Several examples of incipient blow-off phenomena described by the compressible similar laminar boundary-layer equations are considered. An asymptotic technique based on the limit of small wall shear, and the use of a novel form of Prandtl's transposition theorem, leads to a complete analytical description of the blow-off behaviour. Of particular interest are the results for overall boundarylayer thickness, which imply that, for a given large Reynolds number, classical theory fails for a sufficiently small wall shear. A derivation of a new distinguished limit of the Navier–Stokes equations, the use of which will lead to uniformly valid solutions to blow-off type problems for Re → ∞, is included. A solution for uniform flow past a flat plate with classical similarity type injection, based on the new limit, is presented. It is shown that interaction of the injectant layers and the external flow results in a favourable pressure gradient, which precludes the classical blow-off catastrophy.

Amongst the more important laboratory experiments which have produced concentrated vortices in rotating tanks are the sink experiments of Long and the bubble convection experiments of Turner & Lilly. This paper describes a numerical experiment which draws from the laboratory experiments those features which are believed to be most relevant to atmospheric vortices such as tornadoes and waterspouts.

In the numerical model the mechanism driving the vortices is represented by an externally specified vertical body force field defined in a narrow neighbourhood of the axis of rotation. The body force field is applied to a tank of fluid initially in a state of rigid rotation and the subsequent flow development is obtained by solving the Navier–Stokes equations as an initial-value problem.

Earlier investigations have revealed that concentrated vortices will form only for a restricted range of flow parameters, and for the numerical experiment this range was selected using an order-of-magnitude analysis of the steady Navier–Stokes equations for sink vortices performed by Morton. With values of the flow parameters obtained in this way, concentrated vortices with angular velocities up to 30 times that of the tank are generated, whereas only much weaker vortices are formed at other parametric states. The numerical solutions are also used to investigate the comparative effect of a free upper surface and a no-slip lid.

The concentrated vortices produced in the numerical experiment grow downwards from near the top of the tank until they reach the bottom plate whereupon they strengthen rapidly before reaching a quasi-steady state. In the quasi-steady state the flow in the tank typically consists of the vortex at the axis of rotation, strong inflow and outflow boundary layers at the bottom and top plates respectively, and a region of slowly-rotating descending flow over the remainder of the tank. The flow is cyclonic (i.e. in the same sense as the tank) in the vortex core and over most of the bottom half of the tank and is anticyclonic over the upper half of the tank away from the axis of rotation.

Recently Cheng & Akiyama (1970) published a numerical analysis of laminar flow in curved channels of square and rectangular section. Experimental results are presented here for flow in curved channels of square section. The channels were toroidal in shape, and the flow was driven electromagnetically. Various ratios of the channel dimension d to the channel radius of curvature, R, were used to investigate the dependence of friction factor, f, on the Dean number K, and the Reynolds number, Re. For 5 × 102 < K < 7 × 104 the formula (fRe) = 1·51 K½ was found to fit all the results, although R/d was varied from 17·5 down to the low value of 1·75. At lower values of K the analysis of Cheng & Akiyama was approximately validated.

Linear stability theory is applied to the natural convection boundary layer arising from a vertical plate dissipating a uniform heat flux. By using a numerical procedure which is much simpler than those previously employed on this problem, computer solutions are obtained for a much larger range of the Grashof number (G). For a Prandtl number (σ) of 0·733, it is found that, as G → ∞: the effect of temperature coupling vanishes more rapidly than that of viscosity; the upper branch of the neutral curve is oscillatory but does approach a finite non-zero inviscid asymptote. For moderate and large values of σ, a loop appears in the neutral stability curve as a result of the merging of two unstable modes. As σ → ∞, the mode associated with the uncoupled (i.e. Orr–Sommerfeld) problem rapidly becomes less unstable than that arising from the temperature coupling, with the stability characteristics being independent of the thermal capacity of the plate. For small values of σ, only one unstable mode is found to exist with the coupling effect being negligible in the case of large thermal capacity plates but markedly destabilizing when the thermal capacity is small.

