Fluids at supercritical pressure (SCP) exhibit significant real-fluid effects across the pseudo-critical point, which challenges the validity of the existing wall-scaling laws developed under atmospheric pressure condition. This study revisits prior efforts on the temperature-based transformation for the collapse of mean scalar profiles, emphasising the difficulties in accurately describing universal characteristics of thermal boundary layers at SCP. To address this, a novel thermal scaling law using enthalpy transformation is proposed by incorporating the chain rule and heat flux balance. This transformation effectively accounts for variations in the near-wall thermophysical properties associated with the scalar profile while excluding the gradient of isobaric specific heat capacity-related terms. The proposed scaling law demonstrates substantially improved alignment of transformed mean scalar profiles in SCP channel flows at different wall-temperature differences and Reynolds numbers. Additionally, the enthalpy transformation shows superior performance compared with the existing enthalpy–velocity relations, particularly near the heated-wall region where the fluid thermodynamic states undergo the pseudo-boiling process. The present work could facilitate the development of universal wall model in supercritical flows, enabling rapid and reliable heat transfer predictions in practical applications.

Direct numerical simulation is performed to study the effects of spanwise curvature on transitioning and turbulent boundary layers. Turbulent transition is induced with an array of resolved cuboids. Spanwise curvature is prescribed using a novel approach with a body force that is applied orthogonally to the bulk flow to curve the mean free-stream streamlines at a set radius. The flows are analysed in a streamline-aligned coordinate system. Although the radius of curvature is large compared with the size of the boundary layer, its effects on the development of the boundary layer are appreciable. The results indicate that spanwise curvature induces a non-uniform mean secondary flow and alters the structure of turbulence within the boundary layer. Analytical expressions for the crossflow are derived in the viscous sublayer and log layer. These alterations are visible as changes in the distribution of the turbulent stresses and alignment of the vortical structures with the mean flow. These modifications are responsible for a misalignment between the Reynolds stress tensor and the velocity gradient tensor, which has important consequences for the validity of the widely used Boussinesq turbulent viscosity hypothesis in Reynolds-averaged Navier–Stokes models. Spanwise curvature was observed to decrease turbulent kinetic energy. These results have important implications on the development of turbulence in general applications, such as the flow over a prolate spheroid.

Previous studies show that at the small scales of stably stratified turbulence, the scale-dependent buoyancy flux reverses sign, corresponding to a conversion of turbulent potential energy (TPE) back into turbulent kinetic energy (TKE). Moreover, the magnitude of the reverse flux becomes stronger with increasing Prandtl number $\textit{Pr}$. Using a filtering analysis we demonstrate analytically how this flux reversal is connected to the mechanism identified in Bragg & de Bruyn Kops (2024 J. Fluid Mech. vol. 991, A10) that is responsible for the surprising observation that the TKE dissipation rate increases while the TPE dissipation rate decreases with increasing $\textit{Pr}$ in stratified turbulence. The mechanism identified by Bragg & de Bruyn Kops, which is connected to the formation of ramp–cliff structures in the density field, is shown to give the scale-local contribution to the buoyancy flux. At the smallest scales this local contribution dominates and explains the flux reversal, while at larger scales a non-local contribution is important. Direct numerical simulations of three-dimensional statistically stationary, strongly stably stratified turbulence confirm the theoretical analysis, and indicate that, while on average the local contribution only dominates the buoyancy flux at the smallest scales, it remains strongly correlated with the buoyancy flux at all scales. The results show that ramp cliffs are not only connected to the reversal of the local buoyancy flux but also the non-local part. At the small scales (approximately below the Ozmidov scale), ramp structures contribute exclusively to reverse buoyancy flux events, whereas cliff structures contribute to both forward and reverse buoyancy flux events.

