Experiments by Simpson & Linden (1989) have shown that a horizontally nonlinear distribution of fluid density is necessary in order to produce frontogenesis. This paper considers the simplest case of such a nonlinear distribution, a quadratic density distribution in a channel. Two flow models are examined, porous media and Boussinesq. The evolution equations for both these flows can be reduced to one-dimensional systems. An exact solution is derived for porous-media flow with no molecular diffusion. Numerical solutions are shown for the other cases. The porous-media and inviscid/non-diffusive Boussinesq systems exhibit ‘classic’ frontogenesis behaviour: a rapid and intense steepening of the density gradient near the lower boundary while horizontal divergence reduces the upper-boundary density gradient to nearly zero. The viscous Boussinesq system exhibits a more complicated behaviour. In this system, boundary-layer effects force frontogenesis away from the lower boundary and at late times the steepest density gradients are close to mid-channel. One feature of these model systems is that they can exhibit blow-up in finite time. Proof of blow-up is given for the non-diffusive porous media and inviscid/nondiffusive Boussinesq cases. Numerical results indicate that blow-up also occurs for the diffusive porous-media case and that it may occur for the diffusive Boussinesq case. Despite the blow-up we believe that the model solutions can be applied to real situations. To support this a two-dimensional calculation has been made of Boussinesq frontogenesis in a long box. This calculation shows close agreement with the corresponding one-dimensional calculation up to times close to blow-up.

A nine-sensor hot-wire probe is described which is capable of simultaneously measuring the velocity and vorticity vectors with a spatial resolution of about six Kolmogorov microscales just above the viscous sublayer in a thick turbulent boundary layer at a Reynolds number of Rθ = 2685. Results from tests of the probe performance are presented to show that the three velocity components at each of its three arrays are measured with sufficient accuracy to allow determination of velocity gradients and from them the vorticity vector. Measurements with this probe of statistical properties of the velocity and vorticity fields of the turbulent boundary layer are given in Part 2 of this paper. When compared to the results of others, they further demonstrate the capability of this probe to measure simultaneously the velocity and vorticity vectors in turbulent flows of low to moderate Reynolds numbers.

Many of the statistical properties of both the velocity and the vorticity fields of a nominally zero-pressure-gradient turbulent boundary layer at Rδ = 27650 (Rθ = 2685) have been simultaneously measured. The measurements were made with a small nine-sensor hot-wire probe which can resolve the turbulence to within about six Kolmogorov microscales just above the sublayer. The statistical properties of the velocity vector field compare very well with other laboratory measurements and with direct numerical simulations when Reynolds-number dependence is taken into account. The statistical properties of the vorticity field are also in generally good agreement with the few other measurements and with the direct numerical simulations available for comparison. Near the wall, r.m.s. measurements show that the fluctuating spanwise vorticity is the dominant component, but in the outer part of the boundary layer all the component r.m.s. values are nearly equal. R.m.s. measurements of the nine individual velocity gradients show that the gradients normal to the wall of all three velocity components are the largest, with peaks occurring near the wall as expected. Gradients in the streamwise direction are everywhere small. One-dimensional spectra of the vorticity components show the expected shift of the maximum energy to higher wavenumbers compared to spectra of the velocity components at the same location in the flow. The budget of the transport equation for total enstrophy indicates that the viscous dissipation rate is primarily balanced by the viscous diffusion rate in the buffer layer and by the rotation and stretching rate in the logarithmic layer.

A study is undertaken of both parallel flow and cross-flow in the viscous sublayer generated by a fluid streaming along a grooved surface, with the aim of clarifying the phenomena that underlie the reduction of turbulent drag by such surfaces. A quantitative characterization of the effectiveness of different groove profiles in retarding secondary cross-flow is given in terms of the difference of two ‘protrusion heights’. Analytical calculations of limit cases and a boundary-element computer code for the analysis of general profiles are illustrated, and several examples are presented and discussed.

