The stability of small travelling-wave disturbances in the flow over a flat plate is discussed. An iterative method is used to generate an asymptotic series solution in inverse powers of the Reynolds number Rx = Ux/v to the power one half. The neutral-stability boundaries given by the first two terms of this series are obtained and compared with experimental data. It is shown that the parallel flow approximation leads to a valid solution at very large Reynolds numbers.

We present the results of direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up to 120 million particles with Stokes numbers, St, in the range from 0.16 to 3.5 we are able to characterize in full detail the statistics of particle acceleration. We show that: (i) the root-mean-squared acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer value at quite small Stokes numbers; (ii) at a given St the normalized acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently with the trend observed for fluid tracers; (iii) the tails of the probability density function of the normalized acceleration $a/a_{\rm rms}$ decrease with St. Two concurrent mechanisms lead to the above results: preferential concentration of particles, very effective at small St, and filtering induced by the particle response time, that takes over at larger St.

A transformation is described which relates the sound generated by low Mach number flow to the flow vorticity. For compact flow fields the apparent sound source is of quadrupole type and linear in the vorticity and therefore also linear in the flow velocity. This scheme is applied to the sound generated by the interaction of two identical thin vortex rings. Then a flow field with a number of compact vortices is discussed. It is found that each vortex can be replaced acoustically by a dipole related to the impulse of the vortex, plus the quadrupole just mentioned plus a spherically symmetric sound source related to the energy of the vortex. An application to low Mach number free-space turbulence shows that the generated sound is related to the vorticity correlation tensor.

Results from direct numerical simulation (DNS) for three-dimensional Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1/2 and Prandtl number Pr=0.7 are presented. They span five decades of Rayleigh number Ra from 2 × 106 to 2 × 1011. The results are in good agreement with the experimental data of Niemela et al. (Nature, vol. 404, 2000, p. 837). Previous DNS results from Amati et al. (Phys. Fluids, vol. 17, 2005, paper no. 121701) showed a heat transfer that was up to 30% higher than the experimental values. The simulations presented in this paper are performed with a much higher resolution to properly resolve the plume dynamics. We find that in under-resolved simulations the hot (cold) plumes travel further from the bottom (top) plate than in the better-resolved ones, because of insufficient thermal dissipation mainly close to the sidewall (where the grid cells are largest), and therefore the Nusselt number in under-resolved simulations is overestimated. Furthermore, we compare the best resolved thermal boundary layer profile with the Prandtl–Blasius profile. We find that the boundary layer profile is closer to the Prandtl–Blasius profile at the cylinder axis than close to the sidewall, because of rising plumes close to the sidewall.

It has been recognized that the subgrid-scale (SGS) parameterization represents a critical component of a successful large-eddy simulation (LES). Commonly used linear SGS models produce erroneous mean velocity profiles in LES of high-Reynolds-number boundary layer flows. Although recently proposed approaches to solving this problem have resulted in significant improvements, questions about the true nature of the SGS problem in shear-driven high-Reynolds-number flows remain open.

We argue that the SGS models must capture inertial transfer effects including backscatter of energy as well as its redistribution among the normal SGS stress components. These effects are the consequence of nonlinear interactions and anisotropy. In our modelling procedure we adopt a phenomenological approach whereby the SGS stresses are related to the resolved velocity gradients. We show that since the SGS stress tensor is not frame indifferent a more general nonlinear model can be applied to the SGS parameterization. We develop a nonlinear SGS model capable of reproducing the effects of SGS anisotropy characteristic for shear-driven boundary layers. The results obtained using the nonlinear model for the LES of a neutral shear-driven atmospheric boundary layer show a significant improvement in prediction of the non-dimensional shear and low-order statistics compared to the linear Smagorinsky-type models. These results also demonstrate a profound effect of the SGS model on the flow structures.

