Transonic shock buffet is a typical flow instability induced by shock-wave/boundary-layer interaction. It exhibits sustained shock oscillations and lift fluctuations, thereby restricting the flight envelope of aircraft. Previous numerical studies have revealed that evident hysteresis envelopes exist between the onset and extinction boundaries of transonic buffet during angle-of-attack sweeps. Hence accurate characterisation of buffet boundaries under representative dynamic manoeuvres is critical for assessing flight safety and expanding the flight envelope. In this study, a time-varying autoregressive with exogenous input model is established by integrating reduced-order and time-varying modelling to predict the buffet boundary under nonlinear transonic dynamic conditions. Specifically, the nonlinear flow evolution during ascending and descending angle-of-attack sweeps is modelled via piecewise linearisation, and the evolutionary trajectory of system stability is derived from state-space representation to characterise when buffet occurs. The influences of pitching rate and sweep direction on the hysteresis characteristics of the buffet onset boundary are investigated. For rate-dependent hysteresis, a lower pitching rate advances the buffet onset boundary under ascending angle-of-attack sweep, and the boundary gradually converges as the pitching rate further decreases. For path-dependent hysteresis, the ascending sweep yields a higher buffet onset angle of attack than the descending sweep, and such discrepancy diminishes gradually with the reduction of pitching rate. In summary, lowering the pitching rate can effectively suppress both types of dynamic hysteresis characteristics. The proposed method offers engineering guidance for monitoring and early warning of time-varying stability in aerospace and mechanical vibration applications.

Radially spreading extensional flows can become unstable and develop tearing patterns consisting of rifts and tongues. Such patterns have been observed in laboratory experiments with strain-rate-softening (shear-thinning) polymer solutions and in spreading ice shelves. Linear stability analysis predicts that axisymmetric spreading of power-law fluids becomes unstable when the fluid is shear thinning. However, understanding the emergence and evolution of rifts requires analysis of the nonlinear flow regime. In Part 1 of this study we developed a numerical simulation of the full nonlinear equations and showed that Newtonian fluids remain globally stable, consistently with theoretical predictions and experimental observations. Here we extend that framework to explore power-law fluids. We demonstrate that purely viscous shear-thinning fluids can develop rifts, and evolve rift-tongue patterns consistently with theoretical and experimental findings. Starting from an axisymmetric base state, we examine the resulting flow under monochromatic perturbations over a range of amplitudes, wavenumbers, fluid properties, and boundary and initial conditions. We find that for shear-thinning fluids, rift formation depends strongly on perturbation amplitude, while being less sensitive to the perturbation wavenumber. The mechanism reflects competition between a destabilising radial force associated with hoop stresses and a stabilising force arising from radial normal-stress divergence. As the fluid gets more shear thinning, the stabilising contribution weakens, facilitating rift formation. These results provide new insights into the nonlinear mechanisms governing rift formation in polymer solutions and in spreading ice shelves.

We investigate start-up flow from an initially quiescent state in three ordered sphere packings (hexagonal close-packed, face-centred cubic, body-centred cubic) driven by a constant intrinsic pressure gradient using direct numerical simulation. The main finding is that the first nonlinear departure from the initial linear regime occurs at a characteristic inviscid time $\tau _{{inv}}=\sqrt {\rho d/\vert \boldsymbol{\nabla }\langle p\rangle _{\textit{i}}\vert }$ (where $\rho$, $d$, $\boldsymbol{\nabla }\langle p\rangle _{\textit{i}}$ correspond to fluid density, sphere diameter, intrinsic pressure gradient), rather than at a critical instantaneous Reynolds number. At early times, the flow consists of an irrotational outer motion and thin Stokes-type boundary layers near the solid surfaces; nonlinear effects emerge when the convective term in these boundary layers becomes comparable to the other leading-order terms. At the pore scale, the transient evolution is characterised by the growth of thin vorticity layers on the sphere surfaces, their large-scale migration into the pore space after times of order $\tau _{{inv}}$ and the formation of inertial cores. Despite geometric differences, these processes occur in a remarkably similar sequence across all three packings. Vorticity magnitude exhibits laminar boundary-layer scaling with the governing dimensionless parameter Hagen number, whilst in the body-centred cubic sphere pack case a transition towards turbulent-type scaling is observed. These results establish $\tau _{{inv}}$ as a unifying measure for the onset of nonlinearity in strongly accelerated porous media flows, with direct implications for the modelling of unsteady transport in natural and engineered systems.

