1. Introduction
The turbulence in the lower atmosphere is significantly affected by shear, convection, surface roughness and wakes of upstream wind turbines (Wyngaard Reference Wyngaard1992; Amandolèse & Széchényi Reference Amandolèse and Széchényi2004). The turbulence intensity spans
$TI=5\,\%{-}25\,\%$
(Højstrup Reference Højstrup1999; Noda & Flay Reference Noda and Flay1999; Thomsen & Sørensen Reference Thomsen and Sørensen1999; Hand et al. Reference Hand, Kelley and Balas2003). Despite the high
$TI$
, several works report a large region of laminar flow on the suction side of wind turbine blades (up to
$40\,\%$
of the chord) (Madsen et al. Reference Madsen, Bak, Paulsen, Gaunaa, Fuglsang, Romblad, Olesen, Enevoldsen, Laursen and Jensen2010; Reichstein et al. Reference Reichstein, Schaffarczyk, Dollinger, Balaresque, Schülein, Jauch and Fischer2019). This is partly due to the higher relative velocity (
$U_\infty = \sqrt {V_\infty ^2+(\Omega \, r)^2}$
) felt by the blade, where
$V_\infty$
and
$\Omega r$
are the wind and rotation velocities, respectively, reducing the relative
$TI$
(Schaffarczyk et al. Reference Schaffarczyk, Schwab and Breuer2017). The latter found
$TI\approx 2\,\%$
and observed a laminar extent up to
$20\,\%$
of the chord.
The atmospheric turbulence scales help explain the protracted laminar flow. The integral length scale, representing the large eddies, may reach
$\Lambda =27{-}55$
m (IEC 2006; Bertagnolio et al. Reference Bertagnolio, Madsen, Bak, Troldborg and Fischer2015) from wind turbine experiments. The peak in the energy spectrum occurs at
$f=0.01$
Hz, followed by a
$-5/3$
-drop (Schaffarczyk et al. Reference Schaffarczyk, Schwab and Breuer2017). Therefore, large eddies, which are ineffective in exciting the blade boundary layer (Morkovin Reference Morkovin and Wells1969; Saric et al. Reference Saric, Reed and Kerschen2002; Schrader et al. Reference Schrader, Brandt and Henningson2009), carry most of the energy. Reeh (Reference Reeh2014) interpreted the large scales (low frequencies) in the turbulence spectrum as unsteadiness in the angle of attack and pressure distribution. To compute an effective turbulence intensity (
$TI^{N_c}$
) exciting Tollmien–Schlichting (TS) waves on a blade, Madsen et al. (Reference Madsen, Özçakmak, Bak, Troldborg, Sørensen and Sørensen2019) used only the
$100{-}300$
Hz range of the spectrum. Schaffarczyk et al. (Reference Schaffarczyk, Schwab and Breuer2017) highlighted the need for such a cutoff, but its selection criterion is unclear. This consists in selecting the low-frequency cutoff
$N_{c}\gt 0$
in
\begin{equation} TI^{N_c} = \sqrt {\frac {(u_{rms}^{\prime ^{N_c}})^2+(v_{rms}^{\prime ^{N_c}})^2+(w_{rms}^{\prime ^{N_c}})^2}{3}} \stackrel {N_c=0}{=\!\!=} TI, \end{equation}
\begin{equation} u_{rms}^{\prime ^{N_c}} = \sqrt {\frac {1}{N^2}\sum _{N_{c}}^{N/2} |\hat {u}|^2} \stackrel {N_c=0}{=\!\!=} \sqrt {\frac {1}{N}\sum _{0}^{N} u^{\prime ^2}} = u'_{rms}, \end{equation}
where
$u'=U-\overline {U}$
(similarly for
$v'_{rms}$
and
$w'_{rms}$
), with
$U$
and
$\overline {U}$
representing the instantaneous and mean
$x$
velocity,
$\hat {u}$
the single-sided frequency spectrum of
$u'$
, and
$N$
the time-series length; r.m.s. stands for root mean square. The comparison between wind tunnel and atmospheric turbulence from flight data suggests this threshold may be at
$f \approx 300$
Hz, removing the inertial sub-range from the latter (Romblad et al. Reference Romblad, Greiner, Guissart and Würz2022). Obtaining an effective
$TI$
from wind turbine data is particularly interesting for usage with Mack’s empirical relation (Mack Reference Mack1977) between the critical
$N$
factor (
$N_{tr}$
) and
$TI$
given by
\begin{equation} N_{tr}=-8.43-2.40\cdot \ln {\left (TI/100\right )}. \end{equation}
Mack’s correlation is obtained from wind tunnel data of a flat plate under zero pressure gradient and grid turbulence. Here,
$N_{tr}$
is used for transition estimation with the
$e^N$
method (Smith & Gamberoni Reference Smith and Gamberoni1956; van Ingen Reference van Ingen1956). Transition occurs for
$\ln { [A/A(x_0) ]}=N_{tr}$
, where
$A$
is the disturbance amplitude and
$x_0$
is the location of initial growth. This method has been successful for natural or TS-dominated transition (Arnal & Casalis Reference Arnal and Casalis2000; van Ingen Reference van Ingen2008), but bypass transition for
$TI \gtrapprox 0.5\,\%{-}1\,\%$
(Morkovin Reference Morkovin and Wells1969; Reshotko Reference Reshotko1976; Arnal & Juillen Reference Arnal and Juillen1978; Boiko et al. Reference Boiko, Grek, Dovgal and Kozlov2002) and flow separation (Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
) limit its use.
The transition scenario depends on the receptivity process. Linear receptivity may occur for
$TI\lt 3\,\%$
(Brandt et al. Reference Brandt, Schlatter and Henningson2004), where small vortices diffuse into the boundary layer (Bertolotti Reference Bertolotti1997). Shear sheltering prevents high-frequency perturbations from entering the boundary layer (Hunt & Carruthers Reference Hunt and Carruthers1990; Jacobs & Durbin Reference Jacobs and Durbin1998). Nonlinear receptivity due to the interaction of oblique waves in the free stream turbulence (FST) is the primary excitation source for higher
$TI$
(Berlin et al. Reference Berlin, Wiegel and Henningson1999). Blanco et al. (Reference Blanco, Hanifi, Henningson and Cavalieri2024) separated the linear and nonlinear receptivity effects in the Blasius boundary layer, showing that the former generates streaks with energy
$E \propto TI^2$
near the leading edge and the latter creates streaks with
$E \propto TI^4$
further downstream. Faúndez Alarcón et al. (Reference Faúndez Alarcón, Morra, Hanifi and Henningson2022) and Fava et al. (Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
) studied aerofoil receptivity to FST injected outside the boundary layer, showing that optimal perturbation analysis explained the initial disturbance growth for low
$TI$
. Large-scale disturbances can also excite shorter wavelength perturbations through scale conversion/reduction, enabled by fast base-flow variations near the leading edge (Goldstein Reference Goldstein1983; Ruban Reference Ruban1984).
A boundary layer may have a modal instability regime for low
$TI$
characterised by disturbance growth dominated by the eigenmodes (e.g. TS and Kelvin–Helmholtz (KH) modes) of the linearised Navier–Stokes operators (Reed et al. Reference Reed, Saric and Arnal1996). The non-modal scenario (Schmid Reference Schmid2007) is associated with the transient growth of vortical perturbations (Butler & Farrell Reference Butler and Farrell1992), which may lead to streamwise elongated structures denominated streaks (Klebanoff Reference Klebanoff1971). The latter grow via the lift-up mechanism (Landahl Reference Landahl1975, Reference Landahl1980), whose optimal initial disturbances are streamwise vortices (Andersson et al. Reference Andersson, Berggren and Henningson1999; Luchini Reference Luchini2000). The streaks may trigger an inner mode, i.e. the secondary instability of TS waves, with a near-wall critical layer (Vaughan & Zaki Reference Vaughan and Zaki2011). It develops a checkered pattern over the streaks with lambda structures at the intersection of high- and low-speed streaks (Nagarajan et al. Reference Nagarajan, Lele and Ferziger2007; Schlatter et al. Reference Schlatter, Deusebio, de Lange and Brandt2010; Vaughan & Zaki Reference Vaughan and Zaki2011) or hairpin vortices on the sides of the streak (Hack & Zaki Reference Hack and Zaki2014). This mode relies on the excitation near the leading edge (Schrader et al. Reference Schrader, Brandt, Mavriplis and Henningson2010). The increase in the adverse pressure gradient (APG) enhances the amplification of the inner mode and turbulent breakdown (Hack & Zaki Reference Hack and Zaki2014), whereas increasing amplitude and decreasing frequency of streaks have the opposite effect (Vaughan & Zaki Reference Vaughan and Zaki2011). The latter found a maximum phase speed
$c_p=0.54 U_\infty$
for the inner modes.
The outer mode is linked to the secondary instability of the streaks, with a critical layer far from the wall. Its dominance occurs for higher
$TI$
and is associated with bypass transition, being a precursor of turbulent spots (Asai et al. Reference Asai, Minagawa and Nishioka2002; Mans et al. Reference Mans, Kadijk, de Lange and van Steenhoven2005, Reference Mans, de Lange and van Steenhoven2007). It can manifest as spanwise symmetric and antisymmetric oscillations around the streak centreline denominated varicose and sinuous instabilities (Swearingen & Blackwelder Reference Swearingen and Blackwelder1987). The increase in the APG favours a change from the sinuous to the varicose types (Marquillie et al. Reference Marquillie, Ehrenstein and Laval2011). The sinuous instability may be more unstable (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001), triggered for a minimum amplitude of
$0.085 U_\infty$
(Arnal & Juillen Reference Arnal and Juillen1978; Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010; Vaughan & Zaki Reference Vaughan and Zaki2011). Varicose and sinuous outer modes are related to inflectional velocity profiles in the wall-normal and spanwise directions, respectively (Brandt et al. Reference Brandt, Schlatter and Henningson2004). The varicose mode may occur by colliding aligned high- and low-speed streaks, forming a lambda vortex, which was confirmed experimentally by Balamurugan & Mandal (Reference Balamurugan and Mandal2017). This leads to a spatially discontinuous shear layer that allows high-frequency FST disturbances to infiltrate the boundary layer, leading to hairpin vortices and turbulent spots (Brinkerhoff & Yaras Reference Brinkerhoff and Yaras2015). The sinuous instability may arise by a misaligned collision of high- and low-speed streaks (Brandt & de Lange Reference Brandt and de Lange2008) and frequently occurs in the rear part of the former (Mans et al. Reference Mans, de Lange and van Steenhoven2007). Vaughan & Zaki (Reference Vaughan and Zaki2011) found a maximum phase speed of
$0.75 U_\infty$
for the outer modes. For high
$TI$
, inner and outer instabilities may be suppressed, and the turbulent breakdowns may be directly caused by the FST forcing (Zhao & Sandberg Reference Zhao and Sandberg2020).
Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019) showed that an exponential amplification follows the initial slow algebraic growth of streaks in the APG region of a flow with separation. These authors identified transition dominated by KH modes for
$TI=0.1\,\%$
, streaks for
$TI=2\,\%{-}3\,\%$
, and both for intermediate
$TI$
. Istvan & Yarusevych (Reference Istvan and Yarusevych2018) demonstrated that streaks contribute more to the disturbance kinetic energy than KH modes for an aerofoil with flow separation for
$TI\gt 1.99\,\%$
. They noted that high
$TI$
leads to decreased spatial amplification, suggesting that the increased initial disturbance amplitudes in this regime primarily cause earlier transition. Jaroslawski et al. (Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023) indicated that an increase in
$TI$
attenuates the growth of modal instabilities, and the streamwise energy growth shifts from exponential to algebraic for high disturbance levels. Dotto et al. (Reference Dotto, Barsi, Lengani, Simoni and Satta2022) noted that discrete and continuous Orr–Sommerfeld modes were relevant for transition for a zero APG boundary layer for
$TI=2\,\%{-}3\,\%$
.
