shock upstream
motion is represented by phase 8 (not shown), in which the identified low-
momentum wake protrudes even farther into outer flow field than in the
previous phase, and the induced momentum deficit is significantly more severe.
The deficit even results in a small region of reverse flow close to the
airfoil surface with a vertical extent of $y/c\approx 0.025$ above the
trailing edge, comparable to phase 1 (Fig. 3 (a)).

###  3.2 Evolution of the shock-induced wake throughout the buffet cycle

The extent of the turbulent wake varies strongly throughout the buffet cycle
(compare Fig. 5). Its extent is coupled with the direction of motion of the
shock. Since the shock motion is periodic, we expected a phase coupling of the
wake size with the shock position. To verify this assumption, we estimated the
relative size of the turbulent shear layer and wake along the chord length
throughout the buffet cycle. A gradient-based detection was applied to the
time-resolved schlieren sequences to extract the respective edges of the shear
layer. Then, the area enclosed by these edges was calculated and used as an
estimate of the turbulent wake area $A_{w}$. Overall, we evaluated 60 complete
buffet cycles, i.e. over 10,000 images. The time history of the shock position
$x_{s}$ (in relation to the chord length) and separated wake area (normalized
with its temporal mean extent $\overline{A}_{w}$) are shown in Fig. 6.

Figure 6: Temporal development of upstream and downstream shock motion (in
blue) in comparison with the evolution of the highly turbulent wake region (in
red).

The quantities seem to be phase-locked and thus oscillate at an identical
frequency, but at an almost constant small phase shift. The wake area precedes
the shock motion by approximately 0.3491 rad at its most upstream, and by
0.5585 rad at its most downstream location. The wake size reaches its minimum
extent shortly before the shock occupies its most downstream location. The
wake size already begins to increase before the shock passes its reversal
point, and reaches its maximum towards the final phase of the shock upstream
movement. Overall, the wake area almost doubles its size within a full buffet
cycle and performs a pronounced, recurrent flapping motion.

###  3.3 Mean velocity maps in the near wake

Before discussing the strong periodic variation of the wake flow field, we
present ensemble-averaged velocity maps of the streamwise and vertical
velocity components in Figs. 7 and 8, respectively. This global representation
approximates the temporally-averaged flow physics and allows to assess the
integral impact of the buffet state.

Figure 7: Ensemble-averaged streamwise velocity map and profiles at
characteristic chordwise locations in the near wake for fully-established
buffet at M=0.72 and AoA=5 deg.

The mean streamwise contour map (Fig. 7) indicates a large region of
intermittently separated flow: the momentum deficit near the trailing edge
approaches $\overline{u}/u_{\infty}\approx 0$. As will be shown later in the
context of Fig. 10, this region is a result of the vigorous shock dynamics, in
particular phases of upstream shock motion, for which reverse flow extends
over large parts of the rear of the airfoil up until the trailing edge. The
greatest velocity deficit occurs close to the surface and in the inner wake
core region (visible as blue streak with $u\leq 0.25\,u_{\infty}$ in Fig. 7).
The low-momentum dent increases with downstream development. Furthermore, the
wake center shifts slightly upwards.

Figure 8: Ensemble-averaged vertical velocity map and profiles at
characteristic chordwise locations in the near wake for fully-established
buffet at M=0.72 and AoA=5 deg.

The wall-normal velocity field is structured into three characteristic
subdomains (see Fig. 8): a pronounced downwash region along the aft section of
the airfoil, an upwash region downstream of the trailing edge, which emanates
from the pressure side, and a region of almost vanishing $v$-component further
downstream of the airfoil in the upper half of the wake. These regions are
captured by the velocity profiles at chordwise stations $x/c=1.1$ to
$x/c=1.4$. At $0.1\,c$ aft of the trailing edge, a sharp distinction between
up- and downwash is dominant. The upwash from the pressure side seems to
gradually overcompensate the downwash component, yielding a close-to-zero
vertical flow component in the upper half of the captured domain at $x/c=1.4$.

###  3.4 Modification of the wake velocity field by buffet

Despite the considerable level of turbulent perturbations in instantaneous
flow representations, they convey important features characterizing the nature
of the dynamic flow field that are precluded in phase- or ensemble-averaged
velocity contours. These features include turbulent structures, shedding of
vortices, and the overall extent of the turbulent wake.

