The temporal characteristics of fully localised turbulent bands in transitional channel flow remains unclear due to the difficulty in resolving the large length and time scales involved. Here, we tackle this problem by performing statistical lifetime studies in sufficiently large computational domains. The results show signs of stochastic memoryless decay of a fully localised band, suggesting a chaotic-saddle behaviour of the entire band as a coherent entity. Although the mean lifetime of a turbulent band was reported to increase with the band length, our data suggest that it saturates at a certain length. This saturation results in a characteristic lifetime for a fully developed band with a changing length due to the intermittent chipping and decay of turbulence at the upstream end. This memoryless behaviour is observed down to Reynolds number $Re=630$ in our study and we propose that the onset of the memoryless behaviour is in the range of $Re\simeq 620{-}630$. Our data also show that the time it takes for a perturbed flow to enter the saddle, i.e. to start behaving memorylessly, can be thousands of convective time units, which is comparable to the maximum achievable observation time in existing channel set-ups and may pose difficulties for experiments.

In this study, we tackle the challenge of inferring the initial conditions of a Rayleigh–Taylor mixing zone for modelling purposes by analysing zero-dimensional (0-D) turbulent quantities measured at an unspecified time. This approach assesses the extent to which 0-D observations retain the memory of the flow, evaluating their effectiveness in determining initial conditions and, consequently, in predicting the flow’s evolution. To this end, we generated a comprehensive dataset of direct numerical simulations, focusing on miscible fluids with low density contrasts. The initial interface deformations in these simulations are characterised by an annular spectrum parametrised by four non-dimensional numbers. To study the sensitivity of 0-D turbulent quantities to initial perturbation distributions, we developed a surrogate model using a physics-informed neural network (PINN). This model enables computation of the Sobol indices for the turbulent quantities, disentangling the effects of the initial parameters on the growth of the mixing layer. Within a Bayesian framework, we employ a Markov chain Monte Carlo (MCMC) method to determine the posterior distributions of initial conditions and time, given various state variables. This analysis sheds light on inertial and diffusive trajectories, as well as the progressive loss of initial conditions memory during the transition to turbulence. Furthermore, it identifies which turbulent quantities serve as better predictors of Rayleigh–Taylor mixing zone dynamics by more effectively retaining the memory of the flow. By inferring initial conditions and forward propagating the maximum a posteriori (MAP) estimate, we propose a strategy for modelling the Rayleigh–Taylor transition to turbulence.

The evolution of turbulent spots in a flat plate boundary layer is examined using time-resolved tomographic particle image velocimetry (Tomo-PIV) experiments and direct numerical simulation (DNS). The characteristics of flow structures are examined using timelines and material surfaces. Both the numerical and experimental results reveal a notable behaviour in the developmental process of turbulent spots: the development of low-speed streaks at the spanwise edges of turbulent spots, followed by the subsequent formation of hairpin vortices. The behaviour of these low-speed streaks is further investigated using timelines and material surfaces generated for a series of regions and development times. The results indicate that these low-speed streaks exhibit characteristic wave behaviour. The low-speed streaks are observed to lift up as three-dimensional (3-D) waves, with high-shear layers forming at the interface of these waves. These induced high-shear layers become unstable and evolve into vortices, which contribute to the expansion of the turbulent spot. These findings show the significant role of 3-D waves in the development of turbulent spots, supporting the hypothesis that 3-D waves serve as initiators of vortices at the bounding surface of a turbulent spot.

Within the frameworks of the amplitude method and the linear stability theory, a statistical model of the initial stage of laminar–turbulent transition caused by atmospheric particulates (aerosols) penetrating into the boundary layer is developed. The model accounts for the process of boundary layer receptivity to particulates, asymptotic behaviour of unstable wave packets propagating downstream from particle–wall collisions and the amplitude criterion for the transition onset. The resulting analytical relationships can be used for quick predictions of the transition onset on bodies of relatively simple shape, where the undisturbed boundary layer is quasi-two-dimensional. The model allows us to explore the transition onset at realistic distributions of the particle concentration selected based on an analysis of known empirical data. As an example, a 14° half-angle sharp wedge flying in atmosphere at 20 km altitude and Mach number 4 is considered. It is shown that the transition onset corresponds to an N-factor of 15.3 for a flight under normal atmospheric conditions and 12.2 for a flight in a cloud after volcanic eruption. In accordance with physical restrictions, these values are below the upper limit $N\approx 16.8$ predicted for transition due to thermal fluctuations (perfectly ‘clean’ case). Nevertheless, they are significantly greater than $N=10$ which is commonly recommended for estimates of the transition onset in flight.

