We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to a downslope uniform ambient wind aloft. We adopt an extension of Prandtl’s original model for slope flows (Lykosov & Gutman, Izv. Acad. Sci. USSR Atmos. Ocean. Phys., vol. 8 (8), 1972, pp. 799–809) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent dimensionless parameters are sufficient to describe the flow stability. One of these parameters is the stratification perturbation number that we have introduced in Xiao & Senocak (J. Fluid Mech., vol. 865, 2019, R2). The second parameter, which we will henceforth designate the wind forcing number, is hitherto uncharted and can be interpreted as the ratio of the kinetic energy of the ambient wind aloft to the damping due to viscosity and the stabilising effect of the background stratification. For a fixed Prandtl number, stationary transverse and travelling longitudinal modes of instabilities can emerge, depending on the value of the slope angle and the aforementioned dimensionless numbers. The influence of ambient wind forcing on the base flow’s stability is complicated, as the ambient wind can be both stabilising as well as destabilising for a certain range of the parameters. Our results constitute a strong counterevidence against the current practice of relying solely on the gradient Richardson number to describe the dynamic stability of stratified atmospheric slope flows.

Here we provide a theoretical framework revealing that the radius $R_{d}$ of the top droplet ejected from a bursting bubble of radius $R_{b}$ and $Bo\leqslant 0.05$ can be expressed as $R_{d}/R_{b}=K_{b}(1-(Oh/Oh_{c}^{\prime })^{1/2})$ for $Oh\lesssim Oh_{c}^{\prime }$ or as $R_{d}\approx 18\,\unicode[STIX]{x1D707}_{l}^{2}/(\unicode[STIX]{x1D70C}_{l}\unicode[STIX]{x1D70E})$ for $Oh\gtrsim Oh_{c}^{\prime }$, with the numerically fitted constants $K_{b}\approx 0.2$, $Oh_{c}^{\prime }\approx 0.03$, $Oh=\unicode[STIX]{x1D707}_{l}/\sqrt{\unicode[STIX]{x1D70C}_{l}\,R_{b}\,\unicode[STIX]{x1D70E}}\ll 1$ the Ohnesorge number, $Bo=\unicode[STIX]{x1D70C}_{l}\,g\,R_{b}^{2}/\unicode[STIX]{x1D70E}$ the Bond number, and $\unicode[STIX]{x1D70C}_{l}$, $\unicode[STIX]{x1D707}_{l}$ and $\unicode[STIX]{x1D70E}$ indicating the liquid density, dynamic viscosity and interfacial tension coefficient, respectively. These predictions, which do not only have solid theoretical roots but are also much more accurate than the usual 10 % rule used in the context of marine spray generation via whitecaps for $R_{b}\lesssim 1$ mm, agree very well with both experimental data and numerical simulations for the values of $Oh$ and $Bo$ investigated. Moreover, making use of a criterion which reveals the mechanism that controls the growth rate of capillary instabilities, we also explain here why no droplets are ejected from the tip of the fast Worthington jet for $Oh\gtrsim 0.04$. In addition, our results predict the generation of submicron-sized aerosol particles with diameters below 100 nm and velocities ${\sim}\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707}_{l}$ for bubble radii $10~\unicode[STIX]{x03BC}\text{m}\lesssim R_{b}\lesssim 20~\unicode[STIX]{x03BC}\text{m}$, within the range found in natural conditions and in good agreement with experiments – a fact suggesting that our study could be applied in the modelling of sea spray aerosol production.

Stimulation of gas or oil shales by hydraulic fracturing requires injecting water at a very high rate into kilometre-long boreholes, in order to induce sufficient fracture width to place the proppant. Since such high rate of injection implies flow in the turbulent regime, heavy-molecular-weight polymers are added to water to reduce drag and thus drastically lessen the energy required for pumping. Lecampion & Zia (J. Fluid Mech., vol. 880, 2019, pp. 514–550) explore via modelling how the rheology of slickwater – water with a small amount of drag-reducing agents – affects the propagation of a hydraulic fracture. Theoretical models in combination with scaling arguments and numerical simulations indicate that flow in a radial fracture is inherently laminar, with the turbulent regime restricted at most to the first few minutes of injection, for plausible values of rock and fluid parameters and the injection rate.

