In this paper, axial interaction of a vortex ring with a thin circular cylinder has been studied. An apparatus to generate clean vortex rings, free of piston and stopping vortex effects, has been used. Flow visualization and particle image velocimetry (PIV) experiments are carried out to determine and compare the characteristics of free and interacting vortex rings in the Reynolds number (defined with the circulation of the free travelling vortex ring) range of $2270<Re_{\unicode[STIX]{x1D6E4}}<6790$. It is observed that due to the presence of the cylinder, there is an increase in the velocity of the vortex ring. Also, noticeable changes in the characteristic properties of vortex ring such as core circulation, core diameter and ring diameter have been observed. Changes in these parameters are explained by two changes in the flow field between the vortex ring and the cylinder due to axial interactions: (i) displacement of the streamlines and (ii) acceleration in the induced velocity field in this region. These two mutually opposing effects determine the changes in the primary vortex ring properties that take place during interaction. To justify these experimental observations quantitatively, an analytical study of the interaction under an inviscid assumption is performed. The inviscid analysis does predict the increase in velocity during the interaction, but fails to predict the values observed in the present experiments. However, when the theory is used to correct the velocity change through incorporation of the effects of an axisymmetric induced boundary layer region over the cylinder, modelled as an annular vortex sheet of varying strength, the changes in the translational velocities of the vortex rings match closely with the experimental values.

It is well established that when a turbulent flow is subjected to a non-uniform body force, the turbulence may be significantly suppressed in comparison with that of the flow of the same flow rate and hence the flow is said to be laminarised. This is the situation in buoyancy-aided mixed convection when severe heat transfer deterioration may occur. Here we report results of direct numerical simulations of flow with a linear or a step-change profile of body force. In contrast to the conventional view, we show that applying a body force to a turbulent flow while keeping the pressure force unchanged causes little changes to the key characteristics of the turbulence. In particular, the mixing characteristics of the turbulence represented by the turbulent viscosity remain largely unaffected. The so-called flow laminarisation due to a body force is in effect a reduction in the apparent Reynolds number of the flow, based on an apparent friction velocity associated with only the pressure force of the flow (i.e. excluding the contribution of the body force). The new understanding allows the level of the flow ‘laminarisation’ and when the full laminarisation occurs to be readily predicted. In terms of the near-wall turbulence structure, the numbers of ejections and sweeps are little influenced by the imposition of the body force, whereas the strength of each event may be enhanced if the coverage of the body force extends significantly away from the wall. The streamwise turbulent stress is usually increased in accordance with the observation of more and stronger elongated streaks, but the wall-normal and the circumferential turbulent stresses are largely unchanged.

Simultaneous measurements of particle velocity and concentration (number density) in a series of mono-disperse, two-phase turbulent jets issuing from a long, round pipe into a low velocity co-flow were performed using planar nephelometry and digital particle image velocimetry. The exit Stokes number, $Sk_{D}$, was systematically varied over two orders of magnitude between 0.3 and 22.4, while the Reynolds number was maintained in the turbulent regime ($10\,000\leqslant Re_{D}\leqslant 40\,000$). The mass loading was fixed at $\unicode[STIX]{x1D719}=0.4$, resulting in a flow that is in the two-way coupling regime. The results show that, in contrast to all previous work where a single Stokes number has been used to characterise fluid–particle interactions, the characteristic Stokes number in the axial direction is lower than that for the radial direction. This is attributed to the significantly greater length scales in the axial motions than in the radial ones. It further leads to a preferential response of particles to gas-phase axial velocity fluctuations, $u_{p}^{\prime }$, over radial velocity fluctuations, $v_{p}^{\prime }$. This, in turn, leads to high levels of anisotropy in the particle-phase velocity fluctuations, $u_{p}^{\prime }/v_{p}^{\prime }>1$, throughout the jet, with $u_{p}^{\prime }/v_{p}^{\prime }$ increasing as $Sk_{D}$ is increased. The results also show that the region within the first few diameters of the exit plane is characterised by a process of particle reorganisation, resulting in significant particle migration to the jet axis for $Sk_{D}\leqslant 2.8$ and away from the axis for $Sk_{D}\geqslant 5.6$. This migration, together with particle deceleration along the axis, causes local humps in the centreline concentration whose value can even exceed those at the exit plane.

