∎

11institutetext: T C L Fava (Corresponding Author) 22institutetext: D. S.
Henningson 33institutetext: A. Hanifi 44institutetext: KTH Royal Institute of
Technology, Dept. of Engineering Mechanics, FLOW and SeRC, SE-100 44
Stockholm, Sweden, 44email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Numerical investigation of the boundary layer stability on a section of a
rotating wind turbine blade

T. C. L. Fava  D. S. Henningson  A. Hanifi

(31 July 2023)

###### Abstract

Direct numerical simulations (DNS) and linear stability analysis of the flow
on a rotating wind turbine blade section are performed to study the effects of
rotation on laminar-turbulent transition on the suction side of the airfoil. A
chord Reynolds number of $1\times 10^{5}$ and angles of attack ($AoA$) of
$12.8^{\circ}$, $4.2^{\circ}$, and $1.2^{\circ}$ were considered. The first
set of simulations considered Coriolis and centrifugal forces that appear in
the rotating frame of reference. The second one disregarded these effects, but
the same angle of attack and Reynolds number were kept. Rotation effects on
the flow depend on the streamwise pressure gradient and direct/reverse flow
state. Upon separation in the $AoA=12.8^{\circ}$ case, rotation retarded the
flow and rendered the mixed Tollmien-Schlichting/Kelvin-Helmholtz (TS/KH)
instability in the laminar separation bubble (LSB) more unstable. Moreover, an
oblique secondary instability mechanism was more intense in the rotating case,
leading to a rapid breakdown of the rolls to small-scale turbulence. On the
other hand, a sub-harmonic mechanism was dominant in the non-rotating case,
and the rolls required a longer streamwise extent to lose spanwise coherence.
Despite that, rotation accelerated the attached boundary layer upstream of the
LSB, which was subjected to a strong adverse pressure gradient (APG),
stabilizing TS waves and rendering the transition location 3% more downstream
in the rotating case. The reattachment location was shifted 4% downstream in
the rotating case, which was both an effect of transition delay and flow
deceleration by rotation. In the $AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$
cases, rotation decelerated the flow upstream of separation, where a favorable
pressure gradient (FPG) is present. This increased the growth rates of TS
waves in this region. However, rotation slightly accelerated the flow inside
the LSB, rendering the TS/KH mode less unstable. Furthermore, the high
spanwise velocity and inflection crossflow profiles led to the appearance of
stationary and traveling crossflow modes in these cases, which enhanced the
perturbation level in regions of tip flow. Nonetheless, these modes presented
low energy compared to the TS/KH mechanism in the separated shear layer, which
remained the dominant instability. An effect of the crossflow was to generate
coherence in the spanwise component of the TS/KH mechanism, where a wavepacket
was formed close to the location of the inflection point in the spanwise
velocity profiles. Moreover, the secondary instability mechanism was mostly
oblique in the non-rotating case for $AoA=4.2^{\circ}$, whereas it was mainly
sub-harmonic in the rotating one. In the $AoA=1.2^{\circ}$ case, the sub-
harmonic mechanism seemed to dominate both conditions. Modes with negative
spanwise wavenumber were preferentially excited in the rotating case. In both
$AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$ cases, these effects did not affect
the transition location. A finite region of absolute instability was found in
these cases, where rotation generally rendered this mechanism less unstable.

###### Keywords:

Direct numerical simulations Flow instability Rotating wings Separation bubble
instability

††journal: Flow, Turbulence and Combustion

##  1 Introduction

In a pioneering study, Himmelskamp (1947) noted that rotating aircraft
propellers presented modified aerodynamic characteristics compared to static
blades, particularly a higher lift, a phenomenon later called rotational
augmentation. Nevertheless, early theoretical investigations with a blade
modeled as a flat plate indicated reduced rotation influence on the attached
flow region (Fogarty, 1951; McCroskey, 1971). The study of a helical blade by
Banks and Gadd (1963) demonstrated that rotation delays or even suppresses
separation, especially in the inboard region, if the adverse pressure gradient
(APG) is low enough. Horlock and Wordsworth (1965) showed that the crossflow
strongly depends on rotation, and McCroskey and Yaggy (1968) found that this
velocity component is crucial for the separation delay. This phenomenon draws,
in particular, the attention of the wind-energy community due to the important
flow separation over the blades, often used to prevent over-speed in the so-
called stall-regulated wind turbines (Corten, 2001). In this regard, Savino
and Nyland (1985) and Bosschers (1995) confirmed the downstream moving of the
separation point experimentally for increasing rotation rates. Du and Selig
(2000) solved the integral momentum boundary layer equations considering
rotation and also found separation and stall delays and lift enhancement for
decreasing Reynolds ($Re$) and increasing rotation ($Ro_{r}$) numbers and
solidity ($c/r$). Here, $Ro_{r}=\Omega r/V_{\infty}$, where $\Omega$ is the
rotation speed, $r$ is the radius, $V_{\infty}$ is the wind speed, and $c$ the
chord length. Other works also identified the increase in solidity to promote
stronger rotation effects and a direct proportionality between this parameter
and the lift coefficient (Shen and Sørensen, 1999; Chaviaropoulos and Hansen,
2000). Nonetheless, the latter two works, which involved quasi-two-dimensional
computations of a blade section, indicated no change in the separation
location but rather a pressure reduction in this region. Moreover, Dumitrescu
et al. (2007) indicated that the separation location was little affected by
the solidity, but the reattachment point shifted upstream, and the spanwise
velocity increased with $c/r$.

A consequence of these findings is that the inboard region of the blade is
more affected by rotation. This zone, particularly for radial locations below
30% of the rotor radius, presents a marked stall delay and lift increase,
clearly observed in experiments (Ronsten, 1992; Björck, 1995). In regions
where the balance between the Coriolis and centrifugal forces does not occur,
such as in the separated flow, the latter imparts a radial velocity component
to stagnant or reverse flow, known as centrifugal pumping (Herráez et al.,
2014). Indeed, flow visualizations of full-scale wind-turbine rotors indicated
chordwise-oriented flow upstream of separation and after turbulent
reattachment but strongly radial streamlines upon separation (McCroskey, 1973;
Savino and Nyland, 1985; Ronsten, 1992; Corten, 2001; Herráez et al., 2014),
possibly reaching 80% of the azimuthal velocity (Chaviaropoulos and Hansen,
2000). This was also noticed in experiments of rotating propeller blades
(Schülein et al., 2012). The Coriolis force acting on this radial flow towards
the tip tends to accelerate the flow downstream, effectively acting as a
favorable pressure gradient (FPG) (Bosschers, 1995; Breton et al., 2008;
Dumitrescu and Cardos, 2004; Herráez et al., 2014). Chaviaropoulos and Hansen
(2000) and Dumitrescu and Cardos (2004) concluded that the radial component
sucks low-momentum fluid from the separation region, lowering the pressure on
the suction side. This vision is also shared by Schreck et al. (2007), which
concluded that the lift enhancement correlated with the magnitude of the
radial velocity component.

Due to the complexity of experiments and the high computational cost of
numerical investigations, investigations on laminar-turbulent transition in
blades have disregarded rotation effects (Lang et al., 2015). As indicated by
Hernandez et al. (2007), most numerical studies assume the boundary layer to
be entirely turbulent or employ inadequate or oversimplified criteria for the
prediction of laminar-turbulent transition. Hernandez (2012) analyzed the
effects of rotation on transition in a blade by generating a base flow for
stability analysis using integral boundary layer equations. Rotation
stabilized the flow to TS waves upstream of separation and on the pressure
side. Pascal et al. (2018) computed the flow over a rotating fan blade with
Reynolds averaged Navier-Stokes (RANS) equations and performed local linear
stability analysis of the flow profiles. Rotation almost did not change the
growth rates of TS waves but destabilized stationary crossflow modes, which
were too weak to trigger transition. Fava et al. (2021) performed a similar
analysis but considered more general disturbances and concluded that
transition was shifted upstream in regions with strong rotation effects and
three-dimensionality (e.g., the inboard region). Highly oblique modes were
found to play a role in this phenomenon. Jing et al. (2020) performed a direct
numerical simulation (DNS) of a wind-turbine section ranging from 38% to 88%
of the rotor radius at a chord Reynolds number $Re_{c}=3\times 10^{5}$.
Inflectional spanwise velocity profiles were obtained, but they presented a
maximum of 5% of the streamwise velocity, and crossflow instability was not
present. Transition occurred via Tollmien-Schlichting (TS) waves, with waves
slightly misaligned with the spanwise direction. However, these simulations
were performed for near-optimal conditions, reducing separation and rotation
effects. Gross et al. (2012) studied two sections of a rotating wind turbine
blade with DNS at 20% and 80% of the rotor radius, with an angle of attach
($AoA=5^{\circ}$) and showed that rotation triggered stationary and traveling
crossflow modes due to the generation of a crossflow velocity component, which
contributed to an earlier transition, separation delay, and lift increase.
Guntur and Sørensen (2014) indicated that the previous study lacked the
analysis of higher $AoA$, more relevant for wind-turbine applications.
Furthermore, Gross et al. (2012) did not account for rotation terms in the
stability calculations. In their implicit LES of a helicopter blade in yaw,
Wen and Gross (2019) found that the structures shed from the LSB were oblique
and concluded their origin might be crossflow instability. Jing and Ducoin
(2020) employed DNS to analyze a rotating marine propeller and found that the
radial velocity can reach around 14% (50% near the hub upon separation) of the
azimuthal velocity, triggering crossflow instability and transition to
turbulence, similar to the rotating disk. In this regard, the flow was
absolutely unstable in the radial direction and convectively unstable in the
streamwise one. Experiments by Yang et al. (2006) indicated the formation of
discrete quasi-periodic streamwise vortical structures (spanwise cells) in the
separated flow on a rotating hovering blade, which may indicate crossflow
vortices. The PIV measurements of DiOttavio et al. (2008) and Raghav and
Komerath (2014) confirmed these structures. They found that despite being
present in fixed wings under stall (Winkelman and Barlow, 1980; Bippes and
Turk, 1982; Weihs and Katz, 1983; Schewe, 2001), the radial flow generated by
rotation made them more pronounced. Since these cells could be linked with a
steady three-dimensional global mode in the non-rotating case (Theofilis et
al., 2000; Rodríguez and Theofilis, 2010), rotation may affect the global
stability of such flows.

Further results on the effects of rotation on the boundary layer stability and
transition are mainly restricted to canonical configurations, such as rotating
channels (Alfredsson and Persson, 1989; Wall and Nagata, 2006), flat plates
(Potter and Chawla, 1971; Deschamps and Hein, 2018), and disks (Malik et al.,
1981; Hall, 1986; Malik, 1986; Balachandar et al., 1992; Lingwood, 1995,
1996). In the case of rotating disks with axial inflow, which may be a
simplified model for a rotating blade, Hussain et al. (2011) and Deschamps and
Hein (2018) found that the axial inflow was stabilizing vis-à-vis crossflow
instabilities since the crossflow velocity profiles lose their inflectional
character.

From the above, little is known about the impact of rotation on transition
mechanisms on rotating blades (Pascal et al., 2018; Lang et al., 2015;
Deschamps and Hein, 2018). This is partly due to the limited number of studies
carrying out detailed numerical simulations of the transition process on
rotating blades. Furthermore, the available results were performed for near-
optimum operation conditions, which tends to prevent separation and mitigate
the rotation effects (Jing et al., 2020). The lack of parametric change in the
rotation conditions yields another knowledge gap (Guntur and Sørensen, 2014).
For this reason, the current work attempts to shed light on the above
phenomena through the detailed DNS of a rotating wind turbine blade section at
68% of the rotor radius, $Re_{c}=1\times 10^{5}$, and three operational
conditions. The latter consists of three angles of attack, corresponding to
three rotation numbers or tip-speed ratios. Accompanying simulations with the
same parameters but without rotation are also performed for comparison. The
laminar-turbulent transition process is analyzed and compared with linear
stability results. The paper is divided as follows. §2 presents the numerical
setup, including the blade geometry, numerical method for obtaining the flow,
study cases, and temporal and spatial convergence studies. §3 displays the
results. The mean-characterization is performed in §3.1.1, and instantaneous
structures are presented in §3.1.2. Linear primary stability analysis results
are shown in §3.2.1. Further investigation of the flow through Fourier and
SPOD analysis is carried out in §3.2.2. Finally, §4 presents the conclusions
of this work.

##  2 Numerical setup

###  2.1 Blade model

In the following, unless otherwise stated, the variables are non-
dimensionalized with the chord length ($c$), and the resultant free-stream
velocity ($U_{\infty}=\sqrt{V_{\infty}^{2}+(\Omega r)^{2}}$) from the wind
($V_{\infty}$) and rotation ($\Omega r$), where $\Omega$ is the rotation
speed, and $r$ is the radial position of the simulated domain.

The simulation domain consists of a slice of the DTU 10-MW Reference Wind
Turbine (Bak et al., 2012) at 68% of the rotor radius ($R$), whose profile
corresponds to a blend of 96% of FFA-W3-241, and 4% of the FFA-W3-301 airfoils
(Björck, 1990). The blade and the slice simulated are shown in Fig. 1 (a). The
domain has a span width of 10% of the chord length, but no spanwise geometry
variation is considered to allow a periodicity condition in this direction.
The chord Reynolds number was $Re_{c}=U_{\infty}c/\nu=1\times 10^{5}$, where
$\nu$ is the kinematic viscosity, local solidity $c/r=0.055$, and geometric
twist angle $\phi=4.8^{\circ}$. Figure 1 (b) shows the blade cut by a
cylindrical shell with the centerline in the vertical direction aligned with
the rotation axis ($\Omega$ vector). $AoA$ is the angle of attack, and $L$ and
$D$ are the lift and drag forces. The geometries were rotated in the clockwise
direction by $\phi$, as portrayed in Fig. 1 (c), so that the $x$ coordinate of
the simulation frame of reference lies along the airfoil chord. In this way,
the $y$ and $z$ coordinates point in the vertical and out-of-plane directions,
respectively.

Figure 1: (a) DTU 10-MW Reference Wind Turbine blade and domain of simulation
at $r/R=0.68$ ($R$ is the tip radius in meters). (b) Physical frame of
reference. (c) Simulation frame of reference.

