In this paper, weakly nonlinear interactions in a strongly-stratified, inviscid flow are re-examined, taking into account the presence of both internal waves and vortical modes. We use a multiple scale formulation, based on the two characteristic times of the problem. Ertel's potential vorticity motivates a splitting of the velocity into propagating (wave) and non-propagating (vortical) contributions. We focus on the three fundamental interactions: the wave/wave, wave/vortex and vortex/vortex interactions. The oft-studied wave/wave interaction illustrates the difference between potential and vertical vorticities. We then identify two additional resonances for the wave/vortex and vortex/vortex interactions respectively. The wave/vortex resonance provides a mechanism for redistributing energy in spectral space while the vortex/vortex interaction may give rise to an internal wave field.

We consider the waves generated by transcritical flow past a constriction in a channel, or by ships or surface pressure distributions travelling at transcritical speeds. The two-dimensionality of the upstream advancing nonlinear waves, which has been observed both experimentally and numerically by several authors, is described by a modal decomposition of the flow response. We show that the lowest transverse mode may evolve nonlinearly, leading to a two-dimensional response upstream, with the higher transverse modes swept downstream. This description is supported by comparing the initial evolution of the solutions to the corresponding linear and nonlinear problems. Averaging across the channel demonstrates that the three-dimensional problem may be related to the corresponding two-dimensional problem with an additional effective forcing coming from the nonlinear coupling of the higher modes to the lowest two-dimensional mode. This coupling leads to a dependence of the upstream solutions on the channel width as well as the Froude number. Solutions are also obtained for two-layer fluids in which cubic nonlinearity is also important. The inclusion of cubic nonlinearity permits the generation of two-dimensional fronts upstream, and demonstrates that the transition from three- to two-dimensional solutions upstream is not specific to Boussinesq solitary waves.

Low-frequency combustion instabilities are studied in a model ramjet combustor facility. The facility is two-dimensional, and is comprised of a long inlet duct, a dump combustor cavity with variable size capability, and an exhaust nozzle. The flame is observed to be unstable over a wide range of operating conditions. Acoustic pressure and velocity measurements are made at various locations in the system. They show that the inlet duct acts as a long-wavelength acoustic resonator. However, the instability frequency does not lock to any particular value. This result suggests that the instability mechanism is not purely acoustic in nature. Schlieren imaging reveals that the instability is associated with large-scale flame-front motions which are driven by periodic vortex shedding at the instability frequency. Vortices are generated at the dump in phase with the acoustic velocity fluctuations in the inlet duct. The unsteady heat addition process closely follows the vortex history: the vortices form, grow in size, convect through the combustor cavity, impinge on the exhaust nozzle, break down to small scales and burn. C2 and CH radical spectroscopy is used to determine the phase relation between heat release and pressure in the reaction zone. Rayleigh's criterion is thereby shown to be satisfied. Next, the crucial question of how the oscillation frequency is determined is addressed. Inlet velocity and combustor length are systematically varied to assess the role of vortices by modification of their characteristic lifetime. The influence of the acoustic feedback time is also studied by shortening the inlet duct. The results show that the instability frequency is controlled by both vortex kinetics in the combustor and acoustic response of the inlet section. Therefore, the instability may be considered as a mixed acoustic-convective mode. Finally, combining Rayleigh's criterion with a global feedback loop equation, it is found that the resonant frequencies are selected according to the restriction \[ \frac{1}{4N-1} < \frac{\tau_{\rm v}}{\tau_{\rm f}} < \frac{3}{4N-3}, \] where N is the mode of oscillation and τv is the time for vortices to be convected from inlet to exhaust with τf being the feedback time taken for a pressure disturbance to travel up the inlet system and back.

The spectral energy dissipation of finite-depth ocean waves, due to friction in the turbulent bottom boundary layer, is investigated using a formal parameterization of the turbulent stress. This formal parameterization is a generalization from both the eddy-viscosity model and the drag law. The eddy-viscosity model is linear in the random wave phase, whereas the drag law is nonlinear. The phase dependency of the stress is found to determine the form of the dissipation expression. A spectral eddy-viscosity model developed by the author, an eddy-viscosity model based on an ‘equivalent’ monochromatic wave given by Madsen et al. (1989), the drag law as applied by Hasselmann & Collins (1968) and an approximation to the Hasselmann & Collins expression given by Collins (1972) are discussed within the framework of the formal parameterization. Some examples of applications are given.

