The process of internal gravity wave generation by the simple harmonic flow (U = U0, cos ω0t) of a stably stratified fluid (Brunt–Väisälä frequency N) over an obstacle is investigated in some detail. Attention is primarily directed to the behaviour of the solution in various limiting cases, and to estimating the flux of energy into the internal wave field. In general, waves are generated not only at the fundamental frequency ω0, but also at all of its harmonics. But, for values of ω0/N greater than about one half, the waves of fundamental frequency are dominant. For values of ω0/N, less than about one half, the quasi-static approximation, in which the problem is considered as a slowly-varying version of the classical lee wave problem, is found to provide a viable estimate for the wave field. The general solution is found to compare favourably with the limited available experimental data.

It is well known that the small perturbation equation governing steady or mildly unsteady potential flow in a sonic gas jet is nonlinear. However, for a sonic gas jet submerged in a liquid with a disturbance on the gas-liquid interface, it is shown that the transient motion of the gas dominates, and the nonlinear term due to accumulation of disturbances in the basic flow becomes negligible; the condition necessary for the applicability of the linearized governing equation is obtained. It is demonstrated that most gas-jet/liquid systems of physical interest satisfy this condition and that the margin with which this condition is satisfied improves as the wave velocity of the disturbance or, more particularly, as the stagnation pressure or density of the gas for a given gas-liquid system increases. The Kelvin-Helmholtz instability of the gas-liquid interface of a sonic gas jet submerged in a liquid is predominantly governed by the transfer of energy from the gas phase to the liquid layer, both through wave-drag and ‘lift’ components of the pressure perturbation; at and above the cut-off wavenumber, which only exists for very low viscosity liquids owing to the stabilizing effect of surface tension, the pressure perturbation becomes in phase with the wave amplitude. It is shown that for low viscosity liquids the phase angle between the pressure perturbation exerted by the gas phase on the liquid a t the gas-liquid interface and the wave amplitude, which is the measure of the relative effectiveness of the ‘lift’ and wave-drag components of the pressure perturbation, is a function of the density ratio (ratio of gas density a t throat conditions to liquid density). At low density ratios both of these components are operative; however, at high density ratios the wave-drag component becomes dominant. The analysis further shows that the cut-off wave- number and the wavenumber a t maximum instability decrease with increasing density ratio. For highly viscous liquids and liquids having finite viscosity the pressure perturbation is always out of phase with the wave amplitude, and no cut-off wavenumber exists, i.e. the gas-liquid interface is always unstable in spite of the stabilizing effect of viscosity and surface tension.

It was suggested by Lau, Fisher & Puchs (1972) that the basic structure of a ‘turbulent’ round jet might consist, essentially, of an axial array of fairly evenly spaced vortices moving downstream in the mixing region of the jet. The present experimental study is an attempt to establish this hypothesis on a sound footing. The problem which was posed was first to find proof of the existence of a fairly regular pattern in the mixing region, and second to extract detailed information on the component parts of this pattern to identify the nature of the structure.

Hot-wire signals in the mixing layer are known to possess a predominance of spikes. In the region closer to the high velocity side of the layer, these spikes tend to be downward ones whilst in the opposite region, they are upward. These spikes have been attributed to the entrainment mechanism in the mixing layer and had been thought to be random. A closer study of time-history curves of these hotwire signals suggests that they might not be as random as would appear at first glance. A probability analysis was conducted of the time intervals between the successive downward spikes in the u′ signals, and it was found that indeed the highest probability occurred when the time interval corresponded to a frequency equal to the vortex passing frequency.

A time-domain averaging (or eduction) technique was used to try to identify the nature of the flow structure using the spikes to trigger the eduction process. On the basis of these results it would seem that the suggestion of a vortex street is well founded. Furthermore, it appears that, as the individual vortices in the street move downstream, they are continuously transporting fluid masses across the mixing layer, and it is this effect which is producing the Reynolds stresses in the mixing layer, and causing the spikes in the u’ signals in this region.

The evolution of a small disturbance in a three-dimensional steady free-surface flow is investigated. The radius of curvature of the free surface and the length scale characterizing the non-uniformity of the velocity are assumed to be of the same order of magnitude. It is shown that the local rate of growth of the amplitude of the disturbance depends on both the normal pressure gradient (as in the case of Taylor instability) and the rate of strain on the free surface. Application of the theory to rising gaseous bubbles and gravity water waves is discussed.

