The effect of small-amplitude, time-periodic, free-stream disturbances on an otherwise steady axisymmetric boundary layer on a circular cylinder is considered. Numerical solutions to the problem are presented, and an asymptotic solution to the flow, valid far downstream along the axis of the cylinder is detailed. Particular emphasis is placed on the unsteady eigensolutions that occur far downstream, which turn out to be very different from the analogous planar eigensolutions. These axisymmetric eigensolutions are computed numerically and also are described by asymptotic analyses valid for low and high frequencies of oscillation.

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.

In a large-scale process at I.C.I. Mond Division a flocculated suspension of fine solid particles is allowed to settle through an aqueous solution towards the horizontal interface with an underlying denser solution. When the solids have settled into the lower layer they are pumped away and dumped while the clarified upper layer can be safely disposed of or recycled. The passage of the floes through the interface is the rate-determining step in the process, and the analysis of this part of it is the subject of the paper. It is hoped that the analysis may have a much wider application in that the settling process could be used as the basis of a direct laboratory method of measuring properties of floes. At present both the porosity and permeability of flocs are almost impossible to measure directly, but, using the present analysis, the settling of a single floc through two successive density interfaces would provide (in principle) values for both.

A theoretical expression for the slow part (the nonlinear fluctuation part) of the pressure–strain rate is compared with simulations of anisotropic homogeneous flows. The purpose is to determine the quantitative accuracy of the theory and to test its qualitative predictions that the generalized Rotta coefficient, a non-dimensionalized ratio of slow term to kinetic energy anisotropy, varies with direction and can be negative (this is counter to isotropy return). Comparisons are made between theoretical and simulated values of the slow term and of the generalized Rotta coefficients. Also compared to simulations is an extension of the theory to account for non-stationary turbulence fields. The implication of the comparison for two-point closure theories and for Reynolds stress modelling is pointed out.

The flow of a rotating, two-layer fluid system over long ridges of constant cross-section is considered. Homogeneous incompressible fluids of constant, but different, density are confined between two ‘infinite’ horizontal plane surfaces which rotate at a constant angular velocity about a vertical axis. The ridge is located on the lower surface while upstream of the ridge each fluid is in uniform motion in a direction normal to the ridge. Solutions are obtained for both an f-plane and a β-plane under the following restrictions: E [Lt ] 1, Ro ∼ E½, S ∼ O(1), H/L ∼ O(1), d/L ∼ O(1) and h/L ∼ E½ where E is the Ekman number, Ro is the Rossby number, S is a stratification parameter, H/L is the two-fluid depth to ridge width ratio, d/L is the lower fluid depth to ridge width ratio and h/L is the aspect ratio of the ridge. This set of restrictions assures that viscosity is important in considering the dynamics of the system. Furthermore the restrictions are ones that make laboratory experimentation feasible. Solutions are also presented for the non-viscous case (i.e. E = O), and are compared with their viscous counterparts. The importance of viscosity in this physical system is thus demonstrated.

When a horizontal magnetic field B(z) is sheared vertically on a lengthscale L in a diffusionless fluid, critical layers occur at 2c where the local Alfvén speed V(zc) matches the phase speed c of the wave. However, when a vertical field Bz is introduced, all the critical layers disappear. The present study investigates the solution in the neighbourhood of zc when Bz/B is very small, in order to clarify the manner in which the vertical field annihilates the critical layers. It is found that the solution across the critical layer is adjusted in a thin magnetic layer whose thickness is determined by the parameter ε2 (= U/αV, where U, V are measures of the vertical and horizontal components of the Alfvén velocity and α/L is the horizontal wavenumber). The vertical field increases the order of the equation governing the vertical variations of the amplitude of the perturbations from two to four. Within the magnetic layer the two extra Alfvén waves, one upgoing and the other downgoing, interact with those due to the horizontal field to make the solution regular everywhere. The mean vertical wave energy flux varies continuously from one constant value far on one side of the layer to another constant value far on the other side of the layer.

The influence of the vertical field on the resistive instabilities present in its absence is also examined. It is found that the relative importance of resistivity and vertical field is measured by the ratio of the thicknesses of the resistive and magnetic layers. In general, the influence of the vertical field is to suppress resistive instabilities. The slow exchange resistive instabilities are suppressed by the presence of the vertical field if $\epsilon \ges a(S\alpha)^{-\frac{1}{3}}$ while the localized gravitational modes are inhibited for ε ≥ b(α2S)−¼ where a, b are constants whose values depend on the profile of the horizontal field and on the gravitational parameter G; and S is the Lundquist number.

