This paper presents a statistical and topological study of a complex turbulent flow over a backward-facing step by means of direct and large-eddy simulations. Direct simulations are first performed for an isothermal two-dimensional case. In this case, shedding of coherent vortices in the mixing layer is demonstrated. Both direct and large-eddy simulations are then carried out in three dimensions. The subgrid-scale model used is the structure-function model proposed by Métais & Lesieur (1992). Lowstep computations corresponding to the geometry of Eaton & Johnston's (1980) laboratory experiment give turbulence statistics in better agreement with the experimental data than both Smagorinsky's method and K-ε modelling. Furthermore, calculations for a high step show that the eddy structure of the flow presents striking analogies with forced plane mixing layers: large billows are shed behind the step with intense longitudinal vortices strained between them.

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, has been shown to differ from that in still fluid owing to a bias from the particle inertia (Maxey 1987). Previous numerical results for particles in a random flow field, where the flow dynamics were not considered, showed an increase in the average settling velocity. Direct numerical simulations of the motion of heavy particles in isotropic homogeneous turbulence have been performed where the flow dynamics are included. These show that a significant increase in the average settling velocity can occur for particles with inertial response time and still-fluid terminal velocity comparable to the Kolmogorov scales of the turbulence. This increase may be as much as 50% of the terminal velocity, which is much larger than was previously found. The concentration field of the heavy particles, obtained from direct numerical simulations, shows the importance of the inertial bias with particles tending to collect in elongated sheets on the peripheries of local vortical structures. This is coupled then to a preferential sweeping of the particles in downward moving fluid. Again the importance of Kolmogorov scaling to these processes is demonstrated. Finally, some consideration is given to larger particles that are subject to a nonlinear drag force where it is found that the nonlinearity reduces the net increase in settling velocity.

We have studied oscillatory flow through a 180° curved tube with the ratio of tube radius to radius of curvature equal to 1/7. The flow rate is constrained to vary sinusoidally about a non-zero mean at a specified period T, and mean flow rate Q. At a certain parameter range of interest Hamakiotes & Berger (1990) predict that fully developed flow undergoes a period-tripling bifurcation. Our measurements using laser-Doppler velocimetry found no bifurcation. An additional experiment was done to ensure that the flow was fully developed.

A systematic theory is developed to study the nonlinear spatial evolution of the resonant triad in Blasius boundary layers. This triad consists of a plane wave at the fundamental frequency and a pair of symmetrical, oblique waves at the subharmonic frequency. A low-frequency asymptotic scaling leads to a distinct critical layer wherein nonlinearity first becomes important, and the critical layer's nonlinear, viscous dynamics determine the development of the triad.

The plane wave initially causes double-exponential growth of the oblique waves. The plane wave, however, continues to follow the linear theory, even when the oblique waves’ amplitude attains the same order of magnitude as that of the plane wave. However, when the amplitude of the oblique waves exceeds that of the plane wave by a certain level, a nonlinear stage comes into effect in which the self-interaction of the oblique waves becomes important. The self-interaction causes rapid growth of the phase of the oblique waves, which causes a change of the sign of the parametric-resonance term in the oblique-waves amplitude equation. Ultimately this effect causes the growth rate of the oblique waves to oscillate around their linear growth rate. Since the latter is usually small in the nonlinear regime, the net outcome is that the self-interaction of oblique waves causes the parametric resonance stage to be followed by an ‘oscillatory’ saturation stage.

