We analyse the stability of a mushy layer during the directional solidification of a ternary alloy. Our model includes diffusive and convective transport of heat and solutes, coupled by an equilibrium thermodynamic constraint of the ternary phase diagram. The model contains phase change effects due to latent-heat release, solute rejection and background solidification. We identify novel convective instabilities, both direct and oscillatory, which are present under statically stable conditions. We examine these instabilities asymptotically by simplifying to a thin mushy layer with small growth rates. We also discuss numerical results for the full problem, confirming the asymptotic predictions and providing the stability characteristics outside the small-growth-rate approximation. A physical explanation for these instabilities in terms of parcel arguments is proposed, indicating that the instability mechanisms generally involve different rates of solute diffusion, different rates of solute rejection and different background solute distributions induced by the initial alloy composition.

A numerical investigation of unsteady hydrodynamic forces on the particle bed in an oscillatory flow environment is performed by means of direct numerical simulations. Statistical descriptions of drag and lift forces for two particle sizes of diameter 372 and 125 in wall units in a very rough turbulent flow regime are reported. Characterization of unsteady forces in terms of spatial distribution, temporal autocorrelation, force spectrum as well as cross-correlations with measurable flow variables is carried out. Based on the concept of impulse, intermittency in the drag and lift forces is also investigated. Temporal correlations show drag and lift to be positively correlated with a time delay that is approximately equal to the Taylor micro-scale related to the drag/lift fluctuations. The force spectra for drag and lift reveal roughly two scaling regions, $-11/3$ and $-7/3$; the former typically represents turbulence–mean-shear interactions, whereas the latter indicates dominance of turbulence–turbulence interactions. Particle forces are strongly correlated with streamwise velocity and pressure fluctuations in the near-bed region for both flow cases. In comparison to the large-diameter particle case, the spatial extent of these correlations is 2–3 times larger in homogeneous directions for the small sized particle, a feature that is reminiscent of longer near-bed structures. For both large- and small-particle cases, it is shown that the distributions of drag (lift) fluctuations, in particular, peakedness and long tails, match remarkably well with fourth-order Gram–Charlier distributions of velocity (pressure) fluctuations. Furthermore, it is demonstrated that the intermittency is larger in the case of the lift force compared to that for the drag in both flow cases. Distributions of impulse events are heavily and positively skewed and are well described by a generalized extreme value distribution.

The linear stability of the horizontal pipe flow of an equal density oil–water mixture, arranged as a core–annular flow (CAF), is here reconsidered from the point of view of non-modal analysis in order to assess the effects of non-normality of the linearized Navier–Stokes operator on the transient evolution of small disturbances. The aim of this investigation is to give insight into physical situations in which poor agreement occurs between the predictions of linear modal theory and classical experiments. The results exhibit high transient amplifications of the energy of three-dimensional perturbations and, in analogy with single-fluid pipe flow, the largest amplifications arise for non-axisymmetric disturbances of vanishing axial wavenumber. Energy analysis shows that the mechanisms leading to these transient phenomena mostly occur in the annulus, occupied by the less viscous fluid. Consequently, higher values of energy amplifications are obtained by increasing the gap between the core and the pipe wall and the annular Reynolds number. It is argued that these linear transient mechanisms of disturbance amplification play a key role in explaining the transition to turbulence of CAF.

Significant advances have been made in applying ideas from nonlinear dynamical systems theory to flows which exhibit sequences of bifurcations in the transition to turbulence. Moreover, the recent discoveries of finite-amplitude states in linearly stable flows holds great promise for a breakthrough in our understanding transition in shear flows. Tsukahara, Tillmark & Alfredsson (J. Fluid Mech., 2010, this issue, vol. 648, pp. 5–33) study a novel variant of a classical shear flow by adding global rotation. The competition between the induced body force and shear-induced instabilities leads to the discovery of a rich and beautiful tapestry of transition sequences.

