An exploratory computational study of the reflection of an inward-facing conical shock wave from its axis of symmetry is presented. This is related to more complex practical situations in both steady and unsteady flows. The absence of a length scale in the problem studied makes features grow linearly with time. The ensuing flow is related to the Guderley singularity in a cylindrical imploding shock. The problem is explored by making a large number of computations of the Euler equations. Distinct reflection configurations are identified, and the regions of parameter space in which they occur are delineated.

Steady vortex flows past a circular cylinder are obtained numerically as solutions of the partial differential equation Δψ = f(ψ), f(ψ) = ω(1 − H(ω − α)), where H is the Heaviside function. Only symmetric solutions are considered so the flow may be thought of as that past a semicircular bump in a half-plane. The flow is transplanted by the complex logarithm to a semi-infinite strip. This strip is truncated at a finite height, a numerical boundary condition is used on the top, and the difference equations resulting from the five-point discretization for the Laplacian on a uniform grid are solved using Fourier methods and an iteration for the nonlinear equation. If the area of the vortex region is prescribed the magnitude of the vorticity ω is adjusted in an inner iteration to satisfy this area constraint.

Three types of solutions are discussed: vortices attached to the cylinder, vortex patches standing off from the cylinder and strips of vorticity extending to infinity. Three families of each type of solution have been found. Equilibrium positions for point vortices, including the Föppl pair, are related to these families by continuation.

Numerical and analytical solutions to the steady compressible Euler equations corresponding to a compressible analogue of the linear Stuart vortex array are presented. These correspond to a homentropic continuation, to finite Mach number, of the Stuart solution describing a linear vortex array in an incompressible fluid. The appropriate partial differential equations describing the flow correspond to the compressible homentropic Euler equations in two dimensions, with a prescribed vorticity–density–streamfunction relationship. In order to construct a well-posed problem for this continuation, it was found, unexpectedly, to be necessary to introduce an eigenvalue into the vorticity–density–streamfunction equation. In the Rayleigh–Janzen expansion of solutions in even powers of the free-stream Mach number M∞, this eigenvalue is determined by a solvability condition. Accurate numerical solution by both finite-difference and spectral methods are presented for the compressible Stuart vortex, over a range of M∞, and of a parameter corresponding to a confined mass-flow rate. These also confirm the nonlinear eigenvalue character of the governing equations. All solution branches followed numerically were found to terminate when the maximum local Mach number just exceeded unity. For one such branch we present evidence for the existence of a very small range of M∞ over which smooth transonic shock-free flow can occur.

The interaction of two propagating vortex pairs is considered, each pair being initially aligned along the positive principal axis of strain associated with the other. As a preliminary, the action of accelerating strain on a Burgers vortex is considered and the conditions for a finite-time singularity (or ‘blow-up’) are determined. The asymptotic high Reynolds number behaviour of such a vortex under non-axisymmetric strain, and the corresponding behaviour of a vortex pair, are described. This leads naturally to consideration of the interaction of the two vortex pairs, and identifies a mechanism by which blow-up may occur through self-similar evolution in an interaction zone where scale decreases in proportion to (t* − t)1/2, where t* is the singularity time. The relevance of Leray scaling in this interaction zone is discussed.

Data on turbulent mixing and other turbulent-flow phenomena suggest that a (mixing) transition, originally documented to occur in shear layers, also occurs in jets, as well as in other flows and may be regarded as a universal phenomenon of turbulence. The resulting fully-developed turbulent flow requires an outer-scale Reynolds number of Re = Uδ/v [gsim ] 1–2 × 104, or a Taylor Reynolds number of ReT = u′ λT/v [gsim ] 100–140, to be sustained. A proposal based on the relative magnitude of dimensional spatial scales is offered to explain this behaviour.

The theory of intermittency in multiplicative cascades is reviewed, with special, but not exclusive, emphasis on its applications to turbulence. It is noted that, in many physical systems, this theory is incomplete, and two of its limitations are discussed in some detail. It is first argued that large fluctuations will in most cases behave differently from the lower-level background, since the overall mean introduces an intensity scale that breaks self-similarity, and that they will, under the right conditions, evolve into coherent structures decoupled from the rest of the system. The effect of non-local interactions is then addressed. It is shown that the results depend on the nature of the interaction, and that it is possible to generate non-local cascades which are less intermittent, as intermittent, or even more intermittent, than local ones. It is finally stressed that the multiplicative theory of cascades is a kinematic description, and that its relation with the real dynamics is not straightforward.

