Measurements in the neighbourhood of a laminar separation point at a high subcritical Reynolds number are reported. These results are used to test the validity of various theories relating to laminar separation. It is concluded that the boundary-layer equations are valid in the neighbourhood of the separation point without the existence of a singularity.

The velocity field was measured using a dual-beam laser-Doppler anemometer with optical frequency shifting. The wall-shear-stress distribution was measured with a flush-mounted hot-film sensor and the wall-pressure distribution was measured using a strain-gauge pressure sensor. The various terms in the Navier–Stokes equations were evaluated directly from the measurements, permitting the validity of the boundary-layer equations to be established. Proposed solutions for the flow field are compared with the measured flow field.

For a wide gap (R1/R2 = 0.5) and large aspect ratios L/d, axisymmetric Taylor-vortex flow has been observed in experiments up to very high supercritical Taylor (or Reynolds) numbers. This axisymmetric Taylor-vortex flow was investigated numerically by solving the Navier–Stokes equations using a very accurate (fourth-order in space) implicit finite-difference method. The high-order accuracy of the numerical method, in combination with large numbers of grid points used in the calculations, yielded accurate and reliable results for large supercritical Taylor numbers of up to 100Tac (or 10Rec). Prior to this study numerical solutions were reported up to only 16Tac. The emphasis of the present paper is placed upon displaying and elaborating the details of the flow field for large supercritical Taylor numbers. The flow field undergoes drastic changes as the Taylor number is increased from just supercritical to 100Tac. Spectral analysis (with respect to z) of the flow variables indicates that the number of harmonics contributing substantially to the total solution increases sharply when the Taylor number is raised. The number of relevant harmonics is already unexpectedly high at moderate supercritical Ta. For larger Taylor numbers, the evolution of a jetlike or shocklike flow structure can be observed. In the axial plane, boundary layers develop along the inner and outer cylinder walls while the flow in the core region of the Taylor cells behaves in an increasingly inviscid manner.

The evolution equations are obtained which govern the growth of cross-waves generated in a long deep channel by a wavemaker with small amplitude when the waves are modified by finite-amplitude effects. The linearized equations are compared with previous theoretical and experimental work. Some numerical solutions are obtained for illustration.

We extend the simulation techniques of Kraichnan (1970, 1976) to study the effective diffusivity of a scalar field in a turbulent fluid. In our model we have introduced an adjustable helicity parameter and a technique for simulating molecular diffusivity. The results show that for non-helical turbulence the self-consistent perturbation theory of Phythian & Curtis (1978) gives excellent values for the effective diffusivity over a wide range of values for both the molecular diffusivity and the parameters describing the turbulence.

This ceases to be the case immediately the helicity is given a non-zero value. Wide departures are observed between the theoretical calculation and the simulation. Our conclusion is that non-perturbative effects are very important in the presence of helicity.

The paper reports on measurements in the flow generated by a jet issuing from a circular outlet in a wall into a cross-stream along this wall. For the jet-to-crossflow velocity ratios R of 0.5, 1 and 2, the mean and fluctuating velocity components were measured with a three-sensor hot-wire probe. The hot-wire signals were evaluated to yield the three mean-velocity components, the turbulent kinetic energy, the three turbulent shear stresses and, in the case of R = 0.5, the terms in the turbulent-kinetic-energy equation. The results give a quantitative picture of the complex three-dimensional mean flow and turbulence field, and the various phenomena as well as their dependence on the velocity ratio R are discussed in detail.

Steady and oscillating axisymmetric laminar flows are determined by a finite-difference solution of the vorticity and continuity equations for an incompressible fluid contained in a straight concertina-shaped tube far from its ends. In steady flow the size of the wall corrugations is varied as well as the Reynolds number of the flow. In unsteady flow one tube geometry is studied, and the parameters varied are the Reynolds number, the ratio of the mean volume flow rate to its amplitude, and the frequency of oscillation. The analysis produces streamlines, particle paths and the pressure difference across a length of the tube. The resistance to the flow is determined in terms of an equivalent cylindrical tube diameter.