In this paper we consider the flow field generated by a uniform electrostatic field in and about a pair of identical liquid drops immersed in a conducting fluid. It is assumed that the undisturbed electric field is parallel to the line joining the centres of the two drops. The flow field is due to the tangential electric stress over the surfaces of the drops and here this stress and the flow field are expressed in terms of bispherical harmonics. When the distance between the centres of the drops is many drop diameters the tangential electric stress and the flow field in and about one drop are unaffected by the presence of the other drop, as expected. When the distance between the centres of the drops is of the order of two drop diameters there is a substantial modification in the tangential electric stress at the surfaces of the drops and in the induced flow field, especially in the region between the planes through the drop centres perpendicular to the undisturbed electric field.

The Laplace–Young equation is solved for axisymmetric menisci, analytically in terms of elliptic integrals for all possible types of pendular rings and liquid bridges when the effect of gravity is negligible, numerically for selected other cases in order to assess gravity's effect. Meniscus shapes, mean curvatures, areas and enclosed volumes are reported, as are capillary forces. It is shown that capillary attraction may become capillary repulsion when wetting is imperfect. The special configurations of vanishing capillary force and of zero mean curvature are treated. The range of utility of the convenient ‘circle approximation’ is evaluated.

Measurements have been made concerning the fine structure of the turbulence in the part adjacent to the wall of the wall region of a plane turbulent boundary layer. The objective was to gain further information concerning the larger-scale disturbance mechanism which is mainly responsible for the generation of turbulence. Hot-wire anemomet.ry was used and information on the fine structure was obtained by differentiating and filtering the hot-wire signal.

The distributions of the Kolmogorov microscale and of the flatness and skewness factors of the axial fluctuating velocity u and its first and second derivative determined at two Reynolds numbers suggest the existence of Reynolds number similarity. In the region y+ < 15 the flatness and skewness factors of u increase with decreasing y+. At approximately y+ = 15 the flatness factor shows a minimum value, while the skewness factor becomes zero. This location agrees with that where the turbulence intensity u′ has a maximum value. In the outer part of the wall region (y+ > 100) the flatness and skewness factors approach values obtained in shear-free turbulence at the same turbulence Reynolds number.

The fine structure of the turbulence is strongly associated with and dominated by the random, larger-scale, intermittent inrush-ejection cycle. In the viscous sublayer both the fine structure, and the large-scale mechanism of the turbulence are influenced mainly by the inrush phase, while further out in the wall region (y+ > 40) they are influenced by both inrush and ejection. As a result, in the viscous sublayer the average burst periods of the high frequency turbulence components and their flatness factors (of ∂u/∂t and of ∂2u/∂t2) attain values twice those in the outer part.

The change in the mechanism of the fine structure with distance from the wall is clearly demonstrated by the spectra of non-negative variables, i.e. (∂u/∂t)2 and (∂2u/∂t2)2. The spectra agree with each other and decrease with increasing frequency, following a power law as predicted by Gurvich & Yaglom (1967). The power law applies to almost the whole frequency range, when the highest, viscous, frequency range is excluded. However, the exponent is different for the viscous sublayer and the outer part of the wall region. In the buffer layer the spectra have two distinct power-law regions. In the lower frequency range the exponent is the same as that for the viscous sublayer, while in the higher frequency range it is the same as that for the outer part of the wall region.

Measurements of the fine-structure of the energy spectrum of the velocity fluctuations were made in the transition region of a two-dimensional wake. Line and continuous spectra were determined separately with a high-selectivity band-pass filter. The transition was initiated by an external sinusoidal sound. The sound-induced periodic fluctuation and the natural random fluctuation in the wake were added and a system of line and continuous spectra was formed. Higher harmonics of the periodic component were produced by the nonlinear interaction. As a result of the interaction between discrete and continuous components, a continuous spectrum was generated at low wavenumbers. A simple model for the interaction is proposed. The evolution of the spectrum is explained by three fundamental rules concerning the nonlinear interaction between spectral components: (i) the growth of a spectral component is suppressed by the presence of another strong component, (ii) mutual interaction is more effective when the amplitudes of interacting components are closer, and (iii) a stronger interaction takes place between components of closer wavenumbers. The randomization of the regular fluctuation is properly expressed as the growth of the ‘randomness factor’, the ratio of the energy of the random components to the total fluctuation energy.

