A uniform-mean-gradient shear flow was produced using a ten-layer closed-loop water channel, providing long enough dimensionless flow development times (τ = (x/Ū) (∂ Ū/∂z)) for the turbulence to grow. The rate of growth of the turbulence compares well with similar measurements in wind-tunnel-generated uniform shear flows for which the mean shears and centreline velocities are larger by an order of magnitude. Preliminary investigations were undertaken to study the growth of the turbulent intensity as functions of the mean shear, centreline velocity, and initial disturbance lengthscales. Initial disturbance lengthscales were varied by using grids of different mesh sizes.

Turbulent intensities were found to increase nearly linearly with τ. Differences in grid mesh size produce different offsets in the turbulent intensity level, with a larger grid mesh producing a higher positive offset. This offset persists throughout the growth of the turbulent intensity. These observations provide valuable insight in interpreting previous wind-tunnel measurements, in particular the high-shear experiments of Karnik & Tavoularis (1983). Comparison with the theoretical predictions of Tavoularis (1985) allows for an improved universal characterization of evolving turbulence in a uniform mean shear.

The paper presents the asymptotic solution, near a stationary contact line at a plane boundary, for steady viscous incompressible flow of two immiscible liquids. The eigenvalues which determine this Stokes flow are determined by the contact angle α of the more viscous liquid and the ratio μ of the two viscosities. The dominant eigenvalues are found for all values of α and μ. As μ → 0 the results agree with those of Moffatt's (1964) one-phase theory for the case μ = 0 only when α > 81°. For α < 81° the two sets of results are qualitatively different. In particular, the eddy structure corresponding to complex eigenvalues occurs only in the α-range (34°, 81°). As μ increases from 0 to 1, this range steadily decreases to zero, which is located at 60°. The transport of energy across the liquid interface is almost always from the obtuse-angled sector to the acute-angled sector, irrespective of α, μ, and the location of the global power supply.

This paper aims to shed some light on the physical mechanisms involved in flowinduced instabilities of arrays of cylinders in crossflow. In the framework of quasi-steady fluid-dynamic theory, two distinct mechanisms are discussed. The first is similar but not identical to that associated with classical galloping; i.e. it is a negative fluid-dynamic damping mechanism and may obtain even if a single cylinder in the array is free to oscillate with only one degree of freedom. Unlike classical galloping, it is intimately related to the time delay experienced in the wake structure, and hence the fluid forces, adjusting to displacements of the cylinder. The second mechanism is similar to wake flutter; i.e. it is controlled by non-conservative fluid-dynamic stiffness effects and generally requires relative motion between adjacent cylinders in the array, although there is no reason why it should not occur for a single flexible cylinder with two degrees of freedom. The two mechanisms generally coexist, but each is predominant over different ranges of system parameters.

The turbulent boundary layer on a flat plate, with zero pressure gradient, is simulated numerically at four stations between Rθ = 225 and Rθ = 1410. The three-dimensional time-dependent Navier-Stokes equations are solved using a spectral method with up to about 107 grid points. Periodic spanwise and streamwise conditions are applied, and a multiple-scale procedure is applied to approximate the slow streamwise growth of the boundary layer. The flow is studied, primarily, from a statistical point of view. The solutions are compared with experimental results. The scaling of the mean and turbulent quantities with Reynolds number is compared with accepted laws, and the significant deviations are documented. The turbulence at the highest Reynolds number is studied in detail. The spectra are compared with various theoretical models. Reynolds-stress budget data are provided for turbulence-model testing.

A layer of liquid can be supported on the inside or outside of a horizontal rotating cylinder if the viscous forces pulling the liquid around with the cylinder are large enough to overcome the force of gravity. If there are places on the cylinder where the thickness of the layer is larger than a critical value, the excess fluid will run off. For a given maximum thickness the critical condition may be expressed as the minimum speed at which the given layer can be maintained. An approximation of the critical condition using lubrication theory was given by Wallis (1969) and by Deiber & Cerro (1976) for rimming flow and by Moffatt (1977) for coating and rimming flow. Here we address the question of the axial variations of the free surface on the coating layers, and show that they are dominated by the same type of balance between capillarity and centripetal acceleration which determines the shape of rotating drops and bubbles in the absence of gravity. The main results of this paper are the experiments which establish the validity of approximations used to describe the underlying fluid mechanics involved in rimming and coating flows.

In a previous article we introduced a dissipative circular geometry in which stationary states of the shear flow instability were obtained. We show here that the dynamical behaviour of this flow depends strongly on the aspect ratio of the cell. In large cells, where the number of vortices is large, transitions from a mode with m vortices to a mode with (m−1) vortices occur through localized processes. In contrast to that situation, in small cells, transition takes place after a series of bifurcations which correspond to the successive breaking of all the symmetries of the flow.

We show that, provided an adequate forcing term is introduced, a two-dimensional numerical simulation of this flow is sufficient to recover all the dynamical processes which characterize the experimental flow.

