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.

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

Surface roughness significantly impacts transition to turbulence, especially over high-speed, blunt geometries where surface ablation is necessary to mitigate heat loads during atmospheric entry. Inspired by sand-grain roughness experiments performed by Hollis (2017), we perform the first direct numerical simulation of a blunt cylinder in Mach 6 cross-flow with roughness elements distributed along the entire surface. Such simulations aimed to uncover the precise means by which laminar–turbulent transition occurs given the limited measurements attainable from experiments and non-existent high-fidelity simulations. Element heights were held fixed at approximately 35 % boundary-layer thickness, while the relative phasing between streamwise rows was varied. All configurations exhibited convective instabilities in the post-shock subsonic flow driving the transition process, with the mode type being set by the roughness configuration. A fundamental sinuous streak mode dominated the aligned roughness element case, whereas both the staggered and randomly phased cases saw two-dimensional Tollmien–Schlichting (T–S) waves dominating. These instability waves, when grown to sufficient amplitude, triggered the steady streaks seeded by the underlying roughness pattern to begin forming hairpin vortices and breakdown occurred soon thereafter. The roughness arrangement was found to dramatically influence the degree to which the waves were destabilised, as well as the strength of the underlying steady streaks, thereby combining to dictate the position along the surface where laminar–turbulent transition occurred. Since exogenous forcing was not required for the T–S-dominated cases, an acoustic feedback loop is proposed in which upstream-propagating acoustic disturbances generated by turbulence in the subsonic region couple to and sustain the T–S waves.

In real-life tunnel or flight environments, the laminar–turbulent transition of compression ramp flows can be initiated by absolute and/or convective instabilities. Currently, the relative significance of the two instabilities in such flows remains unclear. This paper addresses this issue by decoupling the effects of absolute and convective instabilities, which is an extension of our previous research (Huang et al. Phys. Fluids, vol. 37, 2025, p. 034110). Hypersonic flows over a compression ramp with a 15$^\circ$ ramp angle at Mach number 8.0 and Reynolds number ${{\textit{Re}}_{L}}=9.2\times {{10}^{5}}$ based on the flat plate length are investigated using direct numerical simulation (DNS) and several stability analysis tools. Global stability analysis (GSA) reports that the flow is intrinsically unstable. Without external disturbances, DNS shows a good agreement with GSA regarding the growth rate and the disturbance shape. The flow in DNS is dominated by the most amplified stationary mode and the low-frequency oscillatory mode. However, transition in this case cannot be accomplished in the computational domain purely by the absolute instability. With external forcing, the coexistence of absolute and convective instabilities triggers the transition. For the forcing amplitudes examined in this study, which are representative of moderately to strongly noisy wind tunnel environments, the absolute instability is not significant in affecting the transition onset location. Nonetheless, the interplay between absolute and convective instabilities can profoundly alter the dominant spanwise wavelengths, the pronounced frequency band, the shape of coherent structures, etc. In summary, the interplay between the two types of instabilities recasts the transition scenario, albeit the transition onset location is dominated by the convective instability.

This study employs direct numerical simulations to examine the laminar-to-turbulent transition over a flat plate subjected to a favourable–adverse pressure gradient, incorporating isotropic/anisotropic and negatively skewed rough surfaces. The rough surfaces are positioned upstream of the separation point. By leveraging global stability analysis, inlet/local transient growth analysis and resolvent analysis, the combined effects of rough surface and adverse pressure gradients on disturbance growth mechanisms during transition are elucidated. The findings indicate that isotropic rough surfaces are most likely to trigger global instability, while streamwise modulation demonstrates the strongest inertia in this regard. Convective-type modes dominate near the outlet. As the system approaches the instability threshold, a critical mode emerges in the three-dimensional shear layer downstream of the roughness, characterised by a streamwise velocity peak reminiscent of a laminar separation bubble, with disturbance kinetic energy governed primarily by streamwise shear. The isotropy of the rough surface significantly amplifies linear instability of local travelling waves, transforming the downstream region into both a low-pass filter and an amplifier of high-wavenumber disturbances. Optimal disturbances of local travelling waves in the far downstream exhibit a structure analogous to oblique waves. When harmonic nonlinear forcing is introduced, streamwise modulation emerges as the geometry yielding maximum energy gain. The optimal forcing generated is highly localised in wave packets near the downstream region. Excluding the influence of optimal forcing on the harmonic forcings disrupts the coherence of roughness-induced coherent structures, impeding nonlinear interactions in the downstream region and the overall transition process.

