The Stokes boundary layer (SBL) is the oscillating flow above a flat plate. Its laminar flow becomes linearly unstable at a Reynolds number of $\textit{Re} = U_0 \sqrt {T_0/\nu } \approx 2511$, where $U_0$ is the amplitude of the oscillation, $T_0$ is the period of oscillation and $\nu$ is the fluid’s kinematic viscosity, but turbulence is observed subcritically for $\textit{Re} \gtrsim 700$. The state space consists of laminar and turbulent basins of attraction, separated by a saddle point (the ‘edge state’) and its stable manifold (the ‘edge’). This work presents the edge trajectories for the transitional regime of the SBL. Despite linear dynamics disallowing the lift-up mechanism in the laminar SBL, edge trajectories are dominated by coherent structures as in other canonical shear flows: streaks, rolls and waves. Stokes boundary layer structures are inherently periodic, interacting with the oscillating flow in a novel way: streaks form near the plate, migrate upward at a speed $2\sqrt {\pi }$ and dissipate. A streak-roll-wave decomposition reveals a spatiotemporally evolving version of the self-sustaining process (SSP): (i) rolls lift fluid near the plate, generating streaks (via the lift-up mechanism); (ii) streaks can only persist in regions with the same sign of laminar shear as when they were created, defining regions that moves upward at a speed $2 \sqrt {\pi }$; (iii) the sign of streak production reverses at a roll stagnation point, destroying the streak and generating waves; (iv) trapped waves reinforce the rolls via Reynolds stresses; (v) mass conservation reinforces the rolls. This periodic SSP highlights the role of flow oscillations in sustaining transitional structures in the SBL, providing an alternative picture to ‘bypass’ transition, which relies on pre-existing free stream turbulence and spanwise vortices.

We study the stability of plane Poiseuille flow (PPF) and plane Couette flow (PCF) subject to streamwise system rotation using linear stability analysis and direct numerical simulations. The linear stability analysis reveals two asymptotic regimes depending on the non-dimensional rotation rate ($\textit{Ro}$): a low-$\textit{Ro}$ and a high-$\textit{Ro}$ regime. In the low-$\textit{Ro}$ regime, the critical Reynolds number $\textit{Re}_c$ and critical streamwise wavenumber $\alpha _c$ are proportional to $\textit{Ro}$, while the critical spanwise wavenumber $\beta _c$ is constant. In the high-$\textit{Ro}$ regime, as $\textit{Ro} \rightarrow \infty$, we find $\textit{Re}_c = 66.45$ and $\beta _c = 2.459$ for streamwise-rotating PPF, and $\textit{Re}_c = 20.66$ and $\beta _c = 1.558$ for streamwise-rotating PCF, with $\alpha _c\propto 1/Ro$. Our results for streamwise-rotating PPF match previous findings by Masuda et al. (J. Fluid Mech., vol. 603, 2008, pp. 189–206). Interestingly, the critical values of $\beta _c$ and $\textit{Re}_c$ at $\textit{Ro} \rightarrow \infty$ in streamwise-rotating PPF and PCF coincide with the minimum $\textit{Re}_c$ reported by Lezius & Johnston (J. Fluid Mech., vol. 77, 1976, pp. 153–176) and Wall & Nagata (J. Fluid Mech., vol. 564, 2006, pp. 25–55) for spanwise-rotating PPF at $\textit{Ro}=0.3366$ and PCF at $\textit{Ro}=0.5$. We explain this similarity through an analysis of the perturbation equations. Consequently, the linear stability of streamwise-rotating PCF at large $\textit{Ro}$ is closely related to that of spanwise-rotating PCF and Rayleigh–Bénard convection, with $\textit{Re}_c = \sqrt {Ra_c}/2$, where $Ra_c$ is the critical Rayleigh number. To explore the potential for subcritical transitions, direct numerical simulations were performed. At low $\textit{Ro}$, a subcritical transition regime emerges, characterised by large-scale turbulent–laminar patterns in streamwise-rotating PPF and PCF. However, at higher $\textit{Ro}$, subcritical transitions do not occur and the flow relaminarises for $\textit{Re} \lt Re_c$. Furthermore, we identify a narrow $\textit{Ro}$ range where turbulent–laminar patterns develop under supercritical conditions.