The hot-wire anemometer, used for recording speed variations in turbulent flow, involves in its working principle the unsteady heat transfer from a hot fixed surface to a fluctuating air stream moving past the surface. If the wire is maintained at a constant (high) temperature, the rate of loss of heat from the wire changes with the velocity of the incident stream, and the compensating rate of gain of heat, produced by the Joule heating effect of the electric current, changes, correspondingly. The accompanying change of current can be measured, and used to calculate the varying velocity of the air stream. The hot wire may have a diameter as low as 10−4 in. and the Reynolds number of the flow is then of the order of 0.05 for each ft. per sec of velocity. With low velocities, of the order of 10 or 20 ft./sec, the flow past the wire is in the range of small Reynolds number, and the exact equations of flow may be approximated by simpler equations in the manner of Oseen's theory (Lamb 1932). The approximate equations are not easy to solve when the flow is compressible, as it will be in the presence of the large temperature differences imposed by the heat of the wire. If, however, the temperature differences are assumed to be small, the approximate energy equation is no longer linked with the equations of continuity and momentum, and it may be solved without knowledge of the velocity field. The purpose of this note is to give the solution for the temperature field when a warm circular cylinder or a warm sphere is held at rest in a fluctuating stream.

The formation of a thermocline in a water column, in which shear-free turbulence is generated both at the surface and bottom, and a stabilizing buoyancy flux is imposed at the surface, is studied using a laboratory experiment and a numerical model with the aim of understanding the formation of a tidal front in coastal seas. The results show that the formation of a thermocline, which always occurs in the absence of bottom mixing, is inhibited and the water column maintains a vertically mixed state, when bottom mixing becomes sufficiently strong. It is found from both experimental and numerical results that the criterion for the formation of a thermocline is determined by the balance between the rate of work that is necessary to maintain a mixed state against the formation of stratification by the buoyancy flux and the turbulent kinetic energy flux from the bottom supplied to the depth of thermocline formation. The depth of the thermocline, when it is formed, is found to decrease with bottom mixing.

The unsteady evolution of a boundary layer induced by a rectilinear vortex convecting above a heated surface is considered numerically. This model problem is representative of the types of interactions that can occur when vortices encounter solid surfaces in a wide variety of diverse applications involving high-Reynolds-number and high-Grashof-number flows. It is known that in the case without heat transfer, the vortex-induced boundary layer evolves toward a singularity as it forms a sharp spike that erupts away from the surface. Numerical solutions of the unsteady mixed-convection boundary-layer equations in the Boussinesq limit are obtained in Lagrangian coordinates. Solutions for various values of the inclination angle of the surface and Grashof number show that the coupling between the fluid flow and heat transfer can have a dramatic effect upon the transport of momentum and energy within the boundary layer induced by the vortex. The unsteady eruption convects high-temperature, near-wall fluid away from the surface and causes large gradients in the thermal boundary layer. The buoyancy force acting on the heated boundary-layer fluid can also have a significant impact on the unsteady separation process, either accelerating or delaying it, depending upon the inclination angle of the surface.

The generalized temporal residual mean (TRM-G) framework is reviewed and illustrated using a numerical simulation of vertical shear instability. It is shown how TRM-G reveals the physically relevant amount of diapycnal eddy fluxes and implied diapycnal mixing, and how TRM-G relates to the Osborn–Cox relation, which is often used to obtain observational estimates of the diapycnal diffusivity. An exact expression for the diapycnal diffusivity in the TRM-G is given in the presence of molecular diffusion, based on acknowledging and summing up an entire hierarchy of eddy buoyancy moments. In this revised form of the Osborn–Cox relation, diapycnal diffusivity is related only to irreversible mixing of buoyancy, since all advective and molecular flux terms are converted to dissipation of variance and higher order moments. An approximate but closed analytical expression can be given for the revised Osborn–Cox relation with the caveat that this closed expression implies unphysical cross-boundary rotational fluxes.

