ALBERT

Description: The evolution of two-dimensional regular flows laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle-particle interactions. Flows with large Reynolds and Froude numbers are considered, when effects related to viscous dissipation and gravity are negligible. A Cauchy problem is solved for an initially uniform distribution of particles with Stokes (St) and Reynolds (Rep) numbers of order unity in several types of flows representing steady solutions of the two-dimensional Euler equations. We consider flows in the vicinity of the hyperbolic stagnation point (with a uniform strain and zero vorticity) and the elliptic stagnation point (where vorticity is uniform), a circular vortex (with vorticity depending on the radius) and Stuart vortex flow. Analytical solutions are obtained, for the case of sufficiently small St, describing the accumulation of particles and corresponding modification of the fluid flow. Solutions derived show that the concentration of particles, although remaining uniform, decreases at the elliptic stagnation point and grows at the hyperbolic point. Owing to the coupling between the particulate and fluid dynamics, the flow vorticity is reduced at the elliptic point, while flow strain rate is enhanced at the hyperbolic point. Solutions obtained for the circular vortex show that the accumulation of particles proceeds in the form of a travelling wave. The concentration grows locally, forming the crest of the wave which propagates away from the vortex centre. Owing to the influence of the particulate on the carrier flow, the vorticity is reduced in the vortex centre. At the location of the crest the gradient of the flow grows owing to the drag forces between the fluid and particles and a vorticity peak is generated. Analytical solutions are also obtained for a chain of particle-laden Stuart vortices. Owing to the coupling effects, the concentration is diminished and the vorticity is reduced at the centres of the vortices. A sheet of increased concentration and vorticity is formed extending from the braid region to the periphery of the vortices, and the flow strain in the braid region is enhanced. Results of numerical simulations performed for St = 0.5 show good agreement with analytical solutions. © 1995, Cambridge University Press. All rights reserved.

Description: A well-known technique for metering a multiphase flow is to use small probes that utilize some measurement principle to detect the presence of different phases surrounding their tips. In almost all cases of relevance to the oil industry, the flow around such local probes is inviscid and driven by surface tension, with negligible gravitational effects. In order to study the features of the flow around a local probe when it meets a droplet, we analyse a model problem: the interaction of an infinite, initially straight, interface between two inviscid fluids, advected in an initially uniform flow towards a semi-infinite thin flat plate oriented at 90° to the interface. This has enabled us to gain some insight into the factors that control the motion of a contact line over a solid surface, for a range of physical parameter values. The potential flows in the two fluids are coupled nonlinearly at the interface, where surface tension is balanced by a pressure difference. In addition, a dynamic contact angle boundary condition is imposed at the three-phase contact line, which moves along the plate. In order to determine how the interface deforms in such a flow, we consider the small- and large-time asymptotic limits of the solution. The small-time and linearized large-time problems are solved analytically, using Mellin transforms, whilst the general large-time problem is solved numerically, using a boundary integral method. The form of the dynamic contact angle as a function of contact line velocity is the most important factor in determining how an interface deforms as it meets and moves over the plate. Depending on this, the three-phase contact line may, at one extreme, hang up on the leading edge of the plate or, at the other extreme, move rapidly along the surface of the plate. At large times, the solution asymptotes to an interface configuration where the contact line moves at the far-field velocity. © 1995, Cambridge University Press. All rights reserved.

Description: The impact of a nearly cylindrical water mass on a water surface is studied both experimentally and theoretically. The experiments consist of the rapid release of water from the bottom of a cylindrical container suspended above a large water tank and of the recording of the free-surface shape of the resulting crater with a high-speed camera. A bubble with a diameter of about twice that of the initial cylinder remains entrapped at the bottom of the crater when the aspect ratio and the energy of the falling water mass are sufficiently large. Many of the salient features of the phenomenon are explained on the basis of simple physical arguments. Boundary-integral potential-flow simulations of the process are also described. These numerical results are in fair to good agreement with the observations. © 1995, Cambridge University Press. All rights reserved.

Description: The neutral stability curve for the flat-plate boundary layer has been calculated using the Orr-Sommerfeld equation and compared to those obtained using upper- and lower-branch scalings. The Orr-Sommerfeld results agree well with the lower-branch scaling at Reynolds numbers relevant to experiment, but agree well with the upper-branch scaling only for R3〉 105. It is shown that the critical layer only emerges from the viscous wall layer when Rs〉 105. This suggests that for R3〈 105, when the critical layer lies within the viscous wall layer, the disturbance has a triple-deck structure, even for the upper branch of the neutral curve (which can be reached if the phase jump across the critical layer is retained). The transition from a triple-deck to a five-deck structure with increasing Reynolds number on the upper branch occurs relatively abruptly and can be associated with a square-root branch point in the Tietjens function. Essentially, the lower- and upper-branch scalings pertain to two different modes, the first possessing a triple-deck structure, the second a five-deck structure. The modes are connected at the branch point, and the neutral curves of each mode join to give a single curve close to this branch point. The asymptotic expansions for the upper- and lower-branch neutral curves depend upon the analyticity of the dispersion relationship, and so the proximity of the branch point indicates where these expansions will be liable to inaccuracies. This explains the poor neutral-curve predictions made by five-deck analyses at the Reynolds numbers where transition occurs. © 1995, Cambridge University Press. All rights reserved.

Description: The linearized inertial instability of the parallel shear flow of a viscoelastic liquid is considered. An elastic Rayleigh equation is derived, for high Reynolds numbers and high Weissenberg numbers, and for a viscoelastic liquid whose first normal stress dominates other stresses. The equation is used to investigate the stability of a submerged jet, that may be planar or axisymmetric, having a parabolic velocity profile. The sinuous mode is found to be fully stabilized by sufficiently large elasticity. The varicose mode in the planar case is partially stabilized, being unstable only at longer wavelengths and with a reduced growth rate. An axisymmetric jet, which is stable to varicose perturbations at zero elasticity, is found to be unstable to shortwave disturbances for small non-zero elasticity. This novel instability involves elastic waves in the shear. It is also present in other modes but does not have the fastest growth rate. © 1995, Cambridge University Press. All rights reserved.

Description: Recent studies have demonstrated the strong influence of end effects on low-Reynoldsnumber bluff body wakes, and a number of questions remain concerning the intrinsic nature of three-dimensional phenomena in two-dimensional configurations. Some of them are answered by the present study which investigates the wake of bluff rings (i.e. bodies without ends) both experimentally and by application of the phenomenological Ginzburg–Landau model. The model turns out to be very accurate in describing qualitative and quantitative observations in a large Reynolds number interval. The experimental study of the periodic vortex shedding regime shows the existence of discrete shedding modes, in which the wake takes the form of parallel vortex rings or ‘oblique’ helical vortices, depending on initial conditions. The Strouhal number is found to decrease with growing body curvature, and a global expression for the Strouhal–Reynolds number relation, including curvature and shedding angle, is proposed, which is consistent with previous straight cylinder results. A secondary instability of the helical modes at low Reynolds numbers is discovered, and a detailed comparison with the Ginzburg–Landau model identifies it as the Eckhaus modulational instability of the spanwise structure of the near-wake formation region. It is independent of curvature and its clear observation in straight cylinder wakes is inhibited by end effects. The dynamical model is extended to higher Reynolds numbers by introducing variable parameters. In this way the instability of periodic vortex shedding which marks the beginning of the transition range is characterized as the Benjamin–Feir instability of the coupled oscillation of the near wake. It is independent of the shear layer transition to turbulence, which is known to occur at higher Reynolds numbers. The unusual shape of the Strouhal curve in this flow regime, including the discontinuity at the transition point, is qualitatively reproduced by the Ginzburg–Landau model. End effects in finite cylinder wakes are found to cause important changes in the transition behaviour also: they create a second Strouhal discontinuity, which is not observed in the present ring wake experiments. © 1995, Cambridge University Press. All rights reserved.

Description: The turbulent structure in plane Couette flow at low Reynolds numbers is studied using data obtained both from numerical simulation and physical experiments. It is shown that the near-wall turbulence structure is quite similar to what has earlier been found in plane Poiseuille flow; however, there are also some large differences especially regarding Reynolds stress production. The commonly held view that the maximum in Reynolds stress close to the wall in Poiseuille and boundary layer flows is due to the turbulence-generating events must be modified as plane Couette flow does not exhibit such a maximum, although the near-wall coherent structures are quite similar. For two-dimensional mean flow, turbulence production occurs only for the streamwise fluctuations, and the present study shows the importance of the pressure—strain redistribution in connection with the near-wall coherent events. © 1995, Cambridge University Press. All rights reserved.

Description: According to numerous experimental observations and theoretical models vibrated layers composed of large granules behave like a solid plastic body. In contrast, in this study experimental data are presented that reveal that, for constant vibration amplitudes A ≥ 1 cm with the frequency ω increasing from zero, all layers pass through three vibrational states, with the respective behaviours being as of (i) a solid plastic body, (ii) a liquid, (iii) a gas. In the liquid-like vibrational state transverse waves propagating along the layer width were observed. These waves were shown to be gravitational resonance waves, with the corresponding frequencies well correlated by the known formula for incompressible liquids. In the gas-like vibrational state compression (shock) and expansion waves propagating across the layer height, were observed. A theoretical model for time-periodic collisional vibrational regimes was developed on the basis of the Euler-like equations of a granular gas composed of inelastic spheres. The model shows that the vibrational granular state (bed porosity, shock wave speed, granular pressure and kinetic energy) is inter alia governed by the dimensionless parameter V = (Aω)/(hmg)1/2, with g, hm being the gravitational acceleration and the height of the resting layer, respectively. This is in contrast with the previous studies, where the behaviour of vibrated granular layers was interpreted in terms of the dimensionless acceleration Δ = (Aω2)/g. The proposed model was tested by processing the data obtained from photographs of the particle distribution within vibrated layers. Theoretical predictions of the particle average concentration compared favourably with the experimental data. Other phenomena observed in vibrated granular layers include the formation of caverns, circulatory motion of granules and synchronized periodic motion of two adjacent vibrated layers of different widths. The importance of the observed phenomena in relation to various technological processes involving bulk materials (vibromixing, vibroseparation, etc.) is discussed. © 1995, Cambridge University Press. All rights reserved.