Figure 9: Instantaneous streamwise (left column) and vertical (right column)
velocity maps for characteristic phases of the buffet cycle

The instantaneous vertical velocity contours show a turbulent wake of large
extent in all phases of far upstream shock locations, namely phases 1, 2, 7,
and 8 (see Figs. 9 (b), (d), (n), (p)). In comparison, the flow topology
gradually recovers when the shock progresses towards its most downstream
location (phases 3, 4, and 5 in Figs. 9 (e), (h), and (k). Turbulent
structures are confined to a narrow band, as most clearly visible in phase 5
(Fig. 9 (j)).

The slip line emanating from the triple point location of the lambda shock
(marked with a dashed line in Fig. 9 (i)) is nicely captured in the streamwise
component (left column of Fig. 9), and most clearly reproduced in phases 4, 5,
and 6 (Fig. 9 (g), (i), and (k)) with smaller separation regions. The large
separation in the remaining phases widens the lambda structure and pushes the
triple point upwards.

Phases of pronounced momentum deficit and large-scale separation (phases 1, 2,
7, and 8) show concomitant intense perturbations in the wall-normal velocity.
These manifest themselves in a regular pattern of alternating velocity regions
of opposite sign, forming interlaced structures with positive (upwards)
contributions from the pressure side and negative (downwards) contributions
from the suction side.

The size and intensity of these velocity perturbations vary considerably in
sync with the buffet cycle. Coherent patches of positive or negative wall-
normal velocity fluctuation are largest when the shock is far upstream, and
shrink with downstream shock displacement. The fluctuation magnitude of $\pm
0.25u_{\infty}$ remains almost constant between $x/c=1.0$ and $x/c=1.7$, which
suggests a relevant influence of this phenomenon even farther downstream.

###  3.5 Phase-averaged organization of the wake velocity field

The large-scale variation of the global flow topology associated with the
buffet mode in the present 2D configuration has been discussed in Sec. 3.1. In
good agreement with these prior findings, the streamwise velocity contours in
Fig. 9 clearly indicate the same buffet-governed variation across the near
wake domain. To allow for a conclusive statistical evaluation of the velocity
field and the inherent turbulent quantities, we provide a phase-related
description of the wake velocity field analoguous to the discussion of the
flow along the airfoil suction side following the procedure outlined in Sec.
2.3. Such representions are provided for the streamwise and wall-normal
components in Figs. 10 and 11, respectively.

Figure 10: Phase-averaged streamwise velocity maps and profiles for
characteristic phases of the buffet cycle

The contour maps in Fig. 10 show that the streamwise velocity maps vary
periodically with the buffet cycle. The low-velocity wake core fluctuates
strongly, which is predominantly dictated by the respective location and
direction of motion of the shock wave Schauerte_Schreyer2022a ;
Schauerte_Schreyer2022b . The buffet mechanism leads to a pronounced variation
of the vertical and streamwise extent of the velocity-deficit region, which is
associated with an intense flapping motion of the separated shear layer
Schauerte_Schreyer2022a ; Schauerte_Schreyer2022b .

In the streamwise velocity magnitude, the greatest variation over the buffet
cycle occurs with respect to its distribution the vertical direction (compare
most extreme phases according to Figs. 10 (a) and (e)). In the wall-normal
component, dominant changes are most evident in the streamwise direction (see
Fig. 11) and are attributed to the interplay between up- and downwash from
both sides of the airfoil downstream of the trailing edge.

Figure 11: Phase-averaged vertical velocity maps and profiles for
characteristic phases of the buffet cycle

For the most upstream shock location (phase 1, see Fig. 10 (a)), the severe
flow separation and thickening of the separated shear layer overcompensate the
curvature-related downwash in the rear part of the airfoil. Consequentially, a
region of mild upwash results between $x/c=0.6$ and $x/c=1.0$.

Immediately downstream of the trailing edge, the flow fields from the upper
and lower sides coalesce, and the pressure side induces an upwards deflection
with $v/u_{\infty}\approx 0.1$ adjacent to a region of downwards flow above
the trailing edge. This patch of negative vertical velocity persists until the
chordwise station $x/c\approx 1.4$, beyond which the transverse component $v$
is consistently positive. In phases with upstream shock locations, i.e.
incipient downstream motion (phase 2) or final upstream motion (phase 8), the
downwash along the airfoil suction side is still deficient and less pronounced
compared to phase of widely attached flow (phases 4, 5, and 6).

Regarding the magnitude and spatial distribution, the organization of the
wall-normal velocity maps in phases 3 and 7 are rather similar to the
ensemble-averaged field shown in Fig. 8. This observation emphasizes the
intermittent nature of this configuration.

