# Effect of Tripping and Domain-Width on Transonic Buffet on Periodic NASA-CRM
Airfoils

David J. Lusher111JSPS Research Fellow, Aircraft Lifecycle Innovation Hub,
JAXA Chofu Aerospace Center, Tokyo. Japan Aerospace Exploration Agency (JAXA),
7-44-1 Jindaiji Higashi-machi, Chofu-shi, Tokyo 182-8522, Japan  Andrea
Sansica222Researcher, Aircraft Lifecycle Innovation Hub, JAXA Chofu Aerospace
Center, Tokyo. Japan Aerospace Exploration Agency (JAXA), 7-44-1 Jindaiji
Higashi-machi, Chofu-shi, Tokyo 182-8522, Japan  and Atsushi Hashimoto
333Manager, Aircraft Lifecycle Innovation Hub, JAXA Chofu Aerospace Center,
Tokyo. Japan Aerospace Exploration Agency (JAXA), 7-44-1 Jindaiji Higashi-
machi, Chofu-shi, Tokyo 182-8522, Japan

###### Abstract

Transonic buffet is an instability characterized by shock-oscillations and
separated boundary-layers. High-fidelity simulations have typically been
limited to narrow domains to be computationally feasible, overly constraining
the flow and introducing modelling errors. Depending on the boundary-layer
state upstream of the interaction, different buffet features are observed.
High-fidelity simulations were performed on the periodic NASA-CRM wing at
moderate Reynolds number to assess sensitivity of the two-dimensional
transonic buffet to boundary-layer state and domain width. Simulations were
cross-validated against RANS/URANS and global stability analysis and excellent
agreement was found near the onset. By varying the boundary-layer tripping
amplitude, laminar, transitional, and turbulent buffet interactions were
obtained. All cases consisted of a single shock and low-frequency oscillations
$(St\approx 0.07)$. The transitional interaction also exhibited reduced shock
movement, a $15\%$ increase in $\overline{C_{L}}$, and energy content at
higher frequencies $(St\approx 1.3)$. Span-wise domain studies showed
sensitivity at the shock location and near the trailing edge. We conclude that
the span-width must be greater than the trailing-edge boundary-layer thickness
to obtain span-independent solutions. For largely separated cases, the
sensitivity to span-width increased and variations across the span were
observed. This was found to be associated to a loss of two-dimensionality of
the flow.

## Nomenclature

$c$ = | Airfoil Chord Length
---|---
$u,v,w$ = | Velocity Components
$x_{t}$, $A$, $\sigma$ = | Trip Location, Trip Amplitude, and Gaussian Scaling Factor
$x,y,z$ = | Cartesian Coordinates
$AR$ = | Aspect Ratio of the Airfoil, $AR=L_{z}/c$
$C_{L},C_{D},C_{f}$ = | Lift, Drag, and Skin-friction Coefficients
$C_{Dp},C_{Df}$ = | Pressure and Skin-friction Drag components
$L_{z}$ = | Airfoil Spanwise Length
$M_{\infty}$ = | Freestream Mach number
$Pr$ = | Prandtl number
$Re$ = | Reynolds number
$St$ = | Strouhal Number (non-dimensional frequency)
$T_{\textrm{ref}},T_{\textrm{Suth}}$ = | Reference and Sutherland’s temperature
$\alpha$ = | Angle of Attack (degrees)
$\gamma$ = | Ratio of Specific Heat Capacities
$k_{i},\Phi_{i},\omega_{i}$ = | Tripping Wavenumbers, Phases, and Frequencies
$\rho,p,T,\rho E,\mu$ = | Density, Pressure, Temperature, Total Energy, and Viscosity
$\xi,\eta,\zeta$ = | Curvilinear Coordinates
$\Delta t$ = | Computational Time-step

##  1 Introduction

Transonic buffet is an aerodynamic instability related to shockwave/boundary-
layer interactions (SBLI) [1] and is characterized by periodic self-sustained
shock oscillations and boundary-layer separation [2, 3]. This instability is
relevant at the boundaries of the flight envelope of commercial aircraft,
namely for high speeds and high angles of attack. Transonic shock buffet can
cause large amplitude oscillations in lift and drag, leading to reduced
control and increased fatigue and failure rates [4]. Buffet is known to
consist of both a two-dimensional (2D) chord-wise shock oscillation
instability, and three-dimensional (3D) cross-flow outboard propagating
cellular separation patterns, known as buffet/stall-cells. While these two
instabilities can coexist under specific conditions on 3D idealized wings [5,
6, 7, 8], they have been traditionally categorized based on geometrical
features. The 2D shock-oscillation instability is generally linked to 2D
airfoils and 3D unswept infinite wings [9, 6, 10], and the 3D buffet cells one
is related to 3D swept infinite wings [11, 7, 12, 13, 14], finite wings [15]
and full aircraft [16, 17, 18, 19] configurations. The debate on the
applicability of state-of-the-art solvers for industry relevant
configurations, sustained by communities such as the AIAA Drag Prediction
Workshop [20, 21], is driving the interest towards 3D configurations, hence
with more focus on the buffet cells instability. However, the fundamental
mechanisms of the 2D shock-oscillations are not yet fully understood and
require further scrutiny.

While earlier studies proposed feedback loop models based on upstream and
downstream travelling waves from the trailing edge and shock foot [2] to
explain the 2D shock oscillations, more recent studies have linked the origins
of transonic buffet to a global instability [6, 22]. In this scenario, the
onset of the unsteadiness is the result of a Hopf bifurcation [6], and the
instability is localized to the region around the shock and partially in the
separated shear layer [22]. Despite some rectifications of the feedback loop
model [23, 9, 24], the explanation based on global stability remains the most
popular but does not clarify the full mechanisms that lead to self-sustained
shock oscillations. In this regard, Iwatani et al. [25] recently used
resolvent analysis to suggest that shock induced separation height and
pressure dynamics around the shock-wave both contribute to the self-sustained
oscillations.

To accurately calculate the 2D buffet instability, the prediction of the
separation size is of fundamental importance and a variety of approaches have
been extensively used for this purpose. The computational cost for scale-
resolving methods applied to buffet can be prohibitive, and most numerical
studies have therefore been limited to using low-fidelity approaches such as
those used in Reynolds-Averaged Navier-Stokes (RANS) solvers, or mixed-
fidelity hybrid RANS/Large-Eddy Simulation (LES). Examples of low-
fidelity/reduced-cost methods applied to two-dimensional buffet include:
Unsteady RANS (URANS) [26], modal decomposition and system identification [6,
7, 27, 28] and hybrid methods, such as Detached Detached-Eddy Simulation
(DDES) [29] and Zonal DES (ZDES) [23], among others. Lower-fidelity methods
are generally suitable to predict 2D buffet characteristics, and have shown
good agreement to experiment [23, 9]. However, as shown by Thiery and Coustols
[26], predictions obtained by RANS-based solvers can be sensitive to the
turbulence model used. The review of Giannelis et al. [4] further commented
that URANS simulations exhibit high sensitivity to simulation parameters,
turbulence model, and both the spatial and temporal discretisation methods
used. Finally and just as importantly, RANS models are also known to perform
poorly in the presence of large flow separation. These factors all motivate
the need for high-fidelity studies of buffet, which explicitly resolve the
energy-containing eddies for more accurate predictions of SBLI and unsteady
shock-induced separation [30].

The work of Memmolo et al. [31] compared URANS and DDES methods to scale-
resolving Implicit LES (ILES), for the V2C supercritical laminar wing at
Reynolds number ($Re$, based on free-stream quantities and airfoil chord, $c$)
of $Re=3\times 10^{6}$. ILES [32] (or, alternatively, under-resolved Direct
Numerical Simulation) is the approach taken in the current work. ILES does not
require additional turbulence modelling, as the governing equations are solved
directly, with the numerical dissipation from the shock-capturing scheme and
filters acting as a sub-grid scale model [33, 32, 31, 34]. For high-fidelity
airfoil simulations at both low [10] and high [35] speeds, the size of the
separation is shown to be sensitive to the airfoil span-width ($L_{z}$), that
therefore needs to be appropriately selected. At high angle of attack low-
speed stall, Aihara and Kawai [10] tested domain sensitivity for separated
airfoils at aspect ratios ($AR=L_{z}/c$) of $AR=0.05,0.1,0.2$. It was found
that the narrow domains caused excessive vertical movement of the separated
shear-layer and the overly-constrained wake vortices failed to become fully
three-dimensional, leading to large fluctuations in aerodynamic coefficients
and predictions sensitive to the domain width used. Similarly, the stall
simulations of Turner and Kim [36] tested span-widths up to $100\>\%$ of chord
length ($AR=1$), to assess the effect of domain width on aeroacoustic
predictions. It was shown that even relatively wide $AR=0.2$ domains over-
predicted noise generation by $10\>dB$ compared to $AR=1$. Aside from the
domain-width tests between $AR=0.0365$ and $AR=0.073$ of Garnier and Deck
[35], less attention has been given to domain-width sensitivity in the high-
speed buffet regime, motivating the need for the present work. Most of the
studies in the literature set airfoil span-widths based on general know-how or
referring to studies on similar flow configurations, without directly
addressing the domain width sensitivity problem. Due to computational cost,
these kind of high-fidelity simulations of periodic wings have typically been
limited to narrow aspect ratios. Some relevant examples for 2D buffet studies
used $AR=0.0365-0.073$ [35], $AR=0.065$ [30, 37], $AR=0.05$ [38, 39], and,
most recently, $AR=0.25$ [40]. Garnier and Deck [35] doubled the spanwise
width of their simulations from 3.65% to 7.3% and showed significant reduction
of the pressure fluctuations at the trailing edge [4]. The wider domain
‘better captures trailing edge pressures by allowing three-dimensional
coherent structures to develop’ [4]. This highlights that while scale-
resolving simulations explicitly resolve turbulence and should lead to more
accurate buffet predictions, this cannot be at the expense of overly-narrow
domains which artificially constrain the problem and introduce an alternative
form of modelling error by imposed two-dimensionality on turbulent structures.
Two wide-span high-fidelity outliers are the ILES of Memmolo et al. [31]
($AR=0.5$) and that of Zauner and Sandham [41] ($AR=1$), although the upstream
boundary-layer was un-tripped in these cases.

Another important consideration that affects separation size (and consequently
the selection of an appropriate airfoil domain width) is the nature of the
incoming boundary-layer. In the present work, we categorize the interaction
type by the state of the boundary-layer upstream of the shock-wave. Based on
this definition, transonic buffet can be separated into laminar buffet [42,
41, 38, 40], and turbulent buffet [30, 37]. Turbulent buffet is characterized
by streamwise shock-wave oscillations in the Strouhal number range of
$St=[0.06,0.1]$ [5] ($St$, based on the freestream velocity $U_{\infty}^{*}$
and airfoil chord $c^{*}$). The shock-wave terminates a large region of
supersonic flow on the suction side of the airfoil. A complex inviscid/viscous
coupling between the shock and the shear-layer lead to a periodically
separating and reattaching boundary-layer. In contrast to turbulent buffet,
laminar buffet typically has more extensive regions of flow separation, and
additional higher frequencies present ($St=1-1.2$), which have been observed
both experimentally [43] and computationally [42] at $Re=3\times 10^{6}$.
These higher frequencies have been linked to a breathing of the separation
bubble at the shock foot. Later complementary experiments [44] showed the same
high-frequency instability, accompanied by a low-frequency shock oscillation
with $St=0.05$. The low-frequency shock oscillations were similar to those
observed in the turbulent interaction. In a separate computational study,
Moise et al. [39] simulated cases at a moderate Reynolds number of $5\times
10^{5}$ with both free- and forced-transition, observing similar ranges of
low-frequency peaks for the two-dimensional instability in both cases. Despite
the aforementioned differences between laminar and turbulent buffet, Moise et
al. [39] argued that the underlying mechanism shares the same origin. However,
it should also be noted that in the free-transition simulations of Moise et
al. [38, 39], the laminar interaction was comprised of multiple shock-
wave/expansions across the suction side of the airfoil, which is in contrast
to the single shock-wave configurations found in laminar simulations and
experiments of [43, 42, 44]. The authors of [38] showed that the number of
shocks decreased with increasing Reynolds number, and suggested the
differences in shock topology in the laminar cases may be due to a Reynolds
number effect, consistent with studies at higher Reynolds number [43, 42, 44].
Garbaruk et al. [45] applied RANS and GSA to assess the impact of extended
laminar flow on infinite-span wings at buffet onset. They found that extending
the laminar flow region on the suction side led to a $5-6\%$ increase in lift,
while the boundary-layer state on the pressure side had a negligible effect.
The present study focuses mainly on turbulent transonic buffet, as it is the
most commonly found in experiments and industrial aviation applications [16,
46, 18, 47]. However, the sensitivity and response of buffet to upstream
boundary-layer state are not entirely clear. Given the potential fuel
efficiency improvements associated with the use of laminar wings to meet
climate objectives, laminar and transitional interactions in the transonic
regime are of great importance and require further attention.

