We conduct a numerical study on the drag-reduction mechanism of an opposition- controlled turbulent channel flow from the viewpoint of a symbolic dynamics approach. The effect of the virtual wall formed by the opposition control is maximised at the location of the detection plane $y_d^+ \approx 10$. At this wall-normal location, the local link strength of the self-loop of network nodes representing the negative correlation pattern between the streamwise and wall-normal velocity fluctuations is maximised in the uncontrolled flow. In the controlled case, the multiscale complexity–entropy causality plane and the spatial permutation entropy at $y_d^+ \approx 10$ indicate that the drag-reduction effect is attributed to the reduction of the region where streaks actively coalesce and separate and the suppression of the regeneration cycle in the region near the wall.

We investigate the coupling effects of the two-phase interface, viscosity ratio and density ratio of the dispersed phase to the continuous phase on the flow statistics in two-phase Taylor–Couette turbulence at a system Reynolds number of $6\times 10^3$ and a system Weber number of 10 using interface-resolved three-dimensional direct numerical simulations with the volume-of-fluid method. Our study focuses on four different scenarios: neutral droplets, low-viscosity droplets, light droplets and low-viscosity light droplets. We find that neutral droplets and low-viscosity droplets primarily contribute to drag enhancement through the two-phase interface, whereas light droplets reduce the system's drag by explicitly reducing Reynolds stress due to the density dependence of Reynolds stress. In addition, low-viscosity light droplets contribute to greater drag reduction by further reducing momentum transport near the inner cylinder and implicitly reducing Reynolds stress. While interfacial tension enhances turbulent kinetic energy (TKE) transport, drag enhancement is not strongly correlated with TKE transport for both neutral droplets and low-viscosity droplets. Light droplets primarily reduce the production term by diminishing Reynolds stress, whereas the density contrast between the phases boosts TKE transport near the inner wall. Therefore, the reduction in the dissipation rate is predominantly attributed to decreased turbulence production, causing drag reduction. For low-viscosity light droplets, the production term diminishes further, primarily due to their greater reduction in Reynolds stress, while reduced viscosity weakens the density difference's contribution to TKE transport near the inner cylinder, resulting in a more pronounced reduction in the dissipation rate and consequently stronger drag reduction. Our findings provide new insights into the physics of turbulence modulation by the dispersed phase in two-phase turbulence systems.

We leverage the snap-through response of a bistable origami mechanism to induce a discontinuous evolution of drag with flow speed. The transition between equilibrium states is actuated passively by airflow, and we demonstrate that large shape reconfiguration over a small increment of flow velocity leads to a pronounced and sudden drop in drag. Moreover, we show that systematically varying the geometrical and mechanical properties of the origami unit enables the tuning of this drag discontinuity and the critical speed and loading at which it occurs. Experimental results are supported by a theoretical aeroelastic model, which further guides inverse design to identify the combination of structural origami parameters for targeted drag collapse. This approach sheds light on harnessing origami-inspired mechanisms for efficient passive drag control in a fluid environment, applicable for load alleviation or situations requiring swift transitions in aerodynamic performances.

Reducing drag under high turbulence is a critical but challenging issue that has engendered great concern. This study utilizes hydrophilic tips in superhydrophobic (SHP) grooves to enhance the stability of plastron, which results in a considerable drag reduction ($DR$) up to 62 %, at Reynolds number ($Re$) reaching $2.79 \times 10^{4}$. The effect of the spacing width $w$ of the microgrooves on both $DR$ and flow structures is investigated. Experimental results demonstrate that $DR$ increases as either microgroove spacing $w$ or $Re$ increases. The velocity fields obtained using particle image velocimetry indicate that the air-filled SHP grooves induce a considerable wall slip. This slip significantly weakens the intensity of Taylor rolls, reduces local momentum transport, and consequently lowers drag. This phenomenon becomes more pronounced with increasing $w$. Furthermore, to quantify the multiscale relationship between global response and geometrical as well as driving parameters, $DR\sim (w, \phi _s, Re)$, a theoretical model is established based on angular momentum defect theory and magnitude estimate. It is demonstrated that a decrease in the surface solid fraction can reduce wall shear, and an increase in the groove width can weaken turbulence kinetic energy production, rendering enhanced slip and drag reduction. This research has implications for designing and optimizing turbulent-drag-reducing surfaces in various engineering applications, such as transportation and marine engineering.