In the quest for a physically more realistic transition criterion, the prechaotic bifurcation behaviour of plane Poiseuille flow is studied. Various classes of nonlinear time-periodic equilibrium solutions are computed via Keller's pseudo-arclength continuation method. In particular, attention is focused on three-dimensional nonlinear travelling-wave type secondary bifurcation branches. These saturated equilibrium states originate on the nonlinear primary bifurcation surface from neutral, phase-locked secondary instability modes. Taking advantage of symmetries, only those nonlinear secondary branches which correspond to symmetric and antisymmetric linear secondary instability modes are investigated.

It appears that a new family of secondary bifurcation solutions which contains only even spanwise Fourier modes is particularly important. Dominated largely by the spanwise (0,2) mode and discovered by investigating bicritical secondary bifurcations, the mean quantities of these solutions show a certain resemblance to those observed in transitional flow during the ‘spike’ stage. The friction factor of this new solution branch is in the experimentally observed range and the critical Reynolds number, defined with the mean flow velocity, is reduced to about 1000 in general agreement with experiments.

This is the first part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. Flow is forced by a prescribed jump of composition across the interfaces. Compositional diffusivity is ignored, while thermal diffusivity and viscosity are finite. In this first part, basic-state solutions are presented and discussed for three configurations: a single plane interface, two parallel interfaces and a circular cylindrical interface.

The stability of a single plane interface is then analysed. It is shown that the presence of the compositional jump gives rise to a new type of three-dimensional instability which occurs for any non-zero forcing. This is in contrast to the thermally driven flow adjacent to a rigid wall, which is unstable only for a finite value of the forcing and results in the growth of a two-dimensional perturbation. The timescale for growth of the new instability is given by \[ \tau = \left[\frac{\nu \rho}{(\Delta \rho)_c}\right]^2\frac{\alpha}{\kappa g}\frac{{\rm d}T}{{\rm d}z}, \] where (Δρ)C is the prescribed jump in composition and dT/dz is the imposed temperature gradient.

The influence of thermal diffusion is to enhance instability, while viscosity is stabilizing for nearly all wavenumbers. The interface is unstable for all finite wavenumbers if the Prandtl number is less than 1.472, while regions of stability in the wavenumber plane develop for small horizontal wavenumber and moderate vertical wavenumber for larger values of the Prandtl number. The neutral stability curves are investigated and the maximum growth rate of instability is identified for the whole range of values of the Prandtl number and its properties are elucidated by comparison with previous studies of flows near heated vertical walls.

Turbulence characteristics inside a turbulent spot in plane Poiseuille flow are investigated by analysing a database obtained from a direct numerical simulation. The spot is found to consist of two distinct regions — a turbulent area and a wave area. The flow inside the turbulent area has a strong resemblance to that found in the fully developed turbulent channel. Suitably defined mean and r.m.s. fluctuations as well as the internal shear-layer structures are found to be similar to the turbulent counterpart. In the wave area the inflexional mean spanwise profiles cause a rapid growth of oblique waves, which break down to turbulence. The breakdown process of the oblique waves is reminiscent of the secondary instability observed during transition to turbulence in channel and boundary-layer flows. Other detailed characteristics associated with the Poiseuille spot are presented and are compared with experimental results.

The dynamical behaviour of stretchable, orientable microstructure suspended in a general two-dimensional fluid flow is investigated. The state of the microstructure in question is described by an axial vector; thus the microstructure may consist of axisymmetric particles, droplets of fluid, models of polymer molecules or simply a line element of the fluid itself. A quantitative measure is developed to distinguish conformation(s) (orientations and stretched lengths) of the microstructure that are robust and attractive. This leads to a strong flow criterion for microstructure suspended in unsteady, spatially inhomogeneous flows in which the effects of history-dependence are apparent. The important special case where the influence of the flow on the microstructure is time periodic is considered in some detail, owing to the fact that one can obtain additional results that concern orientation dynamics. Finally, several examples are given which illustrate the application of the present methods and the relevant innovations of the approach. Throughout the analysis, special attention is given to the robustness of the dynamics to changes in the modelling assumptions such as slight three-dimensionality or Brownian diffusion, etc. The results of the study demonstrate that using microdynamical behaviour in steady, homogeneous flows to derive macroscopic properties (such as strong flow criteria) which are then applied to problems in unsteady, spatially inhomogeneous flows can lead to incorrect results. Instead, one must account properly for effects due to the history of the flow.

