2.1. Flow solver

The compressible Navier–Stokes equations are solved. They are supplemented with the Peng–Robinson–Stryjek–Vera (PRSV) equation of state (Stryjek & Vera Reference Stryjek and Vera1986) and the Chung–Lee–Starling model (Chung et al. Reference Chung, Ajlan, Lee and Starling1988) for the transport properties for Novec649 vapour. The ideal gas law and Sutherland's model are used for air flows.

The in-house finite-difference code Musicaa is used to solve the governing equations. The inviscid fluxes are discretized by means of tenth-order centred differences, whereas fourth-order finite differences are used for the visco-thermal fluxes. For a prescribed dispersion error of $5 \times 10^{-4}$, the accuracy limit of the finite-difference scheme is 5.25 points per wavelength. A tenth-order selective filtering is applied to eliminate fluctuations at wavenumbers greater than the finite-difference scheme resolvability. The filter also acts as an implicit LES model, a strategy that has been demonstrated to be effective (Gloerfelt & Cinnella Reference Gloerfelt and Cinnella2019). A four-stage low-storage Runge–Kutta algorithm is used for time integration. To enlarge its stability limit and allow the use of larger time steps, a fourth-order implicit residual smoothing (IRS) method (Cinnella & Content Reference Cinnella and Content2016) is applied. The high-order IRS acceleration, with modifications to enhance robustness (Bienner et al. Reference Bienner, Gloerfelt, Yalçín and Cinnella2023), has been shown to be very efficient in terms of savings of computational time while maintaining a similar accuracy as the explicit method. Periodicity is enforced in the spanwise direction, and adiabatic no-slip conditions are applied at the wall. The non-reflecting conditions of Tam & Dong (Reference Tam and Dong1996) are imposed at the inlet, top and outflow boundaries. Finally, a sponge zone combining grid stretching and a Laplacian filter is added at the outlet.

The solver Musicaa has already been used in previous studies of wall-bounded flows of a heavy fluorocarbon (PP11) in compressible channel flows (Sciacovelli, Cinnella & Gloerfelt Reference Sciacovelli, Cinnella and Gloerfelt2017) and supersonic boundary layers (Sciacovelli et al. Reference Sciacovelli, Gloerfelt, Passiatore, Cinnella and Grasso2020). More recently, the LES strategy and time implication have been assessed carefully for the spatial development of TBLs of Novec649 (Gloerfelt et al. Reference Gloerfelt, Bienner and Cinnella2023). In particular, the fourth-order IRS yielded results in good agreement with the DNS and slightly better than the time-explicit LES, thanks to the less frequent application of the numerical filter.

2.2. Inlet synthetic turbulence

To create turbulent inlet conditions, we use a synthetic flow field based on random Fourier modes (RFMs) (Béchara et al. Reference Béchara, Bailly, Lafon and Candel1994). A homogeneous isotropic turbulent velocity field is generated as the sum of $N$ independent RFMs:

(2.1)\begin{equation} \boldsymbol{u}'_{in}(\boldsymbol{x},t)=\sum_{n=1}^N \hat{u}_n \cos(\boldsymbol{k}_n(\boldsymbol{x}-\bar{\boldsymbol{u}}t)+ \omega_n t+\psi_n)\boldsymbol{a}_n. \end{equation}

Given a logarithmic distribution for the wavenumbers $k_n=|\boldsymbol {k}_n|$, the mode amplitude $\hat {u}_n=\sqrt {2E(k_n)\,\Delta k_n}$ is prescribed from a von Kármán spectrum with Saffman viscous dissipation function and a bottleneck correction (Kang, Chester & Meneveau Reference Kang, Chester and Meneveau2003):

(2.2)\begin{align} E(k) &=1.453\,\frac{(u'_{rms})^2 k^4/k_e^5}{\exp(17/6\log(1+(k/k_e)^2))} \times\exp({-}1.5 c_K (k\eta)^2) \nonumber\\ &\quad \times \left[1+0.522\left(\frac{1}{\rm \pi}\arctan(10\log_{10}(k\eta)+12.58)+ \frac{1}{2}\right)\right] \end{align}

with $k_e=0.747/L_f$, $c_K=1.613$, and $\eta$ the Kolmogorov viscous scale. In (2.1), the phase term $\psi _n$, the wavenumber orientation $\boldsymbol {k}_n/k_n$ and the velocity direction $\boldsymbol {a}_n$ are random variables with given probability density functions. An unfrozen turbulent field is obtained by incorporating the convection velocity $\bar {\boldsymbol {u}}$ and the pulsation $\omega _n$, accounting for the temporal evolution of the perturbations. In the present simulations, we use $N=100$ RFMs. More details on the generation of RFMs can be found in Appendix A. This stochastic velocity $\boldsymbol {u}'_{in}$ is windowed by a van-Driest-type damping function to mimic the exponential damping of continuous Orr–Sommerfeld modes in the boundary layer. We use the function proposed by Pinto & Lodato (Reference Pinto and Lodato2019), defined as

(2.3)\begin{equation} \sigma_{damp}(y) = (1 - \exp({-y/(0.137 \times h)}))^{1000}, \end{equation}

where $h$ corresponds to the height where the coefficient is equal to $0.5$, and needs to be prescribed by the user. The synthetic turbulence is also damped at the upper corner of the inlet to prevent it from being injected too close to the upper boundary condition. It is entered using a modified Tam–Dong boundary condition (see Appendix in Gloerfelt et al. Reference Gloerfelt, Bienner and Cinnella2023). The T3A benchmark case (Roach & Brierley Reference Roach and Brierley1992) is used in Appendix B to validate the strategy based on synthetic turbulence. In particular, the injection height $h$ is a free parameter that can influence the precise location of transition. Nonetheless, FST-induced transition is well reproduced as long as the turbulence is injected above $\delta _{99,in}$. Thereafter, $h$ is kept equal to $1.21\delta _{99,in}$, for all cases (see figure 1a).

Figure 1. (a) Damping function $\sigma _{damp}$, identical among cases, and (b) initial kinetic energy spectra for the inlet synthetic turbulence. Low $L_f$ at $T_u=2.5\,\%$ (), 4 % () and 6.6 % (); high $L_f$ at $T_u=2.5\,\%$ () and 4 % ().

