Turbulence in solid-body rotation is generated by a flow of air passing through a rotating cylinder containing a dense honeycomb structure and a turbulence-producing grid. The velocity field is probed downstream of this device by hot-wire probes. Using the statistical quantities characterizing the fluctuating field, we show that the rotation affects mainly the components normal to the rotation axis and that these effects are triggered when the Rossby numbers constructed from macroscopic turbulent quantities, are less than unity. These results are discussed in the framework of other available experimental results on the subject. A theoretical interpretation, chiefly based on spectral analysis, is then proposed to explain the trends of the observations.

The stability of circular Couette flow with radial heating across a vertically oriented annulus with inner cylinder rotating and outer cylinder stationary is investigated using linear stability theory. Infinite aspect ratio and constant fluid properties are assumed and critical stability boundaries are calculated for a conduction-regime base flow. Buoyancy is included through the Boussinesq approximation and stability is tested with respect to both toroidal and helical disturbances of uniform wavenumber. Symmetries of the linearized disturbance equations based on the sense of radial heating and the sense of cylinder rotation and their effect on the kinematics and morphology of instability waveforms are presented. The numerical investigation is primarily restricted to radius ratios 0.6 and 0.959 at Prandtl numbers 4.35, 15 and 100. The results follow the development of critical stability from Taylor cells at zero heating through a number of asymmetric modes to axisymmetric cellular convection at zero rotation. Increasing the Prandtl number profoundly destabilizes the flow in both wide and narrow gaps and the number of contending critical modes increases with increasing radius ratio. Specific calculations made to compare with the stability measurements of Snyder & Karlsson (1964) and Sorour & Coney (1979) exhibit good agreement considering the idealizations built into the linear stability analysis.

We consider a situation involving a three-dimensional marginal separation, where a (steady) boundary-layer flow is on the verge of separating at a point (located along a line of symmetry/centreline). At this point we include a ‘triple-deck’, thereby permitting a small amount of interaction to occur. Unsteadiness is included within this interaction region through some external means. It is shown that the problem reduces to the solution of a nonlinear, unsteady, partial integro system, which is solved numerically by means of time-marching together with a pseudo-spectral method spatially. A number of solutions to this system are presented which strongly suggest that a breakdown of this system may occur, at a finite spatial position, at a finite time. The structure and details of this breakdown are then described.

The nature of the momentum transport processes responsible for the Reynolds shear stress is investigated using several ensembles of fluid particle paths obtained from a direct numerical simulation of turbulent channel flow. It is found that the Reynolds stress can be viewed as arising from two fundamentally different mechanisms. The more significant entails transport in the manner described by Prandtl in which momentum is carried unchanged from one point to another by the random displacement of fluid particles. One-point models, such as the gradient law are found to be inherently unsuitable for representing this process. However, a potentially useful non-local approximation to displacement transport, depending on the global distribution of the mean velocity gradient, may be developed as a natural consequence of its definition. A second important transport mechanism involves fluid particles experiencing systematic accelerations and decelerations. Close to the wall this results in a reduction in Reynolds stress due to the slowing of sweep-type motions. Further away Reynolds stress is produced in spiralling motions, where particles accelerate or decelerate while changing direction. Both transport mechanisms appear to be closely associated with the dynamics of vortical structures in the wall region.

This paper examines the linear stability of zero-pressure-gradient boundary-layer flow over a class of anisotropically responding compliant walls. The anisotropic wall behaviour is derived from a material anisotropy which is characterized by relatively high tensile and compressive strength along a certain direction, termed the fibre axis. When the material stiffness along the fibre axis is sufficiently high, the resulting correlation between the horizontal and the vertical components of wall displacement induces at the flow–wall interface a Reynolds shear stress of a sign that is predetermined by the angle with which the fibre axis makes with the direction of the flow. The notion that anisotropic surface response could be employed to produce turbulent Reynolds shear stresses of predetermined sign at a surface was first explored by Grosskreutz (1971) in an experimental study on turbulent drag reduction. The present paper examines the implications of this interesting idea in the context of two-dimensional flow stability over anisotropic compliant walls. The study covers single- and two-layer compliant walls using the methodology described in Yeo (1988). The effects of wall anisotropy, as determined by the orientation of the fibre axis and the material stiffness along the fibre axis, on flow stability are examined for a variety of walls. The potential of some anisotropic compliant walls for delaying laminar–turbulent transition is investigated, and the contribution of the anisotropy to transition delay is appraised.

