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.

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.

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.

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 $Pr \gg 1$.

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.

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).

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.

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.