Model-based active flow control requires predictive models that are accurate, stable and fast enough for real-time optimisation. In controlled wake flows, this is often achieved through reduced-order models (ROMs) that first compress high-dimensional velocity snapshots into a latent space and then learn a time-stepping predictor for the dynamics in the latent space. Here, we study how the choice of the spatial encoder affects the predictability of the resulting latent coordinates for wake flows under control inputs. Using two actuated two-dimensional wake configurations, a simplified truck wake and the fluidic pinball, we compare proper orthogonal decomposition (POD) against nonlinear convolutional autoencoders (CAEs) and two types of variational autoencoders for compression, and evaluate several temporal predictors based on long short-term memory networks. The CAEs achieve higher compression efficiency and sharper short-term reconstructions, but they produce a latent dynamics that is more irregular and with broadband spectral content. As a consequence, long-horizon forecasts degrade faster and show a higher probability of catastrophic divergence than POD-based models. The POD yields smoother latent trajectories that are easier to learn and extrapolate, leading to more reliable predictions beyond the short-term regime. These results reveal a clear trade-off between compactness and forecast accuracy, and suggest that the stability of the latent dynamics prediction can outweigh maximal compression. This is particularly relevant for control strategies rooted in forecasts of the dynamics, such as model predictive control and reinforcement learning. The findings provide practical guidance for designing actuation-aware, hardware-feasible predictive ROMs for real-time flow control.

Flow separation in rocket nozzles generates intense side loads that can lead to structural damage affecting space flight safety. Recent studies have hypothesised that a screech-like resonance loop may be responsible for such oscillations in the free shock separation (FSS) regime. In this work, we aim to confirm if screech is indeed present in this flow. Five thrust-optimised parabolic nozzles with different length and area ratios were examined at a nozzle pressure ratio range where FSS was expected to occur. Spectral proper orthogonal decomposition (SPOD) revealed resonance for both $m=0$ and $\,m=1$ azimuthal modes. Further decomposition of the SPOD modes allowed for the identification of downstream- and upstream-travelling waves, whose characteristics match the description of the Kelvin–Helmholtz (KH) instability wave and the guided jet mode (GJM), respectively. Linear stability analysis performed on Reynolds-averaged Navier–Stokes mean-flow fields confirms the existence of these two waves for the conditions studied and provides insight into their dispersion relation. A well-validated screech predictive model was employed and accurately predicted the resonant frequencies. The accurate predictions, combined with the observation of the KH and GJM waves, confirm that screech is active in this flow.

The spectral budget of the vertical heat flux $\langle w\theta \rangle$ is investigated in turbulent Rayleigh–Bénard convection by direct numerical simulation. Eight cases spanning three Rayleigh numbers ($\textit{Ra} = 1\times 10^5$, $6\times 10^6$ and $4\times 10^8$) and up to three geometric aspect ratios each are computed at $\textit{Pr} = 1$. The premultiplied spectra of $\langle w\theta \rangle$ reveal a two-peak structure when both $\textit{Ra}$ and $\varGamma$ are sufficiently large. Production of $\langle w\theta \rangle$ is dominated by temperature fluctuations; production by velocity fluctuations is confined to the near-wall region and the large-scale production that develops at sufficient $\varGamma$ is restricted to the bulk. At large $\varGamma$ an inverse inter-scale cascade transfers $\langle w\theta \rangle$ from intermediate to superstructure scales, the transfer towards the superstructure scale taking place away from the boundary layer. This upscale transfer is specific to the covariance, as the vertical kinetic energy and the temperature variance cascade towards small scales, so that the turbulence selectively transfers the correlation between vertical velocity and temperature towards the superstructure scale, consistent with the strong large-scale coherence of the superstructures. Pressure transport is the only mechanism that delivers large-scale $\langle w\theta \rangle$ towards the wall with appreciable magnitude, yet the pressure–strain term destroys the transported quantity so that the net pressure–temperature interaction is purely negative at all scales. No mechanism in the budget delivers large-scale spectral content into the boundary layer, providing a scale-by-scale explanation for the insensitivity of the Nusselt number to the aspect ratio beyond a moderate threshold.