Wall-resolved large-eddy simulations are used to investigate the effects of elevated surface temperature on the leading edge of a swept wing. Earlier stability analyses demonstrate a strong destabilising effect resulting from surface heating, but do not reveal the ultimate effects on transition and relaminarisation. The current results support the findings from stability analysis with elevated surface temperatures, but show a more complex dependency between instability and turbulence than is observed for adiabatic conditions. For attachment-line Reynolds numbers above the adiabatic relaminarisation threshold, instability leads to fully turbulent flow. At Reynolds numbers below the adiabatic relaminarisation threshold, instability leads to finite-amplitude equilibrium states and intermittent turbulent bursting, but not fully turbulent flow.

Clustering techniques provide a powerful framework for analysing complex flow dynamics while reducing computational costs in large-scale simulations. In this work, we propose a clustering-based approach using vector quantisation principal component analysis (VQPCA) to identify structural sensitivity zones, i.e. regions where the flow is most receptive to perturbations. To the authors’ knowledge, this is the first application of VQPCA to a fluid dynamics problem for identifying flow patterns and dynamically relevant regions. As a fully data-driven technique, it does not rely on adjoint methods and depends exclusively on direct simulation data. This method extracts dominant flow features by partitioning the flow field into regions characterised by their intrinsic dynamics. Its validity is assessed by analysing the wake behind a circular cylinder, where the identified regions show strong similarity with previously established structural sensitivity zones. Robustness is further evaluated under different operating conditions for this configuration. The approach is then applied to the interaction of two planar synthetic jets to investigate more complex flow dynamics and illustrate how the identified regions can guide flow control strategies. These results highlight the potential of clustering-based methods as practical tools for analysing complex flows and identifying regions where small perturbations can strongly influence global dynamics.

This study investigates the effect of a tangential electric field on the curvature singularity encountered by evolving vortex sheets. Weakly nonlinear models are derived by using asymptotic expansions based on the Dirichlet–Neumann operators. A fully nonlinear boundary integral method based on Cauchy’s integral formula and an angle –arclength derivative formulation in a Lagrangian framework is proposed. Linear stability analysis reveals a transition from the unstable regime to a neutrally stable regime at a critical electric field strength. However, both weakly nonlinear models and fully nonlinear simulations suggest that the curvature singularity persists regardless of the value of the electric field strength. The analysis demonstrates that the singularity time obeys two different asymptotic rules. For subcritical values of the tangential electric field and small initial amplitudes, the singularity time scales with the logarithm of the inverse of the amplitude, and monotonically increases with the electric field strength. For supercritical values of the tangential electric field the singular time is asymptotically larger, follows an inverse power law with respect to the initial amplitude and reaches local maxima as the electric field strength varies. Fully nonlinear simulations of the underlying electrodynamic Euler equations confirm that the curvature singularity can be significantly delayed when the electric field strength takes some special values. In the limit of large electric field strengths the numerical simulations indicate that the time to singularity formation is proportional to the inverse square root of the electric field strength.

