A theoretical calculation is made of (the diagonal elements of) pressure-strain-rate calculation ρ0−1〈p[∇u + (∇u)T]〉 for a simple turbulent shear flow. This calculation parallels a previous calculation of the off-diagonal element. The calculation is described as follows. (1) Beginning with the Navier-Stokes equation, an expression for the (diagonal) pressure-strain-rate term is derived analytically in terms of measurable quantities (velocity spectra) - this derivation makes use of a cumulant discard. (2) It is proved that, to lowest order in the spectral anisotropy, the diagonal pressure-strain-rate term is linearly proportional to the diagonal Reynolds-stress elements. (3) A formula is derived for the proportionality constants (Rotta constants) in terms of arbitrary spectra. (4) This formula is used to calculate theoretically the numerical value of Rotta's constant Cii for models of velocity spectra (the variation of Cii with variations of spectral shapes and of Reynolds number are also determined). (5) Deficiencies and limitations of Rotta's model are identified and discussed.

It is found that Rotta's expression for 2ρ0−1〈p∂ui/∂i〉 is only valid for special spectra. Surprisingly large deviations of Rotta's expression from theory are found for a more complex spectra thought to be typical of simple shear flow. In addition, it is found that Cxz is intrinsically and quantitatively different from Cii because the latter depends importantly on the large-wavenumber part of the spectrum (the inertial subrange) whereas the former does not. The numerical ratio Czz/Cxz is calculated theoretically and shown to be about 2 for the zero-moment model. It is concluded that a linear term in the stress anisotropy as proposed by Rotta does not always exist. The deviation of Rotta's model from theory is understood by distinguishing between the spectral anisotropy and the stress anisotropy.

For the zero-moment spectral model, where the Rotta relation is valid, it is found that Cii varies significantly with large Reynolds number but is rather insensitive to the large-wavelength part of the spectrum.

A schlieren system has been arranged to sense the total fluctuation over a cross-section of the flow and thus becomes very sensitive to large-scale azimuthally coherent structures in the flow. For a natural unexcited jet it is found that there is a concentration of the large-scale structure at a characteristic Strouhal number which is not sensitive to the beam thickness and which reduces progressively with distance from the nozzle. This large-scale structure exhibits a coherence of over 70 % with the near-field pressure and convects at between 75 % and 95 % of the jet velocity. The coherence between the potential core-pressure field and the large-scale structure downstream increases rapidly with distance from the nozzle exit plane, rather limited coherence being found at the exit plane for these observations at a jet-exit Mach number Mj = 0·7. Movement of a central microphone from x = 0 to x = 2D introduced a solid centre body over the first 2·5 diameters of flow and gave rise to a set of discrete components in the flow structure in the range 0.6 < S < 1·4.

With harmonic excitation at S = 1·12 a subharmonic at S = 0·55 occurs at x/D = 3 and a second at S = 0.26, x/D = 6. The flow cross-sectional-average sensing thus appears to show up the vortex-pairing mechanism at greater distances from the nozzle than is easily detectable by other means. Under strong impulse excitation a set of discrete components was observed in a transient response extending over times of 400D/Uj. These had a strongest component which decreases more rapidly in Strouhal number with distance than that associated with natural or harmonically excited conditions.

A three-dimensional hydroelastic model of the dynamical motion in the cochlea is analysed. The fluid is Newtonian and incompressible, and the basilar membrane is modelled as an orthotropic elastic plate. Asymptotic expansions are introduced, based on slender-body theory and the relative high frequencies in the hearing range, which reduce the problem to an eigenvalue problem in the transverse cross-section. After this, an example is worked out and a comparison is made with experiment and the earlier low-frequency theory.

Using a variety of flow-visualization techniques, the flow behind a circular cylinder has been studied. The results obtained have provided a new insight into the vortex-shedding process. Using time-exposure photography of the motion of aluminium particles, a sequence of instantaneous streamline patterns of the flow behind a cylinder has been obtained. These streamline patterns show that during the starting flow the cavity behind the cylinder is closed. However, once the vortex-shedding process begins, this so-called ‘closed’ cavity becomes open, and instantaneous ‘alleyways’ of fluid are formed which penetrate the cavity. In addition, dye experiments also show how layers of dye and hence vorticity are convected into the cavity behind the cylinder, and how they are eventually squeezed out.

The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.

A measure of predictability that has many superior features compared to currently used measures is introduced. Through statistical theory it is demonstrated that in inviscid truncated flow this new predictability measure increases monotonically in time while all initial information about the system decays. Under the influence of forcing and viscosity the behaviour of this measure is shown always to satisfy intuitive expectations.