The effects of surface-active agents on drop deformation and breakup in extensional flows at low Reynolds numbers are described. In this free-boundary problem, determination of the interfacial velocity requires knowledge of the distribution of surfactant, which, in turn, requires knowledge of the interfacial velocity field. We account for this explicit coupling of the unknown drop shape and the evolving surfactant distribution. An analytical result valid for nearly spherical distortions is presented first. Finite drop deformation is studied numerically using the boundary-integral method in conjunction with the time-dependent convective–diffusion equation for surfactant transport. This procedure accurately follows interfacial tension variations, produced by non-uniform surfactant distribution, on the evolving interface. The numerical method allows for an arbitrary equation of state relating interfacial tension to the local concentration of surfactant, although calculations are presented only for the common linear equation of state. Also, only the case of insoluble surfactant is studied.

The analytical and numerical results indicate that at low capillary numbers the presence of surfactant causes larger deformation than would occur for a drop with a constant interfacial tension equal to the initial equilibrium value. The increased deformation occurs owing to surfactant being swept to the end of the drop where it acts to locally lower the interfacial tension, which therefore requires increased deformation to satisfy the normal stress balance. However, at larger capillary numbers and finite deformations, this convective effect competes with ‘dilution’ of the surfactant due to interfacial area increases. These two different effects of surface-active material are illustrated and discussed and their influence on the critical capillary number for breakup is presented.

We derive the phase diffusion and mean drift equations for the Oberbeck–Boussinesq equations in large-aspect-ratio containers. We are able to recover all the long-wave instability boundaries (Eckhaus, zigzag, skew-varicose) of straight parallel rolls found previously by Busse and his colleagues. Moreover, the development of the skew-varicose instability can be followed and it becomes clear how the mean drift field conspires to enhance the necking of phase contours necessary for the production of dislocation pairs. We can calculate the wavenumber selected by curved patterns and find very close agreement with the dominant wavenumbers observed by Heutmaker & Gollub at Prandtl number 2.5, and by Steinberg, Ahlers & Cannell at Prandtl number 6.1. We find a new instability, the focus instability, which causes circular target patterns to destabilize and which, at sufficiently large Rayleigh numbers, may play a major role in the onset of time dependence. Further, we predict the values of the Rayleigh number at which the time-dependent but spatially ordered patterns will become spatially disordered. The key difficulty in obtaining these equations is the fact that the phase diffusion equation appears as a solvability condition at order ε (the inverse aspect ratio) whereas the mean drift equation is the solvability condition at order ε2. Therefore, we had to use extremely robust inversion methods to solve the singular equations at order ε and the techniques we use should prove to be invaluable in a wide range of similar situations. Finally, we discuss the introduction of the amplitude as an active order parameter near pattern defects, such as dislocations and foci.

The phenomenon of vortex shedding from a heated/cooled circular cylinder has been investigated numerically in the mixed natural and forced convection regimes. Accuracy of the computation was achieved by the fourth-order Hermitian relation applied to the contravariant velocity components in the convection terms of the vorticity transport equation, and by the far-boundary stream-function condition of an integral-series form developed by the authors. Purely periodic flows at Re = 100, efficiently established through the use of a direct elliptic solver called the SEVP, was found to degenerate into a steady twin-vortex pattern at the critical Grashof number 1500, confirming an earlier experimental observation identified as ‘breakdown of the Kármán vortex street’. Various other buoyancy effects about the heated/cooled cylinder are discussed by means of the flow patterns, the Nusselt number and the drag coefficient curves.

