The laminar viscous flow in the gap between two concentric spheres is investigated for a rotating inner sphere. The solution is obtained by solving the Navier-Stokes equations by means of finite-difference techniques, where the equations are restricted to axially symmetric flows. The flow field is hydrodynamically unstable above a critical Reynolds number. This investigation indicates that the critical Reynolds number beyond which Taylor vortices appear is slightly higher in a spherical gap than for the flow between concentric cylinders. The formation of Taylor vortices could be observed only for small gap widths s ≤ 0·17. The final state of the flow field depends on the initial conditions and the acceleration of the inner sphere. Steady and unsteady flow modes are predicted for various Reynolds numbers and gap widths. The results are in agreement with experiment if certain accuracy conditions of the finite-difference methods are satisfied. It is seen that the equatorial symmetry is of great importance for the development of the Taylor vortices in the gap.

The TURBIT-3 computer code has been used for the direct numerical simulation of Bénard convection in an infinite plane channel filled with air. The method is based on the three-dimensional non-steady-state equations for the conservation of mass, momentum and enthalpy. Subgrid-scale models of turbulence are not required, as calculations with different grids show that the spatial resolution of grids with about 322 × 16 nodes provides sufficient accuracy for Rayleigh numbers up to Ra = 3·8 × 105. Hence this simulation model contains no tuning parameters.

The simulations start from nearly random initial conditions. This has been found to be essential for calculating flow patterns and statistical data insensitive to grid parameters and agreeing with experimental experience. The numerical results show the theoretically predicted ‘skewed varicose’ instability at Ra = 4000. Warm and cold ‘blobs’ are identified as causing temperature-gradient reversals for all the high Rayleigh numbers under consideration. The calculated wavelengths and the corresponding flow regimes observed in the transition range confirm the stability maps determined theoretically. In the turbulent range the wavelengths agree qualitatively with low-aspect-ratio experiments. Accordingly, the Nusselt numbers lie at the upper end of the scatter band of experimental data, as these also depend on the aspect ratio. Appropriately normalized, the velocity and temperature fluctuation peaks are independent of the Rayleigh number. The vertical profiles agree largely with experimental data and, especially in case of temperature statistics, exhibit comparable or less scatter.

Higher-order statistics of the streamwise velocity derivative have been measured on the centre-line of turbulent plane and circular jets. The instrumentation and sources of error are discussed to establish the accuracy of the data and convergence of statistics. The optimum setting for the low-pass filter cut-off was found to be 1·75 times the Kolmogorov frequency fK, in contrast with the majority of previous investigations where it was set equal to fK. The magnitude of the constant μ in Kolmogorov's revised hypothesis is obtained using statistics derived from the instantaneous velocity derivative or its squared value. The correlation and spectrum of fluctuations of the squared velocity derivative and the Reynolds-number variation of the skewness and flatness factors of the velocity derivative are consistent with μ ≃ 0·2, while the most popular value used is 0·5. Second-order moments of the locally averaged dissipation, assumed proportional to the squared streamwise velocity derivative, and breakdown coefficients also suggest a value of μ of about 0·2. Higher-order correlations and spectra of the dissipation are in closer agreement with the Novikov-Stewart or β-model than with Kolmogorov's lognormal model. Higher-order moments of locally averaged values of the dissipation rate are more closely represented by the lognormal than the β-model.

Local measurement of the mass-transfer rate and velocity gradient when an axisymmetric jet impinges on a flat plate was carried out using an electrochemical technique. Local measurement of the surface pressure on the flat plate was carried out separately using piezoelectric pressure transducers. The stagnation-point mass-transfer coefficient reaches a maximum when the flat plate is placed at 6 nozzle diameters from a convergent nozzle. It has been confirmed that the mass transfer to the flat plate for a high Schmidt number is greatly enhanced owing to the velocity-gradient disturbances in the wall region of the boundary layer, while the momentum transfer is insensitive to such disturbances. The relative intensity of the velocity-gradient fluctuations on the wall has an extremely large value at and near to the stagnation point, and decreases downstream, approaching a large constant value.

These velocity-gradient disturbances are not due to the usual interaction of Reynolds stress with the shear stress of the mean flow, but are due to the interaction with the surface-pressure fluctuations converted from the velocity fluctuations of the oncoming jet.

