The violent collapse of bubble clusters is thought to damage adjacent material in both engineering and biomedical applications. Yet the complexities of the root mechanisms have restricted theoretical descriptions to significantly simplified configurations. Reduced-physics models based upon either homogenization or arrays of idealized spherical bubbles do reproduce important gross cluster-scale features. However, these models neglect detailed local bubble–bubble interactions, which are expected to mediate damage mechanisms. To describe these bubble-scale interactions, we simulate the expansion and subsequent collapse of a hemispherical cluster of 50 bubbles adjacent to a plane rigid wall, explicitly representing both the asymmetric dynamics of each bubble within the cluster and the compressible-fluid mechanics of bubble–bubble interactions. Results show that the collapse propagates inward, as visualized in experiments, and that geometric focusing generates high impulsive pressures. This gross behaviour is nearly independent of the specific arrangement of bubbles within the cluster and matches predictions from the corresponding particle and homogenized models we consider. The peak pressure in the detailed simulations is associated with the centremost bubble, which causes a corresponding peak pressure on the nearby wall. However, the peak pressures in all cases are a small fraction – over a factor of ten times smaller in many cases – of those predicted in the corresponding reduced models. This is due to the enhanced focusing in the homogeneous model and the spherical constraint on each bubble in the particle models assessed. These would be important factors to consider in any subsequent predictions of wall damage based upon reduced models.

In this study, shear flows of dry flexible fibres are numerically modelled using the discrete element method (DEM), and the effects of fibre properties on the flow behaviour and solid-phase stresses are explored. In the DEM simulations, a fibre is formed by connecting a number of spheres in a straight line using deformable and elastic bonds. The forces and moments induced by the bond deformation resist the relative normal, tangential, bending and torsional movements between two bonded spheres. The bond or deforming stiffness determines the flexibility of the fibres and the bond damping accounts for the energy dissipation in the fibre vibration. The simulation results show that elastically bonded fibres have smaller effective coefficients of restitution than rigidly connected fibres. Thus, smaller solid-phase stresses are obtained for flexible fibres, particularly with bond damping, compared with rigid fibres. Frictionless fibres tend to align with a small angle from the flow direction as the solid volume fraction increases, and fibre deformation is minimized due to the alignment. However, jamming, with a corresponding sharp stress increase, large fibre deformation and dense contact force network, occurs for fibres with friction at high solid volume fractions. It is also found that jamming is more prevalent in dense flows with larger fibre friction coefficient, rougher surface, larger stiffness and larger aspect ratio.

Bubbles adhered to partially hydrophobic flat surfaces often attain a spherical cap shape with a contact angle much greater than zero. We address the fundamental problem of the diffusion-driven dissolution of a sessile spherical cap bubble (SCB) adhered to a flat smooth surface. In particular, we perform experiments on the dissolution of $\text{CO}_{2}$ bubbles (with initial radii ${\sim}1~\text{mm}$) immersed in air-saturated water adhered to two substrates with different levels of hydrophobicity. It is found that the contact angle dynamics plays an important role in the bubble dissolution rate. A dissolution model for a multicomponent SCB in an isothermal and uniform pressure environment is then devised. The model is based on the quasi-stationary approximation. It includes the effect of the contact angle dynamics, whose behaviour is predicted by means of a simplified model based on the results obtained from adhesion hysteresis. The presence of an impermeable substrate hinders the overall rate of mass transfer. Two approaches are considered in its determination: (a) the inclusion of a diffusion boundary layer–plate interaction model and (b) a finite-difference solution. The model solutions are compared with the experimental results, yielding fairly good agreement.

