The quadratic convection term in the incompressible Navier–Stokes equations is considered as a nonlinear forcing to the linear resolvent operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber–frequency triplets. A framework to quantify the triadic contributions to the forcing and response by each pair of triplets is developed and applied to data from direct numerical simulations of a turbulent channel at $ \textit{Re}_{\tau } \approx 550$. The linear resolvent operator is incorporated to provide the missing link from energy transfer between modes to the effect on the spectral turbulent kinetic energy. The coefficients highlight the importance of interactions involving large-scale structures, providing a natural connection to the modelling assumptions in quasilinear (QL) and generalized QL (GQL) analyses. Specifically, it is revealed that the QL and GQL reductions efficiently capture important triadic interactions in the flow, especially when including of a small number of wavenumbers into the GQL large-scale base flow. Additionally, spatiotemporal analyses of the triadic contributions to a single mode representative of the near-wall cycle demonstrate the spatiotemporal nature of the triadic interactions and the effect of the resolvent operator, which selectively amplifies certain forcing profiles. The tools presented are expected to be useful for improving modelling of the nonlinearity, especially in QL, GQL and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.

In the inclined layer convection system, thermal convection in a Rayleigh–Bénard cell tilted against gravity, the flow is subject to competing buoyancy and shear forces. For varying inclination angle ($\gamma$) and Rayleigh number ($ \textit{Ra}$), a variety of spatio-temporal patterns is observed. We investigate the switching diamond panes (SDP) pattern, observed at $(\gamma , \textit{Ra})\simeq (100^\circ ,10\,000)$, which exhibits large-scale oblique features and is one of the five complex tertiary patterns at Prandtl number $ \textit{Pr}=1.07$. First, we study the linear instability of the secondary-state transverse convection rolls and the five branches including two travelling waves and three periodic orbits, bifurcating simultaneously from it. These non-generic bifurcations arise from the breaking of specific spatial symmetries of transverse rolls, and the resulting bifurcated solutions show large-scale diamond-shaped amplitude modulations. Second, we explore a periodic orbit that captures both the large-scale structure and small-scale defects of modulated rolls. Parametric continuation in $ \textit{Ra}$ reveals the global homoclinic bifurcation via which this periodic orbit emerges. Third, the edge states between two dynamically relevant periodic orbits have been computed. Specifically, additional steady and time-periodic solutions are identified on the basin boundary and their bifurcation structures are analysed. Together, using nonlinear invariant solutions and their bifurcations, we take a further step toward understanding the emergence and dynamics of SDP far from the onset of convection, where linear methods have not been applied successfully.

Direct numerical simulations are performed to elucidate the influence of counter- and co-rotation on turbulent viscoelastic Taylor–Couette flow in the Rossby number range $ \textit{Ro}^{-1}=-0.6$ to $ \textit{Ro}^{-1}=1$. A novel polymer-induced transition pathway is discovered that is fundamentally different from Newtonian flows. In the counter-rotation regime, the neutral surface is elastically modified and separates the turbulent inner-wall region containing chaotic vortices from the relaminarised layer adjacent to the outer cylinder. Strikingly, co-rotation triggers an elasto-rotational instability, which leads to the breakdown of large-scale Taylor vortices into small-scale penetrating structures, thereby preventing the relaminarisation at high co-rotation rates. Examination of turbulence dynamics demonstrates that the structural changes with increasing $ \textit{Ro}^{-1}$ are accompanied by a transition from elasto-inertial to elastically dominated turbulence. Specifically, the polymer stress progressively exceeds the Reynolds stress and monotonically enhances angular momentum transport, which eliminates the optimal transport characteristic found in the Newtonian flow. The elastically dominated nature of the turbulent flow under co-rotation is further corroborated by the more significant elastic production of the turbulent kinetic energy, as well as the monotonic enhancement of the polymer elongation as $ \textit{Ro}^{-1}$ increases. Furthermore, it is indicated that the polymer orientation strongly depends on vortical structures, with enhanced radial alignment occurring in the boundary region between adjacent vortices. This vortex-mediated polymer orientation is crucial for generating substantial polymer shear stress, establishing a direct link between coherent structures and polymer dynamics.