Description: Some methods of formation of preforms for drawing of polarization-maintaining optical fibres are based on utilization of the surface tension of glass in the liquid state. Under the action of surface tension non-circular glass articles begin to flow, which results in formation of an anisotropic internal structure of the preforms. The hydrodynamic analysis of two such methods is given in the paper. Analytical solutions of the Stokes equations with linearized boundary conditions for the corresponding creeping surface-tension-driven flows of liquid glass are obtained. By means of these solutions a processing strategy may be predetermined with a view to a specific internal structure of the fibre, as well as to the required value of birefringence. The theoretical results are compared with experimental data and agreement is fairly good. © 1995, Cambridge University Press. All rights reserved.

Description: A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of order R-1in the boundary-layer Reynolds number R. In Blasius flow, the present approach is consistent with that of Bertolotti et al. (1992) to O(R-1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis to O(R-1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated. However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal order R-1except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism. © 1995, Cambridge University Press. All rights reserved.

Description: The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction—refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory. © 1995, Cambridge University Press. All rights reserved.

Description: The trapping of surface water waves by a thin plate in deep water is reduced to finding non-trivial solutions of a homogeneous, hypersingular integral equation for the discontinuity in velocity potential across the plate. The integral equation is discretized using an expansion-collocation method, involving Chebyshev polynomials of the second kind. A non-trivial solution to the problem is given by the vanishing of the determinant inherent in such a method. Results are given for inclined flat plates, and for curved plates that are symmetric with respect to a line drawn vertically through their centre. Comparisons with published results for horizontal flat plates (in water of finite depth) and for circular cylinders are made. © 1995, Cambridge University Press. All rights reserved.

Description: The unsteady laminar necklace vortex system formed at the junction of a rectangular bluff body and a flat plate was studied experimentally using hydrogen bubble flow visualization and particle image velocimetry (PIV). The vortex system was found to exhibit unsteady behaviour similar to that described by other investigators for cylinder-flat plate junctures, and is characterized by the periodic formation of necklace vortices upstream of the body that subsequently break away and advect towards the block. Detailed analysis of PIV measurements on the plane of symmetry indicates that the dominant mechanism for vorticity balance in the vortex system is the cross-cancellation of the vorticity of the necklace vortex with vorticity of opposite sign generated by interaction of the necklace vortex with the approach surface to the body. © 1995, Cambridge University Press. All rights reserved.

Description: The unsteady nonlinear potential flow induced by a submerged line source or sink is studied by a vortex sheet method, both to trace the free surface evolution and to explore the possible existence of steady-state solutions. Only steady-state flows have been considered by other investigators, and these flows have been insensitive to whether they are generated by a source or sink, except with respect to the flow direction along the streamlines. The time-dependent solution permits an assessment of the stability of previously found steady solutions, and also reveals differences between source and sink flows: for the infinite-depth case, steady stagnation-point-type solutions are found for source flows, even above the critical value of source/sink strength reported by other investigators; for the finite-depth case, steady stagnation-point-type solutions are found both for source flows and sink flows, above the critical value reported by other investigators; finally, it is shown that streamline patterns of steady stagnation-point flows are identical for source and sink flows only in the limiting case of infinite depth. © 1995, Cambridge University Press. All rights reserved.

Description: A non-similar boundary layer theory for air blowing over a water layer on a flat plate is formulated and studied as a two-fluid problem in which the position of the interface is unknown. The problem is considered at large Reynolds number (based on x), away from the leading edge. We derive a simple non-similar analytic solution of the problem for which the interface height is proportional to x1/4and the water and air flow satisfy the Blasius boundary layer equations, with a linear profile in the water and a Blasius profile in the air. Numerical studies of the initial value problem suggest that this asymptotic non-similar air-water boundary layer solution is a global attractor for all initial conditions. © 1995, Cambridge University Press. All rights reserved.

Description: Steady flow with constant circulation into a vertical drain is considered. The precise details of the outflow are simplified by assuming that the drain is equivalent to a distributed volume sink, into which the fluid flows with uniform downward speed. It is shown that a maximum outflow rate exists, corresponding to no fluid circulation and vertical entry into the drain hole. Numerical solutions to the full nonlinear problem are computed, using the method of fundamental solutions. An approximate analysis, based on the use of the shallow-water equations, is presented for flows in which the free surface enters the drain. There is, in addition, a second type of solution, having a stagnation point at the free surface and no fluid circulation. These flows are also computed numerically, and results are presented. © 1995, Cambridge University Press. All rights reserved.

Description: We have experimentally studied the effects of mean strain on the evolution of stably stratified turbulence. Grid-generated turbulence in a stable linear mean background density gradient was passed through a two-dimensional contraction, contracting the stream only in the vertical direction. This induces an increase in stratification strength, which reduces the largest vertical overturning scales allowed by buoyancy forces. The mean strain through the contraction causes, on the other hand, stretching of streamwise vortices tending to increase the fluctuation levels of the transverse velocity components. This competition between buoyancy and vortex stretching dominates the turbulence dynamics inside and downstream of the contraction. Comparison between non-stratified and stratified experiments shows that the stratification significantly reduces the vertical velocity fluctuations. The vertical heat flux is initially enhanced through the contraction. Then, farther downstream the flux quickly reverses, leading to very strong restratification coinciding with an increase in the vertical velocity fluctuations. The vertical heat flux collapses much more rapidly than in the stratified case without an upstream contraction and the restratification intensity is also much stronger, showing values of normalized flux as strong as −0.55. Velocity spectra show that the revival of vertical velocity fluctuations, due to the strong restratification, starts at the very largest scales but is then subsequently transferred to smaller scales. The distance from the turbulence-generating grid to the entrance of the contraction is an important parameter which was varied in the experiments. The larger this distance, the larger the integral length scale can grow, approaching the limit set by buoyancy, before entering the contraction. The evolution of the various turbulence length scales is described. Two-point measurements of velocity and temperature transverse integral scales were also performed inside the contraction. The emergence of ‘zombie’ turbulence, for large buoyancy times, is in good quantitative agreement with the numerical simulations of Gerz & Yamazaki (1993) for stratification number larger than 1. © 1995, Cambridge University Press. All rights reserved.

Description: Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor S2+122; here S and Q are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ∇u. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier-Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ∇u or the complex eigenvalues of ∇u. our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear. © 1995, Cambridge University Press. All rights reserved.

Description: Nonlinear kink instabilities of high-Reynolds-number supersonic shear layers have been studied using high-resolution computer simulations with the piecewise-parabolic-method (PPM). The transition region between the two fluids of the shear layer is spread out over many computational zones to avoid numerical effects introduced on the smallest lengthscales. Mach number, density contrast, and perturbation speed and amplitude were varied to study their effects on the growth of the kink instabilities. In response to a perturbing sound wave, a travelling kink mode grows in amplitude until enough of a disturbance on the shear layer has been created for it to roll up and rapidly grow in thickness. The time it takes for this rapid growth to be initiated is proportional to the initial shear-layer thickness and increases for increasing Mach number or decreasing perturbation amplitude. For equal density, Mach 4 shear layers, perturbed by a sound wave with a 2 % amplitude at the travelling mode velocity, the growth time is rg = (546 ± 24) S/c9 where c is the sound speed and 8 the half-width of the shear layer. © 1995, Cambridge University Press. All rights reserved.

Description: It is well known that the imposition of a static magnetic field tends to suppress motion in an electrically conducting liquid. Here we look at the magnetic damping of liquid-metal flows where the Reynolds number is large and the magnetic Reynolds number is small. The magnetic field is taken as uniform and the fluid is either infinite in extent or else bounded by an electrically insulating surface S. Under these conditions, we find that three general principles govern the flow. First, the Lorentz force destroys kinetic energy but does not alter the net linear momentum of the fluid, nor does it change the component of angular momentum parallel to B. In certain flows, this implies that momentum, linear or angular, is conserved. Second, the Lorentz force guides the flow in such a way that the global Joule dissipation, D, decreases, and this decline in D is even more rapid than the corresponding fall in global kinetic energy, E. (Note that both D and E are quadratic in u.) Third, this decline in relative dissipation, D/E, is essential to conserving momentum, and is achieved by propagating linear or angular momentum out along the magnetic field lines. In fact, this spreading of momentum along the /Mines is a diffusive process, familiar in the context of MHD turbulence. We illustrate these three principles with the aid of a number of specific examples. In increasing order of complexity we look at a spatially uniform jet evolving in time, a three-dimensional jet evolving in space, and an axisymmetric vortex evolving in both space and time. We start with a spatially uniform jet which is dissipated by the sudden application of a transverse magnetic field. This simple (perhaps even trivial) example provides a clear illustration of our three general principles. It also provides a useful stepping-stone to our second example of a steady three-dimensional jet evolving in space. Unlike the two-dimensional jets studied by previous investigators, a three-dimensional jet cannot be annihilated by magnetic braking. Rather, its cross-section deforms in such a way that the momentum flux of the jet is conserved, despite a continual decline in its energy flux. We conclude with a discussion of magnetic damping of axisymmetric vortices. As with the jet flows, the Lorentz force cannot destroy the motion, but rather rearranges the angular momentum of the flow so as to reduce the global kinetic energy. This process ceases, and the flow reaches a steady state, only when the angular momentum is uniform in the direction of the field lines. This is closely related to the tendency of magnetic fields to promote two-dimensional turbulence. © 1995, Cambridge University Press. All rights reserved.

