The development of a two-dimensional viscous incompressible flow generated from a circular cylinder impulsively started into rectilinear motion is studied computationally. An adaptative numerical scheme, based on vortex methods, is used to integrate the vorticity/velocity formulation of the Navier–Stokes equations for a wide range of Reynolds numbers (Re = 40 to 9500). A novel technique is implemented to resolve diffusion effects and enforce the no-slip boundary condition. The Biot–Savart law is employed to compute the velocities, thus eliminating the need for imposing the far-field boundary conditions. An efficient fast summation algorithm was implemented that allows a large number of computational elements, thus producing unprecedented high-resolution simulations. Results are compared to those from other theoretical, experimental and computational works and the relation between the unsteady vorticity field and the forces experienced by the body is discussed.

Head-on collision of two coaxial vortex rings has been studied by joint experimental and numerical investigation. The Reynolds number, ReΓ, based on the initial circulation of the vortex rings, ranged from 400 to 2700. Besides numerical data, the vorticity field was also resolved by a non-intrusive visualization technique, LIPA, which enabled simultaneous measurement of velocities at multiple locations on a plane area. It was found that the enstrophy, rather than circulation, revealed three stages of evolution of the vortex rings prior to their breakdown. These include the free-travelling stage, stage of vortex stretching and the stage of viscous dissipation dominance. The results indicate that it would be incorrect to neglect the viscous effect, in particular, for the latter two stages of flow development. In fact, the rebound behaviour of the vortex rings for lower ReΓ is essentially a viscous phenomenon and is found to be closely related to the dissipation of enstrophy when the vortex rings are brought to interact actively with each other and is also related to the increase of the vorticity core diameter in the stage of dominance of viscous dissipation. Furthermore, an instant dimensionless group, Nt/ReΓ, based on the local vorticity distribution and the radius of a vortex ring, is found to be appropriate to characterize the onset of instability. Our investigation indicates that, in the range of observation, bulging instability will be observed during collision when Nt/ReΓ exceeds a critical value, (Nt/ReΓ)cr, which is a function of the initial core-size of the vortex ring. Comparisons showed that the numerical, measured, and visualization results were in consistent agreement; this not only enables us to assess the range of validity of the axisymmetry assumed for the numerical simulation, but also provides us with a rational basis for further analysis of azimuthal instability.

In a stably stratified shear layer, thin vorticity layers (‘baroclinic layers’) are produced by buoyancy effects and strain in between the Kelvin–Helmholtz vortices. A two-dimensional numerical study is conducted, in order to investigate the stability of these layers. Besides the secondary Kelvin–Helmholtz instability, expected but never observed previously in two-dimensional numerical simulations, a new instability is also found.

The influence of the Reynolds number (Re) upon the dynamics of the baroclinic layers is first studied. The layers reach an equilibrium state, whose features have been described theoretically by Corcos & Sherman (1976). An excellent agreement between those predictions and the results of the numerical simulations is obtained. The baroclinic layers are found to remain stable almost up to the time the equilibrium state is reached, though the local Richardson number can reach values as low as 0.05 at the stagnation point. On the basis of the work of Dritschel et al. (1991), we show that the stability of the layer at this location is controlled by the outer strain field induced by the large-scale Kelvin–Helmholtz vortices. Numerical values of the strain rate as small as 3% of the maximum vorticity of the layer are shown to stabilize the stagnation point region.

When non-pairing flows are considered, we find that only for Re ≤ 2000 does a secondary instability eventually amplify in the layer. (Re is based upon half the initial vorticity thickness and half the velocity difference at the horizontally oriented boundaries.) This secondary instability is not of the Kelvin–Helmholtz type. It develops in the neighbourhood of convectively unstable regions of the primary Kelvin–Helmholtz vortex, apparently once a strong jet has formed there, and moves along the baroclinic layer while amplifying. It next perturbs the layer around the stagnation point and a secondary instability, now of the Kelvin–Helmholtz type, is found to develop there.

