The stability of liquid-film flows is essential in many industrial applications. In the dip-coating process, a liquid film forms over a substrate extracted at a constant speed from a bath. We studied the linear stability of this film considering different thicknesses $\hat {h}$ for four liquids, spanning an extensive range of Kapitza numbers ($Ka$). By solving the Orr–Sommerfeld eigenvalue problem with the Chebyshev–Tau spectral method, we calculated the threshold between growing and decaying perturbations, investigated the instability mechanism, and computed the absolute/convective threshold. The instability mechanism was studied by analysing the perturbations’ vorticity distribution and the kinetic energy balance. It was found that liquids with low $Ka$ (e.g. corn oil, $Ka = 4$) are stable for a smaller range of wavenumbers compared with liquid with high $Ka$ (e.g. liquid zinc, $Ka = 11\,525$). Surface tension has a stabilising and a destabilising effect. For long waves, it curves the vorticity lines near the substrate, reducing the flow under the crests. For short waves, it fosters vorticity production at the interface and creates a region of intense vorticity near the substrate. In addition, we discovered that the surface tension contributes to both the production and dissipation of perturbation's energy depending on the $Ka$ number. Regarding the absolute/convective threshold, we identified a window in the parameter space where unstable waves propagate throughout the entire domain (indicating absolute instability). Perturbations affecting Derjaguin's solution ($\hat {h}=1$) for $Ka<17$ and the Landau–Levich–Derjaguin solution ($\hat {h}=0.945 Re^{1/9}Ka^{-1/6}$), are advected by the flow (indicating convective instability).

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

We investigate effect of porous insert located upstream of the separation edge of a backward-facing step (BFS) in early transitional regime as a function of Reynolds number. This is an example of hydrodynamic system that is a combination of separated shear flow with large amplification potential and porous materials known for efficient flow destabilisation. Spectral analysis reveals that dynamics of BFS is dominated by spectral modes that remain globally coherent along the streamwise direction. We detect two branches of characteristic frequencies in the flow and with Hilbert transform we characterise their spatial support. For low Reynolds numbers, the dynamics of the flow is dominated by lower frequency, whereas for sufficiently large Reynolds numbers cross-over to higher frequencies is observed. Increasing permeability of the porous insert results in decrease in Reynolds number value, at which frequency cross-over occurs. By comparing normalised frequencies on each branch with local stability analysis, we attribute Kelvin–Helmholtz and Tollmien–Schlichting instabilities to upper and lower frequency branches, respectively. Finally, our results show that porous inserts enhance Kelvin–Helmholtz instability and promote transition to oscillator-type dynamics. Specifically, the amplitude of vortical (BFS) structures associated with higher-frequency branch follows Landau model prediction for all investigated porous inserts.

The effects of thermal convection on turbulence in accretion discs, and particularly its interplay with the magnetorotational instability (MRI), are of significant astrophysical interest. Despite extensive theoretical and numerical studies, such an interplay has not been explored experimentally. We conduct linear analysis of the azimuthal version of MRI (AMRI) in the presence of thermal convection and compare the results with our experimental data published before. We show that the critical Hartmann number ($Ha$) for the onset of AMRI is reduced by convection. Importantly, convection breaks symmetry between $m = \pm 1$ instability modes ($m$ is the azimuthal wavenumber). This preference for one mode over the other makes the AMRI wave appear as a ‘one-winged butterfly’.

