Motivated by problems concerning the storage and subsequent escape of the solar magnetic field we have studied how a magnetic layer embedded in a convectively stable atmosphere evolves due to axisymmetric instabilities driven by magnetic buoyancy. The initial equilibrium consists of a toroidal field sheared by a weaker poloidal component. The linear stability problem is investigated for both ideal and resistive MHD, and the nonlinear evolution is followed by numerical integration of the equations of motion. In all cases we found that the instability is greatly affected by the distribution and strength of the poloidal field. In particular, both the horizontal and vertical scales of the motions are controlled by the location of the surface on which the poloidal field vanishes — the resonant surface. In the nonlinear regime, a resonant surface close to the interface between the magnetized and field-free fluid leads to the localization of the instability so that only a fraction of the magnetic region is disrupted by the motions. By contrast, a deeply seated resonant surface leads to the complete disruption of the layer and to the formation of large, helical magnetic fragments whose identity is preserved for the entire simulation.

Laboratory measurements on the instability of axisymmetric capillary surfaces pinned to the corners of annular grooves of rectangular section rotating at constant angular velocity Ω have been conducted. In stable configurations the fluid contact lines remain pinned to the corners of the groove with contact angles θ1,2 relative to the inner and outer vertical walls. Using water as the test fluid in narrow grooves of nearly constant width, the critical frequency Ωc for instability generally decreases with increasing overfill volume ΔV and mean groove radius. Numerical integration of the describing equation gives the shape of the rotating meniscus as a function of five independent parameters. In the range of contact angles θ1, 2 < π, a comparison of experimental results with numerically computed meniscus profiles suggests three mechanisms for contact line movement based on the effective static advancing (θA) and receding (θR) contact angles for liquid pinned to a sharp corner. Measurements of critical frequencies over a wide range of overfill volumes in six different grooves are in favourable agreement with composite regime diagrams for the critical static meniscus configuration. An interesting feature of this system is the existence of a range of overfill volumes inaccessible to experiments conducted by fixing the overfill volume on a stationary disk and subsequently elevating the disk rotation until contact line movement is observed. Numerical studies showing the effects of Bond number, groove curvature and contact angle hysteresis are presented.

Resonant three-wave interactions among capillary–gravity water waves are studied experimentally using a test wavetrain and smaller background waves (noise) generated mechanically in a channel. The spectrum of the background waves is varied from broad-banded to one with discrete components. When the noise spectrum is broad-banded, the test wavetrain amplifies all waves in its low-frequency band of allowable triads B[ell ], as anticipated from RIT (resonant interaction theory). When the noise spectrum has a discrete component in the high-frequency band of allowable triads Bh, the test wavetrain selectively amplifies a triad with two waves from B[ell ], contrary to expectations based on RIT. (Although, in accordance with RIT, no waves in Bh are amplified.) We conjecture that the mechanism for selective amplification comprises a sequence of exceedingly weak, higher-order interactions, normally neglected in RIT. This sequence allows the small amount of energy in a discrete spectral component to cascade to two waves in B[ell ], which then amplify, as anticipated from RIT, and dominate all other waves in B[ell ]. The conjectured sequence of nonlinear interactions is tested using both frequency and wave-vector data, which are obtained by in situ probes and by remote sensing of the water surface with a highspeed imaging system. Our predictions of selective amplification, as well as its absence, are consistent with all of the experiments presented herein and in Part 1. Selective amplification occurs for signal-to-noise (amplitude) ratios as large as 200, and its effects are measurable within ten wavelengths of the wavemaker. When selective amplification occurs, it has a profound impact on the long-time evolution of a ripple wavetrain.

