We perform direct numerical simulations of spiral turbulent Taylor–Couette (TC) flow for $400\leqslant Re_{i}\leqslant 1200$ and $-2000\leqslant Re_{o}\leqslant -1000$, i.e. counter-rotation. The aspect ratio $\unicode[STIX]{x1D6E4}=\text{height}/\text{gap width}$ of the domain is $42\leqslant \unicode[STIX]{x1D6E4}\leqslant 125$, with periodic boundary conditions in the axial direction, and the radius ratio $\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.91$. We show that, with decreasing $Re_{i}$ or with decreasing $Re_{o}$, the formation of a turbulent spiral from an initially ‘featureless turbulent’ flow can be described by the phenomenology of the Ginzburg–Landau equations, similar as seen in the experimental findings of Prigent et al. (Phys. Rev. Lett., vol. 89, 2002, 014501) for TC flow at $\unicode[STIX]{x1D702}=0.98$ an $\unicode[STIX]{x1D6E4}=430$ and in numerical simulations of oblique turbulent bands in plane Couette flow by Rolland & Manneville (Eur. Phys. J., vol. 80, 2011, pp. 529–544). We therefore conclude that the Ginzburg–Landau description also holds when curvature effects play a role, and that the finite-wavelength instability is not a consequence of the no-slip boundary conditions at the upper and lower plates in the experiments. The most unstable axial wavelength $\unicode[STIX]{x1D706}_{z,c}/d\approx 41$ in our simulations differs from findings in Prigent et al., where $\unicode[STIX]{x1D706}_{z,c}/d\approx 32$, and so we conclude that $\unicode[STIX]{x1D706}_{z,c}$ depends on the radius ratio $\unicode[STIX]{x1D702}$. Furthermore, we find that the turbulent spiral is stationary in the reference frame of the mean velocity in the gap, rather than the mean velocity of the two rotating cylinders.

It is widely acknowledged that streamwise ridges on the bed of open-channel flows generate secondary currents (SCs). A recent discovery of meandering long streamwise counter-rotating vortices in open-channel flows, known as very-large-scale motions (VLSMs), raises a question regarding the interrelations between VLSMs and SCs in flows over ridge-covered fully rough beds. To address it, we conducted long-duration experiments using stereoscopic particle image velocimetry, covering a range of ridge spacings ($s$) from ${\approx}0.4$ to ${\approx}4$ flow depths ($H$). For a benchmark no-ridge case, the flow is quasi-two-dimensional in the central part of the channel, exhibiting a strong spectral signature of VLSMs, as expected. With ridges on the bed at $s\lessapprox 2H$, two SC cells are formed between neighbouring ridges and VLSMs are entirely suppressed, suggesting that ridge-induced SCs prevent the formation of VLSMs by absorbing their energy or overpowering their formation. At the same time, velocity auto- and cross-spectra reveal a new feature that can be explained by low-amplitude meandering of the alternating low- and high-momentum flow regions associated with instantaneous manifestations of SCs. Two-point velocity correlations and smooth velocity field reconstructions using proper orthogonal decomposition further support the validity of this effect. Its origin is probably due to the instability related to the presence of inflection points in the spanwise distribution of the streamwise velocity within the SC cells. These results have implications for bed friction in open channels, where the friction factor may increase if depth-scale SCs are present, or decrease under conditions of sub-depth-scale SCs and suppressed VLSMs.

We investigate the interplay between large-scale patterns, so-called superstructures, in the fluctuation fields of temperature $\unicode[STIX]{x1D703}$ and vertical velocity $w$ in turbulent Rayleigh–Bénard convection at large aspect ratios. Earlier studies suggested that velocity superstructures were smaller than their thermal counterparts in the centre of the domain. However, a scale-by-scale analysis of the correlation between the two fields employing the linear coherence spectrum reveals that superstructures of the same size exist in both fields, which are almost perfectly correlated. The issue is further clarified by the observation that, in contrast to the temperature, and unlike assumed previously, superstructures in the vertical-velocity field do not result in a peak in the power spectrum of $w$. The origin of this difference is traced back to the production terms of the $\unicode[STIX]{x1D703}$ and $w$ variance. These results are confirmed for a range of Rayleigh numbers $Ra=10^{5}{-}10^{9}$; the superstructure size is seen to increase monotonically with $Ra$. Furthermore, the scale distribution of the temperature fluctuations in particular is pronouncedly bimodal. In addition to the large-scale peak caused by the superstructures, there exists a strong small-scale peak. This ‘inner peak’ is most intense at a distance of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ from the wall and is associated with structures of size ${\approx}10\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$, where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ is the thermal boundary layer thickness. Finally, based on the vertical coherence relative to a reference height of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$, a self-similar structure is identified in the velocity field (vertical and horizontal components) but not in the temperature.