We investigated the stability of the bottom boundary layer (BBL) beneath periodic internal solitary waves (ISWs) of depression over a flat bottom through two-dimensional direct numerical simulations. We explored the convective versus absolute/global nature of the BBL instability in response to changes in Reynolds number, and the sensitivity of the instability to seeding noise in the front of the ISW – spanning laboratory to geophysical scales. The BBL was laminar at $Re_{ISW}=90$ and convectively unstable at $Re_{ISW}=300$. At laboratory-scale $Re_{ISW}=300$, the convective wave packet was periodically amplified by each successive ISW, until vortex shedding occurred. The associated noise-amplification behaviour potentially explains the discrepancies of the critical $Re_{ISW}$ between the lock–release laboratory experiments and our Dubreil–Jacotin–Long-initialized numerical simulations as the result of the difference in background noise. Instability energy decreased under the front shoulder of the ISW, analogous to flow relaminarization under a favourable pressure gradient. At geophysical-scale $Re_{ISW}=900$, the BBL was initially convectively unstable, and then the instability tracked with the ISW, appearing phenomenologically similar to a global instability. The simulated initial convective instability at both $Re_{ISW}=300$ and $Re_{ISW}=900$ is in agreement with local linear stability analysis which predicts that the instability group speed is always lower than the ISW celerity. Increased free stream perturbations in front of the ISW and larger $Re_{ISW}$ shift the location of vortex shedding (and enhanced bed shear stress) beneath the wave, closer to the ISW trough, thereby potentially changing the location of maximum sediment resuspension, in agreement with field observations at higher $Re_{ISW}$.

The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$. Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$, a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$. The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$. Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$.

A global transient linear stability analysis of the three-dimensional time-dependent flow around an aerofoil undergoing small-amplitude pitching motion is performed using the optimally time-dependent (OTD) framework. The most salient linear instabilities associated with the instantaneous basic state are computed and tracked over time. The resulting OTD modes reflect the variations in the basic state and can be used as predictors of its spatial and temporal evolution, including the formation of a laminar separation bubble and its gradual spanwise modulation via primary global instability, leading to secondary instability and finally rapid breakdown to turbulence. The study confirms and expands upon earlier stability analyses of the same case based on the local properties of spanwise averaged velocity profiles in the bubble that predicted the onset of absolute instability soon followed by rapid breakdown of the separation bubble. The three-dimensional structure of the most unstable OTD mode is extracted, which compares well with both the locally absolutely unstable mode and the evolution of the basic state itself.

We study the stability of a zero-pressure gradient boundary layer subjected to free-stream disturbances by means of local stability analysis. The dataset under study corresponds to a direct numerical simulation (DNS) of a flat plate with a sharp leading edge in realistic wind tunnel conditions, with a turbulence level of 3.45 % at the leading edge. We present a method to track the convective evolution of the secondary instabilities of streaks by performing sequential stability calculations following the wave packet, connecting successive unstable eigenfunctions. A scattered nature, in time and space, of secondary instabilities is seen in the stability calculations. These instabilities can be detected before they reach finite amplitude in the DNS, preceding the nucleation of turbulent spots, and whose appearance is well correlated to the transition onset. This represents further evidence regarding the relevance of secondary instabilities of streaks in the bypass transition in realistic flow conditions. Consistent with the spatio-temporal nature of this problem, our approach allows us to integrate directly the local growth rates to obtain the spatial amplification ratio of the individual instabilities, where it is shown that instabilities reaching an $N$-factor in the range [2.5,4] can be directly correlated to more than 65 % of the nucleation events. Interestingly, it is found that high amplification is not only attained by modes with high growth rates, but also by instabilities with sustained low growth rates for a long time.

This paper reports on the efficacy of the Görtler number in scaling the laminar-turbulent boundary-layer transition on rotating cones facing axial inflow. Depending on the half-cone angle $\psi$ and axial flow strength, the competing centrifugal and cross-flow instabilities dominate the transition. Traditionally, the flow is evaluated by using two parameters: the local meridional Reynolds number $Re_l$ comparing the inertial versus viscous effects and the local rotational speed ratio $S$ accounting for the boundary-layer skew. We focus on the centrifugal effects, and evaluate the flow fields and reported transition points using Görtler number based on the azimuthal momentum thickness of the similarity solution and local cone radius. The results show that Görtler number alone dominates the late stages of transition (maximum amplification and turbulence onset phases) for a wide range of investigated $S$ and half-cone angle ($15^{\circ } \leq \psi \leq 50^{\circ }$), although the early stage (critical phase) seems to be not determined by the Görtler number alone on the broader cones ($\psi =30^{\circ }$ and $50^{\circ }$) where the primary cross-flow instability dominates the flow. Overall, this indicates that the centrifugal effects play an important role in the boundary-layer transition on rotating cones in axial inflow.