2.3. Operating conditions

The operating conditions are given in table 1. The selected configuration for Novec649 vapour corresponds to nominal operating conditions of the CLOWT facility (Reinker et al. Reference Reinker, Hasselmann, aus der Wiesche and Kenig2016), which is a continuously running a pressurized closed-loop wind tunnel using Novec649 in the high subsonic speed range. Two flow speeds are chosen for simulations with air: a low subsonic speed ($M=0.1$) to reproduce an incompressible-like case, and a high subsonic speed ($M=0.9$) corresponding to the Mach number used for Novec649. The fundamental derivative of gas dynamics, defined as $\varGamma =1+({\rho }/{c}){({\partial c}/{\partial \rho })}_s$ ($s$ being the entropy) governs the nonlinear thermodynamic behaviour of dense gases. It is equal to $(\gamma +1)/2$ for a perfect gas. At the conditions chosen for Novec649, $\varGamma$ has a value below 1, meaning that the flow operates in the dense-gas thermodynamic region, often encountered in ORC applications. A preliminary study of oblique transition of a Novec649 boundary layer was carried out for the same thermodynamic conditions (Gloerfelt et al. Reference Gloerfelt, Bienner and Cinnella2023), and assessed the use of PRSV/Chung–Lee–Starling models to describe the fluid behaviour.

Table 1. Thermodynamic and aerodynamic free-stream conditions (subscript $\infty$): $M$ denotes the Mach number, $U$ the velocity, $c$ the sound speed, $p$ the pressure, $T$ the temperature, $\rho$ the density, $\mu$ the dynamic viscosity and $\varGamma$ the fundamental derivative of gas dynamics.

2.4. Set-up of LES

Simulations are initialized with similarity solutions of the compressible laminar boundary layer with zero pressure gradient. The Reynolds number at the inlet is taken as $Re_{x,in} = 10^4$. In the Novec cases, five different FST conditions are analysed. For the lowest value of the integral length scale (hereafter referred to as ‘low-$L_f$’), three turbulent intensities (2.5 %, 4 % and 6.6 %) are tested. The ‘high-$L_f$’ cases consider an integral length scale target multiplied by a factor of ten for two values of $T_{u,in}$ (2.5 % and 4 %). For simulations in air, the low-$L_f$ and high-$L_f$ cases are reproduced at $T_u=4\,\%$ for two values of the Mach number (0.1 and 0.9). The different LES are summarized in table 2, with the corresponding grids and line legends. For the low-$L_f$ cases, the Reynolds number based on the integral length scale, $Re_{L_f}$, is the same as that used to reproduce the T3A experiment in Appendix B, and it is of similar magnitude to that used in most published FST transition experiments (Mans Reference Mans2007; Mandal, Venkatakrishnan & Dey Reference Mandal, Venkatakrishnan and Dey2010) and simulations (Brandt et al. Reference Brandt, Schlatter and Henningson2004; Nagarajan et al. Reference Nagarajan, Lele and Ferziger2007; Pinto & Lodato Reference Pinto and Lodato2019). The effective integral length scale $L_f$ at the inlet is determined by integration of the temporal autocorrelation function $f$ of the velocity signal:

(2.4)\begin{equation} L_f = U_\infty \int_0^{t} f(\tau) \,{\rm d}\tau, \end{equation}

where the upper limit of integration is set as the first zero crossing of $f$ (Kurian & Fransson Reference Kurian and Fransson2009). The Reynolds number based on the integral length scale is defined as $Re_{L_f} = L_f U_\infty \rho _\infty / \mu _\infty$. For the high-$L_f$ cases, the target value $Re_{L_f,t}$ has been multiplied by ten to investigate the effect of large incoming disturbances. The calculated value is $Re_{L_f,c} \sim 14\,000$, corresponding in the end to a factor of seven. The differences in turbulent scales are clearly visible in figure 2. Following Fransson & Shahinfar (Reference Fransson and Shahinfar2020), we also introduce an FST Reynolds number, $Re_{FST}=T_u\times Re_{L_f}$, given in table 2.

Table 2. Computational grid and FST properties of the simulations.

Figure 2. Snapshot of the streamwise velocity in the mid-span plane. Colour scale between $0.9 U_\infty$ (blue) and $1.1 U_\infty$ (red) for (a) low-$L_f$ and (b) high-$L_f$ Novec cases at $T_u=4\,\%$. For the high-$L_f$, high-$T_u$ case in (b), only half of the domain height is shown.

The computational grid resolution needs to comply with two major constraints. The first concerns the resolution of the near-wall boundary-layer flow, for which a target resolution in wall units $\Delta x^+\times \Delta y_{w}^+\times \Delta z^+\sim 30\times 1.0\times 11$ has been determined in a previous study (Gloerfelt et al. Reference Gloerfelt, Bienner and Cinnella2023). The second constraint is the resolution of the free-stream disturbances, which is driven by the values of the RFM wavenumber bounds $k_{min}$ and $k_{max}$. In the high-$L_f$ simulations, $k_{min}$ is such that the crossflow dimensions $L_y$ and $L_z$ are at least ${\sim }10 L_{f}$, which leads to $L_y/\delta ^*_{in} = 1100$ and $L_z/\delta ^*_{in} = 980$. The obtained $L_z$ is halved in the low-$L_f$ simulations, yielding $L_z\sim 50 L_{f}$. In particular, in the low-$L_f$ simulations, $L_z/\delta ^*_{in} = 490$ and $L_y/\delta ^*_{in} = 270$. We then choose $k_{max}\ge 7 k_{L_f}$, which gives a maximal value for the grid spacings to accurately resolve the injected modes (based on the resolvability limit of the current numerical scheme). The resulting resolutions given in table 2 are typical of wall-resolved LES. The streamwise resolution $\Delta x^+\sim 13$–$14$ for the low-$L_f$ cases is dictated by the FST, as shown by the inlet spectra in figure 1(b). On the contrary, when the integral length scale is relatively large, the boundary-layer resolution constraint is more restrictive and the streamwise spacing is $\Delta x^+\sim 27$–$28$. These resolutions have been shown to be sufficient to accurately capture a modal transition and the fully turbulent state of the boundary layer (Gloerfelt et al. Reference Gloerfelt, Bienner and Cinnella2023). To make sure that the receptivity to FST is not affected by grid resolution, the high-$L_f$, high-$T_u$ case at $M=0.9$ in air has been simulated on a grid with a DNS resolution $\Delta x^+\times \Delta y_{w}^+\times \Delta z^+\sim 10.5\times 0.8\times 4.3$, keeping the same RFM wavenumber bounds as the LES. Although transition occurs a little earlier (as also observed e.g. in Sayadi & Moin Reference Sayadi and Moin2012), the results, reported in Appendix C, remain nearly superimposed with DNS in the transitional region, which validates the present LES resolution. Furthermore, a second run of DNS has been performed with $k_{max}$ increased to match the limit imposed by the DNS grid to investigate the influence of an increase of the high-frequency content of the FST on the breakdown. The results are perfectly superimposed on the first DNS, validating the choice $k_{max}\ge 7 k_{L_f}$ used for the LES study. The sensitivity to the choice of $k_{min}$ has also been checked in Appendix D.

