The role of diffusion and transient velocities in the dispersal ofpassive scalars by chaotic advection produced in a low Reynoldsnumber periodic journal-bearing flow is studied numerically andexperimentally. The transient velocity field, which occurs wheneverthe cylinders switch motion, is obtained by solving theNavier–Stokes equations numerically in the eccentric annulus. It isobserved, numerically, that the transient effects, along withdiffusion, significantly enhance the separation of chaoticallyadvected particles even when the Reynolds number is very low.Corresponding experimental observations are found to be in goodqualitative agreement with the numerical results obtained byincluding the effect of transient velocities, which are seen to addto the overall separation of particles.

In this experimental study the mixing of passive scalars that arisesfrom shear-free decaying grid-generated turbulence is examined. Theflow configuration consists of two homogeneous layers of equaldensity separated by a sharp interface between different meanconcentrations of passive scalar. Both layers flow through aturbulence-generating grid. To determine the effect of diffusivitythe scalar was heat in one experiment and salt in a second. TheSchmidt number – the ratio of momentum to species diffusivity – was7 and 700 for heat and salt respectively. Velocity, scalar and fluxfields were mapped and flow visualization was undertaken to studythe flows.

Integral lengthscales and (scalar) flux were determined to beindependent of diffusivity, whereas the scalar Taylor microscalesvaried with Schmidt number,Sc, thusillustrating the disparate effects of diffusivity on large and smallscales. The time series of signals showed a correspondence betweenlocations of wθ extrema and uwextrema. Flux wθ arose from intermittent events;and the magnitude of the time-averaged flux $\overline{w\theta}$ was found todepend on the frequency, rather than on variations in the amplitudeof wθ events.

The results of an experimental study of shear-free decayinggrid-generated turbulence on both sides of a sharp interface betweentwo homogeneous layers of different densities are presented. Theevolution of turbulence and mixing were examined by simultaneouslymapping 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 theSchmidt number Sc =700. Density stratification altered the inertial-buoyancy forcebalance (most simply expressed by Nt, the productof the buoyancy frequency N and turbulent timescalet) so as to attenuate turbulent velocityfluctuations, vertical motions and interfacial convolutions,normalized density fluctuations, vertical flux mass, and meaninterfacial thickness. Vertical velocity fluctuationsw′ were found to increase with distance fromthe interface, whereas the u′-distribution can benon-monotonic. The maximum value of the mass flux,F, was found to be about 0.5 which was lessthan the typical value of 0.7 for thermally stratified wind tunnelexperiments for whichSc = 0.7. Thevertical mass flux can be a combination of down-gradient andcounter-gradient transport with the ratio varying withNt (e.g. at Nt ≈ 5, the fluxis counter-gradient). The flux Richardson numberRf was found toincrease monotonically to values of approximately 0.05.

Considerable confusion surrounds the longstanding question of whatconstitutes a vortex, especially in a turbulent flow. This question,frequently misunderstood as academic, has recently acquiredparticular significance since coherent structures (CS) in turbulentflows are now commonly regarded as vortices. An objective definitionof a vortex should permit the use of vortex dynamics concepts toeduce CS, to explain formation and evolutionary dynamics of CS, toexplore the role of CS in turbulence phenomena, and to developviable turbulence models and control strategies for turbulencephenomena. We propose a definition of a vortex in an incompressibleflow 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 symmetricand antisymmetric parts of the velocity gradient tensor${\bm \Delta}{\bmu}$. This definition captures the pressure minimumin a plane perpendicular to the vortex axis at high Reynoldsnumbers, and also accurately defines vortex cores at low Reynoldsnumbers, unlike a pressure-minimum criterion. We compare ourdefinition with prior schemes/definitions using exact and numericalsolutions of the Euler and Navier–Stokes equations for a variety oflaminar and turbulent flows. In contrast to definitions based on thepositive second invariant of ${\bm\Delta}{\bm u}$ or the complex eigenvalues of${\bm \Delta}{\bmu}$, our definition accurately identifies thevortex core in flows where the vortex geometry is intuitivelyclear.

