The role of diffusion and transient velocities in the dispersal of passive scalars by chaotic advection produced in a low Reynolds number periodic journal-bearing flow is studied numerically and experimentally. The transient velocity field, which occurs whenever the cylinders switch motion, is obtained by solving the Navier–Stokes equations numerically in the eccentric annulus. It is observed, numerically, that the transient effects, along with diffusion, significantly enhance the separation of chaotically advected particles even when the Reynolds number is very low. Corresponding experimental observations are found to be in good qualitative agreement with the numerical results obtained by including the effect of transient velocities, which are seen to add to the overall separation of particles.

In this experimental study the mixing of passive scalars that arises from shear-free decaying grid-generated turbulence is examined. The flow configuration consists of two homogeneous layers of equal density separated by a sharp interface between different mean concentrations of passive scalar. Both layers flow through a turbulence-generating grid. To determine the effect of diffusivity the scalar was heat in one experiment and salt in a second. The Schmidt number – the ratio of momentum to species diffusivity – was 7 and 700 for heat and salt respectively. Velocity, scalar and flux fields were mapped and flow visualization was undertaken to study the flows.

Integral lengthscales and (scalar) flux were determined to be independent of diffusivity, whereas the scalar Taylor microscales varied with Schmidt number, Sc, thus illustrating the disparate effects of diffusivity on large and small scales. The time series of signals showed a correspondence between locations of wθ extrema and uw extrema. Flux wθ arose from intermittent events; and the magnitude of the time-averaged flux $\overline{w\theta}$ was found to depend on the frequency, rather than on variations in the amplitude of wθ events.

The results of an experimental study of shear-free decaying grid-generated turbulence on both sides of a sharp interface between two homogeneous layers of different densities are presented. The evolution of turbulence and mixing were examined by simultaneously mapping the velocity (u, w) and density fields (ρ) and the vertical mass flux $F(=\overline{\rho w}/\rho\prime w\prime)$ together with flow visualization in a low-noise water tunnel. Buoyancy was induced by salinity differences so the value of the Schmidt number Sc = 700. Density stratification altered the inertial-buoyancy force balance (most simply expressed by Nt, the product of the buoyancy frequency N and turbulent timescale t) so as to attenuate turbulent velocity fluctuations, vertical motions and interfacial convolutions, normalized density fluctuations, vertical flux mass, and mean interfacial thickness. Vertical velocity fluctuations w′ were found to increase with distance from the interface, whereas the u′-distribution can be non-monotonic. The maximum value of the mass flux, F, was found to be about 0.5 which was less than the typical value of 0.7 for thermally stratified wind tunnel experiments for which Sc = 0.7. The vertical mass flux can be a combination of down-gradient and counter-gradient transport with the ratio varying with Nt (e.g. at Nt ≈ 5, the flux is counter-gradient). The flux Richardson number Rf was found to increase monotonically to values of approximately 0.05.

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$; here ${\bm {\cal S}}$ and ${\bm \Omega}$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.

We demonstrate experimentally the existence of a family of gravity-induced finiteamplitude water waves that propagate practically without change of form in shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional, and periodic. The basic template of a wave is hexagonal, but it need not be symmetric about the direction of propagation, as required in our previous studies (e.g. Hammack et al. 1989). Like the symmetric waves in earlier studies, the asymmetric waves studied here are easy to generate, they seem to be stable to perturbations, and their amplitudes need not be small. The Kadomtsev–Petviashvili (KP) equation is known to describe approximately the evolution of waves in shallow water, and an eight-parameter family of exact solutions of this equation ought to describe almost all spatially periodic waves of permanent form. We present an algorithm to obtain the eight parameters from wave-gauge measurements. The resulting KP solutions are observed to describe the measured waves with reasonable accuracy, even outside the putative range of validity of the KP model.