The three co-ordinate dimensions of large-scale eddies are calculated from the auto- and spatial correlations of the surface-pressure fluctuations. It is considered that such large-scale eddies play an important role in the production of a velocity-gradient disturbance in the wall region of the boundary layer from the velocity turbulence of the approaching jet.

A difficulty in applying the lubrication approximation to flows where a liquid/air interface forms lies in supplying boundary conditions at the point of formation of the interface that are consistent with the lubrication approximation. The method of matched asymptotic expansions is applied to the flow between partially submerged, counter-rotating rollers, a representative problem from this class, and the lubrication approximation is found to generate the first term of an outer expansion of the problem solution. The first term of an inner expansion describes the two-dimensional flow in the vicinity of the interface, and approximate results are found by the finite-element method. Matching between the inner and outer solutions determines boundary conditions on the pressure and the pressure gradient at the point of formation of the interface which allow the solution to the outer, lubrication flow to be completed.

An experimental study of a steady, incompressible, three-dimensional turbulent boundary layer approaching separation is reported. The flow field external to the boundary layer was deflected laterally by turning vanes so that streamwise flow deceleration occurred simultaneous with cross-flow acceleration. At 21 stations profiles of the mean-velocity components and of the six Reynolds stresses were measured with single- and X-hot-wire probes, which were rotatable around their longitudinal axes. The calibration of the hot wires with respect to magnitude and direction of the velocity vector as well as the method of evaluating the Reynolds stresses from the measured data are described in a separate paper (Müller 1982, hereinafter referred to as II). At each measuring station the wall shear stress was inferred from a Preston-tube measurement as well as from a Clauser chart. With the measured profiles of the mean velocities and of the Reynolds stresses several assumptions used for turbulence modelling were checked for their validity in this flow. For example, eddy viscosities for both tangential directions and the corresponding mixing lengths as well as the ratio of resultant turbulent shear stress to turbulent kinetic energy were derived from the data.

Since the accuracy of mean velocities and Reynolds stresses measured with inclined hot wires depends on the accurate knowledge of the hot-wire cooling law, each hot wire used in the boundary-layer experiment of Müller (1982, hereinafter referred to as I) had to be calibrated individually with respect to the magnitude and direction of the flow vector. In the present paper details of the calibration procedure and an example of calibrated data are reported. The directional hot-wire response was described by an effective cooling velocity, which was then used for the data reduction. The errors in the measured Reynolds stresses evaluated with an empirical cooling law as opposed to the actual one were estimated analytically from the governing equations and were confirmed by corresponding recalculations from the same set of measurements. Additionally, the validity of the conventional linearized method for evaluating the Reynolds stress tensor from the root-expanded equation for the cooling velocity was checked for increasing turbulence levels. In the test measurements all triple velocity correlations, which are usually neglected compared with second-order ones, were measured and taken into account in the data reduction.

In this paper an attempt is made to formulate a model for the mechanism of wall turbulence that links recent flow-visualization observations with the various quantitative measurements and scaling laws established from anemometry studies. Various mechanisms are proposed, all of which use the concept of the horse-shoe, hairpin or ‘A’ vortex. It is shown that these models give a connection between the mean-velocity distribution, the broad-band turbulence-intensity distributions and the turbulence spectra. Temperature distributions above a heated surface are also considered. Although this aspect of the work is not yet complete, the analysis for this shows promise.

We discuss creeping incompressible fluid flow in two-dimensional networks consisting of regular lattice arrays of variable-sized channels and junctions. The intended application is to low-Reynolds-number flow in models of porous media. The flow problem is reduced to an analogue linear-network problem and is solved by numerical matrix inversion. It is found that ‘effective-medium theory’ provides an excellent approximation to flow in such networks. Various qualitative features of such flows are discussed, and an elegant general form for the absolute permeability is derived. The latter, and the effective-medium approximation, are equally applicable to three-dimensional networks.

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

The isopycnal elevation ζ of a stratified incompressible fluid constitutes a generalized co-ordinate in terms of which the dynamics can be described in Hamiltonian form. The description is complete, and no additional variables are necessary, when the isopycnal circulation vanishes and when there are no remote sources of flow. The Hamiltonian, constructed in a co-ordinate system that coincides with the isopycnal surfaces, generates nonlinear equations of motion for ζ and its conjugate variable π which are formally exact for arbitrary displacements of the fluid and which are equivalent at lowest order to the usual linearized equations of internal waves. The appropriate scaling parameters for the nonlinearity are the isopycnal slope [dtri ]ζ and vertical strain ∂zζ, which emerge as the expansion parameters of an explicit powerseries representation of the Hamiltonian.