A study of the structure of stationary, fully and partly dispersed, normal shock waves in steady vapour–droplet, two-phase flow is presented. Pure substances only are considered, but, unlike most previous work, the droplet population is allowed to be polydispersed. It is shown how the effects of thermal relaxation for such a mixture can be elegantly incorporated into the analysis.

Three types of fully dispersed wave are identified. Type I waves are dominated by thermal relaxation and an approximate analytical solution is presented which gives results in close agreement with accurate numerical solutions of the governing equations. The analysis predicts some unexpected behaviour of the thermodynamic variables and demonstrates the correct scaling parameters for such flows. An approximate analysis is also presented for Type II waves, dominated by both velocity and thermal relaxation. Type III waves, where all three relaxation processes are important, are of little practical significance and are only briefly discussed. Partly dispersed waves are also considered and the results of a numerical simulation of the relaxation zone are presented. A linearized solution of this problem is possible but, unlike other relaxing gas flows, does not give good agreement with the more exact numerical calculations.

The apparent discontinuity in the speed of sound in a vapour–droplet mixture as the wetness fraction tends to zero has been responsible for some confusion in the literature. This problem is reconsidered and it is shown that the transition from the two-phase equilibrium to the single-phase frozen shock wave speed is continuous.

The boundary-element method is used to solve Stokes equations for periodic arrays of force-free and torque-free rigid particles. Simple cubic arrays of spheres, spheroids, cubes, and clusters of spheres are subjected to a bulk simple shearing flow. The effective volume-averaged stress tensor for the suspension and the detailed velocity and stress fields throughout the Newtonian suspending fluid are calculated. We find that even crude meshes give very good volume-averaged results, but fine meshes are required to track local minima and maxima in the stress field. For simple cubic arrays of spheres, the boundary-element results are in excellent agreement with the analytical viscosity predictions of Nunan & Keller (1984). Even at the highest concentration of solids studied, no significant normal stress differences were observed, in agreement with Nunan & Keller's results (1984). Up to moderate concentrations of particles, the volume-averaged properties of the suspension display only a weak dependence on the particle geometry. Suspensions of spheroids and cubes behave approximately as suspensions of spheres on the average despite large differences in the local micromechanics of stress and velocity fields. Simple cubic arrays of clusters of spheres tend to behave on a macroscopic level as a cubic array of spheres whose effective volume fraction is about 150% of the total volume fraction of the spheres in the clusters.

Some previously accepted results for the form of one- and two-particle Langrangian turbulence statistics within the inertial subrange are corrected and reinterpreted using dimensional methods and kinematic constraints. These results have a fundamental bearing on the statistical theory of turbulent dispersion.

One-particle statistics are analysed in an inertial frame [Sscr ] moving with constant velocity (which is different for different realizations) equal to the velocity of the particle at the time of labelling. It is shown that the inertial-subrange form of the Lagrangian acceleration correlation traditionally derived from dimensional arguments constrained by the property of stationarity, ${\cal C}_0^{(a)}\overline{\epsilon}/\tau $, where ${\cal C}_0^{(a)}$ is a universal constant, $\overline{\epsilon}$ is the mean rate of dissipation of turbulence kinetic energy and τ is the time lag, is kinematically inconsistent with the corresponding velocity statistics unless ${\cal C}_0^{(a)} = 0$. On the other hand, velocity and displacement correlations in the inertial subrange are non-trivial and the traditional results are confirmed by the present analysis. Remarkably, the universal constant ${\cal C}_0$ which characterizes these latter statistics in the inertial subrange is shown to be entirely prescribed by the inner (dissipation scale) acceleration covariance; i.e. there is no contribution to velocity and displacement statistics from inertial-subrange acceleration structure, but rather there is an accumulation of small-scale effects.

