Spanwise structure and growth mechanisms in a turbulent boundary layer are investigated experimentally. PIV measurements are obtained in the streamwise–spanwise ($x$–$z$)-plane from the buffer layer to the top of the logarithmic region at Re$_\theta = 1015$ and 7705. The dominant motions of the flow are shown to be large-scale regions of momentum deficit elongated in the streamwise direction. Throughout the logarithmic layer, the regions are consistently bordered by vortices organized in the streamwise direction, offering strong support for a vortex packet model. Additionally, evidence is presented for the existence and organization of hairpin vortices in the region $y^+ < 60$. Statistical evidence is also presented for two important aspects of the vortex packet paradigm: vortex organization in the streamwise direction, and the clear association of the hairpin signature with local minima in streamwise velocity. Several spanwise lengthscales are shown to vary linearly with distance from the wall, revealing self-similar growth of spanwise structure in an average sense. Inspection of the data, however, suggests that individual structures do not grow strictly self-similarly in time. It is proposed that additional scale growth occurs by the merging of vortex packets on an eddy-by-eddy basis via a vortex re-connection mechanism similar to that suggested by Wark & Nagib (1989). The proposed mechanism provides a link between vortex-pairing concepts and the observed coalescence of streaky low-speed regions in the inner layer.

Employing a high-speed video system and hydrogen bubble-wire flow visualization, the characteristics of the low-speed streaks which occur in the near-wall region of turbulent boundary layers have been examined for a Reynolds-number range of 740 [les ] Reθ < 5830. The results indicate that the statistics of non-dimensional spanwise streak spacing are essentially invariant with Reynolds number, exhibiting consistent values of $\overline{\lambda^{+}} \approx 100$ and remarkably similar probability distributions conforming to lognormal behaviour. Further studies show that streak spacing increases with distance from the wall owing to a merging and intermittency process which occurs for y+ [simg ] 5. An additional observation is that, although low-speed streaks are not fixed in time and space, they demonstrate a tremendous persistence, often maintaining their integrity and reinforcing themselves for time periods up to an order of magnitude longer than the observed bursting times associated with wall region turbulence production. A mechanism for the formation of low-speed streaks is suggested which may explain both the observed merging behaviour and the streak persistence.

Solutions are analysed from large-eddy simulations of the phase-averaged equations for oceanic currents in the surface planetary boundary layer (PBL), where the averaging is over high-frequency surface gravity waves. These equations have additional terms proportional to the Lagrangian Stokes drift of the waves, including vortex and Coriolis forces and tracer advection. For the wind-driven PBL, the turbulent Langmuir number, Latur = (U∗/Us)1/2, measures the relative influences of directly wind-driven shear (with friction velocity U∗) and the Stokes drift Us. We focus on equilibrium solutions with steady, aligned wind and waves and a realistic Latur = 0.3. The mean current has an Eulerian volume transport to the right of the wind and against the Stokes drift. The turbulent vertical fluxes of momentum and tracers are enhanced by the presence of the Stokes drift, as are the turbulent kinetic energy and its dissipation and the skewness of vertical velocity. The dominant coherent structure in the turbulence is a Langmuir cell, which has its strongest vorticity aligned longitudinally (with the wind and waves) and intensified near the surface on the scale of the Stokes drift profile. Associated with this are down-wind surface convergence zones connected to interior circulations whose horizontal divergence axis is rotated about 45° to the right of the wind. The horizontal scale of the Langmuir cells expands with depth, and there are also intense motions on a scale finer than the dominant cells very near the surface. In a turbulent PBL, Langmuir cells have irregular patterns with finite correlation scales in space and time, and they undergo occasional mergers in the vicinity of Y-junctions between convergence zones.

