We experimentally investigate the self-propulsion of the NACA0015 symmetric airfoil section subject to sinusoidal pitching. A power-spring-based crank–rocker mechanism actuates the airfoil. We employ a rotary apparatus capable of achieving self-propulsion. Self-propulsion across varying pitching frequency, amplitude and pitching-point location is characterised by three distinct scaling relations. These scaling relations relate the self-propulsion Reynolds number ($\textit{Re}_s$) to the frequency Reynolds number ($\textit{Re}_{\!f}$), the pitching amplitude ($\theta _0$) and the pitching-point location ($p$). For $p\lt 0.4$ two scaling relations emerge depending on the pitching amplitude. For $\theta _0 \approx 4.6^\circ$, $\textit{Re}_s \sim (1-2p)(\textit{Re}_{\!f} \theta _0)^{3/2}$ and for $\theta _0 \geqslant 9.4 ^\circ$, up to $\theta _0 = 20.5^\circ$, $\textit{Re}_s \sim (1-2p)^{1/2}\textit{Re}_{\!f}\theta _0^{1/2}$. We call these scaling regimes power scaling and separable scaling, respectively. When pitched near the mid-chord ($p\approx 1/2$), a linear scaling regime emerges with $\textit{Re}_s \sim \textit{Re}_{\!f} \theta _0$. These relations are consistent with the scaling relations derived from balancing inviscid thrust with viscous drag, pressure drag and possibly ’enhanced’ pressure drag for the power, separable and linear regimes, respectively. We observe different vortical patterns in the wake that depend on both the Strouhal number $\textit{St}$ and the normalised distance travelled by the airfoil during a pitching cycle, $U^*_{\textit{BL}}$. The distinct reverse Kármán vortex street, characteristic of self-propelling airfoils, is observed for $\textit{St}$ in the range $0.2$–$0.4$ only when $U^*_{\textit{BL}}$ lies between $0.4$ and $1.0$, with both ranges corresponding to values commonly observed in oscillatory swimmers. These findings provide the necessary experimental foundation for self-propelling rigid pitching airfoils across a wide range of parameters.

This study investigates the flow topology and unsteadiness induced by the interaction of an oblique shock and a supersonic boundary layer developing over an axisymmetric body, using implicit large-eddy simulations. The interaction produces highly three-dimensional flow features, including an open separation region, multiple vortex pairs and strong cross-flow, influenced by the surface curvature and the strong adverse pressure gradient. Three major vortex systems are identified: a primary vortex that detaches from the surface, a secondary vortex near the surface that dominates unsteadiness and a weaker tertiary vortex that dissipates downstream. Turbulence generation mechanisms vary in the azimuthal direction, with production being the dominant source on the windward plane, while turbulent transport contributes mostly in the leeward wake. Windward side unsteady dynamics is dominated by four major frequency bands: a very-low-frequency band ($ \textit{St}_{L} \sim 0.03$, where $St$ is Strouhal number, $St_L$ is the Strouhal number with length of separation bubble as reference length) associated with bubble breathing, low-frequency shear-layer flapping at $ \textit{St}_{L} \sim 0.1$, Kelvin–Helmholtz shedding in the shear layer at $ \textit{St}_{L} \sim 1$ and high-frequency content corresponding to fine-scale convection in the boundary layer. Leeward side dynamics is influenced by the coherent structures generated in the secondary and tertiary vortex pairs, resulting from short-wavelength elliptic instability, leading to vortex meandering. The primary vortex displays a ‘V’-shaped vertical oscillations corresponding to the very-low-frequency band, highlighting the impact of shock-wave/boundary-layer interaction bubble breathing on large-scale vortex oscillations. This low-frequency vortex dynamics has a significant impact on the aerodynamic loading on the cylinder.