The effects of Reynolds number across ${\textit{Re}}=1000$, $2500$, $5000$ and $10\,000$ on separated flow over a two-dimensional NACA0012 airfoil at an angle of attack of $\alpha =14^\circ$ are investigated through biglobal resolvent analysis. We identify modal structures and energy amplifications over a range of frequencies, spanwise wavenumbers, and values of the discount parameter, providing insights across various time scales. Using temporal discounting, we find that the shear-layer dynamics dominates over short time horizons, while the wake dynamics becomes the primary amplification mechanism over long time horizons. Spanwise effects also appear over long time horizons, sustained by low frequencies. The low-frequency and high-wavenumber structures are found to be dominated by elliptic mechanisms within the recirculation region. At a fixed angle of attack and across the Reynolds numbers, the response modes shift from wake-dominated structures at low frequencies to shear-layer-dominated structures at higher frequencies. The frequency at which the dominant mechanism changes is independent of the Reynolds number. Comparisons at a different angle of attack ($\alpha =9^\circ$) show that the transition from wake to shear-layer dynamics with increasing frequency only occurs if the unsteady flow is three-dimensional. We also study the dominant frequencies associated with wake and shear-layer dynamics across the angles of attack and Reynolds numbers, and confirm characteristic scaling laws from the literature.

The effects of high-intensity, large-scale free stream turbulence on the aerodynamic loading and boundary layer flow field development on a NACA 0018 aerofoil model were studied experimentally using direct force measurements and particle image velocimetry at a chord Reynolds number of $7\times 10^4$. An active turbulence grid was used to generate free stream turbulence intensities of up to $16\,\%$ at integral length scales of the order of the aerofoil chord length. Relative to the clean flow condition with a free stream turbulence intensity of $0.1\,\%$, elevated levels of free stream turbulence intensity decrease the lift slope at low angles of attack, and increase the stall angle and maximum lift coefficient. At moderate angles of attack, high-intensity free stream turbulence causes large variations in the location of transition, with laminar flow occasionally persisting over $90\,\%$ of the chord length. At pre-stall angles of attack, high-intensity free stream turbulence causes intermittent massive separation. Variations in the extent of turbulence in the suction surface boundary layer are linked to fluctuations in effective angle of attack, suggesting that the observed variability in transition location is related to large-scale incoming flow disturbances impinging on the aerofoil model. A comparative analysis of the present results and those in previous studies for predominantly smaller integral length scales shows the importance of both the intensity and length scale of free stream turbulence on the flow development over the aerofoil.

We analyse the long-time dynamics of trajectories within the stability boundary between laminar and turbulent square duct flow. If not constrained to a symmetric subspace, the edge trajectories exhibit a chaotic dynamics characterised by a sequence of alternating quiescent phases and intense bursting episodes. The dynamics reflects the different stages of the well-known near-wall streak–vortex interaction. Most of the time, the edge states feature a single streak with a number of flanking vortices attached to one of the four surrounding walls. The initially straight streak undergoes a linear instability and eventually breaks in an intense bursting event. At the same time, the downstream vortices give rise to a new low-speed streak at one of the neighbouring walls, thereby causing the turbulent activity to ‘switch’ from one wall to the other. If the edge dynamics is restricted to a single or twofold mirror-symmetric subspace, the bursting and wall-switching episodes become self-recurrent in time, representing the first periodic orbits found in square duct flow. In contrast to the chaotic edge states in the non-symmetric case, the imposed symmetries enforce analogue bursting cycles to simultaneously appear at two parallel opposing walls in a mirror-symmetric configuration. Both the localisation of turbulent activity to one or two walls and the wall-switching dynamics are shown to be common phenomena in marginally turbulent duct flows. We argue that such episodes represent transient visits of marginally turbulent trajectories to some of the edge states detected here.