Within the framework presented above, the focus in this paper is on
determining the sensitivity of 2D transonic buffet to numerical tripping and
domain width, for cases characterized by different separation sizes. The
objectives of this work are to assess:

  * •

The sensitivity of turbulent transonic buffet to tripping strength and, as a
result, to different separation sizes determined by laminar, transitional, and
turbulent shock-wave/boundary-layer interactions.

  * •

The potential influence of spanwise domain width on the 2D buffet shock-
oscillation instability in high-fidelity 3D (ILES) simulations, for buffet
conditions with moderate and large separations.

  * •

The extent to which low-fidelity industry-relevant RANS/URANS and modal
stability (GSA) methods can correlate to data from high-fidelity simulations
at greatly reduced computational cost.

The current work is organized as follows. Section 2.1 and Section 2.2 describe
the solvers used for ILES (OpenSBLI [48]) and RANS/URANS/GSA (FaSTAR [49]),
respectively. Section 2.3 provides information about the computational meshes
used, with simulation parameters and flow conditions given in Section 2.4.
Section 3 performs initial ILES simulations to identify pre- and post-onset 2D
buffet conditions. This is then compared to steady-RANS, GSA and URANS
predictions in Section 3.1. ILES simulations are then used in Section 4 and
Section 5 to assess sensitivity to tripping strength and span-wise domain
width, respectively. The sensitivity to domain span-width is first determined
at moderate angles of attack (Section 5.1), where the flow is mostly attached
in a time-averaged sense, and then at a higher angle of attack (Section 5.2),
where there is extensive time-averaged flow separation present. Finally,
conclusions are given in Section 6.

##  2 Computational Method

###  2.1 OpenSBLI high-fidelity (ILES) solver

All high-fidelity simulations in this work were performed in OpenSBLI [48], an
open-source high-order compressible multi-block flow solver on structured
curvilinear meshes. OpenSBLI was developed at the University of Southampton
[50, 48] and the Japan Aerospace Exploration Agency (JAXA) [51] to perform
high-speed aerospace research. Written in Python, OpenSBLI utilises symbolic
algebra to automatically generate a complete finite-difference CFD solver in
the Oxford Parallel Structured (OPS) [52] Domain-Specific Language (DSL). The
base governing equations are the non-dimensional compressible Navier-Stokes
equations for an ideal fluid. Applying conservation of mass, momentum, and
energy, in the three spatial directions $x_{i}$ $\left(i=0,1,2\right)$,
results in a system of five partial differential equations to solve. These
equations are defined for a density $\rho$, pressure $p$, temperature $T$,
total energy $E$, and velocity components $u_{k}$ as

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle+\frac{\partial}{\partial x_{k}}\left(\rho u_{k}\right)=0,$ (1)
$\displaystyle\frac{\partial}{\partial t}\left(\rho u_{i}\right)$
$\displaystyle+\frac{\partial}{\partial x_{k}}\left(\rho
u_{i}u_{k}+p\delta_{ik}-\tau_{ik}\right)=0,$ (2)
$\displaystyle\frac{\partial}{\partial t}\left(\rho E\right)$
$\displaystyle+\frac{\partial}{\partial x_{k}}\left(\rho
u_{k}\left(E+\frac{p}{\rho}\right)+q_{k}-u_{i}\tau_{ik}\right)=0,$ (3)

with heat flux $q_{k}$ and stress tensor $\tau_{ij}$ defined as

$q_{k}=\frac{-\mu}{\left(\gamma-1\right)M_{\rm{ref}}^{2}PrRe}\frac{\partial
T}{\partial x_{k}},$ (4) $\tau_{ik}=\frac{\mu}{Re}\left(\frac{\partial
u_{i}}{\partial x_{k}}+\frac{\partial u_{k}}{\partial
x_{i}}-\frac{2}{3}\frac{\partial u_{j}}{\partial x_{j}}\delta_{ik}\right).$
(5)

$Pr$, $Re$, and $\gamma=1.4$ are the Prandtl number, Reynolds number and ratio
of specific heat capacities for an ideal gas, respectively. Support for
curvilinear meshes is provided by using body-fitted meshes with a coordinate
transformation. The equations are non-dimensionalized by a reference velocity,
density and temperature
$\left(U^{*}_{\rm{ref}},\rho^{*}_{\rm{ref}},T^{*}_{\rm{ref}}\right)$. For a
reference Mach number $M_{\rm{ref}}$, the pressure is defined as

$p=\left(\gamma-1\right)\left(\rho E-\frac{1}{2}\rho
u_{i}u_{i}\right)=\frac{1}{\gamma M^{2}_{\rm{ref}}}\rho T,$ (6)

OpenSBLI is explicit in both space and time, with a range of different
discretisation options available to users. Spatial discretisation is performed
in this work by 4 order central differences recast in a cubic split form [53]
to boost numerical stability. Time-advancement is performed by a 4-order
5-stage low-storage Runge-Kutta scheme [54]. Dispersion Relation Preserving
(DRP) filters [55] are applied to the freestream using a targeted filter
approach which turns the filter off in well-resolved regions to further reduce
numerical dissipation [51]. The DRP filters are also only applied once every
25 iterations. Shock-capturing is performed via Weighted Essentially Non-
Oscillatory (WENO) schemes, specifically the 5-order WENO-Z variant by [56].
The effectiveness and resolution of the underlying shock-capturing schemes in
OpenSBLI was assessed for the compressible Taylor-Green vortex case involving
shockwaves and transition to turbulence in [57] and for compressible wall-
bounded turbulence in [58]. The shock-capturing scheme is applied within a
characteristic-based filter framework [59, 60]. The dissipative part of the
WENO-Z reconstruction is applied at the end of the full time-step to capture
shocks based on a modified version of the Ducros sensor [61, 62]. Similar
central plus filter step shock-capturing schemes have been used for a number
of buffet studies, most recently in the turbulent cases of Moise et al. [39].
The numerical methods in OpenSBLI were validated for laminar-transitional
buffet cases compared to literature on the V2C airfoil geometry in [51].

###  2.2 FaSTAR low-fidelity (RANS/URANS/GSA) solver

Comparisons to the high-fidelity ILES data are provided by the FaSTAR
unstructured mesh CFD solver [49, 63] developed at JAXA. A cell-centered
finite volume method is used for the spatial discretisation of the
compressible 3D RANS equations. The numerical fluxes are computed by the
Harten-Lax-van Leer-Einfeldt-Wada (HLLEW) scheme [64] and the weighted Green-
Gauss method is used for the gradient computation [65]. For the mean flow and
transport equations, the spatial accuracy is set to the second and first
order, respectively. The turbulence model selected for the present simulations
is the Spalart-Allmaras turbulence model [66] with rotation correction (SA-R)
[67] and quadratic constitutive relation 2000 version (SA-R-QCR2000) [68]. No-
slip velocity and adiabatic temperature boundary conditions are imposed on the
wing walls; far-field boundary conditions are employed at the outer
boundaries, and the angle of attack is applied to the incoming flow.

To compute the steady solutions, the Lower/Upper Symmetric Gauss-Seidel (LU-
SGS) [69] scheme with a Courant–Friedrichs–Lewy (CFL)-fixed local time step is
used. The CFL number is 50 and all simulations are run until convergence.
Iterative convergence is here defined as a) when the residuals of the state
variables in the $L_{2}$-norm are at least below $10^{-8}$ and b) the
variations of the lift-coefficient are below the fourth decimal digit. To
promote iterative convergence, the selective frequency damping (SFD) method
[70, 71] is used in the LU-SGS pseudo-time integration, as previously done by
Plante et al. [13]. For the unsteady calculations, dual-time stepping [72] is
used to improve accuracy of the implicit time integration method. The LU-SGS
scheme is used for the pseudo-time sub-iterations and the physical time
derivative is approximated by the three-point backward difference.

To perform GSA, the linearized URANS equations need to be derived and solved.
These equations assume the existence of a steady equilibrium solution,
referred to as base flow, to perform the stability calculation. In the URANS-
based GSA framework, the base flow coincides with the (steady) RANS solution.
The details and validation of the solver linearization and Jacobian-free
Newton-Krylov method used for the modal decomposition are thoroughly described
in Sansica and Hashimoto [19].

###  2.3 Geometry and mesh configuration

The selected geometry is the 65% semispan station of the NASA Common Research
Model (CRM) wing, commonly used for turbulent transonic buffet research. Two-
dimensional body-fitted structured meshes are created in Pointwise™. The
three-dimensional mesh is generated by extruding the two-dimensional grid in
the spanwise direction with uniform spacing. Figure 1 shows the meshes used by
the (a) ILES simulations in OpenSBLI, and (b) RANS/URANS/GSA simulations in
FaSTAR, plotted at every 7 and 5 line respectively for visualization purposes.

Figure 1: Visualisation of the mesh configurations for the NASA-CRM-65 [73]
airfoil used in this study at $\alpha=5^{\circ}$. The ILES grid is comprised
of a C-mesh and two wake blocks, with boundary-layer trips placed at $0.1c$.
The URANS grid is comprised of a single block O-mesh. For illustrative
purposes, the ILES and URANS grids are plotted at every 7 and 5 grid line,
respectively.

In the case of OpenSBLI, an airfoil C-mesh is connected to two wake blocks
with a sharp trailing edge configuration. The in-flow boundary is set at a
distance of $25c$ with the outlet $5c$ downstream of the airfoil. The inflow
is set to be uniform $u_{\infty}=1$, with the angle of attack prescribed by
rotating the airfoil within the mesh. For each case, the near-wake mesh is
also slightly modified to take into account the deflection of the wake based
on the AoA. In the $\xi$ and $\eta$ directions clockwise around the airfoil
and normal to the surface, the airfoil and wake blocks have
$\left(2249,681\right)$ and $\left(701,681\right)$ points, respectively.
Around the airfoil, the pressure and suction sides have 500 and 1749 points in
the $\xi$ direction, respectively. The $\xi$ distribution is refined between
$0.3<x<0.6$ on the suction side to improve the resolution at the main shock-
wave and SBLI. The domain width is initially set to a narrow value of $5\%$
chord length with $N_{z}=50$ points to compare to steady RANS predictions.
Upstream of the main shock-wave at $x=0.4$, the grid has wall units of
$\left(\Delta x^{+},\Delta y^{+},\Delta
z^{+}\right)=\left(6.1,2.2,14.8\right)$, and in the attached turbulent region
downstream of the shock reaches a maximum at $x=0.7$ of $\left(\Delta
x^{+},\Delta y^{+},\Delta z^{+}\right)=\left(3.9,1.1,7.6\right)$. In addition
to wall criteria, it is important to maintain good resolution throughout the
entire boundary-layer by applying only weak grid stretching. At $x=0.4$ and
$x=0.7$ there are 80 and 195 points in the boundary-layer, respectively. A
spanwise grid study at buffet conditions is shown in appendix 6.1, with the
medium resolution selected for the cases shown in the rest of this work.
Aerodynamic coefficients, pressure distributions and skin-friction are all
span-averaged unless otherwise mentioned.

In the case of the cell-centered finite volume FaSTAR solver, the blunt
trailing-edge version of the CRM wing is used. The numerical mesh is obtained
by first defining the distribution of cells around the airfoil and then by
normal extrusion to obtain a single block O-grid. The number of cells in the
$\xi$ and $\eta$ directions is $\left(1050,162\right)$. The distribution
around the airfoil consists of $600$ and $400$ cells on suction and pressure
sides respectively, and $50$ cells are used to discretize the blunt trailing
edge. A region of chord-wise width equal to $0.2c$ is refined around the shock
and counts $200$ cells. To account for different shock locations, this
refinement region changes chord-wise position depending on the angle of
attack. The domain boundaries extend to about $100$ chords from the airfoil in
all directions. Unless specified otherwise, all FaSTAR grids are purely 2D
(i.e. only one cell in the spanwise direction).

###  2.4 Flow parameters, computational setup and initial conditions

All simulations were performed at a moderate Reynolds number of $Re=500,000$
based on airfoil chord length and freestream Mach number of $M_{\infty}=0.72$.
The non-dimensional time-step is set as $\Delta t=5\times 10^{-5}$ for all
OpenSBLI ILES cases. The ILES simulations are advanced from uniform flow
conditions for 20 time units until the boundary-layer is fully turbulent and
the buffet unsteadiness fully develops.