The axisymmetric, thermocapillary-driven motion of a pair of non-conducting, spherical drops in near contact is analysed for conditions of small Reynolds and Marangoni numbers. The pairwise motion and an associated contact force are computed by considering touching drops in point contact. Relative motion for nearly touching drops results from the contact force balanced by a lubrication resistance. A new, analytical solution is obtained for the axisymmetric temperature field around an unequal pair of non-conducting, tangent spheres embedded in an ambient temperature gradient. Numerical results for the pairwise migration velocity, contact force, and the relative and individual drop velocities are presented for all size ratios and a wide range of viscosity ratios, and asymptotic formulae are derived for small size ratios. For nearly equisized drops, the ratio of the relative velocity for two drops in near contact to that for widely separated drops is similar for thermocapillary-driven and gravity-driven motion. For small and moderate size ratios, however, this ratio is much larger for thermocapillary-driven relative motion than for gravity-driven relative motion, indicating that the former represents a more efficient coalescence mechanism. An explanation for this finding is provided in terms of the thermocapillary motion of the interface of the larger drop aiding the withdrawal of continuous phase from between the two drops.

A study of the unsteady natural convection in a cavity which was heated by the absorption of radiation entering through part of the surface is reported. While a general scaling analysis is quite complex, involving five separate timescales, most naturally occurring problems fall into just one regime, and it is only this regime which is discussed. To test the scaling, a series of laboratory experiments were performed in which the radiation parameters (surface flux and attenuation coefficient) were varied. The method by which these parameters were determined is also discussed. Numerical experiments were used to extend the parameter range, and the results of all these experiments confirmed the validity of the scaling over a range of parameters.

Direct numerical simulations of fully developed pressure-driven turbulent flow in a rotating channel have been performed. The unsteady Navier–Stokes equations were written for flow in a constantly rotating frame of reference and solved numerically by means of a finite-difference technique on a 128 × 128 × 128 computational mesh. The Reynolds number, based on the bulk mean velocity Um and the channel half-width h, was about 2900, while the rotation number Ro = 2|Ω|h/Um varied from 0 to 0.5. Without system rotation, results of the simulation were in good agreement with the accurate reference simulation of Kim, Moin & Moser (1987) and available experimental data. The simulated flow fields subject to rotation revealed fascinating effects exerted by the Coriolis force on channel flow turbulence. With weak rotation (Ro = 0.01) the turbulence statistics across the channel varied only slightly compared with the nonrotating case, and opposite effects were observed near the pressure and suction sides of the channel. With increasing rotation the augmentation and damping of the turbulence along the pressure and suction sides, respectively, became more significant, resulting in highly asymmetric profiles of mean velocity and turbulent Reynolds stresses. In accordance with the experimental observations of Johnston, Halleen & Lezius (1972), the mean velocity profile exhibited an appreciable region with slope 2Ω. At Ro = 0.50 the Reynolds stresses vanished in the vicinity of the stabilized side, and the nearly complete suppression of the turbulent agitation was confirmed by marker particle trackings and two-point velocity correlations. Rotational-induced Taylor-Görtler-like counter-rotating streamwise vortices have been identified, and the simulations suggest that the vortices are shifted slightly towards the pressure side with increasing rotation rates, and the number of vortex pairs therefore tend to increase with Ro.

We report and discuss the results of a direct Monte Carlo simulation of the flow of a rarefied gas flowing between two cylinders when the inner one rotates. The formation of Taylor vortices is clearly exhibited.

A general one-dimensional model for the steady adiabatic motion of gas-particle mixtures in arbitrarily oriented ducts with gradually varying cross-section and wall friction is presented. The particles are assumed to be incompressible and in thermomechanical equilibrium with a perfect gas phase, and the effects of their finite volume and of gravity are also taken into account.

The equations of motion are written in a form that allows a theoretical analysis of the behaviour of the solutions to be carried out. In particular, the results of the application to the model of a procedure that permits the identification and the topological classification of the singular points of the trajectories representing, in a suitable phase space, the solutions of the set of equations defining the problem are described. This characterization of the singular points is useful in order to overcome difficulties in the numerical integration of the equations.

Subsequently, a geometrical analysis is carried out which allows a study of the signs of the local variations of the flow quantities, and shows that some unusual behaviour may occur if certain geometrical and fluid dynamic conditions are fulfilled. For instance, in an upward motion it is possible to have a simultaneous decrease of velocity, pressure and temperature, while in a downward flow an increase of all these quantities may be found. It is also shown that conditions exist in which expansion and heating of the mixture may take place simultaneously, both in accelerating and decelerating flows.