We explored the laminar boundary-layer dynamics of natural convection from a horizontal cylinder with a linearly varying wall temperature using scaling analysis and direct numerical simulations. The parameter space covers Rayleigh numbers $10^{6}\leq Ra\leq 10^{8}$, Prandtl numbers $2\leq Pr\leq 100$ and wall-temperature-gradient parameters $0\leq s\leq 2$. The imposed temperature gradient couples with the geometric buoyancy projection, producing distinct initial and steady-state regimes. Initially, the thermal response is controlled by radial diffusion, giving $\delta _T\sim Fo^{1/2}$ and $ \textit{Nu}\sim Fo^{-1/2}$, where $Nu$ is the Nusselt number and $ \textit{Fo}$ is the Fourier number, independently of angular position and wall-temperature non-uniformity. At steady state, the boundary layer becomes non-local because fluid arriving at a given angular station carries the cumulative thermal history of the upstream arc. The mean Nusselt number follows $\overline {Nu}_s\propto Ra^{1/4} (1+Pr^{-1})^{-1/8}\varPsi (s)$, where $\varPsi (s)\geq 1$ quantifies the heat-transfer enhancement caused by circumferential temperature redistribution at fixed mean wall temperature. A cumulative-buoyancy coordinate collapses the steady circumferential profiles across $ \textit{Ra}$, $ \textit{Pr}$ and $s$, showing that the steady response is governed by the combined effects of local buoyancy and upstream thermal history.

We investigate how vibration-driven interfacial state transitions regulate heat transport in an immiscible two-phase layer under microgravity. Direct numerical simulations are performed for two immiscible fluid layers subject to a vertical temperature difference and horizontal harmonic vibration, while the Weber number and thermal-expansion contrast are varied systematically. As the Weber number increases, the dispersed phase evolves from a stratified-layer state to an interfacial-bulging state and then to a droplet-dispersed state, and the global Nusselt and Reynolds numbers accordingly separate into three regimes. We identify two critical thresholds associated with the onset of interfacial bulging and droplet detachment; both move to lower Weber number as the thermal-expansion contrast increases. The first threshold is set by the pressure balance between thermovibrational pressure difference and capillary restoring stress, and the second threshold is set by the force balance between the thermovibrational force acting on the lifted liquid volume and the capillary force retaining the remaining neck. When rescaled by the corresponding critical thresholds, the onset of the bulging and detachment transitions collapses across different thermal-expansion contrasts, together with the associated changes in the Nusselt and Reynolds numbers. Our results show that interfacial morphology provides the key link between vibration forcing and transport response, and establishes a compact regime-based framework for regulating microgravity two-phase heat transport.

Transversely segregated longitudinal flow bands, or stripes, can form in rapid thin dense granular flows due to the formation of a series of counter-rotating secondary vortices whose axes of rotation are aligned with the flow direction (Pearse et al. 2026 J. Fluid Mech. 1032, A4). The formation of the secondary vortices is still a subject of active research, but it is likely due to a combination of dilatation and second-normal-stress differences. This paper assumes the existence of the secondary flow and investigates how large and small particles are able to segregate to form stable compositional stripes. In the absence of diffusion, the secondary flow rotates the sharp interface between large particles (above) and small particles (below), around to the vertical, where it breaks to form a series of pure large- and pure small-particle bands that extend through the flow depth, and are separated by a series of breaking-size-segregation waves. Diffusion can compete against this overturning effect. In particular, there is a region of parameter space where the diffusion and the secondary recirculation balance to create a stable wavy interface between large and small particles, which can suppress the development of surface compositional differences. In this regime, a depth-averaged theory for segregation is derived that captures the three-dimensional time-dependent development of the particle-size distribution from a uniformly inversely graded inflow into a series of transversely segregated stripes. The solution is in good quantitative agreement with the evolution of the surface particle-size distribution observed in experiments.

Recent advances in the unsteady potential-flow model have shown promise in providing a low-order theoretical framework for studying insect aerodynamics. Although the assimilation of the discrete-vortex method enables accurate predictions of unsteady aerodynamic forces and wake structure evolution, it remains physically meaningful to seek a minimal representation of the coherent vortices while retaining the essential dynamical coupling between the concentrated vortex core, the feeding shear layer and the wing. The present study explores various reduction strategies in modelling the growth of a leading-edge vortex (LEV) on an accelerating flat plate. The results argue against the use of a single free point vortex, as it completely fails to capture the dynamical effect of vorticity feeding from the wing to the vortex core. By incorporating the momentum conservation of the vorticity-feeding process, both the classical Brown–Michael and the vortex-merge approaches provide notable improvements in predicting lift and wake vortex dynamics. The performance can be further improved by introducing an array of discrete vortices to account for the vorticity-feeding sheet. In particular, a new vortex-merge scheme is proposed to physically determine the vorticity transfer from the feeding sheet to the concentrated vortex core, based on the local rotation strength of the sheet. This yields satisfactory performance in capturing the lift variation, LEV advection and circulation growth rate. Crucially, the proposed physical-merge concept underscores a key insight for low-order modelling of a wing-attached vortex with varying strength: sufficient fidelity is required in resolving both the feeding shear layer and low-rotation vortical structure.