Shock-tube experiments are conducted to examine the initial-amplitude dependence of strong-shock-driven interfacial instability in reflected-shock configurations at Mach numbers ($M_s$) exceeding 3.0. Qualitative observations reveal a marked suppression (enhancement) of bubble (spike) growth under strong shocks, with stronger effects at higher dimensionless initial amplitudes ($ka_0$). The initial-amplitude effect on linear growth exhibits a non-monotonic dependence on $M_s$, contrary to the previously anticipated saturation behaviour, arising from the $M_s$ dependence of compressibility and vorticity-deposition contributions. Compressibility suppresses bubble growth via a shock-proximity mechanism while promoting spike evolution through a newly identified Mach-reflection mechanism, with their competition yielding a saturation dependence on $M_s$. In contrast, the effectiveness of vorticity deposition is modulated by the Mach-reflection structure, with reduced contributions at high $M_s$ due to geometric separation. Based on these insights, an empirical factor is proposed to quantify the initial-amplitude effect across a wide range of $ka_0$ and $M_s$. Dimensionless nonlinear amplitude evolution collapses across diverse $ka_0$ conditions. However, flow evolution depends strongly on $ka_0$, indicating that amplitude alone is insufficient to characterise the dynamics. Reliable prediction of nonlinear bubble (spike) evolution requires accurate representation of the shock-proximity and secondary-compression mechanisms (Mach-reflection and spike-acceleration mechanisms). However, existing nonlinear models fail to satisfy this requirement. The present study provides a foundation for future investigations of interfacial instability under extreme conditions relevant to inertial confinement fusion and astrophysical flows.

In practical combustion systems, co-firing of ammonia (${\rm NH}_3$) with blends of methane (${\rm CH}_4$) and hydrogen (${\rm H}_2$) offers a promising pathway towards low-carbon combustion. However, the nonlinear thermoacoustic oscillations arising from such multi-fuel blends remain poorly understood, particularly in systems supporting intrinsic thermoacoustic (ITA) modes. In this study, we experimentally investigate the effects of varying fuel composition on thermoacoustic instability in a partially premixed swirl-stabilised combustor. Operating at a fixed thermal power ($k=4100\,\mathrm{W}$) and equivalence ratio ($\phi =0.85$), we systematically vary the fuel composition across blends of ${\rm CH}_4$, ${\rm H}_2$ and ${\rm NH}_3$, mapping the resulting dynamical transitions. We show that the system exhibits a sequence of nonlinear states depending on hydrogen enrichment: periodic limit-cycle for pure methane, quasiperiodicity at moderate ${\rm H}_2$ ($P_{\textrm{H}_2} = 0.2$–0.3) and period-3 state at higher enrichment ($P_{\textrm{H}_2} = 0.4$–0.5), all at fixed $P_{\textrm{NH}_3}$. As hydrogen content increases, the system undergoes mode-switching transitions within the ITA mode family, from high-frequency fourth-harmonic ($f_1$) to first-harmonic ($f_2$) oscillations, associated with strong pressure amplitudes and modified flame structures. Conversely, increasing ammonia content at constant $P_{\textrm{H}_2} \lt 0.3$ suppresses oscillation amplitudes across both periodic and quasiperiodic regimes. Our chemiluminescence imaging analysis reveals that the ITA-driven instabilities at its first harmonic ($f_2$) with shorter, flatter, M-shaped flames exhibit reduced flame response time and intensified heat release concentrated near the dump plane, leading to thermoacoustic amplification. These findings demonstrate that mode switching through precise tuning of $P_{\textrm{H}_2}$ and $P_{\textrm{NH}_3}$ offers new insights for the design of low-emission, fuel-flexible combustion systems.

This study investigates the oblique incident wave and the wave crossing in nonlinear wave evolution under variable directional spreading and bathymetry. Building on the depth-modified nonlinear Schrödinger (dNLS) model for a two-dimensional random wavefield, we incorporate wave refraction and symmetric crossing wave systems to numerically simulate realistic ocean conditions more accurately. Monte Carlo simulations are conducted to analyse wave spectrum, surface morphology and statistical moments (skewness and kurtosis). Results show that small incident angles have a very limited effect on extreme events and nonlinear interactions in a single system, but even small-angle wave crossing leads to an increase in the occurrence probability of freak waves from single system to multi-system. As the crossing angle increases, the enhanced dispersion suppresses transverse wave groupiness and reduces the likelihood of extreme events.