In order to investigate turbulent transonic buffet, numerical tripping must be
applied to the oncoming boundary-layer to promote a fast transition to
turbulence upstream of the shockwave. This is achieved by forcing a set of
unstable modes as a time-varying blowing/suction strip near the leading edge
of the airfoil. This type of forcing is commonly used in CFD research as a
method to mimic arrays of tripping dots used in experiments [46, 47]. The
forcing strip is centred around the $0.1c$ location on both the suction and
pressure sides of the airfoil. The forcing is applied to the wall-normal
velocity component, which is then used to set the momentum and total energy on
the wall. Outside of the forcing strip the wall is a standard isothermal no-
slip viscous boundary condition. The forcing is taken to be a modified form of
the one given in [39] as

$\rho
v_{w}=\sum_{i=1}^{3}A\exp\left(-\frac{\left(x-x_{t}\right)^{2}}{2\sigma^{2}}\right)\sin\left(\frac{k_{i}z}{0.05c}\right)\sin\left(\omega_{i}t+\Phi_{i}\right),$
(7)

for simulation time $t$, trip location $x_{t}$, and Gaussian scaling factor
$\sigma=0.00833$. The three modes $\left(0,1,2\right)$ have spatial
wavenumbers of $k_{i}=\left(6\pi,8\pi,8\pi\right)$, phases
$\Phi_{i}=\left(0,\pi,-\pi/2\right)$, and temporal frequencies of
$\omega_{i}=\left(26,88,200\right)$. The tripping strength is initially set to
$7.5\%$ of the freestream ($A=0.075$), to initiate the transition to
turbulence. The sensitivity of the 2D buffet instability to this tripping
strength parameter is investigated in section 4 over a range of $0.5\%$ to
$10\%$ of freestream velocity.

##  3 Identifying buffet conditions on the NASA-CRM profile

High-fidelity simulations are first carried out on narrow-span ($AR=0.05$)
domains to identify the Angle of Attack (AoA, $\alpha$) required to reach
buffet onset for the 2D periodic shock oscillation instability. Figure 2 shows
$x-t$ diagrams of skin-friction for (a) $\alpha=4^{\circ}$ (pre-onset) and (b)
$\alpha=5^{\circ}$ (post-onset) conditions. The diagrams show the time history
of attached ($C_{f}\geq 0$, white) and separated ($C_{f}<0$, blue) flow
regions averaged over the span. The solid black line represents the first
streamwise location of flow separation at that time instance. The side plot
shows the corresponding lift coefficient history. The $C_{L}$ comparison shows
that for the current conditions on the CRM profile, the 2D buffet onset occurs
between $\alpha=[4^{\circ},5^{\circ}]$. Once the AoA is raised sufficiently
high, large periodic oscillations are observed, consistent with previous
buffet studies [4, 39]. At $\alpha=4^{\circ}$, the shock location corresponds
to the vertical dashed red line showing a mean separation location of
$x=0.484$. There is a small region of flow separation at the shock position,
before the flow reattaches at around $x=0.6$. The flow then remains attached
until close to the trailing edge, where it separates due to the curved
geometry at the back of the airfoil. As the angle of attack is increased to
$\alpha=5^{\circ}$, despite a similar mean separation location, the shock and
separation point begin to oscillate periodically across the surface.
Furthermore, the lift coefficient shows oscillations of $\pm 10\%$ of mean
$C_{L}$. At post-onset conditions the region immediately downstream of the
shock-wave ($0.5<x<0.8$) undergoes periodic separation and reattachment as
part of the 2D buffet instability. The lift coefficient plot shows these
correspond to the low- and high-lift phases of the buffet cycle, respectively.

Figure 2: Skin-friction $\left(x-t\right)$ diagrams showing the position of
the shockwave and separated/attached flow at (a) $\alpha=4^{\circ}$ (pre-
onset) and (b) $\alpha=5^{\circ}$ (buffet) conditions at $AR=0.05$. The side
plot shows the lift time-histories.

Figure 3 shows instantaneous Mach number contours for the $\alpha=5^{\circ}$
case at buffet conditions. The solid black line shows the $M=1$ sonic
condition that borders the supersonic flow on the suction side of the
aerofoil. In transonic buffet, the shock-wave oscillates periodically on the
surface of the aerofoil, resulting in large oscillations in lift and drag. The
high- and low-lift phases are shown in Figure 3 (a) and (b), respectively.
During the low-lift phase, the supersonic region decreases in extent as the
shock-wave moves forward and the amount of flow separation increases. The flow
then reattaches and the lift generation increases during the next phase of the
cycle. These periodic oscillations repeat without any external forcing and are
associated with the non-linear saturation of a linear global instability. The
AR005-AoA5 ($AR=0.05$ and $\alpha=5^{\circ}$) case is used as a baseline
narrow domain result at onset conditions. Before proceeding to investigations
of upstream boundary-layer state and buffet with wider span-widths, the next
section first cross-validates the data and 2D buffet onset condition to those
obtained by RANS-based solutions and predictions from GSA.

Figure 3: Illustration of the (a) high- and (b) low-lift phases of the buffet
cycle for the $\alpha=5^{\circ}$ case on the $AR=0.05$ narrow-span domain. The
solid black line outlines the $M=1$ sonic condition.

###  3.1 Cross-validation with low-fidelity methods

Having obtained a baseline ILES case from OpenSBLI [50, 48], this section
compares the data to low-fidelity solutions from the FaSTAR [49, 63] solver.
The aim is both to cross-validate the data before proceeding with the
simulations for the rest of this study, and also to assess how well low-
fidelity RANS/URANS and stability methods agree with the more expensive ILES
approach. We first compare time-averaged pressure distributions from the
narrow-span ILES ($AR=0.05$) to steady RANS solutions at pre- and post-onset.
The RANS solutions are then used as a base flows to perform GSA in the range
$\alpha=\left[4,5^{\circ}\right]$, to obtain a prediction of the buffet onset
angle of attack and related shock-oscillation frequencies. Finally, URANS is
performed on the same configuration to compare the predicted buffet
frequencies to those obtained from both the ILES and GSA approaches.

####  3.1.1 Comparison between RANS and ILES at pre- and post-onset

Figure 4: OpenSBLI (ILES, time-averaged) to FaSTAR (Steady-RANS) comparison of
pressure distribution for (a) $\alpha=4^{\circ}$ (pre-onset) and (b)
$\alpha=5^{\circ}$ (post-onset) conditions.

Figure 4 shows RANS solutions compared to the time- and span-averaged $C_{p}$
distributions for the ILES cases in section 3. At pre-onset conditions
($\alpha=4^{\circ}$) (Figure 4-a), the steady-RANS solution matches the ILES
data very well, despite the two simulations differing on several aspects. As
well as the obvious differences in the numerical approaches, domain topology,
grid resolution and trailing edge geometry, one should be reminded that the
RANS solver assumes the flow to be fully turbulent, whereas the ILES data is
initially laminar and then tripped to turbulence (the tripping in the ILES
simulations is indicated by the wiggles at 10% chord on both pressure and
suction sides). As shown in the previous section, the ILES results at
$\alpha=5^{\circ}$ show the occurrence of the 2D buffet instability. Due to
the steady nature of the calculation, the RANS solution is incapable of
reproducing the unsteadiness and the pressure gradient around the shock is
steeper with respect to the ILES one (figure 4-b), that appears smeared over a
wider $x$-range because of the time-averaging. For the rest of the airfoil,
the agreement between RANS and ILES is again excellent.

####  3.1.2 GSA-based 2D buffet onset prediction

Global stability analysis is performed in the
$\alpha=\left[4,5^{\circ}\right]$ range to determine the onset angle of attack
and shock oscillation frequencies with a tolerance of
$\Delta\alpha=0.1^{\circ}$. For each angle of attack investigated, a base flow
solution is obtained by sufficiently converging the RANS solver. The
linearized URANS solver is then used for the stability calculation and
advanced in time with an integration step of $\Delta t=0.005$ (corresponding
to a dimensional time step $\Delta t^{*}=1.48\times 10^{-3}\>s$) and 40 sub-
iterations for the pseudo-time integration of the dual time stepping method.
The Krylov base used for the stability modal decomposition is constructed by
collecting 128 perturbation fields every $\Delta T_{kryl}=450\times\Delta t$
(Krylov time step). The GSA calculations were restarted from the last
perturbation field of a preliminary simulation and all unstable modes
presented in this section have residuals at least below $10^{-5}$.

Figure 5: Summary of the GSA results indicating (a) the evolution of the
growth rate with the angle of attack and (b) the predicted eigen-spectra (with
frequencies in both dimensional and non-dimensional forms) for the
$\alpha=5^{\circ}$ case. The shaded region indicates stable solutions.

(a)

(b)

Figure 6: The (a) RANS/base flow solution and (b) density eigen-function
corresponding to the temporally amplified mode at $St=0.068$ are plotted for
the $\alpha=5^{\circ}$ case.

Figure 5(a) shows the evolution of the least temporally damped / most
temporally amplified mode for each angle of attack investigated. The first
angle of attack with a temporally amplified mode (or with a positive growth
rate) is found at $\alpha=4.5^{\circ}$, indicating 2D buffet onset. For the
case at $\alpha=5^{\circ}$ previously discussed, the full spectra is reported
in Figure 5(b). A single temporally amplified mode is present at $St=0.068$
($f=16.53\>Hz$), confirming that the $\alpha=5^{\circ}$ case is unstable and
shock oscillations must be present, as shown by the ILES calculations. The
sonic (black) and the zero-streamwise velocity (white) isolines are
superimposed to the Mach number contours in Figure 6(a) for the (unstable)
base flow solution at $\alpha=5^{\circ}$. The density eigen-function
corresponding to the temporally amplified mode at $St=0.068$ is plotted in
Figure 6(b) and shows the typical perturbation distribution responsible for
the 2D buffet shock instability, as seen in the literature [6, 22].

####  3.1.3 Comparison between URANS, GSA and ILES at post-onset conditions

The GSA-based onset prediction determined unstable flow conditions for angles
of attack above $\alpha=4.5^{\circ}$. To verify the stability results, URANS
calculations are carried out at post-onset conditions for the
$\alpha=5^{\circ}$ case. The same time-integration parameters used for the GSA
calculations are here selected. In Figure 7, the URANS result is compared to
ILES data from both the narrow-span ($AR=0.05$) simulation in the previous
section, and one computed on a wider airfoil with $AR=0.50$. The time-
histories of (a) lift coefficient, (b) drag coefficient, and (c) PSD of lift
fluctuations show excellent agreement between the low- and high-fidelity
methods, especially for the wider aspect ratio airfoil (as it will be shown
later, the $AR=0.50$ case is found to be domain span-width independent).
Despite some differences in the mean aerodynamic force values and oscillation
amplitudes, the low-fidelity methods can accurately reproduce the dominant
frequencies at $St\approx 0.077$. As expected, the GSA-predicted frequency is
slightly under-estimated, due to the differences between the (steady) base
flow solution used for the stability analysis and the URANS time-averaged
field far from the onset.

Figure 7: Unsteady OpenSBLI (ILES) data for $\alpha=5^{\circ}$ at $AR=0.05$
and $AR=0.50$, compared to URANS and GSA results from FaSTAR. Showing (a) lift
coefficient, (b) drag coefficient, and (c) PSD of lift fluctuations. The
horizontal dashed lines in the lift and drag coefficient plots indicate time-
averaged values.

####  3.1.4 Discussion on the comparison between low- and high-fidelity
methods

While the low-fidelity methods showed remarkable agreement, it should be
emphasized that the angles of attack here examined fall in a narrow range
around the instability onset. As it will be seen later, when the angle of
attack is further increased, the 2D URANS solutions are no longer
representative of the physics at play and show significant differences to the
ILES results and 3D URANS solutions. Because of the larger separation and
stronger unsteadiness, the quality of the RANS-based results deteriorates when
moving towards higher angles of incidence. Due to the difficulty of obtaining
well converged and representative steady solutions, stability methods also
suffer when applied far from the onset [19]. Last but not least, it should be
emphasized that although the low-fidelity simulations were carefully designed
and set up based on general best practices, potential sensitivity to domain
topology, grid resolution, time integration parameters and turbulence modeling
cannot be entirely ruled out. In this sense, high-fidelity simulations are an
important reference and represent an essential tool when investigating complex
flow phenomena such as turbulent transonic buffet and the range of scales
present. The design of high-fidelity simulations in general is however not
exempt from sources of numerical errors. Sensitivity analyses for the high-
fidelity simulations in this work are presented in the following sections, in
an attempt to provide high-quality data for the characterization of 2D
turbulent transonic buffet.

