Damping mechanisms, damping rates and the dissipative modal structure of internal waves in stratified rotating circular basins are investigated analytically. The damping is shown to be due to a combination of the internal-wave cancelling, where waves emitted by the oscillatory boundary layers destructively interact with the parent wave and drain energy from it, and spin-down modified by the periodicity, where the energy is drained by the sinks and sources at the bottom corner caused by a discontinuity in the Ekman transport. It is shown that super-inertial Poincaré waves and sub-inertial Kelvin waves are damped predominantly by the internal-wave cancelling and modified spin-down, respectively. These processes also modify the internal-wave structure; for super-inertial waves, the boundary-layer-generated waves intensify the interior flow in the lower part of the water column and delay the phase relative to the isopycnal displacements, but for sub-inertial waves, the Ekman pumping and the corner sinks and sources add a horizontal circular flow that slants the crest and trough backwards near the wall.

Packets of hairpin-shaped vortices and streamwise counter-rotating vortex pairs (CVPs) appear to be key structures during the late stages of the transition process as well as in low-Reynolds-number turbulence in wall-bounded flows. In this work we propose a robust model consisting of minimal flow elements that can produce packets of hairpins. Its three components are: simple shear, a CVP having finite streamwise vorticity magnitude and a two-dimensional (2D) wavy (in the streamwise direction) spanwise vortex sheet. This combination is inherently unstable: the CVP modifies the base flow due to the induced velocity forming an inflection point in the base-flow velocity profile. Consequently, the 2D wavy vortex sheet is amplified, causing undulation of the CVP. The undulation is further enhanced as the wave continues to be amplified and eventually the CVP breaks down into several segments. The induced velocity generates highly localized patches of spanwise vorticity above the regions connecting two consecutive streamwise elements of the CVP. These patches widen with time and join with the streamwise vortical elements situated beneath them forming a packet of hairpins. The results of the unbounded (having no walls) model are compared with pipe and channel flow experiments and with a direct numerical simulation of a transition process in Couette flow. The good agreement in all cases demonstrates the universality and robustness of the model.

The nonlinear evolution and breakdown of laminar flow in the boundary layer on a flat plate is examined with the aim of making a closer comparison of theory and experiment than has been attempted previously. The importance of three-dimensionality is emphasized. It is concluded that many features of the nonlinear instability are consistent with existing linear and weakly nonlinear theories even as breakdown is approached. The development of the secondary instability, or ‘spike’, is also considered and suggestions for an improved theory of its growth are made.

An analytical model is presented for the configuration of Mach reflection (MR) due to the interaction of two-dimensional steady supersonic flow over asymmetric wedges. The present asymmetric MR model is an extension of an earlier model for the symmetric MR configuration. The overall Mach reflection (oMR) in the asymmetric wedge configuration is analysed as a combination of upper and lower half-domains of symmetric reflection configurations. Suitable assumptions are made to close the combined set of equations. The subsonic pocket downstream of the Mach stem is taken to be oriented along an average inclination, based on the streamline deflections by the Mach stem at the triple points. This assumption is found to give satisfactory results for all types of oMR configurations. The oMR configuration is predicted for all types of reflections such as direct Mach reflection (DiMR), stationary Mach reflection (StMR) and inverse Mach reflection (InMR). The reflection configuration and Mach stem shape given by the model for various sets of wedge angles, especially those giving rise to InMR, have been predicted and validated with the available numerical and experimental data. The von Neumann criterion for oMR is accurately predicted by this model. The asymmetric Mach stem profile is well captured. The variation of Mach stem height with wedge angle is also analysed and it is found that simplification of an asymmetric MR to a symmetric MR leads to over-prediction of the Mach stem height and hence the extent of the subsonic region.

In this paper we consider the stability of the flow produced by an infinite rotating disk. A large-Reynolds-number asymptotic theory is developed to obtain the non-parallel correction to the local absolute instability (AI) found for this flow by Lingwood (1995), who used the parallel-flow approximation. Our asymptotic theory is based on the inviscid AI underlying the viscous AI and so is expected to give the non-parallel correction to the upper branch of Lingwood's neutral curve for the AI. It is found that non-parallel terms have a destabilizing effect on the AI. Also, it is shown that, although the position of the neutral curve for convective instability is known to depend on choice of measurement quantity, for AI it does not. However, in relating the asymptotic non-parallel results to the numerical parallel results at large Reynolds numbers, it is found that Lingwood's viscous AI does not, after all, asymptote towards the inviscid results. Instead, Lingwood's family of branch points is distinct from a second family of branch points that do asymptote towards the inviscid limit. We show that these two families of branch points are related by a ‘super branch point’ at which three spatial branches connect simultaneously. Lingwood's branch points, in fact, have a viscous long-wave origin, and will therefore be subjected to non-parallel effects that are some power of the Reynolds number larger than if they had been of inviscid origin.

