By generalizing the theory of ‘rapid distortion’ of turbulence developed by Batchelor & Proudman (1954) it is shown in this paper that the turbulent velocity around a bluff body placed in a turbulent flow can be calculated outside and upstream of the regions of separated flow, if the incident turbulent flow satisfies the following conditions: (i) if a/Lx [Lt ] 1 or = O(1), Re−1$u^{\prime}_{\infty}/\overline{u}_{\infty} $ [Lt ] 1 [Lt ] Re½; (ii) if a/Lx [Gt ] 1, Re−1 [Lt ] $u^{\prime}_{\infty}$ [Lt ] 1/(a/Lx) and Re [Gt ] (a/Lx)2, where $Re = \overline{u}_{\infty} a/ν$, is the mean uniform incident velocity, $u^{\prime}_{\infty}$ is the r.m.s. velocity of the homogeneous incident turbulence, a is a transverse dimension of the body (the radius in the case of a circular cylinder), Lx is the integral scale of the incident turbulence and v is the kinematic viscosity.

Detailed calculations are given for the flow around a circular cylinder with particular emphasis on the turbulence very close to the surface. (The results can be generalized to other cylindrical bodies.) Mean-square values and spectra of velocity have been found only in the limiting situations where the turbulence scale is very much larger or smaller than the size of the body, i.e. Lx [Gt ] a or Lx [Lt ] a. But, whatever the value of a/Lx, if the frequency is sufficiently large the results for spectra tend to those of the limiting situation where Lx [Lt ] a. The reason why the turbulence velocities have not been calculated for intermediate values of a/Lx is that closed-form solutions cannot be found and that the computing time then required is quite excessive. However, some computed results are used in the paper to suggest the qualitative behaviour of the turbulence when Lx is of order a. An important result of the theory is that it illuminates and distinguishes between the governing physical processes of distortion of the turbulence by the mean flow, the direct ‘blocking’ of the turbulence by the body, and concentration of vortex lines at the body's surface.

The results of the theory have many applications, for example in calculating turbulent dispersion and fluctuating pressures on the body, as shown elsewhere by Hunt & Mulhearn (1973) and Hunt (1973).

In conclusion the theoretical results are briefly compared with experimental measurements of turbulent flows round non-circular cylinders. A detailed comparison with measurements round circular cylinders will be published later by Petty (1974).

The mean flow development in an initially turbulent boundary layer subjected to a large favourable pressure gradient beginning at a point x0 is examined through analyses expected a priori to be valid on either side of relaminarization. The ‘quasi-laminar’ flow in the later stages of reversion, where the Reynolds stresses have by definition no significant effect on the mean flow, is described by an asymptotic theory constructed for large values of a pressure-gradient parameter Λ, scaled on a characteristic Reynolds stress gradient. The limiting flow consists of an inner laminar boundary layer and a matching inviscid (but rotational) outer layer. There is consequently no entrainment to lowest order in Λ−1, and the boundary layer thins down to conserve outer vorticity. In fact, the predictions of the theory for the common measures of boundary-layer thickness are in excellent agreement with experimental results, almost all the way from x0. On the other hand the development of wall parameters like the skin friction suggests the presence of a short bubble-shaped reverse-transitional region on the wall, where neither turbulent nor quasi-laminar calculations are valid. The random velocity fluctuations inherited from the original turbulence decay with distance, in the inner layer, according to inverse-power laws characteristic of quasi-steady perturbations on a laminar flow. In the outer layer, there is evidence that the dominant physical mechanism is a rapid distortion of the turbulence, with viscous and inertia forces playing a secondary role. All the observations available suggest that final retransition to turbulence quickly follows the onset of instability in the inner layer.

It is concluded that reversion in highly accelerated flows is essentially due to domination of pressure forces over the slowly responding Reynolds stresses in an originally turbulent flow, accompanied by the generation of a new laminar boundary layer stabilized by the favourable pressure gradient.

