Recent work has developed a beautiful model system for studying the energy focusing and heating power of collapsing bubbles. The bubble is effectively one-dimensional and the collapse and heating can be quantitatively measured. Thermal effects are shown to play an essential role in the time-dependent dynamics.

The growth and collapse of a vapour bubble inside a microtube is studied both experimentally and theoretically. The length of the bubble, and the velocity and acceleration of its interface, are obtained from a high-speed image recording (typically 1.25 × 105 fps) for various energy inputs and two tube diameters. To understand the underlying dynamics of the system, two theoretical models are compared with experiment. A model based on a discontinuous time dependence of the vapour pressure inside the bubble is at variance with the data. It proves necessary to account in greater detail for the time dependence of the vapour pressure. A new model is proposed for this purpose which includes heat transfer in addition to inertia and viscous friction. Both the data and the model show that the vapour pressure decreases with time continuously instead of abruptly. The length, velocity and acceleration from the numerical simulations are found to be in good agreement with experimental data. Both the experiments and simulations clearly indicate that thermal effects play an important role throughout the whole growth and collapse process.

Molecular mixing measurements are reported for a high-Schmidt-number (Sc ~ 103), small-Atwood-number (A ≈ 7.5 × 10−4) buoyancy-driven turbulent Rayleigh–Taylor (RT) mixing layer in a water channel facility. Salt was added to the top water stream to create the desired density difference. The degree of molecular mixing was measured as a function of time by monitoring a diffusion-limited chemical reaction between the two fluid streams. The pH of each stream was modified by the addition of acid or alkali such that a local neutralization reaction occurred as the two fluids molecularly mixed. The progress of this neutralization reaction was tracked by the addition of phenolphthalein – a pH-sensitive chemical indicator – to the acidic stream. Accurately calibrated backlit optical techniques were used to measure the average concentration of the coloured chemical indicator. Comparisons of chemical product formation for pre-transitional buoyancy- and shear-driven mixing layers are given. It is also shown that experiments performed at different equivalence ratios (acid/alkali concentrations) can be combined to obtain a mathematical relationship between the coloured product formed and the density variance. This relationship was used to obtain high-fidelity quantitative measures of the degree of molecular mixing which are independent of probe resolution constraints. The dependence of molecular mixing on the Schmidt and Reynolds numbers is examined by comparing the current Sc ~ 103 measurements with previous Sc = 0.7 gas-phase and Pr = 7 (where Pr is the Prandtl number) liquid-phase measurements. This comparison indicates that the Schmidt number has a large effect on the quantity of mixed fluid at small Reynolds numbers Reh < 103. At larger Reynolds numbers, corresponding to later times in this experiment, all mixing parameters indicated a greater degree of molecular mixing and a decreased Schmidt number dependence. Implications for the development and quantitative assessment of turbulent transport and mixing models appropriate for RT instability-induced mixing are discussed.

We present a combined experimental and numerical investigation of a sphere settling in a linearly stratified fluid at small Reynolds numbers. Using time-lapse photography and numerical modelling, we observed and quantified an increase in drag due to stratification. For a salt stratification, the normalized added drag coefficient scales as Ri0.51, where Ri = a3N2/(νU) is the viscous Richardson number, a the particle radius, U its speed, ν the kinematic fluid viscosity and N the buoyancy frequency. Microscale synthetic schlieren revealed that a settling sphere draws lighter fluid downwards, resulting in a density wake extending tens of particle radii. Analysis of the flow and density fields shows that the added drag results from the buoyancy of the fluid in a region of size (ν/N)1/2 surrounding the sphere, while the bulk of the wake does not influence drag. A scaling argument is provided to rationalize the observations. The enhanced drag can increase settling times in natural aquatic environments, affecting retention of particles at density interfaces and vertical fluxes of organic matter.

