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.

Comprehensive results of a joint experimental and computational study of the two-dimensional flow field over flat plate/compression ramp configurations at Mach 14 are presented. These geometries are aimed to simulate, in a simplified manner, the region around deflected control surfaces of hypersonic re-entry vehicles. The test cases considered cover a range of realistic flow conditions with Reynolds numbers to the hinge line varying between 4.5 × 105 and 2.6 × 106 (with a reference length taken as the distance between the leading edge and the hinge line) and a wall-to-total-temperature ratio of 0.12. The combination of flow and geometric parameters gives rise to fully laminar strong shock wave/boundary layer interactions with extensive separation, and transitional interactions with transition occurring near the reattachment point. A fully turbulent interaction is also considered which, however, was only approximately achieved in the experiments by means of excessive tripping of the oncoming hypersonic laminar boundary layer. Emphasis has been placed upon the quality and level of confidence of both experiments and computations, including a discussion on the laminar-turbulent transition process and the associated striation phenomenon. The favourable comparison between the experimental and computational results has profided the grounds for an enhanced understanding of the relevant flow processes and their modelling. Particularly in relation to transitional shock wave/boundary layer interactions, where laminar-turbulent transition is promoted by the adverse pressure gradient and flow concavity in the reattachment region, a method is proposed to compute extreme adverse effects in the interaction region avoiding such inhibiting requirements as transition modelling or turbulence modelling over separated regions.

As discussed in a recent paper by Brasseur & Wei (1994), scale interactions in fully developed turbulence are of two basic types in the Fourier-spectral view. The cascade of energy from large to small scales is embedded within ‘local-to-non-local’ triadic interactions separated in scale by a decade or less. ‘Distant’ triadic interactions between widely disparate scales transfer negligible energy between the largest and smallest scales, but directly modify the structure of the smallest scales in relationship to the structure of the energy-dominated large scales. Whereas cascading interactions tend to isotropize the small scales as energy moves through spectral shells from low to high wavenumbers, distant interactions redistribute energy within spectral shells in a manner that leads to anisotropic redistributions of small-scale energy and phase in response to anisotropic structure in the large scales. To study the role of long-range interactions in small-scale dynamics, Yeung & Brasseur (1991) carried out a numerical experiment in which the marginally distant triads were purposely stimulated through a coherent narrow-band anisotropic forcing at the large scales readily interpretable in both the Fourier- and physical-space views. It was found that, after one eddy turnover time, the smallest scales rapidly became anisotropic as a direct consequence of the marginally distant triadic group in a manner consistent with the distant triadic equations. Because these asymptotic equations apply in the infinite Reynolds number limit, Yeung & Brasseur argued that the observed long-range effects should be applicable also at high Reynolds numbers.

We continue the analysis of forced simulations in this study, focusing (i) on the detailed three-dimensional restructuring of the small scales as predicted by the asymptotic triadic equations, and (ii) on the relationship between Fourier- and physical-space evolution during forcing. We show that the three-dimensional restructuring of small-scale energy and vorticity in Fourier space from large-scale forcing is predicted in some detail by the distant triadic equations. We find that during forcing the distant interactions alter small-scale structure in two ways: energy is redistributed anisotropically within high-wavenumber spectral shells, and phase correlations are established at the small scales by the distant interactions. In the numerical experiments, the long-range interactions create two pairs of localized volumes of concentrated energy in three-dimensional Fourier space at high wavenumbers in which the Fourier modes are phase coupled. Each pair of locally phase-correlated volumes of Fourier modes separately corresponds to aligned vortex tubes in physical space in two orthogonal directions. We show that the dynamics of distant interactions in creating small-scale anisotropy may be described in physical space by differential advection and distortion of small-scale vorticity by the coherent large-scale energy-containing eddies, producing anisotropic alignment of small-scale vortex tubes.