The deformation of a compound capsule (an elastic capsule with a smaller capsule inside) in simple shear flow is studied by using three-dimensional numerical simulations based on a front tracking method. The inner and outer capsules are concentric and initially spherical. Skalak et al.’s constitutive law is employed for the mechanics of both the inner and outer membranes. Our results concerning the deformation of homogeneous capsules (i.e. capsules without the inner capsules) are quantitatively in agreement with the predictions of previous numerical simulations and perturbation theories. Compared to homogeneous capsules, compound capsules exhibit smaller deformation. The deformations of both the inner and outer capsules are significantly affected by the capillary numbers of the inner and outer membranes and the volume ratio of the inner to the outer capsule. When the inner capsule is small, it presents smaller deformation than the outer capsule. However, when the inner capsule is sufficiently large, it can present larger deformation than the outer capsule, even if the inner membrane has much lower capillary number than the outer membrane. The underlying mechanisms are discussed: (i) the inner capsule is deformed by rotational flow with lower rate of strain rather than by simple shear flow that deforms the outer capsule, and thus the inner capsule exhibits smaller deformation; and (ii) when the inner and outer membranes are sufficiently close (i.e. the inner capsule is sufficiently large), the hydrodynamic interaction between the two membranes becomes significant, which is found to inhibit the deformation of the outer capsule but to promote the deformation of the inner capsule.

The scale-dependent turbulent convection velocity of streamwise velocity fluctuations resolved by large eddy simulation is investigated for the first time across the whole profile of a zero-pressure-gradient spatially developing smooth flat plate boundary layer at $\mathit{Re}_{{\it\theta}}=13\,000$. The high Reynolds number and streamwise heterogeneity constraints motivate the derivation of a dedicated new method to assess the frequency-dependent convection velocity from time signals and their local streamwise derivative, using estimates of power spectral densities (PSDs). This method is inspired by del Álamo & Jiménez (J. Fluid Mech., vol. 640, 2009, pp. 5–26), who treated a lower Reynolds number channel flow with a method suited to spectral direct numerical simulations of streamwise homogeneous flows. Reconstruction of the streamwise spectrum from the time spectrum using the scale-dependent convection velocity is illustrated and compared with classical strategies. The new method inherently includes not only the assessment of the validity of Taylor’s hypothesis, whose trend is remarkably consistent with theoretical predictions by Lin (Q. Appl. Maths, vol. X(4), 1953, 154–165), but also the definition of a global convection velocity accounting for any arbitrary frequency band. This global velocity is shown to coincide with a correlation-based method widely used in experiments. In addition to the mathematical least-squares definition of this velocity, new interpretations based on the flow physics and turbulent micro time scales are presented. Further, the group velocity is assessed and its relation to convection is discussed.

In a stable background density gradient, initially turbulent flows eventually evolve into a state dominated by low-Froude-number dynamics and frequently also contain persistent pattern information. Much empirical evidence has been gathered on these latter stages, but less on how they first got that way, and how information on the turbulence generator may potentially be encoded into the flow in a robust and long-lasting fashion. Here an experiment is described that examines the initial stages of evolution in the vertical plane of a turbulent grid-generated wake in a stratified ambient. Refractive-index-matched fluids allow optically based measurement of early ($Nt<2$) stages of the flow, even when there are strong variations in the local density gradient field. Suitably averaged flow measures show the interplay between internal wave motions and Kelvin–Helmholtz-generated vortical modes. The vertical shear is dominant at the wake edge, and the decay of horizontal vorticity is observed to be independent of $\mathit{Fr}$. Stratified turbulence, originating from Kelvin–Helmholtz instabilities, develops up to non-dimensional time $Nt\approx 10$, and the scale separation between Ozmidov and Kolmogorov scales is independent of $\mathit{Fr}$ at higher $Nt$. The detailed measurements in the near wake, with independent variation of both Reynolds and Froude numbers, while limited to one particular case, are sufficient to show that the initial turbulence in a stratified fluid is neither three-dimensional nor universal. The search for appropriately generalizable initial conditions may be more involved than hoped for.