Transition on wind turbine blades was investigated with surface microphones (Özçakmak et al. Reference Özçakmak, Madsen, Sørensen and Sørensen2020). Natural transition was observed in wind-tunnel (Madsen et al. Reference Madsen, Bak, Paulsen, Gaunaa, Fuglsang, Romblad, Olesen, Enevoldsen, Laursen and Jensen2010; Lobo et al. Reference Lobo, Boorsma and Schaffarczyk2018) and field (Troldborg et al. Reference Troldborg, Bak, Madsen and Skrzypinski2013) experiments for low
$TI$
, characterised by a pronounced amplitude peak in the microphone pressure spectrum. Bypass transition occurred when the blade passed through the wake of upstream wind turbines, with the microphone pressure spectrum presenting high energy for low frequencies (
$f\lt 10$
Hz) and no clear peak (Özçakmak et al. Reference Özçakmak, Madsen, Sørensen and Sørensen2020; Lobo et al. Reference Lobo, Özçakmak, Madsen, Schaffarczyk, Breuer and Sørensen2023). Dollinger et al. (Reference Dollinger, Balaresque, Gaudern, Gleichauf, Sorg and Fischer2019) and Reichstein et al. (Reference Reichstein, Schaffarczyk, Dollinger, Balaresque, Schülein, Jauch and Fischer2019) detected wedges aligned with azimuthal direction signalling bypass transition. There are other experimental works on the role of FST on wind turbines, but they do not provide further insight into transition (Døssing Reference Døssing2008; Bertagnolio et al. Reference Bertagnolio, Madsen, Bak, Troldborg and Fischer2015; Schaffarczyk et al. Reference Schaffarczyk, Schwab and Breuer2017; Madsen et al. Reference Madsen, Özçakmak, Bak, Troldborg, Sørensen and Sørensen2019; Reichstein et al. Reference Reichstein, Schaffarczyk, Dollinger, Balaresque, Schülein, Jauch and Fischer2019; Oehme et al. Reference Oehme, Gleichauf, Suhr, Balaresque, Sorg and Fischer2022). Wind-tunnel investigations with grid turbulence and integral length scale
$O(10^{-1})$
m showed an enhancement in the blade performance due to bypass transition and suppression of flow separation (Amandolèse & Széchényi Reference Amandolèse and Széchényi2004; Sicot et al. Reference Sicot, Devinant, Loyer and Hureau2008; Maldonado et al. Reference Maldonado, Castillo, Thormann and Meneveau2015).
The literature on how FST affects transition is dense, but there is limited understanding of how it occurs on wind turbine blades. Pertinent issues include: (i) the assessment of Mack’s correlation; (ii) a cutoff for the size of free stream eddies affecting transition; (iii) a
$TI$
threshold for bypass transition; (iv) a better understanding of receptivity, and the stabilising or destabilising interaction between streaks and modal instabilities. This work attempts to address these questions with detailed wall-resolved large eddy simulation (LES) of a wind turbine aerofoil at a Reynolds number
$Re_c = 10^6$
under FST intensities
$TI=0\,\%$
,
$0.6\,\%$
,
$1.2\,\%$
,
$2.4\,\%$
,
$4.5\,\%$
and
$7\,\%$
. This study extends the results of Lobo et al. (Reference Lobo, Schaffarczyk and Breuer2022) and Fava et al. (Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
) to a realistic
$Re_c$
of wind turbines. The manuscript is divided as follows: § 2 presents the numerical set-up and test cases; § 3 contains the results and is subdivided into several subsections; § 3.1 characterises mean and instantaneous fields; § 3.2 investigates the receptivity process; § 3.3 analyses the evolution of modal and non-modal disturbances, and applies linear stability theory; § 3.4 proposes a low-frequency cutoff for FST; and finally, § 4 presents the conclusions. Appendix A summarises the calculations of coefficients and errors of linear regression of the critical
$N$
factor correlations. Appendix B shows the operators for stability analysis on cross-planes. Appendix C displays the statistics of the time variation of the pressure coefficient due to the FST.
2. Study cases and numerical approach
2.1. Description of flow cases
The flow on the suction side of a 20 % thick aerofoil, employed at a radius of 35 m of the LM45.3p blade of the 2 MW Senvion MM92 wind turbine, is analysed for an angle of attack
$AoA = 4.6^{\circ }$
. The Reynolds number based on the free stream velocity (
$U_{\infty }$
) and chord length (
$c$
) is
$Re_c = 10^6$
. Unless otherwise stated, the dimensional variables (denoted by
$^*$
) are non-dimensionalised by
$U_{\infty }$
and
$c$
. FST intensities
$TI=0\,\%$
,
$0.6\,\%$
,
$1.2\,\%$
,
$2.4\,\%$
,
$4.5\,\%$
and
$7\,\%$
are considered.
Table 1. Kaimal length scales and standard deviation ratios from IEC 61400-1 (IEC 2006).
Two coordinate systems are employed. The
$x$
,
$y$
,
$z$
coordinates are in the chord direction (
$x=0$
at the leading edge and
$x=1$
at the trailing edge), wall-normal direction (
$y$
) and spanwise direction (
$z$
). The
$U$
,
$V$
and
$W$
velocities are in the streamwise (along the aerofoil), wall-normal (
$y$
) and spanwise (
$z$
) directions. The second coordinate system,
$\overline {x}$
,
$\overline {y}$
,
$\overline {z}$
, is only used for the generation of FST. Note that
$\overline {x}$
is aligned with the inflow direction and forms an
$AoA = 4.6^{\circ }$
with
$x$
, whereas
$\overline {z}$
is aligned with
$z$
. Specifically for the generation of FST,
$U$
,
$V$
and
$W$
are in the
$\overline {x}$
,
$\overline {y}$
and
$\overline {z}$
directions. The anisotropic FST is based on the IEC-61400–1 standard (IEC 2006), with integral length scales (
$\Lambda _k$
) and standard deviation (
$\sigma _k$
) ratios given in table 1. They follow a Kaimal spectrum (Kaimal Reference Kaimal1973) represented by (2.1), where
$\bar {U}$
is the hub-height mean velocity:
\begin{equation} E(f) \; = \; \sigma _k ^{2} \frac {4 \Lambda _k / \bar {U}}{(1+6 f \Lambda _k / \bar {U})^{\frac {5}{3}}} \;. \end{equation}
Here,
$\Lambda _{1} = 42$
m for hub heights greater than
$\mathrm {60\,m}$
(IEC 2006). A wall-resolved LES of such large scales is computationally expensive. Therefore, the structures are scaled down, keeping the ratio between the length scales in the three directions, so that they fit within the spanwise width
$L_z = L_z^*/c = 0.06$
(the limiting domain dimension). The maximum energy is located in the spanwise wavenumber
$k_{z} = \sqrt {\pi }/L_z$
, where
$k_{z}$
is a variable. The maximum spanwise wavelength that can be resolved is based on the spanwise dimension such that
$\lambda _z = 2 \pi /k_{z}$
= 0.06. The two-point correlations along the span of the spanwise velocity drop to zero in the range
$l_{corr}=0.012{-}0.019$
considering all studied cases. The fact that
$l_{corr}\lt 0.03=L_z/2$
ensures that the domain is not forcing an artificial spanwise periodicity.
Together with the relations from table 1, it is found that the length scales in the
$\overline {x}$
,
$\overline {y}$
and
$\overline {z}$
directions are 0.211, 0.07 and 0.01 dimensionless units, respectively.
2.2. Details of the numerical method
The Navier–Stokes equations for an incompressible flow are solved using a classical wall-resolved LES methodology with the code
$\mathcal {LESOCC}$
(Breuer Reference Breuer1998, Reference Breuer2000, Reference Breuer2018; Breuer & Schmidt Reference Breuer and Schmidt2019). It employs a finite-volume method on a curvilinear and block-structured grid. The solver is second-order accurate in space and time. The dynamic variant (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Lilly Reference Lilly1992) of the classical Smagorinsky subgrid-scale model (Smagorinsky Reference Smagorinsky1963) is applied, as Sayadi & Moin (Reference Sayadi and Moin2011) showed the former has several advantages necessary for predicting transitional flows. FST is generated with the digital filter method of Klein et al. (Reference Klein, Sadiki and Janicka2003), improved for numerical efficiency by Kempf et al. (Reference Kempf, Wysocki and Pettit2012). The method relies on discrete linear digital non-recursive filters and requires only a few statistical properties for generating FST with proper auto-correlations in time and two-point correlations in space. Realistic cross-correlations between the three velocity components are achieved with the transformation by Lund et al. (Reference Lund, Wu and Squires1998).
The inflow turbulence generator inputs are the mean velocity, Reynolds stresses, and integral time (
$T$
) and length scales (
$\Lambda _{\overline {x}},\Lambda _{\overline {y}},\Lambda _{\overline {z}}$
).Here,
$T$
can be obtained from
$\Lambda _{\overline {x}}$
using Taylor’s frozen turbulence hypothesis. The three normal components of the Reynolds stresses are calculated by first determining the turbulent kinetic energy (TKE or
$k$
) considering isotropic turbulence as
$k = ({3}/{2}) TI^2$
. Next, using the relations between the standard deviations in table 1 and the equation for the turbulent kinetic energy
$k= ({1}/{2}) (\overline {u_{\overline {x}}^{\prime } u_{\overline {x}}^{\prime }}+\overline {u_{\overline {y}}^{\prime } u_{\overline {y}}^{\prime }}+\overline {u_{\overline {z}}^{\prime } u_{\overline {z}}^{\prime }} )$
, the three normal components of the Reynolds stress tensor for anisotropic FST are determined. To allow for a streamwise–spanwise correlation, a non-zero Reynolds shear stress is included. This can be set as
$\overline {u_{\overline {x}}^{\prime } u_{\overline {z}}^{\prime }} = -U_{star}^2$
, where
$U_{star}=0.05$
is the friction or shear velocity, which depends on the ground roughness scale (Jonkman Reference Jonkman2009). Here,
$U_{star}=0.05$
is arbitrarily selected. The direction of the non-zero Reynolds shear stress is chosen to be
$\overline {u_{\overline {x}}^{\prime } u_{\overline {z}}^{\prime }}$
since the spanwise direction (
$z$
) is the component in the direction from the blade root to the tip and the effect of shear within the rotor plane, mainly due to the ground, is important to consider. Other turbulence generators in the wind industry, such as TurbSim (Jonkman Reference Jonkman2009), also consider this component for computing
$U_{star}$
.
The inflow generator only allows one length scale per direction. This drawback is overcome by the superposition of inflow turbulence with different length scales, given by the maximal length scale divided by a factor
$2^{(n-1)}$
, where
$n$
is the index of the superimposed signal. The smallest scales that can be resolved depend on the computational grid. Here,
$n = 6$
is retained. To respect Kolmogorov’s
$k^{-\frac {5}{3}}$
law, the following steps are employed in the scaling.
-
(i) The frequency of each subsequent velocity signal is scaled by a factor of
$2^{(n-1)}$
. -
(ii) Next, since
$k \propto u_i^{\prime 2}$
, the velocity fluctuations are scaled by a factor of
$2^{\frac {1}{2}}$
to represent their contribution to the TKE. This leads to an intermediate scaling factor of
$2^{(n-1) \times (1/2)}$
. -
(iii) Finally,
$k$
should decay according to Kolmogrov’s
$k^{-5/3}$
law and thus, the velocity fluctuations must further be scaled by
$2^{-5/3}$
leading to the final scaling factor to be used, that is,
$2^{(n-1) \times (-5/3 \times 1/2)} = 2^{(-5/6) \times (n-1)}$
.
