Optimal perturbations for the family of three-dimensional boundary layers described by the Falkner–Skan–Cooke similarity solution are obtained using a variational technique in the temporal framework. The disturbances experiencing the most growth take the form of vortices almost aligned with the external streamline at inception and evolve into streaks. In subcritical flows these can attain about twice the transient amplification observed in comparably forced two-dimensional flows. Possible connections between optimal perturbations and exponentially amplified crossflow vortices are explored.

The understanding and quantitative assessment of air flow fields and local micron-particle wall concentrations in tracheobronchial airways are very important for estimating the health risks of inhaled particulate pollutants, developing algebraic transfer functions of global lung deposition models used in dose-response analyses, and/or determining proper drug-aerosol delivery to target sites in the lung. In this paper (Part 1) the theory, model geometries, and air flow results are provided. In a companion paper (Part 2, Comer et al. 2001), the history of particle deposition patterns and comparisons with measured data sets are reported. Decoupling of the naturally dilute particle suspension makes it feasible to present the results in two parts.

Considering a Reynolds number range of 500 [les ] ReD [les ] 2000, it is assumed that the air flow is steady, incompressible and laminar and that the tubular double bifurcations, i.e. Weibel's generations G3–G5, are three-dimensional, rigid, and smooth with rounded as well as sharp carinal ridges for symmetric planar, and just rounded carinas for 90° non-planar configurations. The employed finite-volume code CFX (AEA Technology) and its user-enhanced FORTRAN programs were validated with experimental velocity data points for a single bifurcation. The resulting air flow structures are analysed for relatively low (ReD = 500) and high (ReD = 2000) Reynolds numbers. Sequential pressure drops due to viscous effects were calculated and compared, extending a method proposed by Pedley et al. (1977). Such detailed results for bifurcating lung airways are most useful in the development of global algebraic lung models.

The flow theory and air flow structures in symmetric double-bifurcation airway models assuming steady laminar, incompressible flow, unaffected by the presence of aerosols, has been described in a companion paper (Part 1). The validated computer simulation results showed highly vortical flow fields, especially around the second bifurcations, indicating potentially complex particle distributions and deposition patterns. In this paper (Part 2), assuming spherical non-interacting aerosols that stick to the wall when touching the surface, the history of depositing particles is described. Specifically, the finite-volume code CFX (AEA Technology) with user-enhanced FORTRAN programs were validated with experimental data of particle deposition efficiencies as a function of the Stokes number for planar single and double bifurcations. The resulting deposition patterns, particle distributions, trajectories and time evolution were analysed in the light of the air flow structures for relatively low (ReD1 = 500) and high (ReD1 = 2000) Reynolds numbers and representative Stokes numbers, i.e. StD1 = 0.04 and StD1 = 0.12. Particle deposition patterns and surface concentrations are largely a function of the local Stokes number, but they also depend on the fluid–particle inlet conditions as well as airway geometry factors. While particles introduced at low inlet Reynolds numbers (e.g. ReD1 = 500) follow the axial air flow, secondary and vortical flows become important at higher Reynolds numbers, causing the formation of particle-free zones near the tube centres and subsequently elevated particle concentrations near the walls. Sharp or mildly rounded carinal ridges have little effect on the deposition efficiencies but may influence local deposition patterns. In contrast, more drastic geometric changes to the basic double-bifurcation model, e.g. the 90°-non-planar configuration, alter both the aerosol wall distributions and surface concentrations considerably.

The low-dimensional dynamics of the structures in a turbulent wall flow are studied by means of numerical simulations. These are made both ‘minimal’, in the sense that they contain a single copy of each relevant structure, and ‘autonomous’ in the sense that there is no outer turbulent flow with which they can interact. The interaction is prevented by a numerical mask that damps the flow above a given wall distance, and the flow behaviour is studied as a function of the mask height. The simplest case found is a streamwise wave that propagates without change. It takes the form of a single wavy low-velocity streak flanked by two counter-rotating staggered quasi-streamwise vortices, and is found when the height of the numerical masking function is less than δ+1 ≈ 50. As the mask height is increased, this solution bifurcates into an almost-perfect limit cycle, a two-frequency torus, weak chaos, and full-edged bursting turbulence. The transition is essentially complete when δ+1 ≈ 70, even if the wall-parallel dimensions of the computational box are small enough for bursting turbulence to be metastable, lasting only for a few bursting cycles. Similar low-dimensional dynamics are found in somewhat larger boxes, containing two copies of the basic structures, in which the bursting turbulence is self-sustaining.

Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable travelling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. The travelling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to 50+, 100+ and 250+, respectively. These coherent solutions come in pairs with distinct structure and instabilities. There is a three-dimensional continuum of such exact coherent states.

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincaré's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite.

We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/√d, d being the distance to the characteristic grazing the inner sphere.

We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general reveal an attractor expected at the eigenfrequency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E1/3, E being the Ekman number; for this latter layer, we give its analytical form and show its similarity to vertical 1/3-shear layers of steady flows. Using an inertial wave packet travelling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in [0, 2Ω], contrary to the case of the full sphere (Ω is the angular velocity of the system).

Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers (10−10–10−20), which are out of reach numerically, and this for a wide class of containers.

A family of slowly spatially developing wakes with variable pressure gradient is numerically demonstrated to sustain a synchronized finite-amplitude vortex street tuned at a well-defined frequency. This oscillating state is shown to be described by a steep global mode exhibiting a sharp Dee–Langer-type front at the streamwise station of marginal absolute instability. The front acts as a wavemaker which sends out nonlinear travelling waves in the downstream direction, the global frequency being imposed by the real absolute frequency prevailing at the front station. The nonlinear travelling waves are determined to be governed by the local nonlinear dispersion relation resulting from a temporal evolution problem on a local wake profile considered as parallel. Although the vortex street is fully nonlinear, its frequency is dictated by a purely linear marginal absolute instability criterion applied to the local linear dispersion relation.

Townsend's (1980) model of wind-to-wave energy transfer, which is based on a putative interpolation between an inner, viscoelastic approximation and an outer, rapid-distortion approximation and predicts an energy transfer that is substantially larger (by as much as a factor of three) than that predicted by Miles's (1957) quasi-laminar model, is revisited. It is shown that Townsend's interpolation effectively imposes a rapid-distortion approximation throughout the flow, rather than only in the outer domain, and that his asymptotic (far above the surface) solution implicitly omits one of the two admissible, linearly independent solutions of his perturbation equations. These flaws are repaired, and Townsend's dissipation function is modified to render the transport equation for the perturbation energy of the same form as those for the perturbation Reynolds stresses. The resulting wind-to-wave energy transfer is close to that predicted by Townsend's (1972) viscoelastic model and other models that incorporate the perturbation Reynolds stresses, but somewhat smaller than that predicted by the quasi-laminar model. We conclude that Townsend's (1980) predictions, although closer to observation than those of other models, rest on flawed analysis and numerical error.

Experimental results are reported on non-stationary evolution and interactions of waves forming on water and water–glycerol solution flowing along an inclined plane. A nonlinear wave generation process leads to a large number of solitary humps with a wide variety of sizes. A uorescence imaging method is applied to capture the evolution of film height in space and time with accuracy of a few microns. Coalescence – the inelastic interaction of solitary waves resulting in a single hump – is found to proceed at a timescale correlated to the difference in height between the interacting waves. The correlation indicates that waves of similar height do not merge. Transient phenomena accompanying coalescence are reported. The front-running ripples recede during coalescence, only to reappear when the new hump recovers its teardrop shape. The tail of the resulting solitary wave develops an elevated substrate relative to the front, which decays exponentially in time; both observations about the tail confirm theoretical predictions. In experiments with water, the elevated back substrate is unstable, yielding to a tail oscillation with wavelength similar to that of the front-running ripples. This instability plays a key role in two complex interaction phenomena observed: the nucleation of a new crest between two interacting solitary humps and the splitting of a large hump (that has grown through multiple coalescence events) into solitary waves of similar size.

This paper provides insight to the transient development of density waves generated in gravity-driven flows of granular materials. The evolution of three modes of dominant linear instabilities (predicted in a previous work by Wang, Jackson & Sundaresan 1997) is examined by FFT analysis. For the first symmetric density wave (SDW1) mode, the evolution is governed by the linear instability. The second symmetric density wave (SDW2) mode undergoes a few stages of temporal development; as a result, large particle clusters gradually degenerate into a series of smaller clusters in the flow direction. For the anti-symmetric (ASDW) mode, the corresponding particle distribution shows significant development in the direction perpendicular to the flow. The present study indicates that the wall roughness may affect the structure of the density waves, but these density waves need not be triggered by the wall roughness. All the three modes of instabilities reported in this work are of inertial nature and occur only when the particle–particle collisions are significantly inelastic.

This paper deals with the stability of suspensions modelled as dusty gases for non-uniform profiles of mass fraction of particles. It is known that a stationary uniform fluidized bed may be unstable to small disturbances which grow until a secondary instability develops forming bubbles that rise through the bed. Interactions between particles are difficult to model and this makes it necessary to close the model with some assumptions. However, in dilute fluidized beds which are characterized by a low volume fraction of particles, interactions between particles are negligible and this motivates the study of instabilities of suspensions by means of the dusty gas equations, avoiding the problem of particular closures.

