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 $N$-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.

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.

In the context of linear stability analysis, considering unsteady base flows is notoriously difficult. A generalisation of modal linear stability analysis, allowing for arbitrarily unsteady base flows over a finite time, is therefore required. The recently developed optimally time-dependent (OTD) modes form a projection basis for the tangent space. They capture the leading amplification directions in state space under the constraint that they form an orthonormal basis at all times. The present numerical study illustrates the possibility to describe a complex flow case using the leading OTD modes. The flow under investigation is an unsteady case of the Blasius boundary layer, featuring streamwise streaks of finite length and relevant to bypass transition. It corresponds to the state space trajectory initiated by the minimal seed; such a trajectory is unsteady, free from any spatial symmetry and shadows the laminar–turbulent separatrix for a finite time only. The finite-time instability of this unsteady base flow is investigated using the 8 leading OTD modes. The analysis includes the computation of finite-time Lyapunov exponents as well as instantaneous eigenvalues, and of the associated flow structures. The reconstructed instantaneous eigenmodes are all of outer type. They map unambiguously the spatial regions of largest instantaneous growth. Other flow structures, previously reported as secondary, are identified with this method as relevant to streak switching and to streamwise vortical ejections. The dynamics inside the tangent space features both modal and non-modal amplification. Non-normality within the reduced tangent subspace, quantified by the instantaneous numerical abscissa, emerges only as the unsteadiness of the base flow is reduced.

The actuator line method (ALM) is used extensively in wind turbine and rotor simulations. However, its original uncorrected formulation overestimates the forces near the tip of the blades and does not reproduce well forces on translating wings. The recently proposed vortex-based smearing correction for the ALM is a correction based on physical and mathematical properties of the simulation that allows for a more accurate and general ALM. So far, to correct the forces on the blades, the smearing correction depended on an iterative process at every time step, which is usually slower, less stable and less deterministic than direct methods. In this work, a non-iterative process is proposed and validated. First, we propose a formulation of the nonlinear lifting line that is equivalent to the ALM with smearing correction, showing that the results are practically identical for a translating wing. Then, by linearizing the lifting line method, the iterative process of the correction is substituted by the direct solution of a small linear system. No significant difference is observed in the results of the iterative and non-iterative corrections, in both wing and rotor simulations. Additional contributions of the present work include the use of a more accurate approximation for the velocity induced by a smeared vortex segment and the implementation of a free-vortex wake model to define the vortex sheet, which contribute to the accuracy and generality of the method. The results presented here may motivate the adoption of the ALM by other communities, for example, in fixed-wing applications.

The stability of an incompressible boundary layer flow over a wing in the presence of free stream turbulence (FST) has been investigated by means of direct numerical simulations and compared with the linearised boundary layer equations. Four different FST conditions have been considered, which are characterised by their turbulence intensity levels and length scales. In all cases the perturbed flow develops into elongated disturbances of high and low streamwise velocity inside the boundary layer, where their spacing has been found to be strongly dependent on the scales of the incoming free stream vorticity. The breakdown of these streaks into turbulent spots from local secondary instabilities is also observed, presenting the same development as the ones reported in flat plate experiments. The disturbance growth, characterised by its root mean squares value, is found to depend not only on the turbulence level, but also on the FST length scales. Particularly, higher disturbance growth is observed for our cases with larger length scales. This behaviour is attributed to the preferred wavenumbers that can exhibit maximum transient growth. We study this boundary layer preference by projection of the flow fields at the leading edge onto optimal disturbances. Our results demonstrate that optimal disturbance growth is the main cause of growth of disturbances on the wing boundary layer.