In an attempt to explain the failure of the various pure homogeneous strain experiments to reach equilibrium (and consequently to support the contention of Townsend of an equilibrium structure of the Reynolds stress dependent only on geometry), the nature of the general Reynolds stress-mean velocity relation is examined. It is shown that if homogeneous flows become asymptotically independent of initial conditions, and if the Reynolds stress bearing structure can be characterized by a single time scale (i.e.–at sufficiently high Reynolds number) then these flows behave like classical non-linear viscoelastic media, with the Reynolds stress structure dependent on the (strain-rate) (time scale) product. Thus, the existence of an equilibrium structure implies the existence of an equilibrium time scale and a universal value of the product. The ideas permitting Reynolds stress and mean velocity to be related are applied to the dissipative structure in homogeneous flows, and it is found that in such flow the time scale never ceases to grow, so that these flows can never reach an equilibrium structure. With the aid of an ad-hoc assumption these flows are examined in some detail, and the results of experiments are predicted with considerable accuracy. It is suggested that (inhomogeneous) flows having an equilibrium time scale may, in the homogeneous limit, be expected to display a universal structure. The small departure from universality induced by the large eddies associated with inhomogeneity may be adequately predicted by this same ad-hoc model.

The problem of turbulent-boundary-layer separation due to an adverse pressure gradient is an old but still important problem in many fluid flow devices. Until recent years little quantitative experimental information was available on the flow structure downstream of separation because of the lack of proper instrumentation. The directionally sensitive laser anemometer provides the ability to measure the instantaneous flow direction and magnitude accurately.

The experimental results described here are concerned with a nominally two-dimensional, separating turbulent boundary layer for an airfoil-type flow in which the flow was accelerated and then decelerated until separation. Upstream of separation single and cross-wire hot-wire anemometer measurements are also presented. Measurements in the separated zone with a directionally sensitive laser-anemometer system were obtained for U, V, $\overline{u^2}, \overline{v^2}, - \overline{uv}$, the fraction of time that the flow moves downstream, and the fraction of time that the flow moves away from the wall.

In addition to confirming the earlier conclusions of Simpson, Strickland & Barr (1977) regarding a separating airfoil-type turbulent boundary layer, much new information about the separated region has been gathered. (1) The backflow mean velocity profile scales on the maximum negative mean velocity UN and its distance from the wall N. A U+vs. y+ law-of-the-wall velocity profile is not consistent with this result. (2) The turbulent velocities are comparable with the mean velocity in the backflow, although low turbulent shearing stresses are present. (3) Mixing length and eddy viscosity models are physically meaningless in the backflow and have reduced values in the outer region of the separated flow.

Downstream of fully developed separation, the mean backflow appears to be divided into three layers: a viscous layer nearest the wall that is dominated by the turbulent flow unsteadiness but with little Reynolds shearing stress effects; a rather flat intermediate layer that seems to act as an overlap region between the viscous wall and outer regions; and the outer backflow region that is really part of the large-scaled outer region flow. The Reynolds shearing stress must be modelled by relating it to the turbulence structure and not to local mean velocity gradients. The mean velocities in the backflow are the results of time averaging the large turbulent fluctuations and are not related to the source of the turbulence.

The distance between the separation point and the final approach to a fully developed turbulent mixing layer is found to be of the order of a thousand times the momentum-deficit thickness of the initial boundary layer, whether the latter be laminar or turbulent. There are correspondingly large shifts in the virtual origin of the mixing layer, resulting in spurious Reynolds-number effects which cause considerable difficulties in tests of model jets or blunt-based bodies, and which are probably responsible for the disagreements over the influence of Mach number on the development of free shear layers. These effects are explained.

We present experimental results supporting physics-based ejecta model development, where our main assumption is that ejecta form as a special limiting case of a Richtmyer–Meshkov (RM) instability at a metal–vacuum interface. From this assumption, we test established theory of unstable spike and bubble growth rates, rates that link to the wavelength and amplitudes of surface perturbations. We evaluate the rate theory through novel application of modern laser Doppler velocimetry (LDV) techniques, where we coincidentally measure bubble and spike velocities from explosively shocked solid and liquid metals with a single LDV probe. We also explore the relationship of ejecta formation from a solid material to the plastic flow stress it experiences at high-strain rates () and high strains (700 %) as the fundamental link to the onset of ejecta formation. Our experimental observations allow us to approximate the strength of Cu at high strains and strain rates, revealing a unique diagnostic method for use at these extreme conditions.

This paper describes experiments concerning the structure of large-scale vortices and the unsteady reverse-flow properties in the reattaching zone of a nominally two-dimensional separation bubble formed at the leading edge of a blunt flat plate with right-angled corners. The experiment was performed in a wind tunnel with a constant Reynolds number 2.6 × 104 (based on the main-flow velocity and the thickness of the plate). Split-film probes, being sensitive to instantaneous reversals of flow direction, were extensively employed. An important feature of this study is a judicious use of surface-pressure fluctuations as a conditioning signal to educe the structure of the large-scale vortices.