In the two-particle case the (cross) acceleration covariance is deduced from dimensional arguments to be of the form $\overline{\epsilon}t_1^{-1}{\cal R}_2(t_1/t_2)$ in the inertial subrange. In contrast to the one-particle case this is non-trivial since the two-particle acceleration covariance is non-stationary and there is therefore no condition which constraints [Rscr ]2 to a form which is kinematically inconsistent with the corresponding velocity and displacement statistics. Consequently it is possible for two-particle inertial-subrange acceleration structure to make a non-negligible contribution to relative velocity and dispersion statistics. This is manifested through corrections to the universal constant appearing in these statistics, but does not otherwise affect inertial-subrange structure. Nevertheless, these corrections destroy the simple correspondence between relative- and one-particle statistics traditionally derived by assuming that two-particle acceleration correlations are negligible within the inertial subrange.

A simple analytic expression which is proposed as an example of the form of [Rscr ]2 provides an excellent representation in the inertial subrange of Lagrangian stochastic simulations of relative velocity and displacement statistics.

Results from a wide range of direct numerical simulations of forced-dissipative, differentially rotating two-dimensional turbulence are presented, in order to delineate the broad dependence of flow type on forcing parameters. For most parameter values the energy spectra of simulations forced at low wavenumbers are markedly steeper than the classical k−3 enstrophy inertial-range prediction, and although k−3 spectra can be produced under certain circumstances, the regime is not robust, and the Kolmogorov constant is not universal unless a slight generalization is made in the phenomenology. Long-lived, coherent vortices form in many cases, accompanied by steep energy spectra and a higher than Gaussian vorticity kurtosis. With the addition of differential rotation (the β-effect), a small number of fairly distinct flow regimes are observed. Coherent vortices weaken and finally disappear as the strength of the β-effect increases, concurrent with increased anisotropy and decreased kurtosis. Even in the absence of coherent vortices and with a Gaussian value of the kurtosis, the spectra remain relatively steep, although not usually as steep as for the non-rotating cases. If anisotropy is introduced at low wavenumbers, the anisotropy is transferred to all wavenumbers in the inertial range, where the dynamics are isotropic.

For those simulations that are forced at relatively high wavenumbers, a well resolved and very robust k−5/3 energy inertial range is observed, and the Kolmogorov constant appears universal. The low-wavenumber extent of the reverse energy cascade is essentially limited by the β-effect, which produces an effective barrier in wavenumber space at which energy accumulates, and by frictional effects which must be introduced to achieve equilibrium. Anisotropy introduced at large scales remains largely confined to the low wavenumbers, rather than being cascaded to small scales. When there is forcing at both large and small scales (which is of relevance to the Earth's atmosphere), energy and enstrophy inertial ranges coexist, with an upscale energy transfer and downscale enstrophy transfer in the same wavenumber interval, without the need for any dissipation mechanism between forcing scales.

An inviscid two-dimensional eddy (with maximum circulation $\hat{\Gamma}$ and uniform vorticity $\hat{\zeta}_2 > 0$, surrounded by irrotational fluid is initially located near the edge of a jet, on the other side of which the vorticity $\hat{\zeta}_1(< \hat{\zeta}_2)$ increases. The interaction causes the eddy to move towards the edge and into the shear flow. Eventually the eddy and the ambient (irrotational) fluid are surrounded by the jet fluid. An average entrainment velocity is computed for a variety of relevant conditions, and found to scale mainly with $(\hat{\Gamma}_1)^{\frac{1}{2}}$. The approximate proportionality constant is somewhat larger than the measured value for three-dimensional (turbulent) jets, suggesting that the two-dimensional mechanism may be qualitatively relevant. The results may also be generalized to study the role of mesoscale eddies in the lateral entrainment of ambient fluid into oceanic jets.