###  2.2 Numerical method

The simulations were performed with the incompressible spectral-element
Navier-Stokes solver Nek5000 (Fischer et al., 2008). The rotating reference
frame fixed to the blade was adopted for the computations. The Coriolis and
centrifugal forces were included as volume sources in the momentum equations
and are given by

$\displaystyle f_{x}=-2\Omega_{x}u_{z},$ $\displaystyle
f_{y}=+2\Omega_{y}u_{z},$ (1) $\displaystyle
f_{z}=-r\Omega^{2}+2\Omega_{y}u_{x}-2\Omega_{x}u_{y},$

where $u_{x}$, $u_{y}$, and $u_{z}$ are the velocity components in $x$, $y$,
and $z$. Initial and boundary conditions were computed with RANS simulations
using the solver Fluent (ANSYS, 2013), where Coriolis and centrifugal forces
were also included. The domain size of the RANS computations was five chord
lengths in every direction. The DNS domain extends one-chord length from the
airfoil in every direction except the wake, where the outlet is two-chord
lengths downstream of the trailing edge. The mesh has 246,000 elements, each
discretized with eight Gauss-Lobatto-Legendre (GLL) points in every direction
(seventh-order polynomial approximation), yielding 126 million grid points. A
time step $\Delta t=5\times 10^{-6}$ was used.

A uniform Dirichlet boundary condition for the velocity was imposed on the
inlet of the RANS simulations, consisting of the velocity with magnitude
$U_{\infty}$ and angle $AoA$ on the $xy$ plane. In addition, zero spanwise
velocity was imposed on this boundary, which is far from the blade. A zero-
pressure Neumann boundary condition was enforced on the outlet. The velocity
field computed with RANS was interpolated onto the inlet surface of the DNS
domain and used as a Dirichlet boundary condition. An open boundary condition
where $\left[-p\mathbf{I}+Re_{c}^{-1}\nabla\mathbf{U}\right]\cdot\mathbf{n}=0$
was applied to the outlet ($p$ is pressure, $\mathbf{I}$ the identity matrix,
$\mathbf{U}$ the velocity vector, and $\mathbf{n}$ the normal vector).
Finally, periodic boundary conditions were employed in the spanwise direction
of both RANS and DNS

The spectral element method is known for displaying low noise levels, which
may preclude transition. In order to prevent that, the tripping line described
in Schlatter and Örlü (2012) was applied at 5% of the chord on the suction
side with an unsteady three-dimensional random forcing with an amplitude of
$1\times 10^{-6}$ uniformly distributed over frequency until
$f=630\;U_{\infty}/c$. A fringe was applied to the last 50% of the chord
closest to the outlet to promote an outflow normal to the boundary and without
backflow events. The implicit filter of Negi et al. (2017) was employed,
albeit with a small amplitude corresponding to $2\times 10^{-7}$ in an
explicit filter and acting on the highest 33% of wavenumbers.

###  2.3 Cases of study

Table 1 summarizes the studied cases. $Ro_{r}=(\Omega r)/U_{\infty}$ is the
local radial rotation number, $\lambda=(\Omega R)/V_{\infty}$ is the tip-speed
ratio, $u_{x_{\infty}}$ and $u_{y_{\infty}}$ are the resultant, undisturbed
inflow velocities in the $x$ and $y$ directions, $Ro_{c}$ is the rotation
number based on the chord, which is equivalent to $\Omega$, and $\Omega_{x}$
and $\Omega_{y}$ are the rotation speeds in the $x$ and $y$ directions.

$AoA$ (∘) | $Ro_{r}$ | $\lambda$ | $V_{\infty}$ | $\Omega r$ | $u_{x_{\infty}}$ | $u_{y_{\infty}}$ |  $\Omega$ ($\equiv Ro_{c}$) | $\Omega_{x}$ | $\Omega_{y}$
---|---|---|---|---|---|---|---|---|---
12.8 | 3.1 | 4.6 | 0.30 | 0.95 | 0.9751 | 0.2219 | 0.0523 | 0.0044 | 0.0521
12.8 | 0.0 | 0.0 | 0.30 | 0.95 | 0.9751 | 0.2219 | 0 | 0 | 0
4.2 | 6.3 | 9.3 | 0.16 | 0.98 | 0.9973 | 0.0737 | 0.0542 | 0.0045 | 0.0540
4.2 | 0.0 | 0.0 | 0.16 | 0.98 | 0.9973 | 0.0737 | 0 | 0 | 0
1.2 | 9.4 | 13.9 | 0.11 | 0.99 | 0.9998 | 0.0217 | 0.0546 | 0.0046 | 0.0544
1.2 | 0.0 | 0.0 | 0.11 | 0.99 | 0.9998 | 0.0217 | 0 | 0 | 0

Table 1: Parameters of the studied cases.

###  2.4 Mesh resolution and spatial/temporal convergence

The mesh was generated with the software ICEM (ANSYS, 2017), aiming at an
implicit large eddy simulation (LES) resolution with a well-resolved wall
region. The near-wall grid resolution was assessed using the parameter
$\Delta\xi^{+}_{wall}=\Delta\xi_{wall}/(\nu\sqrt{\tau_{w}})$, where
$\xi=x,y,z$. $\nu$ is the kinematic viscosity, $\tau_{w}$ is the wall stress.
For $\xi=y$, $\Delta y_{wall}$ is taken as the height of the first cell.
Considering all cases and both suction and pressure sides, $\Delta
y^{+}_{wall}<0.4$ for almost the entirety of the domain, except very close to
the trailing edge, where it slightly increases to 0.5. For $\xi=x,z$, since
the GLL points are not uniformly spaced, $\Delta\xi_{wall}$ is taken as the
maximum over each element. The average and minimum $\Delta\xi^{+}_{wall}$ are
related to the maximum by factors of 0.68 and 0.31, respectively.
$\max_{elem.}\Delta x^{+}_{wall}<23$ for the whole airfoil, and
$\max_{elem.}\Delta z^{+}_{wall}<10$, except in the first 4% of the chord,
where $\max_{elem.}\Delta z^{+}_{wall}$ can increase to 15. Note that this is
close to the optimum mesh resolution for LES to match DNS, which is $\Delta
y^{+}_{wall}=0.64$, $\Delta x^{+}_{wall}=18$, $\Delta z^{+}_{wall}=9$, as
found by Negi et al. (2018) and as per simulations of turbulent boundary
layers by Schlatter et al. (2010). The inflation rate of the elements in the
normal direction to the airfoil and streamwise direction in the wake is 1.15.

Furthermore, a grid independence study was performed by simulating the
$AoA=1.2^{\circ}$, $Ro_{r}=0$ case with polynomial order height instead of
seven, yielding 179 million grid points. Figure 2(a) compares the Reynolds
stresses in these two meshes. The results are spatially converged for
polynomial order seven with a maximum difference of 1.5% compared to the finer
grid. Therefore, the order-seven mesh is selected for all computations in this
work. The temporal convergence of mean fields is also assessed concerning the
length of the time series. The mean pressure coefficient ($C_{p}$) averaged
over a time of 5, 7.5, and 10 flowthroughs ($T=5$, 7.5, and 10) are compared
in Fig. 2(b) considering the rotating simulations, i.e., $Ro_{r}\neq 0$. For
$AoA=12.8^{\circ}$ and $AoA=4.2^{\circ}$, the results are converged for
$T=7.5$, although the differences are already minor for $T=5$ in the former
case. For $AoA=1.2^{\circ}$, convergence occurs for $T=5$. The analysis of the
skin friction coefficient ($C_{f}$) led to similar conclusions. Therefore, the
selected averaging time of $T=10$ is enough for the convergence of the mean
fields for all cases. Each simulation required $3.5\times 10^{5}$ CPU hours
for $T=10$.

(a) Reynolds stresses on the suction side for 7th and 8th polynomial orders.
(b) Pressure coefficient on both sides.

Figure 2: Panel (a): spatial convergence of the Reynolds stresses on the
suction side for standard (7th order) and refined (8th order) meshes for
$AoA=1.2^{\circ}$, $Ro_{r}=0$. Panel (b): Temporal convergence of the mean
pressure distribution of the rotating cases.

##  3 Results

Unless otherwise indicated, the following analyses focus on the suction side
of the airfoil, which is more susceptible to separation, transition, and
rotation effects.

###  3.1 General flow characteristics

####  3.1.1 Mean flow

The mean pressure coefficient ($C_{p}$) is displayed in Fig. 3(a). The
$AoA=12.8^{\circ}$ case presents an adverse pressure gradient (APG) almost on
the entirety of the suction side (upper part of the curves) for
$AoA=12.8^{\circ}$, with a peak suction of about $-4$ near the leading edge,
characteristic of wind turbine airfoils under similar $AoA$ (Ronsten, 1992).
Nevertheless, the suction peak is 5.6% more negative for the non-rotating
case, a difference that persists up to the location of the maximum LSB height
for the $Ro_{r}=0$ case, which occurs at $x=0.36$. At this point, the non-
rotating case presents a sharp pressure increase, associated with laminar-
turbulent transition. The pressure rise occurs at $x=0.39$ for the rotating
case, indicating a more downstream transition. This suggests that the Coriolis
force reduces the APG on the suction side, as also noticed by Dumitrescu and
Cardos (2004) and Herráez et al. (2014). However, rotation increases the APG
vis-à-vis the non-rotating case in the region extending from $x=0.42$, close
to the LSB maximum height, to $x=0.5$, 4% downstream of reattachment. This
phenomenon can be attributed to the momentum deficit in this region, which
promotes a reversal in the direction of the Coriolis force that passes to act
as an APG. After turbulent reattachment, there are no significant differences
between the $C_{p}$ curves on the suction side. Considering the
$AoA=4.2^{\circ}$ case, the shape of the pressure distribution does not have a
marked leading-edge suction peak as observed for $AoA=12.8^{\circ}$. The
region under favorable pressure gradient (FPG) on the suction side extends
until $x=0.2$ for the non-rotating case and $x=0.21$ for the rotating one. The
APG is mild downstream of that, indicating the presence of an extended region
of flow separation. The non-rotating case displays a lower $C_{p}$ until the
transition region, around $70\%$ of the chord length, indicating a similar
trend to the $AoA=12.8^{\circ}$ case. Finally, considering the
$AoA=1.2^{\circ}$ case, an FPG acts until $x=0.27$ on the suction, followed by
a low APG leading the flow to reattach around 80% chord. The pressure is
slightly lower for the non-rotating case, consistent with the trend noticed
for higher $AoA$. On the pressure side, the non-rotating cases display a
slightly higher $C_{p}$, but this difference is only significant in the
$AoA=4.2^{\circ}$ case. This suggests that rotation has an opposite effect on
each side of the airfoil, a consequence of the streamwise velocity being
higher than its free-stream value in the accelerating region of the suction
side and lower than it in the decelerating region of the pressure side.
Therefore, the leading-edge region with strong acceleration is especially
susceptible to this effect.

(a) Pressure coefficient on both sides.  (b) Friction coefficient on the
suction side.

Figure 3: Effect of rotation on the mean pressure and skin friction
distributions. The following color code is used:  for $AoA=12.8^{\circ}$,
$Ro_{r}=3.1$;  for $AoA=4.2^{\circ}$, $Ro_{r}=6.3$;  for $AoA=1.2^{\circ}$,
$Ro_{r}=9.4$. Dashed lines correspond to their $Ro_{r}=0$ counterparts.

The friction coefficient ($C_{f}$) on the suction is presented in Fig. 3(b).
The $C_{f}$ is higher in the rotating case for $AoA=12.8^{\circ}$ upstream of
separation, which occurs at $x=0.23$ for both rotating and non-rotating
counterparts. The more elevated $C_{f}$ indicates that the flow is accelerated
in the streamwise direction by rotation in this region. The near-zero negative
$C_{f}$ indicates that separation is mild until $x=0.34$ for $Ro_{r}=0$ and
$x=0.38$ for $Ro_{r}=3.1$. However, downstream of those locations, a stronger
reverse flow occurs, noticed by a sharp drop in $C_{f}$, which is larger for
the rotating case. Although taking place more downstream, the reverse flow is
more intense in the non-rotating case, likely due to the reverse Coriolis
force that acts as an APG in this part of the LSB. This region is highly
unstable, and transition occurs, which leads the flow to reattach at $x=0.46$
and $x=0.42$ for the rotating and non-rotating cases, respectively, followed
by a large increase in $C_{f}$ characteristic of turbulent flow. In regard to
the $AoA=4.2^{\circ}$ case, separation occurs at $x=0.34$ for both rotating
and non-rotating cases, which is mild until around 56% chord. In the region
upstream of this point, the $C_{f}$ curves for both rotating and non-rotating
conditions agree. However, downstream of $x=0.56$, a stronger reverse flow is
present, which occurs earlier for the non-rotating case, displaying a peak
negative $C_{f}$ at $x=0.69$, whereas this occurs at $x=0.72$ for the rotating
case. The flow remains separated until close to the trailing edge of the
airfoil, with the non-rotating case exhibiting more negative $C_{f}$ in most
of this region. Similar trends are noticed in the $AoA=1.2^{\circ}$ case, in
which separation occurs at $x=0.38$ but remains weak until around 65% chord.
Upstream of this location, the $C_{f}$ distributions for $Ro_{r}=0$ and
$Ro_{r}=9.4$ present minimal differences. Stronger separation occurs slightly
upstream for the non-rotating case, with peak negative $C_{f}$ at $x=0.625$,
being 1% of the chord more downstream for the rotating condition. Transition
occurs in this region and leads the flow to reattach at $x=0.79$ for the
rotating case and $x=0.78$ for the non-rotating one, suggesting a slightly
faster breakdown to turbulence in the former case.