The model is applied to the study of upward motion in particular ducts, having converging-diverging and constant-diverging cross-sections; to this end the equations are integrated numerically by using the Mach number as the independent variable. The results show that even limited variations of the duct diameter may give rise to significant qualitative and quantitative variations in the flow conditions inside the duct and in the mass flow rate. Finally, an example is given of a subsonic downward flow in which a simultaneous increase of pressure, temperature and velocity occurs even in the case of a pure perfect gas.

We present an experimental and computational investigation of mixing of a viscoelastic fluid in two-dimensional time-periodic flows generated in an eccentric cylindrical geometry. The objective of the study is to investigate the impact of fluid elasticity on the morphological structures produced by the advection of passive tracers in chaotic flows. The relevant dimensionless numbers that quantify the rheological differences with respect to the Newtonian fluid are the Deborah number (De), defined as the ratio of the fluid timescale to the flow timescale, and the Weissenberg number (We), defined as the product of the fluid timescale and the mean shear rate. The effects of elasticity are investigated in the limit of slow flows, De ≈ 0 and We < 0.1. The experimental window of We is limited to Newtonian behaviour on the low end and the transition to three-dimensional flow on the high end; experiments show that this window is small, 0.02 < We < 0.1. Typical values of the Reynolds number and the Strouhal number are O(0.001) and O(0.1), respectively.

Results from experiments with a constant-viscosity elastic fluid and computations using the upper-convected Maxwell constitutive equation are presented. Even though the streamlines for the elastic flow are nearly indistinguishable from the Newtonian flow, small deviations in the velocity field produce large effects on chaotically advected patterns. Elasticity affects both the asymptotic coverage of a dyed passive tracer and the rate at which the tracer is stretched. In all cases the tracer undergoes exponential stretching, but on a longer timescale as the elasticity increases. According to flow conditions, elasticity might increase or decrease the degree of regularity; however, island symmetry does not seem to be affected. Similar phenomena are observed in both the experiments and computations; therefore, an analysis of the chaotic dynamics of the periodic flow using numerical techniques is possible.

There has been some debate recently on whether the far-wake structure downstream of a cylinder is dependent on, or ‘connected’ with, the precise details of the near-wake structure. Indeed, it has previously been suggested that the far-wake scale and frequency are unconnected with those of the near wake. In the present paper, we demonstrate that both the far-wake scale and frequency are dependent on the near wake. Surprisingly, the characteristic that actually forges a ‘connection’ between the near and far wakes is the sensitivity to free-stream disturbances. It is these disturbances that are also responsible for the regular three-dimensional patterns that may be visualized. Observations of a regular ‘honeycomb’-like three-dimensional pattern in the far wake is found to be caused by an interaction between oblique shedding waves from upstream and large-scale two-dimensional waves, amplified from the free-stream disturbances. The symmetry and spanwise wavelength of Cimbala, Nagib & Roshko's (1988) three-dimensional pattern are precisely consistent with such wave interactions. In the presence of parallel shedding, the lack of a honeycomb pattern shows that such a three-dimensional pattern is clearly dependent on upstream oblique vortex shedding.

With the deductions above as a starting point, we describe a new mechanism for the resonance of oblique waves, as follows. In the case of two-dimensional waves, corresponding to a very small spectral peak in the free stream (fT) interacting (quadratically) with the oblique shedding waves frequency (fK), it appears that the most amplified or resonant frequency in the far wake is a combination frequency fFW = (fK–fT), which corresponds physically with ‘oblique resonance waves’ at a large oblique angle. The large scatter in (fFW/fK) from previous studies is principally caused by the broad response of the far wake to a range of free-stream spectral peaks (fT). We present clear visualization of the oblique wave phenomenon, coupled with velocity measurements which demonstrate that the secondary oblique wave energy can far exceed the secondary two-dimensional wave energy by up to two orders of magnitude. Further experiments show that, in the absence of an influential free-stream spectral peak, the far wake does not resonate, but instead has a low-amplitude broad spectral response. The present phenomena are due to nonlinear instabilities in the far wake, and are not related to vortex pairing. There would appear to be distinct differences between this oblique wave resonance and the subharmonic resonances that have been previously studied in channel flow, boundary layers, mixing layers and airfoil wakes.

