Laboratory experiments on stably stratified grid turbulence have suggested that turbulent diffusivity $\kappa_\rho$ can be expressed in terms of a turbulence activity parameter $\epsilon/\nu N^2$, with different power-law relations appropriate for different levels of $\epsilon/\nu N^2$. To further examine the applicability of these findings to both a wider range of the turbulence intensity parameter $\epsilon/\nu N^2$ and different forcing mechanisms, DNS data of homogeneous sheared stratified turbulence generated by Shih et al. (2000) and Venayagamoorthy et al. (2003) are analysed in this study. Both scalar eddy diffusivity $\kappa_\rho$ and eddy viscosity $\kappa_\nu$ are found to be well-correlated with $\epsilon/\nu N^2$, and three distinct regimes of behaviour depending on the value of $\epsilon/\nu N^2$ are apparent. In the diffusive regime $D$, corresponding to low values of $\epsilon/\nu N^2$ and decaying turbulence, the total diffusivity reverts to the molecular value; in the intermediate regime $I$, corresponding to $7 \,{<} \epsilon/\nu N^2 \,{<}\, 100$ and stationary turbulence, diffusivity exhibits a linear relationship with $\epsilon/\nu N^2$, as predicted by Osborn (1980); finally, in the energetic regime $E$, corresponding to higher values of $\epsilon/\nu N^2$ and growing turbulence, the diffusivity scales with $(\epsilon/\nu N^2)^{1/2}$. The dependence of the flux Richardson number $R_f$ on $\thing$ explains the shift in power law between regimes $I$ and $E$. Estimates for the overturning length scale and velocity scales are found for the various $\epsilon/\nu N^2$ regimes. It is noted that $\epsilon/\nu N^2 \,{\sim}\, \hbox{\it Re}/\hbox{\it Ri}\,{\sim}\,\hbox{\it ReFr}^2$, suggesting that such Reynolds–Richardson number or Reynolds–Froude number aggregates are more descriptive of stratified turbulent flow conditions than the conventional reliance on Richardson number alone.

The equilibrium statistical mechanics of inviscid two-dimensional flow are re-examined both for a continuum truncated at a top wavenumber and for a system of discrete vortices. In both cases, there are negative-temperature equilibria for finite flows. But for spatially infinite flows, there are only positive-temperature equilibria, and both the continuum and discrete system exhibit proper, extensive, thermodynamic limits a t all realizable values of the energy and enstrophy density. The negative-temperature behaviours of the continuum and discrete system are semi-quantitatively the same, except for a supercondensation phenomenon in the discrete case a t the smallest realizable values of negative temperature. The supercondensed states have very large energy and in them all vortex cores of the same sign are clumped within an area small eompared with the mean area per vortex. The approach of the continuum system to absolute equilibrium by enstrophy cascade to high wavenumbers and energy cascade to low wavenumbers is examined. It is argued that the enstrophy cascade is closely analogous to distortion of a passive scalar field by straining of large spatial scale. This implies that high intermittency of spatial derivatives of the vorticity field can develop but that there is no associated change in the previously proposed log-corrected k−1 enstrophy spectrum law. On the other hand, intermittency build-up in the downward energy cascade can result in a change of the exponent in the energy spectrum law to a negative value of smaller magnitude than 5/3. Intermittency effects in the non-equilibrium energy cascade seem a more plausible explanation for vortex clumping observed in recent computer experiments than do the spatially smooth condensation phenomena associated with the negative-temperature absolute equilibria.

The steady-state solution of the non-linear equation \[ h_t + hh_x + h_{xxx} = \delta h_{xx} \] with both damping and dispersion is examined in the phase plane. For small damping an averaging technique is used to obtain an oscillatory asymptotic solution. This solution becomes invalid as the period of the oscillation approaches infinity, and is matched to a straightforward expansion solution. The results obtained are compared with a numerical integration of the equation.

