Low Reynolds number hydrodynamic interactions are generally considered both deterministic and reversible due to their linearity. However, the role of soft interactions in deformable suspensions drives nonlinear effects with ambiguous consequences. On the one hand, nonlinearities can be responsible for soft chaos, i.e. long-time apparent randomisation resulting from sensitivity to initial conditions. On the other hand, they can also drive steady streaming and/or drifting effects leading to alignment and ordering. Here, we conduct a comprehensive study on the binary interaction of elastic capsules positioned in different shear planes using high-fidelity particle-resolved simulations. The effects of alignment angle, inter-surface distance, capillary number and size ratio are systematically explored. Based on interaction stability, three regimes are identified: leapfrog, minuet and a novel capturing regime. Unlike leapfrog and minuet motions, where the satellite capsule ultimately escapes from the reference capsule, the capturing motion forms a stable doublet aligned along the vorticity direction. We reveal that capturing is a gentle interaction, which induces only minimal deformation and stress. The mechanism underlying the capturing regime is attributed to the interplay between periodic oscillations induced by the central capsule and steady drift along the vorticity direction. Harmonic analysis of interaction frequencies further underscores the nonlinearity inherent to this dynamics. Extending beyond binary systems, we show that this mechanism relays into ternary alignment, suggesting a generic route to chain formation, demonstrating that nonlinear hydrodynamic interactions alone can drive spontaneous ordering of deformable particles.

Using linear stability analysis, we study the onset and formation mechanism of wall modes in confined magnetoconvection cells with the degree of confinement characterised by the cell aspect ratio $\varGamma$. We first outline the phase diagram of the dominating factors that determine the critical Rayleigh number $Ra_c$ for the onset of convection in the $\varGamma -Ha$ phase space, with $\textit{Ha}$ being the Hartmann number. Our study shows that $Ra_c$ is primarily determined by geometrical confinement, and bulk convection onset occurs with $Ra_c = 1090 \varGamma ^{-4.0}$ for $\varGamma \lt \varGamma _{c_1} = 1.21 \textit{Ha}^{-0.48}$. No wall modes form and $Ra_c$ depends on the strength of both the confinement and magnetic field for $\varGamma _{c_1} \leqslant \varGamma \lt \varGamma _{c_2} = 4.07 \textit{Ha}^{-0.53}$. For $\varGamma _{c_2}\leqslant \varGamma \lt \varGamma _{c_3}=0.99 \textit{Ha}^{-0.10}$, wall modes emerge and $Ra_c$ drops below the bulk onset Rayleigh number for magnetoconvection. When $\varGamma \geqslant \varGamma _{c_3}$, wall modes become fully developed with an onset Rayleigh number for wall modes $Ra_{c,w} \approx 65 \textit{Ha}^{1.5}$. In this fully developed regime, the radial velocity profile and $Ra_{c,w}$ become independent of $\varGamma$. Through analysing the length scales of wall modes and their interaction with spatial confinement, we show dynamically how wall modes emerge in confined cells: while the first layer with a characteristic length scale $\ell _1 = 1.04 \textit{Ha}^{-0.56}$ forms when $\varGamma \geqslant 5.39 \textit{Ha}^{-0.58}$, the second layer with a characteristic length scale $\ell _2 = 4.94 \textit{Ha}^{-0.56}$ emerges when $\varGamma \geqslant 9.07 \textit{Ha}^{-0.53}$. These scaling relations provide practical guidelines for experimental and numerical studies of the wall-mode dynamics.

Roll patterns on floating ice shelves have been suggested to arise from viscous buckling under compressive stresses. A model of this process is explored, allowing for a power-law fluid rheology for ice. Linear stability theory of uniformly compressing base flows confirms that buckling modes can be unstable over a range of intermediate wavelengths when gravity does not play a dominant role. The rate of compression of the base flow, however, ensures that linear perturbations have wavelengths that continually shorten with time. As a consequence, linear instability only ever arises over a certain window of time $t$, and its strength can be characterised by finding the net amplification factor a buckling mode acquires for $t\to \infty$, beginning from a given initial wavenumber. Bi-axial compression, in which sideways straining flow is introduced to prevent the thickening of the base flow, is found to be more unstable than purely two-dimensional (or uni-axial) compression. Shear-thinning enhances the degree of instability in both uni-axial and bi-axial flow. The implications of the theoretical results for the glaciological problem are discussed.

