The proper orthogonal decomposition technique (Lumley's decomposition) is applied to the turbulent flow in a channel, to extract coherent structures by decomposing the velocity field into characteristic eddies with random coefficients. In the homogeneous spatial directions a generalization of the shot-noise expansion is used to determine the characteristic eddies. In this expansion the Fourier coefficients of the characteristic eddy cannot be obtained from second-order statistics. Three different techniques are used to determine the phases of these coefficients: (i) a technique based on the bispectrum, (ii) a spatial compactness requirement, and (iii) a functional continuity argument. Results from these three techniques are found to be very similar. The implications of these techniques and the shot-noise expansion are discussed in the Appendix. The dominant eddy is found to contribute as much as 76% to the turbulent kinetic energy. In two and three dimensions, the characteristic eddies consist of an ejection region straddled by streamwise vortices which leave the wall in a very short streamwise distance of approximately 100 wall units.

Unsteady flow in a circular conduit with a smooth expansion is studied in detail by numerical integration of the equation of motion in the axisymmetric approximation. The values of governing parameters are chosen to be relevant to medical problems, and the geometry corresponds to a scenario of post-surgical conditions. The flow determined by an oscillatory volume is characterized by a sequence of vortex rings moving in the expanded part of the tube. The development of wall shear stress is governed by the separated translating vorticity which induces an evolving band of large intensity for about a complete oscillation cycle. This influences the dynamics of unsteady separation whose space-time development has revealed features of some generality which have been classified. The time variation of the pressure jump is dominated by inertial effects. The dependence of the details of the flow on the dimensionless parameters has been investigated systematically. The results obtained here have been compared with experimental and numerical studies of similar problems, similarities have been pointed out and differences discussed. Finally, the relevance of these results to physiological applications has been quantified by simulating the flow induced by a pulsatile flow rate.

The fine-scale structure of turbulence in a fully developed turbulent duct flow is examined by considering the three-dimensional velocity derivative field obtained from direct numerical simulations at two relatively small Reynolds numbers. The magnitudes of all mean-square derivatives (normalized by wall variables) increase with the Reynolds number, the increase being largest at the wall. These magnitudes are not consistent with the assumption of local isotropy except perhaps near the duct centre-line. When the assumption of local isotropy is relaxed to one of local axisymmetry, or invariance with respect to rotation about a coordinate axis (here chosen in the streamwise direction), satisfactory agreement is indicated by the data outside the wall region. Support for axisymmetry is demonstrated by anisotropy invariant maps of the dissipation and vorticity tensors. The departure from axisymmetry does not appear to be affected by the Reynolds number. Expressions are proposed for approximations to the average energy dissipation and components of the mean-square vorticity. These proposals should allow these quantities to be measured accurately, at least in the present flow.

Solutions for inviscid rotating flow over a right circular cylinder of finite height are studied, and comparisons are made to quasi-geostrophic solutions. To study the combined effects of finite topography and the variation of the Coriolis parameter with latitude a steady inviscid model is used. The analytical solution consists of one part which is similar to the quasi-geostrophic solution that is driven by the potential vorticity anomaly over the topography, and another, similar to the solution of potential flow around a cylinder, that is driven by the matching conditions on the edge of the topography. When the characteristic Rossby wave speed is much larger than the background flow velocity, the transport over the topography is enhanced as the streamlines follow lines of constant background potential vorticity. For eastward flow, the Rossby wave drag can be very much larger than that predicted by quasi-geostrophic theory. The combined effects of finite height topography and time-dependence are studied in the inviscid initial value problem on the f-plane using the method of contour dynamics. The method is modified to handle finite topography. When the topography takes up most of the layer depth, a stable oscillation exists with all of the fluid which originates over the topography rotating around the topography. When the Rossby number is order one, a steady trapped vortex solution similar to the one described by Johnson (1978) may be reached.

