Axisymmetric flow of viscous incompressible fluid between a rotating porous disk and an impermeable fixed plane is investigated. It is shown that with injection and suction through a porous disk rotating with sufficiently large angular velocity there are many isolated steady self-similar solutions. In the case of suction through a fixed porous disk at a certain Reynolds number there exists bifurcation of the stable rotational regime of flow, implying a spontaneous break of the flow symmetry and an arbitrary rise of the fluid rotation within the framework of self-similarity. This unusual effect is discussed in detail, and the results of a relevant experiment are presented. Another unusual result is the existence of multicellular regimes consistent with suction, when the lift force acting on a rapidly rotating porous disk is anomalously large; in this case some of these regimes are stable relative to self-similar perturbations.

With sufficiently strong suction and rotation the stationary solution with large lift becomes unstable and the regime of self-oscillations arises. Diagrams of the possible stationary flow regimes have been constructed, and the stable ones have been identified. At the limit of vanishing viscosity we find, in the case of the suction, non-classical boundary layers on the solid surfaces characterized by a finite jump of the normal component of the velocity and unlimited tangential components. In this limit, in the interior flow region the singular non-viscous solution with an infinite velocity of rotation arises, while all limited non-singular admissible non-viscous solutions are not stable.

The boundary-integral method is used to calculate the hydrodynamic force and torque on an arbitrary body of revolution whose axis of symmetry is oriented at an arbitrary angle relative to a planar wall in the zero-Reynolds-number limit. The singular solution of the Stokes equations in the presence of a planar wall is used to formulate the integral equations, which are then reduced to a system of linear algebraic equations by satisfying the no-slip boundary conditions on the body surface using the boundary collocation method or weighted residual technique.

Numerical tests for the special case of a sphere moving parallel or perpendicular to a planar wall show that the present theory is accurate to at least three significant figures when compared with the exact solutions for gap widths as small as only one-tenth of the particle radius. Higher accuracy can be achieved and solutions can be obtained for smaller gap widths at the expense of more computation time and larger storage requirements.

The hydrodynamic force and torque on a spheroid with varying aspect ratio and orientation angle relative to the planar wall are obtained. The theory is also applied to study the motion of a toroidal particle or biconcave shaped disc adjacent to a planar wall. The coincidence of the drag and torque of a biconcave-shaped body and a torus having an aspect ratio b/a = 2 with the same surface area shows that in this case the hole of a torus has little influence on the flow field. On the other hand, for an aspect ratio b/a = 10, the effect of the hole is significant. It is also shown that when the body is not very close to the wall, an oblate spheroid can be used as a good approximation of a biconcave-shaped disc.

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on compressible shear layers between two parallel streams. The analysis shows that mean temperature non-uniformities cause nonlinearity to occur at much smaller amplitudes than it does when the flow is isothermal. The nonlinear instability wave growth rate effects are described by an integro-differential equation which bears some resemblance, to the Landau equation in that it involves a cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that inviscid solutions always end in a singularity at a finite downstream distance but that viscosity can eliminate this singularity for certain parameter ranges.

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that they always end in a singularity at a finite downstream distance.

We treat the problem of combined buoyancy-thermocapillary convection in a cavity with a free surface heated differentially in the horizontal. Attention is focused on the structure and strength of the flow for large ΔT, i.e. large Marangoni and Rayleigh numbers. In the combined problem, the boundary-layer scalings for buoyant and thermocapillary convection suggest that in the limit of large ΔT, thermocapillarity will dominate the large-scale flow. Accurate numerical solutions are used to study this question at fixed cavity aspect ratio and Prandtl number, with G = Ra/Ma as a parameter. For G = 1, the flow evolves toward its boundary-layer limit in a fashion identical to that for G = 0, i.e. pure thermocapillary flow. For G = 10, the evolution is from a buoyancy-dominated structure, through a transition, to a thermocapillary-dominated structure. We infer that thermocapillarity will ultimately dominate all such flows at sufficiently large ΔT, for any fixed values of G, the aspect ratio, and the Prandtl number.