In this paper, we investigate to what extent the far-wake ‘signature’ of the near-wake vortex dynamics of a nominally two-dimensional bluff body is affected by the character of the free-stream noise. We confirm the existence of an oblique wave resonance (at frequency, fK–fT), which is caused by nonlinear ‘quadratic’ interactions between primary oblique shedding waves (fK) and secondary two-dimensional waves (fT), which are amplified from free-stream disturbances. In this work, oblique wave resonance is induced by acoustic forcing of two-dimensional waves. The use of acoustic forcing reveals a set of higher-order oblique wave resonances corresponding to frequencies (fK–nfT), where n is an integer. We find from visualization that, even when the secondary two-dimensional waves have the same frequency as the oblique waves, it is the oblique waves that are preferentially amplified. Oblique wave angles up to 74° have been observed. The response of the wake to a large range of forcing frequencies shows a broad region of peak response, centred around F = (fT/fK) = 0.55, and is in reasonable agreement with predictions from linear stability analysis. A similar broad response is found for each of the higher-order oblique wave modes. Simple equations for the oblique waves yield approximate conditions for maximum wake response, with a frequency for peak response given by Fmax = 1/2n = 1/2, 1/4, 1/6,…, and an oblique wave angle given by θmax = 2θK, where θK is the angle of oblique vortex shedding. An increase in forcing amplitude has the effect of bringing the nonlinear wave interactions, leading to oblique wave resonance, further upstream. Paradoxically, the effect of an increase in amplitude (A) of the two-dimensional wave forcing is to further amplify the oblique waves in preference to the two-dimensional waves and, under some conditions, to inhibit the appearance of prominent two-dimensional waves where they would otherwise appear. With a variation in forcing amplitude, the amplitude of oblique wave response is found to be closely proportional to A½. In summary, this investigation confirms the surprising result that it is only through the existence of noise in the free stream that the far wake is ‘connected’ to the near wake.

In this work we consider how nonlinear hydrodynamic effects can lead to a mean force of interaction between two spheres of equal radius a undergoing translational fluctuations parallel or perpendicular to their line of centres. Motivated by amplitudes and Reynolds numbers characteristic of Brownian motion in colloidal systems, nonlinearities due to motion of the boundaries and to inertia throughout the fluid are treated as regular perturbations of the time-dependent Stokes equations. This formulation ultimately leads to a prescription for computing, at leading order, the time-average nonlinear force for the case of pure oscillatory modes – which represents the Fourier decomposition of more general motions. The associated hydrodynamic problems are solved numerically using a least-squares boundary singularity method. Frequency-dependent results over the whole spectrum are presented for a sphere-sphere gap equal to one radius; illustrative calculations are also carried out at other separations. Subsequently we extend the analysis of nonlinear drift to a Langevin equation formulation of the more complex problem of stochastic motion due to thermal fluctuations in the suspending fluid, i.e. Brownian motion. By integrating (numerically) over the spectrum of frequencies, we quantify how the mutual interactions of all translational disturbance modes give rise, on ensemble average, to a stochastic nonlinear force of interaction between the particles. It is particularly interesting that this net interaction – arising from a zero-mean random force – is of O(1) on the Brownian scale kT/a, even though it represents a small O(Re) correction at each frequency of pure oscillations. Finally, we discuss how the presence of stochastic nonlinear drift would lead to non-uniform equilibrium distributions of dilute colloidal suspensions, unless one adds to the random force in the Langevin equation a cancelling non-zero mean component.

