A recent study by Zhang et al. (2024, J. Fluid Mech., vol. 979, A43) introduced an effective control strategy, namely streamwise-uniform spanwise equally distributed injection/suction slots on the pressure (unstable) wall, to enhance passive scalar transport in spanwise rotating plane Poiseuille flows (RPPFs). In this work, we employ direct numerical simulations to further investigate the scalar transport increase rate ($\textit{STI}$) under different slot configurations. Two distinct configurations are investigated, namely uniform-width slots, where injection and suction slots share identical dimensions, and non-uniform-width slots, where their widths vary independently. The former is to examine the effect of slot width, whereas the latter is devoted to distinguishing the individual roles of injection versus suction. While the slot widths change, the root mean square wall-normal velocity is maintained at a fixed minimal value. For uniform configurations, $ \textit{STI}$ increases monotonically with slot width, nearly doubling as the width grows from $\pi /8$ to $\pi /2$. In contrast, non-uniform configurations exhibit a complex, non-monotonic dependence on slot dimensions. Spectral, quadrant and zonal-conditional quadrant analyses reveal that injection and suction slots play distinct roles in modulating near-wall dynamics. Injection enhances ejection events ($Q2$), promoting local plume detachment, and facilitating the formation of large-scale ascending plume currents. Suction, conversely, strengthens sweep events ($Q4$), suppressing plume detachment while intensifying descending currents. This dual mechanism organises turbulent structures into more stable large-scale structures, thereby improving scalar transport efficiency. A decomposition of $ \textit{STI}$ based on clustering analysis confirms that the enhancement stems primarily from increased occurrence possibilities and improved transport capacities of dominant clusters. These findings establish flow stabilisation through selective slot control as an effective mechanism for enhancing passive scalar transport in RPPFs.

Turbulence-induced friction is a significant contributor to energy consumption in the fluid-transport and piping industries. Here, we describe a passive approach to suppress turbulence and reduce friction: we show that a local increase in streamwise flow curvature, combined with changing a circular cross-section to an oval, relaminarises turbulent flow in curved pipes. We exemplify this effect in a $180^{\circ }$ bend at ${\textit{Re}}_D=10\,000$ and $20\,000$ (based on bulk velocity $U_B$ and pipe diameter $D$), well above the limit for sustained turbulence in straight pipes and the linear stability limit in $180^{\circ }$ bends. Curvature inhibits streamwise Reynolds stresses, and cross-sectional modifications weaken the unstable secondary flow, together disrupting the near-wall regeneration cycle and collapsing turbulence. Simulations and experiments confirm that these geometric modifications suppress turbulence and reduce pressure loss by 53 % and 36 % compared with the baseline $180^{\circ }$ bend and a fully developed straight pipe of equal length, respectively. The results establish a passive, mechanism-based route to relaminarisation in curved pipes with implications for energy-efficient control in other wall-bounded flows with curvature.

In a previous study, we have proposed a mechanism for simultaneous reduction of drag and lift by half-rotation at moderately low Reynolds numbers. The axis of rotation ($z$) is perpendicular to both the drag ($x$) and lift ($y$) directions, i.e. the rotation is transverse to the incoming flow direction. Under laminar flow conditions, force-element analysis indicates that a partially rotating sphere can significantly reduce both drag and lift with suppression of vortex shedding. This study extends investigation of the same mechanism of half-rotating a sphere to the turbulence regime at a Reynolds number $Re = 1 \times 10^4$. Similar to the laminar case, half-rotation of the sphere introduces a significant negative velocity drag term, which effectively counteracts the rapid increase in the volume- and surface-vorticity drag terms. Numerical simulations with delayed detached eddy simulation, aided by direct numerical simulation, show that the drag coefficient decreases monotonically with increasing the non-dimensional rotational speed $\alpha$, even becoming negative at $\alpha =10$, while the lift and side-force coefficients remain small for all $\alpha$. However, in contrast to laminar conditions, the turbulent regime is characterised by an earlier onset of shear-layer instabilities, which accelerates the transition of the wake into a fully turbulent state. The relative importance of volume- and surface-vorticity contributions to the drag and lift is the most outstanding difference between the laminar and turbulent flows. In turbulent flow, simultaneous reduction of drag and lift is more pronounced as the contributions of volume- and surface-vorticity lift terms almost cancel each other exactly. These mechanisms and characteristics are systematically compared with those observed in the flow around a fully rotating sphere at the same Reynolds number in terms of vorticity structures, force elements, pressure distributions as well as surface-vorticity distributions.