Unsteady gasdynamic concepts are used to model the piston-driven compression of a confined gas. Perturbation methods, based on the limit of small piston Mach number, are used to construct solutions. The piston Mach number increases smoothly from zero to a maximum value, Mp = O(10−2) during an acoustic time period ta* = O(10−4 s). A linear a coustic field is generated and is represented in terms of an infinite series of Fourier spatial modes. During the longer piston time period tp* = O(10−2 s) the piston moves at constant speed. A multiple-timescale formulation is used to separate the instantaneous acoustic field from the accumulated bulk response of the gas to piston compression. The latter is found to be identical to the classical quasi-static results from equilibrium thermodynamic calculations. Nonlinear effects become important on the piston timescale. Modal interactions are represented by a system of coupled, nonlinear ordinary differential equations for the time-dependent Fourier coefficients. A numerical solution for this system describes the wavefront steepening to form a weak shock and its propagation back and forth repeatedly inside the cylinder.

Calculations are presented for the characteristics of instability waves in the initial mixing region of twin circular supersonic jets. Two models for the basic jet flow are used. In the first, the jets are modelled as two circular vortex sheets. In the second, realistic velocity and density profiles are used. It is shown that the unsteady flow fields of the two jets interact before the time-averaged jets flows have merged. The normal modes or instability waves are classified by their symmetry properties in the twin-jet case and their asymptotic behaviour for large jet separations. Calculations of the growth rates and phase velocities are made for these modes as a function of jet separation and mixing-layer thickness. The associated pressure distributions are also presented. In the realistic jet profile calculations the effect of jet separation is found to be relatively weak. For modes that are even about the symmetry plane between the two jets the pressure levels are found to increase near this plane as the jet separation decreases.

Inviscid–viscous interaction in a high supersonic flow is studied on the triple-deck scales to delineate the wall-temperature influence on the flow structure in a region near a laminar separation. A critical wall-temperature range O(Tw*) is identified, in which the pressure–displacement relation governing the lower deck departs from that of the classical (Stewartson, Messiter, Neiland) formulation, and below which the pressure–displacement relation undergoes still greater changes along with drastic scale changes in the triple deck. The reduced lower-deck problem falls into three domains: (i) supercritical (Tw* [Lt ] Tw), (ii) transcritical (Tw = O(Tw*)) and (iii) subcritical (Tw [Lt ] Tw*). Readily identified is a parameter domain overlapping with the Newtonian triple-deck theory of Brown, Stewartson & Williams (1975), even though the assumption of a specific-heat ratio approaching unity is not required here. Computational study of the compressive free-interaction solutions and solutions for a sharp-corner ramp are made for the three wall-temperature ranges. Finite-difference equations for primitive variables are solved by iterations, employing Newton linearization and a large-band matrix solver. Also treated in the program is the sharp-corner effect through the introduction of proper jump conditions. Comparison with existing numerical results in the supercritical Tw range reveals a smaller separation bubble and a more pronounced corner behaviour in the present numerical solution. Unlike an earlier comparison with solutions by interactive-boundary-layer methods for ramp-induced pressure with separation, the IBL results do approach closely the triple-deck solution at Re = 108 in a Mach-three flow, and the differences at Re = 106 may be attributed in part to the transcritical temperature effect. Examination of the numerical solutions indicates that separation and reattachment on a compressive ramp cannot be effectively eliminated/delayed by lowering the wall temperature, but lowering Tw drastically reduces the triple-deck dimension, and hence the degree of upstream influence.

A numerical solution for the flow field associated with a compressible pair of counter-rotating vortices is developed. The compressible, two-dimensional potential equation is solved utilizing the numerical method of Osher et al. (1985) for flow regions in which a non-zero density exists. Close to the vortex centres, vacuum ‘cores’ develop owing to the existence of a maximum achievable flow speed in a compressible flow field. A special treatment is required to represent these vacuum cores. Typical streamline patterns and core boundaries are obtained for upstream Mach numbers as high as 0.3, and the formation of weak shocks, predicted by Moore & Pullin (1987), is observed.