Distributions of fluctuating-velocity vectors and contour lines of high-frequency turbulent energy in a few space–time domains are presented and discussed. The most economical interpretation of these space-time distributions is that the large-scale vortices in the reattaching zone are hairpin vortices whose configuration is sketched in the text. The unsteady flow in the reattaching zone is mainly governed by two agents; the motion of the large-scale vortices and the low-frequency unsteadiness. The unsteady flow is clarified in terms of the motion (in a space–time domain) of zeros of the longitudinal velocity close to the surface of the plate; the effects of the two agents on this motion are presented separately. On the basis of these results, a mathematical model of the unsteady flow in the reattaching zone is suggested and found to yield good comparison with measured reverse-flow intermittency and frequency of local-flow reversals. It appears that the separation bubble experiences shrinkage and enlargement in connection with the low-frequency unsteadiness and that the speed of shrinkage is much greater than that of enlargement. The strength of the large-scale vortices in the reattaching zone seems to be dependent on the phase of the low-frequency unsteadiness.

A third-order theory has been developed to study capillary instability of a liquid jet. The result shows that the asymmetrical development of an initially sinusoidal wave is a non-linear effect with generation of higher harmonics as well as feedback into the fundamental. The growth of the surface wave is found to depend explicitly on the dimensionless initial amplitude of the disturbance and the dimensionless wave-number k of the wave. For the same initial disturbance, the wave is found to have a maximum growth rate at k = 0·7 in agreement with the linearized theory. For the same wave-number, the growth is proportional to the initial amplitude of the disturbance. The cut-off wave-number and the fundamental frequency (or growth rate for the unstable case) of the wave for a given k are found to be different from the linearized theory. Furthermore, at the cut-off wave-number, the present theory shows the disturbance experiences a growth which is proportional to t2. The excellent agreement between Donnelly & Glaberson's experiment and Rayleigh's linearized theory is found to be due to their method of measurement.

The effects of fluid inertia on the pressure drop required to drive fluid flow through periodic and random arrays of aligned cylinders is investigated. Numerical simulations using a lattice-Boltzmann formulation are performed for Reynolds numbers up to about 180.

The magnitude of the drag per unit length on cylinders in a square array at moderate Reynolds number is strongly dependent on the orientation of the drag (or pressure gradient) with respect to the axes of the array; this contrasts with Stokes flow through a square array, which is characterized by an isotropic permeability. Transitions to time-oscillatory and chaotically varying flows are observed at critical Reynolds numbers that depend on the orientation of the pressure gradient and the volume fraction.

In the limit Re[Lt ]1, the mean drag per unit length, F, in both periodic and random arrays, is given by F/(μU) =k1+k2Re2, where μ is the fluid viscosity, U is the mean velocity in the bed, and k1 and k2 are functions of the solid volume fraction ϕ. Theoretical analyses based on point-particle and lubrication approximations are used to determine these coefficients in the limits of small and large concentration, respectively.

In random arrays, the drag makes a transition from a quadratic to a linear Re-dependence at Reynolds numbers of between 2 and 5. Thus, the empirical Ergun formula, F/(μU) =c1+c2Re, is applicable for Re>5. We determine the constants c1 and c2 over a wide range of ϕ. The relative importance of inertia becomes smaller as the volume fraction approaches close packing, because the largest contribution to the dissipation in this limit comes from the viscous lubrication flow in the small gaps between the cylinders.

In the absence of Brownian motion, inertia and inter-particle forces, two smooth spheres collide in a simple shear flow in a reversible way returning to their initial streamlines. Because the minimum separation during the collision can be less than 10−4 of the radius, quite a small surface roughness can have a significant irreversible effect on the collision. We calculate the change between the initial and final streamlines caused by roughness. Repeated random collisions in a dilute suspension lead to a diffusion of the particles across the streamlines. We calculate the shear-induced diffusivity for both self-diffusion and down-gradient diffusion.

