Homogeneous sheared stratified turbulence was simulated using a DNS code. The initial turbulent Reynolds numbers (Re) were 22, 44, and 89, and the initial dimensionless shear rate (S*) varied from 2 to 16. We found (similarly to Rogers (1986) for unstratified flows) the final value of S* at high Re to be ∼ 11, independent of initial S*. The final S* varies at low Re, in agreement with Jacobitz et al. (1997). At low Re, the stationary Richardson number (Ris) depends on both Re and S*, but at higher Re, it varies only with Re. A scaling based on the turbulent kinetic energy equation which suggests this result employs instantaneous rather than initial values of flow parameters.

At high Re the dissipation increases with applied shear, allowing a constant final S*. The increased dissipation occurs primarily at high wavenumbers due to the stretching of eddies by stronger shear. For the high-Re stationary flows, the turbulent Froude number (Frt) is a constant independent of S*. An Frt-based scaling predicts the final value of S* well over a range of Re. Therefore Frt is a more appropriate parameter for describing the state of developed stratified turbulence than the gradient Richardson number.

The interaction of a dilute dispersed cloud of microbubbles with a planar free-shear layer is investigated experimentally. The emphasis of this study is on the role of the coherent large scales of the flow in the bubble dispersion field and the energy redistribution within the carrier phase. The interphase momentum transfer integrals that appear in the volume-averaged momentum and energy equations account for redistribution of energy from potential to kinetic within the carrier phase. This results from both the hydrostatic and dynamic pressure fields. The energy redistribution within the carrier phase that is associated with the large-scale structures of the flow possesses significant inhomogeneities within the mixing layer. Peaks of enhanced kinetic energy generation are associated with the upwelling regions at the downstream edge of the coherent vortex cores, and weaker peaks of kinetic energy destruction are associated with downwelling regions. The contribution of the quasi-steady drag term to the total energy redistribution is found to be dominant in only a limited region of the flow field.

The trajectory of a laminar buoyant jet discharged horizontally has been studied. The experimental observations were based on the injection of pure water into a brine solution. Under certain conditions the jet has been found to undergo bifurcation. The bifurcation of the jet occurs in a limited domain of Grashof number and Reynolds number. The regions in which the bifurcation occurs has been mapped in the Reynolds number–Grashof number plane. There are three regions where bifurcation does not occur. The various mechanisms that prevent bifurcation have been proposed.

In the presence of background rotation, conventional two-dimensional models of geostrophic flow in a rotating system usually include Ekman friction – associated with the no-slip condition at the bottom – by adding a linear term in the vorticity evolution equation. This term is proportional to E1/2 (where E is the Ekman number), and arises from the linear Ekman theory, which yields an expression for the vertical velocity produced by the thin Ekman layer at the flat bottom. In this paper, a two- dimensional model with Ekman damping is proposed using again the linear Ekman theory, but now including nonlinear Ekman terms in the vorticity equation. These terms represent nonlinear advection of relative vorticity as well as stretching effects. It is shown that this modified two-dimensional model gives a better description of the spin-down of experimental barotropic vortices than conventional models. Therefore, it is proposed that these corrections should be included in studies of the evolution of quasi-two-dimensional flows, during times comparable to the Ekman period.

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.

Baroclinic large-amplitude geostrophic (LAG) models, which assume a leading-order geostrophic balance but allow for large-amplitude isopycnal deflections, provide a suitable framework to model the large-amplitude motions exhibited in frontal regions. The qualitative dynamical characterization of LAG models depends critically on the underlying length scale. If the length scale is sufficiently large, the effect of differential rotation, i.e. the β-effect, enters the dynamics at leading order. For smaller length scales, the β-effect, while non-negligible, does not enter the dynamics at leading order. These two dynamical limits are referred to as strong-β and weak-β models, respectively.

A comprehensive description of the nonlinear dynamics associated with the strong- β models is given. In addition to establishing two new nonlinear stability theorems, we extend previous linear stability analyses to account for the finite-amplitude development of perturbed fronts. We determine whether the linear solutions are subject to nonlinear secondary instabilities and, in particular, a new long-wave–short-wave (LWSW) resonance, which is a possible source of rapid unstable growth at long length scales, is identified. The theoretical analyses are tested against numerical simulations. The simulations confirm the importance of the LWSW resonance in the development of the flow. Simulations show that instabilities associated with vanishing potential- vorticity gradients can develop into stable meanders, eddies or breaking waves. By examining models with different layer depths, we reveal how the dynamics associated with strong-β models qualitatively changes as the strength of the dynamic coupling between the barotropic and baroclinic motions varies.