The renormalization method proposed by Batchelor is used to derive gradient diffusion coefficients in Brownian suspensions of hard spheroidal particles with aspect ratio $\unicode[STIX]{x1D706}$ in the range $1\leqslant \unicode[STIX]{x1D706}\leqslant 3.5$. The theory is based on pairwise steric and hydrodynamic interactions, and the results are therefore valid for dilute suspensions such that $\unicode[STIX]{x1D706}^{2}\unicode[STIX]{x1D719}\ll 1$, where $\unicode[STIX]{x1D719}$ is the particle volume fraction. The driving force for gradient diffusion, i.e. the gradient in chemical potential, is larger for suspensions of spheroidal particles than for spheres at the same volume fraction. The hydrodynamic resistance also increases with aspect ratio, but the increase is weaker than that in the driving force. Consequently, at the same particle volume fraction, the increases in rates of gradient diffusion are greater for spheroidal particles than for spheres. The concentration-dependent gradient diffusion coefficient $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ is shown to be closely approximated by $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706}-1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$, which reduces to the result for spheres when $\unicode[STIX]{x1D706}=1$. Here, $D_{0}$ is the Stokes–Einstein diffusivity of a spherical particle with its radius equal to the longer dimension of the spheroidal particle, and $\unicode[STIX]{x1D709}_{m}D_{0}$ is the orientation-averaged diffusivity of an isolated spheroidal particle.

We present a new method for generating robust guesses for unstable periodic orbits (UPOs) by post-processing turbulent data using dynamic mode decomposition (DMD). The approach relies on the identification of near-neutral, repeated harmonics in the DMD eigenvalue spectrum from which both an estimate for the period of a nearby UPO and a guess for the velocity field can be constructed. In this way, the signature of a UPO can be identified in a short time series without the need for a near recurrence to occur, which is a considerable drawback to recurrent flow analysis, the current state of the art. We first demonstrate the method by applying it to a known (simple) UPO and find that the period can be reliably extracted even for time windows of length one quarter of the full period. We then turn to a long turbulent trajectory, sliding an observation window through the time series and performing many DMD computations. Our approach yields many more converged periodic orbits (including multiple new solutions) than a standard recurrent flow analysis of the same data. Furthermore, it also yields converged UPOs at points where the recurrent flow analysis flagged a near recurrence but the Newton solver did not converge, suggesting that the new approach can be used alongside the old to generate improved initial guesses. Finally, we discuss some heuristics on what constitutes a ‘good’ time window for the DMD to identify a UPO.

We present a theory for the evolution of a one-dimensional, steady-state detonation reaction zone to a two-dimensional reaction zone, when the explosive experiences a sudden loss of side-on confinement as a boundary of the explosive is impulsively withdrawn. Our focus is on condensed-phase explosives, which we describe as having a constant adiabatic gamma equation of state and an irreversible, state-independent reaction rate. We consider two detonation models: (i) the instantaneous reaction heat release Chapman–Jouguet (CJ)-limit and (ii) the spatially resolved reaction heat-release Zel’dovich–von Neumann–Döring (ZND) model, in the limit where only a small fraction of the energy release is resolved (the SRHR-limit). Two competing rarefaction waves are generated by this loss of confinement: (i) a smooth wave coming off the full length of the withdrawn boundary and (ii) a singular fan spreading out from the point where the detonation shock and the withdrawn boundary meet. For the CJ-limit, in all cases the singular rarefaction fan eventually dominates the competition to control the steady-state behaviour. For the SRHR-limit, the spatially resolved heat release moderates this competition. When the withdrawal speed is fast, the rarefaction fan dominates; when the withdrawal speed is slower, the smooth rarefaction eventually dominates, although the flow features a fan at early times. By examining the mathematical properties of the steady two-dimensional fan-based solution, we set down a mechanism for this transition in behaviours.