This study experimentally investigates passive drag reduction on a sphere using azimuthally spaced surface protrusions under subcritical Reynolds numbers, focusing on the effects of the protrusion number at fixed surface coverage. The proposed surface modification strategy, termed partial protrusions, maintains a constant total protruded area while varying the number of protrusions $N$, thereby adjusting their azimuthal spacing. The objective is to determine whether such configurations can outperform the conventional full protrusion, in which protrusions continuously surround the azimuthal direction, and to elucidate the flow mechanisms behind any observed enhancement. Drag and flow field measurements reveal that increasing $N$ significantly improves aerodynamic performance. When $N$ exceeds a certain threshold, the partial-protrusion configuration achieves a greater drag reduction than the full-protrusion case, despite using only half the surface coverage. For low $N$, asymmetric pressure distributions across the protruded and smoothed sides induce unsteady separation delay, leading to shear-layer oscillations and elevated turbulent kinetic energy. As $N$ increases, the azimuthal spacing between protrusions decreases, promoting stable interaction between the two sides and leading to separation delay farther downstream than in the full-protrusion case, along with suppression of flow unsteadiness. These results demonstrate that a well-designed partial-protrusion configuration can outperform the full-protrusion configuration in drag reduction and unsteadiness control, offering new insights into effective passive flow control strategies for bluff body flows.

The compression waves/boundary layer interaction (CWsBLI) in high-speed inlets poses significant challenges for predicting flow separation, rendering traditional shock wave/boundary layer interaction (SWBLI) scaling laws inadequate due to unaccounted effects of the coverage range of compression waves. This study aims to establish a unified scaling framework for CWsBLIs and SWBLIs by proposing an equivalent interaction intensity. Experiments were conducted in a Mach 2.5 supersonic wind tunnel, employing schlieren imaging and pressure measurements to characterise flows induced by curved surfaces at two deflection angles ($10^{\circ }, 12^{\circ }$) and varying coverage ranges of compression waves ($d$). An equivalent transformation method was developed to convert the CWsBLI into an equivalent incident SWBLI (ISWBLI), with interaction intensity derived from pressure gradients considering the coverage range. Key results reveal a critical threshold based on the interaction length of ISWBLI ($L_{\textit{single}}$): when $d \leq L_{\textit{single}}$, the interaction scale remains comparable to ISWBLI; when $d \gt L_{\textit{single}}$, the weakened adverse pressure gradient leads to a reduction in the length scale. The proposed scaling framework unifies the CWsBLIs and SWBLIs, achieving better data collapse compared to the existing methods. This work advances our understanding of complex waves/boundary layer interactions, and provides a prediction method for the length scales of CWsBLIs.

Smooth surface features were recently found to stabilise stationary cross-flow instability (CFI) of swept-wing boundary layers, thus holding potential for passive laminar flow control. Notably, the effect of surface features on the transition location exhibited a significant dependence on the CFI amplitude. In this work, numerical solutions of the harmonic Navier–Stokes (HNS) equations are used to explore the impact of a smooth surface hump on the linear and nonlinear development of stationary CFI under various perturbation amplitudes. Linear simulations identify regions of successive inhibited and enhanced perturbation growth. Despite the recovery of the base flow and perturbation kinetic energy to the reference (i.e. no-hump) state, significantly reduced perturbation growth is observed. The distorted perturbation profile due to the interaction with the hump is postulated to be responsible for this. Increasing the perturbation amplitude results in a response of the flow that is qualitatively similar to the linear case, albeit with increasing local destabilisation of new fundamental (i.e. primary wavelength) structures and higher-order harmonics near the wall. An energy budget analysis reveals that the growth of the fundamental incoming CFI is inhibited through the reduced effectiveness of the lift-up mechanism downstream of the hump. This is preceded by a spatial perturbation shape deformation, governed by (spanwise) transport terms. The results suggest that stabilisation of incoming stationary CFI via smooth surface humps is most effective at low incoming perturbation amplitudes. At higher perturbation amplitudes, newly formed near-wall structures, pre-conditioned by the incoming CFI, overtake the incoming CFI and could anticipate the transition process.