Certain modifications of the Euler equations of fluid motion lead to systems in which the energy decays or grows monotonically, yet which preserve other dynamically important characteristics of the field. In particular, all topological invariants associated with the vorticity field are preserved. In cases where isolated energy extrema exist, a stable steady flow can be found. In two dimensions, highly constrained by vorticity invariants, it is shown that the modified dynamics will lead to at least one non-trivial stationary, generally stable, solution of the equations of motion from any initial conditions. Numerical implementation of the altered dynamics is straightforward, and thus provides a practical method for finding stable flows. The method is sufficiently general to be of use in other dynamical systems.

Insofar as three-dimensional turbulence is characterized by a cascade of energy, but not topological invariants, from large to small scales, the procedure has direct physical significance. It may be useful as a parameterization of the effects of small unresolved scales on those explicitly resolved in a calculation of turbulent flow.

A numerical scheme based on the pseudospectral method has been implemented in order to study three-dimensional convection in a fluid-saturated cube of porous material. With increasing Rayleigh number R, convection first evolves from a symmetric steady state (S) to a partially non-symmetric steady state (S’, physical symmetry in the vertical direction is preserved). The transition Rayleigh number is about 550. At a Rayleigh number of 575 the flow becomes oscillatory P(1) with a single frequency that increases with R. At a value of R between 650 and 680 the oscillation becomes quasi-periodic with at least two fundamental frequencies. It returns to a simply periodic state in a narrow range about R = 725. A further increase of R transforms the simply periodic state again to a quasi-periodic state. The sequence of three-dimensional time-dependent states resembles previously studied two-dimensional cases in that evolution from more complex states to less complex states occurs with increasing R. The partial symmetry breaking prior to the onset of time dependence is unique to the three-dimensional flows, but a dependence of the S → S’ transition on the step size in R suggests the possibility that S → S’ might not occur prior to S → P(1) for sufficiently small steps in R. The quasi-periodic flows sometimes exhibit intermittency, causing difficulty in exactly defining their spectral characteristics.

The interaction of two energy-containing turbulence scales is studied in the absence of mean shear. The flow, a turbulence mixing layer, is formed in decaying grid turbulence in which there are two distinct scales, one on either side of the stream. This is achieved using a composite grid with a larger mesh spacing on one side of the grid than the other. The solidity of the grid, and thus the mean velocity, is kept constant across the entire flow. Since there is no mean shear there is no turbulence production and thus spreading is caused solely by the fluctuating pressure and velocity fields. Two different types of grids were used: a parallel bar grid and a perforated plate. The mesh spacing ratio was varied from 3.3:1 to 8.9:1 for the bar grid, producing a turbulence lengthscale ratio of 2.4:1 and 4.3:1 for two different experiments. For the perforated plate the mesh ratio was 3:1 producing a turbulence lengthscale ratio of 2.2:1. Cross-stream profiles of the velocity variance and spectra indicate that for the large lengthscale ratio (4.3:1) experiment, a single scale dominates the flow while for the smaller lengthscale ratio experiments, the energetics are controlled by both lengthscales on either side of the flow. In all cases the mixing layer is strongly intermittent and the transverse velocity fluctuations have large skewness. The downstream data of the second, third and fourth moments for all experiments collapse well using a single composite lengthscale. The component turbulent energy budgets show the importance of the triple moment transport and pressure terms within the layer and the dominance of advection and dissipation on the outer edge. It is also shown that the bar grids tend toward self-similarity with downstream distance. The perforated plate could not be measured to the same downstream extent and did not reach self-similarity within its measurement range. In other respects the two types of grids yielded qualitatively similar results. Finally, we emphasize the distinction between intermittent turbulent penetration and turbulent diffusion and show that both play an important role in the spreading of the mixing layer.

