Particle-laden supersonic jets are often encountered in advanced engineering applications where a comprehensive control of particle dispersion is crucial. Although particle dispersion has been extensively studied in the past, the local mechanisms that cause the radial particle transport, such that particles leave the jet core, remain unclear in supersonic jets. To this end, we conduct a direct numerical simulation of a confined low Reynolds number, perfectly expanded supersonic jet carrying four different-sized particles. Here, particles and gas are simulated with Lagrangian and Eulerian approaches, and the fluid–particle energy and momentum exchange is modelled with two-way coupling. The initial Stokes number of these particles ranges between $1.5$ and $6.0$. We found that each particle size has a specific axial location, $x_r$, where they start travelling radially. This location is defined by a local Stokes number of approximately ${\textit{St}}^* \approx 0.6$; the delay in particles’ response to the local eddies in a supersonic flow causes their ${\textit{St}}^*$ to drop below unity. The local turbulent structures formed by the jet promote the radial transport of the particles that have similar characteristic time scales. Despite two-way momentum coupling, particles and gas influence each other via different mechanisms. For the considered range of ${\textit{St}}$, particles dominantly influence the fluctuating velocity component of the gas, while gas mainly affects the mean velocity component of the particles. Moreover, the particles’ reaction to the compressibility effects is a direct function of particle inertia, where the probability of finding larger particles in a high-density gradient and dilatation region is higher.

We present an experimental study of convection–evaporation of a pool of water evaporating into a quiescent atmosphere. The temperature difference between the bottom of the pool and the surrounding air, as well as the water layer’s aspect ratio $\varGamma$, are systematically varied. Compared with classical Rayleigh–Bénard convection (RBC), this configuration involves a free-surface mechanical upper boundary and a mixed thermal upper boundary in contact with a poorly conducting air layer: evaporation extracts latent heat from the liquid and injects lighter vapour into the air, while radiation adds further cooling. As a result, neither temperature nor heat flux is fixed at the water–air interface, but they are instead strongly coupled. To characterise the respective contributions of convection, evaporation and radiation, we perform three sets of experiments: convection–evaporation, evaporation without bottom heating and convection without evaporation. High-resolution infrared imaging reveals multiple scales of convection at the surface: small hot plumes, cold sheet-like plumes and a large-scale circulation. The latter is constrained by the tank geometry for $\varGamma \lesssim 12$, but several turbulent superstructures develop for larger $\varGamma$. This is reminiscent of RBC but with different temperature statistics, due to the mixed boundary condition. Scaling laws are derived for interfacial transfers and mean surface temperature. Evaporation dominates heat extraction, accounting for 60 %–70 % of the flux, while radiation contributes 15 %–20 %. The release of vapour further enhances coupling between the liquid and air layers. When evaporation is blocked, radiation becomes dominant (70 %–80 %).These results have important implications for industrial and natural systems.

We consider the problem of a cylindrical (quasi-two-dimensional) droplet impacting on a hard surface. Cylindrical droplet impact can be engineered in the laboratory, and a theoretical model of the system can also be used to shed light on various complex experiments involving the impact of liquid sheets. We formulate a rim-lamella model for the droplet-impact problem. Using Gronwall’s inequality applied to the model, we establish theoretical bounds for the maximum spreading radius $\mathcal{R}_{\textit{max}}$ in droplet impact, specifically $k_1 {\textit{Re}}^{1/3}-k_2(1-\cos \vartheta _a)^{1/2}({\textit{Re}}/{\textit{We}})^{1/2}\leq \mathcal{R}_{\textit{max}}/R_0\leq k_1{\textit{Re}}^{1/3}$, valid for ${\textit{Re}}$ and ${\textit{We}}$ sufficiently large. Here, ${\textit{Re}}$ and ${\textit{We}}$ are the Reynolds and Weber number based on the droplet’s pre-impact velocity and radius $R_0$, $\vartheta _a$ is the advancing contact angle (assumed constant in our simplified analysis) and $k_1$ and $k_2$ are constants. We perform several campaigns of simulations using the volume of fluid method to model the droplet impact, and we find that the simulation results fall within the theoretical bounds.

