We investigate nonlinear energy transfer for channel flows at friction Reynolds numbers $Re_{\tau }=180$ and $590$. The key feature of the analysis is that we quantify the energy transferred from a source mode to a recipient mode, with each mode characterised by a streamwise wavenumber and a spanwise wavenumber. This is achieved through an explicit examination of the triadic interactions of the nonlinear energy transfer term in the spectral turbulent kinetic energy equation. First, we quantify the nonlinear energy transfer gain and loss for individual Fourier modes. The gain and loss cannot be obtained without expanding the nonlinear triadic interactions. Second, we quantify the nonlinear energy transfer budgets for three types of modes. Each type of mode is characterised by a specific region in streamwise–spanwise wavenumber space. We find that a transverse cascade from streamwise-elongated modes to spanwise-elongated modes exists for all three types of modes. Third, we quantify the forward and inverse cascades between resolved scales and subgrid scales in the spirit of large-eddy simulations. For the cutoff wavelength range that we consider, the forward and inverse cascades between the resolved scales and subgrid scales result in a net forward cascade from the resolved scales to the subgrid scales. The shape of the net forward cascade curve with respect to the cutoff wavelength resembles the net forward cascade predicted by the Smagorinsky eddy viscosity.

When one fluid is injected into a confined geometry such as a porous medium filled with another immiscible fluid, even at an extremely low injection speed, rapid filling of several pore spaces accompanied by retraction of multiple fluid–fluid interfaces can be observed. Such processes with fast liquid redistribution within the solid structure, called Haines jumps, are ubiquitous in many multiphase flow systems, which can impact fluid trapping, energy dissipation and hysteretic saturation in various engineering applications. Inspired by this mechanism, here, we propose a dual-channel structure to realise controlled Haines jumps during fluid displacement processes. Via theoretical analysis and numerical simulations, we show that the dynamics of fluid interfaces during Haines jumps can be quantitatively correlated with the driving capillary pressure and dissipating viscous stress, which enables simultaneous determination of the fluid viscosity and interfacial tension in the dual-channel multiphase system.

The small-scale velocity gradient is connected to fundamental properties of turbulence at the large scales. By neglecting the viscous and non-local pressure Hessian terms, we derive a restricted Euler model for the turbulent flow along an undeformed free surface and discuss the associated stable/unstable manifolds. The model is compared with the data collected by high-resolution imaging on the free surface of a turbulent water tank with negligible surface waves. The joint probability density function (p.d.f.) of the velocity gradient invariants exhibits a distinct pattern from the one in the bulk. The restricted Euler model captures the enhanced probability along the unstable branch of the manifold and the asymmetry of the joint p.d.f. Significant deviations between the experiments and the prediction are evident, however, in particular concerning the compressibility of the surface flow. These results highlight the enhanced intermittency of the velocity gradient and the influence of the free surface on the energy cascade.

The concept of vortex lock-in for a single circular cylinder in an oscillating flow, induced through acoustic forcing, is revisited. Multiple cylinder diameters are investigated over a Reynolds number range between 500 and 7200. The lock-in behaviour is investigated quantitatively through hot-wire anemometry and planar particle image velocimetry measurements. The results corroborate previous findings describing the frequency range over which vortex lock-in occurs. It is found that the cylinder location in a standing wave (pressure node or velocity node) had a significant influence on the lock-in behaviour. A novel scaling which captures the onset of vortex lock-in is proposed which demonstrates that the Strouhal number is important in predicting the amplitude of the velocity fluctuations required to induce lock-in. Velocity fields also reveal the existence of bimodal vortex shedding during lock-in. This is confirmed using snapshot proper orthogonal decomposition which demonstrates that symmetric and alternate shedding modes are simultaneously present during lock-in and that symmetric shedding is inherent to the near wake region only. Reduced-order reconstruction of the instantaneous velocity fields confirmed that features associated with the forcing frequency control the shear layer roll-up up to $x/d=2.1$ while the influence of the asymmetric mode is simply to skew the trajectory of the vortex pair. Since vortex roll-up and the cylinder wake ends at $x/d=2.1$, the emergence of spectral content at $0.5f_e$ is attributed to a wavelength doubling measured between the vortical structures in the flow field.

Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. This derivation is further complicated here by the presence of cubic resonances for which a detailed analysis is given. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. It also provides a non-perturbative scheme to reconstruct the ice-sheet deformation from the wave envelope. Linear predictions on the modulational instability of Stokes waves in sea ice are established, and implications for the existence of solitary wave packets are discussed for a range of values of ice compression relative to ice bending. This Dysthe equation is solved numerically to test these predictions. Its numerical solutions are compared with direct simulations of the full Euler system, and very good agreement is observed.

We study the formation of dust-free regions above hot horizontal surfaces of uniform temperature and propose relations for its height in the limit of small particle inertia and gravitational effects. By including particle inertia, thermophoretic, gravitational and viscous effects, we conduct Lagrangian simulations of particle dynamics in a natural convection boundary layer over a horizontal surface. Trajectory analysis of the particles inside the boundary layer on the surface reveals the existence of two separatrices originating from a saddle point, which form the boundary of the dust-free region. These separatrices for low gravitational effects follow the boundary layer thickness, but are of much lower height and also depend on the dimensionless thermophoretic number ($Th$) and Prandtl number ($Pr$). We obtain a relation for the dimensionless height of the dust-free region ($\eta _{df}$) as a function of $Pr$ and $Th$, for low dimensionless gravitational number ($Gn$); the numerical solution of this equation gives us the dust-free region height for any $Th$ and $Pr$. We then obtain scaling laws for $\eta _{df}$ using the boundary layer equations corresponding to the $Pr \gg 1$ and $Pr \ll 1$ cases; these scaling laws are shown to be valid respectively for $Pr>1$ and $Pr<1$, except in the large $\eta$ limit for $Pr>1$, where $\eta$ is the boundary layer similarity variable. We then obtain an empirical relation in this large $\eta$ limit using the numerical solutions of the boundary layer equations for the intermediate $Pr$ case to obtain scaling laws for dust-free region height for the whole range of $Pr \ll 1$ to $Pr \gg 1$.

In recent years, the generalised quasilinear (GQL) approximation has been developed and its efficacy tested against purely quasilinear (QL) approximations. GQL systematically interpolates between QL and fully nonlinear dynamics by employing a generalised Reynolds decomposition. Here, we examine an exact statistical closure for the GQL equations on the doubly periodic $\beta$-plane. Closure is achieved at second order using a generalised cumulant approach which we term GCE2. GCE2 is shown to yield improved performance over statistical representations of purely QL dynamics (CE2) and thus enables direct statistical simulation of complex mean flows that do not entirely fall within the remit of pure QL theory. Despite the existence of an exact closure, GCE2 like CE2 admits the possibility of a rank instability that leads to differences with statistics obtained from GQL. Recognition of this instability is a necessary step before further progress can be made with the GCE2 statistical closure.

An efficient compression scheme for modal flow analysis is proposed and validated on data sequences of compressible flow through a linear turbomachinery blade row. The key feature of the compression scheme is a minimal, user-defined distortion of the mutual distance of any snapshot pair in phase space. Through this imposed feature, the model reduction process preserves the temporal dynamics contained in the data sequence, while still decreasing the spatial complexity. The mathematical foundation of the scheme is the fast Johnson–Lindenstrauss transformation (FJLT) which uses randomized projections and a tree-based spectral transform to accomplish the embedding of a high-dimensional data sequence into a lower-dimensional latent space. The compression scheme is coupled to a proper orthogonal decomposition and dynamic mode decomposition analysis of flow through a linear blade row. The application to a complex flow-field sequence demonstrates the efficacy of the scheme, where compression rates of two orders of magnitude are achieved, while incurring very small relative errors in the dominant temporal dynamics. This FJLT technique should be attractive to a wide range of modal analyses of large-scale and multi-physics fluid motion.