This work uses large-eddy simulations to study the transition of a tulip flame stabilised by bubble vortex breakdown (BVB) mode towards a V flame stabilised by a conical vortex breakdown (CVB). The transition is triggered when the equivalence ratio is increased, resulting in a rise in temperature within the central recirculation zone (CRZ). Simultaneously, the pressure inside the CRZ bubble increases, while the average pressure inside the combustion chamber remains constant. This increase in pressure causes the CRZ bubble to open radially and expand, changing the vortex breakdown mode from BVB to CVB, and the flame shape from a tulip shape to a V shape. A criterion for a limit pressure inside the CRZ was then devised based on the radial momentum equation and the balance between centrifugal force and radial pressure gradient, found to control the radial motion of the CRZ. This criterion helped us to understand the main events of the transition, showing that, once the pressure inside the CRZ exceeds the limit given by the criterion, the flow topology changes from a BVB mode to a CVB mode. This transition highlights the differences between a BVB mode and a CVB mode, showing for the first time that there are characteristic pressure and velocity profiles for each mode in their swirling jets and CRZ. Finally, a significant achievement of this work is the identification of a novel mechanism for the controlled transition of vortex breakdown mode, a combustion-driven transition of vortex breakdown mode.

The modal and non-modal stability of laminar flow in a rectangular microchannel is investigated by incorporating the effects of Coriolis forces due to rotation, cross-sectional aspect ratio and superhydrophobic wall slip. The full Navier–Stokes equations are linearised into modified Orr–Sommerfeld and Squire equations, which are then formulated as an eigenvalue problem using small disturbances of the Tollmien–Schlichting type. These equations are subsequently solved by the spectral collocation method. The transition to instability in rotating microchannel flows, influenced by aspect ratio and slip conditions, is analysed through eigenvalue spectra and neutral stability curves. For non-modal analysis, we express the solution in matrix exponential form and then, using the singular value decomposition method, calculate the maximum energy growth. The study reveals that the flow becomes unstable in the presence of rotation at a critical Reynolds number of $ Re_c \approx 40$ for a low aspect ratio and $ Re_c \approx 50.4$ for a high aspect ratio. We find that instability is more pronounced in spanwise-rotating flows at higher aspect ratios compared with those at lower aspect ratios. Rotation induces disturbances from both walls along the spanwise direction, forming secondary flow structures near the centreline. Furthermore, we examine the influence of anisotropic slip by separately considering streamwise and spanwise slip as limiting cases. The numerical results demonstrate that while streamwise slip has a stabilising effect on rotating flows at small scales, a sufficiently large spanwise slip length can trigger instability at Reynolds numbers lower than those observed in the no-slip case. Rotation has the potential to enhance non-modal transient energy growth, while streamwise slip can effectively suppress this instability. These findings suggest that the onset of instability and transient energy growth can be effectively regulated by adjusting the aspect ratio and spanwise slip of the channel walls.

The dual-tone transition phenomenon and its formation mechanism in the flow around a heptagonal cylinder (side number $N= 7$) are experimentally investigated in depth over a range of free-stream velocities corresponding to Reynolds numbers of the order of $10^4$–$10^5$. Dual tone in this context refers to the emergence of two dominant peaks in the far-field acoustic spectrum when a flow in transition regime passes over a polygonal cylinder in principal orientations. The dual-tone phenomenon is also observed in an $N = 9$ cylinder in the face orientation and an $N = 11$ in the corner orientation, which contrasts with all the other polygonal cylinders of $N\in {\mathbb{Z}}[3,12]$ systematically investigated in the present study, where only a single tonal peak dominates the spectrum, similar to the Aeolian tone observed in the circular cylinder in the subcritical regime. The emergence of the dual tone is found to be responsible for the reduction of far-field noise. Continuous wavelet transform analysis reveals that the occurrence of the two competing tones in the time domain can be empirically modelled by Gaussian distributions. Additional proper orthogonal decomposition based phase averaging using time-resolved planar particle image velocimetry enables coherent vortex structure identification for the two quasi-stable shedding modes, which are responsible for the formation of the two tones. Near-wall flow and pressure fluctuation analysis further confirms that the two tones originate from stochastic shear-layer separation–reattachment switching, thereby generating two patterns of dipole sound sources through distinct vortex formation pathways.