###  3.6 Evolution of turbulent quantities

Considering the strong variations of the turbulent wake across the different
buffet phases, a more detailed analysis of the buffet-related perturbations is
required. The streamwise and transverse turbulent quantities $u_{rms}$ and
$v_{rms}$ as well as the Reynolds shear stresses $Re_{uv}$ were computed for
each phase individually (see Figs. 12 and 13) to assess their huge variations
throughout the buffet cycle and further quantify their spatial structure.

Figure 12: Reynolds shear stress $Re_{uv}$ and its evolution during the buffet
cycle.

The Reynolds shear stress contours in Fig. 12 are consistent with our with
prior findings Schauerte_Schreyer2022a  and agree with our observations above:
intense turbulent mixing coincides with the incidents of large-scale flow
separation, as is most clearly visible in phases 1 and 8 (Figs. 9 (b) and
(p)).

Figure 13: Overview of the turbulent quantities $u_{rms}$ (right column) and
$v_{rms}$ (left column).

These effects are reflected in the shear-stress distributions of the same
phases phases 1 (see Fig. 12 (a)) and 8 (see Fig. 12 (h)), whose respective
maxima are located far above the TE at $y/c=0.1$ (phase 1) and $y/c=0.075$
(phase 8), and the wake influence is still noticeable until $y/c=0.2$ in both
phases. Greatest maximum shear stress values of approx.
$-\overline{u^{\prime}v^{\prime}}/u_{\infty}^{2}=0.04$ occur in phase 1, which
confirms the qualitative observations of the instantaneous vector field in
Figs. 9 (a) and (b). As expected, the weakest shear stress of approx.
$-\overline{u^{\prime}v^{\prime}}/u_{\infty}^{2}=0.015$ occurs in phase 5,
with a vertical peak location of $y/c=0.025.$ This immense variation within
the buffet cycle further illustrates that the wake carries out a strong
flapping motion, as discussed in the context of Fig. 6 in Sec. 3.2 on the
basis of the time history of the wake area estimate.

The streamwise and wall-normal turbulence components $u_{rms}$ and $v_{rms}$
show qualitatively similar distributions, although some distinct features are
visible in the shapes of their profiles (see Fig. 13 left and right columns,
respectively). Both quantities seem to emerge from two major source regions:
from the suction-side shear layer downstream of the shock wave and from the
lower side of the trailing edge. These contributions are reflected in the
respective profiles near the airfoil trailing edge. At $x/c=1.1$, as the flow
of the suction and pressure sides coalesce for phases of far upstream shock
locations (phases 1 and 8, see Figs. 13 (a), (b), (o), and (p)), the effect is
most distinct: both $u_{rms}$ and $v_{rms}$ profiles exhibit two separate peak
contributions (at $y/c\approx 0.1$ and $y/c\approx 0.0$), between which the
turbulence intensity reduces. The distinctness of the peaks is much stronger
for the streamwise $u_{rms}$ component. $u_{rms}$ is much more intense along
the two wake main axes from the airfoil suction and pressure sides, whereas
the wall-normal component is of more even magnitude across the width of the
wake. This effect results in slightly M-shaped profiles for $u_{rms}$, e.g. at
$x/c\geq 1.2$.

For $u_{rms}$, the contribution from the suction dominates: the overall levels
and extent in the $y$-direction at the location of the trailing edge are
greater than the minor contribution from the pressure side. The vertical
fluctuations (right column of Fig. 13) are equally strong on either side.
Furthermore, the wake contribution from the pressure side to the $u_{rms}$
component decays much faster than the contribution to $v_{rms}$ (at approx.
$x/c\leq 1.5$ in phase 1, whereas the contribution in $v_{rms}$ has not yet
decayed at $x/c=1.7$).

###  3.7 Evolution of vortical structures in the wake

We explore the shedding of vortices, expressed by the vorticity, in detail,
since the intermittently strongly separated flow field in transonic buffet is
associated with the formation of vortex streets Zauner2020 . The vorticity
field was computed on the basis of the streamwise and wall-normal components
of instantaneous velocity fields according to eq. 1.

$\omega_{z}=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$ (1)

Fig. 14 displays the instantaneous vorticity distribution for three
characteristic phases of the buffet period: for the most upstream shock
location (a), the most downstream shock position (b), and the final phase of
upstream motion (c). We observe the production of vorticity in the near wake
and throughout the vortex shedding process at the trailing edge, as well as
the strong variation of the regions with intense vorticity coupled with the
buffet phase.