This paper is a continuation of our study on nonlinear processes in large-amplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-β models. These models arise when the intrinsic length scale is large enough so that the dynamics is geostrophic to leading order but not so large that the β-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous analysis of strong-β LAG models in Part 1, we show that the weak-β models allow for vigorous linear baroclinic instability.

For two-layer weak-β LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultraviolet catastrophe. We argue that it is not possible to establish conditions for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable flow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the development of large-amplitude meanders, squirts and eddies is correlated with the presence of these explosively unstable modes.

For two-layer weak-β LAG models in which one of the two layers is substantially thinner than the other, the linear stability problem does not exhibit an ultraviolet catastrophe and it is possible to establish conditions for the nonlinear stability in the sense of Liapunov for steady flows. A finite-amplitude analysis for a marginally unstable flow suggests that nonlinearities act to stabilize eastward and enhance the instability of westward flows. Numerical simulations are presented to illustrate these processes.

In this paper, we are concerned with the effect of fluid elasticity and shear-thinning viscosity on the chaotic mixing of the flow between two eccentric, alternately rotating cylinders. We employ the well-developed h-p finite element method to achieve a high accuracy and efficiency in calculating steady solutions, and a full unsteady algorithm for creeping viscoelastic flows to study the transient process in this periodic viscoelastic flow. Since the distribution of periodic points of the viscoelastic flow is not symmetric, we have developed a domain-search algorithm based on Newton iteration for locating the periodic points. With the piecewise-steady approximation, our computation for the upper-convected Maxwell fluid predicts no noticeable changes of the advected coverage of a passive tracer from Newtonian flow, with elasticity levels up to a Deborah number of 1.0. The stretching of the fluid elements, quantified by the geometrical mean of the spatial distribution, remains exponential up to a Deborah number of 6.0, with only slight changes from Newtonian flow. On the other hand, the shear-thinning viscosity, modelled by the Carreau equation, has a large impact on both the advection of a passive tracer and the mean stretching of the fluid elements. The creeping, unsteady computations show that the transient period of the velocity is much shorter than the transient period of the stress, and from a pragmatic point of view, this transient process caused by stress relaxation due to sudden switches of the cylinder rotation can be neglected for predicting the advective mixing in this time- periodic flow. The periodic points found up to second order and their eigenvalues are indeed very informative in understanding the chaotic mixing patterns and the qualitative changes of the mean stretching of the fluid elements. The comparison between our computations and those of Niederkorn & Ottino (1993) reveals the importance of reducing the discretization error in the computation of chaotic mixing. The causes of the discrepancy between our prediction of the tracer advection and Niederkorn & Ottino's (1993) experiment are discussed, in which the influence of the shear-thinning first normal stress difference is carefully examined. The discussion leads to questions on whether small elasticity of the fluid has a large effect on the chaotic mixing in this periodic flow.

Analytical and numerical methods are applied to investigate the transient evolution of an inviscid bubble in two-dimensional Stokes flow. The evolution is driven by extensional incident flow with a rotational component, such as occurs for flow in a four-roller mill. Of particular interest is the possible spontaneous occurrence of a cusp singularity on the bubble surface. The role of constant as well as variable surface tension, induced by the presence of surfactant, is considered. A general theory of time- dependent evolution, which includes the existence of a broad class of exact solutions, is presented. For constant surface tension, a conjecture concerning the existence of a critical capillary number above which all symmetric steady bubble solutions are linearly unstable is found to be false. Steady bubbles for large capillary number Q are found to be susceptible to finite-amplitude instability, with the dynamics often leading to cusp or topological singularities. The evolution of an initially circular bubble at zero surface tension is found to culminate in unsteady cusp formation. In contrast to the clean flow problem, for variable surface tension there exists an upper bound Qc for which steady bubble solutions exist. Theoretical considerations as well as numerical calculations for Q > Qc verify that the bubble achieves an unsteady cusped formation in finite time. The role of a nonlinear equation of state and the influence of surface diffusion of surfactant are both considered. A possible connection between the observed behaviour and the phenomenon of tip streaming is discussed.

The development of secondary instability on streamwise vortex structures generated in a hypersonic shock layer on a flat plate is experimentally studied for the flow with Mach number M∞ = 21 and unit Reynolds number Re1 = 6 × 105 m−1. The study is performed using the electron-beam method. The generation of weak unsteady vortices and steady streamwise vortex structures with finite-amplitude perturbations imposed onto them is studied in detail. Complex data on the characteristics of density fluctuations developed on quasi-steady and unsteady streamwise vortex structures are obtained. It is shown that the characteristics of the natural fluctuations of density developing in the shock layer on a flat plate are qualitatively similar to density fluctuations induced by weak unsteady vortex perturbations introduced into the shock layer. The possibility of existence of parametric resonance between the fundamental frequency and its harmonic and between harmonics for steady streamwise vortex structure is shown.