Description: The lateral capillary interaction between two particles immersed in a spherical thin liquid film is investigated. The interfacial shape, the lateral capillary force and the interparticle interaction energy are calculated by using the numerical solution of the linearized Laplace equation of capillarity. Orthogonal bipolar coordinates on a sphere (inducing biconical coordinates in space) are introduced as a helpful instrument for solving this problem and other problems of similar geometry. We consider two types of boundary conditions at the particle surfaces: fixed contact angle and fixed contact line. We established that for particles of fixed contact angle the capillary interaction energy depends monotonically on the interparticle distance whereas for particles of fixed contact line the interaction energy exhibits a maximum. The numerical results show that in both cases the capillary interaction is much larger than the thermal energy kT and can induce aggregation and ordering of submicrometre particles. These theoretical findings can be important for understanding the properties of Pickering emulsions (stabilized by particles) and liposomes or biomembranes containing incorporated membrane proteins. © 1995, Cambridge University Press. All rights reserved.

Description: A convex variational principle is used to obtain a generalization of the empirical nonlinear one-dimensional Forchheimer-extended Darcy flow equation to the multidimensional and anisotropic (tensor permeability) case. A modified permeability that is a function of flow velocity (or pressure gradient) is introduced in order to transform the nonlinear flow equation into a pseudo-linear form. Imposing an incompressibility condition on this pseudo-linear equation leads to a flow equation in Euler-Lagrange form which is used to build the corresponding variational principle. It is demonstrated that the variational principle is based on minimizing the power (time rate of doing work) required by the fluid to flow at a certain velocity under a prescribed pressure gradient. A consistent generalization of the Forchheimer equation to the tensor case then follows from the variational principle. The existence and uniqueness of solutions to the nonlinear flow equations might also be demonstrated using the variational principle on a case by case basis, once appropriate boundary conditions are chosen. © 1995, Cambridge University Press. All rights reserved.

Description: Liquid-metal magnetohydrodynamic flow in a system of electrically coupled U-bends in a strong uniform magnetic field is studied. The ducts composing the bends are electrically conducting and have rectangular cross-sections. It has been anticipated that very strong global electric currents are induced in the system, which modify the flow pattern and produce a very high pressure drop compared to the flow in a single U-bend. A detailed asymptotic analysis of flow for high values of the Harmann number (in fusion blanket applications of the order of 103–104) shows that circulation of global currents results in several types of peculiar flow patterns. In ducts parallel to the magnetic field a combination of helical and recirculatory flow types may be present and vary from one bend to another. The magnitude of the recirculatory motion may become very high depending on the flow-rate distribution between the bends in the system. The recirculatory flow may account for about 50 % of the flow in all bends. In addition there are equal and opposite jets at the walls parallel to the magnetic field, which are common to any two bends. The pressure drop due to three-dimensional effects linearly increases with the number of bends in a system and may significantly affect the total pressure drop. To suppress this and some other unwelcome tendencies either the ducts perpendicular to the magnetic field should be electrically separated, or the flow direction in the neighbouring ducts should be made opposite, so that leakage currents cancel each other. © 1995, Cambridge University Press. All rights reserved.

Description: This paper is concerned with the theoretical behaviour of the boundary-layer flow over a disk rotating in otherwise still fluid. The flow is excited impulsively at a certain radius at time t — 0. This paper analyses the inviscid stability of the flow and the stability with viscous, Coriolis and streamline curvature effects included. In both cases, within a specific range of the parameter space, it is shown that the flow is absolutely unstable, i.e. disturbances grow in time at every fixed point in space. Outside this range, the flow is convectively unstable or stable. The absolute or convective nature of the instabilities is determined by examining the branch-point singularities of the dispersion relation. Absolute instability is found for Reynolds numbers above 510. Experimentally observed values for the onset of transition from laminar to turbulent flow have an average value of 513. It is suggested that absolute instability may cause the onset of transition to turbulent flow. The results from the inviscid analysis show that the absolute instability is not caused by Coriolis effects nor by streamline curvature effects. This indicates that this mechanism may be possible on swept wings, where Coriolis effects are not present but the boundary layers are otherwise similar. © 1995, Cambridge University Press. All rights reserved.

Description: The Stokes equation system and Ohm’s law were solved numerically for fluid in periodic bicontinuous porous media of simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) symmetry. The Stokes equation system was also solved for fluid in porous media of SC arrays of disjoint spheres. The equations were solved by Galerkin’s method with finite element basis functions and with elliptic grid generation. The Darcy permeability k computed for flow through SC arrays of spheres is in excellent agreement with predictions made by other authors. Prominent recirculation patterns are found for Stokes flow in bicontinuous porous media. The results of the analysis of Stokes flow and Ohmic conduction through bicontinuous porous media were used to test the permeability scaling law proposed by Johnson, Koplik & Schwartz (1986), which introduces a length parameter A to relate Darcy permeability k and the formation factor F. As reported in our earlier work on the SC bicontinuous porous media, the scaling law holds approximately for the BCC and FCC families except when the porespace becomes nearly spherical pores connected by small orifice-like passages. We also found that, except when the porespace was connected by the small orifice-like passages, the permeability versus porosity curve of the bicontinuous media agrees very well with that of arrays of disjoint and fused spheres of the same crystallographic symmetry. © 1995, Cambridge University Press. All rights reserved.

Description: An experimental investigation of high Mach number free shear layers has been undertaken. The experiments were performed using a Mach 7 gun tunnel facility and a planar duct with injection from the base of a central strut producing a Mach 3 flow parallel to the gun tunnel stream. This configuration is relevant to the development of efficient scramjet propulsion, and the gun tunnel Mach number is significantly higher than the majority of previous supersonic turbulent mixing layer investigations reported in the open literature. Schlieren images and Pitot pressure measurements were obtained at four different convective Mach numbers ranging from 0 to 1.8. Only small differences between the four cases were detected, and the relatively large high-speed boundary layers at the trailing edge of the struct injector appear to strongly influence the shear layer development in each case. The Pitot pressure measurements indicated that, on average, the free shear layers all spread into the Mach 3 stream at an angle of approximately 1.4°, while virtually no spreading into the Mach 7 stream was detected until all of the low-speed stream was entrained. The free shear layers were simulated using a PNS code; however, the experimentally observed degree of spreading rate asymmetry could not be fully predicted with the k-∊ turbulence model, even when a recently proposed compressibility correction was applied. © 1995, Cambridge University Press. All rights reserved.

Description: A theoretical and experimental investigation of drop motion in rotating fluids is presented. The theory describing the vertical on-axis translation of an axisymmetric rigid body through a rapidly rotating low-viscosity fluid is extended to the case of a buoyant deformable fluid drop of arbitrary viscosity. In the case that inertial and viscous effects are negligible within the bulk external flow, motions are constrained to be two-dimensional in compliance with the Taylor-Proudman theorem, and the rising drop is circumscribed by a Taylor column. Calculations for the drop shape and rise speed decouple, so that theoretical predictions for both are obtained analytically. Drop shapes are set by a balance between centrifugal and interfacial tension forces, and correspond to the family of prolate ellipsoids which would arise in the absence of drop translation. In the case of a drop rising through an unbounded fluid, the Taylor column is dissipated at a distance determined by the outer fluid viscosity, and the rise speed corresponds to that of an identically shaped rigid body. In the case of a drop rising through a sufficiently shallow plane layer of fluid, the Taylor column extends to the boundaries. In such bounded systems, the rise speed depends further on the fluid and drop viscosities, which together prescribe the efficiency of the Ekman transport over the drop and container surfaces. A set of complementary experiments is also presented, which illustrate the effects of drop viscosity on steady drop motion in bounded rotating systems. The experimental results provide qualitative agreement with the theoretical predictions; in particular, the poloidal circulation observed inside low-viscosity drops is consistent with the presence of a double Ekman layer at the interface, and is opposite to that expected to arise in non-rotating systems. The steady rise speeds observed are larger than those predicted theoretically owing to the persistence of finite inertial effects. © 1995, Cambridge University Press. All rights reserved.

Description: An analytical model for solving the flow field associated with regular reflections of straight shock waves over porous layers has been developed. The governing equations of the gas inside the porous material were obtained by simplifying the general macroscopic balance equations which were obtained by an averaging process over a representative elementary volume of the microscopic balance equations as originally done by Bear & Bachmat (1990). The analytical predictions of the proposed model were compared to experimental results of Skews (1992) and Kobayashi, Adachi & Suzuki (1993). Very good to excellent agreement was evident. © 1995, Cambridge University Press. All rights reserved.