We combine flow visualisation techniques and particle image velocimetry to experimentally investigate the higher-order transition to elastoinertial turbulence of Boger fluids ($El = 0.11\unicode{x2013}0.34$) in Taylor–Couette flows. The observed route to turbulence is associated with the appearance of chaotic inflow jets, termed flame patterns, for increasing inertia, and stable structures of solitons, known as diwhirls, for decreasing inertia. We also report for the first time spatially and temporally resolved flow fields in the meridional plane for the three characteristic viscoelastic flow regimes (diwhirls, flame patterns and elastoinertial turbulence). We observe in all cases coherent structures of dynamically independent solitary vortex pairs. The stability of these coherent structures is jet-dominated and can be mainly ascribed to the high extension of the polymer chains in the inflow boundaries in the $r$–$z$ plane. Solitary pairs are self-sustained when created through random events and do not split; instead, they merge when moving sufficiently close and annihilate when hoop stresses are not sufficient to sustain them. The highly localised and random events result in highly fluctuating, chaotic flow states. We estimate the decay exponent of spatial power spectral density, illustrating a universal scaling of $-2.5$ for elastoinertial turbulence. Based on our observations and in an effort to unify and combine precedent theories with our results, we suggest a mechanism for the origins of elastoinertial instabilities, accounting for both the effect of elasticity on the vortex formation and the effect of increasing/decreasing inertia on the flow dynamics.

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

Supersonic flow over a hollow cylinder/flare with a free-stream Mach number of 2.25 is numerically investigated in this study. Axisymmetric computational fluid dynamics simulations and global stability analysis (GSA) are performed for a wide range of cylinder radii and flare deflection angles. The onset of incipient and secondary separation is delayed as the cylinder radius is decreased due to the axisymmetric effects. The GSA reveals that a decrease in cylinder radius also postpones the emergence of global instability. The GSA results agree well with the results of direct numerical simulations for a supercritical case in the linear stage. The saturated flow exhibits pairs of unsteady streamwise streaks downstream of reattachment. The criterion of the global stability boundary established for supersonic flow over a compression corner (Hao et al., J. Fluid Mech, vol. 919, 2021, A4) is extended to its axisymmetric counterpart.

Modal and non-modal linear stability analyses are employed to investigate the effect of internal and external heating on disturbance temporal growth for the Darcy–Bénard convection with throughflow. A matrix-forming approach is employed for both purposes, where the generalised eigenvalue problem is built using the generalised integral transform technique. Although the disturbance equations are not self-adjoint, the non-modal analysis indicates that there is no transient growth. Hence, any disturbance growth in time must be induced by modal mechanisms. An absolute instability analysis reveals that viscous dissipation has a destabilising effect and introduces new modes that are eventually destabilised by increasing the Péclet number. Beyond critical values of the Péclet number, where codimension-two absolutely unstable points exist, these modes become more unstable than the classical mode found in the absence of viscous dissipation, which is stabilised by an increasing Péclet number. This internal heating mechanism generated by viscous dissipation is so strong at high enough Péclet numbers that instability becomes possible through heating from above.

The shock wave–turbulent boundary layer interaction over a compression corner is studied using global stability analysis (GSA) and resolvent analysis based on a separation of scales between the low-frequency, large-scale motions and the turbulent fluctuations. The GSA identifies a leading stationary mode, which becomes globally unstable as the ramp angle is beyond a critical value. For globally stable flows, the resolvent analysis captures two-dimensional and three-dimensional local maxima in optimal gain, both of which are due to modal resonance between the forcing and the leading global mode. Notably, the frequency-premultiplied optimal gain associated with two-dimensional disturbances peaks at a low frequency. For different interaction strengths, the peak frequencies collapse onto a universal value of 0.015 when non-dimensionalized using the length of the separation region and the free-stream velocity. A numerical simulation perturbed with the corresponding optimal forcing reveals that the response is in the form of a back-and-forth shock motion.