Many experimenters have used oscillating grids to produce turbulence for various laboratory purposes, especially in studies of mixing, but there have been few direct measurements of the properties of the turbulence itself. In the present paper we report experiments which attempt to relate the turbulent velocity and length scales to the external parameters, the frequency and amplitude, for three forms of grid oscillated in a tank of water. Turbulent velocities have been measured in the absence of a mean flow by using a hot film moved through the fluid to provide its own mean velocity. The output is stored and analysed in a small computer, which rapidly evaluates velocity and length scale statistics from an ensemble of records. The spatial variation of these quantities with distance from the stirrer is of special interest. It agrees with results suggested by an inertial-decay theory, and with previous measurements made by Bouvard & Dumas (1967) using a different form of stirrer. A particular purpose of the work has been to ‘calibrate’ the entrainment experiments of Turner (1968), by providing absolute scales of velocity and length in the fluid near a mixing interface, for the same grid as was used in the earlier experiments. Evidence is presented which suggests that other forms of grid may not be calibrated simply by extrapolating these results.

The incompressible laminar flow in the neighbourhood of the trailing edge of an aerofoil undergoing sinusoidal oscillations of high frequency and low amplitude in a uniform stream is described in the limit as the Reynolds number R tends to infinity. The aerofoil is replaced by a flat plate on the assumption that leadingedge stall does not take place. It is shown that, for oscillations of non-dimensional frequency $O(R^{\frac{1}{4}})$ and amplitude $O(R^{\frac{9}{16}})$, a rational description of the flow at the trailing edge is based on a subdivision of the boundary layer above the plate into five distinct regions. Asymptotic analytic solutions are found in four of these, whilst in the fifth a linearized solution yields an estimate for the viscous correction to the circulation determined by the Kutta condition.

A simple approximation is proposed for the problem of the dispersion of marked particles in an incompressible fluid in random motion when the probability distribution of the velocity field is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. Approximations for the Lagrangian velocity correlation function and the dispersion are given and compared with exact computer calculations of Kraichnan. Agreement is found to be good except for time-independent velocity fields and singular wavenumber spectral functions.

In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order $(\epsilon/\nu)^{\frac{1}{2}}$ (ε is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid). This leads to an estimate that makes the r.m.s. value of du/dt of order $(\epsilon^3/\nu)^{\frac{1}{4}}$. The Eulerian frequency spectrum, however, extends to higher frequencies than its Lagrangian counterpart; this is caused by spectral broadening associated with large-scale advection of dissipative eddies. As a consequence, the r.m.s. value of ∂u/∂t at a fixed observation point is (apart from a numerical factor) $R_{\lambda}^{\frac{1}{2}}$ times as large as the r.m.s. value of du/dt (RΛ is the turbulence Reynolds number based on the Taylor microscale). The results of a theoretical analysis based on these premises agree with data obtained by Comte-Bellot, Shlien and Corrsin. The analysis also suggests that the Eulerian frequency spectrum has a $\omega^{-\frac{5}{3}} $ behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.

This paper suggests a simple way of including gravitational effects in the pres-sure-containing correlations that appear in the equations for the transport of Reynolds stress and heat flux. The predicted changes in structure due to the gravitational field are shown to agree closely with Webster's (1964) measurements in a stably stratified shear flow.

The equilibrium statistical mechanics of inviscid two-dimensional flow are re-examined both for a continuum truncated at a top wavenumber and for a system of discrete vortices. In both cases, there are negative-temperature equilibria for finite flows. But for spatially infinite flows, there are only positive-temperature equilibria, and both the continuum and discrete system exhibit proper, extensive, thermodynamic limits a t all realizable values of the energy and enstrophy density. The negative-temperature behaviours of the continuum and discrete system are semi-quantitatively the same, except for a supercondensation phenomenon in the discrete case a t the smallest realizable values of negative temperature. The supercondensed states have very large energy and in them all vortex cores of the same sign are clumped within an area small eompared with the mean area per vortex. The approach of the continuum system to absolute equilibrium by enstrophy cascade to high wavenumbers and energy cascade to low wavenumbers is examined. It is argued that the enstrophy cascade is closely analogous to distortion of a passive scalar field by straining of large spatial scale. This implies that high intermittency of spatial derivatives of the vorticity field can develop but that there is no associated change in the previously proposed log-corrected k−1 enstrophy spectrum law. On the other hand, intermittency build-up in the downward energy cascade can result in a change of the exponent in the energy spectrum law to a negative value of smaller magnitude than 5/3. Intermittency effects in the non-equilibrium energy cascade seem a more plausible explanation for vortex clumping observed in recent computer experiments than do the spatially smooth condensation phenomena associated with the negative-temperature absolute equilibria.