The resulting fluctuating velocity signal
$u_{{\overline {x}},sum}$
(similarly for
$u_{{\overline {y}},sum}$
and
$u_{{\overline {z}},sum}$
) is obtained by weighing the original signals by this scaling factor for each
$n$
and superimposing them. For obtaining the required
$TI$
and TKE spectrum,
$u_{{\overline {x}},sum}$
is normalised according to
\begin{equation} u_{{\overline {x}},sum} \leftarrow u_{{\overline {x}},sum} / \sqrt {sum_w} \;,\quad sum_w = \sum _n \left [2^{\frac {-5}{6}(n-1)}\right ]^2 \;, \end{equation}
where
$sum_w$
is the total contribution of the weights to the TKE.
For simplicity, the scaling is illustrated with the help of six sinusoidal signals with frequencies
$2^{(n-1)} f$
, where
$f=60$
and
$n=1$
–
$6$
. Figure 1(a) portrays the turbulent kinetic energy spectrum with the scaling as described in the procedure above, where the resulting signal follows Kolmogorov’s
$k^{-5/3}$
law.
Figure 1. Turbulent kinetic energy spectra of the (a) superimposed sinusoidal signals, (b) synthetic inflow turbulence compared with atmospheric measurements from Jeromin et al. (Reference Jeromin, Schaffarczyk, Puczylowski, Peinke and Hölling2014). (c) Variation of the turbulence intensity at
$y=120\delta ^*$
as a function of
$x$
. (d) Local turbulence intensity at
$y=120\delta ^*$
as a function of
$x$
for
$TI=0\,\%$
. (e) Spectrum of perturbation kinetic energy at
$y=120\delta ^*$
at the leading edge for
$TI=0\,\%$
. (f) Ratio of the domain spanwise width to the spanwise integral length scale of the synthetic inflow turbulence with
$x = 0$
corresponding to the leading-edge of the aerofoil.
The inflow turbulence is injected inside the domain, one chord length upstream of the aerofoil (
$x=-1$
), in a region with a streamwise length
$2 \Lambda _{\overline {x}}$
. A special source-term formulation, developed and validated in several studies (Schmidt & Breuer Reference Schmidt and Breuer2017; Breuer Reference Breuer2018; De Nayer et al. Reference De Nayer, Schmidt, Wood and Breuer2018; Breuer & Schmidt Reference Breuer and Schmidt2019), is employed for that. A comparison of the turbulent kinetic energy spectrum of the generated FST with measurements from the free atmosphere is shown in figure 1(b). The measurements are from a two-dimensional (2-D) atmospheric laser cantilever anemometer (ALCA, blue line) and cup anemometers (red symbols) in the lower atmosphere (Jeromin et al. Reference Jeromin, Schaffarczyk, Puczylowski, Peinke and Hölling2014). The power spectral density is normalised by
$\int _{0}^{\infty } S(f) \cdot {\textrm{d}} f= ({1}/{2})\langle \|{\mathbf u}^{\prime} \|^2\rangle$
. The synthetic turbulence agrees with the measured data above a minimum frequency. Figure 1(c) indicates that the turbulence intensity decays from the injection point to
$x=-0.4$
, downstream of which its decrease is not significant, approximately reaching the nominal values of
$TI=0.6\,\%$
,
$1.2\,\%$
,
$2.4\,\%$
,
$4.5\,\%$
and
$7\,\%$
affecting the aerofoil (
$x= 0{-}1$
). Figure 1(d) quantifies the background noise levels for
$TI=0\,\%$
, showing that the local turbulence level spans
$0.1\,\%{-}0.25\,\%$
from the aerofoil leading edge to 40 % of the chord. Figure 1(e) shows the perturbation kinetic energy spectrum at
$y=120\delta ^*$
at the leading edge, which indicates that the disturbances display mainly low frequency (
$f\lt 2$
) and spanwise wavenumber (
$\beta$
), with a predominance of 2-D perturbations.
2.3. Computational domain and grid resolution
The C-type grid included the angle of attack of
$4.6^\circ$
and extended 8 and 15 chord lengths upstream and downstream of the aerofoil, respectively. The spanwise width
$L_z = 0.06$
was selected such that
$L_z/\Lambda _z\gt 8$
, as shown in figure 1(f), respecting the minimum
$L_z/\Lambda _z=6$
proposed by O’Neill et al. (Reference O’Neill, Nicolaides, Honnery and Soria2004) and followed by Faúndez Alarcón et al. (Reference Faúndez Alarcón, Morra, Hanifi and Henningson2022). Although respecting these guidelines from the literature, the limited spanwise width is expected to have some influence on the flow structures. A wider domain enables a higher decorrelation of the structures in the spanwise direction. Conversely, a relatively narrow domain with spanwise periodic boundary conditions may artificially increase the spanwise correlation of the turbulence structures, reducing their spatial heterogeneity. The narrow domain may also constrain global modes arising in laminar separation bubbles (LSBs) to have an artificially high spanwise wavenumber since these modes may typically present large spanwise wavelengths, comparable to those of the length of the LSB (Fava et al. Reference Fava, Massaro, Schlatter, Henningson and Hanifi2024b
). As will be discussed in § 3, the only case with separation is
$TI=0\,\%$
, but the reverse flow is too low to allow a global mode. Finally, the narrow domain limits the number of streaks that can fit the domain side by side, which may preclude or underpredict instabilities related to lateral interactions between streaks.
The time step is
$\Delta t^* \cdot U_{\infty } / c = 3 \times 10^{-6}$
, yielding a maximum Courant number of 0.26. The total run time was
$25\,U_{\infty } / c$
, with statistics collected for the last
$8\,U_{\infty } / c$
, sufficient for the temporal convergence of the Reynolds stresses.
A wall-orthogonal grid was generated with
$y^{+}_{1st}\lt 1.0$
(first cell centre) and expansion factor of 1.05,
$\Delta x^{+}\leqslant 30$
(suction side),
$\Delta x^{+}\leqslant 60$
(pressure side) and
$\Delta z^{+}$
$\leqslant 25$
. This mesh resolution respects the criteria for wall-resolved LES outlined by Piomelli & Chasnov (Reference Piomelli, Chasnov, Hallbäck, Henningson, Johansson and Alfredson1996). On the suction side, the mesh respects the more restrictive criteria proposed by Asada & Kawai (Reference Asada and Kawai2018) for transitional flows with separation and streaks. The dimensionless grid parameters are similar to those in earlier studies at
$Re_c = 1 \times 10^5$
(Lobo et al. Reference Lobo, Schaffarczyk and Breuer2022; Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
). A grid-independence study was conducted with a finer grid with nearly three times the number of points of the standard grid and is available from Lobo (Reference Lobo2023).
3. Results
3.1. General flow analysis
Figure 2 exhibits the spanwise- and time-averaged streamwise velocity (
$\langle U \rangle _{z,t}$
) profiles at
$x=0,0.1,\ldots ,1$
. The edge of the mean LSB, defined as
$(x,y) \in \int _{0}^{y} \langle U\rangle _{z, t}(x,\xi ) \; \textrm{d}\xi =0$
, is shown for
$TI=0\,\%$
in figure 2(a). The LSB lies in the
$x=0.44{-}0.55$
region, with a maximum height
$h_{ {max}}/\delta ^*=0.57$
at
$x=0.53$
and reverse flow of
$-7\,\%$
of
$U_\infty$
. The APG acting downstream of
$x=0.26$
is the main driver of flow separation. The transition location (
$x_{tr}$
) is defined as the streamwise locus of maximum
$\langle u'_{rms} \rangle _z$
. For
$TI=0\,\%$
,
$x_{tr}=0.55$
indicates that flow reattachment occurs nearly immediately after transition. Low levels of FST, such as
$TI=0.6\,\%$
in figure 2(b), are enough to suppress separation unlike found in previous studies at
$Re_c = 10^5$
(Lobo et al. Reference Lobo, Schaffarczyk and Breuer2022; Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
). The spanwise-averaged part of the mean flow distortion (MFD), i.e.
$\langle U \rangle _{z,t}-\langle U \rangle _{z,t_{TI=0.6\,\%}}$
, is also presented for
$TI\gt 0.6\,\%$
. Although the separation suppression is an effect of the MFD (Marxen & Rist Reference Marxen and Rist2010),
$TI=0.6\,\%$
is selected as the base case instead of
$TI=0\,\%$
to isolate the effect of non-modal growth on the mean profiles. The MFD becomes positive near the wall as streamwise momentum is transferred to this region by the lift-up mechanism. Moreover, the MFD increases in the streamwise direction as the streaks grow, reaching a maximum at
$x=0.4$
or
$0.45$
. This maximum increases monotonically with
$TI$
. After transition, the MFD quickly becomes negative as turbulent mixing diffuses momentum across the wall-normal direction.
Figure 2. Wall-normal profiles of spanwise- and time-averaged streamwise velocity (
) and mean streamwise velocity distortion (MFD, 
) for (a)
$TI=0\,\%$
, (b)
$0.6\,\%$
, (c)
$1.2\,\%$
, (d)
$2.4\,\%$
, (e)
$4.5\,\%$
and (f)
$7\,\%$
. The LSB edge (normalised by
$\delta ^*$
) is denoted by 
. The MFD is magnified four times for enhanced visibility.
The capacity of the flow sustaining inflectional instabilities is assessed with the wall-normal location of the inflection point in
$\langle U \rangle _{z,t}$
(
$y_{in}$
) in figure 3(a), non-dimensionalised by the displacement thickness (
$\delta ^*$
). The mean flow is inflectional for
$TI\leqslant 0.6\,\%$
, indicating susceptibility to inflectional instabilities. The farther the inflection point from the wall, the stronger is this type of mechanism. Thus, the flow is more unstable for
$TI=0\,\%$
, where the peak value
$y_{in}/\delta ^*\approx 1$
agrees with Veerasamy et al. (Reference Veerasamy, Atkin and Ponnusami2021) and Jaroslawski et al. (Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023), and occurs at the LSB maximum height location (
$x_{h_{\textit{max}}}=0.53$
). The relatively high values of
$y_{in}/\delta ^*$
for
$TI=0.6\,\%$
also indicate a potential role of an inflectional instability. For
$TI\geqslant 1.2\,\%$
, the mean flow is not inflectional, hence not susceptible to inflectional instability. The evolution of
$\delta ^*$
is shown in figure 3(b). The local maximum in
$\delta ^*$
in the mid-chord region is due to the near-wall mass flux deficit, which promotes an increase in
$\delta ^*$
, followed by transition to turbulence, which locally reduces
$\delta ^*$
via turbulent mixing. The case with flow separation (
$TI=0\,\%$
) has the highest mass flux deficit and, therefore, the most pronounced increase in
$\delta ^*$
. Even though the
$TI=0.6\,\%$
case does not present flow separation, it displays a reduced mass flux deficit due to the APG in the laminar flow region. This leads to a local increase in
$\delta ^*$
, similar to, but less pronounced than,
$TI=0\,\%$
. The increase in
$TI$
reduces the near-wall mass flux deficit, since turbulent mixing brings high-momentum fluid to this region, and the local maximum in
$\delta ^*$
is progressively attenuated.
Figure 3. (a) Wall-normal location of the inflection point in the spanwise- and time-averaged streamwise velocity profiles. (b) Displacement thickness (
$\delta ^*$
).
Figure 4 shows the streamwise velocity fluctuations (
$u'$
) on a wall-parallel plane corresponding to a height of
$\delta ^*$
at 20 % chord and for arbitrary time. For
$TI=0\,\%$
in figure 4(a), 2-D rolls emerge at
$x=0.44$
, which as seen from figure 2(a) lie in the LSB. The separation (S) and reattachment (R) lines are indicated. The rolls are characteristic of a KH instability of the separated shear layer (Jaroslawski et al. Reference Jaroslawski, Forte, Vermeersch, Moschetta and Gowree2023), as confirmed later in the analyses.
Figure 4. Streamwise velocity fluctuations (
$u' = U - \langle U \rangle _{z,t}$
) on a wall-parallel plane at a height of
$\delta ^*$
for (a)
$TI=0\,\%$
, (b)
$0.6\,\%$
, (c)
$1.2\,\%$
, (d)
$2.4\,\%$
, (e)
$4.5\,\%$
and (f)
$7\,\%$
. Here
$S$
and
$R$
denote flow separation and reattachment. The isolines of
$Q$
-criterion (
$Q=250$
) are also shown for
$TI=0.6\,\%$
.
