The effect of an insoluble surfactant on the rheological behaviour of a dilute emulsion is theoretically studied under low-capillary-number conditions. The dynamics depends on three dimensionless time-scale parameters that characterize the strength of the mechanisms that control the magnitude of the distortion of the surfactant distribution on the drop interface. These mechanisms include Marangoni relaxation, drop rotation by the imposed flow, and oscillations of the imposed flow. The interaction of the time scales gives rise to a complex rheological behaviour. The evolution of the system is described by a nonlinear matrix equation derived by expanding the fluid velocity and surfactant distribution in spherical harmonics. Analytical expansions are developed for conditions where the surfactant distribution is only slightly perturbed, which occurs when one of the time-scale parameters is small.

The sound generated by the interaction between a turbulent rotor wake and a stator is modelled by considering the gust response of a cascade of blades in non-uniform, subsonic mean flow. Previous work by Hanson & Horan (1998) that considers a cascade of flat plates at zero incidence is extended to take into account blade geometry and angle of attack. Our approach is based on the work of Peake & Kerschen (1997), who calculate the forward radiation due to the interaction between a single vortical gust and a cascade of flat plates at non-zero angle of attack. The extensions completed in this present paper are two-fold: first we include the effects of small but non-zero camber and thickness; and second we produce uniformly valid approximations which predict the upstream radiation near modal cut-off. The thin-airfoil singularity in the steady flow at each leading edge is crucial in our model of the sound generation. A new analytical expression for the coefficient of this singularity is derived via a sequence of conformal mappings, and it turns out that in our asymptotic limit this is the only quantity which needs to be calculated from the steady flow in order to predict time-averaged noise levels. Once the response to a single gust has been completed, we use Hanson & Horan (1998)'s approach to determine the response to an incident turbulent spectrum, and find that as well as the noise corresponding to the auto-correlation of the gust velocity component normal to the blade, there is also a contribution from the cross-correlation of the normal and tangential velocities. Predictions are made of the effects of blade geometry on the upstream acoustic power level. The blade geometry can have a very significant effect on the noise generated by interaction with a single gust, with changes of up to 10 dB from the flat-plate noise levels. However, once these gust results have been integrated over a full incident turbulence spectrum the effects of the geometry are rather smaller, although still potentially significant, leading to changes of up to about 2 dB from the flat-plate results. The implication of all this is that the blade geometry can have a significant effect on the tonal noise components generated by rotor–stator interaction (i.e. by single harmonic gusts), but that the broadband part of the noise spectrum is relatively unaffected.

Direct numerical simulations of ten turbulent time-evolving strained wakes have been generated using a pseudo-spectral numerical method. In all the simulations, the strain was applied to the same (previously generated) initial developed self-similar wake flow field. The cases include flows in which the wake is subjected to various orientations of the applied mean strain, including both plane and axisymmetric strain configurations. In addition, for one particular strain geometry, cases with differing strain rates were considered. Although classical self-similar analysis does yield a self-similar solution for strained wakes, this solution does not describe the observed flow evolution. Instead, the wake mean velocity profiles evolve according to a different ‘equilibrium similarity solution’, with the strained wake width being determined by the straining in the inhomogeneous cross-stream direction. Wakes that are compressed in this direction eventually exhibit constant widths, whereas wakes in cases with expansive cross-stream strain ultimately spread at the same rate as the distortion caused by the applied strain. The shape of the wake mean velocity deficit profile is nearly universal. Although the effect of the strain on the mean flow is pronounced and rapid, the response of the turbulence to the strain occurs more slowly. Changes in the turbulence intensity cannot keep pace with changes in the mean wake velocity deficit, even for relatively low strain rates.