We study the combined effects of natural convection and rotation on the dissolution of a solute in a solvent-filled circular cylinder. The density of the fluid increases with increasing concentration of the dissolved solute, and we model this using the Oberbeck–Boussinesq approximation. The underlying moving-boundary problem has been modelled by combining the Navier–Stokes equations with the advection–diffusion equation and a Stefan condition for the evolving solute–fluid interface. We use highly resolved numerical simulations to investigate the flow regimes, dissolution rates and mixing of the dissolved solute for $Sc = 1$, $Ra \in [10^5, 10^8]$ and $\varOmega \in [0, 2.5]$. In the absence of rotation and buoyancy, the distance of the interface from its initial position follows a square root relationship with time ($r_d \propto \sqrt {t}$), which ceases to exist at a later time due to the finite-size effect of the liquid domain. We then explore the rotation parameter, considering a range of rotation frequency – from smaller to larger, relative to the inverse of the buoyancy-induced time scale – and Rayleigh number. We show that the area of the dissolved solute varies nonlinearly with time depending on $Ra$ and $\varOmega$. The symmetry breaking of the interface is best described in terms of $Ra/\varOmega ^2$.

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.

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.

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.

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 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.

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.

Lagrangian transit times on basin to planetary scales are controlled by the interplay of multiscale processes. The primary advective time scale is set by throughflow currents, such as interhemispheric western boundary currents. Dispersion by mesoscale eddies introduces fluctuations that erase memory and enhance dispersion, widening the transit-time distribution. The tortuous paths of Lagrangian parcels, particularly within ocean gyres, significantly enhance dispersion beyond the levels attributed to mesoscale eddies alone. Additionally, trapping by ocean gyres leads to multimodal distributions of Lagrangian transit times. These processes are illustrated in three complementary contexts: eddy-permitting ocean state estimates, simplified spatially extended three-dimensional flows and diffusively coupled two-dimensional pipe models.

We study the surfing motion of an active particle along a planar interface, separating a semi-infinite layer of gas from a deep layer of liquid. The interface-trapped particle self-propels, thanks to an uneven distribution of surface tension in its immediate vicinity, which itself results from a non-uniform release of an active agent from the particle’s surface. We use the reciprocal theorem in conjunction with singular perturbation expansions to calculate the leading-order contributions to the propulsion speed of the surfer due to the advective transport of mass and momentum when the Péclet and Reynolds numbers (denoted by $\textit{Pe}$ and $\textit{Re}$, respectively) are small but finite. Assuming that the surface tension varies linearly with the concentration of the agent with a slope of negative $\alpha$, we show, perhaps unexpectedly, that the normalised speed for a purely translating (but otherwise arbitrarily shaped) particle, independent of the agent discharge mechanism, can be expressed as $\mathscr{U} = 1 + \mathscr{A} ( 2 \textit{Pe} \ln \textit{Pe} + \textit{Re} \ln \textit{Re} ) + \mathscr{O}(\textit{Pe}) + \mathscr{O}(\textit{Re})$, where the prefactor $\mathscr{A}$ is positive for negative $\alpha$ and vice versa. For reference, the self-propulsion speed of autophoretic Janus spheres varies with $\textit{Pe}$ as $\mathscr{U} = 1 + \mathscr{B} \, \textit{Pe} + {\cdots}$, where $\mathscr{B}$ is positive when the mobility coefficient of the particle is negative and vice versa. Also, the speed of spherical squirmers changes with $\textit{Re}$ as $\mathscr{U} = 1 + \mathscr{C} \, \textit{Re} + \mathscr{O}(\textit{Re})^2$, with $\mathscr{C}$ being positive for pushers and negative for pullers. Our asymptotic formula reveals that the speed of a Marangoni surfer is a non-monotonic function of the Péclet and Reynolds numbers, hinting at the existence of optimal values for both $\textit{Pe}$ and $\textit{Re}$. The information contained within the multiplier $\mathscr{A}$ also offers guidance for customising the shape of the surfer, as well as the release rate and configuration of the agent, to enhance the self-surfing performance. Our general theoretical analysis is complemented by detailed numerical simulations for a representative spherical surfer. These simulations confirm our theoretical predictions and shed light on the effects of intermediate and large values of $\textit{Pe}$ and $\textit{Re}$ on the performance of Marangoni surfers.

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.

The instabilities of a floating droplet under the action of an inclined temperature gradient in the presence of the spatial modulation of the transverse temperature gradient are investigated. The problem is studied numerically in the framework of the slender droplet approximation and the precursor model. It is shown that the spatial modulation of the transverse component of the Marangoni number is accompanied by the change of the droplet shape and can lead to development of periodic oscillations. In the definite region of parameters, quasi-periodic oscillations accompanied by the creation of pulsating satellites have been obtained. The separation and the recombination of the ‘main’ droplet with the satellites have been observed.

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.