The large eddy simulation (LES) technique will soon be 50 years old. Since Deardorff’s first papers in 1970 introducing this approach, major advances in the theory of LES and its computational implementation have been made and widely adopted. However, in terms of validation, LES studies continue to largely focus on the first- and second-order statistics, which in fact are the same tests that Deardorff conducted 45 years ago. Further advances in LES and wider adoption for new flows require advanced and more challenging tests to be developed and documented to serve as benchmarks. The paper by Stevens, Wilczek & Meneveau (J. Fluid Mech., 2014, vol. 757, pp. 888–907) does precisely that. The authors demonstrate the ability of LES to capture the recently established log-law of streamwise velocity variance and the related log-laws for even-order statistics up to order 10, as well as the departure of these statistics from a Gaussian distribution. The paper also provides key insights into the role of grid resolution on the computed turbulence field.

The flow around a tethered cylinder subjected to an incoming flow transverse to its main axis and confined in a channel is investigated by means of global stability analysis of the full coupled body–fluid problem. When the cylinder is strongly confined (ratio of cylinder diameter to cell height, $D/H=0.66$) we retrieve the confinement-induced instability (CIV) discovered by Semin et al. (J. Fluid Mech., vol. 690, 2011, pp. 345–365), which sets in at a Reynolds number below the vortex-induced vibration threshold. For a moderately confined case ($D/H=0.3$), a new steady static instability is discovered, referred to as confinement-induced divergence (CID). This instability saturates into an asymmetric steady solution through a supercritical pitchfork bifurcation. In addition, the CIV and CID instabilities are studied via a reduced model obtained by considering a quasi-static response of the fluid, allowing for tracing back the physical mechanisms responsible for the instabilities.

A new hybrid model for the dynamics of glaciers, ice sheets and ice shelves is introduced. In this ‘multilayer’ model the domain of ice consists of a pile of thin layers, which can spread out, contract and slide over each other such that the two most relevant types of stresses are accounted for: membrane and vertical shear. Assuming the horizontal velocity field to be vertically piecewise constant in each layer, the model is derived from local depth integrations of the hydrostatic approximation of the Stokes equations. These integrations give rise to interlayer tractions, which can be redefined at zeroth order in the interlayer surface slope by keeping the vertical shear stress components. Furthermore, if the layers are chosen such that they are aligned with the streamlines, then second-order accurate interlayer tractions can replace zeroth-order ones. The final model consists of a tridiagonal system of two-dimensional nonlinear elliptic equations, the size of this system being equal to the number of layers. When running the model for prognostic flowline ISMIP-HOM benchmark experiments, the multilayer solutions show good agreement with the higher-order solutions if no severe depression occurs in the bedrock. As an alternative to three-dimensional models, the multilayer approach offers to glacier and ice sheet modellers a way of upgrading the commonly used shallow shelf approximation model into a mechanically complete but mathematically two-dimensional model.

The stirring of a passive scalar by grid-generated turbulence in the presence of a mean scalar gradient is studied by direct numerical simulations (DNS) for six different grids: one fractal square grid with three fractal iterations, one fractal square grid with four fractal iterations, one fractal I grid and three different regular grids. Our results can be summarised as follows. (i) For all these grids, the turbulence intensity averaged over time and over a plane parallel to the grid takes its peak value when the streamwise position of this plane is between $0.75M_{eff}$ and $1.5M_{eff}$ where $M_{eff}$ is the effective mesh size introduced by Hurst & Vassilicos (Phys. Fluids, vol. 19, 2007, 035103). (ii) Downstream of the location of this peak, the turbulence intensity averaged in this way is greatly enhanced by the fractal grids relative to the regular grids even though the fractal grids have comparable or even lower blockage ratios. The novelty of this result lies in the fact that it concerns turbulence intensities averaged over lateral planes (as well as time). (iii) The pressure drop is about the same across grids of the same blockage ratio whether fractal or not, but the pressure recovery is longer for the fractal grids. (iv) Even so, the fractal grids enhance turbulent scalar fluxes by up to an order of magnitude in the region downstream of the aforementioned peak and they also greatly enhance the streamwise growth of the fluctuating scalar variance in that region. (v) We demonstrate on a simple planar model problem that the cause of this phenomenon lies in the fractality of the grids. (vi) The turbulence scalar flux coefficient is constant far enough downstream of all the present grids and is significantly dependent on the nature and details of the turbulence-generating grid.

