A generation process for the three-dimensional wave which dominates the transition preceded by a Tollmien-Schlichting (T-S) wave is studied both experimentally and numerically in plane Poiseuille flow at a subcritical Reynolds number of 5000. In order to identify the origin of the three-dimensional wave in Nishioka et al.'s laboratory experiment, the corresponding spanwise mean-flow distortion and two-dimensional T-S wave modes are introduced into a parabolic flow as the initial disturbance conditions for a numerical simulation of temporally growing type. Through reproducing the actual wave development into the peak-valley structure, the simulation pinpoints the origin to be the slight spanwise mean-flow distortion in the experimental basic flow. Furthermore, the simulation clearly shows that the growth of the three-dimensional wave requires the vortex stretching effect due to the streamwise vortices, which appear under the experimental conditions only when the amplitude of the two-dimensional T-S wave is above the observed threshold.

Measurements of the rapidly changing gaseous composition in engines at low speed can be made via narrow tubes which convey the gases to monitoring equipment in a less hostile environment. This paper quantifies the extent to which the tube smooths out any changes in concentration. Exact (and approximate) formulae are derived for the temporal variance as weighted double (and single) integrals of the steady flow properties along the tube. Such is the non-uniformity that typically the region near the engine contributes 100 times as much to the spreading as does the region near the monitoring equipment. The advantages of keeping the sampling tubes short and heated are made explicit.

The effect of flow oscillation on the stability of plane channel flow is studied via numerical simulation. For weak oscillation, the ratio of the Stokes layer thickness to the distance from the wall to the critical layer in steady flow provides the best normalization for the mean-flow frequency. Maximum growth rates occur when the instantaneous velocity profile has large regions of positive curvature. The effect of oscillation is generally stabilizing. However, at low frequencies, TS wave energies may vary by 106 in a cycle and irreversible secondary instability may be produced at the peak.

The growth of linear disturbances in the high-Reynolds-number laminar wake of a flat plate aligned with a uniform stream is investigated. The theory is developed rationally by use of appropriate wake profiles which originate at the trailing edge as double Blasius distributions and thereafter satisfy the equations of motion, in contrast to previous theoretical work where model profiles are used. We also emphasize the structures and scales of the instability in order to provide a rational basis for the development of nonlinear analyses as opposed to existing ad hoc ones. Disturbances, in the near wake, respond according to the Rayleigh equation which is considered analytically for short-, long- and neutral-wave solutions. For more general stability characteristics eigensolutions must be obtained numerically. We calculate these at successive wake stations for ‘improved’ basic flow profiles which are obtained as solutions of the wake boundary-layer equations. Our numerical results indicate fairly good agreement with the asymptotic theory and some experimental data (see §7).

The local-energy-transfer (LET) theory was used to calculate freely decaying turbulence for arbitrary initial conditions over a range of microscale-based Reynolds numbers 0.5 [les ] Rλ(tf) [les ] 1009, where tf is the final time of computation. The predicted skewness factor S(Rλ) agreed closely with the results of numerical simulations at low-to-moderate Reynolds numbers and followed the same general trend at larger values of Rλ. It was also found that, for Rλ(tf) [les ] 5, the LET calculation was almost indistinguishable from that of the direct-interaction approximation (DIA), with the difference between the two theories tending to zero as Rλ(tf)∞ 0.

Two-time correlation and propagator (or response) functions were also obtained. Tests of their scaling behaviour suggest that, contrary to general belief, the convective sweeping of the energy-containing range is much less important than the Kolmogorov timescale in determining inertial-range behaviour. This result raises questions about the accepted explanation for the failure of the direct-interaction approximation, thus motivating a discussion about the relevance of random Galilean invariance (RGI). It is argued that, for a properly constructed ensemble of transformations to inertial frames, invariance in every realization necessarily implies RGI. It is suggested that the defects of the direct-interaction approximation can be understood in terms of a failure to renormalize the stirring forces.

A modification is made to the traditional Lagrangian approach to the derivation of the mean EMF of turbulent fluids which allows for finite conductivities. Consideration is confined to the case of homogeneous, isotropic but generally mirrornon-invariant and compressible turbulence. The eddy magnetic diffusivity and the coefficient α of the alpha-effect are expressed in terms of statistical moments of displacements of adjacent particles which undergo convective transport and microscopic diffusion in a turbulent flow. These expressions, being valid for arbitrary conductivities, reproduce known results in the cases of both very large and very small magnetic Reynolds numbers. Difficulties and advantages of the use of the results obtained for evaluations of the mean EMF are discussed.