There is a large amount of experimental and numerical work dealing with dry granular flows (such as sand, glass beads, etc.) that supports the so-called -rheology. The reliability of the -rheology in the case of complex transient flows is not fully ascertained, however. From this perspective, the granular column collapse experiment provides an interesting benchmark. In this paper we implement the -rheology in a Navier–Stokes solver (Gerris) and compare the resulting solutions with both analytical solutions and two-dimensional contact dynamics discrete simulations. In a first series of simulations, we check the numerical model in the case of a steady infinite two-dimensional granular layer avalanching on an inclined plane. A second layer of Newtonian fluid is then added over the granular layer in order to recover a close approximation of a free-surface condition. Comparisons with analytical and semi-analytical solutions provide conclusive validation of the numerical implementation of the -rheology. In a second part, we simulate the unsteady two-dimensional collapse of granular columns over a wide range of aspect ratios. Systematic comparisons with discrete two-dimensional contact dynamics simulations show good agreement between the two methods for the inner deformations and the time evolution of the shape during most of the flow, while a systematic underestimation of the final run-out is observed. The experimental scalings of spreading of the column as a function of the aspect ratio available from the literature are also recovered. A discussion follows on the performances of other rheologies, and on the sensitivity of the simulations to the parameters of the -rheology.

A turbulent field is produced with an oscillating grid in a deep, rotating tank. Near the grid, the Rossby number is kept large, 0(3-33), and the turbulence is locally unaffected by rotation. Away from the grid, the scale of the turbulence increases, the r.m.s. turbulent velocity decreases, and rotation becomes increasingly important. The flow field changes dramatically at a local Rossby number of about 0.20, and thereafter remains independent of depth. The flow consists of concentrated vortices having axes approximately parallel to the rotation axis, and extending throughout the depth of the fluid above the turbulent Ekman layer. The number of vortices per unit area is a function of the grid Rossby number. The local vorticity within cores can be a factor of 50 larger than the tank vorticity 2Ω. The total relative circulation contained in the vortices remains, however, a small fraction of the tank circulation.

The concentrated vortex cores support waves consisting of helical distortions, which travel along the axes of individual vortices. Isolated, travelling waves seem well-described by the vortex-soliton theory of Hasimoto (1972). The nonlinear waves transport mass, momentum and energy from the vigorously turbulent region near the grid to the rotation-dominated flow above. Interactions between waves, which are frequent occurrences, almost always result in a local breakdown of the vortex core, and small-scale turbulence production. Usually the portions of broken core reform within ½−1 rotation periods, but occasionally cores are destroyed and reformed on a much longer timescale.

Direct numerical simulations (DNS) have been performed of turbulent flow in a plane channel with a solid top wall and a permeable bottom wall. The permeable wall is a packed bed, which is characterized by the mean particle diameter and the porosity. The main objective is to study the influence of wall permeability on the structure and dynamics of turbulence. The flow inside the permeable wall is described by means of volume-averaged Navier–Stokes equations. Results from four simulations are shown, for which only the wall porosity ($\epsilon_c$) is changed. The Reynolds number based on the thickness of the boundary layer over the permeable wall and the friction velocity varies from $\hbox{\it Re}_{\tau}^p\,{=}\,176$ for $\epsilon_c\,{=}\,0$ to $\hbox{\it Re}_{\tau}^p\,{=}\,498$ for $\epsilon_c\,{=}\,0.95$. The influence of wall permeability can be characterized by the permeability Reynolds number, $\hbox{\it Re}_K$, which represents the ratio of the effective pore diameter to the typical thickness of the viscous sublayers over the individual wall elements. For small $\hbox{\it Re}_K$, the wall behaves like a solid wall. For large $\hbox{\it Re}_K$, the wall is classified as a highly permeable wall near which viscous effects are of minor importance. It is observed that streaks and the associated quasi-streamwise vortices are absent near a highly permeable wall. This is attributed to turbulent transport across the wall interface and the reduction in mean shear due to a weakening of, respectively, the wall-blocking and the wall-induced viscous effect. The absence of streaks is consistent with a decrease in the peak value of the streamwise root mean square (r.m.s.) velocity normalized by the friction velocity at the permeable wall. Despite the increase in the peak values of the spanwise and wall-normal r.m.s. velocities, the peak value of the turbulent kinetic energy is therefore smaller. Turbulence near a highly permeable wall is dominated by relatively large vortical structures, which originate from a Kelvin–Helmholtz type of instability. These structures are responsible for an exchange of momentum between the channel and the permeable wall. This process contributes strongly to the Reynolds-shear stress and thus to a large increase in the skin friction.