A scale-dependent dynamic subgrid-scale model for large-eddy simulation of turbulent flows is proposed. Unlike the traditional dynamic model, it does not rely on the assumption that the model coefficient is scale invariant. The model is based on a second test-filtering operation which allows us to determine from the simulation how the coefficient varies with scale. The scale-dependent model is tested in simulations of a neutral atmospheric boundary layer. In this application, near the ground the grid scale is by necessity comparable to the local integral scale (of the order of the distance to the wall). With the grid scale and/or the test-filter scale being outside the inertial range, scale invariance is broken. The results are compared with those from (a) the traditional Smagorinsky model that requires specification of the coefficient and of a wall damping function, and (b) the standard dynamic model that assumes scale invariance of the coefficient. In the near-surface region the traditional Smagorinsky and standard dynamic models are too dissipative and not dissipative enough, respectively. Simulations with the scale-dependent dynamic model yield the expected trends of the coefficient as a function of scale and give improved predictions of velocity spectra at different heights from the ground. Consistent with the improved dissipation characteristics, the scale-dependent model also yields improved mean velocity profiles.

The present work is concerned with the flow past spheres in the Reynolds number range 5 × 104 [les ] Re [les ] 6 × 106. Results are reported for the case of a smooth surface. The total drag, the local static pressure and the local skin friction distribution were measured at a turbulence level of about 0·45%. The present results are compared with other available data as far as possible. Information is obtained from the local flow parameters on the positions of boundary-layer transition from laminar to turbulent flow and of boundary-layer separation. Finally the dependence of friction forces on Reynolds number is pointed out.

A general expression is derived for the fluid force on a body of simple shape moving with a velocity v through inviscid fluid in which there is an unsteady non-uniform rotational velocity field u0(x,t) in two or three dimensions. It is assumed that the radius is small compared with the scale over which the strain rate changes, though for the sphere it is also assumed that the changes in the ambient velocity field over the scale of the sphere are small compared with the velocity of the body relative to the flow. Given these approximations it is shown that the effects of the rate of change of the vorticity of the ambient flow is of second order and can be neglected. However the rate of change of the irrotational straining motion is included in the analysis. It is shown that the inertial forces derived by many authors for irrotational flow can be simply added to a generalization of the lift force derived by Auton (1987) in a companion paper. It is shown how this lift force is made up of a rotational and an inertial or added-mass component. For three-dimensional bluff bodies the latter is generally larger (by a factor of three for a sphere), and can be simply calculated from the added-mass coefficient. For illustration, the general expression is used to derive formulae for (i) the motion of a spherical bubble in a steady non-uniform flow to contrast with the motion in an unsteady flow, and (ii) the motion of rigid volumes of neutral density across an inviscid shear flow. These results show how added-mass (and lift) forces lead to different motions for a sphere and a cylinder. The general expression is useful in two-phase flow calculations, and for indicating the forces and motions of ‘lumps of fluid’ in turbulent flows.

The newly developed simulation method known as Stokesian dynamics is used to investigate the rheological behaviour of concentrated suspensions. Both the detailed microstructure (e.g. pair-distribution function) and the macroscopic properties are determined for a suspension of identical rigid spherical particles in a simple shear flow. The suspended particles interact through both hydrodynamic and non-hydrodynamic forces. For suspensions with purely hydrodynamic forces, the increase in the suspension viscosity with volume fraction ϕ is shown to be caused by particle clustering. The cluster formation results from the lubrication forces, and the simulations of a monolayer of spheres show a scaling law for the cluster size: lc ∼ [1 − (ϕ/ϕm)½]−1, where ϕm is the maximum volume fraction that can shear homogeneously. The simulation results suggest that the suspension viscosity becomes infinite at the percolation-like threshold ϕm owing to the formation of an infinite cluster. The predicted simulation viscosities are in very good agreement with experiment. A suspension with short-range repulsive interparticle forces is also studied, and is seen to have a non-Newtonian rheology. Normal-stress differences arise owing to the anisotropic local structure created by the interparticle forces. The repulsive forces also reduce particle clustering, and as a result the suspension is shear-thickening.