An experimental investigation of the flow and acoustic properties of a moderate-Reynolds-number (Re = 70000), Mach number M = 2·1, axisymmetric jet has been performed. These measurements extended the experimental studies conducted previously in this laboratory to a higher-Reynolds-number regime where the flow and acoustic processes are considerably more complex. In fact, mean-flow and acoustic properties of this jet were determined to be closely comparable to published properties of high-Reynolds-number jets.

The major results of the flow-field measurements demonstrate that the jet shear annulus is unstable over a broad frequency range. The initial growth rates and wavelengths of these instabilities as measured by a hot wire were found to be in reasonable agreement with linear stability theory predictions. Also, in agreement with subsonic-jet results, the potential core of the jet was found to be most responsive to excitation at frequencies near a Strouhal number of S = 0·3. The overall development of organized disturbances around S = 0·2 seems to agree in general with calculations performed using the instability theory originally developed by Morris and Tam.

The acoustic near field was characterized in terms of sound-pressure level and directivity for both natural and excited (pure-tone) jets. In addition, propagation direction and azimuthal character of dominant spectral components were also measured. It was determined that the large-scale flow disturbances radiate noise in a directional pattern centred about 30° from the jet axis. The noise from these disturbances appears from simple ray tracing to be generated primarily near the region of the jet where the coherent fluctuations saturate in amplitude and begin to decay. It was also determined that the large-scale components of the near-field sound are made up predominately of axisymmetric (n = 0) and helical (n = ±1) modes. The dominant noise-generation mechanism appears to be a combination of Mach-wave generation and a process associated with the saturation and disintegration of the large-scale instability. Finally, the further development of a noise-generation model of the instability type appears to hold considerable promise.

Oscillations of a cavity shear layer, involving a downstream-travelling wave and associated vortex formation, its impingement upon the cavity corner, and upstream influence of this vortex-corner interaction are the subject of this experimental investigation.

Spectral analysis of the downstream-travelling wave reveals low-frequency components having substantial amplitudes relative to that of the fundamental (instability) frequency component; using bicoherence analysis it is shown that the lowest-frequency component can interact with the fundamental either to reinforce itself or to produce an additional (weaker) low-frequency component. In both cases, all frequency components exhibit an overall phase difference of almost 2kπ(k = 1, 2,…) between separation and impingement. Furthermore, the low-frequency and fundamental components have approximately the same amplitude growth rates and phase speeds; this suggests that the instability wave is amplitude-modulated at the low frequency, as confirmed by the form of instantaneous velocity traces.

At the downstream corner of the cavity, successive vortices, arising from the amplified instability wave, undergo organized variations in (transverse) impingement location, producing a low-frequency component(s) of corner pressure. The spectral content and instantaneous trace of this impingement pressure are consistent with those of velocity fluctuations near the (upstream) shear-layer separation edge, giving evidence of the strong upstream influence of the corner region.

The important role of viscosity in producing second-order Eulerian drift currents in the presence of small-amplitude water waves was first recognized by Longuet-Higgins (1953).

The theoretical and experimental background is first reviewed. It is then shown that, contrary to previous belief, the presence of surface contamination must greatly enhance the drift velocity of short waves. We then solve an initial-value problem for the drift current associated with temporally decaying waves, thereby resolving questions raised by the work of Liu & Davis (1977), whose solution exhibits anomalous singularities. Next, the steady drift velocity of spatially decaying waves is calculated and shown to bear a close resemblance to Longuet-Higgins’ ‘conduction solution’ for unattenuated waves.

Finally, we establish that unidirectional drift currents of both surface and inter-facial waves are sure to be unstable to spanwise-periodic disturbances; the instability mechanism being identical to that first proposed by Craik (1977), and recently developed by Leibovich & Paolucci (1981), to explain the generation of Langmuir circulations.

The long-time evolution of an unstable wave train, consisting of a carrier wave and two 'side-band’ components, is investigated analytically. Mathematical expressions, involving Jacobian elliptic functions, for the wave envelope characteristics are derived. The solution yields the dependence of the long-time evolution on the initial disturbance. Of special interest is the simple formula for the modulation-demodulation recurrence period. The latter is shown to yield results in good agreement with those obtained from numerical solutions of the nonlinear Schrödinger equation.

This paper describes an experimental programme carried out in a laboratory channel with rough and smooth beds, to investigate the interaction between gravity waves and a turbulent current. In particular, changes induced in the mean-velocity profiles, turbulent fluctuations, bed shear stresses and wave attenuation rates are considered for a range of wave heights, keeping the wave period constant. The smooth-boundary tests were carried out as a necessary preliminary to the more-realistic rough-boundary condition.

