2023

[1]Johannes Kleinert

[1]University of Stuttgart, Institute of Aerodynamics and Gas Dynamics (IAG),
Pfaffenwaldring 21, 70569 Stuttgart, Germany

# Wake Tail Plane Interactions for a Tandem Wing Configuration in High-Speed
Stall Conditions

<EMAIL_ADDRESS>Maximilian Ehrle
<EMAIL_ADDRESS>Andreas Waldmann
<EMAIL_ADDRESS>Thorsten Lutz<EMAIL_ADDRESS>stuttgart.de *

###### Abstract

In this work, wake-tail plane interactions are investigated for a tandem wing
configuration in buffet conditions, consisting of two untapered and unswept
wing segments, using hybrid Reynolds-Averaged Navier-Stokes / Large Eddy
Simulations (RANS/LES) with the Automated Zonal Detached Eddy Simulation
(AZDES) method. The buffet on the front wing and the development of its
turbulent wake are characterized, including a spectral analysis of the
fluctuations in the wake and a modal analysis of the flow. The impact of the
wake on the aerodynamics and loads of the rear wing is then studied, with a
spectral analysis of its lift and surface pressure oscillations. Finally, the
influence of the position and the incidence angle of the rear wing is
investigated. For the considered flow conditions, 2D buffet is present on the
front wing. During the downstream movement of the shock, the amount of
separation reaches its minimum and small vortices are present in the wake.
During the upstream movement of the shock, the amount of separation is at its
maximum and large turbulent structures are present accompanied by high
fluctuation levels. A distinct peak in the corresponding spectra can be
associated with vortex shedding behind the wing. The impingement of the wake
leads to a strong variation of the loading of the rear wing. A low-frequent
oscillation of the lift, attributed to the change of the intensity of the
downwash generated by the front segment, can be distinguished from high-
frequent fluctuations that are caused by the impingement of the wake’s
turbulent structures.

###### keywords:

high speed stall, wake tail plane interactions, hybrid RANS/LES, AZDES, tandem
wing, tail buffet

##  1 Introduction

The reliable prediction and control of aircraft flight characteristics and
flow conditions at the edges of the flight envelope is essential for continued
improvement of commercial aviation and for the reduction of fuel consumption.
The envelope boundaries demarcate the limits of safe and economic flight
regimes of a given aircraft, and are typically associated with Mach number and
angle of attack values that should not be exceeded.

One typical phenomenon that limits the flight envelope at transonic Mach
numbers is transonic buffet. This refers to the occurrence of a coupled
periodic oscillation of a shock and the corresponding shock-induced boundary
layer separation over a lifting surface, possibly causing vibrations, which
endanger the integrity of the structure.

Two-dimensional buffet has been widely studied and descriptions of the
phenomenon on airfoils have been published based on computational deck2005 ;
grossi2014  and experimental hartmann2013 ; jacquin2016  investigations. Lee
at al. lee1990  presented a comprehensive model relating the shock motion to
disturbances from the shock propagating downstream and scattering at the
airfoil trailing edge. The roundtrip duration of the downstream propagation
and the scattered upstream propagation corresponds well to the buffet period,
indicating a feedback loop. Jacquin et al. jacquin2009  experimentally
observed buffet periods consistent with this. Hartmann et al. hartmann2013
and Feldhusen et al. feldhusen2018  proposed an additional interaction
mechanism involving propagation along the pressure side surface.

In the context of a conventional aircraft, buffet-induced separation may cause
a wake flow incorporating a large bandwidth of turbulent scales which
propagate downstream. These structures can impinge on the tail plane and cause
undesirable unsteady loads. General insight of such wake-tail plane
interactions is provided by Tan et al. tan2018 . More specifically, Waldmann
et al. waldmann2020  observed strong load fluctuation of the horizontal tail
plane (HTP) due to turbulent wake impingement via Computational Fluid Dynamics
(CFD). The magnitude of turbulent kinetic energy and the extent of the wake
strongly impact the imparted loads. There is a strong variation of the loads
due to angle of attack. For a transonic cruise condition, Illi et al. illi2013
showed that the fundamental frequency of the tail plane load oscillation
correlated to that of the main wing shock movement.

The wake of a wing or airfoil that exhibits transonic buffet or buffeting
includes large-scale fluctuations caused by the periodic shock motion and
small-scale turbulence due to the decay of vortices arising from flow
separation and associated shear layers. Due to this broadband nature of the
wake flow, the impingement of turbulent structures on an airfoil or wing
immersed in a wake is comparable to the situation of the same airfoil or wing
being subjected to an inflow with broadband atmospheric disturbances. Müller
et al. mueller2020  analyzed the impact of broadband turbulence on aircraft
surfaces at flight-relevant subsonic conditions at $Ma_{\infty}=0.25$ and
$Re_{\infty}=11.6\cdot 10^{6}$. They observed that the leading edge and
suction peak areas of airfoils and swept wings are most sensitive to unsteady
pressure fluctuations due to oscillating inflow. Local surface pressure
spectra in these areas correspond to the spectrum of the inflow velocity
fluctuation, correlating inflow turbulence characteristics with load
oscillations over a broad frequency range.

##  2 Motivation and Background

The present results are part of the efforts carried out in sub-project 4 of
the research unit 2895 funded by Deutsche Forschungsgemeinschaft (DFG), whose
goal is the study of aerodynamic phenomena and interactions occurring in high-
speed stall conditions of transport aircraft up to flight Reynolds numbers
lutz2022 . The research unit consists of multiple sub-projects researching
specific aspects of high-speed stall, including transonic buffet on the main
wing’s upper surface, the impact of an Ultra-high Bypass (UHBR) nacelle on
buffet, shock oscillations on the lower wing surface between fuselage and
nacelle, the interaction of the wing’s wake with the tail plane as well as the
development of methods for flow analysis and reduced order modeling. Sub-
project 4 involving the authors’ working group is focused on the analysis of
the development of the wake of the wing at buffet conditions, and on its
interaction with the tail plane.

Several measurement campaigns carried out in the European Transonic Windtunnel
(ETW) support the various computational studies. These involve a realistic
transport aircraft configuration suited to high Reynolds number testing, the
XRF-1 model (”eXternal Research Forum”) designed by Airbus mann2019 . These
experiments include time-resolved pressure-sensitive paint (PSP) of the
surfaces and particle image velocimetry (PIV) measurements of the wake flow,
and enable a combined analysis of the phenomena with as well as a validation
of the numerical simulations. While testing in cryogenic conditions is
necessary to attain flight-relevant Reynolds numbers, it proves rather costly,
which limits the number of flow conditions and test points which can be
realized. Moreover, high Reynolds numbers increase the difficulty of obtaining
detailed optical measurements, i.e. due to very thin boundary layers. For
these reasons, the investigations of the XRF-1 configuration are complemented
by studies using a generic tandem wing configuration in high-speed stall
conditions which creates flow phenomena representative of a wing-wake-tail
plane configuration. This configuration is analyzed experimentally in the
Trisonic Wind Tunnel at the RWTH Aachen University (Rheinisch-Westfälische
Technische Hochschule Aachen) in the context of sub-project 6, at smaller
Reynolds numbers of around two million. In addition to the easier
measurements, the smaller Reynolds numbers also allow for validation and
further development of wall-modeled large eddy simulation (WM-LES) methods
developed in the research unit, enabling them to tackle the high Reynolds
number cases of the XRF-1.

The measurement campaign consists of two different phases. In the first phase,
inflow conditions and angle of attack are chosen so that the front wing
segment exhibits buffet, while the rear wing segment experiences steady flow
conditions, which allows for an unhindered study of the impact of the front
wing’s wake, denoted here as configuration (A). For the second phase of the
measurements, the angle of incidence of the rear wing segment is changed to
increase its loading, provoking buffet also on the latter (configuration (C)).
Therefore, the influence of the turbulent wake inflow on the buffet
characteristics can be studied. Both cases are investigated numerically in
this work by means of hybrid RANS/LES simulations (Reynolds-Averaged Navier-
Stokes / Large Eddy Simulations), presented in section 5.

Thus, a suitable airfoil has to be selected for the rear wing segment in order
to ensure steady flow at the inflow conditions that cause buffet on the front
wing. Therefore, a preliminary computational analysis is performed using
different freely accessible airfoil geometries. This preliminary study
involves simulation methods of varying fidelity, with its results and the
corresponding airfoil selection criteria described in section 3.

The present investigation is centered on the interaction of the turbulent wake
created by the front wing in buffet conditions with the rear wing segment. The
study is conducted using a total of four hybrid RANS/LES simulations of the
tandem configuration with two different vertical positions of the rear wing,
and two different settings of its angle of incidence, as explained in section
4. These incidence settings are selected to create buffet conditions on the
rear wing in the second case, whereas the lower incidence in the first case
leads to smooth flow without buffet.

##  3 Airfoil Selection for the Rear Wing Segment of the Tandem Wing
Configuration

In order to enable a study of its interaction with the front wing’s wake, a
suitable airfoil has to be chosen for the rear wing segment which exhibits
steady flow at the inflow conditions that provoke buffet on the front wing
segment. The geometry of the airfoil needs to be freely accessible so that the
results of this analysis can be subsequently released to a broader research
community. Moreover, the airfoil should ideally resemble one that can be used
for a horizontal tail plane of an actual transport aircraft in terms of its
aerodynamic characteristics. This also means that a reasonable loading that
would be expected for the tail plane of an aircraft in high-speed cruise
conditions should be possible without entering the buffet regime of the
airfoil. A purely subsonic flow around the airfoil without a shock is
preferred, since this simplifies the validation of wall models for the WM-LES
methods developed in the research unit. In addition, requirements regarding
the manufacturing of the wind tunnel model and its instrumentation have to be
taken into account. As the chord length of 75 mm of the model is rather small,
the installation of pressure sensors in the rear section of the wing is
limited due to the available internal space. As a consequence, airfoils with a
higher thickness in the rear part are preferred. The selection criteria for
the airfoil are as follows. First, the geometry of the airfoil should be
openly accessible (criterion 1). Second, the airfoil should exhibit steady
flow conditions without buffet, preferably subsonic flow, for transonic inflow
and low to medium aerodynamic loading (criterion 2). Third, it should be
similar to actual tail plane airfoils of transport aircraft with respect to
aerodynamic characteristics (criterion 3). Finally, it should exhibit
sufficient thickness in the rear section to provide space for instrumentation
(criterion 4).

