A nonlinear integro-differential equation governing finite amplitude wave propagation on concentrated vortices is solved numerically. The solution to the Cauchy problem shows a solitary wave development qualitatively similar to solutions of the Korteweg–de Vries equation. A perturbation solution of the stationary form of the evolution equation confirms the unsteady calculation.

Supersonic laminar boundary-layer equations near the plane of symmetry of a cone at incidence are treated by the similarity method. Numerical integration of differential equations governing such a flow is performed, taking into consideration the temperature dependence of the Prandtl number Pr and viscosity μ throughout the boundary layer. On the leeward side, a detailed consideration of the solutions shows the existence of two solutions up to a critical incidence beyond which it appears that no solution may be found. Calculations carried out for a set of values of the external flow Mach number show up a significant effect of this parameter on the behaviour of the boundary layer.

Symmetric laminar incompressible flow past a parabolic cylinder is considered for all Reynolds numbers. In the limit as the Reynolds number based on nose radius of curvature goes to zero, the solution for flow past a semi-infinite flat plate is obtained. All solutions are found by using an implicit alternating direction method to solve the time-dependent Navier-Stokes equations. The solutions found are compared with various other exact and approximate solutions. Results are presented for skin friction, surface pressure, friction drag and pressure drag. The numerical method developed is of particular interest since it combines the alternating direction method with the implicit method for solving the boundary-layer equations. This leads to fast convergence and may be of use in other problems.

Surface waves in flowing water and their stability are studied. With U(y) denoting the mean velocity and d the depth of water, the following results are obtained: (i) in the plane of the complex wave velocity, c = cr + ici, all eigenvalues c with a positive ci lie within a semicircle which has as its diameter the range of the velocity U(y) of the primary flow, y being the vertical co-ordinate. (ii) If U″(y) does not change sign and U is monotonic in the field of flow, singular neutral modes (for which c = U somewhere in the field of flow) are impossible and the flow is stable. (iii) If U is analytic and U″ vanishes at the point or points where U is equal to the same constant Uc and where U′ is not zero then at least one neutral mode exists with c = Uc, provided U(d) ≠ Uc. (iv) If U is monotonic and U″/(U—c) is finite and non-zero at the critical point (c real), where U″ vanishes, then the neutral mode mentioned in (iii) above is contiguous with unstable modes, (v) If U″ < 0 and U′ [ges ] 0 there are waves with c [les ] U(0), with a finite maximum wavenumber kc corresponding to c = U(0) and with c decreasing monotonically to a finite c0 for k = 0. (vi) If U″ < 0 and U′ [ges ] 0 waves of all wavenumbers can travel with c > U(d). The eigenvalue c for any k is bounded.

Fluid motion driven by the combined effects of a moving wall and natura convection is examined for rectangular cavities with heightlwidth ratios of ½, 1 and 2. The Reynolds number and Prandtl number are held fixed at Re = 100 and Pr = 1; the Grashof number is varied over the range of values Gr = 0, ±104, ±106. Flow and temperature fields obtained from numerical solutions of the Navier-Stokes equations reveal a marked influence of buoyancy for the larger aspect ratios when Gr = ±106 and the dominance of buoyancy for all aspect ratios when Gr = ± 106. Results are compared with earlier work where possible and some observations are offered on the convergence of the numerical solutions.

This paper describes the results of a series of measurements made using a single beam schlieren system to investigate the density fluctuations present in the initial region of a supersonic axisymmetric turbulent jet with a Mach number of 1-82 in the flow a t the nozzle exit. A preheater was used to reduce the difference between the jet static temperature and that of the surrounding air to a relatively low level. The results show that significant density fluctuations are present in the potential core of the jet and that the distribution of fluctuating intensity across the shear layer differs from that obtained with a subsonic jet without preheating.

A series of observations on experimentally produced vortex rings is described. The flow field, ring velocity and growth rate were observed using dye and hydrogen-bubble techniques. It was found that stable rings are formed and grow in such a way that most of their vorticity is distributed throughout a fluid volume which is larger than and moving with the visible dye core.