##  4 Sensitivity of turbulent transonic buffet to numerical tripping strength

The baseline narrow case (AR005-AoA5) at buffet conditions is selected to test
the sensitivity of the solution to different strengths of the tripping, as
defined in equation (7). The aim of this section is to investigate how
sensitive the buffet characteristics are to the upstream boundary-layer state.
In particular, whether or not there are substantial changes in buffet
frequency, aerodynamic coefficients, and shock-structure, as the tripping
amplitude is reduced and the boundary-layer becomes more laminar/transitional.
For fully turbulent transonic buffet on supercritical airfoils in the range of
Mach numbers considered here, a single lambda shock-wave oscillates across the
suction side of the airfoil with low-frequency (Figure 3), as commonly
reported in the literature [9, 30, 28]. In ‘laminar-’ or ‘free-transitional-
buffet’ [42, 41, 38, 39, 40], differences have been reported in the dominant
frequencies, flow structure, and shock topology. Free-transition buffet cases
also typically have far larger regions of flow separation than fully turbulent
configurations, as commonly found in studies of SBLI on different flow
configurations with differing upstream boundary-layer states [74, 75, 76, 77].
Zauner and Sandham [41] and Moise et al. [38, 39] have reported laminar buffet
cases at $Re=5\times 10^{5}$ (equal to the $Re$ in the present study) which
are comprised of a system of multiple shock-wave/expansions which oscillate on
the suction side of the airfoil and interact with one another, which deviates
from the single shock-wave configuration from turbulent buffet. However, the
same authors reported in [38] that the number of observed shock-waves
decreased as the Reynolds number was increased. They suggested that there
could be a critical Reynolds number at which the number of shock-waves reduces
to one, as observed in turbulent buffet. Indeed, this would be consistent with
other laminar buffet simulations [42] and experiments [43, 44] in the recent
literature, which showed laminar buffet with only a single shock-wave at
$Re=3\times 10^{6}$. As we investigate turbulent buffet at moderate $Re$ in
this work, it is of interest to assess how sensitive the configuration is to a
modified upstream boundary-layer state. Table 2 shows a summary of the cases
simulated with six different trip amplitudes, ranging from $A=0.5\%$ to
$A=10\%$ (the amplitude is expressed in percentage of the free-stream velocity
$U_{\infty}$). The baseline AR005-AoA5 case refers to case AR005-AoA5-$A$7.5
in this table, with an amplitude of $A=7.5\%$.

Case | AR | $\alpha$ | Trip Amplitude | Interaction | $\overline{C_{L}}$ | $\overline{C_{Dp}}$ | $\overline{C_{Df}}$ | $\overline{C_{D}}$ | $St$
---|---|---|---|---|---|---|---|---|---
AR005-AoA5-$A$10 (a) | 0.05 | $5^{\circ}$ | $10.0\%\cdot U_{\infty}$ | Turbulent | 0.986 | 0.0498 | 0.0080 | 0.0578 | 0.0778
AR005-AoA5-$A$7.5 (b) | 0.05 | $5^{\circ}$ | $7.5\%\cdot U_{\infty}$ | Turbulent | 0.991 | 0.0501 | 0.0079 | 0.0581 | 0.0778
AR005-AoA5-$A$5 (c) | 0.05 | $5^{\circ}$ | $5.0\%\cdot U_{\infty}$ | Turbulent | 1.000 | 0.0507 | 0.0077 | 0.0584 | 0.0778
AR005-AoA5-$A$2.5 (d) | 0.05 | $5^{\circ}$ | $2.5\%\cdot U_{\infty}$ | Transitional | 1.125 | 0.0584 | 0.0066 | 0.0650 | 0.0667
AR005-AoA5-$A$1 (e) | 0.05 | $5^{\circ}$ | $1.0\%\cdot U_{\infty}$ | Laminar | 1.050 | 0.0532 | 0.0057 | 0.0588 | 0.0778
AR005-AoA5-$A$0.5 (f) | 0.05 | $5^{\circ}$ | $0.5\%\cdot U_{\infty}$ | Laminar | 1.041 | 0.0527 | 0.0056 | 0.0583 | 0.0778
Table 2: Summary of narrow-span $AR=0.05$ cases at different tripping
strengths. For each case, Reynolds and Mach numbers are fixed at $Re=5\times
10^{5}$ and $M_{\infty}=0.72$. Letters refer to the labels used in figure09
figure010. Figure 8: Sensitivity to tripping amplitude for the baseline case
of $M=0.72$, $Re=500,000$, $AR=0.05$, $\alpha=5^{\circ}$. Showing (a) lift
coefficient history (b) PSD of fluctuating lift component (c) time-averaged
pressure coefficient and (d) time-averaged skin-friction.

Figure 8 shows (a) lift coefficient history, (b) PSD of lift coefficient
fluctuations, (c) time-averaged pressure distribution and (d) time-averaged
skin-friction. In Figures 8 (c,d), the variation of trip strength can be seen
in height of the peaks in the pressure coefficient and skin-friction
distributions around $x=0.1$, as the tripping amplitude is varied by a factor
of twenty. Figure 8 (a) shows the effect this has on the variations of lift
coefficient. The first thing to note is that for the three strongest
amplitudes ($A=5\%,7.5\%,10\%$), the buffet characteristics are very similar
and only minor changes on the order of $1\%$ are observed in
$\overline{C_{L}}$. Once the boundary-layer has been tripped to turbulence,
further increases in forcing show a minimal impact on aerodynamic coefficients
and the skin-friction distributions in Figure 8 (d) collapse upstream of the
mean shock location ($x\approx 0.4$). Based on these findings, we select the
$A=7.5\%$ tripping strength throughout the later sections of this work as
representative fully-turbulent buffet conditions. It should be noted that
another confirmation of the turbulent nature of the interaction at this trip
amplitude was previously given in section 3.1 when showing the agreement
between the ILES results and the RANS/URANS simulations, which assumed the
boundary-layer to be fully-turbulent over the entire airfoil.

While the three strongest tripping amplitudes agree well with minimal
differences in aerodynamic coefficients, large variations are observed as the
tripping strength is reduced. At $A=2.5\%$, we observe a 15% increase in mean
$C_{L}$ and a modified low-frequency peak in the PSD given in Figure 8 (b).
Additionally, there are higher frequency peaks visible around $St=1.3$ and
$St=1.9$, not present in the fully-tripped cases. In the pressure distribution
and skin-friction plots in Figures 8 (c,d), there is a large downstream shift
in the mean shock position and a significant region of time-averaged flow
separation around $x=0.6$ on the suction side of the airfoil, distinct from
the higher forcing cases. However, further decreases in tripping amplitude do
not continue a monotonic response in the lift coefficient, with the weaker
cases $(A=0.5\%,1\%)$ having mean lift values and buffet frequencies closer to
the fully turbulent ones. From the skin-friction plot in Figure 8 (d), the
cases can be categorized based on the $C_{f}$ upstream of the shock as laminar
($A=0.5\%,1\%)$, transitional ($A=2.5\%$), and fully-turbulent
($A=5\%,7.5\%,10\%$). This categorization is used in rest of this work.

Figure 9: Skin-friction $\left(x-t\right)$ diagrams of attached/separated flow
for varying tripping strength cases in Table 2. The corresponding time history
of lift coefficient is shown on the side plot.

Figure 9 shows $x-t$ spatio-temporal plots of how the regions of attached and
separated flow vary in time for each of the six tripping strengths. In each
case the corresponding lift coefficient is plotted on the side of the figure.
The black line represents the first streamwise location of flow separation
$(C_{f}<0)$ at each given time instance. The three strongest forcing fully-
turbulent cases (Figures 9 (a)-(c)) are all very similar. All three have
periodic low-frequency separation and reattachment downstream of the main
interaction ($0.5<x<0.9$). The mean separation point is within $1.5\%$ for
each of these cases. The transitional case (Figure 9 (d)), however, shows a
strong deviation from the turbulent topology, with the same high frequency
modulation of the lift coefficient observed on the first separation location.
This suggests that the additional frequencies observed in the PSD in the
$1<S_{t}<5$ range (Figure 8 (b)) are due to the response of the separation
point to the incoming boundary-layer state. As the tripping amplitude is
reduced further to $A=0.5\%,1\%$, clear regions of separation and attachment
are restored. As the boundary-layer becomes more laminar upstream of the
shock-wave, the separation point oscillates with an intermediate frequency in
range $1<St<5$ (Figure 8 (b)). In contrast to the turbulent cases, the weakest
trip case also shows a reattachment line around $x=0.45$ which oscillates at
the same low-frequency as the buffet cycle. The increased values of
$\overline{C_{L}}$ for the laminar and transitional cases relative to the
fully turbulent configuration are consistent with the RANS/GSA findings of
Garbaruk et al. [45].

Figure 10: Instantaneous streamwise density gradient showing boundary-layer
state and shock-wave structure on the suction side of the airfoil at (left)
low- and (right) high-lift phases for different tripping amplitudes of
$A=\left[0.5\%,10\%\right]$.

Figure 10 shows instantaneous contours of density gradient at the main shock-
wave location for the six tripping amplitudes in (a)-(f). The left and right
hand columns of this figure show the low- and high-lift phases of the buffet
cycle, respectively. The three strongest cases (Figures 10 (a)-(c)) all
consist of a clear lambda shock-wave on top of the turbulent boundary-layers.
The upstream and downstream shock-wave positions at the low- and high-lift
phases are very similar for the three strongest tripping amplitudes. As the
tripping strength is reduced in Figures 10 (d)-(f), the lambda shock structure
is seen to partially breakdown. In these cases the boundary-layer upstream of
the main SBLI is no longer turbulent, with the flow observed to rapidly
transition around the shock position. While the turbulent cases (Figures 10
(a)-(c)) all possess a terminating shock-wave approximately perpendicular to
the surface curvature, the laminar cases have a shock-wave that is far more
vertical. Importantly, we note that despite the very low forcing amplitude
($A=0.5\%)$ and essentially laminar boundary-layer upstream of the SBLI, the
interaction still contains only a single main shock-wave, as typically
attributed to fully-turbulent airfoil buffet [9, 30, 28].

The configuration of multiple travelling shock-wave/expansions attributed in
some (but not all) works as representative of ‘free-transitional’ or ‘laminar’
buffet [41, 38, 39] is not observed, even when the upstream boundary-layer in
the present cases is far from being in a turbulent state (Figures 10 (e,f)).
Laminar buffet with a single main shock-wave has been reported both
computationally [42] and experimentally [43, 44], showing shock topologies
closer to those observed for the laminar cases in the present work.
Furthermore, we note the presence of sharp density gradients oriented
vertically above the boundary-layer for the almost laminar cases (Figures 10
(e,f)), which are not observed once the interaction is fully turbulent
(Figures 10 (a-c). This feature is most apparent in Figure 10 (e) at low-lift,
where a secondary weak shock is visible at $x=0.65$. Recent laminar buffet
simulations by Song et al. [40] reported a similar small intermittently-
present terminating shock downstream of the main-shock-wave, for a fully-
untripped OALT25 profile at a higher Reynolds number of $Re=1\times 10^{6}$
with $AR=0.25$. Finally, we note that the distance over which the shock
oscillates between $x_{\textrm{High}}$ and $x_{\textrm{Low}}$ is also
sensitive to the change in upstream boundary-layer state. For the turbulent
cases (Figures 10 (a)-(c)), the main shock-wave traverses over a region of
$\approx 0.13c$ between low- and high-lift phases, compared to the laminar
cases with $\approx 0.1c$. The transitional case shows the least streamwise
shock movement of only $\approx 0.02c$. This lack of shock movement for the
transitional case could partially explain the lower amplitudes observed for
the low-frequency lift coefficient (Figure 8 (a)), relative to the other
tripping amplitudes. The observed oscillations of the separation point and
reduced overall shock-wave movement is similar to the laminar buffet cases of
[42] at the higher Reynolds number of $Re=3\times 10^{6}$. The authors of that
work found lower shock oscillation amplitudes compared to turbulent buffet and
argued that oscillations of the shock-foot played a more prominent role in the
instability. Modal analysis in future work will investigate the transitional
cases in more detail, to understand the role of the shock-foot and separation
point in laminar/transitional buffet.

##  5 Sensitivity of turbulent transonic buffet to span-width

Figure 11: Moderate and largely separated cases for the baseline configuration
($AR=0.05$ and $A=7.5\%$) at angles of attack $\alpha=5$ and $7^{\circ}$.
Showing (a) lift coefficient history and (b) time-averaged skin-friction.