The effect of the phase relation (i.e. phase difference and coupling) between the fundamental and subharmonic modes on the transition to turbulence of a mixing layer is investigated. Experiments are conducted to study the development of the subharmonic and fundamental modes under different phase-controlled excitations. Higher-order spectral moments are used to measure phase differences, levels of phase coupling, and energy transfer rates between the two modes at different downstream locations. Local measurements of the wavenumber–frequency spectra are used to examine the phase-speed matching conditions required for efficient energy transfer. The results show that when the phase coupling between the fundamental and the subharmonic is high, maximum subharmonic growth is found to occur at a critical phase difference close to zero. The subharmonic growth is found to result from a resonant parametric interaction between the fundamental and the subharmonic in which phase-speed matching conditions are satisfied. In contrast, when the phase coupling level is low, the phase difference is irregular and varying, the efficiency of parametric interactions is low, phase-speed matching conditions are not met and subharmonic growth is suppressed.

The quasi-laminar model for the transfer of energy to a surface wave from a turbulent shear flow (Miles 1957) is modified to incorporate the wave-induced perturbations of the Reynolds stresses, which are related to the wave-induced velocity field through the Boussinesq closure hypothesis and the ancillary hypothesis that the eddy viscosity is conserved along streamlines. It is assumed that the basic mean velocity is U(z) = (U*/κ)log(z/z0) for sufficiently large z (elevation above the level interface) and that U(z1) [Gt ] U* for kz1 = O(1), where k is the wavenumber. The resulting vorticity-transport equation is reduced, through the neglect of diffusion, to a modification of Rayleigh's equation for wave motion in an inviscid shear flow. The energy transfer to the surface wave, which comprises independent contributions from the critical layer (where U = c, the wave speed) and the wave-induced Reynolds stresses, is calculated through a variational approximation and, independently, through matched asymptotic expansions. The critical-layer component is equivalent to that for the quasi-laminar model. The Reynolds-stress component is similar to, but differs quantitatively from, that obtained by Knight (1977, Jacobs (1987) and van Duin & Janssen (1992). The predicted energy transfer agrees with the observational data compiled by Plant (1982) for 1 [lsim ] c/U* [lsim ] 20, but the validity of the logarithmic profile for the calculation of the energy transfer in the critical layer for c/U* < 5 remains uncertain. The basic model is unreliable (for water waves) if c/U* [lsim ] 1, but this domain is of limited oceanographic importance. It is suggested that Kelvin–Helmholtz instability of air blowing over oil should provide a good experimental test of the present Reynolds-stress modelling and that this modelling may be relevant in other geophysical contexts.

Compressibility effects within decaying isotropic turbulence and homogeneous turbulent shear flow have been studied using direct numerical simulation. The objective of this work is to increase our understanding of compressible turbulence and to aid the development of turbulence models for compressible flows. The numerical simulations of compressible isotropic turbulence show that compressibility effects are highly dependent on the initial conditions. The shear flow simulations, on the other hand, show that measures of compressibility evolve to become independent of their initial values and are parameterized by the root mean square Mach number. The growth rate of the turbulence in compressible homogeneous shear flow is reduced compared to that in the incompressible case. The reduced growth rate is the result of an increase in the dissipation rate and energy transfer to internal energy by the pressure–dilatation correlation. Examination of the structure of compressible homogeneous shear flow reveals the presence of eddy shocklets, which are important for the increased dissipation rate of compressible turbulence.

The flow at an axisymmetric stagnation point is considered when the outer, inviscid flow is oscillatory with zero mean. It is shown that following the commencement of the flow at an initial instant there is a breakdown of the solution, after a finite time, as fluid erupts from the boundary at the stagnation point. This is interpreted as the origin of the jet-like flow observed in the experiments of Amin & Riley (1990). It is demonstrated that an in-plane, torsional motion of the boundary, either steady or oscillatory, can suppress the singular behaviour.

