Hot-wire measurements of second- and third-order mean products of velocity fluctuations have been made in the flow behind a backward-facing step with a thin, laminar boundary layer at the top of the step. Measurements extend to a distance of about 12 step heights downstream of the step, and include parts of the recirculating-flow region: approximate limits of validity of hot-wire results are given. The Reynolds number based on step height is about 105, the mixing layer being fully turbulent (fully three-dimensional eddies) well before reattachment, and fairly close to self-preservation in contrast to the results of some previous workers. Rapid changes in turbulence quantities occur in the reattachment region: Reynolds shear stress and triple products decrease spectacularly, mainly because of the confinement of the large eddies by the solid surface. The terms in the turbulent energy and shear stress balances also change rapidly but are still far from the self-preserving boundary-layer state even at the end of the measurement region.

Direct numerical simulations (DNS) were performed to investigate the laminar–turbulent transition in a boundary layer on a sharp cone with an isothermal wall at Mach 6 and at zero angle of attack. The motivation for this research is to make a contribution towards understanding the nonlinear stages of transition and the final breakdown to turbulence in hypersonic boundary layers. In particular, the role of second-mode fundamental resonance, or (K-type) breakdown, is investigated using high-resolution ‘controlled’ transition simulations. The simulations were carried out for the laboratory conditions of the hypersonic transition experiments conducted at Purdue University. First, several low-resolution simulations were carried out to explore the parameter space for fundamental resonance in order to identify the cases that result in strong nonlinear interactions. Subsequently, based on the results from this study, a set of highly resolved simulations that proceed deep into the turbulent breakdown region have been performed. The nonlinear interactions observed during the breakdown process are discussed in detail in this paper. A detailed description of the flow structures that arise due to these nonlinear interactions is provided and an analysis of the skin friction and heat transfer development during the breakdown is presented. The controlled transition simulations clearly demonstrate that fundamental breakdown may indeed be a viable path to complete breakdown to turbulence in hypersonic cone boundary layers at Mach 6.

We present simulation results of vortex-induced vibrations of an infinitely long flexible cylinder at Reynolds number Re = 1000, corresponding to a ‘young’ turbulent wake (i.e. exhibiting a small inertial subrange). The simulations are based on a new class of spectral methods suitable for unstructured and hybrid grids. To obtain different responses of the coupled flow–structure system we vary the structure's bending stiffness to model the behaviour of a vibrating inflexible (rigid) cylinder, a cable, and a beam. We have found that unlike the laminar flow previously studied, the amplitude of the cross-flow oscillation is about one diameter for the cable and the beam, close to experimental measurements, but is lower for the rigid cylinder. We have also found that for the latter case the flow response corresponds to parallel shedding, but for the beam and cable with free endpoints a mixed response consisting of oblique and parallel shedding is obtained, caused by the modulated travelling wave motion of the structure. This mixed shedding pattern which alternates periodically along the span can be directly related to periodic spatial variation of the lift force. In the case of structures with pinned endpoints a standing wave response is obtained for the cylinder; lace-like flow structures are observed similar to the ones seen in the laminar regime. Examination of the frequency spectra in the near wake shows that at Re = 1000 all cases follow a −5/3 law in the inertial range, which extends about half a decade in wavenumber. However, these spectra are different in all three cases both in low and high frequencies, with the exception of the beam and cable, for which the high-frequency portion is identical despite the differences in the displacement time history and the large-scale features of the corresponding flow.

High-Reynolds-number data in turbulent boundary layers are analysed to examine statistical moments of streamwise velocity fluctuations ${u}^{\prime } $. Prior work has shown that the variance of ${u}^{\prime } $ exhibits logarithmic behaviour with distance to the surface, within an inertial sublayer. Here we extend these observations to even-order moments. We show that the $2p$-order moments, raised to the power $1/ p, $ also follow logarithmic behaviour according to $\langle \mathop{({u}^{\prime + } ){}^{2p} \rangle }\nolimits ^{1/ p} = {B}_{p} - {A}_{p} \ln (z/ \delta )$, where ${u}^{\prime + } $ is the velocity fluctuation normalized by the friction velocity, $\delta $ is an outer length scale and ${B}_{p} $ are non-universal constants. The slopes ${A}_{p} $ in the logarithmic region appear quite insensitive to Reynolds number, consistent with universal behaviour for wall-bounded flows. The slopes differ from predictions that assume Gaussian statistics, and instead are consistent with sub-Gaussian behaviour.