Experimental observations and linear stability analysis are used to quantitatively describe a purely elastic flow instability in the inertialess motion of a viscoelastic fluid confined between a rotating cone and a stationary circular disk. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the spatially homogeneous azimuthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetric disturbances in the form of spiral vortices that extend throughout the fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-viscosity fluid show that the resulting secondary flow is composed of logarithmically spaced spiral roll cells that extend across the disk in the self-similar form of a Bernoulli Spiral.

Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison. Introduction of a radial coordinate transformation yields an accurate description of the logarithmic spiral instabilities observed experimentally, and substitution into the linearized disturbance equations leads to a separable eigenvalue problem. Experiments and calculations for two different elastic fluids and for a range of cone angles and Deborah numbers are presented to systematically explore the effects of geometric and rheological variations on the spiral instability. Excellent quantitative agreement is obtained between the predicted and measured wavenumber, wave speed and spiral mode of the elastic instability. The Oldroyd-B model correctly predicts the non-axisymmetric form of the spiral instability; however, incorporation of a shear-rate-dependent first normal stress difference via the nonlinear Chilcott–Rallison model is shown to be essential in describing the variation of the stability boundaries with increasing shear rate.

The sedimentation of small particles from a suspension and the concomitant release of light interstitial fluid may constitute a buoyancy source for the development of convective motions. When the dense suspension is emplaced beneath a stratified fluid an intermediate convecting layer between the sedimenting front and the density gradient above gradually grows in depth by erosion of the overlying stratified fluid. Novel laboratory experiments involving sedimentation below a two-layer stratified region show that turbulent mixing and entrainment across the top density interface is significant for a broad range of the Richardson number. A simple theoretical model predicting the rate of erosion of the stratification above the convecting layer agrees well with these experiments. The model is then extended to include the case of an overlying continuous density gradient and compared successfully with both new experimental data and the original data of Kerr (1991). Owing to the effects of dispersion of grain sizes, small particles in the convecting fluid may lower the efficiency of the interfacial mixing by the turbulent eddies.

Our model calculations suggest that turbulent mixing and entrainment driven by sedimentation may be significant in the atmospheric and oceanic contexts, in both of which stratification is weak. Such mixing may also occur in molten magma chambers following the sedimentation of suspended crystals, and in this case it may suppress large-scale overturning events.

The far-field sound generated by compressible co-rotating vortices is computed by direct computation of the unsteady compressible Navier–Stokes equations on a computational domain that extends to two acoustic wavelengths in all directions. The vortices undergo a period of co-rotation followed by a sudden merger. The directly computed far-field sound is compared to the prediction of the acoustic analogy due to Möhring (1978, 1979), a modified form of the analogy developed by Lighthill (1952), and an acoustic analogy derived by Powell (1964). All three predictions are in excellent agreement with the simulation. Results of far-field pressure fluctuations from an acoustically non-compact, co-rotating vortex pair are also presented. In this case, the vortex sound theory over-predicts the sound by 65% in accordance with the analysis of Yates (1978).

In the literature, there are two different asymptotic results for the viscous decay rates of inertial modes in a rotating circular cylinder. In the absence of a viscous corner solution, either result can only be an estimate of the true decay rate. In this note, we numerically calculate the viscous decay rates for some experimentally excited inertial modes (Malkus 1989; Malkus & Waleffe 1991) in order to (i) assist in the interpretation of these experiments and (ii) to assess the usefulness of the two asymptotic estimates available. Our results indicate that the asymptotic estimate due to Kudlick (1966) is more accurate and that the asymptotic regime in which this estimate is useful (accurate to within 10%) can be smaller than is commonly thought.

We investigate an asymptotic model of DiPrima & Stuart (1972b, 1975) describing steady Taylor vortex flow between eccentric cylinders, under the assumption that the eccentricity ε, the clearance ratio δ and the Taylor vortex amplitude A satisfy ε, δ and A small. By solving a boundary value problem for the radial eigenfunctions we numerically obtain the flow field of DiPrima & Stuart and investigate its topology, after correcting higher-order terms to ensure that the flow preserves volume. We find regions of chaotic streamlines at all eccentricities and discuss the reason for their existence. We make an analogy between the full model and a modulated vortex flow field which qualitatively displays the same behaviour.

