Effects of product dissociation and preferential diffusion on the structure, propagation and diffusional-thermal instability of the classical one-dimensional laminar flame have been studied by using activation energy asymptotics. Analytical expressions as functions of dissociation and diffusion parameters have been obtained for such bulk flame parameters as the flame temprature reduction and the propagation rate eigenvalue, and for the dispersion relation governing flame stability. Results on flame propagation show that while under most situations the flame speed is reduced due to product dissociation, it can attain values in excess of the non-dissociative limit for highly mobile product species which can preferentially back diffuse to the upstream portion of the reaction zone where they recombine and release the associated recombination heat. Results on flame stability show that it is promoted in the presence of product dissociation which has a moderating influence on the flame temperature fluctuations, and for highly-mobile product species because of the enhanced burning rate and curvature-induced concentration modification.

Degenerating stable temporal Orr-Sommerfeld eigenmodes are studied for plane Poiseuille flow. The discrete spectrum of the eigenmodes is shown to possess infinitely many degeneracies, each appearing at a certain combination of k (the modulus of resultant wavenumber) and αR (the streamwise wavenumber time the Reynolds number). The eigenmodes are found to degenerate in a specific manner which confines the streamwise phase velocities of the degeneracies to be around $\frac{2}{3}$ of the centreline velocity. The responses of the degeneracies are investigated through the initial-value problem. The responses of the first four symmetric and the first two antisymmetric degeneracies are evaluated numerically for arbitrary initial disturbances expanded in terms of Chebyshev polynomials. The first symmetric and the first antisymmetric degeneracies exhibit temporal growth of the amplitudes in the wavenumber space. The maximum amplitudes are at most 7 times larger than the corresponding initial amplitudes. The amplitudes of the responses of the other four degeneracies decay rapidly owing to their higher damping rates. The time for which the degeneracy-response is in the growing phase is shown to be stretched with increasing Reynolds number. The degeneracies can therefore be active for longer periods of time at larger Reynolds numbers.

The interaction between a freely rising thermal plume and an unheated vertical surface in its neighbourhood has been investigated. The underlying transport mechanisms are of interest from a fundamental standpoint, as well as in a variety of practical problems, such as the cooling of electronic equipment and room fires. A detailed numerical and experimental study of the flow is carried out. The temperature and velocity gradients are expected to be large, particularly near the thermal source. Also, any constraints imposed on the entrainment into the flow in the vicinity of the source are expected to significantly affect the nature of the flow and the interaction. These considerations make it imperative to solve the full governing equations in the interaction region. These equations are solved numerically by finite-difference methods, employing the vorticity-stream function formulation. The important physical variables in the problem are the total thermal energy input by the source, the size of the source, and the distance of the source from the vertical wall which is taken as adiabatic or isothermal in the computation. The flow is found to be strongly deflected towards the vertical surface for the parametric ranges considered. As expected, the diffusion effects in the main flow direction are found to decay downstream and the flow to gradually approach the characteristics of a wall plume resulting from a concentrated line heat source with the same total heat input. Thus, the axial diffusion terms may be neglected far downstream, allowing the flow there to be approximated as a boundary, layer, with the full equations being solved in the interaction region. Finally, an experimental investigation is carried out to characterize the nature of the interaction. The flow is visualized by means of a shadowgraph and the temperature field is measured in the interaction region, downstream of the source. Numerical predictions agree with the experimental results, lending support to the numerical model for this interaction.

In the interior of an extended liquid a cyclic cavitation process can be set up by a focused acoustic wave field. A cluster of supercritical cavities is developed when the tensile strength of the liquid is exceeded. This occurs almost simultaneously in a confined region around the focal point where the sound speed rapidly vanishes when the cavities approach their critical size. As a consequence, the inner boundary condition for the acoustic field, carried by the single-phase liquid, changes from total reflection at the focal point without phase shift, as at a rigid wall, to reflection at the boundary of the cluster, which now forms a low-pressure two-phase kernel embedded in the single-phase liquid. The cluster is compliant to expansion waves but essentially rigid to compression waves. As soon as the cluster is formed its further development is determined jointly by the sound field and by the far field pressure of the liquid. The former first makes the cavities in the cluster grow and then contributes to its collapse, while the latter tends to bring about its collapse from the moment of cluster formation.