This study presents an active flow control framework for fluid–structure interaction (FSI) systems involving a flexible plate in the wake of a cylinder, by integrating Koopman-based reduced-order models (ROMs) with model predictive control (MPC). Specifically, a novel switched-system control strategy is developed, wherein kernel dynamic mode decomposition (DMD) and residual DMD are jointly employed to construct accurate ROMs capable of capturing strongly nonlinear FSI dynamics. This approach ensures accurate low-order representations across multiple control inputs, while suppressing spurious modes. The resulting Koopman ROMs provide fast state predictions over a receding horizon, enabling an MPC optimiser to determine real-time actuation. To improve control performance, resolvent analysis is utilised to optimise the actuator–probe placement. Remarkably, only three strategically placed structural probes are sufficient to capture dominant Koopman modes and enable effective closed-loop control. The proposed framework is then applied to regulate synthetic jets on the cylinder to suppress the plate’s flapping. It successfully stabilises large-amplitude (LAF), small-deflection (SDF) and small-amplitude (SAF) flapping regimes within a unified control strategy. By combining Koopman modal decomposition with an analysis of system energy evolution, we elucidate distinct control mechanisms across these regimes. For LAF and SAF cases, control is achieved primarily through local modulation of existing saturated modes, which requires relatively low actuation energy. In contrast, stabilisation of the SDF case involves the emergence of entirely new Koopman modes that disrupt the original symmetry-breaking dynamics, resulting in increased control input. The framework matches the control performance of reinforcement learning while markedly reducing computational cost.

Aeroacoustic noise generated by wings under the wing-in-ground (WIG) effect is prevalent in various industrial applications, such as WIG vehicles, tower–blade interactions, and slat device noise issues. At chord-based Reynolds number 50 000 and freestream Mach number 0.3, the introduction of an engine jet transforms the separated stall noise of an NACA 4412 aerofoil under the influence of the WIG effect into laminar boundary layer vortex shedding (LBL-VS) noise. This study investigates the underlying mechanisms of this LBL-VS noise. Instead of relying on acoustic analogy, the unapproximated acoustic field is captured with high fidelity using direct numerical simulations. We identify the vorticity transfer process around the trailing edge as a key mechanism in the generation of LBL-VS noise. The results show that as the ground clearance decreases, the overall noise intensity is reduced. When the clearance becomes sufficiently small (10 % chord length), the well-organised vortex structures above the aerofoil break down under high adverse pressure, transitioning into a turbulent state that disrupts the vorticity transfer process. At this clearance, the dominant noise frequency drops from the vortex shedding frequency to an intermittent bursting frequency. This intermittent behaviour arises because only when certain vortices are amplified by acoustic feedback can they shed from the trailing edge, triggering the vorticity transfer process and generating pressure fluctuations. These findings provide new insights into the LBL-VS noise mechanisms under WIG conditions, and can inform strategies for noise reduction in relevant applications.

We integrate a discrete vortex method (DVM) with complex network analysis to strategise dynamic stall mitigation over aerofoils with active flow control. The objective is to inform the actuator placement and the timing to introduce control inputs during the highly transient process of dynamic stall. To this end, we treat a massively separated flow as a network of discrete vortical elements and quantify the interactions among the vortical nodes by tracking the spread of displacement perturbations between each pair of vortical elements using a DVM. This allows us to perform network broadcast mode analysis to identify an optimal set of discrete vortices, the critical timing and the direction to seed perturbations as control inputs. Motivated by the objective of dynamic stall mitigation, the optimality is defined as maximising the reduction of total circulation of the free vortices generated from the leading edge over a prescribed time horizon. We demonstrate the use of the analysis on a two-dimensional flow over a flat plate aerofoil and a three-dimensional turbulent flow over an SD$7003$ aerofoil. The results from the network analysis reveal that the optimal timing for introducing disturbances occurs slightly after the onset of flow separation, before the shear layer rolls up and forms the core of the dynamic stall vortex. The broadcast modes also show that the vortical nodes along the shear layer are optimal for introducing disturbances, hence providing guidance to actuator placement. Leveraging these insights, we perform nonlinear simulations of controlled flows by introducing flow actuation that targets the shear layer slightly after the separation onset. We observe that the network-guided control results in a $21 \,\%$ and $14\,\%$ reduction in peak lift for flows over the flat plate and SD$7003$ aerofoil, respectively. A corresponding decrease in vorticity injection from the aerofoil surface under the influence of control is observed from simulations, which aligns with the objective of the network broadcast analysis. The study highlights the potential of integrating the DVMs with the network analysis to design an effective active flow control strategy for unsteady aerodynamics.