Superhydrophobic (SHPo) surfaces can capture a thin layer of air called a plastron under water to reduce skin friction. Although a ~30 % drag reduction has been recently reported with longitudinal micro-trench SHPo surfaces under a boat and in a towing tank, the results lacked the consistency to establish a clear trend. Designed based on Yu et al. (J. Fluid Mech, vol. 962, 2023, A9), this work develops and tests a series of high-performance SHPo surface coupons that can sustain a pinned plastron underneath a passenger motorboat revamped to reach 14 knots. Importantly, plastrons in a pinned state, not just their existence, are confirmed during flow experiments for the first time. All the drag-reduction data measured on different longitudinal micro-trenches are found to collapse if plotted against slip length in wall units. In comparison, aligned posts and transverse trenches show less and little drag reduction, respectively, confirming the adverse effect of the spanwise slip in turbulent flows. This report not only verifies SHPo surfaces can provide a consistent drag reduction at high speeds in open sea but also shows that one may predict the amount of drag reduction in turbulent flows using the physical slip length obtained for Stokes flows.

The turbulent wake behind a flat-back Ahmed body is investigated using stacked stereoscopic particle image velocimetry. The wake is disturbed by a steady jet from the centre of the base and the effects are quantified for key blowing rates. The unactuated wake exhibits bistable dynamics in the horizontal plane that are completely subdued for the optimal blowing case, yielding a base drag reduction of 9 %. The three-dimensional mean wake is reconstructed and used to evaluate the wake mass fluxes whose equilibrium determines the recirculation length. The results for the unactuated wake show that up to 80 % of replenishment fluid flux entering the recirculation bubble from the free-stream flow is provided through the low-pressure side of the base, where the symmetry-breaking shear layer roll-up occurs near the base. For the optimal blowing configuration, where the wake becomes symmetric, the flux of wake replenishment is severely reduced. This flow configuration results in elongated shear layers on all sides, which terminate the bubble with a roll-up of reduced intensity at a further downstream location. The dominant cause of bubble growth and the accompanying drag reduction is attributed to the momentum of the base blowing, and the new regime is referred to as the ‘favourable momentum regime’. Similar trends are observed when the model is at $5^{\circ }$ yaw where a reduction of drag and yaw-induced asymmetry are obtained. Proper orthogonal decomposition of the wake reveals the coherent structures related to the bistable flow and the symmetric wake under optimal blowing coefficient.

This study explores the implementation of an online control strategy based on dynamic mode decomposition in the context of flow control. The investigation is conducted mainly with a fixed Reynolds number of $Re = 100$, focusing on the flow past a circular cylinder constrained between two walls to mitigate vortex shedding. The control approach involves the activation of two synthetic jets on the cylinder through blowing and suction. Velocity fluctuations in the wake, specifically in the $x$-direction, are harnessed to ascertain the mass flow rate of the jets using the linear quadratic regulator and online dynamic mode decomposition. The study systematically assesses the control performance across various configurations, including different values of the input penalty factor ${\boldsymbol {R}}$, varying numbers of probes and distinct probe arrangement methods. The synthetic jets prove effective in stabilising the separation bubble, and their interaction with the unsteady wake leads to a notable reduction in drag force, its fluctuations and the amplitude of the lift force. Specifically, the mean and standard deviation of the drag coefficient witness reductions of $7.44\,\%$ and $96.67\,\%$, respectively, and the standard deviation of the lift coefficient experiences an impressive reduction of $85.18\,\%$. The robustness of the proposed control method has also been tested on two more complicated cases, involving unsteady incoming flows with multiple frequency components. Comparatively, the methodology employed in this paper yields results akin to those obtained through deep reinforcement learning in terms of control effectiveness. However, a noteworthy advantage lies in the substantial reduction of computational resource consumption, highlighting the efficiency of the proposed approach.