For
$TI=0.6\,\%$
, the isolines of
$Q$
-criterion (
$Q=250$
) are superimposed on the contours of
$u'$
in figure 4(b), indicating spanwise rolls at
$x=0.46{-}0.5$
. However, these are not KH modes since neither instantaneous nor mean flow separation is detected (see figure 2
b). Moreover, they occur downstream of the mean inflectional flow region (
$x=0.27{-}0.46$
), depicted in figure 3(a). Figure 5(a) shows the wall-normal
$|u'|$
profile at
$x=0.54$
,
$z=0.02$
, where these rolls occur, for the same instant as figure 4(b). The profile corresponds to the standard TS waves from Schlatter et al. (Reference Schlatter, Deusebio, de Lange and Brandt2010). However, at
$z=0.06$
, where a streak passes, the profile presents a local maximum around
$y/\delta ^*=2$
, in agreement with the streaky TS waves computed by Schlatter et al. (Reference Schlatter, Deusebio, de Lange and Brandt2010). Cossu & Brandt (Reference Cossu and Brandt2004) attributed these modes to the fundamental secondary instability of the TS waves in the streaky flow. Therefore, the streaky TS waves correspond to the inner modes of Vaughan & Zaki (Reference Vaughan and Zaki2011).
Figure 5. Wall-normal profiles of the absolute value of the instantaneous streamwise velocity fluctuations for (a)
$TI=0.6\,\%$
at
$x=0.5$
and (b)
$TI=1.2\,\%$
and
$2.4\,\%$
at
$x=0.4$
,
$z=0.02$
considering the same time as figure 4. In panel (a), the results are compared with the profiles of TS and streaky TS waves from Schlatter et al. (Reference Schlatter, Deusebio, de Lange and Brandt2010) (circles). In panel (b), the results are compared with the profiles of inner modes from Fava et al. (Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023b
) (circles) for base streaks with amplitudes of
$A_u=5\,\%\,U_\infty$
and
$10\,\%\,U_\infty$
.
The time–space evolution of the streak observed at
$x=0.2$
,
$z=0.01$
for
$TI=0.6\,\%$
in figure 4(b) is investigated. The convection speed of the streak is
$0.67 \, u_e\approx 1$
, where
$u_e$
is the boundary layer edge velocity. Figure 6 shows contour plots of the streamwise velocity fluctuations (
$u'$
) on the cross-sectional planes from the leading edge to
$x=0.2$
. The vectors of fluctuating velocities on this plane, i.e.
$(w',v')$
, are also shown (not to scale). The streak is generated near the leading edge (
$x=0.001$
, panel a), where there is an influx of high-momentum fluid from the top of the boundary layer to the near-wall region, generating high-speed fluctuations in this area for
$z=0-0.04$
. For
$z=0.04{-}0.06$
, the low-speed fluid near the wall is lifted, generating a low-speed distortion. This is typical of the lift-up mechanism responsible for the amplification of the streaks. Furthermore, the diffusion of free stream vortices into the boundary layer is characteristic of linear receptivity (Bertolotti Reference Bertolotti1997). Note that
$\delta ^*$
is nearly constant along
$z$
and only undergoes small modifications further downstream, implying these structures are not due to a temporary thinning (or thickening) of the boundary layer. The streaks quickly grow at
$x=0.002$
(panel b) and
$x=0.038$
(panel c), rising above
$\delta ^*$
in the latter location. Finally, at
$x=0.2$
(panel d), the streaks are fully formed with a spanwise scale within the range of those observed by Brandt et al. (Reference Brandt, Schlatter and Henningson2004), for example, which is
$l_{streaks}=(4{-}7)\delta ^*$
. Moreover, the streamwise vortices characteristic of the lift-up mechanism are present in the vector field.
Figure 6. Contours of streamwise velocity fluctuations on a cross-sectional plane at (a)
$x=0.001$
, (b)
$x=0.002$
, (c)
$x=0.038$
, (d)
$x=0.2$
for
$TI=0.6\,\%$
following the same streak as in figure 4(b). The vectors of fluctuating velocities on this plane are also shown (not to scale). The magenta line indicates
$\delta ^*$
.
The number and amplitude of streaks increase for
$TI=1.2\,\%$
, as shown in figure 4(c). However, the transition line is spanwise uniform without early turbulent spots. The wall-normal
$|u'|$
profile in figure 5(b) agrees with the inner modes computed by Fava et al. (Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023b
) with linear stability analysis for base streaks with an amplitude
$A_u=5\,\%\,U_\infty$
. The
$TI=2.4\,\%$
case in figure 4(d) presents isolated instabilities over individual streaks (e.g. at
$x=0.3$
,
$z=0.04{-}0.06$
). The disturbance profile in figure 5(b) agrees with that from Fava et al. (Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023b
) for base streaks with
$A_u=10\,\%\,U_\infty$
. The outer maximum increases in magnitude compared with the near-wall maximum, suggesting an increased importance of outer modes. However, the near-wall maximum is still
$75\,\%$
of the outer maximum, which shows that inner modes play a significant role in this case. This agrees with Bose & Durbin (Reference Bose and Durbin2016), who found inner modes essential for transition at
$TI\approx 2\,\%$
. The non-uniformity becomes much more pronounced for
$TI=4.5\,\%$
and
$7\,\%$
in figures 4(e) and 4(f), where individual streak breakdowns are the rule, indicating the dominance of the outer modes and bypass transition. Streaky TS waves are not observed. This agrees with Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019), who found the transition dominated by streaks for
$TI\gt 3\,\%$
. Indeed, the increase in the streak amplitude mitigates the inner modes while enhancing the outer ones (Vaughan & Zaki Reference Vaughan and Zaki2011).
3.2. Receptivity
The boundary-layer receptivity to FST is analysed. The spectra of perturbation kinetic energy (
$|k'|$
) at the leading edge, where receptivity mainly occurs, are shown in figure 7. The left and centre columns portray the
$|k'|$
-spectra at
$y=120\delta ^*$
and
$y=\delta ^*$
, respectively. The right column presents the ratio between
$|k'|$
at
$y=\delta ^*$
and
$y=120\delta ^*$
. For
$TI=0\,\%$
, the disturbance amplitude outside the boundary layer is very low (panel a). There is an amplification of three-dimensional disturbances centred around
$f=2.7$
and
$\beta =1057.3$
(
$n_z=L_z \beta /(2 \pi )=10$
) as shown in panel (c), but the disturbance amplitude remains low inside the boundary layer (panel b). This phenomenon may be due to non-modal growth (Schmid Reference Schmid2007) since the mean flow in this region will later be shown to be modally stable (in the local sense) but prone to transient growth. For
$TI=0.6\,\%$
(panels d,e,f) and
$TI=1.2\,\%$
(panels g,h,i), the maximum amplitudes outside and inside the boundary layer occur at low-
$f$
, low-
$\beta$
. However, high-
$f$
, high-
$\beta$
disturbances are mainly excited, as seen in panel (f,i). However, the perturbation amplitudes inside the boundary layer in this zone of the spectrum remain low. This changes for
$TI=2.4\,\%$
(panels j,k,l),
$TI=4.5\,\%$
(panels m,n,o) and
$TI=7\,\%$
(panels p,q,r) for which the regions of high gain overlap with areas of high disturbance amplitude (
$O(10^{-5})$
in the centre column). The excitation of high-frequency vortical disturbances inside the boundary layer is associated with bypass transition (Zaki & Durbin Reference Zaki and Durbin2005). These disturbances are most likely excited by nonlinear interactions (Berlin et al. Reference Berlin, Wiegel and Henningson1999; Schrader et al. Reference Schrader, Brandt and Henningson2009) since shear sheltering attenuates the penetration of high-frequency disturbances into the boundary layer (Durbin Reference Durbin2017).
Figure 7. Spectra of perturbation kinetic energy at
$y=120\delta ^*$
(
$|k'_{120\delta ^*}|$
, left column),
$y=\delta ^*$
(
$|k'_{\delta ^*}|$
, centre column) and
$|k'_{\delta ^*}|/|k'_{120\delta ^*}|$
(right column) at the leading edge (
$x=0$
) for (a,b,c)
$TI=0\,\%$
, (d,e,f)
$0.6\,\%$
, (g,h,i)
$1.2\,\%$
, (j,k,l)
$2.4\,\%$
, (m,n,o)
$4.5\,\%$
and (p,q,r)
$7\,\%$
. The white isoline in the centre column indicates
$|k'_{\delta ^*}|/|k'_{120\delta ^*}|=100$
and the arrow indicates the region with
$|k'_{\delta ^*}|/|k'_{120\delta ^*}|\gt 100$
.
Figure 8(a,b,c) shows the streamwise evolution of
$|u'|$
for three of the highest-amplitude
$(f,\beta )$
fluctuations in the flow. These disturbances have low frequency and they are induced in the boundary layer by the FST, especially near the leading edge, where they grow more rapidly. The lower amplitude of high-frequency fluctuations is possibly related to shear sheltering. The scaling of
$|u'|$
with
$TI$
is assessed in figure 8(d,e,f), where there is good agreement between the scaled curves for
$TI\leqslant 2.4\,\%$
for
$f=2.7$
,
$\beta =104.7$
and
$f=2.7$
,
$\beta =209.4$
(centre and right columns). The scaling is not as good for
$f=1.3$
,
$\beta =104.7$
, but the curves are clustered for
$TI\leqslant 1.2\,\%$
and
$TI\geqslant 4.5\,\%$
, separately. These results suggest that linear receptivity occurs for
$TI\leqslant 1.2\,\%$
and possibly also for
$TI=2.4\,\%$
. Brandt et al. (Reference Brandt, Schlatter and Henningson2004) found that linear receptivity occurred at
$TI\lt 3\,\%$
in Blasius flow, which supports the current results. Figure 8(g,h,i) shows that the curves do not scale well with
$TI^2$
, suggesting that nonlinear receptivity is not dominant in the generation of the leading boundary-layer perturbations. However, lower amplitude disturbances with high
$f$
and
$\beta$
, likely nonlinearly generated, become significant for
$TI\geqslant 2.4\,\%$
, as demonstrated in figure 7. Figure 8(j,k,l) shows the evolution of
$|u'|$
with the streamwise Reynolds number
$Re_x^{1/2}$
(
$\propto$
$Re_{\delta ^*}$
). In most cases, there is a region with a linear trend in the curves, indicating proportionality between
$|u'|$
and
$Re_x^{1/2}$
, which is characteristic of the growth of streaks (Luchini Reference Luchini2000). This linear trend becomes less pronounced for increasing
$TI$
, suggesting that the influence of transient growth lessens.
Figure 8. Streamwise evolution of the wall-normal maximum inside the boundary layer of the (a,b,c) streamwise velocity fluctuations (
$|u'|$
), (d,e,f)
$|u'|/TI$
, (g,h,i)
$|u'|/TI^2$
. (j,k,l) Evolution of
$|u'|$
as a function of
$Re_x^{1/2}$
.
Figure 9. (a) Mean friction coefficient (
$c_f$
). (b) Critical
$N$
factor (
$N_{tr}$
).
Table 2. Mean transition locations.
