Oscillations of low-viscosity drops were studied in the microgravity environment of a Space shuttle flight. From the damped oscillation data, the inviscid frequency shift, due to nonlinearity, has been extracted using a central-averaging scheme. For the classical case of the oscillations of a free low-viscosity drop, it has been found that the frequency shift agrees well with the predictions of the inviscid nonlinear theory of Tsamopoulos & Brown (1983) for ε < 0.3. But for the oscillations of a rotating low-viscosity drop, under acoustic levitation, the frequency shift is smaller, and the percentage of time spent in prolate displacement is significantly less than that for the classical case.

The surface of a turbulent, open-channel flow is often characterized by smooth areas of upwelling, each surrounded by a zone of downwelling marked by short steep waves. The dynamics of short waves on such a downwelling region are investigated and some laboratory experiments are proposed. Assuming that the horizontal strain rate Ω is locally constant, a simple expression is derived for the amplitude a of the short capillary–gravity waves, and hence also for the spectrum of the surface slopes.

This paper presents a linear stability analysis of a rapidly sheared layer of granular material confined between two parallel solid plates. The form of the steady base-state solution depends on the nature of the interaction between the material and the bounding plates and three cases are considered, in which the boundaries act as sources or sinks of pseudo-thermal energy, or merely confine the material while leaving the velocity profile linear, as in unbounded shear. The stability analysis is conventional, though complicated, and the results are similar in all cases. For given physical properties of the particles and the bounding plates it is found that the condition of marginal stability depends only on the separation between the plates and the mean bulk density of the particulate material contained between them. The system is stable when the thickness of the layer is sufficiently small, but if the thickness is increased it becomes unstable, and initially the fastest growing mode is analogous to modes of the corresponding unbounded problem. However, with a further increase in thickness a new mode becomes dominant and this is of an unusual type, with no analogue in the case of unbounded shear. The growth rate of this mode passes through a maximum at a certain value of the thickness of the sheared layer, at which point it grows much faster than any mode that could be shared with the unbounded problem. The growth rate of the dominant mode also depends on the bulk density of the material, and is greatest when this is neither very large nor very small.

A compressible countercurrent shear layer was investigated experimentally by establishing reverse flow around the perimeter of a supersonic jet. Measurements demonstrate that spatial growth rates of the countercurrent shear layer significantly exceed those of the classical coflowing layer at comparable density ratios and levels of compressibility. Experiments also reveal the presence of coherent three-dimensional structures in the countercurrent shear layer at convective Mach numbers where similar structures are not present in coflowing layers. It is argued that these kinematic differences are responsible for the enhanced diffusion of the shear layer with counterflow. The spatio-temporal theory is used to examine the connection between the experimental observations and the existence of a transition from convective to absolute instability in high-speed shear layers.

The development of the two-dimensional flow of a viscous incompressible fluid around a circular cylinder which suddenly starts to move with the velocity U = U0 + U1t + U2t2 is studied. Equations for the flow in terms of the stream function and vorticity in boundary-layer coordinates are presented. A perturbation series solution for small times is developed. The flow for longer times is computed numerically using an accurate implicit time-integration procedure. The numerical method is checked for small times by comparison with the results of the analytical solution. Reynolds numbers R in the range 200 to 104 (based on the diameter of the cylinder) are considered. One particularly interesting result is that for certain values of U1 and U2 at R = 500 and R = 103 it is found that two co-rotating vortices and three co-rotating vortices develop with time in each half of the wake in the two respective cases.

Viscous fingering can occur as a three-dimensional disturbance to plane flow of a hot thermoviscous liquid in a Hele-Shaw cell with cold isothermal walls. This work assumes the principle of exchange of stabilities, and uses a temporal stability analysis to find the critical viscosity ratio and finger spacing as functions of channel length, Lc. Viscous heating is taken as negligible, so the liquid cools with distance (x) downstream. Because the base flow is spatially developing, the disturbance equations are not fully separable. They admit, however, an exact solution for a liquid whose viscosity and specific heats are arbitrary functions of temperature. This solution describes the neutral disturbances in terms of the base flow and an amplitude, A(x). The stability of a given (computed) base flow is determined by solving an eigenvalue problem for A(x), and the critical finger spacing. The theory is illustrated by using it to map the instability for variable-viscosity flow with constant specific heat. Two fingering modes are predicted, one being a turning-point instability. The preferred mode depends on Lc. Finger spacing is comparable with the thermal entry length in a long channel, and is even larger in short channels. When applied to magmatic systems, the results suggest that fingering will occur on geological scales only if the system is about freeze.

