Based on hierarchies of filter lengths, the large eddy decomposition and the related subgrid stresses are recognized to represent generalized central moments for the study and modelling of the different modes composing turbulence. In particular, the subgrid stresses and the subgrid dissipation are shown to be alternative observables for quantitatively assessing the scale-dependent properties of momentum flux (subgrid stresses) and the energy exchange between the large and small scales (subgrid dissipation). In this work we present a theoretical framework for the study of the subgrid stress and dissipation. Starting from an alternative decomposition of the turbulent stresses, a new formalism for their approximation and understanding is proposed which is based on a tensorial turbulent viscosity. The derived formalism highlights that every decomposition of the turbulent stresses is naturally approximated by a general form of turbulent viscosity tensor based on velocity increments which is then recognized to be a peculiar property of small-scale stresses in turbulence. The analysis in a turbulent channel shows the rich physics of the small-scale stresses which is unveiled by the tensorial formalism and usually missed in scalar approaches. To further exploit the formalism, we also show how it can be used to derive new modelling approaches. The proposed models are based on the second- and third-order inertial properties of the grid element. The basic idea is that the structure of the integration volume for filtering (either implicit or explicit) impacts the anisotropy and inhomogeneity of the filtered-out motions and, hence, this information could be leveraged to improve the prediction of the main unknown features of small-scale turbulence. The formalism provides also a rigorous definition of characteristic lengths for the turbulent stresses, which can be computed in every type of computational elements, thus overcoming the rather elusive definition of filter length commonly employed in more classical models. A preliminary analysis in a turbulent channel shows reasonable results. In order to solve numerical stability issues, a tensorial dynamic procedure for the evolution of the model constants is also developed. The generality of the procedure is such that it can be employed also in more conventional closures.

The flow of liquids through paper is challenging to model due to the complexity and disordered layout of the fibre matrix. The expanding use and capability of microfluidic paper-based analytical devices ($\unicode[STIX]{x1D707}$PADs) and their requirement for precision has increased the need to accurately predict the flow of liquids through paper. Many studies have developed models and revealed some of the physical mechanisms responsible for the flow behaviour, but we still lack a complete understanding, particularly in relation to how the fluid fills the various voids with a wide range of shapes and sizes in the fibre matrix of paper. In the featured article, Chang et al. (J. Fluid Mech., vol. 845, 2018, 36–50) used a combined experimental and theoretical approach to uncover the importance of the liquid filling the intra-fibre pores, showed that this results in deviation from the flow behaviour predicted by the Lucas–Washburn equation and developed a model which accounts for this effect.

The development of the flow field of a jet emanating from a point source of momentum in an infinite incompressible fluid of density σ is considered. The flow field is assumed to be due to the application of a constant force F0 at the origin. The problem is formulated in terms of the dimensionless variable λ = (vt)½/r, where v is the kinematic viscosity of the fluid, t the time from the application of the force and r the distance from the origin. At a station r the flow field is dipolar, with the dipole axis in the direction of F0, for all t satisfying the inequalities vt Lt; r2 and F0t2 [Lt ] 4πρr4. Also, at a given time t the streamlines of the developing flow field in a section through the axis of symmetry of the problem form closed loops about a stagnation point. If this occurs at λ = λm, the stagnation point propagates to infinity, along a straight line emanating from the origin, with speed v½/2λmt½, where λm = λm(F0) decreases as F0 increases. The larger F0 is the faster the steady state is established.

We have experimentally examined the effects of a common soluble surfactant on gas bubbles in liquid flows in inclined tubes. Air bubbles of known size (λ = 0.8, 1.0, 1.5) are held stationary under minimum flow conditions in tubes inclined at fixed angles (ω = 25°, 45°, 65°, 90°). Sodium dodecyl sulphate (SDS) is infused into the bulk flow at two bulk concentrations (C = 10% or 100% critical micelle concentration (CMC)). In addition to recording pressure and flow waveforms, we capture video images of bubbles before and during exposure to the surfactant. Modification of the interfacial properties by the surfactant results in extremely dynamic bubble behaviour including interfacial deformation, deformation plus axial translation, and bubble detachment from the wall plus translation. We measure the corresponding time-dependent pressure gradient within the tube. The surfactant mediated responses observed are dependent upon the interrelated effects of C, λ and ω. A high bulk concentration of surfactant may produce more rapid modification of bubble shape and influence wetting, thus increasing the potential for bubble detachment. The likelihood that detachment will occur increases further as bubble volume in increased. In both vertical tubes in which contact forces are absent and in non-vertical tubes, the infusion of surfactant may result in axial translation either in the direction of, or opposite to, the direction of the bulk flow. Critical to the translation and/or detachment of the bubble is the surfactant-mediated modification of contact line mechanics. Contact line velocities corresponding to rates of shrinkage of dewetted surface area are extracted from experimental data. We also explore the potential effects of surfactants on interfacial remobilization. This investigation demonstrates the potential use of surfactants to be used for dislodging dewetted gas bubbles by the intentional manipulation of interfacial properties.

