A theoretical one-dimensional model of penetrative convection in a stable temperature stratification heated from below has been developed in which the partial derivatives of temperature with respect to height and time are assumed to be discontinuous at the interface. As a finite temperature gradient then exists immediately above the interface, molecular diffusion effects at low Pélet number can be included. The results of the numerical-analysis model are used to illustrate the relative contributions of molecular diffusion, interfacial turbulence and the ‘filling’ of the existing temperature stratification by the lower boundary heat flux. Data from low-Pélet-number experiments are used to verify the results of the theoretical model.

The problem of turbulent-boundary-layer separation due to an adverse pressure gradient is an old but still important problem in many fluid flow devices. Until recent years little quantitative experimental information was available on the flow structure downstream of separation because of the lack of proper instrumentation. The directionally sensitive laser anemometer provides the ability to measure the instantaneous flow direction and magnitude accurately.

The experimental results described here are concerned with a nominally two-dimensional, separating turbulent boundary layer for an airfoil-type flow in which the flow was accelerated and then decelerated until separation. Upstream of separation single and cross-wire hot-wire anemometer measurements are also presented. Measurements in the separated zone with a directionally sensitive laser-anemometer system were obtained for U, V, $\overline{u^2}, \overline{v^2}, - \overline{uv}$, the fraction of time that the flow moves downstream, and the fraction of time that the flow moves away from the wall.

In addition to confirming the earlier conclusions of Simpson, Strickland & Barr (1977) regarding a separating airfoil-type turbulent boundary layer, much new information about the separated region has been gathered. (1) The backflow mean velocity profile scales on the maximum negative mean velocity UN and its distance from the wall N. A U+vs. y+ law-of-the-wall velocity profile is not consistent with this result. (2) The turbulent velocities are comparable with the mean velocity in the backflow, although low turbulent shearing stresses are present. (3) Mixing length and eddy viscosity models are physically meaningless in the backflow and have reduced values in the outer region of the separated flow.

Downstream of fully developed separation, the mean backflow appears to be divided into three layers: a viscous layer nearest the wall that is dominated by the turbulent flow unsteadiness but with little Reynolds shearing stress effects; a rather flat intermediate layer that seems to act as an overlap region between the viscous wall and outer regions; and the outer backflow region that is really part of the large-scaled outer region flow. The Reynolds shearing stress must be modelled by relating it to the turbulence structure and not to local mean velocity gradients. The mean velocities in the backflow are the results of time averaging the large turbulent fluctuations and are not related to the source of the turbulence.

The velocity-probability-distribution flatness and skewness factors for u and v are reported for the separating turbulent boundary layer described by Simpson, Chew & Shivaprasad (1981). Downstream of separation the skewness factor for u is negative near the wall, whereas it is positive upstream of separation. The flatness factor for u downstream of separation differs from the upstream behaviour in that it has a local maximum of about 4 at the minimum mean velocity location in the backflow. Both upstream and downstream of separation the skewness factor for v has a profile shape and magnitudes that are approximately the mirror image or negative of the skewness factor for u. The flatness factor for v seems to be affected little by separation.

Examination of the momentum and turbulence-energy equations reveals that the effects of normal stresses are important in a separating boundary layer. Negligible turbulence-energy production occurs in the backflow. Turbulence-energy diffusion is increasingly significant as separation is approached and is the mechanism for supplying turbulence energy to the backflow.

The backflow appears to be controlled by the large-scale eddies in the outer region flow, which provides the mechanism for turbulence-energy diffusion. The backflow behaviour does not appear to be significantly dependent on the far downstream near-wall conditions when the thickness of the backflow region is small compared with the turbulent shear layer thickness.

Simpson, Chew & Shivaprasad (1981a, b) describe many experimentally determined features of a separating turbulent boundary layer. For the same flow, experimental results for the transverse velocity component are presented here. A specially designed directionally sensitive laser anemometer was constructed and used to make measurements in the separated region. Cross-wire hot-wire anemometer measurements were obtained upstream of separation and in the outer region of the separated flow and are in good agreement with the laser anemometer results.