In this section, the sensitivity of transonic buffet to span-wise domain width
is investigated for aspect ratios in the range of $AR=\left[0.025,0.5\right]$.
At the upper end, this is significantly wider than the domains typically used
($AR=0.05-0.065$) in previous high-fidelity studies of transonic buffet [30,
37, 39]. Two post-onset buffet conditions are selected at $\alpha=5$ and
$7^{\circ}$. For both angles of attack, the flow is unsteady, and the shock
oscillates and causes fluctuations in the lift-coefficient histories (Figure
11 (a)). While the differences in dominant frequencies and flow separation
characteristics at different aspect ratios will be discussed later in this
section, it is important to first note Figure 11 (b), that while the boundary-
layer is (in a time-averaged sense) marginally separated at
$\alpha=5^{\circ}$, the separation is far more extensive due to a stronger
shock and increased inclination at $\alpha=7^{\circ}$. A summary of the cases
simulated is shown in Table 3, with their main aerodynamic coefficients shown.

Case | Method | AR | $\alpha$ | $\overline{C_{L}}$ | $\overline{C_{Dp}}$ | $\overline{C_{Df}}$ | $\overline{C_{D}}$ | $St_{2D}$
---|---|---|---|---|---|---|---|---
AR0025-AoA5 | ILES | 0.025 | $5^{\circ}$ | 0.968 | 0.0505 | 0.0078 | 0.0583 | 0.0776
AR005-AoA5 | ILES | 0.05 | $5^{\circ}$ | 0.993 | 0.0504 | 0.0079 | 0.0583 | 0.0778
AR010-AoA5 | ILES | 0.10 | $5^{\circ}$ | 1.004 | 0.0505 | 0.0079 | 0.0584 | 0.0778
AR025-AoA5 | ILES | 0.25 | $5^{\circ}$ | 1.015 | 0.0509 | 0.0079 | 0.0588 | 0.0778
AR050-AoA5 | ILES | 0.50 | $5^{\circ}$ | 1.007 | 0.0506 | 0.0079 | 0.0586 | 0.0778
AR0025-AoA7 | ILES | 0.025 | $7^{\circ}$ | Unstable | Unstable | Unstable | Unstable | Unstable
AR005-AoA7 | ILES | 0.05 | $7^{\circ}$ | 1.019 | 0.0984 | 0.0066 | 0.1049 | 0.0778
AR010-AoA7 | ILES | 0.10 | $7^{\circ}$ | 0.978 | 0.0870 | 0.0066 | 0.0936 | 0.0889
AR025-AoA7 | ILES | 0.25 | $7^{\circ}$ | 0.988 | 0.0865 | 0.0066 | 0.0931 | 0.1001
AR050-AoA7 | ILES | 0.50 | $7^{\circ}$ | 0.988 | 0.0868 | 0.0066 | 0.0934 | 0.1112
Table 3: Summary of the span-width sensitivity cases at moderate AoA (buffet
onset, $5^{\circ}$), and high AoA (buffet, $7^{\circ}$) conditions. For each
case, the tripping amplitude is fixed to $A=7.5\%$ of freestream velocity.

###  5.1 Span-width sensitivity for moderately separated flows

Figure 12 shows the aerodynamic quantities near buffet onset conditions
($\alpha=5^{\circ}$) for $AR=0.025,0.05,0.1,0.25,0.5$. The narrowest domain of
$AR=0.025$ under predicts $\overline{C_{L}}$, and there are large deviations
in distributions of $C_{p}$ and $C_{f}$. While the pressure side of the
airfoil and transition region upstream of the SBLI are essentially insensitive
to domain width, the narrow domain is seen to be insufficient at the shock
location and near the trailing edge. At this AoA, there is a clear sensitivity
to domain width near the trailing edge in both the $C_{p}$ and $C_{f}$
distributions in Figure 12 (c) and (d). There is still a slight over-
prediction of the separated region near the TE at $AR=0.05$, whereas for
$AR=0.1$ and above, the time-averaged distributions converge exactly. Despite
the under-prediction of $\overline{C_{L}}$ for the narrowest two domains, all
five aspect ratios reproduce the same low-frequency buffet peak in the PSD in
figure 12 (b) of $S_{t}\approx 0.078$. The narrowest domain exhibits a
spurious peak at $S_{t}=4.1$ compared to the other aspect ratios where it is
not present. For these buffet cases within $\alpha=0.5^{\circ}$ of onset and
moderate regions of time-averaged flow separation, an aspect ratio of $AR=0.1$
is required to achieve insensitivity to domain width. While the $AR=0.05$ case
generally agrees well for the low-frequency buffet, it does slightly over-
predict separation near the trailing edge as the flow is artificially
constrained to be more two-dimensional than in the wider aspect ratios.

Figure 12: Span-width sensitivity at $\textrm{AoA}=5^{\circ}$. Showing (a)
lift coefficient history (b) PSD of fluctuating lift component (c) time-
averaged pressure coefficient and (d) time-averaged skin-friction.

Figure 13: Spanwise domain width sensitivity for (a) trailing edge streamwise
velocity boundary-layer profiles and (b) RMS of pressure for
$\alpha=5^{\circ}$.

Figure 13 shows the (a) mean streamwise velocity boundary-layer profiles at
$97.5\%$ of chord length and (b) RMS distributions of pressure along the
suction side of the airfoil, for aspect ratios of
$AR=\left[0.025,0.05,0.1,0.25,0.5\right]$. Data to the left of the vertical
dashed $u=0$ line represents regions of reverse flow near the trailing edge.
The wider aspect ratios exhibit far weaker separations compared to the narrow
($AR=0.025,0.05$) cases, both in magnitude and vertical extent. The time-
averaged reverse flow region extends to a distance of $\approx 0.007c$ above
the airfoil surface at $AR=0.1$ and above, compared to $\approx 0.012c$ and
$\approx 0.018c$ for the narrower $AR=0.05$ and $AR=0.025$. The narrow domains
artificially constrain the flow separation to be overly two-dimensional,
leading to the domain-dependent artefacts seen near the trailing edge in the
pressure and skin-friction distributions of Figures 12 (c,d). At $AR=0.1$ and
above the profiles converge and the domain sensitivity is removed.

Figure 13 (b) shows the RMS of pressure fluctuations across the suction side
of the airfoil for each of the aspect ratios. The pressure fluctuations
upstream of the shock-wave $0.2<x<0.35$ collapse for all aspect ratios
considered, indicating that the tripped transition method is insensitive to
the domain width. The pressure rise at the shocked region is similar for all
aspect ratios except the narrowest case ($AR=0.025$), which appears overly
narrow to reproduce the SBLI dynamics captured by the wider simulations. The
main differences appear close to the trailing edge as previously seen. The
three narrowest aspect ratios all overestimate the pressure fluctuations at
the trailing edge compared to the $AR=0.25$ and $AR=0.5$ cases. This is
similar to the findings of Garnier and Deck [35], who observed a significant
reduction in trailing edge pressure fluctuations when increasing the domain
with from $AR=0.0365$ to $AR=0.073$. The simulations of Garnier and Deck [35]
were later reviewed by Giannelis et al. [4], who commented that the wider
domain better captured unsteady pressures at the trailing edge, as three-
dimensional turbulent structures were allowed to develop with a reduced
intensity relative to the two-dimensional structures from the narrow domain.
However, they went on to comment that even at $AR=0.073$ the pressure
fluctuations were still overestimated compared to experiment. This is
consistent with the present cases, which require between $AR=0.1$ and
$AR=0.25$ to remove domain sensitivity at the trailing edge. Given the
critical role that separation height has been argued to have for driving the
underlying self-sustaining buffet mechanism [25], it is essential that
sufficiently wide domains are used even for predictions of the two-dimensional
part of the buffet phenomenon. The next section investigates whether similar
domain sensitivity trends are observed at higher AoA ($\alpha=7^{\circ}$) when
flow separation is even larger.

###  5.2 Span-width sensitivity for largely separated flows

Figure 14: Span-width sensitivity at $\textrm{AoA}=7^{\circ}$. Showing (a)
lift coefficient history (b) PSD of fluctuating lift component (c) time-
averaged pressure coefficient and (d) time-averaged skin-friction.

Figure 15: Spanwise domain width sensitivity for (a) trailing edge streamwise
velocity boundary-layer profiles and (b) RMS of pressure for
$\alpha=7^{\circ}$

In this section the influence of span-width is assessed at a higher angle of
attack of $\alpha=7^{\circ}$, to investigate whether similar domain
sensitivity trends found at $\alpha=5^{\circ}$ are present for buffet in the
presence of more extensive flow separation. As shown in Table 3, the narrowest
domain ($AR=0.025$) proved to be unstable at the higher AoA and was unsuitable
to even provide a solution. The domain width study at $\alpha=7^{\circ}$ was
instead performed at $AR=0.05,0.1,0.25,0.5$. Figure 14 shows the aerodynamic
coefficients for (a) lift, (b) PSD of lift fluctuations, (c) pressure
coefficient, and (d) skin-friction distribution at each of the four aspect
ratios. With respect to the previous moderate angle of attack, at
$\alpha=7^{\circ}$ the shock-wave is stronger, moves upstream, and causes the
flow to largely separate. Unlike the lift-coefficient convergence with
increasing aspect ratio observed at moderate AoA post-onset (Figure 12 (a)),
at $\alpha=7^{\circ}$ no clear convergence is found. However, the time-
averaged $\overline{C_{L}}$ does agree to 3 decimal points at $AR=0.25$ and
$AR=0.5$ despite irregularities between the individual buffet periods. Similar
to the trends observed for moderate AoA buffet cases (Figures 7 and 12),
increasing the aspect ratio leads to increased regularity of the low-frequency
buffet oscillations and reduction in the higher frequency content. While the
results presented here are span-averaged, the same behaviour was observed with
increasing aspect ratio when the aerodynamic coefficients were instead
evaluated over a single spanwise grid location. This suggests that the trend
of more regular low-frequency oscillations is intrinsic to the dynamics on the
wider airfoils and not just an averaging effect.

In the PSD of lift fluctuations in Figure 14 (b), there are low-frequency
peaks visible in the 2D buffet range for each aspect ratio despite the higher-
frequency noise observed in the lift histories (Figure 14 (a)). However,
unlike the lower-AoA buffet case (Figure 12 (b)), the buffet frequency does
not converge exactly with increasing aspect ratio. This suggests that the
buffet dynamics are potentially different between $\alpha=5^{\circ}$ and
$\alpha=7^{\circ}$, beyond just a simple shift in the buffet frequency. In
Figure 14 (c), we see that the narrowest domain of $AR=0.05$ which is similar
to the values commonly used in previous buffet studies (e.g. $AR=0.0365-0.073$
[35], $AR=0.05$ [38, 39], and $AR=0.065$ [30, 37]), vastly over-predicts the
pressure distribution from mid-chord to TE ($0.5<x<1$) at this high AoA.
Elsewhere on the airfoil, all aspect ratios collapse and no domain-width
sensitivity is observed. The $AR=0.1$ case shows a strong improvement, and is
much closer to the wide-span results which converge in the trailing edge
region at $AR=0.25$ and above. Similarly, the skin-friction distributions in
Figure 14 (d) show the inadequacy of the $AR=0.05$ domain at high AoA. The
narrowest two both overestimate the separation size at the trailing edge, most
notably at $AR=0.05$. The $AR=0.05$ case actually reattaches around $x=0.9$
before separating again, a feature not observed on the wider aspect ratios.
This highlights that insufficient spanwise domain width can, in some cases,
lead to opposing flow states, with flow features and energy content that is
only present as a result of modelling error when selecting the airfoil width.

Figure 15 again shows the boundary-layer profiles and RMS of pressure at the
trailing edge, at the higher AoA of $7^{\circ}$. The same, albeit stronger,
trends from the previous cases (Figure 13) are observed once more. The
narrowest $AR=0.05$ case greatly overestimates the size of the separated
region at the aft position of the airfoil both in magnitude and spatial
extent, in contrast to the convergence seen for the wider domains. Finally,
the RMS of pressure fluctuations in Figure 15 (b) shows a similar trend of
sensitivity at the trailing edge for the narrowest domains. In addition to the
trailing edge, at $\alpha=7^{\circ}$ we also observe some sensitivity at the
shock location which will be later clarified.