This study investigates the stability characteristics of rotating-disk boundary layers in rotor–stator cavities under the frameworks of local linear, global linear and global nonlinear analyses. The local linear stability analysis uses the Chebyshev polynomial method, the global linear stability analysis relies on the linearised incompressible Navier–Stokes (N–S) equations and the global nonlinear analysis involves directly solving the complete incompressible N–S equations. In the local linear framework, the velocity profile derived from the laminar self-similar solution on the rotating-disk side of an infinite rotor–stator cavity is mapped to the Bödewadt–Ekman–von Kármán theoretical model to establish a unified analytical framework. For the global stability study, we extend the methodological framework proposed by Appelquist et al. (J. Fluid Mech.,vol 765, 2015, pp. 612–631) for the von Kármán boundary layer, implementing pulsed disturbances and constructing a radial sponge layer to effectively capture the spatiotemporal evolution of perturbation dynamics while mitigating boundary reflection effects. The analysis reveals that the rotating-disk boundary layer exhibits two distinct instability regimes: convective instability emerges at ${\textit{Re}}=r^*/\sqrt {\nu ^*/\varOmega ^*}=204$ (where $r^*$ is the radius, $\nu ^*$ is the kinematic viscosity and $\varOmega ^*$ is the rotation rate of the system) with azimuthal wavenumber $\beta =27$, while absolute instability emerges at ${\textit{Re}}=409.6$ with azimuthal wavenumber $\beta =85$. Under pulsed disturbance excitation, an initial convective instability behaviour dominates in regions exceeding the absolute instability threshold. As perturbations propagate into the sponge layer’s influence domain, upstream mode excitation triggers the emergence of a global unstable mode, characterised by a minimum critical Reynolds number ${\textit{Re}}_{\textit{end}}=484.4$. Further analysis confirms that this global mode is an inherent property of the rotating-disk boundary layer and is independent of the characteristics of the sponge layer. Frequency-domain analysis establishes that the global mode frequency is governed by local stability characteristics at ${\textit{Re}}_{\textit{end}}$, while its growth rate evolution aligns with absolute instability trends. By further incorporating nonlinear effects, it was observed that the global properties of the global nonlinear mode remain governed by ${\textit{Re}}_{\textit{end}}$. The global temporal frequency corresponds to ${\textit{Re}}_{\textit{end}}=471.8$. When ${\textit{Re}}$ approaches 517.2, the spiral waves spontaneously generate ring-like vortices, which subsequently trigger localised turbulence. This investigation provides novel insights into the fundamental mechanisms governing stability transitions in the rotating-disk boundary layer of the rotor–stator cavity.

The simulation of turbulent flow requires many degrees of freedom to resolve all the relevant time and length scales. However, due to the dissipative nature of the Navier–Stokes equations, the long-term dynamics is expected to lie on a finite-dimensional invariant manifold with fewer degrees of freedom. In this study, we build low-dimensional data-driven models of pressure-driven flow through a circular pipe. We impose the ‘shift-and-reflect’ symmetry to study the system in a minimal computational cell (e.g. the smallest domain size that sustains turbulence) at a Reynolds number of 2500. We build these models by using autoencoders to parametrise the manifold coordinates and neural ordinary differential equation to describe their time evolution. Direct numerical simulations (DNSs) typically require of the order of $\mathcal{O}(10^5)$ degrees of freedom, while our data-driven framework enables the construction of models with fewer than 20 degrees of freedom. Remarkably, these reduced-order models effectively capture crucial features of the flow, including the streak breakdown. In short-time tracking, these models accurately track the true trajectory for one Lyapunov time, as well as the leading Lyapunov exponent, while at long-times, they successfully capture key aspects of the dynamics such as Reynolds stresses and energy balance. The model can quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. Additionally, we report new exact coherent states found in the DNS with the aid of these low-dimensional models. This approach leads to the discovery of seventeen previously unknown solutions within the turbulent pipe flow system, notably featuring relative periodic orbits characterised by the longest reported periods for such flow conditions.