Statistical properties of multidimensional Burgers' turbulence evolving in the presence of a force field with random potential, which is delta-correlated in time and smooth in space, are studied in the inviscid limit and at the physical level of rigorousness. The solution algorithm reduces to finding multistream fields describing the motion of an auxiliary gas of interacting particles in a force field. Consequently, the statistical description of forced Burgers' turbulence is obtained by finding the largest possible value of the least action for the auxiliary gas. The exponential growth of the number of streams is found to be a necessary condition for the existence of stationary regimes.

In a conventional hydrostatic thrust bearing, the lubricant is supplied from the centre and flows radially outwards. It has been found that the load capacity of such a bearing decreases with increasing angular speed of the rotor. The bearing fails when a critical rotor speed is reached at which the load capacity becomes zero. In this paper a modified feeding system is suggested in which the lubricant is supplied from a ring-shaped groove situated between the exits at the centre and the edge of the bearing. Analysis shows that, by reversing the flow direction near the centre, the load capacity of a bearing of proper geometry can be made to decrease at a much slower rate than that of a conventional bearing as the rotor speed increases. The non-central feeding system might be considered for use in high-speed thrust bearings.

A theoretical analysis of the distortion of unsteady three-dimensional disturbances in a Hiemenz boundary layer and its effect on the heat transfer enhancement is presented. It is shown that the disturbance length scale is a critical parameter in determining the amplification ratio of the incoming vorticity. For large disturbance length scales, the amplification ratio increases when the length scale decreases, and a maximum value occurs at a length scale close to five times the boundary-layer thickness. The unsteadiness of the disturbances is found to reduce the vorticity amplification, but the effect is second order when the frequency is low compared to the mean flow strain rate. The impinging disturbances induce large-amplitude vorticity of opposite sign at the wall whose magnitude controls the heat transfer enhancement. As an application of the present analysis, a new scaling correlation is derived for stagnation-point heat transfer in the presence of free-stream turbulence. The theoretical correlation, expressed in terms of turbulence intensity, integral length scale and mean flow Reynolds number, agrees reasonably well with recent experimental data.

Nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube of variable cross-section is studied for parameter values where the steady core–annular flow (CAF) is linearly unstable. The simulations are based on a pseudo-spectral numerical method. They are carried out assuming axial symmetry, that the total flow rate remains constant and that all dependent variables are periodic in the axial direction, which includes the minimum necessary number of repeated units so that the obtained solution is independent of this number. The time integration originates with the numerically computed steady CAF or the steady CAF seeded with either the most unstable mode or random small disturbances. Only a limited number of the most interesting cases are presented. For the most part, the values of the majority of the dimensionless parameters are such that oil flows in the centre of the tube driven by an applied pressure gradient against gravity, whereas water is flowing in the annulus. It is shown that, whereas the steady (unstable) solution may indicate that the heavier water flows countercurrently with respect to the oil, the time periodic (observable) solution may indicate the same, albeit at a much smaller core flow rate or that concurrent flow occurs. This is due to the water being trapped between the large-amplitude interfacial waves that are generated and being convected by the oil. It is also shown that increasing the inverse Weber number increases the wave amplitude to the point that the flow of the core fluid may become discontinuous with a mechanism that depends on the viscosity ratio between the two fluids. Increasing the amplitude of the sinusoidal variation of the tube leads to a combination of travelling and standing waves, which interact to produce a time periodic solution with a long period associated with the time it takes the travelling wave to travel through the computational domain and a second much shorter period that is related to their interaction time. Qualitative agreement has been obtained upon comparing our numerical simulations with limited experimental reports, even though the experimental conditions were not identical to those in our model.