Simultaneously fulfilling criteria 1 and 3 is challenging, since airfoils used
in typical recent actual aircraft are mostly confidential. As the state of the
art for jet-powered aircraft wings involves the use of transonic airfoils to
minimize wave drag in cruise, the latter represent suitable candidate
geometries. However, most publicly available transonic airfoils are designed
for wings and therefore exhibit design lift coefficients greater than 0.5
tm85663 ; ar138 ; arccp1386 ; tp2969 , which is above the loading expected for
a tail plane in cruise flight. Such an airfoil would be operated at an off-
design condition if used as an HTP. The tail plane lift coefficient in cruise
can be roughly estimated to be in the range between $-0.1$ and $-0.4$ for
large aircraft based on available information on wing and tail plane geometry,
weight and center of gravity, and cruising speed and altitude taken from the
manufacturers’ websites, (cf. manuals for ”airplane/aircraft characteristics
for airport (and maintenance) planning” of several Airbus and Boeing aircraft,
e.g. manual-737 ; manual-747 ; manual-787 ; manual-a320 ; manual-a330 ;
manual-a380 ). In contrast, the NACA series of airfoils for subsonic
applications provides a vast selection of airfoils, of which many are suited
for low to medium aerodynamic loading. Apart from their well-known application
in general aviation aircraft, modified NACA airfoils have been in fact used
for tail planes of early turboprop- and jet-powered aircraft, e.g. the Fokker
F-27 and F28, the Sud Aviation Caravelle and the Cessna Citation 500
torenbeek1982 . Yet, their critical Mach numbers tend to be lower and the
onset of strong shocks with boundary layer separation typically occurs
earlier, so the range of possible lift coefficients where criterion 2 can be
satisfied needs to assessed. With regard to that criterion, airfoils from the
NACA 6 digit series, which were originally designed for laminar flow, appear
to perform better due to the more aft position of maximum thickness.

Table 1: Airfoils considered in the preliminary analysis Transonic Airfoils | OAT15A, RAE 2822, RAE 5213/5214/5215, NPL 9510,
---|---
| RAE 6-9CK, NASA SC(2)-0410
NACA 4-digit-series | 1410
NACA-5-digit-series | 24010, 25010
NACA 6-digit-series | 63-110/-210/-212/-310/-312, 64-110, 64A109/-110/-111,
| 65-110/-210/-212/-214/-310/-314, 65A210
Research airfoil | HGR-01 (mod.) hgr01
|

The airfoils listed in Table 1 were selected for further analysis, including
airfoils from the NACA 4-, 5- and 6-digit series, the research tail plane
airfoil HGR-01 hgr01  and several widely used transonic airfoils ar138 ;
arccp1386 ; tp2969 ; tm85663 . A numerical study is performed for these
airfoils with respect to criterion 2, investigating the range of lift
coefficients where they exhibit purely subsonic flow, using the MSES
mses_drela  solver toolbox in a first step. MSES solves the full potential
equations or the Euler equations for two-dimensional flows around airfoils
coupled with an integral boundary layer method to account for boundary layers,
allowing a fast computation of lift, drag and pressure distributions for
several airfoils and inflow conditions.

Figure 1: Surface pressure (left) and friction (right) distributions of the
NACA 1410 airfoil, $Re=1.3\cdot 10^{6}$, $Ma=0.72$ obtained from MSES.
Horizontal dashed line denotes critical pressure coefficient $c_{p}^{*}$

Fig. 1 shows the calculated distributions of surface pressure and friction
coefficients using the NACA 1410 airfoil as an example. The airfoil shape is
also depicted here for reference. The inflow Mach number is 0.72, with angles
of attack varying from -0.8∘ to 2.7∘, and lift coefficients between 0.03 to
0.63. The dashed lines denote the lower side of the airfoil. The Reynolds
number is set to 1.3 million corresponding to the value expected for the wind
tunnel tests at that time, and the boundary layer is treated as fully
turbulent. It can be seen that the critical condition, where a local Mach
number of one is attained, is reached at an angle of attack close to 0.2∘ with
a lift coefficient of 0.19. Decreasing the angle of attack below -0.8∘ with
lift coefficients below 0.03, critical flow is reached on the lower side of
the airfoil. At a higher angle of attack to 1∘ (at $c_{l}=0.33$), on the other
hand, a distinct shock forms on the upper side. However, as the shock is
relatively weak, the impact on the boundary layer remains small, as there is
only a small decrease visible in surface friction $c_{f}$. The strengthening
shock then begins to notably influence the boundary layer profile at around
$\alpha=1.5^{\circ}$ and $c_{l}=0.42$, with a decrease in friction of more
than 50%. Boundary layer separation eventually occurs at around
$\alpha=2.7^{\circ}$ and $c_{l}=0.63$. It should be noted here that the
prediction of separation onset using an integral method can only ever be an
approximation. Buffer may occur at even higher angles of attack, however this
can not be accurately modeled using a steady-state flow solver. Even under
this premise, evaluation of transonic buffet occurrence at moderate $\alpha$
can be carried out given the present data. Transonic buffet requires the
presence of both a shock and a boundary layer separation. Operating in the
subsonic regime should provide enough margin to the buffet boundary, as a
small increase in loading may lead to a weak shock, but not necessarily to
separation or buffet. Taking the above into account, the range of lift
coefficients usable for the application, i.e. with purely subsonic flow,
extends from 0.03 to 0.19. It should be noted that the airfoil would be
installed in an inverted manner as an HTP, providing a down force of
$-0.19\leq c_{l}\leq-0.03$. This available range is rather narrow for the
airfoil, as the shock begins to form at a comparably small loading.

Figure 2: Surface pressure (left) and friction (right) distributions of the
NACA 64A110 airfoil, $Re=1.3\cdot 10^{6}$, $Ma=0.72$ (MSES calculation)

A wider range of $c_{l}$ can be achieved with the laminar airfoil NACA 64A110,
as seen in the pressure and friction distributions in Fig. 2 for the same
inflow conditions as above. Purely subsonic flow is present between
$\alpha=-1.5^{\circ}$ and $\alpha=1.3^{\circ}$ with lift coefficients between
-0.13 and 0.32. Here, a distinct shock appears only above $\alpha=2.5^{\circ}$
or $c_{l}=0.52$. Notably, a supersonic plateau forms in the pressure
distribution on the upper side, similar to that of a transonic airfoil, in
contrast to the continuous flow acceleration observed for the NACA 4-digit
airfoil discussed above. This is due to the aft position of maximum thickness
and less curvature on the upper surface. Indeed, the pressure distributions
resemble those of transonic airfoils quite well, with the exception of the
missing aft-loading typical for the latter, cf. Fig. 5. Therefore, criterion 3
can be considered as partially fulfilled, considering that the characteristics
of the front part of the airfoil are of greatest importance for wake
interaction, as it is being directly impinged by turbulent structures. A
decrease in surface friction of more than 50% is found for angles of attack
above approximately $3.1^{\circ}$ or $c_{l}=0.63$ in Fig. 2, and boundary
layer separation does not occur until $\alpha=4^{\circ}$ and $c_{l}=0.77$.
Thus, the usable range of lift coefficients according to criterion 2 is
$-0.13\leq c_{l}\leq 0.32$, which is considerably wider than that of the NACA
1410.

Figure 3: Surface pressure distributions of the HGR-01 research tail plane
airfoil, $Re=1.3\cdot 10^{6}$, $Ma=0.72$ (MSES calculation)

Figure 4: Surface pressure (left) and friction (right) distributions of the
HGR-01 airfoil with reduced camber (1%), $Re=1.3\cdot 10^{6}$, $Ma=0.72$ (MSES
calculation)

The research tail plane airfoil HGR-01, designed by the Technische Universität
Braunschweig for the analysis of a mixed leading-edge trailing-edge stall
behaviour hgr01 , in contrast, exhibits no angle of attack with purely
subsonic flow. When the angle of attack is decreased to reduce the loading of
the upper side, starting from $\alpha=0^{\circ}$, for example, a suction peak
forms on the lower side of the airfoil which becomes supersonic before the
velocity on the upper side falls below $Ma=1$, as shown in Fig. 3.
Furthermore, flow separation on the lower side is reached at $c_{l}=0.01$.
Yet, this is not the case anymore when the airfoil’s camber is scaled down
from 2.5% to 2% chord or less. Here, cambers of 1%, 1.5% and 2% chord have
been investigated. With decreasing camber, the regime of subsonic flow grows.
However, the lift coefficient that can be reached without separation becomes
smaller. For a camber of 1% chord, the range of subsonic flow extends from
$c_{l}=0.01$ to $c_{l}=0.18$, as displayed in Fig. 4. A noticeable shock forms
for $c_{l}=0.31$ and separation is finally reached at $c_{l}=0.62$.

Figure 5: Surface pressure (left) and friction (right) distributions of the
RAE 2822 airfoil, $Re=1.3\cdot 10^{6}$, $Ma=0.72$ (MSES calculation)

As a representative example for the investigated transonic airfoils, the
distributions of pressure and surface friction of the RAE 2822 airfoil are
shown in Fig. 5. Subsonic flow is present between $c_{l}=0.07$ and
$c_{l}=0.37$. The appearance of a strong shock is evident for $c_{l}=0.70$ and
flow separation begins at about $c_{l}=0.85$.

Figure 6: Overview of usable lift coefficients for the analyzed airfoils (MSES
calculation)

The results of the MSES calculations are summarized in Fig. 6 which
illustrates the range of usable lift coefficients for the analyzed airfoils.
Here, the range of purely subsonic flow is indicated in green and the range
where no distinct shock is present is shown in blue. In the yellow and gray
areas, a shock is present but no flow separation is observed. Thus, the upper
end of the region marked with gray indicates the on-set of flow separation
behind the shock. As described above, a further distinction is made between
conditions with a relatively “weak” (marked in yellow) or “strong” (marked in
gray) shock based on the decline of the surface friction behind the shock.
Here, a decrease of more than 50% is used as a threshold for defining a strong
shock. Based on these results, the NACA 64A110 airfoil is selected because of
its wide range of lift coefficients with subsonic flow (down to $c_{l}=0$).
Thus, the latter is chosen for all further investigations.

##  4 Numerical Setup

###  4.1 Geometry of the Tandem Wing Configuration

As described above, the investigated tandem configuration consists of two
straight, untapered and unswept wing segments. The front wing uses the
supercritical OAT15A airfoil, whereas the laminar airfoil NACA 64A110 is
selected for the rear wing segment based on the results of the the work
described in section 3. The former has been the subject of several
experimental and numerical investigations on 2D buffet in the past and is thus
chosen for the analysis as well as other studies within the research unit.