The onset of the vortex-induced vibration (VIV) regime of a flexible cylinder inclined at $80^{\circ }$ within a uniform current is studied by means of direct numerical simulations, at Reynolds number $500$ based on the body diameter and inflow velocity magnitude. A range of values of the reduced velocity, defined as the inverse of the fundamental natural frequency, is examined in order to capture the emergence of the body responses and explore the concomitant reorganization of the flow and fluid forcing. Additional simulations at normal incidence confirm that the independence principle, which states that the system behaviour is determined by the normal inflow component, does not apply at such large inclination angle. Contrary to the normal incidence case, the free vibrations of the inclined cylinder arise far from the Strouhal frequency, i.e. the vortex shedding frequency downstream of a fixed rigid cylinder. The trace of the stationary body wake is found to persist beyond the vibration onset: the flow may still exhibit an oblique component that relates to the slanted vortex shedding pattern observed in the absence of vibration. This flow component which occurs close to the Strouhal frequency, at a high and incommensurable frequency compared to the vibration frequency, is referred to as Strouhal component; it induces a high-frequency component in fluid forcing. The vibration onset is accompanied by the appearance of novel, low-frequency components of the flow and fluid forcing which are synchronized with body motion. This second dominant flow component, referred to as lock-in component, is characterized by a parallel spatial pattern. The Strouhal and lock-in components of the flow coexist over a range of reduced velocities, with variable contributions, which results in a variety of mixed wake patterns. The transition from oblique to parallel vortex shedding that occurs during the amplification of the structural responses, is driven by the opposite trends of these two component contributions: the decrease of the Strouhal component magnitude associated with the progressive disappearance of the high-frequency force component, and simultaneously, the increase of the lock-in component magnitude, which dominates once the fully developed VIV regime is reached and the flow dynamics is entirely governed by wake–body synchronization.

The transport of momentum and heat in the turbulent intermediate wake of a circular cylinder is inherently three-dimensional (3-D). This work aims to gain new insight into the 3-D vorticity structure, momentum and heat transport in this flow. All three components of the velocity and vorticity vectors, along with the fluctuating temperature, are measured simultaneously, at nominally the same point in the flow, with a probe consisting of four X-wires and four cold wires. Measurements are made in the ($x$, $y$) or mean shear plane at $x/d=10$, 20 and 40 at a Reynolds number of $2.5\times 10^{3}$ based on the cylinder diameter $d$ and the free-stream velocity. A phase-averaging technique is developed to separate the large-scale coherent structures from the remainder of the flow. It is found that the effects of vorticity on heat transport at $x/d=10$ and $x/d=20{-}40$ are distinctly different. At $x/d=10$, both spanwise and streamwise vorticity components account significantly for the heat flux. At $x/d=20$ and 40, the spanwise vortex rollers play a major role in inducing the coherent components of the heat flux vector, while the ribs are responsible for the small-scale heat diffusion out of the spanwise vortex rollers. The present data indicate that, if the spanwise-velocity-related terms are ignored, the estimated values of the production can have errors of approximately 22 % and 13 % respectively for the turbulent energy and temperature variance at $x/d=40$, and the errors are expected to further increase downstream. A conceptual model summarizing the 3-D features of the heat and momentum transports at $x/d=10$ is proposed. Compared with the previous two-dimensional model of Matsumura & Antonia (J. Fluid Mech., vol. 250, 1993, pp. 651–668) or MA, the new model provides a more detailed description of the role the rib-like structures undertake in transporting heat and momentum, and also underlines the importance of the upstream half of the spanwise vortex rollers, instead of only one quadrant of these rollers, as in the MA model, in diffusing heat out of the vortex.

Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the $\unicode[STIX]{x1D707}(I)$-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.

This work focuses on the development of a semi-analytical model that is appropriate for the rotational, steady, inviscid, and compressible motion of an ideal gas, which is accelerated uniformly along the length of a right-cylindrical rocket chamber. By overcoming some of the difficulties encountered in previous work on the subject, the present analysis leads to an improved mathematical formulation, which enables us to retrieve an exact solution for the pressure field. Considering a slender porous chamber of circular cross-section, the method that we follow reduces the problem’s mass, momentum, energy, ideal gas, and isentropic relations to a single integral equation that is amenable to a direct numerical evaluation. Then, using an Abel transformation, exact closed-form representations of the pressure distribution are obtained for particular values of the specific heat ratio. Throughout this effort, Saint-Robert’s power law is used to link the pressure to the mass injection rate at the wall. This allows us to compare the results associated with the axisymmetric chamber configuration to two closed-form analytical solutions developed under either one- or two-dimensional, isentropic flow conditions. The comparison is carried out assuming, first, a uniformly distributed mass flux and, second, a constant radial injection speed along the simulated propellant grain. Our amended formulation is consequently shown to agree with a one-dimensional solution obtained for the case of uniform wall mass flux, as well as numerical simulations and asymptotic approximations for a constant wall injection speed. The numerical simulations include three particular models: a strictly inviscid solver, which closely agrees with the present formulation, and both $k$–$\unicode[STIX]{x1D714}$ and Spalart–Allmaras computations.

A stochastic model is presented for the probability density function (p.d.f.) of the liquid velocity fluctuations generated by high-Reynolds-number rising bubbles. It considers three elementary sources of fluctuations: the potential flow disturbance around each bubble; the average bubble wakes, which are assumed to decay exponentially; and the turbulent agitation resulting from the flow instability, which is assumed to be isotropic, homogeneously distributed all over the flow and statistically independent of the two others. The model reproduces well and explains the characteristics of the experimental p.d.f.s: exponential tails, asymmetry of vertical fluctuations and evolution with the gas volume fraction. The model involves two a priori unknown parameters: the volume of the wake and the velocity scale of the turbulent agitation. Because some parts of the probability functions depend only on a single contribution, these two parameters can be uniquely and independently determined from experimental p.d.f.s. This defines an objective method to separate the various kinds of fluctuations and allows one to determine the contribution of each of them to the total agitation.

Filters that act by adsorbing contaminant onto their pore walls will experience a decrease in porosity over time, and may eventually block. As adsorption will generally be greater towards the entrance of a filter, where the concentration of contaminant particles is higher, these effects can also result in a spatially varying porosity. We investigate this dynamic process using an extension of homogenization theory that accounts for a macroscale variation in microstructure. We formulate and homogenize the coupled problems of flow through a filter with a near-periodic time-dependent microstructure, solute transport due to advection, diffusion and filter adsorption, and filter structure evolution due to the adsorption of contaminant. We use the homogenized equations to investigate how the contaminant removal and filter lifespan depend on the initial porosity distribution for a unidirectional flow. We confirm a conjecture made by Dalwadi et al. (Proc. R. Soc. Lond. A, vol. 471 (2182), 2015, 20150464) that filters with an initially negative porosity gradient have a longer lifespan and remove more contaminant than filters with an initially constant porosity, or worse, an initially positive porosity gradient. In addition, we determine which initial porosity distributions result in a filter that will block everywhere at once by exploiting an asymptotic reduction of the homogenized equations. We show that these filters remove more contaminant than other filters with the same initial average porosity, but that filters which block everywhere at once are limited by how large their initial average porosity can be.