Flow resistance reduction, quantified as a change in flow rate with respect to a reference isothermal flow driven by the same pressure gradient, is realizable in a channel flow using a thermal wave applied on the bounding wall. Countercurrent waves provide a resistance-reducing effect at any wave velocity, Reynolds number and wavenumber considered. Cocurrent waves can reduce resistance only if the wave velocity is lower than a certain threshold, and the Reynolds number is larger than a certain threshold, otherwise, such waves increase resistance. The increase of the wave amplitude increases resistance reduction and resistance increase up to a specific limit. It is possible to reduce resistance up to 20 times compared with the isothermal channel using proper waves. It is shown that the same effect is achieved regardless of the waves applied at the upper and lower walls. The wave-modified flows are shown to be stable for the conditions used in this study.

This work examines the control of cross-flow instabilities (CFIs) and laminar–turbulent transition on a swept wing, through the plasma-based base flow modification (BFM) technique. The effect of experimentally derived plasma body forces on the steady boundary layer base flow is explored through numerical simulations. Linear stability theory is subsequently used to predict the net BFM effect on CFIs. Based on these preliminary predictions, experiments are conducted in a low-turbulence wind tunnel where a spanwise-invariant plasma actuator is installed near the wing leading edge and operated at constant input voltage and frequency. Various flow parameters governing the plasma-based BFM technique are investigated, namely the Reynolds number, angle of attack and wavelength of excited stationary CFI modes. Stationary and travelling CFIs are quantified by planar particle image velocimetry while the transition topology and location are recorded by infrared thermography. The results confirm the stabilising effect of BFM on the swept-wing boundary layer. However, the plasma-based BFM is found to render the boundary layer more susceptible to travelling CFIs. In the presence of both net BFM effect and intrinsic plasma unsteady perturbations, the plasma-based BFM technique achieves transition delay with specific combinations of Reynolds number, angle of attack and wavelength of excited stationary CFI modes. The present findings provide insights into the fundamental principles of operating plasma actuators within the context of BFM control.

This study explores the application of a wall-attached ferrofluid film to decrease skin-friction drag in turbulent channel flow. We conduct experiments using water as a working fluid in a turbulent channel flow set-up, where one wall is coated with a ferrofluid layer held in place by external permanent magnets. Depending on the flow conditions, the interface between the two fluids is observed to form unstable travelling waves. While ferrofluid coating has been previously employed in laminar and moderately turbulent flows (Reynolds number $Re<4000$) to reduce drag by creating a slip condition at the fluid interface, its effectiveness in fully developed turbulent conditions, particularly when the interface exhibits instability, remains uncertain. Our primary objective is to assess the effectiveness of ferrofluid coating in reducing turbulent drag with particular focus on scenarios when the ferrofluid layer forms unstable waves. To achieve this, we measure flow velocity using two-dimensional particle tracking velocimetry (2D-PTV), and the interface contour between the fluids is determined using an interface tracking algorithm. Our results reveal the significant potential of ferrofluid coating for drag reduction, even in scenarios where the interface between the surrounding fluid and ferrofluid exhibits instability, with observed drag reduction rates up to 95 %. In particular, waves with an amplitude significantly smaller than a viscous length scale positively contribute to drag reduction, while larger waves are detrimental, because of induced turbulent fluctuations. However, for the latter case, slip outcompetes the extra turbulence so that drag is still reduced.