Secondary instabilities developing from the stationary primary global mode are investigated for an incident shock flow at Mach 2.15. Direct numerical simulations (DNS) and three-dimensional global stability analysis (3-D GSA) are combined to elucidate the evolution and characteristics of the associated secondary branches. Two spanwise domain widths are considered, $\lambda _m$ and $2\lambda _m$, where $\lambda _m$ denotes the spanwise wavelength of the primary global mode. In the $\lambda _m$ domain, a time-periodic unsteadiness emerges as a secondary instability after saturation of the primary global mode, which is identified as a supercritical Hopf bifurcation. In the $2\lambda _m$ domain, an additional secondary growth accompanied by wavelength doubling is observed following the onset of unsteadiness. A corresponding stationary secondary instability with wavelength $2\lambda _m$ is determined to be the mechanism underlying this secondary growth, which is a subcritical pitchfork bifurcation. Overall, DNS and 3-D GSA provide a consistent interpretation of the secondary-instability scenario, and highlight the sensitivity of the observed dynamics to the spanwise domain width.

Direct numerical simulations are performed to investigate the receptivity of a swept-wing boundary layer to unsteady free-stream vortical disturbances in the nose region of the airfoil. An infinite wing with a sweep angle of 35$^\circ$ and at free-stream Mach number of 0.65 has been employed, and single Fourier modes of vortical disturbances are introduced far upstream. It is shown that leading-edge-parallel vortical two-dimensional waves are most receptive, but the induced travelling crossflow modes are typically too small in amplitude to directly trigger transition. The interaction with roughness elements, however, enhances the amplitudes substantially for one type of free-stream disturbances, and travelling-wave dominated transition occurs. The investigation may help in understanding the problem of unreliable transition control by the upstream-flow-deformation or distributed-roughness-element technique in flight, where at the same time a proper working is typically reported in wind-tunnel experiments.

This work investigates the laminar–turbulent transition of an incompressible swept-wing boundary-layer flow interacting with a forward-facing step using direct numerical simulations and two-dimensional linear secondary stability analysis. The focus is placed on the most critical transition scenario, where the step abruptly shifts the transition front upstream. A single-wavelength stationary crossflow perturbation is artificially triggered and allowed to develop upstream of the step. The effect of the amplitude of the incoming perturbation on the salient mechanisms of instability at the step are assessed for a fixed step height. The regime of abrupt laminar–turbulent transition at the step develops only in the case of the largest incoming perturbation amplitude considered, implying that the transition regime due to the step cannot be categorised exclusively based on local Reynolds number and instability wavelength. Step-induced transition is attributed to the stationary deformation of the shear layer immediately downstream of the step, thereby promoting rapid spatial amplification of high-frequency unsteady instabilities. The mechanisms supporting this shear-layer deformation, a mixture of linear and nonlinear growth effects, are examined by deriving energy-balance equations for the stationary spanwise-harmonic perturbation modes.

In hypersonic boundary-layer flows over blunt bodies, laminar–turbulent transition exhibits two distinct regimes: for small nose radii, increased bluntness delays transition; beyond a critical radius, further increasing bluntness reverses this trend. The latter regime corresponds to a bypass transition route, whose onset remains challenging to predict. The primary difficulty lies in capturing the excitation of non-modal streaks in the nose region, which is strongly affected by the bow shock and entropy layer effects. Recently, Zhao & Dong (J. Fluid Mech., 2025, vol. 1013, A44) developed a high-efficiency, high-accuracy shock-fitting harmonic linearised Navier–Stokes (SF-HLNS) approach to quantify the excitation of linear non-modal perturbations. In this paper, we present a predictive framework for bypass transition by integrating the SF-HLNS approach with the nonlinear parabolised stability equations (NPSE) and the bi-global stability analysis (BSA). The NPSE are employed to track the nonlinear evolution of the streaky perturbations up to nonlinear saturation, while the BSA approach is used to capture the high-growth secondary instabilities. By integrating the growth rates of these secondary instabilities from their neutral positions, an amplitude amplification factor is obtained, enabling the prediction of transition onset. Under slow acoustic forcing conditions drawn from wind-tunnel experiments, the present hybrid framework successfully reproduces the transition reversal phenomenon at large nose radii, and yields quantitative agreement with measured transition locations, thereby validating its predictive capability.