In steady flow the onset of flow separation and the growth of the separated region of flow is determined. The equivalent diameter is found to be principally a function of the product of Reynolds number and the non-dimensional pressure difference. This product depends on the height of the wall corrugations and less strongly on Reynolds number and the length of the corrugations. Resistance increases with increasing height of the corrugations. Comparison is made with other computational and experimental values of the pressure difference.

In unsteady flow the mean velocity to amplitude ratio has little effect except on the particle paths. The flow pattern is found to be governed by the Stokes number (radius × (2π/(kinematic viscosity × period))½) and the Reynolds number. There is a region of quasi-steady flow at the time of zero acceleration at maximum flow, but unsteady flow in between. The mixing produced by radial convection is restricted to the outer parts of the tube where the wall is corrugated. In oscillating flow the resistance relative to a cylindrical tube decreases as frequency and Reynolds number increase.

In the medical application of the work the concern is whether sustained stagnant regions occur in the corrugations and whether there is a large change in resistance relative to a cylindrical tube. This part of the investigation was made with an arterial waveform which contained six harmonics. It is found that there are no regions of stagnant fluid in the range of parameters considered. The difference between the variation with the flow parameters of the resistance of the corrugated tube and of a cylindrical tube was found not to be large.

This paper is a study of turbulence near rigid surfaces, in the absence of any mean shear. Different sources of turbulence are considered, including thermal convection and grid turbulence. It is shown that, if a rigid boundary is introduced into the flow, then for short times the linear theory of Hunt & Graham (1978) reveals the common structure of these flows near the boundary, if the parameters used are the rate of energy dissipation per unit mass ε and the distance z from the surface. Over longer times nonlinear effects develop, such as large eddies straining smaller eddies near the boundary. Some new estimates are suggested here and compared with the computations of Biringen & Reynolds (1981) and experiments of Thomas & Hancock (1977).

It is shown that calculations based on the linear theory agree well with many measurements of the vertical profiles of turbulence in thermal convection layers, including those of the vertical variance, the low-frequency end of the spectrum of the vertical turbulence (w), the integral scale of w, and two-point cross-correlations of w. (The latter was a prediction, subsequently tested by atmospheric measurements.) Some discussion of the reasons for this agreement are suggested. The observations of the effects of mean-velocity gradients near the surface are also shown to be consistent with the theoretical arguments proposed here.

Experiments are made in which a stratified shear flow, accelerating from rest and containing a level where the direction of flow reverses, is generated over a rough floor. The roughness elements consist of parallel square bars set at regular intervals normal to the direction of flow. Radiating internal gravity waves are generated in the early stages of flow, whilst flow separation behind the bars produces turbulent mixing regions which eventually amalgamate and entirely cover the floor. This turbulent layer spreads vertically less rapidly than the internal waves. Observed features of the waves are compared with those predicted by a model in which the floor is assumed to be sinusoidal, and fair agreement is found for the amplitude, phase and vertical wavenumber of the waves, even when the latter becomes large.

The rate of spread of the turbulent layer depends on the separation of the bars. Some interaction between the turbulence and the internal waves occurs near the edge of the turbulent layer. Wave-breaking is prevalent and the vertical scale of the waves is affected by turbulent eddies. The radiating internal waves are suppressed by replacing the bars by an array of square cubes, but there is continued evidence of features resembling internal waves near the boundary of the turbulent region. Structures are observed which bear some similarities to those found at the foot of the near-surface mixing layer in a lake.

In this paper three systems of evolution equations are presented which describe the free propagation of long continental shelf waves in the linear and weakly nonlinear regime. Two different degrees of nonlinearity are considered: for the first the Korteweg-de Vries equation is found to govern the dynamics of the system in the case of a single energy-containing mode (theories by Smith 1972; Grimshaw 1977), whereas, for the second nonlinear range, a nonlinear hyperbolic equation is derived. The nonlinear interactions among shelf-wave modes are also considered: they are modelled through nonlinear coupling terms in the evolution equations. This theory allows the timescale for the development of dispersive and nonlinear effects to be determined for each parameter range. The amplitude ranges corresponding to linear and nonlinear shelf waves are evaluated for the Oregon and the East Australian shelves, and some qualitative conclusions on the importance of nonlinear effects are derived. Finally the case of a shelf with longshore variation in topography is analysed and coupling terms in the evolution equations appear. They account for the scattering of energy between the various modes due to the linear and nonlinear interactions of the wave with the topographic changes.