This direct numerical simulation study analyses the transition-to-turbulence in plane Couette flow (PCF) with three-dimensional (3-D) roughness. Square ribs of height $k=0.2h$ (where $h$ is the half-channel height) and a streamwise pitch separation of $\lambda =10k$, classified as k-type roughness, are mounted on the stationary wall. This configuration features alternating rough and smooth zones in the spanwise direction and is referred to as 3-D k-type roughness. We compare the behaviour of 3-D k-type roughness in the transitional regime with that of two-dimensional (2-D) k-type roughness (Gokul & Narasimhamurthy 2024 J. Fluid Mech. vol. 1000, p. A40). The route-to-transition in 3-D k-type roughness confirms the formation of laminar–turbulent patterns, which exist in the transitional Reynolds number range $\textit{Re} \in [325, 350]$. This range interestingly overlaps with those for smooth PCF ($\textit{Re} \in [325, 400]$) and 2-D k-type roughness ($\textit{Re} \in [300, 325]$), thereby indicating that 3-D k-type roughness amalgamates the characteristics of both the rough and smooth zones. In striking contrast to the 2-D k-type roughness, the laminar–turbulent bands in the 3-D k-type configuration are of non-uniform bandwidth. The 3-D k-type roughness surprisingly introduces continuous competition among bands of opposite orientations, a characteristic unique in this case. Due to this competition, the large-scale flow associated with the oblique bands is never fully aligned with the diagonal direction. The smooth zones in the 3-D k-type roughness exhibit complex interactions, evident from oscillatory velocity signals and multiple high-energy peaks in the frequency spectra, which likely contribute to the competing patterns with opposite orientations.

Pulsed gravity currents are generated by the sequential release of dense material into a lighter ambient. We investigate the dynamics of pulsed gravity currents using physical scale experiments, two-dimensional depth-averaged shallow water equation (SWE) based models and three-dimensional lattice Boltzmann method (LBM) simulations. Integrating these results we show for the first time that short duration pulsed releases generate intrusive layers, which accelerate front propagation relative to an instantaneously released current of the same total volume. Conversely, a long delay time between pulses produces a current that propagates slower than an equivalent instantaneous release. This finding is supported by physical experiments and depth-resolving LBM simulations. The depth-resolving simulations show that intrusions in pulsed flows experience less drag resistance than those generated by instantaneous releases. The depth-averaged model considered in the present study does not accurately capture the intrusive flow dynamics of pulsed currents. However, the limitations of the finite-depth SWE model may be mitigated by extensions to incorporate entrainment and density stratification. The results also motivate further research into the impact of buoyancy Reynolds number and channel slope on the propagation of pulsed currents.

We investigate the role of slippery boundaries, quantified by the Navier boundary friction coefficient $\beta$, in regulating heat transport and flow structures in rotating Rayleigh–Bénard convection. Owing to the Ekman pumping effect arising from viscous boundary layers that is intensified with increasing boundary friction, it is found that the properties of global heat transport exhibit two distinct parameter regimes separated by a transitional Rayleigh number ($ \textit{Ra}_t$). In the rotation-dominated regime ($ \textit{Ra} \lt \textit{Ra}_t$), enhanced viscous friction increases the efficiency of Ekman pumping, significantly elevating the Nusselt number and lowering the convection onset threshold. Conversely, in the buoyancy-dominated regime ($ \textit{Ra} \gt \textit{Ra}_t$), boundary-induced viscous dissipation suppresses convective motions, thereby reducing heat transport. Large-scale vortices (LSVs), prevalent under free-slip conditions, progressively dissipate as $\beta$ increases, revealing that viscous friction disrupts the inverse energy cascade from baroclinic to barotropic modes. Through kinetic energy partitioning analysis, the transition between quasi-two-dimensional and three-dimensional turbulent states is identified, with the parameter $\beta _{\textit{cr}}$ following a generic scaling relation on the Prandtl (Pr) and Ekman (Ek) numbers $\beta _{\textit{cr}}\sim \textit{Pr}^{-0.67}\textit{Ek}^{-1.18}$. This relation enables us to predict LSV emergence across different parameter spaces. Furthermore, it is reported that the heat-transport scaling exponent, the convection onset and the partitioning of kinetic energy between barotropic and baroclinic components undergo a smooth flow transition at $\beta _{\textit{cr}}$. These results also indicate a direct correlation between Ekman pumping efficacy and the friction coefficient $\beta$, demonstrating that controlling boundary friction can modulate global transport properties and reshape flow structures.