This paper presents an analytic solution for the sound generated by rotor–stator interaction for aerofoils with small camber and thickness subject to a background flow with small angle of attack. The interaction is modelled as a convected, unsteady vortical or entropic gust incident on an infinite rectilinear cascade of staggered aerofoils in a background flow that is uniform far away from the cascade. Applying rapid distortion theory (RDT) and transforming to an orthogonal coordinate system reduces the cascade of aerofoils to a cascade of flat plates. By seeking a perturbation expansion in terms of the disturbance of the background flow from uniform flow, leading- and first-order governing equations and boundary conditions are obtained for the acoustic potential. The system is then solved analytically using the Wiener–Hopf method. The resulting expression is inverted to give the acoustic potential function in the entire domain, i.e. a solution to the inhomogeneous convected Helmholtz equation with inhomogeneous boundary conditions in a cascade geometry. The solution significantly extends previous analytical work that is restricted to flat plates or only calculates the far-upstream radiation, and as such can give insight into the role played by blade geometry on the acoustic field upstream, downstream and in the important inter-blade region of the cascade. This new solution is validated against solutions that only account for flat plates at zero angle of attack. Various aeroacoustic results, including the scattered pressure, unsteady lift and sound power output, are discussed for a range of geometries and angles of attack.

The drag coefficient of a rigid spherical particle deviates from Stokes law when it is put into a near-critical fluid mixture in the homogeneous phase with the critical composition. The deviation ($\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}_{d}$) is experimentally shown to depend approximately linearly on the correlation length of the composition fluctuation far from the particle ($\unicode[STIX]{x1D709}_{\infty }$), and is suggested to be caused by the preferential adsorption between one component and the particle surface. In contrast, the dependence was shown to be much steeper in the previous theoretical studies based on the Gaussian free-energy density. In the vicinity of the particle, especially when the adsorption of the preferred component makes the composition strongly off-critical, the correlation length becomes very small as compared with $\unicode[STIX]{x1D709}_{\infty }$. This spatial inhomogeneity, not considered in the previous theoretical studies, can influence the dependence of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}_{d}$ on $\unicode[STIX]{x1D709}_{\infty }$. To examine this possibility, we here apply a renormalized local functional theory, describing the preferential adsorption in terms of the surface field. This theory was previously proposed to explain the interaction of walls immersed in a (near-)critical binary fluid mixture. The free-energy density in this theory, coarse-grained up to the local correlation length, has a very complicated dependence on the order parameter, as compared with the Gaussian free-energy density. Still, a concise expression of the drag coefficient, which was derived in one of the previous theoretical studies, turns out to be valid in the present formulation. We show that, as $\unicode[STIX]{x1D709}_{\infty }$ becomes larger, the dependence of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}_{d}$ on $\unicode[STIX]{x1D709}_{\infty }$ becomes distinctly gradual and close to the linear dependence.

Much of our theoretical understanding of statistically stable and unstable flows is from the classical Monin–Obukhov similarity theory: the theory predicts the scaling of the mean flow well, but its prediction of the turbulent fluctuation is far from satisfactory. This study builds on Monin–Obukhov similarity theory and Townsend’s attached-eddy hypothesis. We present a model that connects the mean flow and the streamwise velocity fluctuations in both neutral and unstable boundary-layer flows at both moderate and high Reynolds numbers. The model predictions are compared to direct numerical simulations of weakly unstable boundary layers at moderate Reynolds numbers, and large-eddy simulations of unstable boundary-layer flows at high Reynolds numbers. The flow is shear dominated. The range of stability parameter considered in this work is $L/\unicode[STIX]{x1D6FF}<-0.1$, where $L$ is the Monin–Obukhov length, and $\unicode[STIX]{x1D6FF}$ is the boundary-layer height. Reasonably good prediction of velocity fluctuations based on knowledge of the mean velocity profile is obtained.