By obtaining numerical results out to G ≈ 1010 for the cases σ = 0·733 and 6·7, it becomes possible to attempt to directly relate the theory to the actual observance of turbulent transition. Based upon comparison with available experimental data, empirical correlations are obtained between the linear stability theory and the régimes in which: (i) the boundary layer is first noticeably oscillatory; (ii) the mean (temporal) flow quantities first deviate significantly from those of laminar flow.

It is demonstrated that the basic stratification in a fluid region subject to thermal forcing may be predicted rather simply for a fairly wide class of boundary conditions. Explicit solutions are derived in certain cases. A useful experimental method for maintaining a stratified system with arbitrarily specified vertical variation of density emerges from the analysis. A preliminary laboratory experiment has demonstrated the efficiency of this method. The restrictions on the validity of the theory involve a limitation on the thermal forcing of the fluid, which may be expressed as an upper limit on the thermal conductance of the boundary of the region. Furthermore, the buoyancy frequency characterizing the solution must be sufficiently large to give rise to a boundary-layer-type flow pattern.

The theory of Hunt & Stewartson (1965) for MHD flow in a rectangular duct with conducting walls parallel and non-conducting walls perpendicular to the magnetic field is applied to the problem of electrically driven MHD flow in a rectangular annulus. It is assumed that the Hartmann number M is sufficiently great for secondary flow effects to be negligible. The experiment described here satisfied the conditions of the theory and thus provides a sensitive experimental check on Hunt & Stewartson's theory. The theory is found to agree with the experiments to within the accuracy of the asymptotic theory.

An experimental investigation was conducted to describe the fluid flow about oscillating flat plates and to determine the magnitude and nature of forces acting on the plates at low Reynolds numbers. In the experiment, the Reynolds number was varied from 1·01 to 1057·0; three period parameters, 1·57, 2·07 and 4·71, were applied; two fluids, water and SAE 30 motor oil, and three flat plates of various sizes with or without end plates were used. The analysis of data resulted in graphical presentation of the relationships among the drag coefficient, the Reynolds number and period parameter. The drag coefficient becomes less dependent on the Reynolds number for values greater than 250. The relationship between the drag coefficient and period parameter is pronounced throughout the entire range of the Reynolds number tested.

A solution to the hypersonic small disturbance equations is obtained for a class of two-dimensional bodies supporting logarithmic shock waves by reducing the partial differential equation to an ordinary differential equation. The body shape is calculated and shown not to be logarithmic if γ ≠ 1, where γ is the ratio of the specific heats of the gas.

The behaviour of a strong shock wave, which is initiated by a point explosion and driven continuously outward by an inner contact surface (or a piston), is studied as a problem of multiple time scales for an infinite shock strength, $\dot{y}_{sh}/a_{\infty}\rightarrow \infty $, and a high shock-compression ratio, ρs/ρ∞ ∼ 2γ/(γ − 1) ≡ ε−1 [Gt ] 1. The asymptotic analyses are carried out for cases with planar and cylindrical symmetry in which the piston velocity is a step function of time. The solution shows that the transition from an explosion-controlled régime to that of a reattached shock layer is characterized by an oscillation with slowly-varying frequency and amplitude. In the interval of a scaled time 1 [Lt ] t [Lt ] ε−2/3(1+ν), the oscillation frequency is shown to be (1 + ν) (2π)−1t−½(1−ν) and the amplitude varies as t−¼(3+ν) matching the earlier results of Cheng et al. (1961). The approach to the large-time limit, ε1/(1+ν)t → ∞ is found to involve an oscillation with a much reduced frequency, ¼π(1+ν)ε−½t−1, and with an amplitude decaying more rapidly like ε−⅘t−½(4+3ν); this terminal behaviour agrees with the fundamental mode of a shock/acoustic-wave interaction.