We report on an experimental investigation of temporal, scalar power spectra of round, high Schmidt number (Sc ≃ 1.9 × 103), momentum-dominated turbulent jets, for jet Reynolds numbers in the range of 1.25 × 104 ≤ Re ≤ 7.2 × 104. At intermediate scales, we find a spectrum with a slope (logarithmic derivative) that increases in absolute value with Reynolds number, but remains less than 5/3 at the highest Reynolds number in our experiments. At the smallest scales, our spectra exhibit no k−1 power-law behaviour, but, rather, seem to be approximated by a log-normal function, over a range of scales exceeding a factor of 40, in some cases.

The flow of a continuously stratified fluid over a smooth bottom bump in a channel of finite depth is considered. In the weakly nonlinear-weakly dispersive régime ε = a/h [Lt ] 1, μ = h/l [Lt ] 1 (where h is the channel depth and a, l are the peak amplitude and the width of the obstacle respectively), the parameter A = ε/μp (where p < 0 depends on the obstacle shape) controls the effect of nonlinearity on the steady lee wavetrain that forms downstream of the obstacle for subcritical flow speeds. For A = O(1), when nonlinear and dispersive effects are equally important, the interaction of the long-wave disturbance over the obstacle with the lee wave is fully nonlinear, and techniques of asymptotics ‘beyond all orders’ are used to determine the (exponentially small as μ → 0) lee-wave amplitude. Comparison with numerical results indicates that the asymptotic theory often remains reasonably accurate even for moderately small values of μ and ε, in which case the (formally exponentially small) lee-wave amplitude is greatly enhanced by nonlinearity and can be quite substantial. Moreover, these findings reveal that the range of validity of the classical linear lee-wave theory (A [Lt ] 1) is rather limited.

The formation of separation singularities in solutions of the classical boundary-layer equations is studied numerically and analytically for the case of a two-dimensional incompressible steady flow near a solid surface moving in the direction of the main stream. Unlike the previously studied regime of the incipient separation located at the maximum point in the external pressure distribution, the breakdown in this work occurs under an adverse pressure forcing and involves a regular flow field upstream of the Moore-Rott-Sears point with an algebraic non-analyticity downstream. Small deviations from the precisely regular approach to the singular point are shown to result in an exponential amplification of linear disturbances; in the subsequent nonlinear stage the solution terminates in a finite-distance blow-up singularity or, alternatively, continues in a regular fashion across the singular station. The case of asymptotically small slip velocities is considered and a connection with marginal separation on a fixed wall is discussed.

The effects of compressibility, pitch rate and Reynolds number on the initial stages of two-dimensional unsteady separation of laminar subsonic flow over a pitching NACA-0012 airfoil have been studied numerically. The approach involves the simulation of the flow by solving the two-dimensional unsteady compressible laminar Navier-Stokes equations employing the implicit approximate-factorization algorithm of Beam & Warming and a boundary-fitted C-grid. The algorithm has been extensively validated through comparison with analytical and previous numerical results. The computations display several important trends for the ‘birth’ of the primary recirculating region which is a principal precursor to leading-edge separation. Increasing the non-dimensional pitch rate from 0.05 to 0.2 at a fixed Reynolds number and Mach number delays the formation of the primary recirculating region. The primary recirculating region also forms closer to the leading edge. Increasing the Mach number from 0.2 to 0.5 at a fixed Reynolds number and pitch rate causes a delay in the formation of the primary recirculating region and also leads to its formation farther from the airfoil top surface. The length scale associated with the recirculating regions increases as well. Increasing the Reynolds number from 104 to 105 at a fixed Mach number and pitch rate hastens the appearance of the primary recirculating region. A shock appears on the top surface at a Reynolds number of 105 along with the simultaneous formation of multiple recirculating regions near the leading edge.

