We consider the reflection of oblique compression waves from a two-dimensional, steady, laminar boundary layer on a flat, adiabatic plate at free-stream pressures such that dense-gas effects are non-negligible. The full Navier–Stokes equations are solved through use of a dense-gas version of the Beam–Warming implicit scheme. The main fluids studied are Bethe–Zel'dovich–Thompson (BZT) fluids. These are ordinary gases which have specific heats large enough to cause the fundamental derivative of gasdynamics to be negative for a range of pressures and temperatures in the single-phase vapour regime. It is demonstrated that the unique dynamics of BZT fluids can result in a suppression of shock-induced separation. Numerical tests performed reveal that the physical mechanism leading to this suppression is directly related to the disintegration of any compression discontinuities originating in the flow. We also demonstrate numerically that the interaction of expansion shocks with the boundary layer produces no adverse effects.

A transition scenario initiated by two oblique waves is studied in an incompressible boundary layer. Hot-wire measurements and flow visualizations from the first boundary layer experiment on this scenario are reported. The experimental results are compared with spatial direct numerical simulations and good qualitative agreement is found. Also, quantitative agreement is found when the experimental device for disturbance generation is closely modelled in the simulations and pressure gradient effects taken into account. The oblique waves are found to interact nonlinearly to force streamwise vortices. The vortices in turn produce growing streamwise streaks by non-modal linear growth mechanisms. This has previously been observed in channel flows and calculations of both compressible and incompressible boundary layers. The flow structures observed at the late stage of oblique transition have many similarities to the corresponding ones of K- and H-type transition, for which two-dimensional Tollmien–Schlichting waves are the starting point. However, two-dimensional Tollmien–Schlichting waves are usually not initiated or observed in oblique transition and consequently the similarities are due to the oblique waves and streamwise streaks appearing in all three scenarios.

The flow between parallel walls driven by the time-periodic oscillation of one of the walls is investigated. The flow is characterized by a non-dimensional amplitude Δ and a Reynolds number R. At small values of the Reynolds number the flow is synchronous with the wall motion and is stable. If the amplitude of oscillation is held fixed and the Reynolds number is increased there is a symmetry-breaking bifurcation at a finite value of R. When R is further increased, additional bifurcations take place, but the structure which develops, essentially chaotic flow resulting from a Feigenbaum cascade or a quasi-periodic flow, depends on the amplitude of oscillation. The flow in the different regimes is investigated by a combination of asymptotic and numerical methods. In the small-amplitude high-Reynolds-number limit we show that the flow structure develops on two time scales with chaos occurring on the longer time scale. The chaos in that case is shown to be associated with the unsteady breakdown of a steady streaming flow. The chaotic flows which we describe are of particular interest because they correspond to Navier–Stokes solutions of stagnation-point form. These flows are relevant to a wide variety of flows of practical importance.

The motion of a rigid body in an inviscid incompressible fluid of inhomogeneous density is considered. The size of the body is taken small with respect to the length scale of the density variations; its shape is otherwise arbitrary. The force and the torque acting on the body in an arbitrary motion are derived from Hamilton's principle of least action, thus offering a variational derivation of Kirchhoff's equations, supplemented by the terms due to the density gradient. The force and the torque due to a density gradient are proportional to the gradient and quadratic in the velocity and the angular velocity. The same coefficients are shown to govern both the inertial behaviour of the body, i.e. the response to accelerations, and the effects of density gradients. The free motion of spheres and two-dimensional circular cylinders is shown to obey a condition akin to the Fermat principle in optics.

History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier–Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

We investigate the temporal evolution of the geometrical distribution of a passive scalar injected continuously into the far field of a turbulent water jet at a scale d smaller than the local integral scale of the turbulence. The concentration field is studied quantitatively by a laser-induced- fluorescence technique on a plane cut containing the jet axis. Global features such as the scalar dispersion from the source, as well as the fine structure of the scalar field, are analysed. In particular, we define the volume occupied by the regions whose concentration is larger than a given concentration threshold (support of the scalar field) and the surface in which this volume is enclosed (boundary of the support). The volume and surface extents, and their respective fractal dimensions are measured as a function of time t, and the concentration threshold is normalized by the initial concentration Cs/C0 for different injection sizes d. All of these quantities display a clear dependence on t, d and Cs, and their evolutions rescale with the variable ξ = (ut/d)(Cs/C0), the fractal dimension being, in addition, scale dependent. The surface-to-volume ratio and the fractal dimension of both the volume and the surface tend towards unity at large ξ, reflecting the sheet-like structure of the scalar at small scales. These findings suggest an original picture of the kinetics of turbulent mixing.

