The model proposed by Smith & Stewartson (1973), to describe the separated boundary layer induced by strong injection over a finite length of a flat plate in a supersonic mainstream, is shown to provide the basis for a fully consistent solution of the Navier–Stokes equations for this problem, valid in the limit of infinite Reynolds number. The solution takes the form of asymptotic expansions in each of a large number of overlapping regions of the flow field, which are consistently matched across areas of common validity.

The response of a Pitot probe in a uniform laminar stream is commonly expressed in the form \[ P_s = P+{\textstyle\frac{1}{2}}C_{\rho} U^2, \] where Ps is the probe signal pressure, P is the stream static pressure, U is the stream speed and ρ is the fluid density. It has been found that for ordinary sphere-nosed, round-nosed and square-nosed probes \[ 1-C = K(\sin^2\theta)^m = K(U^2_n/U^2)^m, \] where θ is the angle between the velocity vector and the probe axis, and Un ≡ U sin θ is the transverse velocity component. The parameters m and K are functions of the probe geometry. These formulae also describe the performance in a turbulent stream when the probe is small compared with the turbulence scale. The evaluation of the time-averaged response is treated, and an answer is developed to the question of what it is that a Pitot probe measures in a turbulent stream. In a turbulent shear flow having the properties of a boundary layer, the reference pressure is best taken to be the static pressure at the shear-layer edge. It is shown that round-nosed probes with Di/D≃0·45 and square-nosed probes with Di/D≃0·15 then detect ${\textstyle\frac{1}{2}}\rho\overline{U}^2_x$ with good accuracy, where Di/D is the ratio of the inside and outside diameters of the Pitot tube. When measurements are made with two probes of dissimilar geometry, the differential response can be used to find the mean-square level of the transverse velocity fluctuations. Turbulence levels so measured agree closely with results from hot-wire anemometry.

The onset of convection is observed in a cylindrical annulus which is heated from the outside, cooled from the inside and rotating about its vertical axis of symmetry. The dynamical constraint exerted by the dominating Coriolis force inhibits the instability when the top and bottom boundaries of the annulus are conical so as to make the vertical height vary with distance from the axis. The experimental observations are in good agreement with the theoretical predictions by Busse (1970a). This confirmation indicates the absence of subcritical finite amplitude instabilities and suggests that the annulus experiment provides a close dynamical model for convection in the liquid core of the earth.

Internal Alfvén-gravity waves of small amplitude propagating in a Boussinesq, inviscid, adiabatic, finitely conducting fluid in the presence of a uniform transverse magnetic field in which the mean horizontal velocity U(z) depends on height z only are considered. We find that the governing wave equation is singular only at the Doppler-shifted frequency Ωd = 0 and not at the magnetic singularities Ωd = ± ΩA, where ΩA is the Alfvén frequency. Hence the effect of ohmic dissipation is to prevent the resulting wave equation from having magnetic singularities. Asymptotic solutions of the wave equation, which is a fourth-order differential equation, are obtained. They show the presence of the magnetic Stokes points Ωd = ± ΩA. The interpretation of upward and downward propagation of waves is also discussed.

To study the combined effect of electrical conductivity and the magnetic field on waves at the critical level, we have used the group-velocity approach and found that the waves are transmitted across the magnetic Stokes points but are completely absorbed at the hydrodynamic critical level Ωd = 0. The general expression for the momentum flux is mathematically complicated but will be simplified under the assumption \[ \frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial y^2}\gg \frac{\partial^2h}{\partial z^2}, \] where h is the perturbation magnetic field. In this approximation we find that the momentum flux is not conserved and the waves are completely absorbed at Ωd = 0.

The general theory is applied to a particular problem of flow over a sinusoidal corrugation and asymptotic solutions are obtained by applying the Laplace transformation and using the method of steepest descent.

Wind-tunnel turbulence behind a parallel-rod grid with jets evenly distributed along each rod is nearly isotropic. Homogeneity improvement over prior related experiments was attained by the use of controllable nozzles. Compared with the ‘passive’ case, the downwind-jet ‘active’ grid has a smaller static pressure drop across it and gives a smaller turbulence level at a prescribed distance from it, while the upwind-jet grid gives a larger static pressure drop and larger turbulence level. ‘Counterflow injection’ generates larger turbulence energy and larger scales, both events being evidently associated with instability of the jet system. This behaviour is much like that commonly observed behind passive grids of higher solidities.