It was found that w’2 = v’2 in the outer 90% of the shear layer both upstream and downstream of separation. Features of w’2 profiles in the backflow are related to features of the streamwise velocity component. This behaviour is consistent with the large-scale-structures flow model of a separating boundary layer presented by Simpson et al. (1981a, b).

Large-scale structures supply the mean streamwise backflow. These large-scale structures also transport the turbulence energy to the backflow from the outer flow by turbulent diffusion since advection and production of turbulence kinetic energy are negligible there compared with the dissipation rate. Because of continuity requirements fluid motions toward the wall must be deflected and contribute to streamwise and transverse motions near the wall.

An analysis of equilibrium boundary layers based on the Schofield–Perry defect law, which applies to flow in a moderate to strong adverse pressure gradient, is presented. The conditions derived for self-preserving or equilibrium boundary layers differ from those given by previous analyses based on the usual velocity-defect law. It is shown that twelve observed boundary layers on smooth walls conform to these new conditions for precise equilibrium flow. As the analytical expression for the Schofield–Perry defect law does not vary with pressure gradient, a specific expression for the shear-stress profile in any equilibrium layer can be derived. The predicted shear-stress profiles show good agreement with experimental data. Limits for the flow parameters within which equilibrium layers can exist are derived, and it is shown that observed equilibrium layers fall within these limits. A prediction method for layers in smoothly changing adverse-pressure gradients is outlined and demonstrated using equilibrium data. The unified account of equilibrium flow in adverse pressure gradients presented here is used to resolve some disagreements in the literature concerning existence conditions for equilibrium boundary layers.

A complete description is sought for the two-dimensional laminar flow response of an incompressible boundary layer encountering a hump on an otherwise smooth boundary. Given that the typical Reynolds number Re (based on the development length L* of the boundary layer) is large, the flow characteristics depend on only two parameters, the non-dimensional length and height scales l, h of the obstacle. For short humps of length less than the familiar O(Re−⅜) triple-deck size the critical height scale, which produces a nonlinear interaction and hence the prospect of separation, is of order $Re^{-\frac{1}{2}}l^{\frac{1}{3}}$. For long humps whose length is greater than the triple-deck size the corresponding critical height scale is much bigger, of order $l^{\frac{5}{3}}$. Height scales below critical produce only a weak flow response while height scales above critical force relatively large-scale separated motions to occur. In the paper the flow structures and typical solutions produced by two representative cases, a short obstacle of length comparable with the oncoming boundary-layer thickness and a long obstacle of height comparable with the boundary-layer thickness, are mainly considered. The former case is controlled by the unknown pressure force induced locally in the flow near the hump and by two length scales, that of the hump itself and that of the longer triple deck. The latter case is governed mainly by the inviscid externally produced pressure force. Alternatively, however, all the dominant flow properties in both cases can be obtained as special or limiting solutions of the triple-deck problem. Comparisons between the cases studied are also presented.

Two-dimensional convection in a Boussinesq fluid in the presence of an imposed magnetic field is described in terms of a simplified model, which is exact to second order in the amplitude of the motion and appears to be qualitatively correct for larger amplitudes. If the ratio of the magnetic diffusivity to the thermal diffusivity is sufficiently small and the imposed magnetic field is sufficiently large, convection sets in when r = r(o) as overstable oscillations, which grow in amplitude as the normalized Rayleigh number r is increased. There is also a branch of steady solutions that bifurcates from the static equilibrium at r = r(e) < r(o) and stable steady solutions exist for r > rmin. For certain choices of parameters subcritical steady convection, with rmin < r(e), is found and the oscillatory branch ends on the unstable portion of the steady branch, where the period of the oscillations becomes infinite. In some circumstances there may be a bifurcation from symmetrical to asymmetrical oscillations, followed by a sequence of bifurcations at each of which the period doubles. Other choices of parameters allow only supercritical convection with r increasing monotonically on the steady branch; if convection first appears as overstable oscillations the steady branch is then unstable for r(e) < r < rmin and there is a Hopf bifurcation at r = rmin. This complicated pattern of behaviour is consistent with the results of numerical experiments on the full two-dimensional problem.