3.3. Disturbance evolution and flow stability
3.3.1. Transition to turbulence
Estimations of the mean transition location (
$x_{tr}$
) based on the maximum boundary layer shape factor (
$H$
) are not accurate here as
$H$
does not reach a maximum upon transition, unlike in other works (Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
), especially for high
$TI$
. Three criteria are considered for estimating
$x_{tr}$
. The first two consider it the
$x$
locus of maximum
$\langle u'_{rms} \rangle _{z}$
(
$x_{tr_1}$
) and
$\max _z u'_{rms}$
(
$x_{tr_2}$
). The results summarised in table 2 indicate small differences between the two methods for
$TI\leqslant 1.2\,\%$
. Due to the increased occurrence of spanwise-localised instabilities,
$x_{tr_1}\gt x_{tr_2}$
for
$TI=2.4\,\%{-}4.5\,\%$
as the transition line becomes very spanwise inhomogeneous (see figure 4
e). Nevertheless,
$x_{tr_1}\lt x_{tr_2}$
for
$TI=7\,\%$
since the elevated number of streaks promotes the spanwise homogeneity of the transition line, reducing the probability of a single streak reaching a high amplitude before the onset of the breakdown (see figure 4
f). The disturbance amplitude at
$x_{tr}$
is also listed in table 2, normalised by
$U_\infty$
and the local edge velocity (
$u_e$
). The maximum amplitude of
$0.260U_\infty$
(mean) or
$0.301U_\infty$
(maximum) occurs for
$TI=1.2\,\%$
, the highest
$TI$
for which the mean flow is unstable to TS waves. Interestingly,
$0.260U_\infty$
corresponds to the threshold for the occurrence of an outer secondary instability of the streaks in Blasius flow (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001). Hack & Zaki (Reference Hack and Zaki2014) reported
$u'_{rms}=0.16u_e$
before the breakdown of the streaks, close to the values obtained at
$x_{tr_1}$
for
$TI\geqslant 2.4\,\%$
. Finally, the last criterion for estimating
$x_{tr}$
relies on computing the friction coefficient (
$c_f$
), shown in figure 9(a), and assuming
$x_{tr_3}= ({1}/{2}) (\arg \min _x c_f + \arg \max _x c_f )$
. The rationale is that
$c_f$
grows rapidly upon transition. The
$c_f$
in the leading edge region was excluded since its high values are unrelated to transition. The values obtained with this method are close to
$x_{tr_1}$
, except for
$TI=7\,\%$
. In the remainder of the paper,
$x_{tr}=x_{tr_1}$
.
Figure 9(b) portrays the critical
$N$
factor (
$N_{tr}=N(x_{tr})$
) as a function of
$TI$
, defined as
$N_{tr}=\text {ln }(\max _y \langle k'_{rms} \rangle _z/\max _y \langle k'_{rms_0} \rangle _z)$
, where
$k'_{rms_0}=k'_{rms}(x_0)$
,
$x_0=4\times 10^{-3}$
. Similar to Mack (Reference Mack1977), this definition of
$N_{tr}$
encompasses perturbations unrelated to transition, unlike the
$N$
factor obtained from linear stability theory. However, these two definitions should be close for transition dominated by TS waves, as the disturbances related to the latter tend to stand out above the background noise for low
$TI$
. The figure shows that the
$N$
factor drops rapidly with
$TI$
for
$TI\leqslant 2.4\,\%$
, with a trend well predicted by Mack’s correlation (1.3) (Mack Reference Mack1977). The exception is
$TI=0\,\%$
, whose
$N_{tr}$
is lower than predicted due to flow separation. Mack’s correlation relies on experimental data from zero pressure gradient flat plates under grid turbulence. The close agreement is surprising since the pressure gradient, curvature and FST spectra differ from the experiments. Linear regression of a correlation of the form
$N_{tr}=a-b\ln (TI/100)$
to the
$TI\leqslant 2.4\,\%$
data yields
$a=2.63$
and
$b=1.8$
with errors
$\sigma _a=0.51$
and
$\sigma _b=0.11$
. Appendix A presents details of these calculations. This allows a better fit to the LES data as the expression presents a less steep slope than Mack’s correlation. The low and slightly dropping
$N$
factor obtained from the LES for
$TI\gt 2.4\,\%$
indicates bypass transition. The expression that best fits the
$2.4\,\%\leqslant TI \leqslant 7\,\%$
region has coefficients
$a=1.81$
and
$b=0.9$
with errors
$\sigma _a=0.50$
and
$\sigma _b=0.16$
. Unlike the expression obtained for
$TI\leqslant 2.4\,\%$
, which, together with linear stability theory and the
$e^N$
method, can be used for estimating the transition location, the correlation for
$2.4\,\%\leqslant TI \leqslant 7\,\%$
is merely an indication of the gain of the perturbations in bypass transition. Interestingly, the coefficient
$b$
for
$2.4\,\%\leqslant TI \leqslant 7\,\%$
is half that for
$TI\leqslant 2.4\,\%$
. Note that each correlation was obtained with three data points. A comparison with more data is needed to assess their validity further.
3.3.2. Primary instability and disturbance growth
Local linear stability theory (LST) is employed to help interpret the nonlinear simulations. The formulation of Fava et al. (Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
) is employed where the ansatz
$\phi '=\hat {\phi }(y)\exp {(-i \alpha x + i \omega t + i \beta z)}$
is inserted in the linearised Navier–Stokes equations, with a parallel-flow hypothesis, and solved for
$\hat {\phi }$
and
$\alpha =\alpha _r+i\alpha _i$
. Here,
$\phi '$
represents pressure and velocity disturbances for given
$\omega =2 \pi f$
and
$\beta$
;
$\alpha _r$
is the streamwise wavenumber; and
$\alpha _i$
is the growth rate. The base flow is the spanwise- and time-averaged field for each
$TI$
. Analyses are only performed for
$\beta =0$
(
$n_z=0$
) since these are the most unstable modes in two-dimensional flows (Squire Reference Squire1933), as those for low
$TI$
.
Figure 10. Local spatial stability analysis results for
$\beta =0$
. (a) Frequency envelope of growth rates (solid lines) and
$N$
factor (dashed lines). (b) Neutral curve, where
$-o-$
indicates the most unstable frequency.
Unstable modes with positive group velocity exist. Figure 10(a) presents their frequency envelope of growth rates (solid lines) and
$N$
factors (dashed lines). The maximum amplification occurs on average
$6\,\%$
chord upstream of the transition locations and decreases with
$TI$
due to the spanwise-averaged part of the mean-flow distortion, a phenomenon observed in experiments (Boiko et al. Reference Boiko, Westin, Klingmann, Kozlov and Alfredsson1994; Fransson et al. Reference Fransson, Brandt, Talamelli and Cossu2005) and simulations (Fasel Reference Fasel2002; Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023b
). This effect stabilises the mean flow with respect to two-dimensional TS waves for
$TI\geqslant 2.4\,\%$
, suggesting that the instability of streaks becomes the dominant transition trigger in this range. However, the streaks may have an impact on the growth of the TS waves for lower
$TI$
, distorting the wavefronts and triggering their secondary instability (Kendall Reference Kendall1990; Liu et al. Reference Liu, Zaki and Durbin2008; Vaughan & Zaki Reference Vaughan and Zaki2011; Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2019). This analysis requires the consideration of the spanwise variation of the flow due to the streaks, which is performed in § 3.3.3. The neutral curves in figure 10(b) indicate that the frequency of the most unstable TS waves is little affected by
$TI$
upstream of flow separation. The latter occurs for
$TI=0\,\%$
, leading to the inception of KH modes and a rise in the frequency of the most unstable modes. This differs from the
$Re_c= 10^5$
case, where the KH modes have a lower frequency than the TS waves (Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
). Another reason for the agreement between the neutral curves for
$x\lt 0.4$
is the low MFD in this region (see figure 2). The range of unstable TS waves narrows for
$TI=1.2\,\%$
as the mean flow loses its inflectional character.
Figure 11. Spectra of perturbation kinetic energy at
$y=\delta ^*$
from LES for (a)
$TI=0\,\%$
at
$x=0.53$
, (b)
$TI=0.6\,\%$
at
$x=0.46$
, (c)
$TI=1.2\,\%$
at
$x=0.4$
, (d)
$TI=2.4\,\%$
at
$x=0.38$
, (e)
$TI=4.5\,\%$
at
$x=0.35$
and (f)
$TI=7\,\%$
at
$x=0.3$
.
Figure 11 shows the perturbation kinetic energy spectra close to transition. For
$TI=0\,\%$
, there are high-amplitude disturbances centred around
$f=22.9$
and
$\beta =0$
, as shown in figure 11(a). The maximum
$N$
factor obtained from LST occurs exactly at this frequency, which is a good indication that they represent the same modes, namely inflectional instabilities of the separated shear layer (KH modes). Triadic interactions (Craik Reference Craik1971) between the fundamental (
$f=22.9$
,
$\beta =0$
), oblique (
$f=22.9$
,
$\beta =209.4$
) and streaky (
$f=0$
,
$\beta =209.4$
) modes may be a relevant mechanism due to the observed high amplitude of the fundamental and streaky disturbances. This can correspond to the oblique mechanism described by Marxen et al. (Reference Marxen, Lang, Rist and Wagner2003) and observed by Fava et al. (Reference Fava, Henningson and Hanifi2024a
), which leads to a rapid transition to turbulence, as seen in figure 4(a). The spectrum also shows the excitation of frequency harmonics of the fundamental instability, particularly at
$f=2 f_f=45.8$
. Considering
$TI=0.6\,\%$
in figure 11(b), disturbances centred around
$f=22.9$
present high amplitude, similar to
$TI=0\,\%$
. However, unlike the latter, a larger magnitude occurs for
$\beta =209.4$
instead of
$\beta =0$
. This is expected as separation is suppressed, reducing the growth rates of 2-D inflectional instabilities, while streaks develop a high amplitude. Disturbances with
$f=0$
,
$\beta =209.4$
also display high amplitude. These facts parallel the
$TI=0\,\%$
case and may indicate that vortical disturbances penetrating the boundary layer further excite the oblique mechanism. This mechanism is also possibly present for
$TI=1.2\,\%$
. Although the mean flow is not inflectional, it is unstable to TS waves, as demonstrated with LST, and a region with a large amplitude is present in the spectrum (figure 11
c) near the predicted frequency of these waves and at the same
$\beta$
of the highest amplitude stationary disturbances, related to streaks. Further rises in
$TI$
(figure 11
d,e,f) stabilise two-dimensional disturbances associated with TS waves, and consequently oblique disturbances generated by nonlinear interactions between TS waves and streaks. High-amplitude disturbances are concentrated at low frequencies, corresponding to streaks.
The most energetic structures are extracted with spectral proper orthogonal decomposition (SPOD) of the velocity perturbations (Lumley Reference Lumley1970; Towne et al. Reference Towne, Schmidt and Colonius2018; Schmidt & Colonius Reference Schmidt and Colonius2020). The streaming SPOD algorithm of Schmidt & Towne (Reference Schmidt and Towne2019) is employed, reducing memory requirements as it is unnecessary to store all snapshots simultaneously. The data involve
$6600$
snapshots with a time step of
$9 \times 10^{-4}$
(Nyquist frequency
$f=555.6$
). The snapshots are split into five blocks (five computed SPOD modes) with
$50\,\%$
overlap and a Hamming window. The weight matrix is the distance between two consecutive wall-normal grid points, and the employed norm measures
$k'$
integrated over the
$xy$
plane (Schmidt & Towne Reference Schmidt and Towne2019). The analysed domain stretches from the leading edge (
$x=0$
) to
$x_{tr}$
, excluding the turbulent flow region from the SPOD.