The paper is discussion of measurements of the statistical properties of the pressure field at the wall of turbulent attached shear flows. These measurements have been made only in part by the author. A preliminary discussion is given of the important limitations imposed by the imperfect space resolution of contemporary pressure transducers. There follows a discussion of the appropriate scales of the pressure field. It is shown that measurements of the longitudinal cross-spectral densities lead to similarity variables for the space-time covariance of the pressure and for the corresponding spectra. The existence of these similarity variables may be due to the dispersion of the sources of pressure by the mean velocity gradient. Such a mechanism is illustrated by a simple model. Lateral cross-spectral densities also lead approximately to similarity variables.

Computations based directly upon detailed pressure-velocity correlation measurements by Wooldridge & Willmarth reveal that an important part of the pressure at the wall of a boundary layer is contributed by source terms which are quadratic in the turbulent velocity fluctuations; the interaction of the mean strain rate with normal velocity fluctuations, being in effect limited to a region very near the wall, supplies a dominant contribution only at high frequencies and its scales, downstream convective speed and convective memory are markedly smaller than those of the observed wall pressure.

The inner part of the Law of the Wall region (y* [les ] 100) seems to be substantially free of pressure sources and within that region (a) the pressure can be given in terms of its boundary value, and (b) the local velocity field is dependent upon but unbale to affect appreciably the turbulent pressures.

A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.

In this, the first part of a two-part experimental study of the behaviour of impinging jets, the mean properties of the flow field are established. Velocity profiles are given for three types of jet flow issuing from a circular convergent nozzle. Measured distributions of surface pressure are given which result when the jets impinge both normally and obliquely at various distances on several surface shapes. The pressure distributions are used to compute the radial velocity gradient at the impingement stagnation point. It is found that for normal impingement this gradient correlates with the free jet centreline velocity and half-radius at the same axial location. A fall-off in the correlated value is noted as the impingement is made oblique. Measurements of the azimuthal distribution of momentum flux in the resulting wall jet are also given. The general behaviour of all three types of jet is found to be similar at locations downstream of any local effects due to the shock waves present in the under-expanded types. A special study of the close-range impingement of an under-expanded jet containing a normal shock disk reveals a region of separated flow surrounding the stagnation point. This condition results in a negative value of the radial velocity gradient at the centreline.

Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10−4) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.

Quadrant analysis has been used to investigate the events contributing to the Reynolds shear stress in zero-pressure-gradient turbulent boundary layers over regularly arrayed rough surfaces of several different densities, and over a smooth surface. By partitioning the stress into ejections, sweeps, and inward and outward interactions, it is shown that sweeps account for most of the stress close to rough surfaces, and that the relative magnitude of the sweep component increases both with surface roughness and with proximity to the surface. The sweep-dominated region delineates a ‘roughness sublayer’ with a depth of up to several roughness element heights, in which the turbulence characteristics depend explicitly on the roughness. In the remainder of the inner (or constant-stress) layer, and in the outer layer, the flow obeys familiar similarity laws with respect to surface roughness.

The difference ΔS0 between the fractional contributions of sweeps and ejections to the stress is shown to be well related everywhere to the third moments of the streamwise and normal velocity fluctuations. Experimental proportionalities are established between the third moments and δS0, and are shown to agree with predictions made from cumulant-discard theory.

The time scale for the passage of large coherent structures past a fixed point, T, is assumed proportional to the mean time between occurrences in a specified quadrant of an instantaneous stress u'w’ at least H times the local mean stress u'w’, where H is a threshold level. For both the ejection and sweep quadrants and for any choice of H, it is found that T scales with the friction velocity u* and the boundary-layer thickness δ, such that Tu*/δ is invariant with change of surface roughness.

