A theory is proposed to describe the generation of sound by turbulence at high Mach numbers. The problem is formulated most conveniently in terms of the fluctuating pressure, and a convected wave equation (2.8) is derived to describe the generation and propagation of the pressure fluctuations.

The supersonic turbulent shear zone is examined in detail. It is found that, at supersonic speeds, sound is radiated as eddy Mach waves, and as the Mach number increase, this mechanism of generation becomes increasingly dominant. Attention is concentrated on the properties of the pressure fluctuations just outside the shear zone where the interactions among the weak shock waves have had little effect. An asymptotic solution for large M is derived by a Green's function technique, and it is found that radiation with given frequency n and weve-number K can be associated with a coresponding critical layer within the shear zone.

It is found that $(\overline {p - p_0})^2 $ increases approximately as $\rm {M}^{\frac {3} {2}}$ for M [Gt ] 1 contrasting with the M8 variation found by Lightill for M [Lt ] 1. The acoustic efficiency thus varies as $\rm {M}^-{\frac {3} {2}}$ for M [Gt ] 1, and as M5 for M [Lt ] 1, indicating a maximum acoustic efficiency for Mach numbers near one. The directional distribution of the radiation is discussed and the direction of maximum intensity is shown to move towards the perpendicular to the shear zone as M increases. The predictions of the theory are supported qualitatively by the few available experimental observations.

The effect of Mach number on the evolution of instabilities in the compressible mixing layer is investigated. The full time-dependent compressible Navier–Stokes equations are solved numerically for a temporally evolving mixing layer using a mixed spectral and high-order finite difference method. The convective Mach number Mc (the ratio of the velocity difference to the sum of the free-stream sound speeds) is used as the compressibility parameter. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers (Mc > 0.6) oblique waves grow more rapidly than two-dimensional waves. Simulations are then presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves. A change in the developed large-scale structure is observed as the Mach number is increased, with vortical regions oriented in a more oblique manner at the higher Mach numbers. At convective Mach numbers above unity the two-dimensional instability is found to have little effect on the flow development, which is dominated by the oblique instability waves. The nonlinear structure which develops from a pair of equal and opposite oblique instability waves is found to resemble a pair of inclined A-vortices which are staggered in the streamwise direction. A fully nonlinear computation with a random initial condition shows the development of large-scale structure similar to the simulations with forcing. It is concluded that there are strong compressibility effects on the structure of the mixing layer and that highly three-dimensional structures develop from the primary inflexional instability of the flow at high Mach numbers.

The interaction of solid particles with the temporal features of a turbulent flow has direct relevance to problems in particle and spray combustion and the processing of particulate solids. The object of the present study was to examine the behaviour of particles in a jet dominated by vortex ring structures. An axisymmetric air jet laden with 55 μm glass particles was forced axially with an acoustic speaker to organize the vortex ring structures rolling up in the free shear layer downstream of the nozzle exit. Visualization studies of forced and unforced flow with Reynolds number of the order of 20000 were completed using a pulsed copper vapour laser. Instantaneous photographs and videotapes of strobed forced flow show that particles become clustered in the saddle regions downstream of the vortex rings and are propelled away from the jet axis by the outwardly moving flow in these regions. Phase-averaged spatial distribution of particle number density computed from digitized photographs and phase-averaged particle velocity measurements yield further evidence that local particle dispersion and concentration are governed by convection due to large-scale turbulence structures. The large-scale structures and convection mechanisms were shown to persist for particle-to-air mass loading ratios up to 0.65.

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

Previous measurements of the decay rate of the fluctuation intensity of passive scalars in grid-generated turbulence show large variation. New results presented here show that the decay rate of passive temperature fluctuations produced by heating the grid is a function of the initial temperature fluctuation intensity. Although a full reason for this is wanting, spectra of the temperature fluctuations show that, by varying the heat applied to the grid, the wavenumber of the maximum in the temperature spectrum changes, indicating that the geometry of the thermal fluctuations is being altered in some way. In these experiments the one-dimensional temperature spectrum shows an anomalous $-\frac{5}{3}$ slope. In order to eliminate the dependence of the decay rate of the temperature fluctuations on their intensity, we describe a new way of generating temperature fluctuations by means of placing a heated parallel array of fine wires (a mandoline) downstream from the unheated grid. Results of this experiment show that the decay rate of passive thermal fluctuations is uniquely determined by the wave-number of the initial temperature fluctuations. In this type of flow there appears to be no equilibrium value for the thermal fluctuation decay rate and hence for the mechanical/thermal time-scale ratio since the thermal fluctuation decay rate does not change within the tunnel length, which is the equivalent of nearly one turbulence decay time.