After the initial transient has been discarded, span- and time-averaged quantities are collected over $70\,000$ time steps for the cases at $M=0.9$, and $300\,000$ time steps for those at $M=0.1$. The evolutions of the averaged FST intensity are compared for the five Novec cases in figure 3(a). As expected, when $L_f$ is increased, the decay slope of the FST evolution is decreased. Therefore, in the high-$L_f$ cases, the turbulence intensity remains greater than 3 % and 2 % for the high- and low-$T_u$ cases, respectively, whereas the turbulence intensities for the low-$L_f$ cases are rapidly below $2\,\%$. In order to estimate the homogeneity of the FST, $T_u$ evolutions are plotted in figure 3(b) at three wall-normal locations in the free stream and compared with the averaged one for the low-$L_f$, high-$T_u$ Novec case. The curves match perfectly, meaning that good homogeneity is obtained for the injected synthetic turbulence. Furthermore, to assess the turbulence isotropy, $T_u$ evolutions for the three velocity components are reported in figure 3(c). Slight discrepancies can be observed close to the inlet, but the intensity of the three velocity components stays within 7 % of $T_u$, so the FST can be considered as approximately isotropic.

Figure 3. Evolution of the turbulence intensity for the different Novec cases: (a) low $L_f$ at $T_u=2.5\,\%$ (), 4 % () and 6.6 % (); high $L_f$ at $T_u=2.5\,\%$ () and 4 % (). FST homogeneity (b) and isotropy (c) of the Novec low-$L_f$, high-$T_u$ case. Here, $T_u=\sqrt {(u'^2+v'^2+w'^2)/3}/ U_\infty$, $T_{u,x} = \sqrt {u'^2}/ U_\infty$, $T_{u,y} = \sqrt {v'^2}/ U_\infty$ and $T_{u,z} = \sqrt {w'^2}/ U_\infty$.

3.1. Influence of FST characteristics

The friction coefficient evolutions in the dense-gas boundary layers are reported in figure 4. All the flows transition to a fully turbulent state in the computational domain, except for the low-$L_f$, low-$T_u$ case. Here, onset of transition denotes the location of minimal friction coefficient, and the end of the transition is the location of maximal friction coefficient. As expected, an increase of the inlet turbulence intensity always leads to a faster onset and termination of the transition, for both high- and low-$L_f$ cases. Similarly to Fransson & Shahinfar (Reference Fransson and Shahinfar2020), a double effect is observed by changing $L_f$. For $T_u=2.5\,\%$, an increase of the integral length scale moves the transition region upstream, whereas for $T_u=4\,\%$, an increase of $L_f$ extends the transition region. For $T_u=4\,\%$, the slope of the $C_f$ rise in the transition region with a high $L_f$ is smoother than with a low $L_f$, so that even if the transition onset occurs earlier, the end of the transition region is shifted downstream.

Figure 4. Distribution of the friction coefficient for the different cases with Novec649 vapour: low $L_f$ at $T_u=2.5\,\%$ (), 4 % () and 6.6 % (); high $L_f$ at $T_u=2.5\,\%$ () and 4 % (). The black solid line shows laminar correlation $C_{f,lam}=0.664/\sqrt {Re_x}$.

The streamwise evolutions of the maximum of $u_{rms}$, in figure 5(a), are often correlated with the streak growth. For the low-$L_f$, high-$T_u$ case, a fast increase of $u_{rms,max}$ is observed in the pre-transitional region, i.e. prior to the minimum of skin friction (at $Re_\theta \approx 220$), and can be associated with the transient energy growth in the early streak development. The growth rate then slows down before a secondary growth phase, associated with the eruption of turbulent spots (Nolan & Zaki Reference Nolan and Zaki2013), yields the peak at approximately $Re_\theta =340$. A slower growth of $u_{rms,max}$ in the pre-transitional region is obtained for the low-$L_f$, low-$T_u$ boundary layer, in accordance with the lower FST intensity. An inflection point is visible near $Re_\theta \sim 120$ for the low-$L_f$ cases with $T_u=2.5\,\%$ and 4 %. For the low-$L_f$ case with $T_u=6.6\,\%$, the growth is so fast that no inflection is noticeable before the main peak. For the high-$L_f$ cases, a first peak is visible in the pre-transitional region, just prior to the location of the inflection point of the low-$L_f$ cases. This first local maximum is not linked to the laminar streak growth, and is also present in the results extracted from the database of Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Duràn and Hickey2017). In their simulation, homogeneous isotropic turbulence (HIT) is injected at a height equal to 15$\delta _{99,in}$ with an inlet $Re_\theta =80$, explaining the offset with our curves. As their simulation was characterized by the appearance of low-speed $\Lambda$-shaped structures prior to laminar streaks, the early secondary peak can be associated with the growth of these structures, which could also be present in the high-$L_f$ simulations. In figure 5(b), the height for the maximum of $u_{rms}$ is approximately the same for all cases. The location $y(u_{rms,max})$ moves from values approximately $1.2\delta ^*\unicode{x2013}1.4\delta ^*$ in the pre-transitional region, as predicted by the optimal perturbation theory (Andersson, Berggren & Henningson Reference Andersson, Berggren and Henningson1999; Luchini Reference Luchini2000), towards a low level in the TBL, associated with the near-wall turbulent streaks. The evolutions are also in agreement with low-speed airflow results of Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Duràn and Hickey2017), except for a slightly lower altitude in the laminar region.

Figure 5. Evolution of (a) $u_{rms,max}/U_\infty$ and (b) its height $y(u_{rms,max})/\delta ^*$. Low $L_f$ at $T_u=2.5\,\%$ (), 4 % () and 6.6 % (); high $L_f$ at $T_u=2.5\,\%$ () and 4 % ().