An integral-equation formulation is used to calculate wave forces on bodies having a vertical axis of symmetry. The development enables one to calculate the forces without completely determining the local pressure field, thus offering a considerable reduction of computational effort. Numerical results are presented for a hemisphere at the water surface and vertical circular cylinders.

When a liquid-filled tube of radius b which is spinning about its axis with angular velocity ω is allowed to drain by opening one end, a cavity propagates in the tube with velocity U. This velocity has been measured with the tube in both vertical and horizontal positions for large values of the rat,io ω2b/g of the centrifugal to the gravitational force, g being the gravitational acceleration. It is found that the Rossby number Uωb becomes constant and independent of ω2/g in both cases, taking the value U/ωb = 0.52 in the vertical case and U/Ωb ≃ 0.4 in the horizontal case. Experiments conducted in a tube which is slightly inclined to the horizontal confirm that, although the Rossby number is constant when ω2b/g is large, the value adopted depends upon α, the inclination of the tube to the gravitational field.

The theory of the constant-temperature anemometer has been extended in order to obtain quantitative results for the frequency response. A simple electrical test against which to check the theory has been devised, and the validity of the anemometer equations is demonstrated. Important differences in design philo-sophies and modes of operation are indicated, and results are presented for a design in which high d.c. gain is employed in the servo amplifier. The square-wave response is briefly investigated, and it is concluded that commonly used criteria for determining the frequency response from it should be treated with caution.

To measure fluctuations with a hot-wire anemometer in flows containing both velocity and temperature perturbations, the hot wire must be operated at more than one temperature. Variation of the mean wire temperature causes, in general, a variation in the frequency response, as does variation of the mean flow conditions. It is shown that, by simultaneous variation of the gain of the servo amplifier in the anemometer, the frequency response may be held nearly constant over a useful range of both overheat and flow conditions.

A finite-difference procedure is employed to predict the development of turbulent flow in curved pipes. The turbulence model used involves the solution of two differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. The predicted total-velocity contours for the developing flow in a 180° bend are compared with the experimental data. Predictions of fully developed velocity profiles for long helically wound pipes are also presented and compared with experimental measurements.

A secondary flow is set up when a fluid flows through a stationary curved pipe. The fluid in the middle of the pipe moves outwards and that near the wall inwards. Dean showed that the dynamical similarity of this fully developed flow depends on a non-dimensional parameter $D = 2(a/R)^{\frac{1}{2}}(a\overline{W}/\nu) $, where $\overline{W} $ is the mean velocity along the pipe, v is the coefficient of kinematic viscosity and a is the radius of the pipe, which is bent into a coil of radius R. Dean's analysis was limited to small values of D. Later, Barua developed an asymptotic boundary-layer theory for large values of D and showed for these values of D that the resistance coefficient γc is much larger than that for the corresponding straight pipe. The present work deals with the flow in a curved pipe as it develops from a uniformly distributed velocity at the entrance to a fully developed profile. Barua's results for the fully developed flow are adopted as downstream conditions in the present work. The ratio of the entry lengths of the curved ipe and the corresponding straight one is shown to be proportional to D−1/2 when D is large. Thus, the entry length for a curved pipe is much shorter than that for the corresponding straight pipe.

The interaction between a vibrating structure and an unsteady potential flow-field disturbance induced by the motion of the structure itself is investigated, and is shown to be a significant source of both nonlinear excitation and nonlinear dissipation. An approximate analysis, based on small nonlinear disturbance theory, is presented of the forces that influence the characteristic behaviour of self-excited harmonium reeds vibrating at finite amplitudes. It is demonstrated that the ideas brought forth by this example can be generalized to apply to other flow-induced vibrating systems, regardless of the excitation mechanism, provided that certain basic assumptions about the flow can be made. For the case of the harmonium reed, it is shown that, taken by itself, an account of the feedback forces arising from induced higher-order unsteady disturbances in the surrounding potential flow field is sufficient for predicting the net nonlinear dissipative force that eventually causes the reed to reach and maintain a finite limiting amplitude. In particular, it is demonstrated that the nonlinear energy drain from the motion of the reed is a consequence of the net effect of the higher-harmonic disturbances that are generated near the structure.

A result of the analysis is the development of a functional dependence of the interactive forces on the system geometry and the flow velocity. One of the advantages of obtaining a functional expression is the ability to carry out parametric studies in the context of vibration and noise control.