Resonant interactions between internal gravity waves propagating in a stratified shear flow are considered for the case when the background density and shear flow vary slowly with respect to the waves. In Grimshaw (1988) triad resonances were considered, and interaction equations derived for the case when the resonance conditions are met only on certain space-time surfaces, being resonance sites. Here this analysis is extended to include higher-order resonances, with the aim of studying resonant wave interactions near a critical level. It is shown that a secondary resonant interaction between two incoming waves, in which two harmonic components of one incoming wave interact with a single harmonic component of another incoming wave, produces a reflected wave. This result is shown to agree with the study of Brown & Stewartson (1980, 1982a, b) who obtained this same result by a different approach.

We investigate interactions between interfacial disequilibrium and compositional convection during the freezing of an alloy from below to form a mushy layer. A theoretical model is developed in which a stagnant mushy layer underlies a melt that is convecting vigorously, driven by compositional gradients associated with undercooling at the mush-liquid interface. In a series of laboratory experiments, we measure the interfacial undercooling in aqueous solutions of ammonium chloride contaminated to varying degrees by copper sulphate. It has recently been found (Huppert & Hallworth 1993) that a small amount of copper sulphate added to a solution of ammonium chloride significantly inhibits the formation of chimneys in the mushy layer that forms when the solution is cooled below its liquidus. It is our thesis that this phenomenon can be explained in large part by the consequences of the interactions between compositional convection and interfacial undercooling that are investigated herein. The measured undercooling is a function of the rate of advance of the interface and is found to be a very strong, increasing function of the concentration of copper sulphate in solution. The theoretical model is evaluated using parameter values appropriate to the experimental system and it is found that the transient development of the mushy layer depends significantly on the level of interfacial disequilibrium. In particular, it is predicted that the time taken for the Rayleigh number associated with the mushy layer to reach any particular value increases enormously as the level of interfacial disequilibrium increases and that the Rayleigh number can have an upper bound that is less than the critical value needed for the onset of convection within the mushy layer. This suggests that the formation of chimneys in the mushy layer can be similarly delayed or prohibited, in agreement with the experimental findings of Huppert & Hallworth (1993). Additionally, the model predicts that under certain conditions the solid fraction can increase away from the cooled boundary leading to trapping of the interstitial liquid. The model also describes a mechanism for macrosegregation of alloys cooled and solidified from below.

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimenslons, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular. We present analytical expressions for Γ′, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, Γ′ scales as \[ \Gamma^\prime\propto (1-\eta^{-\frac{1}{2}})(\sin\alpha)(1+M^{-1}+2M^{-2})(M-1)(\gamma^{\frac{1}{2}}/\gamma + 1)\] where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, α is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For α ≤ 30°, the error in this approximation is less than 10% for 1.0 < M ≤ 1.32 for all η > 1, and 5.8 ≤ η ≤ 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code.

We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a ‘near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.

Particle displacement velocimetry is used to measure the velocity and vorticity distributions around an inclined 6: 1 prolate spheroid. The objective is to determine the effects of boundary-layer tripping, incidence angle, and Reynolds number on the flow structure. The vorticity distributions are also used for computing the lateral forces and rolling moments that occur when the flow is asymmetric. The computed forces agree with results of direct measurements. It is shown that when the flow is not tripped, separation causes the formation of a pair of vortex sheets. The size of these sheets increases with increasing incidence angle and axial location. Their orientation and internal vorticity distribution also depend on incidence. Rollup into distinct vortices occurs in some cases, and the primary vortex contains between 20 % and 50 % of the overall circulation. The entire flow is unsteady and there are considerable variations in the instantaneous vorticity distributions. The remainder of the lee side, excluding these vortex sheets, remains almost vorticity free, providing clear evidence that the flow can be characterized as open separation. Boundary-layer tripping causes earlier separation on part of the model, brings the primary vortex closer to the body, and spreads the vorticity over a larger region. The increased variability in the vorticity distribution causes considerable force fluctuations, but the mean loads remain unchanged. Trends with increasing Reynolds number are conflicting, probably because of boundary-layer transition. The separation point moves towards the leeward meridian and the normal force decreases when the Reynolds number is increased from 0.42 × 106 to 1.3 × 106. Further increase in the Reynolds number to 2.1 × 106 and tripping cause an increase in forces and earlier separation.