A pressure-gradient-induced laminar separation bubble (LSB) was examined using wind-tunnel experiments, direct numerical simulations (DNS) and linear local/global stability analysis. The LSB was experimentally generated on a flat plate using the favourable-to-adverse pressure gradient imposed by an inverted modified NACA $64_3-618$ airfoil. Direct numerical simulation was performed using boundary conditions extracted from a steady precursor simulation of the entire flow field. Despite good agreement in the upstream boundary layer with the experiment, DNS exhibited an approximately 25 % longer mean separation bubble, attributed to an earlier onset of transition due to the free-stream turbulence (FST) in the experiment. Introducing a very low level of isotropic FST in the DNS, similar to that measured in the experiment, caused earlier transition, decreased the mean bubble length and led to a remarkably good agreement between the DNS and experiments. Differences were observed for the dominant frequencies in the experiment and DNS, but both were within the band of most amplified frequencies predicted by LST. Proper orthogonal decomposition confirmed that dominant coherent structures from DNS and experiments are related to the inviscid Kelvin–Helmholtz instability and have similar characteristics despite slight differences in frequency. Local and global stability and dynamic mode decomposition analysis corroborated the convective nature of the dominant two-dimensional (2-D) instability and showed that the LSB is globally unstable to a range of 3-D wavenumbers, in agreement with 3-D structures observed in experiments. Results confirmed the strong impact of very low FST levels on the LSB and indicate a close agreement of the time-averaged and instability characteristics between the experiments and DNS.

We revisit viscoelastic Kolmogorov flow to show that the elastic linear instability of an Oldroyd-B fluid at vanishing Reynolds numbers ($Re$) found by Boffetta et al. (J. Fluid Mech., vol. 523, 2005, pp. 161–170) is the same ‘centre-mode’ instability found at much higher $Re$ by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502) in a pipe and by Khalid et al. (J. Fluid Mech., vol. 915, 2021, A43) in a channel. In contrast to these wall-bounded flows, the centre-mode instability exists even when the solvent viscosity vanishes (e.g. it exists in the upper-convective Maxwell limit with $Re=0$). Floquet analysis reveals that the preferred centre-mode instability almost always has a wavelength twice that of the forcing. All elastic instabilities give rise to familiar ‘arrowheads’ (Page et al., Phys. Rev. Lett., vol. 125, 2020, 154501) which in sufficiently large domains and at sufficient Weissenberg number ($W$) interact chaotically in two dimensions to give elastic turbulence via a bursting scenario. Finally, it is found that the $k^{-4}$ scaling of the kinetic energy spectrum seen in this two-dimensional elastic turbulence is already contained within the component arrowhead structures.

A collection of secondary instability calculations in streaky boundary layers is presented. The data are retrieved from well-resolved numerical simulations of boundary layers forced by free-stream turbulence (FST), considering different geometries and FST conditions. The stability calculations are performed before streak breakdown, taking place at various $Rey_x$ the Reynolds number based on the streamwise coordinate. Despite the rich streak population of various sizes, it is found that breaking streaks have similar aspect ratios, independently of the streamwise position where they appear. This suggests that wider streaks will break down further downstream than thinner ones, making the appearance of secondary instabilities somewhat independent of the streak’s wavelength. Moreover, the large difference in the integral length scale among the simulations suggests that this aspect ratio is also independent of the FST scales. An explanation for this behaviour is provided by showing that these breaking streaks are in the range of perturbations that can experience maximum transient growth according to optimal disturbance theory. This could explain why, at a given streamwise position, there is a narrow spanwise wavelength range where streak breakdown is more likely to occur.

