Using complementary numerical approaches at high resolution, we study the late-time behaviour of an inviscid incompressible two-dimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediate-wavenumber spherical harmonics, we find that, contrary to the predictions of equilibrium statistical mechanics, the flow does not evolve into a large-scale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent small-scale vortices interacting with a large-scale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple point-vortex model characterising the interactions between the four main vortices which emerge.

Many flows of electrically conducting fluids can spontaneously generate magnetic fields through the process of dynamo action, but when does a flow produce a better dynamo than another one or when is it simply the most efficient dynamo? Using a variational approach close to that of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find optimal kinematic dynamos within a huge class of stationary and incompressible flows that are confined in a cube. We demand that the magnetic field satisfies either superconducting (T) or pseudovacuum (N) boundary conditions on opposite pairs of walls of the cube, which results in four different combinations. For each of these set-ups, we find the optimal flow and its corresponding magnetic eigenmodes. Numerically, it is observed that swapping the magnetic boundary from T to N leaves the magnetic energy growth nearly unchanged, and both $+\boldsymbol{U}$ and $-\boldsymbol{U}$ are optimal flows for these different but complementary set-ups. This can be related to work by Favier & Proctor (Phys. Rev. E, vol. 88, 2013, 031001). We provide minimal lower bounds for dynamo action and find that no dynamo is possible below an enstrophy (or shear) based magnetic Reynolds number $Rm_{c,min}=7.52{\rm\pi}^{2}$, which is a factor of $16$ above the Proctor/Backus bound.

The coupled dynamics of two conjugated liquid layers of disparate thicknesses, which coat a solid substrate and are subjected to a transverse temperature gradient, is investigated. The upper liquid layer evolves under the short-wavelength Bénard–Marangoni instability, whereas the lower, much thinner film undergoes a shear-driven long-wavelength deformation. Although the lubricating film should reduce the viscous stresses acting on the up to one hundred times thicker upper layer by only 10 %, it is found that the critical Marangoni number of marginal stability may be as low as if a stress-free boundary condition were applied at the bottom of the upper layer, i.e. much lower than the classical value of 79.6 known for a single film. Furthermore, it is experimentally verified that the deformation of the liquid–liquid interface, albeit small, has a non-negligible effect on the temperature distribution along the liquid–gas interface of the upper layer. This stabilizes the hexagonal pattern symmetry towards external disturbances and indicates a two-way coupling of the different layers. The experiments also demonstrate how convection patterns formed in a liquid film can be used to pattern a second conjugated film. The experimental findings are verified by a numerical model of the coupled layers.

In this study, a CFD-based linear dynamics model combined with the direct Computational Fluid Dynamics/Computational Structural Dynamics (CFD/CSD) simulation method is utilized to study the physical mechanisms underlying frequency lock-in in vortex-induced vibrations (VIVs). An identification method is employed to construct the reduced-order models (ROMs) of unsteady aerodynamics for the incompressible flow past a vibrating cylinder at low Reynolds numbers ($Re$). Reduced-order-model-based fluid–structure interaction models for VIV are also constructed by coupling ROMs and structural motion equations. The effects of the natural frequency of the cylinder, mass ratio and structural damping coefficient on the dynamics of the coupled system at $Re=60$ are investigated. The results show that the frequency lock-in phenomenon at low Reynolds numbers can be divided into two patterns according to different induced mechanisms. The two patterns are ‘resonance-induced lock-in’ and ‘flutter-induced lock-in’. When the natural frequency of the cylinder is in the vicinity of the eigenfrequency of the uncoupled wake mode (WM), only the WM is unstable. The dynamics of the coupled system is dominated by resonance. Meanwhile, for relatively high natural frequencies (i.e. greater than the eigenfrequency of the uncoupled WM), the structure mode becomes unstable, and the coupling between the two unstable modes eventually leads to flutter. Flutter is the root cause of frequency lock-in and the higher vibration amplitude of the cylinder than that of the resonance region. This result provides evidence for the finding of De Langre (J. Fluids Struct., vol. 22, 2006, pp. 783–791) that frequency lock-in is caused by coupled-mode flutter. The linear model exactly predicts the onset reduced velocity of frequency lock-in compared with that of direct numerical simulations. In addition, the transition frequency predicted by the linear model is in close coincidence with the amplitude of the lift coefficient of a fixed cylinder for high mass ratios. Therefore, it confirms that linear models can capture a significant part of the inherent physics of the frequency lock-in phenomenon.

