By means of large-eddy simulation, homogeneous turbulence is simulated for neutrally and stably stratified shear flow at gradient-Richardson numbers between zero and one. We investigate the turbulent transport of three passive species which have uniform gradients in either the vertical, downstream or cross-stream direction. The results are compared with previous measurements in the laboratory and in the stable atmospheric boundary layer, as well as with results from direct numerical simulations. The computed and measured flow properties agree with each other generally within the scatter of the measurements. At strong stratification, the Froude number becomes the relevant flow-controlling parameter. Stable stratification suppresses vertical overturning and mixing when the inverse Froude number based on a turn-over timescale exceeds a critical value of about 3. The turbulent diffusivity tensor is strongly anisotropic and asymmetric. However, only the vertical and the cross-stream diagonal components are of practical importance in shear flows. The vertical diffusion coefficient is much smaller than the cross-stream one at strong stratification. This anisotropy is stronger than predicted by second-order closure models. Turbulence fluxes in downstream and cross-stream directions follow classical mixing-length models.

Low-density flow of molecular hydrogen from a small nozzle is studied using numerical and experimental techniques. The conditions in the nozzle indicate that non-equilibrium effects will significantly influence the flow. Therefore, the numerical analysis is undertaken using a Monte Carlo approach. The experimental studies employ spontaneous Raman scattering. Comparisons of the measured data and computed results are made for total number density, rotational temperature, and for the number density of the first rotational level. The numerical results are found to be quite sensitive to the rotational relaxation rate, and a strong degree of thermal non-equilibrium is observed at the exit plane of the nozzle. Comparisons between experiment and analysis permit estimation of the rotational relaxation rate for hydrogen. Investigations are also conducted for expansion of the supersonic jet into a finite back pressure. The interaction of the plume with the chamber background gas is found to form shock waves in both the simulations and experiments. This phenomenon is investigated further by increasing the background pressure. Direct comparison of the simulation results and experimental measurements is very favourable.

We consider Lagrangian stochastic modelling of the relative motion of two fluid particles in the inertial range of a turbulent flow. Eulerian analysis of such modelling corresponds to an equation for the Eulerian probability distribution of velocity-vector increments which introduces a hierarchy of constraints for making the model consistent with results from the theory of locally isotropic turbulence. A nonlinear Markov process is presented, which is able to satisfy exactly, in the statistical sense, incompressibility, the exact results on the third-order structure function, and the experimental second-order statistics. The corresponding equation for the Eulerian probability density of velocity-vector increments is solved numerically. Numerical results show non-Gaussian statistics of the one-dimensional Lagrangian probability distributions, and a complex shape of the three-dimensional Eulerian probability density function. The latter is then compared with existing experimental data.

Building on results from two-dimensional magnetohydrodynamic (MHD) turbulence (Shebalin, Matthaeus & Montgomery 1983), the development of anisotropic states from initially isotropic ones is investigated numerically for fully three-dimensional incompressible MHD turbulence. It is found that when an external d.c. magnetic field (B0) is imposed on viscous and resistive MHD systems, excitations are preferentially transferred to modes with wavevectors perpendicular to B0). The anisotropy increases with increasing mechanical and magnetic Reynolds numbers, and also with increasing wavenumber. The tendency of B0 to inhibit development of turbulence is also examined.

