The solidification of melts can be profoundly influenced by convection. In alloys, compositional convection can be driven by solute gradients generated as one component of the alloy is preferentially incorporated within the solid, even when the thermal field is stabilizing. In this paper, two modes of compositional convection during solidification from below are uncovered using a linear-stability analysis: one, which we shall call the ‘mushy-layer mode’, is driven by buoyant residual fluid within a mushy layer, or porous medium, of dendritic crystals; the other, which we shall call the ‘boundary-layer mode’ is associated with a narrow compositional boundary layer in the melt just above the mush-liquid interface. Either mode can be the first to become unstable depending on the thermodynamical and physical properties of the alloy. The marginally stable eigenfunctions suggest that the boundary-layer mode results in fine-scale convection in the melt above the mushy layer and leaves the interstitial fluid of the mushy layer virtually stagnant. In contrast, the mushy-layer mode causes perturbations to the solid fraction of the mushy layer that are indicative of a tendency to form chimneys, which are vertical channels of reduced or zero solid fraction that have been observed experimentally. Particular attention is focused on the mushy-layer mode and its dependence upon the thermodynamical properties of the alloy. The results of this analysis are used to make a number of interpretations of earlier experimental studies such as the observations that some systems are less prone to form chimneys and that the regions of melt in these systems evolve to supersaturated conditions, while the melt evolves to unsaturated conditions once chimneys have formed. In addition, good quantitative agreement is found between the results of the linear-stability analysis and the experimental results of Tait & Jaupart (1992) for the onset of the mushy-layer mode of convection.

The stability of a sand bed in an alluvial channel is investigated by a two-dimensional mathematical model, based on the vorticity transport equation. The model takes account of the internal friction and describes the non-uniform distribution of the suspended sediment. It turns out that the inclusion of the friction and of a definite model of the sediment transport mechanism leads to results rather different from those obtained previously by potential-flow analysis.

Dense particle suspensions are widely encountered in many applications and in environmental flows. While many previous studies investigate their rheological properties in laminar flows, little is known on the behaviour of these suspensions in the turbulent/inertial regime. The present study aims to fill this gap by investigating the turbulent flow of a Newtonian fluid laden with solid neutrally-buoyant spheres at relatively high volume fractions in a plane channel. Direct numerical simulation (DNS) are performed in the range of volume fractions ${\it\Phi}=0{-}0.2$ with an immersed boundary method (IBM) used to account for the dispersed phase. The results show that the mean velocity profiles are significantly altered by the presence of a solid phase with a decrease of the von Kármán constant in the log-law. The overall drag is found to increase with the volume fraction, more than one would expect if just considering the increase of the system viscosity due to the presence of the particles. At the highest volume fraction investigated here, ${\it\Phi}=0.2$, the velocity fluctuation intensities and the Reynolds shear stress are found to decrease. The analysis of the mean momentum balance shows that the particle-induced stresses govern the dynamics at high ${\it\Phi}$ and are the main responsible of the overall drag increase. In the dense limit, we therefore find a decrease of the turbulence activity and a growth of the particle induced stress, where the latter dominates for the Reynolds numbers considered here.

The process of internal gravity wave generation by the simple harmonic flow (U = U0, cos ω0t) of a stably stratified fluid (Brunt–Väisälä frequency N) over an obstacle is investigated in some detail. Attention is primarily directed to the behaviour of the solution in various limiting cases, and to estimating the flux of energy into the internal wave field. In general, waves are generated not only at the fundamental frequency ω0, but also at all of its harmonics. But, for values of ω0/N greater than about one half, the waves of fundamental frequency are dominant. For values of ω0/N, less than about one half, the quasi-static approximation, in which the problem is considered as a slowly-varying version of the classical lee wave problem, is found to provide a viable estimate for the wave field. The general solution is found to compare favourably with the limited available experimental data.

Thermoelectric currents in the presence of magnetic fields can cause pumping or stirring of liquid-metal coolants in nuclear reactors or stirring of molten metal in industrial metallurgy. The interaction between the thermal and magnetohydrodynamic fields is a mutual one owing to alterations in the thermal convection and to the Peltier and Thomson effects (although these are usually small). This paper sets up the equations of magnetohydrodynamics and thermal convection when coupled by thermoelectricity and solves some illustrative problems in which the thermal field is known ab initio. Examples where the effects are due to either continuous or discontinuous variation of material composition are included. Practical magnitudes are discussed for the case of a fusion-reactor blanket, where the effects are potentially important owing to the unusual thermoelectric power of lithium.