Miniature vortex generators (MVGs) are a promising passive flow control technique for viscous drag reduction by producing large-scale vortical motions that manipulate turbulence structures in turbulent boundary layers (TBLs) without significant device drag. This study conducts hot-wire anemometry experiments to investigate the influence of the Reynolds number and the ratio of MVG height $h$ to TBL thickness $\delta _0$ (MVG height ratio $h/\delta _0$) on turbulence structures. Experiments encompass two MVG height ratios, $h/\delta _0=0.09,\;0.18$, friction Reynolds numbers ranging from ${\textit{Re}}_\tau =400$ to 2000 and measure the velocity information at various downstream stations. Spectral analysis confirms the MVG-induced vortices amplify large-scale structures in the outer region, sustaining up to 100 times the MVG height downstream. The MVGs are also found to attenuate turbulence energy across a wide range of turbulence structures below the amplification location in the logarithmic region, connected with the MVG-induced spanwise motion. Increasing the friction Reynolds number from ${\textit{Re}}_\tau =400$ to $900$ or doubling the MVG height ratio causes the amplified structures to develop into longer motions and move away from the wall, while increasing the turbulence energy attenuation proportion to the log region. Moreover, the energy attenuation amplitude of large-scale structures in the near-wall region increases with a larger MVG height ratio but decreases with increasing Reynolds numbers. The findings indicate that, at friction Reynolds numbers ${\textit{Re}}_\tau \geqslant 900$, MVGs induce spanwise motions that attenuate near-wall structures and modulate large-scale outer motions. The present configuration does not yield a global viscous drag reduction, but the turbulence modulation trends suggest the potential for viscous drag reduction when the MVG configurations are optimised to enhance favourable buffer-layer spanwise motions.

Travelling wave control is a promising technique for reducing turbulent skin friction by suppressing turbulence in boundary layers. This study presents experimental observations of the streamwise evolution of turbulence over a travelling wavy wall. Particle image velocimetry measurements were performed at multiple downstream locations. The travelling wave was generated by oscillating a rubber sheet to attain properties that are known to achieve drag reduction. The results reveal a two-stage process where the drag-reduction mechanism qualitatively changes in the streamwise direction. In the upstream region (up to approximately two wavelengths from the start of control), turbulent fluctuations are rapidly reduced, however, being limited to the near-wall region. Further downstream, the suppression effect diffuses in the wall-normal direction, leading to a modification on the edge of the boundary layer. The diffusion process of the turbulence-suppression effect is consistently interpreted within the framework of an internal boundary layer, whose development follows a power law. Two-point correlation analysis indicates that the wave crests initially disrupt the near-wall streaky structures, and subsequently reorganise into a characteristic state with a shorter streamwise coherence length downstream. While fully developed states have been studied previously, this work presents the streamwise development of turbulence suppression within a finite length, informing the design of practical drag-reduction devices.

