After a careful consideration of the laws of generation, advection, diffusion and dissipation of turbulent kinetic energy proposed by Prandtl (1945) and Emmons (1954), equations of motion and turbulent kinetic energy for the vortex flow near a solid end wall are established. These equations are then evaluated by a numerical procedure. Care is taken to specify boundary conditions such that satisfactory matching of the solution with the main vortex is assured. The agreement between the predicted mean velocity distribution and the experimental data is remarkably good. In addition, several interesting characteristics are predicted by the theory: (i) the vertical distribution of horizontal velocity is oscillatory in the inner region, whilst it is of the ordinary boundary-layer type in the outer region; (ii) the maximum velocity in the boundary layer can exceed that in the main vortex by a considerable amount and (iii) the minimum pressure of the vortex does not occur in the vortex-core root as has been generally believed.

Unsteady separated flow behind an inclined flat plate is numerically studied through the use of the discrete-vortex approximation, in which the shear layers emanating from the edges of the plate are represented by an array of discrete vortices introduced into the flow field at appropriate time intervals at some fixed points near the edges of the plate. The strengths of the nascent vortices are chosen so as to satisfy the Kutta condition at the edges of the plate. Numerical calculations are performed for a plate at 60° incidence impulsively started from rest in an otherwise stationary incompressible fluid, by systematically changing the distance between the location of the nascent vortices and the edges of the plate. The temporal changes in the drag force, the rate of vorticity transport at both edges of the plate and the velocity of the separated shear layers are given together with the flow patterns behind the plate on the basis of this model. The results of the computation show that the vortex street behind the plate inclines as a whole towards the direction of the time-averaged lift force exerted on the plate. It is also predicted from the calculations that the vortex shedding at one edge of the plate will not occur at the mid-interval of the successive vortex shedding at the other edge. The predicted flow patterns are not inconsistent with a few experimental observations based on the flow-visualization technique.

Two-dimensional vortex shedding behind an inclined flat plate in uniform shear flow is numerically investigated by means of an inviscid discrete-vortex approximation. The points of appearance of the vortices are fixed in the plane of the plate at a short distance downstream of the edges of the plate. The strengths of the vortices are determined from the Kutta condition. On the assumption that the steadily periodic flow patterns are independent of initial conditions, the numerical calculations are performed for an inclined flat plate immersed in an incompressible fluid which is set in motion impulsively from rest with the velocity profile of uniform shear flow. The results of analysis show that the Strouhal number of vortex shedding and the time-averaged values of other hydrodynamic characteristics of the flow such as the outer-edge velocity of the separated shear layers, the convective velocity of the shear layers and the drag force exerted on the plate vary closely linearly with the shear parameter of the approaching shear flow. A linear relation between the Strouhal number and the shear parameter is confirmed by an air-tunnel experiment. The effects of the shear parameter on the calculated vortex patterns in the wake are also presented.

This note corrects and augments the results of Blumen, Drazin & Billings (1975) on the linear instability of a shear layer between two streams of an inviscid compressible fluid at the same temperature. The main conclusion of these authors, that the shear layer is unstable to two-dimensional disturbances however large the Mach number, is confirmed in the present paper. Also a third mode of instability is discovered and the character of all three modes revealed by a series of accurate calculations.

The fibre distribution in a sheet of paper, referred to as the formation, is largely the result of turbulence, a stochastic process. Continuous time-series models developed from discrete light-transmission profiles are used to characterize formation. The models are used to obtain explicit expressions for the spectral moments of the profiles. From the moments, estimates of two characteristic lengths of the fibre distribution can be obtained and are interpreted as the average and largest flock size. These lengths are used to develop an index for evaluating the formation of four samples of base sheet paper. The results of this characterization agree with other methods, but this technique has the advantage of providing a physical interpretation of the index.

Earlier investigations by the authors have suggested that coherent structures, in the form of two different arrays of vortices, exist in the initial region of coaxial jets. The present investigation was aimed at obtaining further information on the characteristics of the vortices in coaxial jets. Single- and two-point correlation covariance measurements of the fluctuating pressure and the axial and radial velocity fluctuations have been made in the initial region of coaxial jets. Detailed analysis of the correlograms indicates the combined effect of the two vortex trains on the correlograms. Further, the analysis enables the phase relationship between the pressure and axial and radial velocity fluctuations to be obtained. From the correlation results the phase properties obtained within the whole initial region of coaxial jets are found to agree with the results for single jets. This good agreement supports the formally suggested simple approach to the understanding of the complicated flow in coaxial jets.