We examine three-dimensional (3D) effects on the flapping dynamics of a flag, modelled as a thin membrane, in uniform fluid inflow. We consider periodic spanwise variations of length $S$ (ignoring edge effects), so that the 3D effects are characterized by the dimensionless spanwise wavelength ${\it\gamma}=S/L$, where $L$ is the chord length. We perform linear stability analysis (LSA) to show increase in stability with ${\it\gamma}$, with the purely 2D mode being the most unstable. To confirm the LSA and to study nonlinear responses of 3D flapping, we obtain direct numerical simulations, up to Reynolds number 1000 based on $L$, coupling solvers for the Navier–Stokes equations and that for a thin membrane structure undergoing arbitrarily large displacement. For nonlinear flapping evolution, we identify and characterize the effect of ${\it\gamma}$ on the distinct flag motions and wake vortex structures, corresponding to spanwise standing wave (SW) and travelling wave (TW) modes, in the absence and presence of cross-flow respectively. For both SW and TW, the response is characterized by an initial instability growth phase (I), followed by a nonlinear development phase (II) consisting of multiple unstable 3D modes, and tending, in long time, towards a quasi-steady limit-cycle response (III) dominated by a single (most unstable) mode. Phase I follows closely the predictions of LSA for initial instability and growth rates, with the latter increased for TW due to suppression of restoring forces by the cross-flow. Phase II is characterized by multiple competing flapping modes with energy cascading towards the more unstable mode(s). The wake is characterized by interwoven (SW) and oblique continuous (TW) shed vortices. For phase III, the persistent single dominant mode for SW is the (most unstable) 2D flag displacement with a continuous parallel wake structure; and for TW, the fundamental oblique travelling-wave flag displacement corresponding to the given ${\it\gamma}$ with continuous oblique shedding. The transition to phase III occurs slower for greater ${\it\gamma}$. For the total forces, drag decreases for both SW and TW with decreasing ${\it\gamma}$, while lift is negligible in phase I and II and comparable in magnitude to drag in phase III for any ${\it\gamma}$.

The use of channel pressurisation in drawing microstructured optical fibres (MOFs) potentially allows for fine control of the internal structure of the fibre. By applying extra pressure inside the channels it is possible to counteract the effect of surface tension which would otherwise act to close the channels in the fibre as it is drawn. This paper extends the modelling approach of Stokes et al. (J. Fluid Mech., vol. 755, 2014, pp. 176–203) to include channel pressurisation. This approach treats the problem as two submodels for the flow, one in the axial direction along the fibre and another in the plane perpendicular to that direction. In the absence of channel pressurisation these models decoupled and were solved independently; we show that they become fully coupled when the internal channels are pressurised. The fundamental case of a fibre with an annular cross-section (containing one central channel) will be examined in detail. In doing this we consider both a forward problem to determine the shape of fibre from a known preform and an inverse problem to design a preform such that when drawn it will give a desired fibre geometry. Criteria on the pressure corresponding to fibre explosion and closure of the channel will be given that represent an improvement over similar criteria in the literature. A comparison between our model and a recent experiment is presented to demonstrate the effectiveness of the modelling approach. We make use of some recent work by Buchak et al. (J. Fluid Mech., vol. 778, 2015, pp. 5–38) to examine more complicated fibre geometries, where the cross-sectional shape of the internal channels is assumed to be elliptical and multiple channels are present. The examples presented here demonstrate the versatility of our modelling approach, where the subtleties of the interaction between surface tension and pressurisation can be revealed even for complex patterns of cross-sectional channels.

