In this short note we use the results of numerical solutions of the unsteady Navier–Stokes equations to study the occurrence of separation in unsteady flow. The structure of a flow cycle depends strongly on the Strouhal number, whose value can be classified as small, intermediate or large. In each of these regions different effects are calculated, and we give guidelines for the boundaries of these regions. In particular, we show that the small-Strouhal-number region, in which quasi-steady flow is observed, is very small and decreases rapidly as the peak Reynolds number increases.

The structure of premixed turbulent flames is a problem of fundamental interest in combustion theory. Possible flame geometries have been imagined and diagrams indicating the corresponding regimes of combustion have been constructed on the basis of essentially intuitive and dimensional considerations. A new approach to this problem is described in the present paper. An extended definition of flamelet regimes based on the existence of a continuous active (not quenched) flame front separating fresh gases and burnt products is first introduced. Direct numerical simulations of flame/vortex interactions using the full Navier–Stokes equations and a simplified chemistry model are then performed to predict flame quenching by isolated vortices. The formulation includes non-unity Lewis number, non-constant viscosity and heat losses so that the effect of stretch, curvature, transient dynamics and viscous dissipation can be accounted for. As a result, flame quenching by vortices (which is one of the key processes in premixed turbulent combustion) may be computed accurately. The effects of curvature and viscous dissipation on flame/vortex interactions may also be characterized by the same simulations. The influence of non-unity Lewis number and of thermo-diffusive processes in turbulent premixed combustion is discussed by comparing flame responses for two values of the Lewis number (Le = 0.8 and 1.2). An elementary (‘spectral’) diagram giving the response of one flame to a vortex pair is constructed. This spectral diagram is then used, along with certain assumptions, to establish a turbulent combustion diagram similar to those proposed by Borghi (1985) or Williams (1985). Results show that flame fronts are much more resistant to quenching by vortices than expected from the classical theories. A cut-off scale and a quenching scale are also obtained and compared with the characteristic scales proposed by Peters (1986). Results show that strain is not the only important parameters determining flame/vortex interaction. Heat losses, curvature, viscous dissipation and transient dynamics have significant effects, especially for small scales and they strongly influence the boundaries of the combustion regimes. It is found, for example, that the Klimov–Williams criterion which is generally advocated to limit the flamelet region, underestimates the size of this region by more than an order of magnitude.

The orientation dynamics of small anisotropic tracer particles in turbulent flows is studied using direct numerical simulation (DNS) and results are compared with Lagrangian stochastic models. Generalizing earlier analysis for axisymmetric ellipsoidal particles (Parsa et al., Phys. Rev. Lett., vol. 109, 2012, 134501), we measure the orientation statistics and rotation rates of general, triaxial–ellipsoidal tracer particles using Lagrangian tracking in DNS of isotropic turbulence. Triaxial ellipsoids that are very long in one direction, very thin in another and of intermediate size in the third direction exhibit reduced rotation rates that are similar to those of rods in the ellipsoid’s longest direction, while exhibiting increased rotation rates that are similar to those of axisymmetric discs in the thinnest direction. DNS results differ significantly from the case when the particle orientations are assumed to be statistically independent from the velocity gradient tensor. They are also different from predictions of a Gaussian process for the velocity gradient tensor, which does not provide realistic preferred vorticity–strain-rate tensor alignments. DNS results are also compared with a stochastic model for the velocity gradient tensor based on the recent fluid deformation approximation (RFDA). Unlike the Gaussian model, the stochastic model accurately predicts the reduction in rotation rate in the longest direction of triaxial ellipsoids since this direction aligns with the flow’s vorticity, with its rotation perpendicular to the vorticity being reduced. For disc-like particles, or in directions perpendicular to the longest direction in triaxial particles, the model predicts noticeably smaller rotation rates than those observed in DNS, a behaviour that can be understood based on the probability of vorticity orientation with the most contracting strain-rate eigendirection in the model.