Figure 7: Overview of the analyzed configurations, cf. Table 2

The computational setup includes the front wing segment with a chord length of
$c_{\mathrm{front}}=0.15\,m$ and the rear wing segment with a chord length of
0.075 m, corresponding to the experimental setup of the investigations
performed in the research unit. The OAT15A airfoil used on the front segment
exhibits a blunt trailing edge with a relative thickness of 0.5%. For the NACA
64A110 airfoil applied to the rear segment, the sharp trailing edge has been
slightly modified resulting in a blunt shape for meshing reasons. The trailing
edge thickness is 0.2% chord. The horizontal distance between the trailing
edge of the front wing segment and the leading edge of the rear wing segment
corresponds to $2\,c_{\mathrm{front}}$, i.e. the latter is positioned at
$x/c=3$, as indicated in Fig. 7. As sketched in Fig. 7 and listed in Table 2,
two different vertical positions for the rear wing segment are considered,
denoted as (A) and (B): In configuration (A), the rear segment is located
$1/6\,c_{\mathrm{front}}=0.025\,m$ above the centerline of the front wing
segment, and $1/20\,c_{\mathrm{front}}=0.0075\,m$ above the centerline in
configuration (B), respectively. While the relative size of the rear wing
segment and the horizontal separation between the wing segments are roughly
based on the geometry of actual transport aircraft (cf. e.g. manual-737 ;
manual-a330 ), the vertical position of the rear segment of configuration (A)
is chosen so that it is placed directly in the center of the front wing’s wake
to maximize the interaction effects in order to facilitate the investigation.
It should be noted that the trajectory of the wake varies over the buffet
cycle so that a fixed position will not be exactly in the center of the wake
at every instance in time. In order to analyze the sensitivity with respect to
the vertical position, the position of configuration (B) is slightly offset
from the wake’s center. Since knowledge of the wake’s trajectory is required
for this placement but unknown a priori, simulations of the isolated front
wing segment were performed in advance, denoted as configuration (0).

In addition, two different angles of incidence ($\alpha_{I}$) of the rear wing
segment are investigated. These are realized by rotating the rear segment
around its leading edge. A first setting of $\alpha_{I}=-4^{\circ}$,
corresponding to an angle of $\alpha_{rear}=1^{\circ}$ with regard to the free
stream for the selected inflow angle of $\alpha_{\infty}=5^{\circ}$, leads to
a lift coefficient of about $c_{l}=-0.2$ due to the downwash of the front wing
segment. This is comparable to a typical tail plane lift coefficient in cruise
conditions (cf. section 3). This setting is used in both configurations (A)
and (B). With this $\alpha_{I}$, buffet occurs only on the front wing segment.
It should be noted that the NACA 64A110 airfoil is mounted in an inverted
orientation to represent a typical tail plane section with negative camber
optimized for a negative lift coefficient. The second setting of
$\alpha_{I}=-11^{\circ}$, which corresponds to an angle of
$\alpha_{rear}=-6^{\circ}$ with regard to the free stream, is chosen to
provoke buffet also on the rear wing segment to study the influence of the
turbulent wake inflow on the buffet characteristics. This configuration uses
the same position of the rear segment as configuration (B), and is denoted as
(C). Finally, to allow for a comparison with an undisturbed buffet flow,
simulations of the isolated rear wing segment at an angle of attack of
$\alpha_{rear}=-6^{\circ}$ are performed as a reference (configuration (D)).

Table 2: Overview of the analyzed configurations Identifier | Configuration
---|---
(0) | Isolated front wing segment
(A) | Tandem configuration with rear segment at $z/c=1/6$ and $\alpha_{I}=-4^{\circ}$
(B) | Tandem configuration with rear segment at $z/c=1/20$ and $\alpha_{I}=-4^{\circ}$
(C) | Tandem configuration with rear segment at $z/c=1/20$ and $\alpha_{I}=-11^{\circ}$
(D) | Isolated rear wing segment
|

###  4.2 Computational Grids and Boundary Conditions

The computational grids used for the simulations are hybrid meshes consisting
of hexahedrons in the structured and triangular prisms in the unstructured
regions, respectively, and are created using the meshing software Pointwise.
The grids of all configurations share the same topology, meshing parameters
and cell sizes. Fig. 8 visualizes the mesh for configuration (A). The grids
employ farfield boundary conditions and do not include the wind tunnel
geometry. All boundary layers, as well as the region of the shock above the
front wing segment and the area between the wing segments are discretized with
hexahedrons to assure a high cell quality. This minimizes numerical
dissipation for the resolution of the shock movement and the turbulent
structures in the wake. The height of the first cell layer above all surfaces
is chosen so that a $y^{+}$ of less than 0.4 is achieved as recommended for
the application of Reynolds stress models. A total of 607 points is used the
surface of the front wing segment, with a cell size of 0.4% chord over most of
the airfoil. The surface of the rear wing segment is discretized with 388
points and an average cell size of 0.8% of its chord, respectively. For the
wake region between the two segments, a resolution of 0.7% of the front wing’s
chord is chosen. The spanwise extent of the computational domain is set to
0.0735 m for all cases, which corresponds to 49% of the chord length of the
front wing segment. To allow for the resolution of three-dimensional turbulent
structures present in the wake, the spanwise dimension is discretized with 70
cell layers, created by an extrusion of the sectional 2D meshes in spanwise
direction. Therefore, the cells in the area between the wing segments are
nearly cubic with a resolution of 0.7% of the front wing’s chord. Periodic
boundary conditions are applied at the spanwise boundaries of the domain. The
cylindrical far field boundary has a radius of 50 m, which is equal to about
95 times the overall length of the tandem wing configuration. In total, the
meshes for the different configurations consist of around 16 million points.

Figure 8: Main section of the computational grid used for the hybrid RANS/LES
simulations of the tandem wing configuration (configuration (A))

###  4.3 Flow Conditions

All simulations of the tandem wing configuration are performed for an inflow
Mach number of 0.72 and a Reynolds number of 2 million with respect to
$c_{\mathrm{front}}$ (i.e. 1 million based on $c_{\mathrm{rear}}$), with an
inflow angle of $\alpha_{\infty}=5^{\circ}$ relative to the front segment’s
centerline. These conditions are well inside the buffet regime of the OAT15A
airfoil. The boundary layers of both segments are tripped at 7% of their
respective chord, corresponding to the experimental setup. Tripping is
realized in TAU by switching the production term in the transport equations of
the Reynolds stresses off in the user-defined laminar regions, and on in the
turbulent regions, respectively.

###  4.4 Numerical Setup

The CFD simulations used for further analysis in the present work are
performed with the finite-volume code TAU (version 2018.1.0) schwamborn2008
provided by the German Aerospace Center (Deutsches Zentrum für Luft- und
Raumfahrt, DLR), applying the zonal hybrid RANS/LES method AZDES (Automated
Zonal Detached Eddy Simulation) developed in the authors’ working group at the
Institute of Aerodynamics and Gas Dynamics (IAG) and described by Ehrle et al.
ehrle2020 ; ehrle20202 . This method introduces a fixed, zonal separation of
RANS and LES regions based on geometric parameters and flow properties
obtained during a precursor unsteady RANS (URANS) simulation. This enables a
robust treatment of the attached boundary layer and the region of shock-
boundary layer interaction in RANS mode, as well as an improved resolution of
turbulence in regions of separated flow, when compared to purely geometry-
dependent zonal approaches (e.g. Deck’s Zonal Detached Eddy Simulation (ZDES)
deck2005 ). The scale-resolving behavior is realized by activating the
Detached Eddy Simulation (DES) model (in this case a Delayed Detached Eddy
Simulation (DDES) spalartDDES ) outside of user-defined, zonal RANS areas.

The definition of these RANS areas is controlled by the user based on wall
distance dependent parameters and involves a threshold value for the integral
turbulent length scale $L_{t}$ which is a priori accumulated in the precursor
URANS simulation for the same flow conditions. $L_{t}$ is calculated using the
quantities of the turbulence model and is supposed to give an estimate of the
overall extent of flow separation over time. It is computed as

$L_{t}=\max_{\\{\text{time}\\}}(\sqrt{k_{t}}/(c_{\mu}\,\omega))\text{,}$

where $k_{t}$ is the turbulent kinetic energy, $\omega$ denotes the specific
turbulence dissipation rate estimated by the turbulence model and $c_{\mu}$ is
a model constant. In areas where $L_{t}$ reaches high values, large turbulent
structures are to be expected that can be resolved by the mesh. The threshold
value for $L_{t}$, above which the DES model is activated, denoted as $L_{c}$,
can be adjusted by the user. The latter implies that regions with
$L_{t}<L_{c}$ are marked as RANS zones, whereas areas with $L_{t}>L_{c}$ are
covered in DDES mode in the following actual hybrid RANS/LES simulation. This
is realized ehrle20202  by modifying the hybrid length scale for the hybrid
simulation from the (D)DES length scale

$\displaystyle L_{DES}$
$\displaystyle=\min(L_{RANS},\,C_{DES}\,\Delta)\text{~{}or}$ $\displaystyle
L_{DDES}$
$\displaystyle=L_{RANS}-f_{d}\cdot(L_{RANS}-C_{DES}\,\Delta)\text{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{spalartDDES}{\@@citephrase{(}}{\@@citephrase{)}}}},$

with the RANS length scale $L_{RANS}$ (which is equal to the wall distance in
SA based models or calculated from $k_{t}$ and $\omega$ for others), the
hybrid model constant $C_{DES}$, the filter width $\Delta$ (originally the
largest cell edge’s length) and the DDES shielding function $f_{d}$, to

$L_{AZDES}=L_{RANS}\cdot(1-f_{a})+L_{(D)DES}\cdot f_{a}\text{.}$

Here, $f_{a}$ is a blending function based on the ratio $L_{t}/L_{c}$,

$f_{a}=0.5\cdot(1+\tanh(8\cdot[L_{t}/L_{c}-1]))\text{,}$

i.e. $f_{a}=1$ for $L_{t}/L_{c}\gg 1$ and $f_{a}=0$ for $L_{t}/L_{c}\ll 1$.
Additionally, all regions further away from surfaces than a user-defined wall
distance are forced into DES mode regardless of $L_{t}$.

For the present simulations, a threshold value of $L_{c}/c=11\%$ is selected
in order to enable the switching from RANS to LES mode at the most upstream
position possible, without influencing the periodic shock motion of transonic
buffet at the same time. Regions further away from the airfoil’s surface than
$6\%$ of the chord are forced into DES mode by applying the aforementioned
wall distance dependent parameters. These settings are based on
recommendations of previous buffet studies ehrle2020 ; ehrle20202 ; illi2016
that suggest a threshold length scale $L_{t}$ of at least 6% chord and a DES
switching wall distance of at least 4% chord, respectively, to shield the
region of the shock and attached boundary layers sufficiently. Thus, grid-
induced separation is avoided, which would otherwise lead to a non-physical
dampening of the shock motion. During precursor simulations with different
values for $L_{c}$, the settings above were verified for the present
configuration based on the respective buffet amplitudes observed. The choice
of the threshold length scale is also dependent on the grid resolution. It
should be set according to the size of smallest turbulent eddies that can be
resolved. As the central scheme used in the present work is of second order,
the number of grid points required to resolve a flow structure can roughly be
estimated to be in the range of ten to twenty spalart2001 ; Swanson1992 ;
loewe2016 . Based on the grid resolution of 0.7% chord, this corresponds to 7%
to 14% chord, a range that fits to the value chosen. The resulting zone
division is shown in Fig. 9. The shock (foot) itself and the begin of the
shock-induced separation are shielded and treated in RANS mode, wheres a DDES
is performed beginning in the region above the trailing edge and in the wake
area between the wing segments, as intended within the AZDES approach.