Laminar–turbulent transition in shear flow is complicated and follows many possible routes. In this study, we seek to examine a scenario based on three-dimensional (3-D) waves (Jiang et al., 2020, J. Fluid Mech., vol. 890, A11) in compressible mixing layers, and elucidate the role of 3-D waves in generating streamwise vorticity. The Eulerian–Lagrangian coupled method is used to track the evolution of flow structures. Qualitative evidence shows that localised 3-D waves travel coherently with vortex structures at the early transition stage, which is consistent with the behaviours of 3-D waves in boundary layer transitions. To examine the local flow events surrounding 3-D waves and investigate the cause and effect relationships inherent in wave–vortex interaction, the finite-time Lyapunov exponent and components of the strain rate tensor are integrated into evolving Lagrangian material surfaces. The formation of high-shear layers in the flanks of the 3-D waves is observed, driven by fluid ejection and sweep motions induced by the amplification of 3-D waves. The $\Lambda$-shaped vortices are found born in the vicinity of high-shear regions and then stretched into hairpin-shaped vortices farther downstream. Statistical findings reveal that streamwise vorticity develops concurrently with the significant growth of the oblique mode, while the normal motion of wave structures induces a high strain rate layer in the surrounding region. In addition, conditional statistics underscore the significance of high shear in enstrophy generation. Finally, a conceptual model is proposed to depict the evolution of coherent structures based on the relationship among the 3-D waves, high-shear/strain layers, and $\varLambda$-vortices, providing insights into their collective dynamics within transitional mixing layers.

Interface-resolved direct numerical simulations are performed to investigate bubble-induced transition from a laminar to elasto-inertial turbulent (EIT) state in a pressure-driven viscoelastic square channel flow. The Giesekus model is used to account for the viscoelasticity of the continuous phase, while the dispersed phase is Newtonian. Simulations are performed for both single- and two-phase flows for a wide range of Reynolds (${Re}$) and Weissenberg (${\textit{Wi}}$) numbers. In the absence of any discrete external perturbations, single-phase viscoelastic flow is transitioned to an EIT regime at a critical Weissenberg number ($Wi_{cr})$ that decreases with increasing ${Re}$. It is demonstrated that injection of bubbles into a laminar viscoelastic flow introduces streamline curvature that is sufficient to trigger an elastic instability leading to a transition to an EIT regime. The temporal turbulent kinetic energy spectrum shows a scaling of $-2$ for this multiphase EIT regime, and this scaling is found to be independent of size and number of bubbles injected into the flow. It is also observed that bubbles move towards the channel centreline and form a string-shaped alignment pattern in the core region at the lower values of ${Re}=10$ and ${\textit{Wi}}=1$. In this regime, there are disturbances in the core region in the vicinity of bubbles while flow remains essentially laminar. Unlike the solid particles, it is found that increasing shear-thinning effect breaks up the alignment of bubbles.

Equilibrium, travelling-wave and periodic-orbit solutions of the Navier–Stokes equations provide a promising avenue for investigating the structure, dynamics and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille and pipe flow. However, the organisation of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organisation and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new travelling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one travelling-wave solution whose symmetry involves shifts by one third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behaviour of shear flows.

The vertical heated pipe is widely used in thermal engineering applications, as buoyancy can help drive a flow, but several flow regimes are possible: shear-driven turbulence, laminarised flow and convective turbulence. Steady velocity fields that maximise heat transfer have previously been calculated for heated pipe flow, but were calculated independently of buoyancy forces, and hence independently of the flow regime and time-dependent dynamics of the flow. In this work, a variational method is applied to find an optimal body force of limited magnitude $A_0$ that maximises heat transfer for the vertical arrangement, with the velocity field constrained by the full governing equations. In our calculations, mostly at Reynolds number ${\textit{Re}}=3000$, it is found that streamwise-independent rolls remain optimal, as in previous steady optimisations, but that the optimal number of rolls and their radial position is dependent on the flow regime. Surprisingly, while it is generally assumed that turbulence enhances heat transfer, for the strongly forced case, time dependence typically leads to a reduction. Beyond offering potential improvement through the targeting of the roll configuration for this application, wider implications are that optimisations under the steady flow assumption may overestimate improvements in heat transfer, and that strategies that simply aim to induce turbulence may not necessarily be efficient in enhancing heat transfer either. Including time dependence and the full governing equations in the optimisation is challenging but offers further enhancement and improved reliability in prediction.