The unsteady boundary-layer flow produced by a two-dimensional vortex in motion above an infinite plane wall in an otherwise stagnant fluid is considered in the limit of infinite Reynolds number. This study is part of a continuing investigation into the nature of the physical processes that occur near the surface in transitional and fully turbulent boundary layers. The adverse pressure gradient due to the vortex leads to the development of a zone of recirculation in the viscous flow near the surface, and the boundary-layer flow then focuses rapidly toward an eruption along a band which is very narrow in the stream wise direction. The evolution of the unsteady boundary layer is posed in Lagrangian coordinates and computed using an efficient, factored ADI numerical method. The boundary-layer solution is found to develop a separation singularity and to evolve toward a terminal stage which is generic in two-dimensional unsteady flows. The computed results are compared with the results of asymptotic theory of two-dimensional boundary-layer separation and the agreement is found to be excellent.

The unsteady boundary layer induced by the motion of a rectilinear vortex above an infinite plane wall is calculated using interacting boundary-layer methods. The boundary-layer solution is computed in Lagrangian variables since it is possible to compute the flow evolution accurately in this formulation even when an eruption starts to evolve. Results are obtained over a range of Reynolds numbers, Re. For the limit problem Re → ∞ (studied in Part 1), a singularity develops in the non-interacting boundary-layer solution at finite time. The present results show that the interacting boundary-layer calculations also terminate in a singularity at a time which is earlier than in the limit problem and which decreases with decreasing Reynolds number. The computed results are compared with the length– and timescales predicted by recent asymptotic theories and are found to be in excellent agreement.

When gas bubbles rise in a surface-contaminated liquid, the upper parts of their surfaces may be almost free of contaminant and shear stress, while on the remainder of their surfaces there is enough contaminant to prevent tangential motion. Sadhal & Johnson solved the problem of Stokes flow of a uniform stream past a single spherical bubble. We extend their method to a single bubble in an arbitrary axially symmetric Stokes flow with the aid of an inversion theorem due to Harper. We also investigate in detail the interaction between two bubbles rising in line, and show how the methods can be made to deal with three or more bubbles, or a line of one or more bubbles rising towards a free surface. We show that a pair of bubbles will remain the same distance apart only if there is a certain relationship between the sizes of the caps on the bubbles. The cap sizes will normally be determined by convective diffusion of a surface-active solute from the bulk liquid in which the bubbles rise. The position of the lower bubble vertically under the upper one is stable to small horizontal displacements, but the upper bubble rises faster and so the distance apart gradually increases.

We investigate the stretching and breakup of a drop freely suspended in a viscous fluid undergoing chaotic advection. Droplets stretch into filaments acted on by a complex flow history leading to exponential length increase, folding, and eventual breakup; following breakup, chaotic stirring disperses the fragments throughout the flow. These events are studied by experiments conducted in a time-periodic two-dimensional low-Reynolds-number chaotic flow. Studies are restricted to viscosity ratios p such that 0.01 < p < 2.8.

The experimental results are highly reproducible and illustrate new qualitative aspects with respect to the case of stretching and breakup in linear flows. For example, breakup near folds is associated with a change of sign in stretching rate; this mode of breakup leads to the formation of rather large drops. The dominant breakup mechanism, however, is capillary wave instabilities in highly stretched filaments. Other modes of breakup, such as necking and end-pinching occur as well.

We find that drops in low-viscosity-ratio systems, p < 1, extend relatively little, O (101−102), before they break, resulting in the formation of large droplets that may or may not break again; droplets in systems with p > 1, on the other hand, stretch substantially, O (102–104), before they break, producing very small fragments that rarely break again. This results in a more non-uniform equilibrium drop size distribution than in the case of low-viscosity-ratio systems where there is a succession of breakup events. We find as well that the mean drop size decreases as the viscosity ratio increases.