The instability of an incompressible boundary-layer flow over an infinite swept wing in the presence of disc-type roughness elements and free-stream turbulence (FST) has been investigated by means of direct numerical simulations. Our study corresponds to the experiments by Örlü et al. (Tech. Rep., KTH Royal Institute of Technology, 2021, http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-291874). Here, different dimensions of the roughness elements and levels of FST have been considered. The aim of the present work is to investigate the experimentally observed sensitivity of the transition to the FST intensity. In the absence of FST, flow behind the roughness elements with a height above a certain value immediately undergoes transition to turbulence. Impulse–response analyses of the steady flow have been performed to identify the mechanism behind the observed flow instability. For subcritical roughness, the generated wave packet experiences a weak transient growth behind the roughness and then its amplitude decays as it is advected out of the computational domain. In the supercritical case, in which the flow transitions to turbulence, flow as expected exhibits an absolute instability. The presence of FST is found to have a significant impact on the transition behind the roughness, in particular in the case of a subcritical roughness height. For a height corresponding to a roughness Reynolds number $Re_{hh}=461$, in the absence of FST the flow reaches a steady laminar state, while a very low FST intensity of $Tu =0.03\,\%$ causes the appearance of turbulence spots in the wake of the roughness. These randomly generated spots are advected out of the computational domain. For a higher FST level of $Tu=0.3\,\%$, a turbulent wake is clearly visible behind the element, similar to that for the globally unstable case. The presented results confirm the experimental observations and explain the mechanisms behind the observed laminar–turbulent transition and its sensitivity to FST.

We investigate spanwise-coherent structures in the turbulent flow around airfoils, motivated by their connection with trailing-edge noise. We analyse well-resolved large-eddy simulations (LES) of the flow around NACA 0012 and NACA 4412 airfoils, both at a Reynolds number of 400 000 based on the chord length. Spectral proper orthogonal decomposition performed on the data reveals that the most energetic coherent structures are hydrodynamic waves, extending over the turbulent boundary layers around the airfoils with significant amplitudes near the trailing edge. Resolvent analysis was used to model such structures, using the mean field as a base flow. We then focus on evaluating the dependence of such structures on the domain size, to ensure that they are not an artefact of periodic boundary conditions in small computational boxes. To this end, we performed incompressible LES of a zero-pressure-gradient turbulent boundary layer, for three different spanwise sizes, with the momentum-thickness Reynolds number matching those near the airfoils trailing edge. The same coherent hydrodynamic waves were observed for the three domains. Such waves are accurately modelled as the most amplified flow response from resolvent analysis. The signature of such wide structures is seen in non-premultiplied spanwise wavenumber spectra, which collapse for the three computational domains. These results suggest that the spanwise-elongated structures are not domain-size dependent for the studied simulations, indicating thus the presence of very wide structures in wall-bounded turbulent flows.

An analysis of the statistics of the nonlinear terms in resolvent analysis is performed in this work for turbulent Couette flow at Reynolds number 400. Data from a direct numerical simulation of a minimal flow unit is used to compute the covariance matrix of the velocity. From the same data, we computed the nonlinear terms of the Navier–Stokes equations (treated as forcing), which allowed us to compute the covariance matrix of the forcing. The quantitative relation between the two covariances via the resolvent operator is confirmed here for the first time, accounting for relevant signal processing issues related to the windowing procedure for frequency-domain quantities. Such exact correspondence allowed the eduction of the most relevant force components for the dominant structures in this flow, which participate in the self-sustaining cycle of turbulence: (i) streamwise vortices and streaks, and (ii) spanwise-coherent fluctuations of spanwise velocity. The results show a dominance by a subset of the nonlinear terms for the prediction of the full statistics of streamwise vortices and streaks; a single term is seen to be dominant for spanwise motions. A relevant feature observed in these cases is that the forcing covariance is dominated by its first eigenfunction, showing that nonlinear terms also have a coherent structure at low frequencies in this flow. Different forcing components are also coherent between them, which leads to constructive and destructive interferences that greatly modify the flow response. These are key features of forcing ‘colour’ for the present flow.