The paper presents an approximate analysis for high Hartmann number of the flow of an electrically conducting, incompressible fluid in a duct of square crosssection, having one pair of opposite walls insulating, and the other pair perfectly conducting and inclined at arbitrary orientation to a uniform transverse magnetic field. The flow is considered to be either pressure-driven with the two perfectly conducting electrodes short-circuited together or electrically driven by a potential difference applied between these electrodes in the absence of axial pressure gradient. The paper describes experiments on the pressure-driven, short circuited case using mercury in copper ducts to investigate the variation of the streamwise pressure gradient and of the potential distribution along one insulating wall with orientation, magnetic field and flow rate.

At general orientations the analysis suggests and the experiments confirm the existence of regions of stationary fluid in the corners of the duct, together with viscous shear layers parallel to the magnetic field. For the case in which the electrodes are parallel to the magnetic field the experimental results for the pressure gradient, but not those for the potential distribution, agree reasonably well with Hunt & Stewartson's (1965) asymptotic solution. Both pressure gradient and potential results agree closely with the analysis by Hunt (1965) of the case in which the electrodes are perpendicular to the magnetic field.

In this paper we look at the problem of an undular bore entering still water. The effect of the boundary-layer at the channel bottom is considered. The motion is assumed to be laminar and inviscid, apart from the thin boundary layer, where the normal boundary-layer approximations are used to find the velocity components. The effect on the main flow then appears in the equations of motion as a shear stress.

A doubly-infinite sloping flat plate, initially at rest in a slightly diffusive viscous density-stratified fluid, starts to move impulsively with a constant velocity along the line of greatest slope. The resulting flow is found to be an unsteady motion superimposed on a steady diffusion-induced flow, which is present throughout. Laplace transform methods give solutions which are valid either in an essentially non-diffusive outer layer or in a diffusive inner layer. The impulsive start sets up oscillations in the outer layer. These gradually die out, and a steady diffusive flow develops.

A glass plate was towed vertically through stratified brine, into which aluminium particles were introduced. The flow velocities deduced from the particle motions confirmed the theoretical predictions.

Experiments to determine the value of Taylor's longitudinal diffusivity in turbulent flows in open channels and circular pipes have produced results which are in many cases inconsistent with one another and with theoretical estimates due respectively to Elder (1959) and Taylor (1954). Neither is there general agreement on when Taylor's theory becomes applicable. In an attempt to clarify the discrepancies two well-known sets of experiments by Fischer (1966) and Taylor (1954) are re-examined by using a natural procedure which, it is argued, has certain advantages over more usual methods. It is shown that Fischer's observations in an open channel were not made at a sufficient distance downstream from the point of injection for Taylor's theory to apply but that they are consistent with a description of the early stages of the dispersion process due to Sullivan (1968). It is subsequently argued that these observations suggest that Elder's estimate of the diffusivity is too low for two reasons. The first is the error caused by assuming the existence of an eddy diffusivity calculated by means of Reynolds analogy and the second is the neglect of the viscous sublayer. On the other hand it is shown that some of Taylor's observations in a circular pipe are consistent with his theory, although the values of the diffusivity which best fit the data are about 25% higher than Taylor's estimate, and it is suggested that this is for the same reasons as in the open channel. The paper concludes with a discussion of the effect of the viscous sublayer on the value of the longitudinal diffusivity. Partly on the basis of an approximate model it is argued that theoretical calculations which ignore the viscous sublayer are too low by amounts which depend on the Reynolds and Schmidt numbers and can be of the order of 20%.

The method of multiple scales is used to analyze three non-linear physical systems which support dispersive waves. These systems are (i) waves on the interface between a liquid layer and a subsonic gas flowing parallel to the undisturbed interface, (ii) waves on the surface of a circular jet of liquid, and (iii) waves in a hot electron plasma. It is found that the partial differential equations that govern the temporal and spatial variations of the wave-numbers, amplitudes, and phases have the same form for all of these systems. The results show that the non-linear motion affects only the phase. For the constant wave-number case, the general solution for the amplitude and the phase can be obtained.