The proposed study aims to optimise a real-time opposition control strategy to reduce the intensity of near-wall sweep events by applying a Bayesian optimisation algorithm. The experiments were conducted in a fully turbulent channel flow characterised by a friction Reynolds number of $350$. Sweep events were identified using a gradient-based detection technique and controlled via a wall-normal jet. An open-loop control logic was implemented and the control parameters (frequency, voltage amplitude and delay time) were optimised, within the bounds imposed by the experimental set-up, to bring the maximum sweep events intensity reduction up to $54\,\%$, with a robust cost function. The effects of the control were observed by analysing the conditionally averaged sweep events at various streamwise locations downstream of the actuation point. Moreover, the conditional analysis was applied to the cross-correlation function of velocity signals highlighting the large reduction of the sweep event convection velocity during the blowing phase of the jet. An overall energy increase has been found in the conditionally averaged energy spectra for the controlled case. The analysis of conditionally averaged wavelet spectra revealed that the control, by interrupting the natural evolution of the sweep event, initially leads to a reduction in the energy associated with it, followed by a subsequent increase during the development of the jet-blowing phase.

Bypass transition, momentum and passive scalar transports in an initially laminar low Reynolds number channel flow with a specific roughness morphology are investigated by direct numerical simulations. The roughness elements are square bars of large heights $k$. Turbulence cannot be triggered in an initially laminar flow without external noise, when the bars extend the entire width of the channel. A staggered configuration is necessary to break up the spanwise symmetry, in which case a pseudo-fully developed rough regime sets up and self-sustains near and below the subcritical Reynolds number. The critical parameter is the shift $s$ between two consecutive staggered bars spanning half the width of the channel. A small shift $s/k$ is enough to trigger the turbulent field. Momentum and scalar fields are analysed for different $s/k$ configurations. The Townsend similarity hypothesis postulating that the outer layer is insensitive to the roughness effects, and that the rough- and smooth-wall statistics collapse in the outer layer, holds well for the momentum field despite the large roughness heights. A particular attention is paid to the deviation of the scalar statistics from the Townsend hypothesis. There is a dissimilarity between the fluctuating temperature and the velocity fields. The Reynolds analogy does not hold stricto sensu. Wake-induced terms determined through the double-averaging procedure play an important role in the rough sublayer. For instance, a significative production of the fluctuating spanwise velocity intensity, which is absent in the canonical flow, appears as a wake-induced term at small shifts. This is solely due to the imposed spanwise asymmetry. The nature, the generation and the self-sustaining mechanisms of the coherent structures near and between the roughness elements are analysed in detail in different configurations. There is a substantial increase of the Nusselt number at particularly low Reynolds numbers.

A data assimilation (DA) strategy based on an ensemble Kalman filter (EnKF) is used to enhance the predictive capabilities of scale-resolving numerical tools for the analysis of flows exhibiting cyclic behaviour. More precisely, an ensemble of numerical runs using large-eddy simulations (LES) for a compressible intake flow rig is augmented via the integration of high-fidelity data. This observation is in the form of instantaneous velocity measurements, which are sampled at localised sensors in the physical domain. Two objectives are targeted. The first objective is the calibration of an unsteady inlet condition suitable to capture the cyclic flow investigated. The second objective is the analysis of the synchronisation of the LES velocity field with the available observations. In order to reduce the computational costs required for this analysis, a hyper-localisation procedure (HLEnKF) is proposed and integrated in the library CONES, tailored to perform fast online DA. The proposed strategy performs a satisfactory calibration of the inlet conditions, and its robustness is assessed using two different prior distributions for the free parameters optimised in this task. The DA state estimation is efficient in obtaining accurate local synchronisation of the inferred velocity fields with the observed data. The modal analysis of the kinetic energy field provides additional insight into the improved reconstruction quality of the velocity field. Thus, the HLEnKF shows promising features for the calibration and synchronisation of scale-resolved turbulent flows, opening perspectives of applications for complex phenomena using advanced tools such as digital twins.