Using the theory of functions of a complex variable, in particular the method of conformal mapping, the irrotational and solenoidal flow in two-dimensional radialflow pump and turbine impellers fitted with equiangular blades is analysed. Exact solutions are given for the fluid velocity along straight radial pump and turbine impeller blades, while for logarithmic spiral pump impeller blades solutions are given which hold asymptotically as (r1/r2)n→0, in which r1 is impeller inner radius, r2 is impeller outer radius and n is the number of blades. Both solutions are given in terms of a Fourier series, with the Fourier coefficients being given by the (Gauss) hypergeometric function and the beta function respectively. The solutions are used to derive analytical expressions for a number of parameters which are important for practical design of radial turbomachinery, and which reflect the two-dimensional nature of the flow field. Parameters include rotational slip of the flow leaving radial impellers, conditions to avoid reverse flow between impeller blades, and conditions for shockless flow at impeller entry, with the number of blades and blade curvature as variables. Furthermore, analytical extensions to classical one-dimensional Eulerian-based expressions for developed head of pumps and delivered work of turbines are given.

The most general form for the rapid pressure—strain rate, within the context of classical Reynolds-stress transport (RST) closures for homogeneous flows, is derived, and truncated forms are obtained with the aid of rapid distortion theory. By a classical RST-closure we here denote a model with transport equations for the Reynolds stress tensor and the total dissipation rate. It is demonstrated that all earlier models for the rapid pressure—strain rate within the class of classical Reynolds-stress closures can be formulated as subsets of the general form derived here. Direct numerical simulations were used to show that the dependence on flow parameters, such as the turbulent Reynolds number, is small, allowing rapid distortion theory to be used for the determination of model parameters. It was shown that such a nonlinear description, of fourth order in the Reynolds-stress anisotropy tensor, is quite sufficient to very accurately model the rapid pressure—strain in all cases of irrotational mean flows, but also to get reasonable predictions in, for example, a rapid homogeneous shear flow. Also, the response of a sudden change in the orientation of the principal axes of a plane strain is investigated for the present model and models proposed in the literature. Inherent restrictions on the predictive capability of Reynolds-stress closures for rotational effects are identified.

We attempt to clarify the factors that regulate the propagation and structure of gravity currents through evaluation of idealized theoretical models along with two-dimensional numerical model simulations. In particular, we seek to reconcile research based on hydraulic theory for gravity currents evolving from a known initial state with analyses of gravity currents that are assumed to be at steady state, and to compare these approaches with both numerical simulations and laboratory experiments. The time-dependent shallow-water solution for a gravity current propagating in a channel of finite depth reveals that the flow must remain subcritical behind the leading edge of the current (in a framework relative to the head). This constraint requires that hf/d ≤ 0.347, where hf is the height of the front and d is the channel depth. Thus, in the lock-exchange problem, inviscid solutions corresponding to hf/d = 0.5 are unphysical, and the actual currents have depth ratios of less than one half near their leading edge and require dissipation or are not steady. We evaluate the relevance of Benjamin's (1968) well-known formula for the propagation of steady gravity currents and clarify discrepancies with other theoretical and observed results. From two-dimensional simulations with a frictionless lower surface, we find that Benjamin's idealized flow-force balance provides a good description of the gravity-current propagation. Including surface friction reduces the propagation speed because it produces dissipation within the cold pool. Although shallow-water theory over-estimates the propagation speed of the leading edge of cold fluid in the ‘dam-break’ problem, this discrepancy appears to arise from the lack of mixing across the current interface rather than from deficiencies in Benjamin's front condition. If an opposing flow restricts the propagation of a gravity current away from its source, we show that the propagation of the current relative to the free stream may be faster than predicted by Benjamin's formula. However, in these situations the front propagation remains dependent upon the specific source conditions and cannot be generalized.

A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numerical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol's phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum.

Weakly nonlinear descriptions of axisymmetric cnoidal and solitary waves in vortices recently have been shown to have strongly nonlinear counterparts. The linear stability of these strongly nonlinear waves to three-dimensional perturbations is studied, motivated by the problem of vortex breakdown in open flows. The basic axisymmetric flow varies both radially and axially, and the linear stability problem is therefore nonseparable. To regularize the generalization of a critical layer, viscosity is introduced in the perturbation problem. In the absence of the waves, the vortex flows are linearly stable. As the amplitude of a wave constituting the basic flow increases owing to variation in the level of swirl, stability is first lost to non-axisymmetric ‘bending’ modes. This instability occurs when the wave amplitude exceeds a critical value, provided that the Reynolds number is larger enough. The critical wave amplitudes for instability typically are large, but not large enough to create regions of closed streamlines. Examination of the most-amplified eigenvectors shows that the perturbations tend to be concentrated downstream of the maximum streamline displacement in the wave, in a position consistent with the observed three-dimensional perturbations in the interior of a bubble type of vortex breakdown.