As the vorticity diffuses out of this moving body of fluid into the outer irrotational fluid, it has two effects. It causes some of the fluid, with newly acquired vorticity, to be entrained into the interior of the bubble, while the rest is left behind and accounts for the appearance of ring vorticity in a wake. It was found that the velocity of translation U of these stable rings varies as t−1, at high Reynolds number, where t is the time measured from the start of the motion at a virtual origin at downstream infinity. A simple theoretical model is presented which explains all of these features of the observed stable flow. Rings of even higher Reynolds number become unstable and shed significantly more vorticity into the wake. Under some circumstances a new more stable vortex emerges from this shedding process and continues with less vorticity than before. Eventually, the ring motion ceases as all of its vorticity is deposited into the wake and is spread by viscous diffusion. Observations of the interaction between two nearly identical rings travelling a common path showed that, contrary to popular belief, rings do not pass back and forth through one another, but that the rearward one becomes entrained into the forward one. Only when the rearward ring has a much higher velocity than its partner can it emerge from the joining process and leave a slower-moving ring behind.

The convective motions in a rotating laterally-heated annulus have been studied experimentally. The main feature that differentiates this experiment from others done previously is a rigid lid which is in contact with the fluid. It has been found that the radial temperature differences at which the transition from lower symmetry to the vortex regime occur are different in the cases of positive and negative temperature gradients and that, in particular, the transition is a very strong function of Ω in the case of a negative temperature gradient. It has been observed further that the persisting vortices which form in the annulus always appear at the inner cylinder, whether the radial temperature difference is increased in a quasi-steady way, the radial temperature difference is applied suddenly while the mean temperature is maintained, the heating or cooling is applied suddenly on either the inner or outer cylinder only, thus changing the mean temperature, or whether the rotation of the annulus is started suddenly with the radial temperature difference applied beforehand. This is true regardless of whether the radial temperature gradient is positive or negative and there has not been a single instance in which a persisting vortex developed at the outer cylinder. Neither this observation nor the dependence of the transition temperature difference on Ω seems to be in agreement with the current theoretical descriptions of the annulus motions. On the other hand, the appearance of a warm cell, a counter cell and a cold cell in the time-dependent meridional motions observed in the experiments confirms the results of a numerical study of Williams (1967), which studies the symmetric motions in the annulus.

The growth of small disturbances in a two-dimensional incompressible wake has been investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory wherein small disturbances are considered from both temporal and spatial reference frames. Through a combined stability analysis, in which small disturbances are permitted to amplify in both time and space, the relationship between the disturbance characteristics for the temporal and spatial reference frames is shown. In these analyses a quasi-uniform assumption is adopted to account for the continuously varying mean-velocity profiles that occur behind flat plates and thin airfoils. It is found that the most unstable disturbances in the wake produce transverse oscillations in the mean-velocity profile and correspond to growing waves that have a minimum group velocity.

Experimentally, the downstream development of the wake of a thin airfoil and the wave characteristics of naturally amplifying small disturbances are investigated in a water tank. The disturbances that develop are found to produce transverse oscillations of the mean-velocity profile in agreement with the theoretical prediction. From the comparison of the experimental results with the predictions for the characteristics of the most unstable waves via the temporal and spatial analyses, it is concluded that the stability analysis for the wake is to be considered solely from the more realistic spatial viewpoint. Undoubtedly, this conclusion is also applicable to other highly unstable flows such as jets and free shear layers.

In accordance with the disturbance vorticity distribution as determined from the spatial model, a description of the initial development of a vortex street is put forth that contrasts with the description given by Sato & Kuriki (1961).

A general solution of the linear long-wave equation is obtained for arbitrary ground motion on a uniformly sloping beach. Numerical results are presented for specific shapes and time histories of ground motion. Near-shore large amplitude waves are also investigated using non-linear theory.

A wind-tunnel model was developed to study the two-dimensional turbulent boundary layer in adverse and favourable pressure gradients with out the effects of streamwise surface curvature. Experiments were performed at Mach 4 with an adiabatic wall, and mean flow measurements within the boundary layer were obtained. The data, when viewed in the velocity transformation suggested by Van Driest, show good general agreement with the composite boundary-layer profile developed for the low-speed turbulent boundary layer. Moreover, the pressure gradient parameter suggested by Alber & Coats was found to correlate the data with low-speed results.

Some exact solutions of the steady magnetohydrodynamic equations for a perfectly conducting inviscid self-gravitating incompressible fluid are discussed. It is shown that there exist solutions for which the free surface of the liquid is that of a planetary ellipsoid and rotates with constant angular velocity about its axis. The stability of the equilibrium configuration is not investigated.