In a recent paper, Tatsumi, Kida & Mizushima (1978) have made a numerical study of the quasi-normal Markovian (QNM) equation for homogeneous isotropic incompressible turbulence at Reynolds numbers R up to 800.

Analytical investigations of the QNM equation support the contention of Tatsumi et al. that, at R = ∞, the decay of an initial energy spectrum of the form ka exp (− k2) leads to an initial energy-conserving regularity phase followed by a self-similar decay phase. During the former we give explicit expressions for the enstrophy and skewness. During the latter we show that for 1 < a < 4 the energy follows, for t → ∞, a t−b law with the usual value b = 2(a + 1)/(a + 3); when a [ges ] 4 deviations from Kolmogorov's (1941) $t^{\frac{10}{7}}$ law originate from non-local ‘beating’ interactions between eddies with sizes of the order of the integral scale.

We also show, analytically, that the QNM equation has a k−2, not a $k^{-\frac{5}{3}}$, inertial range and that its dissipation range is of the form k3e−k/kD, rather than e−σk1.5.

Our results are illustrated by numerical integration of the QNM equation for R up to 106 and by comparison with results from the eddy-damped quasi-normal Markovian equation which is known to produce a k−5/3 spectrum.

The interaction of waves on deep water with spatially varying currents may be described by a ray theory, with the wave amplitudes determined by the principle of conservation of wave action (CWA). However, all previous deep water derivations of CWA are restricted to the case of an irrotational current. In this paper, both the ray theory and CWA are derived by a WKB method without the assumption of irrotationality. Also derived is a new equation for a spatially varying phase shift which is not predicted by the usual ray theory, and which, in general, displaces the positions of the wave crests by a distance on the order of a wavelength. This phase shift, which is caused by variations of the current velocity with depth, vanishes in the irrotational case, and so is in accord with the irrotational theory.

In the first part of this paper the nonlinear development of the most unstable mode is numerically studied for a bounded shear layer with a hyperbolic-tangent profile. It is found that the vortex nutation, discovered by Zabusky & Deem (1971) for a jet profile, is a manifestation of strongly coupled oscillations in the vortex amplitude and the phase. In the second part, with the aid of the numerical result we devote ourselves to deriving coupled nonlinear equations that describe the amplitude oscillation, the vortex nutation and the momentum transport. The approximate oscillatory solution for the vortex amplitude and phase in the nonlinear stage is compared with the numerical solution and agreement is found.

The paper contains a detailed study of the inertial instability of a barotropic Bickley jet on the $f$-plane in the continuously stratified primitive equations model, and a comparison of this essentially ageostrophic instability with the classical baroclinic one. Analytical and numerical investigation of the linear stability of the jet in the long-wave sector is performed for a range of Rossby and Burger numbers. The major results are that: (1) the standard symmetric inertial instability, appearing at high enough Rossby numbers, turns out to be the infinite-wavelength limit of an asymmetric inertial instability, this latter having the highest growth rate for a large range of vertical wavenumbers; (2) inertial instability coexists with the standard baroclinic instability, which becomes dominant at small Rossby numbers. Nonlinear saturation of the inertial instability of the jet with a superimposed random small-amplitude perturbation is then studied, using the Weather Research and Forecast model. It is shown that at first stages the inertial instability dominates. It is localized near the maximum of the anticyclonic shear and is associated with the highest attainable value of the vertical wavenumber. The saturation of the inertial instability leads to the homogenization of the geostrophic momentum in the unstable region. At later stages, another baroclinic instability develops, characterized by lower values of the vertical wavenumber. This instability saturates by forming large-scale vortices downstream. It is identified as the leading instability of a marginally inertially stable jet resulting from the initial one through homogenization of the geostrophic momentum. The rough scenario of the evolution of essentially ageostrophic jets is, thus, as follows: the inertial instability rapidly saturates and baroclinic instability takes over. It is shown that reorganization of the flow due to developing instabilities is an efficient source of inertia-gravity waves.