Undulatory locomotion is an archetypal mode of propulsion for natural swimmers across scales. Undulatory swimmers convert transverse body oscillations into forward velocity by a complex interplay between their flexural movements, morphological features and the fluid environment. Natural evolution has produced a wide range of morphokinematic examples of undulatory swimmers that often serve as inspiration for engineering devices. It is, however, unknown to what extent natural swimmers are optimized for hydrodynamic performance. In this work, we reverse-engineer the morphology and gait for fast and efficient swimmers by coupling an evolution strategy to three-dimensional direct numerical simulations of flows at intermediate Reynolds numbers. The fastest swimmer is slender with a narrow tail fin and performs a sequence of C-starts to maximize its average velocity. The most efficient swimmer combines moderate transverse movements with a voluminous head, tapering into a streamlined profile via a pronounced inflection point. These optimal solutions outperform anguilliform swimming zebrafish in both efficiency and speed. We investigate the transition between morphokinematic solutions in the speed–energy space, laying the foundations for the design of high-performance artificial swimming devices.

Direct numerical simulation of highly accelerated turbulent boundary layers (TBLs) reveals that the Reynolds shear stress, $\overline{u^{\prime }v^{\prime }}^{+}$, monotonically decreases downstream and exhibits a logarithmic behaviour (e.g. $-\overline{u^{\prime }v^{\prime }}^{+}=-(1/A_{uv})\ln y^{+}+B_{uv}$) in the mesolayer region (e.g. $50\leqslant y^{+}\leqslant 170$). The thickness of the log layer of $\overline{u^{\prime }v^{\prime }}^{+}$ increases with the streamwise distance and with the pressure gradient strength, extending over a large portion of the TBL thickness (up to 55 %). Simulations reveal that $V^{+}\,\partial U^{+}/\partial y^{+}\sim 1/y^{+}\sim \partial \overline{u^{\prime }v^{\prime }}^{+}/\partial y^{+}$, resulting in a logarithmic $\overline{u^{\prime }v^{\prime }}^{+}$ profile. Also, $V^{+}\sim -y^{+}$ is no longer negligible as in zero-pressure-gradient (ZPG) flows. Other experimental/numerical data at similar favourable-pressure-gradient (FPG) strengths also show the presence of a log region in $\overline{u^{\prime }v^{\prime }}^{+}$. This log region in $\overline{u^{\prime }v^{\prime }}^{+}$ is larger in sink flows than in other spatially developing FPG flows. The latter flows exhibit the presence of a small power-law region in $\overline{u^{\prime }v^{\prime }}^{+}$, which is non-existent in sink flows.

We use direct numerical simulation (DNS) of the Navier–Stokes equations in the low-Mach-number limit to investigate the hydrodynamic instability of a lifted jet diffusion flame. We obtain steady solutions for flames using a finite rate reaction chemistry, and perform a linear global stability analysis around these steady flames. We calculate the direct and adjoint global modes and use these to identify the regions of the flow that are responsible for causing oscillations in lifted jet diffusion flames, and to identify how passive control strategies might be used to control these oscillations. We also apply a local stability analysis to identify the instability mechanisms that are active. We find that two axisymmetric modes are responsible for the oscillations. The first is a high-frequency mode with wavemaker in the jet shear layer in the premixing zone. The second is a low-frequency mode with wavemaker in the outer part of the shear layer in the flame. We find that both of these modes are most sensitive to feedback involving perturbations to the density and axial momentum. Using the local stability analysis, we find that the high-frequency mode is caused by a resonant mode in the premixing region, and that the low-frequency mode is caused by a region of local absolute instability in the flame, not by the interaction between resonant modes, as proposed in Nichols et al. (Phys. Fluids, vol. 21, 2009, article 015110). Our linear analysis shows that passive control of the low-frequency mode may be feasible because regions up to three diameters away from the fuel jet are moderately sensitive to steady control forces.