Figure 9: RANS/LES zone division in the flow field around the front wing
section, together with the averaged streamwise velocity $u$ (left) and the
integral turbulent length scale $L_{t}$ (right) from the precursor URANS
simulation

All simulations are second order accurate in space and time by applying a
central flux approximation for the convective terms and an implicit dual time
stepping scheme based on the second Backward Differentiation Formula (BDF 2)
for time integration, respectively. For the discretization of the turbulence
model equations, a second order Roe scheme is used, and gradients are
evaluated using the Green Gauss Theorem. The quality of the numerical results
heavily depends on the error induced by excessive artificial dissipation.
Therefore, the numerical parameters are chosen to enable a stable, converging
simulation with a minimum of artificial dissipation, as recommended by DLR
TAU2018 ; probst2015 . This is realized by using TAU’s matrix type artificial
dissipation in combination with a hybrid flux blending. 95% matrix dissipation
and a fourth order dissipation coefficient k(4) of 1/1024 are applied the
scale resolving areas. 80% matrix dissipation and k(4)= 1/64 in RANS areas
ensure numerical stability. Furthermore, a low dispersion scheme, as described
by Loewe et al. loewe2016 , is applied in scale resolving areas, which
minimizes numerical dispersion errors in these regions. The locally kinetic
energy conserving skew symmetric scheme kok2009 , which is the preferred
scheme for hybrid RANS/LES computations with low-dissipation requirements, is
chosen for the mean flow fluxes.

A physical time step of $\Delta t=4.4\cdot 10^{-6}\,s$ representing 150 time
steps per convective time unit $\tilde{t}=c_{\mathrm{front}}/u_{\infty}$ is
used, leading to a local Courant-Friedrichs-Lewy (CFL) number of approximately
one in the wake region and a sufficient temporal resolution of the shock
motion. One buffet cycle is resolved with about 1900 time steps. It should be
noted that the time step size is not primarily determined by the shock motion,
which can be captured properly with a coarser temporal resolution, but by the
requirements of the LES in the wake, for which a local CFL number of around
one is generally recommended for accuracy spalart2001 ; kornhaas2008 ;
mocket2014 . The SSG/LRR-$\omega$ Reynolds stress model eisfeld2005  serves as
turbulence model for the RANS regions and as subgrid scale model in the LES
regions. In combination with 150 inner iterations per time step, a sufficient
convergence of the force and moment coefficients within one time step is
achieved. For the inner iterations, the implicit backward Euler scheme is
employed together with a 2v geometrical multigrid method for convergence
acceleration, based on the full approximation storage scheme (FAS) brandt77 .
The corresponding (pseudo) CFL number for the inner iterations is set to 4. On
the coarser grid levels of the multigrid scheme, it is reduced to 2, and
further down to 1 in areas of high pressure gradient to guarantee numerical
stability.

The setup of the time accurate hybrid RANS/LES simulations is started with a
steady state computation with gradually increased angle of attack until
reaching $\alpha=5^{\circ}$. Subsequently, an unsteady RANS simulation is
performed over 50 convective times $\tilde{t}$. This serves as precursor
simulation for the AZDES method to accumulate the turbulent length scale for
the subsequent RANS/LES zone distribution. The ${\Delta}_{max}$-filter width
definition is used. The time series used for statistics and analyses in the
following section comprises around 30 to 40 convective time scales, depending
on the case, which represents two to three buffet cycles on the main wing
segment. The sampling rate corresponds to the applied time step.

##  5 Results

The results presented in the following sections 5.1 and 5.2 correspond to the
simulations with the first setting of the angle of incidence of the rear wing
segment, i.e. the case where buffet occurs only on the front wing segment
(configurations (A) and (B)). First, the transonic buffet and the
corresponding unsteady flow separation from the front wing segment are
characterized in section 5.1. The separated turbulent wake is then
investigated including a spectral analysis of the pressure and velocity
fluctuations. The interaction of the separated wake with the rear wing segment
is analyzed in detail in section 5.2. The corresponding findings for the
second incidence setting (configuration (C)), for which buffet is also present
on the rear segment, are then discussed in section 5.3.

###  5.1 Buffet Flow and Wake Development of the Front Wing Segment

For the considered inflow conditions, a pronounced shock oscillation is
present on top of the upper side of the front wing segment. Accompanying the
movement of the shock, the lift of the front wing segment varies strongly
during the buffet cycle, as displayed in Fig. 10, which shows the development
of the lift coefficient over several buffet periods. A zoomed-in view of one
exemplary cycle is shown on the right. The lift coefficient oscillates between
a minimum of $c_{l}=0.81$ and a maximum of $c_{l}=1.10$ during this cycle,
corresponding to an amplitude of $\hat{c_{l}}=0.145$, and with a frequency of
118.5 Hz or a Strouhal number of $Sr=f\cdot c/U_{\infty}=0.0745$, based on the
chord of the front segment, which is in the range generally found for 2D
buffet. For example, Jacquin et al. found $Sr=0.068..0.075$ jacquin2009 ,
Schrijer et al. reported $Sr=0.071$ schrijer2018 ; aguanno21  and Accorinti et
al. listed $Sr=0.056..0.076$ accorinti2022  for the OAT15A airfoil. Thus, the
buffet period is equal to 13.4 convective time units. The corresponding
experimental investigations performed in the research group (cf. section 2) of
the isolated OAT15A airfoil yield a comparable Strouhal number of $Sr=0.0714$
for the same inflow conditions, as reported by Schauerte et al. schauerte2022
. The mean lift coefficient averaged over one buffet period equals to
$c_{l}=0.94$.

Figure 10: Lift coefficient $c_{l}$ of the front wing segment over several
buffet cycles

Figure 11: Distribution of the pressure (left) and friction (right)
coefficient of the front wing segment at different moments in time during one
buffet cycle

The range of the shock motion is displayed in Fig. 11 (left), which shows the
distribution of the pressure coefficient of the front wing segment at four
distinct moments in time during one buffet period. Here, the most downstream
position of the shock is marked with (I) and its most upstream position with
(III). Snapshots during the upstream and downstream movement of the shock are
denoted as (II) and (IV), respectively. These four moments in time are also
marked in Fig. 10, for reference. Additionally, the mean pressure distribution
averaged over one buffet cycle is included, marked with (av). The shock
position (as defined by the steepest pressure gradient) is found to move
between $x/c=0.28$ and $x/c=0.44$, i.e. a range of 16% chord which is close to
the experimental value measured in the Trisonic Wind Tunnel of 16.5%
schauerte2022 . As the amount of lift generated is largely dominated by the
extent of the low pressure region on the upper side, the most downstream
position of the shock coincides approximately with the maximum, and the most
upstream shock position with the minimum of the lift coefficient,
respectively. The different levels of pressure recovery towards the trailing
edge already indicate different amounts of flow separation for the different
shock positions, which becomes evident in Fig. 11 (right) that shows the
corresponding distributions of the surface friction coefficient. Here, the
upper surface is indicated with solid lines, and the lower surface with dashed
lines. The varying amount of separation can also be seen in Fig. 12, which
shows the Mach number in the flow field around the front wing segment together
with streamlines close to its surface for the four moments in time introduced
above. The flow exhibits a (relatively) small amount of separation when the
shock is located at its most downstream location (I) due to a comparatively
small shock strength. However, during the upstream movement of the shock (II),
the flow behind the shock is completely separated until the trailing edge. As
the shock moves upstream, the velocity of the fluid relative to the shock is
increased, which leads to a greater shock strength, forcing the point of
separation to move forward on the airfoil. When the shock temporarily comes to
a stop at its most upstream position (III), this effect fades, so both the
shock strength and the amount of separation are reduced. The flow begins to
reattach behind a shock-induced separation bubble at around $x/c=0.4$. At this
moment in time, a trailing edge separation is still present in the last ten
percent of the chord. Finally, the velocity of the fluid relative to the shock
is decreased during the downstream movement of the shock (IV). Therefore, the
resulting strength of the shock is reduced allowing for a further reattachment
of the flow during this phase. Whereas the trailing edge separation vanishes
almost completely, the separation bubble behind the shock becomes smaller but
does not disappear fully, however, and quickly grows back towards the trailing
edge when the shock reaches it most downstream location again. Notably, the
boundary layer on the lower surface remains attached at all times, and only
small changes in pressure and surface friction occur during the buffet cycle.
The pronounced variation of the amount of separation within the buffet cycle
strongly influences the characteristics of the wake and its interaction with
the rear wing segment, as discussed later.

Figure 12: Mach number in the flow field surrounding the front wing segment;
at the time of the most downstream shock position (I) (top left), during the
upstream shock motion (II) (top right), at the time of the most upstream shock
position (III) (bottom left), and during the downstream shock motion (IV)
(bottom right). Figure 13: Root mean square (rms) of the pressure fluctuations
on the front wing segment Figure 14: Power spectral density (PSD) of the
surface pressure fluctuations on the front wing segment’s upper side at
$x/c=0.15$, $x/c=0.39$ and $x/c=0.95$

Both shock movement and flow separation induce pressure fluctuations on the
surface of the front wing segment, which are shown in Fig. 13 in terms of root
mean square (rms) values over the chord position, normalized with the dynamic
pressure of the inflow, i.e. $p^{\prime}/q_{\infty}$. The highest fluctuation
levels can be seen between $x/c=0.25$ and $x/c=0.50$, corresponding to the
range of shock motion. Due to the strong pressure gradient at the shock, a
point on the surface experiences a sharp pressure increase or decrease every
time the shock passes, leading to a high temporal variation of the pressure.
In contrast, the pressure fluctuations in the supersonic region upstream of
the shock are comparatively small. The area downstream of the shock, however,
exhibits elevated values that suggest strong pressure disturbances and
turbulent fluctuations in the separated flow. On the lower side that shows no
shock or flow separation, the fluctuations are comparatively small, however,
an increase of unsteadiness is evident close to the trailing edge, which can
be attributed to pressure disturbances propagating from the upper side. A
corresponding power spectral density (PSD), computed with Welch’s method
welch1967 , of the normalized surface pressure fluctuation is displayed in
Fig. 14 for three points on the upper surface, at $x/c=0.15$, $x/c=0.39$ and
$x/c=0.95$, respectively. The first point is located in the region upstream of
the shock at all times, the second one at the position of the rms maximum and
the third one downstream of the shock. It is evident that all spectra are
dominated by the buffet frequency. Corresponding to the small rms of the
pressure fluctuations noted above, the amplitudes are significantly smaller
for the position in front of the shock.