The motions of vortices around single cylinders and around pairs of cylinders in relative sinusoidal flow are investigated in this paper. Using simultaneous flow visualization and force measurements, the vortex motions are related to the fluid-induced lift and in-line forces. For the single cylinder, several repeatable patterns of vortex shedding are identified within particular ranges of flow amplitude. The process of pairing of vortices from a previous half cycle with those in a present half cycle is fundamental to all the patterns. Visualization is shown to be more effective in a reference frame which is fixed with respect to the undisturbed fluid rather than with respect to the cylinders. For this reason, the examples of vortex motions are taken from a rig in which vertical cylinders are oscillated in a tank of fluid. By oscillating a pair of cylinders over a range of gaps, orientations and amplitudes, it is found that the vortex-shedding patterns identified for a single cylinder can synchronize either in phase or in antiphase between the two cylinders. Such observations help to explain how lift and in-line forces are influenced by cylinder proximity and in some cases these forces are significantly magnified. Force coefficients are evaluated for both the single cylinder and the pair of cylinders.

This paper presents the first numerical investigation via direct numerical simulation of the tone noise phenomenon occurring in the flow past laminar airfoils. This phenomenon corresponds to the radiation of discrete acoustic tones in some specific flow conditions, and has received much attention since the 1970s, and several experimental studies have been carried out to identify and understand the underlying physical mechanisms. However, several points remain to be clarified in order to provide a complete explanation of its origin. The flow around a two-dimensional NACA0012 airfoil is considered in order to have a deeper understanding of the tone noise phenomenon. Consistently with previous experimental studies, it is shown that depending on the Reynolds number and angle of attack, two different types of acoustic spectrum are observed: one which exhibits a broadband contribution with a dominant frequency together with a sequence of regularly spaced discrete frequencies, while the other one is only characterized by a simple broadband contribution. The first configuration is typical of the tone noise phenomenon. The present work shows that in this case, the mean flow on the pressure side of the airfoil exhibits a separation bubble near the trailing edge and the main tone frequency is close to the most amplified frequency of the boundary layer. The mechanism proposed in previous works for the main tone generation – which implies the existence of a separation bubble at the pressure side – is therefore validated by numerical simulation. On the other hand, the analysis of the suction side boundary layer reveals that there is no separation and that the most amplified frequency is different from the main tonal one. However, the suction side boundary layer is highly receptive to the tone frequency. Finally, an original explanation for the existence of the secondary discrete frequencies observed in the radiated pressure spectrum is given. They are associated to a bifurcation of the airfoil wake from a symmetric to a non-symmetric vortex pattern. A possible explanation for the existence of this bifurcation is the interaction between the disturbances which are the most amplified by the suction side boundary layer and those originating in the forcing of the suction side flow by the main tone noise mechanism.

The effect of increasing the rate of mixing in turbulent boundary layers in a region of adverse pressure gradient has been investigated experimentally. Only the two-dimensional case was considered. The boundary layer was formed on a flat wall in a special wind tunnel in which a variety of adverse pressure gradients could be obtained. Speeds were low enough to justify the neglect of compressibility. The main objective was to compare the effect of increasing the rate of mixing with the effect of reducing the pressure gradient on boundary-layer development and separation. A Variety of mixing schemes was tried, all of them involving fixed devices arranged in a row on the surface in the region of rising pressure. While these differed considerably in effectiveness, they had a generally similar effect on the flow; and, except for effects arising from changes in displacement and momentum thickness introduced at the devices, their effect on the layer was basically equivalent to that of a decrease in pressure gradient. Apart from forced mixing, the shape of the pressure distribution was found to have a significant effect on displacement and momentum thickness, these being minimized and the wall distance decreased for a given pressure rise by a distribution with an initially steep and progressively decreasing gradient.

In this paper a systematic numerical study of flow past an impulsively started circular cylinder at low and moderate Reynolds numbers using a lattice-Boltzmann algorithmic approach is presented together with an extended volumetric boundary scheme. Results agree well with some well-known previous works. It is demonstrated that in the nearly incompressible limit, this approach is able to provide accurate direct numerical simulations of unsteady flows with curved geometry.

Empirical information is extracted from constant-pressure flows and, on this basis alone, the equations of motion are solved for flows where the pressure gradient parameter, β = δ*(dp/dx)/τ0 is held constant. The experimental defect profiles of Clauser and the near-separating profile of Stratford are predicted quite well.