We develop saddle-point approximations of the probability density function (PDF) for the surface elevation in nonlinear, irregular seas, without any assumptions regarding uni-directionality of the wave field or narrow-bandedness of the spectrum. A new, closed-form approximation of the PDF at third order is derived, accounting for both skewness and kurtosis. The approximate PDF matches known asymptotics of the third-order positive tail exactly and is free of unphysical oscillations in the negative tail, which widely plague existing PDFs describing nonlinear seas. A normalisation procedure is introduced to enforce unit integral, zero mean and unit variance. We further show that the input cumulants can be adjusted to likewise yield prescribed skewness and kurtosis. Comparison of the new third-order PDF against challenging datasets stemming from irregular wave theory, a fully nonlinear wave model as well as laboratory experiments establishes accuracy. In the Appendix we additionally develop a novel perturbation method, which is used to derive the full form of the asymptotic positive and negative tails at fourth and fifth order, including (previously unknown) prefactors. Collectively, it is shown that even-order solutions inevitably involve oscillatory negative tails, whereas odd orders admit non-oscillatory negative-tail solutions.

Hypersonic flows in the transitional regime exhibit strong thermochemical non-equilibrium, where conventional continuum assumptions fail. This paper presents a hybrid kinetic-fluid framework for simulating partially ionised argon plasmas under such conditions. The heavy-particle component is governed by a kinetic equation and solved using the unified gas-kinetic wave–particle (UGKWP) method, which captures flow physics from the rarefied to the continuum regime without being constrained by the mean free path scale. Electrons are modelled as a fluid with a separate temperature equation that accounts for elastic collisions and ionisation energy loss. An electromagnetic-wave solver, coupled with an auxiliary differential equation for the plasma current density, evaluates the wave scattering by the plasma. Validation against shock-tube experiments confirms the accuracy of the present algorithm. Application to a Mach 22 cylinder flow at Knudsen number 0.05 reveals three notable non-equilibrium transport effects – elevated near-wall heavy-particle density enhances ionisation near the stagnation point; non-equilibrium boundary-layer cooling lowers the electron-density peak in the shock layer; and non-equilibrium transport in the rarefied wake yields a higher downstream electron density. The thermal structure predicted by UGKWP differs significantly from that of the continuum gas-kinetic scheme (GKS), underscoring the important role of multiscale methods. These differences in ionisation further lead to distinct electromagnetic scattering patterns. Our results thus establish a link between multiscale non-equilibrium transport, plasma formation and electromagnetic blackout in hypersonic transitional flows.

We consider the generation of extreme waves of flexible features in deep-water surface gravity waves, in the context of the fully nonlinear wave model, i.e. Euler equations of the free surface. This is achieved through a recently developed GenWave approach, coupled with the high-order spectral method to solve the Euler equations. Through an iterative refinement procedure, GenWave progressively refines the initial phases associated with a given wave spectrum, which leads to extreme-wave generation at specified design time and location, with their probability of exceedance quantified. In this study, we consider two types of extreme waves, enabled by the flexibility of GenWave. The first type is a spatially and temporally localised rogue wave, and second is a spatially localised extreme-wave group propagating at a constant speed. The two generated extreme-wave profiles are compared with their prototypes – the Peregrine soliton and regular soliton (or bright soliton) in the context of the nonlinear Schrödinger equation (NLS), with reasonable agreement. For the persistent group, comparison with linear evolution and the NLS identifies nonlinear persistence, faster propagation and bound-wave content beyond the leading-order NLS description.

