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 Λ 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.

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.

The vortex shedding and wake development of a two-dimensional viscous incompressible flow generated by a circular cylinder which begins its rotation and translation impulsively in a stationary fluid is investigated by a hybrid vortex scheme at a Reynolds number of 1000. The rotational to translational speed ratio α varies from 0 to 6. The method used to calculate the flow can be considered as a combination of the diffusion-vortex method and the vortex-in-cell method. More specifically, the full flow field is divided into two regions: near the body surface the diffusion-vortex method is used to solve the Navier–Stokes equations, while the vortex-in-cell method is used in the exterior inviscid domain. Being more efficient, the present computation scheme is capable of extending the computation to a much larger dimensionless time than those reported in the literature.

The time-dependent pressure, shear stress and velocity distributions, the Strouhal number of vortex shedding as well as the mean lift, drag, moment and power coefficients are determined together with the streamline and vorticity flow patterns. When comparison is possible, the present computations are found to compare favourably with published experimental and numerical results. The present results seem to indicate the existence of a critical α value of about 2 when a closed streamline circulating around the cylinder begins to appear. Below this critical α, Kármán vortex shedding exists, separation points can be found, the mean lift and drag coefficients and Strouhal number increase almost linearly with α. Above α ≈ 2, the region enclosed by the dividing closed streamline grows in size, Kármán vortex shedding ceases, the flow structure, pressure and shear stress distributions around the cylinder tend towards self-similarity with increase α, and lift and drag coefficients approach asymptotic values. The optimum lift to drag ratio occurs at α ≈ 2. The present investigation confirms Prandtl's postulation of the presence of limiting lift force at high α, and thus the usefulness of the Magnus effect in lift generation is limited.

The results show that the present method can be used to calculate not only the global characteristics of the separated flow, but also the precise evolution with time of the fine structure of the flow field.

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.

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.

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 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.

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−ε turbulence model, even when a recently proposed compressibility correction was applied.

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-mental 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 B-lines 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.

The modelling of small-amplitude pressure waves in dilute single- or multi-component fogs by means of averaged equations is considered. The problem is cast in a singular-perturbation framework in which the suspended droplets are the singularities. This point of view simplifies the local problem in the vicinity of the droplets. Matching in the overlap region provides the coupling with the averaged fields. Among the advantages of the method is the fact that the leading-order effects are clearly identified. In particular it is shown that, for low-amplitude waves and far below the fluid's critical point, phase change effects only start to be important when the vapour mean free path becomes comparable with the drop radius and dominate for yet smaller drops.

This present method for the derivation of effective equations appears to be of general applicability to a variety of multi-phase situations and is illustrated in detail.

This paper analyses the mass transport velocity in a two-layer system induced by the action of progressive waves. First the movement inside the two layers is obtained. Next the mass transport of spatially decaying waves is calculated by solving the momentum and mass conservation equations in the Lagrangian coordinate system. Two different physical situations are analysed: the first is waves in a closed channel and the second is waves in an unbounded domain, where the steady-state mass flux may be non-zero. The influence of the viscous properties of the lower layer on the mass transport in both layers is studied. Comparison with the experiments of Sakakiyama & Bijker (1989) in a water-mud system shows good agreement. The results show that the mass transport velocity can be quite different from the velocity given by the rigid bed theory, depending on the physical properties of the lower layer.

An evolution equation is derived that describes the nonlinear development of the interface between two viscoelastic fluids flowing, under the action of imposed pressure gradient and gravity, in a vertical channel. The channel walls are kept at different temperatures, resulting in heat transfer across the layers. The equation, based on the lubrication approximation, models the effects of stratifications in density, viscosity, elasticity, shear thinning, and thermal conductivity. It also describes the capillary and thermocapillary effects, as well as the sensitivity of viscosities to temperature. Linear-stability analysis is performed based on the evolution equation to understand the competing effects of viscous, elastic, and Marangoni instabilities. Particular attention is paid to the active control of the interfacial instabilities through the thermocapillarity.

The role of wave-induced separated flow in solute transport above a rippled bed is studied from numerical solutions to the two-dimensional Navier–Strokes equations and the advection-diffusion equation. A horizontal ambient flow that varies sinusoidally in time is imposed far above the bed, and a constant concentration difference between the upper and lower boundaries of computation is assumed. The computed flow field is the sum of an oscillatory rectilinear flow and a vortical flow which is periodic both in time and in the horizontal. Poincaré sections of this flow suggest chaotic mixing. Vertical lines of fluid particles above the crest and above the trough deform into whorls and tendrils, respectively, in just one wave period. Horizontal lines near the bottom deform into Smale horseshoe patterns. The combination of high shear and vortex-induced normal velocity close to the sediment surface results in large net displacements of fluid particles in a period. The resulting advective transport normal to the bed can be higher than molecular diffusion from well within the viscous boundary layer up to a few ripple heights above the bed. When this flow field is applied to the transport equation of a passive scalar, two distinct features – regular temporal oscillations in concentration and a linear time-averaged vertical concentration profile – are found immediately above the bed. These features have also been observed previously in field measurements on oxygen concentration. Advective transport is shown to be dominant even in the region where the time-averaged concentration profile is linear, a region where vertical solute transport has often been estimated using diffusion-type models in many field studies.