We consider a fluid filament on a solid substrate, exposed to localised perturbations that modify its material properties, particularly its viscosity. The considered model geometry and material parameters are motivated by an experimental set-up involving metal filaments subjected to laser heating, which liquefies them, leading to fluid flow while the temperature is above the melting point. The localised perturbations are created by adding disjoint metal pillars, which, due to the effect of ‘thermal crowding’ – meaning increased energy absorption due to the additional deposited metal – modify the local filament properties. Depending on the pillars’ positioning, one could consider them either as ‘thermal scissors’ (splitting the filament at the pillar location into segments) or as the source of the filament’s edge melting, leading to retraction and break-up. A precise understanding of the mechanism underlying the filament’s break-up, supported by efficient simulations, enables rationalising the dynamics and final pattern formation, as well as controlling the size and positioning of the resulting metal particles. In particular, we identify numerically a bifurcation structure in which the positioning and number of pillars lead to a dramatic transition in the final outcome. While we focus on a rather specific set-up, we expect similar mechanisms to be relevant to other systems in which material parameters could be locally modified by externally imposed perturbations.

The dynamics of detonation is governed by intrinsic longitudinal and transverse instabilities. This work fills the research gap of detonation stability at elevated pressure as real gas effects, including the finite molecular volume and inter-molecular attraction forces, become non-negligible. The stability of one-dimensional pulsating and two-dimensional cellular detonations at elevated pressure was investigated using both linear stability analysis and high-fidelity numerical simulations. For pulsating detonations, the neutral stability boundaries derived from the linear stability analysis reveal that the impact of real gas effects on detonation stability depends on the heat content. Specifically, real gas effects stabilise pulsating detonation at high heat content, destabilise it at intermediate levels and exhibit competing influences, i.e. destabilisation from finite molecular volume and stabilisation from the inter-molecular attraction force, at low heat content. For cellular detonations, a stability map is proposed in the Mach number–reduced activation energy plane, demonstrating that real gas effects stabilise detonation and lead to more regular cellular structures or even fully planar detonations. Theoretical predictions show good agreement with numerical simulations. An oscillator model was used to elucidate the interplay between the heat release and shock dynamics. It shows that the stabilising effect of real gas is associated with reduced velocity oscillation amplitude, increased phase lag between heat release and shock speed variations and lower values of the coupling parameter. The comprehensive linear stability analysis with complete real gas models is an effective predictive tool for the detonation dynamics at elevated pressure.

Streamwise vortices are ubiquitous structures on the surfaces of three-dimensional aircrafts and are prone to Z- and Y-mode instabilities. We systematically investigate the nonlinear development of vortex-mode instabilities and resonant interactions employing plane marching nonlinear parabolised stability equations (NPSE3D) and quasi-direct numerical simulation (QDNS). The results suggest the moderately strong subharmonic resonance, and that the detuned interaction between a high-frequency Y-mode and a low-frequency Z-mode may precipitate the transition process. Albeit without any sign of the fundamental resonance, the self-interaction of a Y-mode generates mean flow distortion (MFD) that, in turn, yields the amplitude saturation and causes the hairpin vortex formed by the Y-mode to cascade into smaller scale vortices via vortex reconnection. Sensitivity analysis reveals that MFD exerts a destabilising direct effect on modal instability, but this is offset by its stabilising indirect effect through weakened base-flow shear, ultimately resulting in net stabilisation. Higher-frequency modes display heightened sensitivity to MFD. Moreover, QDNS exhibits a late-stage surface skin friction overshoot, accompanied by an amplification of subharmonic frequencies.

The addition of a small amount of long-chain polymers confers viscoelastic properties to Newtonian flows. The resulting non-Newtonian solution now exhibits a different dynamics, such as enhanced mixing at low Reynolds number, where elastic instabilities can trigger elastic turbulence even though inertial turbulence is absent. Here, we study this phenomenon in viscoelastic planar jets and, in particular, we do it from the perspective of coherent structures to understand how elastic turbulence is triggered and sustained, which remain barely explored in this set-up. We introduce the spatio-temporal Koopman decomposition for extracting the dominant flow patterns, and we compare them with those from Newtonian planar jets at high Reynolds number. Global flow structures are similar between jets, with low-frequency streaks and high-frequency wave packets dominating the turbulent dynamics. However, structures are strikingly different in the near field, where elasticity-driven streaks affect the dynamics in the potential core of the viscoelastic planar jet, modifying the bulk flow and interacting with the flow instability. The analysis of the polymer field reveals stretched polymer filaments and centre-mode structures, which support the implication of the near-field streaks sustaining elastic turbulence in three-dimensional viscoelastic planar jets.