In resolvent analyses of turbulent channel flows it has been common practice to neglect or model the nonlinear forcing term that forms the input of the resolvent. However, the spatiotemporal structure of this term is mostly unknown. Here, this nonlinear forcing term is quantified. The Fourier transform of its two-point space–time correlation, its cross-spectral density (CSD), is computed. The CSD is evaluated for two channel flows at friction Reynolds numbers $Re_{\tau } =179$ and $Re_{\tau } =543$ via direct numerical simulations (DNS). The CSDs are computed for energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. It is found that the forcing is structured and that its solenoidal part, which is the only one affecting the velocity field, is the combination of an oblique streamwise vortical forcing and a streamwise component that counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, with only the most energetic spectral proper orthogonal decomposition (SPOD) modes, leads to the bulk of the response. Moreover, it is found that the nonlinear forcing term has a non-negligible projection onto the linear sub-optimal forcings of resolvent analysis, which demonstrates that the linear optimal forcing is not representative of the nonlinear forcing. Finally, it is clarified that the Cess eddy-viscosity-modelled forcing improves the accuracy of resolvent analysis prediction because the modelled forcing projects onto the linear sub-optimal forcings similarly to DNS data.

Direct numerical simulations, performed with a high-order spectral-element method, are used to study coherent structures in turbulent pipe flow at friction Reynolds numbers $Re_{\tau } = 180$ and $550$. The database was analysed using spectral proper orthogonal decomposition (SPOD) to identify energetically dominant coherent structures, most of which turn out to be streaks and quasi-streamwise vortices. To understand how such structures can be modelled, the linear flow responses to harmonic forcing were computed using the singular value decomposition of the resolvent operator, using the mean field as a base flow. The SPOD and resolvent analysis were calculated for several combinations of frequencies and wavenumbers, allowing the mapping out of similarities between SPOD modes and optimal responses for a wide range of relevant scales in turbulent pipe flows. In order to explore physical reasons behind the agreement between both methods, an indicator of lift-up mechanism in the resolvent analysis was introduced, activated when optimal forcing is dominated by the wall-normal and azimuthal components, and associated response corresponds to streaks of streamwise velocity. Good agreement between leading SPOD and resolvent modes is observed in a large region of parameter space. In this region, a significant gain separation is found in resolvent analysis, which may be attributed to the strong amplification associated with the lift-up mechanism, here understood as nonlinear forcing terms leading to the appearance of streamwise vortices, which in turn form high-amplitude streaks. For both Reynolds numbers, the observed concordances were generally for structures with large energy in the buffer layer. The results highlight resolvent analysis as a pertinent reduced-order model for coherent structures in wall-bounded turbulence, particularly for streamwise elongated structures corresponding to near-wall streamwise vortices and streaks.

The stability of the jet in cross-flow is investigated using a complete set-up including the flow inside the pipe. First, direct simulations were performed to find the critical velocity ratio as a function of the Reynolds number, keeping the boundary-layer displacement thickness fixed. At all Reynolds numbers investigated, there exists a steady regime at low velocity ratios. As the velocity ratio is increased, a bifurcation to a limit cycle composed of hairpin vortices is observed. The critical bulk velocity ratio is found at approximately $R=0.37$ for the Reynolds number $Re_{D}=495$, above which a global mode of the system becomes unstable. An impulse response analysis was performed and characteristics of the generated wave packets were analysed, which confirmed results of our global mode analysis. In order to study the sensitivity of this flow, we performed transient growth computations and also computed the optimal periodic forcing and its response. Even well below this stability limit, at $R=0.3$, large transient growth ($10^{9}$ in energy amplification) is possible and the resolvent norm of the linearized Navier–Stokes operator peaks above $2\times 10^{6}$. This is accompanied with an extreme sensitivity of the spectrum to numerical details, making the computation of a few tens of eigenvalues close to the limit of what can be achieved with double precision arithmetic. We demonstrate that including the meshing of the jet pipe in the simulations does not change qualitatively the dynamics of the flow when compared to the simple Dirichlet boundary condition representing the jet velocity profile. This is in agreement with the recent experimental results of Klotz et al. (J. Fluid Mech., vol. 863, 2019, pp. 386–406) and in contrast to previous studies of Cambonie & Aider (Phys. Fluids, vol. 26, 2014, 084101). Our simulations also show that a small amount of noise at subcritical velocity ratios may trigger the shedding of hairpin vortices.