It is demonstrated that the original Osborn–Cox relation, in which advective and molecular flux terms are simply neglected, is an approximation to the full form valid to first order. In the numerical simulation the original Osborn–Cox relation holds to a surprisingly good approximation despite large advective fluxes of variance and large lateral inhomogeneity in the turbulent mixing.

Sediment transport in rivers consists, at moderate discharge stage, of individual grains that undergo a series of step movements and rest periods (bed-load). Following a large number of grain trajectories in time and space is difficult and the results are affected by bias due to censorship of the time-spatial window. Therefore, the data sets available for the description of the statistics of resting-times, travel-time and lengths of the steps, are still insufficient. In this paper, an innovative experimental methodology has been designed and applied to get data representing the evolution of a bed surface and to support a robust statistical analysis of sediment transport. The methodology is based on image sequences taken of a flat bed made of well-sorted (mono-dispersed) particles. The acquired data are interpreted analytically through equations that describe the effects of grain entrainment and deposition. We show that grains’ displacement have a mean value independent of bed-load rate under low to moderate transport intensity for a given sediment type and bed-slope. Hence, we provide a strong validation of the seminal conjecture of Einstein in his theoretical statistical description of sediment transport. Finally, we describe the probability density functions of the resting-time for a few values of the sediment discharge.

Optical distortion through a plane shear layer between two uniform streams of unequal temperatures is investigated using imaging of scalar mixing by planar laser-induced fluorescence (LIF). Simultaneous fluorescence intensity measurements of disodium fluorescein and rhodamine B (the fluorescence of the latter is temperature dependent) are used to extract the temperature distribution within the plane of a laser sheet. This planar imaging technique allows a direct measurement of the index of refraction field, which is used to determine optical distortions in the imaging plane. While a given ray within the sheet is refracted at a cumulative angle that depends on the local index of refraction gradient along its path, LIF measurements in the cross-stream plane of the flow show substantial optical distortions near the upstream and downstream edges of the primary vortices where the angle between the local temperature (or index) gradient and the wavevector of the incident light is large. These distortions are manifested by the appearance of low- and high-intensity streaks that are advected with the flow and whose characteristic spatial scales significantly diminish with the onset of small-scale (mixing) transition. The formation of secondary (streamwise) vortical structures leads to substantial spanwise variations in index of refraction and consequently to optical distortions that vary with the phase of the base flow and can exceed the distortions in the cross-stream plane. An important aim of the present research is the investigation of the effects of direct actuation of small-scale motions on the mixing and consequently on optical distortion within the shear layer. The actuation suppresses the formation of the large coherent vortical structures and leads to localized enhancement of dissipation and reduction in turbulent kinetic energy. The actuation enhances mixing across the entire width of the forced shear layer and leads to significant reduction of optical distortion, suggesting that active small-scale mixing can play a significant role in the mitigation of aero-optical effects through turbulent shear flows.

The linear stability properties of viscous circular flows in a rotating environment are studied with respect to symmetric perturbations. Through the use of an effective energy or Lyapunov functional, we derive sufficient conditions for Lyapunov stability with respect to such perturbations. For circular flows with swirl velocity V(r) we find that Lyapunov stability is determined by the properties of the function ℱ(r) = (2V/r + f)/Q (with f the Coriolis parameter, r the radius and Q the absolute vorticity) instead of the customary Rayleigh discriminant Φ(r) = (2V/r + f)Q. The conditions for stability are valid for flows with non-zero Q everywhere. Further, the flows are presumed stationary, incompressible and velocity perturbations are required to vanish at rigid boundaries. For Lyapunov stable flows an upper bound for the increase of the total perturbation energy due to transient non-modal growth is derived which is valid for any Reynolds number. The theory is applied to Couette flow and the Lamb–Oseen vortex.