The growth of a single vapour bubble in a saturated pool of boiling water at atmospheric pressure on a transparent heater is investigated experimentally. The study focuses on a several microns thick liquid layer (called a microlayer) that can form between the heater and the bubble. The microlayer profile, the wall temperature distribution and the overall bubble shape are measured simultaneously and synchronously by white light interferometry, infrared thermography and sidewise shadowgraphy. To study the microlayer dynamics for different bubble growth rates, artificial cavities of different sizes were used. These control the wall superheating required for the bubble nucleation, i.e. the nucleation barrier. Both the bubble growth at its inertial stage and the microlayer parameters are found to be almost independent of the applied heat flux and controlled by the nucleation barrier only. The microlayer thickness and its area increase with the nucleation barrier. A model of microlayer formation based on an analogy with the Landau–Levich film deposition is further developed. By using it, the initial microlayer thickness, the time of microlayer formation at a given distance from the bubble centre and the radius of curvature of the bubble foot edge are recovered from the experimental data. The radius of curvature varies in time like the bubble radius, thus suggesting a self-similar bubble growth at its initial inertial stage. The coherency of the above model with the experimental results shows the model’s validity.

An entropy-constrained closure is developed for steady quasi-one-dimensional flow of a calorically perfect ideal gas. Net irreversibility is represented by a prescribed entropy change between the inlet and outlet or, equivalently, by a stagnation pressure ratio together with the stagnation temperature ratio implied by the energy balance equation. The thermodynamic constraint, enforced through Gibbs relation together with conservation of mass and energy, yields an algebraic mapping from the inlet flow state, area ratio and prescribed stagnation property ratios to the outlet flow state. The momentum balance is retained only as an auxiliary post-processing relation and may be used to infer integral quantities such as a net wall force or an equivalent mean wall pressure. For an ideal gas with integer number of molecular degrees of freedom $f$, the resulting equation reduces to a polynomial of degree $f+1$ in the squared outlet Mach number. The $f=3$ case reduces to a quartic, and the $f\to \infty$ limit admits an explicit Lambert-$W$ solution; existence, multiplicity, choking and admissibility conditions consistent with the second law of thermodynamics are obtained for general $f$. Classical isentropic nozzle, Rayleigh and Fanno flows, normal shocks, and simultaneous friction and heat transfer in a constant-area duct are recovered in the respective limiting cases. Comparisons with experimental measurements of nozzle wall pressures, sudden expansion outlet Mach numbers and sudden contraction stagnation pressure ratios show good agreement with predictions across a wide range of inlet Mach numbers and area ratios.

We study the pinch-off dynamics of a fluid surrounded by a significantly more viscous one during fluid–fluid displacement in straight, cylindrical capillary tubes, where the interface evolves under the influence of a moving contact line. We investigate the influence of insoluble surfactants on the dynamics of both the contact line and pinch-off. We focus on a visco-capillary displacement regime where the imposed flow rate exceeds a critical threshold, beyond which the contact-line velocity is governed by the partial wettability of the confined geometry, becoming independent of the flow rate. Under these conditions, the fluid–fluid meniscus forms an advancing axial finger, leaving behind a thin film of the defending fluid. This unstable film retracts along the partially wetting walls, forming a dewetting rim that grows with the steady contact-line motion. Eventually, a surface-tension-driven Rayleigh–Plateau instability dominates, triggering pinch-off at the rim neck. For a surfactant-free interface, our results show that the early-stage evolution of the neck diameter follows a power law $\tau ^\alpha$, where $\tau$ is the time to the pinch-off singularity and the scaling exponent $\alpha$ depends on the contact-line velocity determined by the wettability. Over the range resolved in the present simulations, $\alpha$ decreases from values near $1/2$ at large contact-line velocity towards values close to $1/5$ as the contact-line velocity is reduced. We demonstrate that, in the presence of insoluble surfactants, the contact-line velocity, at a given wettability, scales linearly with surfactant elasticity due to Marangoni stresses along the dewetting rim interface, which affect the timing and location of the pinch-off. Despite these effects, at the early-time regime, the pinch-off dynamics exhibits the same self-similar scaling behaviour as in the surfactant-free case.