Instantaneous top views of the streamwise velocity fluctuations at $y/\delta ^*_{in} = 3.1$ are shown in figure 6, using the same streamwise and spanwise extents to provide an overview of the structures present inside the boundary layer and a first qualitative comparison of the Klebanoff streaks. The streak formation and the presence of turbulent spots are clearly revealed for each case, except for the case $T_u=6.6\,\%$, which is highly disturbed. A first important observation is that the scales associated with Klebanoff modes are different between the low-$L_f$ and high-$L_f$ cases. At constant $L_f$ and to a lesser degree, the spanwise distribution of the laminar streaks is also affected by the FST intensity, in particular for the low-$L_f$ boundary layers. In the low-$L_f$, low-$T_u$ case, the spanwise scale of the Klebanoff modes increase as the boundary layer thickens. Moreover, the presence of low-speed $\Lambda$-shaped structures is clearly revealed in the instantaneous snapshots for the high-$L_f$ cases, as highlighted by the grey circles in the figure. These $\Lambda$-shaped structures bear a strong resemblance to the quasi-periodic spanwise structures described by Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008), and the $\Delta$-shaped structures in Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Duràn and Hickey2017). As in Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008), they are located prior to laminar streaks. For $T_u=2.5\,\%$, a $\Lambda$ structure begins to emerge at the location $(x/\delta ^*_{in},z/\delta ^*_{in})=(75,125)$, at approximately $Re_{\theta }=100$, where the first peak was identified in figure 5. The low-$L_f$, $T_u=6.6\,\%$ case, in figure 6(c) also seems to show several similar structures close to the inlet, but as this case is very disturbed, this is difficult to observe clearly. For the low-$L_f$ cases in figures 6(a,b), the visualizations do not clearly reveal similar structures near the inlet. This point is discussed in details at the end of § 4.2.

Figure 6. Instantaneous streamwise fluctuations in a wall-parallel plane at $y/\delta ^*_{in} = 3.1$. Low-$L_f$ cases (a) $T_u=2.5\,\%$, (b) $T_u=4\,\%$ and (c) $T_u=6.6\,\%$; and high-$L_f$ cases (d) $T_u=2.5\,\%$ and (e) $T_u=4\,\%$. The vertical dashed lines mark the location where $Re_{\theta }=100$. Some occurrences of $\Lambda$-shaped structures are marked by grey ellipses.

3.2. Compressibility and dense-gas effects

For $T_u=4\,\%$, additional simulations are carried out at the same conditions (same resolution) with air at Mach numbers $M=0.1$ and 0.9, to sort out compressibility and non-ideal gas effects on the FST-induced transition. The friction coefficient evolutions show the same trend for low (figure 7a) and high (figure 7b) integral length scales. First, comparing air boundary layers at the two flow speeds, an increase of the Mach number tends to slightly delay the transition and reduces the friction value in the turbulent state due to the friction heating at the wall. As reported by Marensi et al. (Reference Marensi, Ricco and Wu2017) and Ohno et al. (Reference Ohno, Selent, Kloker and Rist2023), temperature fluctuations are enhanced by compressibility, and streamwise velocity fluctuations are consequently reduced. This can be seen in figure 8, where thermal streaks present in the air boundary layer at $M=0.9$ (figure 8a) have an intensity $T_{rms,max}$ (${\sim }3\,\% T_\infty$) significantly higher than at $M=0.1$ (figure 8b). Note that thermal fluctuations for the dense-gas flow at $M=0.9$ are very low due to the high thermal capacity of the organic vapour. As a consequence, the $C_f$ levels of Novec649 simulations in the fully turbulent state collapse on those of air boundary layers at $M=0.1$ or of incompressible simulations (as noted in previous studies such as Gloerfelt et al. Reference Gloerfelt, Bienner and Cinnella2023). However, the $C_f$ curves of Novec simulations in the transitional region show that the use of an organic vapour delays the transition even more than the effects of compressibility. This can be explained by the excitation of internal degrees of freedom of the complex Novec649 molecules, that further reduces streamwise velocity fluctuations. This stabilizing effect is of the same nature as that due to the increase in thermal fluctuations, but is more significant. It was also noted in the boundary layer stability of Gloerfelt et al. (Reference Gloerfelt, Bienner and Cinnella2023), where the neutral curve for Novec649 at $M=0.9$ is contained in that for the same boundary layer in air. The evolutions of $u_{rms,max}$ for low and high $L_f$ are reported in figures 9(a,b), respectively. The hierarchy observed for transition onset is respected, i.e. velocity fluctuations in air at $M=0.1$ are slightly reduced in air at $M=0.9$, and further reduced in Novec at $M=0.9$ for both FST length scales. The curves are, however, very close for a given $L_f$, meaning that the compressibility and non-ideal gas effects are present but secondary compared with the effects of the FST length scale. It can be concluded that the observations made in § 3.1 about the influence of $L_f$ on transition are the same in the air and organic vapour flows.

Figure 7. Distribution of the friction coefficient for the (a) low-$L_f$ and (b) high-$L_f$ cases at $T_u=4\,\%$ for Novec649 at $M=0.9$ (circles) and for air at $M=0.1$ (diamonds) and 0.9 (triangles). The solid black line shows laminar correlation $C_{f,lam}=0.664/\sqrt {Re_x}$.

Figure 8. (a) Instantaneous temperature fluctuations $T'/T_\infty$ at $y/\delta ^*_{in}=3.1$ in the low-$L_f$, high-$T_u$ air $M=0.9$ case. (b) Evolution of $T_{rms,max}/T_\infty$ for the low-$L_f$ cases at $T_u=4\,\%$, for Novec at $M=0.9$ () and for air at $M=0.1$ () and 0.9 ().

Figure 9. Evolution of $u_{rms,max}/U_\infty$ for the (a) low-$L_f$, high-$T_u$ and (b) high-$L_f$, high-$T_u$ cases, for Novec649 at $M=0.9$ (circles) and for air at $M=0.1$ (diamonds) and 0.9 (triangles).

The similarities between the various flows is shown further by the instantaneous views of $u'$ in a wall-parallel plane in figure 10 (see supplementary movies). At a given free stream $L_f$, the laminar streaks retain at first sight similar spanwise distributions and are weakly affected by compressibility or non-ideal gas effects (this will be quantified in § 4.1). For high-$L_f$ cases, the same $\Lambda$-shaped structures are observed in air flows, meaning that their presence is linked to the FST characteristics rather than dense-gas or compressibility effects. Overall, at the selected thermodynamic conditions, the FST-induced transition is only slightly modified by the high-subsonic regime or the organic vapour thermo-physical properties.