This paper presents a detailed investigation of unsteady supersonic and hypersonic flows around spiked-blunt bodies, including the investigation of the effects of the flow field initialization on the flow results. Past experimental research has shown that if the geometry of a spiked-blunt body is such that a shock formation consisting of an oblique foreshock and a bow aftershock appears, then the flow may be unsteady. The unsteady flow is characterized by periodic radial inflation and collapse of the conical separation bubble formed around the spike (pulsation). Beyond a certain spike length the flow is ‘stable’, i.e. steady or mildly oscillating in the radial direction. Both unsteady and ‘stable’ conditions have been reported when increasing or decreasing the spike length during an experimental test and, additionally, hysteresis effects have been observed. The present study reveals that for certain geometries the numerically simulated flow depends strongly on the assumed initial flow field, including the occurrence of bifurcations due to inherent hysteresis effects and the appearance of unsteady flow modes. Computations using several different configurations reveal that the transient (initial) flow development corresponds to a nearly inviscid flow field characterized by a foreshock–aftershock interaction. When the flow is pulsating, the further flow development is not sensitive to initial conditions, whereas for an oscillating or almost ‘steady’ flow, the flow development depends strongly on the assumed initial flow field.

Recent microfluidic experiments by Bremond, Thiam & Bibette (Phys. Rev. Lett., vol. 100, 2008, paper no. 024501), along with simulations by Yoon et al. (Phys. Fluid, vol. 19, 2007, paper no. 102102) and near-contact experiments and simulations by Manica et al. (Langmuir, vol. 24, 2008, pp. 1381–1390), have demonstrated that two droplets can coalesce as they are separating rather than upon their collision. We analyse the experimental microfluidic flow configuration for the approach to contact with a two-dimensional model: we apply a lubrication analysis followed by the method of domain perturbation to determine the droplet deformation as a function of time. We find the approximate shape for the deformed droplet at the time of contact. In particular, for droplets of radius R, moving apart according to h0(t) = h0(0) + αt2, where 2h0(t) is the separation distance, we define a non-dimensional parameter A=4CμR2α1/2/πγ[h0(0)]3/2, where μ is the viscosity of the continuous phase; γ is the interfacial tension; and C depends on the viscosity ratio between the droplets and the continuous phase. Our model suggests that there exists a critical value Acrit = 16/33/2 ≈ 3.0792, below which separation is unlikely to facilitate the coalescence of the droplets. The predictions are in good agreement with available experimental data.

Shear flows of inelastic spheres in three dimensions in the volume fraction range 0.4–0.64 are analysed using event-driven simulations. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution en (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution et (negative of the ratio of post- and pre-collisional velocities perpendicular to the line joining the centres). Here, we have considered both et = +1 and et = en (rough particles) and et = −1 (smooth particles), and the normal coefficient of restitution en was varied in the range 0.6–0.98. Care was taken to avoid inelastic collapse and ensure there are no particle overlaps during the simulation. First, we studied the ordering in the system by examining the icosahedral order parameter Q6 in three dimensions and the planar order parameter q6 in the plane perpendicular to the gradient direction. It was found that for shear flows of sufficiently large size, the system continues to be in the random state, with Q6 and q6 close to 0, even for volume fractions between φ = 0.5 and φ = 0.6; in contrast, for a system of elastic particles in the absence of shear, the system orders (crystallizes) at φ = 0.49. This indicates that the shear flow prevents ordering in a system of sufficiently large size. In a shear flow of inelastic particles, the strain rate and the temperature are related through the energy balance equation, and all time scales can be non-dimensionalized by the inverse of the strain rate. Therefore, the dynamics of the system are determined only by the volume fraction and the coefficients of restitution. The variation of the collision frequency with volume fraction and coefficient of restitution was examined. It was found, by plotting the inverse of the collision frequency as a function of volume fraction, that the collision frequency at constant strain rate diverges at a volume fraction φad (volume fraction for arrested dynamics) which is lower than the random close-packing volume fraction 0.64 in the absence of shear. The volume fraction φad decreases as the coefficient of restitution is decreased from en = 1; φad has a minimum of about 0.585 for coefficient of restitution en in the range 0.6–0.8 for rough particles and is slightly larger for smooth particles. It is found that the dissipation rate and all components of the stress diverge proportional to the collision frequency in the close-packing limit. The qualitative behaviour of the increase in the stress and dissipation rate are well captured by results derived from kinetic theory, but the quantitative agreement is lacking even if the collision frequency obtained from simulations is used to calculate the pair correlation function used in the theory.