The occurrence of modal instabilities (e.g. TS and KH modes) is assessed in figure 12, which shows the isocontours of the real part of the first SPOD mode of streamwise velocity perturbation for
$f=22.8$
,
$\beta =0$
in the left column. These results are compared with those obtained with the parabolised stability equations (PSEs) of Fava et al. (Reference Fava, Henningson and Hanifi2023a
,Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi
b
), shown in the right column. This PSE formulation allows for a streaky base flow and interactions between the streaks and modal instabilities. Such capacity is only used for
$TI=2.4\,\%$
, where a perturbation expansion with
$N_s=4$
spanwise harmonics is considered, and the base flow is a superposition of the spanwise- and time-averaged flow and steady streaks with
$\beta =104.7$
,
$A_u=10\,\%\,U_\infty$
. Figure 12(h) shows the streak profile. In the other cases, the spanwise- and time-averaged flow is assumed to be the base flow and
$N_s=0$
, retrieving the standard PSE of Herbert (Reference Herbert1997). Figure 12(a) shows the results for
$TI=0\,\%$
, indicating the formation of TS waves upstream of the LSB (green line). Near the maximum LSB height, the structure displays a high amplitude lobe inside the LSB, a second lobe at
$\delta ^*$
(black line) and a third lobe above the latter. This indicates a KH mode well predicted by the PSE in figure 12(b). The SPOD mode for
$TI=0.6\,\%$
in figure 12(c) indicates an inflectional TS mode with a local maximum at the inflection point location (
$y=\delta ^*$
) and another further away towards the free stream. The PSE result agrees well with this mode, as shown in figure 12(d). In the
$TI=1.2\,\%$
case in figure 12(e), there is only a region with high amplitude (near-wall lobe) close to the end of the domain (transition location). The PSE predicts a decaying TS wave, as depicted in figure 12(f), which is unsurprising as the mean flow is not inflectional. The
$TI=2.4\,\%$
case is stable to TS waves, considering only the mean flow. Nevertheless, a wavepacket resembling an inflectional TS wave appears in the SPOD mode in figure 12(g). The consideration of a streaky base flow in the PSE, as discussed above, enables an instability with a low near-wall maximum and peak amplitude at the inflection point generated by the base streak, as shown in figure 12(h). This mode seems to display mixed contributions from inner and outer modes, with a predominant contribution from the former for the
$TI=1.2\,\%$
case and the latter for
$TI=2.4\,\%$
.
Figure 12. Real part of the first SPOD mode of streamwise velocity perturbation (normalised) for
$f=22.8$
,
$\beta =0$
for (a)
$TI=0\,\%$
, (c)
$TI=0.6\,\%$
, (e)
$TI=1.2\,\%$
, and (g)
$TI=2.4\,\%$
. PSE results are shown in panels (b), (d), (f) and (h). For
$TI=2.4\,\%$
, the PSE base flow contains a
$u'$
-fluctuation with
$f=0$
,
$\beta =104.7$
,
$A_u=10\,\% U_\infty$
with profile
$U_s$
shown in panel (h). The black and green lines indicate
$\delta ^*$
and the LSB edge.
Figure 13 shows the isocontours of the gain
$G$
for steady disturbances from LES, defined as
$G=E/E_0$
, where
$E=\int _0^{y_{\text {max}}} k'(x,y,\beta ,f) \; \mathrm{d}y$
,
$E_0=E(x_0)$
and
$y_{ {max}}$
is the wall-normal location of the boundary layer edge. Here,
$G$
displays low sensitivity to
$x_0$
for the values assessed and
$x_0=0.005$
is selected. The norm
$E$
is similar to that of optimal perturbation analysis (OPA) (Andersson et al. Reference Andersson, Berggren and Henningson1999; Luchini Reference Luchini2000), which is applied assuming that the mean flow of the
$TI=0\,\%$
case is the base flow where the streaks grow. The isolines of
$G$
from OPA are presented in figure 13(a) with the isocontours of
$G$
from LES for
$TI=0\,\%$
. This panel shows a region of high
$G$
near the leading edge with an initial
$\beta \approx 1500$
, which decays with
$x$
. This region can be associated with the growth of streaks due to the lift-up effect. The maximum
$G\approx 10$
agrees with that predicted by OPA for
$\beta =1256.6$
at
$x=0.07$
. OPA reaches
$G=O(10^{12})$
for
$\beta =2722.7$
further downstream, as also found by Cherubini et al. (Reference Cherubini, Robinet and De Palma2010), due to a high degree of non-normality of the linearised Navier–Stokes operator. Nevertheless, these gains are not attained in the LES. Note that the fact that there is non-modal growth in the
$TI=0\,\%$
case does not mean that streaks participate in the transition process since the background noise is very low. The
$TI=0.6\,\%$
,
$1.2\,\%$
and
$2.4\,\%$
cases in figure 13(b,c,d) also indicate streak growth. The maximum
$G$
is considerably reduced for
$TI=2.4\,\%$
and nearly vanishes for
$TI=4.5\,\%$
and
$7\,\%$
in figure 13(e,f). This suggests that the streaks barely grow due to the lift-up effect for
$TI\gt 2.4\,\%$
since the FST already induces streaks with a large amplitude near the leading edge. This highlights why the perturbation amplitude in the boundary layer scales linearly with
$TI$
for
$TI\leqslant 2.4\,\%$
but sub-linearly otherwise. Lastly, the
$TI\leqslant 1.2\,\%$
cases present a high gain of
$\beta =0$
disturbances near the mid-chord region due to two-dimensional modal instabilities, absent for higher
$TI$
.
Figure 13. Perturbation kinetic energy gain (maximum in the wall-normal direction) from LES for
$f=0$
and (a)
$TI=0\,\%$
, (b)
$TI=0.6\,\%$
, (c)
$TI=1.2\,\%$
, (d)
$TI=2.4\,\%$
, (e)
$TI=4.5\,\%$
and (f)
$TI=7\,\%$
. The
$\times$
markers indicate the
$(x,\beta )$
of the maximum gain of the streaks. The isolines of gain obtained from OPA are shown in black for
$(1,10,100,\ldots)$
.
Figure 14 compares the evolution of
$|u'|$
from LES and OPA. The latter is a linear method, and its results are scaled to match the amplitude of the
$TI=0.6\,\%$
case at a given
$x$
. There is good agreement between the two methods, considering the initial streak growth. This agreement improves with increasing
$f$
and decreasing
$\beta$
, as shown in figure 14(g), where there is a close match until
$x=0.2$
. In general, OPA also predicts well the streak growth for
$TI\leqslant 2.4\,\%$
(the scaled OPA curves for
$TI=1.2\,\%$
and
$2.4\,\%$
are not shown for a cleaner figure). The agreement between OPA and LES is worse for
$TI=4.5\,\%$
and
$7\,\%$
, since the latter displays slower growth, suggesting that the lift-up may not be the leading mechanism of streak generation in these cases. Note that the amplitude obtained from OPA keeps growing downstream, whereas that from LES saturates. Figure 15 also indicates close agreement between the
$|u'|$
profiles from LES and OPA at
$x=0.2$
. The agreement reduces towards the free stream as the LES profiles present a higher amplitude than those from OPA due to the influence of FST.
Figure 14. Wall-normal maximum of the absolute value of streamwise velocity perturbation computed with LES (solid lines) and OPA (circles).
Figure 15. Wall-normal profiles of the absolute value of streamwise velocity perturbation (normalised by the maximum) computed with LES (solid lines) and OPA (circles) at
$x=0.2$
.
3.3.3. Secondary instability and breakdown of streaks
Typical events of instability and nucleation of turbulent spots are studied. Figure 16 shows the
$Q$
-criterion isosurfaces at time
$T=0$
and
$T=0.1$
, preceding and during the turbulent breakdown. The earliest breakdown for
$TI=1.2\,\%$
occurs on a streak near
$z=0$
at
$T=0$
with a wavepacket present in the region denoted by
$W$
in figure 16(a). The streaks travel at a speed
$u_s \approx 0.67 u_e$
, close to the value of Vaughan & Zaki (Reference Vaughan and Zaki2011). Since
$u_s \approx 1$
, the streak is
$\Delta x \approx 0.1$
more downstream at
$T=0.1$
in figure 16(b), where the turbulent spot develops. The rear location of the wavepacket is fixed at
$x=0.38$
, while the streak travels through this point. This suggests an influence of TS waves on the streak breakdown since they reach maximum growth at this location, as depicted in figure 10(a). As seen in other regions along the span, TS waves without streaks lead to a downstream breakdown. The wavepacket disappears after the rear of the streak crosses
$x=0.38$
. For
$TI=2.4\,\%$
, figure 16(c) shows a pair of streaks at
$z=0.055$
at
$T=0$
, with the streamwise location of their trailing edge marked with a grey line. Figure 16(d) indicates that they evolve into a wavepacket downstream of
$x=0.27$
at
$T=0.1$
, suggesting a secondary instability of this dual configuration. The turbulence spot is significantly more upstream than the breakdown in other regions along the span. Regarding
$TI=4.5\,\%$
, streaks centred at
$z=0.04$
at
$T=0$
undergo turbulent breakdown of conical shape at
$T=0.1$
, as shown in figures 16(e) and 16(f). This case presents many narrow streaks, seemingly more susceptible to breakdown, and an elevated spanwise non-uniformity in the transition line.
Figure 16. Analysis of a typical turbulent breakdown of streaks for (a,b)
$TI=1.2\,\%$
, (c,d)
$TI=2.4\,\%$
and (e,f)
$TI=4.5\,\%$
visualised with the isosurfaces of
$Q$
-criterion (
$Q=100$
) coloured by the instantaneous streamwise velocity.
$T$
is the relative time between snapshots and
$T=0$
means different initial times for each
$TI$
. The wavepackets and streaks are shown with the symbols
$W$
and
$S$
. The grey lines in panel (c,d,e,f) indicate the streamwise location of the rear end of the analysed streak.
Figure 17 displays the structures of the modes analysed in figure 16. For
$TI=1.2\,\%$
in figure 17(a), a low-speed streak (grey) lies on the flank of a high-speed streak (black). The former eventually goes above the latter, triggering an instability in the low-speed streak, characterised by spanwise waves. This may indicate TS waves triggered by the wall-normal shear created by the streaks since the mean flow is only weakly unstable. The configuration evolves to a delta shape at the junction of high- and low-speed streaks, characteristic of inner modes (Nagarajan et al. Reference Nagarajan, Lele and Ferziger2007; Vaughan & Zaki Reference Vaughan and Zaki2011), which are predominantly of the varicose type (Dotto et al. Reference Dotto, Barsi, Lengani, Simoni and Satta2022). Even though the overview of the mode structure indicates an inner mode, analyses of the perturbation profiles at
$x=0.38$
,
$z=0.03$
suggest that the wavepacket responsible for the early breakdown in figure 16(a) is an outer mode. This indicates that the inner mode is present more upstream, while the outer mode dominates at the transition point. Regarding
$TI=2.4\,\%$
, figure 17(b) shows that a low-speed streak is beneath a high-speed streak. As the former rises to the top of the boundary layer, an outer varicose instability occurs over the low-speed streak, as shown by Brandt et al. (Reference Brandt, Schlatter and Henningson2004). Structures similar to lambda vortices appear and give rise to a turbulent spot (Perry et al. Reference Perry, Lim and Teh1981; Dotto et al. Reference Dotto, Barsi, Lengani, Simoni and Satta2022). Figure 17(c) presents the results for
$TI=4.5\,\%$
, where a low-speed streak has high-speed streaks on both flanks and beneath it. The low-speed streak develops oscillations in the spanwise and wall-normal directions, corresponding to an outer sinuous instability according to Brandt et al. (Reference Brandt, Schlatter and Henningson2004). The latter typically occurs due to the spanwise shear and inflection point. The low-speed streaks break down first in all cases.
Figure 17. Isosurfaces of streamwise velocity fluctuations showing the instabilities in figure 16. The positive and negative fluctuations are shown in black and grey: (a)
$TI=1.2\,\%$
at
$T=0$
with
$u'=\pm 0.03$
; (b)
$TI=2.4\,\%$
at
$T=0.1$
with
$u'=(-0.1,0.12)$
; (c)
$TI=4.5\,\%$
at
$T=0.1$
with
$u'=(-0.15,0.09)$
. The
$z$
-axis direction in panel (b) is reversed compared with that in panels (a) and (c).
