The snapshot form of the Karhunen-Loéve (K–L) expansion has been applied to twodimensional, two-component hot-wire data from the region of a weakly pertubed free shear layer that includes the first pairing process. Low-level external perturbation was provided by a loudspeaker driven by a computer-generated signal composed of two sine waves of equal amplitude at the frequencies of the naturally developing fundamental instability wave and its first subharmonic, separated by a controllable initial phase angle difference. It was found that a large fraction of the fluctuation energy is carried by the first few modes. A low-dimensional empirical eigenfunction space is obtained which describes the shear-flow coherent structures well. Galerkin projection of the Navier-Stokes equations onto this basis set of principal eigenfunction modes results in a low-order system of dynamical equations, and solution of this system of equations describes the dynamics of the coherent structures associated with eigenfunctions. Finally the simulation, as obtained from the system of dynamical equations, is shown to compare reasonably well with the experiments.

The turbulent flow resulting from a top-hat jet exhausting into a large room was investigated. The Reynolds number based on exit conditions was approximately 105. Velocity moments to third order were obtained using flying and stationary hot-wire and burst-mode laser-Doppler anemometry (LDA) techniques. The entire room was fully seeded for the LDA measurements. The measurements are shown to satisfy the differential and integral momentum equations for a round jet in an infinite environment.

The results differ substantially from those reported by some earlier investigators, both in the level and shape of the profiles. These differences are attributed to the smaller enclosures used in the earlier works and the recirculation within them. Also, the flying hot-wire and burst-mode LDA measurements made here differ from the stationary wire measurements, especially the higher moments and away from the flow centreline. These differences are attributed to the cross-flow and rectification errors on the latter at the high turbulence intensities present in this flow (30% minimum at centreline). The measurements are used, together with recent dissipation measurements, to compute the energy balance for the jet, and an attempt is made to estimate the pressure-velocity and pressure-strain rate correlations.

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of the intruding fluid varies in time according to a power law of the type tα. The resistance of the ambient fluid is taken into account by a boundary condition of the von Kármán type, that depends on a parameter β that is a function of the density ratio of the fluids. The flow is characterized by β, α, and the Froude number [Fscr ]0 near the source. We find four kinds of self-similar solutions: subcritical continuous solutions (Type I), continuous solutions with a supercritical-subcritical transition (Type II), discontinuous solutions (Type III) that have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The current is always subcritical near the front, but near the source it is subcritical ([Fscr ]0 < 1) for Type I currents, and supercritical ([Fscr ]0 > 1) for Types II, III, and IV. Type I solutions have already been found by other authors, but Type II, III, and IV currents are novel. We find the intervals of parameters for which these solutions exist, and discuss their properties. For constant-volume currents one obtains Type I solutions for any β that, when β > 2, have a ‘dry’ region near the origin. For steady inflow one finds Type I currents for O < β < ∞ and Type II and III Currents for and β, if [Fscr ]0 is sufficiently large.

An inviscid fluid in steady two-dimensional motion in a region bounded by a closed streamline must have constant vorticity. We solve here for some such flows where the boundary is in part free, the fluid velocity magnitude being constant on the free boundary. A trivial example of such a flow is a circular cylinder of fluid rotating about its axis as if rigid, for which the whole circular boundary is free, irrespective of its radius. We now ask what happens to that flow when it comes into contact with solid boundaries. There is no steady flow when the contact is with a finite segment of a single plane wall, but a unique solution exists when the rotating fluid mass is in contact with some concave boundaries. Computed results are obtained for vortices lying inside a parabolic dish, or in a corner between two planes.

It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R)½, where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

Biological membranes can be considered two-dimensional fluids with suspended integral membrane proteins (IMPs). We have calculated the effect of hydrodynamic interactions on the various diffusion coefficients of IMPs in lipid bilayers. The IMPs are modelled as hard cylinders of radius a immersed in a thin sheet of viscosity μ and thickness h bounded by a fluid of low viscosity μ′. We have ensemble averaged the N-body Stokes equations to the pair level and have renormalized them following the methods of Batchelor (1972) and Hinch (1977). The lengthscale for the hydrodynamic interactions is λa = μh / μ′, Which is O (100a), and the slow decay of the interactions introduces new features in the renormalizations compared to the analogous analyses for three-dimensional suspensions of spheres.