A directionally sensitive laser anemometer was used to measure horizontal, vertical, and 45° velocity components in the oscillating fluid, and an on-line minicomputer was programmed to produce ensemble averages of velocities, Reynolds stresses and wave-elevation data. The cycle was sampled at 200 separate phase positions, with 180 observations at each position. Measurements were made at up to 30 points in the vertical.

Preliminary tests were carried out on the unidirectional current and on the waves alone. These show that mean-velocity profiles and turbulence parameters of the current agree satisfactorily with previous experiments, and that the waves are approximated closely by Stokes’ second-order theory.

For combined wave and current tests, mean-velocity profiles are generally found to differ from those suggested by a linear superposition of wave and current velocities, a change in boundary-layer thickness being indicated. However, shear stresses at the smooth boundary are found to be described by such a linear addition.

To study effects of flow inhomogeneities on the dynamics of laminar flamelets in turbulent flames, with account taken of influences of the gas expansion produced by heat release, a previously developed theory of premixed flames in turbulent flows, that was based on a diffusive-thermal model in which thermal expansion was neglected, and that applied to turbulence having scales large compared with the laminar flame-thickness, is extended by eliminating the hypothesis of negligible expansion and by adding the postulate of weak-intensity turbulence. The consideration of thermal expansion motivates the formal introduction of multiple-scale methods, which should be useful in subsequent investigations. Although the hydrodynamic-instability mechanism of Landau is not considered, no restriction is imposed on the density change across the flame front, and the additional transverse convection correspondingly induced by the tilted front is described. By allowing the heat-to-reactant diffusivity ratio to differ slightly from unity, clarification is achieved of effects of phenomena such as flame stretch and the flame-relaxation mechanism traceable to transverse diffusive processes associated with flame-front curvature. By carrying the analysis to second order in the ratio of the laminar flame thickness to the turbulence scale, an equation for evolution of the flame front is derived, containing influences of transverse convection, flame relaxation and stretch. This equation explains anomalies recently observed at low frequencies in experimental data on power spectra of velocity fluctuations in turbulent flames. It also shows that, concerning the diffusive-stability properties of the laminar flame, the density change across the flame thickness produces a shift of the stability limits from those obtained in the purely diffusive-thermal model. At this second order, the turbulent correction to the flame speed involves only the mean area increase produced by wrinkling. The analysis is carried to the fourth order to demonstrate the mean-stretch and mean-curvature effects on the flame speed that occur if the diffusivity ratio differs from unity.

Surface-active material is present in most naturally occurring water samples, and it naturally diffuses steadily to free surfaces, where it both reduces the surface tension and gives the surface elastic properties which enable it to resist compression. When the water flows so that the surface layer is trapped and compressed against a fixed shallow-draught barrier the film material makes the surface incompressible, and flow beneath the barrier forms a viscous boundary layer under the film. The stresses associated with this boundary layer are found to distort the surface in the region of the leading edge of the film, giving rise to a phenomenon which is commonly observed in nature and which has been called the Reynolds ridge. This paper describes experimental work on the measurement of the ridge, and compares the results with a theoretical model due to Harper & Dixon. Good agreement is indicated.

An experimental study of the motion of small water droplets in both accelerating and decelerating conditions is presented. Droplets with diameters in the range 115-187μm were exposed to propagating N-waves having strengths smaller than 0.03. Droplet-displacement data were obtained by single-frame stroboscopic photography, at an equivalent framing rate of 4000 pictures per second. The data were fitted by means of best-fit polynomials in time, which were used to obtain drag coefficients in accelerating and decelerating flow conditions. In addition to providing drag data for impulsive-type motions, these data show that the unsteady drag follows two entirely distinct trends. In one, applicable to decelerating relative flows, the unsteady drag is always larger than the steady drag at the same Reynolds number. In the other, applicable to accelerating relative flows, the unsteady drag is always smaller than the corresponding steady value. These trends have not been previously known. They give some support to a mechanism recently proposed (see Temkin & Kim 1980) to explain departures of the drag coefficient for a sphere from its steady value; namely, the changes in size of the recirculating region behind the sphere, relative to its steady counterpart at the same Reynolds number.