The modulation of isotropic turbulence by particles has been investigated using direct numerical simulation (DNS). The particular focus of the present work is on the class of dilute flows in which particle volume fractions and inter-particle collisions are negligible. Gravitational settling is also neglected and particle motion is assumed to be governed by drag with particle relaxation times ranging from the Kolmogorov scale to the Eulerian time scale of the turbulence and particle mass loadings up to 1. The velocity field was made statistically stationary by forcing the low wavenumbers of the flow. The calculations were performed using 963 collocation points and the Taylor-scale Reynolds number for the stationary flow was 62. The effect of particles on the turbulence was included in the Navier–Stokes equations using the point-force approximation in which 963 particles were used in the calculations. DNS results show that particles increasingly dissipate fluid kinetic energy with increased loading, with the reduction in kinetic energy being relatively independent of the particle relaxation time. Viscous dissipation in the fluid decreases with increased loading and is larger for particles with smaller relaxation times. Fluid energy spectra show that there is a non-uniform distortion of the turbulence with a relative increase in small-scale energy. The non-uniform distortion significantly affects the transport of the dissipation rate, with the production and destruction of dissipation exhibiting completely different behaviours. The spectrum of the fluid–particle energy exchange rate shows that the fluid drags particles at low wavenumbers while the converse is true at high wavenumbers for small particles. A spectral analysis shows that the increase of the high-wavenumber portion of the fluid energy spectrum can be attributed to transfer of the fluid–particle covariance by the fluid turbulence. This in turn explains the relative increase of small-scale energy caused by small particles observed in the present simulations as well as those of Squires & Eaton (1990) and Elghobashi & Truesdell (1993).

Upper bounds for the heat flux through a horizontally infinite layer of fluid heated from below are obtained by maximizing the heat flux subject to (a) two integral constraints, the ‘power integrals’, derived from the equations of motion, and (b) the continuity equations. This variational problem is solved completely, for all values of the Rayleigh number R, when only the constrains (a) are imposed, and it is thus shown that the Nusselt number N for any statistically steady convective motion cannot exceed a certain value N1(R), which for large R is approximately (3R/64)½. When (b) is included as a constraint, the variational problem is solved for large R, under the additional hypothesis that the solution has a single horizontal wave number; the associated upper bound on the Nusselt number is (R–248)⅜. The mean properties of this maximizing ‘flow’, in particular the mean temperature and mean square temperature deviation fields, are found to resemble the mean properties of the real flow observed by Townsend; the results thus tend to support Malkus's hypothesis that turbulent convection maximizes heat flux.

This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Kármán vortex street regime that sets in for supercritical Reynolds numbers Re > 46. Barkley (Europhys. Lett. vol.75, 2006, p. 750) has recently studied the linear properties of the associated mean flow, i.e. the flow which is obtained by a time average of this unsteady periodic flow. He observed, thanks to a global mode analysis, that the mean flow is marginally stable and that the eigenfrequencies associated with the global modes of the mean flow fit the Strouhal to Reynolds experimental function well in the range 46 < Re < 180. The aim of this article is to give a theoretical proof of this result near the bifurcation. For this, we do a global weakly nonlinear analysis valid in the vicinity of the critical Reynolds number Rec based on the small parameter ε = Rec−1 − Re−1 ≪ 1. We compute numerically the complex constants λ and μ′ which appear in the Stuart-Landau amplitude equation: dA/dt = ε λA − εμ′ A|A|2. Here A is the scalar complex amplitude of the critical global mode. By analysing carefully the nonlinear interactions yielding the term μ′, we show for the cylinder flow that the mean flow is approximately marginally stable and that the linear dynamics of the mean flow yields the frequency of the saturated Stuart-Landau limit cycle. We will finally show that these results are not general, by studying the case of the bifurcation of an open cavity flow. In particular, we show that the mean flow in this case remains strongly unstable and that the frequencies associated with the eigenmodes do not exactly match those of the nonlinear unsteady periodic cavity flow. It will be demonstrated that two precise conditions must hold for a linear stability analysis of a mean flow to be relevant and useful.