Over the past decade and a half, analyses of the dynamics of fluids containing moving contact lines have specified hydrodynamic models of the fluids in a rather small region surrounding the contact lines (referred to as the inner region) which necessarily differ from the usual model. If this were not done, a singularity would have arisen, making it impossible to satisfy the contact-angle boundary condition, a condition that can be important for determining the shape of the fluid interface of the entire body of fluid (the outer region). Unfortunately, the nature of the fluids within the inner region under dynamic conditions has not received appreciable experimental attention. Consequently, the validity of these novel models has yet to be tested.

The objective of this experimental investigation is to determine the validity of the expression appearing in the literature for the slope of the fluid interface in the region of overlap between the inner and outer regions, for small capillary number. This in part involves the experimental determination of a constant traditionally evaluated by matching the solutions in the inner and outer regions. Establishing the correctness of this expression would justify its use as a boundary condition for the shape of the fluid interface in the outer region, thus eliminating the need to analyse the dynamics of the fluid in the inner region.

Our experiments consisted of immersing a glass tube, tilted at an angle to the horizontal, at a constant speed, into a bath of silicone oil. The slope of the air–silicone oil interface was measured at distances from the contact line ranging between O.O13a. and O.17a, where a denotes the capillary length, the lengthscale of the outer region (1511 μm). Experiments were performed at speeds corresponding to capillary numbers ranging between 2.8 × 10-4 and 8.3 × 10-3. Good agreement is achieved between theory and experiment, with a systematic deviation appearing only at the highest speed. The latter may be a consequence of the inadequacy of the theory at that value of the capillary number.

Accurate numerical simulations of vortex rings impinging on flat boundaries revealed the same features observed in experiments. The results for the impact with a free-slip wall compared very well with previous numerical simulations that used spectral methods, and were also in qualitative agreement with experiments. The present simulation is mainly devoted to studying the more realistic case of rings interacting with a no-slip wall, experimentally studied by Walker et al. (1987). All the Reynolds numbers studied showed a very good agreement between experiments and simulations, and, at Rev > 1000 the ejection of a new ring from the wall was seen. Axisymmetric simulations demonstrated that vortex pairing is the physical mechanism producing the ejection of the new ring. Three-dimensional simulations were also performed to investigate the effects of azimuthal instabilities. These simulations have confirmed that high-wavenumber instabilities originate in the compression phase of the secondary ring within the primary one. The large instability of the secondary ring has been explained by analysis of the rate-of-strain tensor and vorticity alignment. The differences between passive scalars and the vorticity field have been also investigated.

The general incompressible flow uQ(x), quadratic in the space coordinates, and satisfying the condition uQ = · n = 0 on a sphere r = 1, is considered. It is shown that this flow may be decomposed into the sum of three ingredients – a poloidal flow of Hill's vortex structure, a quasi-rigid rotation, and a twist ingredient involving two parameters, the complete flow uQ(x) then involving essentially seven independent parameters. The flow, being quadratic, is a Stokes flow in the sphere.

The streamline structure of the general flow is investigated, and the results illustrated with reference to a particular sub-family of ‘stretch–twist–fold’ (STF) flows that arise naturally in dynamo theory. When the flow is a small perturbation of a flow u1(x) with closed streamlines, the particle paths are constrained near surfaces defined by an ‘adiabatic invariant’ associated with the perturbation field. When the flow u1 is dominated by its twist ingredient, the particles can migrate from one such surface to another, a phenomenon that is clearly evident in the computation of Poincaré sections for the STF flow, and that we describe as ‘trans-adiabatic drift’. The migration occurs when the particles pass a neighbourhood of saddle points of the flow on r = 1, and leads to chaos in the streamline pattern in much the same way as the chaos that occurs near heteroclinic orbits of low-order dynamical systems.

The flow is believed to be the first example ofa steady Stokes flow in a bounded region exhibiting chaotic streamlines.

To reduce the costs of construction, operation, maintenance, energy storage and grid connection, some devices for extracting energy from sea waves are likely to be installed on the coast. We study theoretically a single oscillating water column (OWC) installed at the tip of a long and thin breakwater. The linearized problems of radiation and scattering for a hollow cylinder with an open bottom are then solved by the usual method of eigenfunction expansions and integral equations. Since a thin breakwater is the limit of a wedge, an exact solution for the diffraction by a solid cylinder at the tip of a wedge is derived to facilitate the analysis. Following Sarmento & Falcão (J. Fluid Mech., vol. 150, 1985, pp. 467–485), power takeoff by Wells turbines is modelled by including air compressibility in the chamber above the water surface. The effects of air compressibility on the extraction efficiency is studied. It is shown that for this simple geometry the angle of incidence affects the waves outside the structure but not the extracted power.