We demonstrate experimentally the existence of a family ofgravity-induced finiteamplitude water waves that propagatepractically without change of form in shallow water of uniformdepth. The surface patterns of these waves are genuinelytwo-dimensional, and periodic. The basic template of a wave ishexagonal, but it need not be symmetric about the direction ofpropagation, as required in our previous studies (e.g. Hammacket al. 1989). Like the symmetric waves inearlier studies, the asymmetric waves studied here are easy togenerate, they seem to be stable to perturbations, and theiramplitudes need not be small. The Kadomtsev–Petviashvili (KP)equation is known to describe approximately the evolution of wavesin shallow water, and an eight-parameter family of exact solutionsof this equation ought to describe almost all spatially periodicwaves of permanent form. We present an algorithm to obtain the eightparameters from wave-gauge measurements. The resulting KP solutionsare observed to describe the measured waves with reasonableaccuracy, even outside the putative range of validity of the KPmodel.

Experimental observations and linear stability analysis are used toquantitatively describe a purely elastic flow instability in theinertialess motion of a viscoelastic fluid confined between arotating cone and a stationary circular disk. Beyond a criticalvalue of the dimensionless rotation rate, or Deborah number, thespatially homogeneous azimuthal base flow that is stable in thelimit of small Reynolds numbers and small cone angles becomesunstable with respect to non-axisymmetric disturbances in the formof spiral vortices that extend throughout the fluid sample. Digitalvideo-imaging measurements of the spatial and temporal dynamics ofthe instability in a highly elastic, constant-viscosity fluid showthat the resulting secondary flow is composed of logarithmicallyspaced spiral roll cells that extend across the disk in theself-similar form of a Bernoulli Spiral.

Linear stability analyses are reported for the quasi-linear Oldroyd-Bconstitutive equation and the nonlinear dumbbell model proposed byChilcott & Rallison. Introduction of a radial coordinatetransformation yields an accurate description of the logarithmicspiral instabilities observed experimentally, and substitution intothe linearized disturbance equations leads to a separable eigenvalueproblem. Experiments and calculations for two different elasticfluids and for a range of cone angles and Deborah numbers arepresented to systematically explore the effects of geometric andrheological variations on the spiral instability. Excellentquantitative agreement is obtained between the predicted andmeasured wavenumber, wave speed and spiral mode of the elasticinstability. The Oldroyd-B model correctly predicts thenon-axisymmetric form of the spiral instability; however,incorporation of a shear-rate-dependent first normal stressdifference via the nonlinear Chilcott–Rallison model is shown to beessential in describing the variation of the stability boundarieswith increasing shear rate.

The sedimentation of small particles from a suspension and theconcomitant release of light interstitial fluid may constitute abuoyancy source for the development of convective motions. When thedense suspension is emplaced beneath a stratified fluid anintermediate convecting layer between the sedimenting front and thedensity gradient above gradually grows in depth by erosion of theoverlying stratified fluid. Novel laboratory experiments involvingsedimentation below a two-layer stratified region show thatturbulent mixing and entrainment across the top density interface issignificant for a broad range of the Richardson number. A simpletheoretical model predicting the rate of erosion of thestratification above the convecting layer agrees well with theseexperiments. The model is then extended to include the case of anoverlying continuous density gradient and compared successfully withboth new experimental data and the original data of Kerr (1991).Owing to the effects of dispersion of grain sizes, small particlesin the convecting fluid may lower the efficiency of the interfacialmixing by the turbulent eddies.

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

The far-field sound generated by compressible co-rotating vortices iscomputed by direct computation of the unsteady compressibleNavier–Stokes equations on a computational domain that extends totwo acoustic wavelengths in all directions. The vortices undergo aperiod of co-rotation followed by a sudden merger. The directlycomputed far-field sound is compared to the prediction of theacoustic analogy due to Möhring (1978, 1979), a modified form of theanalogy developed by Lighthill (1952), and an acoustic analogyderived by Powell (1964). All three predictions are in excellentagreement with the simulation. Results of far-field pressurefluctuations from an acoustically non-compact, co-rotating vortexpair are also presented. In this case, the vortex sound theoryover-predicts the sound by 65% in accordance with the analysis ofYates (1978).