The solidification of melts can be profoundly influenced by convection. In alloys, compositional convection can be driven by solute gradients generated as one component of the alloy is preferentially incorporated within the solid, even when the thermal field is stabilizing. In this paper, two modes of compositional convection during solidification from below are uncovered using a linear-stability analysis: one, which we shall call the ‘mushy-layer mode’, is driven by buoyant residual fluid within a mushy layer, or porous medium, of dendritic crystals; the other, which we shall call the ‘boundary-layer mode’ is associated with a narrow compositional boundary layer in the melt just above the mush-liquid interface. Either mode can be the first to become unstable depending on the thermodynamical and physical properties of the alloy. The marginally stable eigenfunctions suggest that the boundary-layer mode results in fine-scale convection in the melt above the mushy layer and leaves the interstitial fluid of the mushy layer virtually stagnant. In contrast, the mushy-layer mode causes perturbations to the solid fraction of the mushy layer that are indicative of a tendency to form chimneys, which are vertical channels of reduced or zero solid fraction that have been observed experimentally. Particular attention is focused on the mushy-layer mode and its dependence upon the thermodynamical properties of the alloy. The results of this analysis are used to make a number of interpretations of earlier experimental studies such as the observations that some systems are less prone to form chimneys and that the regions of melt in these systems evolve to supersaturated conditions, while the melt evolves to unsaturated conditions once chimneys have formed. In addition, good quantitative agreement is found between the results of the linear-stability analysis and the experimental results of Tait & Jaupart (1992) for the onset of the mushy-layer mode of convection.

Bedload sheets are coherent migrating patterns of bed material recently observed both in flume studies and in field streams with beds of coarse sand and fine gravel. This newly recognized feature is inherently associated with the heterogeneous character of the sediment and consists of sorting waves with distinct coarse fronts only one or two coarse grains high.

The question of the formation of bedload sheets poses an interesting and peculiar stability problem for the grain size distribution. Sorting waves are essentially two-dimensional migrating perturbations associated with variations of this distribution. We show that their growth is strictly associated with grain sorting. In fact the latter gives rise to perturbations of bedload transport which drive small perturbations of bottom elevation the amplitude of which scales with grain size. The sorting wave also induces spatial variations of bottom roughness, and consequently alters the fluid motion, which conversely exerts a spatially varying stress on the bed. The feature of bedload sheets which allows them to be distinguished from dunes over beds with coarse sand or fine gravel is then the fact that sorting is the dominant effect controlling their growth, rather than being a relatively small perturbation of the mechanism which gives rise to dunes in the case of uniform sediment.

The requirement that perturbations should not alter the sediment budget leads to an integral condition which gives rise to an integro-differential mathematical problem. With the help of recently developed bedload relationships suitable for mixtures, as well as appropriate modelling of turbulent channel flow over a bed with spatially periodic perturbations of bottom elevation and roughness we are able to derive a general dispersion relation which can be readily solved in terms of undisturbed size densities in the form of sums of Dirac distributions.

Perturbations are found to be unstable within a range of wavenumbers depending on the relative roughness and Froude number. We show that when the effects of perturbations of bottom elevation are neglected the unstable region corresponds to the range of conditions where the bottom stress leads bottom roughness, a range distinct from that which characterizes the formation of dunes. This result is given a physical explanation which depends crucially on the deviation from equal mobility of different grain sizes in the surface layer. The effect of perturbations of bottom elevation is however not negligible when the bottom roughness is fairly large compared to depth. In the latter case perturbations of bottom elevation and of bottom roughness are equally important, and gravel sheets are not easily distinguished from small-amplitude dunes.

Comparison with the field observations of Whiting et al. (1985, 1988) is satisfactory insofar as the bedload sheet mode is unstable under the conditions of the experiments, and the predicted wavelengths fall within the experimental range. The laboratory observations of Kuhnle & Southard (1988), on the other hand, appear to fall within a range of bottom roughness where the observed bedforms do not exhibit features unambiguously distinct from those of small-amplitude dunes.

The effect of rapid mean compression on compressible turbulence at a range of turbulent Mach numbers is investigated. Rapid distortion theory (RDT) and direct numerical simulation results for the case of axial (one-dimensional) compression are used to illustrate the existence of two distinct rapid compression regimes. These regimes – the nearly solenoidal and the ‘pressure-released’ – are defined by a single parameter involving the timescales of the mean distortion, the turbulence, and the speed of sound. A general RDT formulation is developed and is proposed as a means of improving turbulence models for compressible flows. In contrast to the well-documented observation that ‘compressibility’ (measured, for example, by the turbulent Mach number) is often associated with a decrease in the growth rate of turbulent kinetic energy, we find that under rapid distortion compressibility can produce an amplification of the kinetic energy growth rate. We also find that as the compressibility increases, the magnitude of the pressure–dilation correlation increases, in absolute terms, but its relative importance decreases compared to the magnitude of the kinetic energy production.