For large eccentricities, we examine the flow field and the topology of its streamlines, especially where the two-dimensional flow contains a separated region of recirculation. In this case Taylor vortices give rise to transport of fluid particles in and out of the separated region. We find that the onset of Taylor vortices encourages recirculation in the inflow plane, whilst discouraging it in the outflow plane.

A fluid contained between two parallel walls, one of which is at rest and the other moving in the longitudinal direction with a constant velocity, is examined when a standing sound wave is imposed in the transverse direction. Vortical acoustic streaming appears in the region between the walls. The streaming is not affected by the main flow. A qualitative analysis is presented for the Navier–Stokes equations governing the steady-streaming component of the motion. The study considers the case of flow with high streaming Reynolds number and makes an explicit determination of the vorticity in the inviscid core region. The effect of the streaming upon the shear flow in the longitudinal direction is then analysed asymptotically. A periodic structure of the wall shear stress in the transverse direction is detected in which vast areas of vanishing wall shear stress alternate with narrow regions where it increases significantly. A relation expressing the mean wall shear stress in terms of the streaming Reynolds number is derived. Results obtained show that acoustic streaming results in a marked enhancement of the mean wall shear stress at the walls.

The evolution of an internal gravity wave is investigated by direct numerical computations. We consider the case of a standing wave confined in a bounded (square) domain, a case which can be directly compared with laboratory experiments. A pseudo-spectral method with symmetries is used. We are interested in the inertial dynamics occurring in the limit of large Reynolds numbers, so a fairly high spatial resolution is used (1292 or 2572), but the computations are limited to a two-dimensional vertical plane.

We observe that breaking eventually occurs, whatever the wave amplitude: the energy begins to decrease after a given time because of irreversible transfers of energy towards the dissipative scales. The life time of the coherent wave, before energy dissipation, is found to be proportional to the inverse of the amplitude squared, and we explain this law by a simple theoretical model. The wave breaking itself is preceded by a slow transfer of energy to secondary waves by a mechanism of resonant interactions, and we compare the results with the classical theory of this phenomenon: good agreement is obtained for moderate amplitudes. The nature of the events leading to wave breaking depends on the wave frequency (i.e. on the direction of the wave vector); most of the analysis is restricted to the case of fairly high frequencies.

The maximum growth rate of the inviscid wave instability occurs in the limit of high wavenumbers. We observe that a well-organized secondary plane wave packet is excited. Its frequency is half the frequency of the primary wave, corresponding to an excitation by a parametric instability. The mechanism of selection of this remarkable structure, in the limit of small viscosities, is discussed. Once this secondary wave packet has reached a high amplitude, density overturning occurs, as well as unstable shear layers, leading to a rapid transfer of energy towards dissipative scales. Therefore the condition of strong wave steepness leading to wave breaking is locally attained by the development of a single small-scale parametric instability, rather than a cascade of wave interactions. This fact may be important for modelling the dynamics of an internal wave field.

Consider two infinitely long cylinders of different radii with one inside the other but off-centred. The gap between the two cylinders is partially filled with a viscous fluid. As the cylinders rotate with independent velocities U1 and U2, a thin liquid film coats each of their surfaces all the way around except in the region where the viscous fluid completely fills the gap. Interface conditions that connect solutions of averaged equations in the viscous fluid region with solutions in the thin film region are derived. For the two-interface problem analysed here, two types of instabilities occur depending on the amount of viscous fluid between the cylinders. For large fluid volume, the primary supercritical instability occurs when the front interface becomes unstable as the cylinder velocities are increased. For small fluid volume, the back interface passes through the region where the gap width is a minimum to the same side as the front interface. Steady state solutions with straight interface edges exhibit a turning point with respect to the cylinder velocities. The back interface becomes unstable at the turning point; this inverse instability is subcritical.