In this paper, we study the evolution of strongly three-dimensional disturbances which are generated by a point force in a parallel mixing layer. When the input force is a pulse, a wave packet develops whose wavefronts are approximately parallel to the spanwise direction. This is in sharp contrast to a wave packet in a wall boundary layer for which the wavefronts are strongly curved. On the other hand, when the input disturbance is oscillating harmonically in time, a spatially growing instability wave develops in a downstream wedge of (x, z)-space. The size of this wedge, as a function of excitation frequency and velocity ratio, is determined. The receptivity of the shear layer to pulse-type and harmonic excitation is also studied. It is found that the shear layer is especially sensitive to relatively high-frequency forcing on its centreline.

Two-component LDV measurements are obtained over an ellipse at an angle of attack. Detailed information is provided for the stagnation region, the two separation regions and the two free shear layers. Steady and unsteady flow measurements are presented. Periodic disturbances in the oncoming stream are introduced and ensemble-averaged unsteady data are obtained. An integral picture of laminar flow over a lifting body with no sharp edges is thus presented, which is readily available for comparison with asymptotic or numerical calculations.

In this paper we use a boundary-integral technique to numerically investigate the motion of a viscous drop toward a fluid-fluid interface at low Reynolds number. We consider only the case of a drop moving toward its homophase. The solutions include large deformations of both the drop and interface for capillary numbers in the range 0.2 [les ] Ca [les ] 10 and the viscosity ratios between 0.1 [les ] λ [les ] 10, and illustrate the approach toward a film-drainage geometry for a drop which starts at a large distance from an initially undeformed, flat interface. We also consider briefly the effect of starting the drop closer to the interface.

An experimental study has been carried out to examine double-diffusive convection in a porous medium. The experiments were performed in a horizontal layer of porous medium consisting of 3 mm diameter glass beads contained in a box 24 cm × 12 cm × 4 cm high. The top and bottom walls were made of brass and were kept at different constant temperatures by separate baths, with the bottom temperature higher than that of the top. The onset of convection was detected by a heat flux sensor and by the temperature distribution in the porous medium. When the porous medium was saturated with distilled water, the onset of convection was marked by a change in slope of the heat flux curve. The temperature distribution in the longitudinal direction in the middle of the layer indicated a convection pattern consisting of two-dimensional rolls with axes parallel to the short side. This pattern was confirmed by flow visualization. When the porous medium was saturated with salinity gradients of 0.15% cm−1 and 0.225% cm−1, the onset of convection was marked by a dramatic increase in heat flux at the critical ΔT, and the convection pattern was three-dimensional. When the temperature difference was reduced from supercritical to subcritical values, the heat flux curve established a hysteresis loop. Results from linear stability theory, taking into account effects of temperature-dependent viscosity, volumetric expansion coefficients, and a nonlinear basic state salinity profile, are discussed.

The motion of a perfect gas in a closed geometry is studied when it experiences large, transient, spatially non-uniform volumetric heating caused by the passage of energetic particles or intense light through the gas. The spatial non-uniformity of the heating results from the fact that the energy deposition in the gas is characterized by a range, a lengthscale which is inversely proportional to the local gas density. The equations of motion of the gas are acoustically filtered and then specialized to a one-dimensional problem. When written in Lagrangian form, the equations are reduced to a system of ordinary differential equations. Because of the special form of the one-dimensional range-dependent volumetric heating source term, this system can be solved analytically. Limitations on the applicability of this approximate analytical solution are discussed. Numerical simulations of specific cases for which the solution is valid are in agreement with the solution.

The Benjamin & Feir sideband instability of gravity waves over a rigid bed is extended here to the case of wave propagation over a compliant bed. The bed is assumed elastic with inhomogeneous properties, following some vertical stratification profile. The elastic stiffness at the mudline is assumed very small, as is typically the case for upper-stratum gel-like marine mud, so that its effect is assumed comparable in magnitude with that of wave nonlinearity. It is then shown that through the sideband instability, the carrier gravity wave would lose energy to sideband oscillations. These sidebands will be shown to be contaminated with small-amplitude, but very short elastic shear waves. These shear waves will interact with the viscous boundary layer at the bed-water interface, and thus significantly enhance the viscous damping of wave energy in the boundary layer.

Three-dimensional numerical calculations of an inviscid gas past scoops in a gas centrifuge are performed by solving the Euler equations using the explicit Roe upwind scheme with a second-order of accuracy. The scoop is modelled as a cylindrical or a wing-shaped rod attached to a central tube and extending radially outwards, and no inlet flow to the scoop is assumed. The scoops are placed close to the bottom end plate, and there is no covering baffle plate. The numerical grid employed is of the multibox type. The main results are as follows. For a cylindrical scoop, a detached bow shock is formed in front of the scoop. Behind the shock, strong radially inward motion of gas towards the central axis is induced, and it excites an upward flow which becomes a countercurrent. The inward flow just in front of the scoop produces a vortex column in the upstream region of the scoop. For a wing-shaped scoop, an oblique shock attached to the scoop is formed, and an inward flow is induced behind the shock. The shock is not so strong as that in the case of a cylindrical scoop model. The drag coefficient of the wing-shaped scoop is almost one-fourth of that of the cylindrical scoop. The addition theorem of the scoop drag is verified for the wing-shaped scoop.