This study investigates the aerodynamic noise control mechanisms of a forced rotating circular cylinder at a subcritical Reynolds number ($Re = 4.7 \times 10^4$) using large-eddy simulation coupled with the Ffowcs Williams–Hawkings acoustic analogy, validated by wind tunnel experiments. Five representative rotation ratios, $\alpha =0$, $0.5$, $1$, $2$ and $4$, are analysed to relate wall-pressure fluctuations and wake organisation to the radiated noise. At $\alpha =0.5$, the natural shedding tone separates into two components, $St_{01}\approx 0.18$ and $St_{02}\approx 0.23$. Their three-dimensional (3-D) structures exhibit spanwise phase dislocation and reduced coherence, explaining the decrease in tonal radiation despite locally enhanced wall-pressure fluctuations. Coherent shedding partially recovers at $\alpha =1$, whereas $\alpha =2$ gives the strongest tonal-noise reduction among the investigated cases, accompanied by weakened wake fluctuations, spanwise coherence and wall–wake correlation. At $\alpha =4$, the classical Kármán mode is suppressed and localised near-wall helical structures are associated with a broadband elevation near $St\approx 0.1$. The results show that rotation controls noise through changes in both unsteady-loading amplitude and 3-D source organisation.

Large eddy simulation (LES) on a cylindrical grid consisting of approximately 7 billion points is used to analyse the wake flow of a marine propeller equipped with propeller boss cap fins (PBCFs). Computations of the same propeller without this energy saving device (ESD) were also conducted, for comparison purposes. The results of the simulations demonstrate almost an order of magnitude mitigation of the minima of pressure and the maxima of vorticity at the core of the hub vortex. Meanwhile, a significant recovery of both streamwise and azimuthal momentum is achieved, with beneficial effects on the efficiency of propulsion. Additional, smaller vortices are shed by the PBCFs, having a negligible impact on the wake flow, if compared with the benefits allowed by the mitigation of the hub vortex. This is also the case of the turbulent stresses. The only exception is the region just downstream of the propeller plane, upstream of the instability of the hub vortex, where turbulence levels at the core of the hub vortex are still relatively low. However, within a few diameters downstream of the rotor, the instability of the stronger hub vortex shed by the propeller without PBCFs is the source of turbulent stresses more than an order of magnitude higher than those experienced in the wake of the propeller equipped with PBCFs. Meanwhile, the influence of the PBCFs on the outer region of the wake and especially on the tip vortices was found to be practically negligible.

This paper examines how poroelastic backstress induced by fluid leak-off modifies the asymptotic propagation regimes of fluid-driven fractures. We consider a simplified theoretical framework in which the diffusion zone remains small relative to the fracture size, so that Carter-type leak-off can be coupled consistently with one-dimensional fracture models. Within this setting, leak-off-induced pore-pressure diffusion generates a backstress that acts against fracture-opening and alters the classical regime structure. Three typical geometries are studied: plane strain; radial; constant-height fractures. For each geometry, we derive the corresponding backstress-dominated self-similar limit, referred to as the $B$-vertex, and incorporate it into the classical parametric space of hydraulic fracture propagation. The analysis shows that, when poroelastic effects are sufficiently strong, the classical leak-off-dominated asymptotic regimes may cease to exist and the fracture evolves instead towards the backstress-dominated state. Transition solutions connecting the classical limiting regimes to the $B$-vertex are also obtained. The results extend the standard asymptotic classification of hydraulic fractures by identifying a distinct regime controlled by poroelastic backstress. They also provide a simple framework for assessing when backstress is expected to play a leading role in fluid-driven fracture propagation across different geometries and scales.

The evolution of flow structure is fundamental to establishing an accurate framework for predicting turbulent energy dissipation, momentum transfer, and sediment and nutrient transport within and above flexible stem arrays. The dynamics of flexible stems plays a crucial role in shaping these physical features. In this paper, direct numerical simulations are performed to examine flow through a three-dimensional square array of flexible stems, with explicit consideration of flexibility ($K$) effects. The results provide a comprehensive characterisation of the coupled stem dynamics, flow distribution and vortex structures. Three distinct dynamical regimes are identified. For large $K$, the stems behave as nearly rigid, causing the flow to accelerate through the gaps while attenuating behind the stems. At intermediate $K$, a perfect lock-in occurs between the stem’s natural and oscillation frequencies, leading to large-amplitude vibrations and markedly altered flow characteristics. For small $K$, the oscillation amplitude decreases but deformation increases, producing irregular stem dynamics and complex flow features. The flow dynamics underlying these regimes is systematically elucidated. Furthermore, energy dissipation and sediment transport effects are quantified through hydrodynamic drag, bed shear stress, and root mean square bed shear friction distributions. The array with intermediate $K$ exhibits the highest energy dissipation efficiency, surpassing even the rigid configuration. The bed shear stress reaches a peak at the upstream edge of the array, whereas stem oscillation redistributes the flow momentum towards the upper canopy and the overlying flow, reducing velocity fluctuations near the bed. Finally, the mechanisms governing the structural dynamics and hydrodynamic forces are discussed in detail. This study offers new physical insights into flow–structure interactions within flexible stem arrays, and proposes an optimised strategy for improving energy dissipation in coastal protection applications.