The formation of stationary waves in a laboratory flume is described, and possible mechanisms for their production discussed. In particular, an investigation is made of waves on supercritical streams. The mechanism for creating such waves involves frictional action through the boundary layers and an approximate equation describing this process is presented and shown to give qualitative agreement with observation.

Measurements are presented of the pressure fluctuations acting on a stationary squaresection cylinder, with the front face normal to the flow, and one forced to oscillate, transverse to a flow, at amplitudes up to 25% of the length of a side. The range of reduced velocities investigated, 4–13, includes the vortex lock-in regime. At lock-in the amplification of the coefficient of fluctuating lift is found to be much less than that found for a circular cylinder. The variation of the phase angle, between lift and displacement, is also different from that measured on a circular cylinder, and vortex-induced oscillations are possible only at the high-reduced-velocity end of the lock-in range. At reduced velocities sufficiently far below lock-in the natural vortex-shedding mode is suppressed and vortices are found to form over the side faces at the body frequency. Intermittent reattachment occurs over the side faces and, for an amplitude of oscillation equal to 10% of the length of a side face, the time-mean drag coefficient can be reduced to 60% of its fixed-cylinder value.

To evaluate the usefulness of digital image analysis in extracting quantitative information from flow pictures we have studied a 16 mm ciné film of a turbulent mixing layer. A sequence of 373 frames is digitized and analysed to isolate and measure the concentration eddies that constitute the large structure and to follow their individual evolution in time. As a result, statistics are given on the life history of the eddies, the structure of the amalgamation process and the amount of entrainment, as measured by area change, due to amalgamation as compared to the total. It is found that most of the entrainment occurs during the normal life of eddies and not during pairing. Mixing intermittency is computed from the observed shape of the eddies and seen to compare well with previous measurements. The significance of these results in modelling the mixing layer is discussed briefly and some comments are given on the general usefulness of the techniques presented.

A theoretical and numerical investigation of the propagation of one-dimensional waves in a bubbly liquid is presented. A variational formulation of the problem is used that yields both the linear-momentum equation and the equation that describes the oscillations of the bubbles. The compressibility of the liquid is taken into account in the formulation. The thermal dissipation is treated by solving the energy-balance equations simultaneously with the mechanical equations. Solutions are obtained by a finite-difference procedure and are compared to the experimental data of Kuznetsov et al. and Noordzij & van Wijngaarden. In some cases quite good agreement is obtained, but in others substantial errors are found. It is suggested that the observed discrepancies may be due to the breakup of the bubbles in the case of very large amplitude disturbances; the fact that the formulation does not include relative motion between the liquid and the bubbles; and possible non-planarity effects in the experiments.

The equations describing parametric instabilities of a finite-amplitude internal gravity wave in an inviscid Boussinesq fluid are studied numerically. By improving the numerical approach, discarding the concept of spurious roots and considering the whole range of directions of the Floquet vector, Mied's work is generalized to its full complexity. In the limit of large disturbance wavenumbers, the unstable disturbances propagate in the directions of the two infinite curve segments of the related resonant-interaction diagram. They can therefore be classified into two families which are characterized by special propagation directions. At high wavenumbers the maximum growth rates converge to limits which do not depend on the direction of the Floquet vector. The limits are different for both families; the disturbance waves propagating at the smaller angle to the basic gravity wave grow at the larger rate.

Small rigid spherical partials are settling under gravity through Newtonian fluid, and the volume fraction of the particles (ϕ) is small although sufficiently large for the effects of interactions between pairs of particles to be significant. Two neighbouring particles interact both hydrodynamically (with low-Reynolds-number flow about each particle) and through the exertion of a mutual force of molecular or electrical origin which is mainly repulsive; and they also diffuse relatively to each other by Brownian motion. The dispersion contains several species of particle which differ in radius and density.