We next examine the influence of a pairing upon the flow behaviour. We show that this event promotes the occurrence of a secondary Kelvin–Helmholtz instability, which occurs for Re ≥ 400. Moreover, at high Reynolds number (≥ 2000), secondary Kelvin–Helmholtz instabilities develop successively in the baroclinic layer, at smaller and smaller scales, thereby transferring energy towards dissipative scales through a mechanism eventually leading to turbulence. Because the vorticity of such a two-dimensional stratified flow is no longer conserved following a fluid particle, an analogy with three-dimensional turbulence can be drawn.

A biplane grid with a mesh spacing of 10.8 cm was towed horizontally in a towing tank to generate turbulence in a non-stratified fluid and in stratified fluids with different constant density gradients. Turbulence velocity components and density fluctuations were measured using an array of cross-film and conductivity probes. Based on the mesh size of the grid, the nominal values of the (internal) Froude numbers were ∞, 80 and 40, and the corresponding Reynolds number was 4.3 × 104. The decay rates of the (turbulence) kinetic, potential and total energies and the dissipation rates of the kinetic and potential energies were calculated from the experimental data. For each of these quantities, the decay may be represented as a function of the downstream distance raised to a given power. The kinetic energy and its dissipation rate are lower for the stratified cases than for the non-stratified case but are almost compensated for by the corresponding potential energy and its dissipation rate. Our results are consistent with those of direct numerical simulations and agree reasonably well with those obtained in stratified wind and water tunnels. However, the results differ from laboratory results obtained using an optical method to measure the turbulent motion of tracer particles in the wake of a vertically towed grid; these latter results show an abrupt reduction in the decay rate of the turbulence kinetic energy after one Brunt–Väisälä period. A similar trend is also observed in results obtained in facilities with fairly high background turbulence or internal waves. This discrepancy is discussed and an explanation is presented. Furthermore, it is demonstrated that strongly stratified thin sheets with density gradients larger than that of the undisturbed fluid may be generated by local but incomplete mixing. The persistence of such thin sheets is proportional to the Schmidt number (≈ 500) in stratified salt water or the Prandtl number (≈ 0.71) in thermally stratified air.

Measurements have been made of the propulsive effect of supersonic combustion ramjets incorporated into a simple axisymmetric model in a free piston shock tunnel. The nominal Mach number was 6, and the stagnation enthalpy varied from 2.8 to 8.5 MJ kg−1. A mixture of 13% silane and 87% hydrogen was used as fuel, and experiments were conducted at equivalence ratios up to approximately 0.8. The measurements involved the axial force on the model, and were made using a stress wave force balance, which is a recently developed technique for measuring forces in shock tunnels. A net thrust was experienced up to a stagnation enthalpy of 3.7 MJ kg−1, but as the stagnation enthalpy increased, an increasing net drag was recorded. Pilot and static pressure measurements showed that the combustion was supersonic.

The results were found to compare satisfactorily with predictions based on established theoretical models, used with some simplifying approximations. The rapid reduction of net thrust with increasing stagnation enthalpy was seen to arise from increasing precombustion temperature, showing the need to control this variable if thrust performance was to be maintained over a range of stagnation enthalpies. Both the inviscid and viscous drag were seen to be relatively insensitive to stagnation enthalpy, with the combustion chambers making a particularly significant contribution to drag. The maximum fuel specific impulse achieved in the experiments was only 175 s, but the theory indicates that there is considerable scope for improvement on this through aerodynamic design.