Existing theoretical analyses on the Faraday instability in Hele-Shaw cells typically adopt gap-averaged governing equations and rely on Hamraoui’s model coming from molecular kinetics theory, thereby oversimplifying essential transverse information, such as contact line velocity and capillary hysteresis, and conflicting with the unsteady meniscus dynamics. In this paper, a gap-resolved approach is developed by directly modelling the transverse gap flow and the contact angle dynamics, which overcomes the aforementioned limitations, ultimately yielding a modified damping with respect to the static contact angle and hysteresis range. A novel amplitude equation for linear Faraday instability is derived that combines this damping and the gap-averaged counterpart based on the oscillatory Stokes boundary layer, with the viscous dissipation preserved. By means of Lyapunov’s first method, an explicit analytical expression for the critical stability boundary is established. Two series of laboratory experiments are performed that focus, respectively, on evolutions of the lateral meniscus and the longitudinal free surface near the Faraday onset, from which key parameters relevant to the theory are precisely measured. Based on the experimental data, the validity of the proposed mathematical model for addressing the Faraday instability problem in Hele-Shaw cells is confirmed, and the generation and development mechanisms of the onset are clarified. In the asymptotic analysis, the inclusion of contact angle dynamics increases the overall damping and thus partially compensates for the frequency detuning introduced by oscillatory Stokes flow approximation.

A finite wall-mounted cylinder (FWMC) refers to a cylinder mounted to a flat wall, characterised by flow over its free end and at the wall junction. In the present study, finite wall-mounted harbour seal vibrissal cylinders with different spanwise aspect ratios ($\textit{AR} = L/D$, where $L$ is the length of the cylinder and $D$ is its diameter) are examined at a Reynolds number of $\textit{Re} = 12\,000$ based on the cylinder diameter, with an incoming turbulent boundary layer thickness of $\delta /D = 0.16$. The far-field acoustic spectra of the harbour seal vibrissal FWMCs are broadband for all aspect ratios and effectively suppress the tonal peaks observed in the spectra of circular FWMCs. However, the effectiveness of the suppression mechanism diminishes with decreasing aspect ratio. For the shortest harbour seal vibrissal FWMC ($\textit{AR}= 3.2$), the overall sound pressure level exceeds that of its circular counterpart, and the tonal peak is replaced by broadband noise of comparable magnitude. Spanwise-intermittent recirculation zones and cellular vortex shedding are identified in the near wake of the harbour seal vibrissal FWMCs. For all harbour seal vibrissal FWMCs, as well as for circular FWMCs with $\textit{AR}$ = 6.5, hairpin vortex shedding is observed rather than classical von Kármán vortex shedding. This corresponds to the absence of tonal peaks in the far-field acoustic spectra. Wavy separation lines play a crucial role in the formation of hairpin vortices and in inducing phase differences in vortex shedding in the spanwise direction. Concentrated quadrupole streamwise vortical structures periodically appear along the span in the wake of the longest harbour seal vibrissal FWMC, which diminish with $\textit{AR}$ decreasing to $\textit{AR}$ = 6.5. This concentrated quadrupole arrangement of positive–negative streamwise vortices suppresses the vortex shedding observed along the span of circular FWMCs by disrupting the acoustic-coherent structures into smaller, spanwise-wavy organised elements. It also stabilises the flow and balances transverse ($y$) forces.