The nonlinear stability of two-dimensional (2-D) plane Couette flow subject to a constant throughflow is analysed at finite and asymptotically large Reynolds numbers $\textit {Re}$. The speed of this throughflow is quantified by the non-dimensional throughflow number $\eta$. The base flow exhibits a linear instability provided $\eta \gtrsim 3.35$, with multi-deck upper and lower branch structures developing in the limit $1\ll \eta \ll \mathit {O}(\textit {Re})$. This instability provides a springboard for the computation of nonlinear travelling waves which bifurcate subcritically from the linear neutral curve, allowing us to map out a neutral surface at different values of $\eta$. Using strongly nonlinear critical layer theory, we investigate the waves that bifurcate from the upper branch at asymptotically large $\textit {Re}$. This asymptotic structure exists provided the throughflow number is larger than the critical value of $\eta _c\approx 1.20$ and is shown to give quantitatively similar results to the numerical solutions at Reynolds numbers of $\mathit {O}(10^5)$.

We study linear convective instability in a mushy layer formed by solidification of a binary alloy, cooled by either an isothermal perfectly conducting boundary or an imperfectly conducting boundary where the surface temperature depends linearly on the surface heat flux. A companion paper (Hitchen & Wells, J. Fluid Mech., 2025, in press) showed how thermal and salinity conditions impact mush structure. We here quantify the impact on convective instability, described by a Rayleigh number characterising the ratio of buoyancy to dissipative mechanisms. Two limits emerge for a perfectly conducting boundary. When the salinity-dependent freezing-point depression is large versus the temperature difference across the mush, convection penetrates throughout the depth of a high-porosity mush. The other limit, which we will call the Stefan limit, has small freezing-point depression and inhibits convection, which localises at onset to a high-porosity boundary layer near the mush–liquid interface. Scaling arguments characterise variation of the critical Rayleigh number and wavenumber based on the potential energy contained in order-one aspect ratio convective cells over the high-porosity regions. The Stefan number characterises the ratio of latent and sensible heats, and has moderate impact on stability via modification of the background temperature and porosity. For imperfectly conducting boundaries, the changing surface temperature causes stability to decrease over time in the limit of large freezing-point depression, but in the Stefan limit combines with the decreasing porosity to yield non-monotonic variation of the critical Rayleigh number. We discuss the implications for convection in growing sea ice.

Turbulent boundary layers on immersed objects can be significantly altered by the pressure gradients imposed by the flow outside the boundary layer. The interaction of turbulence and pressure gradients can lead to complex phenomena such as relaminarization, history effects and flow separation. The angular momentum integral (AMI) equation (Elnahhas & Johnson, J. Fluid Mech., vol. 940, 2022, A36) is extended and applied to high-fidelity simulation datasets of non-zero pressure gradient turbulent boundary layers. The AMI equation provides an exact mathematical equation for quantifying how turbulence, free-stream pressure gradients and other effects alter the skin friction coefficient relative to a baseline laminar boundary layer solution. The datasets explored include flat-plate boundary layers with nearly constant adverse pressure gradients, a boundary layer over the suction surface of a two-dimensional NACA 4412 airfoil and flow over a two-dimensional Gaussian bump. Application of the AMI equation to these datasets maps out the similarities and differences in how boundary layers interact with favourable and adverse pressure gradients in various scenarios. Further, the fractional contribution of the pressure gradient to skin friction attenuation in adverse-pressure-gradient boundary layers appears in the AMI equation as a new Clauser-like parameter with some advantages for understanding similarities and differences related to upstream history effects. The results highlight the applicability of the integral-based analysis to provide quantitative, interpretable assessments of complex boundary layer physics.

A super-stable granular heap is a pile of grains whose free surface is inclined above the angle of repose, and which forms when particles are poured onto a plane that is confined laterally by frictional sidewalls that are separated by a narrow gap. During continued mass supply, the heap free surface gradually steepens until all the inflowing grains can flow out of the domain. As soon as the supply of grains is stopped, the heap is progressively eroded, and if the base of the domain is inclined above the angle of repose, then all the grains eventually flow out. This phenomenology is modelled using a system of two-dimensional width-averaged mass and momentum balances that incorporate the sidewall friction. The granular material is assumed to be incompressible and satisfy the partially regularized $\mu (I)$-rheology. This is implemented in OpenFOAM$^{\circledR}$ and compared against small-scale experiments that study the formation, steady-state behaviour and drainage of a super-stable heap. The simulations accurately capture the dense liquid-like flows as well as the evolving heap shape. The steady uniform flow that develops along the heap surface has non-trivial inertial number dependence through its depth. Super-stable heaps are therefore a sensitive rheometer that can be used to determine the dependence of the friction $\mu$ on the inertial number $I$. However, these flows are challenging to simulate because the free-surface inertial number is high, and can exceed the threshold for ill-posedness even for the partially regularized theory.