For an anisotropic topographic feature in a large-scale flow, the orientation of the topography with respect to the flow will affect the vorticity production that results from the topography–flow interaction. This in turn affects the amount of form drag that the ambient flow experiences. Numerical simulations and perturbation theory are used to explore these effects of change in topographic orientation. The flow is modelled as a quasi-geostrophic homogeneous fluid on an f-plane. The topography is taken to be a hill of limited extent, with an elliptical cross-section in the horizontal. It is shown that, as a result of a basic asymmetry of the quasi-geostrophic flow, the strength of the form drag depends not only on the magnitude of the angle that the topographic axis makes with the oncoming stream, but also on the sign of this angle. For sufficiently low topography, it is found that a positive angle of attack leads to a stronger form drag than that for the corresponding negative angle. For strong topography, this relation is reversed, with the negative angle then resulting in the stronger form drag.

A procedure based on energy stability arguments is presented as a method for extracting large-scale, coherent structures from fully turbulent shear flows. By means of two distinct averaging operators, the instantaneous flow field is decomposed into three components: a spatial mean, coherent field and random background fluctuations. The evolution equations for the coherent velocity, derived from the Navier–Stokes equations, are examined to determine the mode that maximizes the growth rate of volume-averaged coherent kinetic energy. Using a simple closure scheme to model the effects of the background turbulence, we find that the spatial form of the maximum energy growth modes compares well with the shape of the empirical eigenfunctions given by the proper orthogonal decomposition. The discrepancy between the eigenspectrum of the stability problem and the empirical eigenspectrum is explained by examining the role of the mean velocity field. A simple dynamic model which captures the energy exchange mechanisms between the different scales of motion is proposed. Analysis of this model shows that the modes which attain the maximum amplitude of coherent energy density in the model correspond to the empirical modes which possess the largest percentage of turbulent kinetic energy. The proposed method provides a means for extracting coherent structures which are similar to those produced by the proper orthogonal decomposition but which requires only modest statistical input.

We have experimentally studied the effects of mean strain on the evolution of stably stratified turbulence. Grid-generated turbulence ($Re_{\lambda \leqslant 25}$) in a stable linear mean background density gradient was passed through a two-dimensional contraction, contracting the stream only in the vertical direction. This induces an increase in stratification strength, which reduces the largest vertical overturning scales allowed by buoyancy forces. The mean strain through the contraction causes, on the other hand, stretching of streamwise vortices tending to increase the fluctuation levels of the transverse velocity components. This competition between buoyancy and vortex stretching dominates the turbulence dynamics inside and downstream of the contraction. Comparison between non-stratified and stratified experiments shows that the stratification significantly reduces the vertical velocity fluctuations. The vertical heat flux is initially enhanced through the contraction. Then, farther downstream the flux quickly reverses, leading to very strong restratification coinciding with an increase in the vertical velocity fluctuations. The vertical heat flux collapses much more rapidly than in the stratified case without an upstream contraction and the restratification intensity is also much stronger, showing values of normalized flux as strong as −0.55. Velocity spectra show that the revival of vertical velocity fluctuations, due to the strong restratification, starts at the very largest scales but is then subsequently transferred to smaller scales. The distance from the turbulence-generating grid to the entrance of the contraction is an important parameter which was varied in the experiments. The larger this distance, the larger the integral length scale can grow, approaching the limit set by buoyancy, before entering the contraction. The evolution of the various turbulence length scales is described. Two-point measurements of velocity and temperature transverse integral scales were also performed inside the contraction. The emergence of ‘zombie’ turbulence, for large buoyancy times, is in good quantitative agreement with the numerical simulations of Gerz & Yamazaki (1993) for stratification number larger than 1.