Stationary axi-symmetrical pressurized bells formed by the impact of a liquid jet on a solid disc (so-called Savart water bells) can exhibit uncommon sharp and pointed shapes. They are characterized by two successive inflections of the two-dimensional bell generator profile, corresponding to partially concave bells. We show that this shape is incompatible with the usual assumption that the detail of the flow across the liquid sheet constitutive of the bell is unimportant. We consider the equilibrium of a curved liquid sheet of finite thickness sustaining a pressure difference between both sides, and show that several curvatures of the interface may be a solution under given flow conditions. The inflection of the bell profile is then explained in terms of a spontaneous transition from a ‘negative’ to a ‘positive’ curvature which conserves mass flow, linear and angular momenta. That inflection is also a transition from a super to a subcritical flow (with respect to capillary waves), having the status of a capillary hydraulic jump on a freely suspended sheet, a novel object in fluid mechanics. The azimuthal wrinkles forming at the jump result from the inertial destabilization of the sheet due to the centripetal acceleration fluid particles experience as they flow along the highly curved bell profile in the vicinity of the fold. This finding also explains the singular shape at the edge of freely flapping sheets.

Supersonic laminar boundary-layer equations near the plane of symmetry of a cone at incidence are treated by the similarity method. Numerical integration of differential equations governing such a flow is performed, taking into consideration the temperature dependence of the Prandtl number Pr and viscosity μ throughout the boundary layer. On the leeward side, a detailed consideration of the solutions shows the existence of two solutions up to a critical incidence beyond which it appears that no solution may be found. Calculations carried out for a set of values of the external flow Mach number show up a significant effect of this parameter on the behaviour of the boundary layer.

An examination is undertaken of the validity and limitations of the quasi-steady hypothesis of near-wall turbulence. This hypothesis is based on the supposition that the statistics of the turbulent fluctuations are universal if scaled by the local, instantaneous, wall shear when its variations are determined from footprints of large-scale, energetic, structures that reside in the outer part of the logarithmic layer. The examination is performed with the aid of direct numerical simulation data for a single Reynolds number, which are processed in a manner that brings out the variability of locally scaled statistics when conditioned on the local value of the wall friction. The key question is to what extent this variability is insignificant, thus reflecting universality. It is shown that the validity of the quasi-steady hypothesis is confined, at best, to a thin layer above the viscous sublayer. Beyond this layer, substantial variations in the conditioned shear-induced production rate of large-scale turbulence cause substantial departures from the hypothesis. Even within the wall-proximate layer, moderate departures are provoked by large-scale distortions in the conditioned strain rate that result in variations in small-scale production of turbulence down to the viscous sublayer.

We study the effect of proportional feedback control on the onset and development of finite-wavelength Rayleigh–Bénard–Marangoni (RBM) convection using weakly nonlinear theory as applied to Nield's model, which includes both thermocapillarity and buoyancy but ignores deformation of the free surface. A two-layer model configuration is used, which has a purely conducting gas layer on top of the liquid. In the feedback control analysis, a control action in the form of temperature or heat flux is considered. Both measurement and control action are assumed to be continuous in space and time. Besides demonstrating that stabilization of the basic state can be achieved on a linear basis, the results also indicate that a wide range of weakly nonlinear flow properties can also be altered by the linear and nonlinear control processes used here. These include changing the nature of hexagonal convection and the amount of subcritical hysteresis associated with subcritical bifurcation.

We investigate how the domain depth affects the turbulent behaviour in spatially developing mixing layers by means of large-eddy simulations based on a spectral vanishing viscosity technique. Analyses of spectra of the vertical velocity, of Lumley’s diagrams, of the turbulent kinetic energy and of the vortex stretching show that a two-dimensional behaviour of the turbulence is promoted in spatial mixing layers by constricting the fluid motion in one direction. This finding is in agreement with previous works on turbulent systems constrained by a geometric anisotropy, pioneered by Smith, Chasnov & Waleffe (Phys. Rev. Lett., vol. 77, 1996, pp. 2467–2470). We observe that the growth of the momentum thickness along the streamwise direction is damped in a confined domain. An almost fully two-dimensional turbulent behaviour is observed when the momentum thickness is of the same order of magnitude as the confining scale.

We present a theoretical investigation of the effects of low-frequency vibrations on the motion of two-dimensional droplets on heterogeneous substrates in the presence of gravity and substrate heterogeneities, both chemical and topographical. A combined analytical and numerical approach is undertaken, extending the work of Savva & Kalliadasis (J. Fluid Mech., vol. 725, 2013, pp. 462–491) on inclined heterogeneous substrates to include the effects of substrate vibrations. Via a matching procedure and under the quasi-static assumption, we obtain evolution equations for the moving fronts. These equations are then invoked in a wide variety of case studies. It is demonstrated that vertically vibrated horizontal ratcheted substrates can induce unidirectional motion. For inclined substrates, we focus on a number of qualitative aspects of the peculiar vibration-induced climbing of droplets reported in experiments by Brunet, Eggers & Deegan (Phys. Rev. Lett., vol. 99, 2007, art. 144501). We examine the effects of weak inertia on the dynamics, deduce analytical criteria for the uphill motion in the limit of weak gravitational and vibrational effects, and demonstrate that substrate heterogeneities may be utilised to enhance droplet transport.