The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in a variety of physical systems. An SSD is a dynamical system that evolves a representation of the statistical state of the system. An example of an SSD is the second-order cumulant closure referred to as stochastic structural stability theory (S3T), which has provided insight into the dynamics of wall turbulence, and specifically the emergence and maintenance of the roll/streak structure. S3T comprises a coupled set of equations for the streamwise mean and perturbation covariance, in which nonlinear interactions among the perturbations has been removed, restricting nonlinearity in the dynamics to that of the mean equation and the interaction between the mean and perturbation covariance. In this work, this quasi-linear restriction of the dynamics is used to study the structure and dynamics of turbulence in plane Poiseuille flow at moderately high Reynolds numbers in a closely related dynamical system, referred to as the restricted nonlinear (RNL) system. Simulations using this RNL system reveal that the essential features of wall-turbulence dynamics are retained. Consistent with previous analyses based on the S3T version of SSD, the RNL system spontaneously limits the support of its turbulence to a small set of streamwise Fourier components, giving rise to a naturally minimal representation of its turbulence dynamics. Although greatly simplified, this RNL turbulence exhibits natural-looking structures and statistics, albeit with quantitative differences from those in direct numerical simulations (DNS) of the full equations. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed, the RNL system continues to self-sustain turbulence with qualitatively realistic structure and dynamic properties. RNL turbulence at the Reynolds numbers studied is dominated by the roll/streak structure in the buffer layer and similar very large-scale structure (VLSM) in the outer layer. In this work, diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence are used to demonstrate that the roll/streak dynamics supporting the turbulence in the buffer and logarithmic layer is essentially similar in RNL and DNS.

A theoretical model is developed which illustrates the dynamics of the spontaneous generation of large-scale structures in baroclinically unstable eddying flows. Techniques of asymptotic multiscale analysis are used to identify instabilities resulting from the positive feedback of the background eddies on large-scale perturbations. The novelty of the proposed approach lies in the choice of a dynamically consistent time-dependent background eddy field, which is taken from simulations of baroclinic instability in the Phillips two-layer system. The resulting solutions differ considerably from those of traditional multiscale models, in which the background eddy field is represented by steady analytical patterns. The present formulation makes it possible to (i) test the multiscale theory against the corresponding numerical simulations, (ii) unambiguously interpret the key physical processes at play and (iii) rationalize the emergence of large-scale patterns for certain background parameters. While the proposed approach to multiscale modelling is illustrated on a particular example of the Phillips baroclinic instability model, it is our belief that the presented technique is readily adaptable to a wide range of applications.

In this work, we study the lift force on spherical nanoparticles suspended in a shear flow of rarefied binary gas mixtures. Analytical formulae are developed using the gas kinetic theory by considering non-rigid-body intermolecular interactions between the particle and gas molecules. It has been shown that the lift force formulae can be reduced to those in pure gases. It is also found that the direction of the lift force on nanoparticles in binary gas mixtures can be changed by varying the temperature, gas–particle interaction and/or gas concentrations.

Syringomyelia is a medical condition in which one or more fluid-filled cavities (syrinxes) form in the spinal cord. The syrinxes often form near locations where the spinal subarachnoid space (SSS; the fluid-filled annular region surrounding the spinal cord) is partially obstructed. Previous studies showed that nonlinear interactions between the pulsatile fluid flow in the SSS and the elastic deformation of the tissues surrounding it can generate a fluid pressure distribution that would tend to drive fluid from the SSS into the syrinx if the tissue separating the two regions was porous. This provides a potential explanation for why a partial occlusion of the SSS can induce the growth of an already existing nearby syrinx. We study this hypothesis by analysing the mass transfer between the SSS and the syrinx, using a poroelastic fluid–structure interaction model of the spinal cord that includes a representation of the partially obstructed SSS, the syrinx and the poroelastic tissues surrounding these fluid-filled cavities. Our numerical simulations show that poroelastic fluid–structure interaction can indeed cause an increase (albeit relatively small) in syrinx volume. We analyse the seepage flows and show that their structure can be captured by an analytical model which explains why the increase in syrinx volume tends to be relatively small.