A study of the flow within the critical layer of a forced Rossby-wave is made, using a high-resolution numerical model. The possibility of growth of disturbances through barotropic instability and the extent to which these disturbances modify the subsequent time evolution is of particular interest. The flow is characterized by a parameter μ, equal to the cross-stream lengthscale divided by a downstream wavelength. In the long-wavelength case, μ [Lt ] 1, where there is a clear conceptual division between the instability and the basic flow, the results of the simulation confirm the importance of the growing and saturating disturbances in rearranging the vorticity within the critical layer. When the wavelength is not so long, the distinction between the instability and the straightforward time evolution of the basic flow is less clear. Nonetheless for μ < 0.25 the ultimate evolution is still sensitive to the details of the initial perturbations and in this sense the flow may be regarded as being unstable. The time-integrated absorptivity of the critical layer may be considerably increased by the effects of the instability, sometimes to three or four times that predicted by the Stewartson-Warn-Warn solution. The nature of the flow, at least during the period in which the dynamics are essentially inviscid, does not seem to change when higher harmonics to the forced wave are resonant. The behaviour seen in Béland's (1976) numerical model is re-examined in the light of these findings. A simple model of the redistribution of vorticity by the unstable disturbances is formulated, and its predictions are shown to agree well with the numerical simulations.

Thermal instability of a fluid layer confined between isotropic horizontal solid walls leads to convection cells having no preferred horizontal direction. For thermally anisotropic walls, we find that certain planform orientations are preferred, in that convection sets in at a smaller Rayleigh number (Ra) for some orientations than for others, thus providing a means by which a regular planform may be established in a large-aspect-ratio layer. We consider horizontal layers of two Boussinesq, Newtonian fluids separated by a rigid, thermally anisotropic plate of constant thickness. The upper and lower fluid layers are bounded above and below, respectively, by rigid, thermally anisotropic plates of arbitrary thickness. When the bounding surfaces are thermally anisotropic, the horizontal wavevector (a) of the resulting convective flow has two distinct components. Thus, instead of a neutral curve in the (Ra, a)-plane, there is a neutral surface, and Ra depends on both components of a, or alternatively, on |a| and the planform orientation angle Φrε[0, 2π]. In the isotropic case, the neutral surface is axisymmetric (i.e. invariant with respect to Φr consistent with the known dependence on |a| only.

For anisotropic walls, axisymmetry is replaced by π- periodicity in the Φr direction, corresponding to invariance with respect to a 180° rotation, and the neutral surface has an even number of local minima. We study the dependence of Φr on the middle plate orientation (Φp) Several different Φr−Φp topologies are found. When the number of local minima exceeds two, discontinuous Φr−Φp plots may occur. The dependence of Φr on the thicknesses and conductivities of the plates and fluids and on the orientation of the plates is discussed, with special reference to the transitions between different Φr−Φp topologies.

The effects of a reduction in momentum flux on the downstream flow characteristics of a steady, two-dimensional flow are investigated. In particular, quantities such as the changes in mean depth, mean fluid velocity, mean kinetic and potential energies and the length of the induced downstream wave are examined. This is done with the use of a fourth-order perturbation expansion in wave slope. The results are compared with the conflicting results that had been obtained previously by different authors. Agreement is found with the second-order theory of Benjamin (1970), but the work of Doctors & Dagan (1980) is found to be in error and is corrected.

Experiments have been carried out in a horizontal superposed fluid and porous layer contained in a test box 24 cm × 12 cm × 4 cm high. The porous layer consisted of 3 mm diameter glass beads, and the fluids used were water, 60% and 90% glycerin-water solutions, and 100% glycerin. The depth ratio ď, which is the ratio of the thickness of the fluid layer to that of the porous layer, varied from 0 to 1.0. Fluids of increasingly higher viscosity were used for cases with larger ď in order to keep the temperature difference across the tank within reasonable limits. The top and bottom walls were kept at different constant temperatures. Onset of convection was detected by a change of slope in the heat flux curve. The size of the convection cells was inferred from temperature measurements made with embedded thermocouples and from temperature distributions at the top of the layer by use of liquid crystal film. The experimental results showed (i) a precipitous decrease in the critical Rayleigh number as the depth of the fluid layer was increased from zero, and (ii) an eightfold decrease in the critical wavelength between ď = 0.1 and 0.2. Both of these results were predicted by the linear stability theory reported earlier (Chen & Chen 1988).