The spanwise and time-averaged streamwise velocity field is shown in Fig. 4,
with the displacement thickness ($\delta^{*}$) depicted with a black line and
the edge of the LSB with a gray line. In the $AoA=12.8^{\circ}$ case, one can
notice a stronger reverse flow in the rotating case that reaches -16% of the
free-stream velocity at $x=0.41$ compared to -7% for the non-rotating case at
$x=0.38$ (the LSB is 74.4% higher in the former case). The reverse flow in the
non-rotating case may be enough to trigger absolute instability (Alam and
Sandham, 2000; Rodríguez et al., 2021). These locations are slightly
downstream of the maximum LSB height ($x=0.4$ and $x=0.36$, respectively). The
most intense reverse flow in the rotating case is due to the streamwise
deceleration promoted by the Coriolis force accentuated when the flow velocity
is substantially lower than its free-stream value, such as inside the LSB.
Considering the $AoA=4.2^{\circ}$ case, the LSB consists of three reverse flow
cells located at $x=0.34-0.63$, $x=0.63-0.81$, and $x=0.81-0.99$, with the
center cell presenting the strongest reverse flow of -12% in both cases.
Nonetheless, the location of this maximum is more downstream in the rotating
case ($x=0.71$) than the non-rotating one ($x=0.68$). These locations
correspond approximately to the position of the maximum LSB height. The split
of the reverse flow in separate cells is associated with transition to
turbulence and inception of unsteadiness (Theofilis et al., 2000; Cherubini et
al., 2010). The indentations in the LSB edge are caused by the smaller reverse
flow in the contact regions between two adjacent clockwise-rotating
recirculation cells, where the flow is mostly oriented in the normal
direction. Notice that the LSB height in the rotating case is slightly shorter
(-2.57%); albeit a small reduction, this denotes a different trend from that
observed for $AoA=12.8^{\circ}$. The large $\delta^{*}$ and LSB size for
$AoA=4.2^{\circ}$ is possibly related to weaker instabilities developing on
the separated shear layer, which are not strong enough to make the flow
reattach within a short streamwise extent. This probably results from a weak
APG. Regarding the $AoA=1.2^{\circ}$ case, separation occurs at $x=0.38$, and
two recirculation vortices are formed at $x=0.38-0.66$ and
$x=0.66-0.78(0.79)$. Transition takes place in the first cell, and
reattachment occurs at the end of the second one, located at $x=0.78$ in the
non-rotating case and $x=0.79$ in the rotating one. The maximum reverse flow
is -16% for $Ro_{r}=0$ at $x=0.72$ and -15% for $Ro_{r}=9.4$ at $x=0.71$. The
LSB is only 1.7% taller in the rotating case, indicating that the rotation
effects seem to be less strong for lower $AoA$, despite the higher rotation
speed.

(a) $AoA=12.8^{\circ}$, $Ro_{r}=3.1$.  (b) $AoA=12.8^{\circ}$, $Ro_{r}=0$.
(c) $AoA=4.2^{\circ}$, $Ro_{r}=6.3$.  (d) $AoA=4.2^{\circ}$, $Ro_{r}=0$.  (e)
$AoA=1.2^{\circ}$, $Ro_{r}=9.4$.  (f) $AoA=1.2^{\circ}$, $Ro_{r}=0$.

Figure 4: Spanwise and time-averaged streamwise velocity ($\langle
U\rangle_{z,t}$) contours.  is the displacement thickness ($\delta^{*}$).  is
the edge of the LSB ($(x,y)\in\int_{0}^{y}\langle
U\rangle_{z,t}(x,\xi)\;d\xi=0$, where $\langle\cdot\rangle_{z,t}$ is the
average in span and time). Arrows represent the velocity field.

The mean spanwise velocity field is displayed in Fig. 5 for the rotating
cases. Results for the non-rotating blade are not presented, as the mean
spanwise velocity is zero. In the $AoA=12.8^{\circ}$ case, this variable is
positive on the whole suction side, indicating flow towards the blade root.
However, its magnitude is significantly reduced, particularly in the front
region of the LSB. The low streamwise velocity in this region reduces the
strength of the Coriolis force, pointing towards the blade root, compared to
the centrifugal force. Therefore, the spanwise flow is significantly reduced
but not reversed since the Coriolis force is still dominant. Also, notice that
the spanwise velocity component is significantly low for $y\leq\delta^{*}$ in
the laminar region upstream of the LSB and from the separation point to near
the location of the maximum LSB height. In the turbulent flow region
downstream of the latter location, the low spanwise velocity region is
restricted to an area very close to the wall. The $AoA=4.2^{\circ}$ case
displays a region with negative spanwise velocity, i.e., pointing towards the
blade tip in the region extending from the separation point to slightly
upstream of the maximum height of the first recirculation cell of the LSB. The
maximum tip flow reaches 6% at $x=0.45$, which may be enough to trigger
crossflow instability, especially because the spanwise velocity profile is
inflectional. The region with spanwise velocity deficit occurs for
$y\leq\delta^{*}$ and extends until the end of the first recirculation cell,
approximately at $x=0.63$. For $AoA=1.2^{\circ}$, a tip flow with a maximum
intensity of 5% occurs at $x=0.5$ corresponding to the front part of the first
recirculation cell. In this case, there is also the possibility of crossflow
instability due to the inflectional crossflow velocity profile (Saric et al.,
2003). Similar to the previous cases, the area with a deficit of spanwise
velocity extends from the leading edge to the end of the first recirculation
cell, which in the current case is approximately at $x=0.66$, for
$y\leq\delta^{*}$.

(a) $AoA=12.8^{\circ}$, $Ro_{r}=3.1$.  (b) $AoA=4.2^{\circ}$, $Ro_{r}=6.3$.
(c) $AoA=1.2^{\circ}$, $Ro_{r}=9.4$.

Figure 5: Spanwise and time-averaged spanwise velocity ($\langle
W\rangle_{z,t}$) contours.  is the displacement thickness ($\delta^{*}$).  is
the edge of the LSB ($(x,y)\in\int_{0}^{y}\langle
U\rangle_{z,t}(x,\xi)\;d\xi=0$, where $\langle\cdot\rangle_{z,t}$ is the
average in span and time). Arrows represent the velocity field.

Figure 6 shows the normal profiles of mean streamwise and spanwise velocity
(black and gray solid lines, respectively). Moreover, the difference between
the streamwise velocity profiles in the rotating and non-rotating cases
($\overline{\Delta U}$) is depicted with a blue line. Notice that the scale of
the gray and blue lines is enlarged three times compared to that of the
streamwise velocity. Considering the $AoA=12.8^{\circ}$, $Ro_{r}=3.1$ case,
the flow is streamwise accelerated compared to the non-rotating case upstream
of the LSB, highlighted by the positive values of $\overline{\Delta U}$. The
peak in this variable occurs at $y=\delta^{*}$, the region most affected by
rotation. However, the fact that $\overline{\Delta U}\rightarrow 0$ in the
free-stream indicates that the streamwise velocity in this region is only
mildly affected by rotation. Interestingly, inside the LSB, particularly at
$x=0.4$, there is a strong deceleration of the flow in the rotating case,
demonstrated by the $\overline{\Delta U}<0$ profile at this station. This
effect tends to vanish downstream of the LSB, and the flow is accelerated by
rotation again for $x\geq 0.6$, albeit with a much lower intensity since the
turbulent flow presents a smaller near-wall region with a momentum deficit.
The spanwise velocity profiles do not possess sign inversion, as previously
noticed in the contours of this variable. Nevertheless, the profiles develop a
distinct shape inside the LSB, first with a near-zero spanwise velocity at
$x=0.3$ and then with a clearly inflectional character at $x=0.4$. Regarding
the $AoA=4.2^{\circ}$, $Ro_{r}=6.3$ case, the flow is slightly retarded
upstream of separation and mildly accelerated near the wall in the LSB. This
constitutes a change in the trend with respect to $AoA=12.8^{\circ}$. This may
be connected with the fuller streamwise velocity profiles in the
$AoA=4.2^{\circ}$ case due to the FPG acting for $x<0.21$, which reduces the
near-wall momentum deficit and makes the rotation effects accelerate the flow
downstream in this region preceding separation. At some stations inside the
LSB, the flow is near-stagnant, such as at $x=0.4$, allowing a mild
acceleration of the flow by the Coriolis force. The spanwise velocity profiles
are clearly inflectional for some locations inside the LSB, particularly for
$x=0.4$, $0.5$, and $0.6$, with the first two stations displaying tip flow
($\overline{W}<0$). Regarding the $AoA=1.2^{\circ}$ case, the effects of
rotation on the streamwise velocity are practically zero outside the LSB and
mild inside it. The spanwise velocity profiles indicate near-wall tip flow at
$x=0.5$ and clear inflectional character at $x=0.5$, $0.6$, and $0.7$.

Figure 6: Spanwise and time-averaged profiles of streamwise  ($\langle
U\rangle_{z,t}$) and spanwise  ($\langle W\rangle_{z,t}$) velocity.  is the
difference between the streamwise velocity in the rotating and non-rotating
cases.  is the edge of the LSB ($(x,y)\in\int_{0}^{y}\langle
U\rangle_{z,t}(x,\xi)\;d\xi=0$, where $\langle\cdot\rangle_{z,t}$ is the
average in span and time). The scale of  ($\overline{U}=\langle
U\rangle_{z,t}$) is shown in the upper left corner.  and  are plotted with a
scale three times larger for enhanced visibility.

Table 2 summarizes some characteristics of the LSB and mean flow. $x_{b_{1}}$
and $x_{b_{2}}$ are the leading and trailing edges of the separation region.
$x_{y_{max}}$ and $y_{max}$ are the streamwise location of the separation
bubble maximum height, and the value of this height ($y_{max}/\delta^{*}$ for
the height normalized by the displacement thickness). $x_{a}$ is the position
of the start of the APG. $u_{r_{max}}$ and $w_{r_{max}}$ are the maximum
streamwise and spanwise negative flows, respectively. $x_{u_{r_{max}}}$ and
$x_{w_{r_{max}}}$, in this order, are the streamwise locations where these
maxima occur.

Table 2: Characteristics of the mean flow.

AoA (∘) | $Ro_{r}$ | $x_{b_{1}}$ | $x_{b_{2}}$ | $x_{y_{max}}$ | $y_{max}\times 10^{2}$ | $y_{max}/\delta^{*}$ | $x_{a}$ | $u_{r_{max}}$ | $x_{u_{r_{max}}}$ | $w_{r_{max}}$ | $x_{w_{r_{max}}}$
---|---|---|---|---|---|---|---|---|---|---|---
12.8 | 3.1 | 0.23 | 0.46 | 0.40 | 0.75 | 0.70 | 0.00 | -0.16 | 0.41 | 0 .00 | -
12.8 | 0.0 | 0.23 | 0.42 | 0.36 | 0.43 | 0.54 | 0.00 | -0.07 | 0.38 | - | -
4.2 | 6.3 | 0.34 | 0.99 | 0.70 | 2.65 | 0.51 | 0.21 | -0.12 | 0.71 | -0.06 | 0.45
4.2 | 0.0 | 0.34 | 0.99 | 0.68 | 2.72 | 0.56 | 0.20 | -0.12 | 0.68 | - | -
1.2 | 9.4 | 0.38 | 0.79 | 0.70 | 1.78 | 0.71 | 0.27 | -0.15 | 0.71 | -0.05 | 0.50
1.2 | 0.0 | 0.38 | 0.78 | 0.70 | 1.75 | 0.71 | 0.27 | -0.16 | 0.72 | - | -

Figure 7 shows the Reynolds stresses and the root mean square (RMS) values of
velocity perturbations as a function of the streamwise coordinate.
$\overline{u^{\prime}v^{\prime}}$ (Fig. 7(a)) becomes negative as
instabilities develop in the flow. Yuan et al. (2005) proposed the threshold
$\overline{u^{\prime}v^{\prime}}=-0.001$ to identify transition to turbulence,
which tends to pinpoint the early stages of transition instead of the full
turbulent breakdown. For $AoA=12.8^{\circ}$, this criterion is met at $x=0.33$
in the non-rotating case and $x=0.35$ in the rotating one, indicating a
downstream shift of the transition location by rotation. The streamwise
location of the maximum boundary layer shape factor $H=\delta^{*}/\theta$,
where $\theta$ is the momentum thickness, is also an indicator of the
transition location (Burgmann and Schröder, 2008). These locations are
$x=0.36$ for $Ro_{r}=0$ and $x=0.39$ for $Ro_{r}=3.1$. Therefore, transition
is delayed by 2%-3% by rotation in the $AoA=12.8^{\circ}$ case.
$\overline{u^{\prime}v^{\prime}}$ reaches a more negative peak value in the
rotating case, suggesting a stronger breakdown to turbulence. Regarding
$AoA=4.2^{\circ}$, the $\overline{u^{\prime}v^{\prime}}=-0.001$ threshold is
reached earlier in the rotating case ($x=0.35$) than in the non-rotating one
($x=0.37$), suggesting a slight upstream shift of transition with rotation.
However, the maximum $H$ occurs at $x=0.5$ in both cases, indicating that,
despite a faster beginning of transition in the rotating case, this process is
completed at the same location as in the non-rotating case. Furthermore, the
peak in $\overline{u^{\prime}v^{\prime}}$ is more pronounced in the non-
rotating case this time, indicating a stronger breakdown for this case. The
rotation effects on the $AoA=1.2^{\circ}$ case are qualitatively similar to
those in the $AoA=4.2^{\circ}$ case, although less pronounced. Yuan’s
criterion is met 1% more upstream in the rotating case ($x=0.35$), pointing to
an earlier transition start. Nonetheless, the full turbulent breakdown occurs
at the same place, yielding a transition point at $x=0.7$ in both cases. The
rotating cases also present significantly higher
$\overline{v^{\prime}w^{\prime}}$ component, as presented in Fig. 7(b),
indicating the formation of coherent structures, e.g., streamwise vortices.
This is particularly true for cases with large separation regions, such as
$AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$. Stall cells have been observed
experimentally and numerically in non-rotating (Winkelman and Barlow, 1980;
Bippes and Turk, 1982; Schewe, 2001) and rotating (Yang et al., 2006;
DiOttavio et al., 2008; Raghav and Komerath, 2014) wings. Rodríguez and
Theofilis (2010) indicated that such cells in a non-rotating stalled airfoil
could be associated with a steady three-dimensional global mode, as found by
Theofilis et al. (2000). Another possibility is that coherent structures arise
from crossflow instability (Gross et al., 2012).