Scaling arguments indicate a disparity in timescale between distant triadic interactions and energy-cascading local-to-non-local interactions which increases with scale separation. Consequently, the small scales respond to forcing initially through the distant interactions. However, as energy cascades from the large-scale to the small-scale Fourier modes, the stimulated distant interactions become embedded within a sea of local-to-non-local energy cascading interactions which reduce (but do not eliminate) small-scale anisotropy at later times. We find that whereas the small-scale structure is still anisotropic at these later times, the second-order velocity moment tensor is insensitive to this anisotropy. Third-order moments, on the other hand, do detect the anisotropy. We conclude that whereas a single statistical measure of anisotropy can be used to indicate the presence of anisotropy, a null result in that measure does not necessarily imply that the signal is isotropic. The results indicate that non-equilibrium non-stationary turbulence is particularly sensitive to long-range interactions and deviations from local isotropy.

The breakup mechanism of a capillary jet with thermocapillarity is investigated. Effects of the heat transfer from the liquid to the surrounding ambient, the liquid thermal conductivity, and the temperature-dependent surface tension coefficient on the jet instability and the formation of satellite drops are considered. Two different disturbances are imposed on the jet. In the first case, the jet is exposed to a spatially periodic ambient temperature. In addition to the thermal boundary condition, an initial surface disturbance with the same wavenumber as the thermal disturbance is also imposed on the jet. Both in-phase and out-of-phase thermal disturbances with respect to surface disturbances are considered. For the in-phase thermal disturbances, a parameter set is obtained at which capillary and thermocapillary effects can cancel each other and the jet attains a stable configuration. No such parameter set can be obtained when the thermocapillary flows are in the same direction as the capillary flows, as in the out-of-phase thermal disturbances. In the second case, only an initial thermal disturbance is imposed on the surface of the liquid while the ambient temperature is kept spatially and temporally uniform.

A new frozen-in field w (generalizing vorticity) is constructed for ideal magnetohydrodynamic flow. In conjunction with the frozen-in magnetic field h, this is used to obtain a generalized Weber transformation of the MHD equations, expressing the velocity as a bilinear form in generalized Weber variables. This expression is also obtained from Hamilton's principle of least action, and the canonically conjugate Hamiltonian variables for MHD flow are identified. Two alternative energy-type variational principles for three-dimensional steady MHD flow are established. Both involve a functional R which is the sum of the total energy and another conserved functional, the volume integral of a function Φ of Lagrangian coordinates. It is shown that the first variation δ1R vanishes if Φ is suitably chosen (as minus a generalized Bernoulli integral). Expressions for the second variation δ2R are presented.

The impact of drops impinging one by one on a solid surface is studied experimentally and theoretically. The impact process is observed by means of a charge-coupled-device camera, its pictures processed by computer. Low-velocity impact results in spreading and in propagation of capillary waves, whereas at higher velocities splashing (i.e. the emergence of a cloud of small secondary droplets, absent in the former case) sets in. Capillary waves are studied in some detail in separate experiments. The dynamics of the extension of liquid lamellae produced by an impact in the case of splashing is recorded. The secondary-droplet size distributions and the total volume of these droplets are measured, and the splashing threshold is found as a function of the impact parameters.

The pattern of the capillary waves is predicted to be self-similar. The calculated wave profile agrees well with the experimental data. It is shown theoretically that the splashing threshold corresponds to the onset of a velocity discontinuity propagating over the liquid layer on the wall. This discontinuity shows several aspects of a shock. In an incompressible liquid such a discontinuity can only exist in the presence of a sink at its front. The latter results in the emergence of a circular crown-like sheet virtually normal to the wall and propagating with the discontinuity. It is predicted theoretically and recorded in the experiment. The crown is unstable owing to the formation of cusps at the free rim at its top edge, which results in the splashing effect. The onset velocity of splashing and the rate of propagation of the kinematic discontinuity are calculated and the theoretical results agree fairly well with the experimental data. The structure of the discontinuity is shown to match the outer solution.