Spatial turbulence spectra for high-Reynolds-number shear flows are usually obtained by mapping experimental frequency spectra into wavenumber space using Taylor’s hypothesis, but this is known to be less than ideal. In this paper, we propose a cross-spectral approach that allows us to determine the entire wavenumber–frequency spectrum using two-point measurements. The method uses cross-spectral phase differences between two points – equivalent to wave velocities – to reconstruct the wavenumber–frequency plane, which can then be integrated to obtain the spatial spectrum. We verify the technique on a particle image velocimetry (PIV) data set of a turbulent boundary layer. To show the potential influence of the different mappings, the transfer functions that we obtained from our PIV data are applied to hot-wire data at approximately the same Reynolds number. Comparison of the newly proposed technique with the classic approach based on Taylor’s hypothesis shows that – as expected – Taylor’s hypothesis holds for larger wavenumbers (small spatial scales), but there are significant differences for smaller wavenumbers (large spatial scales). In the range of Reynolds number examined in this study, double-peaked spectra in the outer region of a turbulent wall flow are thought to be the result of using Taylor’s hypothesis. This is consistent with previous studies that focused on examining the limitations of Taylor’s hypothesis (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26). The newly proposed mapping method provides a data-driven approach to map frequency spectra into wavenumber spectra from two-point measurements and will allow the experimental exploration of spatial spectra in high-Reynolds-number turbulent shear flows.

Particle capture, whereby suspended particles contact and adhere to a solid surface (a ‘collector’), is important in a range of environmental processes, including suspension feeding by corals and ‘filtering’ by aquatic vegetation. Although aquatic particles are often considered as perfect tracers when estimating capture efficiency, the particle density ratio (${\it\rho}^{+}$) – the ratio of the particle density to the fluid density – can significantly affect capture. In this paper, we use a numerical analysis of particle trajectories to quantify the influence of ${\it\rho}^{+}$ on particle capture by circular collectors in a parameter space relevant to aquatic systems. As it is generally believed that inertia augments the capture efficiency when the Stokes number ($\mathit{St}$) of the particles exceeds a critical value, we first estimate the critical Stokes number for aquatic-type particles and demonstrate its dependence on both ${\it\rho}^{+}$ and the Reynolds number ($\mathit{Re}$). Second, we analyse how efficiently circular collectors can capture neutrally buoyant (${\it\rho}^{+}=1$), sediment-type (${\it\rho}^{+}=2.6$) and weakly buoyant (${\it\rho}^{+}=0.9$) aquatic particles. Our analysis shows that, for ${\it\rho}^{+}>1$, inertia can either augment or diminish capture efficiency, and inertial effects appear well before the critical Stokes number is reached. The role of particle inertia is maximised at Stokes numbers above the critical value and, for sediment-type particles, can result in as much as a fourfold increase in the rate of capture relative to perfect tracers of the same size. Similar but opposite effects are observed for weakly buoyant particles, where capture efficiency can decrease by 60 % relative to the capture of perfect tracers.

The motion of a liquid in an open cylindrical tank rotating at a constant rate around its vertical axis of symmetry, a configuration called Newton’s bucket, is investigated using a linear stability approach. This flow is shown to be affected by several families of waves, all weakly damped by viscosity. The wave families encountered correspond to: surface waves which can be driven either by gravity or centrifugal acceleration, inertial waves due to Coriolis acceleration which are singular in the inviscid limit, and Rossby waves due to height variations of the fluid layer. These waves are described in the inviscid and viscous cases by means of mathematical considerations, global stability analysis and various asymptotic methods; and their properties are investigated over a large range of parameters $(a,Fr)$, with $a$ the aspect ratio and $Fr$ the Froude number.

The linear response of a superfluid, rotating uniformly in a cylindrical container and threaded with a large number of vortex lines, to an impulsive increase in the angular velocity of the container is investigated. At zero temperature and with perfect pinning of vortices to the top and bottom of the container, we demonstrate that the system oscillates persistently with a frequency proportional to the vortex line tension parameter to the quarter power. This low-frequency mode is generated by a secondary flow analogous to classical Ekman pumping that is periodically reversed by the vortex tension in the boundary layers. We compare analytic solutions to the two-fluid equations by Chandler & Baym (J. Low Temp. Phys., vol. 62, 1986, pp. 119–142) with the spin-up experiments by Tsakadze & Tsakadze (J. Low Temp. Phys., vol. 39, 1980, pp. 649–688) in helium II and find that the frequency agrees within a factor of four, although the experiment is not perfectly suited to the application of linear theory. We argue that this oscillatory Ekman pumping mode, and not Tkachenko modes, provides a natural explanation for the observed oscillation. In neutron stars, the oscillation period depends on the pinning interaction between neutron vortices and flux tubes in the outer core. Using a simplified pinning model, we demonstrate that strong pinning can accommodate modes with periods of days to years, which are only weakly damped by mutual friction over longer time scales.