A uniform magnetic field is switched on at time t = 0 outside a body of conducting fluid. It is assumed that the field strength increases in time in proportion to 1 -e−αt, where α is a constant of the circuit generating the field. Under the assumption of small magnetic Reynolds number and small magnetic Prandtl number the equations governing the diffusion of the field into the fluid are derived and a simple expression is given for the initial vorticity distribution produced in the fluid. The situation in which an initially uniform field is switched off is also considered. It is shown that, for sufficiently symmetrically shaped bodies of fluid, the vorticity generated by the switching-on of the field is the same as that generated by the switching-off. The particular case of an infinitely long circular cylinder of conducting fluid is considered in detail and an explicit expression is derived for the vorticity distribution.

A theoretical method is presented for predicting the deformation and the conditions for breakup of a liquid droplet freely suspended in a general linear shear field. This is achieved by expanding the solution to the creeping-flow equations in powers of the deformation parameter ε and using linear stability theory to determine the onset of bursting. When compared with numerical solutions and with the available experimental data, the theoretical results are generally found to be of acceptable accuracy although, in some cases, the agreement is only qualitative.

Thin viscous jets are considered as they slowly fall, in a state of near-neutral buoyancy, through a liquid. An equation is derived which describes the path of the jet. A small perturbation analysis of nearly vertical jets is carried out, and shows that they are necessarily unstable and will eventually deviate significantly from the vertical. Numerical integration of the nonlinear equation describes the nature of this deviation. These results model some experimental observations made by Taylor (1969).

Effects of moderately large Reynolds numbers R are studied by considering higher order terms in the expansions for turbulent pipe and channel flows for R → ∞. Matched asymptotic expansions using two length scales are employed to emphasize the two-layer structure of turbulent shear flows near solid walls. The effects appear as additional terms in extended forms of the law of the wall, the logarithmic velocity law, the velocity defect law and the logarithmic skinfriction law. These generalizations are critically compared with experimental results for pipe flows of Patel & Head and extremely good agreement is obtained. Also, possible applications are discussed for extending the range of skin-friction and heat-transfer devices which are based on wall similarity.

We have made computational experiments to study the stability and long-time evolution of two-dimensional wakes. We have used the VORTEX code, a finite-difference realization of two-dimensional motions in incompressible inviscid fluids. In the first experiment an initial shear-unstable triangular velocity profile evolves into a non-homogeneous, finite-area, asymmetric vortex array and like-signed regions attract and fuse (or coalesce). Enhanced transport across the profile is due to ‘capture’ and convection of small-scale vortex regions by larger opposite-signed vortex regions. In the following experiments we study the stability of an asymmetric four-vortex finite-area system corresponding to a von Kármán street of point vortices. Here the critical parameter is b/a, the initial transverse-to-longitudinal separation ratio of vortex centres. At \[ b/a = 0.281 \] the four-vortex system is stable and we observe that large-area vortex regions develop elliptical (m = 2), triangular (m = 3), etc. surface modes owing to mutual interactions. At b/a = 0 the measured growth rate is smaller than that for the corresponding von Kármán system and at b/a = 0·6 the measured growth rate is larger. At b/a = 0 one vortex undergoes fission in the high-shear field produced by two nearest-neighbour opposite-signed vortex regions. Heuristic comparisons are made with the two-dimensional tunnel experiments of Taneda and others.

An analytical and experimental investigation of the near and far wake characteristics of a cascade of airfoils is reported in this paper. The measurement of mean velocity, turbulence intensity and Reynolds stress across the wake at several distances downstream of the cascade indicates that the wake is asymmetrical and this asymmetry is maintained even up to 3/4 chord length. Experiments carried out at three incidences reveal that the decay of the wake defect is strongly dependent on the downstream variation of the wake edge velocity. For a cascade, the decay rate of the wake defect is found to be slower than that of a flat plate, cylinder or symmetrical airfoil (at zero incidence). The level of turbulence and Reynolds stresses are found to be high and some comments are made regarding self-preservation and structure of the flow. Semi-theoretical expressions are given for the wake profile, and decay of the velocity defect, turbulence intensity and Reynolds stress.