Direct numerical simulation (DNS) of rotating pipe flows up to $Re_\tau \approx 3000$ is carried out to investigate drag reduction effects associated with axial rotation, extending previous studies carried out at a modest Reynolds number (Orlandi & Fatica, J. Fluid Mech., vol. 343, 1997, pp. 43–72; Orlandi & Ebstein, Intl J. Heat Fluid Flow, vol. 21, 2000, pp. 499–505). The results show that the drag reduction, which we theoretically show to be equivalent to net power saving assuming no mechanical losses, monotonically increases as either the Reynolds number or the rotation number increases, proportionally to the inner-scaled rotational speed. Net drag reduction up to approximately $70\,\%$ is observed, while being far from flow relaminarisation. Scaling laws for the mean axial and azimuthal velocity are proposed, from which a predictive formula for the friction factor is derived. The formula can correctly represent the dependency of the friction factor on the Reynolds and rotation numbers, maintaining good accuracy for low-to-moderate rotation numbers. Examination of the turbulent structures highlights the role of rotation in widening and elongating the small-scale streaks, with subsequent suppression of sweeps and ejections. In the core part of the flow, clear weakening of large-scale turbulent motions is observed at high Reynolds numbers, with subsequent suppression of the outer-layer peak in the pre-multiplied spectra. The Fukagata–Iwamoto–Kasagi decomposition indicates that, consistent with a theoretically derived formula, the outer layer yields the largest contribution to drag reduction at increasingly high Reynolds numbers. In contrast, both the inner and the outer layers contribute to drag reduction as the rotation number increases.

The effectiveness of utilizing heating patterns as a drag-reduction tool in sloping channels is analysed. The usefulness of heating is judged by determining the pressure gradient required to maintain the same flow rate as in the isothermal case. The key to reducing pressure loss is the formation of separation bubbles, although these bubbles are washed away at relatively large Reynolds numbers. The bubbles reduce the direct contact between the stream and the side walls, thereby reducing the friction experienced by the flow. Moreover, the fluid inside the bubbles tends to rotate, a motion provoked by longitudinal temperature gradients. This rotation also seems to reduce the resistance. On the other hand, the existence of the bubbles tends to obstruct the stream, increasing the flow resistance. In general, channels oriented close to horizontal experience a relatively small pressure loss, but this loss grows markedly as the channel inclines towards the vertical. When modest heating is applied, the pressure loss is approximately proportional to the square of the associated Rayleigh number. It is also shown that if the heating wavelength is too short or too long, the heating loses its effectiveness. In certain circumstances, it turns out that the theoretical pressure-gradient reduction achieved by judicious heating is so large that it exceeds the pressure gradient required to drive the flow in the isothermal problem. The conclusion is that in these instances, a pressure gradient of the opposite sign must be applied to prevent flow acceleration.

The practical implementation of machine learning in flow control is limited due to its significant training expenses. In the present study the convolutional neural network (CNN) trained with the data of the restricted nonlinear (RNL) model is used to predict the normal velocity on a detection plane at $y^+=10$ in a turbulent channel flow, and the predicted velocity is used as wall blowing and suction for drag reduction. An active control test is carried out by using the well-trained CNN in direct numerical simulation (DNS). Substantial drag reduction rates up to 19 % and 16 % are obtained based on the spanwise and streamwise wall shear stresses, respectively. Furthermore, we explore the online control of wall turbulence by combining the RNL model with reinforcement learning (RL). The RL is constructed to determine the optimal wall blowing and suction based on its observation of the wall shear stresses without using the label data on the detection plane for training. The controlling and training processes are conducted synchronously in a RNL flow field. The control strategy discovered by RL has similar drag reduction rates with those obtained previously by the established method. Also, the training cost decreases by over thirty times at $Re_{\tau }=950$ compared with the DNS-RL model. The present results provide a perspective that combining the RNL model with machine learning control for drag reduction in wall turbulence can be effective and computationally economical. Also, this approach can be easily extended to flows at higher Reynolds numbers.