A series of experiments are performed in a Hele-Shaw cell, consisting of two parallel closely spaced glass plates. A liquid (oil or water, both of viscosity of 1.0 cP) is injected at a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine. The data presented in this paper are taken mostly at late stages of the fingering process, when the pattern has multiple generations of splitting. Correlations with time are obtained for the finger length and the overall pattern density. The time-and lengthscales have been found for the immiscible case. At the same dimensionless time, immiscible patterns are similar and have the same generation of splitting. The overall density of each pattern decreases with time. The pattern shows fractal behaviour only after a certain number of generations of splitting. The fractal dimension of the immiscible pattern decreases from 1.9 to 1.82 when the pattern goes from the third to the sixth generation of splitting. The fractal dimension of the miscible pattern reaches a constant value after about ten generations of splitting and the fractal dimension ranges from 1.50 to 1.69 for q/Db = 4.8 × 105−7.0 × 106. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers λ/b scales with Ca−0.31 for capillary number Ca ranging from about 8 × 10−4 to 0.05. For miscible fingers, λ/b is insensitive to dispersion and ranges from 5 to 10 for large q/Db. Here D is the molecular diffusion coefficient in glycerine, b the cell gap width and λ the splitting wavelength.

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.

This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

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.

A study has been made of the factors influencing the break-down of sheets formed by the impingement of two liquid jets. It is shown that disintegration generally results from the formation of unstable waves of aerodynamic or hydrodynamic origin. While the characteristics of the former waves are fairly well understood, little is known about the latter. The results of this study indicate that hydrodynamic (or ‘impact’) waves are generated when the Weber number of each jet (ρlDV2 sin 2θ/γ, where ρl is the liquid density, D the jet diameter, V the mean jet velocity, θ the half-angle of impingement, and γ is the surface tension) is above a critical value, and that their formation is independent of the Reynolds number. Drop sizes have been measured and are shown to be critically dependent upon the mechanism of disintegration.

The thermal structure in the boundary layer and its relation to the heat flux from the cooling and evaporating surface of a deep tank of water are investigated. When a deep layer of water in contact with still air above loses heat to the air, the cooled water in a region just under the surface converges along lines and then plunges down in sheets. These sheets of falling water dissipate as they move into the body of the water, which is in turbulent motion. The vertical profiles of the horizontally averaged temperature and its standard deviation agree fairly closely with theoretical profiles based on time averages of the solution to the heat diffusion equation. The differences between observed and thus predicted profile shapes are consistent with the expected effects of the falling cold thermals and the warm return flow, which are neglected in the theories. The profiles of the standard deviation have large values up to the interface and lie between predictions based on boundary conditions of constant surface temperature and constant heat flux, in keeping with the experimental conditions.

The relation between the net heat flux and the temperature difference across the boundary layer is given in non-dimensional form by N = 0[sdot ]156R0[sdot ]33, which is in good agreement with the asymptotic similarity prediction N [vprop ] R1/3 but lower than theoretical calculations of the upper bound of N vs. R.

We study the chaotic mixing in two periodic model flows, the ‘tendril–whorl’ flow and the ‘Aref-blinking-vortex’ flow, with the objective of supplying evidence for the primary mechanisms responsible for mixing in two-dimensional deterministic flows. The analysis is based on tools of dynamical systems theory but it is clear that the mixing problem generates several questions of its own: low periodic points and horseshoes dominate the picture, since we want to achieve mixing quickly; Poincaré sections, popular in dynamical systems analyses, might give misleading information with regard to dispersion at short times. Our analysis shows that both flows are able to stretch and fold material lines well below the lengthscale of the flows themselves. The inner workings of the two systems are revealed by studying the local and global bifurcations. Computations for the blinking-vortex system indicate the existence of an optimum period at which the average efficiency is maximized, whereas the intensity of segregation – a classical parameter in mixing studies – decays rapidly to an asymptotic value in the globally chaotic region. Even though our flows are not turbulent the results might have some implications for pointing to the limits of similar studies in actual turbulent flows (e.g. line stretching).

A super-hydrophobic array of grooves containing trapped gas (stripes) has the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused on idealized cases of stick-perfect slip stripes. Here, we analyse the experimentally more relevant situation of a pressure-driven flow past striped slip-stick surfaces with arbitrary local slip at the gas sectors. We derive approximate formulas for maximal (longitudinal) and minimal (transverse) directional effective slip lengths that are in a good agreement with the exact numerical solution for any surface slip fraction. By representing eigenvalues of the slip length tensor, we obtain the effective slip for any orientation of stripes with respect to the mean flow. Our results imply that flow past stripes is controlled by the ratio of the local slip length to texture size. In the case of a large (compared to the texture period) slip at the gas areas, surface anisotropy leads to a tensorial effective slip, by attaining the values predicted earlier for a perfect local slip. Both effective slip lengths and anisotropy of the flow decrease when local slip becomes of the order of texture period. In the case of a small slip, we predict simple surface-averaged isotropic flows (independent of orientation).