Figure 10. Comparison of air and Novec transitional flows at $T_u=4\,\%$ with (a,c,e) low $L_f$ and (b,d,f) high $L_f$, for (a,b) Novec, (c,d) air $M=0.1$, and (e,f) air $M=0.9$. Instantaneous streamwise fluctuations in a wall-parallel plane at $y/\delta ^*_{in}=3.1$.

4.1. Laminar streaks

As some variability in the streamwise streak sizes was observed in figure 6, the influence of the FST characteristics on their spanwise scale is investigated. The distance $l_z$ that corresponds to a minimum of the spanwise correlation of the streamwise velocity, $R_{uu}$, can be related to the spanwise half-wavelength of the streaks (Fransson & Shahinfar Reference Fransson and Shahinfar2020). Here, $R_{uu}$ is evaluated in the transitional region at the height of the maximum $u_{rms}$, and time averaged. The evolution of $\lambda _z=2 l_z$ for the Novec649 cases, estimated using a cubic interpolation, is reported in figure 11(a). Some important differences between the cases are observed during transition, before all the curves tend to similar values in the TBL, except for the low-$L_f$, low-$T_u$ case, which does not fully transition in the domain. In particular, the latter case exhibits a significant increase in $\lambda _z$ from the inlet, with a spanwise scale eventually larger than the high-$L_f$, high-$T_u$ case. Figure 11(b) compares the Novec cases with air flows at $M=0.1$ and 0.9 for $T_u=4\,\%$. The results show that a similar spanwise scale is achieved for a given $L_f$, indicating that compressibility and dense-gas effects are of secondary importance.

Figure 11. (a) Evolution of spanwise wavelength of the streamwise streaks in the transitional region for the Novec649 cases, and (b) comparison between Novec and Air cases at $T_u=4\,\%$. Novec cases: low $L_f$ at $T_u=2.5\,\%$ (), 4 % () and 6.6 % (); high $L_f$ at $T_u=2.5\,\%$ () and 4 % (). Air $M=0.1$ cases: low $L_f$ () and high $L_f$ (). Air $M=0.9$ cases: low $L_f$ () and high $L_f$ (). The colours are lighter outside the transition region between $C_{f,min}$ and 60 % of $C_{f,max}-C_{f,min}$. In (a), crosses denote the positions $\gamma _{peak}=0.1$ and 0.5 for the Novec cases at $T_u=4\,\%$.

As the minima of the correlation functions vary greatly across the transitional boundary layer for the different cases, the region considered for the evaluation of $\lambda _z$ is restricted between the location of $C_{f,min}$ and the location at 60 % between $C_{f,min}$ and $C_{f,max}$. This region includes the locations where the intermittency function $\gamma _{peak}$ is $10\,\%$ and $50\,\%$ for the $T_u=4\,\%$ cases (see § 4.3 for the definition of $\gamma _{peak}$), as used in Fransson & Shahinfar (Reference Fransson and Shahinfar2020). Along this region, the spanwise distance between streaks remains relatively constant, except for the high-$L_f$, high-$T_u$ case, which exhibits significant variations due to the appearance of $\Lambda$-shaped structures prior to the laminar streaks. The hierarchy between the different cases is more apparent when restricted to this region. At constant $L_f$, an increase of $T_u$ leads to a decrease of $\lambda _{z}$, as in Fransson & Alfredsson (Reference Fransson and Alfredsson2003). Conversely, at constant $T_u$, an increase of $L_f$ leads to an increase of $\lambda _{z}$. Fransson & Shahinfar (Reference Fransson and Shahinfar2020) showed that $\lambda _{z}$ varies monotonically not with $T_u$ and $L_f$, but rather with a combination of these two FST parameters. They proposed an empirical correlation for the spanwise length, which was fitted to their extensive database and was given by

(4.1)\begin{equation} \lambda_{z,FS} = L_{f} T_{u} (\mathcal{D}_1\,Re_{FST}^{{-}1/\sqrt{2}} + \mathcal{D}_2)^2, \end{equation}

with $(\mathcal {D}_1,\mathcal {D}_2)=(186, 0.8)$ to match at ${\pm }10\,\%$ the measured values. The estimator $\lambda _{z,FS}$ is plotted as a function of $Re_{FST} = T_u \times Re_{L_f}$ in figure 12, along with the experimental results of Fransson & Shahinfar (Reference Fransson and Shahinfar2020) at $Re_{FST}$ between 135 and 561 (evaluated at an intermittency level $\gamma _{peak}$ of $10\,\%$). The measurements from Mamidala et al. (Reference Mamidala, Weingärtner and Fransson2022), which contain lower $Re_{FST}$ values (down to 57) and were obtained with an experimental set-up similar to that used by Fransson & Shahinfar (Reference Fransson and Shahinfar2020), are also reported. The mean values of $\lambda _z$ along the evaluation region in the current simulations are also displayed. As noted earlier, the length scale varies with $x$ for the high-$L_f$, high-$T_u$ cases, so to observe its influence on the results, $\lambda _z$ values are plotted with their variations in the inset. Values from other experimental or numerical studies of FST-induced transition are also reported. In the high-$Re_{FST}$ range shown in the inset of figure 12, even taking into account the large $\lambda _z$ variations, the present high-$L_f$ results are in good agreement with the estimator $\lambda _{z,FS}$ for incompressible-like air flows but also for the air and organic vapour boundary layers at $M=0.9$. The results of Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008) are also in fair agreement (${\sim }{-35}\,\%$), showing that the estimator works well for large $Re_{FST}$, at least for the data considered. The majority of data in the literature are concentrated around lower $Re_{FST}$ (${\le }150$) and lower $Re_{L_f}$ (${\le }3000$). In this region, containing only one point from the Fransson & Shahinfar data, the comparison is less convincing. The increase of $\lambda _z/(L_{f}\times T_{u})$ as $Re_{FST}$ decreases is captured correctly by the estimator, but $\lambda _z$ values predicted by (4.1) are twice the spanwise distance obtained in our low-$L_f$ cases. Our results are, however, in good agreement with the more recent experiment of Mamidala et al. (Reference Mamidala, Weingärtner and Fransson2022). Despite some scatter in the results from the literature, the general trend indicates that the correlation is less accurate for low $Re_{FST}$.