Aqueous suspensions of cornstarch abruptly increase their viscosity on raising either shear rate or stress, and display the formation of large-amplitude waves when flowing down inclined channels. The two features have been recently connected using constitutive models designed to describe discontinuous shear thickening. By including time-dependent relaxation and spatial diffusion of the frictional contact density responsible for shear thickening, an analysis of steady sheet flow and its linear stability is presented. The inclusion of such effects is motivated by the need to avoid an ill-posed mathematical problem in thin-film theory and the resulting failure to select any preferred wavelength for unstable linear waves. Relaxation, in particular, eliminates an ultraviolet catastrophe in the spectrum of unstable waves and furnishes a preferred wavelength at which growth is maximized. The nonlinear dynamics of the unstable waves is briefly explored. It is found that the linear instability saturates once disturbances reach finite amplitude, creating steadily propagating nonlinear waves. These waves take the form of a series of steep, shear-thickened steps that translate relatively slowly in comparison with the mean flow.

The effect of geometric twist ($\delta$) of a finite wing of various semi-aspect ratios, on the flow, aerodynamic forces and strength of wing-tip vortex, is investigated. The number of vortex shedding cells increases with increase in $\delta$. In general, the vortex shedding frequency at the root and tip of the wing is approximately the same as that for an untwisted wing. However, close to the $\delta$, where the number of cells changes, the end-cell frequency of the twisted wing undergoes a departure from the value for the untwisted wing. Dislocations at the junction of neighbouring cells are of fork-type for $\delta > -2^\circ$ and of reverse fork-type for $\delta < -2^\circ$. Additional ring-like vortex structures are observed for $\delta =-4^\circ$. Despite a significant effect of the twist on the flow and spanwise variation of the local force coefficients, low to moderate twist of the wing has a relatively minor effect on the span-integrated force coefficients. Larger positive $\delta$, however, results in a significant decrease in the time-averaged force coefficients and rolling moment at the wing root, their unsteadiness and an increase in the strength of the wing-tip vortex. Twist can be utilized as a design parameter for an air vehicle operating at low Reynolds number. Positive twist results in a decrease in unsteadiness in the flow and lower rolling moment at the wing root that can enable lowering the structural weight. Negative twist, on the other hand, weakens the wing-tip vortices that assists in formation and swarm flying by causing lower disturbance to downstream air vehicles.

We model transient mushy-layer growth for a binary alloy solidifying from a cooled boundary, characterising the impact of liquid composition and thermal growth conditions on the mush porosity and growth rate. We consider cooling from a perfectly conducting isothermal boundary, and from an imperfectly conducting boundary governed by a linearised thermal boundary condition. For an isothermal boundary we characterise different growth regimes depending on a concentration ratio, which can also be viewed as characterising the ratio of composition-dependent freezing point depression versus the temperature difference across the mushy layer. Large concentration ratio leads to high porosity throughout the mushy layer and an asymptotically simplified model for growth with an effective thermal diffusivity accounting for latent heat release from internal solidification. Low concentration ratio leads to low porosity throughout most of the mushy layer, except for a high-porosity boundary layer localised near the mush–liquid interface. We identify scalings for the boundary-layer thickness and mush growth rate. An imperfectly conducting boundary leads to an initial lag in the onset of solidification, followed by an adjustment period, before asymptoting to the perfectly conducting state at large time. We develop asymptotic solutions for large concentration ratio and large effective heat capacity, and characterise the mush structure, growth rate and transition times between the regimes. For low concentration ratio the high porosity zone spans the full mush depth at early times, before localising near the mush–liquid interface at later times. Such variation of porosity has important implications for the properties and biological habitability of mushy sea ice.

