The development of a laminar boundary layer upstream of both two- and threedimensional obstacles mounted on a plane wall is considered. The motion is impulsively started from rest, and it is shown that the boundary layer upstream of the obstacle initially develops independently from that on the obstacle itself. Numerical solutions for the unsteady boundary-layer flow on the plane wall are obtained in both Eulerian and Lagrangian coordinates. It is demonstrated that in both situations the flow focuses into a narrow-band eruption characteristic of separation phenomena at high Reynolds number. For the three-dimensional problem, results are obtained on a symmetry plane upstream of the obstacle which indicate the evolution, and subsequent sharp compression, of a spiral vortex in the near-wall flow in a manner consistent with recent experimental studies. The eruptive response of the two-dimensional boundary layer is found to be considerably stronger than the corresponding event in three dimensions. Calculated results for the temperature distribution are obtained for the situation where the wall temperature is constant but different from that of the mainstream. It is shown that a concentrated response develops in the surface heat transfer rate as the boundary layer starts to separate from the surface.

Dusty-gas flow in laminar boundary layer over a body with a curved surface is considered. In addition to Stokes drag, particles experience a centrifugal force and lift which is due to fluid shear. The body size L is taken to be much greater than the relaxation length of the particle velocity due to the action of Stokes drag, & Lambda;st and is of the same order as or less than the relaxation length due to the action of lift. Λsa. Using an asymptotic approach, momentum equations for the particle phase are reduced to an algebraic equation accounting for the variation of lift coefficient with the shear and the slip velocity. Particle velocity and density are computed for the axisymmetric boundary layer in the neighbourhood of the front stagnation point of a blunt body of size much less than Λsa. It is shown that downstream of some point on the wall (the separation point) particle normal velocity becomes non-zero. As a result particle streamlines turn away from the wall, and a particle-free zone arises. The cause of separation is the lift effect; the centrifugal force cannot make the particle flow separate. This conclusion is extended to the case when L ∼ Λsa. The position of separation for the flow past a sphere is evaluated as a function of the ratio of its radius r′ and relaxation length. Dust flow ceases to separate when this value is greater than a critical value r′c /Λsa ≈ 29.2.

The evolution of the coastal upwelling and interfacial instability of a stratified and rotating fluid is studied numerically by using large-eddy simulation. Upwelling is generated near the sidewall of a rotating annulus by the shear at the top. The fluid initially consists of a stably stratified ‘two-layer’ structure with a narrow interface separating the two layers. The large-scale motion of the flow is simulated by solving the time-dependent non-hydrostaic incompressible Navier-Stockes and scalar transport equations while the small-scale motion is represented by a dynamic subgrid-scale model. The upwelling process contains both stable and unstable stratification. The vertical structure of upwelling consists of a persistent primary front, a trailing mixing zone on te shore side of the front, and a temporary secondary front which leads a top inversion layer. The longshore velocity profile has two maxima which occur at the edge of the sidewall boundary layer and at the density front. The upwelled density front is unstable to azimuthal perturbations and baroclinic waves develop and grow to large amplitude. Pairs of cyclonic and anticyclonic waves appear at the front which form ‘jet-streams’. The secondary front is unstable to azimuthal perturbations. Its instability, and the associated drop of the top inversion layer, take the form of radial bands which subsequently break up into isolated patches and eventually sink. The computed values of various upwelling time and length scales are compared to and are in good agreement with past experimental data.

Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance r to their axis of symmetry. This singularity can be regularized through a near-axis boundary layer approximation to the Navier-Stokes equations, as first done by Long for the case of a vortex with potential swirl, v∼r−1. The present work considers the more general situation of a family of self-similar inviscid vortices for which v∼rm−2, where m is in the range 0 n< m < 2. This includes Longs Vortex for the case m =1. The corresponding solutions also exhibit self-similar structure, and have the interesting property of losing existence when the ratio of the inviscid near-axis swirl to axial velocity (the swirl parameter) is either larger (when 1m < 2) or smaller (when 0m < 1) than an m-dependent critical value. This behaviour shows that viscosity plays a key role in the existence or lack of existence of these particular nearly inviscid vortices and supports the theory proposed by Hall and others on vortex breakdown. Comparison of both the critical swirl parameter and the viscous core structure for the present family of vortices with several experimental results under conditions near the onset of vortex breakdown show a good agreement for values of m slightly larger than 1. These results differ strongly from those in the highly degenerate case m =1.