To advance understanding of the influence hill-slope and hill-shape have on neutrally stratified turbulent air flow over isolated forested hills, we interrogate four turbulence-resolving simulations. A spectrally friendly fringe technique enables the use of periodic boundary conditions to simulate flow over isolated two-dimensional (2D) and three-dimensional (3D) hills of cosine shape. The simulations target recently conducted wind tunnel (WT) experiments that are configured to fall outside the regimes for which current theory applies. Simulation skill for flow over isolated 3D hills is demonstrated through matching the canopy and hill configuration with the recently conducted WT experiments and comparing results. The response of the mean and turbulent flow components to 2D versus 3D hills along the hill-centreline are discussed. The phase and amplitude of spatially varying flow perturbations over forested hills are evaluated for flows outside the regime valid for current theory. Flow over isolated 2D forested hills produces larger amplitude vertical motions on a hill’s windward and leeward faces and the speed-up of the mean wind compared with that over isolated 3D forested hills at the hill-centreline. The 3D hills generate surface pressure minima over hill-crests that are only half the magnitude of those over 2D hills. The spatial region over which hill-induced negative pressure drag acts increases with increasing hill steepness. Assumptions in partitioning the flow into an upper layer with an inviscid response to the hill’s pressure field are robust and lead to solid predictions of hill-induced perturbations to the mean flow; however, applying those assumptions to predict the evolution of the turbulent moments only provides approximate explanations at best.

Submesoscale processes, typically shaped by intricate interactions between frontal dynamics and turbulence, have significant impacts on the transport of momentum, heat and biogeochemical tracers in the ocean. This study employs large-eddy simulations to investigate submesoscale frontogenesis and arrest in the ocean surface boundary layer. We compare a single-sided front with a dense filament, which can be viewed as a two-sided front. Both cases exhibit a similar life cycle, including frontogenesis driven by secondary circulation, frontal arrest due to the growth of instability and turbulence, and eventual frontal decay. One major difference is that the filament remains stationary throughout its life cycle, while the front propagates towards the denser side. Another distinction lies in the relative contributions of horizontal and vertical turbulent fluxes. In the filament case, horizontal (cross-front) turbulent flux dominates and effectively counteracts the frontogenetic tendency induced by secondary circulation, leading to frontal arrest. In contrast, both vertical and horizontal turbulent fluxes are crucial for the arrest of the single-sided front. Horizontal shear production is the primary source of turbulence in the filament, associated with the emergence of horizontal coherent eddies and consistent with the characteristics of horizontal shear instability. For the front, the development of horizontal eddies is less pronounced, and vertical shear production plays a more important role. This study reveals the similarities and differences between the dynamics of submesoscale fronts and filaments, as well as the role of turbulence in their evolution, providing insights for improved representation of these processes in ocean models.