In this paper we describe a simple way of modelling boundary-layer effects in analytical flow noise studies. We develop an exact analogy between the real flow and one in which there is a step velocity profile. This profile is intended to model a boundary layer in an idealized way and we recognize it in our Green's function for the problem. We insist that the Green's function is bounded, a step that makes it non-causal and similar to those used in recent jet-noise analogies. We derive an expression for the induced pressure which consists of surface and volume terms, just as in Lighthill's theory, but, because both contain elements to be evaluated in future time, we argue that the turbulence must be able to respond to linear surface stimulus and avoid the otherwise inevitable violation of causality. This novel feature distinguishes our analysis from applications of Lighthill's theory to boundary-layer-induced noise. The response of the turbulence may be large when the surface is driven at low boundary-layer Strouhal number. But it is negligible at high Strouhal number, and in that limit the surface terms are found to depend only on the instantaneous boundary geometry and its rate of change. This leads to a simple expression for ‘boundary-layer fluid loading’, in which the finite boundary-layer scale emerges explicitly. To illustrate the physical consequences of this result, we use it to estimate the impedance of a baffled piston vibrating beneath a boundary layer. Potential theory predicts that flow should destabilize the piston motion while experiments usually indicate the reverse. We find that the boundary layer is responsible for the discrepancy and that experimentally observed behaviour is predicted quite reasonably by our model.

Experimental results are presented concerning the spontaneous oscillations observed when a high-viscosity fluid jet flows vertically against a flat surface. The two jet shapes investigated were the axisymmetric jet and the plane jet. The minimum distance from the jet orifice to the flat surface for which these oscillations are observed, termed the ‘buckling height’, was determined experimentally. The frequency of the subsequent oscillations was also determined. Both were measured as functions of fluid and flow variables. It is found that surface tension effects are the dominant factors influencing the buckling height, while the rate of oscillation is affected by both surface tension effects and by viscous, gravity and inertia effects. The major results are presented in non-dimensional form. Photographs of the buckling phenomenon are provided for representative jet geometries. It is also established experimentally that there is an upper limit to the flow Reynolds number above which buckling does not occur.

Differences in experimental results for surface gravity wave phase speeds obtained by cross spectra and slope-height spectra are compared. It is shown that, for a simple two-dimensional model containing both dispersive and non-dispersive waves, the observed differences can be explained readily. The slope-height technique measures a weighted average of the wavenumber squared; for frequencies sufficiently large compared with the dominant frequency, the computed phase-speed variation with frequency is representative of the wave having the larger wavenumber - the dispersive wave. For the cross-spectral method, it is shown that the small-probe-separation assumption usually employed is not valid for frequencies much larger than the dominant frequency, except at the singular point where both waves have exactly equal spectral densities and the phase function represents an average of the two modes. For all other cases, the phase function approaches that of the wave having the larger spectral density, and essentially ignores the presence of the other wave, even if both modes are relatively close in their contributions to the total spectral density.

An analysis is presented of the dynamics of a small deformable capsule freely suspended in a viscous fluid undergoing shear. The capsule consists of an elastic membrane which encloses another viscous fluid, and it deforms in response to the applied external stresses and the elastic forces generated within the membrane. Equations are derived which give its time-dependent deformation in the limit that the departure of the shape from sphericity is small. The form of the shear flow is arbitrary and a general (two-dimensional) elastic material is considered. Limiting forms are obtained for highly viscous capsules and for membranes which are area-preserving, and earlier results for surface tension droplets and incompressible isotropic membranes are derived as particular cases. Results for the viscosity of a dilute suspension of capsules are also given.

The theoretical prediction for the relaxation rate of the shape is derived for an interface which has elastic properties appropriate for a red-blood-cell membrane, and is compared with experimental observations of erythrocytes.

