The goal of this study is to provide a window into the physics of extinction and reignition via three-dimensional simulations of non-premixed combustion in isotropic decaying turbulence using one-step global reaction and neglecting density variations. Initially non-premixed fields of fuel and oxidant are developing in a turbulent field. Due to straining, the scalar dissipation rate is initially increasing and its fluctuations create extinguished regions on the stoichiometric surface. Later in the process, the stoichiometric surface again becomes uniformly hot. Besides using Eulerian data, this research applies flame element tracking and investigates the time history of individual points (‘flame elements’) along the stoichiometric surface. The main focus of the study is the discussion of the different scenarios of reignition. This paper identifies three major scenarios: independent flamelet scenario, reignition via edge (triple) flame propagation, and reignition through engulfment by a hot neighbourhood. The results give insight into the role different scenarios play in the reignition process, reveal the physical processes associated with each scenario, and provide the relative frequency of reignition for each scenario.

The interior of an insulating cylindrical container is supposed filled with an incompressible, electrically conducting, viscous fluid. An externally applied magnetic field is caused to rotate uniformly about an axis parallel to the cylinder generators (by applying two alternating components out of phase at right angles). Induced currents in the fluid give rise to a Lorentz force which drives a velocity field, which in general may have a steady and a fluctuating component. The particular case of a circular cylindrical container in a transverse magnetic field is studied in detail. Under certain reasonable assumptions, the resulting flow is shown to have only the steady component, and the distribution of this component is determined. Some conjectures are offered about the stability of this flow and about the corresponding flows in cavities of general shape.

The internal gravity wave (IGW) field emitted by a stably stratified, initially turbulent, wake of a towed sphere in a linearly stratified fluid is studied using fully nonlinear numerical simulations. A wide range of Reynolds numbers, $\mathit{Re}= UD/ \nu \in [5\times 1{0}^{3} , 1{0}^{5} ] $ and internal Froude numbers, $\mathit{Fr}= 2U/ (ND)\in [4, 16, 64] $ ($U$, $D$ are characteristic body velocity and length scales, and $N$ is the buoyancy frequency) is examined. At the higher $\mathit{Re}$ examined, secondary Kelvin–Helmholtz instabilities and the resulting turbulent events, directly linked to a prolonged non-equilibrium (NEQ) regime in wake evolution, are responsible for IGW emission that persists up to $Nt\approx 100$. In contrast, IGW emission at the lower $\mathit{Re}$ investigated does not continue beyond $Nt\approx 50$ for the three $\mathit{Fr}$ values considered. The horizontal wavelengths of the most energetic IGWs, obtained by continuous wavelet transforms, increase with $\mathit{Fr}$ and appear to be smaller at the higher $\mathit{Re}$, especially at late times. The initial value of these wavelengths is set by the wake height at the beginning of the NEQ regime. At the lower $\mathit{Re}$, consistent with a recently proposed model, the waves propagate over a narrow range of angles that minimize viscous decay along their path. At the higher $\mathit{Re}$, wave motion is much less affected by viscosity, at least initially, and early-time wave propagation angles extend over a broader range of values which are linked to increased efficiency in momentum extraction from the turbulent wake source.

Laser-Doppler velocity measurements were performed on the entry flow in a 90° bend of circular cross-section with a curvature ratio a/R = 1/6. The steady entry velocity profile was parabolic, having a Reynolds number Re = 700, with a corresponding Dean number κ = 286. Both axial and secondary velocities were measured, enabling a detailed description of the complete flow field. The secondary flow at the entrance of the bend was measured to be directed completely towards the inner bend. Significant disturbance of the axial velocity field was not measured until a downstream distance (aR)½. Maximum secondary velocities were measured at 1.7 (aR)½ downstream from the inlet. The development of the axial flow field can be quite well explained from the secondary velocity field.

A linear theory for steady motions in a rotating stratified fluid is presented, valid under the assumption that ε < E, where ε and E are respectively the Rossby and Ekman numbers. The fact that the stable stratification inhibits vertical motions has important consequences and many features of the dynamics of homogeneous rotating fluids are no longer present. For instance, in addition to the absence of the Taylor-Proudman constraint, it is found that Ekman layer suction no longer controls the interior dynamics. In fact, the Ekman layers themselves are frequently absent. Furthermore, the vertical Stewartson boundary layers are replaced by a new kind of boundary layer whose structure is characteristic of rotating stratified fluids. The interior dynamics are found to be controlled by dissipative processes.