Free-stream perturbations induce large coherent structures, crucial to the laminar–turbulent transition on flight vehicles and turbine blades, significantly affecting drag, heat transfer, noise, etc. This paper computes the optimal free-stream perturbations that undergo the strongest amplification using an adjoint method, specifically for the incompressible flow around a NACA0012 airfoil, in contrast to sharp (or no) leading-edge configurations. These perturbations are examined at different angles of attack (0$^{\circ }$–$6^{\circ }$) and Reynolds numbers (${\textit{Re}} = 1000$–$8000$). At low angles of attack, two extreme points of energy growth emerge in the frequency–spanwise wavenumber plane, both on the axes. One has a spanwise wavenumber of zero, corresponding to two-dimensional unsteady modes in the wake, and the other has a frequency of zero, representing three-dimensional steady boundary-layer modes. The former is driven by the inflection point of the base velocity profile in the wake and is stronger, whereas the latter is related to the lift-up non-modal stability mechanism and directly influences the flow on the airfoil surfaces. The distribution of the three-dimensional steady optimal perturbations, similarly to the thickness of the base flow boundary layer, follows a ${\textit{Re}}^{-0.5}$ scaling law. This scaling enables the projection of optimal profiles obtained at low ${\textit{Re}}$ to high ones for engineering applications, where the direct calculation is prohibitively expensive. Subsequently, the linear optimal perturbation at ${\textit{Re}} = 100\,000$ is obtained, and the generation, nonlinear deformation and breakdown of streaks are examined by direct numerical simulation to illustrate the impact of the inflow perturbations on transition.

This study investigates the response of a Mach 4 flow over blunted plates to free stream disturbances using the linearised Navier–Stokes equations. Six nose radii, ranging from 0.1 to 2 mm, are considered. The free stream conditions match those of the experiments by Lysenko (J. Appl. Mech. Tech. Phys. vol. 31, 1990, pp. 868–873), in which the critical nose radius for transition reversal was found to be approximately 0.5 mm. The current study finds that increasing the nose radius strengthens low-frequency streamwise streaks while stabilising the oblique first mode. This leads to a shift in the dominant instability inside the boundary layer from the oblique first mode to streamwise streaks. For slow acoustic waves, the changeover nose radius is 0.7 mm. For entropy waves, vorticity waves and optimal free stream forcing obtained from resolvent analysis, the changeover nose radius is 0.2 mm. The results reproduce the transition reversal observed in the experiments and suggest that it can occur regardless of the type of free stream disturbances. The streak enhancement is attributed to the increasing magnitude of velocity perturbations near the stagnation line as the nose radius increases. Streamwise vortex-like disturbances of inviscid nature at the stagnation line generate streamwise streaks downstream via the lift-up mechanism. For the most amplified streaks, the ratio of the local boundary-layer thickness to the spanwise wavelength approaches a constant value downstream. This ratio is found to be independent of the nose radius and the type of free stream forcing. For slow acoustic disturbances, the streak strength is found to be insensitive to the angle of incidence.

It is a well-established fact that at a fixed angle of attack, with increasing Reynolds number, ${\textit{Re}}$, a laminar separation bubble (LSB) will shrink and eventually cease to exist, due to increase in the growth rate of hydrodynamic instability modes. However, during this wind tunnel test campaign, we observed that paradoxically the LSB increased in length from ${\textit{Re}}=2\times 10^5$ to ${\textit{Re}}=2.5\times 10^5$. Further diagnostics showed that this was due to a modification of the background forcing environment, often neglected in the literature. The perturbation generated from potential aero–vibroacoustic interaction unexpectedly boosted Tollmien–Schlichting modes and their faster amplification even upstream of the LSB drove the rapid secondary growth and breakdown at ${\textit{Re}}=2\times 10^5$. At ${\textit{Re}}=2.5\times 10^5$, when the lock-in with background forcing modes was not significant, a well-behaved LSB was observed with spatially amplifying Kelvin–Helmhotz modes. This finding raises important questions on the mechanisms entailing the dynamics and instability of LSBs proposed from not only experiments in similar facilities in the literature, but also in other scenarios where background perturbation due vibration, noise and confinement are non-negligible.