Surface waves travelling in water of finite depth may be scattered by a region of undulating bottom topography. The present study is concerned with the idealized two-dimensional situation in which long-crested surface waves are incident upon a patch of long-crested regular bottom ripples. The principal question examined concerns the amount of incident wave energy that is reflected by the ripple patch. Linear perturbation theory is used to show that the reflection coefficient is both oscillatory in the quotient of the length of the patch and the surface wavelength, and also strongly dependent upon the quotient of the surface and bed wavelengths. In particular, there is a Bragg resonance between the surface waves and the ripples, which is associated with the reflection of incident wave energy. A secondary question concerns the nature of the wave field in the immediate vicinity of the ripple patch. In resonant cases, it is shown how the partially standing wave on the upwave side of the ripple patch gives way, in an almost linear manner over the patch itself, to a progressive transmitted wave on the downwave side. The theoretical predictions are compared with an extensive set of laboratory observations made in a wave tank. Comparisons relating both to the reflection coefficient, and also to the wave field over the ripple patch, are shown to give consistently good agreement. Finally, the implications of the results for sediment transport on an erodible bed are examined.

The near- and far-field break-up and atomization of a water jet by a high-speed annular air jet are examined by means of high-speed flow visualizations and phase Doppler particle sizing techniques. Visualization of the jet's near field and measurements of the frequencies associated with the gas–liquid interfacial instabilities are used to study the underlying physical mechanisms involved in the primary break-up of the water jet. This process is shown to consist of the stripping of water sheets, or ligaments, which subsequently break into smaller lumps or drops. An entrainment model of the near-field stripping of the liquid is proposed, and shown to describe the measured liquid shedding frequencies. This simplified model explains qualitatively the dependence of the shedding frequency on the air/water momentum ratio in both initially laminar and turbulent water jets. The role of the secondary liquid break-up in the far-field atomization of the water jet is also investigated, and an attempt is made to apply the classical concepts of local isotropy to explain qualitatively the measurement of the far-field droplet size distribution and its dependence on the water to air mass and momentum ratios. Models accounting for the effect of the local turbulent dissipation rate in the gas on both the break-up and coalescence of the droplets are developed and compared with the measurements of the variation of the droplet size along the jet's centreline. The total flux of kinetic energy supplied by the gas per unit total mass of the spray jet was found to be the primary parameter determining the secondary break-up and coalescence of the droplets in the far field.

The interaction of two pure helical (circularly polarized) velocity waves according to the incompressible Navier–Stokes equation produces modulation products of mixed helicity. In general, the interaction of waves of opposite helicity is stronger than that of waves with the same helicity. The inference is that strong net helicity depresses overall turbulent energy transfer. The conservation laws strongly inhibit energy transfer from higher to lower wavenumbers, when the helicity is large. The absolute equilibrium spectra of velocity and helicity for an inviscid flow system truncated at an upper wavenumber k2 are \[ U(k) = 2\alpha/(\alpha^2-\beta^2k^2),\quad Q(k) = 2\beta k^2/(\alpha^2-\beta^2k^2), \] where the velocity variance and helicity/unit volume are ∫U(k)d3k and ∫Q(k)d3k, respectively. The temperature parameters α and β are constrained by α > 0 and |βk2| < α. There are no analogues of the negative-temperature equilibrium states known for two-dimensional inviscid flow. It is argued that the inertial-range energy cascade in isotropic turbulence driven by helical input should not differ asymptotically from that of non-helical turbulence. The absolute equilibrium distributions suggest that, in contrast to the analogous two-dimensional situation, statistically steady helical input at middle wavenumbers should not produce a significant downward cascade of energy to lower wavenumbers.

The stability of a horizontal layer of fluid heated from above or below is examined for the case of a time-dependent buoyancy force which is generated by shaking the fluid layer, thus causing a sinusoidal modulation of the gravitational field. A linearized stability analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. In this analysis, much emphasis is placed on qualitative results obtained by an approximate solution, which permits a rather complete stability analysis. A useful mechanical analogy is developed by considering the effects of gravity modulation on a simple pendulum. Finally, some effects of finite amplitude flows are considered and discussed.