Laminar helically-symmetric gravity-driven thin-film flow down a helically-wound channel of rectangular cross-section is considered. We extend the work of Stokes et al. (Phys. Fluids, vol. 25 (8), 2013, 083103) and Lee et al. (Phys. Fluids, vol. 26 (4), 2014, 043302) to channels with arbitrary curvature and torsion or, equivalently, arbitrary curvature and slope. We use a non-orthogonal coordinate system and, remarkably, find an exact steady-state solution. We find that the free-surface shape and flow have a complicated dependence on the curvature, slope and flux down the channel. Moderate to large channel slope has a significant effect on the flow in the region of the channel near the inside wall, particularly when the curvature of the channel is large. This work has application to flow in static spiral particle separators used in mineral processing.

Air-assisted primary atomization is investigated in a configuration where liquid is injected in a turbulent gaseous jet flow both within as well as outside of the potential core. Cases are studied where the injection point is moved within the flow to maintain a range of constant gaseous mean velocities but changing local fluctuating velocity root-mean-square (r.m.s.) levels. Over a range of mean conditions, this allows for a systematic understanding of both the effects of gas-phase turbulence and mean shear on primary break-up independently. Extensive data is obtained and analysed from laser Doppler anemometry/phase Doppler anemometry, high-speed microscopic backlit imaging and advanced image processing. It is found that the ratio of the turbulent Weber number $\mathit{We}^{\prime }$ to the mean Weber number $\mathit{We}$ is a relevant parameter as is the turbulence intensity. The primary break-up length is found to be heavily influenced not only by the mean velocity, but also by the turbulence level and the mass fuel to air ratio. Above a particular threshold intensity level the break-up time changes in proportion to the change in the integral time scale of the flow. In addition, it is found that regardless of diameter and turbulent flow conditions at the liquid jet, the final size of ligaments converges to a value which is of the order of the measured primary instability wavelength (${\it\lambda}_{1}$). In contrast, cases of different turbulence intensity show the mean of droplet sizes diverging as the spray is advected downstream and this is because droplets are generated from ligaments, the latter of which are subjected both to Rayleigh–Taylor instabilities and turbulent fluctuations. This contribution, for the first time, examines the theoretical applicability of the Rayleigh–Taylor instability in flows where the turbulence is substantial with respect to the mean flow. It is shown that for high turbulence intensities a full theoretical reconstruction of the measured final droplet size distribution is possible from a probability density function of model Rayleigh–Taylor wavelengths (${\it\lambda}_{RT}$). In agreement with the literature (Varga et al. J. Fluid Mech., vol. 497, 2003, pp. 405–434), mean droplet sizes are found to be equal to a mean theoretical Rayleigh–Taylor wavelength normalized by a particular constant value. This, however, is only true for local turbulence intensities less than ${\sim}25\,\%$, or for ratios of the turbulent Weber number to mean Weber number ($\mathit{We}^{\prime }/\mathit{We}$) of less than ${\sim}6\,\%$. Above this, the normalization value is no longer constant, but increases with $\mathit{We}^{\prime }/\mathit{We}$. Finally, the instability wavelengths can be used as part of an approximation that estimates the total number of objects formed after break-up, where the object number is found to be dictated by a balance of both mean flow conditions and local turbulence.

A general expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in linear shear flow between two parallel walls is derived in terms of the induced stresslets on particles. This formula is applied to $N$-bead rods and to prolate spheroids with the same length, aspect ratio and volume. The effective viscosity of non-Brownian particles in a periodic shear flow is considered here. The oscillating frequency is high enough for the particle orientation and centre-of-mass distribution to be practically frozen, yet small enough for the flow to be quasi-steady. It is known that for spheres, the intrinsic viscosity $[{\it\mu}]$ increases monotonically when the distance $H$ between the walls is decreased. The dependence is more complex for both types of elongated particles. Three regimes are theoretically predicted here: (i) a ‘weakly confined’ regime (for $H>l$, where $l$ is the particle length), where $[{\it\mu}]$ is slightly larger for smaller $H$; (ii) a ‘semi-confined’ regime, when $H$ becomes smaller than $l$, where $[{\it\mu}]$ rapidly decreases since the geometric constraints eliminate particle orientations corresponding to the largest stresslets; (iii) a ‘strongly confined’ regime when $H$ becomes smaller than 2–3 particle widths $d$, where $[{\it\mu}]$ rapidly increases owing to the strong hydrodynamic coupling with the walls. In addition, for sufficiently slender particles (with aspect ratio larger than 5–6) there is a domain of narrow gaps for which the intrinsic viscosity is smaller than that in unbounded fluid.