A two-dimensional model of flow and bed topography is proposed to investigate the effect of sediment heterogeneity on the development of alternate bars. Within the context of a linear stability theory the flow field, the bed topography and the grain size distribution function are perturbed leading to an integro-differential linear eigenvalue problem. It is shown that the selective transport of different grain size fractions and the resulting spatial pattern of sorting may appreciably affect the balance between stabilizing and destabilizing actions which govern bar instability. Theoretical results suggest that sediment heterogeneity leads to a damping of both growth rate and migration speed of bars, while bar wavelength is shortened with respect to the case of uniform sediment. The above findings conform, at least qualitatively, to the experimentally detected reduction of bar height, length and celerity. The observed tendency of coarser particles to pile up towards bar crests is also reproduced by theoretical results.

This paper describes the evolution of an incompressible turbulent boundary layer on the flat wall of an ‘S’-shaped wind tunnel test section under the influence of changing streamwise and spanwise pressure gradients. The unit Reynolds number based on the mean velocity at the entrance of the test section was fixed to 106 m−1, resulting in Reynolds numbers Reδ2, based on the streamwise momentum thickness and the local freestream velocity, between 3.9 and 11 × 103. The particular feature of the experiment is the succession of two opposite changes of core flow direction which causes a sign change of the spanwise pressure gradient accompanied by a reversal of the spanwise velocity component near the wall, i.e. by the formation of so-called cross-over velocity profiles. The aim of the study is to provide new insight into the development of the mean and fluctuating flow field in three-dimensional pressure-driven boundary layers, in particular of the turbulence structure of the near-wall and the cross-over region.

Mean velocities, Reynolds stresses and all triple correlations were measured with a newly developed miniature triple-hot-wire probe and a near-wall hot-wire probe which could be rotated and traversed through the test plate. Skin friction measurements were mostly performed with a wall hot-wire probe. The data from single normal wires extend over wall distances of y+ [gsim ] 3 (in wall units), while the triple-wire probe covers the range y+ [gsim ] 30. The data show the behaviour of the mean flow angle near the wall to vary all the way to the wall. Then, to interpret the response of the turbulence to the pressure field, the relevant terms in the Reynolds stress transport equations are evaluated. Finally, an attempt is made to assess the departure of the Reynolds stress profiles from local equilibrium near the wall.

In this paper we study the acoustic scattering between two flat-plate cascades, with the aim of investigating the possible existence of trapped modes. In practical terms this question is related to the phenomenon of acoustic resonance in turbomachinery, whereby such resonant modes are excited to large amplitude by unsteady processes such as vortex shedding. We use the Wiener–Hopf technique to analyse the scattering of the various wave fields by the cascade blades, and by considering the fields between adjacent blades, as well as between the cascades, we are able to take full account of the genuinely finite blade chords. Analytic expressions for the various scattering matrices are derived, and an infinite-dimensional matrix equation is formed, which is then investigated numerically for singularity. One advantage of this formulation is that it allows the constituent parts of the system to be analysed individually, so that for instance the behaviour of the gap between the blade rows alone can be investigated by omitting the finite-chord terms in the equations. We demonstrate that the system exhibits two types of resonance, at a wide range of parameter values. First, there is a cut-on/cut-off resonance associated with the gap between the rows, and corresponding to modes propagating parallel to the front face of the cascades. Second, there is a resonance of the downstream row, akin to a Parker mode, driven at low frequencies by a vorticity wave produced by trapped duct modes in the upstream row, and at higher frequencies by radiation modes (and the vorticity wave) between the blade rows. The predictions for this second set of resonances are shown to be in excellent agreement with previous experimental data. The resonant frequencies are also seen to be real for this twin cascade system, indicating that the resonances correspond to genuine trapped modes. The analysis in this paper is completed with non-zero axial flow but with zero relative rotation between the cascades – in Part 2 (Woodley & Peake 1999) we will show how non-zero rotation of the upstream row can be included.

In Part 1 (Woodley & Peake 1999) we described a method for predicting the occurrence of resonant states in a system comprising twin cascades in zero relative motion. We now demonstrate how that work can be extended to account for the case of more practical interest, in which the upstream cascade (rotor) is rotating in the transverse direction relative to the downstream cascade (stator). Time periodicity now forces the temporal frequency of any disturbance to be an integer multiple of the rotor passing frequency in the stator frame, and vice versa, and this leads to the requirement to sum over a discrete set of temporal modes, as well as over the spatial modes already described in Part 1. The mechanisms by which temporal and spatial modes are scattered by the blade rows is made clear by the analytical approach adopted here; the scattering of the incident pressure (and, for the stator, vorticity) fields by each row in its own frame is completed using results similar to those presented in Part 1, and the fields in the two frames then matched across the inter-row gap to provide a single matrix equation. Specimen results for the conditioning of this equation are given, and although it seems more difficult to obtain very strong excitation than it was for zero rotation, the significance of Parker resonance of the stator is again apparent.