The present work is an extension of the work of Clauser and Townsend in that a particular form for an effective or eddy viscosity is hypothesized. Here, however, a continuous, and analytically precise family of defect profiles are calculated for the entire range, −0·5 ≤ β ≤ ∞. The solutions span the whole profile with the exception of the viscous sublayer.

A detailed consideration of the viscous sublayer and a comparative examination of various eddy viscosity hypotheses are included in a companion paper.

In a wind tunnel designed for flow-acoustic measurements, the wall-pressure fluctuations beneath a turbulent boundary layer have been investigated. The measurements were carried out with variously sized pressure transducers (19 [les ] d+ [les ] 333) and with an array of four small transducers (separation distance Δx+ = 75). It is shown that the dimensionless diameter d+ = 19 of the transducers is sufficient to resolve the essential structures of the turbulent pressure fluctuations. The power spectrum Φ(ω+) measured with the smallest transducer d+ = 19 partly exhibits power-law decay $\Phi \sim \omega^{\frac{7}{3}}$, which has been theoretically predicted for locally isotropic turbulence. By visual analysis and signal averaging in the time domain, pressure structures with high amplitudes could be detected which have the shape of short wavetrains or pulses. Their characteristic frequency and longitudinal wavelength have the mean values ω+ = 0.52 and λ+ = 145 respectively, and their mean convection velocity amounts to uc/u∞ = 0.53. It was calculated from the measured probability density that these characteristic structures play an important role, although the probability of their occurrence is low. The sources of these wall-pressure structures can be located in the buffer layer of the boundary layer.

Experiments on and calculation methods for flow in straight non-circular ducts involving turbulence-driven secondary motion are reviewed. The origin of the secondary motion and the shortcomings of existing calculation methods are discussed. A more refined model is introduced, in which algebraic expressions are derived for the Reynolds stresses in the momentum equations for the secondary motion by simplifying the modelled Reynolds-stress equations of Launder, Reece & Rodi (1975), while a simple eddy-viscosity model is used for the shear stresses in the axial momentum equation. The kinetic energy k and the dissipation rate ε of the turbulent motion which appear in the algebraic and the eddy-viscosity expressions are determined from transport equations. The resulting set of equations is solved with a forward-marching numerical procedure for three-dimensional shear layers. The model, as well as a version proposed by Naot & Rodi (1982), is tested by application to developing flow in a square duct and to developed flow in a partially roughened rectangular duct investigated experimentally by Hinze (1973). In both cases, the main features of the mean-flow and the turbulence quantities are simulated realistically by both models, but the present model underpredicts the secondary velocity while the Naot-Rodi model tends to overpredict it.

Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α1 and α2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin2ϕ α1,2 or strong if ϕ2 = O(α1,2). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O(α1α2), then is determined from these basic solutions.

Strong interactions are intrinsically nonlinear. It is shown that these interactions are phase-conserving (the sum of the phases of the incoming waves is equal to the sum of the phases of the outgoing waves) if |α2-α1 > (2ϕ)2 but not if |α2-α1| (2ϕ)2 (e.g. the reflexion problem, for which the interacting waves are images and α2 = α1). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech2 profiles yield singular outgoing waves with - csch2 profiles, if \[ \psi_{-}< |\psi| < \psi_{+},\quad{\rm where}\quad\psi_{\pm}={\textstyle\frac{1}{2}}\left|(3\alpha_2)^{\frac{1}{2}}\pm(3\alpha_1)^{\frac{1}{2}}\right|. \]

Regular interactions appear to be impossible within this singular regime, and its end points, |ϕ| = ϕ±, are associated with resonant interactions.

A Boussinesq-type model is derived which is accurate to O(kh)4 and which retains the full representation of the fluid kinematics in nonlinear surface boundary condition terms, by not assuming weak nonlinearity. The model is derived for a horizontal bottom, and is based explicitly on a fourth-order polynomial representation of the vertical dependence of the velocity potential. In order to achieve a (4,4) Padé representation of the dispersion relationship, a new dependent variable is defined as a weighted average of the velocity potential at two distinct water depths. The representation of internal kinematics is greatly improved over existing O(kh)2 approximations, especially in the intermediate to deep water range. The model equations are first examined for their ability to represent weakly nonlinear wave evolution in intermediate depth. Using a Stokes-like expansion in powers of wave amplitude over water depth, we examine the bound second harmonics in a random sea as well as nonlinear dispersion and stability effects in the nonlinear Schrödinger equation for a narrow-banded sea state. We then examine numerical properties of solitary wave solutions in shallow water, and compare model performance to the full solution of Tanaka (1986) as well as the level 1, 2 and 3 solutions of Shields & Webster (1988).