We study the collective behaviour of clusters of cylinders placed in the wake of a fixed cylinder and free to move in a direction perpendicular to that of the incoming flow, with no structural damping or stiffness. We keep the Reynolds number, defined based on the cylinder diameter, at 100 and consider five different configurations for the initial positions of the cluster cylinders: linear, rectangular, V-shaped, triangular and circular. In each configuration, we consider progressively increasing numbers of cylinders in the cluster. We show that overall, the cylinders tend to form final linear configurations, in which, after their transition, the cylinders form one or more lines. Some free-to-move cylinders might take the lead position in some of these linear formations depending on the initial configuration. These steady-state positions are achieved when the mean value of lift that acts on the cylinders becomes negligible. As a byproduct of these reconfigurations, the overall drag force that acts on the collection of cylinders reduces at their final steady-state locations in comparison with their original configurations. The complicated wakes that are observed in the fixed counterparts of these configurations are replaced by a series of vortex rows in the wake of separate lines of cylinders. Reducing the mass ratio allows the cylinders to oscillate about their mean displacement paths, but their transient paths and their final steady-state positions are not affected significantly by the decrease in the mass ratio.

Motivated by the understanding of fluid mechanics behind dry eye syndrome during the winter season, we perform the linear and nonlinear stability analysis of the tear film. To reflect the biological structure of the tear film, we model it as a viscoelastic film on a substrate with an insoluble surfactant at the air–film interface, destabilisation by van der Waals forces, variation of viscosity and elasticity along the film height, and impose a substrate-normal temperature gradient. The air–film interface tension is assumed to decrease linearly with temperature and surfactant concentration. To conduct general linear stability analysis (GLSA), we employ the pseudo-spectral method. The GLSA predicts the existence of a longwave thermocapillary-induced instability, which overcomes the stabilisation by the solutocapillary effect due to the surfactant. The elasticity of the film, van der Waals forces and slip at the film–substrate interface contribute to further destabilisation, while a decrease in viscosity along the film height stabilises the film. The longwave instability is the dominant mode of instability. Thus, we derive longwave evolution equations whose linear stability analysis confirms the predictions of GLSA. The nonlinear analysis of the derived evolution equations, carried out using COMSOL 6.2, demonstrates tear-film rupture due to pure thermocapillary instability for sufficiently high thermal Marangoni numbers. In contrast, diseased eyes suffering from lipid layer dysfunction can undergo tear-film rupture at a much lower temperature difference between the ambient air and the cornea. Consideration of a viscosity decrease with increasing film height delays tear film rupture, while van der Waals forces decrease tear film rupture time. The rupture time range predicted by our model is in good agreement with clinical observations, thereby confirming the impact of the winter season on tear film dynamics.

We investigate Taylor–Couette flow with realistic no-slip boundary conditions at all surfaces through direct numerical simulation (DNS) and theoretical analysis. Imposing physically consistent end-wall conditions at the top and bottom lids significantly alters the flow dynamics compared with that for periodic boundary conditions. We extend the classical angular-momentum-flux framework to account for axial transport, thereby extending the Eckhardt–Grossmann–Lohse model (Eckhardt et al. J. Fluid Mech. vol. 581, 2007, pp. 221–250) to no-slip boundary conditions. A systematic exploration of the parameter space $(\textit{Re}, n)$ uncovers multiple long-lived states with different roll number $n$ configurations at identical Reynolds numbers $\textit{Re}$, giving rise to pronounced hysteresis loops occurring under realistic boundary conditions. Our DNS for no-slip axial endcaps reveals a sequence of structural transitions: as the inner-cylinder Reynolds number increases, the flow evolves from Taylor vortex flow through chaotic wavy vortex flow and turbulent wavy vortex flow to an axisymmetric turbulent Taylor vortex flow. Using modal energy budgets, we identify transition mechanisms and quantify how the accessible phase-space volume and associated roll-specific angular momentum flux depend on control parameters and the specific flow state. Our findings demonstrate the impact of realistic boundary conditions on the dynamics in Taylor–Couette flow and how they change the stability landscape of multiple states. The coexistence of distinct flow patterns and their stability analysis offers promising insights into transition dynamics between laminar and turbulent regimes in closed sheared flows.