We report the normal stresses in a non-Brownian suspension in plane Couette flow determined from Stokesian Dynamics simulations. The presence of normal stresses that are linear in the shear rate in a viscometric flow indicates a non-Newtonian character of the suspension, which is otherwise Newtonian. While in itself of interest, this phenomenon is also important because it is believed that normal stresses determine the migration of particles in flows with inhomogeneous shear fields. We simulate plane Couette flow by placing a layer of clear fluid adjacent to one wall in the master cell, which is then replicated periodically. From a combination of the traceless hydrodynamic stresslet on the suspended particles, the stresslet due to (non-hydrodynamic) inter-particle forces, and the total normal force on the walls, we determine the hydrodynamic and inter-particle force contributions to the isotropic ‘particle pressure’ and the first normal stress difference. We determine the stresses for a range of the particle concentration and the Couette gap. The particle pressure and the first normal stress difference exhibit a monotonic increase with the mean particle volume fraction ϕ. The ratio of normal to shear stresses on the walls also increases with ϕ, substantiating the result of Nott & Brady (1994) that this condition is required for stability to concentration fluctuations. We also study the microstructure by extracting the pair distribution function from our simulations; our results are in agreement with previous studies showing anisotropy in the pair distribution, which is the cause of normal stresses.

The eddy-damped quasi-normal Markovian (EDQNM) turbulence theory has been applied to the covariance spectrum of two passive isotropic scalars with different diffusivities in stationary isotropic turbulence. A rigorous application of EDQNM, which introduces no new modelling assumptions or constants, is shown to yield a covariance spectrum that violates the Cauchy–Schwartz inequality over some of the wavenumbers. One approach to this problem is to derive a model based on a stochastic differential equation, as its presence guarantees realizability. For an isotropic scalar, it is possible to construct a Langevin equation for the Fourier transform of the scalar concentrations that is consistent with each EDQNM scalar autocorrelation spectrum. The Langevin equations can then be used to construct a model for the covariance spectrum that is realizable. However, the resulting covariance transfer term does not properly conserve the scalar covariance, and so the model is still not satisfactory. The problem can be traced to the Markovianization step, which leads to the presence of the scalar diffusivities in the transfer functions in an unphysical fashion. A simple fix is described which reconciles the two approaches and gives conservative, realizable results for all time.

Next, we apply the EDQNM theory to a more general system involving the mixing of anisotropic scalars. Anisotropy in this case results from a uniform mean gradient of the two scalar concentrations in one direction. As with the isotropic scalars, direct application of the EDQNM closure results in a covariance spectrum that violates the Cauchy–Schwartz inequality; however, in this case it is not as simple to construct a Langevin model that reproduces all of the spectral interactions that result from the EDQNM procedure. Nevertheless, we show that the same modification of the inverse time scale as is done for the isotropic scalar results in an anisotropic scalar covariance spectrum that is realizable for all times.

We study symmetric vortex merger in quasi-geostrophic flows using numerical simulations with high vertical resolution. We analyse the effect of varying the vertical aspect ratio of the vortices and compare with the barotropic case. During the merging of potential vorticity cores with small aspect ratio, we observe the birth of secondary instabilities on the filaments. This is a new phenomenon not seen in baroclinic simulations having low vertical resolution. Passive Lagrangian tracers are used to explore the transport of fluid parcels during vortex merger, to provide a detailed view of the flow evolution, and to determine the value of the critical merging distance for baroclinic vortices.

The process which leads to the appearance of three-dimensional vortex structures in the oscillatory flow over two-dimensional ripples is investigated by means of direct numerical simulations of Navier–Stokes and continuity equations. The results by Hara & Mei (1990a), who considered ripples of small amplitude or weak fluid oscillations, are extended by considering ripples of larger amplitude and stronger flows respectively. Nonlinear effects, which were ignored in the analysis carried out by Hara & Mei (1990a), are found either to have a destabilizing effect or to delay the appearance of three-dimensional flow patterns, depending on the values of the parameters. An attempt to simulate the flow over actual ripples is made for moderate values of the Reynolds number. In this case the instability of the basic two-dimensional flow with respect to transverse perturbations makes the free shear layer generated by boundary layer separation become wavy as it leaves the ripple crest. Then the amplitude of the waviness increases and eventually complex three-dimensional vortex structures appear which are ejected in the irrotational region. Sometimes the formation of mushroom vortices is observed.