The collision efficiency in a dilute suspension of sedimenting drops is considered, with allowance for particle Brownian motion and van der Waals attractive force. The drops are assumed to be of the same density, but they differ in size. Drop deformation and fluid inertia are neglected. Owing to small particle volume fraction, the analysis is restricted to binary interactions and includes the solution of the full quasi-steady Fokker—Planck equation for the pair-distribution function. Unlike previous studies on drop or solid particle collisions, a numerical solution is presented for arbitrary Péclet numbers, Pe, thus covering the whole range of particle size in typical hydrosols. Our technique is mainly based on an analytical continuation into the plane of complex Péclet number and a special conformal mapping, to represent the solution as a convergent power series for all real Péclet numbers. This efficient algorithm is shown to apply to a variety of convection—diffusion problems. The pair-distribution function is expanded into Legendre polynomials, and a finite-difference scheme with respect to particle separation is used. Two-drop mobility functions for hydrodynamic interactions are provided from exact bispherical coordinate solutions and near-field asymptotics. The collision efficiency is calculated for wide ranges of the size ratio, the drop-to-medium viscosity ratio, and the Péclet number, both with and without interdroplet forces. Solid spheres are considered as a limiting case; attractive van der Waals forces are required for non-zero collision rates in this case. For Pe [Gt ] 1, the correction to the asymptotic limit Pe → ∞ is O(Pe−1/2). For Pe [Lt ] 1, the first two terms in an asymptotic expansion for the collision efficiency are C/Pe + ½C2, where the constant C is determined from the Brownian solution in the limit Pe → 0. The numerical results are in excellent agreement with these limits. For intermediate Pe, the numerical results show that Brownian motion is important for Pe ≤ O(102). For Pe = 10, the trajectory analysis for Pe → ∞ may underestimate the collision rate by a factor of two. A simpler, approximate solution based on neglecting the transversal diffusion is also considered and compared to the exact solution. The agreement is within 2–3% for all conditions investigated. The effect of van der Waals attractions on the collision efficiency is studied for a wide range of droplet sizes. Except for very high drop-to-medium viscosity ratios, the effect is relatively small, especially when electromagnetic retardation is accounted for.

The dispersion of passive scalars by the steady viscous flow through a twisted pipe is both a simple example of chaotic advection and an elaboration of Taylor's classic shear dispersion problem. In this article we study the statistics of the axial dispersion of both diffusive and perfect (non-diffusive) tracer in this system.

For diffusive tracer chaotic advection assists molecular diffusion in transverse mixing and so diminishes the axial dispersion below that of integrable advection. As in many other studies of shear dispersion the axial distribution ultimately becomes Gaussian as t → ∞. Thus there is a diffusive regime, but with an effective diffusivity that is enhanced above molecular values. In contrast to the classic case, the effective diffusivity is not necessarily inversely proportional to the molecular diffusivity. For instance, in the irregular regime the effective diffusivity is proportional to the logarithm of the molecular diffusivity.

For perfect tracer chaotic advection does not result in a diffusive process, even in the irregular regime in which streamlines wander throughout the cross-section of the pipe. Instead the variance of the axial position is proportional to t in t so that the measured diffusion coefficent diverges like In t. This faster than linear growth of variance is attributed to the trapping of tracer for long times near the solid boundary, where the no-slip condition ensures that the fluid moves slowly. Analogous logarithmic effects associated with the no-slip condition are well known in the context of porous media.

A simple argument, based on Lagrangian statistics and a local analysis of the trajectories near the pipe wall, is used to calculate the constants of proportionality before the logarithmic terms in both the large- and infinite-Péclet-number limits.

The statistical relationship between a passive scalar and its dissipation is important for both a basic understanding of turbulence small-scale properties and for various aspects of turbulent combustion modelling. This problem is studied in two different flows through spectral analysis as well as probability density functions using temperature as a passive scalar. Particular attention is paid to the experimental determination of the three squared derivatives involved in the temperature dissipation. As a first step, it is found that basic properties such as the correlation coefficient between temperature and its dissipation are strongly related to the asymmetry of the scalar fluctuations, so that the usually assumed statistical independence between these variables is not justified. These trends are the same for the two flows investigated here, a boundary layer and a jet. This connection appears to be related to fluctuations of small amplitude for both quantities which are associated with relatively low frequencies lying between the integral scale and the Taylor microscale. In regions where the temperature skewness factor is nearly zero, the correlation coefficient is also very small, and several tests show that the assumption of independence is then fully justified. Thus, the main parameter influencing joint statistics of temperature and its dissipation is the asymmetric feature of temperature fluctuations, but the asymmetry of the longitudinal temperature derivative, which results from the flow boundary conditions and is usually felt through the presence of the so-called temperature ramps, is also involved. Even though the magnitude of the derivative skewness factor is almost uniformly distributed in both flows, the secondary effect becomes the dominant one in flow regions where the influence of the temperature asymmetry is relatively weak.