Flow structure, momentum and heat transport in the wake of two tandem circular cylinders have been experimentally investigated. Measurements were conducted at $x/d\,{=}\, $10, 20 and 30 ($d$ is the cylinder diameter) at a Reynolds number of 7000 using a three-wire (one cross-wire plus a cold wire) probe, in conjunction with a cross-wire. The upstream cylinder was slightly heated. The flow behind two tandem cylinders is conventionally divided into three regimes based on whether the shear layers separated from the upstream cylinder overshoot or reattach on the downstream cylinder before forming a vortex street, or form vortices between the cylinders. The present investigation uncovers two remarkably different flow structures in the reattachment regime, depending on whether the shear layers from the upstream cylinder reattach on the downstream or upstream side of the downstream cylinder. As such, four cylinder centre-to-centre spacing ratios, i.e. $L/d\,{=}\,$1.3, 2.5, 4.0 and 6.0, were examined, each representing one distinct flow structure. The phase-averaged sectional streamlines and vorticity contours display a single vortex street, irrespective of different regimes. However, the detailed flow structure, in particular, the vortex strength, and its downstream development depend upon $L/d$. The cross-stream distributions of the Reynolds stresses and heat fluxes at a given $x/d$ vary from one to another. Such variation is also evident in the coherent contributions to the Reynolds stresses and heat fluxes. The results are connected to different initial conditions for the four flow structures. The momentum and heat transport characteristics are summarized for each flow structure.

A statistical theory is developed for the structure and propagation velocity of premixed flames in turbulent flows with scales large compared with the laminar flame thickness. The analysis, free of usual closure assumptions, involves a regular perturbation for small values of the ratio of laminar flame thickness to turbulence scale, termed the scale ratio ε, and a singular perturbation for large values of the non-dimensional activation temperature β. Any effects of the flame on the flow are considered to be given. In this initial study, molecular coefficients for diffusion of heat and reactants are set equal. The results identify convective-diffusive and reactive-diffusive zones in the flame and predict thickening of the flame by turbulence through streamwise displacement of the reactive-diffusive zone. Profiles for intensities of temperature fluctuations and for streamwise turbulent transport are obtained. A fundamental quantity occurring in the analysis is the longitudinal displacement of the reactive-diffusive zone in an Eulerian frame by turbulent fluctuations, and to first order in the scale ratio this equals the longitudinal displacement of fluid elements in an Eulerian frame by turbulent fluctuations, herein termed simply the Eulerian displacement. To first order in the scale ratio it is found that, if the Eulerian displacement experiences the same type of statistical non-stationarity as the corresponding Lagrangian displacement, then the diffusion approximation is valid for streamwise turbulent transport but the turbulent flame thickens as time increases, while if the Eulerian displacement is statistically stationary then the diffusion approximation necessitates a negative coefficient of diffusion in part of the flame but the flame thickness remains constant. By carrying the analysis to second order in the scale ratio it is shown that the turbulent-flame speed exceeds the laminar-flame speed by an amount proportional to the mean square of the transverse gradient of the Eulerian displacement. This result can be understood from the mechanistic viewpoint of a wrinkled laminar flame in terms of the increase in flame area produced by turbulence. Thus the theory provides a precise statistical quantification of the model of the wrinkled laminar flame for describing structures of turbulent flames.

The results of an experimental study of the interaction between waves and a current propagating in the same direction, have been reported by Kemp & Simons (1982). This paper describes the second part of the study, and considers the case of waves propagating against the current. Tests were performed in a laboratory flume with smooth and rough beds, and velocity measurements were made with a directionally sensitive laser anemometer as described in the previous paper. Analysis, including ensemble averaging of velocities and surface elevation, was performed by an on-line computer.

Results indicate that the rate of wave attenuation is greatly increased by the addition of an opposing current, and reduced by a following current. Wave profiles remain closely described by Stokes second-order theory; orbital velocities are also found to be in agreement with a second-order wave theory modified to take account of the presence of the current.

Certain results described occur regardless of the relative directions of current and wave. Mean velocities in the upper flow increase in the direction of the wave generator for increasing wave height. This suggests that the current is enhancing the wave-induced mass transport. Near the bed the velocity profiles so change that above the rough bed the current is retarded by the wave motion. In the logarithmic layer over the smooth bed velocities are increased with increasing wave height. However, all changes to velocity profiles have to be carefully interpreted, as the sidewall boundary layer decreases in thickness with even the smallest wave superimposed on the current.

Turbulence intensities and Reynolds stresses near the rough bed are increased by the presence of the waves, most strongly in a layer two roughness heights above bed level, where fluctuations are periodic and effected by vortices ejected from the roughness troughs. Above this level, and over the smooth bed, turbulence levels are similar to those for the currents alone.