We have calculated the asymptotic limits for the short- and long-time tracer diffusivities, Ds and Dl, respectively, and for the gradient diffusivity, Dg, for ϕ [Lt ] 1 and λ [Gt ] 1, where ϕ is the IMP area fraction and λ = μh / (μ′a). The diffusivities are \begin{eqnarray*} D_s/D_0 &=& 1-2\phi[1-(1+\ln (2)-9/32)/(\ln(\lambda)-\gamma)], D_l/D_0 &=& D_s/D_0 - 0.07/(\ln(\lambda)-\gamma), D_g/D_0 &=& 1+\phi[-7+(6\ln(2)+7/16+0.37)/(\ln(\lambda)-\gamma)], \end{eqnarray*} where D0 is the diffusivity in the limit of zero area fraction, and γ = 0.577216 is Euler's constant. The results for Dl and Ds differ only slightly. The decrease in Dg/Do as ϕ increases contrasts with the result for spheres for which Dg/Do > 1.

Nonlinear viscous droplet oscillations are analysed by solving the Navier-Stokes equation for an incompressible fluid. The method is based on mode expansions with modified solutions of the corresponding linear problem. A system of ordinary differential equations, including all nonlinear and viscous terms, is obtained by an extended application of the variational principle of Gauss to the underlying hydrodynamic equations. Results presented are in a very good agreement with experimental data up to oscillation amplitudes of 80% of the unperturbed droplet radius. Large-amplitude oscillations are also in a good agreement with the predictions of Lundgren & Mansour (boundary integral method) and Basaran (Galerkin-finite element method). The results show that viscosity has a large effect on mode coupling phenomena and that, in contradiction to the linear approach, the resonant mode interactions remain for asymptotically diminishing amplitudes of the fundamental mode.

This work investigates the two-dimensional flow of a shock wave over a circular light-gas inhomogeneity in a channel with finite width. The pressure gradient from the shock wave interacts with the density gradient at the edge of the inhomogeneity to deposit vorticity around the perimeter, and the structure rolls up into a pair of counter-rotating vortices. The aim of this study is to develop an understanding of the scaling laws for the flow field produced by this interaction at times long after the passage of the shock across the inhomogeneity. Numerical simulations are performed for various initial conditions and the results are used to guide the development of relatively simple algebraic models that characterize the dynamics of the vortex pair, and that allow extrapolation of the numerical results to conditions more nearly of interest in practical situations. The models are not derived directly from the equations of motion but depend on these equations and on intuition guided by the numerical results. Agreement between simulations and models is generally good except for a vortex-spacing model which is less satisfactory.

A practical application of this shock-induced vortical flow is rapid and efficient mixing of fuel and oxidizer in a SCRAMJET combustion chamber. One possible injector design uses the interaction of an oblique shock wave with a jet of light fuel to generate vorticity which stirs and mixes the two fluids and lifts the burning jet away from the combustor wall. Marble proposed an analogy between this three-dimensional steady flow and the two-dimensional unsteady problem of the present investigation. Comparison is made between closely corresponding three-dimensional steady and two-dimensional unsteady flows, and a mathematical description of Marble's analogy is proposed.

Analysis of data from a direct simulation of statistically steady homogeneous turbulence suggests that the vorticity tubes, which constitute the main structure of the vorticity field, are produced by shear instabilities. This is confirmed by a decay calculation, in which vorticity sheets appear at first, and then roll up to form the first tubes. These instabilities seem to be at least as important as vortex stretching in transferring energy from large to small scales.