Radar has been proposed as a way of tracking wake vortices to reduce aircraft spacing and tests have revealed radar echoes from aircraft wakes in clear air. The mechanism causing refractive index gradients in these tests is thought to be the same as that for homogeneous and isotropic atmospheric turbulence in the Kolmogorov inertial range, for which there is a scattering analysis due to Tatarski. In reality, however, the structure of aircraft wakes has a significant coherent part superimposed with turbulence, about whose structure very little is known. This work adopts a picture of a coherent (in fact two-dimensional) wake to perform a scattering analysis and calculate the reflected power. In particular, two simple mechanisms causing refractive index gradients are considered: (A) radial pressure (and therefore density) gradient in a columnar vortex arising from the rotational flow; (B) adiabatic transport of atmospheric fluid within a descending oval surrounding a vortex pair. In the scattering analysis, Tatarski's weak scattering approximation is kept but the usual assumptions of a far field and a uniform incident wave are dropped. Neither assumption is generally valid for a wake that is coherent across the radar beam. For analytical insight, an approximate analysis that invokes, in addition to weak scattering, the far-field and wide cylindrical beam assumptions, is also developed and compared with the more general analysis. Reflectivities calculated for the oval (mechanism B) are within 2–13 dB m2 of the measurements (≈−70 dB m2) of MIT Lincoln Laboratory at Kwajalein atoll. However, the present predictions have a cut-off away from normal incidence which is not present in the measurements. This implies that the two-dimensional picture is not entirely complete. Estimates suggest that the thin layer of vorticity which is baroclinically generated at the boundary of the oval is turbulent and this may account for reflectivity away from normal incidence. The reflectivity of a vortex (mechanism A) is comparable to that of the oval (mechanism B) but occurs at a frequency (about 50 MHz) that is lower than those considered in all the experiments to date. This result may be useful because: (i) existing atmospheric radars (known as ST radars) already operate at this frequency and so the present prediction could be verified; (ii) rain clutter is not a problem at this frequency; (iii) mechanism A is more robust because it is independent of atmospheric stratification.

The stability of a falling liquid curtain is investigated. The sheet of liquid is assumed two-dimensional, driven by gravity and influenced by a compressible cushion of air enclosed on one side of the curtain. The linear stability problem is formulated in the form of an integro-differential eigenvalue problem. Although experimental efforts have consistently reported a peak in the low-frequency range of the spectrum, the linear stability results do not show instabilities at these frequencies. However, a multi-modal approach combined with a projection onto low-frequency modes reveals a dominant and robust instability feature that is in good agreement with experimental measurements. This instability manifests itself as a wave packet, consisting of a linear superposition of linear global modes, that travels down the curtain and causes a strong pressure signal in the enclosed air cushion.

Laminar boundary-layer separation in the supersonic flow past a corner point on a rigid body contour, also termed the compression ramp, is considered based on the viscous–inviscid interaction concept. The ‘triple-deck model’ is used to describe the interaction process. The governing equations of the interaction may be formally derived from the Navier–Stokes equations if the ramp angle θ is represented as θ = θ0Re−1/4, where θ0 is an order-one quantity and Re is the Reynolds number, assumed large. To solve the interaction problem two numerical methods have been used. The first method employs a finite-difference approximation of the governing equations with respect to both the streamwise and wall-normal coordinates. The resulting algebraic equations are linearized using a Newton–Raphson strategy and then solved with the Thomas-matrix technique. The second method uses finite differences in the streamwise direction in combination with Chebychev collocation in the normal direction and Newton–Raphson linearization.

Our main concern is with the flow behaviour at large values of θ0. The calculations show that as the ramp angle θ0 increases, additional eddies form near the corner point inside the separation region. The behaviour of the solution does not give any indication that there exists a critical value θ*0 of the ramp angle θ0, as suggested by Smith & Khorrami (1991) who claimed that as θ0 approaches θ*0, a singularity develops near the reattachment point, preventing the continuation of the solution beyond θ*0. Instead we find that the numerical solution agrees with Neiland's (1970) theory of reattachment, which does not involve any restriction upon the ramp angle.

The objective of the present numerical study is to increase mixing in turbulent flow behind a backward-facing step using a systematic feedback control method. Spatially and temporally varying blowing and suction with zero-net mass flow rate are provided at the step edge, based on the sensing of the spanwise distribution of the wall pressure fluctuations at a downstream location. The cost functional to be increased is the root-mean-square spanwise pressure-gradient fluctuations at the sensing location, which may be associated with mixing behind the backward-facing step. Given the cost functional, the actuation at the step edge is determined through the suboptimal feedback control procedure of Choi et al. (1993). Large-eddy simulations of turbulent flow are conducted at a Reynolds number of 5100 based on the step height and free-stream velocity. The results of suboptimal feedback controls are compared with those of non-feedback single-frequency actuations. In case of the suboptimal control, velocity and vorticity fluctuations substantially increase downstream of the backward-facing step as well as in the recirculation zone. As a result, the reattachment length is significantly reduced, as compared to those of uncontrolled flow and flow with single-frequency actuations. A simple open-loop control method is devised from the suboptimal feedback control result, producing nearly the same mixing enhancement as the feedback control.