Adding polymers to turbulent Newtonian fluid flows can have dramatic effects. A well-known example is a significant drag reduction by flexible polymers in turbulent wall-bounded flows. In numerical studies of polymer drag reduction, polymers are often modelled as dumbbells of two beads connected by a finitely extensible nonlinear elastic (FENE) spring. There are natural queries whether this highly simplified coarse-grained model is adequate for describing a polymer macromolecule in turbulent flows. By carrying out Eulerian–Lagrangian simulations of polymers, described by different models, in a turbulent pipe flow, Serafini et al. (J. Fluid Mech., vol. 987, 2024, R1) have demonstrated that the FENE dumbbell model can accurately capture polymer extension statistics as compared with the realistic Kuhn chain model. Their work further reveals the surprising result that increasing the number of beads in a FENE chain worsens its accuracy in characterizing polymer spatial conformations at large Weissenberg numbers.

The flow resistance, i.e. friction factor times Reynolds number ($\,f\,{Re}$), of longitudinal-fin heat sinks with or without clearance between a shroud and the tips of the fins is an important parameter in thermal design. This is because it dictates the caloric resistance of the heat sink, i.e. change in bulk temperature of the fluid flowing through it. When there is no clearance and the common and oft-valid assumption of negligible fin thickness is invoked, $f\,{Re}$ corresponds to simply that of a rectangular duct. However, with clearance, only numerical results are available as per the well-known study by Sparrow, Baliga and Patankar (ASME J. Heat Transfer, vol. 100, 1978). We develop analytical formulae for $f\,{Re}$ for fully developed flow with clearance. The exact solution is provided by an integral formula derived via conformal mappings. Additionally, simple formulae are derived via asymptotic expansions in three cases: (1) the fin spacing is small compared to the fin height and clearance; (2) the clearance is small compared to the fin spacing, which is small compared to the fin height; (3) the same as case (2) but valid for larger clearances. The different asymptotic formulae are compared to the exact formula, and together cover almost the entire relevant parameter range (for fin spacing and clearance) with errors of less than 15 %. A formula for the limiting case of no clearance is shown to be more accurate, for any fin spacing, than a widely used correlation from the literature.

This study explores the dynamics of flexible ribbons with an added weight $G$ at the tail in uniform flow, considering key parameters like inflow Reynolds number ($Re_u$), mass ratio ($M_t$) and aspect ratio (${A{\kern-4pt}R}$). For two-dimensional ribbons, a simplified theoretical model accurately predicts equilibrium configurations and forces. Inspired by Barois & De Langre (J. Fluid Mech., vol. 735, 2013, R2), we introduce an important control parameter ($C_G$) that effectively collapses normalized forces and angle data. Vortex-induced vibration is observed, and Strouhal number ($St$) scaling laws with $C_G$ are identified. In three-dimensional scenarios, the model effectively predicts lift, but its accuracy in predicting drag is limited to situations with small $Re_u$ values. The flow along the side edges mitigates pressure differences, thereby suppressing vibration and uplift, particularly noticeable in the case of narrow ribbons. This study offers new insights into the dynamics of flexible bodies in uniform flow.

This study proposes a self-learning algorithm for closed-loop cylinder wake control targeting lower drag and lower lift fluctuations with the additional challenge of sparse sensor information, taking deep reinforcement learning (DRL) as the starting point. The DRL performance is significantly improved by lifting the sensor signals to dynamic features (DFs), which predict future flow states. The resulting DF-based DRL (DF-DRL) automatically learns a feedback control in the plant without a dynamic model. Results show that the drag coefficient of the DF-DRL model is 25 % less than the vanilla model based on direct sensor feedback. More importantly, using only one surface pressure sensor, DF-DRL can reduce the drag coefficient to a state-of-the-art performance of approximately 8 % at Reynolds number $(Re) = 100$ and significantly mitigates lift coefficient fluctuations. Hence, DF-DRL allows the deployment of sparse sensing of the flow without degrading the control performance. This method also exhibits strong robustness in flow control under more complex flow scenarios, reducing the drag coefficient by 32.2 % and 46.55 % at $Re =500$ and 1000, respectively. Additionally, the drag coefficient decreases by 28.6 % in a three-dimensional turbulent flow at $Re =10\,000$. Since surface pressure information is more straightforward to measure in realistic scenarios than flow velocity information, this study provides a valuable reference for experimentally designing the active flow control of a circular cylinder based on wall pressure signals, which is an essential step toward further developing intelligent control in a realistic multi-input multi-output system.