This paper investigates the stability of interfacial long waves in two-layer plane Couette flow using a nonlinear, non-local asymptotic model derived from the Navier–Stokes equations and valid for thin upper layers. Non-locality enters through a coupling of the thin and main layers, and crucial inertial effects are retained. The models generically support bistability phenomena observed in experiments where two stable travelling waves, one unimodal and the other bimodal, are recorded at the same lid velocity. In direct comparisons with experiments, the models show remarkable agreement, both qualitatively and quantitatively. The two stable travelling waves are identified and their basins of attraction characterised via large-time computations for different initial conditions. We also identify a new symmetry-breaking travelling-wave branch bifurcating from the bimodal family, compute higher-wavenumber travelling-wave branches and present time-periodic orbits arising via Hopf bifurcation. A symmetry is also presented that links solutions for thin upper layers to those corresponding to thin lower layers. The instability of the two solutions is shown to be identical.

This study presents a combined analytical and experimental investigation into the rising dynamics of Newtonian droplets in viscoelastic fluids, focusing on discontinuous transitions in droplet velocity. Although velocity jumps have been observed in gas bubbles, this work is the first systematic investigation of the phenomenon in Newtonian droplets. Analytically, a triple-perturbation approach is developed using Reynolds (Re), Deborah (De) and capillary (Ca) numbers as expansion parameters. The surrounding viscoelastic medium is modelled using the nonlinear-Giesekus constitutive equation. The analysis reveals that increasing Deborah number intensifies and reorients normal stresses – particularly the polar ($\theta$-direction) and hoop components – along the droplet interface. These stresses migrate rearward with increasing elasticity, resulting in localised elongational flows and concentrated radial stresses near the rear stagnation point, which indicates the onset of elastic instabilities. Experimentally, Newtonian droplets are introduced into a variety of exterior fluids: (i) quasilinear Boger fluids, which are constant-viscosity viscoelastic solutions that isolate elastic effects; and (ii) a strongly nonlinear viscoelastic solution, which exhibits both high elasticity and significant shear-thinning behaviour. Equivalent Newtonian fluids of matched viscosity are also used as controls. Viscoelastic fluids are solutions of polyacrylamide in the water–glycerine mixture. Across all experiments, the dimensionless ranges were $\textit{Re}\approx 6\times 10^{-4}{-}4.59$, $\textit{De}\approx 0.057{-}0.802$ and $\textit{Ca}\approx 0.0026{-}0.638$, framing the scope of applicability. The experimental observations challenge the classical expectation of smooth droplet motion and demonstrate that Newtonian droplets in viscoelastic environments can exhibit sharp, weak or no velocity jumps, depending on the balance between elasticity, viscosity and deformability.

This study develops a novel theoretical model for predicting the nonlinear evolution of Richtmyer–Meshkov instability (RMI) in particle-laden flows at large Stokes numbers. We construct a coupled multiphase potential flow theory framework incorporating two key models: (i) a postshock interface velocity attenuation model based on exponential decay accounting for momentum dissipation and (ii) a unified bubble and spike growth model for multiphase conditions. The multiphase-unified model maintains compatibility with classical single-phase RMI theories in the dilute limit, meanwhile revealing stronger particle-induced nonlinear decay in the amplitude growth rate. Model validation demonstrates good quantitative agreement across key predictive metrics – including dilute to dense particle volume fractions, Atwood numbers, particle sizes and initial perturbation amplitudes. This wide-range predictive capability for nonlinear instability growth may improve theoretical understanding of phenomena relevant to engineering applications. The results reveal that increased particle loading significantly reduces the growth rate of interface disturbances due to enhanced damping effects, leading to the blunting of spikes and flattening of bubbles. Vorticity dynamics analysis further shows that particle-induced vorticity weakens baroclinic production, thereby stabilizing the flow and inhibiting RMI development.