Figures 7(c) and 7(d) present the RMS values of streamwise ($u^{\prime}$) and
spanwise ($w^{\prime}$) velocity perturbations. In the $AoA=12.8^{\circ}$
case, all disturbance components are higher for $Ro_{r}=0$ upstream of
$x=0.43$, indicating that the boundary layer is more unstable in at least part
of this region in the non-rotating case. Downstream of this point,
$w^{\prime}_{RMS}$ develop higher peak values in the rotating case, whereas
the peak in $u^{\prime}_{RMS}$ is comparable to those in the non-rotating
case. This suggests the presence of structures in the $yz$ plane for
$Ro_{r}=3.1$ that could be either crossflow modes or non-linearly/non-modal
generated streamwise vortices (Schmid and Henningson, 2001). A threshold of
$u^{\prime}_{RMS}\approx 40\%$ of the free-stream velocity precedes
transition, even though this value is smaller compared to the local edge
velocity: 25.8% at $x=0.4$ for $Ro_{r}=0$ and 25.9% at $x=0.44$ for
$Ro_{r}=3.1$. Considering $AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$, the RMS
values of all perturbations are higher in the rotating case for a large
portion of the suction side for $AoA=4.2^{\circ}$ and in all this region for
$AoA=1.2^{\circ}$. In particular, $w^{\prime}_{RMS}$ displays considerably
higher values in the rotating case until $x\approx 0.65$. At $x=0.46$, for
example, $w^{\prime}_{RMS}$ is 947% higher in the rotating case compared to
$Ro_{r}=0$ for $AoA=4.2^{\circ}$ and 1316% for $AoA=1.2^{\circ}$. Relative to
the local boundary-layer edge velocity, $w^{\prime}_{RMS}$ corresponds to
11.3% and 3.9% for $AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$, respectively.
Notice that $x=0.46$ is located before transition, suggesting that these
perturbations do not arise from non-linear effects but rather from a possible
crossflow instability. The higher values of $u^{\prime}_{RMS}$ for
$AoA=1.2^{\circ}$ in the rotating case could also indicate the increase in the
amplification of pre-existing instability mechanisms, e.g., TS and KH modes.

(a) max${}_{y}\overline{u^{\prime}v^{\prime}}$.  (b)
max${}_{y}\overline{v^{\prime}w^{\prime}}$.  (c) max${}_{y}u^{\prime}_{RMS}$.
(d) max${}_{y}w^{\prime}_{RMS}$.

Figure 7: Spanwise averaged Reynolds stresses and root mean square (RMS)
values of streamwise, normal, and spanwise velocity perturbations. The
following color code is used:  for $AoA=12.8^{\circ}$, $Ro_{r}=3.1$;  for
$AoA=4.2^{\circ}$, $Ro_{r}=6.3$;  for $AoA=1.2^{\circ}$, $Ro_{r}=9.4$. Dashed
lines correspond to their $Ro_{r}=0$ counterparts.  is the
$\overline{u^{\prime}v^{\prime}}=-0.001$ line.

In order to identify the inflectional character of the flow and the role of
inviscid instabilities, the ratio ($\mathcal{R}$) between the normal locations
of the inflection point ($y_{in}$) in the tangential ($U_{t}$) and crossflow
($U_{cr}$) velocity profiles (Saric et al., 2003), and the locus of maximum
production of perturbation kinetic energy ($y_{max_{\mathcal{P}}}$) is
analyzed. $\mathcal{R}\rightarrow 1$ indicates the dominance of an inviscid
instability mechanism as opposed to a viscous one when $\mathcal{R}\rightarrow
0$ (Diwan and Ramesh, 2009). The tangential velocity profiles are inflectional
on the whole suction side for $AoA=12.8^{\circ}$ due to the strong APG.
However, the non-rotating case reaches $\mathcal{R}=1$ significantly more
upstream ($x=0.12$) than the rotating one ($x=0.32$), indicating that a
predominantly inviscid instability Tollmien-Schlichting (TS) or Kelvin-
Helmholtz (KH) mode develops first in the $Ro_{r}=0$ case, likely due to the
counteraction of the APG by the Coriolis force in this region in the rotating
case. In the $AoA=4.2^{\circ}$ case, $\mathcal{R}=1$ is reached at $x=0.43$
for $Ro_{r}=0$ and $x=0.46$ for $Ro_{r}=6.3$, whereas for $AoA=1.2^{\circ}$
this occurs at $x=0.53$ for $Ro_{r}=0$ and $x=0.54$ for $Ro_{r}=9.4$,
indicating reversed trend to that for higher $AoA$. These results indicate
that an inviscid instability of the tangential velocity profiles plays an
important role in all cases, and rotation considerably shifts downstream this
instability for $AoA=12.8^{\circ}$ and, to a lesser extent, $AoA=4.2^{\circ}$
and moves it slightly upstream for $AoA=1.2^{\circ}$. The crossflow velocity
profiles in the rotating cases become inflectional and thus susceptible to
crossflow instability since the leading edge. However,
$\mathcal{R}\not\rightarrow 1$ indicates that if a crossflow instability
exists, it is not the dominant mechanism in the flow.

Table 3 outlines the streamwise locations where the mean streamwise
($x_{i_{U}}$), tangential ($x_{i_{U_{t}}}$), spanwise ($x_{i_{W}}$), and
crossflow ($x_{i_{U_{cr}}}$) velocity profiles become inflectional. The
transition locations are also summarized, based on the streamwise location of
the $-\overline{uv}/U_{\infty}^{2}=0.001$ threshold (Yuan et al., 2005)
($x_{tr_{1}}$) and the maximum boundary-layer aspect ratio ($x_{tr_{2}}$)
(Burgmann and Schröder, 2008).

Table 3: Characteristics of the inflectional velocity profiles and transition locations. AoA (∘) | $Ro_{r}$ | $x_{i_{U}}$ | $x_{i_{U_{t}}}$ | $x_{i_{W}}$ | $x_{i_{U_{cr}}}$ | $x_{tr_{1}}$ | $x_{tr_{2}}$
---|---|---|---|---|---|---|---
12.8 | 3.1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.35 | 0.39
12.8 | 0.0 | 0.00 | 0.00 | - | - | 0.33 | 0.36
4.2 | 6.3 | 0.20 | 0.22 | 0.07 | 0.00 | 0.35 | 0.50
4.2 | 0.0 | 0.21 | 0.22 | - | - | 0.36 | 0.50
1.2 | 9.4 | 0.26 | 0.29 | 0.08 | 0.00 | 0.40 | 0.59
1.2 | 0.0 | 0.27 | 0.29 | - | - | 0.41 | 0.59

####  3.1.2 Instantaneous flow structures

In order to have a deeper insight into the instantaneous flow structures, Fig.
8 presents the contours of streamwise velocity fluctuations ($u^{\prime}$)
over a near-wall and a longitudinal plane. In the $AoA=4.2^{\circ}$ and
$AoA=1.2^{\circ}$ cases, a plane passing through the shear layer is also
considered. For $AoA=12.8^{\circ}$, spanwise-uniform, streamwise oscillating
structures form downstream of $x\approx 0.35$ in the rotating case, 5%
upstream of the LSB maximum height. These rolls likely correspond to KH rolls
due to the shear layer roll-up (Brinkerhoff and Yaras, 2011). Notice that the
most unstable mode smoothly transition from a TS instability in the attached
shear layer to a KH mode, which occurs close to the LSB maximum height (Diwan
and Ramesh, 2009). The fluctuation amplitude at $x=0.4$ is 36% of $U_{\infty}$
or 21% of the local edge velocity. Transition and breakdown into small-scale
turbulence occur at this location, even though rolls seem to persist further
downstream, as shown in the longitudinal plane and also observed by
Hosseinverdi and Fasel (2019). Streamwise streaks appear downstream of
$x=0.58$, already in the turbulent region. The abrupt breakdown and the
streaks are characteristic of an oblique secondary instability mechanism
(Rist, 2003). However, this requires further analysis. Notice that the streaks
are tilted in the positive $z$ direction, indicating that rotation promotes
asymmetry regarding positive and negative spanwise wavenumbers. Regarding the
non-rotating case, spanwise-uniform structures appear shortly downstream of
separation, at $x\approx 0.25$, much more upstream than in the rotating case.
However, spanwise coherent rolls can be noticed until a more downstream
location ($x\approx 0.45$) compared to the rotating case, suggesting a slower
breakdown. Furthermore, streaks perfectly aligned with the streamwise
direction form shortly downstream of this breakdown. This may indicate a
different secondary instability mechanism compared to the rotating case or an
enhancement of the same mechanism in the latter case.

Considering $AoA=4.2^{\circ}$, featuring a long separation bubble, the primary
instabilities develop mainly on the edge of the shear layer. Over a plane at
this location, rolls are formed downstream of $x=0.6$, probably corresponding
to a KH-dominated instability due to the significant distance from the wall.
According to Jaroslawski et al. (2023), the increased distance of the shear
layer from the wall could foster three-dimensional global instabilities such
as the mechanism found by Rist and Maucher (2002) and Rodríguez et al. (2021).
In the non-rotating case, these structures are spanwise modulated with a
wavelength corresponding to the domain width. In contrast, in the rotating
case, the rolls seem to remain two-dimensional prior to the formation of
turbulent spots. Finally, the $AoA=1.2^{\circ}$ case displays a wavepacket
forming downstream of $x=0.6$ for rotating and non-rotating conditions, also
corresponding to KH rolls. These structures display a spanwise modulation
preceding their breakdown, with the mechanism seemingly the same.

(a) $AoA=12.8^{\circ}$, $Ro_{r}=3.1$.  (b) $AoA=12.8^{\circ}$, $Ro_{r}=0$.
(c) $AoA=4.2^{\circ}$, $Ro_{r}=6.3$.  (d) $AoA=4.2^{\circ}$, $Ro_{r}=0$.  (e)
$AoA=1.2^{\circ}$, $Ro_{r}=9.4$.  (f) $AoA=1.2^{\circ}$, $Ro_{r}=0$.

Figure 8: Contours of streamwise velocity fluctuations ($u^{\prime}=U-\langle
U\rangle_{z,t}$) near the wall, on a longitudinal plane, and, for
$AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$, on a plane parallel to the separated
shear layer.

###  3.2 Flow stability and perturbation evolution

####  3.2.1 Stability analyses

Primary local stability of the mean flow profiles was performed by solving the
spatial stability problem of the linearized Navier-Stokes (LNS) equations with
rotation terms, given by

$\mathbf{\mathcal{A}}\mathbf{\hat{q}}=\alpha\mathbf{\mathcal{B}}\mathbf{\hat{q}},$
(2)

$\mathbf{\mathcal{A}}=\left(\begin{array}[]{cccc}0&\mathcal{D}&i\beta&0\\\
\Xi&-\mathcal{D}\overline{\mathbf{U}}&-2\Omega_{y}\overline{\mathbf{W}}&0\\\
0&\Xi&2\Omega_{x}&-\mathcal{D}\\\
2\Omega_{y}&-\mathcal{D}\overline{\mathbf{W}}-2\Omega_{x}&-i\beta&0\\\
\end{array}\right),\quad\mathbf{\mathcal{B}}=\left(\begin{array}[]{cccc}-i&0&0&0\\\
i\overline{\mathbf{U}}&0&0&i\\\ 0&i\overline{\mathbf{U}}&0&0\\\
0&0&i\overline{\mathbf{U}}&0\\\ \end{array}\right),$ (3)

where
$\Xi=1/Re_{c}(\mathcal{D}-\beta^{2})-i(\beta\overline{\mathbf{W}}-\omega)$,
$\mathcal{D}=\partial/\partial y$, $i=\sqrt{-1}$, $\omega=2\pi f$,
$Re_{c}=U_{\infty}c/\nu$, and
$\mathbf{q^{\prime}}(x,y,z,t)=[u^{\prime}v^{\prime}w^{\prime}p^{\prime}]^{T}=\mathbf{\hat{q}}(y,\beta,\omega)\exp{\left[i(-\omega
t+\alpha x+\beta z)\right]}$ is the vector of velocity and pressure
perturbations. Notice that the assumption of small streamwise variations
allowed dropping high order $\alpha$ terms, reducing the cost of solving the
eigenvalue problem for ample parametric space. The boundary conditions were
$u^{\prime},v^{\prime},w^{\prime},p^{\prime}=0$ at $y=0$ and
$y\rightarrow\infty$. The solution of the stability problem 2 involved the
discretization of the $y$ coordinate with 150 Chebyshev points, which was
proven to yield converged results.

The local growth rates ($-\alpha_{i}$) for rotating and non-rotating cases are
compared in Fig. 9(a). The $AoA=12.8^{\circ}$ non-rotating case presents
higher growth rates upstream of $x=0.26$, a point located 3% downstream of the
separation location. This is likely due to the stabilization of the boundary
layer upstream by the Coriolis force that reduces the APG. However, rotation
passes to decelerate the flow inside the LSB, destabilizing the flow and
leading to higher growth rates. Notice that the small region where the growth
rates of the primary instability in the rotating case are higher than in the
non-rotating case is not enough to counteract the stabilization of the pre-
separation flow. Therefore, transition is more downstream in the rotating
case. For $AoA=4.2^{\circ}$, the first unstable mode appears at $x=0.31$,
slightly upstream of the separation location ($x=0.34$). From $x=0.31$ to
$x=0.42$, corresponding to the first recirculation cell, the growth rates are
slightly higher in the rotating case, a trend that is reversed further
downstream due to the streamwise deceleration of the flow by rotation. Notice
that the regions where rotation destabilizes the flow reverse for the
$AoA=12.8^{\circ}$ and $AoA=4.2^{\circ}$ cases, being the separated region in
the former and the pre-separation one in the latter. The $AoA=1.2^{\circ}$
case follows the same trend as the $AoA=4.2^{\circ}$ one, in which
$-\alpha_{i}$ of the rotating case is higher than the non-rotating one
upstream of separation ($x=0.38$) and the opposite in the LSB.

In order to distinguish if amplification changes by rotation are related to
the modification of the streamwise or spanwise velocity, the stability
analysis of the rotating cases neglecting the spanwise velocity is performed.
The results presented in Fig. 9(b) indicate that the effect of the spanwise
velocity component is negligible in the $AoA=12.8^{\circ}$ case. However, the
spanwise velocity generated by rotation destabilizes the flow in the
$AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$ cases in regions corresponding to
negative near-wall spanwise velocity (tip flow), which are highly inflectional
(Fig. 5). This suggests that crossflow modes render the flow slightly more
unstable in these cases, but a deeper analysis is required to assess the
nature of the instability. Notice that, since $\overline{W}$ is stabilizing,
the higher $-\alpha_{i}$ observed in the non-rotating cases in certain regions
(Fig. 9(a)) must be related to more unstable streamwise velocity profiles.