The purpose of the paper is to derive formulae for the mean velocity of the particles of each species correct to order ϕ, that is, with allowance for the effect of pair interactions. The method devised for the calculation of the mean velocity in a monodisperse system (Batchelor 1972) is first generalized to give the mean additional velocity of a particle of species i due to the presence of a particle of species j in terms of the pair mobility functions and the probability distribution pii(r) for the relative position of an i and a j particle. The second step is to determine pij(r) from a differential equation of Fokker-Planck type representing the effects of relative motion of the two particles due to gravity, the interparticle force, and Brownian diffusion. The solution of this equation is investigated for a range of special conditions, including large values of the Péclet number (negligible effect of Brownian motion); small values of the Ptclet number; and extreme values of the ratio of the radii of the two spheres. There are found to be three different limits for pij(r) corresponding to different ways of approaching the state of equal sphere radii, equal sphere densities, and zero Brownian relative diffusivity.

Consideration of the effect of relative diffusion on the pair-distribution function shows the existence of an effective interactive force between the two particles and consequently a contribution to the mean velocity of the particles of each species. The direct contributions to the mean velocity of particles of one species due to Brownian diffusion and to the interparticle force are non-zero whenever the pair-distribution function is non-isotropic, that is, at all except large values of the Péclet number.

The forms taken by the expression for the mean velocity of the particles of one species in the various cases listed above are examined. Numerical values will be presented in Part 2.

This paper describes an experimental study of the effects of surface roughness on the flow between two concentric cylinders, one of which rotates. The surface roughness has some effects on the coefficient of viscous frictional torque CM in the transient and in the fully developed turbulent region. In the fully developed rough turbulent flow, the value of CM depends on both the Reynolds number Rω and the relative roughness rm/k in the case where the outer cylinder rotates, but CM depends only on rm/k in the case where the inner cylinder rotates. The effect of the surface roughness of the inner cylinder is greater than that of the outer one for both cases.

We propose a simplified theory of a viscous layer in near-wall turbulent flow that determines the mean-velocity profile and integral characteristics of velocity fluctuations. The theory is based on the concepts resulting from the experimental data implying a relatively simple almost-ordered structure of fluctuations in close proximity to the wall. On the basis of data on the greatest contribution to transfer processes made by the part of the spectrum associated with the main size of the observed structures, the turbulent fluctuations are simulated by a three-dimensional running wave whose parameters are found from the problem solution. Mathematically the problem reduces to the solution of linearized Navier-Stokes equations. The no-slip condition is satisfied on the wall, whereas on the outer boundary of a viscous layer the conditions of smooth conjunction with the asymptotic shape of velocity and fluctuation-energy profiles resulting from the dimensional analysis are satisfied. The formulation of the problem is completed by the requirement of maximum curvature of the mean-velocity profile on the outer boundary applied from stability considerations.

The solution of the problem does not require any quantitative empirical data, although the conditions of conjunction were formulated according to the well-known concepts obtained experimentally. As a result, the near-wall law for the averaged velocity has been calculated theoretically and is in good agreement with experiment, and the characteristic scales for fluctuations have also been determined. The developed theory is applied to turbulent-flow calculations in Maxwell and Oldroyd media. The elastic properties of fluids are shown to lead to near-wall region reconstruction and its associated drag reduction, as is the case in turbulent flows of dilute polymer solutions. This theory accounts for several features typical of the Toms effect, such as the threshold character of the effect and the decrease in the normal fluctuating velocity. The analysis of the near-wall Oldroyd fluid flow permits us to elucidate several new aspects of the drag-reduction effect. It has been established that the Toms effect does not always result in thickening of the viscous sublayer; on the contrary, the most intense drag reduction takes place without thickening in the viscous sublayer.

In the present study, it is shown that the spreading rate of a mixing layer can be greatly manipulated at very low forcing level if the mixing layer is perturbed near a subharmonic of the most-amplified frequency. The subharmonic forcing technique is able to make several vortices merge simultaneously and hence increases the spreading rate dramatically. A new mechanism, ‘collective interaction’, was found which can bypass the sequential stages of vortex merging and make a large number of vortices (ten or more) coalesce.

A deeper physical insight into the evolution of the coherent structures is revealed through the investigation of a forced mixing layer. The stability and the forcing function play important roles in determining the initial formation of the vortices. The subharmonic starts to amplify at the location where the phase speed of the subharmonic matches that of the fundamental. The position where vortices are seen to align vertically coincides with the position where the measured subharmonic reaches its peak. This location is defined as the merging location, and it can be determined from the feedback equation (Ho & Nosseir 1981).

The spreading rate and the velocity profiles of the forced mixing layer are distinctly different from the unforced case. The data show that the initial condition has a longlasting effect on the development of the mixing layer.