We develop a numerical model of the interaction between wind and a small-amplitude water wave. The model first calculates the turbulent flows in both the air and water that would be obtained with a flat interface, and then calculates linear perturbations to this base flow caused by a travelling surface wave. Turbulent stresses in the base flow are parameterized using an eddy viscosity derived from a low-turbulent-Reynolds-number κ – ε model. Turbulent stresses in the perturbed flow are parameterized using a new damped eddy viscosity model, in which the eddy viscosity model is used only in inner regions, and is damped exponentially to zero outside these inner regions. This approach is consistent with previously developed physical scaling arguments. Even on the ocean the interface can be aerodynamically smooth, transitional or rough, so the new model parameterizes the interface with a roughness Reynolds number and retains effects of molecular stresses (on both mean and turbulent parts of the flow).

The damped eddy viscosity model has a free constant that is calibrated by comparing with results from a second-order closure model. The new model is then used to calculate the variation of form drag on a stationary rigid wave with Reynolds number, R. The form drag increases by a factor of almost two as R drops from 2 × 104 to 2 × 103 and shows remarkably good agreement with the value measured by Zilker & Hanratty (1979). These calculations show that the damped eddy viscosity model captures the physical processes that produce the asymmetric pressure that leads to form drag and also wave growth.

Results from the numerical model show reasonable agreement with profiles measured over travelling water waves by Hsu & Hsu (1983), particularly for slower moving waves. The model suggests that the wave-induced flow in the water is irrotational except in an extremely thin interface layer, where viscous stresses are as likely to be important as turbulent stresses. Thus our study reinforces previous suggestions that the region very close to the interface is crucial to wind-wave interaction and shows that scales down to the viscous length may have an order-one effect on the development of the wave.

The energy budget and growth rate of the wave motions, including effects of the sheared current and Reynolds number, will be examined in a subsequent paper.

Numerical simulations are employed to study the Brownian motion of a bead-rod polymer chain dissolved in a solvent. An investigation is conducted of the relaxation of the stress for an initially straight chain as it begins to coil.

For a numerical time step δt in the simulations, conventional formulae for the stress involve averaging large ±O(1/(δt)1/2) contributions over many realizations, in order to yield an O(1) average. An alternative formula for the stress is derived which only contains O(1) contributions, thereby improving the quality of the statistics.

For a chain consisting of n rods in a solvent at temperature T, the component of the bulk stress along the initial chain direction arising from tensions in the rods at the initial instant is $k\hat{T}\times n(\frac{1}{3}n^2 + n +\frac{2}{3})$. Thus the bead-rod model yields results very different from other polymer models, such as the entropic spring of Flory (1969), which would assign an infinite stress to a fully aligned chain. For rods of length l and beads of friction factor $\hat{\zeta}$ the stress decays at first on $O(\hat{\zeta}\hat{l}^2/k\hat{T}\times 1/n^2)$ time scales. On longer time scales, this behaviour gives way to a more gradual stress decay, characterized by an $O(k\hat{T}\times n)$ stress following a simple exponential decay with an $O(k\hat{T}/\hat{\zeta}\hat{l}^2\times 1/n^2)$ rate. Matching these two limiting regimes, a power law decay in time t is found with stress $O(k\hat{T}\times n^2\times (k\hat{T}\hat{t}/\hat{\zeta}\hat{l}^2)^{-1/2})$. The dominant physical processes occurring in these separate short, long and intermediate time regimes are identified.

Entrainment of ambient fluid into both two-dimensional and axisymmetric gravity currents is investigated experimentally using a novel neutralization technique. The technique involves the titrative neutralization of an alkaline gravity current which intrudes into and entrains an acidic ambient, and is visualized using a pH indicator solution. Using this technique, we can determine quantitative results for the amount of dilution in the head of the current. The head of the current is able to entrain ambient fluid both by shear instabilities on the current/ambient interface and by over-riding (relatively light) ambient fluid. Guided by our experimental observations, we present two slightly different theoretical models to determine the entrainment into the head of the current as a function of distance from the source for the instantaneous release of a constant volume of fluid in a two-dimensional geometry. By dimensional analysis, we determine from both models that the dimensionless entrainment or dilution ratio, E, defined as the ratio of the volumes of ambient and original fluid in the head, is independent of the initial reduced gravity of the current; and this result is confirmed by our experiments in Boussinesq situations. Our theoretical evaluation of E in terms of the initial cross-sectional area of the current agrees very well with our experimental measurements on the incorporation of an entrainment coefficient α, evaluated experimentally to be 0.063 ± 0.003. We also obtain experimental results for constant-volume gravity currents moving over horizontal surfaces of varying roughness. A particularly surprising result from all the experiments, which is reflected in the theoretical models, is that the head remains essentially unmixed – the entrainment is negligible – in the slumping phase. Thus the heads of gravity currents with identical initial cross-sectional areas but different initial aspect ratios (lock lengths) will begin to be diluted by ambient fluid at different positions and hence propagate at different rates. A range of similar results is determined, both theoretically and experimentally, for the instantaneous release of a fixed volume of (heavy) fluid in an axisymmetric geometry. By contrast, the results of our experiments with gravity currents fed by a constant flux exhibit markedly different entrainment dynamics due to the continual replenishment of the fluid in the head by the constant input of undiluted fluid from the tail.

