Solutions are analysed from large-eddy simulations of the phase-averaged equations for oceanic currents in the surface planetary boundary layer (PBL), where the averaging is over high-frequency surface gravity waves. These equations have additional terms proportional to the Lagrangian Stokes drift of the waves, including vortex and Coriolis forces and tracer advection. For the wind-driven PBL, the turbulent Langmuir number, Latur = (U∗/Us)1/2, measures the relative influences of directly wind-driven shear (with friction velocity U∗) and the Stokes drift Us. We focus on equilibrium solutions with steady, aligned wind and waves and a realistic Latur = 0.3. The mean current has an Eulerian volume transport to the right of the wind and against the Stokes drift. The turbulent vertical fluxes of momentum and tracers are enhanced by the presence of the Stokes drift, as are the turbulent kinetic energy and its dissipation and the skewness of vertical velocity. The dominant coherent structure in the turbulence is a Langmuir cell, which has its strongest vorticity aligned longitudinally (with the wind and waves) and intensified near the surface on the scale of the Stokes drift profile. Associated with this are down-wind surface convergence zones connected to interior circulations whose horizontal divergence axis is rotated about 45° to the right of the wind. The horizontal scale of the Langmuir cells expands with depth, and there are also intense motions on a scale finer than the dominant cells very near the surface. In a turbulent PBL, Langmuir cells have irregular patterns with finite correlation scales in space and time, and they undergo occasional mergers in the vicinity of Y-junctions between convergence zones.

The unsteady flow field of an incompressible viscous fluid around an impulsively started cylinder with slow motion is studied in detail. Integral expressions are derived from the nonlinear vorticity equation, and are solved by the method of matched asymptotic expansions. To complete the matching process five regions are necessary and their regions are essentially governed by the following relations: (i) the initial flow is unsteady Stokes flow (I), (ii) the early transient flow near the cylinder is steady Stokes flow (II), but the far-field flow is unsteady Stokes flow (III), so that Stokes&–Oseen-like matching is necessary, and (iii) as time increases the inertia terms become significant far downstream; thus the far flow is unsteady Oseen flow (IV), but the flow near the cylinder is steady Stokes flow (V), so that the matching of the Stokes–Oseen equations is necessary. The asymptotic analytical solutions are given for five flow fields around a circular cylinder. Also presented are the drag coefficient, the vorticity, and the streamline. The drag coefficient is verified quantitatively by comparing with earlier theories of the initial flow and the steady flow. The streamline patterns calculated show the generation of a circulating zone close to the circular cylinder just as for the transient flow around a sphere, and the difference between two-dimensional and three-dimensional flows is discussed.

Experiments were carried out in a new type of stratified flow facility to study the evolution of turbulence in a mean flow possessing both uniform stable stratification and uniform mean shear.

The new facility is a thermally stratified wind tunnel consisting of ten independent supply layers, each with its own blower and heaters, and is capable of producing arbitrary temperature and velocity profiles in the test section. In the experiments, four different sized turbulence-generating grids were used to study the effect of different initial conditions. All three components of the velocity were measured, along with the temperature. Root-mean-square quantities and correlations were measured, along with their corresponding power and cross-spectra.

As the gradient Richardson number Ri = N2/(dU/dz)2 was increased, the downstream spatial evolution of the turbulent kinetic energy changed from increasing, to stationary, to decreasing. The stationary value of the Richardson number, Ricr, was found to be an increasing function of the dimensionless shear parameter Sq2/∈ (where S = dU/dz is the mean velocity shear, q2 is the turbulent kinetic energy, and ∈ is the viscous dissipation).

The turbulence was found to be highly anisotropic, both at the small scales and at the large scales, and anisotropy was found to increase with increasing Ri. The evolution of the velocity power spectra for Ri [les ] Ricr, in which the energy of the large scales increases while the energy in the small scales decreases, suggests that the small-scale anisotropy is caused, or at least amplified, by buoyancy forces which reduce the amount of spectral energy transfer from large to small scales. For the largest values of Ri, countergradient buoyancy flux occurred for the small scales of the turbulence, an effect noted earlier in the numerical results of Holt et al. (1992), Gerz et al. (1989), and Gerz & Schumann (1991).