Direct numerical simulations are conducted to investigate the transition flow over a flat plate featuring pressure gradients and a three-dimensional rough surface. The rough surface is categorised into nine types based on the effective slope ratio ${E{{S}_{z}}}/{E{{S}_{x}}}$ ($ES_{z}$: spanwise effective slope, $ES_{x}$: streamwise effective slope) and skewness $Sk$, with the embedded boundary method employed for resolving the solid wall. Findings indicate that the influence of ${E{{S}_{z}}}/{E{{S}_{x}}}$ on the streamwise vortex pair counters the effects on the wall-normal shear and the two-dimensional spanwise vortex sheet. Negative skewness alone can stimulate all three components of the hairpin vortex simultaneously. The new formula for predicting the sheltering angle, which incorporates the up-ejecting segment, demonstrates enhanced accuracy in predicting the sheltering area across the entire rough surface, outperforming the previous formulation. The forward displacement relative to the drag peak of the pressure stagnation point along the streamwise direction remains unaffected by the spanwise effective slope and the skewness. In the upper transition region, negative skewness significantly intensifies both the production and dissipation terms of the fluctuating kinetic energy, which correlate with the inviscid instability of the separation flow and the viscous instability induced by the lift-up mechanism. During the early phase of transition, negative skewness is capable of producing linear modes that match the intensity of nonlinear coherent structures at intermediate to high frequencies, exhibiting quasi-orthogonality. During the late transition phase, zero skewness can give rise to linear modes featuring robust quasi-orthogonality at low frequencies.

This study investigates the onset of linear instabilities and their later nonlinear interactions in the shear layer of an initially laminar jet using high-fidelity simulations. We present a quantitative analysis of the vortex-pairing phenomenon by computing the spatial growth rates and energy budget of the dominant frequencies. Compared with a turbulent jet, the hydrodynamic instabilities and vortex pairing are enhanced in an initially laminar jet. Using local linear theory, we identify the fundamental as the frequency with the largest spatial growth rate, and its exponential growth causes the shear layer to roll up into vortices. Visualisations and conditional $x$–$t$ plots reveal that fundamental vortices pair to form subharmonic vortices, which then merge to produce second subharmonic vortices. The energy transfer during this process is evaluated using the spectral turbulent kinetic energy equation, focusing on dominant coherent structures identified through spectral proper orthogonal decomposition. Spectral production and nonlinear transfer terms show that the fundamental frequency gains energy solely from the mean flow, while subharmonics gain energy both linearly from the mean flow and nonlinearly through backscatter from the fundamental frequency. Our results confirm Monkewitz’s theoretical model of a resonance mechanism between the fundamental and subharmonic, which supplies energy to the subharmonic. We highlight the energetic versus dynamical importance of tonal frequencies. The second subharmonic corresponds to the largest spectral peak, while the fundamental, though the fourth largest spectral peak, is dynamically dominant, as it determines all other spectral peaks and supplies energy to the subharmonics through a reverse energy cascade.

Buoyancy-driven exchange flows in geophysical contexts often exhibit significant interfacial turbulence leading to a partially mixed intermediate layer between two counterflowing layers. In this paper we perform a three-layer hydraulic analysis of such flows, highlighting the dynamical importance of the middle mixed layer. Our analysis is based on the viscous, shallow water, Boussinesq equations and includes the effects of mixing as a non-hydrostatic pressure forcing. We demonstrate the superior predictive accuracy of three-layer hydraulics over the more classical two-layer approach by applying it to direct numerical simulation data in stratified inclined duct exchange flows where turbulence is controlled by a modest slope of the duct. The three-layer model predicts a region bounded by two control points in the middle of the duct, linked to the onset of instability and turbulence, whereas a two-layer model only predicts one control point. We show that the nonlinear characteristics of the three-layer model correspond to linear long waves perturbing a three-layer mean flow. We also provide the first evidence of long-wave resonance, as well as resonance between long and short waves, and their connection to turbulence. These results challenge current parameterisations for turbulent transport, which typically overlook long waves and internal hydraulics induced by streamwise variations of the flow.

