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 $\mathit {O}(10^5)$.

Fluid dynamics systems driven by dominant, near-periodic dynamics are common across wakes, jets, rotating machinery and high-speed flows. Traditional modal decomposition techniques have been used to gain insight into these flows, but can require many modes to represent key physical processes. With the aim of generating modes that intuitively convey the underlying physical mechanisms, we propose an intrinsic phase-based proper orthogonal decomposition (IPhaB POD) method, which creates energetically ranked modes that evolve along a characteristic cycle of the dominant near-periodic dynamics. Our proposed formulation is set in the time domain, which is particularly useful in cases where the cyclical content is imperfectly periodic. We formally derive IPhaB POD within a POD framework that therefore inherits the energetically ranked decomposition and optimal low-rank representation inherent to POD. As part of this derivation, a dynamical systems representation is utilized, facilitating a definition of phase within the system's near-periodic cycle in the time domain. An expectation operator and inner product are also constructed relative to this definition of phase in a manner that allows for the various cycles within the data to demonstrate imperfect periodicity. The formulation is tested on two sample problems: a simple, low Reynolds number aerofoil wake, and a complex, high-speed pulsating shock wave problem. The method is compared to space-only POD, spectral POD (SPOD) and cyclostationary SPOD. The method is shown to better isolate the dominant, near-periodic global dynamics in a time-varying IPhaB mean, and isolate the tethered, local-in-phase dynamics in a series of time-varying modes.

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.

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.

We propose a computational framework for simulating the self-similar regime of turbulent Rayleigh–Taylor (RT) mixing layers in a statistically stationary manner. By leveraging the anticipated self-similar behaviour of RT mixing layers, a transformation of the vertical coordinate and velocities is applied to the Navier–Stokes equations (NSE), yielding modified equations that resemble the original NSE but include two sets of additional terms. Solving these equations, a statistically stationary RT (SRT) flow is achieved. Unlike temporally growing Rayleigh–Taylor (TRT) flow, SRT flow is independent of initial conditions and can be simulated over infinite simulation time without escalating resolution requirements, hence guaranteeing statistical convergence. Direct numerical simulations (DNS) are performed at an Atwood number of 0.5 and unity Schmidt number. By varying the ratio of the mixing layer height to the domain width, a minimal flow unit of aspect ratio 1.5 is found to approximate TRT turbulence in the self-similar mode-coupling regime. The SRT minimal flow unit has one-sixteenth the number of grid points required by the equivalent TRT simulation of the same Reynolds number and grid resolution. The resultant flow corresponds to a theoretical limit where self-similarity is observed in all fields and across the entire spatial domain – a late-time state that existing experiments and DNS of TRT flow have difficulties attaining. Simulations of the SRT minimal flow unit span TRT-equivalent Reynolds numbers (based on mixing layer height) ranging from 500 to 10 800. The SRT results are validated against TRT data from this study as well as from Cabot & Cook (Nat. Phys., vol. 2, 2006, pp. 562–568).

The evolutionary process of mixing induced by Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities typically progresses through three stages: initial instability growth, subsequent mixing transition and ultimate turbulent mixing. Accurate prediction of this entire process is crucial for both scientific research and engineering applications. For engineering applications, Reynolds-averaged Navier–Stokes (RANS) simulation stands as the most viable method currently. However, it is noteworthy that existing RANS mixing models are primarily tailored for the fully developed turbulent mixing stage, rendering them ineffective in predicting the crucial mixing transition. To address that, the present study proposes a RANS mixing transition model. Specifically, we extend the idea of the intermittent factor, which has been widely employed to integrate with turbulence models for predicting boundary layer transition, to mixing problems. Based on a high-fidelity simulation of a RT case, the intermittent factor defined based on enstrophy is extracted and then applied to RANS calculations, showing that it is possible to accurately predict mixing transition by introducing the intermittent factor to the turbulence production from the baseline K-L turbulence mixing model. Furthermore, to facilitate practical predictions, a transport equation has been established to model the spatio-temporal evolution of the intermittent factor. Coupled with the K-L model, the intermittent factor provided by the transport equation is applied to modify the Reynolds stress in RANS calculations. Thereafter, the present transition model has been validated in a series of tests, demonstrating its accuracy and robustness in the capturing mixing process in different types and stages of interfacial mixing flows.

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.

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.

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.

The added mass force resulting from the acceleration of a body in a fluid is of fundamental and practical interest in dispersed multiphase flows. Euler–Lagrange (EL) and Euler–Euler (EE) simulations require closure terms for the added mass force in order to accurately couple the conserved variables between phases. Presently, a more thorough understanding of the added mass force in a multi-particle system is developed based on potential flow resulting in a resistance matrix formulation analogous to Stokesian dynamics. This formulation is then used to generate a dataset of added mass resistance matrices for large systems of randomly generated particles. This methodology is used to create a volume fraction corrected binary model for predicting the added mass force in large systems as well as generate statistics of the added mass force in such systems. This work provides clarification to the theory of the added mass force for particle clouds, and modelling options that may be implemented in existing EL and EE codes.

