In an oceanographic setting, the shallow water equations are an asymptotic approximation to the full Euler equations, in the limit ϵ = h0/L → 0, with h0 being the vertical length scale and L a horizontal length scale associated with the fluid layer. However, in arriving at the shallow water equations an additional key step in the derivation is the condition that at some reference time (e.g. t = 0) the thin-layer horizontal vorticity field is identically zero, which corresponds to the horizontal fluid velocity field being independent of the vertical coordinate, z, at t = 0. With this condition in place, the ‘thin-layer equations’ reduce exactly to the shallow water equations. In this paper, we show that this exact condition may be unstable: small, even infinitesimal, perturbations of the thin-layer horizontal vorticity field can grow without bound. When the thin-layer horizontal vorticity grows to be of order 1, the shallow water equations are no longer asymptotically valid as a model for shallow water hydrodynamics, and the ‘thin-layer equations’ must be adopted in their place.

In an experiment on binary alloys directionally solidifying from below, Sample & Hellawell (1984) showed that the plume convection can be successfully prohibited by rotating the cooling tank around an inclined axis. In the present paper we interpret their experimental observation by an analytical approach. Results show that there is a flow induced by the inclination. The induced flow in the fluid layer is a parallel shear flow consisting of three parts: the thermal boundary-layer flow, the solute boundary- layer flow, and the Ekman-layer flow. In the mush, the induced flow is also a parallel flow but of much smaller velocity, consisting of two flows of opposite directions. The induced velocity in the fluid layer increases with inclination angle and decreases with the effective Taylor number Te. The induced velocity in the mush also increases with inclination angle but remains virtually the same on varying the speed of rotation. The linear stability analysis of the mushy layer shows that, due mostly to the reduction of buoyancy, the mush becomes more stable as the inclination angle increases. In the precession-only case, the most-unstable mode of instability is the longitudinal mode, which propagates in a direction perpendicular to the induced flow. In the spin (with or without precession) case, the instability modes propagating in different directions are of equal stability. Because the induced flow changes direction with a frequency equal to the spin angular velocity, the flow scans over all the directions of the system and stabilizes equally the modes in different directions. We conclude on the basis of the present results and from the practical point of view that spin-only rotation is more effective than the precession-only rotation in stabilizing the convection during solidification.

The asymptotic flow structure is considered for a viscous–inviscid conical interaction, in particular that between a swept shock wave and a boundary layer. A flow model is devised based on the three-layer interaction concept. Assuming conicity of the inviscid flow regions, a viscous layer structure is established that is compatible with the inviscid outer flow, and which produces a geometrically conical surface flow pattern. This result is obtained from a dimensional analysis, which reveals similarity of the viscous layer in cross-flow planes at different radial distances from the conical origin. The results of this analysis provide a tool for the quantitative interpretation of surface flow visualizations in terms of the related topological structure of the flow in the cross-flow plane. This method is illustrated by application to the surface flow visualization of a Mach 3 shock-wave/boundary-layer interaction.

A numerical method is developed to study the dynamic behaviour of an encapsulated bubble when the viscous forces of the surrounding liquid are accounted for. The continuity and Navier–Stokes equations are solved for the liquid, whereas the coating is described as a viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed for the spatial discretization of the flow domain surrounding the bubble, with the standard staggered grid arrangement that uses biquadratic and bilinear Lagrangian basis functions for the velocity and pressure in the liquid, respectively, coupled with a superparametric scheme with $B$-cubic splines as basis functions pertaining to the location of the interface. The spine method and the elliptic mesh generation technique are used for updating the mesh points in the interior of the flow domain as the shape of the interface evolves with time, with the latter being distinctly superior in capturing severely distorted shapes. The stabilizing effect of the liquid viscosity is demonstrated, as it alters the amplitude of the disturbance for which a bubble deforms and/or collapses. For a step change in the far-field pressure the dynamic evolution of the microbubble is captured until a static equilibrium is achieved. Static shapes that are significantly compressed are captured in the post-buckling regime, leading to symmetric or asymmetric shapes, depending on the relative dilatation to bending stiffness ratio. As the external overpressure increases, shapes corresponding to all the solution families that were captured evolve to exhibit contact as the two poles approach each other. Shell viscosity prevents jet formation by relaxing compressive stresses and bending moments around the indentation generated at the poles due to shell buckling. This behaviour is conjectured to be the inception process leading to static shapes with contact regions.