###  5.3 Spanwise domain width selection criterion

Figure 16: Spanwise domain width sensitivity for trailing edge streamwise velocity boundary-layer profiles at angles of attack of (a) $\alpha=4^{\circ}$, (b) $\alpha=5^{\circ}$, and (c) $\alpha=7^{\circ}$. The ashed line represents zero crossing for reverse flow region, the dashed-dot line represents $99\%$ of freestream velocity. $\alpha$ | AR | $\overline{\delta}_{99}^{TE}$ | $AR/\overline{\delta}_{99}^{TE}$ | $\alpha$ | AR | $\overline{\delta}_{99}^{TE}$ | $AR/\overline{\delta}_{99}^{TE}$ | $\alpha$ | AR | $\overline{\delta}_{99}^{TE}$ | $AR/\overline{\delta}_{99}^{TE}$
---|---|---|---|---|---|---|---|---|---|---|---
$4^{\circ}$ | 0.025 | 0.050 | 0.500 | $5^{\circ}$ | 0.025 | 0.069 | 0.362 | $7^{\circ}$ | 0.025 | N/A | N/A
$4^{\circ}$ | 0.05 | 0.054 | 0.926 | $5^{\circ}$ | 0.05 | 0.081 | 0.617 | $7^{\circ}$ | 0.05 | 0.117 | 0.427
$4^{\circ}$ | 0.10 | 0.058 | 1.724 | $5^{\circ}$ | 0.10 | 0.092 | 1.087 | $7^{\circ}$ | 0.10 | 0.122 | 0.820
$4^{\circ}$ | 0.25 | 0.058 | 4.310 | $5^{\circ}$ | 0.25 | 0.087 | 2.874 | $7^{\circ}$ | 0.25 | 0.129 | 1.938
$4^{\circ}$ | 0.50 | 0.057 | 8.772 | $5^{\circ}$ | 0.50 | 0.091 | 5.495 | $7^{\circ}$ | 0.50 | 0.129 | 3.876
Table 4: Summary of the span-width selection criterion for cases at pre-onset
($\alpha=4^{\circ}$), post-onset moderately separated ($\alpha=5^{\circ}$),
and post-onset largely separated ($\alpha=7^{\circ}$). For each case, the
tripping amplitude is fixed to $A=7.5\%$ of freestream velocity.

The span-width sensitivity studies are summarized here in the attempt to
determine quantitative criteria for appropriate selection of the spanwise
domain width. Along with the results presented above for moderately and
largely separated flows at post-onset conditions ($\alpha=5^{\circ}$ and
$\alpha=7^{\circ}$, respectively), the pre-onset case at $\alpha=4^{\circ}$
described in section 3 is also reported. Figure 16 shows the time-averaged
boundary-layer profiles near the trailing edge ($x/c=0.975$) for the three
angles of attack and all investigated aspect ratios. As expected, the time-
averaged boundary-layer thickens for increasing angles of attack, and by
inspecting the reverse flow near the wall it is possible to observe the flow
switching from fully-attached to moderately and largely separated. While in
the fully-attached steady pre-onset and moderately separated unsteady post-
onset cases the profiles seem to converge for $AR=0.1$, for the largely
separated case an aspect ratio of at least $AR=0.25$ is needed. The $99\%$
boundary-layer thickness at the trailing edge ($\overline{\delta_{99}^{TE}}$)
for all investigated cases is summarized in Table 4, along with the
$AR/\overline{\delta_{99}^{TE}}$ ratio. This study seems to suggest that to
obtain a spanwise domain width independent solution, the span-width must be at
least greater than the boundary-layer thickness at the trailing edge region,
i.e. $AR/\overline{\delta_{99}^{TE}}>1$. Previous buffet simulations [30] have
used a domain-width set equal to experimental time-averaged boundary-layer
thickness values measured at $75\%$ chord. This is in general agreement with
our findings, although we set our criteria based on boundary-layer thickness
at $97.5\%$ chord where the boundary-layer is thicker. While the spanwise
domain width guidelines given here should be considered appropriate for the
present geometry and flow conditions, there seems to be a general trend which
may also apply to other configurations.

###  5.4 Comparison to 2D and 3D URANS at $AR=1$

Figure 17: Lift coefficient history comparisons for URANS cases performed on
2D and 3D ($AR=1$) configurations. Showing angles of attack of (a)
$\alpha=5^{\circ}$, and (b) $\alpha=7^{\circ}$. Figure 18: Instantaneous
streamwise pressure gradient on the airfoil surface for URANS cases at $AR=1$.
Showing angles of attack of (left) $\alpha=5^{\circ}$, and (right)
$\alpha=7^{\circ}$. The white lines show contours of skin-friction in both
cases.

The domain-width sensitivity studies have shown that while the pressure side
of supercritical airfoils for type II buffet is largely insensitive to domain
width, both the SBLI and separated regions on the suction side show strong
sensitivity to domain width within the range of values typically used in the
literature. At moderate AoA close to buffet onset
$(\alpha_{\textrm{onset}}+0.5^{\circ})$ with minimal flow separation, similar
frequencies are predicted, but narrow-span simulations ($AR=0.05$) under-
predict lift, show less regularity of the buffet cycles, and over-predict
trailing edge separations which are un-physically constrained. At higher AoA
with more extensive flow separation narrow-span simulations are shown to be
highly inadequate. While high-fidelity simulations offer improvements over
low-fidelity methods by reduced modelling and explicit resolution of
turbulence, care must be taken not to artificially constrain the problem with
overly-narrow domains. It is also clear that domain width sensitivity studies
should be performed at representative flow configurations (at the highest AoA
considered in the work), where the span-wise width requirements are at their
most critical.

In this final section we aim to address the disparity in behavior between
moderately separated buffet ($\alpha=5^{\circ}$) which converges with
increasing span width (Figure 12), and the higher AoA case with extensive flow
separation which does not ($\alpha=7^{\circ}$, Figure 14). To do this, we
extend the 2D URANS simulations from Section 3.1.3 to 3D with an even wider
aspect ratio of $AR=1$. The 2D URANS grid is extruded in the spanwise
direction and discretized using 21 cells. Similarly to the 2D unsteady
calculations, the 3D URANS are started from uniform flow conditions. Figure 17
shows the $C_{L}$ histories for 2D and 3D ($AR=1$) simulations at (a)
$\alpha=5^{\circ}$ and (b) $\alpha=7^{\circ}$). For the case with moderate
flow separation, the predictions from the 2D URANS and 3D URANS are identical.
In each case the 2D shock-oscillation buffet mode is reproduced and the
extension to a 3D simulation and a wide domain has no influence during the
initial transient or when the nonlinear dynamics fully saturates and buffet is
fully developed. However, at $\alpha=7^{\circ}$, despite having a similar
buffet frequency and amplitude during the initial transient, the 2D and 3D
URANS calculations diverge. The amplitude of the 3D simulations drops
significantly, and additional frequencies are observed beyond the 2D case.

Figure 19: Three-dimensional variations across the span for the ILES data at
$AR=0.5$. Showing the absolute difference in the streamwise velocity when
averaged over the first and second half of the span individually. Showing the
evolution in time for angles of attack of (a) $\alpha=5^{\circ}$, and (b)
$\alpha=7^{\circ}$.

Figure 18 shows instantaneous streamwise pressure gradient plots on the
suction side of the airfoil for the URANS at $AR=1$, at (left)
$\alpha=5^{\circ}$ and (right) $\alpha=7^{\circ}$. While the lower AoA case
has a shock with an entirely normal orientation to the oncoming flow
regardless of the spanwise position, the high AoA case shows modulation of the
shock-front across the span and is no longer quasi-2D across the periodic
direction. This indicates that the two-dimensionality of the buffet phenomenon
has been lost, and, at this higher AoA, three-dimensional effects such as
buffet/stall-cells (commonly seen with aspect ratio wavelengths around unity
and above [4, 12, 13]), may be starting to affect the role that the spanwise
domain width has. Figure 19 shows $x$-$t$ diagrams for the ILES data on the
widest domain of $AR=0.5$, for the cases of (a) $\alpha=5^{\circ}$, and (b)
$\alpha=7^{\circ}$. At each time instance, the quantity plotted is the
absolute difference between the $u$-velocity component inside the boundary-
layer when averaged over the first and second halves of the span-wise width
($z=0\to\frac{L_{z}}{2}$ and $z=\frac{L_{z}}{2}\to L_{z}$). For a purely two-
dimensional phenomenon with no significant variation across the spanwise
direction, both halves of the span should have the same mean value. In the
moderate AoA case ($\alpha=5^{\circ}$) this is largely true, with only small
variations seen between the mean value in each of the two halves of the
airfoil. However, at the higher AoA this is no longer the case. Significant
variations are now observed between the two halves of the airfoil span. These
are most notable around mid-chord and at the main SBLI. Interestingly, the
peaks of the variation are strongest during the low-lift phases of the buffet
cycle when the shock-wave is at its farthest upstream location, with reduced
activity at the point of maximum lift.

##  6 Conclusions

Turbulent transonic buffet has been investigated with high-fidelity scale-
resolving (ILES) simulations on periodic unswept wing configurations of the
NASA-CRM geometry. Data from the baseline ILES cases at pre- and post-onset
buffet conditions was cross-validated to low-fidelity (RANS/URANS) and global
stability (GSA) methods. Excellent agreement was observed between the methods,
in terms of the average pressure distributions (ILES/RANAS), onset criteria
(ILES, GSA), plus unsteady histories of lift and drag and their low-frequency
oscillations (ILES/URANS).

After validating the initial results, the sensitivity of buffet
characteristics to both upstream boundary-layer state (via wall-tripping of
varying strength) and span-wise domain width were then investigated via ILES.
The role of the boundary-layer state upstream of the main shock-wave for
transonic airfoil buffet was investigated by varying the tripping amplitude
over the range of $0.5\%$ to $10\%$ of freestream velocity. It was found that
buffet with only a single main shock-wave can persist for both laminar and
turbulent boundary-layers with similar low-frequency oscillations and periodic
flow separation and reattachment. In addition to the two limiting regimes
(laminar/turbulent), we show there also exists a transitional interaction for
which the main shock-wave is more stationary and higher frequencies are
present. The laminar and transitional cases exhibit far larger mean flow
separation compared to the turbulent case, but the transitional case does not
share the same separation bubble shape as the laminar ones. The transitional
case exhibited $13.5\%$ and $7\%$ higher mean lift than the turbulent and
laminar cases, respectively.

The role of the span-wise domain extent was assessed at post-onset for a
fully-turbulent case with moderate flow separation ($\alpha=5^{\circ}$), and
post-onset with large flow separation ($\alpha=7^{\circ}$). Domains of up to
$AR=0.5$ were simulated with ILES, up to an order of magnitude wider than what
commonly used in the literature. It was observed that narrow domains
($AR=0.025,0.05$) commonly used in high-fidelity simulations of buffet were
insufficient to provide domain insensitive solutions. At pre- and post-onset
with minimal flow separation, although the main low-frequency buffet peak was
largely insensitive to domain width, at least $AR=0.1$ was necessary for
domain independent amplitudes and pressure/skin-friction distributions. The
main discrepancy was found at the SBLI and at the trailing edge. The
separation at the trailing edge was found to be overly-constrained on narrow
domains, decreasing in extent as the domain was widened, and the flow near the
trailing edge had higher RMS pressure fluctuations compared to solutions that
converged on wider domains. At higher AoA with more extensive flow separation,
even greater span-width sensitivity was observed. Simulations on $AR=0.05$
domains failed to reproduce clear low-frequency buffet peaks, and led to
strongly diverged predictions of pressure distributions and trailing edge
separation. This indicates that high-fidelity airfoil simulations should
assess domain sensitivity on the highest AoA considered, as there is a strong
dependence on increasing AoA where flow separation becomes larger. A summary
of the cases showed that the span-width must be at least greater than the
boundary-layer thickness at the trailing edge to obtain domain-independent
solutions.

Furthermore, unlike the case at moderate AoA close to onset
($\alpha=5^{\circ}$), for which the shock-oscillations remained quasi-2D even
up to $AR=0.5$ (ILES) and $AR=1$ (URANS), additional frequencies and non-2D
features were observed at $\alpha=5^{\circ}$. It’s possible that close to
onset ($\alpha=5^{\circ}$) the problem remains essentially 2D, whereas at
$\alpha=7^{\circ}$ 3D buffet/stall-cell phenomena start appearing, which will
have a spanwise wavelength determined by the aspect ratio used. Instantaneous
surface plots and $z-t$ time histories showed that the moderate AoA case
remains essentially 2D while the higher AoA started to show 3D effects across
the span. The study of 3D buffet effects on wider aspect ratios will be the
topic of a future work.