In turbulent flows subject to strong background rotation, the advective mechanisms of turbulence are superseded by the propagation of inertial waves, as the effects of rotation become dominant. While this mechanism has been identified experimentally (Dickinson & Long, J. Fluid Mech., vol. 126, 1983, pp. 315–333; Davidson, Staplehurst & Dalziel, J. Fluid Mech., vol. 557, 2006, pp. 135–144; Staplehurst, Davidson & Dalziel, J. Fluid Mech., vol. 598, 2008, pp. 81–105; Kolvin et al.Phys. Rev. Lett., vol. 102, 2009, 014503), the conditions of the transition between the two mechanisms are less clear. We tackle this question experimentally by tracking the turbulent front away from a solid wall where jets enter an otherwise quiescent fluid. Without background rotation, this apparatus generates a turbulent front whose displacement recovers the $z(t)\sim t^{1/2}$ law classically obtained with an oscillating grid (Dickinson & Long, Phys. Fluids, vol. 21 (10), 1978, pp. 1698–1701) and we further establish the scale independence of the associated transport mechanism. When the apparatus is rotating at a constant velocity perpendicular to the wall where fluid is injected, not only does the turbulent front become mainly transported by inertial waves, but advection itself is suppressed because of the local deficit of momentum incurred by the propagation of these waves. Scale-by-scale analysis of the displacement of the turbulent front reveals that the transition between advection and propagation is local both in space and spectrally, and takes place when the Rossby number based on the considered scale is of order unity, or equivalently, when the scale-dependent group velocity of inertial waves matched the local advection velocity.

Front-propagating systems provide some of the most fundamental physical examples of interfacial instability and pattern formation. However, their nonlinear dynamics is rarely addressed. Here, we present an experimental study of air displacing a viscous fluid within a collapsed, compliant channel – a model for pulmonary airway reopening. Air injected at a constant flow rate from one end of the liquid-filled, collapsed channel results in the propagation of a reopening finger. Depending on the imposed flow rate, we observe a wide variety of finger-tip morphologies, which occur persistently or evolve transiently as the finger propagates. Persistent fingers are stable in the sense that they propagate with approximately constant bubble pressure. We find that their pressure increases monotonically as a function of bubble speed along two disconnected lines. Although the line associated with higher bubble speed exhibits a minimum pressure value, the low-speed line supports finger propagation down to the smallest flow rates investigated. We present evidence that the lower and higher-speed fingers are dominated by viscous and elastic forces, respectively. We also find a range of bubble speeds separating the two pressure lines where no stable finger propagation is observed. Instead, complex transient dynamics leads to the long-term selection of stable fingers depending on initial conditions. The early transient evolution of these fingers is characterised by an increase in bubble pressure alongside a reduction in bubble speed. We hypothesise the existence of a weakly unstable, steady mode, which orchestrates the transient evolution of the finger towards either low- or high-speed modes of propagation.

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.

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.

The stability of neutrally and non-neutrally buoyant particles immersed in a plane Poiseuille flow of a yield-stress fluid (Bingham fluid) is addressed numerically. Particles being carried by the yield-stress fluid can behave in different ways: they might (i) migrate inside the yielded regions or (ii) be transported without any relative motion inside the unyielded region if the yield stress is large enough compared to the buoyancy stress and the other stresses acting on the particles. Knowing the static stability of particles inside a bath of quiescent yield-stress fluid (Chaparian & Frigaard, J. Fluid Mech., vol. 819, 2017, pp. 311–351), we analyse the latter behaviour when the yield-stress fluid Poiseuille flow is host to two-dimensional particles. Numerical experiments reveal that particles lose their stability (i.e. break the unyielded plug and sediment/migrate) with smaller buoyancy compared to the sedimentation inside a bath of quiescent yield-stress fluid, because of the inherent shear stress in the Poiseuille flow. The key parameter in interpreting the present results is the position of the particle relative to the position of the yield surface in the undisturbed flow (in the absence of any particle): the larger the portion of a particle located inside the undisturbed sheared regions, the more likely is the particle to be unstable. Yet, we find that the core unyielded plug can grow locally to some extent to contain the particles. This picture holds even for neutrally buoyant particles, although they are strictly stable when they are located wholly inside the undisturbed plug. We propose scalings for all cases.