The propagation of plane acoustic waves in smooth pipes and their reflection at open pipe terminations have been studied experimentally. The accuracy of the measurements is determined by comparison of experimental data with results of linear theory for the propagation of acoustic waves in a pipe with a quiescent fluid. The damping and the reflection at an unflanged pipe termination are compared.

In the presence of a fully developed turbulent mean flow the measurements of the damping confirm the results of Ronneberger & Ahrens (1977). In the high-frequency limit the quasi-laminar theory of Ronneberger (1975) predicts accurately the convective effects on the damping of acoustic waves. For low frequencies a simple theory combining the rigid-plate model of Ronneberger & Ahrens (1977) with the theoretical approach of Howe (1984) yields a fair prediction of the influence of turbulence on the shear stress. The finite response time of the turbulence near the wall to the acoustic perturbations has to be taken into account in order to explain the experimental data. The model yields a quasi-stationary limit of the damping which does not take into account the fundamental difference between the viscous and thermal dissipation observed for low frequencies.

Measurements of the nonlinear behaviour of the reflection properties for unflanged pipe terminations with thin and thick walls in the absence of a mean flow confirm the theory of Disselhorst & van Wijngaarden (1980), for the low-frequency limit. It appears however that a two-dimensional theory such as proposed by Disselhorst & van Wijngaarden (1980) for the high-frequency limit underestimates the acoustical energy absorption by vortex shedding by a factor 2.5.

The measured influence of wall thickness on the reflection properties of an open pipe end confirms the linear theory of Ando (1969). In the presence of a mean flow the end correction δ of an unflanged pipe end varies from the value at the high-Strouhal-number limit of δ/a = 0.61, with a the pipe radius, which is close to the value in the absence of a mean flow given by Levine & Schwinger (1948) of δ/a = 0.6133, to a value of δ/a = 0.19 in the low-Strouhal-number limit which is close to the value predicted by Rienstra (1983) of δ/a = 0.26.

The pressure reflection coefficient is found to agree with the theoretical predictions by Munt (1977, 1990) and Cargill (1982b) in which a full Kutta condition is included. The accuracy of the theory is fascinating in view of the dramatic simplifications introduced in the theory. For a thick-walled pipe end and a pipe terminated by a horn the end correction behaviour is similar. It is surprising that the nonlinear behaviour at low frequencies and high acoustic amplitudes in the absence of mean flow does not influence the end correction significantly.

The aero-acoustic behaviour of the pipe end is dramatially influenced by the presence of a horn. In the presence of a mean flow the horn is a source of sound for a critical range of the Strouhal number.

The high accuracy of the experimental data suggests that acoustic measurements can be used for a systematic study of turbulence in unsteady flow and of unsteady flow separation.

We investigate a new mechanism for instability (named divergent instability), characterized by the formation of azimuthal cells, and find it to be a generic feature of three-dimensional steady axisymmetric flows of viscous incompressible fluid with radially diverging streamlines near a planar or conical surface. Four such flows are considered here: (i) Squire–Wang flow in a half-space driven by surface stresses; (ii) recirculation of fluid inside a conical meniscus; (iii) two-cell regime of free convection above a rigid cone; and (iv) Marangoni convection in a half-space induced by a point source of heat (or surfactant) placed at the liquid surface. For all these cases, bifurcation of the secondary steady solutions occurs: for each azimuthal wavenumber m = 2, 3,…, a critical Reynolds number (Re*) exists. The intent to compare with experiments led us to investigate case (iv) in more detail. The results show a non-trivial influence of the Prandtl number (Pr): instability does not occur in the range 0.05 < Pr < 1; however, outside this range, Re*(m) exists and has bounded limits as Pr tends to either zero or infinity. A nonlinear analysis shows that the primary bifurcations are supercritical and produce new stable regimes. We find that the neutral curves intersect and subcritical secondary bifurcation takes place; these suggest the presence of complex unsteady dynamics in some ranges of Re and Pr. These features agree with the experimental data of Pshenichnikov & Yatsenko (Pr = 103).

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t-2.