An inertia–gravity wave (IGW) propagating in a vertically sheared, rotating stratified fluid interacts with the pair of inertial levels that surround the critical level. An exact expression for the form of the IGW is derived here in the case of a linear shear and used to examine this interaction in detail. This expression recovers the classical values of the transmission and reflection coefficients $|T|=\text{e}^{-{\rm\pi}{\it\mu}}$ and $|R|=0$, where ${\it\mu}^{2}=J(1+{\it\nu}^{2})-1/4$, $J$ is the Richardson number and ${\it\nu}$ the ratio between the horizontal transverse and along-shear wavenumbers. For large $J$, a WKB analysis provides an interpretation of this result in term of tunnelling: an IGW incident on the lower inertial level becomes evanescent between the inertial levels, returning to an oscillatory behaviour above the upper inertial level. The amplitude of the transmitted wave is directly related to the decay of the evanescent solution between the inertial levels. In the immediate vicinity of the critical level, the evanescent IGW is well represented by the quasi-geostrophic approximation, so that the process can be interpreted as resulting from the coupling between balanced and unbalanced motion. The exact and WKB solutions describe the so-called valve effect, a dependence of the behaviour in the region between the inertial levels on the direction of wave propagation. For $J<1$ this is shown to lead to an amplification of the wave between the inertial levels. Since the flow is inertially unstable for $J<1$, this establishes a correspondence between the inertial-level interaction and the condition for inertial instability.

This study is focused on two- and three-dimensional incompressible flow past a circular cylinder for Reynolds number $\mathit{Re}\leqslant 1000$. To gain insight into the mechanisms underlying the suppression of unsteadiness for this flow we determine the nonlinear optimal open-loop control driven by surface-normal wall transpiration. The spanwise-constant wall transpiration is allowed to oscillate in time, although steady forcing is determined to be most effective. At low levels of control cost, defined as the square integration of the control, the sensitivity of unsteadiness with respect to wall transpiration is a good approximation of the optimal control. The distribution of this sensitivity suggests that the optimal control at small magnitude is achieved by applying suction upstream of the upper and lower separation points and blowing at the trailing edge. At high levels of wall transpiration, the assumptions underlying the linearized sensitivity calculation become invalid since the base flow is eventually altered by the size of the control forcing. The large-magnitude optimal control is observed to spread downstream of the separation point and draw the shear layer separation towards the rear of the cylinder through suction, while blowing along the centreline eliminates the recirculation bubble in the wake. We further demonstrate that it is possible to completely suppress vortex shedding in two- and three-dimensional flow past a circular cylinder up to $\mathit{Re}=1000$, accompanied by 70 % drag reduction when a nonlinear optimal control of moderate magnitude (with root-mean-square value 8 % of the free-stream velocity) is applied. This is confirmed through linearized stability analysis about the steady-state solution when the nonlinear optimal wall transpiration is applied. While continuously distributed wall transpiration is not physically realizable, the study highlights localized regions where discrete control strategies could be further developed. It also highlights the appropriate range of application of linear and nonlinear optimal control to this type of flow problem.

Motivated by the importance of understanding mountain weather during periods of thermal convection, a laboratory study was conducted to investigate the separation of an upslope (anabatic) flow on a two-dimensional heated mountainous slope in the absence of a background mean flow. Three flow regimes were identified. In the first, at slope angles ${\it\beta}$ larger than a critical value ${\it\beta}_{c}\approx 20^{\circ }$, the separated flow generated a rising plume completely fed by the anterior upslope flow. For this case, a simple model based on a balance between the opposing vorticities of baroclinicity and shear was proposed to predict the location of the separation point relative to the mountain base. The model also predicts the velocity and length scales at separation as well as those of the rising plume after separation. In the second regime, $10^{\circ }<{\it\beta}\leqslant {\it\beta}_{c}$, the volume flow of the separated plume was not fully supplied by the upslope flow, requiring entrainment of additional ambient fluid at the base of the plume source. The third regime occurred when ${\it\beta}\leqslant 10^{\circ }$, wherein the plume almost completely engulfed the slope, similar to a buoyant plume emanating from a source of finite dimensions, thus overshadowing the upslope flow. Measurements of the separation point conducted during the MATERHORN field research program were consistent with the results of the laboratory experiments and modelling.