Linear and nonlinear mechanisms governing the growth of second-mode waves are analysed using a newly derived disturbance energy conservation equation that highlights the physical processes responsible for fluctuation energy production, flux-based transport and destruction. Axisymmetric direct numerical simulations (DNS) data from a Mach 6 hypersonic boundary layer, simulated over a $3^\circ$ half-angle sharp cone at zero angle of attack, is used as a reference. A Legendre polynomial-based forcing methodology is used to trigger transition in the DNS over a range of various amplitude levels and different forcing frequency content. Closure of the disturbance energy budgets is demonstrated numerically using the DNS data. The terms responsible for the amplification of the disturbance are identified, and nonlinear attenuation effects are discussed. We show that the interaction between the entropy/velocity fluctuations and the base temperature gradient governs the second-mode growth in the linear regime. Energy production occurs in the critical layer due to non-isentropic processes and accumulates in acoustic form below the relative sonic line through downward transport. At higher forcing amplitudes, nonlinear spectral broadening is observed, with simultaneous thermoviscous diffusion attenuating the disturbance energy growth. This effect is responsible for the non-monotonic streamwise variation of the wall-pressure spectrum. Phase speed and growth rate analyses, informed by linear stability theory (LST), reveal wave steepening effects preceding this nonlinear attenuation effect. The disturbance energy is observed to match the LST predictions at lower forcing amplitudes, deviating, as expected, at higher amplitudes.

In this work, the stability and transition to turbulence over blunt flat plates with different leading-edge radii are investigated computationally. The benchmark experimental work for comparative studies is conducted by Borovoy et al. (AIAA J., vol. 60, 2022, pp. 497–507). The freestream Mach number is 5, the unit Reynolds number is $6\times 10^7$ m$^{-1}$, and the maximum nose-tip radius 3 mm exceeds the experimental reversal value. High-resolution numerical simulation and stability analysis are performed. Three-dimensional broadband perturbation is added on the far field boundary to initiate the transition. The highlight of this work is that the complete physical process is considered, including the three-dimensional receptivity, linear and nonlinear instabilities, and transition. The experimental reversal phenomenon is reproduced favourably in the numerical simulation for the first time. Linear stability analysis shows that unstable first and second modes are absent in the blunt-plate flows owing to the presence of the entropy layer, although these modes are evident in the sharp-leading-edge case. Therefore, the transition on the blunt plate is due to non-modal instabilities. Numerical results for all the blunt-plate cases reveal the formation of streamwise streaky structures downstream of the nose (stage I) and then the presence of intermittent turbulent spots in the transitional region (stage II). In stage I, a preferential spanwise wavelength approximately 0.9 mm is selected for all the nose-tip radii, and low-frequency components are dominant. In stage II, high-frequency secondary instabilities appear to grow, which participate in the eventual breakdown. By contrast, leading-edge streaks are not remarkable in the sharp-leading-edge case, where transition is induced by oblique first and Mack second modes. The transition reversal beyond the critical nose-tip radius arises from an increasing magnitude of the streaky response in the early stage, while the transition mechanism stays similar qualitatively.

In this paper, we study the rapid transition in Richtmyer–Meshkov instability (RMI) with reshock through three-dimensional double-layer swirling vortex rings. The rapid transition in RMI with reshock has an essential influence on the evolution of supernovas and the ignition of inertial confinement fusion, which has been confirmed in numerical simulations and experiments in shock-tube and high-energy-density facilities over the past few years. Vortex evolution has been confirmed to dominate the late-time nonlinear development of the perturbed interface. However, few studies have investigated the three-dimensional characteristics and nonlinear interactions among vortex structures during the transition to turbulent flows. The coexistence of co-rotating and counter-rotating vortices is hypothesized to induce successive large-scale strain fields, which are the main driving sources for rapid development. The three-dimensional effect is reflected in the presence of local swirling motion in the azimuthal direction, and it decreases the translation velocity of a vortex ring. Large-, middle- and small-scale strain fields are employed to describe the development process of RMI with reshock, e.g. vorticity deposited by the reshock, formation of the coexistence of the co-rotating and counter-rotating vortices, iterative cascade under the amplification of the strain fields and viscous dissipation to internal energy. This provides theoretical suggestions for designing practical applications, such as the estimation of the hydrodynamic instability and mixing during the late-time acceleration phase of the inertial confinement fusion.