The linear stability of boundary-layer flow over compliant or flexible surfaces has been studied by Carpenter & Garrad (1985), Yeo (1988) and others on the assumption of local flow parallelism. This assumption is valid at large Reynolds numbers. Non-parallel effects due to growth of the boundary layer gain in significance and importance as one gets to lower Reynolds number. This is especially so for a compliant surface, which can sustain a variety of wall-related instabilities in addition to the Tollmien—Schlichting instabilities (TSI) that are found over rigid surfaces. The present paper investigates the influence of boundary-layer non-parallelism on the TSI and wall-related travelling-wave flutter (TWF) on compliant layers. Corrections to the growth rate of locally parallel theory for boundary-layer non-parallelism are obtained through a multiple-scale analysis. The results indicate that flow non-parallelism has an overall destabilizing influence on the TSI and TWF. Flow non-parallelism is also found to have a very strong destabilizing effect on the branch of TWF that stretches to low Reynolds number. The results obtained have important implications for the design and use of compliant layers at low Reynolds numbers.

We have conducted further numerical experiments on two-dimensional fully compressible convection in an imposed vertical magnetic field and interpreted the results by reference to appropriate low-order models. Here we focus on streaming instabilities, involving horizontal shear flows, which form an important mechanism for the breakdown of steady convection in relatively weak fields for boxes of sufficiently small aspect ratio. While these shearing modes can arise even in the absence of a field, they will typically lead only to travelling and modulated waves unless there is a field to provide a restoring force. For magnetoconvection a new and dramatic form of pulsating wave appears after a complex sequence of secondary bifurcations.

The melting of a solid in contact with a hot fluid is quantified for the case in which a difference between the densities of the fluid and of the melted solid is able to drive vigorous compositional convection. A scaling analysis is first used to obtain a theoretical expression for the melting rate that is valid for a certain range of Stefan numbers. This expression is then compared with melting velocities measured in laboratory experiments in which ice and wax are melted when they are overlain or underlain by hot aqueous solutions. The melting velocities are consistent with the theoretical expression, and are found to depend on the heats of solution that are released when the melted solids mix with the solutions. The experiments also indicate that, for vigorous convection to occur during the melting of a floor, the unstable compositional buoyancy needs to be at least twice the stabilizing thermal buoyancy.

An important geological situation in which melting occurs is when large volumes of basaltic magma are intruded into the Earth's continental crust. The theoretical and experimental results are used and extended to examine quantitatively the melting of the floor and walls of the magma chamber, and of crustal blocks that fall into the chamber.

The one-dimensional dissolution that occurs when a binary melt is placed above or below a solid of a different composition is examined both theoretically and experimentally. In the case considered, the dissolution is driven by vigorous compositional convection that results from a Rayleigh-Bénard instability of the compositional boundary layer in the vicinity of the dissolving solid. A scaling analysis is used to derive theoretical expressions for both the dissolving velocity and the interfacial fluid concentration. Laboratory experiments are also described in which ice is dissolved when it is overlain or underlain by aqueous solutions. The measured dissolving velocities are consistent with the theoretical expressions, and yield estimates of the critical Rayleigh number for boundary-layer instability. The results of this study are then applied to predict the rate at which dissolution will occur when undersaturated mixed magmas are generated during the periodic replenishment of large basaltic magma chambers in the Earth's crust.

The stability of an axisymmetric vortex with a single radial discontinuity in potential vorticity is investigated in rotating shallow water. It is shown analytically that the vortex is always unstable, using the WKBJ method for instabilities with large azimuthal mode number. The analysis reveals that the instability is of mixed type, involving the interaction of a Rossby wave on the boundary of the vortex and a gravity wave beyond the sonic radius. Numerically, it is demonstrated that the growth rate of the instability is generally small, except when the potential vorticity in the vortex is the opposite sign to the background value, in which case it is shown that inertial instability is likely to be stronger than the present instability.

The flow in a breaking-wave crest is represented by a complex velocity potential on a Riemann surface, satisfying the Bernoulli condition on two free boundaries. The flow is assumed to be stationary in the reference frame which moves with the wave crest, and at large distances approximates Stokes corner flow in the main part of the fluid and a parabolic descending flow in the jet. The interaction of the jet with the rest of the fluid is neglected.