The wake dynamics and regime transitions of a streamwise-rotating angular particle in an inertial flow at moderate Reynolds numbers are investigated by particle-resolved direct numerical simulations, with particular focus on a rotating cube. We first identify the mechanisms of vortex generation on the front edges and the rear face, showing that rotation-induced vortex merging governs several key regime transitions. A previously unreported spearhead regime is uncovered, characterised by a helical near-wake and a steady filamentary far-wake structure. We demonstrate that it originates from the partial merging of edge vortices and the central rotation-induced vortex. Streamwise rotation is found to moderately stabilise the wake, and to induce a quadratic scaling of both the drag coefficient $\overline {C_d}$ and the streamwise torque $\overline {T_x}$ with the rotation rate $\varOmega$. The evolution of the lift coefficient closely mirrors that of the recirculation length $L_r$. For the rotating cube, the streamwise torque data collapse onto a single curve when expressed in terms of the rotation Reynolds number ${\textit{Re}}_{\varOmega }$. Frequency analyses reveal that the fundamental transverse-force frequency is set by the number of rotational symmetry folds $n_{\varOmega }$, while higher harmonics reflect the nonlinear nature of the flow–particle interaction. We further extend the analysis to general angular particles in their dual angular positions. Particles in dual configurations (sRCV, denoting a rotating cube with one vertex facing the incoming flow, and sROF, denoting a rotating octahedron with a front face facing the incoming flow) can display nearly identical regime transitions, as they share the same rotational symmetry folds and comparable vortex-pair strengths. Hydrodynamic responses remain strongly dependent on particle shape and orientation. Their frequency signatures likewise collapse when organised by $n_{\varOmega }$.

We study stirring and dispersion in steady flow through two-dimensional heterogeneous porous media. A fluid is stirred by the porous medium structure that leads to spatial flow variability and the deformation of fluid elements. While dispersion measures the extension of a solute distribution, stretching underpins solute mixing, chemical reactions and biological activity. Despite this, much less is known about fluid stretching in porous media. Its relation to the dispersion process on the one hand and the flow and medium properties on the other remain open questions. To close this gap, we probe the heterogeneity mechanisms that are at the origin of stretching and dispersion of fluid elements, and quantify the upscaled laws that govern their dynamics. Thus we perform detailed numerical simulations of Darcy flow in two-dimensional porous media characterised by a spatially varying hydraulic conductivity. Dispersion is analysed in terms of first-passage time distributions of fluid elements and the temporal evolution of their displacement mean and variance, while deformation is studied in terms of the probability density function and average elongation of fluid elements. We find that the dispersion and stretching behaviours are closely related. The exponents that govern the decay of the first-passage time distributions and evolution of the displacement variances also determine the evolution of the mean elongation. This dynamics is quantified in a single upscaled theoretical framework that bridges between Eulerian flow structure, large-scale dispersion and stretching of fluid elements.

Due to attractive inter-particle forces, cohesive particles suspended in turbulence undergo a complex process of aggregation, breakup and restructuring. Despite a growing body of knowledge on the `flocculation’ of cohesive granular materials suspended in homogeneous isotropic turbulence, little focus has so far been placed on wall-bounded flows where turbulence and shear are inhomogeneous. This study presents a first investigation of a fully developed wall-bounded flow of resolved cohesive particles. Five direct numerical simulations of turbulent channel flows laden with finite-sized particles at successively increasing cohesive strength are performed. A population balance equation (PBE) framework is used to analyse aggregate dynamics. When integrated over the full domain, the PBE is closed by aggregation and breakup alone. However, this balance is found to not hold locally in the wall-normal direction, where regions of net aggregate production and depletion are identified. This imbalance is shown to be compensated by the size-dependent wall-normal transport of aggregates, revealing a mean circulation: larger aggregates are preferentially produced in the channel centre and migrate toward the wall where they break, while smaller aggregates are transported away from the wall, grow and re-enter the cycle.