The stress system near a rigid boundary in a suspension of neutrally buoyant spheres is considered under the assumption of small Reynolds number. The suspending fluid is assumed to be Newtonian and incompressible. An ergodic principle is formulated for parallel mean flows, and the bulk stress is expressed as a surface average. Only dilute suspensions are considered and particle interactions are neglected. A uniform shear flow past a plane wall with a single spherical particle is studied first. A series solution is developed and the mean velocity and stress fields are computed for a force-free and couple-free sphere and also for spheres with couples applied by external means. The translational and angular velocities of the particle and the stress distribution on the surface of the particle are calculated. Properties of dilute suspensions of spheres are found by appropriate surface averages over the solutions for a single particle. The mean stress on a plane parallel to the wall is shown to reduce to the Einstein value when the distance from the boundary is sufficiently large. Mean velocity profiles of the suspension for Couette flow and Poiseuille flow are developed. It is shown that in an average sense particles rotate more slowly than the ambient fluid in a region approximately three sphere radii thick adjacent to the plane wall. But for the suspension as a whole, an apparent slip velocity always develops in this region. This results in an apparent viscosity which is less than the infinite-suspension value of Einstein.

A method for solving quite general three-dimensional incompressible flow problems, in particular those described by the Navier–Stokes equations, is presented. The essence of the method is the expression of the velocity in terms of scalar and vector potentials, which are the three-dimensional generalizations of the two-dimensional stream function, and which ensure that the equation of continuity is satisfied automatically. Although the method is not new, a correct but simple and unambiguous procedure for using it has not been presented before.

In steady axisymmetric flows in a closed swirl chamber one can distinguish between the swirl flow proper, with components normal to the meridian plane, and a secondary flow whose components lie in the meridian plane. One can trace the motion of a particle within the meridian plane. The closed path so obtained will be called a streamline, to be parametrized by a stream function ϕ. One can distinguish between the flow in a boundary layer, where the velocity gradient is large, and a core flow, where the velocity and temperature gradients are relatively small. The present article is concerned with only the core flow. In high Reynolds number flows in which the streamlines are not closed there are three quantities which are constant along the streamlines: the total enthalpy (the right-hand side of Bernoulli's equation), the entropy and the moment of momentum of the particles with respect to the axis of symmetry. These are determined by conditions in the entrance cross-section. In flows with closed streamlines these quantities are ultimately determined by the cumulative effects of viscosity and heat conductivity. Conditions expressing cumulative effects enter the analysis as integrabiliby conditions necessary for the existence of a second approximation in a development of the flow field with respect to the reciprocal of the Reynolds number. They are re-expressed as the balance equations for energy, entropy and angular momentum which are to be satisfied on all surfaces ϕ = constant. One thus obtains an algorithm which leads from expressions for the total enthalpy, the entropy and the angular momentum as functions of ϕ to the residuals in the balance equations, also computed as functions of ϕ. The functions with which this algorithm starts must be chosen such that the residuals in the balance equations become zero. The secondary flow can be arbitrarily slow only if the Prandtl number is ½. At the centre of the secondary motion the balance equations are linearly dependent. This fact introduces an additional free parameter which allows one to compute secondary flows with different speeds. The linearized algorithm has the character of a Fredholm integral equation. This suggests an iterative solution similar to a Neumann series. The particles experience periodic changes of state which can be discussed as thermodynamic cycles. Such an analysis shows that the heat inputs occur on average at lower temperatures than the heat outputs. Besides the work that maintains the swirl motion, which is provided by shear-force components normal to the meridian plane, one therefore needs additional work, provided by the shear-force components within the meridian plane, to maintain a secondary motion. Responsible for this state of affairs is the fact that particles which do not quite maintain adiabatic temperatures move within a field with a large pressure gradient caused by the swirl component. This makes the flow sensitive to disturbances in the energy balance.

Further results from the nonlinear numerical model of the air flow in a deep turbulent boundary layer above water waves described in Gent & Taylor (1976) are presented. The results are calculated with the surface roughness z0 both constant and varying with position along the wave. With the form used when z0 varies, the fractional rate |ζ| of energy transfer per radian advance in phase due to the working of the pressure forces is larger than for z0 constant both when the transfer is from wind to waves and when it is from waves to wind. The latter case occurs when the waves are travelling faster than, or against, the wind. The energy transfer rates are compared with other theoretical predictions and with recent field observations.

Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.

A model of the earth's liquid core is assumed in which the underlying magnetic field and velocity are zonal and axially symmetric. Alfvén waves that vary as ei(kϕ−σt) are considered, where ϕ is the angle of longitude. Buoyancy and Coriolis forces Ω × U are included.

For a wide class of basic states and regions of flow, it is shown that roughly as many of the waves with a given k [ges ] 2 propagate eastwards as propagate westwards. All these waves are neutrally stable. The class of basic states is restricted by certain inequalities involving their velocity, magnetic field and entropy gradient.

it is observed that the known equivalence (Malkus 1967) between Alfvén waves with frequencies σ [Lt ] Ω and inertial waves with frequencies σp which are O(Ω) still holds when buoyancy forces are present. The equivalence requires σp2 to be real. If σp is pure imaginary, as is possible (though perhaps uncommon) in an unstably stratified medium, then the corresponding Alfvén wave is not neutrally stable and travels westwards.