The experimental results are interpreted in terms of a simple model assuming that moderately extended filaments behave passively; this is an excellent approximation especially for low-viscosity-ratio drops. The repetitive nature of stretching and folding, as well as of the breakup process itself, suggests self-similarity. We find that, indeed, upon scaling, the drop size distributions corresponding to different viscosity ratios can be collapsed into a master curve.

The problem of determining the acoustic properties of dilute bubbly liquids is examined using the method of ensemble-averaged equations and pairwise interactions. The phase speed and attenuation of sound waves in the small-amplitude regime are determined as a function of frequency of sound waves including the effects of finite surface tension, small viscosity of the liquid, and non-adiabatic thermal changes, and compared with the experimental data available in the literature. An excellent agreement is found for frequencies smaller than about 1.3 times the natural frequency of the bubbles, but the discrepancy is substantial at larger frequencies.

This paper presents expressions for the harmonic components of the near-field acoustic pressure of a B-bladed unswept single-rotation propeller. These are derived using asymptotic approximations to the standard radiation integrals for steady loading and thickness noise, under the assumption that B is large. The dependence of the pressure on blade operating conditions (both supersonic and subsonic) is described by simple formulae, which provide significant insights into the mechanisms of sound generation by rotating bodies. For supersonic motion, the importance of sources satisfying the Ffowcs Williams & Hawkings sonic condition is demonstrated, whilst for subsonic blades the near-field noise is proved to be tip-dominated. Expressions for the noise (valid from close to the tips right out to infinity) are given in both cases, requiring matching across an Airy function smoothing region when the tips move subsonically. Excellent agreement between the asymptotic formulae and both full numerical evaluations (with a considerable saving in CPU time) and experimental data is achieved.

It is well known that turbulence production in turbulent boundary layers occurs in short, energetic bursts; however, the relationship between the results of pointwise burst detection techniques and the spatial flow structure associated with such burst-type events has not been clearly established. To address this point, a study using VITA detection of burst-type events was done which allows the direct comparison between flow visualization results and quantitative, temporal velocity profile data for a flat-plate turbulent boundary layer. Using automated image processing of hydrogen-bubble flow visualization pictures, temporal velocity profile data are established using a corrected time-of-flight technique for velocity extraction. Using the visualization-derived data, spatial-temporal velocity-derived properties (∂u/∂y, ∂u/∂t, etc.), as well as probe-type burst detection properties are established. In addition, temporal and ensemble-averaged burst-type characteristics are shown to be essentially identical to previous VITA-detected velocity probe results. The VITA approach is extended to establish the spatial extent of burst-type events and the ensemble-averaged spatial-temporal properties associated with VITA-based detection. By use of a regionalized detection procedure, the types of burst-type patterns are categorized and compared with the associated visualization sequences.

The three-dimensional waving plate theory is developed to investigate the swimming performance of fish undulatory motion. In particular, the propulsive effectiveness is discussed. The unsteady potential flow over model rectangular and triangular flexible plates performing a motion which consists of a progressing wave with variable amplitudes is calculated by the vortex ring panel method. It is found that the undulatory motion can reduce three-dimensional effects. It is this important hydrodynamic phenomenon that may be one of the main reasons why such undulation is widely used as the swimming method by a large number of aquatic animals. When the span of the undulating plate is nearly unchanged and the wave amplitude is constant or increases slightly along the chord, and the wavelength is close to the body length, theoretical results show that the swimming performance is best and the flow around the plate has a quasi-two-dimensional property. This swimming method may be observed in many fishes, especially those with an anguilliform mode of propulsion. The modification of the anguilliform mode of propulsion to the carangiform mode is also discussed. It is confirmed that the pronounced necking of the body anterior to the tail, which acts to improve the propulsive performance, is a major morphological adaptation of fishes using the carangiform mode, in which the characteristic nature of flexural movement confined to the rear part of the body is that the amplitude of undulation increases posteriorly and no complete wavelength is at any time apparent.