Application of the kinetic theory of gases to granular flows has greatly increased our understanding of ‘rapid’ granular flows. One of the underlying assumptions is that particles interact only through binary collisions. For a given set of material and flow parameters, as the concentration increases, the transition from a binary collision mode to other modes of interaction occurs. Kinetic theory can no longer be applied. A numerical model is utilized to simulate the mechanical behaviour of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow. There are two objectives: to obtain the ‘empirical’ constitutive law and to gain insight into the mechanisms that operate in the transitional and quasi-static regimes. In a simple shear flow, spatially and temporally averaged dimensionless stresses $\tau^*_{ij} = \tau_{ij}/(\rho_{\rm s}D^2\dot{\gamma}^2)$ are functions of the concentration C, the dimensionless shear rate $B =\dot{\gamma}/(K_n/m)^{\frac{1}{2}}$, and material parameters ζn, Ks/Kn and μ. Here $\dot{\gamma}$ is the shear rate, Kn is the normal stiffness of an assumed viscoelastic contact force model, Ks/Kn is the ratio of tangential to normal stiffness, ζn is the normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass, respectively. The range of B from 0.001 to 0.0707 was investigated for C ranging from 0.5 to 0.9, with material constants fixed as ζn = 0.0709 (corresponding to the restitution coefficient e = 0.8 in binary impacts), Ks/Kn = 0.8 and μ = 0.5. It is found that for lower concentrations (C < 0.75) dimensionless stresses τ*ij are nearly independent of B, while for higher concentrations (C > 0.75) τ*ij monotonically decreases as B increases. Moreover, their relationship in this regime is well approximated by power law: τ*ij ∝ B−n(C). The powers nij range from nearly zero for C = 0.775 (corresponding to the familiar square power dependency of dimensional stresses on the shear rate in the rapid flow regime), to nearly two for C = 0.9 (corresponding to shear-rate independence in quasi-static regime). The intermediate concentration range corresponds to transition. Distinct mechanisms that govern transitional and quasi-static regimes are observed and discussed.

Spanwise vorticity measurements have been performed in zero-pressure-gradient boundary layers over the range 1010 < Rθ < 4850 (Rθ ≡ U∞ θ/ν, where U∞ is the free-stream velocity and θ is the momentum deficit thickness) using a four-wire probe. In addition, experiments quantifying the spatial and temporal resolution required to obtain an accurate statistical representation of the small-scale structure of wall-bounded turbulence were performed. Furthermore, a thorough investigation of statistical convergence for a variety of fluctuating quantities was performed. Comparisons with earlier high-resolution studies indicate that the maximum value of u′/uτ increases with increasing Reynolds number over the given Rθ range (u′ ≡ r.m.s. u, and uτ is the friction velocity). It is suggested that detecting this dependence provides a good measure of probe resolution. In general it was found that statistics of velocity gradients were distinctly more sensitive to finite probe size than velocity statistics. Wire spacing experiments suggest that Wyngaard's (1969) criterion is to a good approximation valid even under anisotropic conditions. Furthermore, it was found that instantaneously spatial averaging of ∂u/∂t caused significant attenuation in the resulting r.m.s., and that this averaging procedure is sensitive to the level of mean shear. A simple method of estimating how noise in the u-velocity signals enters into the ∂u/∂y signals is presented. The convergence study shows that statistical convergence criteria developed from free-shear flows severely underestimates the averaging times required in boundary layers. A table of general convergence criteria is provided.

The impact of a drop on the plane surface of the same liquid is studied numerically. The accuracy of the calculation is substantiated by its good agreement with available experimental data. An attempt is made to explain the recent observation that, in a restricted range of drop radii and impact velocities, small air bubbles remain entrained in the liquid. The implications of this process for the underwater sound due to rain are considered. The numerical approach consists of a new formulation of the boundary-element method which is explained in detail. Techniques to stabilize the calculation in the presence of strong surface-tension effects are also described.

A breakage model for the statistical distribution of the dissipation rate is proposed: this model, B-model, is a modification of the Gurvich & Yaglom model (1967) taking the criticism of Mandelbrot (1974) into account. The B-model uses the beta distribution for the breakage coefficient α. The universal power spectrum of velocity for the B-model has a slightly flatter slope (positive correction) than the ‘−5/3’ in contrast to all other previously proposed models, and this positive correction agrees with a theoretical argument made in Yakhot et al. (1989). The B-model predicts the structure functions of velocity observed by Anselmet et al. (1984) remarkably well without an empirical fit to the data.