Figure 15: Time history of the shock position $x_{s}$ with respect to its mean
position $x_{s,mean}$, simulation and experiment schauerte2022 ; schauerte2023

Figure 16: Phase-averaged flow field surrounding the front wing segment,
simulation (left) and PIV measurement (right) schauerte2022 ; schauerte2023 ;
phase 1: most upstream shock position, phase 3: downstream shock motion, phase
5: most downstream shock position, phase 7: upstream shock motion

A brief comparison of the buffet flow around the front wing segment, as
simulated with the AZDES method, with experimental data from corresponding
measurements of the isolated OAT15A airfoil performed in the Trisonic Wind
Tunnel schauerte2022 ; schauerte2023  is shown in the following. Fig. 15
depicts the time history of the chordwise shock position $x_{s}$ with respect
to its mean position $x_{s,mean}$ as a fraction of the chord length
$c_{front}$ for both the simulation and the experiment, determined by the
maximum of the density gradient at a line $0.1\,c_{front}$ above the airfoil
surface (at $z/c=0.1678$). It is evident that simulation and experiment agree
quite well considering the overall shock motion, its absolute range and
extreme positions. It should also be noted that there is a notable amount of
cycle-to-cycle variation for both cases (please see schauerte2023  for a
longer time series). Whereas the downstream motion is comparable, the
simulated upstream motion consistently appears to be slightly shifted to an
earlier moment in time of the cycle in relation to the experimental data. In
schauerte2022 ; schauerte2023 , PIV measurements of the flow field around the
OAT15A airfoil were performed and analyzed. In order to take the significant
variation of the flow field over the buffet cycle into account, the latter was
divided in eight phases with regard to the shock location relative to its most
upstream and downstream position, and an averaging was performed for these
phases over several buffet periods (for details we refer to schauerte2023 ).
The same phase-averaging is performed with the simulation data in this work,
and compared to the PIV results in Fig. 16 for four selected phases - at the
time of the most upstream and downstream shock position (phase 1 and 5,
respectively) and during the downstream and upstream shock motion (phase 3 and
7, respectively). Again, a good agreement between simulation and experiment is
evident considering the respective shock locations. Additionally, the
curvature of the shock line is matched quite well in the simulation.
Furthermore, the varying extent of the separated flow region behind the shock
is in agreement as well as the flow velocity and direction at the trailing
edge. However, differences inside the separated flow region are visible, where
the flow velocities are smaller or more negative in the simulation than in the
experiment. Also, the thickening of the boundary layer is slightly
overpredicted, especially for the phases with strong separation. Nevertheless,
the overall agreement can be considered satisfactory.

Figure 17: Time-averaged wake behind the front wing segment in terms of Mach
number

Figure 18: Spanwise vorticity in the wake behind the front wing segment at
different moments in time during one buffet cycle

Fig. 17 shows the time-averaged wake downstream of the front wing segment.
Here, the mean Mach number in the flow field is shown, together with time-
averaged streamlines. The separated wake appears as a region of reduced Mach
number downstream of the front wing segment. Remarkably, the wake flow is
orientated almost parallel to the chord close to the trailing edge, despite
the incidence of the inflow of 5∘, which is due to the downwash created by the
front wing segment. Further downstream, at distances greater than roughly one
half of a chord away from the trailing edge ($x/c>1.5$), the trajectory of the
wake is slightly curved upwards in $z$-direction as the strength of the
downwash declines, and the wake flow becomes more and more aligned with the
inflow. Additionally, an increase of the mean velocity in the wake with
increasing distance from the trailing edge is evident, which indicates a
progressive dissipation of the wake. The development of the wake over one
buffet period is depicted in Fig. 18, which shows the instantaneous
dimensionless spanwise vorticity $(c/U_{\infty})\cdot\omega_{y}$ in the flow
field behind the front wing segment for the four moments in time discussed
above. Large turbulent vortices are formed in the separated flow behind the
trailing edge, evolving from the shear layers at the wake boundaries. Notably,
the characteristics of the wake change significantly within the buffet cycle.
During the time when there is only a small amount of separation or attached
flow, the wake appears thin and small vortices are generated, which is the
case for the most downstream shock position (I) and during the downstream
movement of the shock (IV). When the amount of flow separation behind the
shock is large, however, larger vortices are generated from the thick wake,
which is the case for the most upstream position (III) and during the upstream
movement of the shock (II). Strikingly, the amount of separation is largest
during the upstream movement of the shock; yet, the biggest vortices in the
wake are found at a moment later in time when the shock has already reached
its most upstream position (III). This is because of the time that the
separated flow, starting from the shock location, needs to reach the
considered downstream position. The time shift between the flow situation at
the front wing segment and the appearance of the corresponding vortices
increases with increasing distance. It is also evident that the pattern of the
vortices alternates during the buffet period. When the amount of separation is
comparatively small (I)/(II), pairs of alternating vortices can be seen in the
wake, similar to those in the wake of a bluff body. However, when the amount
of separation is large, the pattern seems to be more irregular (III), which is
also the case during the transitional phase (IV). The vortices begin to break
up almost immediately and the pattern becomes increasingly chaotic further
downstream.

Figure 19: Time-averaged streamwise velocity overlaid with the mean wake
centerline position. The data extraction locations at 0.5$c$, 1$c$ and 1.5$c$
behind the trailing edge are indicated with dashed lines. Shown wake data
extends from the front wing’s trailing edge to the rear wing’s leading edge.

Figure 20: Profiles of wake properties extracted at the three positions in the
wake shown in Fig. 19, at 0.5$c$, 1$c$ and 1.5$c$ behind the front wing’s
trailing edge.

The development of the wake velocity characteristics is shown in Fig. 20. The
data is extracted at three locations shown in Fig. 19. The latter also
visualizes the location of the wake centerline, which is computed at every
streamwise plane in the wake (normal to the inflow direction) by locating the
vertical position ($z_{WT}$) of the minimum of the (averaged) axial velocity
($\overline{u})$. For the following discussion, a local wake coordinate system
is introduced, denoted by the subscript ”$WT$” (for ”wind tunnel”), which is
rotated around the spanwise axis (y) compared to the body-fixed coordinate
system shown above in Fig. 12 such that the new x axis $x_{WT}$ is aligned
with the inflow direction, as visible in Fig. 19. The wake velocity deficit is
evident in the shape of $\overline{u}/u_{\infty}$ in Fig. 20, with a sharp
minimum at the upstream location which becomes wider further downstream. The
velocity deficit dissipates and the vertical velocity gradients decrease. The
downward displacement of the velocity minimum between successive positions is
consistent with the centerline shape in Fig. 19, which trends downward due to
the front wing producing lift.

The streamwise normal component of the Reynolds stress
$\overline{u^{\prime}u^{\prime}}/u_{\infty}^{2}$ in Fig. 20 shows two distinct
maxima at the upstream position at 0.5$c$, which is consistent with a strong
velocity deficit bounded by shear layers associated with high speed flow
regions from above and below the wing segment and the wake. The fluctuations
are stronger in the upper portion of the wake. The opposite can be observed
for $\overline{u^{\prime}w^{\prime}}/u_{\infty}^{2}$ in Fig. 20, where the
positive values in the lower wake portion dominate at the upstream position.
These values indicate a strong exchange between the high speed outer flow
below the wake and the low speed wake core. The overall turbulent kinetic
energy $k_{t}$ in Fig. 20 exhibits a maximum near the wake center, which shows
a consistent downward displacement with streamwise distance. All fluctuation
quantities in Fig. 20 as well as the velocity deficit decrease with streamwise
distance in the wake, reflecting momentum exchange and mixing of outer and
inner flow, which eventually leads to the dissipation of the wake.

Figure 21: Lumley’s anisotropy invariant map showing different locations along
the wake centerline. Points are extracted at equidistant streamwise positions,
colors represent streamwise distance from the front wing’s trailing edge.

The anisotropy of the Reynolds stress tensor can give an indication of the
dominant processes occurring in the wake. The anisotropy invariants computed
from eigenvalues of the anisotropy tensor lumley:1977  along the wake
centerline are shown in Lumley’s turbulence triangle in Fig. 21. The latter
visualizes the anisotropy of turbulence by mapping each turbulent state to a
location in an invariant map, depending on the local relation between the
magnitudes of the turbulent fluctuations in the three dimensions in space.
Every (physically) possible state falls inside of a triangle (giving the map
its name), whose borders are indicated here by solid lines. The three corners
of the triangle mark three particular cases: A ”one-component” state, where
fluctuations exist only in one direction (top corner), an ”axisymmetric two-
component” state with fluctuations in two directions of the same magnitude
(left corner), and the isotropic state, which exhibits fluctuations in all
three directions of the same strength (bottom corner). Furthermore, the
borders connecting those corners indicate a ”two-component” state with
fluctuations in only two directions (top border), an ”axisymmetric expansion”
(or ”rod-like”) state, where the fluctuations in a single direction notably
exceed the other two (right border), and an ”axisymmetric contraction” (or
”disc-like”) state, which is characterized by fluctuations in two directions
of the same magnitude that are considerably stronger than those in the third
direction (left border). More details on the theory behind and the involved
algorithms can be found e.g. in lumley:1977 ; choi2001 . Fig. 21 shows a
zoomed-in view of the lower part of the triangle near the state of isotropy,
which is represented by $III_{a}=II_{a}=0$. The dark blue points in the wake
immediately downstream of the trailing edge are located near the center of the
shown view, with the anisotropic state trending toward the right-hand boundary
beginning at about $(x-x_{TE})/c=0.5$. This boundary is representative of
axisymmetric expansion of the tensor, i.e. with one normal component being
significantly larger than the other two, as noted above. In the present case,
the vertical normal stress component
$\overline{w^{\prime}w^{\prime}}/u_{\infty}^{2}$ dominates and is
significantly larger than $\overline{u^{\prime}u^{\prime}}/u_{\infty}^{2}$ or
$\overline{v^{\prime}v^{\prime}}/u_{\infty}^{2}$. This axisymmetric expansion
state remains over a large part of the wake propagation distance. The
dissipation of $\overline{w^{\prime}w^{\prime}}/u_{\infty}^{2}$ causes
movement toward isotropic equilibrium, where all three normal components
approach equality.

Figure 22: Time series of the velocity components and $c_{p}$ at the wake
centerline one chord downstream of the trailing edge, together with the lift
coefficient of the front wing segment.

The wake is characterized by high velocity fluctuations, as shown for a
representative location, one chord behind the trailing edge, in the center of
the wake in Fig. 22 over two buffet periods. In all three components, phases
with comparatively low amplitudes of velocity fluctuations alternate with
phases of high fluctuations during a single buffet cycle. Periods with high
lift correspond to the least amount of separation on the front wing segment,
which correlates with low levels of wake turbulence; and vice versa.