Under Stokesian conditions, a neutrally buoyant non-Brownian spheroid in simple shear flow rotates indefinitely in any of a one-parameter family of closed (Jeffery) orbits characterized by an orbit constant $C$. The limiting values, $C=0$ and $C=\infty$, correspond to spinning and tumbling modes respectively. Hydrodynamics alone does not determine the distribution of spheroid orientations across Jeffery orbits in the absence of interactions, and the rheology of a dilute suspension of spheroids remains indeterminate. A combination of inertia and stochastic orientation fluctuations eliminates the indeterminacy. The steady-state Jeffery-orbit distribution arising from a balance of inertia and thermal fluctuations is shown to be of the Boltzmann equilibrium form, with a potential that depends on $C$, the particle aspect ratio ($\unicode[STIX]{x1D705}$), and a dimensionless shear rate ($Re\,Pe_{r}$, $Re$ and $Pe_{r}$ being the Reynolds and rotary Péclet numbers), and therefore lends itself to a novel thermodynamic interpretation in $C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. In particular, the transition of the potential from a single to a double-well structure, below a critical $\unicode[STIX]{x1D705}$, has similarities to a thermodynamic phase transition, and the small-$C$ and large-$C$ minima are therefore identified with spinning and tumbling phases. The hysteretic dynamics within the two-phase tumbling–spinning envelope renders the rheology sensitively dependent on the precise shear rate history, the signature in simple shear flow being a multivalued viscosity at a given shear rate. The tumbling–spinning transition identified here is analogous to the coil–stretch transition in the polymer physics literature. It should persist under more general circumstances, and has implications for the suspension stress response in inhomogeneous shearing flows.

The steady rotational flow of an inviscid fluid in a two-dimensional channel or a circular tube toward a sink is treated. The velocity distribution at infinity is approximated by a cosine curve (which is nearly parabolic) for the two-dimensional case, and is taken as exactly parabolic for the axisymmetric case. The dependence of vorticity on stream-function is assumed to be everywhere the same as it is for streamlines coming from infinity upstream. The resulting linear equations of motion are solved exactly. The solutions show the rather unusual features of separating streamlines and regions of closed flow (corner eddies).

Drag-reduction experiments with combined injection of high-molecular-weight long-chained polymers and microbubbles were conducted on a 3.1 m long flat plate model in the 1.22 m diameter water tunnel at the Applied Research Laboratory of the Pennsylvania State University. Combined gas injection upstream of polymer injection produced, over a wide range of test conditions, higher levels of drag reduction than those obtained from the independent injection of polymer or microbubbles alone. These increased levels of drag reduction with combined injection were often greater than the product of the drag reductions obtained by the independent constituents, defined as synergy. We speculate that the synergy is a result of the gas-layer-induced extension of the polymer-alone initial diffusion zone in combination with the increased drag reduction by microbubbles. This increased length of the initial zone layer, consistent with high drag reduction, can significantly increase the persistence of the drag reduction and may improve the outlook for practical application.

This work is concerned with the dispersion of a buoyant solute in a straight horizontal pipe of circular cross-section where dispersion is affected by molecular diffusion, the laminar flow along the pipe and density currents. Erdogan & Chatwin (1967) have derived an equation for the mean concentration $\overline{C}$ of a buoyant solute using a relatively simple asymptotic model, and have predicted that the dispersion induced by buoyancy effects depends on the Péclet number of the flow. In this part of this study an approximate expression is derived for $\overline{C}$ from Erdogan & Chatwin's equation, and an asymptotic form is obtained for the second moment of distributions of buoyant solutes. The examination of the second moment leads to a simple, but important, result: the dispersion induced by density currents at large times is small compared with the dispersion induced by density currents at times when transient effects are significant.