In this work, we describe our experimental observations of trapped waves and guided jet waves in schlieren images of jets operating at jet Mach numbers $M_{\!j}=0.9$ and $M_{\!j}=0.983$ produced by circular nozzles. Results from a chevron nozzle are also included as a special case. We establish that: (i) several families of trapped waves are detectable in our experimental images, including higher order axisymmetric and helical modes; (ii) the dispersion relationship of these waves is in agreement with the vortex sheet model of Tam & Hu (J. Fluid Mech., 1989, vol. 201, pp. 447–483) and (iii) the introduction of chevrons does not significantly affect the presence or intensity of the first several peaks associated with the trapped wave modes. Finally, we observe that all detected trapped waves have subsonic group velocities with respect to the quiescent stream.

We develop a theoretical framework for magneto-electrokinetic transport in dilute electrolytes under unidirectional flow and streaming current conditions. Unlike standard magnetohydrodynamics in liquid metals, where charge carriers are electrons and conductivity is typically uniform, electrolytes exhibit spatially varying ion densities and additional forces such as osmotic pressure. In steady state, the coupled Navier–Stokes, Nernst–Planck and Poisson equations become analytically tractable. We derive a Boltzmann-like expression for ion distributions that incorporates both the electrostatic potential and a novel hydrodynamic potential arising from the magnetic field. This potential depends explicitly on the flow profile and is not screened close to the bounding walls. Applying the theory to Poiseuille flow, we derive expressions for the induced Hall potential and demonstrate its near-linear dependence on flow velocity and magnetic field strength, while remaining largely independent of salt concentration. To validate this one-dimensional framework, we analyse the two-dimensional spatial transients near the channel entrance. We demonstrate a separation of length scales: the Hall electric field and space charge adjust to the magnetic field almost instantaneously, whereas the macroscopic salt concentration wake develops over a much longer Graetz length dictated by the Péclet number. We also examine a similar flow with counterions only and no added salt, determining the resulting potential and ion distributions. This work offers new theoretical tools for understanding ion transport in microfluidic and electrochemical systems.

Resolvent analysis effectively characterises flow instabilities and guides active control, but repeated singular-value-decomposition-based (SVD-based) calculations remain computationally expensive in high-dimensional parameter studies. Control-oriented workflows also require efficient mappings from prescribed response fields to compatible forcing fields. To address these needs, we introduce ResFormer, a unified Transformer-based neural operator framework for forward resolvent prediction and response-to-forcing reconstruction in compressible boundary layers. The framework is validated across supersonic and hypersonic regimes ($ \textit{Ma}=2.25$–$5.86$), encompassing laminar, turbulent and controlled base-flow configurations. For forward prediction, ResFormer reconstructs the spatial topology, phase structure and amplitude envelope of both low-frequency Tollmien–Schlichting waves and high-frequency second-mode disturbances over the investigated parameter ranges. For response-to-forcing prediction, ResFormer learns a surrogate mapping from prescribed instability responses to compatible forcing fields. When the complete target response and discrete operator are known, the reference forcing is computed directly as $\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{ideal}}=\boldsymbol{A}_{\omega ,\beta }\hat {\boldsymbol{q}}_{\textit{decay}}$; the physics term therefore acts as an operator consistency regulariser. The reconstructed forcing topology is then reduced to a compact Gaussian wall actuator prior and evaluated using high-fidelity computational fluid dynamics. ResFormer therefore accelerates repeated SVD-based forward and optimal mode evaluations while providing an operator-consistent learned representation for actuator prior identification and surrogate-assisted control design.