The onset of instability in temporally modulated Taylor-Couette flow is considered. Critical Reynolds numbers have been found by computing Floquet exponents. We find that frequency modulation of the inner cylinder introduces a small destabilization, in agreement with the narrow-gap theory of Hall and some recent experiments of Ahlers. We review the previous computational literature on this problem, and find a number of contradictory results: the source of these discrepancies is examined, and a satisfactory resolution is achieved. Nonlinear axisymmetric calculations on the modulated problem have been done with an initial value code using a spectral method with collocation. The results are compared satisfactorily with Ahlers' measurements.

At low modulation frequency, a large destabilization has been observed in past experiments. We show that this cannot be explained on the basis of perfect bifurcation theory: an analysis of the modulated amplitude equation shows that very small imperfections can substantially affect the behaviour at low frequency by giving rise to ‘transient’ vortices at subcritical Reynolds number. We argue that these ‘transient’ vortices are the source of the large destabilization seen in some experiments. Modelling the imperfections in the initial-value code provides additional confirmation of this effect.

Numerical solutions have been obtained for steady streaming flow past an axisymmetric drop over a wide range of Reynolds numbers (0.005 [les ] Re [les ] 250), Weber numbers (0.005 [les ] We [les ] 14), viscosity ratios (0.001 [les ] λ [les ] 1000), and density ratios (0.001 [les ] ζ [les ] 1000). Our results indicate that at lower Reynolds numbers the shape of the drop tends toward a spherical cap with increasing We, but at higher Re the body becomes more disk shaped with increasing We. Unlike the recirculating wake behind an inviscid bubble or solid particle, the eddy behind a drop is detached from the interface. The size of the eddy and the separation distance from the drop depend on the four dimensionless parameters of the problem. The motion of the fluid inside the drop appears to control the behaviour of the external flow near the body, and even for cases when λ and ζ [Lt ] 1 (a ‘real’ bubble), a recirculating wake remains unattached.

The installation of large-eddy breakup devices (LEBUs) or ‘flow manipulators’ in a turbulent boundary layer over a rigid plane surface is known to lead to reductions in skin-friction coefficient, turbulence intensity and fluctuating Reynolds stress. We investigate the effect of such devices on the surface pressure spectrum and the far-field sound radiation. A model problem, in which a two-dimensional elliptical vortex is convected past a LEBU, is solved analytically in the low-Mach-number limit. The main noise source mechanisms are identified in this idealized problem and we go on to obtain scaling laws for the sound produced by a turbulent boundary-layer flow over a LEBU. The introduction of a LEBU reduces the strength of the Lighthill quadrupole source terms, but it produces an additional dipole source. However, the pressure fluctuations in this dipole field decay rapidly with distance from a LEBU, and we find that an array of LEBU's could have a beneficial effect on the flow noise for radian frequencies which are large in comparison with c/30Δ, where c is the sound speed and Δ denotes the boundary-layer thickness. At lower frequencies the LEBUs are predicted to increase the flow noise.

The effect of initial flow conditions on the wake of an airfoil is examined in an experiment which uses the ‘strip heater’ technique to externally force the airfoil wake. The strip heaters are used to introduce waves into the top and bottom boundary layers of a thin symmetric airfoil which are subsequently amplified and introduced to the wake. The evolution and interaction of the waves in the wake is the primary interest of this study. A linear stability analysis is applied to the mean velocity profiles in order to understand the frequency selection process in the wake. It is seen that the mean velocity profile adjusts itself in order to become more receptive to the forced frequency of oscillation, resulting in the suppression of previously existing frequencies. The amplitude of oscillations in the wake can be controlled by varying the phase relation between two input signals. In this respect, cancellation and enhancement of the oscillations is possible. The linear stability analysis is applied to the cancellation/enhancement flow to verify the level of cancellation achieved. The receptivity of the system to external forcing is established. A substantial reduction in drag is achieved for forcing frequencies near the centre of the receptivity range.

This paper summarizes experimental studies of incompressible elliptic jets of different aspect ratios and initial conditions, and effects of excitations at selected frequencies and amplitudes. Elliptic jets are quite different from the extensively studied plane and circular jets - owing mainly to the fact that the azimuthal curvature variation of a vortical structure causes its non-uniform self-induction and hence complex three-dimensional deformation. Such deformation, combined with properly selected excitation can substantially alter entrainment and other turbulence phenomena, thus suggesting preference for the elliptic shape in many jet applications. The dominance of coherent structures in the jet far field is evident from the finding that switching over of the cross-section shape continues at least up to 100 equivalent diameters De. The locations and the number of switchovers are strongly dependent on the initial condition, on the aspect ratio, and, when excited, on the Strouhal number and the excitation level. We studied jets with constant exit momentum thickness θe, all around the perimeter, thus separating the effects of azimuthal variations of θe, (typical of elliptic jets) and of the shear-layer curvature. Also investigated are the instability characteristics, and enhanced entrainment caused by bifurcation as well as pairing of vortical structures. We discuss shear-layer and jet- column domains, and find the latter to be characterized by two modes : the preferred mode and the stable pairing mode - similar to those found in circular jets -both modes scaling on the newly-defined lengthscale De. The paper documents some time- average measurements and their comparison with those in circular and plane jets.