A series of experiments are performed in a Hele-Shaw cell, consisting of two parallel closely spaced glass plates. A liquid (oil or water, both of viscosity of 1.0 cP) is injected at a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine. The data presented in this paper are taken mostly at late stages of the fingering process, when the pattern has multiple generations of splitting. Correlations with time are obtained for the finger length and the overall pattern density. The time-and lengthscales have been found for the immiscible case. At the same dimensionless time, immiscible patterns are similar and have the same generation of splitting. The overall density of each pattern decreases with time. The pattern shows fractal behaviour only after a certain number of generations of splitting. The fractal dimension of the immiscible pattern decreases from 1.9 to 1.82 when the pattern goes from the third to the sixth generation of splitting. The fractal dimension of the miscible pattern reaches a constant value after about ten generations of splitting and the fractal dimension ranges from 1.50 to 1.69 for q/Db = 4.8 × 105−7.0 × 106. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers λ/b scales with Ca−0.31 for capillary number Ca ranging from about 8 × 10−4 to 0.05. For miscible fingers, λ/b is insensitive to dispersion and ranges from 5 to 10 for large q/Db. Here D is the molecular diffusion coefficient in glycerine, b the cell gap width and λ the splitting wavelength.

The two-dimensional sloshing of a fluid in a horizontal circular cylindrical container and the three-dimensional sloshing of a fluid in a spherical container are considered. The linearized theory of water waves is used to determine the frequencies of free oscillations under gravity of an arbitrary amount of fluid in such tanks. Special coordinate systems are used and the problems are formulated in terms of integral equations which are solved numerically for the eigenvalues. Detailed tables of the sloshing frequencies are presented for a range of fill-depths of the containers.

The problem of calculating nonlinear two-dimensional free-surface potential flow about a pair of counter-rotating point vortices rising under their own influence towards a free surface is solved numerically. The two vortices are inserted into fluid which is initially at rest. A boundary/integral-equation method is used to obtain free-surface elevations and streamlines about the rising pair of vortices for several vortex strengths. The paths of the two vortices are compared with those of a counter-rotating vortex pair under a rigid wall.

The linear stability characteristics of non-isothermal flow in vertical annuli has been determined for two geometries. The analysis demonstrates that a fully developed mixed-convection flow in a vertical annulus is unstable in certain regions of an appropriate parameter space. Consequently, parallel countercurrent flows, predicted by previous numerical models and commonly used by engineers, are often physically unrealizable and can be observed experimentally only in special circumstances. In addition, it is found that the most unstable disturbances are often asymmetric in the parameter range of practical interest. The instability behaviour was also found to depend on whether the inner or outer cylinder was heated.

A wide-angle model for water-wave propagation on an irregular bathymetry is developed based on the linear mild-slope equation. The spectral model decomposes the incident wavetrain into directional modes, or an angular spectrum. The effect of the bottom topography is shown to force the generation of additional directional wave modes. Nonlinearity is incorporated in the model by correcting the wave parameters iteratively using an empirical nonlinear dispersion relationship which is approximately valid over the entire range of water depths.

Numerical examples are presented for waves incident on a transverse bar field, a laboratory experiment involving wave focusing over an elliptic shoal on a sloping beach for which detailed measurements are available and for waves focusing behind a circular shoal resting on a flat botom. The application of the model is limited to cases in which the model domain is rectangular and the depth variation in the lateral direction is small if waves of large incident angle are modelled.

The stability of core-annular flow (CAF) in pipes is analysed using the linear theory of stability. Attention is confined to the potentially stable case of lubricated pipelining with the less viscous liquid, say water, in the annulus. The effects of surface tension and density are included, but gravity is excluded. We find upper and lower branches of the neutral curve in a Reynolds number (ℝ) vs. wavenumber (α) plane. A window of parameters is identified in which CAF is stable to small disturbances. When ℝ is below the lower critical value, CAF is destabilized by surface tension and long waves break up into slugs and bubbles. The sizes of slugs and bubbles of oil in water observed by Charles, Govier & Hodgson (1961) are given by the wavelength of the fastest growing long wave. This long-wave instability is a capillary instability, modified by shear, which reduces to Rayleigh's instability in the appropriate limit. At higher ℝ, the capillary instability is stabilized by shear. At yet higher ℝ, above the upper critical value, the flow is unstable to generally shorter waves which leads to emulsification, water droplets in oil. The theory agrees with experiments. The analysis seems to be applicable to the design of lubricated pipelines; for example, there is an optimum viscosity ratio for stability, greater stability can be obtained by using heavy liquid as a lubricant when the flow is unstable to capillary modes on the lower branch and by using light liquids when the flow is unstable to emulsifying disturbances on the upper branch.