Townsend’s attached-eddy hypothesis models the logarithmic region of high-Reynolds-number wall turbulence as a random superposition of wall-attached, self-similar eddies whose sizes obey a scale-invariant population law. Building on the statistical framework of Woodcock & Marusic (2015 Phys. Fluids, vol. 27, no. 1, p. 015104), the present work (i) poses an inverse problem to infer the ideal single-eddy contribution (influence) functions for the mean velocity and Reynolds stresses from direct numerical simulations, (ii) uses these inferred kernels to guide a minimal Biot–Savart-consistent hairpin-type eddy built from Rankine vortex rods with an inviscid image system and (iii) introduces and infers a spectral influence kernel that maps a self-similar eddy footprint to its one-dimensional energy spectrum. The influence-kernel viewpoint yields an explanation for the emergence of the linear part of the energy spectrum, provides a clear scale-by-scale decomposition and helps rationalize why simple eddy templates can reproduce a broad set of log-layer statistics once the mean-flow anchoring is fixed. Closed-form expressions for the mean influence function and the Fourier-space streamwise velocity of a general straight-segment hairpin family with image are derived, revealing a mean-variance duality: for a rectangular hairpin, the horizontal head determines the entire mean kernel while the inclined legs dominate the spectral energy. It is shown that the rectangular hairpin yields a precise plateau in the influence function, resulting in a logarithmic mean velocity profile and desirable properties for the streamwise fluctuations. The model is further extended by calibrating the eddy population density as a function of scale, yielding accurate descriptions of mean velocity and streamwise variance across $\textit{Re}_{\tau }=6000{-}20\,000$.

This study examines the role of buoyancy-induced baroclinic vorticity production in the three-dimensional (3-D) wake transition behind an infinitely long heated square cylinder subjected to aiding and cross-flow buoyancy, using direct numerical simulation (DNS). A non-Oberbeck–Boussinesq (NOB) model is employed, requiring the solution of the full compressible Navier–Stokes equations to capture baroclinic effects under large-scale heating regimes. Simulations are carried out at a Reynolds number of $\textit{Re} = 300$ and a low Mach number ($M = 0.1$) to minimise compressibility effects, with air (Prandtl number $\textit{Pr} = 0.71$) as the working fluid. Previous studies have demonstrated that baroclinic torque significantly enhances the strength of 3-D wake structures, thereby promoting an earlier onset of three-dimensionality compared with the unheated case. However, these effects have been predominantly investigated in cross-buoyancy configurations, and the underlying mechanisms remain insufficiently understood. In the present study, the contributions of streamwise and spanwise baroclinic torque are systematically analysed to elucidate the mechanisms governing flow instability and organisation in thermally influenced wakes. The findings reveal a complex interplay between baroclinic torque and buoyancy, indicating that increasing heating levels weaken the 3-D wake under aiding-buoyancy conditions while strengthening it in cross-buoyancy configurations. Furthermore, the production of spanwise baroclinic torque, whether aligned with or opposed to local vortices, increases flow complexity and influences vortex shedding intensity and vortex dislocations. It also governs the contrasting wake behaviour, resulting in a symmetric wake under aiding buoyancy and an asymmetric wake under cross-buoyancy.

In the past, various flow control methods have been used to enhance the growth rate of the supersonic mixing layer (SML), yet suppression strategies remain underexplored. The flow fields of the free SML (FSML) and the SML controlled by the micro vortex generator (MVG) array (CSML) were obtained via flow visualisation and large-eddy simulations, which show that the CSML’s growth rate is substantially lower. The $Q$-criterion analysis confirms that under MVG control, the dominant vortical structures in the initial stage of SML transition from spanwise to streamwise vortices. Circulation analysis shows that streamwise circulation $\varGamma _x$ and transverse circulation $\varGamma _y$ are significantly enhanced in the CSML, thereby facilitating energy redistribution. Dynamic mode decomposition reveals that the first two dominant modes of the FSML exhibit spanwise periodic structures. However, the dominant modes exhibit ‘comb-like’ patterns in the initial stage of the CSML, while the spanwise vortices regain dominance further downstream. The interaction between streamwise and spanwise vortices dominates the spatial growth of the CSML, which is in contrast to the sole dominance of Kelvin–Helmholtz vortices in the FSML. By suppressing the Reynolds normal stress $\langle {v'v'} \rangle$ and restructuring the turbulent kinetic energy (TKE) transport process, MVGs enhance the central amplitudes of TKE transport terms, and narrow their spanwise distribution width. Therefore, the diffusion of TKE towards the high- and low-speed flows is inhibited, thereby reducing the SML’s growth rate. This study offers a viable approach and insights for applications that require inhibiting the SML’s growth rate.