We report on localised patches of cellular hexagons observed on the surface of a magnetic fluid in a vertical magnetic field. These patches are spontaneously generated by jumping into the neighbourhood of the unstable branch of the domain-covering hexagons of the Rosensweig instability upon which the patches equilibrate and stabilise. They are found to coexist in intervals of the applied magnetic field strength parameter around this branch. We formulate a general energy functional for the system and a corresponding Hamiltonian that provide a pattern selection principle allowing us to compute Maxwell points (where the energy of a single hexagon cell lies in the same Hamiltonian level set as the flat state) for general magnetic permeabilities. Using numerical continuation techniques, we investigate the existence of localised hexagons in the Young–Laplace equation coupled to the Maxwell equations. We find that cellular hexagons possess a Maxwell point, providing an energetic explanation for the multitude of measured hexagon patches. Furthermore, it is found that planar hexagon fronts and hexagon patches undergo homoclinic snaking, corroborating the experimentally detected intervals. Besides making a contribution to the specific area of ferrofluids, our work paves the ground for a deeper understanding of homoclinic snaking of two-dimensional localised patches of cellular patterns in many physical systems.

Blindfolded harbour seals are able to use their uniquely shaped whiskers to track vortex wakes left by moving animals and identify objects that passed by 30 s earlier, an impressive feat, as the flow features have velocities as low as $1~\text{mm}~\text{s}^{-1}$. The seals sense while swimming, hence their whiskers are sensitive enough to detect small-scale changes in the flow, while rejecting self-generated flow noise. Here we identify and illustrate a novel flow mechanism, causing a large-amplitude ‘slaloming’ whisker response, which allows artificial whiskers with the identical unique undulatory geometry as those of the harbour seal to detect the features of minute flow fluctuations when placed within wakes. Whereas in open water the whisker responds with very low-amplitude vibration, once within a wake, it oscillates with large amplitude and, importantly, its response frequency coincides with the Strouhal frequency of the upstream cylinder, making the detection of an upstream wake and an estimation of the size and shape of the wake-generating body possible. This mechanism has some similarities with the flow mechanisms observed in actively controlled propulsive foils within upstream wakes and trout swimming behind bluff cylinders in a stream, but there are also differences caused by the unique whisker morphology, which enables it to respond passively and within a much wider parametric range.

Aerodynamic performance and agility during flapping flight are determined by the combination of wing shape and kinematics. The degree of morphological and kinematic optimization is unknown and depends upon a large parameter space. Aimed at providing an accurate and computationally inexpensive modelling tool for flapping-wing aerodynamics, we propose a novel CFD (computational fluid dynamics)-informed quasi-steady model (CIQSM), which assumes that the aerodynamic forces on a flapping wing can be decomposed into quasi-steady forces and parameterized based on CFD results. Using least-squares fitting, we determine a set of proportional coefficients for the quasi-steady model relating wing kinematics to instantaneous aerodynamic force and torque; we calculate power as the product of quasi-steady torques and angular velocity. With the quasi-steady model fully and independently parameterized on the basis of high-fidelity CFD modelling, it is capable of predicting flapping-wing aerodynamic forces and power more accurately than the conventional blade element model (BEM) does. The improvement can be attributed to, for instance, taking into account the effects of the induced downwash and the wing tip vortex on the force generation and power consumption. Our model is validated by comparing the aerodynamics of a CFD model and the present quasi-steady model using the example case of a hovering hawkmoth. This demonstrates that the CIQSM outperforms the conventional BEM while remaining computationally cheap, and hence can be an effective tool for revealing the mechanisms of optimization and control of kinematics and morphology in flapping-wing flight for both bio-flyers and unmanned aerial systems.