Measurements of a wall jet in a self-preserving pressure gradient are described. The quantities measured with a linearized hot-wire anemometer were the mean velocity, the turbulence stresses, triple and quadruple velocity correlations, intermittency and spectra of the longitudinal turbulence intensity. The turbulence, as well as the mean flow, reached a self-preserving state in which the ratio of the maximum velocity to the free-stream velocity was 2·65. Skin friction was also measured using the razor-blade technique in the viscous sublayer and buffer region. The values of the constants in the logarithmic law of the wall are found to be similar to those in boundary-layer and pipe flows. The skin-friction coefficient is slightly lower than that found for the wall jet in still air (Guitton 1970), but close to the formula of Bradshaw & Gee (1962) for the wall jet in an external stream with zero pressure gradient.

A balance of the terms in the turbulence energy equation is presented and discussed. The shearing stress is not zero at the point of maximum velocity but is of opposite sign to that at the wall and hence the contribution of this stress to turbulence production is negative in the outer part of the boundary-layer region. However, the total turbulence production remains positive because the contribution of the normal stresses is positive and slightly larger. The pressure—velocity gradient correlations are evaluated by difference from the Reynolds stress equations and are compared with the theoretical model of Hanjalić & Launder (1972b). Agreement is quite good in the outer region of the wall jet. The above model is also compared with the triple velocity correlations and again found to be in fair agreement.

The first part of the paper is an extension of the statistical theory of turbulent diffusion to the problem of turbulent diffusion from a point or line source in an inhomogeneous straining flow such as flow round a two-dimensional body. The new effects which have to be considered are the convergence and divergence of mean streamlines, the inhomogeneity of the mean and turbulent velocities, and the presence of a boundary. We assume that the upstream turbulence intensity $u^{\prime}_{\infty}/\overline{u}_{\infty} $ is weak, i.e. $u^{\prime}_{\infty}/\overline{u}_{\infty} $ [Lt ] 1, and that molecular diffusion is negligible, i.e. $(\overline{u}_{\infty}/u^{\prime}_{\infty})^2[D/(a\overline{u}_{\infty})] $ [Lt ] 1, D and a being the molecular diffusivity and the scale of the obstacle respectively. The theory predicts the mean-square dispersion of the plume about the mean streamline through the source in terms of the Lagrangian statistics of the turbulence. Making the further assumption that the scale LE of the turbulence is large enough to satisfy the condition that ($(u^{\prime}_{\infty}/\overline{u}_{\infty}) $) (a/LE) [Lt ] 1, it is shown that the turbulent dispersion can be calculated in terms of the Eulerian statistics, which can either be measured or in some cases calculated. In the second part we analyse diffusion from various sources in potential flows over two-dimensional obstacles assuming constant (or variable) eddy diffusivity, and compare the results with those of the more rigorous statistical analysis for sources one or two diameters upwind of the obstacle. However, unlike the statistical analysis, this eddy-diffusivity analysis can also be extended to calculate diffusion from sources placed some distance upwind of an obstacle, and an example is given of how this analysis may be applied to calculating concentrations on hills due to distant sources.

This is a study of turbulence which results from Kelvin—Helmholtz instability at the interface between two miscible fluids in a two-dimensional shear flow in the laboratory. The growth of two-dimensional ‘billows’, their disruption by turbulence, and the eventual decay of this turbulence and the re-establishment of a gravitationally and kinematically stable interface are described. Continuous measurements of density and horizontal velocity from both fixed and vertically moving probes have been made, and the records obtained are presented, together with photographs showing the simultaneous appearance of the flow, which serve to identify the physical nature of events seen in the records. The measurements show how the fine-structure of the density field described in earlier experiments is related to velocity fluctuations. The vertical length scales of the final mean velocity and density structure are found to be different, and to depend on the Richardson number at which instability first occurred. The eventual Richardson number at the centre of layer is, however, not dependent on the initial Richardson number and has a value of about one third. The implications of these results to the eddy diffusion coefficients, to the energy exchange, and to turbulence in the ocean and the atmosphere are discussed.