Convection driven by radial and normal gravity forces in a rotating, horizontal cylinder is examined. The cylinder is subjected to uniform volumetric heating and constant-temperature wall cooling. The parameters are the radial-gravity and normal-gravity Rayleigh numbers, Rar and Rag (with Rar, Rag ≤ 106), the rotational Reynolds number, Re = 2Ω r02/v (0 ≤ Re ≤ 250), and the Prandtl number (Pr = 7). Critical conditions for the radial-gravity rest state correspond to a two-cell flow in the azimuthal plane with Rar,c = 13738. Finite-amplitude transient and steady flows are obtained with a Galerkin finite element method for Rayleigh number ratios in the range 0.1 ≤ Rar/Rag ≤ 100. When radial gravity dominates the flows tend to be multicellular and, during transients, initial high-wavenumber forms evolve to lower-wavenumber forms. When normal gravity dominates the flows are bicellular. When radial and normal gravity forces are comparable, in the presence of rotation, complex time-dependent motions occur and the largest rates of fluid circulation and heat transfer are observed.

We investigate the transition to turbulence in a free shear layer which contains a thin central region of stable density stratification. The fluid is assumed to possess Prandtl number significantly larger than unity, and the flow may exhibit either Holmboe or Kelvin–Helmholtz (KH) instability, depending upon the intensity of the stratification. A sequence of two-dimensional nonlinear numerical simulations of flows near the KH–Holmboe transition (i.e. having bulk Richardson numbers near 1/4) clearly illustrates the structural relationship between Holmboe and Kelvin–Helmholtz waves. The time-dependent nonlinear wave states delivered by the simulations are subjected to a three-dimensional normal-mode stability analysis in order to discover the physical processes that might drive the flow towards a turbulent state. Strong secondary instability is found to persist up to large spanwise wavenumbers, with no indication of a preferred lengthscale. These results indicate that secondary instability may lead the flow directly into the turbulent state.

When raindrops with a diameter of the order of 1 mm hit a plane water surface they entrain air bubbles that radiate noise in the course of volume oscillations. The paper presents a model of the underwater noise of rain produced by this process. The depth of submergence, radius, and initial energy of the entrained bubbles are obtained numerically for a number of drop sizes. The bubbles are assumed to radiate as dipoles, and the total underwater noise is calculated by integrating over the size of the entraining rain drops. The results are compared both with laboratory experiments of single-drop impacts and field data of rain noise. It is found that the model gives somewhat larger bubbles than are observed experimentally. As a consequence, the characteristic spectral peak of rain is predicted to occur at a somewhat lower frequency than found in experiment. However the level of the peak is in reasonable agreement with data. The amount of noise due to the process of drop impact itself is also estimated and found to be several orders of magnitude lower than the data. Therefore, in spite of some deficiencies of the model and of the computational results, the proposed mechanism for the underwater noise of rain is strongly supported by this study.

Compressible linear stability theory for axisymmetric flows is presented. The theory is applied to flow past a cylinder and a sharp cone at a Mach number of 5 with adiabatic wall conditions. The effect of transverse curvature and body divergence is studied. It is found that transverse curvature has a stabilizing influence on axisymmetric (first and second mode) disturbances while it has a destabilizing influence on the asymmetric (oblique first mode) disturbances. The body divergence effects are stabilizing for both symmetric and asymmetric disturbances. Comparisons made with the results of planar stability theory show that, for a cylinder, curvature effects become more pronounced with increasing distance along the cylinder. For a sharp cone, these effects become less significant further away from the cone tip since the body radius increases faster than the growth of the boundary layer. The effect of cone angle on stability is also studied.