When a liquid stream is injected into a gaseous atmosphere, it destabilizes and continuously passes through different states characterized by different morphologies. Throughout this process, the flow dynamics may be different depending on the region of the flow and the scales of the involved liquid structures. Exploring this multi-scale, multi-dimensional phenomenon requires some new theoretical tools, some of which need yet to be elaborated. Here, a new analytical framework is proposed on the basis of two-point statistical equations of the liquid volume fraction. This tool, which originates from single phase turbulence, allows us notably to decompose the fluxes of liquid in flow–position space and scale space. Direct numerical simulations of liquid–gas turbulence decaying in a triply periodic domain are then used to characterize the time and scale evolution of the liquid volume fraction. It is emphasized that two-point statistics of the liquid volume fraction depend explicitly on the geometrical properties of the liquid–gas interface and in particular its surface density. The stretch rate of the liquid–gas interface is further shown to be the equivalent for the liquid volume fraction (a non-diffusive scalar) of the scalar dissipation rate. Finally, a decomposition of the transport of liquid in scale space highlights that non-local interactions between non-adjacent scales play a significant role.

The effects of dispersion acting on gravity currents propagating through porous media are considered theoretically and experimentally. We exploit the large aspect ratio of these currents to formulate a depth-averaged model of the evolution of the mass and buoyancy. Dispersion, acting predominantly at the interface between the current and the ambient, is velocity dependent and acts to entrain fluid into the gravity current, in direct analogy to turbulent mixing. Here, we show that when the gravity current is fed by a constant buoyancy and mass flux the buoyancy of the current is self-similar and recovers the classical solution for gravity currents in porous media. In contrast, the profile and the depth-averaged concentration of the current evolve in a non-self-similar manner. The total volume of the current increases with time as $t^{1/3}$ due to this dispersive entrainment. We test our theoretical predictions using a suite of laboratory experiments in which the evolution of the concentration within the current was mapped using a dye-attenuation technique. These experimental results show good agreement with the early-time limits of our theoretical model, and in particular accurately predict the evolution of the depth-averaged concentration profile. These results suggest that mixing within porous media may be modelled using an effective dispersive entrainment, the magnitude of which may be set by the underlying structure of the porous medium.

As an important dimensionless parameter for the vortex formation process, the general form of the formation time defined by Dabiri (Annu. Rev. Fluid Mech., vol. 41, 2009, pp. 17–33) is refined so as to provide better normalization for various vortex generator configurations. Our proposed definition utilizes the total circulation over the entire flow domain rather than that of the forming vortex ring alone. It adopts an integral form by considering the instantaneous infinitesimal increment in the formation time so that the effect of temporally varying properties of the flow configuration can be accounted for properly. By including the effect of buoyancy, the specific form of the general formation time for the starting forced plumes with negative and positive buoyancy is derived. A theoretical prediction based on the Kelvin–Benjamin variational principle shows that the general formation time manifests the invariance of the critical time scale, i.e. the formation number, under the influence of a source–ambient density difference. It demonstrates that the general formation time, based on the circulation production over the entire flow field, could take into account the effect of various vorticity production mechanisms, such as from a flux term or in a baroclinic fluid, on the critical formation number. The proposed definition may, therefore, serve as a guideline for deriving the specific form of the formation time in other types of starting/pulsatile flows.

Richtmyer–Meshkov instability of the SF6 gas layer surrounded by air is experimentally investigated. Using the soap film technique, five kinds of gas layer with two sharp interfaces are generated such that the development of each individual interface is highlighted. The flow patterns are determined by the amplitudes and phases of two corrugated interfaces. For a layer with both interfaces planar, the interface velocity shows that the reflected rarefaction waves from the second interface accelerate the first interface motion. For a layer with the second interface corrugated but the first interface planar, the reflected rarefaction waves make the first interface develop with the same phase as the second interface. For a layer with the first interface corrugated but the second interface planar, the rippled shock seeded from the first interface makes the second interface develop with the same phase as the first interface and the layer evolves into an ‘upstream mushroom’ shape. For two interfaces corrugated with opposite (the same) phase but a larger amplitude for the first interface, the layer evolves into ‘sinuous’ shape (‘bow and arrow’ shape, which has never been observed previously). For the interface amplitude growth in the linear stage, the waves’ effects are considered in the model to give a better prediction. In the nonlinear stage, the effect of the rarefaction waves on the first interface evolution is quantitatively evaluated, and the nonlinear growth is well predicted. It is the first time in experiments to quantify the interfacial instability induced by the rarefaction waves inside the heavy gas layer.

