The three-dimensional turbulent field of a passive scalar has been mapped quantitatively by obtaining, effectively instantaneously, several closely spaced parallel two-dimensional images; the two-dimensional images themselves have been obtained by laser-induced fluorescence. Turbulent jets and wakes at moderate Reynolds numbers are used as examples. The working fluid is water. The spatial resolution of the measurements is about four Kolmogorov scales. The first contribution of this work concerns the three-dimensional nature of the boundary of the scalar-marked regions (the ‘scalar interface’). It is concluded that interface regions detached from the main body are exceptional occurrences (if at all), and that in spite of the large structure, the randomness associated with small-scale convolutions of the interface are strong enough that any two intersections of it by parallel planes are essentially uncorrelated even if the separation distances are no more than a few Kolmogorov scales. The fractal dimension of the interface is determined directly by box-counting in three dimensions, and the value of 2.35 ± 0.04 is shown to be in good agreement with that previously inferred from two-dimensional sections. This justifies the use of the method of intersections. The second contribution involves the joint statistics of the scalar field and the quantity χ* (or its components), χ* being the appropriate approximation to the scalar ‘dissipation’ field in the inertial–convective range of scales. The third aspect relates to the multifractal scaling properties of the spatial intermittency of χ*; since all three components of χ* have been obtained effectively simultaneously, inferences concerning the scaling properties of the individual components and their sum have been possible. The usefulness of the multifractal approach for describing highly intermittent distributions of χ* and its components is explored by measuring the so-called singularity spectrum (or the f(α)-curve) which quantifies the spatial distribution of various strengths of χ*. Also obtained is a time sequence of two-dimensional images with the temporal resolution on the order of a few Batchelor timescales; this enables us to infer features of temporal intermittency in turbulent flows, and qualitatively the propagation speeds of the scalar interface. Finally, a few issues relating to the resolution effects have been addressed briefly by making point measurements with the spatial and temporal resolutions comparable with the Batchelor lengthscale and the corresponding timescale.

We experimentally investigate dispersion from a heated line source placed in the central region of a turbulence mixing layer. Recently described by Veeravalli & Warhaft (1989) the mixing layer has no mean shear and consists of gradients in the velocity variance and scale; it is formed from a composite grid of constant solidity from which two distinct velocity scales are formed, one on either side of the stream. Mixing is effected by intermittent turbulent penetration and diffusion. The dispersion measurements were carried out in the convective regime where both plume flapping and fine-scale turbulent mixing play a role, the latter becoming more dominant as the plume evolves. The mean and variance temperature profiles are strongly skewed (with larger tails on the low turbulence side of the flow) in the earlier stages of the plume development. Here, in the convective range, the median and peak of the mean plume are deflected toward the large-scale region. As the flow evolves the profiles become more symmetrical but as the plume enters the turbulent diffusive stage there is evidence that the profiles again became asymmetric but now with longer tails in the high turbulence side of the flow (owing to the higher diffusivity). The temperature variance and heat flux budgets are highly asymmetric but tend to exhibit many of the characteristics of the budget of a line source in decaying homogeneous grid turbulence which is also presented here. However, a distinct region of negative production (counter-gradient heat flux) is found in the temperature variance budget and this is shown to be a consequence of the asymmetry of the transverse velocity probability density function in the mixing layer. Temperature spectra, both of the time series and of the intermittency function, across the plume are described. They are shown to peak at high wavenumbers in the centre and edge of the plume and at lower wavenumbers in the intermediate region. Their form is shown to change as the plume develops fine-scale structure and flapping becomes less important.