Description: A potential function/stream function formulation is introduced for the solution of the fully 3-D inverse potential ‘target pressure’ problem. In the companion paper (Part 1) it is seen that the general 3-D inverse problem is ill-posed but accepts as a particular solution elementary streamtubes with orthogonal cross-section. Under this simplification, a novel set of flow equations was derived and discussed. The purpose of the present paper is to present the computational techniques used for the numerical integration of the flow and geometry equations proposed in Part 1. The governing flow equations are discretized with centred finite difference schemes on a staggered grid and solved in their linearized form using the preconditioned GMRES algorithm. The geometry equations which form a set of first-order o.d.e.s are integrated numerically using a second-order-accurate space marching scheme. The resulting computational algorithm is applied to a double turning duct and a 3-D converging-diverging nozzle ‘reproduction’ test case. © 1995, Cambridge University Press. All rights reserved.

Description: The planar flow arising in an initially quiescent viscous fluid under the action of a localized dipolar-type forcing has been studied analytically and experimentally. The force dipole, with non-dimensional forcing amplitude Re, brings net zero momentum into the fluid and gives rise to the formation of a quadrupolar vortex: a system of two dipolar vortices moving apart. Experimentally, the action of a force dipole was modelled by a vertical cylinder oscillating horizontally in the shallow upper layer of a two-layer fluid. Two cases were studied: single quadrupoles and an array of quadrupoles. It is found that single quadrupoles develop in a self-similar manner: the length L and the translation velocity U of the quadrupolar vortex change with time as L - t1/2and U - t-1/2. These quantities are characterized by non-dimensional functions ai{Re) and fi (Re), respectively, which have been determined theoretically for small Revalues and experimentally for ite-values in the range 160-2200. To produce an array of quadrupoles an array of oscillating vertical rods was used. Two stages in the flow evolution were studied experimentally: the initial stage, when the interactions between the quadrupoles are weak, and the intermediate stage when the interactions play an essential role and the flow is (two-dimensionally) turbulent. It is found that at both stages the width H of the region with intense vortical motions increases with time as H - t1!2. A theoretical explanation of the experimental results is given. © 1995, Cambridge University Press. All rights reserved.

Description: An analysis is given for the electro-kinetic transport properties in a system consisting of a line of identical spheres placed equidistantly with their centres on the axis of a cylindrical tube containing a viscous fluid. Both the spheres and the wall of the tube are charged and a two-species symmetrical electrolyte with valence Z is present in the system. As a result of the charges on the surface of the spheres and on the surface of the tube electrical double layers will develop. When an electrical field is applied to the system an electrokinetic motion is induced. We will use the thin double layer theory (Dukhin & Derjaguin 1974; O’Brien 1983), valid for sufficiently high electrolyte concentration and where the polarization of the electrical double layer is included. Using a multipole expansion an infinite set of linear equations for the multipoles will be derived from which the electro-kinetic transport coefficients may be determined. These coefficients depend on the system parameters, such as the radius of the tube R, the radius of the sphere a, the separation between the spheres d, the Debije radius k-1, the zeta-potentials of the spheres ζpand of the wall of the tube ζwand the valency Z of the electrolyte. From these coefficients a relation is found between the pressure drop Δp per unit length and the drag force D on the spheres on one side and with the velocity U of the spheres, the total discharge Q and the applied electrical field E0on the other side. For some values for the system parameters we have numerically solved the infinite set of linear equations by truncation and calculated the transport coefficients. We have also calculated the streamlines for some situations. The plots of these streamlines show that depending on the conditions on the system vortices may appear. © 1995, Cambridge University Press. All rights reserved.

Description: The orientation of an ellipsoid falling in a viscoelastic fluid is studied by methods of perturbation theory. For small fall velocity, the fluid’s rheology is described by a second-order fluid model. The solution of the problem can be expressed by a dual expansion in two small parameters: the Reynolds number representing the inertial effect and the Weissenberg number representing the effect of the non-Newtonian stress. Then the original problem is split into three canonical problems: the zeroth-order Stokes problem for a translating ellipsoid and two first-order problems, one for inertia and one for second-order rheology. A Stokes operator is inverted in each of the three cases. The problems are solved numerically on a three-dimensional domain by a finite element method with fictitious domains, and the force and torque on the body are evaluated. The results show that the signs of the perturbation pressure and velocity around the particle for inertia are reversed by viscoelasticity. The torques are also of opposite sign: inertia turns the major axis of the ellipsoid perpendicular to the fall direction; normal stresses turn the major axis parallel to the fall. The competition of these two effects gives rise to an equilibrium tilt angle between 0° and 90° which the settling ellipsoid would eventually assume. The equilibrium tilt angle is a function of the elasticity number, which is the ratio of the Weissenberg number and the Reynolds number. Since this ratio is independent of the fall velocity, the perturbation results do not explain the sudden turning of a long body which occurs when a critical fall velocity is exceeded. This is not surprising because the theory is valid only for slow sedimentation. However, the results do seem to agree qualitatively with ‘shape tilting’ observed at low fall velocities. © 1995, Cambridge University Press. All rights reserved.

Description: Care is needed with algorithms for computer simulations of the Brownian motion of complex systems, such as colloidal and macromolecular systems which have internal degrees of freedom describing changes in configuration. Problems can arise when the diffusivity or the inertia changes with the configuration of the system. There are some problems in replacing very stiff bonds by rigid constraints. These problems and their resolution are illustrated by some artificial models; firstly in one dimension, then in the neighbourhood of an ellipse in two dimensions and finally for the trimer polymer molecule. © 1995, Cambridge University Press. All rights reserved.

Description: In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.

Description: The influence of different boundary conditions applied in the contact line region on the outer meniscus shape is analysed by means of a finite-element numerical simulation of the steady movement of a liquid-gas meniscus in a capillary tube. The free-surface steady shape is obtained by solving the unsteady creeping-flow approximation of the Navier-Stokes equations starting from some initial shape. Comparisons of the outer solutions obtained using two different inner models, together with that published by Lowndes (1980), indicate the relative insensitivity of the outer solution to the type of model utilized in the contact line region.

Description: A series of laboratory experiments was performed to investigate the overall mixing characteristics of oscillatory stratified flow past an isolated topography. The experiments were conducted by oscillating a right-circular cylinder in an otherwise quiescent linearly stratified fluid contained in a rectangular basin. The mixing was largely confined to the turbulent ‘core’ region around the cylinder. This mixed fluid was then injected into the fluid interior of the basin by numerous intrusive tongues. These intrusions were accompanied by return currents of unmixed stratified fluid into the turbulent core. The overall effect of this mixing process was to increase the potential energy of the fluid in the basin. An expression is derived to relate the rate of change of potential energy of the system to the basin-averaged buoyancy flux. This formula was then used to calculate the mean buoyancy flux from measurements of the rate of change of potential energy of the fluid system. Basin-averaged diapycnal eddy diffusivities for the experiments were evaluated and the results were found to be in good agreement with the predictions of a heuristic model based on the energetics of the mixing. Observations on the spreading of intrusions and the evolution of the density field are also presented. © 1995, Cambridge University Press. All rights reserved.

Description: Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented. A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematic problem. A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion. Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations. © 1995, Cambridge University Press. All rights reserved.

Description: An experimental and theoretical investigation of low Reynolds number, high subsonic Mach number, compressible gas flow in channels is presented. Nitrogen, helium, and argon gases were used. The channels were microfabricated on silicon wafers and were typically 100 μm wide, 104 μm long, and ranged in depth from 0.5 to 20 μm. The Knudsen number ranged from 10-3 to 0.4. The measured friction factor was in good agreement with theoretical predictions assuming isothermal, locally fully developed, first-order, slip flow. © 1995, Cambridge University Press. All rights reserved.

Description: In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl number Pr takes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh number R and the Ekman number E for the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions - the critical parameters for the onset of convection and the corresponding flow and temperature structure - are obtained and a new asymptotic relation between R and E is derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of small E and Pr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached. © 1995, Cambridge University Press. All rights reserved.

Description: A procedure based on energy stability arguments is presented as a method for extracting large-scale, coherent structures from fully turbulent shear flows. By means of two distinct averaging operators, the instantaneous flow field is decomposed into three components: a spatial mean, coherent field and random background fluctuations. The evolution equations for the coherent velocity, derived from the Navier-Stokes equations, are examined to determine the mode that maximizes the growth rate of volume-averaged coherent kinetic energy. Using a simple closure scheme to model the effects of the background turbulence, we find that the spatial form of the maximum energy growth modes compares well with the shape of the empirical eigenfunctions given by the proper orthogonal decomposition. The discrepancy between the eigenspectrum of the stability problem and the empirical eigenspectrum is explained by examining the role of the mean velocity field. A simple dynamic model which captures the energy exchange mechanisms between the different scales of motion is proposed. Analysis of this model shows that the modes which attain the maximum amplitude of coherent energy density in the model correspond to the empirical modes which possess the largest percentage of turbulent kinetic energy. The proposed method provides a means for extracting coherent structures which are similar to those produced by the proper orthogonal decomposition but which requires only modest statistical input. © 1995, Cambridge University Press. All rights reserved.

