Ionic surfactants are commonly employed to modify the rheological properties of fluids, particularly in terms of surface viscoelasticity. Concurrently, external electric fields can significantly impact the dynamics of liquid threads. A key question is how ionic surfactants affect the dynamic behaviour of threads in the presence of an electric field? To investigate this, a one-dimensional model of a liquid thread coated with surfactants within a radial electric field is established, employing the long-wave approximation. We systematically investigate the effects of dimensionless parameters associated with the surfactants, including surfactant concentration, dilatational Boussinesq number ${\textit{Bo}}_{\kappa \infty }$ and shear Boussinesq number ${\textit{Bo}}_{\mu \infty }$. The results indicate that increasing the surfactant concentration and the two Boussinesq numbers reduces both the maximum growth rate and the dominant wavenumber. In addition, both the electric field and surfactants mitigate the breakup of the liquid thread and the formation of satellite droplets. At low applied electric potentials, the surface viscosity induced by surfactants predominantly governs this suppression. Surface viscosity suppresses the formation of satellite droplets by maintaining the neck point at the centre of the liquid thread within a single disturbance wavelength. When the applied potential is high, the electric stress has two main effects: the external electric field exerts a normal pressure on the liquid thread surface, suppressing satellite droplet formation, while the internal electric field inhibits liquid drainage. Surface viscosity further stabilizes the system by suppressing flow dynamics during this process.

The triadic interactions and nonlinear energy transfer are investigated in a subsonic turbulent jet at $Re = 450\,000$. The primary focus is on the role of these interactions in the formation and attenuation of streaky structures. To this end, we employ bispectral mode decomposition, a technique that extracts coherent structures associated with dominant triadic interactions. A strong triadic correlation is identified between Kelvin–Helmholtz (KH) wavepackets and streaks: interactions between counter-rotating KH waves generates streamwise vortices, which subsequently give rise to streaks through the lift-up mechanism. The most energetic streaks occur at azimuthal wavenumber $m = 2$, with the dominant contributing triad being $[m_1, m_2, m_3] = [1, 1, 2]$. The spectral energy budget reveals that the net effect of nonlinear triadic interactions is an energy loss from the streaks. As these streaks convect downstream, they engage in further nonlinear interactions with other frequencies, which drain their energy and ultimately lead to their attenuation. Further analysis identifies the dominant scales and direction of energy transfer across different spatial regions of the jet. While the turbulent jet exhibits a forward energy cascade in a global sense, the direction of energy transfer varies locally: in the shear layer near the nozzle exit, triadic interactions among smaller scales dominate, resulting in an inverse energy cascade, whereas farther downstream, beyond the end of the potential core, interactions among larger scales prevail, leading to a forward cascade.

The nonlinear Tollmien–Schlichting waves mechanism of subcritical transitional flow in quasi-two-dimensional flow and two-dimensional (2-D) plane Poiseuille flow have been investigated (Camobreco et al. 2023 J. Fluid Mech., vol. 963, p. R2; Huang et al. 2024 J. Fluid Mech., vol. 994, p. A6). However, the subcritical transitional flow threshold has remained unsolved for 2-D shear flows since the problem was proposed in Trefethen et al. (1993 Science vol. 261, no. 5121, pp. 578–584). In this study, we proposed a theoretical analysis based on the nonlinear non-modal analysis and asymptotic analysis to quantify the scaling law for subcritical transitional flow of 2-D plane Poiseuille flow. The subcritical transitional flow induced by the critical disturbance experiences the nonlinear edge state with invariant disturbance kinetic energy (Huang et al. 2024 J. Fluid Mech. vol. 994, p. A6). Consequently, the required magnitude along with the edge state is predicted by asymptotic analysis, and the a priori threshold is achieved theoretically. All stages are validated by the numerical minimal seeds of different channels. The proposed theory predicts that the scaling laws are $O(Re^{-11/3})$ and $O(\textit{Re}^{-7/3})$ for the critical disturbances and their edge state, respectively. While the numerical thresholds of the subcritical transitional flow are $ \textit{Re}^{-11/3 \pm 0.06}$ and $ \textit{Re}^{-7/3 \pm 0.05}$, respectively.