This work experimentally investigates Richtmyer–Meshkov (RM) turbulence driven by a cylindrical converging shock using time-resolved planar laser-induced fluorescence. An automatically retractable plate is developed to generate membraneless yet sharp interfaces characterised by random short-wavelength perturbations and controllable long-wavelength components. Five initial interfaces are examined: one with random short-wavelength perturbations, and four with these random perturbations superimposed with single-mode long-wavelength perturbations of controlled amplitude and wavenumber. The shocked interface in the converging configuration exhibits strongly unsteady motion: it initially moves inwards uniformly, then decelerates, moves outwards following reshock, and finally decelerates again with noticeable oscillations. The interface initially develops with a well-defined morphology, but quickly transitions to turbulent mixing after reshock. Both reshock and Rayleigh–Taylor instability, unique to converging RM flows, significantly influence the mixing layer evolution. The mixing width exhibits significantly higher post-reshock growth rates compared with the planar counterpart. Scalar mixedness analysis indicates that the random-perturbation case achieves significantly higher homogenisation efficiency than single-mode cases, whereas the mean scalar dissipation rate follows a similar temporal pattern regardless of initial morphology. The initial perturbation spectrum strongly influences the timing of turbulent transition: the high-wavenumber single-mode case reaches the empirical threshold earliest, followed by the random-perturbation case, while the low-wavenumber single-mode cases exhibit substantially delayed or incomplete transition. Scalar energy spectra further highlight that initial perturbations critically affect the efficiency of turbulent cascade and the small-scale energy distribution.

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

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a channel is investigated using asymptotic and numerical methods, building on the linear framework developed by Ricco & Alvarenga (2021 J. Fluid Mech., vol. 927, A18). The focus is on low-frequency disturbances that induce streamwise-elongated structures at Reynolds numbers for which the entrance flow is locally stable according to classical linear stability theory. The perturbation flow along the channel entrance is generated by free-stream vortical disturbances located at the channel inlet. These disturbances are symmetric or antisymmetric with respect to the centreplane and their amplitude is sufficiently intense to provoke nonlinear interactions within the channel. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations. Combined with physically realistic initial conditions, the resulting initial-boundary-value problems are solved numerically using a streamwise integration method. A parametric study is conducted to elucidate how the nonlinear channel flow is influenced by the Reynolds number and the inlet-disturbance properties, i.e. the amplitude and the streamwise, wall-normal and spanwise wavelengths. Nonlinearity is found to stabilise the intense algebraic growth and to drive the formation of elongated channel-entrance structures that span the entire cross-section. These structures, characterised by low- and high-speed regions and streamwise vortices, meander along the streamwise direction and persist even when the base flow is fully developed. They exhibit a half-turn rotational symmetry with respect to the vortex centres. These properties emerge downstream regardless of the symmetry of the initial perturbation flow, provided nonlinear interactions are sufficiently intense. The occurrence of travelling waves is detected sufficiently downstream, and their similarity to those found in the fully developed region by other researchers is discussed. Our results show good agreement with theoretical predictions, numerical results and experimental measurements for both the mean flow and the perturbation flow.

Recent studies have shown that, in coastal waters where water depth decreases significantly due to rapid bathymetric changes, the non-equilibrium dynamics (NED) substantially increases the occurrence probability of extreme (rogue) waves. Nevertheless, research on depth-induced NED has been predominantly confined to unidirectional irregular waves, while the role of directionality remains largely unexplored. The scarce studies on multidirectional waves mainly rely on numerical simulations and have yielded conflicting results. In this work, we report on an experimental investigation of wave directionality on the depth-induced non-equilibrium wave statistics. High-order statistical moments, skewness and kurtosis, are used as proxies for the non-equilibrium wave response. Our results indicate that the directional spreading has a minor effect on decreasing the maximum values of these statistical moments. In contrast, the incidence direction plays a significant role in the non-equilibrium wave response, which is attributed to the effective bottom slope.