Description: The evolution of an internal gravity wave is investigated by direct numerical computations. We consider the case of a standing wave confined in a bounded (square) domain, a case which can be directly compared with laboratory experiments. A pseudo-spectral method with symmetries is used. We are interested in the inertial dynamics occurring in the limit of large Reynolds numbers, so a fairly high spatial resolution is used (1292 or 2572), but the computations are limited to a two-dimensional vertical plane. We observe that breaking eventually occurs, whatever the wave amplitude: the energy begins to decrease after a given time because of irreversible transfers of energy towards the dissipative scales. The life time of the coherent wave, before energy dissipation, is found to be proportional to the inverse of the amplitude squared, and we explain this law by a simple theoretical model. The wave breaking itself is preceded by a slow transfer of energy to secondary waves by a mechanism of resonant interactions, and we compare the results with the classical theory of this phenomenon: good agreement is obtained for moderate amplitudes. The nature of the events leading to wave breaking depends on the wave frequency (i.e. on the direction of the wave vector); most of the analysis is restricted to the case of fairly high frequencies. The maximum growth rate of the inviscid wave instability occurs in the limit of high wavenumbers. We observe that a well-organized secondary plane wave packet is excited. Its frequency is half the frequency of the primary wave, corresponding to an excitation by a parametric instability. The mechanism of selection of this remarkable structure, in the limit of small viscosities, is discussed. Once this secondary wave packet has reached a high amplitude, density overturning occurs, as well as unstable shear layers, leading to a rapid transfer of energy towards dissipative scales. Therefore the condition of strong wave steepness leading to wave breaking is locally attained by the development of a single small-scale parametric instability, rather than a cascade of wave interactions. This fact may be important for modelling the dynamics of an internal wave field. © 1995, Cambridge University Press. All rights reserved.

Description: The far-field sound generated by compressible co-rotating vortices is computed by direct computation of the unsteady compressible Navier-Stokes equations on a computational domain that extends to two acoustic wavelengths in all directions. The vortices undergo a period of co-rotation followed by a sudden merger. The directly computed far-field sound is compared to the prediction of the acoustic analogy due to Mohring (1978, 1979), a modified form of the analogy developed by Lighthill (1952), and an acoustic analogy derived by Powell (1964). All three predictions are in excellent agreement with the simulation. Results of far-field pressure fluctuations from an acoustically non-compact, co-rotating vortex pair are also presented. In this case, the vortex sound theory over-predicts the sound by 65 % in accordance with the analysis of Yates (1978). © 1995, Cambridge University Press. All rights reserved.

Description: For an anisotropic topographic feature in a large-scale flow, the orientation of the topography with respect to the flow will affect the vorticity production that results from the topography-flow interaction. This in turn affects the amount of form drag that the ambient flow experiences. Numerical simulations and perturbation theory are used to explore these effects of change in topographic orientation. The flow is modelled as a quasi-geostrophic homogeneous fluid on an /-plane. The topography is taken to be a hill of limited extent, with an elliptical cross-section in the horizontal. It is shown that, as a result of a basic asymmetry of the quasi-geostrophic flow, the strength of the form drag depends not only on the magnitude of the angle that the topographic axis makes with the oncoming stream, but also on the sign of this angle. For sufficiently low topography, it is found that a positive angle of attack leads to a stronger form drag than that for the corresponding negative angle. For strong topography, this relation is reversed, with the negative angle then resulting in the stronger form drag. © 1995, Cambridge University Press. All rights reserved.

Description: Instead of considering just the vertically averaged current and the vertically averaged concentration, a multi-mode model is derived in which more of the vertical structure can be computed directly rather than being lumped into a dispersion coefficient. Test cases, of laminar flows, are used to quantify the accuracy of the lowest non-trivial truncation (two modes) in replicating both the flow and the dispersion process. © 1995, Cambridge University Press. All rights reserved.

Description: Surface-tension-driven convection in a planar fluid layer is studied by numerical simulation of the three-dimensional time-dependent governing equations in the limit of infinite Prandtl number. Emphasis is placed on the spatial scale of weakly supercritical flows and on the generation of small-scale structures in strongly supercritical flows. The decrease of the size of weakly supercritical hexagonal convection cells that we find is in agreement with experimental results. In the case of high Marangoni number, discontinuities of the temperature gradient are formed between convection cells, producing a universal spectrum E - k-3 of the two-dimensional surface temperature field. The possibility of experimental verification is discussed on the basis of shadowgraph images calculated from the predicted hydrodynamic fields. © 1995, Cambridge University Press. All rights reserved.

Description: Direct numerical simulation of turbulent homogeneous shear flow is performed in order to clarify compressibility effects on the turbulence growth in the flow. The two Mach numbers relevant to homogeneous shear flow are the turbulent Mach number Mt and the gradient Mach number Mg. Two series of simulations are performed where the initial values of Mg and Mt are increased separately. The growth rate of turbulent kinetic energy is observed to decrease in both series of simulations. This ‘stabilizing’ effect of compressibility on the turbulent energy growth rate is observed to be substantially larger in the DNS series where the initial value of Mg is changed. A systematic comparison of the different DNS cases shows that the compressibility effect of reduced turbulent energy growth rate is primarily due to the reduced level of turbulence production and not due to explicit dilatational effects. The reduced turbulence production is not a mean density effect since the mean density remains constant in compressible homogeneous shear flow. The stabilizing effect of compressibility on the turbulence growth is observed to increase with the gradient Mach number Mg in the homogeneous shear flow DNS. Estimates of Mg for the mixing layer and the boundary layer are obtained. These estimates show that the parameter Mg becomes much larger in the high-speed mixing layer relative to the high-speed boundary layer even though the mean flow Mach numbers are the same in the two flows. Therefore, the inhibition of turbulent energy production and consequent 6 stabilizing ’ effect of compressibility on the turbulence (over and above that due to any mean density variation) is expected to be larger in the mixing layer relative to the boundary layer, in agreement with experimental observations. © 1995, Cambridge University Press. All rights reserved.

Description: The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed. © 1995, Cambridge University Press. All rights reserved.

Description: The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios Γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on Γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1,2,3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with Γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Description: Since Bödewadt's (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bödewadt's self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savaş. (1983, 1987) has studied the end wall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bödewadt layers. For certain conditions, Savaş has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bödewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further. Here, the impulsive spin-down problem is examined numerically for Savaş' (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.

Description: It is well known that subgrid models such as Smagorinsky's cannot be used for the spatially growing simulation of the transition to turbulence of flat-plate boundary layers, unless large-amplitude perturbations are introduced at the upstream boundary: they are over-dissipative, and the flow simulated remains laminar. This is also the case for the structure-function model (SF) of Métais & Lesieur (1992). In the present paper we present a sequel to this model, the filtered-structure-function (FSF) model. It consists of removing the large-scale fluctuations of the field before computing its second-order structure function. Analytical arguments confirm the superiority of the FSF model over the SF model for large-eddy simulations of weakly unstable transitional flows. The FSF model is therefore used for the simulation of a quasi-incompressible (M∞ = 0.5) boundary layer developing spatially over an adiabatic flat plate, with a low level of upstream forcing. With the minimal resolution 650 × 32 × 20 grid points covering a range of streamwise Reynolds numbers Rex1 ε [3.4 × 105, 1.1 × 106], transition is obtained for 80 hours of time-processing on a CRAY 2 (whereas DNS of the whole transition takes about ten times longer). Statistics of the LES are found to be in acceptable agreement with experiments and empirical laws, in the laminar, transitional and turbulent parts of the domain. The dynamics of low-pressure and high-vorticity distributions is examined during transition, with particular emphasis on the neighbourhood of the critical layer (defined here as the height of the fluid travelling at a speed equal to the phase speed of the incoming Tollmien–Schlichting waves). Evidence is given that a subharmonic-type secondary instability grows, followed by a purely spanwise (i.e. time-independent) mode which yields peak-and-valley splitting and transition to turbulence. In the turbulent region, flow visualizations and local instantaneous profiles are provided. They confirm the presence of low- and high-speed streaks at the wall, weak hairpins stretched by the flow and bursting events. It is found that most of the vorticity is produced in the spanwise direction, at the wall, below the high-speed streaks. Isosurfaces of eddy viscosity confirm that the FSF model does not perturb transition much, and acts mostly in the vicinity of the hairpins.

Description: Results of numerical computations are presented of time-dependent three-dimensional convection flows in a horizontal layer heated from below which evolve from the oscillatory blob instability of steady two-dimensional rolls. It is shown that the heat transport is typically increased in the transition to blob convection. Oscillatory blob convection exists in the forms of standing or travelling blob convection. The latter type of solution represents the stable form bifurcating supercritically at the Rayleigh number R11 for the onset of the oscillatory blob instability. In contrast to standing blob convection travelling blob convection exhibits a mean flow. © 1995, Cambridge University Press. All rights reserved.

Description: Thermohaline convection in a salt water loop is discussed. Fluid temperature is affected by relaxation on the loop surface and fluid salinity by a freshwater flux through the loop surface. In addition, other boundary conditions on salinity, such as the equivalent virtual salt flux or salinity relaxation condition, are examined and the dynamic role of diffusion in thermohaline convection is analysed. Both analytical and numerical analyses indicate that the system behaviour depends sensitively on the nature of the salinity boundary condition. For the saline-only loop model, analysis indicates that perturbations are advected by the mean flow, and flow stability is independent of the strength of the boundary forcing. In the full thermohaline loop problem, the virtual salt flux formulation accurately mirrors the freshwater flux results when the system is in the thermal mode. However, these formulations can differ substantially when the system is in the haline mode, especially in the strongly forced, weakly diffusive limit. For both types of loop configuration, salinity profiles governed by freshwater flux have scales determined by the internal parameters, while virtual salt flux profiles necessarily reflect the lengthscales of applied boundary conditions. Negative salinities can also appear under virtual salt flux owing to the inaccuracies inherent in the approximation, while freshwater flux ensures positive-definite salinity values. Our analysis supports the use of the physically more accurate freshwater flux boundary conditions when simulating thermohaline circulation. © 1995, Cambridge University Press. All rights reserved.