The heating of particles in a dilute suspension, for instance by radiation, chemical reactions or radioactivity, leads to local temperature fluctuations in the fluid due to the non-uniformity of the disperse phase. In the presence of a gravity field, the fluid is set in motion by the resulting buoyancy forces. When the particle density is different than that of the fluid, the fluid motion alters the spatial distribution of the particles and possibly strengthens their concentration inhomogeneities. This in turn causes more intense local heating. Direct numerical simulations in the Boussinesq limit show this feedback loop. Various regimes are identified depending on the particle inertia. For very small particle inertia, the macroscopic behaviour of the system is the result of many thermal plumes that are generated independently of each other. For significant particle inertia, clusters of particles are observed and their dynamics controls the flow. The emergence of very intermittent turbulent fluctuations shows that the flow is influenced by the larger structures (turbulent convection) as well as by the small-scale dynamics that affect particle segregation and thus the flow forcing. Assuming thermal equilibrium between the particles and the fluid (i.e. infinitely fast thermal relaxation of the particle), we investigate the evolution of statistical observables with the change of the main control parameters (namely the particle number density, the particle inertia and the domain size), and propose a scaling argument for these trends. Concerning the energy density in the spectral space, it is observed that the turbulent energy and temperature spectra follow a power law, the exponent of which varies continuously with the Stokes number. Furthermore, the study of the spectra of the temperature and momentum forcing (and thus of the concentration/temperature and velocity/temperature correlations) gives strong support to the proposed feedback loop mechanism. We then discuss the intermittency of the flow, and analyse the effect of relaxing some of the simplifying assumptions, thus assessing the relevance of the original studied configuration.

We investigate mechanisms governing the initial growth and nonlinear evolution of interfacial waves in horizontal two-fluid plane Couette–Poiseuille flows. Nonlinear coupling of the Kelvin–Helmholtz interfacial instability with resonant wave interactions has been shown to be capable of rapidly generating long waves through the transfer of energy from linearly unstable short waves to stable long-wave components within the context of potential flow theory. The objective of this work is to determine whether that coupled mechanism persists in laminar and turbulent viscous flows. Utilizing both theoretical and computational methods, we analyse the initial Orr–Sommerfeld instability to quantify the frequencies and growth/decay rates of each wave mode for two-fluid laminar and turbulent channel flows. The obtained dispersion relation allows for the identification of resonant and/or near-resonant triads among (unstable and damped) wave components in an interfacial wave spectrum. We perform direct numerical simulations (DNS) of the two-phase Navier–Stokes equations with a fully nonlinear interface to formally establish the validity of our theoretical predictions for viscous flows. DNS results show the existence of a nonlinear energy cascade from unstable short- to damped long-wavelength waves due to resonant subharmonic and/or triadic interactions in both laminar Couette and turbulent Poiseuille flows. Spectral analysis of the interfacial evolution confirms that the combined instability–resonance mechanism persists in the presence of viscosity despite being derived under the assumption of potential flow theory. Finally, we perform a detailed examination of experimentally measured wave power spectra from Jurman et al. (J. Fluid Mech., vol. 238, 1992, pp. 187–219) and carry out a numerical sensitivity study of the flow conditions to demonstrate and verify the existence of the coupled instability–resonance mechanism in physical systems. Our analysis accurately predicts the initial instability and the resulting nonlinear energy cascade through subharmonic and triadic interfacial wave resonances.