The effect of initial conditions on the decay exponent and coefficient and virtual origin in the decay power-law form for the variation of the variance of the turbulent velocity downstream of biplane grids constructed of rods of both round and square cross-section is determined. This effect is determined for data obtained as part of the present study as well as from previous studies. These studies cover a Reynolds number range from 6000 to 68000, mesh sizes of 2.54 and 5.08 cm, and solidities of 0.34 and 0.44.

It is shown that the choice of the virtual origin and the use of data in the non-homogeneous portion of the flow can have a significant influence on the value of the parameters in the decay power-law. Criteria are developed to identify the nearly homogeneous and isotropic portion of the flow. These criteria include low values of the velocity skewness, constancy of the skewness of the velocity derivative and balance of the turbulent kinetic energy equation.

Results based on data selected by means of these criteria show that the decay exponent and virtual origin are independent of initial conditions such as Reynolds number, mesh size, solidity, and rod shape and surface roughness with values of respectively 1.30 and 0. In contrast and as expected, the decay coefficient is found to be a function of these initial conditions. Thus, the downstream variation of the variance of the turbulent velocity is universally self-similar.

The paper is a continuation of work published in Boubnov & Golitsyn (1986). We present new measurements of the temperature and velocity field patterns and their statistical characteristics. This allows us to classify regimes of convection in a plane rotating horizontal fluid layer in terms of Rayleigh and Taylor numbers. Within the irregular regimes geostrophic convection is found for which the Rossby number is much less than unity.

In the regular regimes the mean temperature profiles are linear with height in the bulk of the fluid, the gradient being dependent mainly on rotation rate Ω and fluid depth h. These together with some dimensional arguments lead to the heat transfer relationship Nu ∝ Ra3Ta−2 between Nusselt, Rayleigh and Taylor numbers. Experimental results by Rossby (1969) and theoretical work by Chan (1974) and Riahi (1977) suggested this dependence. The dependence on ωτ of the temperature power spectrum normalized by the variance was found to be universal at higher frequencies for all irregular convective motions, where τ is the timescale of the thermal boundary layer for cases with a small influence of rotation and with τ about three times larger (in numerical coefficient) for geostrophic convection. For irregular geostrophic regimes it is found that the temperature variance depends on rotation rate and heat flux, and is inversely proportional to the buoyancy parameter.

Horizontal and vertical components of the velocity fields were measured for regular as well as irregular regimes, confirming, especially for geostrophic convection, the theoretical results by Golitsyn (1980). In conclusion some geophysical applications are briefly mentioned.

The unsteady viscous flow induced by a deformable gas bubble approaching or receding away from a rigid boundary is investigated for moderate Reynolds numbers. The full Navier–Stokes equations were solved by means of a finite-element method. The bubble is driven by the buoyancy force. The performance of the numerical scheme is displayed for two different configurations of the flow: the bubble moves (i) in the half-space bounded by a rigid plate; (ii) in a spherical container filled with viscous fluid. Results are obtained for the evolution of the flow pattern and bubble shape for a number of values of Reynolds and Eötvös numbers: 2.2 × 10−3 < [Rscr ] < 60, 1 < [Escr ] < 360. The influence of specific values of [Rscr ][Escr ] and wall curvature on the shape of the deformable interface is thoroughly investigated. Several physical effects are included in our theory: dimpling and film formation; appearance of a concavity at the rear of the bubble for intermediate Reynolds numbers; and elongation of the bubble receding from the wall. Where possible comparisons are carried out with other experimental or numerical investigations. The good agreement achieved confirms the reliability of the numerical technique developed, of the results presented and the conclusions.