The internal waves produced by a moving body are generally longer in the direction of motion than the corresponding surface waves. This difference is accentuated when the density variation is slight and the body velocity is large in which case a very long towing tank may be required for the simulation of a steady-state condition. The following theoretical study of transient waves is intended as a step in relating test conditions and requisite towing-tank sizes.

A source–sink pair travelling for a finite time is used to represent the restricted motion of a body in a tank. The approximate length and volume of the body are fixed, but its precise shape (somewhat irregular and slightly time dependent) is assumed to be of secondary importance and is not calculated here. The density-stratified fluid is assumed to have a constant Brunt–Väisälä frequency.

A solution in the form of a triple sum over the tank eigenfunctions applies quite generally for the internal wave system (neglecting surface waves and the potential-flow-type solution near the body). Examples covering the large-scale structure of the flow field have been solved for two values of an approximate similarity parameter. The value of the similarity parameter indicates how closely steady-state conditions are approached. The first (larger) value chosen produces a well-defined quasi-steady state near the body with transient fluctuations of the order of ± 10%. The second (smaller) value gives a poorly defined quasisteady state with fluctuations of the order of ±50%. More elaborate studies varying the tank length, width and depth could be made by programming the calculations.

The effect of a collapsing wake has not been considered here, but might possibly be treated by similar methods.

The boundary-layer flow over a circular cylinder at a Reynolds number of 1·06 × 105 has been studied both experimentally and theoretically. The investigation was designed to concentrate on the self-induced oscillations occurring in the flow; at this Reynolds number, these oscillations have generally been ignored heretofore. In the experimental part of the investigation both the inviscid flow and boundary-layer flow reversals were measured as functions of time. The theoretical part of the study started with the measured inviscid flow and calculated all the boundary-layer characteristics. The boundary-layer calculations themselves revealed some very interesting fine-scale structure of the flow, which strongly indicated that the vanishing of wall shear does not signal the onset of separation for unsteady flow. In general, the agreement between the theoretical calculations and the experimental results was excellent and the unsteady component of this supposedly steady flow was found to be very significant.

The solution to the steady problem of an inviscid jet emerging from a symmetric nozzle of slowly varying profile is sought as an asymptotic series in the wall slope. The expansion of the solution in the region near the nozzle lip is singular at infinity, so that a matched expansion technique is evolved to solve the problem. To the order to which the solution is obtained in the present paper, the jet contraction ratio is shown to be the same as that from a nozzle formed by two inclined planes with inclination angle the same as the exit slope of the nozzle. Composite expansions are formed and used to check the consistency of expansion and matching procedures.

An analysis is made of the flow within a three-dimensional explosion, or spark, created in a gas absorbing energy from a steady conical beam of radiation with nearly spherical symmetry. The radiation, typically from an array of lasers with a common focus, is assumed to be very intense, and absorbed immediately behind an outwardly advancing strong shock. The resulting self-similar flow has previously been studied for spherical symmetry; somewhat improved calculations for that case are presented here.

Departures of the laser power from spherical uniformity, which would result from practical problems of arrangement, are conveniently represented by an ascending series of Legendre polynomials in the polar angle. For non-uniformities of small amplitude, first-order perturbations of the flow field are analysed in detail. Self-similarity is shown to be retained, for zero counter-pressure and power constant with time.