Flows produced by a circular cylinder undergoing oscillatory rotation and translation in a quiescent fluid have been studied via direct numerical simulations. The incompressible Navier–Stokes equations were solved for large dimensionless time windows using an immersed boundary method with adaptive Cartesian grid refinement. Parametric studies were conducted in two dimensions on the Reynolds number, Keulegan–Carpenter number and phase shift. In addition to the previously reported net thrust case (Blackburn et al., Phys. Fluids, vol. 11, 1999, pp. 4–6), the study catalogued the appearance of several streaming jet regimes with varying deflection angles, deflected and horizontal vortex shedding regimes, and a double mirrored jet regime with varying inter-jet angles, as well as several chaotic cases. Visualizations are presented to clarify each observed flow regime and to illustrate the parameter space. Connections are drawn between these canonical bluff-body deflected wakes and a similar phenomenon observed in aerofoils oscillating at high reduced frequencies in a cross-flow. Also, the discovery of the streaming jet regimes with varying deflection angles opens the door for using these flows as a low-Reynolds-number propulsive mechanism requiring only a two-degree-of-freedom actuator. Simulation results suggest that the flow phenomena observed in two dimensions persist in three dimensions, despite spanwise fluctuations.

We explore how the rheology of dense granular flows is affected by the presence of sidewalls. The study is based on discrete element method simulations of plane-shear flows between two rough walls, prescribing both the normal stress and the shear rate. Results confirm previous observations for different systems: large layers near the walls develop where the local viscosity is not constant, but decreases when approaching the walls. The size of these layers can reach several dozen grain diameters, and is found to increase when the flow decelerates, as a power law of the inertial number. Two non-local models are found to adequately explain such features, namely the kinetic elasto-plastic fluidity (KEP) model and the eddy viscosity model (EV). The analysis of the internal kinematics further shows that the vorticity and its associated length scale may be a key component of these non-local behaviours.

Quantitative results for the linear stability of planar Stokes layers subject to small, high-frequency perturbations are obtained for both a narrow channel and a flow approximating the classical semi-infinite Stokes layer. Previous theoretical and experimental predictions of the critical Reynolds number for the classical flat Stokes layer have differed widely with the former exceeding the latter by a factor of two or three. Here it is demonstrated that only a 1 % perturbation, at an appropriate frequency, to the nominal sinusoidal wall motion is enough to result in a reduction of the theoretical critical Reynolds number of as much as 60 %, bringing the theoretical conditions much more in line with the experimentally reported values. Furthermore, within the various experimental observations there is a wide variation in reported critical conditions and the results presented here may provide a new explanation for this behaviour.

This study presents a detailed scaling analysis quantifying the transient behaviour of natural convection in a reservoir model induced by iso-flux surface heating. It is found that horizontal conduction, which has often been neglected in previous analyses, plays an important role in the development of the flow. Depending on the Rayleigh number, three possible pathways through which the flow develops towards the final steady state are identified. A thermal boundary layer initially grows downwards from the surface. When the thermal boundary layer reaches the sloping bottom and becomes indistinct, a horizontal temperature gradient establishes due to the increasing water depth in the offshore direction. A flow is then driven towards the offshore direction by a buoyancy-induced horizontal pressure gradient, which convects away the heat input from the water surface. On the other hand, the horizontal temperature gradient also conducts heat away. The flow behaviour is determined by the interaction between the horizontal conduction and convection. An interesting flow feature revealed by the present scaling analysis is that the region across which the thermal boundary layer encompasses the full water depth shrinks over time at a certain stage of the flow development. The shrinking process eventually stops when this region coincides with a conduction-dominated subregion. The present scaling results are verified by corresponding numerical simulations.