The adjustment of rotating free-surface flow over a step-like escarpment abutting a vertical wall is discussed in the context of the shallow-water equations. The problem is simplified by considering an escarpment of small fractional depth, so that on the slow topographic timescale the initial, fast Poincaré and Kelvin wave adjustment of the free surface is effectively instantaneous, and further simplified by considering the surface displacement to be small compared with the escarpment height so that particle velocities are negligible during the topographic adjustment. Direct solution of the resulting linear system is not straightforward as arbitrarily small-scale motions are generated at sufficiently large times. The problem is reduced by a Green's function technique to one spatial dimension and the wall boundary layers resolved by introducing a scaling based on previously obtained limit solutions. Solutions verify the information-propagation arguments of Johnson (1985) and Gill et al. (1986) and also show interchange of fluid across the escarpment as eddies formed as the current crosses the step travel along the step with shallow water to their right. The pattern of evolution of the system is independent of the direction of the flow, depending solely on the sign of the topographic step. If the escarpment is such that topographic waves travel away from the wall, then a tongue of fluid moves outward along the step: the initial jet along the wall is diverted to flow parallel to, rather than across, the step. If waves travel towards the wall then the current is pinched into the wall and fluid crosses the escarpment in a thinning jet.

Solutions are obtained for hypersonic viscous interaction along a flat plate in the presence of strong boundary-layer blowing, with inverse-square-root injection velocity, for laminar flow over a cold wall and with a power-law viscosity–temperature relation. In the strong-interaction region, self-similarity is preserved if the blowing is such that the thicknesses of the inviscid shock layer, viscous shear layer, and inviscid blown layer all have the same order of magnitude. The weak-interaction region is also considered, and an approximate interpolation is used to join the solutions for the surface pressure. Certain difficulties in asymptotic matching are discussed, and the extension to flow past a thin wedge is shown.

Several wave patterns can be produced by the interaction of a shock wave with a fluid interface. Regular wave patterns have previously been explained by a shock-polar analysis. We focus on an irregular wave pattern that typically occurs when a shock passes from a medium of high to low acoustic impedance. Curvature of either the shock front or contact causes the flow to bifurcate from a locally self-similar quasi-stationary shock diffraction, to an unsteady anomalous reflection. We show that the anomalous reflection wave pattern can be explained with a modified shock-polar analysis in which the geometric node velocity is replaced by a downstream boundary condition. Anomalous reflection is analogous to the transition from a regular to a Mach reflection when the reflected wave is a rarefaction instead of a shock. These bifurcations have been incorporated into a front-tracking code that provides an accurate description of wave interactions. Numerical results for two illustrative cases are described: a planar shock passing over a bubble, and an expanding shock impacting a planar contact.

A thin liquid layer flowing down an inclined plane exhibits a long-wave interfacial instability at a critical value of the Reynolds number. Past work on this problem has shown that heating or cooling the layer does not significantly change the characteristics of this instability. We show that this is not correct when the Prandtl number of the liquid is large and that both heating and cooling from below can destabilize the layer depending on the interfacial heat-transfer conditions. The mechanism for this unstable behaviour involves the direct expansion of the liquid as it experiences a temperature perturbation produced by the deformation of the interface. When the layer is heated from below, this additional effect changes the critical angle at which longitudinal, buoyancy-driven rolls are preferred relative to the long-wave interfacial instability.

A solution of decaying two-dimensional turbulence at large Reynolds number is analysed by means of an automated vortex census. The census identifies the flow structures which approximately conform to the idealized shape of an isolated, coherent vortex. It also determines vortex characteristics, such as amplitude, size, radial profile, and deformation from the ideal axisymmetric shape. The distributions of these characteristics within the vortex population are examined, as are their time evolutions. Interpretation of these distributions is made with reference to both the random initial conditions for the solution and the dynamical processes of vortex emergence, survival, and interaction.