The slip line originating from the triple point of the lambda shock structure
is well-resolved by the in-plane vorticity maps derived from PIV: it is
visible as a thin streak of moderately positive vorticity that follows the
vertical extent of the wake and is almost parallel to the edges of the
turbulent wake (see e.g. Fig. 14 (b)).

Figure 14: Representative instantaneous vorticity contour maps for shock most
upstream (a) and downstream (b) locations, and final upstream motion (c).

We observe coherent vortical structures across all phases of the buffet cycle,
and they follow the expected cyclic variation. The contribution from the lower
half of the wake is consistently positive, and negative vorticity is shed
along the upper half. This tendency was previously observed by Szubert et al.
Szubert2015  based on consecutive snapshots covering one vortex shedding
period.

As discussed above, buffet phase 1 captures the most upstream shock locations
and is associated with massive flow separation with a vertical extent of the
turbulent wake of almost $0.2\,c$. The corresponding vorticity map in Fig. 14
(a) reveals two aspects of its macroscopic organization: first, Kelvin-
Helmholtz-type instabilities develop downstream of the shock wave up to about
$x/c=1.2$; second, a pronounced vortex roll-up is observed around the trailing
edge and behind. We used an identical scaling of the colorbar as Szubert et
al. Szubert2015  and observe almost perfect agreement with their case in terms
of vorticity distribution and magnitude.

The quantitative differences between buffet phases are quite severe. In
contrast to the widely separated shear layer and large coherent vortical
structures in the wake of phase 1, phase 5 exhibits small-scale vortical
structures confined to a narrow band. The vortical structures protrude less
far into the vertical direction, which results in an almost sharp separation
between positive contributions from the lower side of the wake and negative
contributions from the upper side (compare Figs. 14 (a) and (b)). This
observation is in good agreement with the findings of Epstein et al.
Epstein1988 , that an unseparated transonic flow past an airfoil exhibits
considerably reduced vortex shedding.

The massive thickening of the boundary layer in the aft section of the airfoil
results in a pronounced de-cambering effect. Hence, the periodic variation
between fully-separated and mostly reattached flow throughout a buffet cycle
results in a continuous alteration of the integral circulation around the
airfoil and therefore in continuous shedding of vorticity aft of the trailing
edge. This effect presumably also contributes to the strongly alternating
vortex-shedding intensity.

###  3.8 Spectral analysis of the vortex shedding

The vortex cores of the von Kármán vortex street induced upon the interaction
of the flow with the trailing edge are resolved by alternating dark and bright
spots in the Schlieren visualizations shown in Fig. 15. By tracking the time
history of the dark/bright transition, we analyze the spectral content of the
vortex shedding quantitatively. As visible in the detail views at the trailing
edge (lower row of Fig. 15), both the distinctness and spatial extent of the
resolved vortex cores decrease from phases 1 – 8 (Fig. 15 (a)-(c)). Assuming
that the density gradient scales with the vortex strength, the strength
decreases with increasing chordwise shock location. This observation is in
good agreement with the shrinking of the vortex street (see Fig. 14).

Figure 15: Schlieren visualizations of the vortex shedding downstream of the
trailing edge for shock locations: (a) most upstream, (b) incipient downstream
motion, and (c) most downstream shock position. Second row: zooms of the
trailing-edge region; vortex shedding indicated with arrows.

Szubert et al. Szubert2015  pointed out that events of massive flow separation
– not necessarily driven by transonic buffet – may induce von Kármán-like
vortex shedding in the vicinity of the trailing edge and the near wake. With
pressure sensors placed along the airfoil upper surface and in the near wake,
they captured strong secondary fluctuations (in addition to the primary buffet
oscillation) over a broad range of chordwise stations. These fluctuations –
attributed to a von Kármán mode – reach their maximum at $x/c=1.20$, remain
pronounced until $x/c=1.5$, and only decay beyond one chord length downstream
of the trailing edge Szubert2015 . In the wall-pressure spectra, Szubert et
al. Szubert2015  observed distinct contributions at higher frequency
($St_{c}=2.5$) close to the trailing edge in addition to the characteristic
low-frequency ($St_{c}=0.075$) footprint of shock buffet. Szubert et al.
Szubert2015  verified the relation of the $St_{c}=2.5$ peak with the vortex
shedding event on the basis of series of instantaneous vorticity maps with
adequate temporal resolution.

