A collection of secondary instability calculations in streaky boundary layers is presented. The data are retrieved from well-resolved numerical simulations of boundary layers forced by free-stream turbulence (FST), considering different geometries and FST conditions. The stability calculations are performed before streak breakdown, taking place at various $Rey_x$ the Reynolds number based on the streamwise coordinate. Despite the rich streak population of various sizes, it is found that breaking streaks have similar aspect ratios, independently of the streamwise position where they appear. This suggests that wider streaks will break down further downstream than thinner ones, making the appearance of secondary instabilities somewhat independent of the streak's wavelength. Moreover, the large difference in the integral length scale among the simulations suggests that this aspect ratio is also independent of the FST scales. An explanation for this behaviour is provided by showing that these breaking streaks are in the range of perturbations that can experience maximum transient growth according to optimal disturbance theory. This could explain why, at a given streamwise position, there is a narrow spanwise wavelength range where streak breakdown is more likely to occur.

This study employs spectral proper orthogonal decomposition (SPOD) on direct numerical simulation data from a low-pressure turbine (LPT) operating under high freestream turbulence levels. The impacts of upstream wakes on the transition process are assessed by considering both cases with and without wakes, modeled by a moving cylinder placed upstream of the LPT blade. In the absence of upstream wakes, the SPOD eigenvalues decreases almost monotonically as frequency increases. At high frequencies, the spectra reveal a broadband interval with minimal elevation, corresponding to the Kármán vortex streets formed downstream of the blade's trailing edge. The SPOD modes in this inflow condition show fully attached boundary layers across the entire blade, suggesting that the boundary layers may be transitional. When subjected to upstream wakes, however, the SPOD spectra display several intense peaks linked to the wake passage frequencies. The associated SPOD modes reveal turbulent spots and lambda vortices on the rear suction side of the blade, typical indicators of turbulent boundary layers. Between the fundamental passage frequency and its harmonics, a series of tones emerge, representing the Doppler-shifted wakes. Triadic interactions between modes involving upstream wakes and their translation induce a cascade of these intermediate components, as verified by the bispectrum map. The SPOD modes capture interactions of structures carried by upstream wakes and the freestream flow with the blade boundary layers, manifested as low- and high-velocity streaks whose breakdown promotes the transition. High-frequency modes describe coherent structures break down into the vortex streets at the trailing edge.

Wall-resolved large eddy simulations of the flow on a rotating wind turbine blade section are conducted to study the rotation effects on laminar-turbulent transition on the suction surface. A chord Reynolds number of 1x10(5) and angles of attack (AoA) of 12.8 degrees, 4.2 degrees, and 1.2 degrees are considered. Simulations with and without rotation are performed for each AoA. For AoA=12.8 degrees, rotation increases the reverse flow from 7% of the free-stream velocity in the non-rotating case to 16% of it in the rotating case in the laminar separation bubble (LSB), triggering an oblique instability mechanism in the latter, leading to a faster breakdown to small-scale turbulence. However, rotation delays transition and reattachment in 3%-4% of the chord due to the acceleration of the boundary layer upstream of the LSB, which is subject to a strong adverse pressure gradient (APG), stabilizing Tollmien-Schlichting (TS) waves. Regarding AoA=4.2 degrees and 1.2 degrees, rotation slightly decelerates the attached boundary layer since the APG is very mild but accelerates the separated flow downstream, stabilizing Kelvin-Helmholtz (KH) modes. This mitigates the oblique instability mechanism and slows down the breakdown of KH vortices in the rotating case. In these cases, the transition location is little affected by rotation, possibly due to a rotation-independent absolute instability. Rotation also generates a spanwise tip-flow in the LSB for AoA=4.2 degrees and 1.2 degrees, which is highly unstable and triggers stationary and traveling crossflow modes. Nevertheless, the amplitudes of these modes remain too low to trigger transition.