A new model of the mean temperature profiles within incompressible wall-bounded turbulence is developed based on a relationship between the momentum and thermal eddy diffusivities for a wide range of Prandtl numbers ($ \textit{Pr}$). Flow media with $ \textit{Pr}$ and Schmidt numbers ($ \textit{Sc}$) other than unity are of great engineering importance, and the proposed model for passive scalar mean profiles is able to suitably account for $ \textit{Pr}$ and $ \textit{Sc}$ variations and their effects. Existing models have a higher degree of error at the lower $ \textit{Pr}$ ranges due to significant inaccuracies of the modelled thermal eddy diffusivity in this region. Considering incompressible wall-bounded turbulence at $0.007 \leqslant \textit{Pr} \leqslant 10$, the proposed methodology reduces the average error to just around $4\,\%$ across all cases considered, lower than previously proposed models due to its ability to capture low $ \textit{Pr}$ behaviour, showcasing a new temperature–velocity relationship that can account for variations in $ \textit{Pr}$ for all the cases considered.

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.

We model filtration of a feed solution, containing both small and large foulant particles, by a membrane filter. The membrane interior is modelled as a network of pores, allowing for the simultaneous adsorption of small particles and sieving of large particles, two fouling mechanisms typically observed during the early stages of commercial filtration applications. In our model, first-principles continuum partial differential equations model transport of the small particles and adsorptive fouling in each pore, while sieving particles are assumed to follow a discrete Poisson arrival process with a biased random walk through the pore network. Our goals are to understand the relative influences of each fouling mode and highlight the effect of their coupling on the performance of filters with a pore-size gradient (specifically, we consider a banded filter with different pore sizes in each band). Our results suggest that, due to the discrete nature of pore blockage, sieving alters qualitatively the rate of the flux decline. Moreover, the difference between sieving-particle sizes and the initial pore size (radius) in each band plays a crucial role in indicating the onset and disappearance of sieving–adsorption competition. Lastly, we demonstrate a phase transition in the filter lifetime as the arrival frequency of sieving particles increases.

We investigate the oscillation of the large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection by combining laboratory experiments and numerical simulations. The experiments are conducted mainly in a laterally confined rectangular cell, and the flow fields are measured by particle image velocimetry in the vertical mid-plane. It is found that the velocity field exhibits not only the commonly observed oscillation frequencies $(f\tau _o=1,2)$ but also higher-order harmonic frequencies $(f\tau _o\gt 2)$, where $\tau _o$ is the turnover time of the LSC. Spectral proper orthogonal decomposition (SPOD) reveals that the coherent structures underlying these frequencies display an approximate azimuthal periodicity when viewed in polar coordinates. These structures can be interpreted as travelling waves, consistent with the advected-oscillation picture proposed by Brown & Ahlers (2009 J. Fluid Mech., vol. 638, pp. 383–400). A data-driven resolvent analysis, used here as an effective input–output model inferred from the experimental data, yields response modes and gain peaks that are consistent with the SPOD results, and support the interpretation that the dominant oscillations are selected through linear amplification about the mean circulation. This perspective is further supported by direct numerical simulation in a rectangular cell and by velocity measurements in a cylindrical cell. These findings provide an additional viewpoint for understanding the oscillation mechanism of the LSC in turbulent Rayleigh–Bénard convection.