If the turbulent kinetic energy is approximated as an inverse power law in distance, the (positive) exponent decreases with increasing (downwind or upwind) jet strength, corresponding to slower absolute decay rates. No peculiar decay behaviour occurs when the jet grid is ‘self-propelled’ (zero net average force), or when the static pressure drop across it is zero.

The injection does not change the general behaviour of the energy spectra, although the absolute spectra change inasmuch as the turbulence kinetic energy changes.

Consistency and uniqueness questions raised by both the 1941 and 1962 Kolmogorov inertial-range theories are examined. The 1941 theory, although unlikely from the viewpoint of vortex-stretching physics, is not ruled out just because the dissipation fluctuates; but self-consistency requires that dissipation fluctuations be confined to dissipation-range scales by a spacewise mixing mechanism. The basic idea of the 1962 theory is a self-similar cascade mechanism which produces systematically increasing intermittency with a decrease of scale size. This concept in itself requires neither the third Kolmogorov hypothesis (log-normality of locally averaged dissipation) nor the first hypothesis (universality of small-scale statistics as functions of scale-size ratios and locally averaged dissipation). It does not even imply that the inertial range exhibits power laws. A central role for dissipation seems arbitrary since conservation alone yields no simple relation between the local dissipation rate and the corresponding proper inertial-range quantity: the local rate of energy transfer. A model rate equation for the evolution of probability densities is used to illustrate that even scalar nonlinear cascade processes need not yield asymptotic log-normality. The approximate experimental support for Kolmogorov's hypothesis takes on added significance in view of the wide variety of a priori admissible alternatives.

If the Kolmogorov law $E(k) \propto k^{-\frac{5}{3}-\mu}$ is asymptotically valid, it is argued that the value of μ depends on the details of the nonlinear interaction embodied in the Navier–Stokes equation and cannot be deduced from overall symmetries, invariances and dimensionality. A dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equation but which has radically different inertial-range behaviour.

We consider axisymmetric steady centrifugally driven thermal convection in a compressible fluid in a rapidly rotating circular cylinder. The Boussinesq approximation is not used, because it is not valid for the case of practical interest. We clarify the importance of the effect of the flow-induced volume change of a fluid particle, and propose a widely applicable method of solution.

Data are presented for the extensional flow of suspensions containing 0.1−1 % of fibres by volume. The aspect ratio of the fibres was varied from 280 to 1260. The observed stress levels were between one and two orders of magnitude greater than in shearing flow, in agreement with the a priori predictions of Batchelor (1971).

Kolmogorov's ‘third hypothesis’ asserts that in intermittent turbulence the average ε of the dissipation ε, taken over any domain D, is ruled by the lognormal probability distribution. This hypothesis will be shown to be inconsistent, save under assumptions that are extreme and unlikely. Further, a widely used justification of lognormality, due to Yaglom and based on probabilistic argument involving a self-similar cascade, will be discussed. In this model, lognormality indeed applies strictly when D is ‘an eddy’, typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when D consists of a few such eddies. On the other hand, the experimental situation is described better by considering averages taken over essentially one-dimensional domains D. The first purpose of this paper is to carry out Yaglom's cascade argument, labelled as ‘microcanonical’, for such averaging domains. The second is to replace Yaglom's model by a different, less constrained one, based upon the concept of ‘canonical cascade’. It will be shown, both for one-dimensional domains in a microcanonical cascade, and for all domains in canonical cascades, that in every non-degenerate caJe the distribution of ε differs from the lognormal distribution. Depending upon various parameters, the discrepancy may be either moderate, or considerable, or even extreme. In the latter two cases, high-order moments of E turn out to be infinite. This avoids various paradoxes (to be explored) that are present in Kolmogorov's and Yaglom's approaches. The third purpose is to note that high-order moments become infinite only when the number of levels of the cascade tends to infinity, meaning that the internal scale η tends to zero. Granted the usual value of η, this number of levels is actually small, so the representativity of the limit is questionable. This issue was investigated through computer simulation. The results bear upon the question of the extent to which Kolmogorov's second hypothesis applies in the face of intermittency. The fourth purpose is as follows. Yaglom noted that the cascade model predicts that dissipation only occurs in a portion of space of very small total volume. In order to describe the structure of this portion of space, it will be shown useful to introduce the concept of the ‘intrinsic fractional dimension’ A of the carrier of intermittent turbulence. The fifth purpose is to study the relations between the parameters ruling the distribution of η, and those ruling its spectral and dimensional properties. Both conceptually and numerically, these various parameters turn out to be distinct, which opens up several problems for empirical study.