The stability of two-dimensional thermal convection in an infinite-Prandtl-number fluid layer with zero-stress boundaries is investigated using numerical calculations in three-dimensional rectangles. At low Rayleigh numbers (Ra < 20000) calculations of the stability of two-dimensional rolls to cross-roll disturbances are in agreement with the predictions of Bolton & Busse for a fluid with a large but finite Prandtl number. Within the range 2 × 104 < Ra [les ] 5 × 105, steady rolls with basic wavenumber α > 2.22 (aspect ratio < 1.41) are stable solutions. Two-dimensional rolls with basic wavenumber α < 1.96 (aspect ratio > 1.6) are time dependent for Ra > 4 × 104. For every case in which the initial condition was a time-dependent large-aspect-ratio roll, two-dimensional convection was found to be unstable to three-dimensional convection. Time-dependent rolls are replaced by either bimodal or knot convection in cases where the horizontal dimensions of the rectangular box are less than twice the depth. The bimodal planforms are steady states for Ra [les ] 105, but one case at Ra = 5 × 105 exhibits time dependence in the form of pulsating knots. Calculations at Ra = 105 in larger domains resulted in fully three-dimensional cellular planforms. A steady-state square planform was obtained in a 2.4 × 2.4 × 1 rectangular box. started from random initial conditions. Calculations in a 3 × 3 × 1 box produced steady hexagonal cells when started from random initial conditions, and a rectangular planform when started from a two-dimensional roll. An hexagonal planform started in a 3.5 × 3.5 × 1 box at Ra = 105 exhibited oscillatory time dependence, including boundary-layer instabilities and pulsating plumes. Thus, the stable planform in three-dimensional convection is sensitive to the size of the rectangular domain and the initial conditions. The sensitivity of heat transfer to planform variations is less than 10%.

Calculations are carried out of the shape of gravity and gravity–capillary waves on deep water in the presence of a thin sheet of uniform vorticity which models the effect of a wind drift layer. The dependence of the fluid speed at the wave crest is determined and compared for gravity waves with the theory of Banner & Phillips (1974). It is found that this theory underestimates the retardation due to drift and tendency to break. The retardation disappears when capillary forces are significant, but in this case it is found that there can be a significant alteration of the wave shape.

The detonation stability problem is studied by a normal mode approach which greatly simplifies the calculation of linear instability of detonation in contrast to the Laplace transform procedure used by Erpenbeck. The method of solution, for an arbitrary parameter set, is a shooting method which can be automated to generate easily the required information about instability. The condition on the perturbations applied at the end of the reaction zone is shown to be interpreted as either a boundedness condition or an acoustic radiation condition. Continuous and numerically exact neutral stability curves and boundaries are given as well as growth rates and eigenfunctions which are calculated for the first time. Our calculations include the Chapman–Jouguet (CJ) case which presents no special difficulty. We give representative results for our detonation model and summarize the one-dimensional stability behaviour in parameter space. Comparison with previous results for the neutral stability boundaries and approximations to the unstable discrete spectrum are given. Parametric studies of the unstable, discrete spectrum's dependence on the activation energy and the overdrive factor are given with the implications for interpreting the physical mechanism of instability observed in experiments. This first paper is restricted to the case of one-dimensional linear instability. Extensions to transverse disturbances will be treated in a sequel.

A uniformly valid theory (all wavelengths and angles of incidence) for the diffraction of sea waves by a slender body, correct to second order in the slenderness parameter, has been derived for the shallow-water limit. This theory is now extended to the finite water depth case, with the same results and accuracy.

To design self-cooled liquid metal blankets for fusion reactors, one must know about the behaviour of MHD flows at high Hartmann numbers. In this work, finite difference codes are used to investigate the influence of Hartmann number M, interaction parameter N, wall conductance ratio c, and changing magnetic field, respectively, on the flow.

As liquid-metal MHD flows are characterized by thin boundary layers, resolution of these layers is the limiting issue. Hartmann numbers up to 103 are reached in the two-dimensional case of fully developed flow, while in three-dimensional flows the limit is 102. However, the calculations reveal the main features of MHD flows at large M. They are governed by electric currents induced in the fluid. Knowing the paths of these currents makes it possible to predict the flow structure.

Results are shown for two-dimensional flows in a square duct at different Hartmann numbers and wall conductivities. While the Hartmann number governs the thickness of the boundary layers, the wall conductivities are responsible for the pressure losses and the structure of the flows. The most distinct feature is the side layers where the velocities can exceed those at the centre by orders of magnitude.