To date, the influence of nonlinear stratifications and two layer stratifications on internal wave propagation has been studied for two-dimensional wave fields in a Cartesian geometry. Here, we use a novel wave generator configuration to investigate transmission in nonlinear stratifications of an axisymmetric internal wave. We demonstrate that, despite the additional geometric complexity, with associated features such as an inhomogeneous spatial distribution of the energy flux, results for plane waves can be generalised to axisymmetric wave fields. Two configurations are studied, both theoretically and experimentally. In the case of a free incident wave, a transmission maximum is found in the vicinity of evanescent frequencies. In the case of a confined incident wave, resonant effects, in the sense of constructive interference, lead to enhanced transmission rates from an upper layer to a layer below. We consider the oceanographic relevance of these results by applying them to an example oceanic stratification, finding that there can be real-world implications.

How non-spherical particles orient as they settle in a flow has important practical implications in a number of scientific and engineering problems. In a quiescent fluid, a slowly settling particle orients so that it settles with its broad side first. This is an effect of the torque due to convective inertia of the fluid that is set in motion by the settling particle, which maximises the drag experienced by the particle. Turbulent fluid-velocity gradients, on the other hand, tend to randomise the particle orientation. Recently the settling of non-spherical particles in turbulence was analysed neglecting the effect of convective fluid inertia, but taking into account the effect of the turbulent fluid-velocity gradients on the particle orientation. These studies reached the opposite conclusion, namely that the particle tends to settle with its narrow edge first, therefore minimising the drag on the particle. Here, we consider both effects, the convective inertial torque as well as the torque due to fluctuating fluid-velocity gradients. We ask under which circumstances either one or the other dominates. To this end we estimate the ratio of the magnitudes of the two torques. Our estimates suggest that the fluid-inertia torque prevails in high-Reynolds-number flows. In this case non-spherical particles tend to settle with orientations maximising drag. But when the Reynolds number is small, then the torque due to fluid-velocity gradients may dominate, causing the particle to settle with its narrow edge first, minimising the drag.

Fluid flow through pipe-like conduits embedded in viscously deformable material occurs in many natural systems, including magma transport in the Earth’s mantle and channelized water flow beneath glaciers. Here, we present and explore a model of fluid flow in viscously deformable conduits that unifies previously published models of magmatic and glacial systems. Previous results for magmatic systems have demonstrated the existence of solitary wave solutions for the case of laminar flow in Newtonian conduits. Here we extend these models to allow turbulent fluid flow in power-law materials consistent with models used in subglacial hydrology. The generalized model encompasses both laminar and turbulent fluid flow, and the solid matrix may deform according to any power-law rheology. A quasilinear approximation of the governing equations is introduced, along with an initial condition that develops into a perfect step shock. This initial condition is used in numerical solution of the full nonlinear system where a dispersive wave train forms at shock time. We show that solitary wave solutions exist for all parameters. Rheology-dependent flattening of the wave peaks is investigated. In the limit of a perfectly plastic matrix, the solitary waves approach square waves asymptotically. Motivated by subglacial hydrology models, we study the effect of discharge-dependent melting on evolution of the solitary waves. We find that melting focuses at the wave peaks, causing the waves to grow and accelerate over time.

A statistical model is developed to reconstruct space–time energy spectra in turbulent flows from a non-extensive dataset comprising a time series of velocity fluctuations at a finite number of measurement points. This model is based on a higher approximation of energetic flow structures and developed by using local modulated waves. As a result, it can correctly predict the mean wavenumbers and spectral bandwidths. In contrast, Taylor’s frozen-flow hypothesis incorrectly predicts the spectral bandwidths to be zero, and the local wavenumber model significantly under-predicts the spectral bandwidths. An analytical example is formulated to illustrate the present model, and datasets from direct numerical simulations of turbulent channel flows are used to validate this model. The present statistical model is also discussed in terms of the dominating processes of temporal decorrelation in turbulent flows.