Shock buffet on wings encountered in edge-of-the-envelope transonic flight remains an unresolved and disputed flow phenomenon, challenging both fundamental fluid mechanics and applied aircraft aerodynamics. Its dynamics is revealed through the interaction of spanwise shock-wave oscillations and intermittent turbulent boundary-layer separation. Resulting unsteady aerodynamic loads, and their mutual working with the flexible aircraft structure, need to be accounted for in establishing the safe flight envelope. The question of global instability leading to this flow unsteadiness is addressed herein. It is shown for the first time on an industrially relevant configuration that the dynamics of a single unstable oscillatory eigenmode plays a prominent role in near-onset shock buffet on a quasi-rigid wing. Its three-dimensional spatial structure, previously inferred both from experiment and time-marching simulation, describes a spanwise-localised pocket of shear-layer pulsation synchronised with an outboard-propagating shock oscillation. The results also suggest that the concept of a critical global shock-buffet mode commonly reported for two-dimensional aerofoils also applies to three-dimensional finite and swept wings, albeit different modes at play. Specifically, the modern wing design, NASA Common Research Model, with publicly available geometry and experimental data for code validation is studied at a free-stream Mach number of 0.85 with Reynolds number per reference chord of $5.0\times 10^{6}$ and varying angle of attack between 3. 5° and 4. 0° targeting the instability onset. Strouhal number at instability onset just above 3. 7° is approximately 0.39. At the same time, a band of eigenmodes shows reduced decay rate in the Strouhal-number range of 0.3 to 0.7, with additional unstable oscillatory modes appearing beyond onset. Importantly, those emerging modes seem to discretise the continuous band of medium-wavelength modes, as recently reported for infinite swept wings using stability analysis, hence generalising those findings to finite wings. Through conventional time-marching unsteady simulation it is explored how the critical linear eigenmode feeds into the nonlinearly saturated limit-cycle oscillation near instability onset. The established numerical strategy, using an iterative inner–outer Krylov approach with shift-and-invert spectral transformation and sparse iterative linear solver, to solve the arising large-scale eigenvalue problem with an industrial Reynolds-averaged Navier–Stokes flow solver means that such a practical non-canonical test case at a high-Reynolds-number condition can be investigated. The numerical findings can potentially be exploited for more effective unsteady flow analysis in future wing design and inform routes to flow control and model reduction.

The stability of a uniform flow above an erodible bed composed of a bimodal mixture of sediments is investigated by means of linear analysis. Results show that, for any given set of the flow and sediment parameters, two distinct modes of instability arise, each one characterized by its own wave speed, growth rate and longitudinal wavelength, each one involving spatial variations of both grain size density and bed elevation. Although at a linear level no information on the amplitude of the perturbations is gathered, the analysis of the eigenvectors associated with the two modes of instability allows for an easy classification in terms of the relative amplitudes of the perturbations of bed elevation and size density. One eigenvalue is shown to be associated with the modifications of bed forms induced by the presence of the heterogeneous mixture, such as the local accumulation of finer and coarser material along the unit wavelength, the other with the formation of the low-amplitude sorting waves known as bedload sheets. In the present unidirectional shallow-water framework, only the sorting wave is found to be unstable, since dunes and antidunes, the relevant bed forms for this case, require a more refined rotational flow model in order to become unstable. On the other hand, the simple flow model adopted allows for the formulation of an algebraic eigenvalue problem that can be solved analytically, allowing for a deep insight into the mechanisms that drive both instabilities.

Turbulent flow over a surface with streamwise-elongated rough and smooth stripes is studied by means of direct numerical simulation (DNS) in a periodic plane open channel with fully resolved roughness. The goal is to understand how the mean height of roughness affects the characteristics of the secondary flow formed above a spanwise heterogeneous rough surface. To this end, while the statistical properties of roughness texture as well as the width and spacing of the rough stripes are kept constant, the elevation of the smooth stripes is systematically varied in different simulation cases. Utilizing this variation, three configurations – representing protruding, recessed and an intermediate type of roughness – are analysed. In all cases, secondary flows are present and the skin friction coefficients calculated for all the heterogeneous rough surfaces are meaningfully larger than what would result from the area-weighted average of those of homogeneous smooth and rough surfaces. This drag increase appears to be linked to the strength of the secondary flow. The rotational direction of the secondary motion is shown to depend on the relative surface elevation. The present results suggest that this rearrangement of the secondary flow is linked to the spatial distribution of the spanwise-wall-normal Reynolds stress component, which carries opposing signs for protruding and recessed roughness.

