The problem of solidification or melting under the action of a forced hydrodynamic flow is considered. In the appropriate parameter régime, the problem admits a formulation in terms of analytic functions. It is shown that a crystal with a parabolic tip propagates without change of shape at a steady velocity. Some novel explicit solutions are presented.

The flow of an incompressible viscous fluid past a sphere is investigated numerically and experimentally over flow regimes including steady and unsteady laminar flow at Reynolds numbers of up to 300. Flow-visualization experiments are used to validate the numerical results and to provide additional insight into the behaviour of the flow. Near-wake visualizations are presented for both steady and unsteady flows. Calculations for Reynolds numbers of up to 200 show steady axisymmetric flow and compare well with previous experimental and numerical observations. For Reynolds numbers of 210 to 270, a steady non-axisymmetric regime is found, also in agreement with previous work. To advance the basic understanding of this transition, a symmetry breaking mechanism is proposed based on a detailed analysis of the calculated flow field.

Unsteady flow is calculated at Reynolds numbers greater than 270. The results at a Reynolds number of 300 show a highly organized periodic flow dominated by vortex shedding. An analysis of the calculated vortical structure of the wake reveals a sequence of shed hairpin vortices in combination with a sequence of previously unidentified induced hairpin vortices. The numerical results compare favourably with experimental flow visualizations which, interestingly, fail to reveal the induced vortices. Based on the deduced symmetry-breaking mechanism, an analysis of the unsteady kinematics, and the experimental results, a mechanism driving the transition to unsteady flow is proposed.

The screech noise generation process from supersonic underexpanded jets, issuing from a sonic nozzle at pressure ratios of 2.4 and 3.3 (fully expanded Mach number, Mj=1.19 and 1.42), was investigated experimentally. The extremely detailed data provide a fresh, new look at the screech generation mechanism. Spark schlieren visualization at different phases of the screech cycle clearly shows the convection of the organized turbulent structures over a train of shock waves. The potential pressure field (hydrodynamic fluctuations) associated with the organized structures is fairly intense and extends outside the shear layer. The time evolution of the near-field pressure fluctuations was obtained from phase-averaged microphone measurements. Phase-matched combined views of schlieren photographs and pressure fluctuations show the sound generation process. The individual compression and rarefaction parts of the sound waves are found to be generated from similar hydrodynamic fluctuations. A partial interference between the upstream-propagating sound waves and the downstream-propagating hydrodynamic waves is found to be present along the jet boundary. The partial interference manifests itself as a standing wave in the root-mean-square pressure fluctuation data. The standing wavelength is found to be close to, but somewhat different from, the shock spacing. An outcome of the interference is a curious ‘pause and go’ motion of the sound waves along the jet periphery. Interestingly, a length scale identical to the standing wavelength is found to be present inside the jet shear layer. The coherent fluctuations and the convective velocity of the organized vortices are found to be modulated periodically, and the periodicity is found to match with the standing wavelength distance rather than the shock spacing. The reason for the appearance of this additional length scale, different from the shock spacing, could not be explained. Nevertheless, it is demonstrated that an exact screech frequency formula can be derived from the simple standing wave relationship. The exact relationship shows that the correct spacing between the sources, for a point source model similar to that of Powell (1953), should be a standing wavelength (not the shock spacing).

We present the results of an experimental and numerical study of the effects of a steady magnetic field on sidewall convection in molten gallium. The magnetic field is applied in a direction which is orthogonal to the main flow which reduces the convection and good agreement is found for the scaling of this effect with the relevant parameters. Moreover, qualitatively similar changes in the structure of the bulk of the flow are observed in the experiment and the numerical simulations. In particular, the flow is restricted to two dimensions by the magnetic field, but it remains different to that found in two-dimensional free convection calculations. We also show that oscillations found at even greater temperature gradients can be suppressed by the magnetic field.

We analyse and improve a recently-proposed two-phase flow model for the statistical evolution of two-fluid mixing. A hyperbolic equation for the volume fraction, whose characteristic speed is the average interface velocity v*, plays a central role. We propose a new model for v* in terms of the volume fraction and fluid velocities, which can be interpreted as a constitutive law for two-fluid mixing. In the incompressible limit, the two-phase equations admit a self-similar solution for an arbitrary scaling of lengths. We show that the constitutive law for v* can be expressed directly in terms of the volume fraction, and thus it is an experimentally measurable quantity. For incompressible Rayleigh–Taylor mixing, we examine the self-similar solution based on a simple zero-parameter model for v*. It is shown that the present approach gives improved agreement with experimental data for the growth rate of a Rayleigh–Taylor mixing layer.

Closure of the two-phase flow model requires boundary conditions for the surfaces that separate the two-phase and single-phase regions, i.e. the edges of the mixing layer. We propose boundary conditions for Rayleigh–Taylor mixing based on the inertial, drag, and buoyant forces on the furthest penetrating structures which define these edges. Our analysis indicates that the compatibility of the boundary conditions with the two-phase flow model is an important consideration. The closure assumptions introduced here and their consequences in relation to experimental data are compared to the work of others.