We develop a simple model for the behaviour of an inviscid stratified shear flow with a thin mixed layer of intermediate fluid. We find that the flow is simultaneously unstable to oscillatory disturbances that are a generalization of those discussed by Holmboe (1962), purely unstable modes analogous to those considered by Taylor (1931), and a new type of oscillatory disturbance at large wavelength. The relative significance of these different types of instability depends on the ratio R of the depth of the intermediate layer to the depth of the shear layer. For small values of R, the new type of oscillatory wave has both the largest growthrate for given bulk Richardson number Ri0, and is also primarily unstable to disturbances propagating at an angle to the mean flow, i.e. such modes violate the conditions of Squire's theorem (1933), and thus the assumption of initial two dimensionality of such flows is invalid. For intermediate values of R, the Holmboe-type modes and the Taylor-types modes may have wavelengths and phase speeds conducive to the formation of a resonant triad over a wide range of Ri0. Thus the presence of an intermediate layer in a stratified shear flow markedly changes its stability properties.

The investigation is concentrated on two important quantities – the Strouhal number and the mean base suction coefficient, both measured at the mid-span position. Reynolds numbers from about 50 to 4 × 104 were investigated. Different aspect ratios, at low blockage ratios, were achieved by varying the distance between circular end plates (end plate diameter ratios between 10 and 30). It was not possible, by using these end plates in uniform flow and at very large aspect ratios, to produce parallel shedding all over the laminar shedding regime. However, parallel shedding at around mid-span was observed throughout this regime in cases when there was a slight but symmetrical increase in the free-stream velocity towards both ends of the cylinder. At higher Re, the results at different aspect ratios were compared with those of a ‘quasi-infinite cylinder’ and the required aspect ratio to reach conditions independent of this parameter, within the experimental uncertainties, are given. For instance, aspect ratios as large as L/D = 60–70 were needed in the range Re ≈ 4 × 103–104. With the smallest relative end plate diameter and for aspect ratios smaller than 7, a bi-stable flow switching between regular vortex shedding and ‘irregular flow’ was found at intermediate Reynolds number ranges in the subcritical regime (Re ≈ 2 × 103).

We consider the decay of an initial discontinuity in a polytropic gas in a Minkowski space–time (the special relativistic Riemann problem). In order to get a general analytical solution for this problem, we analyse the properties of the relativistic flow across shock waves and rarefactions. As in classical hydrodynamics, the solution of the Riemann problem is found by solving an implicit algebraic equation which gives the pressure in the intermediate states. The solution presented here contains as a particular case the special relativistic shock-tube problem in which the gas is initially at rest. Finally, we discuss the impact of this result on the development of high-resolution shock-capturing numerical codes to solve the equations of relativistic hydrodynamics.

A numerical program is developed to simulate an assembly of inelastic frictional spheres inside a control volume undergoing rapid shearing motion induced by the top and bottom moving periodic boundaries. A sticking–sliding collision model is used to emulate binary collisions of real particles. After the flow has reached a steady state, ensemble averages of macroscopic properties such as translational and rotational granular temperatures, and kinetic and collisional stresses at different solids concentrations are obtained. The present results are compared with previous theoretical, numerical and experimental works, and favourable agreement is found among them. The simulation results show that the stresses are anisotropic and decrease with decreasing coefficient of restitution and increasing friction coefficient. At high solids fraction, above about 0.5, there exists a critical concentration where the layering effects of particles, the formation of high-density microstructures and the increase in correlation of particle velocities are the major causes of abrupt changes in flow properties.

The effective viscosity μe for the Brinkman–Forchheimer flow (BFF) model has been determined experimentally for steady flow through a wall-bounded porous medium. Nuclear magnetic resonance (NMR) techniques were used to measure non-invasively the ensemble-average velocity profile of water flowing through a tube filled with an open-cell rigid foam of high porosity (ϕ = 0.972). By comparing these data with the BFF model, for which all remaining parameters were measured independently, it was determined that μe = (7.5+3.4−2.4)μf, where μf was the viscosity of the fluid. The Reynolds number, based upon the square root of the permeability, was 17.