The three-dimensional resonant interaction of a plane Tollmien-Schlichting wave, having a frequency f1, with a pair of oblique waves having frequencies ½ f1, was observed and studied experimentally. In the initial stages, the interaction proved to be a parametric resonance, resulting in the amplification of small random priming (background) oscillations of frequency ½ f1, and of a packet of low-frequency oscillations. The resonant interaction of waves in a boundary layer was investigated also by introducing a priming oscillation with frequency f’ = ½ f1 + Δf for different values of the frequency detuning Δf. The importance of the discovered wave interaction in boundary-layer transition is demonstrated. Causes of realization of different types of laminar-flow breakdown are discussed.

The problem of acoustic radiation generated by a spatially growing instability wave of a supersonic two-dimensional mixing layer is studied. It is shown that at high supersonic Mach numbers the classical locally parallel-flow hydrodynamic stability theory as well as the more recent theories based on the method of multiple scales (e.g. Saric & Nayfeh 1975; Crighton & Gaster 1976; Plaschko 1979; Tam & Morris 1980) would fail to give even a first-order instability wave solution. Physically, at these high flow speeds the radiated sound field is no longer an insignificant part of the total phenomenon. The disturbances associated with the flow-instability process now extend from the mixing layer all the way to the far field. The problem is therefore global in nature. Methods of solution which are predicated on local approximations such as the classical locally parallel-flow hydrodynamic-stability theory or the method of multiple scales are hence inappropriate and inapplicable. A global solution based on the method of matched asymptotic expansions is constructed. The outer solution is valid outside the mixing layer. It provides a mathematical description of the radiated acoustic field and the pressure near field. The near field in this case consists of both the acoustic and the hydrodynamic (non-propagating) fluctuation components. The inner solution is valid inside and in the immediate vicinity of the mixing layer. Physically it represents the instability wave of the flow. Matching is carried out according to the intermediate matching principle of Van Dyke (1975) and Cole (1968). Matching terms to order unity gives the basic instability-wave solution. Matching terms to the next order gives the instability- and acoustic-wave amplitude equation. For low-Mach-number flows it is found that the present results agree with the multiple-scales solution of Tam & Morris (1980).

A solution describing the spatial evolution of small-amplitude instability waves and their associated sound field of axisymmetric supersonic jets is found using the method of matched asymptotic expansions (see Part 1, Tam & Burton 1984). The inherent axisymmetry of the problem allows the instability waves to be decomposed into azimuthal wave modes. In addition, it is found that because of the cylindrical geometry of the problem the gauge functions of the inner expansion, unlike the case of two-dimensional mixing layers, are no longer just powers of ε. Instead they contain logarithmic terms. To test the validity of the theory, numerical results of the solution are compared with the experimental measurements of Troutt (1978) and Troutt & McLaughlin (1982). Two series of comparisons at Strouhal numbers 0.2 and 0.4 for a Mach-number 2.1 cold supersonic jet are made. The data compared include hot-wire measurements of the axial distribution of root-mean-squared jet centreline mass-velocity fluctuations and radial and axial distributions of near-field pressure-level contours measured by microphones. The former is used to test the accuracy of the inner (or instability-wave) solution. The latter is used to verify the correctness of the outer solution. Very favourable overall agreements between the calculated results and the experimental measurements are found. These very favourable agreements strongly suggest that the method of solution developed in Part 1 paper is indeed valid. Furthermore, they also offer concrete support to the proposition made previously by a number of investigators that instability waves are important noise sources in supersonic jets.

The stability of a decelerating boundary-layer flow is investigated experimentally and numerically. Experimentally, a flat plate having a Blasius boundary layer is decelerated in an 18 m towing tank. The boundary layer becomes unstable to two-dimensional waves, which break down into three-dimensional patterns, hairpin vortices, and finally turbulent bursts when the vortices lift off the wall. The unsteady boundary-layer equations are solved numerically to generate instantaneous velocity profiles for a range of boundary and initial conditions. A quasi-steady approximation is invoked and the stability of local velocity profiles is determined by solving the Orr–Sommerfeld equation using Chebyshev matrix methods. Comparisons are made between the numerical predictions and the experimentally observed instabilities.