We show in this work that suspensions blown to regions of higher concentration are unstable to two-dimensional disturbances. An equation which governs this instability is obtained. A physical mechanism is proposed to explain this instability and it is related to the Rayleigh–Taylor one in the limit of long characteristic lengthscales associated with the concentration profile. Finally, the evolution of this instability is followed using a fully nonlinear numerical code showing the formation of streamers and clusters of particles.

A numerical investigation is presented of kinematic dynamo action in a dynamically driven fluid flow. The model isolates basic dynamo processes relevant to field generation in the Solar tachocline. The horizontal plane layer geometry adopted is chosen as the local representation of a differentially rotating spherical fluid shell at co-latitude ϑ; the unit vectors [xcirc ], ŷ and zˆ point east, north and vertically upwards respectively. Relative to axes moving easterly with the local bulk motion of the fluid the rotation vector Ω lies in the (y, z)-plane inclined at an angle ϑ to the z-axis, while the base of the layer moves with constant velocity in the x-direction. An Ekman layer is formed on the lower boundary characterized by a strong localized spiralling shear flow. This basic state is destabilized by a convective instability through uniform heating at the base of the layer, or by a purely hydrodynamic instability of the Ekman layer shear flow. The onset of instability is characterized by a horizontal wave vector inclined at some angle ∈ to the x-axis. Such motion is two-dimensional, dependent only on two spatial coordinates together with time. It is supposed that this two-dimensionality persists into the various fully nonlinear regimes in which we study large magnetic Reynolds number kinematic dynamo action.

When the Ekman layer flow is destabilized hydrodynamically, the fluid flow that results is steady in an appropriately chosen moving frame, and takes the form of a row of cat's eyes. Kinematic magnetic field growth is characterized by modes of two types. One is akin to the Ponomarenko dynamo mechanism and located close to some closed stream surface; the other appears to be associated with stagnation points and heteroclinic separatrices.

When the Ekman layer flow is destabilized thermally, the well-developed convective instability far from onset is characterized by a flow that is intrinsically time-dependent in the sense that it is unsteady in any moving frame. The magnetic field is concentrated in magnetic sheets situated around the convective cells in regions where chaotic particle paths are likely to exist; evidence for fast dynamo action is obtained. The presence of the Ekman layer close to the bottom boundary breaks the up–down symmetry of the layer and localizes the magnetic field near the lower boundary.

This paper presents the results of a theoretical study of the interactions between a laminar, premixed flame front and a plane acoustic wave. Its objective is to elucidate the processes that damp or drive acoustic waves as they interact with flames. Using linear analysis, the characteristics of the acoustic field, the flame's movement and wrinkling in response to acoustic perturbations, and the acoustic energy that is produced or dissipated at the flame are calculated. These calculations show that the net acoustic energy flux out of the flame is controlled by competing acoustic energy production and dissipation processes. Energy is added to the acoustic field by unsteady heat release processes resulting from the unsteady flux of unburned reactants through the flame by fluctuations in the flame speed or density of the unburned reactants. Energy is dissipated by the transfer of acoustic energy into fluctuations in vorticity that are generated at the flame front because of the misaligned fluctuating pressure and mean density gradients (i.e. the baroclinic vorticity production mechanism). The paper concludes by showing how these results can be generalized to determine the response of planar flames to an arbitrarily complex acoustic field. The principal contribution of this work is its demonstration that the excitation of vorticity and fluctuations in the flame speed have significant qualitative and quantitative affects on the interactions between flames and acoustic waves.

The paper explores, using different levels of turbulence closure, the computed behaviour of the three-dimensional turbulent wall jet in order to determine the cause of the remarkably high lateral rates of spread observed in experiments. Initially, to ensure accurate numerical solution, the equations are cast into the form appropriate to a self-similar shear flow thereby reducing the problem to one of two independent variables.

Our computations confirm that the strong lateral spreading arises from the creation of streamwise vorticity, rather than from anisotropic diffusion. The predicted ratio of the normal to lateral spreading rates is, however, very sensitive to the approximation made for the pressure–strain correlation. The version that, in other flows, has led to the best agreement with experiments again comes closest in calculating the wall jet, although the computed rate of spread is still some 50% greater than in most of the measurements. Our subsequent calculations, using a forward-marching scheme show that, because of the strong coupling between axial and secondary flow, the flow takes much longer to reach its self-preserving state than in a two-dimensional wall jet. Thus, it appears very probable that none of the experimental data are fully developed.