The global linear stability analysis for the magnetohydrodynamic liquid metal flow past an insulated sphere subjected to a constant streamwise magnetic field is investigated in the range of the Reynolds number $Re\leq 400$ and the interaction number $N\leq 40$ coupled with direct numerical simulations, where $N$ stands for strength of the electromagnetic force. The stability of the steady axisymmetric base flow to independent time-azimuthal modes is discussed. Five critical curves associated with various wake transitions are obtained in the $\{Re, N\}$ phase diagram. These critical curves reveal the stabilising effect of a weak magnetic field, the destabilising effect of a strong magnetic field and re-stabilising effect of a much stronger magnetic field. To explore the impact of the magnetic field on flow instability, a sensitivity analysis utilizing an adjoint method is performed for the first regular bifurcation. Sensitivity functions of growth rate to base-flow modifications and Lorentz force are defined to identify the region that has the most significant influence on flow instability, such as the recirculation region responsible for the stabilising effect at a weak magnetic field and the shear layer region responsible for the destabilising effect at a strong magnetic field. Furthermore, a competition between the stabilising and shear destabilising effects of the magnetic field is discussed. This analysis provides valuable insights into the non-monotonic effect of the magnetic field on flow instability.

Shock buffet on wings is a phenomenon caused by strong shock-wave/boundary-layer interaction resulting first in self-sustained flow unsteadiness and eventually in a detrimental structural response called buffeting. While it is an important aspect of wing design and aircraft certification, particularly for modern transonic air transport, not all of the underlying multidisciplinary physics is thoroughly understood. Building upon a single-discipline shock-buffet stability study, this work now investigates the impact of an elastic structure in these extreme flow conditions. Specifically, a triglobal stability analysis of a fluid–structure coupled system is presented, utilising the implicitly restarted Arnoldi method with a sparse iterative Krylov solver and novel preconditioner. Asymmetry resulting from a static aeroelastic simulation based on a finite-element model of the underlying geometry in a wind tunnel modifies the global modes of the earlier fluid-only symmetric full-span analysis. A flutter stability analysis at wind-tunnel flow conditions below shock-buffet onset finds no instability in the structural degrees-of-freedom, whereas in shock-buffet flow with globally unstable fluid modes additional marginally unstable structural (and fluid) modes emerge. The developed stability tool for coupled analysis is instrumental in identifying those physically relevant and strongly coupled modes where a standard pk-type (p being eigenvalue and k reduced frequency) flutter analysis fails. With the complementary computation of adjoint eigenmodes, the core of the instability is pinpointed to a relatively small wing area which may help to effect the control and delay of this detrimental transonic unsteadiness. We contribute to the question on how the presence of the elastic wing structure impacts on the otherwise pure aerodynamic three-dimensional shock-buffet dynamics.

The primary instability of the steady two-dimensional flow past rectangular cylinders moving parallel to a solid wall is studied, as a function of the cylinder length-to-thickness aspect ratio ${A{\kern-4pt}R} =L/D$ and the dimensionless distance from the wall $g=G/D$. For all ${A{\kern-4pt}R}$, two kinds of primary instability are found: a Hopf bifurcation leading to an unsteady two-dimensional flow for $g \ge 0.5$, and a regular bifurcation leading to a steady three-dimensional flow for $g < 0.5$. The critical Reynolds number $Re_{c,2\text{-}D}$ of the Hopf bifurcation ($Re=U_\infty D/\nu$, where $U_\infty$ is the free stream velocity, $D$ the cylinder thickness and $\nu$ the kinematic viscosity) changes with the gap height and the aspect ratio. For ${A{\kern-4pt}R} \le 1$, $Re_{c,2\text{-}D}$ increases monotonically when the gap height is reduced. For ${A{\kern-4pt}R} >1$, $Re_{c,2\text{-}D}$ decreases when the gap is reduced until $g \approx 1.5$, and then it increases. The critical Reynolds number $Re_{c,3\text{-}D}$ of the three-dimensional regular bifurcation decreases monotonically for all ${A{\kern-4pt}R}$, when the gap height is reduced below $g < 0.5$. For small gaps, $g < 0.5$, the hyperbolic/elliptic/centrifugal character of the regular instability is investigated by means of a short-wavelength approximation considering pressureless inviscid modes. For elongated cylinders, ${A{\kern-4pt}R} > 3$, the closed streamline related to the maximum growth rate is located within the top recirculating region of the wake, and includes the flow region with maximum structural sensitivity; the asymptotic analysis is in very good agreement with the global stability analysis, assessing the inviscid character of the instability. For cylinders with $AR \leq 3$, however, the local analysis fails to predict the three-dimensional regular bifurcation.