The collapsing ‘Lissajous-elliptic’ (LE) vortex ring is examined via quantifications of Single- and multi-filament Biot-Savart numerical simulations. In the single-filament simulations, parametric studies show simple relationships between the collapse boundary and the impulse and energy invariants. Collapse becomes non-monotonic in time, for a sufficiently small initial core ‘radius’. Self-similar, singular-like behaviour of the off-filament strain-rate growth has been observed in a small interval, just prior to core overlapping. The computation of the strain-rate eigenvalues and vortex stretching in a diagnostics box surrounding the collapse region yields patterns observed previously in continuum simulations. New diagnostics are presented, including line densities of the energy and the linear and angular momentum, all of which approach zero in the collapse region of the ring. These diagnostics may provide critical parameters for initiating surgery in a topology-changing algorithm. Our multi-filament simulations exhibit layer-like vortex regions and a ‘torus’-shaped vortex stretching pattern observed previously in continuum periodic-domain simulations of vortex reconnection. Quantifications in a cross-section of the collapse region indicate that the circulation tends to concentrate in the head or frontside of the convecting dipolar structure. This is also the location of the incipient ‘bridge’ which is evolving from the weak filaments that have been convected from the initially outer-vortex regions. The formation of this smaller scale vortex structure exhibits the largest vorticity amplification in the variable-core model simulations.

This study investigated particle behaviour in the stagnation zone of natural and forced round impinging air jets using flow visualization, image analysis, and particle image velocimetry. The jet Reynolds number was 21000, and the nozzle to plate spacing was five diameters. Small mass loadings of glass beads with inertial time constants τp of 1.7 and 7 ms were examined. The Stokes number associated with the mean flow Stm = τpU0/D ranged from 0.6 to 2.4, and the Stokes number associated with vortices in the forced flow St′ = τpf ranged from 0.3 to 1.25 where f is the vortex passage frequency. Particle velocities near the wall deviated strongly from fluid velocities, resulting in rebound and non-Stokesian effects (i.e. significant particle Reynolds numbers Rep). The deceleration associated with rebounding caused long particle residence times in the stagnation zone and significant increases in particle number density above the plate. Rebound height and the height of the region of particle accumulation were well correlated and increased with Stm. Particles associated with lower Stm were accelerated in the radial direction more quickly, not only because of their decreased inertia, but also because of the larger fluid velocties encountered. Shear layer vortices produced spatial variations in particle concentration in the free jet which caused number density near the plate to fluctuate with time. The vortices had little effect on particle motion near the stagnation point, however. Only particles in the vicinity of vortex cores felt the influence of the vortex-induced velocity field. Hence, particle motion in the stagnation zone was most dependent on the mean flow (and thus Stm).

Numerical simulations are presented of the long time behaviour of viscous columnar vortices subject to non-uniform axial stretching. The relevant result is that the vortices reach a steady state even when the axial average of the strain is zero, such that they are being compressed during half of their extent. The structure of the flow is analysed and shown to range from local Burgers equilibrium to massive separation. For an intermediate range of Reynolds numbers the vortices are more or less uniform and compact, and it is suggested that this condition is related to the strong vortices observed in turbulent flows. The reason for the survival of the vortices under compression is traced to induced axial pressure gradients and to the viscous cancellation of outgoing vorticity. Theoretical analyses of the linear Burgers’ regime and of the onset of separation are presented and compared to the numerical experiments. The results are related to the observation of intermittency in turbulence, and shown to be consistent both with the observed scaling of vortex diameter, and with the lack of intermittency of the velocity signal.

Previous work on the correlation of dissociative non-equilibrium effects on the flow field in front of blunt bodies considered the dependence of the dimensionless shock stand-off distance on the dimensionless dissociation rate immediately after the normal shock in the simple case of a diatomic gas with only one reaction. In this paper, the correlation is corrected to take into account the additional parameter of the dimensionless free-stream kinetic energy, and extended to the case of complex gas mixtures with many species and many reactions, by introducing a new reaction rate parameter that has a clear physical meaning, and leads to an approximate theory for the stand-off distance. Extensive new experimental results and numerical computations of air, nitrogen and carbon dioxide flow over spheres were obtained over a large range of total enthalpy. The results comprise surface heat flux measurements and differential interferograms. Both experimental results and numerical computations substantiate the approximate theory.