Premixed H2/O2/N2 flames propagating in two-dimensional turbulence have been studied using direct numerical simulations (DNS: simulations in which all fluid and thermochemical scales are fully resolved). Simulations include realistic chemical kinetics and molecular transport over a range of equivalence ratios Φ (Φ = 0.35, 0.5, 0.7, 1.0, 1.3). The validity of the flamelet assumption for premixed turbulent flames is checked by comparing DNS data and results obtained for steady strained premixed flames with the same chemistry (flamelet ‘library’). This comparison shows that flamelet libraries overestimate the influence of stretch on flame structure. Results are also compared with earlier zero-chemistry (flame sheet) and one-step chemistry simulations. Consistent with the simpler models, the turbulent flame with realistic chemistry aligns preferentially with extensive strain rates in the tangent plane and flame curvature probability density functions are close to symmetric with near-zero means. For very lean flames it is also found that the local flame structure correlates with curvature as predicted by DNS based on simple chemistry. However, for richer flames, by contrast to simple-chemistry results with non-unity Lewis numbers (ratio of thermal to species diffusivity), local flame structure does not correlate with curvature but rather with tangential strain rate. Turbulent straining results in substantial thinning of the flame relative to the steady unstrained laminar case. Heat-release and H2O2 contours remain thin and connected (‘flamelet-like’) while species including H-atom and OH are more diffuse. Peak OH concentration occurs well behind the peak heat-release zone when the flame temperature is high (of the order of 2800 K). For cooler and leaner flames (about 1600 K and for an equivalence ratio below 0.5) the OH radical is concentrated near the reaction zone and the maximum OH level provides an estimate of the local flamelet speed as assumed by Becker et al. (1990).

The effects of thermal conditions on the patterns of two-dimensional Boussinesq convection are studied by numerical integration. The adopted thermal conditions are (i) the heat fluxes through both upper and lower boundaries are fixed, (ii) the same as (i) but with internal cooling, (iii) the temperature on the lower boundary and the heat flux through the upper boundary are fixed, (iv) the same as (iii) but with internal cooling, and (v) the temperatures on both upper and lower boundaries are fixed. The numerical integrations are performed with Ra = 104 and Pr = 1 over the region whose horizontal and vertical lengths are 8 and 1, respectively.

The results confirm that convective cells with the larger horizontal sizes tend to form under the conditions where the temperature is not fixed on any boundaries. Regardless of the existence of internal cooling, one pair of cells spreading all over the region forms in the equilibrium states. On the other hand, three pairs of cells form and remain when the temperature on at least one boundary is fixed. The formation of single pairs of cells appearing under the fixed heat flux conditions shows different features with and without internal cooling. The difference emerges as the appearance of a phase change, whose existence can be suggested by the weak nonlinear equation derived by Chapman & Proctor (1980).

We study the average mobilities and long-time self-diffusion coefficients of a suspension of bimodally distributed spherical particles. Stokesian dynamics is used to calculate the particle trajectories for a monolayer of bimodal-sized spheres. Hydrodynamic forces only are considered and they are calculated using the inverse of the grand mobility matrix for far-field many-body interactions and lubrication formulae for near-field effects. We determine both the detailed microstructure (e.g. the pair-connectedness function and cluster formation) and the macroscopic properties (e.g. viscosity and self-diffusion coefficients). The flow of an ‘infinite’ suspension is simulated by considering 25, 49, 64 and 100 particles to be one ‘cell’ of a periodic array. Effects of both the size ratio and the relative fractions of the different-sized particles are examined. For the microstructures, the pair-connectedness function shows that the particles form clusters in simple shearing flow due to lubrication forces. The nearly symmetric angular structures imply the absence of normal stress differences for a suspension with purely hydrodynamic interactions between spheres. For average mobilities at infinite Péclet number, Ds0, our simulation results suggest that the reduction of Ds0 as concentration increases is directly linked to the influence of particle size distribution on the average cluster size. For long-time self-diffusion coefficients, Ds∞, we found good agreement between simulation and experiment (Leighton & Acrovos 1987 a; Phan and Leighton 1993) for monodispersed suspensions. For bimodal suspensions, the magnitude of Ds∞, and the time to reach the asymptotic diffusive behaviour depend on the cluster size formed in the system, or the viscosity of the suspension. We also consider the effect of the initial configuration by letting the spheres be both organized (size segregated) and randomly placed. We find that it takes a longer time for a suspension with an initially organized structure to achieve steady state than one with a random structure.