In the literature, there are two different asymptotic results for theviscous decay rates of inertial modes in a rotating circularcylinder. In the absence of a viscous corner solution, either resultcan only be an estimate of the true decay rate. In this note, wenumerically calculate the viscous decay rates for someexperimentally excited inertial modes (Malkus 1989; Malkus &Waleffe 1991) in order to (i) assist in the interpretation of theseexperiments and (ii) to assess the usefulness of the two asymptoticestimates available. Our results indicate that the asymptoticestimate due to Kudlick (1966) is more accurate and that theasymptotic regime in which this estimate is useful (accurate towithin 10%) can be smaller than is commonly thought.

We investigate an asymptotic model of DiPrima & Stuart(1972b, 1975) describing steady Taylor vortexflow between eccentric cylinders, under the assumption that theeccentricity ε, the clearance ratio δ and the Taylor vortexamplitude A satisfy ε, δ and Asmall. By solving a boundary value problem for the radialeigenfunctions we numerically obtain the flow field of DiPrima &Stuart and investigate its topology, after correcting higher-orderterms to ensure that the flow preserves volume. We find regions ofchaotic streamlines at all eccentricities and discuss the reason fortheir existence. We make an analogy between the full model and amodulated vortex flow field which qualitatively displays the samebehaviour.

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

A fluid contained between two parallel walls, one of which is at restand the other moving in the longitudinal direction with a constantvelocity, is examined when a standing sound wave is imposed in thetransverse direction. Vortical acoustic streaming appears in theregion between the walls. The streaming is not affected by the mainflow. A qualitative analysis is presented for the Navier–Stokesequations governing the steady-streaming component of the motion.The study considers the case of flow with high streaming Reynoldsnumber and makes an explicit determination of the vorticity in theinviscid core region. The effect of the streaming upon the shearflow in the longitudinal direction is then analysed asymptotically.A periodic structure of the wall shear stress in the transversedirection is detected in which vast areas of vanishing wall shearstress alternate with narrow regions where it increasessignificantly. A relation expressing the mean wall shear stress interms of the streaming Reynolds number is derived. Results obtainedshow that acoustic streaming results in a marked enhancement of themean wall shear stress at the walls.

The evolution of an internal gravity wave is investigated by directnumerical computations. We consider the case of a standing waveconfined in a bounded (square) domain, a case which can be directlycompared with laboratory experiments. A pseudo-spectral method withsymmetries is used. We are interested in the inertial dynamicsoccurring in the limit of large Reynolds numbers, so a fairly highspatial resolution is used (1292 or 2572), butthe computations are limited to a two-dimensional verticalplane.

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

The maximum growth rate of the inviscid wave instability occurs inthe limit of high wavenumbers. We observe that a well-organizedsecondary plane wave packet is excited. Its frequency is half thefrequency of the primary wave, corresponding to an excitation by aparametric instability. The mechanism of selection of thisremarkable structure, in the limit of small viscosities, isdiscussed. Once this secondary wave packet has reached a highamplitude, density overturning occurs, as well as unstable shearlayers, leading to a rapid transfer of energy towards dissipativescales. Therefore the condition of strong wave steepness leading towave breaking is locally attained by the development of a singlesmall-scale parametric instability, rather than a cascade of waveinteractions. This fact may be important for modelling the dynamicsof an internal wave field.

Consider two infinitely long cylinders of different radii with oneinside the other but off-centred. The gap between the two cylindersis partially filled with a viscous fluid. As the cylinders rotatewith independent velocities U1 andU2, a thin liquid film coats each oftheir surfaces all the way around except in the region where theviscous fluid completely fills the gap. Interface conditions thatconnect solutions of averaged equations in the viscous fluid regionwith solutions in the thin film region are derived. For thetwo-interface problem analysed here, two types of instabilitiesoccur depending on the amount of viscous fluid between thecylinders. For large fluid volume, the primary supercriticalinstability occurs when the front interface becomes unstable as thecylinder velocities are increased. For small fluid volume, the backinterface passes through the region where the gap width is a minimumto the same side as the front interface. Steady state solutions withstraight interface edges exhibit a turning point with respect to thecylinder velocities. The back interface becomes unstable at theturning point; this inverse instability is subcritical.