The present work reports some new aspects of non-axisymmetric gravity currents obtained from laboratory experiments, fully resolved simulations and box models. Following the earlier work of Zgheib et al. (Theor. Comput. Fluid Dyn., vol. 28, 2014, pp. 521–529) which demonstrated that gravity currents initiating from non-axisymmetric cross-sectional geometries do not become axisymmetric, nor do they retain their initial shape during the slumping and inertial phases of spreading, we show that such non-axisymmetric currents eventually reach a self-similar regime during which (i) the local front propagation scales as $t^{1/2}$ as in circular releases and (ii) the non-axisymmetric front has a self-similar shape that primarily depends on the aspect ratio of the initial release. Complementary experiments of non-Boussinesq currents and top-spreading currents suggest that this self-similar dynamics is independent of the density ratio, vertical aspect ratio, wall friction and Reynolds number $\mathit{Re}$, provided the last is large, i.e. $\mathit{Re}\geqslant O(10^{4})$. The local instantaneous front Froude number obtained from the fully resolved simulations is compared to existing models of Froude functions. The recently reported extended box model is capable of capturing the dynamics of such non-axisymmetric flows. Here we use the extended box model to propose a relation for the self-similar horizontal aspect ratio ${\it\chi}_{\infty }$ of the propagating front as a function of the initial horizontal aspect ratio ${\it\chi}_{0}$, namely ${\it\chi}_{\infty }=1+(\ln {\it\chi}_{0})/3$. The experimental and numerical results are in good agreement with the proposed relation.

In this paper we study the spatial evolution of zero-pressure-gradient (ZPG) turbulent boundary layers from their origin to a canonical high-Reynolds-number state. A prime motivation is to better understand under what conditions reliable scaling behaviour comparisons can be made between different experimental studies at matched local Reynolds numbers. This is achieved here through detailed streamwise velocity measurements using hot wires in the large University of Melbourne wind tunnel. By keeping the unit Reynolds number constant, the flow conditioning, contraction and trip can be considered unaltered for a given boundary layer’s development and hence its evolution can be studied in isolation from the influence of inflow conditions by moving to different streamwise locations. Careful attention was given to the experimental design in order to make comparisons between flows with three different trips while keeping all other parameters nominally constant, including keeping the measurement sensor size nominally fixed in viscous wall units. The three trips consist of a standard trip and two deliberately ‘over-tripped’ cases, where the initial boundary layers are over-stimulated with additional large-scale energy. Comparisons of the mean flow, normal Reynolds stress, spectra and higher-order turbulence statistics reveal that the effects of the trip are seen to be significant, with the remnants of the ‘over-tripped’ conditions persisting at least until streamwise stations corresponding to $Re_{x}=1.7\times 10^{7}$ and $x=O(2000)$ trip heights are reached (which is specific to the trips used here), at which position the non-canonical boundary layers exhibit a weak memory of their initial conditions at the largest scales $O(10{\it\delta})$, where ${\it\delta}$ is the boundary layer thickness. At closer streamwise stations, no one-to-one correspondence is observed between the local Reynolds numbers ($Re_{{\it\tau}}$, $Re_{{\it\theta}}$ or $Re_{x}$ etc.), and these differences are likely to be the cause of disparities between previous studies where a given Reynolds number is matched but without account of the trip conditions and the actual evolution of the boundary layer. In previous literature such variations have commonly been referred to as low-Reynolds-number effects, while here we show that it is more likely that these differences are due to an evolution effect resulting from the initial conditions set up by the trip and/or the initial inflow conditions. Generally, the mean velocity profiles were found to approach a constant wake parameter ${\it\Pi}$ as the three boundary layers developed along the test section, and agreement of the mean flow parameters was found to coincide with the location where other statistics also converged, including higher-order moments up to tenth order. This result therefore implies that it may be sufficient to document the mean flow parameters alone in order to ascertain whether the ZPG flow, as described by the streamwise velocity statistics, has reached a canonical state, and a computational approach is outlined to do this. The computational scheme is shown to agree well with available experimental data.