This work addresses the transition from 2D steady to 2D unsteady laminar flow for a fully developed regime in a symmetric wavy channel geometry. We investigate the existence and characteristics of the spatio-temporal structure of the fully developed unsteady laminar flow for those particular geometries for which the steady flow presents a periodic variation of the main stream velocity component. We perform a 2D global linear stability analysis of the fully developed steady laminar flow, and we show that, for all the geometries studied, the transition is triggered by a Hopf bifurcation associated with the breaking of the symmetries and the invariance of the steady flow. Critical Reynolds numbers, most unstable modes and their characteristics are presented for large ranges of the geometric parameters, namely wavenumber ${\it\alpha}$ from 0.3 to 5 and amplitude from 0 (straight channel) to 0.5. We show that it is possible to define geometries for which the wavenumber is proportional to the most unstable mode wavenumber for the critical Reynolds number. From this modal study we address a weakly nonlinear stability analysis with a view to obtaining the Landau coefficient $g$, and then the sub- or supercritical nature of the first bifurcation characterising the transition. We show that a critical geometric amplitude beyond which the first bifurcation is supercritical is associated with each geometric wavenumber.

We examine two-dimensional flows of a viscous liquid on an inclined plate. If the upstream depth $h_{-}$ of the liquid is larger than its downstream depth $h_{+}$, a smooth hydraulic jump (bore) forms and starts propagating down the slope. If the inclination angle of the plate is small, the bore can be described by the so-called lubrication theory. In this work we demonstrate that bores with $h_{+}/h_{-}<(\sqrt{3}-1)/2$ either are unstable or do not exist as steady solutions of the governing equation (physically, these two possibilities are difficult to distinguish). The instability/evolution occurs near a stagnation point and, generally, causes overturning – sometimes on the scale of the whole bore, sometimes on a shorter, local scale. The overturning occurs because the flow advects disturbances towards the stagnation point and, thus, ‘compresses’ them, increasing the slope of the free surface. Interestingly, this effect is not captured by the lubrication theory, which formally yields a well-behaved stable solution for all values of $h_{+}/h_{-}$.

Motivated by the process of $\text{CO}_{2}$ convective mixing in porous media, here we study the formation of rock-dissolution patterns that arise from geochemical reactions during Rayleigh–Bénard–Darcy convection. Under the assumption of instantaneous chemical equilibrium, we adopt a formulation of the local reaction rate as a function of scalar dissipation rate, a measure that depends solely on flow and transport, and chemical speciation, which is a measure that depends only on the equilibrium thermodynamics of the chemical system. We use high-resolution simulations to examine the interplay between the density-driven hydrodynamic instability and the rock dissolution reactions, and analyse the impact of geochemical reactions on the macroscopic mass exchange rate. We find that dissolution of carbonate rock initiates in regions of locally high mixing, but that the geochemical reaction shuts down significantly earlier than shutdown of convective mixing. This early shutdown feature reflects the important role that chemical speciation plays in this hydrodynamics–reaction coupled process. Finally, we extend our analysis to three dimensions and explore the morphology of dissolution patterns in three dimensions.

Linear stability of magnetoconvection of a ferromagnetic fluid contained between two infinite differentially heated non-magnetic plates in the presence of an oblique uniform external magnetic field is studied in zero gravity conditions. The thermomagnetic convection that arises is caused by the spatial variation of magnetisation occurring due to its dependence on the temperature. The critical values of the governing parameters at which the transition between motionless and convective states is observed are determined for various field inclination angles and for fluid magnetic parameters that are consistently chosen from a realistic experimental range. It is shown that, similar to natural paramagnetic fluids, the most prominent convection patterns align with the in-layer component of the applied magnetic field but in contrast to such paramagnetic fluids the instability patterns detected in ferrofluids can be oscillatory. It is also found that, contrary to paramagnetic fluids, the stability characteristics of magnetoconvection in ferrofluids depend on the magnitude of the applied field which becomes an additional parameter of the problem. This is shown to be due to the nonlinearity of the magnetic field distribution within the ferrofluid.

Results from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$–$1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl–Blasius–Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$-power of a wind-based Reynolds number, where the ‘wind’ of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the ‘bulk’ or ‘background’ dissipation of the GL theory, is found to obey the Kolmogorov–Obukhov–Corrsin scaling for fully developed turbulence. In contrast to Rayleigh–Bénard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