The distribution of relative velocities between colliding particles in shear flows of inelastic spheres is analysed in the volume fraction range 0.4–0.64. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution en (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution et (negative of the ratio of post- and pre-collisional velocities perpendicular to line joining the centres).

The distribution of pre-collisional normal relative velocities (along the line joining the centres of the particles) is found to be an exponential distribution for particles with low normal coefficient of restitution in the range 0.6–0.7. This is in contrast to the Gaussian distribution for the normal relative velocity in an elastic fluid in the absence of shear. A composite distribution function, which consists of an exponential and a Gaussian component, is proposed to span the range of inelasticities considered here. In the case of rough particles, the relative velocity tangential to the surfaces at contact is also evaluated, and it is found to be close to a Gaussian distribution even for highly inelastic particles.

Empirical relations are formulated for the relative velocity distribution. These are used to calculate the collisional contributions to the pressure, shear stress and the energy dissipation rate in a shear flow. The results of the calculation were found to be in quantitative agreement with simulation results, even for low coefficients of restitution for which the predictions obtained using the Enskog approximation are in error by an order of magnitude. The results are also applied to the flow down an inclined plane, to predict the angle of repose and the variation of the volume fraction with angle of inclination. These results are also found to be in quantitative agreement with previous simulations.

A linear study is carried out for the axisymmetric and non-axisymmetric instability of a viscous coaxial jet in a radial electric field. The outer liquid is considered to be a leaky dielectric and the inner a perfect dielectric. The generalized eigenvalue problem is solved and the growth rate of disturbance is obtained by using Chebyshev spectral collocation method. The effects of the radial electric field, liquid viscosity, surface tension as well as other parameters on the instability of the jet are investigated. The radial electric field is found to have a strong destabilizing effect on non-axisymmetric modes, especially those having smaller azimuthal wavenumbers. The helical mode becomes prevalent over other modes when the electric field is sufficiently large. Non-axisymmetric modes with high azimuthal wavenumbers may be the most unstable at zero wavenumber. Liquid viscosity has a strong stabilizing effect on both the axisymmetric and non-axisymmetric instability. Relatively, the helical instability is less suppressed and therefore becomes predominant at high liquid viscosity. Surface tension promotes the instability of the para-sinuous mode and meanwhile suppresses the helical and the other non-axisymmetric modes in long wavelength region.

Differential solar heating can result from shading by rooted emergent aquatic plants, producing a temperature difference between vegetated and unvegetated regions of a surface water body. This temperature difference will promote an exchange flow between the vegetation and open water. Drag associated with the submerged portion of the plants modifies this exchange, specifically, changing the dominant velocity scale. Scaling analysis predicts several distinct flow regimes, including inertia-dominated, drag-dominated and energy-limiting regimes. After a constant heat source is initiated, the flow is initially inertial, but quickly transitions to the drag-dominated regime. The energy-limiting regime is not likely to occur in the presence of rooted vegetation. Laboratory experiments describe the exchange flow and confirm the scaling analysis. Particle Imaging Velocimetry (PIV) was used to quantify the velocity field. Once the exchange flow enters the drag-dominated regime, the intrusion velocity uV is steady. The intrusion velocity decreases with increasing density of vegetation. The thickness of the intruding layer is set by the length scale of light penetration.