The ability of streamwise-travelling waves of spanwise velocity to reduce the turbulent skin-friction drag is assessed in the compressible regime. Direct numerical simulations are carried out to compare drag reduction in subsonic, transonic and supersonic channel flows. Compressibility improves the benefits of the travelling waves, in a way that depends on the control parameters: drag reduction becomes larger than the incompressible one for small frequencies and wavenumbers. However, the improvement depends on the specific procedure employed for comparison. When the Mach number is varied and, at the same time, wall friction is changed by the control, the bulk temperature in the flow can either evolve freely in time until the aerodynamic heating balances the heat flux at the walls, or be constrained such that a fixed percentage of kinetic energy is transformed into thermal energy. Physical arguments suggest that, in the present context, the latter approach should be preferred. This provides a test condition in which the wall-normal temperature profile more realistically mimics that in an external flow, and also leads to a much better scaling of the results, over both the Mach number and the control parameters. Under this comparison, drag reduction is only marginally improved by compressibility.

We investigate the three-dimensional (3-D) flow around and through a porous screen for various porosities at high Reynolds number $Re = {O}(10^4)$. Historically, the study of this problem has been focused on two-dimensional cases and for screens spanning completely or partially a channel. Since many recent problems have involved a porous object in a 3-D free flow, we present a 3-D model initially based on Koo & James (J. Fluid Mech., vol. 60, 1973, pp. 513–538) and Steiros & Hultmark (J. Fluid Mech., vol. 853, 2018 pp. 1–11) for screens of arbitrary shapes. In addition, we include an empirical viscous correction factor accounting for viscous effects in the vicinity of the screen. We characterize experimentally the aerodynamic drag coefficient for a porous square screen composed of fibres, immersed in a laminar air flow with various solidities and different angles of attack. We test various fibre diameters to explore the effect of the space between the pores on the drag force. Using PIV and hot wire probe measurements, we visualize the flow around and through the screen, and in particular measure the proportion of fluid that is deviated around the screen. The predictions from the model for drag coefficient, flow velocities and streamlines are in good agreement with our experimental results. In particular, we show that local viscous effects are important: at the same solidity and with the same air flow, the drag coefficient and the flow deviations strongly depend on the Reynolds number based on the fibre diameter. The model, taking into account 3-D effects and the shape of the porous screen, and including an empirical viscous correction factor that is valid for fibrous screens may have many applications including the prediction of water collection efficiency for fog harvesters.

Direct numerical simulations are performed to investigate the characteristics of a turbulent core–annular flow with water-lubricated high viscosity oil in a horizontal pipe. Six different superficial velocity ratios ($\,j_w/j_o = 0.057\unicode{x2013}0.41$) are examined by changing the water superficial velocity $j_w$ for a fixed oil superficial velocity $j_o$. The pressure drops in the pipe and the shapes of the phase interface agree well with those from previous experiments. The oil core flow is almost a plug flow, and the gaps between the phase interface and pipe wall are narrow and wide near the upper and lower surfaces of the pipe, respectively, due to the buoyancy. Within a narrow gap, water is confined mostly in a valley region of the wavy phase interface and hardly goes through its crest. On the other hand, water near the phase interface at a wide gap convects downstream almost at the core speed and the flow near the wall is similar to that of single-phase wall-bounded turbulent flow. The annular flow is characterized by three different regimes depending on the clearance Reynolds number ($Re_c$) based on the core velocity and local gap size: laminar Couette flow driven by the core for $Re_c \le 600$, transitional flow for $600 < Re_c < 2500$ and turbulent flow for $Re_c \ge 2500$. The minimum pressure drop occurs at $j_w/j_o = 0.11$ in the early transition regime. For all $j_w/j_o$ considered, the negative lift force acting on the core comes from the pressure force which balances the buoyancy.