Gravity wave radiation by vortical flows in the f-plane shallow-water equations is investigated by direct nonlinear numerical simulation. The flows considered are initially parallel flows, consisting of a single strip in which the potential vorticity differs from the background value. The flows are unstable to the barotropic instability mechanism, and roll up into a train of vortices. During the subsequent evolution of the vortex train, gravity waves are radiated. In the limit of small Froude number, the gravity wave radiation is compared with that predicted by an appropriately modified version of the Lighthill theory of aerodynamic sound generation. It is found that the gravity wave field agrees well with that predicted by the theory, provided typical lengthscales of vortical motions are well within one deformation radius.

It is found that the nutation time for vortices in the train increases rapidly with increasing Froude number in cases where the potential vorticity in the vortices is of the same sign as the background value, whereas the nutation time is almost independent of Froude number in cases where the potential vorticity in the vortices is zero or of opposite sign to the background. Consequently, in the former cases, the unsteadiness of the flow decreases with increasing Froude number, so the effect of the inertial cutoff frequency is increased, leading to an optimal Froude number for gravity wave radiation, above which the intensity of the radiated waves decreases as the Froude number is further increased. It is proposed that the existence of a finite range of interaction within the vortices, for flows with positive vortex potential vorticity, may account for the strong dependence of nutation time on Froude number in those cases. The interaction scale within the vortices becomes infinite in the limit of zero vortex potential vorticity, and so the arguments do not apply in those cases.

This paper describes a combined computational and experimental study of the flow between contra-rotating disks for – 1 ≤ Γ ≤ 0 and Reϕ = 105, where Γ is the ratio of the speed of the slower disk to that of the faster one and Reϕ is the rotational Reynolds number of the faster disk. For Γ = 0, the rotor-stator case, laminar and turbulent computations and experimental measurements show that laminar Batchelor-type flow occurs: there is radial outflow in a boundary layer on the rotating disk, inflow on the stationary disk and a rotating core of fluid between. For Γ = – 1, the laminar computations produce Batchelor-type flow: there is radial outflow on both disks and inflow in a free shear layer in the mid-plane, on either side of which is a rotating core of fluid. The turbulent computations and the velocity measurements for Γ = – 1 show Stewartson-type flow: radial outflow occurs in laminar boundary layers on the disks and inflow occurs in a non-rotating turbulent core between the boundary layers. For intermediate values of Γ, transition from Batchelor-type flow to Stewartson-type flow is associated with a two-cell structure, the two-cells being separated by a streamline that stagnates on the slower disk; Batchelor-type flow occurs radially outward of the stagnation point and Stewartson-type flow radially inward. The turbulent computations are mainly in good agreement with the measured velocities for Γ = 0 and Γ = – 1, where either Batchelor-type flow or Stewartson-type flow occurs; there is less good agreement at intermediate values of Γ, particularly for Γ = – 0.4 where the double transition of Batchelor-type flow to Stewartson-type flow and laminar to turbulent flow occurs in the two-cell structure.