(a) Rotating versus non-rotating cases.  (b) Rotating cases with and without
$W$.

Figure 9: Local growth rate from spatial stability analysis. The following
color code is used:  for $AoA=12.8^{\circ}$, $Ro_{r}=3.1$;  for
$AoA=4.2^{\circ}$, $Ro_{r}=6.3$;  for $AoA=1.2^{\circ}$, $Ro_{r}=9.4$. Dashed
lines correspond to their $Ro_{r}=0$ counterparts. Circle lines denote
rotating cases ($Ro_{r}\neq 0$) with $W=0$ in the stability analysis.

In the $AoA=12.8^{\circ}$ case, the maximum amplification occurs for $f=15$,
$\beta=10$ in the rotating condition, whereas the corresponding locus is
$f=12$, $\beta=0$ in the non-rotating condition. The shift from $\beta=0$ to
$\beta=10$ results from the spanwise flow in the $+z$ direction in the
rotating case. Notice that $\beta=0$ perturbations are the most unstable in
two-dimensional flows, like the non-rotating case (Schmid and Henningson,
2001). The rotation terms in the stability equations have limited effects on
the results. The modes responsible for this instability are of mixed nature
containing both the influence of TS waves near the wall and KH instability
close to the inflection point location. The relative importance of each
mechanism varies between cases and with the streamwise location, but
essentially the further away the shear layer lies from the wall, the more
preponderant the KH mechanism (Diwan and Ramesh, 2009), such as in the
$AoA=4.2^{\circ}$ case. The most unstable mode will be denominated TS/KH
instability. The most crucial role of rotation is to generate a near-wall peak
in the spanwise velocity perturbation ($w^{\prime}$) of the TS/KH mechanism.
The normal perturbation profiles of the mode with maximum amplification for
$AoA=12.8^{\circ}$, $Ro_{r}=3.1$ are presented in Fig. 11(a). The $u^{\prime}$
profile displays a near-wall peak corresponding to the TS mechanism. This is
followed by a phase jump and a second amplitude maximum at the location of the
inflection point of the mean streamwise velocity profile (red dot),
characteristic of a KH instability (Rist, 2003). It is also important to
analyze the propagation angle
$\Psi=\arctan(\beta/\alpha_{r})-\arctan(\overline{W}_{e}/\overline{U}_{e})$,
i.e., the angle between the wavevector ($\alpha_{r},\beta$) and the inviscid
streamline ($\overline{U}_{e},\overline{W}_{e}$) in the $xz$ plane. For
$Ro_{r}=3.1$, $\Psi=[-98.6^{\circ},81.1^{\circ}]$ and for $Ro_{r}=0$,
$\Psi=[-85.1^{\circ},85.1^{\circ}]$. Essentially, $\Psi<0$ is mostly located
on the $\beta<0$ semi-plane and vice-versa. An exception occurs when modes
display $\alpha_{r}<0$, which may allow $\Psi<0$ modes on the $\beta>0$ semi-
plane. Notice that $\Psi$ reaches more negative values for the rotating case.
Borodulin et al. (2019) found traveling crossflow modes with
$\Psi=[84^{\circ},87^{\circ}]$ or $\Psi=[-97^{\circ},-92^{\circ}]$ in a swept
wing, whereas stationary crossflow modes were clustered at $\Psi\pm
89^{\circ}$. Therefore, rotation may enable crossflow modes, but the growth
rates were low compared to the TS/KH instability. The reason for that may be
the not very inflectional spanwise velocity profiles, similar to the rotating
disk with axial inflow (Hussain et al., 2011; Deschamps and Hein, 2018).

For $AoA=4.2^{\circ}$, Fig. 10(a) indicates that the highest $-\alpha_{i}$
occurs at $f=9$ and $\beta=30$ at $x=0.46$ in the rotating condition,
corresponding to a TS/KH instability. The propagation angle is
$\Psi=[-100.4^{\circ},77.2^{\circ}]$, as shown in Fig. 10(b), where this
variable is shown for unstable modes only. Low-frequency $\beta\neq 0$ modes
become unstable in the rotating case, including stationary disturbances, which
is more visible for $\beta>0$, although there is a small unstable region close
to the $f=0$ axis for $\beta<0$. These modes present high $|\Psi|$ but low
growth rates and may be crossflow modes. The profiles and contours of the
stationary mode for $\beta=188.5$ ($\Psi=-95.5^{\circ}$, $-\alpha_{i}=3.7$)
are presented in Fig. 11(b). The eigenfunction and contours of spanwise
velocity disturbances agree with those from the literature for a crossflow
mode (Saric et al., 2003; Borodulin et al., 2019). In the non-rotating case,
the maximum amplification occurs at $f=10$ and $\beta=0$. The low-frequency
unstable region with $\beta\neq 0$ is absent in this case, and
$\Psi=[-83.7^{\circ},83.7^{\circ}]$, with much smaller $|\Psi|$ than the
rotating case. This supports the appearance of crossflow modes due to
rotation. At last, the $AoA=1.2^{\circ}$ case indicates maximum growth at
$f=11$, $\beta=20$ for $Ro_{r}=9.4$, and $f=11$, $\beta=0$ for $Ro_{r}=0$,
both consisting of a TS/KH mode. Furthermore,
$\Psi=[-101.1^{\circ},83.5^{\circ}]$ in the former and
$\Psi=[-84.2^{\circ},84.2^{\circ}]$, also suggesting the appearance of
crossflow vortices in the rotating cases. However, unlike the
$AoA=4.2^{\circ}$ case, only non-stationary vortices are present, similar to
the $AoA=12.8^{\circ}$ case. Their growth rates are low compared to the TS/KH
mode.

(a) Local growth rate.  (b) Propagation angle.

Figure 10: Contours of local growth rate ($-\alpha_{i}$) and propagation angle
($\Psi$) from local spatial stability analysis for $AoA=4.2^{\circ}$,
$Ro_{r}=6.3$, at $x=0.46$.

(a) TS/KH mode.  (b) Crossflow mode.

Figure 11: Perturbation profiles and contours of (a) TS/KH mode for
$AoA{=}12.8^{\circ}$, $Ro_{r}{=}3.1$, $f{=}15$, $\beta{=}10$, $x{=}0.33$, and
(b) crossflow mode for $AoA{=}4.2^{\circ}$, $Ro_{r}{=}6.3$, $f{=}0$,
$\beta{=}188.5$, $x{=}0.46$ from local spatial stability analysis. The red dot
on the profile of the left panel indicates the normal location of the
inflection point in the mean streamwise velocity.

The $N$ factor is a more relevant parameter to predict transition (Arnal and
Casalis, 2000). For clean environments, $N=9$ is often assumed as a reliable
threshold for natural transition. This parameter was computed with the
parabolized stability equations (PSE) code NOLOT (Hanifi et al., 1994). Here
the $N$ factor is based on the total kinetic energy of the perturbations. The
results for $\beta=0$ are presented in Fig. 12, as a function of the frequency
and streamwise coordinate. For $AoA=12.8^{\circ}$, the maximum $N$ factor is
$N=12.1$ for $f=16$ at $x=0.43$ in the rotating case and $N=11.6$ for $f=17$
at $x=0.4$ in the non-rotating one. At those locations, the streamwise
wavelength $\lambda_{x}=2\pi/\alpha_{r}$ is 0.05, agreeing with that of the
rolls in Fig. 8(b). The results also indicate the stabilization of high-
frequency TS waves close to the leading edge. For $AoA=4.2^{\circ}$, the
rotating case reaches $N=3.8$ for $f=7$ at $x=0.54$, whereas the non-rotating
case presents $N=4.4$ also for $f=7$ but at $x=0.59$. This low $N$ factor
could be due to an early start of secondary instability or the inception of
absolute or global instabilities (Huerre and Monkewitz, 1990; Theofilis et
al., 2000), not captured by the PSE method. Finally, considering
$AoA=1.2^{\circ}$, the maximum $N$ factor is $8.1$ for $f=8.6$ in the rotating
case and $N=10.7$ for $f=9.5$ in the non-rotating one, which occurs 1%
downstream of the transition location ($x=0.59$). In summary, the $N$ factor
reaches higher maximum values in the rotating case for $AoA=12.8^{\circ}$ and
in the non-rotating case otherwise. Note that the transition location is more
upstream (Table 3) in the rotating case for $AoA=12.8^{\circ}$. This is not a
contradiction, as the higher $N$ factor in the rotating case only occurs more
downstream than the transition location in the non-rotating condition, as
rotation stabilizes the boundary layer upstream of the LSB and destabilizes it
in the region of strong reverse flow.

(a) $AoA=12.8^{\circ}$, $Ro_{r}=3.1$.  (b) $AoA=12.8^{\circ}$, $Ro_{r}=0$.
(c) $AoA=4.2^{\circ}$, $Ro_{r}=6.3$.  (d) $AoA=4.2^{\circ}$, $Ro_{r}=0$.  (e)
$AoA=1.2^{\circ}$, $Ro_{r}=9.4$.  (f) $AoA=1.2^{\circ}$, $Ro_{r}=0$.

Figure 12: Contours of $N$ factor obtained with PSE for $\beta=0$ using
($\langle\mathbf{U}\rangle_{z,t}$) as the base flow.  are the isolines of the
real part of the streamwise wavenumber ($\alpha_{r}$), and  denotes the loci
of maximum amplification. The white line corresponds to $N=9$ (threshold for
natural transition).

Sufficiently high reverse flows can cause the boundary layer to behave as an
oscillator in the so-called absolute instability (Huerre and Monkewitz, 1990).
Thresholds for its inception may be reverse flows above 20% (Hammond and
Redekopp, 1998), $15\leavevmode\nobreak\ \%-20\leavevmode\nobreak\ \%$ (Alam
and Sandham, 2000), and $6\leavevmode\nobreak\ \%-8\leavevmode\nobreak\ \%$
(Rodríguez et al., 2013; Hosseinverdi and Fasel, 2013; Rodríguez et al.,
2021). Nevertheless, other parameters, such as the Reynolds number and the
distance of the inflection point from the wall, also play a role (Rist, 2003;
Avanci et al., 2019). The cusp map (Kupfer et al., 1987), i.e., the Orr-
Sommerfeld/Squire mapping $\alpha\rightarrow\omega$, where
$\alpha\in\mathbb{C}$, was obtained to assess whether convective/absolute
instabilities exist. The anstaz for the normal velocity ($v^{\prime}$) and
vorticity ($\eta^{\prime}$) perturbations is
$\mathbf{q^{\prime}}(x,y,z,t)=[v^{\prime}\eta^{\prime}]^{T}=\mathbf{\hat{q}}(y,\beta,\omega)\exp{\left[i(-\omega
t+\alpha x+\beta z)\right]}$. Thus, a cusp in the $\omega$ trajectories in the
$\omega_{i}>0$ semi-plane indicates an absolute instability. Figure 13(a)
shows the complex $\alpha$ plane mapped on $\omega$. No unstable cusp exists
for $AoA=12.8^{\circ}$, suggesting that the LSB is convectively stable. For
$AoA=4.2^{\circ}$, two unstable cusps with frequencies $f=1.2$ and $f=2.4$ at
$x=0.57$ were found in the rotating case (Fig. 13(c)), whereas only one
appears in the non-rotating one, at $f=1.7$. The absolutely unstable region
for $AoA=4.2^{\circ}$ is $x=0.54-0.62$ in the first condition (Fig. 13(b)) and
$x=0.57-0.63$ for $Ro_{r}=0$, with the growth rates being slightly higher in
the latter scenario. These LSBs are likely globally unstable as a finite
region of absolute instability suggests the occurrence of global instability
(Huerre and Monkewitz, 1990). The results for $AoA=1.2^{\circ}$, at $x=0.66$,
indicate an unstable cusp at $f=8$ in the rotating case (Fig. 13(e)), and at
$f=5.4$ in the non-rotating one. The rotating case region of absolute
instability is $x=0.61-0.68$, and $x=0.60-0.68$ for the non-rotating one. The
absolute growth rates are lower in the rotating case for $x\leq 0.64$,
possibly due to the streamwise acceleration promoted by rotation. Notice that
absolute instability may be related to the split of the LSB in more than one
recirculation cell (Cherubini et al., 2010).

(a) Complex $\alpha$ plane.  (b) Growth rate of the absolute instability.  (c)
$\alpha\rightarrow\omega$, $AoA=4.2^{\circ}$, $Ro_{r}=6.3$, $x=0.57$.  (d)
$\alpha\rightarrow\omega$, $AoA=4.2^{\circ}$, $Ro_{r}=0$, $x=0.57$.  (e)
$\alpha\rightarrow\omega$, $AoA=1.2^{\circ}$, $Ro_{r}=9.4$, $x=0.66$.  (f)
$\alpha\rightarrow\omega$, $AoA=1.2^{\circ}$, $Ro_{r}=0$, $x=0.66$.

Figure 13: Complex mapping $\alpha\rightarrow\omega$ via the Orr-Sommerfeld
operator. The $\omega$ image for $AoA=4.2^{\circ}$ (panels (c) and (d)) were
obtained at $x=0.57$ and for $AoA=1.2^{\circ}$ (panels (e) and (f)) at
$x=0.66$. The unstable cusp is shown with a circle in both $\omega$ (panels
(c), (d), (e), and (f)) and $\alpha$ (panel (a)) planes. The imaginary part of
$\omega$, corresponding to the growth rate of the absolute instability, is
shown in panel (b).