Wall-bounded turbulent flows are chaotic and multiscale, rendering fast prediction at high Reynolds numbers computationally prohibitive in applications such as wind farms. Classical data assimilation is based on repeated solutions of the governing equations and, thus, inherits this cost. Generative models learn the probability distribution of flow states, enabling scalable probabilistic reconstruction. Our generative framework couples a variational autoencoder with a diffusion transformer to generate four-dimensional spatiotemporal samples. Bayesian conditioning enables data assimilation without retraining and allows statistical constraints to be imposed through sampling. The framework is applied to a subdomain of turbulent plane Couette flow, where the corresponding direct numerical simulations in this generation region requires $O(10^6)$ spatial degrees of freedom. Using $O(10)$ latent spatial degrees of freedom, the model achieves a compression ratio of $O(10^5)$, which is one to two orders of magnitude above prior reports. It reproduces single-point statistics up to fourth order and the energy spectra, as well as the intermittency and phase-sensitive structure captured by velocity-increment and $Q$–$R$ statistics. Two assimilation scenarios demonstrate that, when observations are statistically consistent with the prior, conditional diffusion models with the proposed sampling strategy preserve complex turbulent statistics in the posterior. However, enforcing these constraints while preserving physical fidelity and sample diversity introduces an inherent trade-off. Excessive conditioning can distort the learned prior, paralleling limitations of classical ensemble-based data assimilation, where this can likewise degrade the prior covariance. These results highlight both the promise of diffusion models as probabilistic surrogates and the challenges of conditioning them.

A body-forcing strategy is proposed to model the effects of surface roughness on heat transfer in a turbulent flow. The model introduces a source term into the temperature transport equation that depends on a characteristic velocity scale, the local wall-relative temperature and the roughness density. We test it in direct numerical simulation (DNS) of a turbulent open channel at a constant Prandtl number of 0.71, and through extensive comparison with roughness-resolved DNS data, we show that the instantaneous local friction velocity is the appropriate velocity scale. Despite its simple formulation, it accurately captures the salient physics of rough-wall heat transfer, such as the breakdown of the Reynolds analogy, the characteristic plateau of the temperature roughness function and the fully rough behaviour of the ratio of roughness lengths of momentum and heat transfer. An appropriate decomposition of the total heat flux reveals that the scaling of the conductive and body-forcing Stanton numbers is similar to that of the exposed and sheltered Stanton numbers from fully resolved roughness simulations, respectively. This suggests that the primary effect of roughness on heat transfer can be effectively captured so long as the surface heat flux is modelled correctly in a statistical sense, even if the precise contribution from individual roughness elements is not accurately resolved. Finally, the body-forcing model can be used over regular no-slip smooth walls, bypassing the need to resolve the exact details of the roughness geometry.

We investigate the gas flow induced in a closed, two-dimensional square cavity by imposed differences in the temperatures of its walls. The problem is studied over the entire range of gas rarefaction rates. Building upon the known free-molecular description, we derive an asymptotic solution at high yet finite Knudsen numbers. The latter is derived based on the Bhatnagar–Gross–Krook (BGK), Shakhov and full hard-sphere (HS) Boltzmann collision operator. These are complemented by direct simulation Monte Carlo and particle scheme calculations at arbitrary rarefaction rates, together with a thermodynamic ‘pure conduction’ continuum solution at vanishing $\textit{Kn}$. The gas is found at an expectedly macroscopically quiescent state in both limiting states: the collisionless free-molecular limit, where heat is transported ballistically, and the leading-order Navier–Stokes–Fourier continuum limit, in which heat is transported by pure conduction. At large yet finite Knudsen numbers, however, even a small number of molecular collisions suffices to generate vortical flow, which magnitude vanishes as the free-molecular limit is approached. Analysis of the BGK asymptotic correction identifies the microscopic origin and rotation direction of this circulation. The asymptotic results are found in excellent agreement with the numerical calculations through $O(1)$ Knudsen numbers. The variation of the gas total kinetic energy with gas rarefaction is analysed, exhibiting a maximum at $\sim O(10^{-1})$ Knudsen numbers. The maximum location and magnitude are quantified. The quantitative differences between the HS, BGK and Shakhov model predictions are discussed.

Mean flow and instability measurements of a nominally axisymmetric, transitional separation bubble arising from a flare-induced shock-wave/boundary-layer interaction were conducted on a cone–flare geometry in hypersonic flow. Simultaneous infrared thermography and high-speed, high-resolution background-oriented schlieren measurements provided a global, time-resolved characterisation of both the mean and unsteady flow. Experiments were performed in the AFOSR–Notre Dame Large Mach-6 Quiet Tunnel under conventional noise at free-stream unit Reynolds numbers ranging from $5.8\times 10^6$ to $12.3\times 10^6$ m$^{-1}$. The model comprised a $7^\circ$ half-angle circular cone, a $20^\circ$ half-angle flare and three nose radii. Complementary laminar and turbulent base flows, convective and global stability analyses (GSAs) and direct numerical simulations (DNSs) were carried out at the experimental conditions. With inflow forcing of disturbances, the DNS reproduces the experimentally observed flare-heating trends. The flow exhibits both convective and global instabilities, with the latter giving rise to stationary and oscillatory three-dimensional disturbances within the recirculation bubble. Thermal streaks observed in global heating measurements provide experimental evidence of these instabilities, which is further corroborated by the onset of three-dimensionality in DNS. Quantitative agreement of streak wavelengths and their spatial prevalence across experiments, GSAs and DNSs confirms the recirculation bubble’s global instability is the primary driver of the reattachment streaks in this flow, providing direct experimental evidence of this mechanism.