Experiments are performed on axisymmetric spreading of viscous drops on glass plates. Two liquids are investigated: silicone oil (M-100), which spreads to ‘infinity’, and paraffin oil, which spreads to a finite-radius steady state. The experiments with silicone oil partly recover the behaviour of previous workers’ data; those experiments with paraffin oil provide new data. It is found that gravitational forces dominate at long enough times while at shorter times capillary forces dominate. When the plate is heated or cooled with respect to the ambient gas, thermocapillary forces generate flows that alter the spreading dynamics. Heating (cooling) the plate is found to retard (augment) the spreading. Moreover, in case of partial wetting, the drop radius finally approached is smaller (larger) for a heated (cooled) plate. These data are all new. All these observations are in good quantitative agreement with the related model predictions of Ehrhard & Davis (1991). A breakdown of the axisymmetric character of the flow is observed only for very long times and/or very thin liquid layers.

The quasi-laminar model for the transfer of energy to a surface wave from a turbulent shear flow (Miles 1957) is modified to incorporate the wave-induced perturbations of the Reynolds stresses, which are related to the wave-induced velocity field through the Boussinesq closure hypothesis and the ancillary hypothesis that the eddy viscosity is conserved along streamlines. It is assumed that the basic mean velocity is U(z) = (U*/κ)log(z/z0) for sufficiently large z (elevation above the level interface) and that U(z1) [Gt ] U* for kz1 = O(1), where k is the wavenumber. The resulting vorticity-transport equation is reduced, through the neglect of diffusion, to a modification of Rayleigh's equation for wave motion in an inviscid shear flow. The energy transfer to the surface wave, which comprises independent contributions from the critical layer (where U = c, the wave speed) and the wave-induced Reynolds stresses, is calculated through a variational approximation and, independently, through matched asymptotic expansions. The critical-layer component is equivalent to that for the quasi-laminar model. The Reynolds-stress component is similar to, but differs quantitatively from, that obtained by Knight (1977, Jacobs (1987) and van Duin & Janssen (1992). The predicted energy transfer agrees with the observational data compiled by Plant (1982) for 1 [lsim ] c/U* [lsim ] 20, but the validity of the logarithmic profile for the calculation of the energy transfer in the critical layer for c/U* < 5 remains uncertain. The basic model is unreliable (for water waves) if c/U* [lsim ] 1, but this domain is of limited oceanographic importance. It is suggested that Kelvin–Helmholtz instability of air blowing over oil should provide a good experimental test of the present Reynolds-stress modelling and that this modelling may be relevant in other geophysical contexts.

The mixing produced by dropping a horizontal grid through a sharp density interface is examined experimentally. The fraction of the available kinetic energy used to mix the fluid, the flux Richardson number Rf, is measured as a function of the overall Richardson number Ri0. It is found that Rf increases from zero as Ri0 does, reaches a maximum, and then decreases with further increase in Ri0. The final interface thickness is found to be a decreasing function of Ri0, and the equivalent vertical diffusion coefficient is calculated. Qualitative observations of the flow show that the final stages of the decay of motion in the interface are characterized by pancake-shaped modes with large horizontal and small vertical scales.

In treating unsteady particle motions in creeping flows, a quasi-steady approximation is often used, which assumes that the particle's motion is so slow that it is composed of a series of steady states. In each of these states, the fluid is in a steady Stokes flow and the total force and torque on the particle are zero. This paper examines the validity of the quasi-steady method. For simple cases of sedimenting spheres, previous work has shown that neglecting the unsteady forces causes a cumulative error in the trajectory of the spheres. Here we will study the unsteady motion of solid bodies in several more complex flows: the rotation of an ellipsoid in a simple shear flow, the sedimentation of two elliptic cylinders and four circular cylinders in a quiescent fluid and the motion of an elliptic cylinder in a Poiseuille flow in a two-dimensional channel. The motion of the fluid is obtained by direct numerical simulation and the motion of the particles is determined by solving their equations of motion with solid inertia taken into account. Solutions with the unsteady inertia of the fluid included or neglected are compared with the quasi-steady solutions. For some flows, the effects of the solid inertia and the unsteady inertia of the fluid are importanty quantitatively but not qualitatively. In other cases, the character of the particles' motion is changed. In particular, the unsteady effects tend to suppress the periodic oscillations generated by the quasi-steady approximation. Thus, the results of quasi-steady calculatioins are never uniformly valid and can be completely misleading. The conditions under which the unsteady effects at small Reynolds numbers are important are explored and the implications for modelling of suspension flows are addressed.