The evolutionary process of mixing induced by Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities typically progresses through three stages: initial instability growth, subsequent mixing transition and ultimate turbulent mixing. Accurate prediction of this entire process is crucial for both scientific research and engineering applications. For engineering applications, Reynolds-averaged Navier–Stokes (RANS) simulation stands as the most viable method currently. However, it is noteworthy that existing RANS mixing models are primarily tailored for the fully developed turbulent mixing stage, rendering them ineffective in predicting the crucial mixing transition. To address that, the present study proposes a RANS mixing transition model. Specifically, we extend the idea of the intermittent factor, which has been widely employed to integrate with turbulence models for predicting boundary layer transition, to mixing problems. Based on a high-fidelity simulation of a RT case, the intermittent factor defined based on enstrophy is extracted and then applied to RANS calculations, showing that it is possible to accurately predict mixing transition by introducing the intermittent factor to the turbulence production from the baseline K-L turbulence mixing model. Furthermore, to facilitate practical predictions, a transport equation has been established to model the spatio-temporal evolution of the intermittent factor. Coupled with the K-L model, the intermittent factor provided by the transport equation is applied to modify the Reynolds stress in RANS calculations. Thereafter, the present transition model has been validated in a series of tests, demonstrating its accuracy and robustness in the capturing mixing process in different types and stages of interfacial mixing flows.

Turbulent mixing is a pivotal phenomenon in fusion research with profound implications for energy gain. A Reynolds-averaged Navier–Stokes model capable of predicting realistic mixing transition processes is of significant importance for fusion applications, yet such a model remains elusive. This work addresses the limitations of prevalent global transition criteria, proposing a new idea to quantify local transition characteristics based on the mixing state, recognizing its direct relevance to fusion reaction rates. We delve into the description and analysis of the spatiotemporal evolution of the mixing state and its interplay with the transition process. Then, a local transition indicator is developed and compared with conventional global criteria using the large-eddy simulation (LES) of Rayleigh–Taylor turbulent mixing. Building upon this foundation, we introduce a novel eddy viscosity model based on the local transition indicator. A posterior assessment using LES data validates that it significantly outperforms standard gradient transport models during the transition stage. Consequently, we integrate this new eddy viscosity model with the Besnard–Harlow–Rauenzahn model to construct a comprehensive transition model, which demonstrates reasonably good performance in comparison with LES results. This work paves the way for future research in developing advanced modelling strategies that can effectively address the complexities of transitional flows in fusion engineering applications.

We study the generation, nonlinear development and secondary instability of unsteady Görtler vortices and streaks in compressible boundary layers exposed to free-stream vortical disturbances and evolving over concave, flat and convex walls. The formation and evolution of the disturbances are governed by the compressible nonlinear boundary-region equations, supplemented by initial and boundary conditions that characterise the impact of the free-stream disturbances on the boundary layer. Computations are performed for parameters typical of flows over high-pressure turbine blades, where the Görtler number, a measure of the curvature effects, and the disturbance Reynolds number, a measure of the nonlinear effects, are order-one quantities. At moderate intensities of the free-stream disturbances, increasing the Görtler number renders the boundary layer more unstable, while increasing the Mach number or the frequency stabilises the flow. As the free-stream disturbances become more intense, vortices over concave surfaces no longer develop into the characteristic mushroom-shaped structures, while the flow over convex surfaces is destabilised. An occurrence map identifies Görtler vortices or streaks for different levels of free-stream disturbances and Görtler numbers. Our calculations capture well the experimental measurements of the enhanced skin friction and wall-heat transfer over turbine-blade pressure surfaces. The time-averaged wall-heat transfer modulations, termed hot fingers, are elongated in the streamwise direction and their spanwise wavelength is half of the characteristic wavelength of the free-stream disturbances. Nonlinearly saturated disturbances are unstable to secondary high-frequency modes, whose growth rate increases with the Görtler number. A new varicose even mode is reported, which may promote transition to turbulence at the stem of nonlinear streaks.

Rotor-stator flows are known to exhibit instabilities in the form of circular and spiral rolls. While the spirals are known to emanate from a supercritical Hopf bifurcation, the origin of the circular rolls is still unclear. In the present work we suggest a quantitative scenario for the circular rolls as a response of the system to external forcing. We consider two types of axisymmetric forcing: bulk forcing (based on the resolvent analysis) and boundary forcing using direct numerical simulation. Using the singular value decomposition of the resolvent operator the optimal response is shown to take the form of circular rolls. The linear gain curve shows strong amplification at non-zero frequencies following a pseudo-resonance mechanism. The optimal energy gain is found to scale exponentially with the Reynolds number $Re$ (for $Re$ based on the rotation rate and interdisc spacing $H$). The results for both types of forcing are compared with former experimental works and previous numerical studies. Our findings suggest that the circular rolls observed experimentally are the effect of the high forcing gain together with the roll-like form of the leading response of the linearised operator. For high enough Reynolds number it is possible to delineate between linear and nonlinear responses. For sufficiently strong forcing amplitudes, the nonlinear response is consistent with the self-sustained states found recently for the unforced problem. The onset of such non-trivial dynamics is shown to correspond in state space to a deterministic leaky attractor, as in other subcritical wall-bounded shear flows.