The yield conditions for the displacement of three-dimensional fluid droplets adhering to a plane solid boundary in pressure-driven flows are studied through a series of numerical computations. The study considers low-Reynolds-number flows between two parallel plates and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, capillary number Ca, advancing and receding contact angles, θA and θR, and dimensionless plate separation H/h (where H is the plate spacing and h is the unperturbed droplet height). This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(λ, H/h, θA, Δθ) where Δθ = θA − θR is the contact angle hysteresis. The numerical solutions are based on an efficient, three-dimensional Newton method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. As the viscosity of a droplet increases, the critical flow rate decreases, facilitating the displacement. This is consistent with our previous results for shear flows (Dimitrakopoulos & Higdon 1997, 1998), which represent the limit of infinite plate spacing. As the plate spacing is reduced, the critical flow rate increases until a maximum value is reached. Further reduction in the plate spacing decreases the critical flow rate. The effects of both viscosity ratio and plate separation are much more pronounced for high contact angles. Inviscid droplets (or bubbles) show behaviour dramatically different from that of viscous droplets. For these droplets, a significantly higher flow rate is required for drop displacement, but this critical flow rate decreases monotonically as the distance between the plates decreases. In the Appendix, we clarify the necessary conditions for low-Reynolds-number flows past low viscosity droplets or bubbles.

We consider the evaporation and subsequent burning of thin films of liquid fuels. Previous studies on liquid films, with and without evaporation, have primarily considered the gas phase to be passive. The new element in this study is the introduction of combustion and the examination of both the liquid and gas phases and their effect on the film's behaviour. For the case of a liquid film burning in quiescent air we show that the problem can be simplified to a single nonlinear evolution equation for the film thickness. All remaining variables, which are simply expressed in terms of the function describing the instantaneous position of the liquid–vapour interface, are subsequently determined. This equation is then solved in order to understand the dynamics of the film in the presence of evaporation and combustion.

The planar configuration is discussed first. Predictions for the total evaporation time are obtained, along with the time history of the film thickness, the interfacial surface temperature, the flame standoff distance and its temperature, and the mass burning rate. The dependence of the burning characteristics on the fuel and oxidizer Lewis numbers, which measure the relative importance of thermal and molecular diffusivities, is also determined. Second, we analyse the case of a non-planar interface, where temperature variations along the film's surface cause fluid motion in the liquid that could either dampen or amplify spatial non-uniformities. We show that, while thermocapillarity has the tendency to destabilize the planar interface, combustion acts to reduce this effect. In particular, when the heat release by combustion is substantial, all disturbances are obliterated, the film remains nearly planar and the burning occurs along nearly horizontal surfaces.

The location of the asymptotic virtual origin of positively buoyant turbulent plumes with a deficit of initial momentum flux when compared with equivalent pure plumes is investigated. These lazy plumes are generated by continuous steady releases of momentum, buoyancy and volume into a quiescent uniform environment from horizontal sources (at z = 0) of finite area, and are shown to be equivalent to the far-field flow above point source pure plumes, of buoyancy only, rising from the asymptotic virtual source located below the actual source at z = −zavs.

An analytical expression for the location of the asymptotic virtual source relative to the actual source of the lazy plume is developed. The plume conservation equations are solved for the volume flow rate, and the position of the asymptotic virtual origin is deduced from the scaling for the volume flow rate at large distances from the source.

The displacement zavs of the asymptotic virtual origin from the actual origin scales on the source diameter and is a function of the source parameter Γ ∝ Qˆ20Fˆ0/Mˆ5/20 which is a measure of the relative importance of the initial fluxes of buoyancy Fˆ0, momentum Mˆ0, and volume Qˆ0 in the plume. The virtual origin correction developed is valid for Γ > 1/2 and is therefore applicable to lazy plumes for which Γ > 1, pure plumes for which Γ = 1, and forced plumes in the range 1/2 < Γ < 1. The dimensionless correction z*avs decreases as Γ increases, and for Γ [Gt ] 1, z*avs → 0.853Γ−1/5. Comparisons made between the predicted location of the asymptotic virtual origin and the location inferred from measurements of lazy saline plumes in the laboratory show close agreement.

The roll-up of an initially spherical vortex sheet into a vortex ring is computed using the vortex blob method. The ring sheds about 30% of its circulation into a tail which, in turn, rolls up into a ring that sheds circulation. The process repeats itself at smaller and smaller scales in a self-similar manner. The relation between the vortex shedding and the energy of the vortices is investigated. In contrast, an initially cylindrical vortex sheet rolls up into a vortex pair that sheds essentially no circulation.