The information exchange dynamics between different spatial scales of a turbulent field is experimentally investigated, with particular reference to the energy cascade process. This is done by use of the collective light scattering (CLS) diagnostic, a new optical setting tuned for the observation of atomic number density fluctuations at macroscopic scales. A two-channel scattering device is described that provides simultaneously the fluctuations at two different spatial scales. The turbulent system is an axisymmetric air jet. With this system, time correlations between signal amplitudes for two different scales show a characteristic time width. This time does not depend on the observed scales: it is close to the turbulence production time. Nevertheless, signal amplitude autocorrelation shows a shorter characteristic time. The autocorrelation time scaling law behaves almost like the eddy turnover time from Kolmogorov theory.

Oblique transition was experimentally investigated in a Blasius boundary layer formed on a flat plate. This transition mechanism was provoked by exciting a pair of oppositely oriented oblique Orr–Sommerfeld (O–S) modes given by (ω/ωts, ±β/βts) = (1, ±1) in the frequency-wavenumber (spanwise) space. Surface waviness with height Δh and a well-defined wavenumber spectrum that is synchronized with the neutral O–S wavenumber at Branch I, (αw, ±βw) = (αts,I, ±βts,I), was used to provide a steady velocity perturbation in the near-wall region. A planar downstream-travelling acoustic wave of amplitude ε was created to temporally excite the flow near the resonance frequency, ωts(= 2πfo), of an unstable eigenmode corresponding to kts = kw (where k =±[α2+β2]1/2). Possible mechanisms leading to laminar-to-turbulent breakdown were examined for various forcing combinations, εΔh. For small values of εΔh, a peak-valley structure corresponding to a spanwise wavenumber of 2βw was observed. As expected, the maximum r.m.s. narrow-band streamwise velocity fluctuations, ut(fo), occur at peak locations, which correspond to regions with mean streamwise velocity, U, deficits. For the largest value of εΔh, significant mean-flow distortion was observed in the spanwise profiles of U. Large spanwise velocity gradients, [mid ]dU/dζ[mid ], exist between peaks and valleys and appear to generate an explosive growth in the velocity fluctuations. The maximum values of ut no longer occur at peak locations of the stationary structure but at locations of spanwise inflection points. The magnitude of ut scales with [mid ]dU/dζ[mid ]. A nonlinear interaction of two non-stationary modes was conjectured as a possible mechanism for the enhancement of the streak ampliflcation rate.

The relative motion of drops in shear flows is responsible for collisions leading to the creation of larger drops. The collision of liquid drops in a gas is considered here. The drops are small enough for the Reynolds number to be low (negligible fluid motion inertia), yet large enough for the Stokes number to be possibly of order unity (non-negligible inertia in the motion of drops). Possible concurrent effects of Van der Waals attractive forces and drop inertia are taken into account.

General expressions are first presented for the drag forces on two interacting drops of different sizes embedded in a general linear flow field. These expressions are obtained by superposition of solutions for the translation of drops and for steady drops in elementary linear flow fields (simple shear flows, pure straining motions). Earlier solutions adapted to the case of inertialess drops (by Zinchenko, Davis and coworkers) are completed here by the solution for a simple shear flow along the line of centres of the drops. A solution of this problem in bipolar coordinates is provided; it is consistent with another solution obtained as a superposition of other elementary flow fields.

The collision efficiency of drops is calculated neglecting gravity effects, that is for strongly sheared linear flow fields. Results are presented for the cases of a simple linear shear flow and an axisymmetric pure straining motion. As expected, the collision efficiency increases with the Stokes numbers, that is with drop inertia. On the other hand, the collision efficiency in a simple shear flow becomes negligible below some value of the ratio of radii, regardless of drop inertia. The value of this threshold increases with decreasing Van der Waals forces. The concurrence between drop inertia and attractive van der Waals forces results in various anisotropic shapes of the collision cross-section. By comparison, results for the collision efficiency in an axisymmetric pure straining motion are more regular. This flow field induces axisymmetric sections of collision and strong inertial effects resulting in collision efficiencies larger than unity. Effects of van der Waals forces only appear when one of the drops has a very low Stokes number.