Moore's (1978) equation for following the evolution of a thin layer of uniform vorticity in two dimensions is extended to the case of a non-uniform, instantaneously known, vorticity distribution, using the method of matched asymptotic expansions. In general, the vorticity distribution satisfies a boundary-layer equation. This has a similarity solution in the case of a vortex layer of small thickness in a viscous fluid. Using this solution, an equation of motion of a diffusing vortex sheet is obtained. The equation retains the simplicity of Birkhoff's integro-differential equation for a vortex sheet, while incorporating the effect of viscous diffusion approximately. The equation is used to study the growth of long waves on a Rayleigh layer.

Recent experimental observation of supercooling in large hypersonic wind tunnels using pure nitrogen identified a broad range of non-equilibrium metastable vapour states of the flow in the test cell. To investigate this phenomenon a number of real-gas effects are analysed and compared with predictions made using the ideal-gas equation of state and equilibrium thermodynamics. The observed limit on the extent of supercooling is found to be at 60% of the temperature difference from the sublimation line to Gibbs’ absolute limit on phase stability. The mass fraction then condensing is calculated to be 12–14%. Included in the study are virial effects, quantization of rotational and vibrational energy, and the possible role of vibrational relaxation and freezing in supercooling. Results suggest that use of the supercooled region to enlarge the Mach–Reynolds number test envelope may be practical. Data from model tests in supercooled flows support this possibility.

An analytical theory of barotropic β-plane vortices is presented in the form of an asymptotic series based on the inverse of vortex nonlinearity. In particular, a solution of the initial value problem is given, in which the vortex is idealized as a radially symmetric function of arbitrary structure. Motion of the vortex is initiated by its interaction with the so-called ‘β-gyres’ which, in turn, are generated by the vortex circulation. Comparisons of analytical and numerical predictions for vortex motion are presented and demonstrate the utility of the present theory for times comparable to the ‘wave’ timescale. The latter exceeds the temporal limit derived from formal considerations. The properties of the far-field planetary wave radiation are also computed.

This theory differs from previous calculations by considering more general initial vortex profiles and by obtaining a more complete solution for the perturbation fields. Vortex trajectory predictions accrue error systematically by integrating vortex propagation rates which are too strong. This appears to be connected to higher-order planetary wave radiation effects.

We consider the inviscid stability of the Batchelor (1964) vortex in a compressible flow. The problem is tackled numerically and also asymptotically, in the limit of large (azimuthal and streamwise) wavenumbers, together with large Mach numbers. The nature of the solution passes through different regimes as the Mach number increases, relative to the wavenumbers. At very high wavenumbers and Mach numbers, the mode which is present in the incompressible case ceases to be unstable, whilst a new ‘centre mode’ forms, whose stability characteristics are determined primarily by conditions close to the vortex axis. We find that generally the flow becomes less unstable as the Mach number increases, and that the regime of instability appears generally confined to disturbances in a direction counter to the direction of the rotation of the swirl of the vortex.

Throughout the paper comparison is made between our numerical results and results obtained from the various asymptotic theories.

Weakly nonlinear capillary—gravity waves of frequency ω and wavenumber k that are induced on the surface of a liquid in a square cylinder that is subjected to the vertical displacement a0 cos 2ωt are studied on the assumptions that: 0 < δ < ka0, where δ is the linear damping ratio; the dominant modes are cos kx and cos ky, where x and y are Cartesian coordinates in a horizontal plane. The formulation extends those of Simonelli & Gollub (1989), Feng & Sethna (1989), Nagata (1989, 1991) and Umeki (1991) by incorporating capillarity, cubic forcing and cubic damping. The results are also applicable to a laterally unbounded fluid, but the basic symmetry then is hypothetical rather than imposed by the boundaries. Canonical evolution equations for the modal amplitudes are determined from an average Lagrangian. The fixed points of the evolution equations comprise: (i) the null solution; (ii) an orthogonal pair of rolls described by either cos kx or cos ky; (iii) an orthogonal pair of squares described by either cos kx + cos ky or cos kx – cos ky; (iv) coupled-mode solutions for which both modes are active and neither in phase nor in antiphase. The solutions for squares are isomorphic to those for rolls through a linear transformation of the coefficients in the Hamiltonian. The fixed points for rolls and squares lie on separate loci in an energy–frequency plane that intersect the null solution at a pair of pitchfork bifurcations, one of which is definitely supercritical and the other of which may be either subcritical or supercritical. The parametric domain of the various solutions includes subdomains in which squares/rolls are stable/unstable and conversely. In the limiting case of deep-water capillary waves in the threshold domain 0 < ka0 — δ < 8δ3/9 all of the rolls and coupled-mode solutions are unstable, while squares are stable except for fixed points between the subcritical bifurcation (if it exists) and the corresponding turning point.