Pulsatile fluid flows through straight pipes undergo a sudden transition to turbulence that is extremely difficult to predict. The difficulty stems here from the linear Floquet stability of the laminar flow up to large Reynolds numbers, well above experimental observations of turbulent flow. This makes the instability problem fully nonlinear and thus dependent on the shape and amplitude of the flow perturbation, in addition to the Reynolds and Womersley numbers and the pulsation amplitude. This problem can be tackled by optimising over the space of all admissible perturbations to the laminar flow. In this paper, we present an adjoint optimisation code, based on a GPU implementation of the pseudo-spectral Navier–Stokes solver nspipe, which incorporates an automatic, optimal checkpointing strategy. We leverage this code to show that the flow is susceptible to two distinct instability routes: one in the deceleration phase, where the flow is prone to oblique instabilities, and another during the acceleration phase with similar mechanisms as in steady pipe flow. Instability is energetically more likely in the deceleration phase. Specifically, localised oblique perturbations can optimally exploit nonlinear effects to gain over nine orders of magnitude in energy at a peak Reynolds number of ${\textit{Re}}_{\textit{max}}\approx 4000$. These oblique perturbations saturate into regular flow patterns that decay in the acceleration phase or break down to turbulence depending on the flow parameters. In the acceleration phase, optimal perturbations are substantially less amplified, but generally trigger turbulence if their amplitude is sufficiently large.

Ice shelves that spread into the ocean can develop rifts that can trigger iceberg calving and enhance ocean-induced melting. Fluid mechanically, this system is analogous to an extensionally dominated radial spreading of a non-Newtonian fluid into a relatively inviscid and denser ambient fluid. Laboratory experiments have shown that rift patterns can emerge when the spreading fluid is shear thinning. Linear stability analysis supports these findings, predicting that while the instability mechanism is active in Newtonian fluids, it is suppressed by stabilising secondary-flow cellular vortices. Here, we explore the fully nonlinear evolution of a radially spreading Newtonian fluid, assessing whether large-amplitude perturbations could drive an instability. We use a quasi-three-dimensional numerical simulation that solves the full nonlinear shallow-shelf approximation, tracing the evolving fluid front, and validate it with known axisymmetric solutions and predictions from linear-stability theory. We find that large-amplitude perturbations induce nonlinear effects that give rise to non-axisymmetric patterns, including cusp-like patterns along the fluid front and complex secondary-flow eddies, which have neither been predicted theoretically nor observed experimentally. However, despite these nonlinear effects, large-amplitude perturbations alone are insufficient to induce rift-like patterns in Newtonian fluids. Strain-rate peaks at the troughs of the fluid front suggest that shear-thinning fluids may become more mobile in these regions, potentially leading to rift formation. This coincides with the likely weakening of stabilising forces as the fluid becomes more shear-thinning. These findings elucidate the critical role of nonlinear viscosity on the formation of rift-like patterns, which is the focus of Part 2 of this study.

The nonlinear growth of perturbations in hydrodynamic interfacial instabilities can be of particular importance in both scientific research (e.g. supernova explosion) and engineering applications (e.g. inertial confinement fusion). One of the most significant issues in these instabilities is the long-time nonlinear bubble evolution of a single-mode Rayleigh–Taylor instability (RTI), which remains as an unsolved and challenging problem since Taylor’s seminal work more than seven decades ago. Introduced in this paper is an analytical model for the long-time evolution of bubble velocity, curvature and vorticity, which is established by considering the vorticity accumulation around the bubble in a bilaterally rotational flow system under the classical planar potential-flow theory framework. The proposed theoretical model incorporates not only the classical linear, nonlinear and quasi-steady stages, but the late re-acceleration stage. Meanwhile, the new model can capture the phenomenon of secondary velocity saturation following the stage of bubble re-acceleration. The good agreement between the present model and numerical simulations for all density ratios and dimensions confirms that the accumulation in vorticity tends to break the early stage buoyancy-drag equilibrium mechanism and leads to the establishment of a new equilibrium in the late-stage RTI.

Asymmetries and anisotropies are widespread in biological systems, including in the structure and dynamics of cilia and eukaryotic flagella. These microscopic, hair-like appendages exhibit asymmetric beating patterns that break time-reversal symmetry needed to facilitate fluid transport at the cellular level. The intrinsic anisotropies in ciliary structure can promote preferential beating directions, further influencing their dynamics. In this study, we employ numerical simulation and bifurcation analysis of a mathematical model of a filament driven by a follower force at its tip to explore how intrinsic curvature and direction-dependent bending stiffnesses impact filament dynamics. Our results show that while intrinsic curvature is indeed able to induce asymmetric beating patterns when filament motion is restricted to a plane, this beating is unstable to out-of-plane perturbations. Furthermore, we find that a three-dimensional whirling state seen for isotropic filament dynamics can be suppressed when sufficient asymmetry or anisotropy are introduced. Finally, for bending stiffness ratios as low as 2, we demonstrate that combining structural anisotropy with intrinsic curvature can stabilise asymmetric beating patterns, highlighting the crucial role of anisotropy in ciliary dynamics.