An accurate representation of the motion-causing buoyancy force, in the vicinity of the maximum-density condition in water at low temperatures, is determined by using a very accurate and quite simple equation of state for both pure and saline water. The resulting general laminar boundary-layer flow equations admit similarity solutions and, in the absence of salinity diffusion, require only one additional new dimensionless variable, called R. For flows of constant heat content in an unstratified ambient fluid, similarity is found only when the far-field temperature is at the temperature of maximum density. The three flow situations considered here are two above a horizontal line heat source and one above a point heat source. The first flow is a freely rising plane plume and the second is a wall plume (flow over a vertical, adiabatic surface with a horizontal line source imbedded in it). The third flow is the freely rising axisymmetric plume. These are the models in laminar theory of many processes which arise in cold water. A primary objective here is to calculate the effect of using a nonlinear density relation for water, which is much more accurate at low temperatures than the conventional linear one used in ‘classical’ analyses. The downstream variations of the temperature and velocity fields are found to be very different from those for flows where the effect of a density extremum is not included.

The problem of describing an unsteady cylindrical pipe flow with one-dimensional equations is investigated, and an exact method for obtaining a closure relationship is proposed for the transient shear stress in a laminar flow submitted to an arbitrary transient pressure gradient. Extensive comparisons are given for a step or a harmonic pressure gradient between the approximate solution derived from this method, some results of the literature and exact solutions of the Navier–Stokes equations.

A differencing scheme known as the ‘Fluid-in-Cell’ method has been used in the numerical simulation of a choked jet of air impinging on a flat plate. Before sonic conditions are applied at the nozzle exit, the field of interest is at rest at the ambient pressure and temperature. The instantaneous application of sonic conditions at the nozzle exit is tantamount to the sudden appearance of a normal shock wave whose strength is determined by the experimental conditions.

The results of the simulation describe the decay of the initial shock wave and its reflection at the plate; the formation of a second shock wave and its merging with the reflected shock giving rise to a detached shock wave which oscillates; and the growth and subsequent motion of a toroidal vortex that is generated between this shock wave and the plate. The results show clearly how the flow field which has been observed in physical experiments under stable operating conditions is developed.

Consider a rigid body which is performing simple harmonic oscillations of small amplitude in the free surface of deep water under gravity. Under certain geometrical conditions on ∂D, the wetted surface of the body, it is known that the linear boundary-value problem [Pscr ] for a corresponding velocity potential ϕ is uniquely solvable at all frequencies. The usual method for solving [Pscr ] is to derive a Fredholm integral equation of the second kind over ∂D. There are two familiar ways of doing this: (i) represent ϕ as a distribution of simple wave sources over ∂D, leading to an integral equation for the unknown source strength; (ii) apply Green's theorem to ϕ and a simple wave source; when the field point lies on ∂D, this gives an integral equation for the boundary values of ϕ. It is well known that both of these integral equations have unique solutions, except at the same infinite discrete set of frequencies (the irregular frequencies).

In this paper, we shall describe an alternative method for solving [Pscr ]: when the field point, in (ii), lies inside the body, we obtain an integral relation. If the simple wave source has a suitable bilinear expansion, this integral relation may be reduced to an infinite set of equations for the boundary values of ϕ. These equations, called the ‘null-field equations for water waves’, appear to be new; equations of this type were first obtained by Waterman for electromagnetic and acoustic scattering problems. The required bilinear expansion has been given by Ursell (1981) for two dimensions, and is given here for three dimensions. Using these, we show that the null-field equations always have a unique solution – irregular frequencies do not occur. This result is proved here for water waves in two and three dimensions. Similar results may be obtained for water of constant finite depth.

The thermal non-equilibrium boundary layer at the end wall of a shock tube in methane combustion initiated by a reflected shock, is investigated theoretically and experimentally. Time-dependent boundary conditions are caused by the shock–boundary-layer interaction and the combustion process outside the boundary layer, and are taken into account. Space- and time-resolved density measurements using a focused laser beam are in good agreement with results from numerical computation and show existence of similarity solutions for short and long times. Differences, so far unexplained, occur between measured and predicted heat fluxes to the wall.