A crossing-shock-wave/turbulent-boundary-layer interaction is investigated using the k–ε turbulence model with a new low-Reynolds-number model based on the approach of Saffman (1970) and Speziale et al. (1990). The crossing shocks are generated by two wedge-shaped fins with wedge angles α1 and α2 attached normal to a flat plate on which an equilibrium supersonic turbulent boundary layer has developed. Two configurations, corresponding to the experiments of Zheltovodov et al. (1994, 1998a, b), are considered. The free-stream Mach number is 3.9, and the fin angles are (α1, α2) = (7°, 7°) and (7°, 11°). The computed surface pressure displays very good agreement with experiment. The computed surface skin friction lines are in close agreement with experiment for the initial separation, and are in qualitative agreement within the crossing shock interaction region. The computed heat transfer is in good agreement with experiment for the (α1, α2) = (7°, 7°) configuration. For the (α1, α2) = (7°, 11°) configuration, the heat transfer is significantly overpredicted within the three-dimensional interaction. The adiabatic wall temperature is accurately predicted for both configurations.

Drag reduction due to dilute addition of high polymers has been known for about fifty years. In spite of this long history, many aspects of the problem remain poorly understood. Two of its features for pipe flow are considered here in the context of the elastic theory of de Gennes: the onset of drag reduction and the so-called maximum drag reduction (MDR) asymptote. A cautious conclusion is that a version of the theory agrees qualitatively with existing experiments.

Philip Saffman made valuable theoretical contributions to different areas of low-Reynolds-number hydrodynamics. Three themes are selected for discussion here: (i) the lift force on a sphere in a shear flow at small, but finite Reynolds number, (ii) Brownian motion in thin liquid films, and (iii) particle motion in rapidly rotating flows. In addition, brief descriptions are given of some of Saffman's other contributions including dispersion in porous media, the average velocity of sedimenting suspensions, and compressible low-Reynolds-number flows.

Convection effects in the melt of a vertical Bridgman furnace, used for solidifying a dilute binary alloy, are known to cause significant, and undesirable, non-uniformity in the alloy. We have found previously that non-axisymmetry significantly degrades the performance of the furnace at large Rayleigh number, Ra, and small Biot number, [Bscr ]. There have been a number of studies on improvement of the alloy quality by the introduction of additional forces into the melt flow – magnetic forces or d'Alembert forces due to various sorts of acceleration of the ampoule. In this paper, we explore the effects on the radial segregation generated by rotating the ampoule about its vertical axis. We determine that the magnitude of segregation is proportional to the product of [Bscr ] and the thickness of the thermal layer on the crystal–melt interface. As the rotation, as measured by a Taylor number, [Tscr ], increases beyond O(Ra1/3), the thermal layer thickens and so the segregation increases. Finally, at [Tscr ] = O(Ra1/2), the thermal adjustment occurs on outer scales, and hence the solutal concentration increases to O([Bscr ]). Hence rotation about the vertical axis actually degrades performance!

By adapting a new mathematical approach to the problem of steady free-surface Euler flows with surface tension recently devised by the present author, it is demonstrated that exact solutions for steady, free-surface multipole-driven Hele-Shaw flows with surface tension can be constructed using similar methods. Moreover, a (one-way) mathematical transformation between exact solutions to the two distinct free-boundary problems is identified: known exact solutions for free-surface Euler flows with surface tension are shown to automatically generate steady quadrupolar-driven Hele-Shaw flows (with non-zero surface tension) existing in exactly the same domain with the same free surface. This correspondence highlights the essential dynamical differences between the two physical problems. Using the transformation, the exact Hele-Shaw analogues of all known exact solutions for free-surface Euler flows (including Crapper's classic capillary water wave solution) are catalogued thereby producing many previously unknown exact solutions for steady Hele-Shaw flows with capillarity. In particular, this paper reports what are believed to be the first known exact solutions for Hele-Shaw flows with surface tension in a doubly-connected fluid region.

We discuss some techniques for finding explicit solutions to immiscible two-phase flow in a Hele-Shaw cell, exploiting properties of the Schwartz function of the interface between the fluids. We also discuss the question of the well-posedness of this problem.

Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidification. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity, what is then the role played by small surface tension in the dynamic formation and selection of these patterns? What is the asymptotic behaviour of the interface in the limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension problem and the singular nature of surface tension pose challenging difficulties in the investigation of these questions. Here, we design a novel numerical method that greatly reduces the impact of noise, and allows us to accurately capture and identify the singular contributions of extremely small surface tensions. The numerical method combines the use of a compact interface parametrization, a rescaling of the governing equations, and very high precision. Our numerical results demonstrate clearly that the zero-surface-tension limit is indeed singular. The impact of a surface-tension-induced complex singularity is revealed in detail. The singular effects of surface tension are first felt at the tip of the interface and subsequently spread around it. The numerical simulations also indicate that surface tension defines a length scale in the fingers developing in a later stage of the interface evolution.

In this paper, we review some aspects of viscous fingering in a Hele-Shaw cell that at first sight appear to defy intuition. These include singular effects of surface tension relative to the corresponding zero-surface-tension problem both for the steady and unsteady problem. They also include a disproportionately large influence of small effects like local inhomogeneity of the flow field near the finger tip, or of the leakage term in boundary conditions that incorporate realistic thin-film effects. Through simple explicit model problems, we demonstrate how such properties are not unexpected for a system approaching structural instability or ill-posedness.