Existence of edge waves travelling along three-dimensional periodic structures is considered within the linear water-wave theory. A condition ensuring the existence is derived by analysing the spectrum of a suitably defined trace operator. The sufficient condition is a simple inequality comparing a weighted volume integral, taken over the submerged part of an element in the infinite array of identical obstacles, to the area of the free surface pierced by the obstacle. Various examples are given, and the results are extended to edge waves along periodic coastlines and over a periodically varying ocean floor.

The analytical characteristic method is an effective method for computing non-linear effects in inviscid supersonic flow problems. Although only linear equations have to be solved, the results are essentially nonlinear, in the sense that the functional relations between physical state variables and space co-ordinates are nonlinear in the small perturbation parameter introduced, like the thickness ratio or incidence of a wing. This holds even for the first-order approximation of the method.

In the case of two-dimensional (plane or axisymmetric) flow the independent variables are characteristic co-ordinates, i.e. they are chosen so as to be constant along corresponding characteristic lines. The space co-ordinates are considered as dependent variables. In three dimensions there is no unique definition of a characteristic co-ordinate system, because the manifold of characteristic surfaces or bi-characteristics is larger than is necessary for defining a co-ordinate system. The success of a three-dimensional analytical characteristic method, however, depends on the proper choice of the co-ordinate system.

The present analytical Characteristic method for three-dimensional flow is based on the fact that three-dimensional flow behaves locally like axisymmetric flow if it is considered in the osculating plane. The corresponding ‘distance from the axis’ is a function of space depending on the flow field. No change of pressure occurs normal to the osculating plane and in isentropic flow no change of speed either. Therefore no co-ordinate perturbation is performed in this normal direction. In the osculating plane the analytical characteristic methodis applied locally as in axisymmetric flow. In the large the space co-ordinates are obtained by integration along the main bi-characteristics.

As an example the flow field on the suction side of a flat delta wing with sub-sonic leading edges is computed. As a main result one obtains shock waves in the neighbourhood of the leading edges following the expansion.

Artificial acceleration and deceleration of the transition process in a two-dimensional wake were attempted. The wake was produced behind a thin aerofoil placed parallel to uniform flow. The sound from a loudspeaker introduced into the wake acted as an artificial disturbance. Various kinds of sound were tested and the effect on the transition was judged by the energy spectrum. Sinusoidal sound of the frequency of the maximum growth rate in the linear region decelerates the transition, whereas sound of a different frequency accelerates it. Sound of two or four frequencies is more effective in accelerating the transition when the frequencies are properly chosen. White noise from the loudspeaker is not effective, but a two-peak noise specially designed for strong nonlinear interaction is the most effective in accelerating the transition process. These results can be explained by two empirical properties of the nonlinear interaction: the growth suppression induced by a large amplitude fluctuation and the stronger interaction between fluctuations of closer amplitudes.

In this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number . When , the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

We report on turbulent Rayleigh–Bénard convection of a nematic liquid crystal while it undergoes a transition from the nematic to the isotropic phase in a cylindrical convection cell with a height equal to twice the diameter (aspect ratio $\Gamma = 0. 50$). The difference between the top and bottom plate temperature $ \mathrm{\Delta} T= {T}_{b} - {T}_{t} $ was held constant, while the average temperature ${T}_{m} = ({T}_{b} + {T}_{t} )/ 2$ was varied. There was a significant increase of the transported heat when the phase transition temperature ${T}_{NI} $ was between ${T}_{b} $ and ${T}_{t} $. Measurements of the temperatures along the sidewall of the sample as a function of ${T}_{m} $ showed several ranges with qualitatively different behaviour of quantities such as the time-averaged sidewall temperature, temperature gradient, or temperature fluctuations. We interpret these different ranges in terms of properties of the thermal boundary layers close to the top and bottom plates whose stability and nature depends on the location within the sample of ${T}_{NI} $.