We develop a mathematical model for the spreading of a thin volatile liquid droplet on a uniformly heated surface. The model accounts for the effects of surface tension, evaporation, thermocapillarity, gravity and disjoining pressure for both perfectly wetting and partially wetting liquids. Previous studies of non-isothermal spreading did not include the effects of disjoining pressure and therefore had to address the difficult issue of imposing proper boundary conditions at the contact line where the droplet surface touches the heated substrate. We avoid this difficulty by taking advantage of the fact that dry areas on the heated solid surface are typically covered by a microscopic adsorbed film where the disjoining pressure suppresses evaporation. We use a lubrication-type approach to derive a single partial differential equation capable of describing both the time-dependent macroscopic shape of the droplet and the microscopic adsorbed film; the contact line is then defined as the transition region between the two. In the framework of this model we find that both evaporation and thermocapillary stresses act to prevent surface-tension-driven spreading. Apparent contact angle, defined by the maximum interfacial slope in the contact-line region, decays in time as a droplet evaporates, but the rate of decay is different from that predicted in earlier studies of evaporating droplets. We attribute the difference to nonlinear coupling between different physical effects contributing to the value of the contact angle; previous studies used a linear superposition of these effects. We also discuss comparison of our results with experimental data available in the literature.

Governing equations for a mushy layer are analysed in the asymptotic regime Rm [Gt ] 1, where Rm is an appropriately defined Rayleigh number. A model is proposed in which there is downward flow everywhere in the mushy layer except in and near localized chimneys, which are characterized by having zero solid fraction. Upward, convective flow within the chimneys is driven by compositional buoyancy. The radius of each chimney is determined locally by thermal balances within a boundary layer that surrounds it. Simple solutions are derived to determine the structure of the mushy layer away from the immediate vicinity of chimneys in order to demonstrate the gross effects of convection upon the solidification within the layer.

A photochromic tracer method has been used to record pulsatile flow velocity profiles simultaneously at three axial locations along a flow channel. Two major advantages of this multiple-trace method are that it enables velocity data to be acquired in an efficient non-invasive manner and that it provides a detailed description of the spatial relationship of the flow field. The latter is found to be particularly useful in the investigation of transitional type flows; for example, in describing coherent flow structures. Studies of the flow patterns in tubes with mild to moderate degrees of vessel constriction were performed using a 2.9 Hz sinusoidal flow superimposed on a steady flow (frequency parameter of 7.5; mean and modulation Reynolds numbers of 575 and 360, respectively). With mild constrictions (< 50% area reduction), isolated regions of vortical and helical structures were observed primarily during the deceleration phase of the flow cycle and in the vicinity of the reattachment point. As expected, these effects were accentuated when the constriction was asymmetric. For moderate constrictions (50%–80%), transition to turbulence was triggered just before peak flow through the breakdown of waves and streamwise vortices that were shed in the high-shear layer. During this vortex generation phase of the flow cycle, the wall shear stress fluctuated quite intensely, especially in the vicinity of the reattachment point, and its instantaneous value increased by at least a factor of eight. Such detailed descriptions of the transition to turbulence and of the spatial and temporal variation of the wall shear stress, particularly near the reattachment point, have not been previously reported for pulsatile flow through constricted tubes. The observed wall shear stress variations support a proposal by Mao & Hanratty (1986) of an interaction of the imposed flow oscillation with the turbulent fluctuations within the viscous sublayer.

This paper deals with a survey of mean flow and fluctuating quantities in a turbulent boundary layer developing on a smooth wall in a pressure domain P(x), where both dP/dx and d2P/dx2 are positive (increasingly adverse). The two-dimensional nature of the flow field was checked by momentum balance, as well as velocity traverses either side of the working section centre-line. Using the integrated form of the momentum integral equation, it was found that the skinfriction term and the summed momentum and pressure terms differed by at most 19%; but for the majority of measuring points they differed by less than 14%. The off-centre-line velocity profiles were indistinguishable from those taken on the centre-line. The flow field was also surveyed for fluctuating components $(\overline{u^2_1})^{\frac{1}{2}}, (\overline{u^2_2})^{\frac{1}{2}}, (\overline{u^2_3})^{\frac{1}{2}}$, and $\overline{u_1u_2}$, as well as for u1 spectra. Wherever possible, the results were compared with existing models of boundary-layer development. These comparisons indicated that the only all-embracing model for boundary-layer development is the law of the wall.

The purpose of this study is to generalize the experimental method for deciding when a fluid motion can be considered turbulent. The rationale advanced for processing a transducer signal, which indicates the intermittent nature of a turbulent field, is given a probabilistic outlook. In addition an improved detector function is introduced which uses information from longitudinal and lateral fluctuation components.