In this paper, we present a stability analysis of passive suppression devices for the vortex-induced vibration (VIV) in the laminar flow condition. A data-driven model reduction approach based on the eigensystem realization algorithm is used to construct a reduced-order model in a state-space format. From the stability analysis of the coupled system, two modes are found to be dominant in the phenomenon of self-sustained VIV: namely, the wake mode, with frequency close to that of the wake flow behind a stationary cylinder; and the structure mode, with frequency close to the natural frequency of the elastically mounted cylinder. The present study illustrates that VIV can be suppressed by altering the structure mode via shifting of the eigenvalues from the unstable to the stable region. This finding is realized through the simulations of passive control devices, such as fairings and connected-C devices, wherein the presence of appendages breaks the self-sustenance of the wake–body interaction cycle. A detailed proper orthogonal decomposition analysis is employed to quantify the effect of a fairing on the complex interaction between the wake features. From the assessment of the stability characteristics of appendages, the behaviour of a connected-C device is found to be similar to that of a fairing, and the trajectories of the eigenspectrum are nearly identical, while the eigenspectrum of the cylinder–splitter arrangement indicates a galloping behaviour at higher reduced velocities. Finally, we introduce a stability function to characterize the influence of geometric parameters on VIV suppression.

Unsteady thin-aerofoil theory is a low-order method for calculating the forces and moment developed on a camber line undergoing arbitrary motion, based on potential-flow theory. The vorticity distribution is approximated by a Fourier series, with a special ‘$A_{0}$’ term that is infinite at the leading edge representing the ‘suction peak’. Though the integrated loads are finite, the pressure and velocity at the leading edge in this method are singular owing to the $A_{0}$ term. In this article, the principle of matched asymptotic expansions is used to resolve the singularity and obtain a uniformly valid first-order solution. This is performed by considering the unsteady thin-aerofoil theory as an outer solution, unsteady potential flow past a parabola as an inner solution, and by matching them in an intermediate region where both are asymptotically valid. Resolution of the leading-edge singularity allows for derivation of the velocity at the leading edge and location of the stagnation point, which are of physical and theoretical interest. These quantities are seen to depend on only the $A_{0}$ term in the unsteady vorticity distribution, which may be interpreted as an ‘effective unsteady angle of attack’. The leading-edge velocity is proportional to $A_{0}$ and inversely proportional to the square root of leading-edge radius, while the chordwise stagnation-point location is proportional to the square of $A_{0}$ and independent of the leading-edge radius. Closed-form expressions for these in simplified scenarios such as quasi-steady flow and small-amplitude harmonic oscillations are derived.

Design of isolated microparticles which stabilize at the centreline of a channel flow is examined numerically, for moderate Reynolds numbers ($10\leqslant Re\leqslant 80$). This problem is motivated by the need for the design of shaped particle carriers for use in next generation microfluidic cell analysis devices. Stability metrics for particles with arbitrary shapes are formulated based on linear stability theory. Particle shape is parametrized by a compact, non-uniform rational B-spline-based representation. Shape design is posed as an optimization problem and solved using adaptive Bayesian optimization. We focus on designing particles for stability at the channel centreline robust to perturbations. Our results indicate that centreline focusing particles are families of characteristic ‘fish’/‘bottle’/‘dumbbell’-like shapes, exhibiting fore–aft asymmetry. A parametric exploration is then performed to identify stable particle designs at different $k$ values (particle chord-to-channel width ratio) and $Re$ values ($0.1\leqslant k\leqslant 0.4,10\leqslant Re\leqslant 80$). Particles at high $k$ values and $Re$ values are highly stabilized when compared to those at low $k$ values and $Re$ values. We validate the performance of designed particles to perturbations in flow using detailed fluid–structure interaction simulations over different $k$ values and $Re$ values. We identify a basin of attraction around the centreline, within which any arbitrary release results in rotationally stable centreline focusing. We find that this basin spans larger release angle ranges and lateral locations (tending to the channel width) for narrower channels. This effectively standardizes the notion of global focusing using the current stability paradigm in narrow channels, which eliminates the need for an independent design for global focusing in such configurations. The framework detailed in this work is illustrated for two-dimensional cases and is generalizable to stability in three-dimensional flow fields. The current formulation is agnostic to $Re$ and particle/channel geometry, which indicates substantial potential for integration with imaging flow cytometry tools and microfluidic biosensing assays.