Finite-amplitude spiral vortex flows are obtained numerically for the Taylor–Couette system in the narrow limit of the gap between two concentric rotating cylinders. These spiral vortex flows bifurcate from circular Couette flow before axisymmetric Taylor vortex flow sets in when the ratio $\mu$ of the angular velocities of the outer to the inner cylinder is less than −0.78, consistent with the results of linear stability analysis by Krueger et al. (J. Fluid Mech., vol. 24, 1966, pp. 521–538), while the boundary of existence of spiral vortex flows is determined not by the linear critical point, but by the saddle-node point of the subcritical spiral vortex flow branch for $\mu \lessapprox -0.75$, when the axial wavenumber $\beta =2.0$. It is found that the nonlinear spiral vortex flows exhibit the mean flow in the axial direction as well as in the azimuthal direction, and that the profiles of both mean-flow components are asymmetric about the centre plane between the gap.

Pendant drops appear in many engineering applications, such as inkjet printing and optical tensiometry, and they have also been the subject of studies of droplet–particle interaction. While the hydrostatics of pendant drops has been studied extensively, the influence of external flow disturbances has received limited attention. This research aims to incorporate aerodynamic factors into the understanding of pendant drop behaviour. Employing a simplified model, an irrotational flow aligned with the drop’s axis is derived from a distribution of singularity elements within the drop. The drop’s equilibrium shape is then determined using a numerical model that couples the flow field with the Young–Laplace equation. The model’s predictions are compared to droplet images captured via high-speed shadowgraph in a vertical wind tunnel, showing good agreement with the experimentally observed shapes. Additionally, under certain flow conditions, the drop exhibits instability in the form of periodic pendulum-like motion. This instability was linked to two distinct critical drop heights, and the corresponding stability criterion was mathematically derived from the numerical model. Our theoretical and experimental findings provide the first quantitative description of the equilibrium shape and stability criterion of pendant drops under the influence of external flow.

The complex behaviour of air–liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder et al. (1986, Phys. Rev. A, vol. 34, 5175) to present a systematic investigation of the dynamics that results when a small air bubble is placed at the tip of a steadily propagating air finger in a horizontal Hele-Shaw channel. The system can exhibit steady and oscillatory behaviours, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger’s tip. At fixed flow rate, the oscillations commence when the bubble size is increased sufficiently that the decreased curvature of the bubble tip in the horizontal plane matches that of the finger tip. This causes the axial pressure gradient along the bubble to vanish, thus rendering the bubble susceptible to lateral perturbations. Differing time scales for finger and bubble restoral allow sustained oscillations to develop in the finger–bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a single finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie disorder in viscous fingering.

The Lamb–Oseen vortex is a model for practical vortical flows with a finite vortex core. Vortices with a Lamb–Oseen vortex velocity profile are stable according to the Rayleigh criterion in an infinite domain. Practical situations introduce boundary conditions over finite domains. Direct numerical simulations are performed on the evolution of perturbations to a viscous Lamb–Oseen vortex with uniform inlet axial velocity in a pipe of finite length. Linear stability boundaries are determined in the $(\textit{Re},\omega )$ plane. For a given swirl ratio $\omega$, the flow is found to become linearly unstable when the Reynolds number $\textit{Re}$ is above a critical value. The complete evolution history of the flow is followed until it reaches its final state. For small swirl ratios, the axisymmetric mode is linearly unstable and evolves to a final steady axisymmetric but non-columnar accelerated flow state after nonlinear saturation. For large swirl ratios, the spiral mode is linearly unstable. The spiral mode is found to force growth of an axisymmetric component due to nonlinear interaction. The flow evolves to a final unsteady spiral vortex breakdown state after it undergoes nonlinear saturation. The energy transfer between the mean flow and perturbations is studied by the Reynolds–Orr equation. The pressure work at the exit of the finite pipe is a major source of energy production. Finite-domain boundary conditions also modify the perturbation mode shapes, which can render the vortex core from absorbing energy to producing energy, and thus lead to instabilities. As the pipe length increases, the stability behaviour of the flow is found to approach that predicted by the classical Rayleigh criterion.