The linear stability of Burgers and Lamb–Oseen vortices is addressed when the vortex of circulation Γ and radius δ is subjected to an additional strain field of rate s perpendicular to the vorticity axis. The resulting non-axisymmetric vortex is analysed in the limit of large Reynolds number RΓ=Γ/v and small strain s[Lt ]Γ/δ2 by considering the approximations obtained by Moffatt et al. (1994) and Jiménez et al. (1996) for each case respectively. For both vortices, the TWMS instability (Tsai & Widnall 1976; Moore & Saffman 1975) is shown to be active, i.e. stationary helical Kelvin waves of azimuthal wavenumbers m=1 and m=−1 resonate and are amplified by the external strain in the neighbourhood of critical axial wavenumbers which are computed. The additional effects of diffusion for the Lamb–Oseen vortex and stretching for the Burgers vortex are proved to limit in time the resonance. The transient growth of the helical waves is analysed in detail for the distinguished scaling s∼Γ/ (δ2R1/2Γ). An amplitude equation describing the resonance is obtained and the maximum gain of the wave amplitudes is calculated. The effect of the vorticity profile on the instability characteristic as well as of a time-varying stretching rate are analysed. In particular the stretching rate maximizing the instability is calculated. The results are also discussed in the light of recent observations in experiments and numerical simulations. It is argued that the Kelvin waves resonance mechanism could explain various dynamical behaviours of vortex filaments in turbulence.

In this paper the structure of the Taylor meniscus and emitted jet is studied by perturbation methods in the limit of low flow rates. An asymptotic system of governing equations is derived from the basic equations of electrohydrodynamics. They rigorously take into account the inertia and viscosity of the liquid as well as the surface ion mobility. The solutions to the asymptotic equations in the meniscus, jet and surrounding gas regions are found, matched with each other, and applied to study distributions of electric and hydrodynamic variables. Such an approach allows the liquid velocity, surface charge, and meniscus-jet radius as well as electric potential inside and outside the liquid to be calculated. We also derive the theoretical dependences of the current carried by the jet and its diameter on the liquid properties and flow rate. These dependences are consistent with the scaling laws found experimentally by Fernández de la Mora & Loscertales (1994) and data obtained by Chen & Pui (1997).

The nonlinear evolution of deep-water wave groups, which are initiated by unstable three-wave systems, have been observed in a large wave tank (50 m long, 4.2 m wide, 2.1 m deep), equipped with a programmable, high-resolution wave generator. A large number of experiments were conducted (over 80 cases) for waves 1.0–4.0 m long, initial steepness ε=0.10–0.28, and normalized sideband frequency differences, δω=δω, 0.2–1.4. Using an array of eight high-resolution wave wires distributed in range (up to 43 m fetch), spectral evolution was studied in detail including the effect of background disturbances on the evolution. Minimizing those, new observations were made which extend the pioneering work of Lake et al. (1977) and of Melville (1982). Foremost, near recurrence without downshifting was observed without breaking, despite a significant but reversible energy transfer to the lower sideband at peak modulation; complete recurrence was prevented by the spreading of discretized energy to higher frequencies. Strong breaking was found to increase the transfer of energy from the higher to the lower sideband and to render that transfer irreversible. The end state of the evolution following strong breaking is an effective downshifting of the spectral energy, where the lower and the carrier wave amplitudes nearly coincide; the further evolution of this almost two-wave system was not studied here. Breaking during strong modulation was observed not only for the fastest growing initial condition, but over a wide parameter range. An explanation of the sideband behaviour in both the breaking and non-breaking case was given based on wave energy and momentum considerations, including the separate effects of energy and momentum loss due to breaking, and transfer to discretized higher frequencies throughout the spectra. Attention was drawn to the latter, which was almost universally observed.

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t=0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

We present direct numerical simulations of the spatial development of normal mode perturbations to boundary layers with Falkner–Skan velocity profiles. Values of the pressure gradient parameter considered range from very small, i.e. nearly flat-plate conditions, to relatively large values corresponding to incipient separation. In almost all cases, we find that the most effective perturbation is one composed of a plane wave and a pair of oblique waves inclined at equal and opposite angles to the primary flow direction. The frequency of the oblique waves is half that of the fundamental plane wave and because the conditions for resonance are satisfied exactly, all modes share a common critical layer, thus facilitating a strong interaction.

The oblique waves initially undergo a parametric type of subharmonic resonance, but in accordance with recent analyses of non-equilibrium critical layers, the system subsequently becomes fully coupled. From that point on, the amplification of all modes, including the plane wave, substantially exceeds the predictions of linear stability theory. Good agreement is obtained with the experimental small pressure gradient results of Corke & Gruber (1996). Our growth rates are slightly larger flowing to slight differences in initial conditions (e.g. the angle of inclination of the oblique waves).

The spectral element method was used to discretize the Navier–Stokes equations and the preconditioned conjugate gradient method was used to solve the resulting system of algebraic equations. At the inflow boundary, Orr–Sommerfeld modes were employed to provide the initial forcing, whereas the buffer domain technique was used at the outflow boundary to prevent convective wave reflection or upstream propagation of spurious information.