The behaviour of a nominally two-dimensional ventilated partial cavity was examined over a wide range of size scales and flow speeds to determine the influence of Froude, Reynolds, and Weber number on the cavity shape, dynamics, and gas entrainment rate. Two geometrically similar experiments were conducted with a 14:1 length scale ratio. The results were compared to a two-dimensional semi-analytical model of the cavity flow, and Froude scaling was found to be sufficient to match basic cavity shapes. However, the air flux required to maintain a stable cavity did not scale with Froude number alone, as the dynamics of the cavity closure changed with increasing Reynolds number. The required air flux differed over one order of magnitude between the lowest and highest Reynolds number flows. But, for sufficiently high Reynolds numbers, the rate of scaled entrainment appeared to approach Reynolds number independence. Modest changes in surface tension of the small-scale experiment suggested that the Weber number was important only at the lowest speeds and smaller length scale. Otherwise, the Weber numbers of the flows were sufficiently high to make the effects of interfacial tension negligible. We also observed that modest unsteadiness in the inflow to the large-scale cavity led to a significant increase in the required air flux needed to maintain a stable cavity, with the required excess gas flux nominally proportional to the flow’s perturbation amplitude. Finally, discussion is provided on how these results relate to model testing of partial cavity drag reduction (PCDR) systems for surface ships.

An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.

Autorotation of an elliptic cylinder about an axis fixed perpendicular to a parallel flow is explained in this paper by means of numerical solutions of the Navier-Stokes equations. Potential-flow theory predicts, for constant angular velocity, half a period in which a torque supports rotation and half a period in which it opposes rotation, with vanishing torque in the average. This balance is disturbed by viscous-flow effects in such a way that, for a given angular velocity, vortex shedding either damps rotation or, under certain conditions, favours rotation. The proper interplay of those conditions, which include synchronization of vortex shedding and rate of rotation, results in auto-rotation. The numerical results for Re [les ] 400 are compared with experimental data for Re = 90000 from the literature. The agreement of the force coefficients and the large-scale flow patterns is surprisingly good.

This study investigates how and why dynamical self-similarities emerge with increasing Reynolds number within the canonical wall flows beyond the transitional regime. An overarching aim is to advance a mechanistically coherent description of turbulent wall-flow dynamics that is mathematically tractable and grounded in the mean dynamical equations. As revealed by the analysis of Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst. A, vol. 24, 2009, pp. 781–807), the equations that respectively describe the mean dynamics of turbulent channel, pipe and boundary layer flows formally admit invariant forms. These expose an underlying self-similar structure. In all cases, two kinds of dynamical self-similarity are shown to exist on an internal domain that, for all Reynolds numbers, extends from $O(\nu / {u}_{\tau } )$ to $O(\delta )$, where $\nu $ is the kinematic viscosity, ${u}_{\tau } $ is the friction velocity and $\delta $ is the half-channel height, pipe radius, or boundary layer thickness. The simpler of the two self-similarities is operative on a large outer portion of the relevant domain. This self-similarity leads to an explicit analytical closure of the mean momentum equation. This self-similarity also underlies the emergence of a logarithmic mean velocity profile. A more complicated kind a self-similarity emerges asymptotically over a smaller domain closer to the wall. The simpler self-similarity allows the mean dynamical equation to be written as a closed system of nonlinear ordinary differential equations that, like the similarity solution for the laminar flat-plate boundary layer, can be numerically integrated. The resulting similarity solutions are demonstrated to exhibit nearly exact agreement with direct numerical simulations over the solution domain specified by the theory. At the Reynolds numbers investigated, the outer similarity solution is shown to be operative over a domain that encompasses ${\sim }40\hspace{0.167em} \% $ of the overall width of the flow. Other properties predicted by the theory are also shown to be well supported by existing data.

The evaporation of sessile drops in quiescent air is usually governed by vapour diffusion. For contact angles below , the evaporative flux from the droplet tends to diverge in the vicinity of the contact line. Therefore, the description of the flow inside an evaporating drop has remained a challenge. Here, we focus on the asymptotic behaviour near the pinned contact line, by analytically solving the Stokes equations in a wedge geometry of arbitrary contact angle. The flow field is described by similarity solutions, with exponents that match the singular boundary condition due to evaporation. We demonstrate that there are three contributions to the flow in a wedge: the evaporative flux, the downward motion of the liquid–air interface and the eigenmode solution which fulfils the homogeneous boundary conditions. Below a critical contact angle of , the evaporative flux solution will dominate, while above this angle the eigenmode solution dominates. We demonstrate that for small contact angles, the velocity field is very accurately described by the lubrication approximation. For larger contact angles, the flow separates into regions where the flow is reversing towards the drop centre.