The three-dimensional results are also for a square duct. The main interest here is to investigate the redistribution of the fluid in a region where the magnetic field changes. Large axial currents are induced leading to the ‘M-shaped’ velocity profiles characteristic of MHD flow. So-called Flow Channel Inserts (FCI), of great interest in blanket design, are investigated. They serve to decouple the load carrying wall from the currents in the fluid. The calculations show that the FCI is indeed a suitable measure to reduce the pressure losses in the blanket.

This paper examines the sound generation process when a supersonic aerofoil cuts through a steady jet flow. It is shown that the principal sound is generated by the leading edge of the aerofoil when it interacts with the streaming jet. To the leading order in terms of the jet velocity, no trailing-edge sound is generated. This is not the result of the cancellation of a trailing-edge sound by that from vortex shedding through the imposition of the Kutta condition. Instead, the null acoustic radiation from the trailing edge is entirely because, to the leading order, there is no interaction between the trailing edge and the jet. The effect of the trailing edge is to diffract sound waves generated by the leading edge. It is shown that the diffracted field (as well as the incident field) is regular at the trailing edge and the issue of satisfying the Kutta condition does not arise during the diffraction process. Thus, there is no extra vortex shedding from the trailing edge owing to its interaction with the flow, apart from those resulting from the discontinuity across the aerofoil, generated by the flow-leading edge interaction. This is in sharp contrast to the case of subsonic aerofoils where the removal of the singularity in the diffracted field at the trailing edge through the imposition of the Kutta condition results in vortex shedding from the sharp edge and energy exchange between the sound field and the vortical wake.

Penetration of high-speed targets, such as missiles or satellites, involves a glancing impact resulting from the velocities of the target and the projectile not necessarily being parallel to one another. Glancing is different from oblique impact in that here the target is in lateral motion relative to the projectile (at speeds that may be comparable).

The effect of the transverse motion of the target on the penetration performance is analysed by means of a hydrodynamic model. As is usually done, both the projectile and target materials are taken to behave as ideal fluids, owing to the large stresses obtained upon impact. A complete description of the flow field is obtained for the planar case, making use of a transformation to the hodograph plane. Owing to the lack of symmetry, the solution is obtained indirectly from the known solution for asymmetrically impinging jets, obtaining a determinate solution by a limit process where the jet representing the target fluid becomes infinitely wide.

The variations of the rate of penetration and the streamline pattern with the respective ratios of target to projectile speeds and densities are studied. It is found that the relative lateral motion of the target causes a decrease in the rate of penetration (in comparison to normal impact). The analysis is utilized to obtain an estimate for the total depth of penetration (relative to the projectile width). Comparison with experimental data of yawed impact available in the open literature shows good agreement with the theoretical predictions when the latter are modified to account for the deceleration of a finite-length projectile.

The axisymmetric deformation of a red blood cell placed in a uniaxial straining Stokes flow is considered. The cell is modelled as a fluid capsule that contains a Newtonian fluid, and is bounded by an area-preserving membrane with negligible resistance to bending. First, it is theoretically demonstrated that spheroidal cells with isotropic membrane tension constitute stationary configurations. To compute transient cell deformations, a numerical procedure is developed based on the boundary-integral method for Stokes flow. Calculations show that initially prolate or oblate cells with isotropic membrane tension deform into stationary spheroids. Cells with a highly oblate initial shape may develop a persistent small pocket along their axis during the deformation. The shear elasticity of the membrane prevents folding, but may cause the formation of sharp corners and concave regions along the cell contour. A decrease in the membrane shear elasticity results in substantial increase in the magnitude of the transient and asymptotic membrane tensions. The maximum strain rate below which a red blood cell remains intact is estimated to be ex = 105 s−1.

Global interaction of the boundary layer separating from an obstacle with resulting open/closed wakes is studied for a thin airfoil in a steady flow. Replacing the Kutta condition of the classical theory is the breakaway criterion of the laminar triple-deck interaction (Sychev 1972; Smith 1977), which, together with the assumption of a uniform wake/eddy pressure, leads to a nonlinear equation system for the breakaway location and wake shape. The solutions depend on a Reynolds number Re and an airfoil thickness ratio or incidence τ and, in the domain $Re^{\frac{1}{16}}\tau = O(1)$ considered, the separation locations are found to be far removed from the classical Brillouin–Villat point for the breakaway from a smooth shape. Bifurcations of the steady-state solution are found among examples of symmetrical and asymmetrical flows, allowing open and closed wakes, as well as symmetry breaking in an otherwise symmetrical flow. Accordingly, the influence of thickness and incidence, as well as Reynolds number is critical in the vicinity of branch points and cut-off points where steady-state solutions can/must change branches/types. The study suggests a correspondence of this bifurcation feature with the lift hysteresis and other aerodynamic anomalies observed from wind-tunnel and numerical studies in subcritical and high-subcritical Re flows.