## Appendix

###  6.1 Sensitivity to spanwise grid resolution

Figure 20: Sensitivity to spanwise grid resolution for the baseline case of
$M=0.72$, $Re=500,000$, $AR=0.05$, $\alpha=5^{\circ}$. Showing (a) lift
coefficient history (b) PSD of fluctuating lift component (c) time-averaged
pressure coefficient and (d) time-averaged skin-friction.

Figure 21: Spanwise grid resolution sensitivity for (a) trailing edge
streamwise velocity boundary-layer profiles and (b) RMS of pressure
fluctuations.

Given the large number of cases in the present study with aspect ratios up to
an order of magnitude wider than commonly used for high-fidelity buffet
simulations (i.e. $AR=0.5$ compared to commonly used $AR=0.05$ [38, 39] or
$AR=0.065$ [30, 37]), it was necessary to a identify a moderate spanwise grid
resolution that could reproduce the main buffet dynamics while still remaining
computationally feasible for the wider aspect ratios. This is of particular
relevance in the case of buffet, where long time integration is required to
capture multiple cycles of the low-frequency phenomenon. With this in mind,
Figure 20 shows aerodynamic coefficients for the $AR=0.05$ baseline case at
$\alpha=5^{\circ}$, with four spanwise grid resolutions of
$N_{z}=\left[25,50,75,100\right]$. Despite the factor of four difference in
spanwise resolution, all resolutions tested reproduce the low-frequency buffet
oscillation, with peaks in the same Strouhal number range (figure 20 (b)). The
main difference in the lift is an over-prediction of $\overline{C_{L}}$ for
the coarsest $N_{z}=25$ grid. The medium resolution ($N_{z}=50$) follows the
finer meshes well, with a slight over-prediction of $\overline{C_{L}}$ by
$2.8\%$. Similarly, the pressure coefficient in Figure 20 (c) shows a large
downstream shift of pressure gradient near mid-chord at $N_{z}=25$, and a much
smaller one at $N_{z}=50$, close to the finer mesh results. While the coarsest
grid also diverges from the finer solutions on the pressure side, transition
location, and near the trailing edge, the medium resolution agrees well in
these areas. In a similar vein, the $N_{z}=50$ resolution matches the finer
mesh skin-friction distributions very well in Figure 20 (d), in terms of the
shock location, SBLI separation and reattachment locations, and trailing edge
separation. Figure 21 shows the same cases for the (a) mean streamwise
velocity boundary-layer profile at $97.5\%$ chord, and (b) RMS of pressure
fluctuations on the suction side of the airfoil. In both cases the coarsest
$N_{z}=25$ case shows to be insufficient, prediction attached flow in the
region that is separated at the finer resolutions, and large disparity in
$p_{rms}$. In contrast, the medium $N_{z}=50$ resolution matches the behaviour
well both in terms of boundary-layer profile and $p_{rms}$ compared to the
finer meshes. Importantly, the separation magnitude and size in figure 21 (a)
are actually slightly larger on the refined grids, suggesting that the
separation based domain sensitivity conclusions reported in section
LABEL:sec:moderate_separation and section LABEL:sec:large_separation are
likely to be more prevalent on finer mesh solutions closer to a full DNS. For
these reasons, the $N_{z}=50$ medium resolution is selected to perform the
wide-aspect ratio sensitivity tests in this work.

## Funding Sources

Dr. David J. Lusher is funded by the Japan Society for the Promotion of
Science (JSPS), on a postdoctoral fellowship awarded to the JAXA Chofu
Aerospace Center. Additional funding was provided by a JSPS KAKENHI grant
award (22F22059). The numerical work on global stability in FaSTAR was
partially supported by the JSPS KAKENHI Grant-in-Aid for Early-Career
Scientists 20K14953 awarded to Dr. Andrea Sansica.

## Acknowledgments

Computational time was provided by the JAXA JSS3 supercomputing facility and
associated support staff, and the Fugaku supercomputer at RIKEN on projects
hp220195, hp220226. The authors thank Dr. Markus Zauner for the help with
initial testing on the URANS setup.