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body force. This system admits vertically invariant solutions that satisfy the 2D Navier–Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations. We first consider the 3D rotating Navier–Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value $Ro_{c}(Re)$ of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on $Ro_{c}(Re)$ and therefore determine regions of the parameter space $(Re,Ro)$ where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier–Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number $Ro^{(f)}$ is lower than a Grashof-number-dependent threshold value $Ro_{c}^{(f)}(Gr)$. We then consider the fully nonlinear 3D rotating Navier–Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value $Ro_{\mathit{abs}}^{(f)}(Gr)$ of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude). These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number $Re$ and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone–anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.

Superhydrophobic surfaces can entrap gas pockets within their grooves when submerged in water. Such a mixed-phase boundary is shown to result in an effective slip velocity on the surface, and has promising potential for drag reduction and energy-saving in hydrodynamic applications. The target flow regime, in most such applications, is a turbulent flow. Previous analyses of this problem involved direct numerical simulations of turbulence with the superhydrophobic surface modelled as a flat boundary, but with a heterogeneous mix of slip and no-slip boundary conditions corresponding to the surface texture. Analysis of the kinematic data from these simulations has helped to establish the magnitude of drag reduction for various texture topologies. The present work is the first investigation that, alongside a kinematic investigation, addresses the robustness of superhydrophobic surfaces by studying the load fields obtain from data from direct numerical simulations (DNS). The key questions at the focus of this work are: does a superhydrophobic surface induce a different pressure field compared to a flat surface? If so, how does this difference scale with system parameters, and when does it become significant that it can deform the air–water interface and potentially rapture the entrapped gas pockets? To this end, we have performed DNS of turbulent channel flows subject to superhydrophobic surfaces over a wide range of texture sizes spanning values from $L^{+}=6$ to $L^{+}=155$ when expressed in terms of viscous units. The pressure statistics at the wall are decomposed into two contributions: one coherent, caused by the stagnation of slipping flow hitting solid posts, and one time-dependent, caused by overlying turbulence. The results show that the larger texture size intensifies the contribution of stagnation pressure, while the contribution from turbulence is essentially insensitive to $L^{+}$. The two-dimensional stagnation pressure distribution at the wall and the pressure statistics in the wall-normal direction are found to be self-similar for different $L^{+}$. The scaling of the induced pressure and the consequent deformations of the air–water interface are analysed. Based on our results, an upper bound on the texture wavelength is quantified that limits the range of robust operation of superhydrophobic surfaces when exposed to high-speed flows. Our results indicate that when the system parameters are expressed in terms of viscous units, the main parameters controlling the problem are $L^{+}$ and a Weber number based on inner dimensions; We obtain good collapse when all our results are expressed in wall units, independently of the Reynolds number.

A more poetic long title could be ‘A voyage from the shifting grounds of existing data on zero-pressure-gradient (abbreviated ZPG) turbulent boundary layers (abbreviated TBLs) to infinite Reynolds number’. Aided by the requirement of consistency with the Reynolds-averaged momentum equation, the ‘shifting grounds’ are sufficiently consolidated to allow some firm conclusions on the asymptotic expansion of the streamwise normal stress $\langle uu\rangle ^{+}$, where the $^{+}$ indicates normalization with the friction velocity $u_{{\it\tau}}$ squared. A detailed analysis of direct numerical simulation data very close to the wall reveals that its inner near-wall asymptotic expansion must be of the form $f_{0}(y^{+})-f_{1}(y^{+})/U_{\infty }^{+}+\mathit{O}(U_{\infty }^{+})^{-2}$, where $U_{\infty }^{+}=U_{\infty }/u_{{\it\tau}}$, $y^{+}=yu_{{\it\tau}}/{\it\nu}$ and $f_{0}$, $f_{1}$ are $\mathit{O}(1)$ functions fitted to data in this paper. This means, in particular, that the inner peak of $\langle uu\rangle ^{+}$ does not increase indefinitely as the logarithm of the Reynolds number but reaches a finite limit. The outer expansion of $\langle uu\rangle ^{+}$, on the other hand, is constructed by fitting a large number of data from various sources. This exercise, aided by estimates of turbulence production and dissipation, reveals that the overlap region between inner and outer expansions of $\langle uu\rangle ^{+}$ is its plateau or second maximum, extending to $y_{\mathit{break}}^{+}=\mathit{O}(U_{\infty }^{+})$, where the outer logarithmic decrease towards the boundary layer edge starts. The common part of the two expansions of $\langle uu\rangle ^{+}$, i.e. the height of the plateau or second maximum, is of the form $\,A_{\infty }-B_{\infty }/U_{\infty }^{+}+\cdots \,$with $A_{\infty }$ and $B_{\infty }$ constant. As a consequence, the logarithmic slope of the outer $\langle uu\rangle ^{+}$ cannot be independent of the Reynolds number as suggested by ‘attached eddy’ models but must slowly decrease as $(1/U_{\infty }^{+})$. A speculative explanation is proposed for the puzzling finding that the overlap region of $\langle uu\rangle ^{+}$ is centred near the lower edge of the mean velocity overlap, itself centred at $y^{+}=\mathit{O}(\mathit{Re}_{{\it\delta}_{\ast }}^{1/2})$ with $\mathit{Re}_{{\it\delta}_{\ast }}$ the Reynolds number based on free stream velocity and displacement thickness. Finally, similarities and differences between $\langle uu\rangle ^{+}$ in ZPG TBLs and in pipe flow are briefly discussed.