Low-frequency phenomena in an incompressible pressure-induced laminar separation bubble (LSB) on a flat plate is investigated using direct numerical simulation. The LSB configuration of Spalart and Strelets (J. Fluid Mech., vol. 403, 2000, pp. 329–349) is used. Wall pressure spectra indicate low-frequency-flapping $(St \sim 0.08)$ and high-frequency-shedding $(St \sim 1.52)$ regimes. Conditional velocity averages based on the fraction of reversed flow reveal the low frequency as an expansion/contraction of the LSB. While the high frequency only exhibits exponential growth within the LSB up to breakdown of the spanwise rollers, the low frequency and velocity fluctuations exhibit exponential growth upstream of separation. Instantaneous flow fields reveal large streamwise streaky structures forming within the LSB and extending past reattachment, much like high and low speed streaks in turbulent boundary layers. A predominance of sweep-like events ($Q4$) is observed during contraction and of ejection-like events ($Q2$) during expansion. These motions appear as dominant low-frequency modes in three-dimensional proper orthogonal and dynamic mode decompositions, exhibiting spatial amplification from separation to reattachment. The advection of a group of spanwise alternating streaky structures past the LSB results in an overall contraction after which the bubble expands to its ‘unforced’ state in the absence of the streaks. The low frequency then corresponds to the time it takes for streaks to form, amplify and advect past the LSB from separation to reattachment. This behaviour is linked to the mean flow deformation reported by Marxen and Rist (J. Fluid Mech., vol. 660, 2010, pp. 37–54), where the presence of streaks results in reduced mean bubble size. The formation of these streaky structures, in the absence of free stream turbulence, may be attributed to an absolute instability of the LSB due to the development of a secondary bubble within the primary.

Three-dimensional numerical simulations of hypersonic boundary layer transition delay due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) were carried out on a 7$^{\circ {}}$ half-angle cone for unit Reynolds numbers $Re_m=2.43 \times 10^6$–$6.40\times 10^6\ \text {m}^{-1}$, at the free-stream Mach number $M_\infty =7.4$, for both sharp and 2.5 mm nose tip radii. A broadband time-domain impedance boundary condition was used to model the acoustic effects of the porous surface on the flow field. A quasi-spectral sub-filter-scale dynamic closure was adopted to stabilize the computations upon turbulent breakdown under extreme cooling conditions, with wall-to-adiabatic temperature ratio of $T_{w}/ T_{ad} \simeq 0.08 $, while accurately recovering the growth rates of the unstable modes present in the early transition stages. Good agreement is observed with the reference experimental data, both in terms of the predicted extent of the transition delay and the measured second-mode frequency spectrum. The latter is strongly modulated by the formation of near-wall low-temperature three-dimensional streaks. Pressure disturbances concentrate in corridors of locally thickened boundary layer, with frequencies lower than what predicted by linear theory. Here, trapped wavetrains are formed, which can persist long into the turbulent region. Finally, it is shown that the presence of a porous wall simply shifts the onset of turbulence downstream, without affecting its structure.

In this paper, we study the effect of lateral wall vibrations on the excitation and evolution of non-modal perturbations in hypersonic boundary layers subject to low-frequency freestream vortical disturbances (FSVDs). A novel, high-efficiency numerical approach, combining the harmonic weakly nonlinear Navier–Stokes and nonlinear parabolised stability equation approaches, is developed, which is sufficient to accommodate both the rapid distortion of the perturbation in the leading-edge vicinity and the nonlinear development of finite-amplitude high-order harmonics in the downstream region. The boundary-layer response to low-frequency FSVDs shows a longitudinal streaky structure, for which the temperature perturbation shows much greater magnitude than the streamwise velocity perturbation. The lateral vibration induces a Stokes layer solution for the spanwise velocity perturbation, which interacts with the FSVD-induced perturbations and leads to a suppression of the non-modal perturbation and an enhancement of the downstream modal perturbation. The new perturbations excited by the FSVD–vibration interaction strengthen as the vibration intensifies, and they could become comparable with the FSVD-induced perturbations in downstream locations at a high vibration intensity, indicating a remarkable modification of the streaky structure and its instability property. Secondary instability (SI) analyses based on the streaky base flow indicate that the vibration could enhance or suppress the SI modes, depending on their initial phases over the vibration period. Overall, the average effect is that the low-frequency and high-frequency SI modes are stabilised and destabilised by the vibration, respectively. Since the high-frequency SI modes undergo higher amplifications, the subsequent bypass transition is likely to be promoted by relatively strong vibrations.