This paper extends the two-layer high-level Green–Naghdi (HLGN) internal-wave model to study boundary time-varying problems, involving moving bottom or surface disturbances. The equations for the two-layer HLGN model with time-varying boundaries are presented, accompanied by a time-domain algorithm for solving these equations. The wave profiles predicted by the HLGN model for internal waves generated by boundary disturbances, whether occurring at the bottom or at the surface, show excellent agreement with results obtained by the fully nonlinear potential-flow (FNPF) solution. For internal waves generated by a surface moving disturbance, the results obtained by the HLGN model show good agreement with the experimental observations and the FNPF solution, including the relationships between the disturbance speed and the resulting wave amplitude and phase speed. Furthermore, the HLGN model is applied to analyse the evolution of the wave profiles and speed generated by the surface disturbance with different moving speeds. In addition, the extended HLGN model incorporates background linear shear currents to examine internal waves generated by a moving bottom disturbance with a linear shear current. The results reveal that background vorticity exerts a pronounced modulation effect on the wave profile and velocity field. Counter-flow narrows the waves and increases their phase speed, whereas co-flow broadens the waves and enhances their amplitude.

This work proposes a data-driven explicit algebraic stress-based detached-eddy simulation (DES) method. Despite the widespread use of data-driven methods in model development for both Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulations (LES), their applications to DES remain limited. The challenge mainly lies in the absence of modelled stress data, the requirement for proper length scales in RANS and LES branches, and the maintenance of a reasonable switching behaviour. The data-driven DES method is constructed based on the algebraic stress equation. The control of RANS/LES switching is achieved through the eddy viscosity in the linear part of the modelled stress, under the $\ell ^2-\omega$ DES framework. Three model coefficients associated with the pressure–strain terms and the LES length scale are represented by a neural network as functions of scalar invariants of velocity gradient. The neural network is trained using velocity data with the ensemble Kalman method, thereby circumventing the requirement for modelled stress data. Moreover, the baseline coefficient values are incorporated as additional reference data to ensure reasonable switching behaviour. The proposed approach is evaluated on two challenging turbulent flows, i.e. the secondary flow in a square duct and the separated flow over a bump. The trained model achieves significant improvements in predicting mean flow statistics compared with the baseline model. This is attributed to improved predictions of the modelled stress. The trained model also exhibits reasonable switching behaviour, enlarging the LES region to resolve more turbulent structures. Furthermore, the model shows satisfactory generalization capabilities for both cases in similar flow configurations.

We present a linear stability analysis of two-dimensional magnetoconvection considering the effects of spatial confinement (characterised by the aspect ratio $\varGamma$) and magnetic field (characterised by the Hartmann number $\textit{Ha}_{i=x,y,z}$ with subscript representing its direction). It is found that when the magnetic field is perpendicular to the convection domain ($y$-direction), it does not affect the onset of convection due to zero Lorentz force. With a magnetic field in the $z$ (vertical) or $x$ (horizontal) directions, the onset of convection is delayed, resulting in a larger critical Rayleigh number $Ra_c$ for the onset of convection. We outline phase diagrams showing the dominating factors determining $Ra_c$. When $\varGamma \leqslant 0.83\textit{Ha}_z^{-0.5}$ for vertical and $\varGamma \leqslant 0.66\textit{Ha}_x^{-1.01}$ for horizontal magnetic field, $Ra_c$ is mainly determined by the geometrical confinement with $Ra_c=502\varGamma ^{-4.0}$. When $\varGamma \geqslant 2^{1/6}\pi ^{1/3}\textit{Ha}_z^{-1/3}$ for vertical and $\varGamma \geqslant 5$ for the horizontal magnetic field, $Ra_c$ is mainly determined by the magnetic field with $Ra_c=\pi ^2\textit{Ha}^2$. In the intermediate regime, both the magnetic field and spatial confinement determine $Ra_c$, and a horizontal magnetic field is found to suppress convection more than a vertical magnetic field. In addition, under a horizontal magnetic field, there exists a subregime characterised by $Ra_c = 9.9\,\varGamma ^{-2.0} \textit{Ha}_x^2$, which is explained by a theoretical model. The magnetic field also modifies the length scale $\ell$. For a vertical magnetic field, $\ell$ decreases with increasing $\textit{Ha}_z$, following $\ell =2^{1/6}\pi ^{1/3}\textit{Ha}^{-1/3}$. For a horizontal magnetic field, when $\varGamma \lt 0.62\textit{Ha}_x^{0.47}$, the flow is a single-roll structure with $\ell$ being the width of the domain. The study thus shed new light on the interplay between magnetic field and spatial confinement.

