The steady axisymmetric flow in and around a deformable drop moving under the action of gravity along the axis of a vertical tube at intermediate Reynolds number is studied by solving the nonlinear free-boundary problem using a Galerkin finite-element method. For the case where the drop and suspending liquid have the same viscosity, the ratio of the densities is 6/5 or 5/6, and the radius of the tube is equal to twice the radius of a sphere having the drop volume, four significant results are apparent in the computations. First, we compute drops showing much more deformation, and in particular the development of considerably more non-convexity, than those found in previous calculations for non-zero Reynolds number. The degree of non-convexity typically grows with the Reynolds number. Secondly, external recirculation zones can be attached to or disjoint from the drop. We find when there is a single external recirculation zone, that is disjoint (as found by Dandy & Leal), it can attach to the drop as the Reynolds number is increased. As the Reynolds number further increases, this is immediately followed by division of the drop into two adjacent recirculating regions. Thirdly, we sometimes find two recirculation zones in the suspending liquid. Finally, the drag coefficient, axis ratio, and normalized interfacial and frontal areas of the drop can vary non-monotonically with the Weber number, exhibiting as many as four local extrema. The results are compared to previous theoretical and experimental work, and implications for drop motion and heat and mass transfer are discussed.

The ‘almost-highest wave’, which is the asymptotic form of the flow in a steep irrotational water wave of less than the limiting height, was recently shown to be unstable to infinitesimal disturbances (see Longuet-Higgins & Cleaver 1994). It was also shown numerically that the lowest eigenfrequency is asymptotic to that of the lower superharmonic instability of a progressive wave in deep water (Longuet-Higgins, Cleaver & Fox 1994). In the present paper these calculations are revised, indicating the presence of more than one such instability, in agreement with recent calculations on steep periodic and solitary waves (Longuet-Higgins & Tanaka 1997).

The nonlinear development of the fastest-growing instability is also traced by a boundary-integral time-stepping method and the initial, linear growth rate is confirmed. The subsequent, nonlinear stages of growth depend as expected on the sign of the initial perturbation. Perturbations of one sign lead to the familiar overturning of the wave crest. Perturbations of the opposite sign lead to a smooth transition of the wave to a lower progressive wave having nearly the same total energy, followed by a return to a wave of almost the initial wave height. This appears to be the beginning of a nonlinear recurrence phenomenon.

The forms of the superharmonic instabilities of irrotational surface waves on deep water are calculated for wave steepnesses up to 99.9% of the limiting value. It is found that as the limiting wave steepness is approached the rates of growth of the lowest two unstable modes (n=1 and 2) increase according to the asymptotic law suggested by the theory of the almost-highest wave (Longuet-Higgins & Cleaver 1994; Longuet-Higgins, Cleaver & Fox 1994; Longuet-Higgins & Dommermuth 1997). Moreover, each eigenfunction becomes concentrated near the wave crest, with a horizontal scale proportional to the local radius of curvature at the crest. These are therefore ‘crest instabilities’ in the original sense.

Similar calculations are carried out for the normal-mode instabilities of solitary waves in shallow water, at steepnesses up to 99.99% of the limiting steepness. Similar conclusions are found to apply, though with greater accuracy.

In significantly underexpanded jets, screech inherently ceases to exist. This paper studies screech cessation in a supersonic rectangular jet and provides an explanation for its occurrence. Experimental data are presented for fully expanded Mach numbers, Mj, ranging from 1.1 to 1.9. Screech becomes unsteady beyond Mj=1.65 and ceases to exist beyond Mj=1.75. The reason for this cessation has remained a mystery, and this paper examines three suspects: (i) the theory of a frequency mismatch between screech tones and the band of the most-amplified jet instability waves, (ii) the notion that Mach disk formation disrupts the shock-cell structure and weakens the screech-producing shocks, and (iii) the idea that acoustic feedback and receptivity diminish at high levels of underexpansion. A thorough interrogation of experimental data shows that (i) is not the main cause of screech cessation here, (ii) plays an insignificant role, and (iii) appears to have been largely responsible for screech cessation. Cessation occurs because feedback to the jet lip is diminished due to excessive expansion of the jet boundary. Further, since the jet lip now reflects and scatters low intensity sound, the end result is poor receptivity at the initial shear layer. This theory is substantiated by the re-activation of screech when the nozzle lip thickness is made larger than the expanded jet boundary. Finally, increasing lip thickness is seen to produce a systematic shift (to higher Mj) of the onset of cessation. The results of this study are of direct relevance to the sonic fatigue problem in aircraft structures, because understanding screech helps prevent such damage.