This paper summarizes results of our investigations on: optimization of conditional sampling technique for eduction of naturally occurring large-scale structures in an axisymmetric mixing layer, comparison of the natural structure with that induced via controlled excitation, and the sensitivity of the educed structure to the excitation amplitude and of the natural coherent structure to Reynolds number and initial condition. Measurements include sectional-plane contours of various structure properties; however, coherent vorticity is the principal measure used in these considerations. All plausible alternative triggering criteria, based on reference velocity signals from fixed probes, were considered in order to arrive at the best practical eduction technique. It is shown that the simple criterion of triggering on the positive peaks of the longitudinal velocity signal derived from the high-speed edge of the mixing layer results in the optimum eduction. The characteristics of the natural structures, educed by the optimum detection criterion, are found to be independent of ReD over the measurement range 5.5 × 104−8 × 105. A mild dependence on the initial condition (viz laminar vs. turbulent) is observed, the structures being more disorganized for the initially laminar boundary-layer case. The educed natural structures agree well with those induced by controlled sinusoidal excitation at low excitation levels; higher levels, however, produce considerably stronger structures.

Two simple turbulent shear flows, namely a surface jet and a mixing layer, under the influence of stable gravity stratification, were investigated experimentally. The shear flows were generated in the laboratory by letting fresh water flow over saline water in a two-dimensional channel. Velocity and salinity measurements were made using a hot-film probe and a single-electrode conductivity probe. The experimental results for the two flows were correlated each using a different set of length and velocity scales. The initial development of the flows was relatively unaffected by the stable stratification. As the shear flows grew in thickness, they were observed to have a tendency to approach a ‘neutrally stable state’ in which the turbulent motion neither extracted energy from nor lost energy to the mean flow. The gradient Richardson number in this neutrally stable state was found to have the critical value predicted by linear inviscid stability theory. The decay of turbulent intensity in the longitudinal direction was observed to follow a power-law relationship similar to the one obtained by Comte-Bellot & Corrsin (1966) for the decay of grid-generated turbulence.

The flow between a shallow rotating cone and a stationary plate has been investigated using flow visualization, hot-film heat-transfer probes, and measurements of the torque required to rotate the cone against the retardation of the viscous fluid that fills the device. Theory appropriate to these experiments is also presented.

An expansion of the Navier–Stokes equations is performed for small values of the single parameter $\tilde{R} = r^2\omega\alpha^2/12\nu $. (Here r is the local radius, ω the angular velocity of the cone, α([Lt ] 1) is the angle between the cone and plate, and v is the fluid kinematic viscosity.) The measurements at low rotational speeds describe a simple linear velocity profile as predicted for the laminar flow of a Newtonian fluid. At larger rotational speeds, strong secondary flows are observed. There is agreement between the laminar theory and the measured streamline angles and shear stresses for values of $\tilde{R} < 0.5$. Turbulence is observed for $\tilde{R} \gtrsim 4$.

Under the auspices of the Engineering Foundation, financial support of the National Science Foundation, and the cochairmanship of the authors, a conference on ‘Double-Diffusive Convection’ was held from 14–18 March 1983 in Santa Barbara. The conference attracted more than seventy scientists and engineers working in various disciplines, and 45 talks were presented. There was an ad hoc film session which ended with a demonstration of a laboratory experiment on crystallization in a double-diffusive system.

Liquid bridges appear in a large variety of industrial processes such as the so-called floating-zone technique, used in recent years in crystal growth and in purification of high-melting-point materials.

In this paper the dynamics of axisymmetric, slender, viscous liquid bridges having volume close to the cylindrical one, and subjected to a small gravitational field parallel to the axis of the liquid bridge, is considered within the context of one-dimensional theories. Although the dynamics of liquid bridges has been treated through a numerical analysis in the inviscid case, numerical methods become inappropriate to study configurations close to the static stability limit because the evolution time, and thence the computing time, increases excessively. To avoid this difficulty, the problem of the evolution of these liquid bridges has been attacked through a nonlinear analysis based on the singular perturbation method and, whenever possible, the results obtained are compared with the numerical ones.