Longuet-Higgins's (1976) analysis of energy transfer within a narrow spectrum of gravity waves with approximately uncorrelated phases is generalized to accommodate capillarity and weak damping. The analysis is based on the corresponding generalization of Zakharov's (1968) evolution equation for weakly nonlinear, deep-water gravity-wave packets. The results for a symmetric normal spectrum are expressed in terms of elliptic integrals and depend, after appropriate scaling, on a single similarity parameter and on the sign of the curvature of the linear dispersion relation. Energy transfer is away from the peak of that spectrum if kl* < 0.393, where k is the wavenumber and l* is the capillary length (2.8 mm for water), but may be towards the peak if 0.343 < kl* < 0.707 (4.5 cm > 2π/k > 2.5 cm for water). The formulation is based on energy exchange through resonant quartets and is not valid in the neighbourhood of kl* = 0.707; at which the second harmonic of a capillary-gravity wave resonates with its fundamental (Wilton's ripples). The modulational instability of a weakly damped capillary-gravity wave is examined in an Appendix.

In this paper, we study the receptivity of a typical free shear layer to pulse-type and periodic excitation. We do this by solving the initial-value problem completely and studying its long-time behaviour. This leads to a wave packet for the pulse. By the superposition of many wave packets, we generate a spatial instability mode when the flow is convectively unstable. This establishes a general and simple relationship between the receptivities for pulse-type and sinusoidal excitations. We find that a shear layer is very receptive to high-frequency disturbances that are generated near the centreline of the layer.

An axisymmetric capsule, consisting of an incompressible liquid droplet, surrounded by an infinitely thin elastic membrane having a Mooney constitutive behaviour, is suspended into another incompressible Newtonian liquid subjected to an elongational shear flow. The motion and the deformation of the capsule are determined numerically by means of a boundary-integral technique. It is thus possible to reach large deformations, and to study the influence of the initial geometry of the particle, as well as that of the constitutive behaviour of the membrane. In all cases considered here, it appears that there exists a critical value of the non-dimensional shear rate (the capillary number) above which no steady solution can be obtained, and where the capsule continuously deforms. This phenomenon is interpreted as the outset of burst. The model shows also the importance of the sphericity index for the determination of the overall capsule deformability.

It is shown that the accumulation of small-amplitude gasdynamic perturbations is able to accelerate the process of self-ignition of a homogeneous explosive mixture. For analytical simplicity, the chemical kinetics are represented by a one-step irreversible reaction with a large activation energy. A one-dimensional piston-cylinder geometry is considered, which allows a slow compression of the gas during the induction period. It is assumed that the rates of temperature increase due to the reaction and due to the compression are comparable. A two-timescale asymptotic expansion for small Mach numbers is performed considering the distinguished limit ME = O(1), where E is the non-dimensional activation energy, and M is the Mach number based on a characteristic speed of the gas motion. Gasdynamic-chemical interaction is observed at the second order, where the secular equations describing the evolution on the slow timescale explicitly show a cumulative influence.

The method of domain perturbations is used to study the problem of a nearly spherical bubble in an inviscid, axisymmetric straining flow. Steady-state shapes and axisymmetric oscillatory motions are considered. The steady-state solutions suggest the existence of a limit point at a critical Weber number, beyond which no solution exists on the steady-state solution branch which includes the spherical equilibrium state in the absence of flow (e.g. the critical value of 1.73 is estimated from the third-order solution). In addition, the first-order steady-state shape exhibits a maximum radius at θ = ⅙π which clearly indicates the barrel-like shape that was found earlier via numerical finite-deformation theories for higher Weber numbers. The oscillatory motion of a nearly spherical bubble is considered in two different ways. First, a small perturbation to a spherical base state is studied with the ad hoc assumption that the steady-state shape is spherical for the complete Weber-number range of interest. This analysis shows that the frequency of oscillation decreases as Weber number increases, and that a spherical bubble shape is unstable if Weber number is larger than 4.62. Secondly, the correct steady-state shape up to O(W) is included to obtain a rigorous asymptotic formula for the frequency change at small Weber number. This asymptotic analysis also shows that the frequency decreases as Weber number increases; for example, in the case of the principal mode (n = 2), ω2 = ω00(1−0.31W), where ω0 is the oscillation frequency of a bubble in a quiescent fluid.

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. $$Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}.$$ This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.

Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.

Settling characteristics of bidisperse suspensions containing light and heavy particles in inclined channels have been studied both experimentally and theoretically. The suspension is relatively dilute with a total volume fraction of no more than 0.16. In this dilute range the flow-visualization experiments indicate the formation of distinct zones with clear interfaces between them. There is no evidence of lateral segregation of particles. The convection currents formed near an inclined boundary are transferred to different zones depending on the relative concentration of the two species. Based on the flow-visualization experiments, the well known Ponder-Nakamura-Kuroda (PNK) model has been adopted to predict the settling characteristics of bidisperse suspensions.