The flow within an oscillatory boundary layer, which approximates the flow generated by propagating sea waves of small amplitude close to the bottom, is simulated numerically by integrating the Navier–Stokes and continuity equations. The bottom is made up of spherical particles, free to move, which mimic sediment grains. The approach allows one to fully resolve the flow around the particles and to evaluate the forces and torques that the fluid exerts on their surface. Then, the dynamics of sediments is explicitly computed by means of the Newton–Euler equations. For the smallest value of the flow Reynolds number presently simulated, the flow regime turns out to fall in the intermittently turbulent regime such that turbulence appears when the free-stream velocity is close to its largest value but the flow recovers a laminar-like behaviour during the remaining phases of the cycle. For the largest value of the Reynolds number, turbulence is significant during almost the whole flow cycle. The evaluation of the sediment transport rate allows one to estimate the reliability of the empirical predictors commonly used to estimate the amount of sediments transported by sea waves. For large values of the Shields parameter, the sediment flow rate during the accelerating phases does not differ from that observed during the decelerating phases. However, for relatively small values of the Shields parameter, the amount of moving particles depends not only on the bottom shear stress but also on flow acceleration. Moreover, the numerical results provide information on the role that turbulent eddies have on sediment dynamics.

Body-inclusive large-eddy simulations of disk wakes are performed for a homogeneous fluid and for different levels of stratification. The Reynolds number is 5 × 104 and the Froude number ($Fr$) takes the values of $\infty$, 50, 10 and 2. In the axisymmetric wake of a disk with diameter $L_{b}$ in a homogeneous fluid, it is found that the mean streamwise velocity deficit ($U_{0}$) decays in two stages: $U_{0}\propto x^{-0.9}$ during $10<x/L_{b}<65$ and, subsequently, $U_{0}\propto x^{-2/3}$. Consequently, none of the simulated stratified wakes is able to exhibit the classical 2/3 decay exponent of $U_{0}$ in the interval before buoyancy effects set in. Stratification affects the wake within approximately one buoyancy time scale, after which, we find three regimes: weakly stratified turbulence (WST), intermediately stratified turbulence (IST) and strongly stratified turbulence (SST). WST begins when the turbulent Froude number ($Fr_{h}$) decreases to $O(1)$, spans $1\lesssim Nt_{b}\lesssim 5$ and, while the mean flow is strongly affected by buoyancy in WST, turbulence is not. During IST, which commences at $Nt_{b}\approx 5$ when $Fr_{h}=O(0.1)$, the mean flow has arrived into the non-equilibrium (NEQ) regime with $U_{0}\propto x^{-0.18}$, but the turbulence state is still in transition, as indicated by progressively increasing turbulence anisotropy. When $Fr_{h}\sim O(0.01)$ at $Nt_{b}\approx 20$, the wake transitions into SST, where the turbulent vertical Froude number ($Fr_{v}$) asymptotes to a $O(1)$ constant. There is strong anisotropy ($u_{z}^{\prime }\ll u_{h}^{\prime }$), and both $u_{h}^{\prime }$ and $U_{0}$ satisfy $x^{-0.18}$ decay, signifying the arrival of the NEQ regime for both turbulence and mean flow. Turbulence is patchy and temporal spectra are broadband in the SST wake. The wake height decreases as $L_{V}\sim O(U_{0}/N)$ in IST/SST. Energy budgets reveal that stratification prolongs wake life during WST/early-IST by both an energy transfer from mean potential energy to mean kinetic energy and reduction of turbulent production. In the late-IST/early-SST stages, production is enhanced and, additionally, there is injection from turbulent potential energy slowing down turbulent kinetic energy (TKE) decay. Only in the SST stage, when NEQ is realized for both the mean and turbulence, does the turbulent buoyancy flux become negative again, acting as a sink of TKE.