An investigation of the local receptivity of a Blasius boundary layer to a harmonic vortical disturbance is presented as a step towards understanding boundary-layer receptivity to free-stream turbulence. Although there has been solid experimental verification of the linear theory describing acoustic receptivity of boundary layers, this was the first experimental verification of the mechanism behind local receptivity to a convected disturbance. The harmonic wake from a vibrating ribbon positioned upstream of a flat plate provided the free-stream disturbance. Two-dimensional roughness elements on the surface of the plate acted as a local receptivity site. Hot-wire measurements in the boundary layer downstream of the roughness confirmed the generation of Tollmien–Schlichting (TS) instability waves by an outer-layer interaction between the long-wavelength convected disturbance and the short-scale mean-flow distortion due to the roughness. The characteristics of the instability waves were carefully measured to ensure that their behaviour was correctly modelled by linear stability theory. This theory was then used to determine the immeasurably small initial wave amplitudes resulting from the receptivity process, from wave amplitudes measured downstream. Tests were performed to determine the range of validity of the linear assumptions made in current receptivity theories. Experimental data obtained in the linear regime were then compared to theoretical results of other authors by expressing the experimental data in the form of an efficiency function which is independent of the free-stream amplitude, roughness height and roughness geometry. Reasonable agreement between the experimental and theoretical efficiency functions was obtained over a range of frequencies and Reynolds numbers.

The rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water were investigated experimentally and numerically. We developed an experimental system that uses a charged-coupled device (CCD) camera coupled with a microscope to track the rising bubble. By precisely measuring the bubble size and rising speed, we were able to accurately estimate the drag coefficient and the Sherwood number for the dissolution rate of gas bubbles at Reynolds numbers below 100 in the transient regime, where the bubble changes from behaving as a fluid sphere to behaving as a solid particle. We also numerically estimated the drag coefficient and Sherwood number of the ‘stagnant cap model’ by directly solving the coupled Navier–Stokes and convection–diffusion equations. We compared our experimental results with our numerical results and proposed equations for estimating the drag coefficient and Sherwood number of the bubble affected by contamination and clarified that the gas–liquid interface of the carbon dioxide bubble in water is immobile. We also show that the experimental and numerical results are in good agreement and the stagnant cap model can explain the mechanism of the transient process where the bubble behaviour changes from that of a fluid sphere to that of a solid particle.

Transport of tracers subject to mass transfer reactions in single rock fractures is investigated. A Lagrangian probabilistic model is developed where the mass transfer reactions are diffusion into the rock matrix and subsequent sorption in the matrix, and sorption on the fracture surface as well as on gauge (infill) material in the fracture. Sorption reactions are assumed to be linear, and in the general case kinetically controlled. The two main simplifying assumptions are that diffusion in the rock matrix is one-dimensional, perpendicular to the fracture plane, and the tracer is displaced within the fracture plane by advection only. The key feature of the proposed model is that advective transport and diffusive mass transfer are related in a dynamic manner through the flow equation. We have identified two Lagrangian random variables τ and β as key parameters which control advection and diffusive mass transfer, and are determined by the flow field. The probabilistic solution of the transport problem is based on the statistics of (τ, β), which we evaluated analytically using first-order expansions, and numerically using Monte Carlo simulations. To study (τ, β)-statistics, we assumed the ‘cubic law’ to be applicable locally, whereby the pressure field is described with the Reynolds lubrication equation. We found a strong correlation between τ and β which suggests a deterministic relationship β∼τ3/2; the exponent 3/2 is an artifact of the ‘cubic law’. It is shown that flow dynamics in fractures has a strong influence on the variability of τ and β, but a comparatively small impact on the relationship between τ and β. The probability distribution for the (decaying) tracer mass recovery is dispersed in the parameter space due to fracture aperture variability.

If a sill-enclosed basin, connected to a large reservoir, is suddenly subjected to a de-stabilizing surface buoyancy flux, it will first mix vertically by turbulent convection before the resulting lateral buoyancy gradient generates a horizontal exchange flow across the sill. We present a study which examines the unsteady adjustment of such a basin under continued steady forcing. It is shown, through theoretical development and laboratory experimentation, that two consecutive unsteady regimes characterized by different dynamic balances are traversed as the flow approaches a steady state.

Once established the exchange flow is controlled at the sill crest where it is hydraulically critical. In the absence of a lateral contraction, the single control at the sill crest allows a range of submaximal exchange states with the flow at the sill being dependent not only on the forcing and geometrical parameters but also on mixing conditions within the basin which are, in turn, dependent on the sill exchange. The sill–basin system is therefore strongly coupled although it remains isolated from the external reservoir conditions by a region of internally supercritical flow. Results from the laboratory experiments are used to demonstrate the link between the forcing and the exchange flow at the sill. Steady-state measurements of the interior mean velocity and buoyancy fields are also compared with previous analytical models.