The effect of oscillatory forcing as a dynamic stabilization or destabilization mechanism for two-layer plane Couette–Poiseuille flow at low Reynolds number is studied using numerical and asymptotic methods. The flow is driven by the relative planar motion of the upper boundary and a pressure gradient in the streamwise direction. Both driving forces are composed of a steady part and small-amplitude time-periodic fluctuations. An asymptotic expansion for the growth rates for small amplitudes is developed and the correction terms are quadratic in the amplitudes. The modulations to the steady flow can have either a stabilizing or destabilizing influence depending upon the conditions of flow. Complete stabilization is possible for certain flows which are otherwise unstable owing to the viscosity stratification across the interface. The combined pressure and velocity fluctuations can have an opposite effect on the flow stability to that induced by the separate time-periodic forcing mechanisms.

Unsteady heat or mass transport from a particle with an arbitrary shape suspended in a fluid of infinite expanse is considered in the limit of small Péclet numbers where diffusion is dominant. In a frame of reference in which the particle appears to be stationary, the velocity of the fluid is uniform or varies in a linear manner with respect to the spatial coordinates, with an arbitrary time dependence. The temperature or concentration of a species at the surface of the particle is held at a certain constant value, whereas that at infinity is held at another constant value. Two particular problems are considered, both to leading order with respect to the Péclet number: (a) the rate of transport from a particle that is introduced suddenly into a steady flow near the steady state; and (b) the average rate of transport from a particle that is suspended in a time-periodic flow. The theory uses the method of matched asymptotic expansions and employs the Green's function of the convection–diffusion equation for a generally unsteady uniform or linear flow. The Green's function is derived in closed form by first performing a transformation to a Lagrangian framework. In the first problem of transient transport, it is found that the functional form of the rate of transport near the steady state is affected strongly by the structure of the incident flow: the decay in uniform or elongational flow is exponential, whereas the decay in simple shear flow is algebraic. In the second problem of transport in a periodic flow, it is found that the value of a properly defined frequency parameter has a strong influence on the mean rate of transport, for all types of flow. The oscillation induces convective mixing and thereby reduces the mean rate of transport by a substantial factor. The ability of the theory to describe another situation of heat or mass transport considered by Pedley is also discussed.

Diverse subsonic initial-boundary-value problems (flows in a closed volume initiated by blowing or suction through permeable walls, flows with continuously distributed sources, viscous flows with substantial heat fluxes, etc.) are considered, to show that they cannot be solved by using the classical theory of incompressible fluid motion which involves the equation div u = 0. Application of the most general theory of compressible fluid flow may not be best in such cases, because then we encounter difficulties in accurately resolving the complex acoustic phenomena as well as in assigning the proper boundary conditions. With this in mind a new non-local mathematical model, where div u ≠ 0 in the general case, is proposed for the simulation of unsteady subsonic flows in a bounded domain with continuously distributed sources of mass, momentum and entropy, also taking into account the effects of viscosity and heat conductivity when necessary. The exclusion of sound waves is one of the most important features of this model which represents a fundamental extension of the conventional model of incompressible fluid flow. The model has been built up by modifying both the general system of equations for the motion of compressible fluid (viscous or inviscid as required) and the appropriate set of boundary conditions. Some particular cases of this model are discussed. A series of exact time-dependent solutions, one- and two-dimensional, is presented to illustrate the model.

It is well known that most gas fluidized beds of particles bubble, while most liquid fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson (1995), through direct numerical integration of the volume-averaged equations of motion for the fluid and particles, that this distinction is indeed accounted for by these equations, coupled with simple, physically credible closure relations for the stresses and interphase drag. The aim of the present study is to investigate how the model equations afford this distinction and deduce an approximate criterion for separating bubbling and non-bubbling systems. To this end, we have computed, making use of numerical continuation techniques as well as bifurcation theory, the one- and two-dimensional travelling wave solutions of the volume-averaged equations for a wide range of parameter values, and examined the evolution of these travelling wave solutions through direct numerical integration. It is demonstrated that whether bubbles form or not is dictated by the value of Ω = (ρsv3t/Ag) 1/2, where ρs is the density of particles, vt is the terminal settling velocity of an isolated particle, g is acceleration due to gravity and A is a measure of the particle phase viscosity. When Ω is large (> ∼ 30), bubbles develop easily. It is then suggested that a natural scale for A is ρsvtdp so that Ω2 is simply a Froude number.

The dependence of spin-up in a rectangular tank on deformation of the free surface is investigated experimentally. The results agree with earlier experimental and numerical data about the motion of vortices over topography. However, the presence of sidewalls appears to interact with the vortex drift induced by the surface topography. This combined effect provides a qualitative explanation for the observed behaviour of individual vortices. In particular, in the presence of free-surface deformation, cyclonic vortices in an elongated rectangle tend to drift away from the centre of the tank, so that their merging in the centre is discouraged.