A systematic study of two-dimensional, turbulent, small-deficit wakes was carried out to determine their structure and the universality of their self-preserving states. Various wake generators, including circular cylinders, a symmetrical airfoil, a flat plate, and an assortment of screens of varying solidity, were studied for a wide range of downstream distances. Most of the generators were tailored so that their drag coefficients, and therefore their momentum thicknesses, were identical, permitting comparison at identical Reynolds numbers and aspect ratios. The flat plate and airfoil had a small, trailing-edge flap which could be externally driven to introduce forced sinuous oscillations into the wake. The results indicate that the normalized characteristic velocity and length scales depend on the initial conditions, while the shape of the normalized mean velocity profile is independent of these conditions or the nature of the generator. The normalized distributions of the longitudinal turbulence intensity, however, are dependent on the initial conditions.

Linear inviscid stability theory, in which the divergence of the mean flow is taken into account, predicts quite well the amplification and the transverse distributions of amplitudes and phases of externally imposed sinuous waves on a fully developed turbulent wake generated by a flat plate. There is a strong indication that the large structures observed in the unforced wake are related to the two-dimensional instability modes and therefore can be modelled by linear stability theory. Furthermore, the interaction of the two possible modes of instability may be responsible for the vortex street-type pattern observed visually in the small-deficit, turbulent wake.

A simple model for the experimentally observed instability of the vortex ring to azimuthal bending waves of wavelength comparable with the core size is presented. Short-wave instabilities are discussed for both the vortex ring and the vortex pair. Instability for both the ring and the pair is predicted to occur whenever the self-induced rotation of waves on the filament passes through zero. Although this does not occur for the first radial bending mode of a vortex filament, it is shown to be possible for bending modes with a more complex radial structure with at least one node at some radius within the core. The previous work of Widnall & Sullivan (1973) is discussed and their experimental results are compared with the predictions of the analysis presented here.

We study the organization of turbulence in supersonic boundary layers through large-scale direct numerical simulations (DNS) at , and momentum-thickness Reynolds number up to (corresponding to ) which significantly extend the current envelope of DNS in the supersonic regime. The numerical strategy relies on high-order, non-dissipative discretization of the convective terms in the Navier–Stokes equations, and it implements a recycling/rescaling strategy to stimulate the inflow turbulence. Comparison of the velocity statistics up to fourth order shows nearly exact agreement with reference incompressible data, provided the momentum-thickness Reynolds number is matched, and provided the mean velocity and the velocity fluctuations are scaled to incorporate the effects of mean density variation, as postulated by Morkovin’s hypothesis. As also found in the incompressible regime, we observe quite a different behaviour of the second-order flow statistics at sufficiently large Reynolds number, most of which show the onset of a range with logarithmic variation, typical of ‘attached’ variables, whereas the wall-normal velocity exhibits a plateau away from the wall, which is typical of ‘detached’ variables. The modifications of the structure of the flow field that underlie this change of behaviour are highlighted through visualizations of the velocity and temperature fields, which substantiate the formation of large jet-like and wake-like motions in the outer part of the boundary layer. It is found that the typical size of the attached eddies roughly scales with the local mean velocity gradient, rather than being proportional to the wall distance, as happens for the wall-detached variables. The interactions of the large eddies in the outer layer with the near-wall region are quantified through a two-point amplitude modulation covariance, which characterizes the modulating action of energetic outer-layer eddies.

A detailed equation is proposed for the force exerted on a sphere that accelerates rectilinearly in an otherwise still fluid. In addition to the buoyant force, the fluid exerts forces that depend on (a) the velocity of the sphere, (b) the acceleration of the sphere and (c) the history of the motion. The equation reduces to the known theoretical solution for low velocity and large acceleration.

The proposed equation was tested and found most satisfactory for a particular case in which the velocity was not small, viz. the case of simple harmonic motion along a straight line. The acceleration (added mass) and history coefficients in the equation were evaluated experimentally. They were found to depend on the ratio of the convective acceleration to the local acceleration as measured by the parameter V2/aD, in which V, a and D are the velocity, acceleration and diameter of the sphere, respectively. The Reynolds numbers varied from 0 to 62 during the tests.

In the present study the mechanisms of evolution of propeller tip and hub vortices in the transitional region and the far field are investigated experimentally. The experiments involved detailed time-resolved visualizations and velocimetry measurements and were aimed at examining the effect of the spiral-to-spiral distance on the mechanisms of wake evolution and instability transition. In this regard, three propellers having the same blade geometry but different number of blades were considered. The study outlined dependence of the wake instability on the spiral-to-spiral distance and, in particular, a streamwise displacement of the transition region at the increasing inter-spiral distance. Furthermore, a multi-step grouping mechanism among tip vortices was highlighted and discussed. It is shown that such a phenomenon is driven by the mutual inductance between adjacent spirals whose characteristics change by changing the number of blades.