It is shown that the expressions for the correlation tensors of homogeneous axisymmetric turbulence can be considerably simplifed compared to previous analyses of Batchelor (1946) and Chandrasekhar (1950). Representations of the axisymmetric two-point correlations tensors are found, such that each measurable correlation corresponds to a single scalar function, and moreover such that the equations of continuity relating different tensor components to each other take the most simple form. Reflectional symmetry in planes normal to but not in planes through the axis of symmetry is demanded, which allows a full description of states with rotation about the axis of symmetry. The second and third-order velocity correlation tensors and the first-order pressure velocity correlation tensor are analysed with the new method. Small separation expansions of the correlation functions yield the quantities which have to be measured to determine various terms in the governing equations for the Reynolds stresses and the dissipation tensor. A scalar Poisson equation for the pressure-strain is derived. and the single-pint solution is written as a sum of integrals over measurable correlation functions. The simplified analysis can be of great experimental importance. It reveals in a simple way how a full experimental picture of homogeneous axisymmetric turbulence can be obtained by measuring components of the velocity at two points at variable distance from each other on a line perpendicular to the mean flow in a wind tunnel. By using the Fouricr Besscl transform it is also shown that the three-demensional energy, transfer, and pressure stain spectra can be extracted from such measurements.

The properties of a relatively uncommon regime of fluid dynamics, low Mach number compressible flow are investigated. This regime, which is characterized by an exceptionally large channel aspect ratio L/d ∼ 106 leads to highly subsonic flows in which friction dominates inertia. Even so, because of the large aspect ratio, finite pressure, temperature, and density gradients are required, implying that compressibility effects are also important. Analytical results are presented which show, somewhat unexpectedly, that for forced channel flow, steady-state solutions exist only below a critical value of heat input. Above this value the flow reverses against the direction of the applied pressure gradient causing fluid to leave both the inlet and outlet implying that the related concepts of a steady-state friction factor and heat transfer coefficient have no validity.

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.

Statistical analysis was performed for interpreting the dissipation correlations in turbulent wall-bounded flows. The fundamental issues related to the formulation of the closure assumptions are discussed. Using the two-point correlation technique, a distinction is made between the homogeneous and inhomogeneous parts of the dissipation tensor. It is shown that the inhomogeneous part contributes half of the dissipation rate at the wall and vanishes remote from the wall region. The structure of an analytically derived equation was analysed utilizing the results of direct numerical simulations of turbulent channel flow at low Reynolds number.

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t-2.

In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as O(1000), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the non-orthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille–Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal.

When a body of fluid with a vertical salinity gradient is heated from a vertical sidewall instabilities are sometimes observed. The linear stability of this basic state has been investigated by Kerr (1989). This linear theory predicts the onset of instability well when compared with experimental results; however, the form of the observed nonlinear instabilities does not coincide with the linear predictions (cf. Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986; Tanny & Tsinober 1988). In this paper we investigate some of the nonlinear aspects of the problem. A weakly nonlinear analysis reveals that the bifurcation into instability is subcritical, and that the initial trend along this branch of solutions is towards the co-rotating cells observed in experiments. The heating levels for which instabilities are absent are investigated by the use of energy stability analysis. This yields a weak result for arbitrary disturbances, showing that disturbances will decay for sufficiently low wall heating. This bound is greatly strengthened by imposing a vertical periodicity on the lengthscale proposed by Chen et al.

The process of alignment, a new fundamental interaction between vortices in a stratified and rapidly rotating fluid, is defined and studied in detail in the context of the two-layer quasi-geostrophic model. Alignment occurs when two vortices in different density layers coalesce by reducing their horizontal separation. It is found that only vortices whose radii are comparable with or larger than the Rossby deformation radius can align. In the same way as the merger process (in a single two-dimensional layer) is related to the reverse energy cascade of two-dimensional turbulence, geostrophic potential vorticity alignment is related the barotropic-to-baroclinic energy cascade of geostrophic turbulence in two layers. It is also shown how alignment is intimately connected with the existence of two-layer doubly connected geostrophic potential vorticity equilibria (V-states), for which the analysis of the geometry of the stream function in the corotating frame is found to be a crucial diagnostic. The finite-area analogues of the hetons of Hogg & Stommel (1985) are also determined: they consist of a propagating pair of opposite-signed potential vorticity patches located in different layers.

Depending on the flow conditions, the liquid film in annular two-phase flow in coiled tubes may be pushed towards the outer or the inner side by the centrifugal force. It is important to understand the mechanism of this ‘film inversion’ in order to develop a predictive model for the film thickness distribution. In this paper, this phenomenon is studied analytically, and a new criterion, based on the secondary flow in the thin liquid film, is proposed to predict its occurrence. The criterion shows good agreement with available experimental data. It is suggested that the analytical model can readily be extended to predict the distribution of the film thickness and film flow rate in coiled tubes.