We use weakly nonlinear analysis via a two-parameter expansion to study bifurcation of conduction into cellular convection of an internally heated fluid in a porous medium that forms a horizontal layer between two isothermal walls. The Darcy–Boussinesq model of convection is enhanced by including two nonlinear terms: (i) quadratic (Forchheimer) drag; and (ii) hydrodynamic dispersion enhancement of the thermal conductivity described by a weak linear relationship between effective conductivity and local amplitude of filtration velocity. The impact of the second term on the shape of the bifurcation curve for two-dimensional rolls is profound in the presence of uniform volumetric heating. The resulting bifurcation structure is unlike any pitchfork bifurcations typical of the classical Bénard problem. Although direct experimental validation of the novel bifurcation is not available, we would like to register it as an alternative or a supplement to models of small imperfections, and as an attempt to account for the scatter of observed critical values for the first bifurcation.

A complete analysis of the one-dimensional vibratory instability of planar flames of premixed gases propagating in tubes is provided. The driving mechanism results from unsteady coupling between flame structure and acoustic waves through temperature fluctuations. In certain conditions, the strength of such an instability will be proved to be sufficiently strong to produce large-amplitude fluctuations as soon as the flame has travelled a distance of the order of the acoustic wavelength. Stability limits and total amplification of an initial perturbation are computed in the framework of the simple flame mode of a one-step exothermic reaction governed by an Arrhenius law with an activation energy much larger than the thermal energy. Diffusive and thermal effects within the flame are included with a Lewis number different from unity. Damping mechanisms associated with viscous and thermal dissipation at the walls, as well as with loss of acoustic energy by sound radiation from the open end of the tube, are retained. In ordinary conditions, for a reactive mixture with an effective Lewis number close to unity, the predicted instability is weak. In the framework of the simplified flame model used here, islands of strong instabilities are predicted to occur at low Mach numbers for Lewis numbers larger than unity.

The interaction between the solidification and convection that occurs when a melt is cooled from above is investigated in a series of three papers. In these papers we consider a two-component melt that partially solidifies to leave a buoyant residual fluid. The solid forms a mushy layer of dendritic crystals, the interstices of which accommodate the residual fluid. The heat extraction through the upper boundary, necessary to promote solidification, drives convection at high Rayleigh numbers in the melt below the mushy layer. The convection enhances the heat transfer from the melt and alters the rate of solidification. In this paper the various phenomena are studied in a series of laboratory experiments in which ice is frozen from aqueous solutions of isopropanol. The experiments are complemented by the development of a general theoretical model in which the mush is treated as a continuum phase with thermodynamic properties that are functions of the local solid fraction. The model, which is based upon principles of equilibrium thermodynamics and local conservation of heat and solute, produces results in good agreement with the experimental data. Careful comparisons between this theory and experiments suggest the need to explore non-equilibrium effects, which are investigated in Parts 2 and 3.

We consider the resonant excitation of surface waves inside a rectangular wave tank of arbitrary water depth with a flap-type wavemaker on one side. Depending on the length and width of the tank relative to the sinusoidal forcing frequency of the wave paddle, three classes of resonant mechanisms can be identified. The first two are the well-known synchronous, resonantly forced longitudinal standing waves, and the subharmonic, parametrically excited transverse (cross) waves. These have been studied by a number of investigators, notably in deep water. We rederive the governing equations and show good comparisons with the experimental data of Lin & Howard (1960). The third class is new and involves the simultaneous resonance of the synchronous longitudinal and subharmonic cross-waves and their internal interactions. In this case, temporal chaotic motions are found for a broad range of parameter values and initial conditions. These are studied by local bifurcation and stability analyses, direct numerical simulations, estimations of the Lyapunov exponents and power spectra, and examination of Poincaré surfaces. To obtain a global criterion for widespread chaos, the method of resonance overlap (Chirikov 1979) is adopted and found to be remarkably effective.