A general mathematical model which describes the motion of an interface between immiscible viscous fluids along a smooth homogeneous solid surface is examined in the case of small capillary and Reynolds numbers. The model stems from a conclusion that the Young equation, σ1 cos θ = σ2 − σ3, which expresses the balance of tangential projection of the forces acting on the three-phase contact line in terms of the surface tensions σi and the contact angle θ, together with the well-established experimental fact that the dynamic contact angle deviates from the static one, imply that the surface tensions of contacting interfaces in the immediate vicinity of the contact line deviate from their equilibrium values when the contact line is moving. The same conclusion also follows from the experimentally observed kinematics of the flow, which indicates that liquid particles belonging to interfaces traverse the three-phase interaction zone (i.e. the ‘contact line’) in a finite time and become elements of another interface – hence their surface properties have to relax to new equilibrium values giving rise to the surface tension gradients in the neighbourhood of the moving contact line. The kinematic picture of the flow also suggests that the contact-line motion is only a particular case of a more general phenomenon – the process of interface formation or disappearance – and the corresponding mathematical model should be derived from first principles for this general process and then applied to wetting as well as to other relevant flows. In the present paper, the simplest theory which uses this approach is formulated and applied to the moving contact-line problem. The model describes the true kinematics of the flow so that it allows for the ‘splitting’ of the free surface at the contact line, the appearance of the surface tension gradients near the contact line and their influence upon the contact angle and the flow field. An analytical expression for the dependence of the dynamic contact angle on the contact-line speed and parameters characterizing properties of contacting media is derived and examined. The role of a ‘thin’ microscopic residual film formed by adsorbed molecules of the receding fluid is considered. The flow field in the vicinity of the contact line is analysed. The results are compared with experimental data obtained for different fluid/liquid/solid systems.

We present a study of the rheological and optical behaviour of Kramers bead–rod chains in dilute solution using stochastic computer simulations. We consider two model linear flows, steady shear and uniaxial extensional flow, in which we calculate the non-Newtonian Brownian and viscous stress contribution of the polymers, their birefringence and a stress-optic coefficient. We have developed a computer algorithm to differentiate the Brownian from the viscous stress contributions which also avoids the order (δt)−1/2 noise associated with the Brownian forces. The strain or shear rate is made dimensionless with a molecular relaxation time determined by simulated relaxation of the birefringence and stress after a strong flow is applied. The characteristic long relaxation time obtained from the birefringence and stress are equivalent and shown to scale with N2 where N is the number of beads in the chain. For small shear or extension rates the viscous contribution to the effective viscosity is constant and scales as N. We obtain an analytic expression which explains the scaling and magnitude of this viscous contribution. In uniaxial extensional flow we find an increase in the extensional viscosity with the dimensionless flow strength up to a plateau value. Moreover, the Brownian stress also reaches a plateau and we develop an analytic expression which shows that the Brownian stress in an aligned bead–rod chain scales as N3. Using scaling arguments we show that the Brownian stress dominates in steady uniaxial extensional flow until large Wi, Wi ≈ 0.06N2, where Wi is the chain Weissenberg number. In shear flow the viscosity decays as Wi−1/2 and the first normal stress as Wi−4/3 at moderate Wi. We demonstrate that these scalings can be understood through a quasi-steady balance of shear forces with Brownian forces. For small Wi (in shear and uniaxial extensional flow) and for long times (in stress relaxation) the stress-optic law is found to be valid. We show that the rheology of the bead–rod chain can be qualitatively described by an equivalent FENE dumbbell for small Wi.