Large-eddy simulation has been used to model turbulent channel flow over a range of surfaces featuring a prominent spatial heterogeneity; the flow streamwise direction is aligned relative to the heterogeneity at a range of angles, defined herein with $\unicode[STIX]{x1D703}$. Prior work has established that a sharp roughness heterogeneity orthogonal to the flow streamwise direction ($\unicode[STIX]{x1D703}=0$) induces formation of an internal boundary layer, which originates at the heterogeneity and thickens in the downflow direction before being homogenized via ambient shear. In contrast, more-recent studies have shown that a sharp roughness heterogeneity parallel to the flow streamwise direction ($\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$) induces streamwise-aligned, Reynolds-averaged secondary cells, where the spacing between adjacent surface heterogeneities regulates the spatial extent of secondary cells. No prior study has addressed intermediate (oblique) cases, $0\leqslant \unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x03C0}/2$. Results presented herein show that the momentum penalty exhibits a nonlinear dependence upon obliquity, where internal boundary layer-like flow processes persist over a range of obliquity angles before abruptly vanishing for spanwise roughness heterogeneity ($\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$). This result manifests itself within effective roughness lengths recovered a posteriori: the traditional approach to roughness modelling – predicated upon dependence with surface geometric arguments including height root-mean-square, skewness, frontal- and plan-area index, effective slope. and combinations thereof – is insufficient. A revised model incorporating dependence upon roughness frontal area index and flow-heterogeneity obliquity angle is able to accurately predict effective roughness length a priori.

A model of the growth of an ice lens in a saline porous medium is developed. At high lens growth rates the pore fluid becomes supercooled relative to its equilibrium Clapeyron temperature. Instability occurs when the supercooling increases with distance away from the ice lens. Solute diffusion in the pore fluid significantly enhances the instability. An expression for the segregation potential of the soil is obtained from the condition for marginal stability of the ice lens. The model is applied to a clayey silt and a glass powder medium, indicating parameter regimes where the ice lens stability is controlled by viscous flow or by solute diffusion. A mushy layer, composed of vertical ice veins and horizontal ice lenses, forms in the soil in response to the instability. A marginal equilibrium condition is used to estimate the segregated ice fraction in the mushy layer as a function of the freezing rate and salinity.

Active droplets swim as a result of the nonlinear advective coupling of the distribution of chemical species they consume or release with the Marangoni flows created by their non-uniform surface distribution. Most existing models focus on the self-propulsion of a single droplet in an unbounded fluid, which arises when diffusion is slow enough (i.e. beyond a critical Péclet number, $Pe_{c}$). Despite its experimental relevance, the coupled dynamics of multiple droplets and/or collision with a wall remains mostly unexplored. Using a novel approach based on a moving fitted bi-spherical grid, the fully coupled nonlinear dynamics of the chemical solute and flow fields is solved here to characterise in detail the axisymmetric collision of an active droplet with a rigid wall (or with a second droplet). The dynamics is strikingly different depending on the convective-to-diffusive transport ratio, $Pe$: near the self-propulsion threshold (moderate $Pe$), the rebound dynamics is set by chemical interactions and is well captured by asymptotic analysis; in contrast, for larger $Pe$, a complex and nonlinear combination of hydrodynamic and chemical effects set the detailed dynamics, including a closer approach to the wall and a velocity plateau shortly after the rebound of the droplet. The rebound characteristics, i.e. minimum distance and duration, are finally fully characterised in terms of $Pe$.