A flagellated, bottom-heavy micro-organism's swimming direction in a shear flow is determined from a balance between the gravitational and viscous torques (gyrotaxis). Hitherto, the cell has been assumed to be a spheroid and the flagella have been neglected. Here we use resistive-force theory to calculate both the magnitude and the direction of a biflagellate cell's swimming velocity and angular velocity relative to the fluid when there is an arbitrary linear flow far from the cell. We present an idealized model for the flagellar beat but, in calculating the velocity of the fluid relative to an element of a flagellum, the presence of the cell body is not neglected. Results are given for the case of a spherical cell body whose flagella beat in a vertical plane, when the ambient linear flow is in the same vertical plane. Results show that resistive-force theory can be used for organisms where the cell body has significant effect on the flow past the flagella and that the viscous torque on the flagella is a significant term in the torque balance equations. A model is presented for the calculation of a cell's velocity and angular velocity in a shear flow which is valid up to high magnitudes of rate of strain or vorticity. The main application of the results will be to modify a recent continuum model for suspensions of gyrotactic micro-organisms (Pedley & Kessler 1990).

Based on linear potential theory, the general three-dimensional problem of steady supersonic flow inside quasi-cylindrical ducts is formulated as an initial-boundary-value problem for the wave equation, whose general solution arises as an infinite double series of the Fourier–Bessel type. For a broad class of solutions including the general axisymmetric case, it is shown that the presence of a discontinuity in wall slope leads to a periodic singularity pattern associated with non-uniform convergence of the corresponding series solutions, which thus are unsuitable for direct numerical computation. This practical difficulty is overcome by extending a classical analytical method, viz. Kummer's series transformation. A variety of elementary flow fields is presented, whose complex cellular structure can be qualitatively explained by asymptotic laws governing the propagation of small perturbations on characteristic surfaces.

The modification of the mean and fluctuating characteristics of a flat-plate boundary layer subjected to nearly isotropic free stream turbulence (FST) is studied experimentally using hot-wire anemometry. The study is focussed on the region upstream of the transition onset, where the fluctuations inside the boundary layer are dominated by elongated flow structures which grow downstream both in amplitude and length. Their downstream development and scaling are investigated and the results are compared with those obtained by previous authors. This allows some conclusions about the parameters which are relevant for the modelling of the transition process. The mechanisms underlying the transition process and the relative importance of the Tollmien–Schlichting wave instability in this flow are treated in an accompanying paper (part 2 of the present report).

The natural occurrence of Tollmien-Schlichting (TS) waves has so far only been observed in boundary layers subjected to moderate levels of free stream turbulence (Tu < 1%), owing to the difficulty in detecting small-amplitude waves in highly perturbed boundary layers. By introducing controlled oscillations with a vibrating ribbon, it is possible to study small-amplitude waves using phase-selective filtering techniques. In the present work, the effect of TS-waves on the transition is studied at Tu = 1.5%. It is demonstrated that TS-waves can exist and develop in a similar way as in an undisturbed boundary layer. It is also found that TS-waves with quite small amplitudes are involved in nonlinear interactions which lead to a regeneration of TS-waves in the whole unstable frequency band. This results in a significant increase in the number of turbulent spots, which promote the onset of turbulence.

The existence of an organized streamwise vortical structure, which is superimposed on the well known coherent spanwise vorticity in nominally two-dimensional free shear layers, has been studied extensively. In the presence of stratification, however, buoyancy forces contribute to an additional mechanism for the generation of streamwise vorticity. As the spanwise vorticity layer rolls up and pulls high-density fluid above low-density fluid, a local instability results. The purpose of the current investigation is to force the three-dimensional instability in the stratified shear layer. In this manner, we experimentally observe the effect of buoyancy on the streamwise vortex tube evolution, the evolution of the buoyancy-induced instability, and the interaction between these two vortical structures. A simple numerical model is proposed which captures the relevant physics of the flow evolution. It is found that, depending on the location, streamwise vortices resulting from vortex stretching may be weakened or enhanced by the stratification. Buoyancy-induced vortex structures are shown to form where the unstable part of the interface is tilted by the streamwise vortex tubes. These vortices strengthen initially, then weaken downstream, the timescale for this process depending upon the degree of stratification. For initial Richardson numbers larger than about 0.03, the baroclinically weakened vortex tubes eventually disappear as the flow evolves downstream and the baroclinically generated vortices dominate the three-dimensional flow structure.