The characteristics of a turbulent spot propagating in a laminar boundary layer subjected to a self-similar adverse pressure gradient (defined by a Falkner–Skan parameter β = -0.1) were investigated experimentally. It was observed that some small differences in the normalized shape of the undisturbed velocity profile caused by the pressure gradient had a major influence on the spreading rate of the spot at comparable Reδ*. The rate of spread of the spot in the spanwise direction was affected most dramatically by the pressure gradient where the average angle at which the tips of the spots moved outward relative to the plane of symmetry was 21°. It was noted that the strength and duration of the disturbance initiating the spots had an effect on their spanwise rate of spread. For example, a strong impulsive disturbance and a disturbance caused by a stationary three-dimensional roughness generated spots which spread at a much smaller rate. The interaction of the spot with the wave packet existing beyond its tip was enhanced by the adverse pressure gradient because the Reynolds number of the surrounding boundary layer was everywhere supercritical. Thus, the maximum linear amplification rate in this case is approximately four times larger than in Blasius flow. Some features of the breakdown and their relationship to the shape and the perturbation velocities in the spot are discussed. The normalized length of the calmed region relative to the length of the spot is enhanced by the adverse pressure gradient and by an increase in the intensity of the disturbance.

Kinetic theory and numerical simulations are used to explore the dynamics of a dilute gas–solid suspension subject to simple shear flow. The particles experience a Stokes drag force and undergo solid-body interparticle collisions. Two qualitatively different steady-state behaviours are possible: an ignited state, in which the variance of the particle velocity is very large; and a quenched state, in which most of the particles follow the local fluid velocity. Theoretical results for the ignited state are obtained by perturbing from a Maxwell distribution, while predictions for the quenched state result from consideration of the collision of particles that initially move with the fluid. A composite theory, which includes effects of collisions driven by both the mean shear and the velocity fluctuations, predicts the existence of multiple steady states. Dynamic simulations and calculations using the direct-simulation Monte Carlo method confirm the result that, for certain volume fractions and shear rates, either the quenched or ignited state can be achieved depending on the initial velocity variance.

Simulations are also performed for particles experiencing a nonlinear drag force. Both the theory of rapid granular flow, which neglects drag, and our theory for the ignited state with linear drag predict that the particle velocity variance can grow without bound as ϕ → 0, where ϕ is the volume fraction. The nonlinear drag force eliminates the divergence and leads to a particle velocity variance that will always decrease with decreasing volume fraction in the limit ϕ → 0.

Beginning with the known relationship between the pressure structure function and the fourth-order two-point correlation of velocity derivatives, we obtain a new theory relating the pressure structure function and spectrum to fourth-order velocity structure functions. This new theory is valid for all Reynolds numbers and for all spatial separations and wavenumbers. We do not use the joint Gaussian assumption that was used in previous theory. The only assumptions are local homogeneity, local isotropy, incompressibility, and use of the Navier–Stokes equation. Specific formulae are given for the mean-squared pressure gradient, the correlation of pressure gradients, the viscous range of the pressure structure function, and the pressure variance. Of course, pressure variance is a descriptor of the energy-containing range. Therefore, for any Reynolds number, the formula for pressure variance requires the more restrictive assumption of isotropy. For the case of large Reynolds numbers, formulae are given for the inertial range of the pressure structure function and spectrum and of the pressure-gradient correlation; these are valid on the basis of local isotropy, as are the formulae for mean-squared pressure gradient and the viscous range of the pressure structure function. Using the experimentally verified extension to fourth-order velocity structure functions of Kolmogorov's theory, we obtain r4/3 and k−7/3 laws for the inertial range of the pressure structure function and spectrum. The modifications of these power laws to account for the effects of turbulence intermittency are also given. New universal constants are defined; these require experimental evaluation. The pressure structure function is sensitive to slight departures from local isotropy, implying stringent conditions on experimental data, but applicability of the previous theory is likewise constrained. The results are also sensitive to compressibility.