####  3.2.2 Spectral analysis

The spectra of the growth rate
$\sigma=|u^{\prime}|^{-1}\frac{\partial|u^{\prime}|}{\partial x}$ at several
streamwise slices are presented in Fig. 14. Disturbances with frequency $f$
and spanwise wavenumber $\beta$ will be labeled with the pair
$(f/f_{f},n_{z})$, where $f_{f}$ is the fundamental instability frequency, to
be identified in each case, and $n_{z}=\beta L_{z}/(2\pi)$, where $L_{z}$ is
the domain width. For $AoA=12.8^{\circ}$, rotating case, TS waves display high
$\sigma$ for $f=23$, $\beta=0$ at $x=0.25$. The nature of this instability can
be confirmed by linear stability analysis, as done before, or by noting that
$F=\frac{2\pi f\nu 10^{6}}{\overline{U}_{e}^{2}}=663.5$ and
$Re_{\delta^{*}}=\frac{\overline{U}_{e}\delta^{*}}{\nu}=444.1$ at $x=0.15$ for
$f=23$ is close to the most unstable TS waves in a Falkner-Skan profile under
APG (Obremski et al., 1969). At $x=0.25$, several modes are excited over the
$\beta=0$ axis due to flow separation, and non-linearly generated modes appear
for $\beta=\pm 314$ ($n_{z}=\pm 5$). The mode $(1,0)$ ($f_{f}=15$),
corresponding to the most-unstable TS/KH instability, is also present for
$x\geq 0.25$ but displays lower $\sigma$ than secondary disturbances. At
$x=0.33$ and $x=0.35$, the $(0,\pm 2)$ modes, representing streamwise
vortices, present the highest $\sigma$. Moreover, the growth of the $(1,0)$
and $(1,\pm 2)$ modes become apparent at $x=0.35$, where $(1,-2)$ grows more
than $(1,2)$, explaining the tilted streaks in the simulation. This confirms
the presence of Marxen’s oblique mechanism (Marxen et al., 2003) involving the
triad $(1,0)$, $(0,\pm 2)$, and $(1,\pm 2)$. The second-generation modes
$(0,\pm 2)$ possibly result from the oblique mechanism involving $(1,\pm 1)$
waves (Rist, 2003). The lack of symmetry around $\beta=0$ is due to the
rotation-generated spanwise flow that renders one side of the spectra more
unstable. At $x=0.38$, shortly before transition ($x=0.39$), the amplitude of
the $(1,0)$ mode becomes substantial, and the sub-harmonic mode $(0.5,-2)$
develops the highest $\sigma$. In the non-rotating case (Figs. 14(b)), due to
more unstable TS waves near the leading edge compared to the rotating case,
which triggers early non-linear interactions, the spectrum at $x=0.15$ is
already noisy. At $x=0.3$, the TS/KH mode with $f_{f}=12$ presents high
$\sigma$, and, at $x=0.35$, preceding transition ($x=0.36$), the non-linearly
generated oblique mode $(1,\pm 1)$ shows an important amplification, with the
generation of sub-harmonics. Marxen’s oblique mechanism involving $(\cdot,\pm
2)$ modes seems not dominant in the non-rotating case.

The results for $AoA=4.2^{\circ}$ indicate that disturbances with
$f\lessapprox 5$ display significant $\sigma$ in the rotating case, especially
for $\beta>0$, for $x\leq 0.48$. The TS/KH instability appears at
$f_{f}=13.5$, $\beta=0$, for $x\leq 0.46$. At $x=0.48$, the maximum $\sigma$
shifts to the $(1,1)$ mode. Linear stability analysis predicts a shift of the
most unstable TS/KH mode to $\beta=30$ due to rotation and not to
$\beta=62.8$. Therefore, the $(1,1)$ mode is likely to be non-linearly
generated. Transition occurs at $x=0.5$, noticed by the high $\sigma$,
especially for low frequencies and $\beta<0$. At this location, $(1,\pm 2)$
and $(0.5,\pm 2)$ modes are also excited, especially for $\beta<0$. The lack
of symmetry around the $\beta=0$ axis is again an effect of rotation. In the
non-rotating case, the $f=13.5$, $\beta=0$ mode displays the highest $\sigma$
for $x\leq 0.46$, similarly to the rotating case, corresponding to the TS/KH
instability. For $0.44\leq x\leq 0.48$, two lobes appear corresponding to the
$(1,\pm 2)$ modes. Unlike the rotating case, these modes present the highest
$\sigma$ and are symmetrical around the $\beta=0$ axis. At $x=0.48$, the sub-
harmonics $(0.5,\pm 2)$ also develop important amplification, and transition
is triggered. Marxen’s oblique mechanism seems only important in the non-
rotating case. Lastly, the $AoA=1.2^{\circ}$ case displays a high $\sigma$ at
$f_{f}=9.6$, $\beta=0$ for $0.44\leq x\leq 0.56$, agreeing with the frequency
of the most amplified TS/KH instability predicted with linear stability
analysis. This mode is present in the rotating and non-rotating cases. In the
rotating case, low-frequency, $\beta\neq 0$ disturbances present high $\sigma$
in this $x$ range, especially for $\beta>0$, possibly corresponding to
crossflow modes. At $x=0.56$, the sub-harmonic mode $(0.5,\pm 2)$ appears in
$\sigma$, where the $(0.5,-2)$ is more amplified. This leads to a transition
scenario where particularly low-frequency, $\beta<0$ disturbances grow first,
noticed in the heterogeneity in $|u^{\prime}|$ at $x=0.56$ around the
$\beta=0$ axis, similarly to $AoA=4.2^{\circ}$. The non-rotating condition
presents higher $\sigma$ for the TS/KH mode. At $x=0.52$, two lobes are formed
at $\beta=\pm 125.7$, suggesting the generation of $(1,\pm 2)$ modes, besides
the presence of high $\sigma$ for $(0,\pm 2)$, symmetric around the $\beta=0$
axis. This indicates the occurrence of Marxen’s mechanism. Sub-harmonic modes
$(0.5,\pm 2)$ appear at $x=0.56$, triggering transition. Similar to
$AoA=4.2^{\circ}$, the oblique mechanism involving $(\cdot,\pm 2)$ waves
appears not dominant in the rotating case.

(a) $AoA=12.8^{\circ}$, $Ro_{r}=3.1$.  (b) $AoA=12.8^{\circ}$, $Ro_{r}=0$.
(c) $AoA=4.2^{\circ}$, $Ro_{r}=6.3$.  (d) $AoA=4.2^{\circ}$, $Ro_{r}=0$.  (e)
$AoA=1.2^{\circ}$, $Ro_{r}=9.4$.  (f) $AoA=1.2^{\circ}$, $Ro_{r}=0$.

Figure 14: Growth rate ($\sigma$) spectra at several streamwise stations.

The streamwise evolution of $|u^{\prime}|$ for the main modes is shown in Fig.
15. For $AoA=12.8^{\circ}$, the primary instability, consisting of the TS/KH
mode, presents a lower amplitude before transition with a more downstream
maximum in the rotating case. Due to the higher growth rates of the $(1,0)$
mode near the leading edge, all secondary modes also display growth in this
region. However, in the rotating case, these modes only develop higher
amplitudes for $x\geq 0.23$, i.e., after separation. The secondary modes with
peak amplitude correspond to the sub-harmonic disturbances $(0.5,\pm 1)$,
followed by the second-generation modes $(0.5,\pm 2)$. These modes present
higher maximum amplitude in the rotating case, which could explain the faster
breakdown of the TS/KH rolls. The more pronounced decay of the $(1,0)$ mode
after its peak value indicates a faster loss of spanwise coherence.

The $AoA=4.2^{\circ}$ results indicate that the $(0,\pm 2)$ modes undergo
strong amplification starting at $x=0.25$. These modes could be generated via
an oblique mechanism involving $(1,\pm 1)$ modes (Rist, 2003). The amplitude
of the non-rotating disturbances is generally higher considering both primary
and secondary modes. The $(0,\pm 1)$ modes reach much higher amplitude in the
non-rotating and are responsible for the spanwise modulation of the TS/KH
rolls, as noticed in the simulations. On the other hand, the sub-harmonic mode
$(0.5,-1)$ is the strongest in the non-rotating case. In the $AoA=1.2^{\circ}$
case, the $(1,0)$ mode displays amplification, particularly downstream of
$x=0.4$, with initially higher amplitude for the non-rotating case supplanted
by the rotating case downstream of $x=0.5$. The streamwise vortices $(0,\pm
1)$ and $(0,\pm 2)$ present growth since the leading edge, similar to the
$AoA=4.2^{\circ}$ case, and become the most important modes in the flow in a
region prior to the saturation of the primary modes. After that, the sub-
harmonic $(0,\pm 1)$ and $(0,\pm 2)$ modes have the highest amplitude,
spanwise modulating the rolls, as seen in the simulations.

Figure 15: Amplitude spectra from numerical simulation as a function of the
streamwise coordinate. Panel (a): $AoA=12.8^{\circ}$, $f_{f}=15$ for
$Ro_{r}=3.1$ and $f_{f}=12$ for $Ro_{r}=0$. Panel (b): $AoA=4.2^{\circ}$,
$f_{f}=13.5$. Panel (c): $AoA=1.2^{\circ}$, $f_{f}=9.6$. $\beta=0$ in all
cases. Solid lines correspond to the rotating case and dashed line to the non-
rotating one.

Spectral proper orthogonal decomposition (SPOD) (Towne et al., 2018; Schmidt
and Colonius, 2020) was employed to visualize the modal structures. The SPOD
modes denoted by $\Psi_{k}$, $k=1,2,...,N$, where $N$ is the number of SPOD
modes, have a corresponding eigenvalue
$\lambda_{1}\geq\lambda_{2},...,\geq\lambda_{N}$, which is a measure of the
total perturbation kinetic energy contained in each mode. These modes are
orthogonal for given $f$ and $\beta$ and sorted in decreasing energy.
Therefore, the first modes describe the leading structures contributing to the
flow dynamics. Moreover, these modes evolve coherently in space and time
(Towne et al., 2018). In the current analysis, 2,000 snapshots spaced with a
time step $\Delta t=5\times 10^{-3}$ are divided into ten blocks with $80\%$
overlap. The convergence of the method was evaluated by comparing the leading
SPOD modes obtained with 75% and the full-time series, indicating minimal
deviations.

The first SPOD mode for frequencies close to the most amplified TS/KH
instability is presented in Fig. 16. For $AoA=12.8^{\circ}$, the wavepacket
initiates just downstream of the separation location. A phase shift in the
mode of streamwise velocity perturbation occurs at $y\approx\delta^{*}$, which
separates the near-wall region with viscous instability from the inviscid
instability area further into the free-stream. The normal velocity
perturbation mode presents its peak amplitude at $y=\delta^{*}$. Overall the
modes for streamwise and normal velocity perturbations are very similar in the
rotating and non-rotating cases. However, the spanwise velocity perturbation
mode presents a coherent wavepacket in the rotating case, starting at the
front part of the LSB. The peak amplitude of this mode occurs at the location
of the inflection point of the mean spanwise velocity profiles, situated above
$y=\delta^{*}$. The instability engendered by this inflection point
contributes little to the total perturbation kinetic energy, which can also be
seen by the low relative energy of $Re(\Psi_{W_{1}})$ compared to the other
components. In the non-rotating case, this coherent structure is not present.
Regarding $AoA=4.2^{\circ}$, the TS/KH mode wavepacket is also formed.
Nonetheless, the mode lies farther away from the wall due to the thick shear
layer and tends to a pure KH instability from a free shear layer. While the
peak amplitudes occur at $y=\delta^{*}$ in the rotating case, this occurs
close to the LSB edge in the non-rotating one. Notice that this wavepacket is
only formed downstream of where $y_{in}/y_{max_{\mathcal{P}}}\rightarrow 1$
for the mean streamwise velocity profile, clearly indicating its inviscid
character. The $Re(\Psi_{W_{1}})$ mode is also coherent in the rotating case
with peak amplitude at the location of the inflection point in the mean
spanwise velocity profiles. Moreover, the amplitude of this mode is higher
than in the $AoA=12.8^{\circ}$ case. The non-rotating case does not display
coherent $Re(\Psi_{W_{1}})$. Finally, the $AoA=1.2^{\circ}$ case presents a
mode that is qualitatively similar between the rotating and non-rotating
cases, considering $Re(\Psi_{U_{1}})$ and $Re(\Psi_{V_{1}})$, also
corresponding to a TS/KH instability. Regarding $Re(\Psi_{W_{1}})$, even
though apparently less coherent, $Re(\Psi_{W_{1}})$ presents higher amplitude
than for higher $AoA$. The coherence of this mode increases with frequency.

(a) $AoA=12.8^{\circ}$, $f=16$, $Ro_{r}=3.1$.  (b) $AoA=12.8^{\circ}$, $f=16$,
$Ro_{r}=0$.  (c) $AoA=4.2^{\circ}$, $f=10$, $Ro_{r}=6.3$.  (d)
$AoA=4.2^{\circ}$, $f=10$, $Ro_{r}=0$.  (e) $AoA=1.2^{\circ}$, $f=10$,
$Ro_{r}=9.4$.  (f) $AoA=1.2^{\circ}$, $f=10$, $Ro_{r}=0$.

Figure 16: Real part of the first SPOD mode of streamwise, normal, and
spanwise velocity perturbations for $\beta=0$.  and  indicate contour levels
0.001 and -0.001.  is the displacement thickness ($\delta^{*}$).  is the edge
of the LSB.

Figure 17 shows the first SPOD modes for the $AoA=4.2^{\circ}$ case in
rotating condition for $\beta=188.5$ ($n_{z}=3$). Figure 17(a) shows the
results for $f=2$, which presents the characteristics of a traveling crossflow
mode with vortices nearly aligned with the streamwise direction on a plane
parallel to the airfoil (left panels). $\Psi=-93^{\circ}$, as shown in the
panel for $Re(\Psi_{V_{1}})$, in the range of crossflow modes. The contours on
the spanwise plane (right panels) indicate that the SPOD mode consists of
counter-rotating vortices with peak amplitude at $y/\delta^{*}\approx 0.7$ for
$Re(\Psi_{U_{1}})$ and $Re(\Psi_{W_{1}})$ and $y/\delta^{*}\approx 0.5$ for
$Re(\Psi_{V_{1}})$), the same locations predicted by linear stability
analysis. This further indicates that these are crossflow vortices. Notice
that their energy is much lower than that of the maximum amplified TS/KH
waves, suggesting that they do not act as the primary instability driver but
rather influence the secondary instability process. The SPOD modes for $f=0$,
portrayed in Fig. 17(b), display an angle of $\Psi=-97.7^{\circ}$ relative to
the inviscid streamline, which suggests these are stationary crossflow modes.
The amplitude of these modes is higher than that of stationary crossflow
vortices. It increases with $x$ but is probably insufficient for causing
transition alone since the transition location occurs at $x=0.5$. The results
on the spanwise plane do not indicate so clearly a crossflow mode such as in
the $f=2$ case since the maxima of $Re(\Psi_{U_{1}})$ and $Re(\Psi_{W_{1}})$
occur closer to the wall than expected. This can be due to the non-linear
forcing of these modes by the TS/KH instability. Nevertheless, there are still
energetic structures for these modes at $y/\delta^{*}\approx 0.7$, where
stability analysis indicates the maximum amplitude takes place. The modes
found here were only present in the rotating case, proving they are engendered
by rotation. Traveling crossflow vortices were also observed for
$AoA=12.8^{\circ}$ and $AoA=1.2^{\circ}$ in the rotating condition.