The inviscid instability of a longitudinal vortex structure within a steady boundary layer is investigated. The instability has wavelength comparable with the boundary-layer thickness so that a quasi-parallel approach to the instability problem can be justified. The generalization of the Rayleigh equation to such a flow is obtained and solved for the case when the vortex structure is induced by curvature. Two distinct modes of instability are found; these modes correspond with experimental observations on the breakdown process for Görtler vortices.

The changes in turbulence in a flow over a two-dimensional curved hill, described in Part 1 (Baskaran, Smits & Joubert 1987), are analysed in the light of transport equations for the turbulent kinetic energy, $\frac{1}{2}\overline{q^2}$, and the primary shear stress, $-\overline{uv}$, in order to infer the way in which the extra strain rates due to streamline curvature and the streamwise pressure gradient contribute to the changes. Interaction between the two extra strain rates is also considered. The triple correlation data presented here are consistent with the fact already established in Part 1 that the upwind boundary-layer structure bifurcates to form two distinct turbulent zones over the hill, namely, an internal boundary layer and an external free turbulent flow. The source terms in the transport equations imply that the effects of streamline curvature and streamwise pressure gradient are felt differently on $\overline{q^2}$ and $-\overline{uv}$. The present experimental results show that the shear stress is more sensitive to streamline curvature than is the turbulent kinetic energy. The anisotropy parameter, $\overline{u^2}/\overline{v^2}$, plays a major role in determining the difference in the behaviour of $\overline{q^2}$ and $-\overline{uv}$ under the influence of streamline curvature. The distribution of turbulent lengthscales follows the general formulae suggested by Bradshaw (1969) for streamline curvature of either sign. The pressure-strain redistribution term deduced from the experimental data is in good agreement with the model of Zeman & Jensen (1987) for flows over hills. The influence of streamwise pressure gradient enters through the normal stress production terms, which appears only in the transport equation for $\overline{q^2}$. The transport terms are found to be affected by streamline curvature. To the thin shear layer approximation, the interaction between streamline curvature and streamwise pressure gradient appears to be weak.

Numerical experiments have been performed on normal shock waves with Monte Carlo Direct Simulations (MCDS's) to investigate the validity of continuum theories at very low Mach numbers. Results from the Navier—Stokes and the Burnett equations are compared to MCDS's for both hard-sphere and Maxwell gases. It is found that the maximum-slope shock thicknesses are described equally well (within the MCDS computational scatter) by either of the continuum formulations for Mach numbers smaller than about 1.2. For Mach numbers greater than 1.2, the Burnett predictions are more accurate than the Navier—Stokes results. Temperature—density profile separations are best described by the Burnett equations for Mach numbers greater than about 1.3. At lower Mach numbers the MCDS scatter is too great to differentiate between the two continuum theories. For all Mach numbers above one, the shock shapes are more accurately described by the Burnett equations.

The effect of small-amplitude, time-periodic, free-stream disturbances on an otherwise steady axisymmetric boundary layer on a circular cylinder is considered. Numerical solutions to the problem are presented, and an asymptotic solution to the flow, valid far downstream along the axis of the cylinder is detailed. Particular emphasis is placed on the unsteady eigensolutions that occur far downstream, which turn out to be very different from the analogous planar eigensolutions. These axisymmetric eigensolutions are computed numerically and also are described by asymptotic analyses valid for low and high frequencies of oscillation.

We present a theory for the decay of the relative motion of a homogenous fluid with a free surface in a rotating cylindrical tank with a flat bottom, induced by an abrupt change in the angular velocity. We then describe a set of laboratory experiments designed to test the predictions of the theory.

At low rates of rotation the dynamics of the adjustment is well understood and measurements have verified the established theoretical results that the motion decays exponentially, with a timescale proportional to the rotation period divided by the square root of the Ekman number, and that the relative vorticity remains independent of radius. At higher rotation rates, however, the curvature and motion of the free surface complicate the dynamics, and have hindered the development of a more general theory.