Turbulent airflow over a Stokes water-wave train of small amplitude is studied numerically based on the two-equation closure model of Saffman & Wilcox (1974) together with appropriate boundary conditions on the wave surface. The model calculates, instead of assuming, the viscous sublayer flow, and it is found that the energy transfer between wind and waves depends significantly on the flow being hydraulically rough, transitional or smooth. Systematic computations have yielded a simple approximate formula for the fractional rate of growth per radian \[ \zeta = \delta_{\rm i}\frac{\rho}{\rho_{\rm w}}\left(\frac{U_{\lambda}}{c}-1 \right)^2, \] with δi = 0.04 for transitional or smooth flow and δi = 0.06 for rough flow, where ρ is density of air, ρw that of water, Uλ wind speed at one wavelength height and c the wave phase velocity. This formula is in good agreement with most existing data from field experiments and from wave-tank experiments. In the case of waves travelling against wind, the corresponding values are δi = −0.024 for transitional and smooth flow, and δi = −0.04 for rough flow.

Noise generation mechanisms for a perfectly expanded supersonic Mach number $M=1.8$ turbulent jet impinging on a $45^{\circ }$ inclined plate are investigated for a Reynolds number of $1.6\times 10^{6}$ employing a large-eddy simulation. Excellent comparisons with experimental acoustic far-field measurements and pressure measurements on the impingement plate are obtained. Two local maxima are identified in the far-field overall sound pressure levels in the $75^{\circ }$ and $120^{\circ }$ observer directions, which are associated with different noise generation mechanisms. The peak frequencies in the spectra with Strouhal numbers of $St=0.2$ for $75^{\circ }$ and $St=0.5$ for $120^{\circ }$ match the experimental measurements. The jet-impingement region generates pressure waves that propagate predominantly in the $120^{\circ }$ observer direction. The noise generation in this region is attributed to vortex stretching and tearing during shear-layer impingement, and shock oscillations that are induced by the motion of downstream convected vortical flow structures. The second peak in the overall sound pressure distribution at $75^{\circ }$ is associated with noise sources located in the wall jet. The noise generation in the wall jet is associated with supersonically convecting large-scale coherent flow structures that also interact with tail shocks in the wall jet causing large localized pressure fluctuations. Strongly coherent flow structures are identified by applying proper orthogonal decomposition (POD) to the unsteady flow field. The frequency characteristics of the most energetic POD modes are distinctly different based on which energy norm is chosen. The most energetic entropy-based POD modes contain a peak frequency of approximately $St=0.4{-}0.6$, while the most energetic turbulent kinetic-energy-based POD modes appear to be dominated by lower-frequency content. The causality method, based on Lighthill’s acoustic analogy, is used to link the acoustic noise signature to the relevant physical mechanisms in the source region. A differentiation is made between the application of normalized and non-normalized cross-correlation functions for noise source identification and characterization.

Oscillatory translational and rotational motions of small particles in viscous fluids are studied for two cases: (i) circular disks and (ii) nearly spherical particles. For circular disks, four motions are treated: broadside and edgewise oscillatory translations and out-of-plane and in-plane oscillatory rotations. In each case the unsteady Stokes equations are reduced to dual integral equations and solved exactly for all frequencies. Streamline portraits of the flow fields are used to understand the evolution of the velocity and pressure fields. The motions of nearly spherical particles are then studied using the reciprocal theorem. Asymptotic formulae for the hydrodynamic resistance tensors are derived and discussed.