Three-dimensional numerical solutions have been obtained for steady, linear shear flow past a fixed, heated spherical particle over a wide range of Reynolds number (0.1 [les ] R [les ] 100) and dimensionless shear rates (0.005 [les ] α [les ] 0.4). The results indicate that at a fixed shear rate, the dimensionless lift coefficient is approximately constant over a wide range of intermediate Reynolds numbers, and the drag coefficient also remains constant when normalized by the known values of drag for a sphere in uniform flow. At lower values of the Reynolds number, the lift and drag coefficients increase sharply with decreasing R, with the lift coefficient being directly proportional to R−½. For the range of shear rates studied here, the rate of heat transfer to the particle surface was found to depend only on the Reynolds number, that is, it was insensitive to the shear rate. The dimensionless rate of heat transfer, the Nussel number Nu, was seen to increase monotonically with R.

The linear-eddy model of turbulent mixing represents a spatially developing flow by simulating the time development along a comoving transverse line. Along this line, scalar quantities evolve by molecular diffusion and by randomly occurring spatial rearrangements, representing turbulent convection. The modelling approach, previously applied to homogeneous turbulence and to planar shear layers, is generalized to axisymmetric flows. This formulation captures many features of jet mixing, including differential molecular diffusion effects. A novel experiment involving two unmixed species in the nozzle fluid is proposed and analysed.

The ribbing instability, an extremely common cause of non-uniform liquid films in coating operations, is investigated both theoretically and experimentally. The Navier–Stokes system for the two-dimensional flow in symmetric film-splitting in forward roll coating is solved by finite-element analysis. Stability of the flow with respect to three-dimensional disturbances is examined by applying linear stability theory in a consistent finite-element approach, taking Fourier components in the transverse direction. The resulting generalized asymmetric eigenproblem is solved for the growth rates of disturbances as functions of wavenumber. The theory accurately predicts the critical capillary number and wavenumber at the transition to large-amplitude ribs. A sensitive experimental technique for detecting the ribs was developed that relies on low-angle reflection of a focused strip of white light off the meniscus between the rolls. This allowed detection of much smaller amplitude ribs, and much smaller critical capillary numbers were measured. The results indicate that the transition to ribbing is an imperfect bifurcation due to end effects, and clarify earlier discordances in the literature.

Large-amplitude axisymmetric waves on columnar vortices, thought to be related to flow structures observed in vortex breakdown, are found as static bifurcations of the Bragg–Hawthorne equation. Solutions of this equation satisfy the steady, axisymmetric, Euler equations. Non-trivial solution branches bifurcate as the swirl ratio (the ratio of azimuthal to axial velocity) changes, and are followed into strongly nonlinear regimes using a numerical continuation method. Four types of solutions are found: multiple columnar solutions, corresponding to Benjamin's ‘conjugate flows’, with subcritical–supercritical pairing of wave characteristics; solitary waves, extending previously known weakly nonlinear solutions to amplitudes large enough to produce flow reversals similar to the breakdown transition; periodic wavetrains; and solitary waves superimposed on the conjugate flow that emerge from the periodic wavetrain as the wavelength or amplitude becomes sufficiently large. Weakly nonlinear soliton solutions are found to be accurate even when the perturbations they cause are fairly strong.

The interaction between a regular wavetrain and a current possessing an arbitrary distribution of vorticity, in two dimensions, is considered for waves of finite amplitude. A numerical model is constructed, primarily for use in the finite depth regime, extending the work of Dalrymple (1973, 1977) and this is used to predict the wavelength and the particle velocities under the waves. These predictions agree very well with experimentally obtained data and the importance of the vorticity in the wave–current interaction is clarified. Amplitude and wavelength modulations are considered for finite amplitude waves on a slowly varying irrotational current; moderate agreement is found between theory and experiment.