For a two-dimensional potential flow, Föppl obtained the equilibrium positions for a symmetric vortex pair behind a circular cylinder in a uniform oncoming flow. In this article it is shown that such an equilibrium is in general possible for a vortex in a stagnation flow (e. g. in a corner). Furthermore it is found that a vortex near such an equilibrium position will rotate with a definite frequency around this equilibrium. Expressions are derived for the frequencies associated with the closed orbits of the vortices in the case of equilibrium of a vortex in a stagnation flow and for the equilibrium of the symmetric vortex pair behind a circular cylinder in oncoming flow. For the large-amplitude case the vortex trajectories are claculated using a fifth-order Runge-Kutta integration method. The analysis is then extended to the case of a simple wing-body combination in a cross-flow such as arises for a slender aircraft at an angle of attack with vortices generated by strakes or at the front part of the body. At the wint-body junctions the motions of the vortices may be periodic, quasi-periodic or the vortices may be swept away, depending on the initial conditions.

This paper concerns a steady liquid-metal flow through an expansion or contraction with electrically insulated walls, with rectangular cross-sections and with a uniform, transverse, externally applied magnetic field. One pair of duct walls is parallel to the applied magnetic field, and the other pair diverges or converges symmetrically about a plane which is perpendicular to the field. The magnetic field is assumed to be sufficiently strong that inertial effects can be neglected and that the well-known Hartmann-layer solution is valid for the boundary layers on the walls which are not parallel to the magnetic field. A general treatment of three-dimensional flows in constant-area ducts is presented. An error in the solution of Walker et al. (1972) is corrected. A smooth expansion between two different constant-area ducts is treated. In the expansion the flow is concentrated inside the boundary layers on the sides which are parallel to the magnetic field, while the flow at the centre of the duct is very small and may be negative for a large expansion slope. In each constant-area duct, the flow evolves from a concentration near the sides at the junction with the expansion to the appropriate fully developed flow far upstream or downstream of the expansion. The pressure drop associated with the three-dimensional flow increases as the slope. increases.

In wall turbulence, it is widely accepted that the coherent motions determine the essential features of turbulent transport phenomena. In the present study, we have refined a trajectory-based detection algorithm for coherent motions and have investigated the relationship between coherent motions and scalar (heat) transfer from a structural point of view, i. e. trajectory analysis of the VITA heat transfer events, extraction of key flow modules and the relevant heat transport, and the prediction of passive scalar transfer by means of an autoregressive (AR) model. As a result, it is shown that the phase relationship of fluctuating velocity components dominates the essential characteristics of the transport processes of heat and momentum in wall turbulence and there exist distinct differences in individual correspondence between the coherent motions and heat transport processes, neither of which can be revealed by the widely used VITA technique. Also, the AR model is shown to provide good time-series predictions for turbulent heat transfer associated with coherent structures near the wall.

A study of compressible supersonic turbulent flow in a plane channel with isothermal walls has been performed using direct numerical simulation. Mach numbers, based on the bulk velocity and sound speed at the walls, of 1.5 and 3 are considered; Reynolds numbers, defined in terms of the centreline velocity and channel half-width, are of the order of 3000. Because of the relatively low Reynolds number, all of the relevant scales of motion can be captured, and no subgrid-scale or turbulence model is needed. The isothermal boundary conditions give rise to a flow that is strongly influenced by wall-normal gradients of mean density and temperature. These gradients are found to cause an enhanced streamwise coherence of the near-wall streaks, but not to seriously invalidate Morkovin's hypothesis : the magnitude of fluctuations of total temperature and especially pressure are much less than their mean values, and consequently the dominant compressibility effect is that due to mean property variations. The Van Driest transformation is found to be very successful at both Mach numbers, and when properly scaled, statistics are found to agree well with data from incompressible channel flow results.

The present paper addresses some topical issues in modelling compressible turbulent shear flows. The work is based on direct numerical simulation (DNS) of two supersonic fully developed channel flows between very cold isothermal walls. Detailed decomposition and analysis of terms appearing in the mean momentum and energy equations are presented. The simulation results are used to provide insights into differences between conventional Reynolds and Favre averaging of the mean-flow and turbulent quantities. Study of the turbulence energy budget for the two cases shows that compressibility effects due to turbulent density and pressure fluctuations are insignificant. In particular, the dilatational dissipation and the mean product of the pressure and dilatation fluctuations are very small, contrary to the results of simulations for sheared homogeneous compressible turbulence and to recent proposals for models for general compressible turbulent flows. This provides a possible explanation of why the Van Driest density-weighted transformation (which ignores any true turbulent compressibility effects) is so successful in correlating compressible boundary-layer data. Finally, it is found that the DNS data do not support the strong Reynolds analogy. A more general representation of the analogy is analysed and shown to match the DNS data very well.