Figure 12. Spanwise wavelength of the streamwise streaks as a function of $Re_{FST}$. Novec cases: low $L_f$ at $T_u=2.5\,\%$ (red square), at $T_u=4\,\%$ (blue circle) and at $T_u=6.6\,\%$ (olive triangle); high $L_f$ at $T_u=2.5\,\%$ (yellow square) and at $T_u=4\,\%$ (purple circle) cases. At $T_u=4\,\%$, air $M=0.1$ for low-$L_f$ (blue diamond) and high-$L_f$ (magenta diamond) cases, and air $M=0.9$ for low-$L_f$ (blue down triangle) and high-$L_f$ (purple down triangle) cases. Regarding the literature results, $\lambda _z$ was either directly available (Jacobs & Durbin Reference Jacobs and Durbin2001; Brandt et al. Reference Brandt, Schlatter and Henningson2004; Mans Reference Mans2007; Ovchinnikov et al. Reference Ovchinnikov, Choudhari and Piomelli2008; Fransson & Shahinfar Reference Fransson and Shahinfar2020; Mamidala, Weingärtner & Fransson Reference Mamidala, Weingärtner and Fransson2022) or evaluated from $R_{uu}$ profiles (Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010; Muthu et al. Reference Muthu, Bhushan and Walters2021). The vertical bars in the inset show the variability in the transition region.

Turbulent breakdowns due to low-speed streak instabilities are observed in all the simulations. As the FST levels are relatively high, the streaks rapidly become distorted, making it difficult to distinguish clearly between the sinuous and varicose modes. Also, knowing that the symmetric and antisymmetric modes have been shown to combine in the case of streak transient growth (Hoepffner et al. Reference Hoepffner, Brandt and Henningson2005) or unsteady base streaks (Vaughan & Zaki Reference Vaughan and Zaki2011), no attempt has been made to classify the streak instabilities. An example of each mode is given below for the high-$L_f$, low-$T_u$ Novec case to show that despite the competition with another breakdown mechanism and relatively wider streaks, both modes are active with the larger FST length scale. An example of a sinuous breakdown is illustrated in figure 13 using three successive three-dimensional (3-D) views of the low-speed streak before the breakdown. In the first view, two quasi-streamwise vortices are observed on each side of the low-speed streak. In particular, the streak is flanked by a high-speed streak on only one side. This is similar to the one-sided sinuous mode described in Brandt et al. (Reference Brandt, Schlatter and Henningson2004), where a high-speed streak approaches a low-speed region on one side. In figure 13(c), the low-speed streak begins to be fully disrupted, prior to the emergence of a turbulent spot. A varicose-like breakdown is shown in figure 14. In figure 14(a), vortices, identified by the Q-criterion, are characterized by alternating V and $\Lambda$ structures joining in the middle of the low-speed streak, as described by Brandt et al. (Reference Brandt, Schlatter and Henningson2004). The $\Lambda$ vortices evolve into hairpin vortices in figure 14(b), and the low-speed streak is disrupted symmetrically relative to the centre of the streak.

Figure 13. Observation of a sinuous-like breakdown in the high-$L_f$, low-$T_u$ case at three successive times ($t^* = (t-t_0) U_\infty / \delta ^*_{in}$, with $t_0$ the time of the first snapshot). Isosurfaces of the streamwise velocity fluctuations: $u'=0.16 U_\infty$ (white), $-0.16 U_\infty$ (green). Isosurfaces of the Q-criterion are also depicted in light brown.

Figure 14. Observation of a varicose-like breakdown in the high-$L_f$, low-$T_u$ case at two successive times ($t^* = (t-t_0) U_\infty / \delta ^*_{in}$, with $t_0$ the time of the first snapshot). Isosurfaces of the streamwise velocity fluctuations: $u'=0.16 U_\infty$ (white), $-0.16 U_\infty$ (green). Isosurfaces of the Q-criterion are also depicted in light brown.

4.2. $\Lambda$-shaped structures

An important difference between the low-$L_f$ and high-$L_f$ cases is the presence of low-speed $\Lambda$-shaped structures in the latter. The formation of these $\Lambda$-shaped structures is illustrated in figure 15 through 3-D views in the high-$L_f$, low-$T_u$ Novec case. First, several quasi-spanwise vortices appear close to the wall due to the interaction of the boundary layer with the FST. Similarly to Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008), these structures are stretched in the streamwise direction, leading to the formation of $\Lambda$ vortices. The strong $\Lambda$ vortex in the upper right corner is associated with a low-speed patch of velocity that evolves further downstream in the $\Lambda$ structure. The $\Lambda$ vortex in the bottom of the images is linked to a one-leg $\Lambda$, whose head then connects to a downstream low-speed streak.

Figure 15. The $\Lambda$ structure formation in the high-$L_f$ Novec case with $T_u=2.5\,\%$ at four successive times ($t^* = (t-t_0) U_\infty / \delta ^*_{in}$, with $t_0$ the time of the first snapshot). Isosurfaces of the streamwise velocity fluctuations ($u'=0.12 U_\infty$ in white, $-0.12 U_\infty$ in green), and isosurface of the Q-criterion in light brown.

The breakdown of a low-speed $\Lambda$ structure is illustrated in figures 16(a,c,e). The $\Lambda$ vortex, flanking the low-speed structures, evolves into a hairpin vortex, the tips of which turn into the characteristic $\Omega$ shape. As observed in figures 16(b,d,f) in an $xy$-plane located in the middle of the low-speed structure, the breakdown imprints both the spanwise and wall-normal velocity components in the upper half region of the boundary layer, consistent with the observations in Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008) or more generally with descriptions in the literature of horseshoe vortices evolution and breakdown (see e.g. Bake, Meyer & Rist Reference Bake, Meyer and Rist2002).

Figure 16. (a,c,e) Observations of a $\Lambda$ structure undergoing turbulent breakdown in the high-$L_f$, high-$T_u$ case, with $z/\delta ^*_{in} \in [-11,1]$, at three successive times ($t^* = (t-t_0) U_\infty /\delta ^*_{in}$, with $t_0$ the time of the first snapshot). Isosurfaces of the streamwise velocity fluctuations ($u'=0.2 U_\infty$ in white, $-0.2 U_\infty$ in green), and isosurface of the Q-criterion in light brown. (b,d,f) Wall-normal (upper images) and spanwise (lower images) components of velocity fluctuations at the $z$ position denoted by the transparent white plane on the associated images (a,c,e), with levels ${\pm }0.07 U_\infty$ from black to white, and $x$- and $y$-axes aspect ratios 1.4, 1.2 and 1.0 for (b,d,f), respectively.