The translational and rotational dynamics of oblate spheroidal particles suspended in a directly simulated turbulent channel flow have been examined. Inertial disk-like particles exhibited a significant preferential orientation in the plane of the mean shear. The rotational inertia about the symmetry axis of the disk-like particles hampered the spin-up of the flattest particles to match the mean flow vorticity. The influence of the particle shape on the orientation and rotation diminished as the translational inertia increased from Stokes number 1 to 30. An isotropization of both orientation and rotation could be observed in the core region of the channel. The translational motion of the oblate spheroids had a weak dependence on the aspect ratio. We therefore concluded that inertial particles sample nearly the same flow field irrespective of shape. Nevertheless, the orientation and rotation of disk-like particles turned out to be qualitatively different from the dynamics of fibre-like particles.

An experimental study of the microwave backscatter and acoustic radiation from breaking waves is reported. It is found that the averaged microwave and acoustic measurements correlate with the dynamics of wave breaking. Both the mean-square acoustic pressure and the backscattered microwave power correlate with the wave slope and dissipation, for waves of moderate slope (S < 0.28). The backscattered power and the mean-square pressure are also found to correlate strongly with each other. As the slope and wavelength of the breaking wave packet is increased, both the backscattered power and the mean-square pressure increase. It is found that a large portion of the backscattered microwave power precedes the onset of sound production and visible breaking. This indicates that the unsteadiness of the breaking process is important and that the geometry of the wave prior to breaking may dominate the backscattering. It is observed that the amount of acoustic energy radiated by an individual breaking wave scaled with the amount of mechanical energy dissipated during breaking. These laboratory results are compared to the field experiments of Farmer & Vagle (1988), Crowther (1989) and Jessup et al. (1990).

Velocity measurements using hot wires are performed across a high-Reynolds-number turbulent plane wake, with the aim of studying the subgrid-scale (SGS) stress and its modelling. This quantity is needed to close the filtered Navier–Stokes equations used for large-eddy simulation (LES) of turbulent flows. Comparisons of various globally time-averaged quantities involving the measured and modelled SGS stress are made, with special emphasis on the SGS energy dissipation rate. Experimental constraints require the analysis of a one-dimensional surrogate of the SGS dissipation. Broadly, the globally averaged results show that all models considered, namely the Smagorinsky and similarity models, as well as the dynamic Smagorinsky model, approximately reproduce profiles of the surrogate SGS dissipation. Some discrepancies near the outer edge of the wake are observed, where the Smagorinsky model slightly overpredicts, and the similarity model underpredicts, energy dissipation unless the filtering scale is about two orders of magnitude smaller than the integral length scale.

A more detailed comparison between real and modelled SGS stresses is achieved by conditional averaging based on particular physical phenomena: (i) the outer intermittency of the wake, and (ii) large-scale coherent structures of the turbulent wake. Thus, the interaction of the subgrid scales with the resolved flow and model viability can be individually tested in regions where isolated mechanisms such as outer intermittency, vortex stretching, rotation, etc., are dominant. Conditioning on outer intermittency did not help to clarify observed features of the measurements. On the other hand, the large-scale organized structures are found to have a strong impact upon the distribution of surrogate SGS energy dissipation, even at filter scales well inside the inertial range. The similarity model is able to capture this result, while the Smagorinsky model gives a more uniform (i.e. unrealistic) distribution. Both dynamic Smagorinsky and similarity models reproduce realistic distributions, but only if all filter levels are contained well inside the inertial range.