The effects of wall temperature on hypersonic boundary layer transition are investigated by analysing the kinematics (acoustic ray trajectories) and mechanics (fluctuation energy production and transport) of second-mode instabilities. The disturbance energy formulation is taken from Roy & Scalo (J. Fluid Mech. 2025, vol. 1007, A49). Flow conditions are taken from a Mach 6 boundary layer over a $3^\circ$ cone, with varying degrees of wall-to-adiabatic temperature ratios, $\varTheta =T_w/T_{\textit{ad}}=0.25{-}1.75$. Boundary layer-resolved axisymmetric direct numerical simulations with companion Laguerre polynomials-based linear stability theory provide the supporting numerical datasets. It was found that second-mode instabilities comprise two decks, separated by the pressure node location $(y=y_\pi )$. The upper deck ($y\gt y_\pi$) is characterised by temperature ($T^{\prime}$) and density ($\rho'$) fluctuations working with in-phase wall-normal velocity fluctuations ($v'$) to sustain the total disturbance energy production term, $-(\rho _0 v'T'\partial T_0/\partial y+\rho ' u_0 v' \partial u_0/\partial y$), which peaks at the generalised inflection point $y=y_i$. The downward-oriented energy flux peaks below the critical layer, $y\lt y_c$, and sustains acoustic energy accumulation in the lower deck. Effective energy transfer requires the streamwise and wall-normal fluxes to maintain a $90^\circ$ phase difference. This is satisfied especially for colder walls, whereas heated walls yield out-of-phase $v'$–$T'$ and in-phase pressure ($p'$) – streamwise velocity ($u'$) fluctuations, reducing the disturbance energy production and discouraging the coupling between the two decks. Ray tracing reveals the trajectory of purely acoustic wave paths emanating from the wall, as trapping occurs below the generalised inflection line $(y_i)$, governed by the mean flow velocity gradients $(\partial u_0/\partial y)$.

The effects arising from interactions between two identical starting jets on their propulsive characteristics have been investigated numerically for different dimensionless distances $S/D$ (the distance between two jet axes normalised by nozzle diameter, from 1.1 to 4) and stroke ratios $L/D$ (the length-to-diameter ratio of jet column, from 2 to 5). The two jets are arranged in parallel and initiated simultaneously with identical conditions. Their leading vortical structures evolve from an axisymmetric to a plane-symmetric configuration, with deceleration in regions where the two jet wakes approach each other. The generation of axial thrust is affected, primarily dominated by variations in the pressure thrust component. This results from the combination of the mutually induced pressure (MIP) and the coupling effects of vortex rings (CEVR for $S/D\gt1.5$ and CEVR-R for $S/D\lt1.5$). The MIP governs the fluctuations introduced into thrust development, while CEVR (CEVR-R) is responsible for the reduction in average thrust. These effects become more pronounced as $S/D$ decreases, but remain almost unaffected by $L/D$. Adjusting the acceleration and deceleration rates of the velocity program shows limited effects on either the thrust fluctuations or the average thrust reduction. Furthermore, the interaction induces two lateral force components with equal magnitude but opposite directions on the outer walls of the two nozzles, with their magnitude exceeding $15\,\%$ of the axial thrust. The introduction of an additional vertical wall within the nozzle exit plane effectively eliminates the lateral force. However, it consequently enhances both the average thrust reduction and the thrust fluctuations induced by the interaction.

We carry out an experimental study of granular flow in a quasi-two-dimensional wedge-shaped hopper, with glass front and back walls, using videography, along with image analysis and particle tracking. Results are presented for different orifice sizes and roughnesses of the sidewalls for nearly spherical glass and steel particles of different sizes. The data for the radial velocity in the hopper (wedge angle $2\theta _w$) are well described by $v_r(r,\theta )=v_{r0}(r)[1-F(r)(\theta /\theta _w)^2],$ in cylindrical coordinates $(r,\theta )$, with the origin at the apex of the wedge. The centreline velocity is given by $v_{r0}=(a_0/r+a_1)$, and the effective wall friction by $F=(b_0+b_1r)$, where $a_0$ and $a_1$ increase with orifice width, while $b_0$ increases with roughness. For the smooth wall system, we obtain $F\in (0,1)$, however, for the rough walls $F\gt 1$ for most cases, with the velocity at the wall being zero, and a few layers of slow-moving particles adjacent to the wall. The mass flow rate scaled by the particle density and the radial velocity profile are independent of the particle density, for a threefold increase in the density, implying insignificant inertial effects. Discrete element method simulations are carried out using glass particles for a system of the same size as the experimental hopper, with the simulation parameters calibrated to closely match the experimental results. The simulation results indicate that the variation in the direction normal to the plane of the flow is small and the radial velocity profiles without the front and back walls are similar to the experimental profiles.

