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.

Resolvent analysis has found applications in several areas of fluid mechanics, providing physical insight into both laminar and turbulent flows. In spite of such fact, the global (3D) resolvent is computationally expensive, which limits the size of the domain and the Reynolds number of the flows which can be considered. In this work, we derive a parabolic resolvent approach, which enables a significant increase in the computational efficiency of the calculation, for streaky structures in boundary layer flows. The computational speedup depends on the size of the problem and could be of more than one order of magnitude for the same accuracy as the global calculation. The method is derived based on an optimiza-tion method via the Lagrange multipliers over the linearized boundary layer equations and it is coupled to a Krylov-Arnoldi decomposition to the computation of suboptimals. The application of the method is exemplified for two problems: a Falkner-Skan boundary layer, where we obtain trends for both the optimals and suboptimals, and a turbulent boundary layer, where characteristics such as the double peak in the spectrum and the characteristic inner and outer length scales can be recovered when a variable eddy viscosity is considered. In both cases, a scaling is found for the dominant gain, given in terms of the fourth power of the Reynolds number, defined in terms of the relevant scale for the problem, the displacement thickness, and the modified Rotta-Clauser parameter for the laminar and turbulent boundary layers, respectively. For the laminar case, we further demonstrate that a forcing limited to the free-stream region is capable of generating streaky structures inside the boundary layer, a relevant feature for free-stream turbulence-induced transition.

This investigation deals with the numerical implementation of a data-driven method for reactive control of the boundary-layer over a NACA0008 airfoil. The aim of this work is to evaluate the performance of controller in damping the flow disturbances and its efficiency in delaying laminar-turbulent transition. We focus our attention on the bypass transition scenario caused by free-stream turbulence. In this scenario, the perturbations in the wing boundary-layer develop into streaky structures. We show that this data-driven method is effective in decreasing the wall shear stress and disturbance energy at the objective location, and this damping is sustained downstream of the objective location. However, further downstream, the fluctuations grow again reaching amplitudes similar to those in the uncontrolled case.

This work deals with the closed-loop control of streaky structures induced by free-stream turbulence (FST), at the levels of 3.0% and 3.5 %, in a zero-pressure-gradient transitional boundary layer, by means of localized sensors and actuators. A linear quadratic Gaussian regulator is considered along with a system identification technique to build reduced-order models for control. Three actuators are developed with different spatial supports, corresponding to a baseline shape with only vertical forcing, and to two other shapes obtained by different optimization procedures. A computationally efficient method is derived to obtain an actuator that aims to induce the exact structures that are inside the boundary layer, given in terms of their first spectral proper orthogonal decomposition (SPOD) mode, and an actuator that maximizes the energy of induced downstream structures. All three actuators lead to significant delays in the transition to turbulence and were shown to be robust to mild variations in the FST levels. Integrated total drag reductions observed were up to 21% and 19% for turbulence intensity levels of 3.0% and 3.5 %, respectively, depending on the considered actuator. Differences are understood in terms of the SPOD of actuation and FST-induced fields along with the causality of the control scheme when a cancellation of disturbances is considered along the wall-normal direction. The actuator optimized to generate the leading downstream SPOD mode, representing the streaks in the open-loop flow, leads to the highest transition delay, which can be understood due to its capability of closely cancelling structures in the boundary layer.

Three methods are evaluated to estimate the streamwise velocity fluctuations of a zero-pressure-gradient turbulent boundary layer of momentum-thickness-based Reynolds number up to using as input velocity fluctuations at different wall-normal positions. A system identification approach is considered where large-eddy simulation data are used to build single and multiple-input linear and nonlinear transfer functions. Such transfer functions are then treated as convolution kernels and may be used as models for the prediction of the fluctuations. Good agreement between predicted and reference data is observed when the streamwise velocity in the near-wall region is estimated from fluctuations in the outer region. Both the unsteady behaviour of the fluctuations and the spectral content of the data are properly predicted. It is shown that approximately 45 % of the energy in the near-wall peak is linearly correlated with the outer-layer structures, for the reference case. These identified transfer functions allow insight into the causality between the different wall-normal locations in a turbulent boundary layer along with an estimation of the tilting angle of the large-scale structures. Differences in accuracy of the methods (single- and multiple-input linear and nonlinear) are assessed by evaluating the coherence of the structures between wall-normally separated positions. It is shown that the large-scale fluctuations are coherent between the outer and inner layers, by means of an interactions which strengthens with increasing Reynolds number, whereas the finer-scale fluctuations are only coherent within the near-wall region. This enables the possibility of considering the wall-shear stress as an input measurement, which would more easily allow the implementation of these methods in experimental applications. A parametric study was also performed by evaluating the effect of the Reynolds number, wall-normal positions and input quantities considered in the model. Since the methods vary in terms of their complexity for implementation, computational expense and accuracy, the technique of choice will depend on the application under consideration. We also assessed the possibility of designing and testing the models at different Reynolds numbers, where it is shown that the prediction of the near-wall peak from wall-shear-stress measurements is practically unaffected even for a one order of magnitude change in the corresponding Reynolds number of the design and test, indicating that the interaction between the near-wall peak fluctuations and the wall is approximately Reynolds-number independent. Furthermore, given the performance of such methods in the prediction of flow features in turbulent boundary layers, they have a good potential for implementation in experiments and realistic flow control applications, where the prediction of the near-wall peak led to correlations above 0.80 when wall-shear stress was used in a multiple-input or nonlinear scheme. Errors of the order of 20 % were also observed in the determination of the near-wall spectral peak, depending on the employed method.