Turbulence accounts for most of the energy losses associated with the pumping of fluids in pipes. Pulsatile drivings can reduce the drag and energy consumption required to supply a desired mass flux, when compared with steady driving. However, not all pulsation waveforms yield reductions. Here, we compute drag- and energy-optimal driving waveforms using direct numerical simulations and a gradient-free black-box optimisation framework. Specifically, we show that Bayesian optimisation is vastly superior to ordinary gradient-based methods in terms of computational efficiency and robustness, due to its ability to deal with noisy objective functions, as they naturally arise from the finite-time averaging of turbulent flows. We identify optimal waveforms for three Reynolds numbers and two Womersley numbers. At a Reynolds number of $8600$ and a Womersley number of 10, optimal waveforms reduce total energy consumption by 22 % and drag by 37 %. These reductions are rooted in the suppression of turbulence prior to the acceleration phase, the resulting delay in turbulence onset, and the radial localisation of turbulent kinetic energy and production towards the pipe centre. Our results pinpoint that the predominant, steady operation mode of pumping fluids through pipes is far from optimal.

Wall-based active and passive flow control for drag reduction in low-Reynolds-number (${\textit{Re}}$) turbulent flows can lead to three typical phenomena: (i) attenuation or (ii) amplification of the near-wall cycle, and (iii) generation of spanwise rollers. The present study conducts direct numerical simulations of a low ${\textit{Re}}$ turbulent channel flow and demonstrates that each flow response can be generated with a wall transpiration at two sets of spatial scales, termed streak and roller scales. The effect of the transpiration is controlled by its relative phase to the background flow, which can be parametrised by the wall pressure. Streak scales (i) attenuate the near-wall cycle if transpiration and wall pressure are approximately in-phase or (ii) amplify it otherwise, and (iii) roller scales energise spanwise rollers when transpiration and wall pressure are out-of-phase. Conditions for establishing these robust phase relations are derived from the analytical solution to the pressure Poisson equation and rely on splitting the pressure into its fast, slow and Stokes component. The importance of each condition depends on the relative magnitude of the pressure components, which is significantly altered by the transpiration. The analogy in flow response suggests that transpiration with the two scale families and their phase relations to the wall pressure represent fundamental building blocks for flows over tailored surfaces including riblets, porous and permeable walls.

We address the Reynolds number dependence of the turbulent skin-friction drag reduction induced by streamwise-travelling waves of spanwise wall oscillations. The study relies on direct numerical simulations of drag-reduced flows in a plane open channel at friction Reynolds numbers in the range $1000 \leqslant Re_\tau \leqslant 6000$, which is the widest range considered so far in simulations with spanwise forcing. Our results corroborate the validity of the predictive model proposed by Gatti & Quadrio (J. Fluid Mech. vol. 802, 2016, pp. 553–558): regardless of the control parameters, the drag reduction decreases monotonically with $Re$ at a rate that depends on the drag reduction itself and on the skin-friction of the uncontrolled flow. We do not find evidence in support of the results of Marusic et al. (Nat. Commun. vol. 12, no. 1, 2021, pp. 5805), which instead report by experiments an increase of the drag reduction with $Re$ in turbulent boundary layers, for control parameters that target low-frequency, outer-scaled motions. Possible explanations for this discrepancy are provided, including obvious differences between open channel flows and boundary layers, and possible limitations of laboratory experiments.