The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit $E\ll 1$, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the $E^{1/2}$ Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder $\mathscr{C}$, is geostrophic. Stewartson identified a complicated nested layer structure on $\mathscr{C}$, which comprises relatively thick quasigeostrophic $E^{2/7}$- (inside $\mathscr{C}$) and $E^{1/4}$- (outside $\mathscr{C}$) layers. They embed a thinner ageostrophic $E^{1/3}$ shear layer (on $\mathscr{C}$), which merges with the inner sphere Ekman layer to form the $E^{2/5}$-equatorial Ekman layer of axial length $E^{1/5}$. Under appropriate scaling, this $E^{2/5}$-layer problem may be formulated, correct to leading order, independent of $E$. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate $z$) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite $z$ to account for the far-field behaviour. Adopting $z^{-1}$ as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

Simulated upwelling fronts have been generated around the outer edge of a cylindrical tank filled with a two-layer fluid system and driven by a surface stress. Initially, an axisymmetric front was observed which subsequently became unstable to small baroclinic eddies. These eddies continued to grow until they reached an equilibrium size. Under some circumstances, cyclonic eddies pinched-off from the fully developed front and moved away from the mean position of the front into the fluid interior. Streak photographs of the fully developed flow field were digitized to generate a velocity field interpolated on to a regular grid. A direct two-dimensional Fourier transform was performed on the turbulent kinetic energy field deduced from such images and one-dimensional energy E(k) spectra were extracted. Consistent $k^{-\frac{5}{3}}$ energy spectra were found at lower wavenumber, k and approximately k−5.5 spectra at higher k. In any given experiment, the two spectral slopes meet close to a wavenumber kw = 2π/λw (where λw is the mean diameter of a frontal eddy and kw is the associated wavenumber). According to classical theories, kw would be the input wavenumber, and the range of k with a $k^{-\frac{5}{3}}$ spectrum would correspond to an inverse energy cascade range; this yielded a Kolmogorov constant (C) that varied within the limits 2.8 [les ] C [les ] 3.8. The approximately k−5.5 range, which is much steeper than that predicted by the original statistical theories, is nevertheless consistent with those found frequently in numerical experiments.

The spectral slopes inferred from particle dispersion methods and from one-dimensional Fourier transforms of the longitudinal velocity correlations were compared with the results obtained above and in previous laboratory experiments. In general, the global energy spectra are consistent with an interpretation of the fluid dynamics as being that of two-dimensional turbulence. This in turn implies that known properties of such flows may be invoked to explain the appearance of a number of naturally occurring phenomena in coastal upwelling fronts.

The internal-wave system is calculated for a body oscillating transversely, and translating uniformly, through an infinite stratified fluid of constant Brunt-Väisälä frequency. A linearized, time-dependent analysis is used, in which the vertical displacement of a fluid element is the basic dependent variable. Axisym-metric slender-body theory for a homogeneous fluid is used to determine the time-dependent source and dipole distributions required to represent the motion of the body for excitation of the internal waves. The equations are solved by Fourier-transform techniques; and the internal-wave amplitude is evaluated in the far field by the method of stationary phase. The surfaces of constant phase are found to change character as the ratio of the oscillation frequency ω of the body to the Brunt-Väisälä frequency N varies through unity. Along preferred directions, the amplitude of the internal waves is found to decay inversely with distance to the $\frac{5}{6}$ power, whereas, for uniform translation, the amplitude of the internal waves falls off inversely with distance from the body. An asymptotic expression for the amplitude in preferred directions is calculated for several values of the ratio ω/N.

Hypersonic small-disturbance theory is extended to consider the problem of dusty-gas flow past thin two-dimensional bodies. The mass fraction of suspended particles is assumed to be sufficiently large that the two-way interaction between particle phase and gas phase must be considered. The system of eight governing equations is further reduced by considering the Newtonian approximation γ → 1 and M∞ → ∞. The Newtonian theory up to second order is studied and the equations are solved for the case of a thin wedge at zero angle of attack. Expressions for the streamlines, dust-particle paths, shock-wave location and all flow variables are obtained. It is seen that the presence of the dust increases the pressure along the wedge surface and tends to bend the shock wave towards the body surface. Other effects of the interaction of the two phases are also discussed.

A series of laboratory experiments was carried out to examine the interaction between stratification and turbulence at an inversion layer, with the objective of gaining insight into certain wave–turbulence encounters in the atmosphere. A three-layer stratified fluid system, consisting of a (thick) strongly stratified inversion layer, sandwiched between an upper turbulent layer and a lower non-turbulent weakly stratified layer, was employed. Oscillating-grid-induced shear-free turbulence was used in the upper layer. During the experiments, a fourth (interfacial) layer developed in the region between the inversion and the turbulent layer, and most of the wave–turbulence interactions were confined to this layer. Detailed measurements of the vertical velocity structure, internal-wave parameters and mixing characteristics were made in the stratified layers and, whenever possible, the results were compared to available theoretical predictions.