Development of coherent structures in the separated shear layer and wake of an airfoil in low-Reynolds-number flows was studied experimentally for a range of airfoil chord Reynolds numbers, 55 × 103 ≤ Rec ≤ 210 × 103, and three angles of attack, α = 0°, 5° and 10°. To illustrate the effect of separated shear layer development on the characteristics of coherent structures, experiments were conducted for two flow regimes common to airfoil operation at low Reynolds numbers: (i) boundary layer separation without reattachment and (ii) separation bubble formation. The results demonstrate that roll-up vortices form in the separated shear layer due to the amplification of natural disturbances, and these structures play a key role in flow transition to turbulence. The final stage of transition in the separated shear layer, associated with the growth of a sub-harmonic component of fundamental disturbances, is linked to the merging of the roll-up vortices. Turbulent wake vortex shedding is shown to occur for both flow regimes investigated. Each of the two flow regimes produces distinctly different characteristics of the roll-up and wake vortices. The study focuses on frequency scaling of the investigated coherent structures and the effect of flow regime on the frequency scaling. Analysis of the results and available data from previous experiments shows that the fundamental frequency of the shear layer vortices exhibits a power law dependency on the Reynolds number for both flow regimes. In contrast, the wake vortex shedding frequency is shown to vary linearly with the Reynolds number. An alternative frequency scaling is proposed, which results in a good collapse of experimental data across the investigated range of Reynolds numbers.

Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lower Re and its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier–Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The critical Re for the onset of convective instability is compared with earlier results from local analysis. Good agreement is found.

The normal impact of a drop of yield-stress fluid on a flat rigid surface is investigated experimentally. Using different model fluids (polymer microgels, clay suspensions) and impacted surfaces (partially wettable, super-hydrophobic), we find a rich variety of impact regimes from irreversible viscoplastic coating to giant elastic spreading and recoil. A minimal model of inertial spreading, taking into account an elasto-viscoplastic rheology, allows explaining in a single framework the different regimes and scaling laws. In addition, semi-quantitative predictions for the spread factor are obtained when the measured rheological parameters of the fluid (elasticity, yield stress, viscosity) are injected into the model. Our study offers a means to probe the short-time rheology of yield-stress fluids and highlights the role of elasticity on the unsteady hydrodynamics of these complex fluids. Movies are available with the online version of the paper (go to journals.cambridge.org/flm).

Loitsyanky's integral I = − ∫ r2〈u ⋅u′〉dr is known to be approximately conserved in certain types of fully developed, isotropic turbulence, and its near conservation controls the rate of decay of kinetic energy. Landau suggested that this integral is related to the angular momentum H = ∫ (x × u)dV of some large volume V of the turbulence, according to the expression I = 〈H2〉/V. He also suggested that the approximate conservation of I is related to the principle of conservation of angular momentum. However, Landau's analysis can be criticized because, formally, it applies only to inhomogeneous turbulence evolving in a closed domain. So how are we to interpret the near conservation of I? And what is its relationship, if any, to angular momentum conservation? We show that the key to extending Landau's analysis to strictly homogeneous turbulence is to rewrite Loitsyansky's integral in terms of the vector potential of the velocity field, i.e. I = 6 ∫〈A ⋅ A′〉dr, where ∇ × A = u. This yields I = 6〈[∫VAdV]2〉/V for any large spherical volume V of radius R. Crucially, J = 3∫VAdV can be rewritten as the weighted integral of the angular momentum density throughout all space. This fundamentally changes the way in which we interpret the dynamical behaviour of I. For example, we show that the conservation of 〈J2〉/V, and hence of I, which occurs when the long-range correlations are weak, is a direct consequence of the decorrelation of the flux of angular momentum out through a spherical control surface S and the local angular momentum in the vicinity of S. Thus, within the framework of strictly homogeneous turbulence, we provide the first self-consistent interpretation of Loitsyanky's integral in terms of angular momentum conservation. We also show that essentially the same ideas carry over to certain types of anisotropic turbulence, such as magnetohydrodynamic (MHD), rotating and stratified turbulence. This is important because conservation of angular momentum, which manifests itself in the form of a Loitsyansky-like invariant, places a fundamental restriction on the way in which the integral scales can evolve in such turbulence. This, in turn, controls the rate of decay of energy. We illustrate this by deriving new decay laws for MHD and stratified turbulence. The MHD decay laws are consistent with the available numerical evidence, but further study is required to verify, or otherwise, the predictions for stratified turbulence.