Flow of a fluid with a strongly temperature-dependent viscosity in a finite-length slot is analysed as a model of magma flow in dikes. The slot walls are held at a fixed temperature, thus cooling and increasing the viscosity of the fluid as it moves along the gap. Poiseuille flow and temperature advection, averaged across the slot, are used to study the stability of this basic one-dimensional flow to lateral perturbations. A linear stability analysis shows that for sufficiently strong cooling and viscosity increase with decreasing temperature, the flow is unstable to a fingering instability. Warm fluid is focused into relatively fast flowing zones and suffers only modest cooling, while cold, slow flowing regions experience more cooling and an increase in viscosity, which acts to locally clog the slot. The necessary condition for instability is the presence of multiple solutions for velocity (fast, intermediate and slow branches) in the basic one-dimensional flow. The intermediate branch, where the thermal adjustment lengthscale is comparable to the slot length, is unstable and the analysis indicates that the instability continues onto the slow branch. The parametric regions of instability and the growth rates are dependent on the choice of boundary conditions at the slot entrance (i. e. the magma source): either uniform flux, or uniform pressure. The latter case is the more geophysically realistic and has the larger unstable region and growth rates. Numerical solutions of the nonlinear equations show that at finite-amplitude the hot, low-viscosity, fast-flowing fingers continue to speed up, while the slow, cold regions continue to cool and slow down. At the slot exit fluid issues from the gap in isolated hot, low-viscosity spouts separated by zones of cold, nearly still fluid. Application of the model to geophysical settings indicates that the instability is expected for realistic parameter values. The model may help explain the observed focusing of fissure eruptions.

A theoretical description is given of pressure-driven viscous flow of an initially hot fluid through a planar channel with cold walls. The viscosity of the fluid is assumed to be a function only of its temperature. If the viscosity variations caused by the cooling of the fluid are sufficiently large then the relationship between the pressure drop and the flow rate is non-monotonic and there can be more than one steady flow for a given pressure drop. The linear stability of steady flows to two-dimensional and three-dimensional disturbances is calculated. The region of instability to two-dimensional disturbances corresponds exactly to those flows in which an increase in flow rate leads to a decrease in pressure drop. At higher viscosity contrasts some flows are most unstable to three-dimensional (fingering) instabilities analogous, but not identical, to Saffman-Taylor fingering. A cross-channel-averaged model is derived and used to investigate the finite-amplitude evolution.

Scattering properties of an incident field upon a periodic array of identical rectangular barriers, each extending throughout the water depth, are calculated based on a Galerkin approximation to an integral representation of the problem derived using the linear theory of water waves. The method incorporates full multi-modal scattring using the linear theory of water waves. The method incorporates full multi-modal scattering using a matrix formulation and is equivalent to a corresponding two-dimensional acoustics problem also discussed.

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.

Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.

If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.

In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

We consider inviscid incompressible flow about an infinite non-slender flat delta wing with leading-edge separation modeled by symmetrical conical vortex sheets. A similarity solution for the three dimensional steady velocity potential Φ is sought with boundary conditions to be satisfied on the line which is the intersection of the wing sheet surface with the surface of the unit sphere. A numerical approach is developed based on the construction of a special boundary element or ‘winglet’ which is effectively a Green function for the projection of ∇2Φ = 0 onto the spherical surface under the similarity ansatz. When the wing semi-apex angle γo is fixed satisfaction of the boundary conditions of zero normal velocity on the wing and zero normal velocity and pressure continuity across the vortex sheet then leads to a nonlinear eigenvalue problem. A method of ensuring a condition of zero lateral force on a lumped model of the inner part of the rolled-up vortex sheet gives a closed set of a equations which is solved numerically by Newton's method. We present and discuss the properties of solutions for γ0 in the range 1.30 < γ <89.50. The dependencies of these solutions on γ0 differs qualitatively from predictions of slender-body theory. In particular the velocity field is in general not conical and the similarity exponent must be calculated as part of the global eigenvalue problem. This exponent, together with the detailed flow field including the position and structure of the separated vortx sheet, depend only on γ0. In the limit of small γ0, a comparison with slender-body theory is made on the basis of an effective angle of incidence.

An experimental investigation of mode-2 (’lump-Like’) Solitary waves propagaling on a thin interface between two deep layers of different densities is presented. Small-and large-amplitude waves behaved differently: small waves carried energy and momentum, whereas sufficiently large waves also carried mass. Weakly nonlinear theory anticipated the result for amplitudes a/h [les ] 0.5 but did not provide even a qualitative description of the large-amplitude waves. In particular, the prediction that for waves to maintain permanent form their wavelength must decrease with increasing amplitude failed; instead the wavelength of large waves was observed to increase with increasing amplitude. Furthermore, whilst the waves were expected to emerge from interactions along their precollision trajectories, the large waves actually suffered a backward shift.