Particle velocity autocorrelations of single spherical beads (46·5 μhollow glass, 87 μ glass, 87 μ corn pollen, and 46·5 μ copper) were measured in a grid-generated turbulence. The hollow glass beads were small and light enough to behave like fluid points; the other types had significant inertia and ‘crossing trajectories’ effects. The autocorrelations decreased much faster for heavier particles, in contradiction to previous experimental results. The integral scale for the copper beads was 1/3 of that for the hollow glass beads. The particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale. The data generated by the hollow glass beads can be used to estimate Lagrangian fluid properities. The Lagrangian time integral scale is approximated by L/u′, where L is the Eulerian integral scale and u′ is the turbulence intensity.

Experimental results concerning the supercritical behaviour of instabilities occurring in the thermal convective flow between inclined plates are presented. For inclination δ between 90° and 10° (δ = 0° is the vertical orientation), with the bottom plate hotter, it is known that the primary instabilities are longitudinal convective rolls with axes oriented up the slope. These rolls are found to become wavy at a supercritical Rayleigh number Ra(δ, Pr), with an upslope wavelength which seems to be related directly to the wavelength of the original rolls. Measurements indicate that these transitions take place at Reynolds numbers which are probably too small for the process to be attributed to a shear instability. It is thought that the cross-slope derivative of the buoyancy force, which exists because of the tilted geometry, is important in generating the required vorticity. The transition is crucial to the development of violent unsteadiness. The breaking of the waves leads to turbulence at much lower Rayleigh numbers than those required in convection between horizontal plates. The transition to wavy vortices appears to be very similar to that which occurs for Taylor vortices in cylindrical Couette flow.

Measurements of turbulence energy diffusion and the spectral distributions of stress components in the core of turbulent pipe flow are presented. The results tend to confirm the proposal of Bradshaw (1967a, b) that an inertial subrange in the spectra can exist at quite modest laboratory Reynolds numbers. They also illuminate the inconsistencies in Laufer's (1954) measurements of dissipation and suggest that the fitting of a −5/3 power law to the spectra may well provide the most accurate method of determining dissipation for Re [gsim ] 105.

Measurements in the wake behind turbulent jets exhausting from a solid surface into a cross-wind indicate that vortex shedding occurs as in the case of flow past solid bluff bodies. The Strouhal numbers for flow past a circular and a blunt jet are in qualitative agreement with those for corresponding solid bodies, provided that the width of the spreading jet some distance from the surface is used rather than the jet exit plane dimension.

Convection experiments described by Tritton & Zarraga (1967) with electrolytically heated fluid layers were renewed in order to investigate the reported phenomena, which were hitherto unknown and which contradicted a corresponding theory of Roberts. While the apparatus was essentially unchanged, provisions were incorporated to study the possible influence of several flow and equipment parameters on the convection pattern. With the exception of the temperature dependence of the electric conductivity, the new experiments displayed no essential effects of the convection parameters. Experiments with shallow fluid layers revealed a clear co-orientation of the convection flows with the electric current and a strong time dependence of the hexagonal patterns. Experiments with deeper fluid layers exhibited a considerably diminished time and direction dependence of the convection flow, and a significant reduction of the dilation of the cells. Based on these observations, it is concluded that no drastic differences between theory and experiments, and between internal and external heating, exist, provided the heating is sufficiently uniform.

Space-time correlation measurements in the roughly isotropic turbulence behind a regular grid spanning a uniform airstream give the simplest Eulerian time correlation if we choose for the upstream probe signal a time delay which just ‘cancels’ the mean flow displacement. The correlation coefficient of turbulent velocities passed through matched narrow-band niters shows a strong dependence on nominal filter frequency (∼ wave-number at these small turbulence levels). With plausible scaling of the time separations, a scaling dependent on both wave-number and time, it is possible to effect a good collapse of the correlation functions corresponding to wave-numbers from 0·5 cm−1, the location of the peak in the three-dimensional spectrum, to 10 cm−1, about half the Kolmogorov wave-number. The spectrally local time-scaling factor is a ‘parallel’ combination of the times characterizing (i) gross strain distortion by larger eddies, (ii) wrinkling distortion by smaller eddies, (iii) convection by larger eddies and (iv) gross rotation by larger eddies.