The stress tensor is calculated for dilute active suspensions composed of colloidal Janus particles propelled by self-diffusiophoresis and powered by a chemical reaction. The Janus particles are assumed to be spherical and made of catalytic and non-catalytic hemispheres. The reaction taking place on the catalytic part of each Janus particle generates local molecular concentration gradients at the surface of the particle and, thus, an interfacial velocity slippage between the fluid and the particle, which is the propulsion mechanism of self-diffusiophoresis. In the dilute-system limit, the contributions of the suspended particles to the stress tensor are calculated by solving the linearised chemohydrodynamic equations for the fluid velocity and the molecular concentrations around every Janus particle considered as isolated and far apart from each other. The results are the following. First, the well-known Einstein formula for the effective shear viscosity of colloidal suspensions is recovered, including the effect of a possible uniform Navier slip length. Next, two further contributions are obtained, which depend on the molecular concentrations of the fuel and product species of the reaction, on the concentration gradients, and on the orientation of the Janus particles. The second contribution is caused by simple diffusiophoresis, which already exists in passive suspensions with global concentration gradients and no reaction. The third contribution is due to the self-diffusiophoresis generated by the chemical reaction, which arises in active suspensions. The calculation gives quantitative predictions based on the geometry of the Janus particles and on the constitutive properties of the fluid and the fluid–solid interfaces.

Cavitation inception in the wake of propulsor systems often arises from the interaction between multiple vortices. We use large-eddy simulation (LES) to study cavitation during the canonical interaction of a pair of unequal strength counter-rotating vortices generated in the wake of a hydrofoil pair at a chord-based Reynolds number ($ \textit{Re}$) of $1.7 \times 10^6$. The simulations reproduce the experimental observations by Knister et al. (In 33rd Symposium on Naval Hydrodynamics, Osaka, Japan, 2020) of spatially and temporally intermittent inception events occurring in the weaker vortex. Sinusoidal instabilities representing the Crow instability develop on the weaker vortex beyond one chord length downstream of the hydrofoils, causing it to bend and wrap around the stronger vortex. The inviscid stretching causes a significant reduction of the weaker core pressure and inception occurs as it approaches close to the stronger core. These intermittent inception events correspond to $3{-}4$ fold pressure reduction from the unperturbed value, with the instantaneous pressures reaching $40\,\%{-}60\,\%$ lower than the mean minimum pressure. However, the loss of circulation (${\gt} 20\,\%$) in both cores during the later stages of interaction reduces the possibility of further inception events. Statistical analysis reveals that inception occurs once per Crow cycle and is most likely to occur near the central regions of the Crow wavelength. Conditional averages show that the axial stretching is non-uniform along the weaker vortex axis, with the stretching intensities in the central regions being four times larger than the wavelength-averaged value. Probability distribution analysis shows that only a small portion of the weaker core experiences inception pressures and these regions have relatively lower axial stretching intensities compared with the bulk of the core.

We present a study on the melting dynamics of neighbouring ice bodies by means of idealised simulations, focusing on collective effects, with the goal of obtaining fundamental insight into how collective interactions influence the melting of ice. Two neighbouring (vertically or horizontally aligned), square-shaped and equally sized ice objects (size of the order of centimetres) are immersed in quiescent fresh water at a temperature of ${20}\,^\circ \textrm {C}$. By performing two-dimensional direct numerical simulations, and using the phase-field method to model the phase change, the collective melting of these objects is studied. When the objects are horizontally aligned, no significant influence of the neighbouring object on the melting time is observed. On the other hand, when vertically aligned, although the melting of the upper object is mostly unaffected, the melting time and the morphology of the lower ice body strongly depends on the initial inter-object distance. We report that the melting of the bottom object can be enhanced by more than 10 %, or delayed more than 20 %, displaying a non-monotonic dependence on the initial object size. We show that this behaviour results from a non-trivial competition between layering of cold fluid, which lowers the heat transfer, and convective flows, which favour mixing and heat transfer. For this melting in mixed convection, we were able to collapse our data onto a single curve.