The impact of two-dimensional (2-D) periodic forcing on transition dynamics in laminar separation bubbles (LSBs) generated on a flat plate is investigated experimentally. Laminar separation is caused by the favourable-to-adverse pressure gradient under an inverted modified NACA $64_3\text{-}618$ and periodic disturbances are generated by an alternating current dielectric barrier discharge plasma actuator located near the onset of the adverse pressure gradient. Surface pressure and time-resolved particle image velocimetry measurements along the centreline and several wall-parallel planes show significant reductions in bubble size with active flow control. Periodic excitation leads to amplification of the Kelvin–Helmholtz (K–H) instability resulting in strong 2-D coherent roller structures. Spanwise modulation of these structures is observed and varies with the forcing amplitude. Intermediate forcing amplitudes result in periodic spanwise deformation of the mean flow at large wavelength ($\lambda _z/L_{b,5kVpp} \approx 0.76$). For high-amplitude forcing, the spanwise modulation of the mean flow agrees with the much smaller wavelength of the difference interaction of two oblique subharmonic modes ($\lambda _z/L_{b,5kVpp} \approx 0.24$). Modal decomposition shows nonlinear interaction of the forced 2-D mode leading to growth of subharmonic and harmonic content, and the observation of several half-harmonics ($[n+1/2]f_{\textit{AFC}}$) at intermediate forcing amplitudes. Strongest amplitudes of the 2-D mode and delay of transition downstream of the time-averaged reattachment are observed for the intermediate forcing amplitudes, previously only observed in numerical simulations. Consistent with numerical results, further increase of the forcing amplitude leads to rapid breakdown to turbulence in the LSB. This suggests that the most effective exploitation of the K–H instability for transition delay is connected to an optimal (moderate) forcing amplitude.

We use direct numerical simulations to examine the onset of stratified turbulence triggered by the zigzag instability recently identified in columnar Taylor–Green vortices (Guo et al. 2024, J. Fluid Mech., vol. 997, A34) and its role in layer formation within the flow. The study focuses on Froude numbers $0.125 \leqslant \textit{Fr} \leqslant 2.0$ and Reynolds numbers ${\textit{Re}}$ ranging from 800 to 3200. The breakdown of the freely evolving vortex array is driven by local density overturns, combining shear and convective mechanisms initiated by the primary zigzag instability. Our results show a linear relationship between the peak buoyancy Reynolds number ${{\textit{Re}}}_b^{\star }$, driven by the zigzag instability, and ${\textit{Re}}\, {\textit{Fr}}^2$. When the flow does not exhibit local shear or convective instability, the value of ${{\textit{Re}}}_b^{\star }$ falls below unity. Both density and momentum layers arise from the zigzag instability: horizontal velocity layers are strong and persistent, while density layers are weaker and more transient. The vertical scale of the mean shear layers increases with ${\textit{Fr}}$ for ${\textit{Fr}} \leqslant 1$, shows weak dependence on ${\textit{Re}}$, and agrees well with the length scale associated with the fastest-growing linear mode of the zigzag instability. Further analysis in the sorted buoyancy coordinate highlights the role of density overturns caused by the zigzag instability in forming buoyancy layers during the transition to turbulence.

Optimal transitional mechanisms are analysed for an incompressible shear layer developing over a short, pressure gradient-induced laminar separation bubble (LSB) with peak reversed flow of 2 %. Although the bubble remains globally stable, the shear layer destabilises due to the amplification of external time- and spanwise-periodic disturbances. Using linear resolvent analysis, we demonstrate that the pressure gradient modifies boundary layer receptivity, shifting from Tollmien–Schlichting (T-S) waves and streaks in a zero-pressure-gradient environment to Kelvin–Helmholtz (K-H) and centrifugal instabilities in the presence of the LSB. To characterise the nonlinear evolution of these disturbances, we employ the harmonic-balanced Navier–Stokes (N-S) framework, solving the N-S equations in spectral space with a finite number of Fourier harmonics. Additionally, adjoint optimisation is incorporated to identify forcing disturbances that maximise the mean skin friction drag, conveniently chosen as the cost function for the optimisation problem since it is commonly observed to increase in the transitional stage. Compared with attached boundary layers, this transition scenario exhibits both similarities and differences. While oblique T-S instability is replaced by oblique K-H instability, both induce streamwise rotational forcing through the quadratic nonlinearity of the N-S equations. However, in separated boundary layers, centrifugal instability first generates strong streamwise vortices due to multiple centrifugal resolvent modes, which then develop into streaks via lift-up. Finally, we show that the progressive distortion and disintegration of K-H rollers, driven by streamwise vortices, lead to the breakdown of large coherent structures.