(a) Traveling crossflow mode for $f=2$.  (b) Stationary crossflow mode.

Figure 17: Real part of the first SPOD mode of streamwise, normal, and
spanwise velocity perturbations for $\beta=188.5$, $AoA=4.2^{\circ}$,
$Ro_{r}=6.3$ at $x=0.49$ and $y=0.0045$ ($0.2\delta^{*}$). The isocontours of
amplitude 0.001 and -0.001 are marked with  and  , respectively.
$\overrightarrow{k}$ is the wavevector and $\overrightarrow{U_{e}}$ is the
velocity vector at the inviscid streamline.

##  4 Conclusions

Direct numerical simulations (DNS) of the flow over a section of a rotating
wind turbine blade are performed to study the effects of rotation on the flow
stability and laminar-turbulent transition on the suction side. The
computations were carried out in the rotating frame of reference, including
centrifugal and Coriolis forces as source terms in the momentum equations. The
angles of attack ($AoA$) of $12.8^{\circ}$, $4.2^{\circ}$, and $1.2^{\circ}$
were selected to assess the role of different flow separation conditions,
which attempts to overcome the shortcomings pointed out by Guntur and Sørensen
(2014) in the study by Gross et al. (2012). Three accompanying simulations
with the same $AoA$ were also conducted without incorporating centrifugal and
Coriolis force and with zero spanwise velocity to emulate corresponding non-
rotating cases. All simulations considered a chord Reynolds number of $1\times
10^{5}$. The results were analyzed using Fourier analysis and spectral proper
orthogonal decomposition (SPOD). Moreover, primary stability analyses were
performed to assess the roles of convective and absolute instabilities.

The effects of rotation depend on the difference between the local and free-
stream velocities. Rotation accelerates the attached flow downstream in
regions under a strong adverse pressure gradient (APG), counteracting the
latter. On the other hand, in the case of a streamwise accelerating flow,
i.e., under a favorable pressure gradient (FPG), rotation decelerates the
flow, acting as an APG. Nevertheless, the effects are the opposite when
reversal flow occurs, such as in laminar separation bubbles (LSBs). Upon
sufficiently high reverse flow, the Coriolis force further decelerates the
stream, increasing the LSB compared to the non-rotating case. These results
are valid when the spanwise velocity is positive towards the root. One may
expect the reverse of them under the occurrence of tip flow.

The $AoA=12.8^{\circ}$ case presents a strong (APG) since the leading edge and
the rotation effects tend to counteract that by accelerating the flow
downstream prior to separation. Inside the LSB, rotation strongly decelerates
the flow increasing the reverse flow from -7% in the non-rotating case to -16%
in the rotating one. Separation occurs at the same location in the rotating
and non-rotating cases, but reattachment occurs 4% more downstream in the
rotating condition, and the LSB height is 74.7% higher. These rotation-induced
flow modifications stabilize the boundary layer upstream of separation
concerning Tollmien-Schlichting (TS) waves. Nevertheless, inside the LSB,
particularly near the maximum LSB height, rotation increases the growth rates
of the mixed Tollmien-Schlichting/Kelvin-Helmholtz (TS/KH) instability that
develops in the separated shear layer. The secondary instability of these
rolls occurs via Marxen’s oblique mechanism (Marxen et al., 2003) involving
the triad $(1,0)$, $(0,\pm 2)$, and $(1,\pm 2)$, followed by a sub-harmonic
growth of $(0.5,\pm 1)$ modes. Both mechanisms are enhanced in the rotating
case, which leads to a rapid breakdown of the rolls to small-scale turbulence.
This is characteristic of an oblique mechanism (Rist, 2003). On the other
hand, the non-rotating case displays a predominantly sub-harmonic secondary
instability, with the dilation of the streamwise wavelength, which leads to a
slower breakdown of the two-dimensional structures. Another effect of rotation
is rendering the above modes with $\beta<0$ dominant in the secondary
instability process. Due to the stabilization of the attached boundary layer
upstream and the increase in the reverse flow, transition is 3% more
downstream in the rotating case. The spanwise flow in the rotating case does
not enhance primary instabilities, albeit coherent structures in the spanwise
velocity appear in the TS/KH wavepacket. Reattachment occurs 4% more
downstream in the rotating case.

Considering the $AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$ cases, due to the
prolonged region with FPG upstream of separation, the flow is decelerated
there, rendering TS waves more unstable. However, this trend is reversed after
separation, stabilizing the mixed TS/KH mode in the separated shear layer. Due
to the higher rotation rate, a flow directed towards the blade tip appears in
the fore part of the LSB. These crossflow velocity profiles are inflectional,
and stationary crossflow modes emerge in the $AoA=4.2^{\circ}$ case and
traveling ones in both $AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$ cases,
rendering the flow more unstable. Nonetheless, the occurrence of radial flow
may reduce the growth rates of these modes, similar to the observed for a
rotating disk with axial inflow (Hussain et al., 2011; Deschamps and Hein,
2018). Therefore, the low growth rates of these modes do not compensate for
the stabilization of the TS/KH mode by rotation inside the LSB. Despite a
higher initial growth in the rotating case, the amplitude and growth rates of
the primary instability are higher in the non-rotating one close to
transition. In both cases, the streamwise vortices $(0,\pm 1)$ and $(0,\pm 2)$
present a high amplitude. This is particularly true in the non-rotating case
for $AoA=4.2^{\circ}$, where these modes modulate the spanwise rolls and lead
to their breakdown. The rotating case presents lower amplitudes of these
modes, and the spanwise modulation of the two-dimensional rolls is less
visible. The sub-harmonic modes $(0.5,-1)$ is more dominant in the rotating
case, representing a change in the trend noticed for $AoA=12.8^{\circ}$, where
the sub-harmonic mechanism was more relevant in the non-rotating case and the
oblique one in the rotating condition. The secondary instability in the
$AoA=1.2^{\circ}$ case is predominantly sub-harmonic in both conditions but
with the predominance of $\beta<0$ modes in the rotating case. The spanwise
flow sets a preferential direction for disturbance growth and leads to a lack
of symmetry around the $\beta=0$ axis. The coherent structure in the spanwise
velocity formed together with the TS/KH mode is stronger for
$AoA=1.2^{\circ}$, probably due to the higher rotation rate. However, no
change in the transition location is noticed between rotating and non-rotating
conditions for $AoA=4.2^{\circ}$ and $AoA=1.2^{\circ}$, when measured as the
location of the maximum boundary-layer shape factor. A finite region of
absolute instability exists between the first two recirculation cells in these
cases. This may indicate the global instability of the flow (Huerre and
Monkewitz, 1990). Rotation rendered the absolute instabilities less unstable
in the $AoA=4.2^{\circ}$ case, possibly due to the streamwise acceleration of
the flow inside the LSB. For the $AoA=1.2^{\circ}$ case, this effect also
occurred in the first recirculation cell, but rotation enhanced the absolute
instability in the second one due to a stronger reverse flow.

###### Acknowledgements.

This research was supported by StandUp for Energy and StandUp for Wind. The
computations were performed on resources from the Swedish National
Infrastructure for Computing (SNIC) at the PDC Center for High-Performance
Computing at the Royal Institute of Technology (KTH), the National
Supercomputer Centre at Linköping University, and the High-Performance
Computing Center North (HPC2N) at Umeå University.

## Compliance with Ethical Standards

The authors declares that they have no conflict of interest.