Figure 23: Power spectral densities of the velocity components $u,v,w$. Colors
correspond to streamwise position, as in Fig. 20

Spectral densities of the time signals of the velocity components at three
positions in the wake center at 0.5$c$, 1$c$ and 1.5$c$ are shown in Fig. 23.
The high frequency region is focused upon, as the short time series does not
permit a high resolution of the region of the buffet oscillation around
$Sr\approx 0.075$. Therefore, a variance-reducing averaging using Welch’s
method welch1967  and 5 overlapping segments is used for power spectral
density estimation. The high frequency content decreases with streamwise
distance, especially in the streamwise and vertical components. Apart from
high amplitudes at low frequencies near the frequency resolution limit at
$Sr=0.1$ caused by the shock oscillation, there is a distinct peak at about
$Sr\approx 2.4$ in the spectrum of the axial velocity $u$ at the upstream
position at $0.5c$. This spectral feature dissipates downstream and is not
discernible in the spectra of the spanwise velocity $v$. The spectra of the
vertical velocity $w$, however, consistently show this peak at all three
positions. This high-frequent wake fluctuation propagates consistently over
significant distances in the wake. The spanwise velocity oscillation in the
center panel of Fig. 23 does not show characteristic peaks to a degree similar
to the other two components. The spectra generally reflect the insights gained
via the anisotropy invariants, which showed that the streamwise and vertical
fluctuations predominate in the flow behind the front wing trailing edge, with
$\overline{w^{\prime}w^{\prime}}/u_{\infty}^{2}$ remaining dominant
thereafter. While the Reynolds and anisotropy tensors contain information
which is integrated over the entire frequency range, Fig. 23 shows that the
redistribution and shifts in anisotropy invariant space occur predominantly
between $Sr=2$ and $Sr=4$.

Fig. 22 shows how the spectral characteristics change over time during the
buffet phases. Fourier transformation based methods cannot resolve this when
applied to the entire time series. Future work on the interpretation of
spectral characteristics of the wake may involve wavelet transformation in
order to isolate temporal variation in the spectra, as the dynamics in the
wake are strongly dependent on the buffet phase.

Especially the spectrum of the vertical velocity $w$ shows strong fluctuations
in the wake at frequencies comparatively high in relation to the buffet
frequency in the region of $Sr\approx 1.5$ to $Sr\approx 8$, that can be
attributed to the vortices or turbulent structures present in the wake. The
broadband characteristic in the spectrum mirrors the variation of the size and
frequency of those structures. As the pairs of counter-rotating vortices
forming from the shear layers are accompanied by the alternation of positive
and negative vertical velocities, their footprint is most notable in the
$w$-spectrum.

#### Modal Analysis of the Wake Flow

Modal analyses of the flow field using Proper Orthogonal Decomposition (POD)
make it possible to distinguish different flow phenomena and their respective
temporal behavior. The POD technique decomposes (the unsteady component of) a
time-dependent flow field into different spatial modes with corresponding
amplitude signals (or temporal coefficients), based on a singular value
decomposition of the data set. Unlike a (spatial) Fourier transformation which
breaks down the field into Fourier modes, a POD results in an optimal basis of
orthogonal spatial functions for the given flow field data. This decomposition
is optimal in the least squares sense, e.g. the modes are chosen optimal to
capture the kinetic energy if the velocity field is used as input, for
example. The resulting modes then represent the major time-dependent and
coherent features of the flow field, like vortex streets for example,
depending on the respective case. More details on POD theory and the involved
algorithms can be found e.g. in taira2017 . The time series of three POD
coefficients of the horizontal velocity obtained in the midplane are shown in
Fig. 24(a). The first mode pair exhibits a clearly periodic shape and can be
associated with the shock motion of the transonic buffet present on the front
wing segment. Spectral densities of the time signals of the POD modes are
depicted in Fig. 24(b). As already mentioned earlier, buffet occurs at a
distinct frequency of $\mathrm{Sr\approx 0.075}$, which corresponds to
$\mathrm{f\approx 120Hz}$ and is the dominant frequency of the first POD mode.

(a) Time series of selected POD mode coefficients

(b) Spectral densities of selected POD modes

Figure 24: Time series and spectra of selected POD mode coefficients in the
spanwise midplane of the front wing segment

(a) POD mode 1

(b) POD mode 6

Figure 25: POD modes in the spanwise midplane of the front wing segment

The time series of the POD modes 4 and 6, as shown in Fig. 24(a), also yield
some kind of periodicity with the buffet frequency. Here, oscillations of high
frequency occur during the upstream motion of the shock during each buffet
cycle. During the downstream motion, when the flow is mainly attached, the
amplitudes are significantly lower, as described above. These modes can be
associated with flow separation behind the shock and the resulting wake
motion. Fig. 24(b) shows that these modes yield distinct spectral peaks at
$\mathrm{Sr=2.0..2.6}$, which agrees well with the peaks of the velocity
components in the wake shown in Fig. 23. Contours of the first and sixth POD
mode of the horizontal velocity component are shown in Fig. 25. The first mode
exhibits clear maxima with changing sign in the shock region, which shows its
main connection to the shock motion. The regularly spaced regions of positive
and negative values of the sixth POD mode in the wake can be attributed to the
shedding of wake vortices already noticed above that mainly takes place during
the upstream motion of the shock. The corresponding frequency range falls in
the region already discussed for the spectra of $w$ in Fig. 23.

###  5.2 Interaction of the Wake with the Rear Wing Segment

The following section discusses the wake interactions in the configurations
(A) and (B), for the first setting of the angle of incidence of the rear wing
segment ($\alpha_{I}=-4^{\circ}$). Buffet occurs only on the front wing
segment in these configurations, which strongly impacts the aerodynamics of
the rear wing segment.

Figure 26: Lift coefficient of the rear wing segment over two buffet periods,
configuration (A)

The wake impingement causes a strong variation of the rear wing segment’s
loading, as shown in Fig. 26. The figure displays the evolution of the lift
coefficient of the rear wing segment over two buffet periods. At first look, a
relatively low-frequent oscillation of the loading can be seen which shows a
period that is equal to the buffet period of the front wing segment. This low-
frequent oscillation dominates the load fluctuation with a variation of the
lift coefficient between approximately $c_{l,min}=-0.3$ and $c_{l,max}=0.2$,
corresponding to an amplitude of $\hat{c_{l}}=0.25$. This is caused by the
temporal variation of the downwash intensity behind the front wing segment
during the buffet cycle. With the oscillation of lift of and circulation
around the front wing segment, the induced vertical velocity downstream from
trailing edge oscillates too. This induces a variation of the effective angle
of attack of the rear wing segment. Consequently, the lift coefficient of the
rear wing segment reaches its minimum shortly after the lift coefficient of
the front wing segment reaches its maximum value, considering the time delay
of the propagation of the downwash in downstream direction. The same
correlation holds between the time of the lift maximum of the rear and the
lift minimum of the front wing segment.

Figure 27: Instantaneous spanwise vorticity field during vortex impingement
onto the rear wing segment, configuration (A)

The low-frequent load oscillation at the rear wing segment is superposed by
relatively high-frequent oscillations of varying amplitude and frequency.

These high-frequent oscillations are caused by the impingement of the
turbulent structures or vortices in the wake discussed in section 5.1 upon the
rear wing segment. As an example, such a vortex impingement is depicted in
Fig. 27, which displays the instantaneous dimensionless spanwise vorticity in
the immediate vicinity of the rear wing segment. A turbulent vortex pair,
indicated by the region of strongly positive and negative vorticity,
respectively, impinges on the leading edge of the rear wing segment at the
depicted moment in time. Additionally, a second turbulent structure can be
identified on the upper side close to the trailing edge, which appears notably
stretched and distorted due to the interaction with the wing segment.

The impact of the turbulent vortices leads to a change of both the effective
angle of attack and the effective inflow velocity for the rear wing segment,
resulting in changes to the acting forces. Furthermore, the smaller turbulent
structures also distort the local pressure distribution on the surface as they
pass, contributing to the load fluctuation.

As evident from Fig. 26, the load oscillations caused by the vortex
impingement exhibit significant amplitudes reaching up to $\hat{c_{l}}=0.2$,
which is comparable to the amplitude of the low-frequent oscillation.
Moreover, the frequencies and especially the amplitudes of the high-frequent
oscillations vary significantly within one buffet period. Phases with a low
level of fluctuation alternate with phases of high fluctuation, which is
explained by the alternation of comparatively weak fluctuations of velocity
and pressure in the wake at times when only small vortices are present, and
strong fluctuations, respectively, accompanied by the presence of large
vortices. Consequently, the vortex-induced force oscillations are stronger
when the vortices impinge on the rear segment that have been generated
upstream at a moment earlier in time, when the amount of separation on the
front wing segment has been large. It should be noted here that there is a
time delay between the formation of the vortices and their impingement due to
their propagation time through the wake, which can be roughly approximated by
$D/U_{\infty}$, with $D$ being the distance between the wing segments and the
free stream velocity $U_{\infty}$. As the mean convective velocity in the wake
is smaller than $U_{\infty}$, however, the actual time delay proves to be
slightly larger.

Figure 28: Power spectral density of the lift coefficient of the rear wing
segment, configuration (A))

The contribution of the impinging turbulent structures to the rear wing lift
oscillation is also evident when looking at the spectral distribution of the
latter, in form of the PSD of the lift coefficient shown in Fig. 28. Beside
the low-frequent oscillation caused by the change of the downwash magnitude
around the buffet frequency, broadband spectral content contributions are
visible at higher frequencies around $Sr\approx 1$ to $Sr\approx 4$ due to the
vortex impingement. The peak is around $Sr\approx 2.5$ which corresponds to
the peak frequency identified in the wake, cf. Fig. 23.

The interaction with the wake generates pressure fluctuations on the surface
of the rear wing segment. Fig. 29 shows the evolution of the pressure
coefficient for three locations on the lower surface of the rear wing segment,
at $x/c=0.05$, $x/c=0.2$ and $x/c=0.6$. Both the low-frequent and the high-
frequent components discussed above can be identified. Additionally, the
intensity of the fluctuations decreases with increasing downstream position,
as the turbulent structures are dissipated during the course of their
interaction with the rear wing segment.

Figure 29: Time series of the pressure coefficient for different locations on
the rear wing segment’s surface, configuration (A) Figure 30: Power spectral
density of the surface pressure for different locations on the rear wing
segment’s surface, configuration (A)

The spectral distribution of the pressure fluctuations for the three locations
on the surface is displayed in Fig. 30 in terms of the power spectral density
(PSD) of the pressure coefficient. As for the lift coefficient, the
fluctuations of the surface pressure are dominated by low-frequent
oscillations at the front wing buffet frequency, which is related to the
variation of the downwash as described above. Likewise, the increased level
evident at high frequencies between $Sr\approx 1$ and $Sr\approx 5$
corresponds to the impingement of vortices of varying size and frequency.

This correlation is also supported by the comparison of the surface pressure
spectra with the corresponding spectra in the wake (cf. Figures 23 and 24(b)).
They appear very similar in shape and exhibit the same features, i.e. the high
amplitudes in the low frequency range at the buffet frequency and the
broadband signal in the high frequency range with a dominant peak
corresponding to the turbulent fluctuations. The turbulent structures
themselves cause the pressure fluctuations on the rear wing segment. The
comparison also reveals and underlines the direct correlation between the wake
fluctuations and the variation of the rear wing segment loading. A similar
relationship has been reported by Müller et al. mueller2020  for the case of
atmospheric turbulence.

