The axisymmetric boundary-layer separation of an incompressible impulsively started flow in a wavy-walled tube is analysed at moderate to high values of the Reynolds number. The investigation is carried out by numerical integration of either the Navier–Stokes equations or Prandtl's asymptotic formulation of the boundary-layer problem. The presence of an adverse pressure gradient induces reverse flow at the tube wall independently of the Reynolds number; its occurrence can be predicted by a timescale analysis. Following that, the viscous calculations show different dynamics depending on the Reynolds number. As the Reynolds number increases, the boundary layer has in a well-defined internal structure where longitudinal lengthscales become comparable with the viscous one. Thus the boundary-layer scaling fails locally, with a minimum of pressure inside the boundary layer itself. The formation of the primary recirculation is well captured by the asymptotic model which, however, is not able to describe the roll-up of the vortex structure inside the recirculating region. This inadequacy appears well before the flow evolves to the characteristic terminal singularity usually assumed as foreshadowing the vortex shedding phenomenon. The outcomes are compared with the existing results of analogous problems giving an overall agreement but improving, in some cases, the physical picture.

Prandtl–Meyer flows with heat addition from homogeneous condensation not exceeding a critical value (subcritical flows) are investigated by an asymptotic method in the double limit of a large nucleation time followed by a small droplet growth time. The physically distinct condensation zones, with detailed analytical structure, are displayed along streamlines and the flow field in each zone is determined utilizing the asymptotic solution of the rate equation along streamlines. In particular the nucleation wave front, which corresponds to states of maximum nucleation along streamlines, is accurately located independently of the particular condensation model employed. Results obtained using the classical nucleation equation together with the Hertz–Knudsen droplet growth law show, despite qualitative agreement, considerable differences between the nucleation wave fronts and measured onset conditions for the experiments of Smith (1971), because of intersecting characteristics in the heat addition zones. This shows the necessity of including an embedded oblique shock wave in the expansion fan of corner expansion flows for the cases investigated.

The flow in a rotating cylinder driven by the differential rotation of its top endwall is studied by numerically solving the time-dependent axisymmetric Navier–Stokes equations. When the differential rotation is small, the flow is well described in terms of similarity solutions of individual rotating disks of infinite radius. For larger differential rotations, whether the top is co-rotating or counter-rotating results in qualitatively distinct behaviour. For counter-rotation, the boundary layer on the top endwall separates, forming a free shear layer and this results in a global coupling between the boundary layer flows on the various walls and a global departure from the similarity flows. At large Reynolds numbers, this shear layer becomes unstable. For a co-rotating top, there is a qualitative change in the flow depending on whether the top rotates faster or slower than the rest of the cylinder. When the top rotates faster, so does the bulk of the interior fluid, and the sidewall boundary layer region where the fluid adjusts to the slower rotation rate of the cylinder is centrifugally unstable. The secondary induced meridional flow is also potentially unstable in this region. This is manifested by the inflectional radial profiles of the vertical velocity and azimuthal vorticity in this region. At large Reynolds numbers, the instability of the sidewall layer results in roll waves propagating downwards.

We investigate with direct numerical simulations the onset of unsteadiness, the route to chaos and the dynamics of fully chaotic natural convection in an upright square air-filled differentially heated cavity with adiabatic top and bottom walls. The numerical algorithm integrates the Boussinesq-type Navier–Stokes equations in velocity–pressure formulation with a Chebyshev spatial approximation and a finite-difference second-order time-marching scheme. Simulations are performed for Rayleigh numbers up to 1010, which is more than one order of magnitude higher than the onset of unsteadiness. The dynamics of the time-dependent solutions, their time-averaged structure and preliminary results concerning their statistics are presented. In particular, the internal gravity waves are shown to play an important role in the time-dependent dynamics of the solutions, both at the onset of unsteadiness and in the fully chaotic regime. The influence of unsteadiness on the local and global heat transfer coefficients is also examined.

The propagation at high Reynolds number of a heavy, two-dimensional gravity current of given initial volume at the base of a uniform flow is considered. An experimental setup is described for which a known volume of fluid is rapidly introduced halfway down a 9 m channel in which there is a uniform flow of water. The density excess of the released fluid is produced by either dissolving salt or suspending particles in water. The upstream and downstream propagation of the current was measured for different initial salt concentrations, particle sizes and concentrations. A simple box model for the motion of and deposit from the gravity current is constructed. The analytical results obtained compare well with our numerical solutions of one-layer and two-layer models incorporating the appropriate shallow-water equations. Both sets of results are in very good agreement with the experimental data.