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.

The current work presents a realizable method to control streaky disturbances in boundary layer flows and delay transition to turbulence by means of active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed where streaks are excited in the boundary layer by means of a high level of free-stream turbulence. The occurring disturbances are measured by means of localized wall-shear-stress sensors and damped out using near-wall actuators, which resemble ring plasma actuators. Each actuator is powered by a time-varying signal whose amplitude is computed by processing signals from the sensors. The processed signal is the result of two control laws: the linear quadratic Gaussian regulator (LQG) and the inverse feed-forward control technique (IFFC). The use of the first control method, LQG, requires a state-space representation of the system dynamics, so the flow is described by means of a linear time-invariant operator that captures only the most relevant information of the dynamics and results in a reduced-order model (ROM). The ROM is computed by means of the eigensystem realization algorithm (ERA), which is based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when considering free-stream turbulence because of the high dimensionality of the input forcing needed for a precise description of such a phenomenon. Here, a new method to identify the relevant system dynamics and generate the needed impulse responses is proposed, based on additional shear-stress measurements in an upstream location. Transfer functions between such measurements and other downstream sensors are obtained and allow the derivation of the ERA system, in a data-driven approach that would be realizable in experiments. Finally, in order to discuss the advantages of the LQG based on the ROM and analyse its performance, the implemented LQG is compared to the IFFC, which consists of wave cancellation. The work (i) presents a systematic and straightforward way to deal with high-dimensional disturbances in order to build ROMs for a feasible control technique, and (ii) shows that even when considering practical constraints, such as the type and size of actuators and sensors, it is possible to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in the literature.

Recent progress in understanding subcritical transition to turbulence is based on the concept of the edge, the manifold separating the basins of attraction of the laminar and the turbulent state. Originally developed in numerical studies of parallel shear flows with a linearly stable base flow, this concept is adapted here to the case of a spatially developing Blasius boundary layer. Longer time horizons fundamentally change the nature of the problem due to the loss of stability of the base flow due to Tollmien–Schlichting (TS) waves. We demonstrate, using a moving box technique, that efficient long-time tracking of edge trajectories is possible for the parameter range relevant to bypass transition, even if the asymptotic state itself remains out of reach. The flow along the edge trajectory features streak switching observed for the first time in the Blasius boundary layer. At long enough times, TS waves co-exist with the coherent structure characteristic of edge trajectories. In this situation we suggest a reinterpretation of the edge as a manifold dividing the state space between the two main types of boundary layer transition, i.e. bypass transition and classical transition.

In this study the origins of premature transition due to oversuction in boundary layers are studied. An infinite row of circular suction pipes that are mounted at right angles to a flat plate subject to a Blasius boundary layer is considered. The interaction between the flow originating from neighbouring holes is weak and for the parameters investigated, the pipe is always found to be unsteady regardless of the state of the flow in the boundary layer. A stability analysis reveals that the appearance of boundary layer transition can be associated with a linear instability in the form of two unstable eigenmodes inside the pipe that have weak tails, which extend into the boundary layer. Through an energy budget and a structural sensitivity analysis, the origin of this flow instability is traced to the structures developing inside the pipe near the pipe junction. Although the amplitudes of the modes in the boundary layer are orders of magnitude smaller than the corresponding amplitudes inside the pipe, a Koopman analysis of the data gathered from a nonlinear direct numerical simulation confirms that it is precisely these disturbances that are responsible for transition to turbulence in the boundary layer due to oversuction.

The ability of linear stochastic response analysis to estimate coherent motions is investigated in turbulent channel flow at the friction Reynolds number $\text{Re}_{\unicode[STIX]{x1D70F}}=1007$. The analysis is performed for spatial scales characteristic of buffer-layer and large-scale motions by separating the contributions of different temporal frequencies. Good agreement between the measured spatio-temporal power spectral densities and those estimated by means of the resolvent is found when the effect of turbulent Reynolds stresses, modelled with an eddy-viscosity associated with the turbulent mean flow, is included in the resolvent operator. The agreement is further improved when the flat forcing power spectrum (white noise) is replaced with a power spectrum matching the measures. Such a good agreement is not observed when the eddy-viscosity terms are not included in the resolvent operator. In this case, the estimation based on the resolvent is unable to select the right peak frequency and wall-normal location of buffer-layer motions. Similar results are found when comparing truncated expansions of measured streamwise velocity power spectral densities based on a spectral proper orthogonal decomposition to those obtained with optimal resolvent modes.

