Using direct numerical simulations (DNS), we explore local and volume-integrated measures of turbulence and mixing in breaking internal waves on slopes. We consider eight breaking wave cases with a range of normalized pycnocline thicknesses $k\unicode[STIX]{x1D6FF}$, where $k$ is the horizontal wavenumber and $\unicode[STIX]{x1D6FF}$ is the pycnocline thickness, but with similar incoming wave properties. The energetics of wave breaking is quantified in terms of local turbulent dissipation and irreversible mixing using the method of Scotti & White (J. Fluid Mech., vol. 740, 2014, pp. 114–135). Local turbulent mixing efficiencies are calculated using the irreversible flux Richardson number $R_{f}^{\ast }$ and are found to be a function of the turbulent Froude number $Fr_{k}$. Volume-integrated measures of the turbulent mixing efficiency during wave breaking are also made, and are found to be functions of $k\unicode[STIX]{x1D6FF}$. The bulk turbulent mixing efficiency ranges from 0.25 to 0.37 and is maximized when $k\unicode[STIX]{x1D6FF}\approx 1$. In order to connect local and bulk mixing efficiency measures, the variation in the bulk turbulent mixing efficiency with $k\unicode[STIX]{x1D6FF}$ is related to the turbulent Froude number at which the maximum total mixing occurs over the course of the breaking event, $Fr_{k}^{max}$. We find that physically, $Fr_{k}^{max}$ is controlled by the vertical length scale of billows at the interface during wave breaking.

The characteristic function defining the eigenvalues of the Orr-Sommerfeld equation is discussed and it is shown how the expected analytic properties of this function can be exploited to generate series expansions defining eigenvalues within the circle of convergence. This technique is applied to the modes arising in the Blasius flat-plate boundary layer (treated as a parallel flow), for which the complex wavenumber α can be expanded as a convergent power series in the complex frequency parameter β in various regions of the β plane. Such power series are effectively equivalent to Fourier expansions and the properties of the latter are used to find the coefficients.

A square-root singularity in the relationship between α and β is found and it is shown how α can, nevertheless, be expressed in terms of β as the sum of one regular series and the square root of a second regular series. The loci of the real and imaginary parts of α have been computed from these series and show the behaviour in the neighbourhood of the branch point.

The series description provides a particularly simple and rapid method of evalauting eigenvalues and their derivatives within any given region.

Studies on the $\alpha $- and $\beta $-phenomena, terms coined by Bouard & Coutanceau (J. Fluid Mech., vol. 101, 1980, pp. 583–607) for the flow past an impulsively started circular cylinder, have been confined only to the very early stages of the flow. In this paper, besides making a comprehensive in-depth analysis of these phenomena for a much longer period of time, we report the existence of some tertiary vortex phenomena for the first time, which we term the sub-$\alpha $- and sub-$\beta $-phenomena. The mechanism of unsteady flow separation at high Reynolds numbers for the flow past a circular cylinder developed in the last two decades has been used to understand these flow phenomena. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier–Stokes equations for the range of Reynolds number $500\leq \mathit{Re}\leq 10\hspace{0.167em} 000$. We specifically choose $\mathit{Re}= 5000$ to describe the interplay among the primary, secondary and tertiary vortices leading to these interesting vortex dynamics. We also report a $\beta $-like phenomenon which is very similar to the $\beta $-phenomenon, but slightly differs in details. We offer a new perception of the $\alpha $-phenomenon by defining its existence in a strong and weak sense along with a clearer characterization of the $\beta $-phenomenon. Apart from numerical computation, a detailed theoretical characterization using topological aspects of the boundary layer separation leading to the secondary and tertiary vortex phenomena has also been carried out. We compare our numerical results with established experimental and numerical results wherever available and an excellent match with the experimental results is obtained in all cases.

The effect of solute concentration on hindered diffusion of sphere-like colloidal solutes in stiff polymer hydrogels is examined theoretically and experimentally. In the theoretical development, it is shown that the presence of the gel fibres enhances the effect of concentration on the thermodynamic driving force for gradient diffusion, while simultaneously reducing the effect of concentration on the hydrodynamic drag. The result is that gradient diffusion depends more strongly on solute concentration in gels than it does in pure solution, by an amount that depends on the partition coefficient and hydraulic permeability of the gel–solute system. Quantitative calculations are made to determine the concentration-dependent diffusivity correct to first order in solute concentration. In order to compare the theoretical predictions with experimental data, rates of diffusion have been measured for nonionic micelles and globular proteins in solution and agarose hydrogels at two gel concentrations. The measurements were performed by using holographic interferometry, through which one monitors changes in refractive index as gradient diffusion takes place within a transparent gel. If the solutes are modelled as spheres with short-range repulsive interactions, then the experimentally measured concentration dependence of the diffusivities of both the protein and micelles is in good agreement with the theoretical predictions.

A generalized solution for pressure-driven flow through a permeable rotating inner cylinder in an impermeable concentric outer cylinder, a situation used commercially in rotating filtration, is challenging due to the interdependence between the pressure drop in the axial direction and that across the permeable inner cylinder. Most previous approaches required either an imposed radial velocity at the inner cylinder or radial throughflow with both the inner and outer cylinders being permeable. We provide an analytical solution for rotating Couette–Poiseuille flow with Darcy's law at the inner cylinder by using a small parameter related to the permeability of the inner cylinder. The theory works for suction, injection and even combined suction/injection, when the axial pressure drop in the annulus is such that the transmembrane pressure difference reverses sign along the axial extent of the system. Corresponding numerical simulations for finite-length systems match the theory very well.