We investigate the influence of porous walls on the linear temporal stability of compressible boundary layers with adiabatic wall temperature, focusing on effects arising from variations in the porous wall configuration. Assuming an exponential base-flow profile and a regularly structured wall with cylindrical pores as a model assumption allows for an analytic solution of the underlying inviscid compressible Rayleigh equation in terms of the general Heun function, thereby reducing the problem to an algebraic eigenvalue equation. This enables the analysis of eigenmodes across a wide Mach number range and broad variations in the porous-wall parameters, porosity $\phi$ and layer thickness $h$, revealing a novel delineation of parameter regimes associated with different effects on the stability characteristics. We find that porous walls with very small porosities $\phi$ allow damping, whereas higher $\phi$ lead to destabilisation. The character of destabilisation is largely determined by the layer thickness: for larger $h$, long-wave, three-dimensional first-mode instabilities dominate, while small $h$ favour short-wave, two-dimensional second-mode instabilities. For intermediate $h$, the most unstable mode can be a two-dimensional first mode, in contrast to its usual three-dimensional character. Further, we show that the layer thickness $h$ causing strongest destabilisation at a given Mach number decreases with increasing $M$. The destabilising effect of porous walls also results in inviscid instabilities occurring at lower and even subsonic Mach numbers, provided $\phi$ and $h$ exceed certain thresholds. The threshold for $h$ is larger at lower $M$. Furthermore, increasing $h$ allows acoustically radiating supersonic instabilities with significant growth rates at lower Mach numbers.

We present a perturbation-based framework that captures buoyancy effects on modal instabilities in stratified boundary-layer flows within the fully compressible, non-Oberbeck–Boussinesq formulation. Treating the Richardson number as a small parameter and recasting the stability problem into an adjoint-residual form, we derive a first-order correction for the eigenvalues using only the neutrally buoyant eigenvalue problem. The framework applies to both ideal-gas and non-ideal fluid boundary layers and eliminates the need to re-solve the eigenvalue problem at each stratification level at minimal computational cost. For ideal-gas boundary layers, the framework accurately predicts how stable and unstable stratification modifies Tollmien–Schlichting waves, from growth rates and eigenfunctions to $N$-factors, across a wide range of Prandtl numbers, temperature ratios and Mach numbers. Notably, the buoyancy sensitivity varies strongly with Prandtl number, revealing that for a given Richardson number, buoyancy can switch from destabilising to stabilising depending on the fluid. Beyond ideal-gas conditions, we apply the first-order buoyancy correction to strongly stratified boundary layers with supercritical fluids, where the phase relationship between density and velocity perturbations determines whether buoyancy stabilises or destabilises the underlying instability. The resulting $N$-factors demonstrate, for the first time, that buoyancy significantly affects transition predictions under pseudo-boiling conditions.

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

This study experimentally examines how free stream turbulence (FST) alters roughness-induced transition by exposing cylindrical roughness elements of diameter-to-height ratios $\eta =1,2,3$ to three FST levels ($ \textit{Tu}\approx 0.05\,\%$, $1.15\,\%$, $1.55\,\%$). Although the roughness Reynolds number $ \textit{Re}_{\textit{kk}}$ is varied solely through the free stream velocity, a detailed characterisation of the incoming FST and resulting Klebanoff modes shows that their variation remains moderate, enabling a controlled assessment of FST effects. The FST consistently promotes earlier transition for all $\eta$ at subcritical global-instability Reynolds numbers, but its influence on the underlying instability mechanisms depends strongly on $\eta$. For the ‘thick’ cylinders ($\eta =2,3$), the global instability remains remarkably robust: the dominant varicose mode and its frequency scaling show virtually no sensitivity to FST. In contrast, the ‘thin’ cylinder ($\eta =1$) exhibits an extreme susceptibility: already moderate FST suppresses the sinuous mode predicted by global stability theory at supercritical Reynolds numbers. These findings provide a clear, experimentally grounded explanation for why global stability predictions often fail to materialise the structures in real, turbulence-contaminated environments and establish FST as a decisive factor governing whether, when and how roughness-induced instabilities manifest.