The secondary stability analysis over cross-sectional planes of the instantaneous flows preceding the breakdowns discussed above follows Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015). A 2-D, local eigenvalue problem is justified as the streamwise flow variations are much slower than those in the wall-normal and spanwise directions. The spatial approach was employed, where downstream-propagating waves with a frequency
$f \in \mathbb {R}$
are amplified with spatial growth rate
$-\alpha _i$
. Appendix B presents further details. Figure 18(a) shows the time evolution of the growth rates at
$x=0.4$
for
$TI=1.2\,\%$
. This location is near the wavepacket trailing edge, fixed in space while the streak travels. The growth rates increase with time until
$T=0$
, followed by a drop due to the ensuing transition. Figure 18(b) presents the spectrum of perturbation kinetic energy from the LES at
$x=0.4$
for
$TI=1.2\,\%$
and indicates energetic disturbances close to the frequency predicted by the secondary stability analysis (LST) at peak amplification (
$f=120$
) at
$T=0$
. Figure 18(c) exhibits the growth rates on a plane travelling with the streak for
$TI=2.4\,\%$
. The amplification increases as the streak moves downstream, with an exceptionally high value at
$x=0.3$
,
$T=0.1$
, where the most unstable mode frequency drops, suggesting a mechanism change compared with more upstream locations. The spectrum in figure 18(d) displays several regions with highly energetic disturbances near the frequency of peak amplification (
$f=120$
) at
$x=0.3$
,
$T=0.1$
. Figure 18(e) portrays the growth rates for
$TI=4.5\,\%$
, revealing a rise in amplification as the streaks move downstream. At
$x=0.3$
,
$T=0.1$
, the frequency of the most unstable mode is
$f=115$
, which also lies near regions with high energy in the spectrum in figure 18(f). The fact that the frequencies of the most unstable modes in the stability analysis lie close to regions with high energy in the LES spectrum suggests that the investigated breakdowns may be general events.
Figure 18. Time evolution of the growth rates obtained with cross-sectional secondary linear stability theory (LST) (a) at
$x=0.4$
for
$TI=1.2\,\%$
, and following a streak for (c)
$TI=2.4\,\%$
and (e)
$TI=4.5\,\%$
. Spectra of perturbation kinetic energy at
$y=\delta ^*$
from LES (b) at
$x=0.4$
for
$TI=1.2\,\%$
, and at
$x=0.3$
for (d)
$TI=2.4\,\%$
and (f)
$TI=4.5\,\%$
.
Figure 19. Wall-normal profiles of the absolute value of streamwise velocity perturbation computed with LES (solid lines) and secondary stability analysis over the cross-planes considering the most unstable frequency (circles) at (a)
$z=0.005$
for
$TI=1.2\,\%$
, (b)
$z=0.052$
for
$TI=2.4\,\%$
and (c)
$z=0.041$
for
$TI=4.5\,\%$
.
Figure 19 shows the comparison between the profiles of the streamwise velocity perturbations extracted from the LES and those predicted by secondary stability analysis, considering the most unstable mode at each location and time. Panel (a) presents the results for
$TI=1.2\,\%$
. At
$T=-0.25$
, the profile at
$x=0.13$
indicates a streak, whereas that at
$x=0.38$
corresponds to a TS wave, well predicted by LST. This streak arrives at
$x=0.38$
at
$T=0$
, where the profile displays the characteristics of an outer mode in agreement with that obtained with secondary stability analysis for
$f=120$
. The phase speed is
$c_p=0.73 \, u_e$
, close to the
$c_p=0.75 \, u_e$
value obtained by Vaughan & Zaki (Reference Vaughan and Zaki2011) for outer modes. Moreover, the peak amplitude occurs at the location of the inflection point in the wall-normal profile of streamwise velocity, indicating that this instability is of the varicose type. Less amplified modes in the secondary stability analysis present
$c_p=0.54 \, u_e$
, which agrees with the phase speed of inner modes by Vaughan & Zaki (Reference Vaughan and Zaki2011). The streak has a lower initial amplitude at
$x=0.13$
,
$T=0$
. In this case, the mode at
$x=0.38$
,
$T=0.25$
presents an increased near-wall maximum due to the inner mode contribution and a reduced maximum due to the outer mode contribution.
Figure 19(b) presents the results for
$TI=2.4\,\%$
. At
$x=0.2$
,
$T=0$
, and
$x=0.25$
,
$T=0.05$
, the perturbation profiles correspond to outer modes. The secondary stability results predict these modes well and indicate a phase speed
$c_p=0.69 \, u_e$
, in the range the literature provides for outer modes. The maximum amplitude occurs near the location of the maximum wall-normal shear and inflection point (
$y/\delta ^*=1.6$
), which lie on the low-speed streak. Although the spanwise inflection point is near the mode maximum amplitude location, the spanwise shear is low compared with the wall-normal shear. These facts suggest a varicose type of instability. However, the mode characteristics change considerably at
$x=0.3$
,
$T=0.1$
, where the peak amplitude moves closer to the wall (
$y/\delta ^*=0.56$
), and the frequency of the most unstable mode according to LST drops relative to the outer modes upstream. The profile and phase speed
$c_p=0.54 \, u_e$
are typical of an inner mode (Vaughan & Zaki Reference Vaughan and Zaki2011). The inner mode at
$x=0.3$
,
$T=0.1$
is driven by the spanwise shear and presents a maximum amplitude at the location of the inflection point in the spanwise profile of streamwise velocity. These factors suggest a change of the most unstable mode from an outer varicose instability to an inner sinuous mode as the streak travels downstream due to the subsiding APG.
Figure 19(c) displays the results for
$TI=4.5\,\%$
. The profile at
$x=0.2$
,
$T=0$
indicates an outer mode well predicted by secondary stability analysis. The mode type remains the same at
$x=0.25$
,
$T=0.05$
and
$x=0.3$
,
$T=0.1$
, with phase speed
$c_p= 0.59 \, u_e$
. Furthermore, the maximum disturbance amplitude lies near the maximum spanwise shear and inflection point, suggesting a sinuous mode. The predominance of outer modes for
$TI=4.5\,\%$
agrees with the fact that the MFD stabilises the inner mode for increasing
$TI$
. This also indicates the occurrence of bypass transition.
3.4. Remarks on a low-frequency cutoff of the free stream turbulence
The results in § 3.2 indicate that the FST with scales employed here (
$\Lambda = O(10^{-2}{-}10^{-1})$
) effectively generate boundary-layer perturbations. However, due to these scales being smaller than those of atmospheric turbulence on real blades (
$\Lambda = O(10^1)$
(IEC 2006; Bertagnolio et al. Reference Bertagnolio, Madsen, Bak, Troldborg and Fischer2015), a cutoff scale, above which FST does not significantly influence transition, cannot be numerically established.
Low wavenumbers and, by Taylor’s frozen turbulence hypothesis, low frequencies in the FST are associated with large eddies or gusts, which modify the flow by a change in
$AoA$
and
$Re_c$
(Reeh Reference Reeh2014; Reeh & Tropea Reference Reeh and Tropea2015). Generally, one is typically interested in studying statistically stationary transition processes for a given flow. Unsteady effects are not substantial for
$\pi f^* c/U_\infty \lt 0.05$
(Leishman Reference Leishman2016), yielding
$f\lt 0.016=f_{c_1}$
here, where
$f_{c_1}$
can be considered the first cutoff frequency. The maximum atmospheric turbulence energy occurs at
$f^*=0.01$
Hz or
$f=f^*c/U_\infty =O(10^{-3})$
since typically,
$c/U_\infty =O(10^{-1})$
s in wind turbines. Thus, the blade sees a steady flow regarding the most energetic eddies in the atmospheric turbulence and
$f_{c_1}$
is a first frequency cutoff. Indeed, there is no significant time variation in the pressure distribution due to FST for
$TI\leqslant 2.4\,\%$
, as shown in Appendix C, which is the range of
$TI$
where TS waves may be relevant.
A higher frequency cutoff
$f_{c_2}$
for natural transition is estimated by noting that TS waves develop for
$2 \pi f^* \nu 10^6/U_\infty ^2 \geqslant F_L$
, where
$F_L=F_L(Re_c,\partial p/\partial x)$
is the lowest unstable reduced frequency (Schmid & Henningson Reference Schmid and Henningson2001). Moreover,
$F_L=F_L(TI)$
, where
$F_L$
is a monotonically increasing function of
$TI$
(see figure 10
b). Therefore, results for
$TI=0\,\%$
provide a lower bound for
$F_L$
, as sought for estimating
$f_{c_2}$
. The inequality is rewritten as
\begin{equation} f \geqslant f_{c_2} = (2\pi )^{-1} 10^{-6}\, F_L \, Re_c. \end{equation}
The latter can be written in terms of the streamwise wavelength (
$\lambda _x$
) as
\begin{equation} \lambda _x \leqslant 2 \pi 10^6 c_s F_L^{-1} Re_c^{-1}, \end{equation}
where
$c_s$
is the phase speed of TS waves. Due to scale reduction, the maximum wavelength of fluctuations exciting TS waves should be retro-estimated at the leading edge (
$\lambda _{LE}$
). Goldstein (Reference Goldstein1983) proposed a model for scale reduction for a flat plate in which
\begin{equation} \lambda _x \propto \left [2 \pi f^* x^* / U_\infty \right ]^{-1/2}. \end{equation}
The right-hand side of (3.3) evaluated at
$x^*=c$
gives
$10^3 F_L^{-1/2} Re_c^{-1/2}$
, which is equivalent to the maximum factor of reduction of
$\lambda _x$
along the aerofoil. Dividing both sides of (3.2) by this factor yields
\begin{equation} \lambda _{LE} \leqslant 2 \pi 10^3 c_s F_L^{-1/2} Re_c^{-1/2}=\lambda _{c_2}, \end{equation}
or in terms of the cutoff wavenumber,
\begin{equation} k_{c_2} = 2 \pi /\lambda _{c_2} = 10^{-3} c_s^{-1} F_L^{1/2} Re_c^{1/2}. \end{equation}
There are two unknowns in (3.5). The first is
$c_s$
, which may be taken as a lower bound for the phase speed (e.g.
$c_s=c_s^*/U_\infty =0.36$
). The second is
$F_L$
, a more complicated quantity to estimate as previous knowledge of the neutral curve is necessary. The naive assumption is to consider a fit for
$F_L$
extracted from the neutral curve (Schmid & Henningson Reference Schmid and Henningson2001) for Blasius flow (Blasius Reference Blasius1913), which disregards flow acceleration and curvature. The expression is given by
\begin{equation} F_L = 8.5775 \times 10^7 Re_x^{-1.1243}, \end{equation}
valid for
$Re_x=9\times 10^4$
–
$ 9\times 10^6$
. For
$Re_c=1\times 10^6$
,
$F_L=15.4$
and
$\lambda _{c_2}=0.58$
. The more refined method is to obtain the neutral curves of the wind turbine flow with linear stability analysis, as shown in figure 10(b). Due to an LSB for
$TI=0\,\%$
, which is not present for higher
$TI$
,
$F_L$
was extracted for
$TI=0.6\,\%$
, yielding
$F_L=17.7$
and
$\lambda _{c_2}=0.54$
. These results are close to those obtained by assuming the neutral curve for Blasius flow. Interestingly, the largest integral length scale of the FST
$\Lambda _{\overline {x}}=0.211$
is lower than
$\lambda _{c_2}=0.54$
, which may explain the efficiency of the FST in exciting the boundary layer. Note that the developed cutoff wavenumber estimate is only valid for TS waves and is based on the two-dimensional Goldstein model. Validation against experimental transition data is necessary to assess its validity in a complex flow under FST.
4. Conclusions
Laminar–turbulent transition on the suction surface of a section of the LM45.3p blade (
$20\,\%$
thickness), with chord Reynolds number of
$10^6$
and angle of attack of
$4.6^\circ$
, was studied with wall-resolved large eddy simulation (LES). The blade was subject to anisotropic free stream turbulence (FST) with turbulence intensities
$TI=0$
%, 0.6%, 1.2%, 2.4%, 4.5% and 7 %. The upper bound is selected based on the experimental observation by Özçakmak et al. (Reference Özçakmak, Madsen, Sørensen and Sørensen2020). The integral length scales correspond to scaled-down values from atmospheric measurements.