Both the theoretical predictions and the experiments show that at high rotation rates the decay of the relative vorticity is independent of radius and exponential in time, but with a decay timescale, τe, that increases linearly with the rotational Froude number F, i.e. $\tau_{\rm e} = 1+\frac{1}{16} F$. An analysis of the vorticity dynamics during spin-up indicates that, near the centre of the tank, this simple behaviour is the result of vigorous competition between the rate of vortex line stretching by Ekman-layer pumping and surface deformation. Near the boundary, these mechanisms cooperate, but are partially offset by the stretching produced by the secondary radial circulation.

Chemical reactions in shock waves can be strongly affected by minute impurity concentrations. Thus it is not adequate to take into account the additional impurity electron production in relaxation studies simply by global adjustment of the atom—atom excitation cross-section constant to the measured electron density.

A definite improvement, however, can only be achieved if the ionization relaxation model is extended to include all relevant impurity atom reactions. Consequently we treated the real test gas as a mixture of krypton and impurity carbon atoms. For the carbon model it is important to take the lower real excitation levels into consideration. Carrying out a sensitivity analysis we were able to reduce the number of reactions substantially. A comparison with experimental electron density profiles yielded 3.0 × 10−6 m2/J for the Kr—Kr excitation cross-section constant as well as values for the C—Kr constants.

For a temperature of about 8000 K and an impurity concentration of about 40 p.p.m. it is shown that the impurity reactions dominate the electron production in the initial relaxation zone. This effect causes a pronounced decrease of the relaxation time with increasing concentration.

By comparing computational results of the Kr–C model with those of the simplistic pure Kr model it is possible to explain the dependence of the Kr–Kr excitation cross-section constant on the impurity concentration and the plasma temperature.

A decomposition of turbulent velocity fields into modes that exhibit both localization in wavenumber and physical space is performed. We review some basic properties of such a decomposition, the wavelet transform. The wavelet-transformed Navier—Stokes equations are derived, and we define new quantities such as e(r, x), t(r, x) and π(r, x) which are the kinetic energy, the transfer of kinetic energy and the flux of kinetic energy through scale r at position x. The discrete version of e(r, x) is computed from laboratory one-dimensional velocity signals in a boundary layer and in a turbulent wake behind a circular cylinder. We also compute e(r, x), t(r, x) and π(r, x) from three-dimensional velocity fields obtained from direct numerical simulations. Our findings are that the localized kinetic energies become very intermittent in x at small scales and exhibit multifractal scaling. The transfer and flux of kinetic energy are found to fluctuate greatly in physical space for scales between the energy containing scale and the dissipative scale. These fluctuations have mean values that agree with their traditional counterparts in Fourier space, but have standard deviations that are much larger than their mean values. In space (at each scale r), we find exponential tails for the probability density functions of these quantities. We then study the nonlinear advection terms in more detail and define the transfer T(r|r′, x) between scale r and all scales smaller than r′, at location x. Then we define πsg(r, x), the flux of energy caused only by the scales smaller than r, at x, and find negative values for πsg(r, x), at almost 50% of the physical space at every scale (backscatter). We propose the inclusion of local backscatter in the phenomenological cascade models of intermittency, by allowing some energy to flow from small to large scales in the context of a multiplicative process in the inertial range.

The stability and bifurcations associated with the loss of azimuthal symmetry of planar flows of a viscous incompressible fluid, such as vortex-source and Jeffery–Hamel flows, are studied by employing linear, weakly nonlinear and fully nonlinear analyses, and features of new solutions are explained. We address here steady self-similar solutions of the Navier–Stokes equations and their stability to spatially developing disturbances. By considering bifurcations of a potential vortex-source flow, we find secondary solutions. They include asymmetric vortices which are generalizations of the classical point vortex to vortical flows with non-axisymmetric vorticity distributions. Another class of solutions we report relates to transition trajectories that connect new bifurcation-produced solutions with the primary ones. Such solutions provide far-field asymptotes for a number of jet-like flows. In particular, we consider a flow which is a combination of a jet and a sink, a tripolar jet, a jet emerging from a slit in a plane wall, a jet emerging from a plane channel and the reattachment phenomenon in the Jeffery–Hamel flow in divergent channels.