A study is made of uniform flow past a semi-infinite flat plate with a similarity injection distribution of boundary-layer magnitude. Attention is focused on a solution at exactly the critical injection rate for which classical boundary-layer theory predicts the blow-off singularity. Following a description of the more recent interaction analyses which also fail at the critical rate, a new theory is developed which leads to physically meaningful results. In particular, it is shown that the non-monotonic variation in wall shear with increasing injection rate near the critical value, noted by Klemp & Acrivos (1972), is real. A delicate interplay of weak pressure interactions and viscous effects is shown to be responsible for this surprising phenomenon.

General dimensional and similarity arguments are applied to derive a heat and mass transfer law for fully turbulent flow along a rough wall. The derivation is quite analogous to Millikan's (1939) derivation of a skin-friction law for smooth-and rough-wall flows and to the derivation of the heat and mass transfer law for smooth-wall flows by Fortier (1968a, b) and Kader & Yaglom (1970, 1972).

The equations derived for the heat or mass transfer coefficient (Stanton number) Ch and Nusselt number Nu include the constant term β of the logarithmic equation for the mean temperature or concentration of a diffusing substance. This term is a function of the Prandtl number, the dimensionless height of wall protrusions and of the parameters describing the shapes and spatial distribution of the protrusions. The general form of the function β is roughly estimated by a simplified analysis of the eddy-diffusivity behaviour in the proximity of the wall (in the gaps between the wall protrusions). Approximate values of the numerical coefficients of the equation for β are found from measurements of the mean velocity and temperature (or concentration) above rough walls. The equation agrees satisfactorily with all the available experimental data. It is noted that the results obtained indicate that roughness affects heat and mass transfer in two ways: it produces the additional disturbances augmenting the heat and mass transfer and simultaneously retards the fluid flow in the proximity of the wall. This second effect leads in some cases to deterioration of heat and mass transfer from a rough wall as compared with the case of a smooth wall at the same values of the Reynolds and Prandtl numbers.

A fluid layer that lines the inner surface of a circular tube has motion induced by axial surface-tension gradients. The lubrication equations for the system are analysed and it is found that even for thin layers the motions differ markedly from those in planar layers. The planar case serves as a class of outer solutions. These approximate solutions are modified by a boundary-layer correction where the mean surface tension is important.

The distortion of a gas bubble rising steadily in an inviscid incompressible liquid of infinite extent under the action of surface tension forces is investigated theoretically using an appropriate extension of the tensor virial theorem. A convenient parameter for distinguishing the bubble shape is the Weber number W. The virial method leads to an expression relating W and the axis ratio χ, of the transverse and longitudinal axes of the bubble. To first order in W, this relation agrees with the linear theory established by Moore (1959). Also, comparison of the results with his (1965) approximate theory reveals similar features and excellent agreement up to χ = 2. In particular, it confirms his prediction of the existence of a maximum Weber number. Although the present work does not consider the stability of these bubbles, it is interesting to note that the maximum value of 3.271 attained by W differs only by about 2.8% from the critical Weber number obtained by Hartunian & Sears (1957) for the onset of instability.

An approximate method for the study of slightly distorted spheroidal gas bubbles is also formulated and the resulting boundary-value problem solved numerically. The theory is then extended to include gravity. The joint effect of surface tension as well as gravitational forces has not been included in earlier theories. The shapes of the bubbles are traced and compared with the unperturbed spheroids. Comparisons for the velocity of bubble rise are made between the present predictions and some experimental results. In particular the results are compared with recent experimental data for the motion of gas bubbles in liquid metals.

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to be more stable than that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

A numerical solution is presented for the flow of a uniform stream past a semi-infinite heated flat plate at whose surface the heat flux remains constant. The buoyancy forces oppose the free-stream motion and separation occurs. An examination of the singularities in the skin-friction and heat-transfer coefficients suggests, rather surprisingly, a behaviour as (ξs−ξ)⅗ at separation.