Description: A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region. © 1995, Cambridge University Press. All rights reserved.

Description: The interaction between a zero-pressure-gradient turbulent boundary layer and a pair of strong, common-flow-down, streamwise vortices with a sizeable velocity deficit is studied by large-eddy simulation. The subgrid-scale stresses are modelled by a localized dynamic eddy-viscosity model. The results agree well with experimental data. The vortices drastically distort the boundary layer, and produce large spanwise variations of the skin friction. The Reynolds stresses are highly three-dimensional. High levels of kinetic energy are found both in the upwash region and in the vortex core. The two secondary shear stresses are significant in the vortex region, with magnitudes comparable to the primary one. Turbulent transport from the immediate upwash region is partly responsible for the high levels of turbulent kinetic energy in the vortex core; its effect on the primary stress (úú) is less significant. The mean velocity gradients play an important role in the generation of (úú) in all regions, while they are negligible in the generation of turbulent kinetic energy in the vortex core. The pressure - strain correlations are generally of opposite sign to the production terms except in the vortex core, where they have the same sign as the production term in the budget of (úú). The results highlight the limitations of the eddy-viscosity assumption (in a Reynolds-averaged context) for flows of this type, as well as the excessive diffusion predicted by typical turbulence models.

Description: The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

Description: The effects of molecular diffusivities of heat and mass on the counter-gradient scalar and momentum transfer in strongly stable stratification are experimentally investigated in unsheared and sheared stratified water mixing-layer flows downstream of turbulence-generating grids. Experiments are carried out in two kinds of stably stratified water flows. In the case of thermal stratification, the difference between the turbulent fluxes of an active scalar (heat with the Prandtl number of Pr ≈ 6) and a passive scalar (mass with the Schmidt number of Sc ≈ 600) is investigated. In the case of salt stratification, the effects of the molecular diffusion of the active scalar (salt) with a very high Schmidt number of Sc ≈ 600 on the counter-gradient scalar transfer is studied. Comparisons of the effects of molecular diffusivities are also made between thermally stratified water and air (Pr ≈ 0.7) flows. Further, the effects of mean shear on the counter-gradient scalar and momentum transfer are investigated for both stratified cases. Instantaneous temperature, concentration and streamwise and vertical velocities are simultaneously measured using a combined technique with a resistance thermometer, a laser-induced fluorescence method, and a laser-Doppler velocimeter with high spatial resolution. Turbulent scalar fluxes, joint probability density functions, and cospectra are estimated.The results of the first case show that both active heat and passive mass develop counter-gradient fluxes but that the counter-gradient flux of passive mass is about 10% larger than that of active heat, mostly due to molecular diffusion effects at small scales. The counter-gradient scalar transfer mechanism in stable stratification can be explained by considering the relative balance between the available potential energy and the turbulent kinetic energy as in Schumann (1987). In thermally and salt-stratified water mixing-layer flows with the active scalars of high Prandtl and Schmidt numbers, the buoyancy-induced motions with finger-like structures first contribute to the counter-gradient scalar fluxes at small scales, and then the large-scale motions, which bring fluid back to its original levels, generate the counter-gradient fluxes at large scales. The contribution of the small-scale motions to the counter-gradient fluxes in stratified water flows is quite different from that in stratified air flows. The higher Prandtl or Schmidt number of the active scalar generates both the stronger buoyancy effects and the longer time-oscillation period of the counter-gradient scalar fluxes. The time-oscillation occurs at large scales but the counter-gradient fluxes at small scales persist without oscillating. The mean shear acts to reduce the counter-gradient scalar and momentum transfer at large scales, and therefore the counter-gradient fluxes in sheared stratified flows can be seen only in very strong stratification. The behaviour of the counter-gradient momentum flux in strong stratification is quite similar to that of the counter-gradient scalar flux.

Description: The improvement of heat transfer conditions in liquid-metal magnetohydrodynamic (MHD) flows is of prime importance for self-cooled fusion blanket design concepts. For many years the research was based on stationary inertialess assumptions since it was expected that time-dependent inertial flows would be suppressed by strong electromagnetic damping, especially in the extreme range of fusion relevant parameters. In the present analysis the stationary inertialess assumptions are abandoned. Nevertheless, the classical ideas usually used to obtain inertialess asymptotic solutions are drawn on. The basic inertial equations are reduced to a coupled two-dimensional problem by analytical integration along magnetic field lines. The magnetic field is responsible for a quasi-two-dimensional flow; the non-uniform distribution of the wall conductivity creates a wake-type profile, the MHD effect reducing to a particular forcing and friction. The solution for the two-dimensional variables, the field aligned component of vorticity, the stream function, and the electric potential are obtained by numerical methods. In a flat channel with non-uniform electrical wall conductivity, time-dependent solutions similar to the Kármán vortex street behind bluff bodies are possible. The onset of the vortex motion, i.e. the critical Reynolds number depends strongly on the strength of the magnetic field expressed by the Hartmann number. Stability analyses in viscous hydrodynamic wakes often use the approximation of a unidirectional flow which does not take into account the spatial evolution of the wake. The present problem exhibits a wake-type basic flow, which does not change along the flow path. It represents, therefore, an excellent example to which the simple linear analysis on the basis of Orr-Sommerfeld-type equations applies exactly. Once unstable, the flow first exhibits a regular time periodic vortex pattern which is rearranged further downstream. One can observe an elongation, pairing, or sometimes more complex merging of vortices. All these effects lead to larger flow structures with lower frequencies. The possibility for a creation and maintenance of time-dependent vortex-type flow pattern in MHD flows is demonstrated.

Description: Experiments have been performed, in capillary tubes, on the displacement of a viscous fluid (glycerine) by a less viscous one (a glycerine-water mixture) with which it is miscible in all proportions. A diagnostic measure of the amount of viscous fluid left behind on the tube wall has been found, for both vertical and horizontal tubes, as a function of the Péclet (Pe) and Atwood (At) numbers, as well as a parameter that is a measure of the relative importance of viscous and gravitational effects. The asymptotic value of this diagnostic quantity, for large Pe and an At of unity, has been found to agree with that found in immiscible displacements, while the agreement with the numerical results of Part 2 (Chen & Meiburg 1966), over the whole range of At, is very good. At values of the average Pe greater than 1000 a sharp interface existed so that it was possible to make direct comparisons between the present results and a prior experiment with immiscible fluids, in particular an effective surface tension at the diffusing interface could be evaluated. The effect of gravity on the amount of viscous fluid left on the tube wall has been investigated also, and compared with the results of Part 2. A subsidiary experiment has been performed to measure both the average value of the diffusion coefficient between pure glycerine and several glycerine-water mixtures, in order to be able to calculate a representative Péclet number for each experiment, and the local value as a function of the local concentration of glycerine, in the dilute limit.

Description: Thermoacoustic refrigeration occurs in periodic flow in a duct with heat transfer within the fluid and to the tube. This study considers the periodic limit cycle with large pressure oscillations that is obtained in a tube when prescribed, phase-shifted, periodic velocities at the tube ends, at frequencies lower than acoustic eigenmodes, sweep a length comparable to the tube length. The temperature differences between the two ends are of arbitrary magnitude, heat transfer in the transverse direction within the fluid is assumed to be very effective and the thermal mass of the wall is large. The geometry is two-dimensional, axisymmetric, and conduction is accounted for, not only in the fluid, but also with and within the tube wall. A perturbation solution valid in a local near-isothermal limit determines the equilibrium longitudinal temperature profile that is reached at the periodic regime, the pressure field including longitudinal gradients, and the longitudinal enthalpy flux. Results are presented for tubes open at both ends and also with one end closed. In the latter case, a singularity occurs in the temperature at the closed end, with behaviour identical to Rott's result for acoustic flow with small pressure amplitude. Other new results obtained for tubes open at both ends show that when velocities at both ends are in opposite phase, internal singularities in the temperature profiles may occur.

Description: Following the study in Part 1 of cross-flow and other non-symmetric effects on vortex/wave interactions in boundary layers, the present Part 2 applies the ideas of Part 1 and related works to an incident axisymmetric flow supplemented by a small swirl or azimuthal velocity. This is with a view to possibly increasing understanding of vortex breakdown. The wave components involved are predominantly inviscid Rayleigh-like ones. The presence of the swirl leads to extra features and complications associated mainly with extra logarithmic contributions but for the dominant interactions essentially the same equations as in Part 1 are found. These dominant nonlinear interactions must be based on azimuthal wavenumbers of ± 1 in the case of the Squire jet with swirl. In contrast to Part 1, which consisted mainly of an analysis of the quasi-bounded solutions, a representative set of numerical solutions of the full integro-differential amplitude equations is presented, for realistic axial and swirl velocity profiles. The work points also to the influence of further increases in the incident swirl.

Description: An asymptotic analysis for the long-time unsteady laminar far wake of a bluff body due to a step change in its travelling velocity from U1 to U2 is presented. For U1 ≥ 0 and U2 〉 0, the laminar wake consists of a new wake of volume flux Q2 corresponding to the current velocity U2, an old wake of volume flux Q1 corresponding to the original velocity U1, and a transition zone that connects these two wakes. The transition zone acts as a sink (or a source) of volume flux (Q2 - Q1) and is moving away from the body at speed U2. Streamwise diffusion is negligible in the new and old wakes but a matched asymptotic expansion that retains the streamwise diffusion is required to determine the vorticity transport in the transition zone. A source of volume flux Q2 located near the body needs to be superposed on the unsteady wake to form the global flow field around the body. The asymptotic predictions for the unsteady wake velocity, unsteady wake vorticity, and the global flow field around the body agree well with finite difference solutions for flow over a sphere at finite Reynolds numbers. The long-time unsteady flow structures due to a sudden stop (U2 = 0) and an impulsive reverse (U1 U2 〈 0) of the body are analysed in detail based on the asymptotic solutions for the unsteady wakes and the finite difference solutions. The elucidation of the long-time behaviour of such unsteady flows provides a framework for understanding the long-time particle dynamics at finite Reynolds number.