The toe pads of frogs consist of soft hexagonal structures and a viscous liquid contained between and within the hexagonal structures. It has been hypothesized that this configuration creates adhesion by allowing for long-range capillary forces, or, alternatively, by allowing for exit of the liquid and thus improving contact of the toe pad. In this work, we suggest interaction between viscosity and elasticity as a mechanism to create temporary adhesion, even in the absence of capillary effects or van der Waals forces. We initially illustrate this concept experimentally by a simplified configuration consisting of two surfaces connected by a liquid bridge and elastic springs. We then utilize poroelastic mixture theory and model frogs’ toe pads as an elastic porous medium, immersed within a viscous liquid and pressed against a rigid rough surface. The flow between the surface and the toe pad is modelled by the lubrication approximation. Inertia is neglected and analysis of the elastic–viscous dynamics yields a governing partial differential equation describing the flow and stress within the porous medium. Several solutions of the governing equation are presented and show a temporary adhesion due to stress created at the contact surface between the solids. This work thus may explain how some frogs (such as the torrent frog) maintain adhesion underwater and the reason for the periodic repositioning of frogs’ toe pads during adhesion to surfaces.

We report upon laboratory experiments and numerical simulations examining the evolution of an interfacial internal solitary wave incident upon a triangular ridge whose peak lies below the interface. If the ridge is moderately large, the wave is observed to shoal and break similar to solitary waves shoaling upon a constant slope, but interfacial waves are also observed to transmit over and reflect from the ridge. In laboratory experiments, by measuring the interface displacement as it evolves in time, we measure the relative transmission and reflection of available potential energy after the incident wave has interacted with the ridge. The numerical simulations of laboratory- and ocean-scale waves measure both the available potential and kinetic energy to determine the partition of incident energy into that which is transmitted and reflected. From shallow-water theory, we define a critical amplitude, $A_{c}$, above which interfacial waves are unstable. The transmission is found to decrease from one to zero as the ratio of the incident wave amplitude to $A_{c}$ increases from less than to greater than one. Empirical fits are made to analytic curves through measurements of the transmission and reflection coefficients.

The interaction between a steady current and propagating surface waves is investigated by means of a perturbation approach, which assumes small values of the wave steepness and considers current velocities of the same order of magnitude as the amplitude of the velocity oscillations induced by wave propagation. The problems, which are obtained at the different orders of approximation, are characterized by a further parameter which is the ratio between the thickness of the bottom boundary layer and the length of the waves and turns out to be even smaller than the wave steepness. However, the solution is determined from the bottom up to the free surface, without the need to split the fluid domain into a core region and viscous boundary layers. Moreover, the procedure, which is employed to solve the problems at the different orders of approximation, reduces them to one-dimensional problems. Therefore, the solution for arbitrary angles between the direction of the steady current and that of wave propagation can be easily obtained. The theoretical results are compared with experimental measurements; the fair agreement found between the model results and the laboratory measurements supports the model findings.

The non-axisymmetric structure of an unstable Stewartson shear layer generated via a differential rotation between flush disks and a cylindrical enclosure is investigated numerically using both three-dimensional direct numerical simulation and a quasi-two-dimensional model. Previous literature has only considered the depth-independent quasi-two-dimensional model due to its low computational cost. The three-dimensional model implemented here highlights the supercritical instability responsible for the polygonal deformation of the shear layer in the linear and nonlinear growth regimes and reveals that linear stability analysis is capable of accurately determining the preferred azimuthal wavenumber for flow conditions near the onset of instability. This agreement is lost for sufficiently forced flows where nonlinear effects encourage the coalescence of vortices towards lower-wavenumber structures. Time-dependent flows are found for large Reynolds numbers defined based on the Stewartson layer thickness and azimuthal velocity differential. However, this temporal behaviour is not solely characterized by Reynolds number but is rather a function of both the Rossby and Ekman numbers. At high Ekman and Rossby numbers, unsteady flow emerges through a small-scale azimuthal destabilization of the axial jets within the Stewartson layers; at low Ekman numbers, unsteady flow emerges through a modulation in the strength of one of the axial vortices rolled up by non-axisymmetric instability of the Stewartson layer.