This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debye's approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Keller's geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargill's meridional rays, is summarized.

Oscillations of a tank at a near-resonant frequency have been shown to produce a response which changes from a ‘hard-spring’ to a ‘soft-spring’ response as the depth passes through a critical value. This paper investigates the transition region and it is shown, using a symbolic manipulator, that in fact the large-amplitude response is that of a soft spring on either side of this critical depth.

Linear birefringence measurements of dilute and semi-dilute polyisobutylene solutions following flow through a disordered fixed fibre bed of 2.47% solids volume fraction provide both transient and steady measurements of chain deformation. Our results indicate that the flexible polyisobutylene polymers undergo a large conformation change, stretching in the direction of the average flow. This occurs even though the average flow in the bed is a plug flow which would not cause any polymer stretch by itself. The polymer stretch or conformation change increases with the number of chain interactions with bed fibres and ultimately reaches a steady-state value that can be correlated with the pore-size Deborah number (i.e. a characteristic polymer relaxation time divided by a characteristic flow time in the bed pore). Large changes in the polymer conformation are noted for values of the Deborah number, De > 5. In addition, the time to steady state scales with the characteristic flow time within a pore over a large range of Deborah numbers. The pressure drop across the fibre bed was also measured simultaneously with the birefringence measurement and was found to be directly proportional to the birefringence throughout the range of De investigated. Thus, we show empirically, for the first time, that chain elongation, which produces normal stress anisotropy within the fluid, is directly responsible for the increased flow resistance. These findings are then analysed in the light of recent theories for the response of polymer molecules in fixed bed flow fields (Shaqfeh & Koch 1992). It is shown that our results are consistent with the interpretation that these flows are stochastic strong flows, which create an apparent ‘coil-stretch’ transition. After extending the theory of Shaqfeh & Koch to account for the specifics in the experiments, including the bed geometry and statistics as well as the polydispersity of the polymer solutions, it is shown that the theory can predict most of the experimental results both qualitatively and quantitatively.

In an earlier paper (Davey 1978) the first author investigated the linear stability of flow in a straight pipe whose cross-section was an ellipse, of small ellipticity e, by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. That paper contained some serious errors which we correct herein. We show analytically that for the most important mode n = 1, for which the circular problem has a double eigenvalue c0 as the ‘swirl’ can be in either direction, the ellipticity splits the double eigenvalue into two separate eigenvalues c0 ± e2c12, to leading order, when the cross-sectional area of the pipe is kept fixed. The imaginary part of c12 is non-zero and so the ellipticity always makes the flow less stable. This specific problem is generic to a much wider class of fluid dynamical problems which are made less stable when the symmetry group of the dynamical system is reduced from S1 to Z2.

In the Appendix, P. G. Drazin describes simply the qualitative structure of this problem, and other problems with the same symmetries, without technical detail.

We investigate the weakly nonlinear spatial evolution of helical disturbances of an axisymmetrical jet which are the analogue of three-dimensional disturbances, such as a single oblique wave (the wave vector is directed at an angle to the main flow velocity) in plane-parallel flows. It is shown that when a supercriticality is large enough, the perturbation amplitude A grows in the streamwise direction (along z) explosively: A ∼ (z0 — z)−5/2, though more slowly than in the case of essentially three-dimensional disturbances in the form of a pair of oblique waves (A ∼ (z0 — z)−3; Goldstein & Choi 1989). The nonlinearity needed for such a growth, is due equally to the cylindricity of shear layer and to the spatial character of the evolution (in the temporal problem the ‘evolution’ contribution is absent). At a smaller supercriticality, the evolution equation has a non-local (integral in z) nonlinearity, unusual for the regime of a viscous critical layer. Scenarios of disturbance development for different levels of supercriticality are studied, with proper account taken of viscous broadening of the flow.