Novel acoustic liner designs often incorporate new materials with non-uniform impedance distributions. Therefore, new methods are required for their modelling and analysis. In this paper, a theoretical model is developed to investigate the scattering of sound waves from an axially symmetrical flow duct with a semi-infinite, azimuthally non-uniform acoustic lining on the duct wall. More specifically, the incorporation of Fourier series expansions into the Wiener–Hopf method leads to an analytical model with a matrix kernel, which is further factorised by using the pole-removal method to obtain a closed-form solution. A new mathematical method is developed to solve the residues associated with the pole-removal technique. The proposed model has been verified and validated by comparing with corresponding computational results. In addition to shedding light on the possible physical effect of azimuthally non-uniform liners along with an axial hard–soft interface, the current model enhances the theoretical modelling capability for a complicated set-up of practical importance, and can be used to investigate new liner designs for passive noise control in flow ducts.

The problem of a two-dimensional inviscid compressible bubble evolving in Stokes flow is considered. By generalizing the work of Tanveer & Vasconcelos (1995) it is shown that for certain classes of initial condition the quasi-steady free boundary problem for the bubble shape evolution is reducible to a finite set of coupled nonlinear ordinary differential equations, the form of which depends on the equation of state governing the relationship between the bubble pressure and its area. Recent numerical calculations by Pozrikidis (2001) using boundary integral methods are retrieved and extended. If the ambient pressures are small enough, it is shown that bubbles can expand significantly. It is also shown that a bubble evolving adiabatically is less likely to expand than an isothermal bubble.

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear.

This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models.

Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

Homogeneous, approximately isotropic turbulence at two Taylor-scale Reynolds numbers, Rλ=50, 190, with a mean transverse temperature gradient is passed through an axisymmetric contraction. The effects of the straining on the velocity field, and on the passive scalar field, are investigated within the contraction as are the effects of releasing the strain in the post-contraction region. Components of the fluctuating velocity and scalar gradient covariance are measured in order to understand their relation to the large-scale anisotropy of the flow. The scale-dependent spectral evolution of the scalar is also determined. A tensor model is constructed to predict the evolution of the fluctuating scalar gradient covariance. The model constants are determined in the post-contraction relaxation region, where the flow geometry does not vary. The model is shown to perform well throughout the flow, even in the contraction in which the geometry varies. Rapid distortion theory is applied to the scalar field in the contraction, and its solutions are compared to the experimental results.

Spatial length scales of the rate of dissipation, χ, of fluctuations of a conserved scalar, Z, are inferred numerically using a DNS database of a turbulent planar jet flame. The Taylor-scale Reynolds numbers lie in the range of 38 to 58 along the centreline of the simulated jet flame. Three different methods are used to study the spatial length scales associated with χ. First, analysis of the one-dimensional dissipation spectra shows an expected Reδ−3/4 (Kolmogorov) scaling with the outer-scale Reynolds number, Reδ. Secondly, thin sheet-like three-dimensional scalar dissipation structures have been investigated directly. Such structures were identified within the computational domain using level-sets of the χ-field, and their thicknesses were subsequently computed. The study shows, in accordance with experimental studies, that the captured dissipation-layer thickness also shows a Kolmogorov scaling with Reδ. Finally, spatial filters of varying widths were applied to the instantaneous Z field in order to model the averaging effect that takes place with some experimental measurement techniques. The filtered scalar dissipation rate was then calculated from the filtered scalar field. The peaks in the instantaneous filtered χ-profiles are observed to decrease exponentially with increasing filter width, yielding estimates of the true value of χ. Unlike the dissipation length scales obtained from the spectral analysis and the level-set method, the length-scale estimates from the spatial-filtering method are found to be proportional to Reδ−1. This is consistent with the small-scale intermittency of χ which cannot be captured by techniques that just resolve the conventional Batchelor/Obukhov–Corrsin scale. These results have implications when considering resolution requirements for measuring scalar dissipation length scales in experimental flows.

Wind tunnel experiments are performed on a body of revolution with aft-section designed to be an axisymmetric Stratford ramp (i.e. the flow over the ‘ramp’ experiences an adverse pressure gradient that causes it to be continuously on the verge of separation). Digital particle image velocimetry (DPIV) measurements over the ramp reveal a thick boundary layer that is characterized by self-similar velocity profiles with a large wake component and organized vorticity structures. The mean skin friction quickly drops to a value near zero.