When a body of fluid with a vertical salinity gradient is heated from a single vertical wall, instabilities have sometimes been observed experimentally (Thorpe, Hutt & Soulsby 1969; Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986). We present a linear stability analysis for this configuration and show that for strong salinity gradients the stability of the fluid to infinitesimal disturbances is governed by a single non-dimensional parameter, \[ Q = \frac{(1-\tau)^6g(\alpha\Delta T)^6}{\nu\kappa_Sl^2(-\beta\overline{S}_z)^5} \] where g is the acceleration due to gravity, α the coefficient of thermal expansion, β the density change due to a unit change in the salinity, ΔT the change of temperature at the wall, $\overline{S}_z$ the vertical salinity gradient, l the horizontal lengthscale (kTt)½, v the kinematic viscosity, (κTt)½ the diffusivity of heat, kS the diffusivity of salt and τ the salt/heat diffusivity ratio. This non-dimensional parameter is related to the Rayleigh number, however, it involves two different lengthscales; the penetration depth of the thermal front, l, and the height by which a heated element of fluid would rise in the salinity gradient, $g\alpha \Delta T/(- \beta \overline{S}_z)$. This analysis is valid when the ratio of the vertical lengthscale to the horizontal lengthscale is small. This analysis gives good agreement with the published experimental results.

The problem of the hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas is treated following the singular perturbation scheme of Sone & Onishi (1978), valid at small, but finite, particle Knudsen numbers. In this method the solution to the linearized BGKW transport equation governing the gas molecular motion consists of two parts: one describing a Knudsen layer where the actual microscopic boundary conditions are applied and the other describing a Hilbert region where the Stokes equations of continuum hydrodynamics hold. The Knudsen-layer solution establishes the ‘slip’ boundary conditions for the Stokes equations. Here we clearly distinguish between particle ‘slip’ due to the type of boundary conditions and particle ‘slip’ due to lengthscale effects as measured by the Knudsen number. The present analysis has been carried out to first order in particle Knudsen number for the case of diffuse reflective molecular boundary conditions. General relationships between the first- and zero-order velocity fields, both of which are written in the form of Lamb's (1932) solution to the Stokes equation, are established. It is illustrated how these general relationships can be used to determine the force and torque acting on a single sphere translating and rotating in a slightly rarefied gas. Finally, we have treated the two-sphere problem in a slightly rarefied gas using the twin multipole expansion method of Jeffrey & Onishi (1984). Here again, general relationships are established between the solutions of the first-order fluid velocity field and the zero-order velocity field, the latter being shown to recover Jeffrey & Onishi's results for stick boundary conditions. These general relationships are subsequently used to determine the complete resistance and mobility matrices of the two-sphere system. The symmetric properties of the resistance and mobility matrices are demonstrated for slip boundary conditions, in agreement with the general proof of Landau & Lifshitz (1980) and Bedeaux, Albano & Mazur (1977).

Three-dimensional laminar flow profiles were measured at each of three axial stations downstream from the inlet of a slowly tapering tube of constant flow cross-section. It was found that strong inlet wall shear induces significant secondary swirl, whose downstream growth is initially consistent with an inviscid model of vorticity transfer. This secondary flow may act to retard the onset of local flow separation by convecting axial momentum into the boundary layer adjacent to the diverging wall of the tube. These observations could account for the similar geometric shape of major bronchial airways: the suppression of flow separation in pulmonary bifurcations would promote moire effective airway ventilation in the human lung.