Modelling of turbulent passive-scalar diffusion is studied using the statistical results from a two-scale direct-interaction approximation. In this model, the mean scalar, the scalar variance and the dissipation rate of scalar variance constitute fundamental diffusion quantities. The turbulent scalar flux is written in the form of an anisotropic eddy-diffusivity representation. This representation, paving the way for explaining anisotropic heat transport, is tested against typical experimental data. The present model equation for the dissipation rate of scalar variance also gives a theoretical justification for the existing equations that are adopted in the second-order models.

A theoretical study is made of the behaviour of clusters of spheres falling in a viscous fluid under the assumptions that: (a) intertial effects are negligible, (b) the distance between any two spheres is larg compared with their radii. The equations of motion are derived and solved for a number of particular cases and the results compared with the experimental observations of the same motions reported in the preceding paper (Jayaweera, Mason & Slack 1964). For three or four spheres, initially in a horizontal line, the theory is in general agreement with the experiments. Three spheres forming an isosceles triangle are shown to oscillate about the horizontal and about the equlateral shape, so that this theory is unable to explain the observed tendency for three to six spheres to form a regular horizontal polygon. The stability of the steady configuration of n spheres at the vertices of a regular horizontal polygon is examined and it is found that the configuration is only stable for n < 7, which explains why this configuration is not observed for more than six spheres.

In this article, we present a modelling approach for rapid dynamic wetting based on the phase field theory. We show that in order to model this accurately, it is important to allow for a non-equilibrium wetting boundary condition. Using a condition of this type, we obtain a direct match with experimental results reported in the literature for rapid spreading of liquid droplets on dry surfaces. By extracting the dissipation of energy and the rate of change of kinetic energy in the flow simulation, we identify a new wetting regime during the rapid phase of spreading. This is characterized by the main dissipation to be due to a re-organization of molecules at the contact line, in a diffusive or active process. This regime serves as an addition to the other wetting regimes that have previously been reported in the literature.

This study revisits the problem of a steadily propagating semi-infinite hydraulic fracture in which the processes of toughness-related energy release, viscous dissipation and leak-off compete on multiple length scales. This problem typically requires the solution of a system of integro-differential equations with a singular kernel, which is complicated by the need to capture extremely disparate length scales. In this study the governing equations are rewritten in the form of one non-singular integral equation. This reformulation enables the use of standard numerical techniques to capture the complete multiscale behaviour accurately and efficiently. This formulation also makes it possible to approximate the problem by a separable ordinary differential equation, whose closed-form solution captures the multiscale behaviour sufficiently accurately to be used in practical applications. We also consider a similar reformulation of the equations governing the propagation of a buoyancy-driven semi-infinite hydraulic fracture. The resulting numerical solution is able to capture the near-tip multiscale behaviour efficiently and agrees well with published solutions calculated in the large-toughness limit.

In order to determine the heat transfer inside a TIG (tungsten/inert gas) weld pool, it is necessary to have a good understanding of the flow patterns of the liquid metal. The principal force driving the fluid motion is the electromagnetic j × B force due to the current from the welding arc and its self-magnetic field. In this paper we consider the flow of a viscous incompressible conducting fluid in a hemispherical container due to various distributions of the electric current. The problem is posed as a time-dependent problem and is solved numerically using the Du Fort–Frankel leap-frog method. Results are presented for currents of 100 A flowing through the weld pool. This is a typical current for TIG welding, and corresponds to a Reynolds number in the range 200 < Re < 600. Previous solutions of the problem were restricted to low Reynolds numbers, i.e. low currents.

Dilute aqueous solutions of long alcohol chains were recently found to cause a quadratic dependence of surface tension on the temperature without affecting other bulk properties of the liquid: σ = σ0 + αQ(T − T0)2, αQ > 0. The impact of such Marangoni instability on the behaviour of a thin liquid layer is studied in this work. We derive an equation describing a nonlinear spatiotemporal evolution of a thin film. The behaviour of the perturbed film in the absence of gravity, critically depends on whether the temperature T0, yielding a minimal surface tension, is attained on the surface of the film. When this is the case, a qualitatively new behaviour is observed: perturbations of the film interface may evolve into continuous steady patterns that do not rupture. Otherwise, the observed patterns due to the linear and quadratic Marangoni effects are qualitatively similar and result in the rupture of the film into separate drops.

Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.