Combined thermocapillary–buoyancy convection in a thin rectangular geometry is investigated experimentally, with an emphasis on the generation of hydrothermal-wave instabilities. For sufficiently thin layers, pure hydrothermal waves are observed, and are found to be oblique as predicted by a previous linear-stability analysis (Smith & Davis 1983). For thicker layers, both a steady multicell state and an oscillatory state are found to exist, but the latter is not in the form of a pure hydrothermal wave.

Hydrothermal-wave instabilities in thermocapillary convection are known to produce undesirable effects when they occur during the float-zone crystal-growth process, and perhaps in other situations. Suppression of the hydrothermal-wave instability produced in the model system of Part 1 (Riley & Neitzel 1998) is demonstrated through the sensing of free-surface temperature perturbations and the periodic addition of heat at the free surface along lines parallel to the crests of the hydrothermal waves.

An important criterion in the development of modern aeroengines is the identification of the dominant noise sources under typical aircraft take-off and approach conditions, and also in ground-based tests in which the engine is stationary. In this paper, we develop a theoretical model for unsteady distortion noise, which results from the interaction of ingested atmospheric turbulence with the rotating fan, with a view to providing a better understanding of the important physical mechanisms in this particular aspect of sound generation. The theory, developed in the frequency domain, is applicable for any arbitrary spectral form of atmospheric turbulence upstream of the fan, and as a simple model we take the von Kármán spectra for isotropic turbulence. The key fluid dynamical process in unsteady distortion is the deformation of turbulent eddies into long, narrow filaments as they enter the engine, due to the strong streamtube contraction experienced by the steady, non-uniform mean flow generated by the fan. Simple models of the steady flow fields are provided for both open and ducted rotor geometries. The distorted turbulent field at the fan face can be obtained using rapid distortion theory, and considerable simplification is made here by noting that the number of blades in typical aeroengine fans is large, allowing the application of asymptotic analysis and the derivation of closed-form expressions for those parts of the turbulence spectrum at the fan face which dominate the radiation. The unsteady forces exerted on the rotating fan blades are then calculated via a strip-theory approach. The resulting sound scattered to the far field is then evaluated using asymptotic theory for open and ducted rotors. Results are presented in the form of frequency spectra for the turbulent field at the fan face, the blade forces and the radiated sound for typical testing and aircraft operating conditions. High tonal noise levels are obtained under static conditions, whereas the sound is generally broadband in flight. The dependence on turbulence parameters such as the integral lengthscale is highlighted.

A rotational shear flow is examined in the bounded parallel-plate geometry for a four-constant Oldroyd-type fluid which has a constant viscosity, and constant first and second normal stress coefficients. A new type of Galerkin spectral technique is introduced to solve the resulting two-dimensional stiff boundary value problem. We show that even a small second normal stress difference can lead to a significant increase (nearly 100%) in the stability of the base torsional flow. Beyond a critical Deborah number the secondary flow, in the form of travelling waves, appears to be confined between two critical radii, in qualitative agreement with the experimental results of Byars et al. (1994). The mechanism behind this instability is investigated for dilute polymer solutions.

This paper describes the development of a numerical model for studying the evolution of a wave train, shoaling and breaking in the surf zone. The model solves the Reynolds equations for the mean (ensemble average) flow field and the k–ε equations for the turbulent kinetic energy, k, and the turbulence dissipation rate, ε. A nonlinear Reynolds stress model (Shih, Zhu & Lumley 1996) is employed to relate the Reynolds stresses and the strain rates of the mean flow. To track free-surface movements, the volume of fluid (VOF) method is employed. To ensure the accuracy of each component of the numerical model, several steps have been taken to verify numerical solutions with either analytical solutions or experimental data. For non-breaking waves, very accurate results are obtained for a solitary wave propagating over a long distance in a constant depth. Good agreement between numerical results and experimental data has also been observed for shoaling and breaking cnoidal waves on a sloping beach in terms of free-surface profiles, mean velocities, and turbulent kinetic energy. Based on the numerical results, turbulence transport mechanisms under breaking waves are discussed.

Solutions are determined for normally incident non-breaking linear gravity waves in a perfect fluid over a porous plane bed of arbitrary slope α both with and without bed friction. For simplicity, computations are restricted to α=π/2m, m∈N. Modifications to wave height transformations due to percolation and to friction are determined for a variety of slopes and coefficients. The effect on the reflection coefficient Rf is also studied and excellent qualitative agreement is found with recent work on damping and reflection by permeable structures. In particular, for a choice of parameters, the Rf response is determined in closed form.