Many thermoacoustic systems exhibit rich nonlinear behaviour. Recent studies show that this nonlinear dynamics can be well captured by low-order time domain models that couple a level set kinematic model for a laminar flame, the $G$-equation, with a state-space realization of the linearized acoustic equations. However, so far the $G$-equation has been coupled only with straight ducts with uniform mean acoustic properties, which is a simplistic configuration. In this study, we incorporate a wave-based model of the acoustic network, containing area and temperature variations and frequency-dependent boundary conditions. We cast the linear acoustics into state-space form using a different approach from that in the existing literature. We then use this state-space form to investigate the stability of the thermoacoustic system, both in the frequency and time domains, using the flame position as a control parameter. We observe frequency-locked, quasiperiodic and chaotic oscillations. We identify the location of Neimark–Sacker bifurcations with Floquet theory. We also find the Ruelle–Takens–Newhouse route to chaos with nonlinear time series analysis techniques. We highlight important differences between the nonlinear response predicted by the frequency domain and the time domain methods. This reveals deficiencies with the frequency domain technique, which is commonly used in academic and industrial studies of thermoacoustic systems. We then demonstrate a more accurate approach based on continuation analysis applied to time domain techniques.

We present a detailed analysis of temperature statistics in an oceanographic observational dataset. The data are collected using a moored array of thermistors, $100~\text{m}$ tall and starting $5~\text{m}$ above the bottom, deployed during four months above the slopes of a Seamount in the north-eastern Atlantic Ocean. Turbulence at this location is strongly affected by the semidiurnal tidal wave. Mean stratification is stable in the entire dataset. We compute structure functions, of order up to 10, of the distributions of temperature increments. Strong intermittency is observed, in particular, during the downslope phase of the tide, and farther from the solid bottom. In the lower half of the mooring during the upslope phase, the temperature statistics are consistent with those of a passive scalar. In the upper half of the mooring, the temperature statistics deviate from those of a passive scalar, and evidence of turbulent convective activity is found. The downslope phase is generally thought to be more shear-dominated, but our results suggest on the other hand that convective activity is present. High-order moments also show that the turbulence scaling behaviour breaks at a well-defined scale (of the order of the buoyancy length scale), which is however dependent on the flow state (tidal phase, height above the bottom). At larger scales, wave motions are dominant. We suggest that our results could provide an important reference for laboratory and numerical studies of mixing in geophysical flows.

Classical scaling arguments of Kolmogorov, Oboukhov and Corrsin (KOC) are evaluated for turbulence strongly influenced by stable stratification. The simulations are of forced homogeneous stratified turbulence resolved on up to $8192\times 8192\times 4096$ grid points with buoyancy Reynolds numbers of $\mathit{Re}_{b}=13$, 48 and 220. A simulation of isotropic homogeneous turbulence with a mean scalar gradient resolved on $8192^{3}$ grid points is used as a benchmark. The Prandtl number is unity. The stratified flows exhibit KOC scaling only for second-order statistics when $\mathit{Re}_{b}=220$; the $4/5$ law is not observed. At lower $\mathit{Re}_{b}$, the $-5/3$ slope in the spectra occurs at wavenumbers where the bottleneck effect occurs in unstratified cases, and KOC scaling is not observed in any of the structure functions. For the probability density functions (p.d.f.s) of the scalar and kinetic energy dissipation rates, the lognormal model works as well for the stratified cases with $\mathit{Re}_{b}=48$ and 220 as it does for the unstratified case. For lower $\mathit{Re}_{b}$, the dominance of the vertical derivatives results in the p.d.f.s of the dissipation rates tending towards bimodal. The p.d.f.s of the dissipation rates locally averaged over spheres with radius in the inertial range tend towards bimodal regardless of $\mathit{Re}_{b}$. There is no broad scaling range, but the intermittency exponents at length scales near the Taylor length are in the range of $0.25\pm 0.05$ and $0.35\pm 0.1$ for the velocity and scalar respectively.