Interfacial interactions between gas bubbles and the free surface are a hallmark of flows involving aqueous foams. In practice, bubble foams commonly arise from processes such as breaking waves at the ocean–atmosphere interface, plunging liquid jets and the effervescence of carbonated liquids. Once generated, bubbles within foam layers remain afloat at the free surface for finite durations before finally bursting into a fine spray of droplets. While the birth and bursting of bubble foams have received considerable attention, the understanding of floating bubbles is limited mainly to a single bubble. To build on this, in this article, we undertake numerical simulations of two or more floating bubbles in various canonical settings to examine their geometry and self-organising nature, with implications for real-world phenomena such as ocean spray production. Under lateral confinement, floating bubbles are prone to form vertically stacked layers. To this end, we analyse the geometry of coaxial pairs of floating bubbles and link geometrical differences between single and coaxial bubbles to various aspects of the ensuing bursting stage. Furthermore, we extend the existing theory of isolated floating bubbles to obtain unified analytical expressions for the shape parameters of single and coaxial bubbles of small sizes. Next, we investigate a pair of side-by-side floating bubbles, which serves as a minimal configuration to understand the formation of bubble rafts through self-organisation. We discover that Bond numbers in the range $10\leqslant \textit{Bo}\leqslant 50$ are more favourable for raft formation due to pronounced capillary attraction. The time required for two floating bubbles to assemble through capillary attraction grows exponentially with their initial separation. We also develop a linear model to capture the evolution of bubble spacing during capillary migration at low Bond numbers. Lastly, we extend the two-bubble configuration and showcase the emergent dynamics of a swarm of floating bubbles in mono- and bilayer configurations.

This work presents wavepacket models for supersonic round twin jets operating at perfectly expanded conditions, computed via plane-marching parabolised stability equations based on mean flows obtained from the compressible Reynolds-averaged Navier–Stokes (RANS) equations. High-speed schlieren visualisations and non-time-resolved PIV measurements are performed to obtain experimental datasets for validating the modelling strategy. The RANS solutions are found to be in good quantitative agreement with the particle image velocimetry (PIV) mean-flow measurements, confirming the ability of the approach to capture the interaction between jets at the mean-flow level. The obtained wavepackets consist of toroidal and flapping fluctuations of the twin-jet system, and show similarities with those of single axisymmetric jets. However, for the case of closely spaced jets, they exhibit deviations in the phase speed of structures travelling in the outer mixing layer and those travelling in the inner one, leading to different non-axisymmetric behaviours. In particular, toroidal twin-jet wavepackets feature tilted ring-like structures with respect to the jet axis, while flapping twin-jet wavepackets are distorted and lose the clean chequerboard pattern typically observed in $m = 1$ modes in axisymmetric jets. A quantitative comparison of the modelled wavepackets with experimentally educed coherent structures is performed in terms of their structural agreement measured through an alignment coefficient, providing a first validation of the modelling strategy. Alignment coefficients are found to be particularly high in the intermediate range of studied frequencies.

Nucleation phenomena associated with cloud cavitation about a three-dimensional (3-D) NACA$\,$16-029 hydrofoil are explored experimentally in a cavitation tunnel where susceptible free stream nuclei are absent. Microbubble nuclei are found to be intrinsically generated by cavity collapse and become sequestered in the low-momentum separated region ahead of the cavity leading edge. Nuclei dynamics upstream of a shedding sheet cavity was investigated using high-speed photography. Measurements were performed at zero incidence for cavitation numbers in the range of $0.55 \gt \sigma \gt 0.45$, and chord-based Reynolds numbers of $ \textit{Re} = 0.75\times 10^6$ and $ \textit{Re} = 1.5\times 10^6$. Nuclei are generated each shedding cycle due to cavity breakup from condensation shock-wave phenomena. These nuclei may undergo immediate activation or transport due to pressure gradients, local re-circulation and jetting. Some nuclei remain upstream of the cavity leading edge over multiple cycles. Several phenomena influence this behaviour, including cyclical variation of the boundary layer properties with each shedding cycle. A major conclusion of the work is that these nuclei are produced in a self-sustaining manner from near surface, small scale, interfacial or viscous phenomena rather than from surface or free stream nuclei. Additionally, these experiments reveal the low-momentum region upstream of the cavity to be above vapour pressure, despite the meta-stable tension developed in the boundary layer further upstream of the cavity.