Edge waves travelling along a straight coastline are examined in the case when the water depth approaches a constant value at large distances from the coast. Only the fundamental mode is examined in the limit as the ratio of the water depth at infinity to the edge-wave wavelength tends to zero. Two comparison theorems are used to obtain upper and lower bounds for the dispersion relation. A long-wave approximation procedure is used to obtain the leading terms in the dispersion relation for a wide class of bottom topographies. The results obtained are compared with an exact result for the case when the bottom topography is a rectangular step.

We consider the low Rossby number (R) flow of a stratified fluid in a long rotating channel, for which the bottom elevation varies in the downstream direction. The quasi-geostrophic response is shown to be singular at the side walls of the channel, and thus an ageostrophic analysis is necessary even for vanishingly small R. Part of the ageostrophic steady-state response is a modified quasi-geostrophic perturbation trapped near the bottom. A second component which is present even as R approaches zero is an internal Kelvin wave whose vertical wavelength adjusts so that the wave remains stationary with respect to the channel bottom and which propagates energy and momentum to infinite heights in an unbounded channel. The case of a bounded layer of fluid is also considered, and the resonance conditions are given. We also calculate the flow field when the bottom elevation varies in the cross-stream direction. We conclude that stagnation or flow reversal can be caused either by the modified quasi-geostrophic component or by the Kelvin wave and estimate the critical condition by an extrapolation of the perturbation velocity computed from linear theory.

The steady flow in and around a deformable liquid sphere moving in an unbounded viscous parabolic flow and subject to an external body force is calculated for small values of the ratio of the Weber number to the Reynolds number in the creeping-flow regime. It is found that, in addition to the drag force, the drop experiences a force orthogonal to the undisturbed flow direction. When the body force is absent (neutrally buoyant drop), this lift force tends to drive the drop inwards to the axis, where the undisturbed flow velocity is maximum, i.e., towards a position of lower velocity gradient. In the case for which the parabolic flow profile is a Poiseuille flow profile, the lift force is given by the expression. \[ {\bf F}_1 =-6\pi\mu\epsilon U_0\frac{\alpha +\frac{2}{3}}{\alpha + 1}\bigg(\frac{a}{R_0}\bigg)^4{\bf b}F[1+o(\epsilon)]. \] Here a is the radius of the undeformed sphere, R0 is the radial distance from the position of maximum undisturbed flow U0 at the profile axis to the position of zero flow, ε is the ratio of the Weber number to the Reynolds number, given by ε=μU0T−1, where μ is the external fluid viscosity and T is the surface tension of the drop, α is the ratio of the drop and external fluid viscosities, b is the radial vector from the flow axis to the centre of mass of the drop, and F is a function of α and a dimensionless parameter dependent on the body force that is determined in the analysis. Reasonable agreement is found between the observations by Goldsmith & Mason (1962) of the axial drift of liquid drops in Poiseuille flow and the predictions of the theory herein.

The method of energy is used to develop two stability criteria for a large class of modulated Bénard problems. Both criteria give stability limits which hold for disturbances of arbitrary amplitude. The first of these, designated as strong global stability, requires the energy of all disturbances to decay monotonically and exponentially in time. Application of this criterion results in a prediction of Rayleigh numbers below which the diffusive stagnant solution to the Bous-sinesq equations is unique. The second criterion requires only that disturbances decay asymptotically to zero over many cycles of modulation, and is a weaker concept of stability. Computational results using both criteria are given for a wide range of specific cases for which linear asymptotic stability results are available, and it is seen that the energy and linear limits often lie close to one another.

The equations of conservation of momentum, energy and mass together with the equations of state are examined for free convection from a vertical paraboloid. A transformation due to Saville & Churchill is applied to the first- and second-order boundary-layer equations, which are then solved using series about the stagnation point, using asymptotic series far up the body and in between by a method due to Merk. The second-order outer inviscid flow is given in terms of infinite integrals as a solution of Laplace's equation in paraboloidal co-ordinates.

Eight second-order effects are distinguished, depending on longitudinal and transverse curvatures, the displacement flow, heat flux into the boundary layer and the variation of density, viscosity, thermometric conductivity and the coefficient of expansion with temperature. Expressions for the skin friction, heat-transfer coefficient and various flux thicknesses are obtained and a comparison of the second-order effects is made.