Interestingly, these structures are often organized obliquely in an $xz$-plane, as shown in figure 17(a). A similar organization can be observed in figure 16 of Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008) (not commented by the authors). As the $\Lambda$ structures break down into turbulence, an oblique turbulent band may form eventually. In the high-$L_f$, high-$T_u$ case, on a total of nine relatively well-defined oblique bands, the angle is estimated between $25^\circ$ and $40^\circ$, with an average value at approximately $30^\circ$. This point is discussed further in § 5.

Figure 17. Observation of $\Lambda$ structures in the high-$L_f$, high-$T_u$ Novec case: (a) organized in a oblique manner, and (b) leading to a turbulence band. Wall-normal velocity fluctuations ($v' = \pm 0.01 U_\infty$ , from black to white). Isolines of the Q-criterion projection around the wall-normal position of the plane plotted in red.

As reported in § 3, a first peak marks the evolution of $u_{rms,max}$ (see figure 5) in the large-scale FST simulations around the position where $\Lambda$ structures emerge, which is related to their development. Similarly, the inflection points, observed on $u_{rms,max}$ evolution for the low-$L_f$ cases at $T_u=2.5\,\%$ and 4 %, are an indication that similar structures may also be present in these simulations. However, no low-speed $\Lambda$-shaped structures are observed by inspection of snapshots of the streamwise fluctuations (e.g. figure 6). A more careful examination of 2-D and 3-D visualizations of the low-$L_f$, high-$T_u$ case also failed to reveal the presence of low-speed $\Lambda$-shaped structures, but did reveal the presence of spanwise and $\Lambda$ vortices close to the inlet, such as in figure 18. Relative to the high-$L_f$ simulations, the vortices are smaller and may not be strong enough to be associated with the low-speed $\Lambda$ structures.

Figure 18. Comparison of $\Lambda$ structures between (a) low-$L_f$ and (b) high-$L_f$ cases at $T_u=4\,\%$ for $x/\delta ^*_{in} \in [33,181]$ and $z/\delta ^*_{in} \in [-230,230]$. Isosurfaces of the streamwise velocity fluctuations ($u'=0.22 U_\infty$ in white, $-0.22 U_\infty$ in green), and isosurface of the Q-criterion in light brown.

4.3. Competition between the two breakdown mechanisms

To get more insights into the competition between these two breakdown scenarios, a laminar–turbulent discrimination is applied on the low-$L_f$ and high-$L_f$ cases in Novec at $T_u=4\,\%$ over the transitional region, to identify turbulent spot inception locations. The discrimination algorithm is adapted from Durovic (Reference Durovic2022) and applied on volume snapshots, allowing the spatial separation into laminar and turbulent regions in the boundary layer. A detailed description is provided in Appendix E. In particular, the peak of the intermittency distribution inside the boundary layer at each streamwise location can be retrieved, as plotted in figure 19(a). The intermittency function corresponds to the probability of the flow being turbulent at a given location. A dimensionless streamwise coordinate $\xi =(x-x_{tr})/\Delta x_{tr}$ ($x_{tr}$ being the position where $\gamma _{peak} = 0.5$, and $\Delta x_{tr}$ being the distance between $x_{\gamma _{peak} = 0.1}$ and $x_{\gamma _{peak} = 0.9}$) is used as in Fransson & Shahinfar (Reference Fransson and Shahinfar2020). The intermittency curves are in relatively good agreement with the correlation of Fransson & Shahinfar (Reference Fransson and Shahinfar2020), despite the presence of the two competing mechanisms in the high-$L_f$, high-$T_u$ case. The conditioned $u_{rms,max}$ evolution is plotted in figure 19(b) for the low-$L_f$, high-$T_u$ case, and in figure 19(c) for the high-$L_f$, high-$T_u$ case. The laminar-conditioned curve in the low-$L_f$, high-$T_u$ case is initially coincident with the unconditioned statistics before deviating around $Re_\theta \sim 250$. It does not exhibit the secondary growth observed in the unconditioned curve, related to the eruption of turbulent spots (Nolan & Zaki Reference Nolan and Zaki2013). By contrast, in the high-$L_f$, high-$T_u$ case, the laminar-conditioned $u_{rms}$ quickly deviates from the unconditioned data. This early deviation can be attributed to the turbulent breakdown of low-speed $\Lambda$-shaped structures near the inlet, which is consistent with the earlier transition onset reported on the $C_f$ evolution in figure 4 and the extension of the transitional region. While characterizing turbulence in transitional boundary layers with inlet $T_u$ of 2.5 % and 3 %, Marxen & Zaki (Reference Marxen and Zaki2019) reported for the standard-averaged $u_{rms,max}$ a peak larger than both the laminar- and turbulent-averaged values that the authors linked to the contribution of the change in mean velocity to the total stress. Peaks were also present in the different simulations of Nolan & Zaki (Reference Nolan and Zaki2013) with $T_u$ of 3 %, except for their case that transitioned the fastest. Interestingly, such peak is, however, absent in figure 19, but is present on the high-$L_f$, low-$T_u$ case (not shown here for brevity), which corresponds to a lower $T_u=2.5\,\%$. This could be a consequence of the higher $T_u$ in our low-$L_f$, high-$T_u$ and high-$L_f$, high-$T_u$ simulations, and/or the fact that these cases transition relatively quickly.

Figure 19. (a) Peak intermittency evolutions for the low-$L_f$, high-$T_u$ (solid blue) and high-$L_f$, high-$T_u$ (dashed purple) cases, and correlation of Fransson & Shahinfar (Reference Fransson and Shahinfar2020) (). Evolution of $u_{rms,max}/U_\infty$ for unconditioned and conditioned statistics for (b) low-$L_f$, high-$T_u$ and (c) high-$L_f$, high-$T_u$ cases in Novec.