We examine the stability of a suspension of swimming bacteria in a Newtonian medium. The bacteria execute a run-and-tumble motion, runs being periods when a bacterium on average swims in a given direction; runs are interrupted by tumbles, leading to an abrupt, albeit correlated, change in the swimming direction. An instability is predicted to occur in a suspension of ‘pushers’ (e.g. E. Coli, Bacillus subtilis, etc.), and owes its origin to the intrinsic force dipoles of such bacteria. Unlike the dipole induced in an inextensible fibre subject to an axial straining flow, the forces constituting the dipole of a pusher are directed outward along its axis. As a result, the anisotropy in the orientation distribution of bacteria due to an imposed velocity perturbation drives a disturbance velocity field that acts to reinforce the perturbation. For long wavelengths, the resulting destabilizing bacterial stress is Newtonian but with a negative viscosity. The suspension becomes unstable when the total viscosity becomes negative. In the dilute limit (nL3 ≪ 1), a linear stability analysis gives the threshold concentration for instability as (nL3)crit = ((30/Cℱ(r))(DrL/U)(1 + 1/(6τ Dr)))/(1−(15𝒢(r)/Cℱ(r))(DrL/U)(1 + 1/(6τ Dr))) for perfectly random tumbles; here, L and U are the length and swimming velocity of a bacterium, n is the bacterial number density, Dr characterizes the rotary diffusion during a run and τ−1 is the average tumbling frequency. The function ℱ(r) characterizes the rotation of a bacterium of aspect ratio r in an imposed linear flow; ℱ(r) = (r2 −1)/(r2 + 1) for a spheroid, and ℱ(r) ≈ 1 for a slender bacterium (r ≫ 1). The function 𝒢(r) characterizes the stabilizing viscous response arising from the resistance of a bacterium to a deforming ambient flow; 𝒢(r) = 5π/6 for a rigid spherical bacterium, and 𝒢(r)≈ π/45(ln r) for a slender bacterium. Finally, the constant C denotes the dimensionless strength of the bacterial force dipole in units of μU L2; for E. Coli, C ≈ 0.57. The threshold concentration diverges in the limit ((15𝒢(r)/Cℱ(r)) (DrL/U)(1 + 1/(6τ Dr))) → 1. This limit defines a critical swimming speed, Ucrit = (DrL)(15𝒢(r)/Cℱ(r))(1 + 1/(6τ Dr)). For speeds smaller than this critical value, the destabilizing bacterial stress remains subdominant and a dilute suspension of these swimmers therefore responds to long-wavelength perturbations in a manner similar to a suspension of passive rigid particles, that is, with a net enhancement in viscosity proportional to the bacterial concentration.

On the other hand, the stability analysis predicts that the above threshold concentration reduces to zero in the limit Dr → 0, τ → ∞, and a suspension of non-interacting straight swimmers is therefore always unstable. It is then argued that the dominant effect of hydrodynamic interactions in a dilute suspension of such swimmers is via an interaction-driven orientation decorrelation mechanism. The latter arises from uncorrelated pair interactions in the limit nL3 ≪ 1, and for slender bacteria in particular, it takes the form of a hydrodynamic rotary diffusivity (Dhr); for E. Coli, we find Dhr = 9.4 × 10−5(nUL2). From the above expression for the threshold concentration, it may be shown that even a weakly interacting suspension of slender smooth-swimming bacteria (r ≫ 1, ℱ(r) ≈ 1, τ → ∞) will be stable provided Dhr > (C/30)(nUL2) in the limit nL3 ≪ 1. The hydrodynamic rotary diffusivity of E. Coli is, however, too small to stabilize a dilute suspension of these swimmers, and a weakly interacting suspension of E. Coli remains unstable.