In this work, we study the effectiveness of the time-localised principal resolvent forcing mode at actuating the near wall cycle of turbulence. This mode is restricted to a wavelet pulse and computed from a singular value decomposition of the windowed wavelet-based resolvent operator (Ballouz et al. 2024b, J. Fluid Mech. vol. 999, A53) such that it produces the largest amplification via the linearised Navier–Stokes equations. We then inject this time-localised mode into the turbulent minimal flow unit at different intensities, and measure the deviation of the system’s response from the optimal resolvent response mode. Using the most energetic spatial wavenumbers for the minimal flow unit – i.e. constant in the streamwise direction and once-periodic in the spanwise direction – the forcing mode takes the shape of streamwise rolls and produces a response mode in the form of streamwise streaks that transiently grow and decay. Though other works such as Bae et al. (2021 J. Fluid Mech. vol. 914, A3) demonstrate the importance of principal resolvent forcing modes to buffer layer turbulence, none instantaneously track their time-dependent interaction with the turbulence, which is made possible by their formulation in a wavelet basis. For initial times and close to the wall, the turbulent minimal flow unit matches the principal response mode well, but due to nonlinear effects, the response across all forcing intensities decays prematurely with a higher forcing intensity leading to faster energy decay. Nevertheless, the principal resolvent forcing mode does lead to significant energy amplification and is more effective than a randomly generated forcing structure and the second suboptimal resolvent forcing mode at amplifying the near-wall streaks. We compute the nonlinear energy transfer to secondary modes and observe that the breakdown of the actuated mode proceeds similarly across all forcing intensities: in the near-wall region, the induced streak forks into a structure twice-periodic in the spanwise direction; in the outer region, the streak breaks up into a structure that is once-periodic in the streamwise direction. In both regions, spanwise oscillations account for the dominant share of nonlinear energy transfer.

The effects of surface roughness in the transitionally rough regime on the overlying near-wall turbulence are modelled using quasi-linear approximations proposed recently: minimal quasi-linear approximation (MQLA) (Hwang & Ekchardt, 2020, J. Fluid Mech., vol. 894, A23), data-driven quasi-linear approximation (DQLA) (Holford et al., 2024, J. Fluid Mech., vol. 980, A12) and a newly established variant of MQLA (M2QLA, minimal two-mode quasi-linear approximation). The transpiration-resistance model (TRM) for boundary conditions is applied to account for the surface roughness (Lācis et al., 2020, J. Fluid Mech., vol. 884, A21). It is shown that many essential near-wall turbulence statistics are fairly well captured by the quasi-linear approximations in a wide range of slip and transpiration lengths for the TRM boundary conditions. In particular, the virtual origins and the resulting roughness functions are well predicted, showing good agreement with those from previous direct numerical simulations (DNS) in mild roughness cases. The DQLA and M2QLA, which incorporate streamwise-dependent Fourier modes in the approximations, are also shown to perform a little better than MQLA, especially with DQLA reproducing the two-dimensional energy spectra qualitatively consistent with the DNS. Finally, with a computational cost much lower than DNS, it is shown that the proposed quasi-linear approximation frameworks offer an efficient tool to rapidly explore the roughness effects within a large parameter space.

We explore a reduced-order model (ROM) of plane Couette flow with a view to performing near-wall turbulence control. The ROM is derived through Galerkin projections of the incompressible Navier–Stokes system onto a basis of controllability modes. Such ROMs were found to reproduce key aspects of turbulence dynamics in Couette flow with only a few hundred degrees of freedom, and here we use them to devise a control strategy. We consider a ROM with an extra forcing term whose structure is given by a combination of eigenfunctions of a linear viscous diffusion equation, optimised in order to minimise the total fluctuation energy. The optimisation is performed at Reynolds numbers $Re=1000, 2000, 3000$, and produces a novel control mechanism wherein the optimal forcing leads the flow to laminarisation in all cases. The forcing acts by reducing the shear in a large portion of the channel, hindering the main energy input mechanism. The forced flow possesses a new laminar solution which is linearly stable at $Re=1000$ and unstable at higher $Re$, but whose transient growth of streaky structures is substantially lower than that of laminar Couette flow, leading the flow to full laminarisation when the forcing is removed. Forcings optimised in the ROM are subsequently applied in direct numerical simulations (DNS). The same control mechanisms are observed in the DNS, where laminarisation is also achieved. We show that the ROMs provide an effective framework to design turbulence control strategies, despite the high degree of truncation, which opens up interesting possibilities for turbulence control.