An approximate expression is given for the thickness of weak fully dispersed shock waves. Using available data on the thermodynamic properties of air, it is shown that shocks of the strength expected in sonic bangs are fully dispersed. Estimated relaxation times for dry and humid air lead to wide variations in possible thickness, varying from millimetres to metres.

Addition of particles or droplets to turbulent liquid flows or addition of droplets to turbulent gas flows can lead to modulation of turbulence characteristics. Corresponding observations have been reported for very small particle or droplet volume loadings ${\Phi }_{v} $ and therefore may be important when simulating such flows. In this work, a modelling framework that accounts for preferential concentration and reproduces isotropic and anisotropic turbulence attenuation effects is presented. The framework is outlined for both Reynolds-averaged Navier–Stokes (RANS) and joint probability density function (p.d.f.) methods. Validations are performed involving a range of particle and flow-field parameters and are based on the direct numerical simulation (DNS) study of Boivin, Simonin & Squires (J. Fluid Mech., vol. 375, 1998, pp. 235–263) dealing with heavy particles suspended in homogeneous isotropic turbulence (Stokes number $\mathit{St}= O(1{\unicode{x2013}} 10)$, particle/fluid density ratio ${\rho }_{p} / \rho = 2000$, ${\Phi }_{v} = O(1{0}^{- 4} )$) and the experimental investigation of Poelma, Westerweel & Ooms (J. Fluid Mech., vol. 589, 2007, pp. 315–351) involving light particles ($\mathit{St}= O(0. 1)$, ${\rho }_{p} / \rho \gtrsim 1$, ${\Phi }_{v} = O(1{0}^{- 3} )$) settling in grid turbulence. The development in this work is restricted to volume loadings where particle or droplet collisions are negligible.

Flows in channels exposed to spatially distributed heating were investigated. Such flows are of interest as theoretical analyses suggest that heating leads to the reduction of pressure losses. A special apparatus providing the means for the creation of well-controlled spatially periodic heating with the desired intensity as well as precise control of the flow rate in flows with small Reynolds numbers was constructed. The apparatus works with air and provides optical access to the flow interior. The relevant theory has been generalized to handle the temperature fields measured in the experiments. The experiments were carried out for Reynolds numbers $Re<20$ and at a single Rayleigh number based on the peak-to-peak temperature difference and channel half-height of $Ra_{p}=3500$. Flow visualization and particle image velocimetry measurements demonstrated the formation of two-dimensional steady rolls whose size was dictated by $Re$, with the largest rolls observed for the smallest $Re$ and the roll size being gradually reduced as $Re$ increased until their complete elimination at the largest $Re$ used in the experiment. An excellent agreement between the theoretically and experimentally determined complex flow fields was found. Wall shear stresses extracted from the velocity measurements agree with their theoretical counterparts within the expected accuracy. The agreement between the experimental and theoretical velocity fields and their unique relation with the corresponding pressure fields indirectly verify the heating-induced pressure-gradient-reducing effect.

We investigate the effects of an insulating lid of variable spatial extent on convection in the stagnant-lid regime under thermally steady-state conditions. Using a combination of laboratory experiments, numerical simulations and scaling analyses we characterize the qualitative structure and quantitative heat transfer properties of flows in terms of the fractional extent L of an insulating lid applied at the cold boundary, the thermal resistance of the lid, the magnitude of the temperature dependence of the fluid viscosity Λ and the effective Rayleigh number Rae for the composite system. A partial insulating lid has two main effects: (i) To increase the mean interior temperature and reduce the average viscosity of the system, which enhances fluid motions, and (ii) to impart a lateral asymmetry to the thermal structure of the cold boundary that leads, in turn, to lateral temperature gradients that drive an overturning flow. Consequently, whereas flow in the uninsulated stagnant-lid regime is in the form of ‘small-scale’ rising and sinking thermals, there is an additional ‘large-scale’ circulation in the presence of partial insulation. The structure, wavelength and heat transfer properties of this large-scale stirring depends on L, Λ and Rae. For given Rae – Λ conditions we find optimal values of L at which there occur well-defined maxima in the rate of overturn, the local heat flux carried into the uninsulated part of the cold boundary and in the global average heat flux Nu carried across the system. Whereas both the rate of overturning and local heat flux are associated with the largest lateral temperature gradients, the optimal basal heat flux depends also on a tradeoff with the fractional surface area of the lid. Remarkably, maximal values of the global heat flux can significantly exceed that of the uninsulated stagnant-lid case. The occurrence of such maxima is insensitive to the mechanical boundary conditions applied and is not strongly influenced by lid shape. However, the magnitude and location of optimal heat fluxes depends in a complicated way on the lid surface area and shape, as well as the structure of the hot and cold boundary layers and the wavelength of the large-scale flow.