This study deals with the transition toward quasi-periodicity of buoyant convection generated by a horizontal temperature gradient in a three-dimensional parallelepipedic cavity with dimensions $4\times 2\times 1$ (length $\times$ width $\times$ height). Numerical continuation techniques, coupled with an Arnoldi method, are used to locate the steady and Hopf bifurcation points as well as the different steady and periodic flow branches emerging from them for Prandtl numbers ranging from 0 to 0.025 (liquid metals). Our results highlight the existence of two steady states along with many periodic cycles, all with different symmetries. The bifurcation scenarios consist of complex paths between these different solutions, giving a succession of stable flow states as the Grashof number is increased, from steady to periodic and quasi-periodic. The change of these scenarios with the Prandtl number, in connection with the crossing of bifurcation points, was carefully analysed.

Perforated screens are often deployed to attenuate aerodynamic sound in heat-exchanger cavities and other ducts conveying mean flow. The dissipation is caused by vorticity production in the perforations. This mechanism is investigated theoretically for the case of a thin rigid plate lying along the centreline of a duct and having a single transverse slot, a configuration that is to be studied experimentally at the Institute of Sound and Vibration Research of Southampton University. Time-harmonic acoustic waves are incident on the slot in the presence of equal parallel mean flows on either side of the plate. A linearized theory of unsteady shearing flow over a slot, which incorporates the influence of vorticity ejection into the flow, is used to examine mean-flow/acoustic energy exchanges. According to this theory acoustic energy is absorbed provided that the Strouhal number based on slot width and mean-flow velocity is sufficiently small. At higher frequencies there exists an infinite set of discrete frequency intervals within which there is a net production of acoustic energy at the slot at the expense of the kinetic energy of the mean flow.

The problem considered here is to determine the shape of a symmetric two-dimensional plate so that the drag of this plate in infinite cavity flow is a minimum. With the flow assumed steady and irrotational, and the effects due to gravity ignored, the drag of the plate is minimized under the constraints that the frontal width and wetted arc-length of the plate are fixed. The extremization process yields, by analogy with the classical Euler differential equation, a pair of coupled nonlinear singular integral equations. Although analytical and numerical attempts to solve these equations prove to be unsuccessful, it is shown that the optimal plate shapes must have blunt noses. This problem is next formulated by a method using finite Fourier series expansions, and optimal shapes are obtained for various ratios of plate arc-length to plate width.

The Lagally theorem describes the unsteady hydrodynamic force on a rigid body exhibiting arbitrary motion in an inviscid and incompressible fluid by the properties of the singularities employed to generate the flow and the body motion and to meet the boundary condition. So far, only sources and dipoles have been considered, and the present work extends the theorem to include free vortices in a two-dimensional flow. The present extension is validated by reproducing the system dynamics or the force evolution of three literature problems: (i) a free cylinder interacting with a free vortex; (ii) the moving Föppl problem; and (iii) a cylinder in constant normal approach to a fixed identical cylinder. This work further extends the bifurcation analysis on the moving Föppl problem by including the solid-to-liquid density ratio as a new parameter, in addition to the system total impulse and the vortex strength. We then apply the theorem to the problem where a moving Föppl system is made to approach a fixed or a free neutrally buoyant target cylinder of identical size from far away. The force developed on each cylinder is examined with respect to the vortex pair configuration and the target mobility. When approaching a fixed target, a greater force is developed if the vortex pair has stronger circulation and larger structure. The mobility of the target cylinder, however, can modify the hydrodynamic force by reducing its magnitude and reversing the force ordering with respect to the vortex pair configuration found for the case with fixed target. Possible mechanisms for such a change of force nature are given based on the currently derived equation of motion.

A hypersonic gun tunnel has been used to measure the heat transfer to a sharpedged flat plate inclined at various incidences to generate local Mach numbers from 3 to 9. The measurements have been compared with a number of theoretical estimates by plotting the Stanton number against the energy-thickness Reynolds number. The prediction giving the most reasonable agreement throughout the above Mach number range is that due to Fernholz (1971).

The values of the skin-friction coefficient derived from velocity profiles and Preston tube data are also given.