Laboratory experiments were conducted to study the interaction between two downward propagating internal wave rays with identical properties but opposite horizontal phase velocities. The intersection of the rays produced a velocity field with stagnation points, and these points propagated vertically upwards within the intersection region. Nonlinear non-resonant interactions between the two rays produced evanescent modes, with frequencies greater than the ambient buoyancy frequency, trapped within the intersection region. These evanescent modes provided a mechanism whereby energy could accumulate locally and, even though the vertical wavelength of the primary resultant wave remained the same, the local isopycnal displacements increased in time. Eventually, the isopycnals were forced to overturn in the region just above the stagnation points by the variation with depth in the local horizontal strain rate.

The gravitationally unstable overturning ultimately broke down releasing its available potential energy and generating turbulence within the intersection region. The results showed that the release of available potential energy was disrupted by the wave motions and even the dissipative scales were directly affected by the ambient stratification and the background wave motion. The distribution of the centred displacement scales was highly skewed towards the Kolmogorov scale and the turbulent Reynolds number Ret was low. Thus, the net buoyancy flux was very small and almost all turbulent kinetic energy was dissipated over the parameter range investigated. The results also showed that for such dissipative events the square of the strain Froude number (ε/νN20) and the turbulent Reynolds number Ret can be less than one.

Landau & Lifshitz showed that Kolmogorov's E∼t−10/7 law for the decay of isotropic turbulence rests on just two physical ideas: (a) the conservation of angular momentum, as expressed by Loitsyansky's integral; and (b) the removal of energy from the large scales via the energy cascade. Both Kolmogorov's original analysis and Landau & Lifshitz's reinterpretation in terms of angular momentum are now known to be flawed. The existence of long-range velocity correlations means that Loitsyansky's integral is not an exact representation of angular momentum, nor is it strictly conserved. However, in practice the long-range velocity correlations are weak and Loitsyansky's integral is almost constant, so that the Kolmogorov/Landau model provides a surprisingly simple and robust description of the decay. In this paper we redevelop these ideas in the context of MHD turbulence. We take advantage of the fact that the angular momentum of a fluid moving in a uniform magnetic field has particularly simple properties. Specifically, the component parallel to the magnetic field is conserved while the normal components decay exponentially on a time scale of τ=ρ/σB2. We show that the counterpart of Loitsyansky's integral for MHD turbulence is ∫x2⊥Q⊥dx, where Qij is the velocity correlation. When the long-range correlations are weak this integral is conserved. This provides an estimate of the rate of decay of energy. At low values of magnetic field we recover Kolmogorov's law. At high values we find E∼t−1/2, which is a result derived earlier by Moffatt. We also show that ∫x2⊥Q∥dx decays exponentially on a time scale of τ. We interpret these results in terms of the behaviour of isolated vortices orientated normal and parallel to the magnetic field.

It has been recognized that the subgrid-scale (SGS) parameterization represents a critical component of a successful large-eddy simulation (LES). Commonly used linear SGS models produce erroneous mean velocity profiles in LES of high-Reynolds-number boundary layer flows. Although recently proposed approaches to solving this problem have resulted in significant improvements, questions about the true nature of the SGS problem in shear-driven high-Reynolds-number flows remain open.

We argue that the SGS models must capture inertial transfer effects including backscatter of energy as well as its redistribution among the normal SGS stress components. These effects are the consequence of nonlinear interactions and anisotropy. In our modelling procedure we adopt a phenomenological approach whereby the SGS stresses are related to the resolved velocity gradients. We show that since the SGS stress tensor is not frame indifferent a more general nonlinear model can be applied to the SGS parameterization. We develop a nonlinear SGS model capable of reproducing the effects of SGS anisotropy characteristic for shear-driven boundary layers. The results obtained using the nonlinear model for the LES of a neutral shear-driven atmospheric boundary layer show a significant improvement in prediction of the non-dimensional shear and low-order statistics compared to the linear Smagorinsky-type models. These results also demonstrate a profound effect of the SGS model on the flow structures.