## References

  * Himmelskamp (1947) H. Himmelskamp, Profile investigations on a rotating airscrew, Ph.D. thesis, Völkenrode MAP, Göttingen, Germany, 1947.
  * Fogarty (1951) L. E. Fogarty,  The laminar boundary layer on a rotating blade,  Journal of the Aeronautical Sciences 18 (1951) 247–252. doi:10.2514/8.1921.
  * McCroskey (1971) W. J. McCroskey, Measurements of boundary layer transition, separation and streamline direction on rotating blades, Technical Report, NASA TN D-6321, 1971.
  * Banks and Gadd (1963) W. H. H. Banks, G. E. Gadd,  Delaying effect of rotation on laminar separation,  AIAA Journal 1 (1963) 941–941. doi:10.2514/3.1687.
  * Horlock and Wordsworth (1965) J. H. Horlock, J. Wordsworth,  The three-dimensional laminar boundary layer on a rotating helical blade,  Journal of Fluid Mechanics 23 (1965) 305. doi:10.1017/S0022112065001374.
  * McCroskey and Yaggy (1968) W. J. McCroskey, P. F. Yaggy,  Laminar boundary layers on helicopter rotors in forward flight.,  AIAA Journal 6 (1968) 1919–1926. doi:10.2514/3.4901.
  * Corten (2001) G. P. Corten, Flow separation on wind turbine blades, Ph.D. thesis, Utretcht University, Utretcht, 2001.
  * Savino and Nyland (1985) J. M. Savino, T. W. Nyland, Wind turbine flow visualisation studies, Technical Report, NASA Lewis Research Center, Cleveland, OH, 1985.
  * Bosschers (1995) J. Bosschers, Influence of blade rotation on the sectional aerodynamics of rotational blades, Technical Report, National Aerospace Laboratory NLR, Amsterdam, 1995\.
  * Du and Selig (2000) Z. Du, M. Selig,  The effect of rotation on the boundary layer of a wind turbine blade,  Renewable Energy 20 (2000) 167–181. doi:10.1016/S0960-1481(99)00109-3.
  * Shen and Sørensen (1999) W. Z. Shen, J. N. Sørensen,  Quasi-3D Navier–Stokes model for a rotating airfoil,  Journal of Computational Physics 150 (1999) 518–548. doi:10.1006/jcph.1999.6203.
  * Chaviaropoulos and Hansen (2000) P. K. Chaviaropoulos, M. O. L. Hansen,  Investigating three-dimensional and rotational effects on wind turbine blades by means of a quasi-3D Navier-Stokes solver,  Journal of Fluids Engineering 122 (2000) 330–336. doi:10.1115/1.483261.
  * Dumitrescu et al. (2007) H. Dumitrescu, V. Cardoş, A. Dumitrache,  Modelling of inboard stall delay due to rotation,  Journal of Physics: Conference Series 75 (2007) 012022. doi:10.1088/1742-6596/75/1/012022.
  * Ronsten (1992) G. Ronsten,  Static pressure measurements on a rotating and a non-rotating 2.375 m wind turbine blade. Comparison with 2D calculations,  Journal of Wind Engineering and Industrial Aerodynamics 39 (1992) 105–118. doi:10.1016/0167-6105(92)90537-K.
  * Björck (1995) A. Björck, Dynamic stall and three-dimensional effects, Technical Report, FFA TN 1995-31, Bromma, Sweden, 1995.
  * Herráez et al. (2014) I. Herráez, B. Stoevesandt, J. Peinke,  Insight into rotational effects on a wind turbine blade using Navier–Stokes computations,  Energies 7 (2014) 6798–6822. doi:10.3390/en7106798.
  * McCroskey (1973) W. J. McCroskey, Recent developments in rotor blade stall, Technical Report, AGARD CP 111, 1973\.
  * Schülein et al. (2012) E. Schülein, H. Rosemann, S. Schaber,  Transition detection and skin friction measurements on rotating propeller blades,  in: 28th Aerodynamic Measurement Technology, Ground Testing, and Flight Testing Conference, American Institute of Aeronautics and Astronautics, Reston, Virigina, 2012. doi:10.2514/6.2012-3202.
  * Breton et al. (2008) S.-P. Breton, F. N. Coton, G. Moe,  A study on rotational effects and different stall delay models using a prescribed wake vortex scheme and NREL phase VI experiment data,  Wind Energy 11 (2008) 459–482. doi:10.1002/we.269.
  * Dumitrescu and Cardos (2004) H. Dumitrescu, V. Cardos,  Rotational Effects on the Boundary-Layer Flow in Wind Turbines,  AIAA Journal 42 (2004) 408–411. doi:10.2514/1.9103.
  * Schreck et al. (2007) S. J. Schreck, N. N. Sørensen, M. C. Robinson,  Aerodynamic structures and processes in rotationally augmented flow fields,  Wind Energy 10 (2007) 159–178. doi:10.1002/we.214.
  * Lang et al. (2015) W. Lang, A. D. Gardner, S. Mariappan, C. Klein, M. Raffel,  Boundary-layer transition on a rotor blade measured by temperature-sensitive paint, thermal imaging and image derotation,  Experiments in Fluids 56 (2015) 118. doi:10.1007/s00348-015-1988-5.
  * Hernandez et al. (2007) G. G. M. Hernandez, J. N. Sørensen, W. Z. Shen,  3D boundary layer study on a rotating wind turbine blade,  Journal of Physics: Conference Series 75 (2007) 012032. doi:10.1088/1742-6596/75/1/012032.
  * Hernandez (2012) G. G. M. Hernandez, Laminar-turbulent transition on wind turbines, Ph.D. thesis, DTU, Lyngby, 2012\.
  * Pascal et al. (2018) L. Pascal, R. Barrier, G. Billonet, J. Marty,  Linear stability analysis in rotating frames and its application to fan blade transition prediction,  in: 2018 Applied Aerodynamics Conference, American Institute of Aeronautics and Astronautics, Reston, Virginia, 2018. doi:10.2514/6.2018-4215.
  * Fava et al. (2021) T. Fava, M. Lokatt, N. Sørensen, F. Zahle, A. Hanifi, D. Henningson,  A simplified model for transition prediction applicable to wind-turbine rotors,  Wind Energy Science 6 (2021). doi:10.5194/wes-6-715-2021.
  * Jing et al. (2020) Z. Jing, A. Ducoin, C. Braud,  Direct numerical simulation of transitional boundary layers on a horizontal axis wind turbine blade,  Journal of Physics: Conference Series 1618 (2020) 052042. doi:10.1088/1742-6596/1618/5/052042.
  * Gross et al. (2012) A. Gross, H. F. Fasel, T. Friederich, M. J. Kloker,  Numerical investigation of rotational augmentation for S822 wind turbine airfoil,  Wind Energy 15 (2012). doi:10.1002/we.540.
  * Guntur and Sørensen (2014) S. Guntur, N. N. Sørensen,  Comments on the research article by Gross et al. (2012),  Wind Energy 17 (2014) 1985–1987. doi:10.1002/we.1674.
  * Wen and Gross (2019) G. Wen, A. Gross,  Numerical investigation of deep dynamic stall for a helicopter blade section,  AIAA Journal 57 (2019) 1434–1451. doi:10.2514/1.J057686.
  * Jing and Ducoin (2020) Z. Jing, A. Ducoin,  Direct numerical simulation and stability analysis of the transitional boundary layer on a marine propeller blade,  Physics of Fluids 32 (2020) 124102. doi:10.1063/5.0030070.
  * Yang et al. (2006) J. Yang, B. Ganesh, N. Komerath,  Radial flow measurements downstream of forced dynamic separation on a rotor blade,  in: 36th AIAA Fluid Dynamics Conference and Exhibit, American Institute of Aeronautics and Astronautics, Reston, Virigina, 2006. doi:10.2514/6.2006-3377.
  * DiOttavio et al. (2008) J. DiOttavio, K. Watson, J. Cormey, S. Kondor, N. Komerath,  Discrete structures in the radial flow over a rotor blade in dynamic stall,  in: 26th AIAA Applied Aerodynamics Conference, American Institute of Aeronautics and Astronautics, Reston, Virigina, 2008. doi:10.2514/6.2008-7344.
  * Raghav and Komerath (2014) V. Raghav, N. Komerath,  Velocity measurements on a retreating blade in dynamic stall,  Experiments in Fluids 55 (2014) 1669. doi:10.1007/s00348-014-1669-9.
  * Winkelman and Barlow (1980) A. E. Winkelman, J. B. Barlow,  Flowfield model for a rectangular planform wing beyond stall,  AIAA Journal 18 (1980) 1006–1008. doi:10.2514/3.50846.
  * Bippes and Turk (1982) H. Bippes, M. Turk,  Half model testing applied to wings above and below stall,  in: Recent Contributions to Fluid Mechanics, Springer Berlin Heidelberg, Berlin, Heidelberg, 1982, pp. 22–30. doi:10.1007/978-3-642-81932-2{\\_}3.
  * Weihs and Katz (1983) D. Weihs, J. Katz,  Cellular patterns in poststall flow over unswept wings,  AIAA Journal 21 (1983) 1757–1759. doi:10.2514/3.8321.
  * Schewe (2001) G. Schewe,  Reynolds-number effects in flow around more-or-less bluff bodies,  Journal of Wind Engineering and Industrial Aerodynamics 89 (2001) 1267–1289. doi:10.1016/S0167-6105(01)00158-1.
  * Theofilis et al. (2000) V. Theofilis, S. Hein, U. Dallmann,  On the origins of unsteadiness and three-dimensionality in a laminar separation bubble,  Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 358 (2000) 3229–3246. doi:10.1098/rsta.2000.0706.
  * Rodríguez and Theofilis (2010) D. Rodríguez, V. Theofilis,  Structural changes of laminar separation bubbles induced by global linear instability,  Journal of Fluid Mechanics 655 (2010) 280–305. doi:10.1017/S0022112010000856.
  * Alfredsson and Persson (1989) P. H. Alfredsson, H. Persson,  Instabilities in channel flow with system rotation,  Journal of Fluid Mechanics 202 (1989) 543–557. doi:10.1017/S002211208900128X.
  * Wall and Nagata (2006) D. P. Wall, M. Nagata,  Nonlinear secondary flow through a rotating channel,  Journal of Fluid Mechanics 564 (2006) 25. doi:10.1017/S0022112006001157.
  * Potter and Chawla (1971) M. C. Potter, M. D. Chawla,  Stability of boundary layer flow subject to rotation,  Physics of Fluids 14 (1971) 2278. doi:10.1063/1.1693328.
  * Deschamps and Hein (2018) X. Deschamps, S. Hein,  Extension of the PSE code NOLOT for transition analysis in rotating reference frames,  2018, pp. 179–188. doi:10.1007/978-3-319-64519-3{\\_}16.
  * Malik et al. (1981) M. R. Malik, S. P. Wilkinson, S. A. Orszag,  Instability and transition in rotating disk flow,  AIAA Journal 19 (1981) 1131–1138. doi:10.2514/3.7849.
  * Hall (1986) P. Hall,  An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc,  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 406 (1986) 93–106. doi:10.1098/rspa.1986.0066.
  * Malik (1986) M. R. Malik,  The neutral curve for stationary disturbances in rotating-disk flow,  Journal of Fluid Mechanics 164 (1986) 275–287. doi:10.1017/S0022112086002550.
  * Balachandar et al. (1992) S. Balachandar, C. L. Streett, M. R. Malik,  Secondary instability in rotating-disk flow,  Journal of Fluid Mechanics 242 (1992) 323–347. doi:10.1017/S0022112092002398.
  * Lingwood (1995) R. J. Lingwood,  Absolute instability of the boundary layer on a rotating disk,  Journal of Fluid Mechanics 299 (1995) 17–33. doi:10.1017/S0022112095003405.
  * Lingwood (1996) R. J. Lingwood,  An experimental study of absolute instability of the rotating-disk boundary-layer flow,  Journal of Fluid Mechanics 314 (1996) 373–405. doi:10.1017/S0022112096000365.
  * Hussain et al. (2011) Z. Hussain, S. J. Garrett, S. O. Stephen,  The instability of the boundary layer over a disk rotating in an enforced axial flow,  Physics of Fluids 23 (2011) 114108. doi:10.1063/1.3662133.
  * Bak et al. (2012) C. Bak, R. D. Bitsche, A. Yde, T. Kim,  Light Rotor: The 10-MW reference wind turbine,  in: Proceedings of EWEA 2012 - European Wind Energy Conference & Exhibition, 2012.
  * Björck (1990) A. Björck, Coordinates and calculations for the FFA-W1-xxx, FFA-W2-xxx and FFA-W3-xxx series of airfoils for horizontal axis wind turbines, Technical Report, FFA TN 1990-15, 1990.
  * Fischer et al. (2008) P. F. Fischer, J. W. Lottes, S. G. Kerkemeier, Nek5000 web page, 2008\.
  * ANSYS (2013) ANSYS, ANSYS Fluent User’s Guide 15, 2013\.
  * Schlatter and Örlü (2012) P. Schlatter, R. Örlü,  Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects,  Journal of Fluid Mechanics 710 (2012) 5–34. doi:10.1017/jfm.2012.324.
  * Negi et al. (2017) P. S. Negi, P. Schlatter, D. S. Henningson, A re-examination of filter-based stabilization for spectral-element methods, Technical Report, Royal Institute of Technology, Stockholm, 2017.
  * ANSYS (2017) ANSYS, Ansys ICEM CFD User’s Manual 18.2, 2017.
  * Negi et al. (2018) P. Negi, R. Vinuesa, A. Hanifi, P. Schlatter, D. Henningson,  Unsteady aerodynamic effects in small-amplitude pitch oscillations of an airfoil,  International Journal of Heat and Fluid Flow 71 (2018) 378–391. doi:10.1016/j.ijheatfluidflow.2018.04.009.
  * Schlatter et al. (2010) P. Schlatter, Q. Li, G. Brethouwer, A. V. Johansson, D. S. Henningson,  Simulations of spatially evolving turbulent boundary layers up to,  International Journal of Heat and Fluid Flow 31 (2010) 251–261. doi:10.1016/j.ijheatfluidflow.2009.12.011.
  * Alam and Sandham (2000) M. Alam, N. D. Sandham,  Direct numerical simulation of ’short’ laminar separation bubbles with turbulent reattachment,  Journal of Fluid Mechanics 410 (2000). doi:10.1017/S0022112099008976.
  * Rodríguez et al. (2021) D. Rodríguez, E. M. Gennaro, L. F. Souza,  Self-excited primary and secondary instability of laminar separation bubbles,  Journal of Fluid Mechanics (2021). doi:10.1017/jfm.2020.767.
  * Cherubini et al. (2010) S. Cherubini, J.-C. Robinet, P. De Palma,  The effects of non-normality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble,  Physics of Fluids 22 (2010) 014102. doi:10.1063/1.3276903.
  * Saric et al. (2003) W. S. Saric, H. L. Reed, E. B. White,  Stability and transition of three-dimensional boundary layers,  Annual Review of Fluid Mechanics 35 (2003).
  * Yuan et al. (2005) W. Yuan, M. Khalid, J. Windte, U. Scholz, R. Radespiel,  An investigation of low-Reynolds-number flows past airfoils,  in: 23rd AIAA Applied Aerodynamics Conference, American Institute of Aeronautics and Astronautics, Reston, Virigina, 2005. doi:10.2514/6.2005-4607.
  * Burgmann and Schröder (2008) S. Burgmann, W. Schröder,  Investigation of the vortex induced unsteadiness of a separation bubble via time-resolved and scanning PIV measurements,  Experiments in Fluids 45 (2008). doi:10.1007/s00348-008-0548-7.
  * Schmid and Henningson (2001) P. J. Schmid, D. S. Henningson, Stability and transition in shear flows, volume 142, Springer New York, New York, NY, 2001. doi:10.1007/978-1-4613-0185-1.
  * Diwan and Ramesh (2009) S. S. Diwan, O. N. Ramesh,  On the origin of the inflectional instability of a laminar separation bubble,  Journal of Fluid Mechanics 629 (2009). doi:10.1017/S002211200900634X.
  * Brinkerhoff and Yaras (2011) J. R. Brinkerhoff, M. I. Yaras,  Interaction of viscous and inviscid instability modes in separation-bubble transition,  Physics of Fluids 23 (2011). doi:10.1063/1.3666844.
  * Hosseinverdi and Fasel (2019) S. Hosseinverdi, H. F. Fasel,  Numerical investigation of laminar-turbulent transition in laminar separation bubbles: The effect of free-stream turbulence,  Journal of Fluid Mechanics 858 (2019). doi:10.1017/jfm.2018.809.
  * Rist (2003) U. Rist,  Instability and transition mechanisms in laminar separation bubbles,  in: VKI/RTO-LS ”Low Reynolds Number Aerodynamics on Aircraft Including Applications in Emerging UAV Technology, von Karman Institute, Rhode-Saint-Genese, Belgium, 2003, pp. 1–29.
  * Jaroslawski et al. (2023) T. Jaroslawski, M. Forte, O. Vermeersch, J.-M. Moschetta, E. R. Gowree,  Disturbance growth in a laminar separation bubble subjected to free-stream turbulence,  Journal of Fluid Mechanics 956 (2023) A33. doi:10.1017/jfm.2023.23.
  * Rist and Maucher (2002) U. Rist, U. Maucher,  Investigations of time-growing instabilities in laminar separation bubbles,  European Journal of Mechanics, B/Fluids 21 (2002). doi:10.1016/S0997-7546(02)01205-0.
  * Borodulin et al. (2019) V. I. Borodulin, A. V. Ivanov, Y. S. Kachanov, D. A. Mischenko, R. Örlü, A. Hanifi, S. Hein,  Experimental and theoretical study of swept-wing boundary-layer instabilities. Unsteady crossflow instability,  Physics of Fluids 31 (2019). doi:10.1063/1.5094609.
  * Arnal and Casalis (2000) D. Arnal, G. Casalis,  Laminar-turbulent transition prediction in three-dimensional flows,  Progress in Aerospace Sciences 36 (2000) 173–191. doi:10.1016/S0376-0421(00)00002-6.
  * Hanifi et al. (1994) A. Hanifi, D. Henningson, S. Hein, F. P. Bertolotti, M. Simen, Linear nonlocal instability analysis - the linear NOLOT code, Technical Report, FFA TN 1994-54, Bromma, Sweden, 1994.
  * Huerre and Monkewitz (1990) P. Huerre, P. A. Monkewitz,  Local and global instabilities in spatially developing flows,  Annual Review of Fluid Mechanics 22 (1990). doi:10.1146/annurev.fl.22.010190.002353.
  * Hammond and Redekopp (1998) D. Hammond, L. Redekopp,  Local and global instability properties of separation bubbles,  European Journal of Mechanics - B/Fluids 17 (1998) 145–164. doi:10.1016/S0997-7546(98)80056-3.
  * Rodríguez et al. (2013) D. Rodríguez, E. M. Gennaro, M. P. Juniper,  The two classes of primary modal instability in laminar separation bubbles,  Journal of Fluid Mechanics 734 (2013). doi:10.1017/jfm.2013.504.
  * Hosseinverdi and Fasel (2013) S. Hosseinverdi, H. Fasel,  Direct numerical simulations of transition to turbulence in two-dimensional laminar separation bubbles,  in: 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 2013, 2013\. doi:10.2514/6.2013-264.
  * Avanci et al. (2019) M. P. Avanci, D. Rodríguez, L. S. B. Alves,  A geometrical criterion for absolute instability in separated boundary layers,  Physics of Fluids 31 (2019). doi:10.1063/1.5079536.
  * Kupfer et al. (1987) K. Kupfer, A. Bers, A. K. Ram,  The cusp map in the complex-frequency plane for absolute instabilities,  Physics of Fluids 30 (1987) 3075. doi:10.1063/1.866483.
  * Obremski et al. (1969) H. J. Obremski, M. V. Morkovin, M. Landahl, A portfolio of stability characteristics of incompressible boundary layers, Technical Report, AGARD-R-134, 1969.
  * Marxen et al. (2003) O. Marxen, M. Lang, U. Rist, S. Wagner,  A combined experimental/numerical study of unsteady phenomena in a laminar separation bubble,  in: Flow, Turbulence and Combustion, volume 71, 2003. doi:10.1023/B:APPL.0000014928.69394.50.
  * Towne et al. (2018) A. Towne, O. T. Schmidt, T. Colonius,  Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis,  Journal of Fluid Mechanics 847 (2018). doi:10.1017/jfm.2018.283.
  * Schmidt and Colonius (2020) O. T. Schmidt, T. Colonius,  Guide to spectral proper orthogonal decomposition,  AIAA Journal 58 (2020). doi:10.2514/1.J058809.