Large-eddy simulations of open-channel flow are conducted to investigate how the channel aspect ratio (), defined as the width ($B$) to depth ($h$) ratio (), affects secondary currents (SCs), large-scale motions (LSMs) and very-large-scale motions (VLSMs). Four cases spanning are considered. For wide-channel flows (), certain flow features dominate depending on the distance from the sidewall, which are: (i) secondary current instability (SCI) within ${\sim} 0.6h$ of the sidewalls; (ii) VLSMs and LSMs from distance $2.4h$ onwards from the sidewalls; and (iii) prevalence of LSMs between (i) and (ii). In the SCI-dominated region (i), two time-averaged SC cells appear, one near the bottom wall–bed corner (BV) and another near the surface (SV). The size and position of the BV and SV oscillate quasi-periodically with a wavelength ${\sim} 6h$, reflecting the instability of the secondary currents. In the VLSM/LSM-dominated region (ii), pre-multiplied spectra of the streamwise velocity fluctuations peak at ${\sim} 3h$ and ${\sim} 20h$, consistent with the presence of LSMs and VLSMs. In the LSM-dominated region (iii), velocity spectra exhibit a LSM prevalent peak at ${\sim} 3h$. The near-sidewall SCI region (i) and central VLSM/LSM region (ii) are phase independent in their oscillations, likely due to a barrier effect of LSM-dominated region (iii) between them. For , the VLSM/LSM region (ii) disappears, while the other two regions exhibit distinct alterations. At , the SCI-dominated region resembles that for , while VLSMs in the LSM-dominated region have a remarkably shorter wavelength (${\sim} 10h$). At a lower (), VLSMs are completely absent in the LSM-dominated region. Based on these results, a schematic framework of the interrelations between SCs and VLSMs across different channel is proposed.

Process-based studies of the transition to turbulence beneath internal solitary waves (ISWs) have typically assumed a laminar bottom boundary layer (BBL) and two-dimensional flow. In natural environments, however, ISWs often propagate over turbulent three-dimensional BBLs. Here, we perform three-dimensional high-resolution spectral numerical simulations of periodic ISWs of depression at momentum-thickness Reynolds numbers representative of laboratory ($Re_{ISW}=300$) and lake ($Re_{ISW}=900$) scale conditions to examine how ambient turbulence modifies the transition to turbulence beneath ISWs. When the ambient flow is laminar, this transition is governed by a modal instability. At $Re_{ISW}=300$, the instability generates localised vortices that subsequently break down, while at $Re_{ISW}=900$ it produces sustained vortex shedding and a turbulent wake in the lee of the wave. In contrast, when the ambient flow is turbulent, the modal pathway is bypassed. Free-stream turbulence excites near-wall streamwise streaks (Klebanoff modes) beneath the front shoulder of the wave, which amplify, undergo secondary instability and break down into turbulent bursts. The resulting turbulence develops upstream of the modal-instability region. With increasing Reynolds number and stronger ambient turbulence, the onset of turbulence shifts progressively toward the ISW trough. These results show that ambient turbulence triggers a bypass mechanism fundamentally different from the modal-instability-driven transition in laminar environments. By suppressing the modal instability and shifting turbulence generation closer to the wave trough, bypass transition provides an explanation for field observations that cannot be reconciled with laminar-transition models. Accurate prediction of near-bed turbulence and sediment transport in natural environments therefore requires accounting for bypass transition.