According to the state of incoming boundary layer, shock wave/boundary layer interaction can be classified into three groups: shock wave/laminar boundary layer interaction (SLBLI), shock wave/transitional boundary layer interaction (STrBLI) and shock wave/turbulent boundary layer interaction (STBLI). Although STBLI has been comprehensively studied, the difference between SLBLI and STrBLI is still not fully understood, especially in the framework of laminar–turbulent transition. In this work, a compression-ramp flow at Mach 6.5 is studied using experiment, direct numerical simulations (DNS) and stability analysis. Two distinct transition scenarios are revealed. On the one hand, without introducing external disturbances in the DNS, transition to turbulence is observed downstream of reattachment. This transition process is excited by the global instability in the separation bubble. On the other hand, the experiment captures a much smaller separation bubble under the same flow conditions. When realistic disturbances are superimposed on the freestream of DNS, a good agreement between the DNS and the experiment is achieved. The first mode is shown to be responsible for the transition. It is found that in the first transition scenario, the transition zone is located downstream of the reattachment region, and as a result, the size of the separation bubble is nearly not affected by the transition. This identifies an SLBLI. In the second transition scenario, however, the reattachment region is fully covered by the transition zone, accompanied by the shrinking of the separation bubble. This features an STrBLI. Therefore, the relative position between the reattachment region and the transition zone can be used to identify whether a flow belongs to SLBLI or STrBLI.

Görtler vortices developing over a concave wall support rapidly oscillating wavelike disturbances through secondary instabilities. Although experiments indicate that the finite-amplitude evolution of these waves acts as a precursor to turbulence transition, accurate and efficient prediction has remained out of reach. We overcome this limitation by using the parabolised coherent structures (PCS) method of Song & Deguchi (2025 J. Fluid Mech., vol. 1025, A42), which incorporates the nonlinear vortex-wave interaction into a standard spatial-marching approach. Our computational results agree well with the wave amplitude and displacement thickness observed in the widely known experiments of Swearingen & Blackwelder (1987 J. Fluid Mech., vol. 182, pp. 255–290).

We show that both temporal and spatial symmetry breaking in canonical K-type boundary layer transition arise as organised structures with quantifiable energetic pathways rather than unstructured noise. Before the skin-friction maximum, the flow is described by a periodic, spanwise-symmetric fundamental harmonic response (FHR) to the Tollmien–Schlichting wave. The FHR is spatially compact, produces hairpin packets and remains fully harmonic despite a turbulence-like appearance, thereby delimiting the deterministic regime. Past this point, a distinct regime change occurs: a hierarchy of quasi-periodic and aperiodic structures emerges, followed shortly by anti-symmetric structures that develop similarly despite no anti-symmetric inputs. We identify these structures as symmetry-decomposed spectral and space–time proper orthogonal modes that resolve the progression from deterministic harmonics to broadband dynamics. We introduce inter-modal and inter-symmetry energy budgets derived from symmetry-decomposed Navier–Stokes equations. They reveal a directed energy transfer from the FHR into the leading temporal and spatial symmetry breaking modes and, subsequently, into broadband residual fluctuations, showing that broadband dynamics grow only once inter-modal transfer is active, while inter-symmetry transfer also strongly amplifies broadband anti-symmetric fluctuations once asymmetry is present. These key insights support a view of laminar–turbulent transition as a sequence of symmetry breaking events, energetically driven by dominant space–time modes that route energy from harmonic flow to broadband turbulence.

In this paper, the transition processes induced by three-dimensional wavy wall roughnesses with two different distribution topologies (staggered type-S and aligned type-A) are studied at Mach 5.92 by direct numerical simulations. For the first time, the effects of the two distribution topologies on transition are investigated. It is found that the type-S roughness can induce transition significantly earlier – about 34.5 % earlier than that of the type-A roughness under the conditions in this paper. Both the type-S and type-A roughnesses can induce counter-rotating pairs of streamwise vortices. A ‘staggered-enhancing’ mechanism for the vortices is discovered in the type-S roughness, which results in significantly stronger vortices than in the type-A case. The enhanced vortices in turn produce stronger shear layers, which is the key factor leading to the stronger transition-induced ability of the type-S roughness. Then, based on the spectral proper orthogonal decomposition, the linear and nonlinear instability characteristics of the two roughnesses are investigated. For type-S roughness, its downstream linear instability is dominated by the low-frequency shear layer instability near 50 kHz. Once the linear fluctuation amplitude saturates, the nonlinear breakdown is triggered from the shear layer. For type-A roughness, its downstream linear instability is co-dominated by two modes: the low-frequency shear layer instability near 23 kHz, and the high-frequency Mack’s second mode above 100 kHz. Both modes exhibit significantly lower growth rates than the dominant shear layer instability in the type-S case, ultimately delaying the transition onset compared to type-S roughness.