We investigate the spreading of falling ambient-temperature Newtonian drops after their normal impact on a quartz plate covered with a thin layer of liquid nitrogen. As a drop expands, liquid nitrogen evaporates, generating a vapour film that maintains the drop in levitation. Consequently, the latter spreads in inverse Leidenfrost conditions. Three drop-spreading regimes are observed: (i) inertio-capillary, (ii) inertio-viscous, and (iii) inertio-viscous-capillary. In the first regime, although the drop expansion is essentially driven by a competition between inertial and capillary stresses, it is also affected by viscous effects emerging from the vapour film, which ultimately favours the development of a shear flow within the drop. Interestingly, vapour film effects become marginal in both the second and third regimes, allowing the drop to undergo biaxial extension primarily. More specifically, in the inertio-viscous scenario, the expansion is driven by the balance between inertial and biaxial extensional viscous stresses in the drop. Finally, inertia, capillarity and drop viscosity are all relevant in the third regime. These physical mechanisms are underlined through a mixed approach combining experiments with multiphase three-dimensional numerical simulations in light of spreading dynamics analyses, energy transfer and scaling laws. Our results are rationalized in a two-dimensional diagram linking the drops’ maximum expansion and spreading time with the observed spreading regimes through a single dimensionless parameter given by the square root of the capillary number (the ratio of the viscous stress to the capillary stress).

When atmospheric storms pass over the ocean, they resonantly force near-inertial waves (NIWs), internal waves with a frequency close to the local Coriolis frequency $f$. It has long been recognised that the evolution of NIWs is modulated by the ocean's mesoscale eddy field. This can result in NIWs being concentrated into anticyclones which provide an efficient pathway for NIW propagation to depth. Here we analyse the eigenmodes of NIWs in the presence of mesoscale eddies and heavily draw on parallels with quantum mechanics. Whether the eddies are effective at modulating the behaviour of NIWs depends on the wave dispersiveness $\varepsilon ^2 = f\lambda ^2/\varPsi$, where $\lambda$ is the deformation radius and $\varPsi$ is a scaling for the eddy streamfunction. If $\varepsilon \gg 1$, NIWs are strongly dispersive, and the waves are only weakly affected by the eddies. We calculate the perturbations away from a uniform wave field and the frequency shift away from $f$. If $\varepsilon \ll 1$, NIWs are weakly dispersive, and the wave evolution is strongly modulated by the eddy field. In this weakly dispersive limit, the Wentzel–Kramers–Brillouin approximation, from which ray tracing emerges, is a valid description of the NIW evolution even if the large-scale atmospheric forcing apparently violates the requisite assumption of a scale separation between the waves and the eddies. The large-scale forcing excites many wave modes, each of which varies on a short spatial scale and is amenable to asymptotic analysis analogous to the semi-classical analysis of quantum systems. The strong modulation of weakly dispersive NIWs by eddies has the potential to modulate the energy input into NIWs from the wind, but we find that this effect should be small under oceanic conditions.

We present an experimental study on the drag reduction by polymers in Taylor–Couette turbulence at Reynolds numbers ($Re$) ranging from $4\times 10^3$ to $2.5\times 10^4$. In this $Re$ regime, the Taylor vortex is present and accounts for more than 50 % of the total angular velocity flux. Polyacrylamide polymers with two different average molecular weights are used. It is found that the drag reduction rate increases with polymer concentration and approaches the maximum drag reduction (MDR) limit. At MDR, the friction factor follows the $-0.58$ scaling, i.e. $C_f \sim Re^{-0.58}$, similar to channel/pipe flows. However, the drag reduction rate is about $20\,\%$ at MDR, which is much lower than that in channel/pipe flows at comparable $Re$. We also find that the Reynolds shear stress does not vanish and the slope of the mean azimuthal velocity profile in the logarithmic layer remains unchanged at MDR. These behaviours are reminiscent of the low drag reduction regime reported in channel flow (Warholic et al., Exp. Fluids, vol. 27, no. 5, 1999, pp. 461–472). We reveal that the lower drag reduction rate originates from the fact that polymers strongly suppress the turbulent flow while only slightly weaken the mean Taylor vortex. We further show that polymers steady the velocity boundary layer and suppress the small-scale Görtler vortices in the near-wall region. The former effect reduces the emission rate of both intense fast and slow plumes detached from the boundary layer, resulting in less flux transport from the inner cylinder to the outer one and reduces energy input into the bulk turbulent flow. Our results suggest that in turbulent flows, where secondary flow structures are statistically persistent and dominate the global transport properties of the system, the drag reduction efficiency of polymer additives is significantly diminished.

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 stability and dynamics of solitary waves propagating along the surface of an inviscid ferrofluid jet in the absence of gravity are investigated analytically and numerically. For the axisymmetric geometry, the problem is shown to be a conservative system with total energy as the Hamiltonian; however, one of the canonical variables differs from those in the classic water-wave problem in the Cartesian coordinate system. The Dirichlet–Neumann operator appearing in the kinetic energy is then expanded as a Taylor series, described in homogeneous powers of the surface displacement. Based on the further analysis of the Dirichlet–Neumann operator, a systematic procedure is proposed to derive reduced model equations of multiple scales in various asymptotic limits from the full Euler equations in the Hamiltonian/Lagrangian framework. Particularly, a fully dispersive model arising from retaining terms valid up to the quartic order in the series expansion of the kinetic energy, which results in quadratic and cubic algebraic nonlinearities in Hamilton's equations and henceforth is abbreviated as the cubic full-dispersion model, is proposed. By comparing bifurcation curves and wave profiles of various types of axisymmetric solitary waves among different model equations, the cubic full-dispersion model is found to agree well with the full Euler equations, even for waves of considerably large amplitudes. The stability properties of axisymmetric solitary waves subjected to longitudinal disturbances are verified with the newly proposed model. Our analytical results, consistent with Saffman's theory, indicate that in the axisymmetric cylindrical system, the stability exchange subjected to superharmonic perturbations also occurs at the stationary point of the speed-energy bifurcation curve. A series of numerical experiments for the stability and dynamics of solitary waves are performed via the numerical time integration of the model equation, and collision interactions between stable solitary waves show non-elastic features.

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.

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.

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.

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.