We consider the dynamics of two equal, spherical liquid drops of Newtonian fluid, brought into contact at a negligible speed, and merging driven by surface tension. The effects of an outer fluid or that of gravity are disregarded, so that the entire motion is determined uniquely by the Ohnesorge number $\textit{Oh }= \eta /\sqrt {\rho \gamma R}$, where $\eta$ is the viscosity, $\rho$ the density, $\gamma$ the surface tension and $R$ the drop radius. Using methods of matched asymptotics, in the limit that the minimum radius $r_0$ of the bridge connecting the drops is much smaller than the initial drop radius, we compute the dynamics for $r_0(t)$, as well as the shape of the gap between the drops. For very early times, the translational motion of the drops along the axis of symmetry is arrested by inertia, which affects the shape of the scaling function describing the gap on the scale of $r_0$. For $r_0/R \approx \sqrt {\ln \textit{Oh}}/\textit{Oh}$, the motion crosses over to an entirely viscous similarity solution, passing through a sequence of time-dependent scaling functions, which we compute. Our results agree well with full numerical simulations of the Navier–Stokes equation, as well as experiment.

We investigate the presence of intermittency, a well-known phenomenon in Newtonian turbulence, in viscoelastic Taylor–Couette flow at fixed inertia, (Reynolds number, $\textit {Re}$ = 120). Using particle image velocimetry measurements in the r–z plane for a broad range of elasticities, $\textit{El}$ = 0–0.64, we reveal a dual role of viscoelasticity, even at moderate $\textit {Re}$: it can destabilise the flow and trigger turbulence at moderate elasticity but suppress turbulence intensity at higher values. Gradual turbulence suppression is evident through reduced velocity fluctuations, longer integral time scales proportional to the polymer relaxation time and weakened coherent structures as the flow approaches elasto-inertial turbulence. Higher-order statistics indicate that elasticity increases spatial and temporal intermittency while altering the dynamics: spatially, through a reduced viscous range and enhanced large-scale decorrelation; temporally, via a characteristic time scale consistent with polymer-driven state switching rather than a classical cascade. This spatio-temporal intermittency manifests as alternating active/hibernating turbulence cycles. During hibernation, the flow reorganises into a state distinct from active turbulence, characterised by low velocity fluctuations and Reynolds stresses, suppressing not only turbulence but also linearly unstable Newtonian states such as Taylor vortices. Rather than being featureless, this state exhibits weakened coherent structures in the bulk of the flow and sustained near-wall dynamics, where elastic stresses are likely generated. The present results demonstrate that hibernation is not exclusive to Newtonian turbulence, but also emerges in viscoelastic flows under conditions where inertia alone is insufficient to trigger or sustain turbulent motion.

We present a theoretical and experimental study of the pressure-driven flow of a suspension of non-Brownian, negatively buoyant particles through a tall horizontal rectangular duct. We address two limits of the flow, namely the cases of smooth and rough side walls, corresponding to traction-free and no-slip boundary conditions on the suspension, respectively. In both cases, any solution to the system can be fully classified by the dimensionless fluid pressure gradient and the average solid fraction in the duct. In the case of smooth side walls, the flow becomes horizontally uniform, and admits semi-analytical solutions, which we compute and find asymptotic expressions for in the limits of large and small pressure gradient. We investigate the results of modifying this system with a Coulomb friction condition on the side walls. With rough side walls, the flow adopts both cross-duct and vertical dependence. We study the limit in which cross-duct shear stress dominates, and the solid fraction is nearly uniform across the duct, to avoid cross-duct buoyancy gradients, classifying all solutions. We carry out laboratory experiments with suspensions of glass beads to compare with the two models, and find more favourable qualitative agreement with the smooth-side-wall model, modified by a Coulomb friction condition.

We consider the flow of a gravity current released from a lock inside a tank which performs horizontal oscillations (equivalent to internal pressure-gradient pulses), in the buoyancy–inertia (high-Reynolds-number) regime. We develop a shallow-water model which provides clear-cut insights and simple predictions of the propagation, height and velocity fields as a function of time, for a wide range of parameters. Comparisons with the simulations of Bingol et al. (2025 J. Fluid. Mech., vol. 1002, A14) support the approximate analysis. Moreover, we point out a new effect, called pulse-buoyancy, which was not covered by these simulation. The clear-cut predictions of the model are amenable to straightforward experimental verification, and suggest further numerical work for an as yet unexplored parameter range.