Description: A theory and simulation code are developed to study non-steady sources as means to control sonic booms of supersonic aircraft. A key result is that the source of sonic boom pressure is not confined to the length of the aircraft but occupies an extensive segment of the flight path. An aircraft in non-steady flight functions as a synthetic aperture antenna, generating complex acoustic waves with no simple relation to instantaneous volume or lift distributions. The theory applies linear acoustics to slender non-steady sources but requires no farfield approximation. The solution for pressure contains a term not seen in Whitham's theory for sonic booms of distant supersonic aircraft. The term describes a pressure field that decays algebraically behind the Mach cone and, in the case of steady flight, integrates to a ground load equal to the weight of the aircraft. The algebraic term is separate from those that describe the sonic boom. Two non-steady source phenomena are evaluated: periodic velocity changes (surge), and periodic longitudinal lift redistribution (slosh). Surge can attenuate a sonic boom and covert it into prolonged weak reverberation, but accelerations needed to produce the phenomenon seem too large for practical use. Slosh may be practical and can alter sonic booms but does not, on average, result in boom attenuation. The conclusion is that active sonic boom abatement is possible in theory but maybe not practical.

Description: The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics. The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg-de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg-de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg-de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant. Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.

Description: Experiments and simulations of a travelling wave state of incompressible Newtonian flow in a curved duct of square cross-section are presented. The travelling wave mode develops from the well-documented steady four-cell flow state and is characterized by oscillations of the two Dean vortices near the centre of the outer wall. The oscillations were induced by a carefully positioned pin at 5° from the inlet of the curved section along the symmetry line of the cross-section. It was shown that the travelling wave state is characteristic for curved duct flow and that the pin made it possible to observe the oscillations within the 270° long curved duct. Travelling waves were observed at flow rates above Dn = 170 (Dn = Re/(R/a)l/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension. The curvature ratio, R/a, is 15.1). If no other disturbances are imposed, the oscillations are the result of the selective amplification of random disturbances in the flow, leading to a broad frequency spectrum. The travelling wave was found to lock in to an imposed periodic disturbance at a selected frequency. The flow structure of the locked state was investigated in detail, using flow visualization and a one-component laser Doppler anemometer to measure streamwise or spanwise velocities. Direct numerical simulations using the package CFDS-FLOW3D are in very good agreement with the experiments and confirm the existence of a fully developed, streamwise-periodic travelling wave state. The inflow region between the two Dean vortices, which transports low-speed fluid away from the outer wall, creates strongly inflectional spanwise profiles of the streamwise velocity. Similarities with twisting vortices in a curved channel and sinuous oscillations of Görtler vortices show that the travelling waves observed here result from a secondary shear instability of these spanwise inflectional profiles.

Description: The anisotropic behaviour of density-gradient fluctuations in stably stratified grid turbulence and the consequences for simplified (isotropic) estimates of scalar dissipation rates χ were experimentally studied in a thermally stratified wind tunnel at moderate Reynolds numbers (Reλ ≃ 20). Strong stable stratifications were attained, with Brunt-Väisälä frequency N as high as 4 rad s-1. The correlation method was used to estimate the mean-square cross-stream and streamwise density gradients. Cross-stream gradients were measured using two cold wires. The mean-square vertical gradients were found to become larger than the streamwise gradients by as much as a factor of 2.2 for the largest dimensionless buoyancy times (Nt = 7). This corresponds to a 40% error in the scalar dissipation estimates based on ∂θ/∂x alone, and assuming the validity of the isotropic relations. Gradient spectral relations show that this buoyancy-induced anisotropy persists at all length scales. Better closure of the scalar variance balance was attained than in previously reported measurements by other researchers. This is attributed to our use of cold-wire temperature sensors having larger length-to-diameter ratio than used in the previous measurements.

Description: A two-dimensional Boussinesq equation, utt-uxx+ 3(u2)xx - uxxxx - uyy = 0, is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation. A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

Description: Individual travelling cavitation bubbles were examined as they interacted with the flow over a two-dimensional hydrofoil. Each bubble was produced from a single nucleus created upstream of the hydrofoil, and the flow near the hydrofoil was visualized using particle imaging velocimetry (PIV). Travelling bubbles were observed to generate a local region of turbulence as they passed close to an unstable laminar boundary layer. By producing a locally turbulent region, the bubbles could temporarily sweep away a portion of attached cavitation at the foil midchord. Also, the bubbles were observed to strongly interact with a turbulent boundary layer, producing local regions of patch cavitation.

Description: The problem of gas motion in a tube closed at one end and driven at the other by an oscillating piston is studied theoretically. When the piston vibrates with a finite amplitude at the first acoustic resonance frequency, periodic shock waves are generated, travelling back and forth in the tube. A perturbation method, based on a small Mach number, M and a global mass conservation condition, is employed to formulate a solution of the problem in the form of two standing waves separated by a jump (shock front). By expanding the equations of motion in a series of a small parameter ε = M1/2, all hydrodynamic properties are predicted with an accuracy to second-order terms, i.e. to ε2. It is found that the first-order solution coincides with the previous theories of Betchov (1958) and Chester (1964), while additional terms predict a non-homogeneous time-averaged pressure along the tube. This prediction compares favourably with experimental results from the literature. The importance of the phenomenon is discussed in relation to different transport processes in resonance tubes.

Description: Two-dimensional mixing driven by an instability mechanism which is concentrated near one of the boundaries is considered, with particular application to Langmuir circulations driven by a wave spectrum. The question of how to define the equivalent of the Rayleigh number is attacked using the energy balance equations and simple truncated models of the instability. Given a particular horizontal wavelength for the disturbance, the strength of the forcing on the cells, and thus the growth rate, is determined by a tradeoff between maximizing the depth-averaged forcing and maximizing the depth of penetration. As a result of this tradeoff, long-wavelength cells grow more slowly, but penetrate more deeply and have a larger equivalent Rayleigh number. At finite amplitude, these long-wavelength cells come to dominate the flow field. The depth of penetration of, and density transport accomplished by, Langmuir cells is considered as a function of the mean stratification and diffusion. An application to oceanic mixed layers is considered assuming the Mellor-Yamada 2 1/2-level turbulence closure model to define the background level of turbulent mixing. For many realistic cases, Langmuir cells are predicted to dominate the vertical transport of momentum and density.

Description: A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.

Description: A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant - unlike the deep-water case - in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities 'contaminate' most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves 'close' to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d = 1; the study is performed down to depth d = 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of shortcrested wave fields will not be observable, although most of the short-crested wave fields will be.

Description: In a series of laboratory experiments the growth of double-diffusive salt fingers from an initial configuration of two homogeneous reservoirs with salt in the lower and sugar in the upper layer was investigated. For most of the experiments the stability ratio was between 2.5 and 3, where the latter value is at the upper limit (the ratio of salt to sugar diffusivities) for which fingers can exist. In these experiments long slender fingers are generated at the interface. Essentially all theories or physical bases for models of salt fingers presuppose such a configuration of long fingers. Our measurements show that the length of fingers at high stability ratio increases with time like i'/2, with a coefficient that is consistent with the diffusive spread of the faster diffusing component (salt). When the initial stability ratio is closer to unity, fingers penetrate into the reservoirs very rapidly carrying with them large anomalies of salt and sugar which give rise to convective overturning of the reservoirs. The convection sweeps away the ends of the fingers, and when it is intense enough (as it is when the sugar anomaly is large) it can reduce the finger height to a value less than the width. After this initial phase the finger length grows linearly with time as has been found in previous studies. These results show that salt fingers can evolve in quite different ways depending on the initial stability ratio and must cast doubt on the use of simple similarity arguments to parameterize the heat and salt fluxes produced by fingers.

Description: We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure-velocity correlation to the single-point pressure-strain tensor, is also derived. This law shows that the two-point pressure-velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity-enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with εω 2/3 in the enstrophy inertial range, εω being the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to the k-1 law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).

Description: Simulations of decaying two-dimensional turbulence suggest that the one-point vorticity density has the self-similar form ρω ∼ t f(ωt) implied by Batchelor's (1969) similarity hypothesis, except in the tails. Specifically, similarity holds for |ω| 〈 ωm, while ρω falls off rapidly above. The upper bound of the similarity range, ωm, is also nearly conserved in high-Reynolds-number hyperviscosity simulations and appears to be related to the average amplitude of the most intense vortices (McWilliams 1990), which was an important ingredient in the vortex scaling theory of Carnevale et al. (1991). The universal function f also appears to be hyperbolic, i.e. f(x) ∼ c/2|x|1+qc, for |x| 〉 x*, where qc = 0.4 and x* = 70, which along with the truncated similarity form implies a phase transition in the vorticity moments matrix presented between the self-similar 'background sea' and the coherent vortices. Here cq and c are universal. Low-order moments are therefore consistent with Batchelor's similarity hypothesis whereas high-order moments are similar to those predicted by Carnevale et al. (1991). A self-similar but less well-founded expression for the energy spectrum is also proposed. It is also argued that ωs = x*/t represents 'mean sea-level', i.e. the (average) threshold separating the vortices and the sea, and that there is a spectrum of vortices with amplitudes in the range (ωs, ωm). The total area occupied by vortices is also found to remain constant in time, with losses due to mergers of large-amplitude vortices being balanced by gains due to production of weak vortices. By contrast, the area occupied by vortices above a fixed threshold decays in time as observed by McWilliams (1990).