We address the question of stability of the Euler flow with elliptical streamlines in a rotating frame, interacting with uniform external magnetic field perpendicular to the plane of the flow. Our motivation for this study is of astrophysical nature, since many astrophysical objects, such as stars, planets and accretion discs, are tidally deformed through gravitational interaction with other bodies. Therefore, the ellipticity of the flow models the tidal deformations in the simplest way. The joint effect of the magnetic field and the Coriolis force is studied here numerically and analytically in the limit of small elliptical (tidal) deformations (ζ ≪ 1), using the analytical technique developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). We find that the effect of background rotation and external magnetic field is quite complex. Both factors are responsible for new destabilizing resonances as the vortex departs from axial symmetry (ζ ≪ 1); however, just like in the non-rotating case, there are three principal resonances causing instability in the leading order. The presence of the magnetic field is very likely to destabilize the system with respect to perturbations propagating in the direction of the magnetic field if the basic vorticity and the background rotation have opposite signs (i.e. for anticyclonic background rotation). We present the dependence of the growth rates of the modes on various parameters describing the system, such as the strength of the magnetic field (h), the inverse of the Rossby number (ℛv), the ellipticity of the basic flow (ϵ) and the direction of propagation of modes (ϑ). Our analytical predictions agree well with the numerical calculations.

The extent or existence of similarities between fully developed turbulent pipes and channels, and in zero-pressure-gradient turbulent boundary layers has come into question in recent years. This is in contrast to the traditionally accepted view that, upon appropriate normalization, all three flows can be regarded as the same in the near-wall region. In this paper, the authors aim to provide clarification of this issue through streamwise velocity measurements in these three flows with carefully matched Reynolds number and measurement resolution. Results show that mean statistics in the near-wall region collapse well. However, the premultiplied energy spectra of streamwise velocity fluctuations show marked structural differences that cannot be explained by scaling arguments. It is concluded that, while similarities exist at these Reynolds numbers, one should exercise caution when drawing comparisons between the three shear flows, even near the wall.

This paper is a direct sequel to Bewley & Aamo (J. Fluid Mech., vol. 499, 2004, pp. 183–196). It was conjectured in that paper, based on the numerical evidence available at that time, that the minimum drag of a constant mass flux channel flow might in fact be that of the laminar flow. This conjecture turned out to be false; Min et al. (J. Fluid Mech., vol. 558, 2006, 309318) discovered a curious control strategy which in fact reduces the time-averaged drag to sub-laminar levels. The present paper establishes rigorously that the power of the control input applied at the walls is always larger than the power saved (due to drag reduction below the laminar level) for any possible control distribution, including that proposed by Min et al. (2006), thus establishing that, energetically (that is accounting for the power saved due to drag reduction and the power exerted by application of the control), the optimal control solution is necessarily to relaminarize the flow.

We consider the solution in the time domain of the two-dimensional water-wave scattering by fixed bodies, which may or may not intersect with the free surface. We show how the problem with arbitrary initial conditions can be found from the single-frequency solutions using a generalized eigenfunction expansion, required because the operator has a continuous spectrum. From this expansion we derive simple formulas for the evolution in time of the initial surface conditions, and we present some examples of numerical calculations.

The generalized temporal residual mean (TRM-G) framework is reviewed and illustrated using a numerical simulation of vertical shear instability. It is shown how TRM-G reveals the physically relevant amount of diapycnal eddy fluxes and implied diapycnal mixing, and how TRM-G relates to the Osborn–Cox relation, which is often used to obtain observational estimates of the diapycnal diffusivity. An exact expression for the diapycnal diffusivity in the TRM-G is given in the presence of molecular diffusion, based on acknowledging and summing up an entire hierarchy of eddy buoyancy moments. In this revised form of the Osborn–Cox relation, diapycnal diffusivity is related only to irreversible mixing of buoyancy, since all advective and molecular flux terms are converted to dissipation of variance and higher order moments. An approximate but closed analytical expression can be given for the revised Osborn–Cox relation with the caveat that this closed expression implies unphysical cross-boundary rotational fluxes.

It is demonstrated that the original Osborn–Cox relation, in which advective and molecular flux terms are simply neglected, is an approximation to the full form valid to first order. In the numerical simulation the original Osborn–Cox relation holds to a surprisingly good approximation despite large advective fluxes of variance and large lateral inhomogeneity in the turbulent mixing.