For the first five harmonics in power distortion, the resulting fourth-order system of equations is solved numerically for profiles of velocity components, density and pressure, and for shock shape. Results are presented graphically. These solutions are singular near the focus, but are nevertheless fully determined. In the limit of large wavenumber, the core of the flow has vanishing tangential velocity and pressure perturbations, and hence the governing equations are only of second order, except presumably in a boundary layer appearing near the shock.

Study of the nonlinear case of large wavenumber along the axis of symmetry shows that the singularity at the focus reflects the existence of a ‘forbidden zone’ whose extent depends on the degree of asymmetry. It is argued that this zone is one within which diffusional processes must dominate.

Turbulent entrainment across a density interface is studied in a laboratory wind–wave tank in which a stably stratified system consisting of two homogeneous fluid layers is introduced. The results indicate that the rate of change of the potential energy of the mixing layer is proportional to the rate of work done by the wind. However, only a very small fraction of the work done by the wind is used for interfacial mixing or developing a seasonal thermocline. A formula relating the entrainment rate to the density stratification and the wind-friction velocity is derived from the experimental results.

This paper describes how the flows around two circular cylinders, displaced in a plane normal to the free stream, interact as the two bodies are brought close together. Surface pressure measurements at a Reynolds number of 2·5 × 104, based on the diameter of a single cylinder, show the presence of a mean repulsive force between the cylinders. An instability of the flow was found when the gap between the cylinders was in the range between one diameter and about 0·1 of a diameter. Correlation measurements of hot-wire outputs indicate how mutual interference influences the formation of vortex streets from the two cylinders. Spanwise correlation measurements show that the correlation length doubles as the cylinders are brought into contact.

The lift and drag forces were measured on both a single circular cylinder and tandem circular cylinders in uniform flow at Reynolds numbers from 40 to 104, to investigate the stability of an oscillating cylinder. A cylinder (the downstream one in the tandem case) was made to oscillate in either the transverse or longitudinal direction (perpendicular or parallel to the stream). In the case of a single cylinder, its oscillation causes the so-called synchronization in a frequency range around the Strouhal frequency (transverse mode) or double the Strouhal frequency (longitudinal mode). The aerodynamic damping for transverse oscillation becomes negative in the synchronization range. In the case of tandem cylinders, at low Reynolds numbers in the pure Kármán range synchronization was observed to occur only when the downstream cylinder oscillated inside the vortex-formation region of the upstream one, and at high (low subcritical) Reynolds numbers synchronization occurred irrespective of the cylinder spacing in either oscillating mode. In the tandem case, too, the transverse oscillation of the downstream cylinder becomes unstable in the range of synchronization.

Investigations are made of the wave motion which arises near resonance in a tube with an applied transverse magnetic field filled with a highly electrically conducting gas and closed by two rigid walls. The wave motion is driven by the sinusoidal radiative flux emitted by one of the walls as a consequence of its oscillatory temperature; the other wall is taken to be a perfect reflector of thermal radiation. The effects of radiative transfer are treated by the use of the differential approximation. The analysis leads to the same formal governing integral equation for the solution as arises in the ordinary gasdynamic case in the absence of electromagnetic effects. Within a narrow frequency band around resonance the theory predicts the occurrence of magnetogasdynamic shock waves which become dispersed as the thermal radiation is strengthened and may be totally dispersed to leave a continuous, periodic, but not necessarily sinusoidal, wave motion. The effect of the magnetic field is to delay the onset of dispersion.

The laser-Doppler ambiguities predicted by George & Lumley have been verified experimentally for turbulent pipe flows. Experiments were performed at Reynolds numbers from 5000 to 15000 at the centre-line and near the wall. Ambiguity levels were measured from power spectral densities of FM demodulated laser signals and compared with calculations based on the theory. The turbulent spectra for these water flows after accounting for the ambiguity were equivalent to hot-film measurements at similar Reynolds numbers. The feasibility of laser-Doppler measurements very close to the wall in shear flows is demonstrated.