The instability of a rotating-magnetic-field-driven liquid metal flow in a finite cylinder with respect to infinitesimal azimuthally periodic perturbations is studied numerically. This instability is observed to set in prior its axisymmetric counterpart with relatively low frequency at diameter-to-height ratios between 0.5 and 2. The axisymmetric and three-dimensional instabilities have similar characteristic features. The instability originates in the cross-section of the horizontal and vertical rotating boundary layers and excites inertial waves in the inviscid core.

We focus on the early evolution of energy E and enstrophy Z when the dissipation grows in significance from negligible to important. By considering a sequence of viscous shell model solutions we find that both energy and dissipation are continuous functions of time in the inviscid limit. Inviscidly, Z takes only a finite time t* to diverge, where t* depends on initial conditions. For viscous solutions, Z peaks long after t*, but the inflection point for Z(t) provides an excellent approximation to t*. Near t*, all of our high Reynolds number solutions obey the formula ναdZ/dt = F(νβZ). Neither the function F nor the constants α and β depend on initial conditions. We use F to obtain the inviscid limit. The energy spectrum remains concave down on double logarithmic scales until t*. At t*, the spectrum becomes algebraic at high wavenumbers, i.e. E(k, t*) ∼ C0kα. Crucially, the spectral slope σ is steeper than −5/3. Thus, we conclude that the inviscid singularity at t* is not associated with the establishment of a semi-infinite Kolmogorov range. For viscous solutions, the −5/3 range builds gradually after t* starting from high wavenumbers, and Z peaks when the inertial range reaches the integral scale. Thus, the formation of the inertial range is a viscous process in our shell model.

This paper describes an exploration of the behaviour and properties of swirling flow through a constant-diameter pipe. The experiments reveal a complicated transition process as the swirl intensity Ω is increased at fixed pipe Reynolds number Re ≈ 4900. For Ω [les ] 1.09, the vortex was steady, laminar, axisymmetric, and developed slowly with streamwise distance. The upstream velocity profiles were similar to those commonly appearing in the literature in similar apparatus. Spiral vortex breakdown appeared in the test section for 1.09 [les ] Ω [les ] 1.31 and was associated with a localized transition from jet-like to wake-like mean axial velocity profiles. Further increase in Ω caused the breakdown to move upstream of the test section. Downstream, the core of the post-breakdown flow was unsteady and recovered toward jet-like profiles with streamwise distance. At Ω = 2.68, a global transition occurred in which the mean axial velocity profiles suddenly developed an annular axial velocity deficit. At the same time, disturbances began to appear in the outer flow. Further increase in Ω eventually led to an annulus of reversed axial flow and a completely unsteady vortex.

A model of single-room displacement ventilation is extended to a space consisting of two chambers of equal height connected by two openings. Individually, both chambers have displacement ventilation in this geometrical arrangement, but the space itself is not connected to the outside. Thus we are considering ventilation of two chambers in the interior of a building, such as an office connected to an internal atrium. Theoretical analysis and experimental results are presented in this paper. The experiments use salt solutions to simulate thermal forcing in buildings and the theoretical analysis is based on plume theory. The two chambers have a time-dependent interaction resulting from changing stratification in the two chambers. We concentrate here on a small chamber with an internal heat source connected to a large unheated chamber, and show that the time variation is determined by the size of the larger chamber. We discuss the implications of these results for building ventilation design and control.