Figure 31: Root mean square (rms) of the pressure fluctuations on the rear
wing segment, configuration (A))

Comparing the spectra at the different chordwise locations, a decrease of the
spectral density over the whole spectrum can be seen for increasing chord
position, which indicates the dissipation of the turbulent structures due to
the interaction with the rear wing segment. This effect also becomes apparent
in the decreasing rms of the pressure fluctuations with increasing downstream
position, as shown in Fig. 31. As the vortices impinge on the leading edge of
the rear wing segment, the highest fluctuation levels are found there,
continuously decreasing towards the trailing edge. Furthermore, the effect of
the wake on the upper (or pressure) side is a smaller than the impact on the
lower (or suction) side, as both the flow velocity and the rate of change of
the pressure with angle of attack are higher on the latter.

#### Influence of the Position of the Rear Wing Segment

The influence of the position of the rear wing segment on the wake interaction
phenomena is shown in Fig. 32, which displays the time series of the lift
coefficient for both positions. Here, the $c_{l}$ for the higher vertical
position ($z/c=1/6$) is marked in black, which corresponds to the
configuration considered above (A), and the lower position ($z/c=1/20$) in
blue, i.e. configuration (B). Generally, it is evident that the effect of the
wake on the load fluctuation is qualitatively the same for both positions of
the rear wing. Both the low-frequent lift variation caused mainly by the
downwash and the high-frequent oscillations due to the impact of the wake’s
vortices are visible, as discussed above, which is expected as the lower
position of configuration (B) is still well inside the region covered by the
wake.

Figure 32: Lift coefficient of the rear wing segment over two buffet periods
for configurations (A) and (B) Figure 33: Root mean square (rms) of the
pressure fluctuation on the rear wing segment for configurations (A) and (B)

However, qualitative differences can be seen. Because the rear segment is
positioned slightly below the center of the wake for configuration (B), the
amplitudes of the load fluctuation are smaller for this case. Notably, the
average lift is also lower due to the smaller velocity deficit at this
position, which leads to a higher effective dynamic pressure of the flow that
increases the - negative - loading of the rear segment. The impact on the
surface pressure fluctuations is shown in Fig. 33. Again, the characteristic
remains qualitatively identical; however, the fluctuation levels are reduced
for configuration (B). Although the wake interactions are qualitatively
similar, in general, the quantitative differences considering the small shift
between both positions.

###  5.3 Wake Interactions for Buffet Flow at the Rear Wing Segment

In the following, the wake interaction phenomena of the tandem wing
configuration with the adapted setting of the angle of incidence of the rear
wing segment ($\alpha_{I}=-11^{\circ}$, configuration (C)) are presented. To
ensure that buffet is also present on the rear wing segment under the
considered inflow conditions, unsteady RANS simulations of the isolated rear
segment (configuration (D)) were performed in advance, increasing the angle of
attack until buffet onset. A pronounced shock oscillation was found for
$\alpha_{rear}=6^{\circ}$, which has consequently been analysed with hybrid
simulations. The buffet cycle and the corresponding variation of lift and flow
separation of the rear segment with the NACA64A110 airfoil are similar to the
case analyzed above in section 5.1 for the front segment exhibiting the OAT15A
airfoil. A lift amplitude of $\hat{c_{l}}=0.16$ is found with a buffet
frequency of 244 Hz, which corresponds to a Strouhal number of 0.077 with
regard to the rear segment’s chord, or $Sr=0.154$ in relation to the front
wing segment, respectively. Whereas the reduced frequency - in relation to the
respective chord - is similar to that of the OAT15A airfoil, the absolute
frequency is higher because of the smaller chord length.

Figure 34: Lift coefficient of the rear wing segment over several buffet
periods for configurations (C) and (D)

Changing the angle of incidence of the rear wing segment to
$\alpha_{I}=-11^{\circ}$ for configuration (C) leads to the same angle of
attack relative to the free inflow as in configuration (D), which allows for a
comparison to analyze the wake impact under buffet conditions. However, the
airfoil is mounted upside down in the tandem setup, thus creating a down
force. The time series of the lift coefficient for both configurations is
depicted in Fig. 34. Here, the lift of the isolated segment (D) is shown pre-
multiplied by -1 to allow for a better comparison. Notably, the harmonic
oscillation of the lift of the uninfluenced buffet seen for configuration (D)
is disrupted by the impact of the vortices of the wake. The characteristic is
more closer to the one found for configurations (A) and (B), as the wake
impact dominates the aerodynamics under the analyzed conditions. An
unambiguous periodicity of the loading can not be determined. Yet, lift maxima
reoccur after 9.6 to 14.3 dimensionless time units, corresponding to a
frequency range of $f=110..165$ Hz or a Strouhal number of $Sr=0.07..0.1$,
which is close to the buffet frequency of the front wing segment.

Figure 35: Power spectral density (PSD) of the lift coefficient for
configurations (B), (C) and (D)

A spectral analysis of the lift oscillations, depicted in Fig. 35, confirms
that the shock oscillation of the unhindered buffet at $Sr=0.154$ is
suppressed and the lift oscillations are dominated by the wake fluctuations at
$Sr\approx 1..4$ as for the case of $\alpha_{I}=-4^{\circ}$ in configuration
(B).

##  6 Conclusion

The characteristics of the turbulent wake downstream from a wing segment in
buffet conditions, and the interaction phenomena occurring between the wake
and a rear wing segment were investigated for a tandem wing configuration
using hybrid RANS/LES simulations with the AZDES method. The configuration
consists of two straight, untapered and unswept wing segments, creating flow
phenomena representative of a wing-wake-tail plane configuration. A suitable
airfoil for the rear wing was selected based on a preliminary study using the
MSES potential solver toolbox.

For the considered flow conditions, a pronounced shock oscillation is present
on the front wing segment, with a lift amplitude of $\hat{c_{l}}=0.145$ and a
buffet frequency of $f=118.5$ Hz, corresponding to a Strouhal number of
$Sr=0.0745$, which is comparable to previous experimental investigations of 2D
buffet jacquin2009  and recent measurements of the research unit schauerte2022
. The separated flow behind the shock forms a turbulent wake that impinges on
the rear wing segment positioned downstream. Turbulent vortices develop behind
the front wing segment and high levels of velocity and pressure fluctuations
can be seen in the wake. It is found that the characteristics of the wake vary
strongly within the buffet cycle. During the downstream movement of the shock,
the amount of shock-induced separation reaches its minimum, the wake is
comparably thin, and small, regular alternating vortices are present in the
wake. In this phase, the corresponding fluctuations of velocity and pressure
also reach their minimum. In contrast, during the upstream movement of the
shock, the amount of separation is at its maximum, the wake exhibits its
largest extent, and large vortices are forming in the wake, which break into
more irregular, chaotic turbulent structures. In this phase, the velocity and
pressure fluctuations reach their maximum level, with the vertical
fluctuations $w^{\prime}$ being the largest reaching up to a half of the
inflow velocity $u_{\infty}$. All fluctuation quantities as well as the
velocity deficit decrease with streamwise distance in the wake, reflecting
momentum exchange and mixing of outer and inner flow, which eventually leads
to the dissipation of the wake. Apart from high amplitudes at low frequencies
caused by the shock oscillation with $Sr=0.0745$, a high level of high-
frequent fluctuations is found in the range of $Sr\approx 1.5$ to $Sr\approx
8$ that can be attributed to the vortices or turbulent structures present in
the wake. A distinct peak at about $Sr\approx 2.4$ can be associated with the
shedding of wake vortices, identified by means of a modal analysis of the flow
field using Proper Orthogonal Decomposition (POD).

The aerodynamics and loading of the rear wing segment are strongly influenced
by the interaction with the unsteady turbulent wake which results from the
buffet occurring on the front wing segment. The impingement of the wake causes
a strong variation of the loading, with a lift amplitude of $\hat{c_{l}}=0.25$
for configuration (A). A comparably low-frequent oscillation of the lift
coefficient, attributed to the change of intensity of the downwash caused by
the front segment, can be distinguished from fluctuations of high frequency.
The latter are caused by the impingement of the vortices and turbulent
structures in the wake and reach amplitudes comparable to those of the low-
frequent oscillation. As the amplitude of the velocity and pressure
fluctuations varies strongly within the buffet cycle depending on the phase of
the buffet, the same holds true for the loading fluctuations caused by the
wake impingement. The same oscillating characteristic is also found for the
pressure fluctuations on the rear segment’s surface. The similarity of the
spectra in the wake and on the rear wing segment show that the turbulent
fluctuations in the wake correlate with the pressure fluctuations on the rear
wing segment, which is consistent with the findings reported by Müller et al.
mueller2020  for the case of atmospheric turbulence. The highest levels of
fluctuations are found on the leading edge, where the vortices directly impact
on the surface.

Comparing the different vertical positions (A) and (B) of the rear wing
segment, it is found that the effect of the wake on the load fluctuation is
qualitatively the same for both positions. As the rear segment is positioned
slightly below the center of the wake for configuration (B), the amplitude of
the load fluctuation is smaller for this case. Those quantitative differences
are notable considering the the small vertical shift between both positions.

Changing the angle of incidence of the rear wing segment so that a relative
angle of attack of $\alpha_{rear}=6^{\circ}$ is reached, buffet occurs for the
isolated rear wing segment (D) with a frequency of $Sr=0.154$. In the tandem
configuration (C), however, the shock oscillation on the rear segment is
suppressed by the wake impact, and the lift oscillations are dominated by the
wake fluctuations at $Sr\approx 1..4$ as for the case of
$\alpha_{I}=-4^{\circ}$ in configuration (B).

Acknowledgments

The authors gratefully acknowledge the Deutsche Forschungsgemeinschaft DFG
(German Research Foundation) for funding this work in the framework of the
research unit FOR 2895. The authors also would like to thank ONERA for
providing the geometry of the OAT15A airfoil, as well as the
Höchstleistungsrechenzentrum Stuttgart (HLRS) for providing computation time
on the Hawk cluster.

## Declarations

Conflict of Interest

Co-author Thorsten Lutz is Guest Editor of the special issue of the CEAS
Aeronautical Journal in which this paper will be included. The authors have no
other conflicts of interest nor competing interests to declare that are
relevant to the content of this article.

Funding

The research leading to these results received funding from the Deutsche
Forschungsgemeinschaft DFG (German Research Foundation) in the framework of
the research unit FOR 2895 (Unsteady flow and interaction phenomena at high-
speed stall conditions), subproject TP4, under the grant number LU809/11-1.
The computation time needed for performing the numerical simulations is
provided by the Höchstleistungsrechenzentrum Stuttgart (HLRS) as part of
SCBOPT project.