Applying a similar analysis to the present data base yields the spectrum shown
in Fig. 16. We observe high-frequency contributions peaking at
$2800\,\textrm{Hz}$, but with overall enhanced intensity between
$2000\,\textrm{Hz}$ and $4000\,\textrm{Hz}$. The peak frequency corresponds to
$St_{c}=1.8$, and is thus of the same order of magnitude as the study reported
above Szubert2015 . We also observe a dominant buffet peak at
$f_{b}=113\,\textrm{Hz}$, or $St_{c}=0.071$ (not displayed here to improve the
visibility of the higher-frequency contributions). Considering the otherwise
flat power spectrum, the pronounced bump can thus be attributed to the vortex
shedding process and the incipient formation of a von Kármán vortex street.

Figure 16: Power spectral density distribution of the von Kármán vortex
shedding at the trailing edge.

Also Zauner and Sandham Zauner2020  showed similar alternating large-scale
structures (as shown in the present study in instantaneous v velocity (see
Fig. 9) and vorticity plots (see Fig. 14)) based on DMD modes representing
both density and streamwise velocity in the wake field between $x/c=0.8$ and
$x/c=2.0$. The associated dynamics were centered around $St_{c}=1.8$, which is
a similar range as observed in the present case ($St_{c}=1.3$ to
$St_{c}=1.7$). This agreement is satisfactory, considering the slightly higher
Strouhal numbers reported for the overall airfoil configuration in Zauner2020
, which also includes slightly higher values for the buffet-related peak
(St=0.12 vs. St=0.07)).

##  4 Conclusions

Transonic buffet strongly affects the aerodynamic performance of passenger
aircraft. Due to large-scale shock oscillations, the flow on the suction side
of the airfoil is subjected to large-scale intermittent flow separation. Via
the wake of the airfoil or wing, also aerodynamic components downstream of the
wing, e.g. the tail plane, are strongly affected by this intermittent flow
field. To shed light on the flow structure and characteristics impinging on
the tail plane, we analyzed the effects of the buffet cycle on the wake in
detail.

The buffet-dominated flow fields along the OAT15A-airfoil suction side and in
the airfoil wake were presented and discussed in a complementary way to
elucidate the coupling between the shock-induced unsteadiness, the
disturbances incited at the trailing edge, and the vortical structures in the
wake. The flow field on the airfoil suction side undergoes a strong periodic
and global variation throughout the buffet cycle, and also the wake (shown
until $x/c=1.7$) is dominated by pronounced periodicity. Owing to the large-
scale shock displacement coupled with an intermittent separation, the flow
topology both along the airfoil and in the near wake is modified on a global
scale. The associated fluctuations remain at a constantly high level
throughout the entire studied domain ($x/c=1.7$) and are suspected to persist
also farther downstream.

On the basis of phase-averaged streamwise velocity maps, we showed the flow’s
periodic variation, which is synchronized with the buffet cycle: the low-
velocity wake core fluctuates strongly, dictated by the respective location
and direction of motion of the shock wave. The wake is dominated by a
pronounced variation of the vertical and streamwise extent of the velocity
deficit associated with the flapping motion of the separated shear layer. Each
buffet phase has a characteristic footprint of the flow topology, which is
defined by the turbulent intensity and effective angular evolution of the main
wake axis.

Downstream of the trailing edge, vortical structures form and are organized as
large coherent streaks in phases where the wake is largest. They decrease
substantially for phases of reattached flow. Positive vortical motion emanates
at the lower edge of the wake and is intertwined with negative vorticity from
the upper region. Consequently, Kármán vortex-street-like configurations form.

The low-frequency buffet unsteadiness modulates the vortex shedding, as it
drastically modifies the circulation around and past the airfoil. Spectral
analyses of the vortex shedding revealed that the associated frequencies are
one order of magnitude larger than those of the buffet mode
($St=\mathcal{O}({1})$) and $St=\mathcal{O}({0.1})$, respectively). These
structures and associated frequencies will affect any downstream aerodynamic
devices, such as the tail plane, on which the wake of the airfoil impinges. It
is therefore to expect that the flow around the tail plane is also strongly
periodic, with varying turbulent intensities associated with distinctly
dominating peak frequencies in the range of the shock-buffet ($St_{c}=0.07$)
and vortex shedding frequencies ($St_{c}=1.8$).

## Acknowldgements

The authors gratefully acknowledge the German Research Foundation (Deutsche
Forschungsgemeinschaft DFG) for funding this work in the framework of the
research unit FOR 2895 (project number 406435057). The authors thank ONERA for
providing the OAT15A airfoil geometry for our wind tunnel model. We also
gratefully acknowledge the contribution and support of Nick Capellmann and the
workshop team of the Institute of Aerodynamics during the manufacturing
process of the wind tunnel model.

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