The relationship between the upstream boundary layer and the low-frequency, large-scale unsteadiness of the separated flow in a Mach 2 compression ramp interaction is investigated by performing wide-field particle image velocimetry (PIV) and planar laser scattering (PLS) measurements in streamwise–spanwise planes. Planar laser scattering measurements in the upstream boundary layer indicate the presence of spanwise strips of elongated regions of uniform momentum with lengths greater than 40δ. These long coherent structures have been observed in a Mach 2 supersonic boundary layer (Ganapathisubramani, Clemens & Dolling 2006) and they exhibit strong similarities to those that have been found in incompressible boundary layers (Tomkins & Adrian 2003; Ganapathisubramani, Longmire & Marusic 2003). At a wall-normal location of y/δ=0.2, the inferred instantaneous separation line of the separation region is found to oscillate between x/δ=−3 and −1 (where x/δ=0 is the ramp corner). The instantaneous spanwise separation line is found to respond to the elongated regions of uniform momentum. It is shown that high- and low-momentum regions are correlated with smaller and larger size of the separation region, respectively. Furthermore, the instantaneous separation line exhibits large-scale undulations that conform to the low- and high-speed regions in the upstream boundary layer. The low-frequency unsteadiness of the separation region/shock foot observed in numerous previous studies can be explained by a turbulent mechanism that includes these elongated regions of uniform momentum.

We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.

Experimental and theoretical studies were undertaken to investigate the particulatefree secondary flow interaction in the wire-type electrostatic precipitator. The secondary flow generated by the corona discharge is not negligible, and strong flow interactions take place owing to the induced circulatory cells. The calculated numerical results demonstrate close agreement with experiment.

Experiments on transition to turbulence in a purely oscillatory pipe flow were performed for values of the Reynolds number Rδ, defined using the Stokes-layer thickness δ = (2ν/ω)½ and the cross-sectional mean velocity amplitude Û, from 19 to 1530 (or for values of the Reynolds number Re, defined using the pipe diameter d and Û, from 105 to 5830) and for values of the Stokes parameter λ = ½d(ω/2ν)½ (ν = kinematic viscosity and ω = angular frequency) from 1·35 to 6·19. Three types of turbulent flow regime have been detected: weakly turbulent flow, conditionally turbulent flow and fully turbulent flow. Demarcation of the flow regimes is possible on Rλ, λ or Re, λ diagrams. The critical Reynolds number of the first transition decreases as the Stokes parameter increases. In the conditionally turbulent flow, turbulence is generated suddenly in the decelerating phase and the profile of the velocity distribution changes drastically. In the accelerating phase, the flow recovers to laminar. This type of partially turbulent flow persists even at Reynolds numbers as high as Re = 5830 if the value of the Stokes parameter is high.

This paper describes the flow structure observed over a bell-shaped hill with height h (the profile of which is the reciprocal of a fourth-order polynomial) when it was placed first in a large towing tank containing stratified saline solutions with uniform stable density gradients and second in an unstratified wind tunnel. (A similarly shaped model hill was also studied in a small towing tank.) Observations were made at values of the Froude number F (≃ U/Nh) in the range 0·1 to 1·7 and at F = ∞, where U is the towing speed and N is the Brunt-Väisälä frequency, and at values of the Reynolds number from 400 to 275000. For F ≲ 0·4, the observations verify Drazin's (1961) theory for low-Froude-number flow over three-dimensional obstacles and establish limits of applicability. For Froude numbers of the order of unity, it is found that a classification of the lee-wave patterns and separated-flow regions observed in two-dimensional flows also appears to apply to three-dimensional hills.

Flow-visualization techniques were used extensively in obtaining both qualitative and quantitative information on the flow structure around the hill. Representative photographs of dye tracers, potassium permanganate dye streaks, shadowgraphs, surface dye smears, and hydrogen-bubble patterns are included here. While emphasis is centred on obtaining a basic understanding of the flow around three-dimensional hills, the results are applicable to the estimation of air pollutant dispersion around hills.

A two-dimensional analysis, based on linear surface-wave theory, is developed for an oscillating-water-column wave-energy device in water of arbitrary constant depth. The immersed part of the structure is assumed of shallow draught except for a submerged vertical reflecting wall. Both the cases of linear and nonlinear power take-off are considered. The results show that air compressibility can be important in practice, and its effects may in general be satisfactorily represented by linearization. The analysis indicates that using a turbine whose characteristic exhibits a phase difference between pressure and flow rate may be a method of strongly reducing the chamber length and turbine size with little change in the capability of energy extraction from regular waves. It was found in two examples of devices with strongly nonlinear power take-off that the maximum efficiency is only marginally inferior to what can be achieved in the linear case.

It is shown that the pressure signal measured at the outer edge of a jet mixing layer is entirely hydrodynamic in nature and provides a good measure of the large-scale structure of the turbulent flow. Measurement of the pressure signal provides a unique opportunity to utilize proper orthogonal decomposition (POD) to deduce the streamwise structure. Since pressure is a scalar, a significant reduction in the numerical and experimental complexity inherent in the analysis of velocity vector fields results.