Preliminary experimental results are presented which describe the interaction of an internal-wave field with a steady shearing motion. The results are primarily qualitative and presented in the form of photographs of shadowgraph images. Several internal-wave sources are used, and both critical- and non-critical-layer flows are examined. The results of these observations are interpreted in terms of several existing theories. For critical-layer flows the primary result is that virtually none of the internal-wave momentum flux penetrates the critical-level region, and under certain conditions a critical-layer instability develops resulting in the generation of turbulence. Such wave-induced turbulence is also observed for certain non-critical-layer flows and is believed to be the result of a convective instability.

A uniform, lateral heat flux applied to a body of fluid with a vertical solute gradient, causes the horizontal growth of cellular convection. This paper deals with a series of experimental studies of the phenomenon, which yielded the following results. (1) The third non-dimensional parameter besides the Prandtl number and the diffusivity ratio, τ, in this particular double-diffusive convection was found to be π3 = − α(q/k)/β(dS/dz) in which q, k, −(dS/dz) are the lateral heat flux, the thermal conductivity of the fluid and the initial solute gradient respectively, and where α = −ρ−1(∂ρ/∂ T), and β = ρ−1(∂ρ/∂ S), ρ being the density. The existence of a critical value of π3 above which cellular convection occurs has been confirmed in this study. (2) When the solute used was varied (i.e. with τ different), the corresponding shift in the critical value of π3 was also observed indicating that the critical values of π3 are 0·13 for CuSO4 (τ = 3·5 × 10−3), 0·28 for common salt (τ = 9 × 10−3) and 0·76 for HCl (τ = 24 × 10−3). (3) The measured vertical height of a cell, h, when normalized with respect to a characteristic length L(= [vk/gα(q/k)]1/4 where v and k are the kinematic viscosity and the thermal diffusivity, respectively), increased steadily with π3, and for a given value of π3, h/L decreased with an increase in τ. (4) Studies of the shadowgraph pictures indicated that the initially developed roll cells quickly merge to form layers of outward-growing convection cells.

This note has two aspects. The first concerns comparisons between the theoretical predictions of Smith (1979b) and the calculations of Fornberg (1980) for flow past a circular cylinder: there is quite good agreement overall, in such quantities as the drag, the front stagnation point pressure, the eddy pressure and the skin friction. The second aspect concerns comments and reservations on the calculated eddy lengths and discrepancies at higher Reynolds numbers.

A rough-wall turbulent boundary layer which grows into a temperature interface situated at the outer edge is investigated experimentally. Reynolds numbers based on boundary-layer thickness δ range from 4 × 103 to 104. Overall Richardson numbers Ri*, defined in terms of the friction velocity and the boundary-layer thickness, are in the range 0 ≤ Ri* [lsim ] 80. Measurements of the mean profiles and of the variances of the velocity fluctuations show that the interface acts in some respects like a moving wall: the velocity profile tends towards a turbulent Couette-type profile and the longitudinal r.m.s. turbulent velocity begins to be amplified at the base of the interfacial layer and reaches a maximum in about the centre. Time-lag correlations of fluctuating quantities taken just above the centre of the interfacial layer have a behaviour characteristic of internal waves, namely a 90° phase lag between vertical velocity and temperature fluctuations. These waves occur as short wave packets and propagate mainly horizontally. On the base of the interface the correlations exhibit the usual symmetric behaviour.

The normalized entrainment velocity ue/u* decreases when Ri* increases but does not follow a power law in Ri*. This is consistent with momentum balance, which indicates that ue/u* also depends on the mean-flow Richardson number Ri0 = gΔTδ/TU02 and on the change in momentum and temperature defect. Momentum balance also shows that, when the undisturbed flow has zero pressure gradient, the boundary layer is expected to separate owing to entrainment when Ri0 ≅ 0.5