Obtaining accurate field statistics continues to be one of the major challenges in turbulence theory and modelling. From the various existing modelling approaches, multifractal models have been successful in capturing intermittency in velocity gradient and increment distributions. Moreover, superstatistical models from non-equilibrium statistical mechanics have shown the capacity to model probability density functions (p.d.f.s) of various statistical turbulent quantities as ensembles of simpler stochastic processes. Here, we present an approach that generates field statistics in the form of a characteristic functional by promoting a model for multifractal increment statistics to an ensemble of Gaussian fields. By carefully designing the correlation function and the corresponding weight of each subensemble, we are able to define a functional that exhibits multifractal two-point inertial-range and dissipation-range statistics, and that blends into realistic large-scale behaviour. Additionally, the method is capable of producing multifractal statistics with any of the widely used singularity spectra. We characterise the fidelity of our approach through comparisons to literature results from direct numerical simulations. Overall, our framework thereby bridges between three different perspectives: superstatistics, multifractals and functional approaches to turbulence.

In this paper, we investigate the influences of wall temperature on compressible turbulent boundary layers at free-stream Mach number 6.0 and moderate Reynolds numbers. The findings demonstrate that turbulent statistics, including the logarithmic scaling of Reynolds stresses in the overlap region, the presence of very-large-scale motions in the outer layer, and their superposition on near-wall turbulence, exhibit qualitative invariance across varying wall temperatures. However, the reduced scale separation between near-wall small-scale motions and outer-layer large-scale motions leads to a contraction in the vertical extent of Reynolds stresses adhering to the logarithmic law. Very-large-scale motions are attenuated in viscous units but intensified in global scalings due to the lower free-stream Reynolds number, following approximately the power law. Their superposition effects on near-wall turbulence, when weighted by density, show only weak dependence on wall temperature. Conversely, the modulation of near-wall velocity and temperature fluctuations by very-large-scale motions diminishes with decreasing wall temperature. Through detailed analysis of spanwise spectra in the outer region, a distinct inertial subrange obeying the classical $-5/3$ scaling law is identified, alongside a universal scaling over the dissipative range. These observations suggest that the mechanisms governing energy cascade processes and the associated small-scale turbulent fluctuations remain independent of wall temperature variations.

The first bifurcation of the flow around a spheroid is analysed using global stability analysis to understand the development of flow asymmetry despite the symmetry of the configuration. The base flow, perturbations, adjoint modes, vorticity fields, structural sensitivity and skin friction lines are analysed to characterise the flow. The study of aspect ratio effects at zero angle of attack establishes that the structure of the asymmetry is the same whether the axisymmetric body is bluff or more streamlined. Stability of the flow field around the 6 : 1 spheroid is then investigated for angles of attack $\alpha \in [0{-}90]^\circ$ as a function of Reynolds number. Comparison of the low angle of attack results with the DARPA SUBOFF experiments of Ashok et al. (J. Fluid Mech., 2015, vol. 774, pp. 416–442) shows that the asymmetry observed in the experiments is similar to the global mode predicted by the stability calculations. It is conjectured that the experimental asymmetry is triggered by the weak cross-stream circulating flow. The leading eigenmode is a stationary asymmetric mode in the angle of attack $\alpha \in [0{-}65]^\circ$ range, while above $\alpha =65^\circ$, the leading mode is an oscillatory shedding mode. The rapid decrease in the critical Reynolds number between cases $\alpha =49.25^\circ$ and $49.5^\circ$ is attributed to coexisting symmetric flow states that have different susceptibilities to asymmetry; there exists a symmetric stationary mode that does not become unstable first, but appears to be the difference between the base flows at the same Reynolds number at the two angles of attack. The change from a stationary to an oscillatory instability between $\alpha =65^\circ$ and $70^\circ$ is linked to the ability/inability of the vortex sheets to roll up and reattach to the body in the former/latter cases, respectively. The difference in the separation patterns and the similarity between the eigenmodes indicate that asymmetry of the flow field is governed by the same mechanism across a wide angle of attack range, regardless of whether the flow is like a bluff body wake, a streamlined body wake or a vortex wake. Previous studies have argued that the asymmetry of vortex pairs emerges either because of a vortex instability or a separation/reattachment related mechanism; since the development of the asymmetry cannot be linked to specific features in the separation pattern in the investigated configurations, our results support the former argument.