The POD streamwise eigenfunctions show that the structure associated with any frequency–azimuthal mode number combination displays the general characteristics of amplification–saturation–decay of an instability wave, all within about three wavelengths. High-frequency components saturate early in x and low-frequency components saturate further downstream, indicative of the inhomogeneous character of the flow in the streamwise direction. Application of the POD technique allows the phase velocity to be determined taking into account the inhomogeneity of the flow in the streamwise direction. The phase velocity of each instability wave (POD eigenvector) is constant and equal to 0.58Uj, indicating that the jet structure is non-dispersive.

Using the shot-noise decomposition, a characteristic event is constructed. This event is found to contain evidence of both pairings and triplings of vortex structures. The tripling results in a rapid increase in the first asymmetric (m=1) component. On average, pairing occurs once every four Uj/D while tripling occurs once every 13Uj/D.

The stability of non-Newtonian jets was investigated. A linearized stability analysis shows that a liquid column of a viscoelastic fluid exhibits more rapid growth of axisymmetric wave disturbances than a Newtonian fluid of the same zero shear viscosity. This result is independent of the form of constitutive equation chosen. Experiments in weakly elastic fluids confirm this expectation, whereas data on fluids with more pronounced elastic properties indicate that non-linear phenomena are dominating. The disturbances appear as a series of droplets connected by random lengths of threads, which thin with distance and eventually lead to jet breakup. Even in dilute viscoelastic solutions, jet breakup does not occur by the growth of clearly defined waves.

The suggestion made by the authors in a previous paper (Hurle & Jakeman 1969) that the Soret effect could give rise to overstable solutions of the thermosolutal Rayleigh–Jeffreys problem is investigated theoretically and experimentally.

Oscillatory instability is shown to occur in initially homogeneous layers of water-methanol mixtures when they are heated from below. This instability triggers a finite-amplitude steady mode. The magnitude and sign of the Soret coefficient was changed by varying the composition of the mixture; as predicted, overstable modes were observed when the sign of the coefficient was such as to produce a stabilizing contribution to the density gradient. The observed critical Rayleigh numbers and temporal frequencies are consistent with theory.

Weis-Fogh (1973) proposed a new mechanism of lift generation of fundamental interest. Surprisingly, it could work even in inviscid two-dimensional motions starting from rest, when Kelvin's theorem states that the total circulation round a body must vanish, but does not exclude the possibility that if the body breaks into two pieces then there may be equal and opposite circulations round them, each suitable for generating the lift required in the pieces’ subsequent motions! The ‘fling’ of two insect wings of chord c (figure 1) turning with angular velocity Ω generates irrotational motions associated with the sucking of air into the opening gap which are calculated in § 2 as involving circulations −0·69Ωc2 and + 0.69Ωc2 around the wings when their trailing edges, which are stagnation points of those irrotational motions, break apart (position (f)). Viscous modifications to this irrotational flow pattern by shedding of vorticity at the boundary generate (§ 3) a leading-edge separation bubble, and tend to increase slightly the total bound vorticity. Its role in a three-dimensional picture of the Weis-Fogh mechanism of lift generation, involving formation of trailing vortices at the wing tips, and including the case of a hovering insect like Encarsia formosa moving those tips in circular paths, is investigated in § 4. The paper ends with the comment that the far flow field of such very small hovering insects should take the form of the exact solution (Landau 1944; Squire 1951) of the Navier-Stokes equations for the effect of a concentrated force (the weight mg of the animal) acting on a fluid of kinematic viscosity v and density p, whenever the ratio mg/pv2 is small enough for that jet-type induced motion to be stable.

A chemical explosive mode analysis (CEMA) was developed as a new diagnostic to identify flame and ignition structure in complex flows. CEMA was then used to analyse the near-field structure of the stabilization region of a turbulent lifted hydrogen–air slot jet flame in a heated air coflow computed with three-dimensional direct numerical simulation. The simulation was performed with a detailed hydrogen–air mechanism and mixture-averaged transport properties at a jet Reynolds number of 11000 with over 900 million grid points. Explosive chemical modes and their characteristic time scales, as well as the species involved, were identified from the Jacobian matrix of the chemical source terms for species and temperature. An explosion index was defined for explosive modes, indicating the contribution of species and temperature in the explosion process. Radical and thermal runaway can consequently be distinguished. CEMA of the lifted flame shows the existence of two premixed flame fronts, which are difficult to detect with conventional methods. The upstream fork preceding the two flame fronts thereby identifies the stabilization point. A Damköhler number was defined based on the time scale of the chemical explosive mode and the local instantaneous scalar dissipation rate to highlight the role of auto-ignition in affecting the stabilization points in the lifted jet flame.