For
$TI=0\,\%$
, a laminar separation bubble (LSB) forms in the mid-chord region, and transition ensues via the breakdown of Kelvin–Helmholtz (KH) vortices. The lift-up mechanism is very robust at
$Re_c = 10^6$
so that even low levels of FST (e.g.
$TI=0.6\,\%$
) suppress the LSB due to the strong streak growth, unlike at
$Re_c = 10^5$
(Fava et al. Reference Fava, Lobo, Nogueira, Schaffarczyk, Breuer, Henningson and Hanifi2023c
). Two-dimensional Tollmien–Schlichting (TS) waves play a clear role in transition for
$TI=0.6\,\%$
. Considering
$TI=1.2\,\%$
and
$2.4\,\%$
, the flows become weakly unstable or stable to TS waves in the mean sense. The distortions generated by the streaks are essential for triggering instabilities leading to transition in these cases. The analysis of breakdown events indicates transition via inner and outer varicose modes for
$TI=1.2\,\%$
and
$2.4\,\%$
, driven by the wall-normal shear and inflectional velocity profile. However, the latter case also displays inner sinuous modes related to the spanwise shear and inflectional velocity profile preceding transition. In summary, the following conclusions can be drawn regarding
$0 \lt TI \leqslant 2.4\,\%$
:
-
(i) a linear receptivity occurs, and optimal perturbation theory describes the initial streak growth and profiles well;
-
(ii) transition occurs via two-dimensional TS waves for
$TI=0.6\,\%$
, and via predominantly varicose inner and outer modes for
$TI=1.2\,\%$
and
$TI=2.4\,\%$
; -
(iii) Mack’s correlation between the critical
$N$
factor and
$TI$
displays good agreement with the simulations; -
(iv) the critical disturbance kinetic energy is proportional to
$TI^{-1.80\pm 0.11}$
.
The
$TI=2.4\,\%$
case represents an intermediate state towards bypass transition. For
$TI\geqslant 2.4\,\%$
, the FST induces high-frequency boundary layer perturbations, although nonlinear receptivity of the leading streaks is not observed. Particularly for
$TI\geqslant 4.5\,\%$
, the streaks are generated with a high amplitude near the leading edge and grow little due to lift-up. The subsequent conclusions can be drawn for these cases:
-
(i) the scaling of the leading streamwise velocity perturbations with
$TI$
is sub-linear, although high-frequency disturbances are possibly nonlinearly generated in the boundary layer; -
(ii) bypass transition occurs with the dominance of breakdowns via outer sinuous modes, driven by the spanwise shear and inflectional velocity profile;
-
(iii) the turbulent breakdown is highly non-uniform in the spanwise direction;
-
(iv) the critical disturbance kinetic energy is proportional to
$TI^{-0.90\pm 0.16}$
, an exponent half of that of the modal regime.
The inception of bypass transition between
$TI=2.4\,\%$
and
$4.5\,\%$
agrees with
$TI=3\,\%$
found by Brandt et al. (Reference Brandt, Schlatter and Henningson2004) for the Blasius boundary layer and Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019) for a flat plate with flow separation. Finally, an estimate for a low-frequency/low-wavenumber cutoff for the turbulence spectrum affecting transition via TS waves is proposed. This allows for obtaining an effective
$TI$
from atmospheric turbulence measurements, compatible with wind tunnel data and Mack’s correlation.
Funding.
This work was possible with funding from StandUp for Wind, and HPC resources provided by the North-German Supercomputing Alliance (HLRN) and the Swedish National Infrastructure for Computing (SNIC).
Declaration of interests.
The authors report no conflict of interest.
Author contributions.
B.A.L. ran the simulations. T.C.L.F developed post-processing tools and carried out the analyses of the results with B.A.L. T.C.L.F. and B.A.L. wrote the paper with feedback from A.P.S., M.B., D.S.H. and A.H.
Data availability statement.
Data are available upon request.
Appendix A. Calculation of coefficients and errors of the
$N$
factor correlations
The calculation of the regression coefficients was performed according to Neter et al. (Reference Neter, Kutner, Nachtsheim and Wasserman1996) as
\begin{equation} (a,b)^T = ({{\mathbf X}^T\mathbf X})^{-1} {\mathbf X}^{T} {\mathbf N}, \end{equation}
where
${\mathbf N}$
is the vector of critical
$N$
factors from the LES and
\begin{eqnarray} {\mathbf X}=\left ( \begin{array}{cccc} 1 & -\ln {(TI_1/100)} \\ \vdots & \vdots \\ 1 & -\ln {(TI_n/100)}\\ \end{array} \right ). \end{eqnarray}
The errors of the coefficients
$a$
and
$b$
were estimated as
\begin{equation} (\sigma _a,\sigma _b)^T = \text {diag}\left [\sqrt {\hat {\sigma }^2 ({{\mathbf X}^T\mathbf X})^{-1}}\right ], \end{equation}
\begin{equation} \hat {\sigma }^2 = \frac {\sum _{i=1}^n (N_i-\tilde {N}_i)^2}{n-2}, \end{equation}
where
$N$
and
$\tilde {N}$
are the critical
$N$
factors from the LES and correlations, respectively. The procedure is applied to the low-
$TI$
range, which contains the points
$TI=0.6\,\%$
,
$1.2\,\%$
and
$2.4\,\%$
, yielding the coefficients
$a=2.63$
and
$b=1.8$
with corresponding errors
$\sigma _a=0.51$
and
$\sigma _b=0.11$
. For the high-
$TI$
range, containing the points
$TI=2.4\,\%$
,
$4.5\,\%$
and
$7\,\%$
, the coefficients are
$a=1.81$
and
$b=0.9$
with errors
$\sigma _a=0.50$
and
$\sigma _b=0.16$
, respectively.
Appendix B. Operators for secondary stability analysis
Figure 20. Mean pressure distribution on the suction side (black line) for (a)
$TI=0\,\%$
, (b)
$0.6\,\%$
, (c)
$1.2\,\%$
, (d)
$2.4\,\%$
, (e)
$4.5\,\%$
and (f)
$7\,\%$
. The shaded regions indicate one and two standard deviations around the mean.
The secondary stability analysis of the streaks was carried out with the two-dimensional eigenvalue problem over the cross-sectional planes (Siconolfi et al. Reference Siconolfi, Camarri and Fransson2015). The method was obtained by introducing the ansatz
${\mathbf q}^{\prime}={\hat {\mathbf q}}(y,z)\exp {(i \alpha x - i \omega t)}$
in the linearised Navier–Stokes equations (equations for the momentum in
$x$
,
$y$
,
$z$
, and continuity equation) with non-local terms dropped, where
${\mathbf q}^{\prime}(x,y,z,t)=[u^{\prime} \; v^{\prime} \; w^{\prime} \; p^{\prime}]^{T}$
is the vector of perturbations of pressure (
$p'$
), and streamwise (
$u'$
), wall-normal (
$v'$
) and spanwise (
$w'$
) velocities. Here,
$\alpha =\alpha _r+i\alpha _i$
, where
$\alpha _r$
is the streamwise wavenumber and
$\alpha _i$
is the spatial growth rate. Additionally,
$\omega =2 \pi f \in \mathbb {R}$
is the angular frequency, an input in the spatial framework employed here. The resulting generalised eigenvalue problem, solved for
$\alpha$
,
${\hat {\mathbf q}}$
and
$\alpha {\hat {\mathbf q}}$
, is given by
\begin{eqnarray} \left ( \begin{array}{cccc}\boldsymbol{ \mathcal {A}} &\quad \mathbf {0} \\ \mathbf {0} &\quad \mathbf {I}\\ \end{array} \right )\left ( \begin{array}{cccc} {\hat{ \mathbf{q}}} \\ \alpha {\hat{ \mathbf{q}}} \\ \end{array} \right )=\alpha \left ( \begin{array}{cccc} \boldsymbol{ \mathcal {B}} &\quad \boldsymbol{ \mathcal {C}} \\ \mathbf {I} &\quad \mathbf {0}\\ \end{array} \right )\left ( \begin{array}{cccc} {\hat{ \mathbf{q}}} \\ \alpha {\hat{ \mathbf{q}}} \end{array} \right ), \end{eqnarray}
\begin{eqnarray}\boldsymbol{\mathcal{A}} = \left( \begin{array}{cccc}\mathcal{A}_{11} & \overline{U}_y & \overline{U}_z & 0\\0 & \mathcal{A}_{22} & \overline{V}_z & \mathcal{D}_y\\0 & \overline{W}_y & \mathcal{A}_{33} & \mathcal{D}_z\\0 & \mathcal{D}_y & \mathcal{D}_z & 0 \\\end{array} \right),\end{eqnarray}
\begin{align}\boldsymbol{\mathcal{B}} = -\left( \begin{array}{cccc}i \overline{U} & 0 & 0 & i\\0 & i \overline{U} & 0 & 0\\0 & 0 & i \overline{U} & 0\\i & 0 & 0 & 0 \\\end{array} \right), \quad \boldsymbol{\mathcal{C}} = -\frac {1}{Re_c}\left ( \begin{array}{cccc} 1 &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 \\ \end{array} \right ), \end{align}
\begin{eqnarray} & \mathcal{A}_{11} = \overline{V}\mathcal{D}_y + \overline{W}\mathcal{D}_z - \frac{1}{Re_c}\left(\mathcal{D}_{yy} + \mathcal{D}_{zz} \right) - i \omega , \end{eqnarray}
\begin{eqnarray} & \mathcal{A}_{22} = \overline{V}\mathcal{D}_y + \overline{W}\mathcal{D}_z - \frac{1}{Re_c}\left(\mathcal{D}_{yy} + \mathcal{D}_{zz} \right) + \overline{V}_y - i \omega , \end{eqnarray}
\begin{eqnarray} & \mathcal{A}_{33} = \overline{V}\mathcal{D}_y + \overline{W}\mathcal{D}_z - \frac{1}{Re_c}\left(\mathcal{D}_{yy} + \mathcal{D}_{zz} \right) + \overline{W}_z - i \omega , \end{eqnarray}
where
$\mathcal {D}_y$
and
$\mathcal {D}_z$
are the derivatives in the
$y$
and
$z$
directions, obtained with a fourth-order finite-difference approximation;
$\mathbf {I}$
and
$\mathbf {0}$
are the identity and null matrices. The vector of base-flow variables is given by
${\overline {\mathbf q}}=[\overline {U} \; \overline {V} \; \overline {W} \; \overline {P}]^T$
, where
$\overline {P}$
is the pressure, and
$\overline {U}$
,
$\overline {V}$
and
$\overline {W}$
are the streamwise, wall-normal and spanwise velocities. Subscripts
$_y$
and
$_z$
indicate derivatives in the
$y$
and
$z$
directions, respectively. Here,
$i=\sqrt {-1}$
is the imaginary unity;
$Re_c$
is the Reynolds number. The boundary conditions are
$u',v',w'=0$
at
$y=0$
(wall),
$y\rightarrow \infty$
(free stream). Furthermore, a periodic boundary condition is imposed in the spanwise direction, such that
${\mathbf q}^{\prime} (x,y,z,t) = \mathbf {q'}(x,y,z+L_z,t)$
, where
$L_z$
is the spanwise width. The grid comprises 150 and 96 points in the
$z$
and
$y$
directions, respectively.
Appendix C. Non-stationarity of the pressure distributions due to the FST
Figure 20 shows the statistics of the pressure distribution on the aerofoil suction side. The mean distribution is shown with a black line, whereas the regions with one and two standard deviations around the mean are shown with shades of grey. There is little non-stationarity for
$TI\leqslant 2.4\,\%$
. However, the
$TI=7\,\%$
case presents a high degree of non-stationarity, with significant oscillations in the angle of attack and Reynolds number. However, this is not enough to cause instantaneous flow separation.
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