The transient deformation of liquid capsules enclosed by incompressible membranes whose mechanical properties are dominated by isotropic tension is studied as a model of red blood cell deformation in simple shear flow. The problem is formulated in terms of an integral equation for the distribution of the tension over the cell membrane which is solved using a point-wise collocation and a spectral-projection method. The computations illustrate the dependence of the deformed steady cell shape, membrane tank-treading frequency, membrane tension, and rheological properties of a dilute suspension, on the undeformed cell shape. The general features of the evolution of two-dimensional cells are found to be similar to those of three-dimensional cells, and the corresponding membrane tank-treading frequency and maximum tension are seen to attain comparable values. The numerical results are compared with previous asymptotic analyses for small deformations and available experimental observations, with satisfactory agreement. An estimate of the maximum shear stress for membrane breakup and red blood cell hemolysis is deduced on the basis of the computed maximum membrane tension at steady state.

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.

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.

The temporal growth of Görtler vortices and the associated secondary instability mechanisms are investigated numerically in the case of an asymptotic suction boundary layer on a curved plate. Highly inflectional velocity profiles are generated in both the spanwise and vertical directions. The inflectional velocity profile develops earlier in the spanwise direction. There exist two distinct modes of instability that eventually lead to the breakdown of Görtler vortices: the sinuous mode and the varicose mode. The temporal secondary instability analysis of the three-dimensional inflectional velocity profile reveals that the sinuous mode becomes unstable earlier than the varicose mode. The sinuous mode is shown to be primarily related to shear in the spanwise direction, ∂U/∂z, and the varicose mode to shear in the vertical direction, ∂U/∂y.

Three-dimensional solutions have been obtained for the steady simple shear flow over a spherical particle in the intermediate Reynolds number range 0.1 [les ] Re [les ] 100. The shear flow was generated by two walls which move at the same speed but in opposite directions, and the particle was located in the middle of the gap between the walls. The particle-wall interaction is treated by introducing a fully three-dimensional Chimera or overset grid scheme. The Chimera grid scheme allows each component of a flow to be accurately and efficiently treated. For low Reynolds numbers and without any wall influence we have verified the solution of Taylor (1932) for the shear around a rigid sphere. With increasing Reynolds numbers the angular velocity for zero moment for the sphere decreases with increasing Reynolds number. The influence of the wall has been quantified with the global particle surface characteristics such as net torque and Nusselt number. A detailed analysis of the influence of the wall distance and Reynolds number on the surface distributions of pressure, shear stress and heat transfer has also been carried out.

The hydrodynamic influence of deformable porous surface layers on the motion of a rigid sphere falling in a narrow cylindrical tube filled with a stationary Newtonian fluid is studied using lubrication theory. The porous layers on both the surface of the tube and the sphere are modelled as binary mixtures of solid and liquid components. The sphere is placed at an arbitrary position in the tube and is free to rotate. Effects of the clearance between the sphere and the tube, the eccentricity of the position of the sphere and the properties of the surface layers on the velocity and rotation of the sphere are studied. It is found that, when the lengthscale on which the velocity varies within the porous layer is much smaller than the clearance, the effects of the porous layer can be represented by an equivalent slip boundary condition, the slip velocity at the boundary being proportional to the local shear rate. The slip velocities have a strong influence on the motion of the sphere when the clearance is small. For a given clearance and slip parameters, both the falling and rotation velocities of the sphere increase with the sphere eccentricity. The shear stresses on the surfaces of both the tube and the sphere are greatly reduced when slip boundary conditions are applied, as is the pressure gradient in the region between the sphere and the tube wall. This work could have some relevance to the creeping motion of blood cells in the microcirculation where the glycocalyx, a polysaccharide-rich layer, covers the external surfaces of both endothelial and red blood cells.