We consider a sequence of boundary-value problems for the acoustic wave equation, with the pressure specified on the boundary as a function of space and time, and simulating features of the pressure field measured just outside a turbulent shear layer supporting large-scale coherent structures. The boundary pressure field has the form of a travelling subsonic plane wave, modulated by a large-scale envelope function. Three models for the envelope distribution are studied in detail, and the particular features which they exhibit are shown to be representative of large classes of amplitude functions.

We start by looking at the hydrodynamic near field of the boundary pressure fluctuations, over spatial regions throughout which the motion can be taken as incompressible. Very close to the boundary, the pressure fluctuations decay exponentially with transverse distance, while at sufficiently large distances from the whole wave packet on the boundary, the pressure fluctuations have a dipole algebraic decay. We investigate the transition from exponential to algebraic decay, and find that it is effected through quite a complicated multilayer structure which depends crucially on the detailed form of the envelope.

Acoustic fields are then determined both from exact solutions to the wave equation, and from matching arguments. In some cases, where the boundary source is compact, the distant acoustic fields have a simple compressible dipole type of behaviour. In other cases, however, when the boundary source is non-compact, the acoustic field has a superdirective character, the angular variation being described by exponentials of cosines of the angle with the streamwise direction. It is shown how the superdirective acoustic sources are completely compatible with the features of the inner incompressible field, and a criterion for the occurrence of the superdirective acoustic fields will be given. Superdirective fields of this kind have been observed in measurements by Laufer & Yen (1983) on a low-speed round jet of Mach number 0.1, and the general relation of our results to those experiments is explained.

The different modifications of the models of two-phase multicomponent filtration (Collins 1961; Nikolaevsky et al. 1968) enable one to study the dynamics of filtration flows, taking into account phase transitions. The equations of multicomponent filtration are quite complicated and only in a few individual cases do they allow for an exact solution. The most frequently used of these appears to be the solution of the stationary problem of the flow of a multicomponent mixture toward a well or a system of wells (Khristianovich 1941). In the present paper we show that at certain values of pressure, temperature and composition of the multicomponent mixture a stationary solution of the problem may not exist. The absence of a stationary solution is related to the possibility of a spatially homogeneous solution losing its stability under a perturbation (Mitlin 1986a, 1987b). We obtain an analytical criterion for instability. As an illustration, we present the results of the numerical solution of the planar linear problem of the evolution of a multicomponent system whose pressure and composition are perturbed with respect to their constant values, which are equal at both ends. We have done a numerical analysis of the plane-radial problem of the operation of a gas–condensate well with different mass fluxes, applying the conditions of a real deposit. There are several ranges of flux where the flow becomes pulsating. It is shown that the time within which the stationary solution sets in is a non-monotonic function of flux and on approaching the stability limit diverges in inverse proportion to the undercriticality of debit. We have analysed the connection between the observed instabilities and the thermodynamics of two-phase multicomponent mixtures. It is shown that the instabilities are associated with the system entering the region of retrograde condensation. We discuss the relation of retrograde phenomena to the effect of negative volume of heavy components and, as a consequence, to the negative compressibility of an individual volume of a two-phase mixture moving in a porous medium. It is shown that the observed autowave modes are relaxation oscillations in a distributed system. By using the method of perturbations in the interphase equilibrium time, we have analysed the loss of stability in a more general – non-equilibrium – model. We show that the instabilities are generated according to the Landau–Hopf scenario and calculate the period of auto-oscillations. The one-mode approximation of the theory leads to the Van der Pol equation. In conclusion we present an experimental confirmation of the theory.