A solution for decaying geostrophic turbulence at large Reynolds number is analysed by means of an automated vortex census. The census identifies the flow structures that approximately conform to the idealized shape of an isolated, coherent geostrophic vortex. It also determines vortex characteristics, such as amplitude, size, and shape. The distributions of these characteristics within the vortex population are examined, as are their time evolutions. These distributions are interpreted with reference to the dynamical processes of vortices, and they are compared to analogous distributions from a solution for two-dimensional turbulence.

Kinetic theory for the drag and thermal polarization of a spherical particle in a low-speed flow of a rarefied gas is presented. The problem is solved on the basis of the linearized kinetic equation (Shakhov 1974) with the correct Prandtl number, $Pr = \frac{2}{3}2$, for monatomic gas. The integral-moment method of solution for arbitrary values of the Knudsen number is employed. The possibility of arbitrary energy, and tangential and normal momentum accommodation of gas molecules on the particle surface is taken into account in the boundary condition. The particle–gas heat conductivity ratio Λ is assumed to be arbitrary.

Numerical results for the isothermal drag, radiometric force affecting a non-uniformly heated particle in a rarefied gas, and temperature drop between the ends of the particle diameter owing to its thermal polarization in a gas flow have been obtained. The analytical expressions approximating the numerical calculations for the whole range of Knudsen numbers are given. The results obtained are compared to the available theoretical and experimental data.

In this paper the effect of a small-amplitude progressive wave on the laminar boundary layer on a semi-infinite flat plate, due to a uniform supersonic free-stream flow is considered. The perturbation to the flow caused by the wave divides into two streamwise zones. In the first, relatively close to the leading edge of the plate, on a transverse scale comparable with the boundary-layer thickness, the perturbation flow is described by a form of the unsteady linearized compressible boundary-layer equations. In the free stream, this component of flow is governed by the wave equation, the solution of which provides the outer velocity conditions for the boundary layer. This boundary-layer system is solved numerically, and also the asymptotic structure in the far downstream limit is studied. This reveals a breakdown and a subsequent second streamwise zone, where the flow disturbance is predominantly inviscid. The two zones are shown to match in a proper asymptotic sense.

We consider instabilities in a fluid with a constant density gradient that is subject to arbitrarily oriented oscillatory accelerations. With the Boussinesq approximation and for the case of an unbounded fluid, transformation to Lagrangian coordinates allows the reduction of the problem to an ordinary differential equation for each three-dimensional wavenumber. The problem has three parameters: the non-dimensional amplitude R of the base-state oscillation, the non-dimensional level of background steady acceleration, which for some cases can be represented in terms of a local (in time) Richardson number Ri, and the Prandtl number Pr. Some general bounds on stability are derived. For Pr = 1 closed-form solutions are found for impulse (delta function) accelerations and a general asymptotic solution is constructed for large R and general imposed accelerations. The asymptotic solution takes advantage of the fact that at large R wave growth is concentrated at ‘zero points’. These are times when the effective vertical wavenumber passes through zero. Kelvin–Helmholtz instabilities are found to dominate at low R while Rayleigh–Taylor instabilities dominate at high. At high R, the uniform shear of the Kelvin–Helmholtz case tends to distort and weaken instability waves. With unsteady flows, Ri = ¼ is no longer an instability limit. Significant instabilities have been found for sinusoidal forcing for Ri up to 0.6.

Comparisons of numerical and experimental results for transient two-dimensional natural convection initiated by instantaneously heating and cooling the opposing vertical walls of a square cavity containing a stationary and isothermal fluid are presented. The good comparisons indicate that the simulation is capturing the important features of the flow. Several features are identified and discussed in detail; in particular, the presence of travelling wave instabilities on the vertical-wall boundary layers and horizontal intrusions, the existence of a rapid flow divergence in the region of the outflow of the intrusions, and the presence of cavity-scale oscillations, caused by the interaction of the intrusions with the opposing vertical boundary layer. The utilization of both numerical and experimental investigations has allowed a more complete exploitation of the available resources than would have been possible had each been conducted separately.