For an anisotropic topographic feature in a large-scale flow, theorientation of the topography with respect to the flow will affectthe vorticity production that results from the topography–flowinteraction. This in turn affects the amount of form drag that theambient flow experiences. Numerical simulations and perturbationtheory are used to explore these effects of change in topographicorientation. The flow is modelled as a quasi-geostrophic homogeneousfluid on an f-plane. The topography is taken to bea hill of limited extent, with an elliptical cross-section in thehorizontal. It is shown that, as a result of a basic asymmetry ofthe quasi-geostrophic flow, the strength of the form drag dependsnot only on the magnitude of the angle that the topographic axismakes with the oncoming stream, but also on the sign of this angle.For sufficiently low topography, it is found that a positive angleof attack leads to a stronger form drag than that for thecorresponding negative angle. For strong topography, this relationis reversed, with the negative angle then resulting in the strongerform drag.

A procedure based on energy stability arguments is presented as amethod for extracting large-scale, coherent structures from fullyturbulent shear flows. By means of two distinct averaging operators,the instantaneous flow field is decomposed into three components: aspatial mean, coherent field and random background fluctuations. Theevolution equations for the coherent velocity, derived from theNavier–Stokes equations, are examined to determine the mode thatmaximizes the growth rate of volume-averaged coherent kineticenergy. Using a simple closure scheme to model the effects of thebackground turbulence, we find that the spatial form of the maximumenergy growth modes compares well with the shape of the empiricaleigenfunctions given by the proper orthogonal decomposition. Thediscrepancy between the eigenspectrum of the stability problem andthe empirical eigenspectrum is explained by examining the role ofthe mean velocity field. A simple dynamic model which captures theenergy exchange mechanisms between the different scales of motion isproposed. Analysis of this model shows that the modes which attainthe maximum amplitude of coherent energy density in the modelcorrespond to the empirical modes which possess the largestpercentage of turbulent kinetic energy. The proposed method providesa means for extracting coherent structures which are similar tothose produced by the proper orthogonal decomposition but whichrequires only modest statistical input.

We have experimentally studied the effects of mean strain on theevolution of stably stratified turbulence. Grid-generated turbulence($Re_{\lambda \leqslant25}$) in a stable linear mean backgrounddensity gradient was passed through a two-dimensional contraction,contracting the stream only in the vertical direction. This inducesan increase in stratification strength, which reduces the largestvertical overturning scales allowed by buoyancy forces. The meanstrain through the contraction causes, on the other hand, stretchingof streamwise vortices tending to increase the fluctuation levels ofthe transverse velocity components. This competition betweenbuoyancy and vortex stretching dominates the turbulence dynamicsinside and downstream of the contraction. Comparison betweennon-stratified and stratified experiments shows that thestratification significantly reduces the vertical velocityfluctuations. The vertical heat flux is initially enhanced throughthe contraction. Then, farther downstream the flux quickly reverses,leading to very strong restratification coinciding with an increasein the vertical velocity fluctuations. The vertical heat fluxcollapses much more rapidly than in the stratified case without anupstream contraction and the restratification intensity is also muchstronger, showing values of normalized flux as strong as −0.55.Velocity spectra show that the revival of vertical velocityfluctuations, due to the strong restratification, starts at the verylargest scales but is then subsequently transferred to smallerscales. The distance from the turbulence-generating grid to theentrance of the contraction is an important parameter which wasvaried in the experiments. The larger this distance, the larger theintegral length scale can grow, approaching the limit set bybuoyancy, before entering the contraction. The evolution of thevarious turbulence length scales is described. Two-pointmeasurements of velocity and temperature transverse integral scaleswere also performed inside the contraction. The emergence of‘zombie’ turbulence, for large buoyancy times, is in goodquantitative agreement with the numerical simulations of Gerz &Yamazaki (1993) for stratification number larger than 1.