We study the collapse of a transient cavity of air in water created by the impact of a solid body. Experimentally, we characterize the dynamics of the cavity from its creation (t = 0) until it collapses (t = τ) in the limit where inertia dominates viscous and capillary effects. Theoretically, we find in this regime an approximate analytical solution which describes the time evolution of the shape of the cavity. This theoretical solution predicts the existence of two different types of cavities that we also observe experimentally.

Previous experimental studies are reviewed and those whose data are deemed reliable are identified. New experiments at larger scale are described and the results are reported. These are combined with the reliable previous studies to form a data set covering heights from 3.66 to 50 m and gasflow rates from 0.0002 to 0.59 normal m3/s. These wide-ranging data are combined with an integral theory for bubble plumes to determine functional relationships between local plume properties and the entrainment coefficient and the fraction of the momentum flux that is carried in the turbulent velocity fluctuations. These relationships together with the integral theory provide a set of equations that are suitable for numerical solution for the mean flow properties of any round bubble plume. Examples of the numerical solutions are presented and a comparison of one of these with existing experimental data is given. The relationships between the local plume properties and the entrainment coefficient and the momentum flux carried by the turbulence are interpreted to provide a qualitative understanding of the parameters involved and their influences on the plume.

The expansion method introduced by Friedrichs (1948) for the systematic development of shallow-water theory for water waves of large wavelength was used by Keller (1948) to obtain the first approximation for the finite-amplitude solitary wave of Boussinesq (1872) and Rayleigh (1876), as well as for periodic waves of permanent type, corresponding to the cnoidal waves of Korteweg & de Vries (1895).

The present investigation extends Friedrich's method so as to include terms up to the fourth order from shallow-water theory for a flat horizontal bottom, and thereby obtains the complete second approximations to both cnoidal and solitary waves. These second approximations show that, unlike the first approximation, the vertical motions cannot be considered as negligible, and that the pressure variation is no longer hydrostatic.

Richtmyer–Meshkov instability is investigated for negative Atwood number and two-dimensional sinusoidal perturbations by comparing experiments, numerical simulations and analytic theories. The experiments were conducted on the NOVA laser with strong radiatively driven shocks with Mach numbers greater than 10. Three different hydrodynamics codes (RAGE, PROMETHEUS and FronTier) reproduce the amplitude evolution and the gross features in the experiment while the fine-scale features differ in the different numerical techniques. Linearized theories correctly calculate the growth rates at small amplitude and early time, but fail at large amplitude and late time. A nonlinear theory using asymptotic matching between the linear theory and a potential flow model shows much better agreement with the late-time and large-amplitude growth rates found in the experiments and simulations. We vary the incident shock strength and initial perturbation amplitude to study the behaviour of the simulations and theory and to study the effects of compression and nonlinearity.

This paper studies several geometric aspects of the Poisson and Gaussian random fields approximating Burgers k−2 and Kolmogorov $k^{-\frac{5}{3}}$ homogeneous turbulence. In particular, simulated sample scalar iso-surfaces (e.g. surfaces of constant temperature or concentration) are exhibited, and their relative degrees of wiggliness are shown to be best characterized by saying that the corresponding fractal dimensions are respectively equal to 3−½ and $3-\frac{1}{3}$.

A stochastic model, implemented as a Monte Carlo simulation, is used to compute statistical properties of velocity and scalar fields in stationary and decaying homogeneous turbulence, shear flow, and various buoyant stratified flows. Turbulent advection is represented by a random sequence of maps applied to a one-dimensional computational domain. Profiles of advected scalars and of one velocity component evolve on this domain. The rate expression governing the mapping sequence reflects turbulence production mechanisms. Viscous effects are implemented concurrently.

Various flows of interest are simulated by applying appropriate initial and boundary conditions to the velocity profile. Simulated flow microstructure reproduces the −5/3 power-law scaling of the inertial-range energy spectrum and the dissipation-range spectral collapse based on the Kolmogorov microscale. Diverse behaviours of constant-density shear flows and buoyant stratified flows are reproduced, in some instances suggesting new interpretations of observed phenomena. Collectively, the results demonstrate that a variety of turbulent flow phenomena can be captured in a concise representation of the interplay of advection, molecular transport, and buoyant forcing.