We investigate the cavity formation during the impact of spheres and cylinders into a liquid pool by using a combination of experiments, simulations and theoretical analysis, with particular interest in contact-line pinning and its relation with the subsequent cavity evolution. The flows are simulated by a Navier–Stokes diffuse-interface solver that allows for moving contact lines. On the basis of agreement on experimentally measured quantities such as the position of the pinned contact line and the interface shape, we investigate flow details that are not accessible experimentally, identify the interface regions in the cavity formation and examine the geometric effects of impact objects. We connect wettability, inertia, geometry of the impact object, interface bending and contact-line position with the contact-line pinning by analysing the force balance at a pinned meniscus, and the result compares favourably with those from simulations and experiments. In addition to adjusting the interface bending, the object geometry also has a significant effect on the magnitude of low pressure in the liquid and the occurrence of flow separation. As a result, it is easier for an object with sharp edges to generate a cavity than a smooth object. A theoretical model based on the Rayleigh–Besant equation is developed to provide a quantitative description of the radial expansion of the cavity after the pinning of the contact line. The accuracy of the solution is greatly affected by the geometrical information on the interface connected to the pinned meniscus, showing the dependence of the global cavity dynamics on the local flows around the pinned contact line. Vertical ripple propagation on the cavity wall is found to follow the dispersion relation for the perturbation evolution on a hollow jet.

We explore the feasibility of a novel method for the regulation of heat transfer across a cavity, by using a controllable yield stress in order to suppress the convective heat transfer. Practically, this type of control can be actuated with electro-rheological or magneto-rheological fluids. We demonstrate that above a given critical yield stress value only static steady regimes are possible, i.e. a purely conductive unyielded fluid fills the cavity. We show that this limit is governed by a balance of yield stress and buoyancy stresses, here described by $B$. With proper formulation the critical state can be described as a function of the domain geometry, and is independent of other dimensionless flow parameters (Rayleigh number, $\mathit{Ra}$, and Prandtl number, $\mathit{Pr}$). On the theoretical side, we examine the conditional stability of the static regime. We derive conservative conditions on disturbance energy to ensure that perturbations from a static regime decay to zero. Assuming stability, we show that the kinetic energy of the perturbed field decays to zero in a finite time, and give estimates for the stopping time, $t_{0}$. This allows us to predict the response of the system in suppressing advective heat transfer. The unconditional stability is also considered for the first time, illustrating the role of yield stress. We focus on the hydrodynamic characteristics of Bingham fluids in transition between conductive and convective limits. We use computational simulations to resolve the Navier–Stokes and energy equations for different yield stresses, and for different imposed controls. We show that depending on the initial conditions, a yield stress less than the critical value can result in temporary arrest of the flow. The temperature then develops conductively until the fluid yields and the flow restarts. We provide estimates of the hydrodynamic timescales of the problem and examples of flow transitions. In total, the theoretical and computational results establish that this methodology is feasible as a control, at least from a hydrodynamic perspective.