The problem of whether a spontaneous singularity can occur in finite time in an incompressible inviscid fluid flow is addressed. As suggested by previous numerical simulations, candidate flows are restricted to be invariant under the octahedral group of symmetries and to have a compact vortex tube in the fundamental domain. It is shown that in such a flow the image vorticity contributes strongly to the axial strain rate on the fundamental in a way which is only weakly proportional to the curvature of the vortex lines. Analysis of a model flow shows that axial strain rate scales as the inverse square of the distance to the origin, and that the velocity field forms a topological trap in which the vortex tube is accelerated towards the origin – a degenerate critical point. Evidence from simulations supports these findings. These features suggest that linear strain rate/vorticity coupling can occur in a finite-time pointwise collapse of such symmetric flows.

Studying particle-laden oscillatory channel flow constitutes an important step towards understanding practical application. This study aims to take a step forward in our understanding of the role of turbulence on fine-particle transport in an oscillatory channel and the back effect of fine particles on turbulence modulation using an Eulerian–Eulerian framework. In particular, simulations presented in this study are selected to investigate wave-induced fine sediment transport processes in a typical coastal setting. Our modelling framework is based on a simplified two-way coupled formulation that is accurate for particles of small Stokes number (St). As a first step, the instantaneous particle velocity is calculated as the superposition of the local fluid velocity and the particle settling velocity while the higher-order particle inertia effect neglected. Correspondingly, only the modulation of carrier flow is due to particle-induced density stratification quantified by the bulk Richardson number, Ri. In this paper, we fixed the Reynolds number to be ReΔ = 1000 and varied the bulk Richardson number over a range (Ri = 0, 1 × 10−4, 3 × 10−4 and 6 × 10−4). The simulation results reveal critical processes due to different degrees of the particle–turbulence interaction. Essentially, four different regimes of particle transport for the given ReΔ are observed: (i) the regime where virtually no turbulence modulation in the case of very dilute condition, i.e. Ri ~ 0; (ii) slightly modified regime where slight turbulence attenuation is observed near the top of the oscillatory boundary layer. However, in this regime a significant change can be observed in the concentration profile with the formation of a lutocline; (iii) regime where flow laminarization occurs during the peak flow, followed by shear instability during the flow reversal. A significant reduction in the oscillatory boundary layer thickness is also observed; (iv) complete laminarization due to strong particle-induced stable density stratification.

The lift forces on rigid spheres entrained in a vortex and a linear shear flow are computed using a direct numerical simulation. The sphere Reynolds number is in the range 10 to 100. The lift coefficient in a vortex is shown to be nearly two orders of magnitude higher than that in a shear flow. The inviscid mechanism is shown to be inadequate to account for the enhanced lift force. The effect of free rotation of the sphere is also shown to be too small to account for the enhanced lift force. Flow structure around the sphere is studied to explain the generation of the strong lift force in a vortex.

Strong near-shore earthquakes are the most frequent sources of tsunamis in many oceans of the world. In the framework of the nonlinear shallow-water theory, the initial sea-surface tsunami elevation is assumed to equal the sea-floor co-seismic displacement produced by the seismic event. This is quantified by means of the analytical formulas due to Okada (1985, 1992), dealing with seismic faults buried in an elastic medium. In this work the propagation of tsunamis is studied along two-dimensional profiles on an idealized constant-slope sea bed, an approximation that allows one to reduce the governing nonlinear equations to a linear problem by means of the classical Carrier & Greenspan (1958) approach. We introduce an analytical solution that is sufficiently general to account for initial conditions associated with paradigmatic cases of sea-bottom deformations produced by near-shore earthquakes, such as subsidence or uplift of the coastal area, and can be also used to treat more complex deformations. The main result is that the amplification of the tsunami height at the coast is found to range between approximately 1 and 2. The amplification is around 1 for tsunamis induced by earthquakes with their epicentre inland and tends to grow as the fault moves seaward. We restrict our analysis to earthquakes that dislocate the shore region. Within the class of sources that we consider, the tsunamis that are most amplified are the ones having initial profiles with a crest–trough–crest system or conversely with a trough–crest–trough system. The bottom slope is found to have no effect on tsunami run-ups and run-downs, but to influence tsunami periods and tsunami speed remarkably. Breaking analysis shows that wave breaking does not occur if the initial wave height is less than 8–9 m, and that the simplest sea-level profiles, which are associated with earthquakes with their epicentre on land, are not expected to break even if their initial height exceeds 19 m.