The sensitivity of the boundary layer to the degree of severity of the adverse pressure gradient is investigated by testing two additional ramps; one is slightly more conservative (i.e. less steep) than the Stratford ramp while the other is slightly more radical (i.e. steeper). In comparison to the Stratford ramp, the conservative ramp is characterized by a thinner boundary layer, with velocity profiles that start attached and gradually develop a large wake component, a much more gradual drop in the skin friction, and vorticity that is concentrated very close to the wall. On the other hand, the boundary layer over the radical ramp is unsteady and separates intermittently. Measurements of the drag force on each of the three bodies confirm that the Stratford ramp experiences the least amount of drag.

Finally, additional data are gathered on the windward and leeward sides of the Stratford ramp when subjected to a small angle of attack. This case exhibits a more complex flow structure: the flow remains attached over the windward side of the ramp while separating over the leeward side.

The drag of a rigid object is expected to increase with flow velocity. For wide ranges of velocities commonly encountered, the drag increases as the square of the relative velocity of the fluid. This strong dependence of the load on the velocity accounts for specific survival strategies adopted by passive living systems such as plants in wind or algae in marine environments: through elastic reconfiguration, the drag on plants is reduced when compared to a rigid configuration and the velocity exponent for the drag is typically found to be between 1 and 1.5. In this work, a membrane configuration is presented that exhibits a drag force that is almost independent of the free-stream velocity. This surprising result is shown to be remarkably robust as it is experimentally observed for a range of geometries. This study opens the way for the design of devices subjected to a drag that is independent of the flow velocity. This possibility constitutes a key point in various fields involving flexible structures that are towed or subjected to wind.

The propagation of a two-dimensional weakly nonlinear wavefront into a polytropic gas in a uniform state and at rest has been studied. Successive positions of the wavefront and the distribution of amplitude on it are obtained by solving a system of conservation forms of the equations of weakly nonlinear ray theory (WNLRT) using a TVB scheme based on the Lax–Friedrichs flux. The predictions of the WNLRT are found to be qualitatively quite different from the predictions of the linear theory. The linear wavefronts leading to the formation of caustics are replaced by nonlinear wavefronts with kinks. By varying the initial shape of the wavefront and the amplitude distribution on it, the formation and separation of kinks on the wavefront has been studied.

The linear hydrodynamic stability problem for plane Poiseuille flow of a dilute suspension of rigid fibres is solved numerically. The constitutive equation given by Batchelor (1970a, b, 1971) is used to model the rheological properties of the suspension. The resulting eigenvalue problem is shown to be singular. The appropriate contour in the complex plane is determined by considering an initial-value problem. It is shown that, for a fixed, but not too large, inclination of the wave front to the mean flow, the fibres cause the critical Reynolds number to increase monotonically with the product of the volume fraction of the fibres and the square of their aspect ratio. The stabilizing influence of the fibres seems to vanish for large wave inclination angles.

We develop an analysis of the two-dimensional cascade of a tracer (passive or active Lagrangian-conserved scalar), locally in space and time, and establish connections with the modelling of turbulent mixing. We define a local scale-to-scale flux of tracer variance based on the dynamics of tracer increments. This flux reduces at small scales to the production or destruction rate of tracer gradients by stirring as a function of their local orientation with respect to the compressional axis of the strain-rate tensor. The local detailed budget of tracer variance on which this approach is based is compared to the global statistical budget expressed by Yaglom's equation. The spatial pattern of the local transfers produced by a numerical simulation as well as their statistical distribution are discussed.

We then address the problem of the parameterization of turbulent mixing. We consider an anisotropic tensor diffusivity proportional to the velocity gradients. In this model the tracer dissipation involves the axes of the strain-rate tensor and we shall refer to it as strain diffusivity. We show analytically that it locally matches the scale-to-scale flux through the cutoff scale. This matching is studied numerically in decaying two-dimensional turbulence. A comparison is made with eddy diffusivity and hyperdiffusivity. The presence of a numerical instability and ways to suppress it are discussed from numerical and fundamental points of view.