Hypersonic transition studies on systems sustaining multimodal dynamics are critical to understanding aerothermal loading on flight-relevant configurations. The present work evaluates transition mechanisms in hypersonic boundary layers over a cone–cylinder–flare geometry, and its sensitivity to free stream disturbance amplitudes, using a global linear stability approach and direct numerical simulations (DNS). Under relatively quiet conditions, the flow field resembles the laminar solution, consisting of a large separation zone over the cylinder–flare junction. Linear analysis identifies multiple convective instabilities including, oblique first modes and two-dimensional second modes over the cone segment, and shear layer instabilities over the separation zone. This separation zone also supports a stationary global instability, producing streamwise streaks with an azimuthal wavenumber, $m=21$, which eventually drives transition as captured in the DNS. Conversely, at higher disturbance amplitudes, the largely attached boundary layer transitions through a bypass mechanism, involving intermodal interactions between low-frequency streaks, and first mode instabilities. The resulting upstream shift in transition onset leads to a significant rise in both steady and unsteady surface loading. Peak thermal loading under quiet conditions displays the signature of the linear global instability over the flare, whereas that under noisier environments is dominated by an imprint of unsteady Görtler vortices over the cylinder–flare junction.

This work explores the use of a shallow surface hump for passive control and stabilisation of stationary crossflow (CF) instabilities. Wind tunnel experiments are conducted on a spanwise-invariant swept-wing model. The influence of the hump on the boundary layer stability and laminar–turbulent transition is assessed through infrared thermography and particle image velocimetry measurements. The results reveal a strong dependence of the stabilisation effect on the amplitude of the incoming CF disturbances, which is conditioned via discrete roughness elements at the wing leading edge. At a high forcing amplitude, weakly nonlinear stationary CF vortices interact with the hump and result in an abrupt anticipation of transition, essentially tripping the flow. In contrast, at a lower forcing amplitude, CF vortices interact with the hump during linear growth. Notable stabilisation of the primary CF disturbance and considerable transition delay with respect to the reference case (i.e. without hump) is then observed. The spatial region just downstream of the hump apex is shown to be key to the stabilisation mechanism. In this region, the primary CF disturbances rapidly change spanwise orientation and shape, possibly driven by the pressure gradient change-over caused by the hump and the development of CF reversal. The amplitude and shape deformation of the primary CF instabilities are found to contribute to a long-lasting suboptimal growth downstream of the hump, eventually leading to transition delay.

In the dynamical systems approach to turbulence, unstable periodic orbits (UPOs) provide valuable insights into system dynamics. Such UPOs are usually found by shooting-based Newton searches, where constructing sufficiently accurate initial guesses is difficult. A common technique for constructing initial guesses involves detecting recurrence events by comparing past and future flow states using their $L_2$-distance. An alternative method uses dynamic mode decomposition (DMD) to generate initial guesses based on dominant frequencies identified from a short time series, which are signatures of a nearby UPO. However, DMD struggles with continuous symmetries. To address this drawback, we combine symmetry-reduced DMD (SRDMD) introduced by Marensi et al. (2023, J. Fluid Mech., vol. 954, A10), with sparsity promotion. This combination provides optimal low-dimensional representations of the given time series as a time-periodic function, allowing any time instant along this function to serve as an initial guess for a Newton solver. We also discuss how multi-shooting methods operate on the reconstructed trajectories, and we extend the method to generate initial guesses for travelling waves. We demonstrate SRDMD as a method complementary to recurrent flow analysis by applying it to data obtained by direct numerical simulations of three-dimensional plane Poiseuille flow at the friction Reynolds number $\textit{Re}_\tau \approx51$ ($\textit{Re}=802$), explicitly taking a continuous shift symmetry in the streamwise direction into account. The resulting unstable relative periodic orbits cover relevant regions of the state space, highlighting their potential for describing the flow.