The separation of a two-dimensional laminar boundary layer under the influence of a suddenly imposed external adverse pressure gradient was studied by time-accurate numerical solutions of the Navier–Stokes equations. It was found that a strong adverse pressure gradient created periodic vortex shedding from the separation. The general features of the time-averaged results were similar to experimental results for laminar separation bubbles. Comparisons were made with the ‘steady’ separation experiments of Gaster (1966). It was found that his ‘bursting’ occurs under the same conditions as our periodic shedding, suggesting that bursting is actually periodic shedding which has been time-averaged. The Strouhal number based on the shedding frequency, local free-stream velocity, and boundary-layer momentum thickness at separation was independent of the Reynolds number and the pressure gradient. A criterion for onset of shedding was established. The shedding frequency was the same as that predicted for the most amplified linear inviscid instability of the separated shear layer.

The Taylor–Saffman problem concerns the fingering instability which develops when one liquid displaces another, more viscous, liquid in a porous medium, or equivalently for Newtonian liquids, in a Hele-Shaw cell. Recent experiments with Hele-Shaw cells using non-Newtonian liquids have shown striking qualitative differences in the fingering pattern, which for these systems branches repeatedly in a manner resembling the growth of a fractal. This paper is an attempt to provide the beginnings of a hydrodynamical theory of this instability by repeating the analysis of Taylor & Saffman using a more general constitutive model. In fact two models are considered; the Oldroyd ‘Fluid B’ model which exhibits elasticity but not shear thinning, and the Ostwald–de Waele power-law model with the opposite combination. Of the two, only the Oldroyd model shows qualitatively new effects, in the form of a kind of resonance which can produce sharply increasing (in fact unbounded) growth rates as the relaxation time of the fluid increases. This may be a partial explanation of the observations on polymer solutions; the similar behaviour reported for clay pastes and slurries is not explained by shear-thinning and may involve a finite yield stress, which is not incorporated into either of the models considered here.

The evolution of infinitesimal material line and surface elements in homogeneous isotropic turbulence is studied using velocity-gradient data generated by direct numerical simulations (DNS). The mean growth rates of length ratio (l) and area ratio (A) of material elements are much smaller than previously estimated by Batchelor (1952) owing to the effects of vorticity and of non-persistent straining. The probability density functions (p.d.f.'s) of l/〈l〉 and A/〈A〉 do not attain stationarity as hypothesized by Batchelor (1952). It is shown analytically that the random variable l/〈l〉 cannot be stationary if the variance and integral timescale of the strain rate along a material line are non-zero and DNS data confirm that this is indeed the case. The application of the central limit theorem to the material element evolution equations suggests that the standardized variables $\hat{l}(\equiv (\ln l - \langle \ln l\rangle)/({\rm var} l)^{\frac{1}{2}})$ and Â(≡(ln A − 〈ln A〉)/(var A)½) should attain stationary distributions that are Gaussian for all Reynolds numbers. The p.d.f.s of $\hat{l}$ and  calculated from DNS data appear to attain stationary shapes that are independent of Reynolds number. The stationary values of the flatness factor and super-skewness of both $\hat{l}$ and  are in close agreement with those of a Gaussian distribution. Moreover, the mean and variance of ln l (and ln A) grow linearly in time (normalized by the Kolmogorov timescale, τη), at rates that are nearly independent of Reynolds number. The statistics of material volume-element deformation are also studied and are found to be nearly independent of Reynolds number. An initially spherical infinitesimal volume of fluid deforms into an ellipsoid. It is found that the largest and the smallest of the principal axes grow and shrink respectively, exponentially in time at comparable rates. Consequently, to conserve volume, the intermediate principal axis remains approximately constant.

The performance of the stochastic model of Girimaji & Pope (1990) for the velocity gradients is also studied. The model estimates of the growth rates of 〈ln l〉 and 〈ln A〉 are close to the DNS values. The growth rate of the variances are estimated by the model to within 17%. The stationary distributions of $\hat{l}$ and  obtained from the model agree very well with those calculated from DNS data. The model also performs well in calculating the statistics of material volume-element deformation.