Direct numerical simulations are performed for turbulent forced convection in a half-channel flow with a wall oscillating either as a spanwise plane oscillation or to generate a streamwise travelling wave. The friction Reynolds number is fixed at $Re_{\tau _0} = 590$, but the Prandtl number $Pr$ is varied from 0.71 to 20. For $Pr\gt 1$, the heat transfer is reduced by more than the drag, 40 % compared with 30 % at $Pr=7.5$. This outcome is related to the different responses of the velocity and thermal fields to the Stokes layer. It is shown that the Stokes layer near the wall attenuates the large-scale energy of the turbulent heat flux and the turbulent shear stress, but amplifies their small-scale energy. At higher Prandtl numbers, the thinning of the conductive sublayer means that the energetic scales of the turbulent heat flux move closer to the wall, where they are exposed to a stronger Stokes layer production, increasing the contribution of the small-scale energy amplification. A predictive model is derived for the Reynolds and Prandtl number dependence of the heat-transfer reduction based on the scaling of the thermal statistics. The model agrees well with the computations for Prandtl numbers up to 20.

We investigate the drag reduction effects by two representative blowing/suction-based control methods having different drag reduction mechanisms, i.e. the opposition control and uniform blowing (UB), in a bump-installed turbulent channel flow through direct numerical simulations. We consider two different bulk Reynolds numbers ${\textit {Re}}_b = 5600$ and $12\,600$, and bump heights $h^+ \approx 20$ and $40$. In the opposition-controlled case, the friction drag reduction effect in the case with a bump is similar to that in the case without a bump, while the control effect on the pressure drag is hardly observed. The total drag reduction rate decreases for the higher bump height because the ratio of the pressure drag to the total drag increases as the bump height. In the UB case, UB at $0.1\,\%$ or $0.5\,\%$ of the bulk-mean velocity is imposed on the lower wall with a bump, while the same amount of uniform suction (US) is applied on the upper flat wall to keep the mass flow rate. Although the total friction drag increases due to a detrimental effect of US on the upper wall, the wall-normal motions due to the existence of a bump on the lower wall are suppressed by the UB, so that the pressure drag is decreased, unlike the opposition-controlled case. Due to the difference in the inherent drag reduction mechanisms, the flow separation in the region behind the bump is enhanced by the opposition control, while suppressed by UB.

Spanwise wall forcing in the form of streamwise-travelling waves is applied to the suction side of a transonic airfoil with a shock wave to reduce aerodynamic drag. The study, conducted using direct numerical simulations, extends earlier findings by Quadrio et al. (J. Fluid Mech. vol. 942(R2), 2022, pp. 1–10) and confirms that the wall manipulation shifts the shock wave on the suction side towards the trailing edge of the profile, thereby enhancing its aerodynamic efficiency. A parametric study over the parameters of wall forcing is carried out for the Mach number set at 0.7 and the Reynolds number at 300 000. Similarities and differences with the incompressible plane case are discussed; for the first time, we describe how the interaction between the shock wave and the boundary layer is influenced by flow control via spanwise forcing. With suitable combinations of control parameters, the shock is delayed, which results in a separated region whose length correlates well with friction reduction. The analysis of the transient process following the sudden application of control is used to link flow separation with the intensification of the shock wave.