The boundary-layer stability on a section of a rotating wind turbine blade with an FFA-W3 series aerofoil at a chord Reynolds number of $3 \times 10^5$, with varying rotation and radii, is studied with direct numerical simulations and linear stability analyses. Low rotation does not significantly affect transition in the outboard blade region. The relative insensitivity to rotation is due to a laminar separation bubble near the leading edge, spanwise-deformed by a primary self-excited instability, promoting the secondary absolute instability of the Kelvin–Helmholtz (KH) vortices and rapid transition. Moderate increases in rotation, or moving inboard, stabilise the flow by accelerating the attached boundary layer and possibly inducing competition between cross-flow and KH modes. This delays separation and transition. Initially, for high rotation rates or radial locations close to the hub, transition is delayed. Nevertheless, strong stationary and travelling cross-flow modes are eventually triggered, spanwise modulating the KH rolls and shifting the transition line close to the leading edge. Cross-flow velocities as high as $56\,\%$ of the free stream velocity directed towards the blade tip are reached at the transition location. For radial locations farther from the hub, the effective angle of attack is decreased, and cross-flow transition occurs at lower rotation rates. The advance or delay of the transition line compared with a non-rotating configuration depends on the competing rotation effects of stabilising the attached boundary layer and triggering cross-flow modes in the separation flow region.

Several transition scenarios are present in a hypersonic compression-ramp flow. In our previous work (Cao et al., J. Fluid Mech., vol. 941, 2022, p. A8), a complete transition process induced by the global instability of a compression-ramp flow was revealed. In a globally stable flow, however, the transition to turbulence can be promoted by convective instabilities, which is the focus of this work. The same flow conditions as in our previous work (Mach number 7.7, Reynolds number $8.6\times 10^5$ based on the flat-plate length) are considered here. Owing to a smaller ramp angle, a weakly separated flow forms on the compression ramp, which supports no global instability. Resolvent analysis identifies low-frequency streamwise streaks as the optimal response of base flow to upstream forcing. Local stability analysis reveals Mack's second mode in the boundary layer downstream of reattachment. By introducing random disturbances upstream of separation in direct numerical simulations, we observe breakdown to turbulence downstream of reattachment. Two transition scenarios are revealed, and they are highly dependent on the amplitude of upstream disturbances. For a large amplitude, strong streamwise streaks develop near the reattachment region, which break down to turbulence quickly. However, when the disturbance amplitude is reduced, the second-mode instability dominates the transition to turbulence.

Isolated-roughness-induced transitions controlled by local wall heating strips are studied via direct numerical simulation and BiGlobal linear stability analysis. The transition mechanisms are studied first with different wall temperatures. The separated shear layer–counter-rotating vortex system is found to be the main source for transitions. Symmetric and antisymmetric modes are observed in the wake, and the former is dominant. The local wall heating strip can delay the transition, and this effect is enhanced with higher heating temperature, wider strip and a combination of upstream and downstream control strips. The upstream strip lifts up the inlet flow and weakens the wake system in an indirect manner. The antisymmetric mode gradually vanishes, while the symmetric mode always exists but becomes weaker. The downstream strip exhibits a more effective transition delay by directly weakening the separated shear layer and vortex system in the wake. Vorticity transport analysis suggests that the downstream strip increases dissipation for streamwise vorticity and transfers it into wall-normal and spanwise vorticity. BiGlobal analyses indicate that the downstream strip shows less influence on the peak growth rate of the symmetric mode but significantly shrinks its unstable region. Analyses of the disturbance energy production indicate that the upstream strip wakens the wall-normal and spanwise shear at the same time, but the downstream strip mainly wakens the wall-normal one. More simulations are performed with different roughness heights, point-source disturbance and different roughness shapes. The results show that the current method remains effective enough in delaying transitions at a wide range of conditions.