Numerical simulations and theoretical analysis are conducted to investigate the Atwood-number dependence of perturbation evolution in a shocked heavy fluid layer. For layers without reverberating waves, a higher Atwood number of one interface significantly enhances its coupling effect on the perturbation growth at the opposite interface. A theoretical model incorporating the startup, linear and nonlinear stages is developed to predict the interface mixing width. Dimensionless formulae are derived, identifying eight distinct modulation regimes of multi-interface instability. When reverberating waves are present, the individual effects of the upstream ($A_1$) and downstream ($A_2$) Atwood numbers are examined. The model is further modified to account for additional reverberating waves required at higher $A_2$ values for accurate amplitude prediction. Both theory and simulations demonstrate that perturbation growth at one interface can be actively controlled by adjusting the Atwood number of the opposite interface. These findings provide insights for mitigating instabilities in applications such as inertial confinement fusion through appropriate material selection.

Negatively electrified liquid cone jets supported on capillary tubes with 30–36 $ \,\unicode{x03BC} \text{m}$ tip diameters are investigated in vacuo with four ionic liquids (ILs) selected for their high electrical conductivity and low viscosity. All four use the same cation 1-ethyl-3-methylimidazolium$^+$ ($ \text{EMI}^+$), paired with the four anions $\textrm{SCN}^-$, $\text{N}(\text{CN})_2^-$, $\text{C}(\text{CN})_3^-$ and $ \text{BF}_4^-$. Purely ionic (PI) emissions are not unambiguously achieved with any of the four, but are closely approached by all under a broad range of conditions. In this unusual quasi-ionic (QI) regime, drops contribute minimally to the current ($\sim$ 0.5 %–3 %) but substantially to the mass flow. A sharp QI$\rightarrow$PI transition below a critical liquid flow rate has been demonstrated for capillary emitters by Caballero-Perez et al. (2025) J. Propul. Power, for 1-butyl-3-methylimidazolium-C(CN)$_3$ (BMI-C(CN)$_3$) by using 15 $\unicode{x03BC}$m capillary tips able to stabilise unusually small liquid flow rates. None of their other 3 ILs achieves the QI regime, indicating the singularity of BMI-C(CN)$_3$ and our four ILs. We focus on the peculiarities of the QI regime, the likely mechanism for the QI$\rightarrow$PI transition and argue that ILs reaching the QI regime will probably also attain the PI regime when sprayed from sufficiently small capillary tips. Paradoxically, while high conductivity and low viscosity appear to favour the QI mode, for a liquid operating in this regime, inverting these properties by lowering the emitter temperature appears to better approach the PI regime.

We investigate and model the initiation of motion of a single particle on a structured substrate within an oscillatory boundary layer flow, following a mechanistic approach. By deterministically relating forces and torques acting on the particle to the instantaneous ambient flow, the effects of flow unsteadiness are captured, revealing rich particle dynamics. Laboratory experiments in an oscillatory flow tunnel characterise the initiation and early stages of motion, with particle imaging velocimetry measurements yielding the flow conditions at the motion threshold. The experiments validate and complement results from particle-resolved direct numerical simulations, combining an immersed boundary method with a discrete element method that incorporates a static friction contact model. Within the parameter range just above the motion threshold, the mobile particle rolls without sliding over the substrate, indicating that motion initiation is governed by an unbalanced torque rather than a force. Both experimental and numerical results show excellent agreement with an analytical torque balance including hydrodynamic torque derived from the theoretical Stokes velocity profile, and contributions of lift, added mass and externally imposed pressure gradient. In addition to static and rolling particle states, we identify a wiggling regime where the particle moves but does not leave its original pocket. Our deterministic approach enables prediction of the phase within the oscillation cycle at which the particle starts moving, without relying on empirical threshold estimates, and can be extended to a wide range of flow and substrate conditions, as long as turbulence is absent and interactions with other mobile particles are negligible.

We investigate the linear stability of a Plateau border and the existence of solitary waves. Firstly, we formulate a new non-orthogonal coordinate system that describes the specific geometry of a Plateau border. Within the framework of this coordinate system, the equations of motion for the fluid potential and free surface are derived. By performing a linear stability analysis we find that the Plateau border is stable under small perturbations. Next, a weakly nonlinear theory is developed, leading to a Korteweg–de Vries equation for the free surface profile. Our weakly nonlinear evolution equation predicts depression solitary waves, such as those observed by Bouret et al. (Phys. Rev. Fluids., 2016, vol. 1 no. 4, p. 043902).