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 $Re_{\unicode[STIX]{x1D703}}\simeq 8200$, 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 $Re_{\unicode[STIX]{x1D703}}=4430$. 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.

We investigate theoretically and experimentally the stability of two interlaced helical vortices with respect to displacement perturbations having wavelengths that are large compared to the size of the vortex cores. First, existing theoretical results are recalled and applied to the present configuration. Various modes of unstable perturbations, involving different phase relationships between the two vortices, are identified and their growth rates are calculated. They lead to a local pairing of neighbouring helix loops, or to a global pairing with one helix expanding and the other one contracting. A relation is established between this instability and the three-dimensional pairing of arrays of straight parallel vortices, and a striking quantitative agreement concerning the growth rates and frequencies is found. This shows that the local pairing of vortices is the driving mechanism behind the instability of the helix system. Second, an experimental study designed to observe these instabilities in a real flow is presented. Two helical vortices are generated by a two-bladed rotor in a water channel and characterised through dye visualisations and particle image velocimetry measurements. Unstable displacement modes are triggered individually, either by varying the rotation frequency of the rotor, or by imposing a small rotor eccentricity. The observed unstable mode structure, and the corresponding growth rates obtained from advanced processing of visualisation sequences, are in good agreement with theoretical predictions. The nonlinear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent breakup of the vortices, global pairing results in a leapfrogging phenomenon, which temporarily restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.

With the motivation of determining the critical roughness size, a global stability and sensitivity analysis of a three-dimensional Falkner–Skan–Cooke (FSC) boundary layer with a cylindrical surface roughness is performed. The roughness size is chosen such that breakdown to turbulence is initiated by a global version of traditional secondary instabilities of the cross-flow (CF) vortices instead of an immediate flow tripping at the roughness. The resulting global eigenvalue spectra of the systems are found to be very sensitive to numerical parameters and domain size. This sensitivity to numerical parameters is quantified using the $\unicode[STIX]{x1D700}$-pseudospectrum, and the dependency on the domain is analysed through an impulse response, structural sensitivity analysis and an energy budget. It is shown that while the frequencies remain relatively unchanged, the growth rates increase with domain size, which originates from the inclusion of stronger CF vortices in the baseflow. This is reflected in a change in the rate of advective energy transport by the baseflow. It is concluded that the onset of global instability in a FSC boundary layer as the roughness height is increased does not correspond to an immediate flow tripping behind the roughness, but occurs for lower roughness heights if sufficiently long domains are considered. However, the great sensitivity results in an inability to accurately pinpoint the exact parameter values for the bifurcation, and the large spatial growth of the disturbances in the long domains eventually becomes larger than can be resolved using finite-precision arithmetic.

A reactive control technique with localised actuators and sensors is used to delay the transition to turbulence in a flat-plate boundary-layer flow. Through extensive direct numerical simulations, it is shown that an adaptive technique, which computes the control law on-line, is able to significantly reduce skin-friction drag in the presence of random three-dimensional perturbation fields with linear and weakly nonlinear behaviour. An energy budget analysis is performed in order to assess the net energy saving capabilities of the linear control approach. When considering a model of the dielectric-barrier-discharge (DBD) plasma actuator, the energy spent to create appropriate actuation force inside the boundary layer is of the same order as the energy gained from reducing skin-friction drag. With a model of an ideal actuator a net energy gain of three orders of magnitude can be achieved by efficiently damping small-amplitude disturbances upstream. The energy analysis in this study thus provides an upper limit for what we can expect in terms of drag-reduction efficiency for linear control of transition as a means for drag reduction.