The present work studies the problem of the flow field around the inlet section in the case of a plane slider bearing with upstream free surface. In order to determine the pressure head build-up, the inlet region of the bearing can be studied by considering a rigid plane sheet (pad) parallel to a plane wall (guide) which slides at a steady speed in a viscous incompressible fluid. The mathematical model described is based on the application of the theory of analytic functions. Through Dini's equation, which relates the real and the complex part of any analytic function, defined on the boundary of a domain, it is possible to determine the distribution of pressure on this boundary and the geometry of the free surface. In this way the problem is formally solved; however, it is difficult to obtain the solution because the thickness of the free surface is unknown. It would be possible to adopt an iterative method but the great difficulty associated with this method of solution induced the authors to follow a simplified technique. Through this study it was possible to determine the pressure build-up at the inlet; this quantity is the essential boundary condition to evaluate the actual load capacity of a slider bearing.

In 1921 G. I. Taylor introduced (with little discussion) the notion that the dispersion of a conserved passive scalar in a turbulent flow is determined by the motion of fluid particles (independent of the molecular diffusivity). Here, a hypothesis of diffusivity independence is introduced, which provides a sufficient condition for the validity of Taylor's approach. The hypothesis, which is supported by DNS data, is that, at high Reynolds number, the mean of the scalar conditional on the velocity is independent of the molecular diffusivity. From this hypothesis it is shown that (at high Reynolds number) the conditional Laplacian of the scalar is zero. This new result has several significant implications for models of turbulent mixing, and for the scalar flux. Primarily, a model of turbulent scalar mixing that is independent of velocity is inconsistent with the hypothesis, and gives rise to a spurious source or (more likely) sink of the scalar flux.

An asymptotic analysis of two-dimensional free-surface cusps associated with flows at low Reynolds numbers is presented on the basis of a model which, in agreement with direct experimental observations, considers this phenomenon as a particular case of an interface formation–disappearance process. The model was derived from first principles and earlier applied to another similar process: the moving contact-line problem. As is shown, the capillary force acting on a cusp from the free surface, which in the classical approach can be balanced by viscous stresses only if the associated rate of dissipation of energy is infinite, in the present theory is always balanced by the force from the surface-tension-relaxation ‘tail’, which stretches from the cusp towards the interior of the fluid. The flow field near the cusp is shown to be regular, and the surface-tension gradient in the vicinity of the cusp, caused and maintained by the external flow, induces and is balanced by the shear stress. Existing approaches to the free-surface cusp description and some relevant experimental aspects of the problem are discussed.

The effects of a favourable pressure gradient (K[les ]4×10−6) and of the Reynolds number (862[les ]Reδ2[les ]5800) on the mean and fluctuating quantities of four turbulent boundary layers were studied experimentally and are presented in this paper and a companion paper (Part 2). The measurements consist of extensive hot-wire and skin-friction data. The former comprise mean and fluctuating velocities, their correlations and spectra, the latter wall-shear stress measurements obtained by four different techniques which allow testing of calibrations in both laminar-like and turbulent flows for the first time. The measurements provide complete data sets, obtained in an axisymmetric test section, which can serve as test cases as specified by the 1981 Stanford conference.

Two different types of accelerated boundary layers were investigated and are described: in this paper (Part 1) the fully turbulent, accelerated boundary layer (sometimes denoted laminarescent) with approximately local equilibrium between the production and dissipation of the turbulent energy and with relaxation to a zero pressure gradient flow (cases 1 and 3); and in Part 2 the strongly accelerated boundary layer with ‘inactive’ turbulence, laminar-like mean flow behaviour (relaminarized), and reversion to the turbulent state (cases 2 and 4). In all four cases the standard logarithmic law fails but there is no single parametric criterion which denotes the beginning or the end of this breakdown. However, it can be demonstrated that the departure of the mean-velocity profile is accompanied by characteristic changes of turbulent quantities, such as the maxima of the Reynolds stresses or the fluctuating value of the skin friction.

The boundary layers described here are maintained in the laminarescent state just up to the beginning of relaminarization and then relaxed to the turbulent state in a zero pressure gradient. The relaxation of the turbulence structure occurs much faster than in an adverse pressure gradient. In the accelerating boundary layer absolute values of the Reynolds stresses remain more or less constant in the outer region of the boundary layer in accordance with the results of Blackwelder & Kovasznay (1972), and rise both in the vincinity of the wall in conjunction with the rising wall shear stress and in the centre region of the boundary layer with the increase of production.

This is an experimental investigation of two turbulent boundary layers (cases 2 and 4) where the streamwise negative pressure gradient changes mean properties of the flow, e.g. mean velocity profiles and skin friction, so that they display laminar-like behaviour. The maximum acceleration parameter K[les ]4×10−6 and the starting value of the Reynolds number is 862 or 2564. Relaminarization occurs in both boundary layers as a gradual change of the turbulence properties and is not catastrophic. Retransition, however, is a fast process due to the remaining turbulence structure and may be compared with bypass transition. Together with an extensive investigation of the turbulence structure as in the companion paper, Part 1, which describes two cases (1 and 3) of boundary layers which remain turbulent, spectra and integral length scales for all four boundary layers are discussed.