A nonlocal theory for stress in bound suspensions of rigid, slender fibres is developed and used to predict the rheology of dilute, rigid polymer suspensions when confined to capillaries or fine porous media. Because the theory is nonlocal, we describe transport in a fibre suspension where the velocity and concentration fields change rapidly on the fibre's characteristic length. Such rapid changes occur in a rigidly bound domain because suspended particles are sterically excluded from configurations near the boundaries. A rigorous no-flux condition resulting from the presence of solid boundaries around the suspension is included in our nonlocal stress theory and naturally gives rise to concentration gradients that scale on the length of the particle. Brownian motion of the rigid fibres is included within the nonlocal stress through a Fokker–Planck description of the fibres’ probability density function where gradients of this function are proportional to Brownian forces and torques exerted on the suspended fibres. This governing Fokker–Planck probability density equation couples the fluid flow and the nonlocal stress resulting in a nonlinear set of integral-differential equations for fluid stress, fluid velocity and fibre probability density. Using the method of averaged equations (Hinch 1977) and slender-body theory (Batchelor 1970), the system of equations is solved for a dilute suspension of rigid fibres experiencing flow and strong Brownian motion while confined to a gap of the same order in size as the fibre's intrinsic length. The full solution of this problem, as the fluid in the gap undergoes either simple shear or pressure-driven flow, is solved self-consistently yielding average fluid velocity, shear and normal stress profiles within the gap as well as the probability density function for the fibres’ position and orientation. From these results we calculate concentration profiles, effective viscosities and slip velocities and compare them to experimental data.

A well-known technique for metering a multiphase flow is to use small probes that utilize some measurement principle to detect the presence of different phases surrounding their tips. In almost all cases of relevance to the oil industry, the flow around such local probes is inviscid and driven by surface tension, with negligible gravitational effects. In order to study the features of the flow around a local probe when it meets a droplet, we analyse a model problem: the interaction of an infinite, initially straight, interface between two inviscid fluids, advected in an initially uniform flow towards a semi-infinite thin flat plate oriented at 90° to the interface. This has enabled us to gain some insight into the factors that control the motion of a contact line over a solid surface, for a range of physical parameter values.

The potential flows in the two fluids are coupled nonlinearly at the interface, where surface tension is balanced by a pressure difference. In addition, a dynamic contact angle boundary condition is imposed at the three-phase contact line, which moves along the plate. In order to determine how the interface deforms in such a flow, we consider the small- and large-time asymptotic limits of the solution. The small-time and linearized large-time problems are solved analytically, using Mellin transforms, whilst the general large-time problem is solved numerically, using a boundary integral method.

The form of the dynamic contact angle as a function of contact line velocity is the most important factor in determining how an interface deforms as it meets and moves over the plate. Depending on this, the three-phase contact line may, at one extreme, hang up on the leading edge of the plate or, at the other extreme, move rapidly along the surface of the plate. At large times, the solution asymptotes to an interface configuration where the contact line moves at the far-field velocity.

A theory for the reflection-transmission problem of linear water waves in shallow channels with large-amplitude, rapidly varying topographies is given in Nachbin & Papanicolaou (1992b). However, it is very difficult to extract quantitative information from the theory in the large-amplitude regime. In this work, theoretical parameters are evaluated numerically, through the use of a numerical Schwarz–Christoffel transformation and of a Monte Carlo simulation. This enables the theory to be applied to its full extent. As a result we calculate the localization length for any given type of random bottom topography. Additionally, the numerical conformal mapping provides further insight into depth effects arising from potential theory. Statistical results, for numerically generated reflected waves, are in very good agreement with the theory for both piecewise-linear and piecewise-constant topographies of large amplitude.

A general closed form for the mobility tensor of a sphere moving in a fluid in stationary homogeneous flow is derived using the induced force method up to the first order in the square root of the Reynolds number based on the velocity gradient of the unperturbed flow. The closed form for the mobility tensor is valid for the time-dependent case as well as for the stationary case. As a special case, we calculate it explicitly for a simple shear flow. The result for the x, z-component for the stationary case, which gives the lift force, agrees with the value calculated by Saffman.

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.