The goal of this work is to analyse how solid body rotation affects forced turbulence enclosed within solid boundaries, and to compare it to results of the experiment performed by Hopfinger et al. (1982). In order to identify various mechanisms associated with rotation, confinement, and forcing, a numerical pseudo-spectral code is used for performing direct numerical simulations. The geometry is simplified with respect to the experimental one. First, we are able to reproduce the linear regime, as propagating inertial waves that undergo reflections at the walls. Second, the Ekman pumping phenomenon, proportional to the rotation rate, is identified in freely decaying turbulence, for which the evolution of the flow bounded by walls is compared to the evolution of unbounded homogeneous turbulence. Finally we introduce a local forcing on a plane in physical space, for simulating the effect of an oscillating grid, so that diffusive turbulence is created, and we examine the structuring of the flow under the combination of the linear and nonlinear mechanisms. A transition to an almost two-dimensional state is shown to occur between the region close to the forcing and an outer region in which vortices appear, the number of which depends on the Reynolds and Rossby numbers. In this region, the anisotropy of turbulence is examined, and the numerical predictions are shown to reproduce many of the most important features present in the experimental flow.

Three-dimensional large-amplitude oscillations of a mercury drop were obtained by electrical excitation in low gravity using a drop tower. Multi-lobed (from three to six lobes) and polyhedral (including tetrahedral, hexahedral, octahedral and dodecahedral) oscillations were obtained as well as axisymmetric oscillation patterns. The relationship between the oscillation patterns and their frequencies was obtained, and it was found that polyhedral oscillations are due to the nonlinear interaction of waves.

A mathematical model of three-dimensional forced oscillations of a liquid drop is proposed and compared with experimental results. The equations of drop motion are derived by applying the variation principle to the Lagrangian of the drop motion, assuming moderate deformation. The model takes the form of a nonlinear Mathieu equation, which expresses the relationships between deformation amplitude and the driving force's magnitude and frequency.

Viscous liquid drops undergoing forced oscillations have been shown to exhibit hysteretic deformation under certain conditions both in experiments and by solution of simplified model equations that can only provide a qualitative description of their true response. The first hysteretic deformation results for oscillating pendant drops obtained by solving the full transient, nonlinear Navier–Stokes system are presented herein using a sweep procedure in which either the forcing amplitude A or frequency Ω is first increased and then decreased over a given range. The results show the emergence of turning-point bifurcations in the parameter space of drop deformation versus the swept parameter. For example, when a sweep is carried out by varying Ω while holding A fixed, the first turning point occurs at Ω ≡ Ωu as Ω is being increased and the second one occurs at Ω ≡ Ωl < Ωu as Ω is being decreased. The two turning points shift further from each other and toward lower values of the swept parameter as Reynolds number Re is increased. These turning points mark the ends of a hysteresis range within which the drop may attain either of two stable steady oscillatory states – limit cycles – as identified by two distinct solution branches. In the hysteresis range, one solution branch, referred to as the upper solution branch, is characterized by drops having larger maximum deformations compared to those on the other branch, referred to as the lower solution branch. Over the range Ωl [les ] Ω [les ] Ωu, the sweep procedure enables detection of the upper solution branch which cannot be found if initially static drops are set into oscillation as in previous studies of forced oscillations of supported and captive drops, or liquid bridges. The locations of the turning points and the associated jumps in drop response amplitudes observed at them are studied over the parameter ranges 0.05 [les ] A [les ] 0.125, 20 [les ] Re [les ] 40, and gravitational Bond number 0 [les ] G [les ] 1. Critical forcing amplitudes for onset of hysteresis are also determined for these Re values. The new findings have important ramifications in several practical applications. First, that Ωu − Ωl increases as Re increases overcomes the limitation which is inherent to the current practice of inferring the surface tension and/or viscosity of a bridge/drop liquid from measurement of its resonance frequencies (Chen & Tsamopoulos 1993; Mollot et al. 1993). Moreover, that the value of A for onset of hysteresis can be as low as 5% of the drop radius, or lower, has important implications for other free-surface flows such as coating flows.

This paper considers the structure of steady-state solutions of the Swift–Hohenberg equation describing convection in shallow rectangular-planform containers heated from below. The lateral dimensions of the planform are assumed to be much larger than the characteristic wavelength of convection. Results are restricted to patterns composed of rolls orthogonal to the sides of the rectangle in which case convection sets in at a critical value of the Rayleigh number in the form of rolls parallel to the shorter sides. This primary bifurcation from the conductive state of no motion produces a solution which subsequently undergoes a secondary bifurcation in which the low-amplitude motion near the shorter sides is replaced locally by cross-rolls perpendicular to the sides. This results in the formation of grain boundaries (or domain boundaries) within the fluid which mark the division between the different roll orientations.

With increasing Rayleigh number the grain boundaries approach the sides of the rectangle and a boundary-layer structure is formed. In the present paper the method of matched asymptotic expansions is used to determine this boundary-layer structure and to predict the location of the grain boundaries. An interesting feature of the solution is that the grain boundaries develop significant curvature and bend into the corners of the rectangle, where the local solution is also determined.

The results are compared with numerical computations of the secondary solution branch and with previous numerical and experimental work.