The results of an experimental study carried out to investigate the structure of turbulence near a shear-free density interface are presented. The experimental configuration consisted of a two-layer fluid medium in which the lower layer was maintained in a turbulent state by an oscillating grid. The measurements included the root-mean-square (r.m.s.) turbulent velocities, wavenumber spectra, dissipation of turbulent kinetic energy and integral lengthscales. It was found that the introduction of a density interface to a turbulent flow can strongly distort the structure of turbulence near the interface wherein the horizontal velocity components are amplified and the vertical component is damped. The modification of r.m.s velocities is essentially limited to distances smaller than about an integral lengthscale. Inspection of spectra shows that these distortions are felt only at small wavenumbers of the order of the integral scale and a range of low-wavenumbers of the inertial subrange; the distortions become pronounced as the interface is approached. Comparison of the horizontal velocity data with the rapid distortion theory (RDT) analyses of Hunt & Graham (1978) and Hunt (1984) showed a qualitative agreement near the interface and a quantitative agreement away from the interface. On the other hand, the RDT predictions for the vertical component were in general agreement with the data. The near-interface horizontal velocity data, however, showed quantitative agreement with a model proposed by Hunt (1984) based on nonlinear vortex dynamics near the interface. The effects due to interfacial waves appear to be important for distances less than about 10% of the integral lengthscale. As a consequence of the non-zero energy flux divergence, the introduction of a density interface to oscillating grid turbulence increases the rate of dissipation in the turbulent layer except near the interface, where a sharp drop occurs. The present measurements provide useful information on the structure of turbulence in shear-free boundary layers, such as atmospheric and oceanic convective boundary layers, thus improving modelling capabilities for such flows.

A simple, dynamically consistent model of mixing and transport in Rossby-wave critical layers is obtained from the well-known Stewartson–Warn–Warn (SWW) solution of Rossby-wave critical-layer theory. The SWW solution is thought to be a useful conceptual model of Rossby-wave breaking in the stratosphere. Chaotic advection in the model is a consequence of the interaction between a stationary and a transient Rossby wave.

Mixing and transport are characterized separately with a number of quantitative diagnostics (e.g. mean-square dispersion, lobe dynamics, and spectral moments), and with particular emphasis on the dynamics of the tracer field itself. The parameter dependences of the diagnostics are examined: transport tends to increase monotonically with increasing perturbation amplitude whereas mixing does not. The robustness of the results is investigated by stochastically perturbing the transient-wave phase speed. The two-wave chaotic advection model is contrasted with a stochastic single-wave model. It is shown that the effects of chaotic advection cannot be captured by stochasticity alone.

Direct numerical simulation and inviscid linear analysis are used to study the interaction of a normal shock wave with an isotropic turbulent field of vorticity and entropy fluctuations. The role of the upstream entropy fluctuations is emphasized. The upstream correlation between the vorticity and entropy fluctuations is shown to strongly influence the evolution of the turbulence across the shock. Negative upstream correlation between u′ and T′ is seen to enhance the amplification of the turbulence kinetic energy, vorticity and thermodynamic fluctuations across the shock wave. Positive upstream correlation has a suppressing effect. An explanation based on the relative effects of bulk compression and baroclinic torque is proposed, and a scaling law is derived for the evolution of vorticity fluctuations across the shock. The validity of Morkovin's hypothesis across a shock wave is examined. Linear analysis is used to suggest that shock-front oscillation would invalidate the relation between urms and Trms, as expressed by the hypothesis.

The concentration field of mixing layers subject to stabilizing and destabilizing streamwise curvature was investigated at post-mixing-transition conditions. A set of operating conditions was implemented, identical to those at which straight layers were previously investigated in the same facility, in order to compare the effects of hydrodynamic instabilities upon scalar mixing. Quantitative imaging of planar laser-induced fluorescence was used for (i) passive scalar measurements, and (ii) chemical product measurements. Similar to the straight mixing layer, the results for the curved layers show that beyond the mixing transition the layer continues to evolve, and undergoes a small change in its scalar structure. At conditions just past the mixing transition both stable and unstable layers have average mixed-fluid compositions which are uniform across the layer, and average chemical product concentration profiles which are symmetric. At more fully developed conditions, the scalar field evolved: the average mixed-fluid concentration developed a small lateral variation, while the chemical product concentration profiles became asymmetric. Similar to the straight layer, the mixture-fraction PDF is believed to be of the tilted type for the most fully developed layer examined, with the marching PDF being a poor representation. Consistent with previous investigations, the growth rate of the unstable layer was found to be higher than that of straight or stable layers. The most important result is that the measured mixing efficiency of all the layers (curved and straight) was found to be the same: both the total mixed-fluid composition, and the volume fraction of mixed fluid were the same for all unstable, stable, and straight layers. The amount of mixed fluid (and of chemical product formed) was larger for the unstable layer, but always in a fixed proportion to the layer's thickness. The lack of increase in the mixing efficiency for the unstable layer is surprising, given that previous hydrodynamic measurements had shown enhanced turbulent transport for the unstable case. Thus, for all liquid shear layers studied, the rate of scalar mixing appears to be directly proportional to the entrainment rate (which essentially determines the layer's growth rate), and not to any hydrodynamic measures.