Eastward-flowing zonal currents on a thin rotating shell, such as a planetary atmosphere or ocean, have integral properties analogous to open channel flows, the latitudinal width of the zonal current being the analogue of the depth of an open channel flow. The purpose here is to apply the formalism and some of the concepts of open channel flow hydraulics to zonal flows and demonstrate the results with laboratory experiments. In particular a critical relationship is found between a representative zonal velocity, U, and the half-width of the current, a. A dimensionless parameter (U/βa2), the Froude/Rossby number, is found analogous to the Froude number of open channel flow. Westward-flowing currents do not have an equivalent analogue.

We investigate numerically the transition from laminar to chaotic flow of a Boussinesq fluid with Pr = 0.71 in two-dimensional closed, differentially heated, vertical cavities having aspect ratios near unity. The cavities have rigid conducting sidewalls, and rigid insulating top and bottom walls. The physical nature of the resulting flow is a function of the aspect ratio and Rayleigh number.

It is shown that an oscillatory approach to steady-state, oscillatory instabilities, quasi-periodic flow, and chaotic flow exist for the flow regimes investigated. We find that for aspect ratios of approximately three or larger the the first transition from steady-state is due to instability of the sidewall boundary layers, while for small aspect ratios, but larger than ½, it is due to internal waves near the departing corners. For both instabilities we obtain the critical Rayleigh number as a function of aspect ratio and write expressions relating the fundamental frequencies of the oscillatory flow to the Rayleigh number and aspect ratio. When Ra is increased significantly above the first critical value, the flow becomes complex since both types of instabilities can be present. With a further increase in Rayleigh number the flow becomes chaotic and eventually turbulent. The above results are illustrated for different Rayleigh numbers and aspect ratios using time histories, spectral analysis, and streamlines at different values of time.

The distortion by large-scale random motions of small-scale turbulence is investigated by examining a model problem. The changes in energy spectra, velocity and vorticity moments, and anisotropy of small-scale turbulence are calculated over timescales short compared with the timescale of small-scale turbulence by applying rapid distortion theory with a random distortion matrix for different initial conditions: irrotational or rotational, and isotropic or anisotropic large-scale turbulence with or without mean strain, and isotropic or anisotropic small-scale turbulence.

We have obtained the following results: (1) Irrotational random strains broaden the small-scale energy spectrum and transfer energy to higher wavenumbers. (2) The rotational part of the large-scale strain is important for reducing anisotropy of turbulence rather than transferring energy to higher wavenumbers. (3) Anisotropy in small-scale turbulence is reduced by large-scale isotropic turbulence. The reduction of anisotropy of the velocity field depends on the initial value of the velocity anisotropy tensor of the small-scale velocity field ui defined by $\overline{u_iu_j}/\overline{u_lu_l}-\frac{1}{3}\delta_{ij}$, and also on the anisotropy of the distribution of the energy spectrum in wavenumber space. The reduction in anisotropy of the vorticity field ωi depends only on the vorticity anisotropy tensor. (4) The pressure-strain correlation is calculated for the change in Reynolds stress of the anisotropic small-scale turbulence. The correlation is proportional to time and depends on the difference between the velocity and wavenumber anisotropy tensors. These results (which are exact for small time) differ significantly from current turbulence models. (5) The effect of large-scale anisotropic turbulence on isotropic small-scale turbulence is calculated in general. Results are given for the case of axisymmetric large scales and are compared with the observed behaviour of small-scale turbulence near interfaces. (6) When a mean irrotational straining motion is applied to turbulence with distinct large-scale and small-scale components in their velocity field, the large-scale irrotational motions combine with the mean straining to increase further the anisotropy of the vorticity of the small scales, but the large-scale rotational motions reduce the small-scale anisotropy. For isotropic straining motion, the latter is weaker than the former. After the mean distortion ceases, both kinds of large-scale straining tend to reduce the anisotropy. This also has implications for modelling the rate of reduction of anisotropy.