In a backward boundary layer the fluid has, in the mathematical model, been flowing along a solid wall through an infinite distance. The co-ordinate distance x along the boundary is measured upstream, and the velocity U of the flow outside the boundary layer is taken as negative. The main application is to flow in converging passages.

The existence of similar solutions is considered, with emphasis on the correct asymptotic behaviour for large values of the stretched co-ordinate normal to the wall. This emphasis is shown to be necessary in considering backward boundary layers.

For two-dimensional flow in converging passages the requirement that a boundary layer should be possible for vanishingly small viscosity with a potential core flow is shown to lead directly to Hamel's spirals as the shape of the boundary streamlines.

Flow in axisymmetric converging passages is considered. For flow in a cone there is no limit as the viscosity tends to zero, and no potential core flow with a boundary layer is possible. The nature of a solution of the Navier-Stokes equations for laminar flow is considered.

Two-dimensional turbulence is investigated experimentally in thin liquid films. This study shows the spontaneous formation of couples of opposite-sign vortices in von Kármán wakes. The structure of these couples, their behaviour and their role in turbulent flows is then studied using both a numerical simulation and laboratory results.

The theoretical study presented in this paper was inspired by the recent report (Krämer 1960) of experiments showing that considerable reductions in the drag of an underwater solid body were achieved by covering it with a skin of flexible material; apparently this effect was due to the boundary layer being stabilized in the presence of the skin, so that transition to a turbulent condition of flow was prevented or at least delayed. The stability problem for flow past a flexible boundary is here formulated in a general way which allows a full exploration of the possibility of a stabilizing effect without the need to assign specific properties to the flexible medium; the collective properties of possible boundaries are represented by a ‘response coefficient’ α (a sort of ‘effective compliance’) measuring the deflexion of the surface under a travelling sinusoidal distribution of pressure.

A remarkably simple analytical connexion is established between the present general problem and the corresponding stability problem for the boundary layer on a rigid plane wall, and hence many details of the existing theory of hydrodynamic stability are immediately useful. However, the presence of the flexible boundary admits possible modes of instability additional to those which already exist when the boundary is rigid, and clearly every mode must be considered with regard to practical measures for stabilization—that is to say, it might be useless to inhibit one mode by a device which lets in another. What is believed to be an essentially complete interpretation of the over-all possibilities is deduced on recognizing three more or less distinct forms of instability. The first comprises waves resembling the unstable waves which can arise in the presence of a rigid boundary, but now being modified by the effects of flexibility. These waves tend to be stabilized when the boundary has a compliant response to them, which means the respective wave velocity has to be less than the velocity of free surface waves on the boundary; but it is found that the effect of internal friction in the flexible medium is actually destabilizing. The second form of instability is essentially a resonance effect and comprises waves travelling at very nearly the velocity of free surface waves. These waves can only be excited when the latter velocity falls below the free-stream velocity; they are scarcely affected by the viscosity of the fluid since the ‘wall friction layer’ is largely cancelled, so that damping due to the medium itself becomes the only stabilizing factor. The third form is akin to Kelvin–Helmholtz instability.

This interpretation of the theoretical results seems to point to the essential factors in the operation of a flexible skin as a stabilizing device, and accordingly in the concluding secttion of the paper two alternative sets of criteria are proposed each of which would provide a logical basis for designing such a device. The principle of the first alternative explains the success of Gamer's invention, but the second appears equally promising and the relative advantages of the two can really be proved only by further experiment.

Kolmogoroff's (1941) theory of local isotropy and universal similarity predicts that all turbulent velocity spectra are reducible to a single universal curve for the highest wave-numbers and that under certain conditions dimensional analysis may be used to predict spectral shapes. Identical arguments predict that the fine structure of conserved dynamically passive scalar fields mixed by turbulence will also be universally similar.

A single-electrode conductivity probe in a bridge circuit was used to measure the spectra and decay of a random homogeneous field of concentration and temperature behind a grid, and a Lintronic constant-temperature hot-film anemometer was used to measure the decay of velocity field. These experimental measurements of absolute turbulent velocity, temperature, and concentration spectra in salt water are here compared with the general predictions of universal similarity and local isotropy theories, as well as a prediction by Batchelor (1959) of the exact large wave-number spectral form for scalar mixing at high Schmidt number (v[Gt ]D. The spectral shapes are found to have the predicted similarity forms, and the data are consistent with Batchelor's predicted spectrum of the scalar field.