The unsteady flow past a circular cylinder which starts translating and rotating impulsively from rest in a viscous fluid is investigated both theoretically and experimentally in the Reynolds number range 103 [les ] R [les ] 104 and for rotational to translational surface speed ratios between 0.5 and 3. The theoretical study is based on numerical solutions of the two-dimensional unsteady Navier–Stokes equations while the experimental investigation is based on visualization of the flow using very fine suspended particles. The object of the study is to examine the effect of increase of rotation on the flow structure. There is excellent agreement between the numerical and experimental results for all speed ratios considered, except in the case of the highest rotation rate. Here three-dimensional effects become more pronounced in the experiments and the laminar flow breaks down, while the calculated flow starts to approach a steady state. For lower rotation rates a periodic structure of vortex evolution and shedding develops in the calculations which is repeated exactly as time advances. Another feature of the calculations is the discrepancy in the lift and drag forces at high Reynolds numbers resulting from solving the boundary-layer limit of the equations of motion rather than the full Navier–Stokes equations. Typical results are given for selected values of the Reynolds number and rotation rate.

Continuous-phase properties were studied for homogeneous dilute particle-laden flows caused by nearly monodisperse glass particles falling in a stagnant water bath. Test conditions included 0.5, 1.0 and 2.0 mm diameter particles (yielding particle Reynolds numbers based on terminal velocities of 38, 156 and 545) with particle volume fractions less than 0.01%. Measurements included mean and fluctuating velocities, as well as temporal spectra and spatial correlations of velocity fluctuations in the streamwise and cross-stream directions, using a two-point phase-discriminating laser velocimeter. Flow properties were also analysed using a stochastic method involving linear superposition of randomly-arriving particle velocity fields.

For present test conditions, liquid velocity fluctuations varied solely as a function of the rate of dissipation of particle energy in the liquid. The flows were highly anisotropic with streamwise velocity fluctuations being roughly twice cross-stream velocity fluctuations. Correlation coefficients and temporal spectra were independent of both particle size and the rate of dissipation of particle energy in the liquid. The temporal spectra indicated a large range of frequencies even though particle Reynolds numbers were relatively low, since both mean and fluctuating velocities in the particle wakes contributed to the spectra because particle arrivals were random. The theory predicted many of the features of the flows reasonably well but additional information concerning the mean and turbulent structure of the wakes of freely moving particles having moderate Reynolds numbers in turbulent environments is needed to address deficiencies in predictions of integral scales and streamwise spatial correlations.

Turbulent dispersion of particles in their self-generated homogeneous turbulent field was studied both experimentally and theoretically. Measurements involved nearly monodisperse spherical glass particles (nominal diameters of 0.5, 1.0 and 2.0 mm) falling with uniform particle number fluxes in a nearly stagnant water bath. Particle Reynolds numbers based on terminal velocities were 38, 156, and 545 for the three particle sizes. The flows were dilute with particle volume fractions less than 0.01%. Measurements included particle motion calibrations, using motion-picture shadowgraphs; and streamwise and cross-stream mean and fluctuating particle velocities, using a phase-discriminating laser velocimeter. Liquid-phase properties were known from earlier work. Particle properties were predicted based on random-walk calculations using statistical time-series methods to simulate liquid velocities along the particle path.

Calibrations showed that particle drag properties were within 14% of estimates based on the standard drag correlation for spheres, however, the particles (particularly the 1.0 and 2.0 mm diameter particles) exhibited self-induced lateral motion even in motionless liquid due to eddy-shedding and irregularities of shape. Particle velocity fluctuations were primarily a function of the rate of dissipation of kinetic energy in the liquid since this variable controls liquid velocity fluctuations. Streamwise particle velocity fluctuations were much larger than cross-stream particle velocity fluctuations (2–5:1) largely due to varying terminal velocities caused by particle size variations. Cross-stream particle and liquid velocity fluctuations were comparable owing to the combined effects of turbulent dispersion and self-induced motion. Predicted mean and fluctuating particle velocities were in reasonably good agreement with the measurements after accounting for effects of particle size variations and self-induced motion. However, the theory must be extended to treat self-induced motion and to account for observations that this motion was affected by the turbulent environment.