We study flows generated within a two-dimensional corner by the chemical activity of the confining boundaries. Catalytic reactions at the surfaces induce diffusio-osmotic motion of the viscous fluid throughout the domain. The presence of chemically active sectors can give rise to steady eddies reminiscent of classical Moffatt vortices, which are mechanically induced in similar confined geometries. In our approach, an exact analytical solution of the diffusion problem in a wedge geometry is derived and coupled to the diffusio-osmotic slip-velocity formulation, yielding the stream function of associated Stokes flow. In selected limiting cases, simple closed-form expressions provide clear physical insight into the underlying mechanisms. Our results open new perspectives for the design of microscale mixing strategies in dead-end pores and cornered microfluidic channels, and offer benchmarks for numerical simulations of confined (diffusio-)osmotic systems.

This work presents an analytical solution for the steady laminar wake generated by a finite wall segment acting as a sink for heat or mass transfer. The classical Lévêque solution is extended to include the wake region downstream of the active surface by employing Laplace transform methods to couple Dirichlet and Neumann boundary value problems through convolution identities. This yields a unified closed-form expression for the scalar field that reduces to the Lévêque result above the sink and provides a new analytical expression for the wake region. Numerical simulations confirm the analytical solution, with errors decreasing systematically under mesh refinement. The derived expressions enable direct calculation of scalar recovery at any point in the wake, providing essential information for designing segmented systems where wake interference between adjacent active elements must be predicted. The solution also serves as a benchmark for numerical methods solving mixed boundary value problems in convective transport.

This research investigates the hydrodynamics of a physical boundary transition from free slip to no slip, which usually occurs in ice-jams, large wood and debris accumulation in free-surface flows. Using direct numerical simulation coupled with a volume penalisation method, a series of numerical simulations is performed for an open-channel flow covered with a layer of floating spherical particles, replicating the laboratory set-up of Yan Toe et al. (2025 J. Hydraul. Eng., vol. 151, 04025010). Flow transition from the open channel to the closed channel induces a new boundary-layer development at the top surface, accompanied by a flow separation and an increased bottom shear stress that enhances particle mobility at the bottom. Analysis of a fully developed flow in an asymmetric roughness channel (rough surface at the top boundary and smooth surface at the bottom boundary) also shows that the vertical position of maximum velocity is higher than the position of zero Reynolds shear stress, which supports the experimental observation of Hanjalić & Launder (J. Fluid Mech., vol. 51, 1972, pp. 301–335), demonstrating the shortcoming of traditional turbulence closure models such as the $k{-}\varepsilon$ model. Finally, the stagnation force acting on a particle at the leading edge of the accumulation layer is compared with the analytical prediction of Yan Toe et al. Understanding the flow transition improves the prediction of the stability threshold of the accumulation layer and design criteria for debris-collection devices.

Experimental deep reinforcement learning (DRL) control of a turbulent boundary layer is conducted for the first time at $Re_\tau$ = 1196, with the aim of friction-drag reduction. Two hot films, an impinging plasma jet actuator array and two wall hot wires act as the state detector, flow disturber and reward evaluator, respectively. The control law parametrised by a radial basis function network is executed in real time on a field programmable gate array and optimised using a classical value-based algorithm (deep Q-network). Results show that DRL control requires only 30 s to train a closed-loop control law with satisfactory drag-reduction performance. Compared with open-loop control where only fine-tuned periodical forcing can reduce the friction drag, the experimental efficiency is improved significantly. Proper setting of the hyper-parameters is crucial in DRL. Particularly, the reward time delay and control frequency need to match the convection time scale and the characteristic frequency of the turbulent boundary layer. The optimal DRL control setting achieves 6.7 % relative drag reduction, almost three times that of the best open-loop control (2.3 %). Physically, plasma actuation induces alternating low-speed and high-speed zones that confine the sidewise motion of turbulent streaks. The final control law optimised by DRL can be simplified as a threshold control, firing the plasma actuator after perceiving a streak burst event and a long-lasting high-speed zone. Control benefits are attributed to the increase in the occurrence probability of high-reward states and the elevation of mean reward at different clusters.