This paper treats the two-dimensional steady flow of a viscous incompressible fluid driven through a channel bounded by two walls which are the radii of a sector and two arcs (the ‘inlet’ and ‘outlet’), with the same centre as the sector, at which inflow and outflow conditions are imposed. The computed flows are related to both a laboratory experiment and recent calculations of the linearized ‘spatial’ modes of Jeffery–Hamel flows. The computations, at a few values of the angle between the walls of the sector and several values of the Reynolds number, show how the first bifurcation of the flow in a channel is related to spatial instability. They also show how the end effects due to conditions at the inlet and outlet of the channel are related to the spatial modes: in particular, Saint-Venant's principle breaks down when the flow is spatially unstable, there being a temporally stable steady flow for which small changes at the inlet or outlet create substantial effects all along the channel. The choice of a sector as the shape of the channel is to permit the exploitation of knowledge of the spatial modes of Jeffery–Hamel flows, although we regard the sector as an example of channels with walls of moderate curvature.

This paper deals with the self-induced translation of intense vortices on the β-plane in the framework of the multi-layer quasi-geostrophic approximation. An analytical theory is presented and compared to numerical experiments. To predict the vortex trajectories, we consider initially monopolar vortices, with a core of piecewise-constant potential vorticity, and calculate the evolution of the dipolar circulation which advects the vortex core. This multi-layer model yields analytical solutions for a period while the Rossby wave radiation is small.

The development of the dipolar circulation and corresponding vortex translation are described as the results of three effects. The first and second are similar to what was found in earlier studies with a one-layer model: advection of the planetary vorticity by the symmetric vortex circulation, and horizonal deformations of the vortex core. In addition, when stratification is taken into account, the vertical tilting of the vortex core also plays a role. This third effect is here represented by the relative displacement of potential vorticity contours in different layers.

Examples are given for one-, two- and three-layer models and compared with numerical simulations. It is found that the analytical predictions are good for several Rossby wave periods.

This paper is concerned with the motion of small gas bubbles, equivalent diameter about 1.0 mm, in isotropic turbulent flows. Data on the mean velocity of rise and the dispersion of the bubbles have been obtained numerically by simulating the turbulence as a sum of Fourier modes with random phases and amplitudes determined by the Kraichnan and the von Kármán–Pao energy-spectrum functions, and by calculating the bubble trajectories from a reasonably well-established equation of motion. The data cover the range β[les ]1, where β is the ratio between the turbulence intensity and the velocity of rise of the bubbles in still fluid. An approximate analysis based on the assumption that β is small yields results that compare favourably with the numerical data, and clarifies the important role played by the lift forces exerted by the fluid.

The receptivity to fundamental and subharmonic secondary instabilities is analysed for two-dimensional boundary layers. Fundamental modes are excited by the direct scattering of Tollmien–Schlichting (TS) waves over surface variations. The excitation of subharmonic modes stems from the combined scattering of acoustic free-stream disturbances and TS waves over surface variations. The surface variations are localized in their streamwise extent and are the result of roughness or suction. The velocity field is expanded in terms of small parameters characterizing the acoustic disturbance and the surface variation. The TS wave is included as part of the base flow leading to a non-homogeneous system with periodic coefficients governing the receptivity. The receptivity amplitudes show a strong dependence on the TS-wave amplitude, and for subharmonic modes a strong dependence on the TS-wave phase at the location of the surface variation. The receptivity analysis shows a significant bias toward fundamental modes of secondary instability for larger TS-wave amplitudes – except for conditions of extremely high free-stream sound level. A combination of receptivity results and stability results suggests a bias toward subharmonic modes for TS-wave amplitudes below 0.5% and toward fundamental modes for TS-wave amplitudes above 0.5% (normalized by the local edge velocity).