The paper presents a method that removes the effects of all irregular frequencies in boundary-integral equations governing the interaction of regular waves with floating bodies of general geometry. A modified integral equation is obtained by the linear superposition of the classical Green equation and its normal derivative with respect to the field point. The selection of a purely imaginary constant of proportionality ensures the removal of all irregular frequencies in the continuous problem and the appropriate selection of its magnitude eliminates their undesirable effects in its numerical implementation. Computations are presented of the added-mass and damping coefficients and exciting forces on a sphere and a truncated vertical cylinder, illustrating the effectiveness of the method.

The influence of stabilizing buoyancy forces on the spectral characteristics and spectral energy transfer of grid-generated turbulence was studied in a ten-layer closed-loop stratified water channel. The results are compared to the limiting ideal cases of the three-dimensional turbulence and two-dimensional turbulence theories. The velocity power spectra evolve from a classical isotropic shape to a shape of almost k−2 after the suppression of the net vertical mixing. This final spectral shape is rather different from the k−3 to k−4 predicted by the theory of two-dimensional turbulence and could result from the interaction between small-scale internal waves and quasi-two-dimensional turbulent structures as well as some Doppler shift of advected waves. Several lengthscales are derived from the cospectra of the vertical velocity and density fluctuations and compared with the buoyancy, overturning and viscous lengthscales measured in previous studies, e.g. Stillinger, Helland & Van Atta (1983) and Itsweire, Helland & Van Atta (1986). The smallest turbulent scale, defined when the buoyancy flux goes to zero, can be related to the peak of the cospectra of the buoyancy flux. This new relationship can be used to provide a measure of the smallest turbulent scale in cases where the buoyancy flux never goes to zero, i.e. a growing turbulent stratified shear flow. Finally, the one-dimensional energy transfer term computed from the bispectra shows evidence of a reverse energy cascade from the small scales to the large scales far from the grid where buoyancy forces dominate inertial forces. The observed reverse energy transfer could be produced by the development of quasi-two-dimensional eddies as the original three-dimensional turbulence collapses.

Large-scale oceanic flows, ranging from large estuaries to basin-scale abyssal circulation, can be driven by imposed lateral density gradients and are significantly influenced by the Earth's rotation. Some essential features of such flows have been incorporated into a laboratory model by differentially heating and cooling the vertical endwalls of a shallow rectangular cavity, rotating about a vertical axis through its centre. In this paper, the flow observed after this system had been allowed to develop to a steady mean state is described. Measurements were made of the velocity and temperature fields, as well as the variability in temperature field associated with eddy motions. The temperature data were also used to calculate the potential vorticity distribution within the cavity. These results indicate that the heated and cooled end-walls acted not only as continuous sources of buoyancy, but also of anomalous potential vorticity. It is shown that the potential vorticity served as an indicator and tracer of the laboratory flow. The structure and location of boundary currents were largely determined by nonlinear potential vorticity dynamics which resulted in a mean circulation consisting of two basin-scale counter-rotating gyres. This flow differed markedly from the initial boundary currents set up by Kelvin waves.

We present the results of a numerical and experimental study of the five-cell flow in the Taylor vortex problem. There is evidence for a new five-cell solution which is different in nature to the previously recorded solution. We find that there is an exchange in stability between the two solutions which we explain in terms of bifurcation theory. Streamline plots of the numerical results are compared with photographs of the observed flow. The agreement between the calculations and experiments is good. We use the Schaeffer homotopy to study the new solution in the periodic model. The results show that the new solution is not connected in any continuous manner to the trivial Couette flow solution.

Numerical solutions have been obtained for three-dimensional buoyant flow of air under laminar conditions in a slender, square-section cavity lying parallel to the gradient vector of the temperature field in which it is embedded. Velocity and temperature profiles in the cavity are presented in support of a flow model in which primary and secondary circulation are reconciled by an advective mechanism in the central region. The effect of the temperature gradient, represented by a Rayleigh number, is explored when the cavity is horizontal. The effect of inclining the cavity above and below this horizontal position is also explored. Comparisons are made with related work on cylindrical and two-dimensional rectangular cavities.