We consider the size spectrum of entrained bubbles under strong free-surface turbulence (SFST). We investigate the entrainment bubble-size spectrum per unit (mean) interface area, ${\mathcal{N}}_{a}^{E}(r)$, with dimension length$^{-3}$, and develop a physical/mechanistic model for ${\mathcal{N}}_{a}^{E}(r)$ through energy arguments. The model obtains two distinct regimes of ${\mathcal{N}}_{a}^{E}(r)$, separated by bubble-size scale $r_{0}$. For bubble radius $r>r_{0}$, the effects of gravity $g$ dominate those of the surface tension force $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}$, and ${\mathcal{N}}_{a}^{E}(r)\propto g^{-1}\unicode[STIX]{x1D716}^{2/3}r^{-10/3}$, where $\unicode[STIX]{x1D716}$ is the turbulence dissipation rate. For $r<r_{0}$, surface tension is more important and ${\mathcal{N}}_{a}^{E}(r)\propto (\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C})^{-1}\unicode[STIX]{x1D716}^{2/3}r^{-4/3}$. From the model, we show that $r_{0}\approx r_{c}=1/2\sqrt{\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}g}$, the capillary length scale, and not the generally assumed Hinze scale $r_{H}$. For an air–water interface and Earth gravity, $r_{c}\approx$ 1.5 mm. The model provides an $\unicode[STIX]{x1D716}$–$r$ entrainment regime map that identifies a critical dissipation rate $\unicode[STIX]{x1D716}_{cr}$ (constant for given $g$ and $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}$) above which there is appreciable air entrainment, thus separating SFST and weak FST. We confirm the theoretical model and its predictions using two-phase, high-fidelity direct numerical simulations of a canonical FST flow using the conservative volume-of-fluid method: the respective power laws of ${\mathcal{N}}_{a}^{E}(r)\propto r^{-10/3}$ and $r^{-4/3}$ for $r>r_{0}$ and $r<r_{0}$; the value $r_{0}=r_{c}$; the scaling ${\mathcal{N}}_{a}^{E}(r)\propto \unicode[STIX]{x1D716}^{2/3}$; and the predictions of the $\unicode[STIX]{x1D716}$–$r$ entrainment regime map.

Flow separation and its control have been the subject of intensive research for decades. Flow separation occurs when the boundary layer loses contact with the associated confining wall, which is usually caused by a pressure gradient acting against the local flow direction. Numerous strategies exist to control flow separation, and in this study we demonstrate experimentally that vertical flow separation over steep slopes in shallow free-surface flows may be suppressed by contracting the flow horizontally upstream of the slope. We found that, unexpectedly, introducing lateral non-uniformity in the upstream flow field could suppress vertical flow separation for steep slopes up to 1 in 2. This study reveals the possibility of two different flow states over steep slopes; (i) a vertically attached flow combined with horizontal convergence, and (ii) a vertically detached flow combined with horizontal divergence. A detailed analysis of the dynamics of the two different flow states is presented. Although a predictive relation determining the transition point between the two flow states was not found in the current study, the observed phenomena were shown to be strongly related to the magnitude of the lateral gradient at the upstream edge of the slope. The results demonstrate a significant influence of the vertical flow state – separated or attached – on the shear stress at the confining boundaries of the flow.

The flow within an infinitely long cylinder exhibiting solid-body rotation (SBR) is impulsively stopped. The complete decay of the initial SBR is captured by means of direct numerical simulations for a wide range of Reynolds numbers ($Re$). Five distinct stages are identified during the decay process according to their flow structure and their underlying mechanisms of kinetic-energy dissipation. Initially, the laminar boundary layer undergoes a primary centrifugal instability, which causes the formation of coherent Taylor rolls. The flow then becomes turbulent, once the Taylor rolls are corrupted by secondary instabilities. Within the turbulent stage, two phases are distinguished. In the first turbulent phase, the SBR core is still intact and turbulence is sustained. The mean velocity profile is well described by the superposition of a near-wall region, a retracting SBR core and an intermediate region of constant angular momentum. In the latter region, the magnitude of angular momentum in viscous units $l^{+}(Re)$ is approximately constant in time. In the second turbulent phase, the SBR core breaks down, turbulence starts to decay exponentially and the kinetic energy of the mean flow decays logarithmically. Eventually, the flow relaminarises and the velocity profile of the analytical solution for purely laminar decay is recovered, albeit at an earlier temporal instant due to the net effect of transition and turbulence.

An asymptotic model is derived for the competitive diffusion-limited evaporation of multiple thin sessile droplets under the assumption that the droplets are well separated. Exact solutions of the model are obtained for a pair of and for a polygonal array of identical droplets, and the model is found to perform well even outside its formal range of validity, up to and including the limit of touching droplets. The shielding effect of droplets on each other is demonstrated, and the model is used to investigate the effect of this shielding on droplet evolutions and lifetimes, as well as on the coffee-ring effect. The theoretical predictions of the model are found to be in good agreement with recent experimental results for seven relatively closely spaced droplets, suggesting that the model could be a useful tool for studying a wide range of other droplet configurations.