Vortex ‘shedding’ behind circular cylinders can be altered and suppressed altogether (or ‘controlled’) over a limited range of Reynolds numbers, by a proper placement of a second, much smaller, cylinder in the near wake of the main cylinder. This new and dramatic suppression of vortex ‘shedding’ is the subject of this paper. Details of the phenomenon are documented through parallel experimental and numerical investigations, including flow visualization. Temporal growth rate measurements of the velocity fluctuations reveal that the presence of the smaller cylinder reduces the growth rate of the disturbances leading to vortex ‘shedding’, and that its suppression, accompanied by the disappearance of sharp spectral peaks, coincides with negative temporal growth rates. It is argued that the presence of the secondary cylinder has the effect of altering the local stability of the flow by smearing and diffusing concentrated vorticity in the shear layers behind the body; a related effect is that the secondary cylinder diverts a small amount of fluid into the wake of the main cylinder. A unified explanation of the formation and suppression of the vortex street is attempted, and it is suggested that the vortex ‘shedding’ is associated with temporally unstable eigenmodes which are heavily weighted by the near field. It is also shown that absolute instability is relevant, up to a point, in explaining vortex shedding, whose suppression can similarly be associated with altering the instability in the near wake region from absolute to convective.

Short gravity waves, when superposed on much longer waves of the same type, have a tendency to become both shorter and steeper at the crests of the longer waves, and correspondingly longer and lower in the troughs. In the present paper, by taking into account the non-linear interactions between the two wave trains, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated. The results differ in some essentials from previous estimates by Unna. The variation in energy of the short waves is shown to correspond to work done by the longer waves against the radiation stress of the short waves, which has previously been overlooked. The concept of the radiation stress is likely to be valuable in other problems.

The present paper addresses some topical issues in modelling compressible turbulent shear flows. The work is based on direct numerical simulation (DNS) of two supersonic fully developed channel flows between very cold isothermal walls. Detailed decomposition and analysis of terms appearing in the mean momentum and energy equations are presented. The simulation results are used to provide insights into differences between conventional Reynolds and Favre averaging of the mean-flow and turbulent quantities. Study of the turbulence energy budget for the two cases shows that compressibility effects due to turbulent density and pressure fluctuations are insignificant. In particular, the dilatational dissipation and the mean product of the pressure and dilatation fluctuations are very small, contrary to the results of simulations for sheared homogeneous compressible turbulence and to recent proposals for models for general compressible turbulent flows. This provides a possible explanation of why the Van Driest density-weighted transformation (which ignores any true turbulent compressibility effects) is so successful in correlating compressible boundary-layer data. Finally, it is found that the DNS data do not support the strong Reynolds analogy. A more general representation of the analogy is analysed and shown to match the DNS data very well.

Frictionless flows with finite voticity are usually made determinate by the imposition of boundary conditions specifying the distribution of vorticity ‘at infinity’. No such boundary conditions are available in the case of flows with closed streamlines, and the velocity distributions in regions where viscous forces are small (the Reynolds number of the flow being assumed large) cannot be made determinate by considerations of the fluid as inviscid. It is shown that if the motion is to be exactly steady there is an integral condition, arising from the existence of viscous forces, which must be satisfied by the vorticity distribution no matter how small the viscosity may be. This condition states that the contribution from viscous forces to the rate of change of circulation round any streamline must be identically zero. (In cases in which the vortex lines are also closed, there is a similar condition concerning the circulation round vortex lines.)

The inviscid flow equations are then combined with this integral condition in cases for which typical streamlines lie entirely in the region of small viscous forces. In two-dimensional closed flows, the vorticity is found to be uniform in a connected region of small viscous forces, with a value which remains to be determined—as is done explicitly in one simple case—by the condition that the viscous boundary layer surrounding this region must also be in steady motion. Analogous results are obtained for rotationally symmetric flows without azimuthal swirl, and for a certain class of flows with swirl having no interior boundary to the streamlines in an axial plane, the latter case requiring use of the fact that the vortex lines are also closed. In all these cases, the results are such that the Bernoulli constant, or ‘total head’, varies linearly with the appropriate stream function, and the effect of viscosity on the rate of change of vorticity at any point vanishes identically.