Adverse pressure gradient (APG) turbulent boundary layers (TBL) require an understanding of the details of the pressure gradient, or history effect, to characterize the associated variation of spatiotemporal turbulent statistics. The streamwise-varying mean pressure gradient is reflected in the streamwise developing mean flow field and thus resolvent analysis, which captures the amplification of the Navier–Stokes equations linearized about the turbulent mean, can be used to understand linear amplification in APG TBLs. In particular, by using a biglobal approach in which the amplification is characterized by a temporal frequency and spanwise wavenumber, the streamwise and wall-normal inhomogeneities of the APG TBL can be resolved and related to the APG history. The linear response is able to identify multiscale phenomena, identifying a near-wall peak with $\lambda _{z}^+\approx 100$ for zero pressure gradient TBLs and mild to moderate APG TBLs as well as large-scale modes whose amplification increases with APG strength and Reynolds number. It is shown that the monotonic growth in the turbulent statistics with increasing APG is reflected in the linear growth in the associated resolvent amplification. Collapse in the Reynolds stresses is obtained through an augmented hybrid velocity scale, which replaces the local APG strength measure in the hybrid velocity scale presented in Romero et al. (Intl J. Heat Fluid Flow, vol. 93, 2022, 108885) with a velocity that encapsulates the pressure gradient history. While this resolvent approach is applicable to any APG TBL, it is shown from a scaling analysis of the linearized Navier–Stokes equations that the linear growth observed in the resolvent amplification with the history effect is limited to near-equilibrium APG TBLs.

The amplitude modulation coefficient, $R$, that is widely used to characterize nonlinear interactions between large- and small-scale motions in wall-bounded turbulence is not compatible with detecting the convective nonlinearity of the Navier–Stokes equations. Through a spectral decomposition of $R$ and a simplified model of triadic convective interactions, we show that $R$ suppresses the signature of convective scale interactions, but is strongly influenced by linear interactions between large-scale motions and the background mean flow. We propose an additional coefficient that is specifically designed for the detection of convective nonlinearities, and we show how this new coefficient, $R_T$, quantifies the turbulent kinetic energy transport involved in turbulent scale interactions and reveals a classical energy cascade across widely separated scales.

We study the instability of a dusty simple shear flow where the dust particles are distributed non-uniformly. A simple shear flow is modally stable to infinitesimal perturbations. Also, a band of particles remains unaffected in the absence of any background flow. However, we demonstrate that the combined scenario – comprising a simple shear flow with a localized band of particles – can exhibit destabilization due to their two-way interaction. The instability originates solely from the momentum feedback from the particle phase to the fluid phase. Eulerian–Lagrangian simulations are employed to illustrate the existence of this instability. Furthermore, the results are compared with a linear stability analysis of the system using an Eulerian–Eulerian model. Our findings indicate that the instability has an inviscid origin and is characterized by a critical wavelength below which it is not persistent. We have observed that increasing particle inertia dampens the unstable modes, whereas the strength of the instability increases with the strength of the coupling between the fluid and particle phases.

Liquid flowing down a fibre readily destabilises into a train of beads, commonly called a bead-on-fibre pattern. Bead formation results from capillary-driven instability and gives rise to patterns with constant velocity and time-invariant bead frequency $f$ whenever the instability is absolute. In this study, we develop a scaling law for $f$ that relates the Strouhal number $St$ and capillary number $Ca$ for Ostwaldian power-law liquids with Newtonian liquids recovered as a limiting case. We validate our proposed scaling law by comparing it with prior experimental data and new experimental data using xanthan gum solutions to produce a low capillary number $Ca \leq 10^{-2}$ regime. The experimental data encompasses both Ostwaldian and Newtonian flow, as well as symmetric and asymmetric patterns, and we find the data collapses along the predicted trend across seven orders of magnitude in $Ca$. Our proposed scaling law is a powerful tool for studying and applying bead-on-fibre flows where $f$ is critical, such as heat and mass transfer systems.

The multipolar spherical vortex solutions to the Euler and Navier–Stokes equations in background cylindrical flow with swirl admit an additional background divergent radial flow with arbitrary time-dependent amplitude. In this case the radial wavenumber $k$, fundamental frequency $\omega$ and overall amplitude $U$ of the multipolar mode superposition become time-dependent and related functions. Assumption of an additional constraint, as a constitutive equation defining the time evolution of the spatially homogeneous divergence of the background flow, is required for the time evolution of the total flow to be completely evaluated from the initial conditions. It is found that flow compression implies an increase of the absolute values of the fundamental frequency $\omega$ and overall velocity amplitude $U$ of the oscillations.