The solution is obtained by means of a conformal transformation from a bounded, teardrop-shaped domain, using a Faber polynomial expansion. The Bernoulli condition is applied at a number of discrete points on the boundaries, and the resulting nonlinear equations for the expansion coefficients are solved iteratively. The resulting surface form is similar to that obtained by laboratory experiments and time-dependent numerical simulations of waves up to the point of breaking, with a stagnation point at the top of the crest, an overturning loop with major axis $\ap 8g^{-\frac{1}{3}}\Psi^{\frac{2}{3}}$, and a maximum acceleration of ≈ 5.4 g, where g is the gravitational acceleration and ψ is the flux in the jet.

When a rotating fluid over sloping topography is heated from below and/or cooled from above, horizontal temperature gradients develop which drive convection cells aligned with isobaths. We refer to these cells as topographic Hadley cells. Laboratory experiments reveal that sinking occurs in small cyclonic vortices situated in relatively shallow regions. This is balanced by slower upwelling in adjacent deeper regions. The cross-isobath motions which connect the upwelling and downwelling are accelerated by Coriolis forces, resulting in strong jets which follow isobathic contours. For anticlockwise rotation, the surface jets keep the shallows to their left when looking in the direction of flow, which is opposite to both Kelvin and Rossby wave propagation. The width of the jets scales with the Rossby deformation radius and if this is much less than the width of the slope region then a number of parallel jets form. Motions on the deeper side of the jets where the flow is accelerating are adequately described by linear inviscid theory. However, the strong shears generated by this acceleration lead to baroclinic instability. The resulting cross-stream momentum fluxes broaden and flatten the velocity profile, allowing the flow on the shallow side of the jet to decelerate smoothly before sinking. Topographic Hadley cells are dynamically similar to terrestrial atmospheric Hadley cells and may also be relevant to the zonal jet motions observed on Jupiter and Saturn. It is also suggested that in coastal seas they may represent an important mode of heat (or salt) transfer where surface cooling (or evaporation) drives convection.

The equilibrium dynamics in a homogeneous forced-dissipative f-plane shallow-water system is investigated through numerical simulations. In addition to classical two-dimensional turbulence, inertio-gravity waves also exist in this system. The dynamics is examined by decomposing the full flow field into a dynamically balanced potential-vortical component and a residual ‘free’ component. Here the potential-vortical component is defined as part of the flow that satisfies the gradient-wind balance equation and that contains all the linear potential vorticity of the system. The residual component is found to behave very nearly as linear inertio-gravity waves. The forcing employed is a mass and momentum source balanced so that only the large-scale potential-vortical component modes are directly excited. The dissipation is provided by a linear relaxation applied to the large scales and by an eighth-order linear hyperdiffusion. The statistical properties of the potential-vortical component in the fully developed flow were found to be very similar to those of classical two-dimensional turbulence. In particular, the energy spectrum of the potential-vortical component at scales smaller than the forcing is close to the ∼ k−3 expected for a purely two-dimensional system. Detailed analysis shows that the downscale enstrophy cascade into any wavenumber is dominated by very elongated triads involving interactions with large scales. Although not directly forced, a substantial amount of energy is found in the inertio-gravity modes and interactions among inertio-gravity modes are principally responsible for transferring energy to the small scales. The contribution of the inertio-gravity modes to the flow leads to a shallow tail at the high-wavenumber end of the total energy spectrum. For parameters roughly appropriate for the midlatitude atmosphere (notably Rossby number ∼ 0.5), the break between the roughly ∼ k−3 regime and this shallower regime occurs at scales of a few hundred km. This is similar to the observed mesoscale regime in the atmosphere. The nonlinear interactions among the inertio-gravity modes are extremely broadband in spectral space. The implications of this result for the subgrid-scale closure in the shallow-water model are discussed.

It is shown that realizability of second-moment turbulence closure models can be established by finding a Langevin equation for which they are exact. All closure models currently in use can be derived formally from the type of Langevin equation described herein. Under certain circumstances a coefficient in that formalism becomes imaginary. The regime in which models are realizable is, at least, that for which the coefficient is real. The present method does not imply unrealizable solutions when the coefficient is imaginary, but it does guarantee realizability when the coefficient is real; hence, this method provides sufficient, but not necessary, conditions for realizability. Illustrative computations of homogeneous shear flow are presented. It is explained how models can be modified to guarantee realizability in extreme non-equilibrium situations without altering their behaviour in the near-equilibrium regime for which they were formulated.