In order that it be correctly characterized, irreversible turbulent mixing in stratified fluids must distinguish between adiabatic ‘stirring’ and diabatic ‘mixing’. Such a distinction has been formalized through the definition of a diapycnal diffusivity, $K_{{\it\rho}}$ (Winters & D’Asaro, J. Fluid Mech., vol. 317, 1996, pp. 179–193) and an appropriate mixing efficiency, $\mathscr{E}$ (Caulfield & Peltier, J. Fluid Mech., vol. 413, 2000, pp. 1–47). Equivalent attention has not been paid to the definitions of a corresponding momentum diffusivity $K_{m}$ and hence an appropriately defined turbulent Prandtl number $\mathit{Pr}_{t}=K_{m}/K_{{\it\rho}}$. In this paper, the diascalar framework of Winters & D’Asaro (1996) is first reformulated to obtain an ‘Osborn-like’ formula in which the correct definition of irreversible mixing efficiency $\mathscr{E}$ is shown to replace the flux Richardson number which Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89) assumed to characterize this efficiency. We advocate the use of this revised representation for diapycnal diffusivity since the proposed reformulation effectively removes the simplifying assumptions on which the original Osborn formula was based. We similarly propose correspondingly reasonable definitions for $K_{m}$ and $\mathit{Pr}_{t}$ by eliminating the reversible component of the momentum production term. To explore implications of the reformulations for both diapycnal and momentum diffusivity we employ an extensive series of direct numerical simulations (DNS) to investigate the properties of the shear-induced density-stratified turbulence that is engendered through the breaking of a freely evolving Kelvin–Helmholtz wave. The DNS results based on the proposed reformulation of $K_{{\it\rho}}$ are compared with available estimations due to the mixing length model, as well as both the Osborn–Cox and the Osborn models. Estimates based upon the Osborn–Cox formulation are shown to provide the closest approximation to the diapycnal diffusivity delivered by the exact representation. Through compilation of the complete set of DNS results we explore the characteristic dependence of $K_{{\it\rho}}$ on the buoyancy Reynolds number $\mathit{Re}_{b}$ as originally investigated by Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193–214) in their idealized study of homogeneous stratified and sheared turbulence, and show that the validity of their results is only further reinforced through analysis of the turbulence produced in the more geophysically relevant Kelvin–Helmholtz wave life-cycle ansatz. In contrast to the results described by Shih et al. (2005) however, we show that, besides $\mathit{Re}_{b}$, a vertically averaged measure of the gradient Richardson number $\mathit{Ri}_{b}$ may equivalently characterize the turbulent mixing at high $\mathit{Re}_{b}$. Based on the dominant driving processes involved in irreversible mixing, we categorize the intermediate (i.e. $\mathit{Re}_{b}=O(10^{1}{-}10^{2})$) and high (i.e. $\mathit{Re}_{b}>O(10^{2})$) range of $\mathit{Re}_{b}$ as ‘buoyancy-dominated’ and ‘shear-dominated’ mixing regimes, which together define a transition value of $\mathit{Ri}_{b}\sim 0.2$. Mixing efficiency varies non-monotonically with both $\mathit{Re}_{b}$ and $\mathit{Ri}_{b}$, with its maximum (on the order of 0.2–0.3) occurring in the ‘buoyancy-dominated’ regime. Unlike $K_{{\it\rho}}$ which is very sensitive to the correct choice of $\mathscr{E}$ (i.e. $K_{{\it\rho}}\propto \mathscr{E}/(1-\mathscr{E})$), we show that $K_{m}$ is almost insensitive to the choice of $\mathscr{E}$ (i.e. $K_{m}\propto 1/(1-\mathscr{E})$) so long as $\mathscr{E}$ is not close to unity, which implies $K_{m}\approx \mathit{Ri}_{b}\mathit{Re}_{b}$ for the entire range of $\mathit{Re}_{b}$. The turbulent Prandtl number is consequently shown to decrease monotonically with $\mathit{Re}_{b}$ and may be (to first order) simply approximated by $\mathit{Re}_{b}$ itself. Assuming $\mathit{Pr}_{t}=1$, or $\mathit{Pr}_{t}=10$ (as is common in large-scale numerical models of the ocean general circulation), is also suggested to be a questionable assumption.