Convection driven by radial and normal gravity forces in a rotating, horizontal cylinder is examined. The cylinder is subjected to uniform volumetric heating and constant-temperature wall cooling. The parameters are the radial-gravity and normal-gravity Rayleigh numbers, Rar and Rag (with Rar, Rag ≤ 106), the rotational Reynolds number, Re = 2Ω r02/v (0 ≤ Re ≤ 250), and the Prandtl number (Pr = 7). Critical conditions for the radial-gravity rest state correspond to a two-cell flow in the azimuthal plane with Rar,c = 13738. Finite-amplitude transient and steady flows are obtained with a Galerkin finite element method for Rayleigh number ratios in the range 0.1 ≤ Rar/Rag ≤ 100. When radial gravity dominates the flows tend to be multicellular and, during transients, initial high-wavenumber forms evolve to lower-wavenumber forms. When normal gravity dominates the flows are bicellular. When radial and normal gravity forces are comparable, in the presence of rotation, complex time-dependent motions occur and the largest rates of fluid circulation and heat transfer are observed.

General solutions for the linearized three-dimensional unsteady motion of a relaxing gas, including the effects of direct heat addition, are obtained using a Green's function technique. The possibility of direct heat addition into each separate mode of energy storage must be catered for, and the solutions have some bearing on the question of the heating of a gas by radiation.

Examples are given of initial-value and heat-source solutions in an unbounded domain and the boundary-value problem is exemplified by a study of the motion created by a spherical piston.

The phenomenon of steady cellular convection in a rotating fluid in which the stratification is statically unstable is well known. If the temperature field also varies in the horizontal direction a thermal wind is generated which can diminish the amplitude of the cells, or if inertial effects are strong enough, confine them to a narrow vertical band. On the other hand, in an enclosed container resonance can occur which increases the amplitude of the cells beyond the scope of a linear theory.

The diffraction of an initially one-dimensional plane expansion wave over a 90° corner was explored using experiment and numerical simulation. Unlike studies of shock diffraction, expansion wave diffraction has hardly been documented previously. The planar expansion wave was produced in a shock tube by bursting a diaphragm. Two independent parameters were identified for study: (i) the initial diaphragm shock tube pressure ratio, which determines the strength (pressure ratio) of the expansion, and (ii) the position of the diaphragm from the apex of the 90° corner, which determines the width of the wave. The experimentation only considered variation in the shock tube pressure ratio whereas the simulation varied both parameters. A Navier–Stokes solver with Menter’s shear stress transport $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k\mbox{--}\omega $ turbulence model was found to adequately model the overall flow field. A number of major flow features were identified occurring in the vicinity of the corner. These were: a shear layer that originated by flow separation at the apex of the corner; a vortex within a separation bubble that remained attached to the wall, in sharp contrast to what happens in the shock wave diffraction case, where the vortex convects downstream; and a reflected compression wave arising from perturbation signals generated by the diffraction. For a narrow-width expansion wave existing prior to diffraction, the reflected compression wave steepens into an outwardly propagating, weak cylindrical shock wave. Regions of supersonic flow are identified surrounding the bubble and can extend downstream depending on the pressure ratio. Another major flow feature identified in some cases was an oblique shock located near the separation bubble. A large wake region is evident immediately downstream of the bubble and appears to consist of two distinct layers. The experimental results showed large-scale turbulent structures within the separation bubble, and shear layer instability and vortex shedding from the separation bubble were also evident.

We study so-called wave boundary layers (BLs) arising in a stably stratified fluid at large times. The BL is a narrow domain near the surface and/or bottom of the fluid; with increasing time, gradients of buoyancy and horizontal velocity in the BL grow sharply and the BL thickness tends to zero. The non-stationary BL can arise both as a result of linear evolution of the initial perturbation and under the action of an external force (tangential stress exerted on the fluid surface in our case). We analyse both the variants and find that the ‘forced’ BLs are much more intense than the ‘free’ ones. In the ‘free’ BLs all fields are bounded and the gradients of buoyancy and horizontal velocity grow linearly in time, whereas in the ‘forced’ BL only the vertical velocity is bounded and the buoyancy and horizontal velocity grow linearly in time. As to the gradients in the ‘forced’ BL, the vertical velocity gradient grows in time linearly and the gradients of buoyancy and horizontal velocity grow quadratically. In both of the cases we determine exact solutions in the form of expansions in the vertical wave modes and find asymptotic solutions valid at large times. The comparison between them shows that the asymptotic solutions approximate the exact ones fairly well even for moderate times.