Rotor–stator interactions in turbomachines are characterised by a complex interplay of hydrodynamic instabilities, acoustic pressure waves and receptivity mechanisms, as well as the collision of coherent structures with the blade geometry. An unsteady dual analysis of self-excited instabilities and flow interactions, exemplified by a simple model compressor stage under subsonic conditions, is proposed and presented. Using a low-dissipation sliding-plane implementation, instability-resolving nonlinear-adjoint looping simulations provide detailed sensitivity information that allows for the dissection of the full flow into sub-components linked to distinct flow phenomena. This sensitivity information further links observed flow behaviour to its hydrodynamic or acoustic origin, thereby laying the foundation for a cause-and-effect analysis and for flow control.

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.

We study the stationary, intermittent and nonlinear dynamics of nominally ideally expanded, natural and forced supersonic twin-rectangular turbulent jets using spectral modal decomposition. We decompose large-eddy simulation data into four reflectional symmetry components about the major and minor axes. In the natural jet, spectral proper orthogonal decomposition (SPOD) uncovers two resonant instabilities antisymmetric about the major axis. Known as screech tones, the more energetic of the two is a steady flapping instability, while the other is an intermittent double-flapping instability. We test the hypothesis that symmetry breaking can be leveraged for control design. Time-periodic forcing symmetric about the major and minor axes is implemented using a plasma actuation model, and succeeds in removing screech from a different symmetry component. We investigate the spectral peaks of the forced jet using an extension of bispectral mode decomposition (BMD), where the bispectrum is bounded by unity and which conditionally recovers the SPOD. We explain the appearance of harmonic peaks as three sets of triadic interactions between reflectional symmetries, forming an interconnected triad network. BMD modes of active triads distil coherent structures comprising multiple coupled instabilities, including Kelvin–Helmholtz, core and guided-jet modes (G-JM). Downstream-propagating core modes can be symmetric or antisymmetric about the major axis, whereas upstream-propagating G-JM responsible for screech closure (Edgington-Mitchell et al. J. Fluid Mech.945, 2022, p. A8) are antisymmetric only. The dependence of G-JM on symmetry hence translates from the azimuthal symmetry of the round jet to the dihedral group symmetry of the twin-rectangular jet, and explains why the twin jet exhibits antisymmetric but not symmetric screech modes.

Interactions between shock waves and gas bubbles in a liquid can lead to bubble collapse and high-speed liquid jet formation, relevant to biomedical applications such as shock wave lithotripsy and targeted drug delivery. This study reveals a complex interplay between acceleration-induced instabilities that drive jet formation and radial accelerations causing overall bubble collapse under shock wave pressure. Using high-speed synchrotron X-ray phase contrast imaging, the dynamics of micrometre-sized air bubbles interacting with laser-induced underwater shock waves are visualised. These images offer full optical access to phase discontinuities along the X-ray path, including jet formation, its propagation inside the bubble, and penetration through the distal side. Jet formation from laser-induced shock waves is suggested to be an acceleration-driven process. A model predicting jet speed based on the perturbation growth rate of a single-mode Richtmyer–Meshkov instability shows good agreement with experimental data, despite uncertainties in the jet-driving mechanisms. The jet initially follows a linear growth phase, transitioning into a nonlinear regime as it evolves. To capture this transition, a heuristic model bridging the linear and nonlinear growth phases is introduced, also approximating jet shape as a single-mode instability, again matching experimental observations. Upon piercing the distal bubble surface, jets can entrain gas and form a toroidal secondary bubble. Linear scaling laws are identified for the pinch-off time and volume of the ejected bubble relative to the jet’s Weber number, characterising the balance of inertia and surface tension. At low speeds, jets destabilise due to capillary effects, resulting in ligament pinch-off.

When a liquid film on a horizontal plate is driven in motion by a shear stress, surface waves are easily generated. This paper studies such flow at moderate Reynolds numbers, where the surface tension and inertial force are equally important. The governing equations for two-dimensional flows are derived using the long-wave approximation along with the integral boundary-layer theory. For small disturbances, the dispersion relation and neutral curves are determined by the linear stability analysis. For finite-amplitude perturbations, the numerical simulation suggests that the oscillations generated by the perturbation in a certain place continuously spread to the surrounding areas. When the effects of surface tension and gravity reach equilibrium, steady-state solutions will emerge, which include two cases: solitary waves and periodic waves. The former have heteroclinic trajectories between two stationary points, while the latter include five patterns at different parameters. In addition, there are also periodic waves that do not converge after a long period of time. During these evolution processes, strange attractors appear in the phase space. By examining the Poincaré section and the sensitivity to initial values, we demonstrate that these waves can be divided into two types: quasi-periodic and chaotic solutions. The specific type depends on parameters and initial conditions.