The stability of quasi-geostrophic barotropic fields induced by localized finite-amplitude potential vorticity sources or topographies which depend only on the zonal direction is investigated both analytically and numerically. The analytical study is for weak forcing. It demonstrates that the field induced by the topography is stable whereas the field induced by the potential vorticity source can be unstable. The growth rate is exponential and is a function of both nonlinearity and friction. In the absence of friction the flow field is always unstable. The instability takes the form of a current whose meridional wavenumber is that of a stationary Rossby wave. In the zonal direction the current exhibits long-scale oscillations and exponential decay. The numerical computations which are for strong forcing verify all the indications of the asymptotic study. They show a rapid exponential growth of a non-propagating but oscillatory wave packet whose location is fixed relative to the forcing. The zonal scale of the packet is that of a stationary Rossby wave. For weak forcing the instability can be responsible for changing the flow field from one quasi-steady state to another where the energy extraction takes place in the region of the source. It is efficient for potential vorticity sources whose length scale is comparable to the length scale of stationary Rossby waves. In agreement with the asymptotic study, fields induced by strong topographic forcing are found to be stable. The asymptotic analysis is also applied to baroclinic flows where the investigation is performed in the framework of the two-layer model. It is demonstrated that the same mechanism which operates in barotropic systems can also destabilize baroclinic flows which possess subcritical shears.

Turbulence measurements in dispersed-bubble two-phase pipe flow, using laser velocimetry techniques, are presented. The turbulence-intensity measurements show a strong dependence on the quality of the flow. A theoretical basis for the prediction of turbulence levels in two-phase flows is proposed. The approach is applied to dispersed-gas/liquid (bubbly) and solid/gas (particulate) two-phase flows, for which experimental data are available, with excellent results.

The nonlinear double-diffusive convection in a Boussinesq fluid with stable constant vertical solute gradient, and bound by two differentially heated rigid inclined parallel plates is considered. The analysis was carried out by a Galerkin method for the cases when the angle of inclination was 0°, −45° and +45° (positive angle denotes heating from below, and negative angle denotes heating from above). The counter-rotating cells predicted by the linear theory merge into single cells with the same sense of rotation within a very short period of time even under slightly supercritical conditions. This is consistent with the experimental observations. Furthermore, as observed in the experiments, the evolution of instability is more rapid when heating is from above than when heating is from below. Our results for a salt-heat system are in excellent agreement with those based on the limiting case of Lewis number → 0 and Schmidt number → ∞.

The aerodynamic noise generated by a subsonic jet impinging on a flat plate is studied from measurements of near-field and surface-pressure fluctuations. The far-field noise measured at 90° to the jet axis is found to be generated by two different physical mechanisms. One mechanism is the impinging of the large coherent structures on the plate, and the other is associated with the initial instability of the shear layer. These two sources of noise radiate to the far field via different acoustical paths.

The cross-section shape and stability of a liquid cylinder moving perpendicularly to its axis in a gaseous medium is studied. Such liquid cylinders are formed during the break-up process of thin, rapidly moving liquid sheets, appearing in spray and atomization processes. The equilibrium shape is affected mainly by two factors: the dynamic-pressure distribution in the gas flow and the surface tension on the liquid boundary. The former tends to distort the liquid cross-section into an oval shape while the latter tends to restore the circular cross-section.

A series expansion for the shape of the cylinder cross-section was determined by assuming incompressible potential flow, neglecting the effects of body forces and internal circulation in the liquid.

The stability analysis shows that in the range of low Weber numbers the cylinder break-up is due to the divergence of varicose perturbations. The wavenumber of the most rapidly growing perturbation, its rate of growth and the maximal wavenumber for which varicose instability occurs, are all found to decrease as the Weber number grows, owing to a pressure distribution caused by the varicose distortion, which tends to reduce these perturbations.

Mixing-layer development is investigated in laboratory experiments of salt-stratified solutions which are cooled from above or heated from below through the imposition of isothermal boundaries. A Mach-Zehnder interferometer is used to infer salt and density distributions within stable regions of the solution and to determine the extent of mixing-layer development. In both heating from below and cooling from above, this development differs significantly from that which has been observed for constant heating from below. Although the formation of a secondary mixed layer is observed, it does not lead to the development of additional mixed layers. Instead, the secondary layer eventually recedes, and the existence of a single mixed layer is restored. This behaviour is due to the isothermal boundary and the effect which it has no decreasing the heat transfer to or from the solution with increasing time. Once the condition of a single mixed layer is restored, extremely large (stable) density gradients develop in the boundary layer separating the mixed and stable regions, and subsequent growth of the mixed layer is slow. In cooling from above, mixing-layer development depends strongly on whether the isothermal boundary is in direct contact with the solution or separated by an air space.