Direct numerical simulation (DNS) of an impinging jet flow with a nozzle-to-plate distance of two jet diameters and a Reynolds number of 10 000 is carried out at high spatial resolution using high-order numerical methods. The flow configuration is designed to enable the development of a fully turbulent regime with the appearance of a well-marked secondary maximum in the radial distribution of the mean heat transfer. The velocity and temperature statistics are validated with documented experiments. The DNS database is then analysed focusing on the role of unsteady processes to explain the spatial distribution of the heat transfer coefficient at the wall. A phenomenological scenario is proposed on the basis of instantaneous flow visualisations in order to explain the non-monotonic radial evolution of the Nusselt number in the stagnation region. This scenario is then assessed by analysing the wall temperature and the wall shear stress distributions and also through the use of conditional averaging of velocity and temperature fields. On one hand, the heat transfer is primarily driven by the large-scale toroidal primary and secondary vortices emitted periodically. On the other hand, these vortices are subjected to azimuthal distortions associated with the production of radially elongated structures at small scale. These distortions are responsible for the appearance of very high heat transfer zones organised as cold fluid spots on the heated wall. These cold spots are shaped by the radial structures through a filament propagation of the heat transfer. The analysis of probability density functions shows that these strong events are highly intermittent in time and space while contributing essentially to the secondary peak observed in the radial evolution of the Nusselt number.

It has previously been shown that the transient flow in a channel following a step increase of Reynolds number from 2800 to 7400 (based on channel half-height and bulk velocity) is effectively a laminar–turbulent bypass transition even though the initial flow is turbulent (He & Seddighi, J. Fluid Mech., vol. 715, 2013, pp. 60–102). In this paper, it is shown that the transient flow structures exhibit strong contrasting characteristics in large and small flow perturbation scenarios. When the increase of Reynolds number is large, the flow is characterized by strong elongated streaks during the initial period, followed by the occurrence and spreading of isolated turbulent spots, as shown before. By contrast, the flow appears to evolve progressively and the turbulence regeneration process remains largely unchanged during the flow transient when the Reynolds number ratio is low, and streaks do not appear to play a significant role. Despite the major apparent differences in flow structures, the transient flow under all conditions considered is unambiguously characterized by laminar–turbulent transition, which exhibits itself clearly in various flow statistics. During the pre-transition period, the time-developing boundary layers in all the cases show a strong similarity to each other and follow closely the Stokes solution for a transient laminar boundary layer. The streamwise fluctuating velocity also shows good similarity in the various cases, irrespective of the appearance of elongated streaks or not, and the maximum energy growth exhibits a linear rate similar to that in a spatially developing boundary layer. The onset of transition is clearly definable in all cases using the minimum friction factor, and the critical time thus defined is strongly correlated with the free-stream turbulence in a power-law form.

The collision of a liquid drop against a small target results in the formation of a thin liquid sheet that extends radially until it reaches a maximum diameter. The subsequent retraction is due to the air–liquid surface tension. We have used a time- and space-resolved technique to measure the thickness field of this class of liquid sheet, based on the grey-level measurement of the image of a dyed liquid sheet recorded using a high-speed camera. This method enables a precise measurement of the thickness in the range $10{-}450~{\rm\mu}\text{m}$, with a temporal resolution equal to that of the camera. We have measured the evolution with time since impact, $t$, and radial position, $r$, of the thickness, $h(r,t)$, for various drop volumes and impact velocities. Two asymptotic regimes for the expansion of the sheet are evidenced. The scalings of the thickness with $t$ and $r$ measured in the two regimes are those that were predicted by Rozhkov et al. (Proc. R. Soc. Lond. A, vol. 460, 2004, pp. 2681–2704) for the short-time regime and Villermaux and Bossa (J. Fluid Mech., vol. 668, 2011, pp. 412–435) for the long-time regime, but never experimentally measured before. Interestingly, our experimental data also provide evidence for the existence of a maximum of the film thickness $h_{max}(r)$ at a radial position $r_{h_{max}}(t)$ corresponding to the cross-over of these two asymptotic regimes. The maximum moves with a constant velocity of the order of the drop impact velocity, as expected theoretically. Thanks to our visualization technique, we also provide evidence of an azimuthal thickness modulation of the liquid sheets.

For contact line motion where the full Stokes flow equations hold, full matched asymptotic solutions using slip models have been obtained for droplet spreading and more general geometries. These solutions to the singular perturbation problem in the slip length, however, all involve matching through an intermediate region that is taken to be separate from the outer–inner regions. Here, we show that the intermediate region is in fact an overlap region representing extensions of both the outer and the inner region, allowing direct matching to proceed. In particular, we investigate in detail how a previously seen result of the matching of the cubes of the free surface slope is justified in the lubrication setting. We also extend this two-region direct matching to the more general Stokes flow case, offering a new perspective on the asymptotics of the moving contact line problem.