The interactions between small dense particles and fluid turbulence have been investigated in a downflow fully developed channel in air. Particle velocities of, and fluid velocities in the presence of, 50 μm glass, 90 μm glass and 70 μm copper spherical beads were measured by laser Doppler anemometry, at particle mass loadings up to 80%. These particles were smaller than the Kolmogorov lengthscale of the flow and could respond to some but not all of the scales of turbulent motion. Streamwise mean particle velocity profiles were flatter than the mean fluid velocity profile, which was unmodified by particle loading. Particle velocity fluctuation intensities were larger than the unladen-fluid turbulence intensity in the streamwise direction but were smaller in the transverse direction. Fluid turbulence was attenuated by the addition of particles; the degree of attenuation increased with particle Stokes number, particle mass loading and distance from the wall. Turbulence was more strongly attenuated in the transverse than in the streamwise direction, because the turbulence energy is at higher frequencies in the transverse direction. Streamwise turbulence attenuation displayed a range of preferred frequencies where attenuation was strongest.

The problem of transient natural convection in a cavity of aspect ratio A [les ] 1 with differentially heated end walls is considered. Scale analysis is used to show that a number of initial flow types are possible, collapsing ultimately onto two basic types of steady flow, determined by the relative value of the non-dimensional parameters describing the flow. A number of numerical solutions which encompass both flow types are obtained, and their relationship to the scale analysis is discussed.

A normal magnetic field has a destabilizing influence on a flat interface between a magnetizable and a non-magnetic fluid. Stabilizing influences are provided by interfacial tension and gravity if the lighter fluid is uppermost. The critical level of magnetization for onset of the instability is derived for a fluid having a non-linear relation between magnetization and magnetic induction. Experiments using a magnetizable fluid, which contains a colloidal suspension of ferromagnetic particles, at interfaces with air and water are made and cover a wide range of density differences. Measurements confirm the prediction for critical magnetization, and it was found that, after onset, the interface took a new form in which the elevation had a regular hexagonal pattern. The pattern was highly stable, and the measured spacing of peaks agreed reasonably with that derived from the critical wave-number for the instability of a flat interface.

Turbulence in the convective boundary layer (CBL) uniformly heated from below and topped by a layer of uniformly stratified fluid is investigated for zero mean horizontal flow using large-eddy simulations (LES). The Rayleigh number is effectively infinite, the Froude number of the stable layer is 0.09 and the surface roughness height relative to the height of the convective layer is varied between 10−6 and 10−2. The LES uses a finite-difference method to integrate the three-dimensional grid-volume-averaged Navier–Stokes equations for a Boussinesq fluid. Subgrid-scale (SGS) fluxes are determined from algebraically approximated second-order closure (SOC) transport equations for which all essential coefficients are determined from the inertial-range theory. The surface boundary condition uses the Monin–Obukhov relationships. A radiation boundary condition at the top of the computational domain prevents spurious reflections of gravity waves. The simulation uses 160 × 160 × 48 grid cells. In the asymptotic state, the results in terms of vertical mean profiles of turbulence statistics generally agree very well with results available from laboratory and atmospheric field experiments. We found less agreement with respect to horizontal velocity fluctuations, pressure fluctuations and dissipation rates, which previous investigations tend to overestimate. Horizontal spectra exhibit an inertial subrange. The entrainment heat flux at the top of the CBL is carried by cold updraughts and warm downdraughts in the form of wisps at scales comparable with the height of the boundary layer. Plots of instantaneous flow fields show a spoke pattern in the lower quarter of the CBL which feeds large-scale updraughts penetrating into the stable layer aloft. The spoke pattern has also been found in a few previous investigations. Small-scale plumes near the surface and remote from strong updraughts do not merge together but decay while rising through large-scale downdraughts. The structure of updraughts and downdraughts is identified by three-dimensional correlation functions and conditionally averaged fields. The mean circulation extends vertically over the whole boundary layer. We find that updraughts are composed of quasi-steady large-scale plumes together with transient rising thermals which grow in size by lateral entrainment. The skewness of the vertical velocity fluctuations is generally positive but becomes negative in the lowest mesh cells when the dissipation rate exceeds the production rate due to buoyancy near the surface, as is the case for very rough surfaces. The LES results are used to determine the root-mean-square value of the surface friction velocity and the mean temperature difference between the surface and the mixed layer as a function of the roughness height. The results corroborate a simple model of the heat transfer in the surface layer.