The influence of the underlying flow topology on the shape and size of sub-Kolmogorov droplets dispersed in a turbulent flow is of considerable interest in many industrial and scientific applications. In this work we study the deformation and orientation statistics of sub-Kolmogorov droplets dispersed into a turbulent Taylor–Couette flow. Along with direct numerical simulations (DNS) of the carrier phase and Lagrangian tracking of the dispersed droplets, we solve a phenomenological equation proposed by Maffettone and Minale (J. Non-Newtonian Fluid Mech., vol. 78, 1998, pp. 227–241) to track the shape evolution and orientation of approximately $10^{5}$ ellipsoidal droplets. By varying the capillary number $Ca$ and viscosity ratio $\hat{\unicode[STIX]{x1D707}}$ of the droplets we find that they deform more with increasing capillary number $Ca$ and this effect is more pronounced in the boundary layer regions. This indicates that along with an expected capillary number effect there is also a strong correlation between spatial position and degree of deformation of the droplet. Regardless of the capillary number $Ca$, the major axis of the ellipsoids tends to align with the streamwise direction and the extensional strain rate eigendirection in the boundary layer region while the distribution is highly isotropic in the bulk due to the strong mixing provided by the large-scale vortical structures. When the viscosity ratio between the droplet and the carrier fluid is increased we find that there is no preferential stretched axis which is due to the increased influence of rotation over stretching and relaxation. Droplets in high viscosity ratio systems are thus less deformed and oblate (disk-like) as compared to highly deformed prolate (cigar-like) droplets in low viscosity ratio systems.

The equations for three-dimensional hydrostatic Boussinesq dynamics are equivalent to a variational principle that is closely analogous to the variational principle for classical electrodynamics. Inertia–gravity waves are analogous to electromagnetic waves, and available potential vorticity (i.e. the amount by which the potential vorticity exceeds the potential vorticity of the rest state) is analogous to electric charge. The Lagrangian can be expressed as the sum of three parts. The first part corresponds to quasi-geostrophic dynamics in the absence of inertia–gravity waves. The second part corresponds to inertia–gravity waves in the absence of quasi-geostrophic flow. The third part represents a coupling between the inertia–gravity waves and quasi-geostrophic motion. This formulation provides the basis for a general theory of inertia–gravity waves interacting with a quasi-geostrophic mean flow.

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

The buoyancy- and capillary-driven counter-current flow of $\text{CO}_{2}$ and brine through and around a semi-permeable layer is studied both numerically and theoretically. The continuities of the capillary pressure and the total flux at the interface between the permeable matrix and layer control the $\text{CO}_{2}$ saturation discontinuity at the interface and the balance between the buoyant and capillary diffusion fluxes on each side of the interface. This interface process is first studied in a one-dimensional (1-D) vertical column geometry using the concept of extended capillary pressure and a graphical representation of the continuity conditions in the ($S_{L}$, $S_{U}$) plane, where $S_{L}$ and $S_{U}$ are the lower and upper saturation traces at the interface, respectively. In two dimensions, we heuristically extend the two-phase gravity current model to the case where the current is bounded by a semi-permeable layer. Consequently, the current is not saturated with $\text{CO}_{2}$, and its saturation and shape are derived from the flux and capillary pressure continuity conditions at the interface. This simplified model, which depends on $\text{CO}_{2}$ saturation only, is compared to fine grid simulations in the capillary-free and gravity-dominant cases. A good agreement is obtained in the second case; the current geometrical characteristics are accurately described. In the capillary-free case, we demonstrate that the local total velocity, which is, on average, zero because the flow is counter-current, must be considered in the total flux at the interface to obtain the same level of agreement.

It is remarked that fluxes in conservation laws, such as the Reynolds stresses in the momentum equation of turbulent shear flows, or the spectral energy flux in anisotropic turbulence, are only defined up to an arbitrary solenoidal field. While this is not usually significant for long-time averages, it becomes important when fluxes are modelled locally in large-eddy simulations, or in the analysis of intermittency and cascades. As an example, a numerical procedure is introduced to compute fluxes in scalar conservation equations in such a way that their total integrated magnitude is minimised. The result is an irrotational vector field that derives from a potential, thus minimising sterile flux ‘circuits’. The algorithm is generalised to tensor fluxes and applied to the transfer of momentum in a turbulent channel. The resulting instantaneous Reynolds stresses are compared with their traditional expressions, and found to be substantially different. This suggests that some of the alleged shortcomings of simple subgrid models may be representational artefacts, and that the same may be true of the intermittency properties of the turbulent stresses.