The presence of large-scale coherent structures can significantly impact the dynamics of a turbulent flow field and the behaviour of a flame stabilized in that flow. The goal of this study is to analyse how increasing free-stream turbulence can change the response of the flow to longitudinal acoustic excitation of varying amplitudes. We study the flow in the wake of a cylindrical bluff body at both non-reacting and reacting conditions, as the presence of a flame can significantly alter the global stability of the flow. The frequency of longitudinal acoustic excitation is set to match the natural frequency of anti-symmetric vortex shedding for this configuration and we vary the free-stream turbulence using perforated plates upstream of the bluff body. The results show that varying the level of free-stream turbulence can influence not only the amplitude of the coherent flow response, but also the symmetry of vortex shedding in the presence of longitudinal acoustic excitation. Increasing the turbulence intensity can fundamentally change the structure of the time-averaged flow and can directly impact the coherent flow response in two ways. First, increasing turbulence intensity can enhance the amplitude of the natural anti-symmetric vortex shedding mode in the wake. Second, increasing turbulence intensity weakens the symmetric response of the flow to the longitudinal acoustic excitation. In the non-reacting and reacting conditions, both symmetric and anti-symmetric modes are present and are characterized using a spectral proper orthogonal decomposition. We see evidence of interaction between the symmetric and anti-symmetric modes, which leads to an interference pattern in the coherent vorticity response in the shear layers. We conclude by presenting a conceptual model for the influence that turbulence has on these flows.

In this paper, the possibility of absolute instability in a plane unidirectional jet is analysed. We consider a parametrized family of velocity profiles with variable inflection point location and shear layer thickness. Using the inviscid saddle-point analysis, we show that absolute instability can occur in the case of a sufficiently low velocity at the inflection point or a sufficiently thin shear layer. Then we proceed to the viscous analysis and find the critical Reynolds numbers separating the zones of convective and absolute instability. We obtained a minimum value $Re = 315$. As an independent verification of the theoretical results, we conduct a direct numerical simulation of the evolution of a localized pulse perturbation in the framework of the linearized Navier–Stokes equations. The calculated absolute/convective instability boundary is in a good agreement with theoretical results of the saddle-point analysis.

Thin-film flow down a fibre exhibits rich dynamics and is relevant to applications such as desalination, fibre coating and fog harvesting. These flows are subject to instabilities that result in dynamic bead-on-fibre patterns. We perform an experimental study of shear-thinning flow down fibres using 20 different xanthan gum solutions as our working liquid. The bead-on-fibre morphology can be oriented either symmetrically or asymmetrically on the fibre, and this depends upon the surface tension, fibre diameter and liquid rheology, as defined by the Ostwald power-law index. For highly shear-thinning liquids, it is possible for the pattern to be complex and exhibit simultaneously both asymmetric large beads and symmetric small beads in the isolated and convective flow regimes. We quantify the transition between flow regimes and bead dynamics for the asymmetric morphology, and compare with Newtonian flow, as it depends upon the experimental parameters. Finally, the dimensionless bead frequency is shown to scale with the Bond number for all of our experimental data (symmetric and asymmetric).