Opposition control (OC) is a reactive flow-control approach that mitigates the near-wall fluctuations by imposing blowing and suction at the wall, being opposite to the off-wall observations. We carried out high-resolution large-eddy simulations to investigate the effects of OC on turbulent boundary layers (TBLs) over a wing at a chord-based Reynolds number (${Re}_c$) of $200 \ 000$. Two cases were considered: flow over the suction sides of the NACA0012 wing section at an angle of attack of $0^{\circ }$, and the NACA4412 wing section at an angle of attack of $5^{\circ }$. These cases represent TBLs subjected to mild and strong non-uniform adverse pressure gradients (APGs), respectively. First, we assessed the control effects on the streamwise development of TBLs and the achieved drag reduction. Our findings indicate that the performance of OC in terms of friction-drag reduction significantly diminishes as the APG intensifies. Analysis of turbulence statistics subsequently reveals that this is directly linked to the intensified wall-normal convection caused by the strong APG: it energizes the control intensity to overload the limitation that guarantees drag reduction. The formation of the so-called virtual wall that reflects the mitigation of wall-normal momentum transport is also implicitly affected by the pressure gradient. Control and pressure-gradient effects are clearly apparent in the anisotropy invariant maps, which also highlight the relevance of the virtual wall. Finally, spectral analyses indicate that the wall-normal transport of small-scale structures to the outer region due to the APG has a detrimental impact on the performance of OC. Uniform blowing and body-force damping were also examined to understand the differences between the various control schemes. Despite the distinct performance of friction-drag reduction, the effects of uniform blowing are akin to those induced by a stronger APG, while the effects of body-force damping exhibit similarities to those of OC in terms of the streamwise development of the TBL although there are differences in the turbulent statistics. To authors’ best knowledge, the present study stands as the first in-depth analysis of the effects of OC applied to TBL subjected to non-uniform APGs with complex geometries.

Self-similar fractal tree models are numerically investigated to elucidate the drag coefficient, non-equilibrium dissipation behaviour and various turbulence statistics of fractal trees. For the simulation, a technique based on the lattice Boltzmann method with a cumulant collision term is used. Self-similar fractal tree models for aerodynamic computations are constructed using parametric L-system rules. Computations across a range of tree-height-based Reynolds numbers $Re_H$, from 2500 to 120 000, are performed using multiple tree models. As per the findings, the drag coefficients ($C_D$) of these models match closely with those of the previous literature at high Reynolds numbers ($Re_H \geqslant 60\,000$). A normalization process that collapses the turbulence intensity across various tree models is formulated. For a single tree model, a consistent centreline turbulence intensity trend is maintained in the wake region beyond a Reynolds number of 60 000. The global and local isotropy analysis of the turbulence generated by fractal trees indicates that, at high Reynolds numbers ($Re_H \geqslant 60\,000$), the distant wake can be considered nearly locally isotropic. The numerical results confirm the non-equilibrium dissipation behaviour demonstrated in previous studies involving space-filling fractal square grids. The non-dimensional dissipation rate $C_\epsilon$ does not remain constant; instead, it becomes approximately inversely proportional to the local Taylor-microscale-based Reynolds number, $C_\epsilon \propto 1/Re_\lambda$. We find significant one-point inhomogeneity, production and transverse transport of turbulent kinetic energy within the non-equilibrium dissipation near wake region.

Deep reinforcement learning (DRL) is employed to develop control strategies for drag reduction in direct numerical simulations of turbulent channel flows at high Reynolds numbers. The DRL agent uses near-wall streamwise velocity fluctuations as input to modulate wall blowing and suction velocities. These DRL-based strategies achieve significant drag reduction, with maximum rates $35.6\,\%$ at $Re_{\tau }\thickapprox 180$, $30.4\,\%$ at $Re_{\tau }\thickapprox 550$, and $27.7\,\%$ at $Re_{\tau }\thickapprox 1000$, outperforming traditional opposition control methods. An expanded range of wall actions further enhances drag reduction, although effectiveness decreases at higher Reynolds numbers. The DRL models elevate the virtual wall through blowing and suction, aiding in drag reduction. However, at higher Reynolds numbers, the amplitude modulation of large-scale structures significantly increases the residual Reynolds stress on the virtual wall, diminishing the drag reduction. Analysis of budget equations provides a systematic understanding of the underlying drag reduction dynamics. The DRL models reduce skin friction by inhibiting the redistribution of wall-normal turbulent kinetic energy. This further suppresses the wall-normal velocity fluctuations, reducing the production of Reynolds stress, thereby decreasing skin friction. This study showcases the successful application of DRL in turbulence control at high Reynolds numbers, and elucidates the nonlinear control mechanisms underlying the observed drag reduction.