Based on the laminar–turbulent discrimination, the turbulent spots are tracked in the Novec cases with $T_u=4\,\%$. First, using a snapshot of the $xz$-plane from the binary segmentation, an estimation of the $xz$ location is obtained for each spot. Then the evolution of the spot footprint in the different directions is reconstructed, and the 3-D locations of the spots are determined. A total of $N_s=327$ and 381 spots are counted for the low-$L_f$, high-$T_u$ and high-$L_f$, high-$T_u$ cases, respectively. When divided by the spanwise extent of the computational domain, the comparison of $N_s/L_z$ clearly shows a more intense turbulent spot production in the low-$L_f$, high $T_u$ case, consistent with its shorter transition length. The streamwise and wall-normal histograms of spot inception locations are reported in figure 20. The histograms are normalized so that the sum of the bins is equal to 100 %. The standard shape for spot nucleation rate is recovered for the streamwise distributions of the low-$L_f$ simulation (Kreilos et al. Reference Kreilos, Khapko, Schlatter, Duguet, Henningson and Eckhardt2016; Dellacasagrande et al. Reference Dellacasagrande, Lengani, Simoni, Pralits, Durovic, Hanifi and Henningson2021). Increasing $L_f$ gives a more spread distribution, in line with the smoother and wider transition length in the high-$L_f$ case. Interestingly, the streamwise distribution in the high-$L_f$ case has two maxima, one of which is very close to the inlet and contributes to the spreading of the transition region. The second maximum can be related to the turbulent breakdown of laminar streaks, while the first is due to the presence of $\Lambda$-shaped turbulent spot precursors. Due to their large number, spots are not directly categorized as being linked to streak instability or $\Lambda$ structures. Instead, the first three bars of the spot inceptions streamwise location in the high-$L_f$ case, which encompass the first peak and represent spots associated mostly with $\Lambda$-shaped turbulent spot precursors, are represented in orange. Conversely, the spots in green are mostly associated with streak instability. Overall, we observe that $\Lambda$ precursors account for about a third of the total number of spots. No significant differences are observed between the simulations for their wall-normal distribution (figure 20b). The maxima of the distributions, associated with streak instabilities, are located at approximately $y/\delta _{99} \sim 0.4$–0.5, i.e. slightly lower than the value (${\sim }0.55$) in the ZPG case of Nolan & Zaki (Reference Nolan and Zaki2013). This reveals that the outer instability is dominant in the simulations, and that increasing $L_f$ does not tend to promote inner instabilities (Vaughan & Zaki Reference Vaughan and Zaki2011). Similarly, the location of the peak of the wall-normal distribution of the orange spots, associated with $\Lambda$ structures in the high-$L_f$, high-$T_u$ case, is located in the outer region (${\sim }0.6$), which is consistent with the observations of Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008) and the turbulent breakdown mechanism described in figure 16.

Figure 20. Normalized distributions of (a) streamwise and (b) wall-normal positions of the spot inceptions. The solid lines in (a) represent the intermittency function, and the horizontal dashed lines in (b) denote the boundary-layer thickness $\delta _{99}$. In the high-$L_f$, high-$T_u$ case, the first three bars of the spot inceptions streamwise location in (a), which are associated with the first peak, are represented in orange in order to observe their vertical position in the boundary layer in (b).

The competition between these two mechanisms is analysed further by looking at the distribution of the laminar streaks amplitude. Using a detection algorithm (Nolan & Zaki Reference Nolan and Zaki2013), the streamwise streaks are identified in the laminar region (see Appendix E), and represented as a collection of points, each characterized by a position and a value of $u'$. In figure 21, the repartitions of the streak amplitude $u'$ along the streamwise direction, smoothed by a 2-D kernel density estimator and normalized by the integral, are shown for the $T_u=4\,\%$ Novec cases. A distribution similar to that in Nolan & Zaki (Reference Nolan and Zaki2013) is obtained for the low-$L_f$, high-$T_u$ case, where only the classical streak instability mechanism is present. The distribution of streak amplitudes spreads out quickly in the pre-transitional region to reach high intensities, which then remain relatively constant. In the two cases, very intense low-speed and high-speed streaks are present, with the 1 % most intense streaks (marked by the 99 % line) reaching amplitudes as high as 40 % of $U_\infty$. Even if the values of $u_{rms,max}$ (represented by the cyan dashed line) generally underestimate the low-speed streak amplitudes (Nolan & Zaki Reference Nolan and Zaki2013), the median distribution and the $u_{rms,max}$ evolution are in fairly good agreement. The amplitudes of the low-speed and high-speed streaks in closest proximity of the turbulent spot locations (see figure 20) are identified by black circles in figure 21. In the low-$L_f$ case, the turbulent spots are associated with streak amplitudes $u'/U_\infty \sim 0.2$–0.4, consistent with the values found in Nolan & Zaki (Reference Nolan and Zaki2013). The vast majority of the associated streaks lie within the median and the 99th percentile of the streak population, well above the laminar-conditioned $u_{rms,max}$ evolution, which peaks at approximately 0.15 (see figure 19). The same observations can be made in the high-$L_f$ simulation for the turbulent spots located beyond $Re_x=80 \times 10^3$, which are mainly associated with streak instabilities. Closer to the inlet, however, the amplitude distribution shows a different shape in the high-$L_f$ case. The median distribution of the low-speed streaks (denoted by 50 %) quickly reaches a first level and then rises to a second level along the transition region. As the low-speed $\Lambda$ shapes are captured by the detection algorithm, the initial small peak in the pre-transitional region is reflected in the median amplitude distribution. This gives an indication of how intense the low-speed $\Lambda$ structures leading to turbulent breakdowns can be, as the turbulent spots located below $Re_x = 80 \times 10^3$ are associated with amplitudes greater than $0.4 U_\infty$ for a non-negligible fraction. Moreover, while $\approx$10 % of turbulent spots can be linked to the most extreme 1 % of streak population amplitudes in the low-$L_f$, high-$T_u$ case, this percentage increases to 16 % in the high-$L_f$, high-$T_u$ case, mostly due to the $\Lambda$ structures, which tend to favour such events.

Figure 21. The 2-D density function of the streak amplitude, plotted between the inlet and the station where $\gamma _{peak}=90\,\%$. Darker colours denote higher density; colour levels between 0 and $3 \times 10^{-7}$ (low-$L_f$, high-$T_u$) and $2.6 \times 10^{-7}$ (high-$L_f$, high-$T_u$). Black circles indicate the positive and negative streak amplitudes in closest vicinity to the turbulent spot inception locations.