We consider the response of the hydrodynamic drag on a body in rectilinear motion to a change in the speed between two steady states, from U1 to U2 [ges ] 0. We consider situations where the body generates no lift, such as occur for bodies with an axis of symmetry aligned with the motion. At large times, the laminar wake consists of two quasi-steady regions – the new wake and the old wake – connected by a transition zone that is convected downstream with the mean speed U2. A global mass balance indicates the existence of a sink flow centred on the transition zone, and this is responsible for the leading-order behaviour of the unsteady force at long times. For the case of U1 [ges ] 0, the force is shown to decay algebraically with the inverse square of time for any finite Reynolds number (Re), and this result is also shown to hold for non-rectilinear motions. A recent analysis for small Reynolds number including terms to O(Re) (Lovalenti & Brady 1993 a) has indicated that the force decays as the inverse square of time for motion started from rest, but decays exponentially for a change between two positive velocities. The former result is found to be correct, but the exponential decay at O(Re) in the latter case is superseded at large times by the inverse-square time decay which is shifted to O(Re2) because the wake flux is nearly constant for small Re. The cases of reversed flow (U1 < 0) and stopped flow (U2 = 0) are treated separately, and it is shown that the transient force is dominated by the effects of the old wake, leading to a slower decay as the simple inverse of time. The force is determined by the far regions of the flow field and so the results are valid for any (symmetric) particle, bubble or drop and (in an average sense) for any Re, provided τ ma {Re, Re−1}, where the time τ is made dimensionless with the convection timescale. The analytical results are compared to detailed numerical calculations for transient flow over spherical particles and bubbles and compelling agreement is observed. These are believed to be the first calculations which adequately resolve the transient far wake behind a bluff body at long times. The asymptotic result for the force is applied to determine that the approach to terminal velocity of a body in free fall is also as the inverse square of time.

We solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations of motion of helium II for the first time and investigate the configuration of quantized vortex lines in Taylor–Couette flow. The results are interpreted in terms of quantities which can be observed by measuring the attenuation of second sound. Comparison is made with existing experimental results.

The ubiquitous presence of river drainage basins in the terrestrial environment suggests that distributed overland flow generated by rainfall tends to spontaneously organize itself into dendritic systems of discrete channels. Several recent numerical models describe the evolution of complete drainage basins from the initial condition of rainfall on a flat, tilted plateau, the surface of which has been provided with random elevation perturbations. These analyses model overland flow via the assumption of a perfect balance between gravitational and frictional terms, i.e. in terms of normal flow.

Linear stability analysis applied to the normal flow model has been shown, however, to fail to select a wavelength corresponding to a finite distance of separation between incipient basins. This suggests that the normal flow model may not be a sufficient basis for studying drainage basin development, especially at the finest scales of morphologic significance.

Here the concept of a threshold condition for bed erosion is combined with an analysis of the full equations of shallow overland flow in order to study wavelength selection. Classical linear stability analysis is shown to be inadequate to analyse the problem at the level of inception. An alternative linear analysis of bed perturbations based on the threshold condition is developed, and shown to lead to the selection of finite wavelength of the correct order of magnitude.

The analysis here is driven from the upstream direction in that bed erosion is first caused only when sufficient flow has gathered from upstream due to rainfall. A downstream-driven theory of incipient channelization that is not necessarily dependent upon rainfall is presented in Izumi (1993), and is presently in preparation for publication.

A resonant subharmonic interaction between two axisymmetric travelling waves was induced in the shear layer of an axisymmetric jet by controlled sinusoidal perturbations with two frequencies separated by one octave. Wherever the two excited waves are non-dispersive and the fundamental is close to its linear neutral point the two waves may interact in a manner that enhances the amplification rate of the subharmonic wave train. The amplified subharmonic will exceed the fundamental's level to become the dominant instability component. The initial phase difference between the subharmonic and the fundamental plays an important role in the amplification of the subharmonic. For specific phase angles between the two excited waves a suppression of the subharmonic may be observed. The influence of other initial parameters such as amplitude ratio, overall forcing level, excitation frequency and flow conditions at the nozzle (i.e. the initial turbulence level and the initial momentum thickness) was also investigated. An increase in the combined forcing level reduces the effect of the initial phase difference on the amplification of the subharmonic. Stronger excitation moves the location at which the two waves are locked in space further upstream while the effect of the initial phase difference decreases. The energy transfer to the subharmonic wave has been analysed by estimating the production terms. The results clearly indicate that most of the energy for the resonant growth of the subharmonic comes directly from the mean flow. The fundamental wave acts as a catalyst, as long as the resonance conditions are satisfied, enhancing the rate of energy transfer from the mean flow to the subharmonic.