The effect of gravity on the dynamics of non-isothermic ultra-thin two-layer films is studied in this paper. The joint action of disjoining pressure and thermocapillary forces is taken into account. The problem is considered in a long-wave approximation. The linear stability of a quiescent state and thermocapillary flows is investigated. It has been found that the influence of the upper fluid density is significantly stronger than that of the difference of fluid densities. Nonlinear flow regimes are studied by means of numerical simulations. The gravity can lead to the formation of stripes or holes instead of droplets. The two-dimensional wavy patterns are replaced by one-dimensional waves with the fronts inclined or transverse to the direction of the horizontal temperature gradient.

We present a new approach to model coastal waves in the shoaling and surf zones. The model can be described as a depth-averaged large-eddy simulation model with a cutoff in the inertial subrange. The large-scale turbulence is explicitly resolved through an extra variable called enstrophy while the small-scale turbulence is modelled with a turbulent-viscosity hypothesis. The equations are derived by averaging the mass, momentum and kinetic energy equations assuming a shallow-water flow, a negligible bottom shear stress and a weakly turbulent flow assumption which is not restrictive in practice. The model is fully nonlinear and has the same dispersive properties as the Green–Naghdi equations. It is validated by numerical tests and by comparison with experimental results of the literature on the propagation of a one-dimensional solitary wave over a mild sloping beach. The wave breaking is characterized by a sudden increase of the enstrophy which allows us to propose a breaking criterion based on the new concept of virtual enstrophy. The model features three empirical parameters. The first one governs the turbulent dissipation and was found to be a constant. The eddy viscosity is determined by a turbulent Reynolds number depending only on the bottom slope. The third parameter defines the breaking criterion and depends only on the wave initial nonlinearity. These dependences give a predictive character to the model which is suitable for further developments.

We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continuum field, and the cycle is repeated. The target applications for this type of method are transition-regime gas flows for which standard continuum models (e.g. Navier–Stokes equations) cannot be used, but solution of Boltzmann's equation is prohibitively expensive. The use of molecular-level data to close the conservation laws significantly extends the range of applicability of the continuum conservation laws. In this study, the CM3 is used to perform two proof-of-principle calculations: a low-speed Rayleigh flow and a thermal Fourier flow. Velocity, temperature, shear-stress and heat-flux profiles compare well with direct-simulation Monte Carlo solutions for various Knudsen numbers ranging from the near-continuum regime to the transition regime. We discuss algorithmic problems and the solutions necessary to implement the CM3, building upon the conceptual framework of the heterogeneous multiscale methods.

We study turbulent channel flow of a binary mixture of finite-sized neutrally buoyant rigid particles by means of interface-resolved direct numerical simulations. We fix the bulk Reynolds number and total solid volume fraction, $Re_{b}=5600$ and $\unicode[STIX]{x1D6F7}=20\,\%$, and vary the relative fraction of small and large particles. The binary mixture consists of particles of two different sizes, $2h/d_{l}=20$ and $2h/d_{s}=30$ where $h$ is the half-channel height and $d_{l}$ and $d_{s}$ the diameters of the large and small particles. While the particulate flow statistics exhibit a significant alteration of the mean velocity profile and turbulent fluctuations with respect to the unladen flow, the differences between the mono-disperse and bi-disperse cases are small. However, we observe a clear segregation of small particles at the wall in binary mixtures, which affects the dynamics of the near-wall region and thus the overall drag. This results in a higher drag in suspensions with a larger number of large particles. As regards bi-disperse effects on the particle dynamics, a non-monotonic variation of the particle dispersion in the spanwise (homogeneous) direction is observed when increasing the percentage of small/large particles. Finally, we note that particles of the same size tend to cluster more at contact whereas the dynamics of the large particles gives the highest collision kernels due to a higher approaching speed.

We model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

Many of the statistical properties of both the velocity and the vorticity fields of a nominally zero-pressure-gradient turbulent boundary layer at Rδ = 27650 (Rθ = 2685) have been simultaneously measured. The measurements were made with a small nine-sensor hot-wire probe which can resolve the turbulence to within about six Kolmogorov microscales just above the sublayer. The statistical properties of the velocity vector field compare very well with other laboratory measurements and with direct numerical simulations when Reynolds-number dependence is taken into account. The statistical properties of the vorticity field are also in generally good agreement with the few other measurements and with the direct numerical simulations available for comparison. Near the wall, r.m.s. measurements show that the fluctuating spanwise vorticity is the dominant component, but in the outer part of the boundary layer all the component r.m.s. values are nearly equal. R.m.s. measurements of the nine individual velocity gradients show that the gradients normal to the wall of all three velocity components are the largest, with peaks occurring near the wall as expected. Gradients in the streamwise direction are everywhere small. One-dimensional spectra of the vorticity components show the expected shift of the maximum energy to higher wavenumbers compared to spectra of the velocity components at the same location in the flow. The budget of the transport equation for total enstrophy indicates that the viscous dissipation rate is primarily balanced by the viscous diffusion rate in the buffer layer and by the rotation and stretching rate in the logarithmic layer.

A liquid jet falling by gravity ultimately destabilizes by capillary forces. Accelerating as it falls, the jet thins and stretches, causing a capillary instability to develop on a spatially varying substrate. We discuss quantitatively the interplay between instability growth, jet thinning and longitudinal stretching for two kinds of perturbations, either solely introduced at the jet nozzle exit or affecting the jet all along its length. The analysis is conducted for any values of the liquid properties for a sufficiently large flow rate. In all cases, we determine the net gain of the most dangerous perturbation for all downstream distances, thus predicting the jet length, the wavelength at breakup and the resulting droplet size.