A set of hydrodynamic equations has been applied to processes occurring in non-conductive fluids placed into magnetic fields. The equations are valid for equilibrium magnetization within the framework of a continuous medium. The ranges of physical parameters have been evaluated for which magnetization of a fluid should be taken into account in problems concerning the determination of equilibrium forms, and flows and their stability. The conclusion has been drawn that magnetization of natural fluids in these problems must be taken into consideration for fluids in high (exceeding 1 T) magnetic fields. As examples, the solutions of several typical problems concerning the equilibrium surface of capillary fluids and their stability in external magnetic fields have been considered. Hydrodynamic effects related to magnetization of natural capillary fluids in high magnetic fields are studied with due regard for the above solutions. Hydrodynamic effects in para- and diamagnetic fluids have been studied and the common and distinctive features of these effects discussed.

Three-dimensional weakly nonlinear Rayleigh-Taylor instability is analysed. The stability of a confined inviscid liquid and an overlying gas with density much less than that of the liquid is considered. An asymptotic solution for containers of arbitrary cross-sectional geometry, valid up to order ε3 (where ε is the root-mean-squared initial surface slope) is obtained. The solution is evaluated for the rectangular and circular geometries and for various initial modes (square, hexagonal, axisymmetric, etc.). It is found that the hexagonal and axisymmetric instabilities grow faster than any other shapes in their respective geometries. In addition it is found that, sufficiently below the cutoff wavenumber, instabilities that are equally proportioned in the lateral directions grow faster than those with longer, thinner shape. However, near the cutoff wavenumber this trend reverses with instabilities having zero aspect ratio growing faster than those with aspect ratio near 1.

Three-dimensional Rayleigh-Taylor instability, induced by accelerating a small volume of water down a vertical tube using air pressure, is investigated. Two geometries are studied: a 15.875 cm circular tube and a 12.7 cm square tube. Runs were made with initial disturbances in the form of standing waves forced by shaking the test section in a lateral direction. Accelerations ranging from 5 to 10 times gravitational acceleration and wavenumbers from 1 cm−1 to 8 cm−1 are studied. The resulting instability was recorded and later analysed using high-speed motion picture photography. Measurements of the growth rate are found to agree well with linear theory. In addition, good qualitative agreement between photographs and three-dimensional surface plots of the weakly nonlinear solution of Part 1 of this series (Jacobs & Catton 1988) is obtained.

The results of an experimental programme designed to investigate turbulent dispersion of a continuous contaminant source in a wide channel are presented. Both two-dimensional vertical dispersion and the determination of the lateral diffusion coefficient are described. The eigenfunction solution to the turbulent diffusion equation, presented in Nokes et al. (1984) and discussed in greater detail in Nokes (1985), is strongly supported by the results of vertical mixing described here. A variety of source locations are examined in this study and the location of the ideal source, predicted by theory, is verified by the experimental results. For the two smooth-bed flows investigated the depth-averaged values of εz, deduced from the rates of lateral spreading of the plume, lie at the lower end of the range of values obtained by other researchers. Considering only the results obtained in wide channels, the authors demonstrate that previously published values of the lateral diffusion coefficient, non-dimensionalized by the shear velocity u* and the flow depth d are independent of all flow parameters except the friction factor f = 8u*/ū where ū is the mean velocity in the flow. Indeed, above a value of f = 0.055 εz/u*d is also found to be independent of f, and takes a value of 0.134. A brief mathematical analysis of the three-dimensional mixing processes in the near-source region is presented, and utilized to investigate the coupling between the lateral and vertical diffusion processes in this region. Based on these mathematical arguments the experimental results imply that the vertical and lateral diffusion processes are essentially uncoupled in the near-source zone, and thus the lateral diffusivity and longitudinal velocity have similar vertical dependence.

The effects of a free surface on the spin-up of a homogeneous fluid are studied, both analytically and experimentally. The analysis is carried out in cylindrical geometry and shows that the spin-up process is strongly modified as the rotational Froude number F = 4ω2L2/gH becomes large. The dynamic effect of the free surface causes delayed response outside a sidewall boundary layer of thickness LF−½. The timescale in the slowly decaying core is larger than the usual spin-up time by a factor of order F. A set of laboratory experiments using a cylinder with a parabolic bottom were carried out in order to test the theory. Reasonable agreement is found in all the experiments except close to the centre where an interesting deviation was observed, especially in cases corresponding to smaller Froude numbers. The deviation consisted of an anticyclonic vortex at the centre. It is shown that this phenomenon might be explained by Lagrangian mean motion resulting from inertial oscillations. In fact, the analysis shows that this motion produces a singular vortex at the centre.

Solidification of a double-diffusive liquid from a vertical wall in a rectangular cavity is investigated using both mathematical and physical modelling. A two-dimensional mathematical model is developed to simulate the process of solidification in a two-component liquid containing a denser solute. Observations on the solidification behaviour of aqueous sodium carbonate solutions are used to verify the results of the mathematical analysis. The theory and experiments provide a clear picture of the role of double-diffusion in producing vertical compositional and density stratification in an initially homogeneous liquid during solidification. The development of horizontally oriented convection cells in the stratified liquid is correlated with the magnitude of the destabilizing lateral temperature difference across the liquid region.

For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes to the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.