The steady flow in rectangular cavities is investigated both numerically and experimentally. The flow is driven by moving two facing walls tangentially in opposite directions. It is found that the basic two-dimensional flow is not always unique. For low Reynolds numbers it consists of two separate co-rotating vortices adjacent to the moving walls. If the difference in the sidewall Reynolds numbers is large this flow becomes unstable to a stationary three-dimensional mode with a long wavelength. When the aspect ratio is larger than two and both Reynolds numbers are large, but comparable in magnitude, a second two-dimensional flow exists. It takes the form of a single vortex occupying the whole cavity. This flow is the preferred state in the present experiment. It becomes unstable to a three-dimensional mode that subdivides the basic streched vortex flow into rectangular convective cells. The instability is supercritical when both sidewall Reynolds numbers are the same. When they differ the instability is subcritical. From an energy analysis and from the salient features of the three-dimensional flow it is concluded that the mechanism of destabilization is identical to the destabilization mechanism operative in the elliptical instability of highly strained vortices.

We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary–gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.

A nonlinear evolution equation that describes the propagation of a premixed flame in a closed tube has been derived from the general conservation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flame front. The two important parameters in this equation are the tube's aspect ratio and the Markstein parameter. A linear stability analysis indicates that when the Markstein parameter α is above a critical value αc the planar flame is the stable equilibrium solution. For α below αc the planar flame is no longer stable and there is a band of growing modes. Numerical solutions of the full nonlinear equation confirm this conclusion. Starting with random initial conditions the results indicate that, after a short transient, a at flame develops when α>αc and it remains flat until it reaches the end of the tube. When α<αc, on the other hand, stable curved flames may develop down the tube. Depending on the initial conditions the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip shape characterized by a sharp indentation at the centre of the tube pointing toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evolves from an ignition source, a tulip flame evolves downstream. In accord with experimental observations the tulip shape forms only after the flame has travelled a certain distance down the tube, it does not form in short tubes and its formation depends on the mixture composition. While the initial deformation of the flame front is a direct result of the hydrodynamic instability, the actual formation of the tulip flame results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.

The yield conditions for the displacement of fluid droplets from solid boundaries are studied through a series of numerical computations. The study includes gravitational and interfacial forces, but is restricted to two-dimensional droplets and low-Reynolds-number flow. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, Bond number Bd, capillary number Ca and contact angles θA and θR. The yield conditions for drop displacement are calculated and the critical shear rates are presented as functions Ca(λ, Bd, θA, Δθ) where Δθ=θA−θR is the contact angle hysteresis. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surface profiles. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While excellent agreement is found in the joint asymptotic limits Δθ[Lt ]θA[Lt ]1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. Gravitational forces normal to the solid boundary have a significant effect on the displacement process, reducing the critical shear rate for viscous drops and increasing the rate for inviscid droplets. The low-viscosity limit λ→0 is shown to be a singular limit in the lubrication theory, and the proper scaling for Ca at small λ is identified.

The concentration distribution of massive dilute species (e.g. aerosols, heavy vapours, etc.) carried in a gas stream in non-isothermal boundary layers is studied in the large-Schmidt-number limit, Sc[Gt ]1, including the cross-mass-transport by thermal diffusion (Ludwig–Soret effect). In self-similar laminar boundary layers, the mass fraction distribution of the dilute species is governed by a second-order ordinary differential equation whose solution becomes a singular perturbation problem when Sc[Gt ]1. Depending on the sign of the temperature gradient, the solutions exhibit different qualitative behaviour. First, when the thermal diffusion transport is directed toward the wall, the boundary layer can be divided into two separated regions: an outer region characterized by the cooperation of advection and thermal diffusion and an inner region in the vicinity of the wall, where Brownian diffusion accommodates the mass fraction to the value required by the boundary condition at the wall. Secondly, when the thermal diffusion transport is directed away from the wall, thus competing with the advective transport, both effects balance each other at some intermediate value of the similarity variable and a thin intermediate diffusive layer separating two outer regions should be considered around this location. The character of the outer solutions changes sharply across this thin layer, which corresponds to a second-order regular turning point of the differential mass transport equation. In the outer zone from the inner layer down to the wall, exponentially small terms must be considered to account for the diffusive leakage of the massive species. In the inner zone, the equation is solved in terms of the Whittaker function and the whole mass fraction distribution is determined by matching with the outer solutions. The distinguished limit of Brownian diffusion with a weak thermal diffusion is also analysed and shown to match the two cases mentioned above.