## References

  * Dolling [2001] Dolling, D., “Fifty years of shock-wave/boundary-layer interaction research: what next?” _AIAA Journal_ , Vol. 39(8), 2001, pp. 1517–1531.
  * Lee [1990] Lee, B. H. K., “Oscillatory shock motion caused by transonic shock boundary-layer interaction,” _AIAA Journal_ , Vol. 28(5), 1990, pp. 942–944.
  * Lee [2001] Lee, B. H. K., “Self-sustained shock oscillations on airfoils at transonic speeds,” _Progress in Aerospace Sciences_ , Vol. 37, 2001, pp. 147–196.
  * Giannelis et al. [2017] Giannelis, N. F., Vio, G. A., and Levinski, O., “A review of recent developments in the understanding of transonic shock buffet,” _Progress in Aerospace Sciences_ , Vol. 92, 2017, pp. 39–84.  https://doi.org/10.1016/j.paerosci.2017.05.004.
  * Plante [2020] Plante, F., “Towards understanding stall cells and transonic buffet cells,” Ph.D. thesis, Ecole Polytechnique, Montreal (Canada), 2020.
  * Crouch et al. [2009] Crouch, J. D., Garbaruk, A., Magidov, D., and Travin, A., “Origin of Transonic Buffet on Aerofoils,” _Journal of Fluid Mechanics_ , Vol. 628, 2009, pp. 357–369.
  * Crouch et al. [2019] Crouch, J. D., Garbaruk, A., and Strelets, M., “Global instability in the onset of transonic-wing buffet,” _Journal of Fluid Mechanics_ , Vol. 881, 2019, pp. 3–22.
  * Paladini et al. [2019] Paladini, E., Beneddine, S., Dandois, J., Sipp, D., and Robinet, J.-C., “Transonic buffet instability: From two-dimensional airfoils to three-dimensional swept wings,” _Phys. Rev. Fluids_ , Vol. 4, 2019, p. 103906\.  10.1103/PhysRevFluids.4.103906, URL https://link.aps.org/doi/10.1103/PhysRevFluids.4.103906.
  * Jacquin et al. [2009] Jacquin, L., Molton, P., Deck, S., Maury, B., and Soulevant, D., “Experimental study of shock oscillation over a transonic supercritical profile,” _AIAA Journal_ , Vol. 47, 2009, pp. 1985–1994.
  * Aihara and Kawai [2023] Aihara, A., and Kawai, S., “Effects of Spanwise Domain Size on LES-Predicted Aerodynamics of Stalled Airfoil,” _AIAA Journal_ , Vol. 61, No. 3, 2023, pp. 1440–1446.  10.2514/1.J062375.
  * Iovnovich and Raveh [2015] Iovnovich, M., and Raveh, D. E., “Numerical study of shock buffet on three-dimensional wings,” _AIAA Journal_ , Vol. 53(2), 2015, pp. 449–463.
  * Plante et al. [2020] Plante, F., Dandois, J., and Laurendeau, E., “Similarities between cellular patterns occurring in transonic buffet and subsonic stall,” _AIAA Journal_ , Vol. 58, No. 1, 2020, pp. 71–84.  10.2514/1.J058555.
  * Plante et al. [2021] Plante, F., Dandois, J., Beneddine, S., Laurendeau, E., and Sipp, D., “Link between subsonic stall and transonic buffet on swept and unswept wings: from global stability analysis to nonlinear dynamics,” _Journal of Fluid Mechanics_ , Vol. 908, 2021.  10.1017/jfm.2020.848.
  * Sansica et al. [2022a] Sansica, A., Hashimoto, A., Koike, S., and Kouchi, T., “Side-Wall Effects on the Global Stability of Swept and Unswept Supercritical Wings at Buffet Conditions,” _AIAA SCITECH 2022 Forum, AIAA Paper 2022-1972_ , 2022a.  10.2514/6.2022-1972.
  * Ohmichi et al. [2018] Ohmichi, Y., Ishida, T., and Hashimoto, A., “Modal decomposition analysis of three-dimensional transonic buffet phenomenon on a swept wing,” _AIAA Journal_ , Vol. 56, No. 10, 2018, pp. 3938–3950.  10.2514/1.J056855.
  * Masini et al. [2020] Masini, L., Timme, S., and Peace, A. J., “Analysis of a civil aircraft wing transonic shock buffet experiment,” _Journal of Fluid Mechanics_ , Vol. 884, 2020.
  * Timme [2020] Timme, S., “Global instability of wing shock-buffet onset,” _Journal of Fluid Mechanics_ , Vol. 885, 2020.
  * Sugioka et al. [2021] Sugioka, Y., Nakakita, K., Koike, S., Nakajima, T., Nonomura, T., and Asai, K., “Characteristic unsteady pressure field on a civil aircraft wing related to the onset of transonic buffet,” _Experiments in Fluids_ , , No. 62:20, 2021.
  * Sansica and Hashimoto [2023] Sansica, A., and Hashimoto, A., “Global Stability Analysis of Full-Aircraft Transonic Buffet at Flight Reynolds Numbers,” _AIAA Journal_ , Vol. 61, No. 10, 2023, pp. 4437–4455.  10.2514/1.J062808.
  * Tinoco [2019] Tinoco, E. N., “An Evaluation and Recommendations for Further CFD Research Based on the NASA Common Research Model (CRM) Analysis from the AIAA Drag Prediction Workshop (DPW) Series,” _NASA/CR-2019-220284_ , 2019, pp. 13–17.
  * Tinoco et al. [2018] Tinoco, E. N., Brodersen, O. P., Keye, S., Laflin, K. R., Feltrop, E., Wahls, R. A., Morrison, J. H., Vassberg, J. C., Mani, M., Rider, B., Hue, D., Roy, C. J., Mavriplis, D. J., and Murayama, M., “Summary of Data from the Sixth AIAA CFD Drag Prediction Workshop: CRM Cases 2 to 5,” _Journal of Aircraft_ , Vol. 55, No. 4, 2018, pp. 1352–1379.  10.2514/1.C034409.
  * Sartor et al. [2015] Sartor, F., Mettot, C., and Sipp, D., “Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile,” _AIAA Journal_ , Vol. 53(7), 2015, pp. 1980–1933.
  * Deck [2005] Deck, S., “Numerical simulation of transonic buffet over a supercritical airfoil,” _AIAA Journal_ , Vol. 43(7), 2005, pp. 1556–1566.
  * Hartmann et al. [2013] Hartmann, A., Feldhusen, A., and Shroder, W., “On the interaction of shock waves and sound waves in transonic buffet flow,” _AIAA Journal_ , Vol. 28, No. 942, 2013.  10.1063/1.4791603.
  * Iwatani et al. [2023] Iwatani, Y., Asada, H., Yeh, C.-A., Taira, K., and Kawai, S., “Identifying the Self-Sustaining Mechanisms of Transonic Airfoil Buffet with Resolvent Analysis,” _AIAA Journal_ , Vol. 61, No. 6, 2023, pp. 2400–2411.  10.2514/1.J062294.
  * Thiery and Coustols [2006] Thiery, M., and Coustols, E., “Numerical prediction of shock induced oscillations over a 2D airfoil: Influence of turbulence modelling and test section walls,” _International Journal of Heat and Fluid Flow_ , Vol. 27, 2006, pp. 661–670.
  * Poplinger et al. [2019] Poplinger, L., Raveh, D. E., and Dowell, E. H., “Modal analysis of transonic shock buffet on 2D airfoil,” _AIAA Journal_ , Vol. 57, No. 7, 2019\.
  * Sansica et al. [2022b] Sansica, A., Loiseau, J.-C., Kanamori, A., M. Hashimoto, and Robinet, J.-C., “System Identification of Two-Dimensional Transonic Buffet,” _AIAA Journal_ , Vol. 60, No. 5, 2022b.
  * Grossi et al. [2014] Grossi, F., Braza, M., and Hoarau, Y., “Prediction of Transonic Buffet by Delayed Detached-Eddy Simulation,” _AIAA Journal_ , Vol. 52, 2014, pp. 2300–2312.
  * Fukushima and Kawai [2018] Fukushima, Y., and Kawai, S., “Wall-modeled Large-Eddy Simulation of transonic airfoil buffet at high Reynolds number,” _AIAA Journal_ , Vol. 56(6), 2018, pp. 1–18.
  * Memmolo et al. [2018] Memmolo, A., Bernardini, M., and Pirozzoli, S., “Scrutiny of buffet mechanisms in transonic flow,” _International Journal of Numerical Methods for Heat and Fluid Flow_ , Vol. 28, 2018, pp. 1031–1046.
  * Grinstein et al. [2007] Grinstein, F. F., Margolin, L. G., and Rider, W. J., _Implicit large eddy simulation_ , Vol. 10, Cambridge university press Cambridge, 2007.
  * Garnier et al. [1999] Garnier, E., Mossi, M., Sagaut, P., Comte, P., and Deville, M., “On the Use of Shock-Capturing Schemes for Large-Eddy Simulation,” _Journal of Computational Physics_ , Vol. 153, No. 2, 1999, pp. 273–311.  https://doi.org/10.1006/jcph.1999.6268.
  * Fu [2023] Fu, L., “Review of the high-order TENO schemes for compressible gas dynamics and turbulence,” _Archives of Computational Methods in Engineering_ , Vol. 30, No. 4, 2023, pp. 2493–2526.
  * Garnier and Deck [2013] Garnier, E., and Deck, S., _Large-Eddy Simulation of transonic buffet over a supercritical airfoil_ , Turbulence and Interactions (ed. M. Deville, T.-H. Le and P. Sagaut). Springer, Berlin, Heidelberg, 2013.
  * Turner and Kim [2020] Turner, J. M., and Kim, J. W., “Effect of spanwise domain size on direct numerical simulations of airfoil noise during flow separation and stall,” _Physics of Fluids_ , Vol. 32, No. 6, 2020, p. 065103.  10.1063/5.0009664.
  * Nguyen et al. [2022] Nguyen, N. C., Terrana, S., and Peraire, J., “Large-Eddy Simulation of Transonic Buffet Using Matrix-Free Discontinuous Galerkin Method,” _AIAA Journal_ , Vol. 60, No. 5, 2022, pp. 3060–3077.  10.2514/1.J060459.
  * Moise et al. [2022] Moise, P., Zauner, M., and Sandham, N. D., “Large-eddy simulations and modal reconstruction of laminar transonic buffet,” _Journal of Fluid Mechanics_ , Vol. 944, 2022, p. A16.  10.1017/jfm.2022.471.
  * Moise et al. [2023] Moise, P., Zauner, M., Sandham, N. D., Timme, S., and He, W., “Transonic Buffet Characteristics Under Conditions of Free and Forced Transition,” _AIAA Journal_ , Vol. 61, No. 3, 2023, pp. 1061–1076.  10.2514/1.J062362.
  * [40] Song, H., Wong, M. L., Ghate, A. S., and Lele, S. K., _Numerical study of transonic laminar shock buffet on the OALT25 airfoil_ , ????, Chaps. AIAA 2024-2148.  10.2514/6.2024-2148.
  * Zauner and Sandham [2020] Zauner, M., and Sandham, N. D., “Wide domain simulations of flow over an unswept laminar wing section undergoing transonic buffet,” _Phys. Rev. Fluids_ , Vol. 5, 2020, p. 083903.  10.1103/PhysRevFluids.5.083903, URL https://link.aps.org/doi/10.1103/PhysRevFluids.5.083903.
  * Dandois et al. [2018] Dandois, J., Mary, I., and Brion, V., “Large-eddy simulation of laminar transonic buffet,” _Journal of Fluid Mechanics_ , Vol. 850, 2018, p. 156–178.  10.1017/jfm.2018.470.
  * Brion, V. et al. [2017] Brion, V., Dandois, J., Abart, J.-C., and Paillart, P., “Experimental analysis of the shock dynamics on a transonic laminar airfoil,” , 2017.  10.1051/eucass/2016090365, URL https://doi.org/10.1051/eucass/2016090365.
  * Brion et al. [2020] Brion, V., Dandois, J., Mayer, R., Reijasse, P., Lutz, T., and Jacquin, L., “Laminar buffet and flow control,” _Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering_ , Vol. 234, No. 1, 2020, pp. 124–139.  10.1177/0954410018824516.
  * Garbaruk et al. [2021] Garbaruk, A., Strelets, M., and Crouch, J. D., “Effects of Extended Laminar Flow on Wing Buffet-Onset Characteristics,” _AIAA Journal_ , Vol. 59, No. 8, 2021, pp. 2848–2854.  10.2514/1.J060707.
  * Sugioka et al. [2018] Sugioka, Y., Koike, S., Nakakita, K., Numata, D., Nonomura, T., and Asai, K., “Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint,” _Experiments in Fluids_ , Vol. 59, No. 108, 2018.
  * Sugioka et al. [2022] Sugioka, Y., Kouchi, T., and Koike, S., “Experimental comparison of shock buffet on unswept and 10-deg swept wings,” _Experiments in Fluids_ , Vol. 63, No. 132, 2022.
  * Lusher et al. [2021] Lusher, D. J., Jammy, S. P., and Sandham, N. D., “OpenSBLI: Automated code-generation for heterogeneous computing architectures applied to compressible fluid dynamics on structured grids,” _Computer Physics Communications_ , Vol. 267, 2021, p. 108063.  https://doi.org/10.1016/j.cpc.2021.108063.
  * Hashimoto et al. [2012] Hashimoto, A., Murakami, K., Aoyama, T., Ishiko, K., Hishida, M., Sakashita, M., and Lahur, P., “Toward the Fastest Unstructured CFD Code ’FaSTAR’,” _50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition_ , , No. AIAA 2012-1075, 2012.  10.2514/6.2012-1075.
  * Lusher et al. [2018] Lusher, D. J., Jammy, S. P., and Sandham, N. D., “Shock-wave/boundary-layer interactions in the automatic source-code generation framework OpenSBLI,” _Computers & Fluids_, Vol. 173, 2018, pp. 17 – 21.
  * Lusher et al. [2023] Lusher, D. J., Zauner, M., Sansica, A., and Hashimoto, A., “Automatic Code-Generation to Enable High-Fidelity Simulations of Multi-Block Airfoils on GPUs,” _AIAA Scitech 2023 Forum, AIAA SciTech Forum_ , 2023.
  * Reguly et al. [2014] Reguly, I. Z., Mudalige, G. R., Giles, M. B., Curran, D., and McIntosh-Smith, S., “The OPS Domain Specific Abstraction for Multi-block Structured Grid Computations,” IEEE Press, 2014, pp. 58–67.
  * Coppola et al. [2019] Coppola, G., Capuano, F., Pirozzoli, S., and de Luca, L., “Numerically stable formulations of convective terms for turbulent compressible flows,” _Journal of Computational Physics_ , Vol. 382, 2019, pp. 86–104.  https://doi.org/10.1016/j.jcp.2019.01.007.
  * Carpenter and Kennedy [1994] Carpenter, M. H., and Kennedy, C. A., “Fourth-order 2N-storage Runge-Kutta schemes,” _NASA Langley Research Center_ , 1994.
  * Bogey and Bailly [2004] Bogey, C., and Bailly, C., “A family of low dispersive and low dissipative explicit schemes for flow and noise computations,” _Journal of Computational Physics_ , Vol. 194, No. 1, 2004, pp. 194–214.  https://doi.org/10.1016/j.jcp.2003.09.003.
  * Borges et al. [2008] Borges, R., Carmona, M., Costa, B., and Don, W. S., “An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws,” _Journal of Computational Physics_ , Vol. 227, No. 6, 2008, pp. 3191–3211.
  * Lusher and Sandham [2021] Lusher, D. J., and Sandham, N. D., “Assessment of low-dissipative shock-capturing schemes for the compressible Taylor–Green vortex,” _AIAA Journal_ , Vol. 59, No. 2, 2021, pp. 533–545.
  * Hamzehloo et al. [2021] Hamzehloo, A., Lusher, D. J., Laizet, S., and Sandham, N. D., “On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high-speed compressible flows,” _International Journal for Numerical Methods in Fluids_ , Vol. 93, No. 1, 2021, pp. 176–196.  https://doi.org/10.1002/fld.4879.
  * Yee et al. [1999] Yee, H., Sandham, N., and Djomehri, M., “Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters,” _Journal of Computational Physics_ , Vol. 150, No. 1, 1999, pp. 199–238.  https://doi.org/10.1006/jcph.1998.6177.
  * Yee and Sjögreen [2018] Yee, H. C., and Sjögreen, B., “Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows,” _Computers & Fluids_, Vol. 169, 2018, pp. 331–348.
  * Ducros et al. [1999] Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C., and Poinsot, T., “Large-Eddy Simulation of the Shock/Turbulence Interaction,” _Journal of Computational Physics_ , Vol. 152, No. 2, 1999, pp. 517 – 549.
  * Bhagatwala and Lele [2009] Bhagatwala, A., and Lele, S. K., “A modified artificial viscosity approach for compressible turbulence simulations,” _Journal of Computational Physics_ , Vol. 228, No. 14, 2009, pp. 4965–4969.  https://doi.org/10.1016/j.jcp.2009.04.009.
  * Ishida et al. [2016] Ishida, T., Ishiko, K., Hashimoto, A., Aoyama, T., and Takekawa, K., “Transonic buffet simulation over supercritical airfoil by unsteady-FaSTAR code,” _AIAA Paper 2016-1310_ , 2016.
  * Obayashi and Guruswamy [1995] Obayashi, S., and Guruswamy, G. P., “Convergence Acceleration of an Aeroelastic Navier-Stokes Solver,” _AIAA Journal_ , Vol. 33, 1995, pp. 1134–1141.
  * Mavriplis [2003] Mavriplis, D. J., “Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured,” _16th AIAA Computational Fluid Dynamics Conference, AIAA paper 2003-3986_ , 2003.  10.2514/6.2003-3986.
  * Spalart and Allmaras [1992] Spalart, P. R., and Allmaras, S. R., “A One-Equation Turbulence Model for Aerodynamic Flows,” _30th Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings_ , 1992.  10.2514/6.1992-439.
  * Dacles-Mariani et al. [1995] Dacles-Mariani, J., Zilliac, G. G., Chow, J. S., and Bradshaw, P., “Numerical/Experimental Study of a Wingtip Vortex in the Near Field,” _AIAA Journal_ , Vol. 33, 1995, pp. 1561–1568.  10.2514/3.12826.
  * Spalart [2000] Spalart, P. R., “Strategies for Turbulence Modelling and Simulation,” _International Journal of Heat and Fluid Flow_ , Vol. 21, 2000, pp. 252–263.  10.1016/S0142-727X(00)00007-2.
  * Sharov and Nakahashi [1998] Sharov, D., and Nakahashi, K., “Reordering of hybrid unstructured grids for lower-upper symmetric Gauss-Seidel computations,” _AIAA Journal_ , Vol. 36, 1998, pp. 484–486.  10.2514/2.392.
  * Akervik et al. [2006] Akervik, E., Brandt, L., Henningson, D. S., Hoepffner, J., Marxen, O., and Schlatter, P., “Steady solutions of the Navier-Stokes equations by selective frequency damping,” _Physics of Fluids_ , Vol. 18, 2006, p. 068102\.  10.1063/1.2211705.
  * Richez et al. [2016] Richez, F., Lequille, M., and Marquet, O., “Selective frequency damping method for steady RANS solutions of turbulent separated flows around an airfoil at stall,” _Computers and Fluids_ , Vol. 132, 2016, pp. 51–61.  10.1016/j.compfluid.2016.03.027.
  * Visbal and Gordnier [2000] Visbal, M. R., and Gordnier, R., “A high-order flow solver for deforming and moving meshes,” _Fluids 2000 Conference and Exhibit, AIAA Paper 2000-2619_ , 2000.  10.2514/6.2000-2619.
  * CRM [[Accessed 19 October 2023]] “CRM.65.airfoil sections [Online],” _Available:https://commonresearchmodel.larc.nasa. gov/crm-65-airfoil-sections_ , [Accessed 19 October 2023].
  * Sandham et al. [2014] Sandham, N., Schülein, E., Wagner, A., Willems, S., and Steelant, J., “Transitional shock-wave/boundary-layer interactions in hypersonic flow,” _Journal of Fluid Mechanics_ , Vol. 752, 2014, p. 349–382.  10.1017/jfm.2014.333.
  * Sansica et al. [2014] Sansica, A., Sandham, N. D., and Hu, Z., “Forced response of a laminar shock-induced separation bubble,” _Physics of Fluids_ , Vol. 26, No. 9, 2014, p. 093601.  10.1063/1.4894427, URL https://doi.org/10.1063/1.4894427.
  * Sansica et al. [2016] Sansica, A., Sandham, N. D., and Hu, Z., “Instability and low-frequency unsteadiness in a shock-induced laminar separation bubble,” _Journal of Fluid Mechanics_ , Vol. 798, 2016, p. 5–26.  10.1017/jfm.2016.297.
  * Lusher and Sandham [2020] Lusher, D. J., and Sandham, N. D., “Shock-Wave/Boundary-Layer Interactions in Transitional Rectangular Duct Flows,” _Flow, Turbulence and Combustion_ , 2020.  10.1007/s10494-020-00134-0.