We consider the special case of vorticity, an active scalar in two dimensions. When applied to vorticity, models affect the energy budget. The two-dimensional inverse energy cascade requires that parameterizations conserve energy and we show that strain diffusivity conserves energy. We finally study the sensitivity of the large scales of the flow to the operator used on vorticity in forced stationary simulations. Strain diffusivity is found to produce more realistic spectral features than hyperviscosity.

The paper is concerned with analytical study of inertia-gravity waves in rotating density-stratified ideal fluid confined in a spherical shell. It primarily aims at clarifying the possible role of these motions in deep ocean mixing. Recently, it was found that on the ‘non-traditional’ β-plane inertia-gravity internal waves can propagate polewards beyond their inertial latitude, where the wave frequency equals the local Coriolis parameter, by turning into subinertial modes trapped in the narrowing waveguides around the local minima of buoyancy frequency N. The behaviour of characteristics was established: wave horizontal and vertical scales decrease as the wave advances polewards and tend to zero at a latitude corresponding to an attractor of characteristics. However, the basic questions about wave evolution, its quantitative description and the possibility of its reflection from the critical latitude remain open. The present work addresses these issues by studying the linear inviscid evolution of finite bandwidth wavepackets on the ‘non-traditional’ β-plane past the inertial latitude for generic oceanic stratification. Beyond the inertial latitude, the wave field is confined in narrowing waveguides of three distinct generic types around different local minima of the buoyancy frequency. In the oceanic context, the widest is adjacent to the flat bottom, the thinnest is the upper mixed layer, and the middle one is located between the seasonal and main thermocline. We find explicit asymptotic solutions describing the wave field in the WKB approximation. As a byproduct, the conservation of wave action principle is explicitly formulated for all types of internal waves on the ‘non-traditional’ β-plane. The wave velocities and vertical shear tend to infinity and become singular at the attractor latitude or its vicinity for both monochromatic and finite bandwidth packets. We call this phenomenon singular focusing. These WKB solutions are shown to remain valid up to singularity for the bottom and mid-ocean waveguides. The main conclusion is that even in the inviscid setting the wave evolution towards smaller and smaller horizontal and vertical scales is irreversible: there is no reflection. For situations typical of deep ocean, a simultaneous increase in wave amplitude and decrease of vertical scale causes a sharp increase of vertical shear, which may lead to wave breaking and increased mixing.

A semi-analytical numerical study was performed to simulate the development of a vortex ring in a stratified and/or shearing environment. Practical applications of results of this study can be found in turbulence modelling and in studies of plumes and wakes. The objective is to follow exactly the evolution of a vortex ring so that the three-dimensional vortex-stretching mechanisms due to stratification and the shear effects, respectively, can be understood.

The basic formulation consists of the solution of the vorticity equation in a stratified medium. The approach adopted is unique in that discrete vortex elements are used and arbitrary nonlinear interactions are allowed (therefore three-dimensional effects) among various vorticity generators. One of the two fundamental assumptions in this approach is that the vorticity is allowed to be generated only along the density discontinuity. The second assumption is that, while the vorticity carried by the vortex ring is modelled by vortex elements tangential to the vortex loop (which was a vortex ring initially), the vorticity generated by stratification effects is modelled by long vortex lines parallel to the axis of the vortex ring. This limits the validity of the present calculation to high Froude number flow.

Numerical stability is guaranteed by the finite core radius for each discrete vortex element and uniform spacing between them; the former is determined by consideration of the momentum integral over the vortex-ring plane. The latter is determined by a cubic spline interpolation method which conserves the circulation and centroids of the vorticity. The velocity of each vortex element is determined by the discretized Biot-Savart law, and motion of the vortex loop is calculated by a predicator-corrector time integration method.

Calculations were carried out for both momentum-carrying and momentumless vortex rings. A particular two-dimensional case gives good agreement with Kármán's theory. The evolution of the vortex loop reveals a process in which only the vorticity normal to the stratification is conserved; the remaining vorticity is dissipated through a simulated viscous dissipation. Evolution of a vortex loop on a shear layer reveals a vortex-loop rotation rate equal to the velocity shear, and a twisting motion due to the Magnus force which can lead to the turbulence energy cascade phenomenon. Numerical results demonstrate effects of each individual vorticity source and observed phenomena can be explained.