Rayleigh–Taylor instability experiments are performed using both immiscible and miscible incompressible liquid combinations having a relatively large Atwood number of $A\equiv ({\it\rho}_{2}-{\it\rho}_{1})/({\it\rho}_{2}+{\it\rho}_{1})=0.48$. The liquid-filled tank is attached to a test sled that is accelerated downwards along a vertical rail system using a system of weights and pulleys producing approximately $1g$ net acceleration. The tank is backlit and images are digitally recorded using a high-speed video camera. The experiments are either initiated with forced initial perturbations or are left unforced. The forced experiments have an initial perturbation imposed by vertically oscillating the liquid-filled tank to produce Faraday waves at the interface. The unforced experiments rely on random interfacial fluctuations, resulting from background noise, to seed the instability. The main focus of this study is to determine the effects of forced initial perturbations and the effects of miscibility on the growth parameter, ${\it\alpha}$. Measurements of the mixing-layer width, $h$, are acquired, from which ${\it\alpha}$ is determined. It is found that initial perturbations of the form used in this study do not affect measured ${\it\alpha}$ values. However, miscibility is observed to strongly affect ${\it\alpha}$, resulting in a factor of two reduction in its value, a finding not previously observed in past experiments. In addition, all measured ${\it\alpha}$ values are found to be smaller than those obtained in previous experimental studies.

Dissipation elements are identified for various direct numerical simulation (DNS) fields of homogeneous shear turbulence. The fields are those of the fluctuations of a passive scalar, of the three components of velocity and vorticity, of the second invariant of the velocity gradient tensor, turbulent kinetic energy and viscous dissipation. In each of these fields trajectories starting from every grid point are calculated in the direction of ascending and descending gradients, reaching a local maximum and minimum point, respectively. Dissipation elements are defined as spatial regions containing all the grid points from which the same pair of minimum and maximum points is reached. They are parameterized by the linear length between these points and the difference of the field variable at these points.

In analysing the changes that occur during one time step in the linear length as well as in the number of grid points contained in the elements, it is found that rapid splitting and attachment processes occur between elements. These processes are much more frequent than the previously identified processes of cutting and reconnection. The model for the length-scale distribution function that had previously been proposed is modified to include these additional processes. Comparisons of the length-scale distribution function for the various fields with the proposed model show satisfactory agreement.

The conditional mean difference of the field variable at the minimum and maximum points of dissipation elements is calculated for the passive scalar field and the three components of velocity. While the conditional mean difference follows the 1/3 inertial-range Kolmogorov scaling for the passive scalar field, the scaling exponent differs from the 1/3 law for each of the three components of velocity. This is thought to be due to the relatively high shear rate of the DNS calculations.

The conditional mean viscous dissipation shows, differently from all other field variables analysed, a pronounced dependence on the linear length of elements. This is explained by intermittency. This finding is used to evaluate the production and the dissipation term of the empirically derived ϵ-equation that is often used in engineering calculations.

Mean velocity and mean turbulent field measurements are performed for the case of a three-dimensional turbulent boundary layer on an axially rotated cylinder. The cylinder model consists of two parts: a stationary section followed by a spinning afterbody. Techniques of hot-wire anemometry are employed, which yield complete mean velocity and turbulence measurements in skewed flows. The general behaviour of the three-dimensional boundary layer is first discussed: two asymptotic layers analogous to the two-dimensional wall and defect layers are observed; they are shown to evolve from the equations of mean motion. The hypothesis of scalar eddy viscosity is investigated in the light of these results. Using conventional length scale assumptions together with the Reynolds stress tensor equations, a prediction of curvature effects in the law of the wall region is developed; a result in the present case is a smaller slope of the semi-logarithmic portion of the law of the wall, No assumptions over and above those necessary for plane, two-dimensional flow are required for this analysis. The geometry of the model is such that a rapid change in mean rate of strain occurs along the streamlines. From the history of the components of the $\overline{u_iu_j}$ tensor, it is possible to draw some fundamental conclusions concerning the dynamics of the energy dissipation, diffusion and redistribution processes.