Slide coating is a means of rapidly depositing multilayered liquid films of precise thickness and uniformity, as in manufacture of photographic products. Liquid is metered through one or more slots onto the inclined surface of the coating die, flows down that face and across a gap onto a fast-moving smooth surface. In this paper the steady, two-dimensional slide coating flow of a Newtonian liquid is analysed by solving the full Navier–Stokes system with the Galerkin/finite-element technique, spine parametrization of free surfaces and full Newton iteration. The lower meniscus in the gap is assumed to remain pinned at the die edge and the wetting-line singularity on the surface being coated is relieved by introducing dynamic-slip and contact-angle parameters. Results include the effects of several design and operating parameters on free-surface profiles and details of the flow field; these are presented by means of contours of kinematic and dynamic variables and local force balances over subdomains. The profiles show standing waves on the slide, rapid film thinning just before the gap, and exponential approach to the final film thickness on the web. As Reynolds number is raised and/or web speed is lowered several recirculation regions are predicted, deleterious features that have also been detected in experiments.

The separation of a binary gas mixture for high feed flow rates such that the E¼ vertical shear layer is nonlinear is considered. Numerical solutions for the velocity field and the temperature within the centrifuge are computed using the method originally described by Bennetts & Hocking (1973). These solutions are inputs to the separation problem which is characterized by a concentration boundary layer also of width of O(E¼). Results are presented for a wide range of parameters and the effect of thermal drive strength is examined in detail. Surprisingly the numerical results indicate that the analytical solution for the separation factor given by Conlisk, Foster & Walker (1983) may be used far outside its strict asymptotic region of validity.

Measurements are presented of the mean pressures around rectangular and D-section cylinders, with a flat front face normal to the flow, forced to oscillate transversely at an amplitude of 10% of the length of the front face. The ratio of depth (streamwise dimension) to height (cross-stream dimension) of the cross-section ranges from 0.2 to 1.0 for rectangular cylinders and from 0.5 to 1.5 for D-section cylinders. The range of reduced velocities investigated, 3 to 11, includes the vortex-resonance region. When increasing the depth, an oscillating bluff cylinder shows a critical depth where base suction attains a peak. The value of a critical depth is lowered with decreasing reduced velocity. In particular, an extraordinarily low critical depth with a very high base suction is obtained on cylinders oscillating at vortex resonance. For cylinders with depths beyond the critical, a reattachment-type pressure distribution is established on the afterbody due to the shear-layer/edge direct interaction. The shear-layer/edge direct interaction can also occur on oscillating cylinders with a fixed splitter plate. At low reduced velocities, the reattachment-type pressure distributions on cylinders with and without a splitter plate are similar except for the mean level. A remark is made on the critical geometry of bluff bodies under various flow conditions.

The propagation of a train of short, small-amplitude, internal waves through a long, finite-amplitude, two-dimensional, internal wave is studied. An exact solution of the equations of motion for a Boussinesq fluid of constant density gradient is used to describe the long wave, and its distortion of the density gradient as well as its velocity field are accounted for in determining the propagation characteristics of the short waves. To illustrate the magnitude of the effects on the short waves, particular numerical solutions are found for short waves generated by an idealized flow induced by a long wave adjacent to sloping, sinusoidal topography in the ocean, and the results are compared with a laboratory experiment. The theory predicts that the long wave produces considerably distortion of the short waves, changing their amplitudes, wavenumbers and propagation directions by large factors, and in a way which is generally consistent with, but not fully tested by, the observations. It is suggested that short internal waves generated by the interaction of relatively long waves with a rough sloping topography may contribute to the mixing observed near continental slopes.

A detailed wind tunnel study has been carried out on the near-wall turbulence structure over smooth and riblet wall surfaces under zero pressure gradient. Time-average quantities as ‘well as conditionally sampled profiles were obtained using hotwire/film anemometry, along with a simultaneous flow visualization using the smoke-wire technique and a sheet of laser light. The experimental results indicated a significant change of the structure in the turbulent boundary layer near the riblet surface. The change was confined within a small volume of the flow close to the wall surface. A conceptual model for the sequence of the bursts was then proposed based on an extensive study of the flow visualization, and was supported by the results of conditionally sampled velocity fields. A possible mechanism of turbulent drag reduction by riblets is discussed.