The coalescence and breakup of drops are classic examples of flows that feature singularities. The behaviour of viscoelastic fluids near these singularities is particularly intriguing – not only because of their added complexity, but also due to the unexpected responses they often exhibit. In particular, experiments have shown that the coalescence of viscoelastic sessile drops can differ significantly from that of their Newtonian counterparts, sometimes resulting in a sharply distorted interface. However, the mechanisms driving these differences in dynamics, as well as the potential influence of the contact angle are not fully known. Here, we study two different flow regimes effectively induced by varying the contact angle and demonstrate how that leads to markedly different coalescence behaviours. We show that the coalescence dynamics is effectively unaltered by viscoelasticity at small contact angles. The Deborah number, which is the ratio of the relaxation time of the polymer to the time scale of the background flow, scales as $\theta ^3$ for $\theta \ll 1$, thus rationalising the near-Newtonian response. On the other hand, it has been shown previously that viscoelasticity dramatically alters the shape of the interface during coalescence at large contact angles. We study this large contact angle limit using two-dimensional numerical simulations of the equation of motion. We show that the departure of the coalescence dynamics from the Newtonian case is a function of the Deborah number and the elastocapillary number, which is the ratio between the shear modulus of the polymer solution and the characteristic stress in the fluid.

Boundary-layer instability and transition control have drawn extensive attention from the hypersonic community. The acoustic metasurface has become a promising passive control method, owing to its straightforward implementation and lack of requirement for external energy input. Currently, the effects of the acoustic metasurface on the early and late transitional stages remain evidently less understood than the linear instability stage. In this study, the transitional stage of a flat-plate boundary layer at Mach 6 is investigated, with a particular emphasis on the nonlinear mode–mode interaction. The acoustic metasurface is modelled by the well-validated time-domain impedance boundary condition. First, the resolvent analysis is performed to obtain the optimal disturbances, which reports two peaks corresponding to the oblique first mode and the planar Mack second mode. These two most amplified responses are regarded as the dominant primary instabilities that trigger the transition. Subsequently, both optimal forcings are introduced upstream in the direct numerical simulation, which leads to pronounced detuned modes before breakdown. The takeaway is that the location of the acoustic metasurface is significant in minimising skin friction and delaying transition onset simultaneously. The bispectral mode decomposition results reveal the dominant energy-transfer routine along the streamwise direction – from primary modes to low-frequency detuned modes. By employing the acoustic metasurface, the nonlinear triadic interaction between high- and low-frequency primary modes is effectively suppressed, ultimately delaying transition onset, whereas the late interaction related to lower-frequency detuned modes is reinforced, promoting the late skin friction. The placement of the metasurface in the linearly unstable region of the second mode delays the transition, which is due to the suppressed streak in the oblique breakdown scenario. However, in the late stage of the transition, the acoustic metasurface induces an undesirable increment of skin friction overshoot due to the augmented shear-induced dissipation work, which mainly arises from reinforced detuned modes related to the combination resonance. Meanwhile, by restricting the location of the metasurface upstream of the overshoot region, this undesirable augmentation of skin friction can be eliminated. As a result, the reasonable placement of the metasurface is crucial to damping the early instability while causing less negative impacts on the late transitional stage.

This work investigates the weakly nonlinear dynamics of internal shear layers and the mean zonal flow induced by the longitudinal libration of an inner core within a spherical shell. Building on the work of He et al. (2022 J. Fluid Mech., vol. 939, p. A3), which focused on linear dynamics, we adopt a similar set-up to explore the nonlinear regime using both asymptotic theory and numerical computations, with Ekman numbers as low as $E=10^{-10}$. A specific forcing frequency of $\widehat {\omega }=\sqrt {2}\widehat {\varOmega }$, where $\widehat {\varOmega }$ denotes the rotation rate, is introduced to generate a closed rectangular path of characteristics for the inertial wave beam generated at the critical latitude. Our approach extends previous results by Le Dizès (2020 J. Fluid Mech., vol. 899, p. A21) and reveals that nonlinear interactions are predominantly localised around regions where the wave beam reflects on the boundary. We derive specific scaling laws governing the nonlinear interactions: the width of the interaction region scales as $E^{1/3}$ and the amplitude of the resulting mean zonal flow scales as $E^{1/6}$ in general. However, near the rotation axis, where the singularity of the self-similar solution becomes more pronounced, the amplitude exhibits a scaling of $E^{-1/2}$. In addition, our study also examines the nonlinear interactions of beams that are governed by different scaling laws. Through comparison with numerical results, we validate the theoretical predictions of the asymptotic framework, observing good agreement as the Ekman number decreases.