Description: Motivated by the study of blood flow in the major coronary arteries, which are situated on the outer surface of the pumping heart, we analyse flow of an incompressible Newtonian fluid in a tube whose curvature varies both along the tube and with time. Attention is restricted to the case in which the tube radius is fixed and its centreline moves in a plane. Nevertheless, the governing equations are very complicated, because the natural coordinate system involves acceleration, rotation and deformation of the frame of reference, and their derivation forms a major part of the paper. Then they are applied to two particular, relatively simple examples: a tube of uniform but time-dependent curvature; and a sinuous tube, representing a small-amplitude oscillation about a straight pipe. In the former case the curvature is taken to be small and to vary by a small amount, and the solution is developed as a triple power series in mean curvature ratio δ0, curvature variation ε and Dean number D. In the latter case the Reynolds number is taken to be large and a linearized solution for the perturbation to the flow in the boundary layer at the tube wall is obtained, following Smith (1976a). In each case the solution is taken far enough that the first non-trivial effects of the variable curvature can be determined. Results are presented in terms of the oscillatory wall shear stress distribution and, in the uniform curvature case, the contribution of steady streaming to the mean wall shear stress is calculated. Estimation of the parameters for the human heart indicates that the present results are not directly applicable, but point the way for future work.

Description: The aim of the paper is to present a new transport process which is likely to have great importance for understanding the internal constitution of the stars.In order to set the problem in context, we first give a short presentation of the physical properties of the Sun and stars, described usually under the names Standard Solar Model or Standard Stellar Models (SSM). Next we show that an important shortcoming of SSM is that they do not explain the age dependence of the lithium deficiency of stars of known age: stars of galactic clusters and the Sun. It was suggested a long time ago that the presence of a macroscopic diffusion process in the radiative zone should be assumed, below the surface convective zone of solar-like stars. It is then possible for the lithium present in the convective zone to be carried to the thermonuclear burning level below the convective zone. The first assumption was that differential rotation generates turbulence and therefore that a turbulent diffusion process takes place. However, this model predicts a lithium abundance which is strongly rotation dependent, contrary to the observations. Furthermore, as the diffusion coefficient is large all over the radiative zone, it prevents the possibility of gravitational separation by diffusion and consequently leads to the impossibility of explaining the difference in helium abundance between the surface and the centre of the Sun. The consequence is obviously that we need to take into account another physical process.Stars having a mass M 〈 1.3M[odot ] have a convective zone which begins close to the stellar surface and extends down to a depth which is an appreciable fraction of the stellar radius. In the convective zone, strong stochastic motions carry, at least partially, heat transfer. These motions do not vanish at the lower boundary and generate internal waves into the radiative zone. These random internal waves are at the origin of a diffusion process which can be considered as responsible for the diffusive transport of lithium down to the lithium burning level. This is certainly not the only physical process responsible for lithium deficiency in main sequence stars, but its properties open the way to a completely consistent analysis of lithium deficiency.The model of generation of gravity waves is based on a model of heat transport in the convective zone by diving plumes. The horizontal component of the turbulent motion at the boundary of the convective zone is assumed to generate the horizontal motion of internal waves. The result is a large horizontal component of the diffusion coefficient, which produces in a short time an horizontally uniform chemical composition. It is known that gravity waves, in the absence of any dissipative process, cannot generate vertical mixing. Therefore, the vertical component of the diffusion coefficient is entirely dependent on radiative damping. It decreases quickly in the radiative zone, but is large enough to be responsible for lithium burning.Owing to the radial dependence of velocity amplitude, the diffusion coefficient increases when approaching the stellar centre. However, very close to the centre, nonlinear dissipative and radiative damping of internal waves become large and the diffusion coefficient vanishes at the very centre.

Description: Axisymmetric gravity currents that result when a dense suspension intrudes under a lighter ambient fluid are studied theoretically and experimentally. The dynamics of and deposition from currents flowing over a rigid horizontal surface are determined for the release of either a fixed volume or a constant flux of a suspension. The dynamics of the current are assumed to be dominated by inertial and buoyancy forces, while viscous forces are assumed to be negligible. The fluid motion is modelled by the single-layer axisymmetric shallow-water equations, which neglect the effects of the overlying fluid. An advective transport equation models the distribution of particles in the current, and this distribution determines the local buoyancy force in the shallow-water equations. The transport equation is derived on the assumption that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through a viscous sublayer at the bottom of the current. No adjustable parameters are needed to specify the theoretical model. The coupled equations of the model are solved numerically, and it is predicted that after an early stage both constant-volume and constant-flux, particle-driven gravity currents develop an internal bore which separates a supercritical particle-free region upstream from a subcritical particle-rich region downstream near the head of the current. For the fixed-volume release, an earlier bore is also predicted to occur very shortly after the initial collapse of the current. This bore transports suspended particles away from the origin, which results in a maximum in the predicted deposition away from the centre.To test the model several laboratory experiments were performed to determine both the radius of an axisymmetric particle-driven gravity current as a function of time and its deposition pattern for a variety of initial particle concentrations, particle sizes, volumes and flow rates. For the release of a fixed volume and of a constant flux of suspension, the comparisons between the experimental results and the theoretical predictions are fairly good. However, for the current of fixed volume, we did not observe the bore predicted to occur shortly after the collapse of the current or the resulting maximum in deposition downstream of the origin. This is unlike the previous study of Bonnecaze et al. (1993) on two-dimensional currents, in which a strong bore was observed during the slumping phase. The radial extent R of the deposit from a fixed-volume current is accurately predicted by the model, and for currents whose particles settle sufficiently slowly, we find that R = 1.9(g′0V3 / v2s)1/8, where V is the volume of the current, vs is the settling velocity of a particle in the suspension and g’0 is the initial reduced gravity of the suspension.

Description: The problem of determining the particle-phase stress in potential flow has been examined recently using two different procedures by Sangani & Didwania (1993a) and by Bulthuis (Appendix C of Zhang & Prosperetti 1994). The present study corrects errors in the expression given by Sangani & Didwania, recasts the expression given by Bulthuis in a form suitable for computation, and shows the equivalence of the results obtained by the two methods. © 1995, Cambridge University Press. All rights reserved.

Description: Beyond a short transition region near the inlet, waves on a falling film evolve into distinct pulse-like solitary waves that dominate all subsequent interfacial dynamics. Numerical and physical experiments indicate that these localized structures can attract and repel each other. Attractive interaction through the capillary ripples of the pulses causes two pulses to coalesce into a bigger pulse which accelerates and precipitates further coalescence. This binary interaction between an ‘excited’ pulse after coalescence and its smaller front neighbour is the key mechanism that drives the observed wave dynamics. From symmetry arguments, two dominant modes for a solitary pulse are obtained and used to develop an inelastic coherent structure theory for binary interaction between an excited pulse and its front neighbour. The theory offers a simple dynamical system that quantitatively describes the binary interaction and promises to elucidate the complex wave dynamics on a falling film. © 1995, Cambridge University Press. All rights reserved.

Description: To determine a suitable boundary-condition model for the contact line in oscillatory flow, an upright plate, oscillated vertically with sinusoidal motion in dye-laden water with an air interface, is considered experimentally. Constrained by the desirability of a two-dimensional flow field, eight frequencies in the 1-20 Hz range, each with seven different stroke amplitudes (0.5-6 mm) are chosen. The Reynolds number varies from 1.6 to 1878.3 in the experiments, large relative to the Reynolds number in the conventional uni-directional contact-line experiments (e.g. Dussan V.'s 1974 experiments). To facilitate prediction, a high-speed video system is used to record the plate displacement, the contact-line displacement, and the dynamic behaviour of the contact angle. Several interesting contact-line phenomena are shown in the present results. An expression for A, the dimensionless capillary coefficient, is formulated such that the dynamic behaviour at the contact line is predicted reasonably well. A particle-tracking-velocimetry (PTV) technique is used to detect particle trajectories near the plate such that the boundary condition along the entire plate can be modelled. Two sets of PTV experiments are conducted. One set is for stick contact-line motion, the other set is for stick-slip contact-line motion. The results from the PTV experiments show that a vortex is formed near the meniscus in the stick-slip contact-line experiments; however, in the stick contact-line experiments, no such vortex is present. Using the present experimental results, a model is developed for the boundary condition along the vertically oscillating vertical plate. In this model, slip occurs within a specific distance from the contact line while the flow obeys the no-slip condition outside this slip region. Also, the mean slip length is determined for each experimental stroke amplitude. © 1995, Cambridge University Press. All rights reserved.

Description: Strongly nonlinear vortex-Tollmien-Schlichting-wave interaction equations are derived for the case where the undisturbed motion represents the developing flow in a circular pipe. The effect upon the equations of moving the wave input position further downstream is investigated and the development of the flow is found to be accelerated by increasing the size of the wave disturbance. Numerical solutions of the three-dimensional interaction equations are presented and indicate that the form of interaction considered here appears to promote the three-dimensionality as the flow develops downstream. It is shown that one of the interactions considered here can develop within an initially two-dimensional Blasius boundary layer.