A matched-asymptotic analysis has been carried out for an axisymmetric convection cell in the case of stress-free boundaries. This problem differs from that of two-dimensional convection rolls mainly through the special role played by the central plume. The radius, of order ε, of the latter depends on the Rayleigh number R through the relationship $\epsilon^4(-\ln \epsilon) = R^{\frac{2}{3}}$. The plume velocity is independent of height at lowest order and its magnitude exceeds by a factor (− ln ε)½ the strength, of order $R^{\frac{2}{3}}$, of the core flow. As a result of these properties the central plume is governed by advection, in contrast to the perimeter plume which is affected by conduction as well. This asymmetry is reflected in the different thickness of the horizontal thermal boundary layers and gives rise to the deviation of the core temperature from the mean value of the top and bottom temperatures. This deviation is positive (negative) for the case of a falling (rising) central plume. While the core flow is driven mainly by the perimeter plume the fraction of the heat flux carried by the central plume is always above three-quarters and increases as the radius-to-height-ratio λ decreases.

The search for a statistical stability criterion characterizing steady-state turbulent shear flows ‘has led us to the study of a function proportional to the ratio of the fluctuation dissipation rate and the dissipation rate of the mean. The first Euler–Lagrange equations for an upper bound of this function have optimal solutions with the observed scaling laws and an asymptotic velocity defect for turbulent channel flow. As in the case of maximum transport, the optimal solution for the model equations is a discrete spectrum of streamwise vortices. It is shown how these solutions can be brought closer to the realized flow with additional constraints on the smallest vortex.

The effect of gyrotaxis on the linear stability of a suspension of swimming, negatively buoyant micro-organisms is examined for a layer of finite depth. In the steady basic state there is no bulk fluid motion, and the upwards swimming of the cells is balanced by diffusion resulting from randomness in their shape, orientation and swimming behaviour. This leads to a bulk density stratification with denser fluid on top. The theory is based on the continuum model of Pedley, Hill & Kessler (1988), and employs both asymptotic and numerical analysis. The suspension is characterized by five dimensionless parameters: a Rayleigh number, a Schmidt number, a layer-depth parameter, a gyrotaxis number G, and a geometrical parameter measuring the ellipticity of the micro-organisms. For small values of G, the most unstable mode has a vanishing wavenumber, but for sufficiently large values of G, the predicted initial wavelength is finite, in agreement with experiments. The suspension becomes less stable as the layer depth is increased. Indeed, if the layer is sufficiently deep an initially homogeneous suspension is unstable, and the equilibrium state does not form. The theory of Pedley, Hill & Kessler (1988) for infinite depth is shown to be appropriate in that case. An unusual feature of the model is the existence of overstable or oscillatory modes which are driven by the gyrotactic response of the micro-organisms to the shear at the rigid boundaries of the layer. These modes occur at parameter values which could be realized in experiments.

A new set of balanced equations (to be called planetary semi-geostrophic equations) for planetary-scale flow is derived from Hamilton's principle and constitutes a natural generalization of the semi-geostrophic equations of motion. Analogues of the global conservation of energy and the Lagrangian conservation of potential vorticity follow automatically by introducing approximations directly into the Hamiltonian in such a way that time and particle label symmetries are preserved. Two approximations are required: first, the kinetic energy associated with the component of velocity parallel to the axis of rotation is neglected; and secondly, the Lagrangian rate of change of the wind and pressure gradient directions (when projected onto the equatorial plane) must be small compared with twice the angular rotation rate of the system. Although the first of these approximations entails some loss of accuracy for application to the terrestrial atmosphere it is not nearly as severe as that for the Phillips type II geostrophic equations in which all of the kinetic energy is omitted from the Hamiltonian. The resulting equations take exactly the same form as the f-plane semi-geostrophic equations apart from a modification to the pseudo-density appearing in the continuity equation. They are also amenable to the geostrophic momentum coordinate transformation – a device which has had considerable impact on the theory of atmospheric fronts. In order to assess the accuracy of the equations, three different linearized eigenvalue problems on the sphere are solved and compared with the equivalent primitive equation problems. Eigenmodes are least accurate for high-zonal-wavenumber disturbances with grave meridional structure. Stationary, baroclinic planetary waves with zonal wavenumber less than ≈ 7 are shown to be accurately treated. The equations also support equatorially trapped Kelvin and Rossby modes which are accurate in the long-wave limit for meteorologically relevant equivalent depths.