The spatial and temporal dynamics of separated flow induced by a fence with a splitter plate in uniform flow at moderate Reynolds numbers were investigated using time-resolved particle image velocimetry. The Reynolds numbers, based on the free-stream velocity and the fence height above the splitter plate, were 2700, 5500 and 7500. The mean separated flow is characterised by a large primary clockwise recirculating vortex above the splitter plate and a smaller counter-rotating secondary vortex in the corner of the fence and splitter plate. Guided by different vorticity thickness growth rates, the separated shear layer is partitioned into four regions, whose dynamics are governed by the initial roll-up, shear layer stabilisation, wall-induced large-scale vortex breakdown and post-reattachment recovery. Streamwise momentum transport, entrainment velocity, convective velocity, spectral proper orthogonal decomposition, as well as spatial autocorrelation and whole-field integral time scales are used to examine the dynamics of the shear layer across these four regions. Spectral analysis reveals a distinct frequency migration along the shear layer, wherein the Kelvin–Helmholtz frequency near the fence transitions to the vortex shedding frequency near reattachment through successive vortex pairing events. Analysis of instantaneous reverse flow areas demonstrates that the second half of the primary vortex dominates the global unsteadiness of the separation bubble, whereas the first half exhibits strong coupling between the primary and secondary vortices. Conditional averaging analysis indicates that flapping motion significantly modulates turbulence production, however, the dynamic importance of the mean velocity gradients and Reynolds stresses varies across the four regions.

A self-consistent saturation model for the prediction of aeroacoustic limit cycles emerging in turbulent low-Mach cavity flows (Reynolds number $Re=\mathcal{O}(10^5)$, Mach number $M\simeq 0.2$) is proposed. It predicts the nonlinear interactions between the acoustic modes of a deep rectangular cavity and the hydrodynamic instabilities of the turbulent shear layer that forms over its opening due to the presence of a grazing flow. The model is based on the triple decomposition of the flow variables and the compressible Navier–Stokes equations. At each step of the iterative process, the nonlinear eigenvalue problem associated with perturbations around the mean flow is updated with the steady component of the forcing from the unstable eigenmode’s Reynolds stress. The iterations are performed until the dominant eigenmode becomes marginally stable, i.e. its growth rate vanishes. The evolution of the coherent velocity fluctuations as a function of the oscillation amplitude is in good qualitative agreement with previously published compressible large eddy simulations. Furthermore, the predictions of the frequency and amplitude of the aeroacoustic limit-cycle oscillations are validated against the ones obtained from a low-order model, whose parameters were adjusted to reproduce the experimental measurements of the deep cavity whistling.

Radially spreading extensional flows can become unstable and develop tearing patterns consisting of rifts and tongues. Such patterns have been observed in laboratory experiments with strain-rate-softening (shear-thinning) polymer solutions and in spreading ice shelves. Linear stability analysis predicts that axisymmetric spreading of power-law fluids becomes unstable when the fluid is shear thinning. However, understanding the emergence and evolution of rifts requires analysis of the nonlinear flow regime. In Part 1 of this study we developed a numerical simulation of the full nonlinear equations and showed that Newtonian fluids remain globally stable, consistently with theoretical predictions and experimental observations. Here we extend that framework to explore power-law fluids. We demonstrate that purely viscous shear-thinning fluids can develop rifts, and evolve rift-tongue patterns consistently with theoretical and experimental findings. Starting from an axisymmetric base state, we examine the resulting flow under monochromatic perturbations over a range of amplitudes, wavenumbers, fluid properties, and boundary and initial conditions. We find that for shear-thinning fluids, rift formation depends strongly on perturbation amplitude, while being less sensitive to the perturbation wavenumber. The mechanism reflects competition between a destabilising radial force associated with hoop stresses and a stabilising force arising from radial normal-stress divergence. As the fluid gets more shear thinning, the stabilising contribution weakens, facilitating rift formation. These results provide new insights into the nonlinear mechanisms governing rift formation in polymer solutions and in spreading ice shelves.

The axial flow through a thin gap between two concentric cylinders is investigated using direct numerical simulation (DNS) of the Navier–Stokes equations. It is shown that by imposing a stationary, harmonic, azimuthal velocity at the inner cylinder wall, drag reduction is always attained while power saving is limited to a portion of the explored amplitude–axial wavelength space. The largest values of drag reduction and power saving are $44\,\%$ and $11\,\%$, respectively, obtained for different amplitude–axial wavelength pairs. It has been verified that the effect of such a control on a convex surface aligns well with the known concept in plane channel flows of depressing in turn the pressure–strain, the radial Reynolds stress, the turbulent shear stress and ultimately the viscous drag. A direct connection between the wall motion, through the Stokes layer, and the pressure–strain reduction has been identified as the root cause of the passive flow control effectiveness.