The turbulent flow in a duct with an abrupt increase in its cross-sectional area is very unstable and exhibits strong oscillations. Such a flow is investigated experimentally using a simple device — a pipe–collar nozzle, composed of a convergent nozzle and a pipe lengthened by a collar with an enlarged cross-section. The oscillating flow characteristics are considered over a wide range of flow velocities including both subsonic and transonic flow regimes. Two modes of symmetric and one of asymmetric oscillations are distinguished. For the first mode of symmetric oscillation the jet is separated in each phase of the oscillation cycle, whereas for the second the jet alternately separates and reattaches to the collar wall. Both modes of symmetric oscillations are controlled by shear-layer instability of the collar flow. For low velocities and short collars the oscillation is also affected by the organ-pipe resonance in the preceding pipe. The asymmetric oscillation appears at relatively large flow Mach numbers. This mode exhibits a radial resonance of the jet.

Premixed turbulent flame propagation is analysed under the assumptions of stationarity and transverse homogeneity by expansions for small values of the ratio of the turbulence intensity to the laminar burning velocity. For large Zel'dovich numbers, the effects of diffusive—thermal phenomena within the flame, gas expansion, buoyancy and Lewis and Prandtl numbers different from unity are taken into account under the constraint that turbulence scales are large compared with the laminar flame thickness. A general formulation is given, involving solutions through Fourier decompositions. Parametric results for turbulent burning velocities are obtained, and the evolution of components of turbulent kinetic energies through the flame is calculated. It is shown how buoyancy counteracts the tendency for gas expansion to increase transverse components of the turbulent kinetic energy, pressure fluctuations and vorticity generation across the wrinkled flame. Strong readjustments in components of the turbulent kinetic energy are shown to occur in the downstream hydrodynamic zone. It is established that, with the effects of the hydrodynamic zones fully taken into account, the flame can induce anisotropy in initially isotropic turbulence such that the final velocity fluctuations exhibit higher intensities in the longitudinal mode than in transverse modes, while the enhanced vorticity fluctuations are entirely transverse.

Laboratory experiments were carried out to investigate the effects of rotation on turbulent convection. The experimental facility was a bottom-heated, water-filled, cubical tank mounted on a turntable. The investigations were performed over a wide range of bottom buoyancy fluxes q0 and rotation rates Ω, including Ω = 0; q0 and Ω were held constant during each experiment. The depth of the water column H was fixed for the entire experimental programme. For the non-rotating experiments, the r.m.s. velocity fluctuations were found to scale well with the convective velocity $w_* = (q_0 H)^{\frac{1}{3}}$, while the mean and r.m.s. fluctuations of buoyancy were found to scale with q0/w*. The spectra of temperature fluctuations were measured and were used to assess the applicability of two types of scaling, one of which is advanced in the present study.

For the rotating experiments, the convective-layer growth is affected by the rotation at a height hc ≈ 4.5(q0Ω−3)½. The r.m.s. horizontal velocity of the rotationally affected mixed layer is uniform throughout the mixed layer and is given by $(\overline{u^{\prime 2}})^{\frac{1}{2}}_{\rm r}\approx 1.7(q_0\Omega^{-1})^{\frac{1}{2}}$. The time growth law of the mixed-layer thickness hr, when hr > hc, is given by hr ≈ 0.7(q0Ω−3)½Ωt, where t is the time. The rotational effects become important when the Rossby number is given by $Ro = (\overline{u^{\prime 2}})^{\frac{1}{2}}_{\rm r}/\omega l_{\rm r}\approx 1.5$, where the integral lengthscale is estimated as lr ≈ 0.25hc. The mean buoyancy gradient in the mixed layer was found to be much higher than in the corresponding non-rotating case, and the r.m.s. fluctuations and mean buoyancies were found to scale satisfactorily with (q0Ω)½. A spectral form for the temperature fluctuations in rotating convection is also proposed and is compared to the experimental results.