## References

  *   * (1) Deck, S.: Numerical Simulation of Transonic Buffet over a Supercritical Airfoil. AIAA Journal 43(7) (2005). https://doi.org/10.2514/1.9885
  * (2) Grossi, F., Braza, M., Hoarau, Y.: Prediction of Transonic Buffet by Delayed Detached-Eddy Simulation. AIAA Journal 52(10) (2014). https://doi.org/10.2514/1.J052873
  * (3) Hartmann, A., Feldhusen, A., Schröder, W.: On the interaction of shock waves and sound waves in transonic buffet flow. Physics of Fluids 25(026101) (2013). https://doi.org/10.1063/1.4791603
  * (4) Jacquin, L., Brion, V., Molton, P., Sipp, D., Dandois, J., Deck, S., Sartor, F., Coustols, E., Caruana, D.: Testing in Aerodynamics Research at ONERA: the Example of the Transonic Buffet. AerospaceLab Journal 12 (2016). https://doi.org/10.12762/2016.AL12
  * (5) Lee, B.H.K.: Oscillatory shock motion caused by transonic shock boundary layer interaction. AIAA Journal 28(5) (1990). https://doi.org/10.2514/3.25144
  * (6) Jacquin, L., Molton, P., Deck, S., Maury, B., Soulevant, D.: An Experimental Study of Shock Oscillation over a Transonic Supercritical Profile. AIAA Journal 47(9) (2009). https://doi.org/10.2514/1.30190
  * (7) Feldhusen-Hoffmann, A., Statnikov, V., Klaas, M., Schröder, W.: Investigation of shock-acoustic-wave interaction in transonic flow. Experiments in Fluids 59(15) (2018). https://doi.org/10.1007/s00348-017-2466-z
  * (8) Tan, K.B., Wang, P.C., Srigrarom, S.: Low-Speed Post-Stall Wing Wake Impingement on Horizontal Stabilizer of the Common Research Model. In: 2018 Aviation Technology, Integration, and Operations Conference, Atlanta, Georgia, USA (2018). https://doi.org/10.2514/6.2018-3898
  * (9) Waldmann, A., Konrath, R., Lutz, T., Krämer, E.: Unsteady Wake and Tailplane Loads of the Common Research Model in Low Speed Stall. In: Dillmann, A., Heller, G., Krämer, E., Wagner, C., Tropea, C., Jakirlić, S. (eds.) New Results in Numerical and Experimental Fluid Mechanics XII. DGLR 2018\. Notes on Numerical Fluid Mechanics and Multidisciplinary Design vol. 142. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-25253-3_2
  * (10) Schulte am Hülse, S.A., Fingskes, C., Lutz, T., Krämer, E.: Transonic Tail Buffet Simulations for the Common Research Model. In: 31st AIAA Applied Aerodynamics Conference, San Diego, California, USA (2013). https://doi.org/10.2514/6.2013-2510
  * (11) Müller, J., Lutz, T., Krämer, E.: Numerical Simulation of the Swept FNG Wing in Atmospheric Turbulence. In: AIAA Aviation 2020 Forum. https://doi.org/10.2514/6.2020-2778
  * (12) Lutz, T., Kleinert, J., Waldmann, A., Koop, L., Yorita, D., Dietz, G., Schulz, M.: Research Initiative for Numerical and Experimental Studies on High-Speed Stall of Civil Aircraft. Journal of Aircraft (Nov. 2022). https://doi.org/10.2514/1.C036829
  * (13) Mann, A., Thompson, G., White, P.: Civil Aircraft Wind Tunnel Feature Rich Testing at the Edge of the Envelope. In: 54th 3AF International Conference on Applied Aerodynamcis, Paris, France (2019)
  * (14) Jenkins, R.V.: Reynolds number tests of an NPL 9510 airfoil in the Langley 0.3-meter transonic cryogenic tunnel. Technical Report TM-85663, NASA (1983)
  * (15) Barche, J.: Experimental Data Base for Computer Program Assessment - Report of the Fluid Dynamics Panel Working Group 04. Technical Report AR-138, AGARD (1979)
  * (16) Wilby, P.G.: The Design and Aerodynamic Characteristics of the RAE 5215 Aerofoil. Technical Report C.P. No. 1386, R.A.E. (1977)
  * (17) Harris, C.D.: NASA supercritical airfoils: A matrix of family-related airfoils. Technical Report TP-2969, NASA (1990)
  * (18) 737 Airplane Characteristics for Airport Planning (D6-58325-6) (2013)
  * (19) 747-400 Airplane Characteristics for Airport Planning (D6-58326-1) (2002)
  * (20) 787 Airplane Characteristics for Airport Planning (D6-58333) (2015)
  * (21) A320 Aircraft Characteristics for Airport and Maintenance Planning (Revision No. 35 - Dec 01/18) (2018)
  * (22) A330 Aircraft Characteristics for Airport and Maintenance Planning (Revision No. 25 - Jul 01/18) (2018)
  * (23) A380 Aircraft Characteristics for Airport and Maintenance Planning (Revision No. 16 - Feb 01/19) (2019)
  * (24) Torenbeek, E.: Synthesis of Subsonic Airplane Design. Springer, Dordrecht (1982). https://doi.org/10.1007/978-94-017-3202-4
  * (25) Wokoeck, R., Krimmelbein, N., Ortmanns, J., Ciobaca, V., Radespiel, R., Krumbein, A.: RANS Simulation and Experiments on the Stall Behaviour of an Airfoil with Laminar Separation Bubbles. In: 44th AIAA Aerospace Sciences Meeting and Exhibit. https://doi.org/%****␣ceas-2023-kleinert-clean-arxiv.bbl␣Line␣375␣****10.2514/6.2006-244
  * (26) Drela, M.: A User’s Guide to MSES 3.05 (2007)
  * (27) Schwamborn, D., Gardner, A.D., von Geyr, H., Krumbein, A., Lüdecke, H., Stürmer, A.: Development of the TAU-Code for Aerospace Applications. In: 50th NAL International Conference on Aerospace Science and Technology, Bangalore, India (2008)
  * (28) Ehrle, M., Waldmann, A., Lutz, T., Krämer, E.: Simulation of transonic buffet with an automated zonal DES approach. CEAS Aeronautical Journal 11 (2020). https://doi.org/10.1007/s13272-020-00466-7
  * (29) Ehrle, M., Waldmann, A., Lutz, T., Krämer, E.: An Automated Zonal Detached Eddy Simulation Method for Transonic Buffet. In: Hoarau, Y., Peng, S.-H., Schwamborn, D., Revell, A., Mockett, C. (eds.) Progress in Hybrid RANS-LES Modelling, pp. 271–281. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-27607-2_22
  * (30) Spalart, P.R., Deck, S., Shur, M.L., Squires, K.D., Strelets, M.K., Travin, A.: A New Version of Detached-eddy Simulation, Resistant to Ambiguous Grid Densities. Theoretical and Computational Fluid Dynamics 20(181) (2006). https://doi.org/10.1007/s00162-006-0015-0
  * (31) Schulte am Hülse, S.A.: Simulation transsonischen Buffets an Transportflugzeugen mittels hybrider RANS-/LES Verfahren. PhD thesis, Institute of Aerodynamics and Gas Dynamics, University of Stuttgart (2016)
  * (32) Spalart, P., Streett, C.: Young-Person’s Guide to Detached-Eddy Simulation Grids. NASA CR-2001-211032 (2001)
  * (33) Swanson, R.C., Turkel, E.: On central-difference and upwind schemes. Journal of Computational Physics 101(2), 292–306 (1992). https://doi.org/10.1016/0021-9991(92)90007-L
  * (34) Löwe, J., Probst, A., Knopp, T., Kessler, R.: Low-Dissipation Low-Dispersion Second-Order Scheme for Unstructured Finite Volume Flow Solvers. AIAA Journal 54(10), 2961–2971 (2016). https://doi.org/10.2514/1.J054956
  * (35) Deutsches Zentrum für Luft- und Raumfahrt: DLR TAU-Code User Guide (2018)
  * (36) Probst, A., Reuß, S.: Scale-resolving simulations of wall-bounded flows with an unstructured compressible flow solver. In: Girimaji, S., Haase, W., Peng, S.-H., Schwamborn, D. (eds.) Progress in Hybrid RANS-LES Modelling, pp. 481–491. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15141-0_39
  * (37) Kok, J.C.: A high-order low-dispersion symmetry-preserving finite-volume method for compressible flow on curvilinear grids. Journal of Computational Physics 228(18) (2009). https://doi.org/10.1016/j.jcp.2009.06.015
  * (38) Kornhaas, M., Sternel, D.C., Schäfer, M.: In: Meyers, J., Geurts, B.J., Sagaut, P. (eds.) Quality and Reliability of Large-Eddy Simulation. Influence of Time Step Size and Convergence Criteria on Large Eddy Simulations with Implicit Time Discretization, pp. 119–130. Springer, Dordrecht (2008). https://doi.org/10.1007/978-1-4020-8578-9_10
  * (39) Mockett, C., Fuchs, M., Thiele, F.: Progress in DES for wall-modelled LES of complex internal flows. Computers and Fluids 65, 44–55 (2012). https://doi.org/10.1016/j.compfluid.2012.03.014
  * (40) Eisfeld, B., Brodersen, O.: Advanced Turbulence Modelling and Stress Analysis for the DLR-F6 Configuration. In: 23rd AIAA Applied Aerodynamics Conference, Toronto, Canada (2005). https://doi.org/10.2514/6.2005-4727
  * (41) Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation 31(138), 333–390 (1977). https://doi.org/10.2307/2006422
  * (42) Schrijer, F.F.J., Solana Perez, R., van Oudheusden, B.W.: Investigation of transonic buffet using high speed PIV. In: 5th International Conference on Experimental Fluid Mechanics (ICEFM), Munich, Germany, 2018
  * (43) D’Aguanno, A., Schrijer, F., Oudheusden, B.: Experimental investigation of the transonic buffet cycle on a supercritical airfoil. Experiments in Fluids 62 (2021). https://doi.org/10.1007/s00348-021-03319-z
  * (44) Accorinti, A., Baur, T., Scharnowski, S., Kähler, C.J.: Experimental Investigation of Transonic Shock Buffet on an OAT15A Profile. AIAA Journal 60(11), 6289–6300 (2022). https://doi.org/10.2514/1.J061135
  * (45) Schauerte, C., Schreyer, A.-M.: Characterization of Shock Buffet on a Supercritical 2D Airfoil in Transonic Flow. In: 20th International Symposium on Applications of Laser and Imaging Techniques to Fluid Mechanics (2022)
  * (46) Welch, P.D.: The use of the fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Transactions on Audio Electroacoustics 15(2), 70–73 (1967). https://doi.org/10.1109/TAU.1967.1161901
  * (47) Schauerte, C., Schreyer, A.-M.: Experimental analysis of transonic buffet conditions on a two-dimensional supercritical airfoil. AIAA Journal (2023). https://doi.org/10.2514/1.J062349
  * (48) Lumley, J.L., Newman, G.R.: The return to isotropy of homogeneous turbulence. Journal of Fluid Mechanics 82(1), 161–178 (1977). https://doi.org/10.1017/S0022112077000585
  * (49) Choi, K.-S., Lumley, J.L.: The return to isotropy of homogeneous turbulence. Journal of Fluid Mechanics 436, 59–84 (2001). https://doi.org/10.1017/S002211200100386X
  * (50) Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V., Ukeiley, L.S.: Modal analysis of fluid flows: An overview. AIAA Journal 55(12), 4013–4041 (2017). https://doi.org/10.2514/1.J056060