Numerical studies on the statistical properties of irregular waves in finite depth have to date been based on models founded on weak nonlinearity; as a consequence, only lower-order (usually third-order) nonlinear interactions have thus far been investigated. The present study performs numerical simulations with a fully nonlinear, spectrally accurate model to investigate the statistics of irregular, unidirectional wave fields in finite water depth initially given by a Texel, Marsen and Arsloe spectrum. A series of random unidirectional wave fields are considered, covering a wide range of water depth. The wave spectrum and statistical properties, including the probability density function of the surface elevation, exceedance probability of wave crests and occurrence probability of extreme (rogue) waves, are investigated. The importance of full nonlinearity in comparison with third-order results is likewise evaluated. The results show that full nonlinearity increases kurtosis and enhances the occurrence probability of large wave crests and rogue waves substantially, in both deep water and finite water depth. Therefore, we propose that full nonlinearity may contribute significantly to the formation of rogue waves. Furthermore, to account for the effects of higher-order nonlinearity on modulational instability, we analyse the relationship between the Benjamin–Feir index (BFI) and maximal excess kurtosis. Our results show a strong linear relationship i.e. $({\mathcal{K}}_{max}-3)\propto {\textrm{BFI}}$, in contrast to $({\mathcal{K}}_{max}-3)\propto {\textrm{BFI}}^2$ based on the assumptions of weak nonlinearity, a narrow-banded spectrum and deep-water conditions. Above, $\mathcal{K}_{max}$ is the maximal kurtosis.

The reactivity of transverse waves in detonations of methane, oxygen and nitrogen are experimentally assessed using MHz rate schlieren and chemiluminescence imaging. In these highly unstable mixtures, the mode of wave propagation is more complex than what is described by the cellular instability model that is conventionally used for weakly unstable mixtures. Behind the low-speed leading shock in unstable waves, the processed gas remains essentially unreacted until transverse waves reach this region. In highly unstable waves, the transverse waves have a range of reactivity, that is rates of reaction in the flow immediately behind the wave. In this study, we present examples of transverse waves for near-limit detonations and analyse four cases in detail. In some cases, these waves appear to be essentially non-reactive or cause very slow reaction. In other cases, the transverse waves can be highly reactive. In the most extreme example, the transverse wave is propagating at the Chapman–Jouguet speed with a small reaction zone, i.e. a transverse detonation. A reactive oblique shock model is used to approximate the triple-point configuration of this case as a double-Mach reflection, which shows good agreement with the images. The reaction evolution along path lines is analysed using detailed reaction mechanisms and considerations about flow-field unsteadiness. Length scales of the energy release and expansion processes within the reaction zone region are used to explain the observed modes of wave propagation and interaction.

Uniform arrays of particles tend to cluster as they sediment in viscous fluids. Shape anisotropy of the particles enriches this dynamics by modifying the mode structure and the resulting instabilities of the array. A one-dimensional lattice of sedimenting spheroids in the Stokesian regime displays either an exponential or an algebraic rate of clustering depending on the initial lattice spacing (Chajwa et al. 2020 Phys. Rev. X vol. 10, pp. 041016). This is caused by an interplay between the Crowley mechanism, which promotes clumping, and a shape-induced drift mechanism, which subdues it. We theoretically and experimentally investigate the sedimentation dynamics of one-dimensional lattices of oblate spheroids or discs and show a stark difference in clustering behaviour: the Crowley mechanism results in clumps comprising several spheroids, whereas the drift mechanism results in pairs of spheroids whose asymptotic behaviour is determined by pair–hydrodynamic interactions. We find that a Stokeslet, or point-particle, approximation is insufficient to accurately describe the instability and that the corrections provided by the first reflection are necessary for obtaining some crucial dynamical features. As opposed to a sharp boundary between exponential growth and neutral eigenvalues under the Stokeslet approximation, the first-reflection correction leads to exponential growth for all initial perturbations, but far more rapid algebraic growth than exponential growth at large dimensionless lattice spacing $\tilde {d}$. For discs with aspect ratio $0.125$, corresponding to the experimental value, the instability growth rate is found to decrease with increasing lattice spacing $\tilde {d}$, approximately as $\tilde {d}^{ -4.5}$, which is faster than the $\tilde {d}^{-2}$ for spheres (Crowley 1971 J. Fluid Mech. vol. 45, pp. 151–159). It is shown that the first-reflection correction has a stabilising effect for small lattice spacing and a destabilising effect for large lattice spacing. Sedimenting pairs predominantly come together to form an inverted ‘T’, or ‘$\perp$’, which our theory accounts for through an analysis that builds on Koch & Shaqfeh (1989 J. Fluid Mech. vol. 209, pp. 521–542). This structure remains stable for a significant amount of time.

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.