Experimental studies have shown that convergent–divergent nozzles, when run at low pressure ratios, often undergo a flow resonance accompanied by emission of acoustic tones. The phenomenon, different in characteristics from conventional ‘screech’ tones, is addressed in this paper. Unlike screech, the resonant frequency (fN) increases with increasing supply pressure. There is a ‘staging’ behaviour; odd-harmonic stages resonate at lower pressures while the fundamental occurs in a wide range of higher pressures corresponding to a ‘fully expanded Mach number’ (Mj) around unity. Within a stage, fN varies approximately linearly with Mj; the slope of the variation steepens when the angle of divergence of the nozzle is decreased. Based on the data, correlation equations are provided for the prediction of fN. A companion computational study captures the phenomenon and predicts the frequencies, including the stage jump, quite well. While the underlying mechanisms are not completely understood yet, it is clear that the unsteadiness of a shock occurring within the divergent section plays a direct role. The shock drives the flow downstream like a vibrating diaphragm, and resonance takes place similarly to the (no-flow) acoustic resonance of a conical section having one end closed and the other end open. Thus, the fundamental is accompanied by a standing one-quarter wave within the divergent section, the next stage by a standing three-quarter wave, and so on. The distance from the foot of the shock to the nozzle exit imposes the pertinent length scale. The principal trends in the frequency variation are explained qualitatively from the characteristic variation of that length scale. A striking feature is that tripping of the nozzle's internal boundary layer tends to suppress the resonance. It is likely that the trip effect occurs due to a break in the azimuthal coherence of the unsteady flow.

We derive the asymptotic spectrum (as the Ekman number E → 0) of axisymmetric inertial modes when the problem is restricted to two dimensions. We show that the damping rate of such modes scales with the square root of the Ekman number and that the width of the shear layers of the eigenfunctions scales with E1/4. The eigenfunctions obey a Schrödinger equation with a quadratic potential; we provide the analytical expression for eigenvalues (frequency and damping rate). These results validate the picture that attractors act like a potential well, trapping inertial waves which resist confinement owing to viscosity. Using three-dimensional numerical solutions, we show that the results can be applied to equatorially trapped modes in a thin spherical shell; in fact, these two-dimensional solutions give the first step (the zeroth order) of a perturbative approach to three-dimensional solutions in a spherical shell. Our method is applicable in a straightforward way to any other container where bi-dimensionality dominates.

Taylor–Couette flow in the presence of a magnetic field is a problem belonging to classical hydromagnetics and deserves to be more widely studied than it has been to date. In the nonlinear regime the literature is scarce. We develop a formulation suitable for solution of the full three-dimensional nonlinear hydromagnetic equations in cylindrical geometry, which is motived by the formulation for the magnetic field. It is suitable for study at finite Prandtl numbers and in the small Prandtl number limit, relevant to laboratory liquid metals. The method is used to determine the onset of axisymmetric Taylor vortices, and finite-amplitude solutions. Our results compare well with existing linear and nonlinear hydrodynamic calculations and with hydromagnetic experiments.

Spatio-temporal characteristics of wall pressure fluctuations in separated and reattaching flows over a backward-facing step were investigated through an extensive pressure-velocity joint measurement with an array of microphones. The experiment was performed in a wind tunnel with a Reynolds number of 33 000 based on the step height and the free-stream velocity. Synchronized wavelet maps showed the evolutionary behaviour of pressure fluctuations and gave further insight into the modulated nature of large-scale vortical structures. To see the relationship between the flow field and the relevant spatial mode of the pressure field, a new kind of wavenumber filtering, termed ‘spatial box filtering’ (SBF), was introduced and examined. The vortical flow field was reconstructed using every single-point velocity measurement by means of the conditional average based on the SBF second mode of pressure fluctuations. The flow field showed a well-organized spanwise vortical structure convected with a speed of 0.6U0 and a characteristic ‘sawtooth’ pattern of the unsteady trace of reattachment length. In addition to the coherent vortical structures, the periodic enlargement/shrinkage process of the recirculation region owing to apping motion was analysed. The recirculation region was found to undergo an enlargement/shrinkage cycle in accordance with the lowpass-filtered component of pressure fluctuations. In addition, such modulatory behaviour of the vortical structure as the global oscillation phase was discussed in connection with the conditionally averaged flow field.

Traditionally, trends of universal turbulence statistics are presented versus Rλ, which is the Reynolds number based on Taylor's scale λ and the root-mean-square (rms) of a component of velocity urms. λ and urms, and hence Rλ, do not have the attribute of universality. The ratio of rms fluid-particle acceleration to the rms of the acceleration caused by the viscous force, Ra, is an alternative to Rλ. Ra is a Reynolds number. It is composed of statistics of the small scales of turbulence. It can be evaluated with single-wire hot-wire anemometry. Like Rλ, it can be partially evaluated by means of flow similarity. Direct measurement of Ra is challenging; therefore, Ra is not a replacement for Rλ. For isotropic turbulence the relationship between Ra and Rλ is given. Anisotropic turbulence is discussed.