The paper is a first attempt at theoretical investigation of the experimentally observed enigmatic phenomenon of surface manifestations of bottom topography reproducing well the image of the relief despite several kilometres of ocean depth. Both satellite observations and direct measurements have been repeatedly reported in the last two decades. We suggest a possible mechanism for these manifestations in a large scale range (of the order 5 × 101–103 km), based on the hydrodynamic theory of quasi-geostrophic stratified flow over topography on a β-plane.

The classical theories of quasi-geostrophic flow over topography on a β-plane do not include vertical shear, and it is well-known that the disturbance caused by topography cannot reach the surface of a stratified ocean unless the stratification or current velocity is unrealistic. The new element changing the situation qualitatively is the taking into account of the influence of near-bottom and near-surface boundary layers, where flow velocities, velocity gradients and stratification can significantly exceed the corresponding values for the flow in the main body. The asymptotic solution derived shows the considerable increase of the normal mode amplitude towards the boundaries. Thus, this specific distortion of the eigenmode structure results in effective forcing of the modes by topography and, on the other hand, leads to pronounced disturbances in the fields of near-surface characteristics. The mechanism effectiveness is demonstrated by the fact that the surface disturbance amplitude normally significantly exceeds the corresponding value for the barotropic current equal to the maximum of the shear flow. A remarkable feature of the solution is that the Green's function is strongly localized in the horizontal plane for a wide range of relevant parameters, thus leading to the close resemblance of surface patterns and bottom relief. To get a better understanding of the quantitative characteristics of the mechanism, the dependence of the effect on the parameters of an N-layer model was studied in detail. The amplification of the surface manifestations due to the presence of boundary layers can reach several orders of magnitude and thus make the manifestations easily observable. The surface temperature anomalies due to topography were estimated and found to be observable under favourable conditions.

Viscous flow through three-dimensional models of fibrous porous media is analysed. The periodic models are based on ordered networks of cylindrical fibres on regular cubic lattices. Numerical solutions are obtained using the spectral boundary element method introduced by Muldowney & Higdon (1995). Results are presented for solid volume fractions ranging from extreme dilution to near the maximum volume fraction for permeable media. Theoretical estimates are derived using slender-body theory and lubrication approximations in the appropriate asymptotic regimes. Comparisons are made with model predictions based on properties of two-dimensional media (Jackson & James 1986), and with results for disordered dispersions of prolate spheroids (Claeys & Brady 1993b).

The subject of total temperature separation in jets was treated in Fox et al. (1993) for subsonic jets. When we extended this study to the case of supersonic jets, we found the presence of a different mechanism of cooling, an effect which does not appear to have been known in the past. Named the ‘shock-induced total temperature separation’, this cooling can be of much greater magnitude than the subsonic cooling treated previously; it is caused by the interaction of convected vortical structures near the jet exhaust with the shock structure of the supersonic jet.

In studying this phenomenon, we focus our attention on overexpanded jets exiting a convergent-divergent nozzle. The theoretical results for the shock-induced cooling which are based on a linearized, unsteady supersonic analysis are shown to agree favourably with experiments.

When an impingement plate is inserted, the shock-induced cooling would manifest itself as wall cooling, whose magnitude is significantly larger than the subsonic counterpart. This has implications for heat transfer not only in jets, but wherever vortical structures may interact with shock waves.