Under the action of free-stream turbulence (FST), elongated streamwise streaky structures are generated inside the boundary layer, and their amplitude and wavelength are crucial for the transition onset. While turbulence intensity is strongly correlated with the transitional Reynolds number, characteristic length scales of the FST are often considered to have a slight impact on the transition location. However, a recent experiment by Fransson and Shahinfar [J. Fluid Mech. 899, A23 (2020)] shows significant effects of FST scales. They found that, for higher free-stream turbulence levels and larger integral length scales, an increase in the length scale postpones transition, contrary to established literature. Here, by performing well-resolved numerical simulations, we aim at understanding why the FST integral length scale affects the transition location differently at low- and high turbulence levels. We found that the integral length scales in Fransson and Shahinfar's experiment are so large that the introduced wide streaks have substantially lower growth in the laminar region, upstream of the transition to turbulence, than the streaks induced by smaller integral length scales. The energy in the boundary layer subsequently propagate to smaller spanwise scales as a result of the nonlinear interaction. When the energy has reached smaller spanwise scales, larger amplitude streaks results in regions where the streak growth are larger. It takes longer for the energy from wider streaks to propagate to the spanwise scales associated with the breakdown to turbulence, than for those with smaller spanwise scales. Thus, there is a faster transition for FST with lower integral length scales in this case.

Thin airfoil dynamic stall at moderate Reynolds numbers is typically linked to the sudden bursting of a small laminar separation bubble close to the leading edge. Given the strong sensitivity of laminar separation bubbles to external disturbances, the onset of dynamic stall on a NACA0009 airfoil section subject to different levels of low-amplitude free stream disturbances is investigated using direct numerical simulations. The flow is practically indistinguishable from clean inflow simulations in the literature for turbulence intensities at the leading edge of Tu = 0.02 %. At slightly higher turbulence intensities of Tu = 0.05 %, the bursting process is found to be considerably less smooth and strong coherent vortex shedding from the laminar separation bubble is observed prior to the formation of the dynamic stall vortex (DSV). This phenomenon is considered in more detail by analysing its appearance in an ensemble of simulations comprising statistically independent realisations of the flow, thus proving its statistical relevance. In order to extract the transient dynamics of the vortex shedding, the classical proper orthogonal decomposition method is generalised to include time in the energy measure and applied to the time-resolved simulation data of incipient dynamic stall. Using this technique, the dominant transient spatiotemporally correlated features are distilled and the wave train of the vortex shedding prior to the emergence of the main DSV is reconstructed from the flow data exhibiting dynamics of large-scale coherent growth and decay within the turbulent boundary layer.

The linear temporal stability of the fully developed pulsatile flow in a torus with high curvature is investigated using Floquet theory. The baseflow is computed via a Newton-Raphson iteration in frequency space to obtain basic states at supercritical Reynolds numbers in the steady case for two curvatures, δ=0.1 and 0.3, exhibiting structurally different linear instabilities for the steady flow. The addition of a pulsatile component is found to be overall stabilizing over a wide range of pulsation amplitudes, in particular for high values of curvature. The pulsatile flows are found to be at most transiently stable with large intracyclic growth rate variations even at small pulsation amplitudes. While these growth rates are likely insufficient to trigger an abrupt transition at the parameters in this work, the trends indicate that this is indeed likely for higher pulsation amplitudes, similar to pulsatile flow in straight pipes. At the edge of the considered parameter range, subharmonic eigenvalue orbits in the local spectrum of the time-periodic operator, recently found in pulsating channel flow, have been confirmed also for pulsatile flow in toroidal pipes, underlining the generality of this phenomenon.

Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free-stream turbulence are considered in the present work. Within an input-output model, where nonlinear terms of the incompressible Navier-Stokes equations are treated as an external forcing, we manage to separate inputs related to perturbations coming through the intake of the numerical domain, whose evolution represents a linear mechanism, and the volumetric nonlinear forcing due to triadic interactions. With these, we perform the full reconstruction of the statistics of the flow, as measured in the simulations, to quantify pairs of wavenumbers and frequencies more affected by either linear or nonlinear receptivity mechanisms. Inside the boundary layer, different wavenumbers at near-zero frequency reveal streaky structures. Those that are amplified predominantly via linear interactions with the incoming vorticity occur upstream and display transient growth, while those generated by the nonlinear forcing are the most energetic and appear in more downstream positions. The latter feature vortices growing proportionally to the laminar boundary layer thickness, along with a velocity profile that agrees with the optimal amplification obtained by linear transient growth theory. The numerical approach presented is general and could potentially be extended to any simulation for which receptivity to incoming perturbations needs to be assessed.

A global transient linear stability analysis of the three-dimensional time-dependent flow around an aerofoil undergoing small-amplitude pitching motion is performed using the optimally time-dependent (OTD) framework. The most salient linear instabilities associated with the instantaneous basic state are computed and tracked over time. The resulting OTD modes reflect the variations in the basic state and can be used as predictors of its spatial and temporal evolution, including the formation of a laminar separation bubble and its gradual spanwise modulation via primary global instability, leading to secondary instability and finally rapid breakdown to turbulence. The study confirms and expands upon earlier stability analyses of the same case based on the local properties of spanwise averaged velocity profiles in the bubble that predicted the onset of absolute instability soon followed by rapid breakdown of the separation bubble. The three-dimensional structure of the most unstable OTD mode is extracted, which compares well with both the locally absolutely unstable mode and the evolution of the basic state itself.

We study the stability of a zero-pressure gradient boundary layer subjected to free-stream disturbances by means of local stability analysis. The dataset under study corresponds to a direct numerical simulation (DNS) of a flat plate with a sharp leading edge in realistic wind tunnel conditions, with a turbulence level of 3.45 % at the leading edge. We present a method to track the convective evolution of the secondary instabilities of streaks by performing sequential stability calculations following the wave packet, connecting successive unstable eigenfunctions. A scattered nature, in time and space, of secondary instabilities is seen in the stability calculations. These instabilities can be detected before they reach finite amplitude in the DNS, preceding the nucleation of turbulent spots, and whose appearance is well correlated to the transition onset. This represents further evidence regarding the relevance of secondary instabilities of streaks in the bypass transition in realistic flow conditions. Consistent with the spatio-temporal nature of this problem, our approach allows us to integrate directly the local growth rates to obtain the spatial amplification ratio of the individual instabilities, where it is shown that instabilities reaching an -factor in the range [2.5,4] can be directly correlated to more than 65 % of the nucleation events. Interestingly, it is found that high amplification is not only attained by modes with high growth rates, but also by instabilities with sustained low growth rates for a long time.

The boundary-layer stability on a section of a rotating wind turbine blade with an FFA-W3 series aerofoil at a chord Reynolds number of 3 × 105, with varying rotation and radii, is studied with direct numerical simulations and linear stability analyses. Low rotation does not significantly affect transition in the outboard blade region. The relative insensitivity to rotation is due to a laminar separation bubble near the leading edge, spanwise-deformed by a primary self-excited instability, promoting the secondary absolute instability of the Kelvin–Helmholtz (KH) vortices and rapid transition. Moderate increases in rotation, or moving inboard, stabilise the flow by accelerating the attached boundary layer and possibly inducing competition between cross-flow and KH modes. This delays separation and transition. Initially, for high rotation rates or radial locations close to the hub, transition is delayed. Nevertheless, strong stationary and travelling cross-flow modes are eventually triggered, spanwise modulating the KH rolls and shifting the transition line close to the leading edge. Cross-flow velocities as high as 56 % of the free stream velocity directed towards the blade tip are reached at the transition location. For radial locations farther from the hub, the effective angle of attack is decreased, and cross-flow transition occurs at lower rotation rates. The advance or delay of the transition line compared with a non-rotating configuration depends on the competing rotation effects of stabilising the attached boundary layer and triggering cross-flow modes in the separation flow region.