In recent years there has been significant progress made towards understanding the large-scale structure of wall-bounded shear flows. Most of this work has been conducted with turbulent boundary layers, leaving scope for further work in pipes and channels. In this article the structure of fully developed turbulent pipe and channel flow has been studied using custom-made arrays of hot-wire probes. Results reveal long meandering structures of length up to 25 pipe radii or channel half-heights. These appear to be qualitatively similar to those reported in the log region of a turbulent boundary layer. However, for the channel case, large-scale coherence persists further from the wall than in boundary layers. This is expected since these large-scale features are a property of the logarithmic region of the mean velocity profile in boundary layers and it is well-known that the mean velocity in a channel remains very close to the log law much further from the wall. Further comparison of the three turbulent flows shows that the characteristic structure width in the logarithmic region of a boundary layer is at least 1.6 times smaller than that in a pipe or channel.

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters, J a ‘typical’ Richardson number, α the (real) wavenumber and c the complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first finds c given α and J, the second finds α and J when c ≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes, but not always, the stability boundary in the α, J plane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

The inertial migration of a small sphere in a Poiseuille flow is calculated for the case when the channel Reynolds number is of order unity. The equilibrium position is found to move towards the wall as the Reynolds number increases. The migration velocity is found to increase more slowly than quadratically. These results are compared with the experiments of Segré & Silberberg (1962 a, b).

The flow of a viscous fluid through axially symmetric pipes and symmetrical channels is investigated under the assumption that the Reynolds number is small enough for the Stokes flow approximations to be made. It is assumed that the cross-section of the pipe or channel varies sinusoidally along the length. The flow is produced by a prescribed pressure gradient and by the variation in cross-section that occurs during the passage of a prescribed sinusoidal peristaltic wave along the walls. The theory is applied in particular to two extreme cases, peristaltic motion with no pressure gradient and flow under pressure along a pipe or channel with fixed walls and sinusoidally varying cross-section. Perturbation solutions are found for the stream function in powers of the ratio of the amplitude of the variation in the pipe radius or channel breadth to the mean radius or breadth respectively. These solutions are used to calculate, in particular, the flux through the pipe or channel for a given wave velocity in the first case and for a given pressure gradient in the second case. With a suitable notation it is possible to combine the analysis required for the two cases of pipe and channel flow.

The three-dimensional transition of the flow behind a bluff body is studied, with an emphasis placed on the evolution of large-scale structures in the wake. It has previously been found that there are two fundamental modes of three-dimensional vortex shedding in the wake of a circular cylinder (each mode being dependent on the range of Reynolds number), with a spanwise lengthscale of the same order as the primary streamwise wavelength of the vortex street. However. it is shown in the present study that the wake transition also involves the appearance of large-scale spot-like ‘vortex dislocations’, that grow downstream to a size of the order of 10–20 primary wavelengths. Vortex dislocations are generated between spanwise vortex-shedding cells of different frequency. The presence of these dislocations explains the large intermittent velocity irregularities that were originally found by Roshko (1954) and later by Bloor (1964) to characterize transition. The presence of these vortex dislocations in wake transition is largely responsible for the break-up to turbulence of the wake as it travels downstream.

In order to study their evolution in detail, dislocations have been (passively) forced to occur at a local spanwise position with the use of a small ring disturbance. It is found that ‘two-sided’ dislocations are stable in a symmetric in-phase configuration, and that they induce quasi-periodic velocity spectra and (beat) dislocation-frequency oscillations in the near wake. Intrinsic to these dislocations is a mechanism by which they spread rapidly in the spanwise direction, involving helical twisting of the vortices and axial core flows. This is felt to be a fundamental mechanism by which vortices develop large-scale distortions in natural transition. As the wake travels downstream, the energy at the low dislocation frequency decays slowly (in contrast to the rapid decay of other frequencies), leaving the downstream wake dominated by the large dislocation structures. Distinct similarities are found between the periodic forced dislocations and the intermittent dislocations that occur in natural transition. Further similarities of dislocations in different types of flow suggest that vortex or phase dislocations could conceivably be a generic feature of transition in all shear flows.