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters, J a ‘typical’ Richardson number, α the (real) wavenumber and c the complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first finds c given α and J, the second finds α and J when c ≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes, but not always, the stability boundary in the α, J plane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

This paper presents some new measurements which have been made on a drag-reducing polymer solution in pipe flow. A novel type of laser dopplermeter, which has been developed by the author, is briefly described and the measurements which have been obtained are given. These results and their implications are then discussed in terms of conventional models for turbulent flow in a pipe. These suggest that the polymer has very little effect upon the turbulent core of the flow, but thickens and stabilizes the viscous sublayer. The turbulent intensity inside the sublayer is unchanged but, owing to its thickening, the velocity fluctuations just outside are greater. There is not a general suppression of turbulence within the sublayer although well inside the sublayer the spanwise velocity component is found to be reduced.

Plasma motions in the gaps between successive red cells in narrow-capillary blood flow are obtained in an idealized model, using a series of eigenfunctions to represent the disturbance to a basic Poiseuille flow created by the cells. The flow is matched, in the narrow entry and exit regions, to the lubrication flow in the constricted zone around the red cell (Fitz-Gerald 1969). Basically, the circulating toroidal motion predicted by Prothero & Burton (1961) is obtained in a reference frame in which the cells are considered stationary. Small secondary circulations are also found near the axis and close to the red cells, whose intensity is controlled by the amount of leakback past the cells. Zones of high shear are found along the capillary wall and in some cases on part of the red-cell face; implications of this for mass transport are discussed (see §4). Because of the unusual behaviour of the slowest-decaying dominant eigenfunction circulation and wall shear increase as the cell spacing decreases, contrary to expectation, until the spacing becomes very small indeed.

Stokes flow through a random, moderately dense bed of spheres is treated by a generalization of Brinkman's (1947) method, which is applicable to both stationary beds and suspensions. For stationary beds, Darcy's law with a permeability result similar to Brinkman's is derived. For suspensions an effective viscosity μ/(1–2·60ψ) is found, where ψ is the volume fraction of spheres. Also, an expression for the settling velocity is derived.

A general method for studying two-dimensional problems in hydrodynamic stability is presented and applied to the classical problem of predicting instability in plane Poiseuille flow. The disturbance stream function is expanded in a Fourier series in the axial space dimension which, on substitution into the Navier-Stokes equation, leads to a system of parabolic partial differential equations in the coefficient functions. An efficient, stable and accurate numerical method is presented for solving these equations. It is demonstrated that the numerical process is capable of accurate reproduction of known results from the linear theory of hydrodynamic stability.

Disturbances that are stable according to linear theory are shown to become unstable with the addition of finite amplitude effects. This seems to be the first work of quantitative value for disturbances of moderate and larger amplitudes. A relationship between critical amplitude and Reynolds number is reported, the form of which indicates the existence of an absolute critical Reynolds number below which an arbitrary disturbance cannot be made unstable, no matter how large its initial amplitude. The critical curve shows significantly less effect of amplitude than do those obtained by earlier workers.

The intrinsic equation governing the curvature K and the torsion τ of an isolated very thin vortex filament without stretching in an incompressible inviscid fluid is reduced to a non-linear Schrödinger equation \[ \frac{{\rm l}}{i}\frac{\partial \psi}{\partial t} = \frac{\partial^2\psi}{\partial s^2}+{\textstyle\frac{1}{2}}(|\psi|^2+A)\psi, \] where t is the time, s the length measured along the filament, ψ is the complex variable \[ \psi = \kappa\exp\left(i\int_0^{s}\tau \,ds\right) \] and is a function oft. It is found that this equation yields a solution describing the propagation of a loop or a hump of helical motion along a line vortex, with a constant velocity 2τ. The relation to the system of intrinsic equations derived by Betchov (1965) is discussed.

The mean distribution of concentration of a solute over the cross-section of a tube through which a solvent is flowing is given approximately by the solution of an equation of the heat-conduction type (Taylor 1953, 1954). One solution of this equation is a Gaussian curve but observed distributions are normally not Gaussian. On the basis of an equation derived by Chatwin (1970) it is shown here that the deviation of an observed distribution from the Gaussian curve with the same mean and variance is not determined by the heat-conduction equation, although the practical importance of this remark may be small if the initial distribution of solute is greatly elongated along the tube axis.

The paper presents the results of a detailed experimental examination of fully developed asymmetric flow between parallel planes. The asymmetry was introduced by roughening one of the planes while the other was left smooth; the ratio of the shear stresses at the two surfaces was typically about 4:1.

The main emphasis of the research has been on establishing the turbulence structure, particularly in the central region of the channel where the two dissimilar wall flows (generated by the smooth and rough surfaces) interact. Measurements have included profiles of all non-zero double and triple velocity correlations; spectra of the same correlations at several positions in the channel; skewness and flatness factors; and lateral two-point space correlations of the streamwise velocity fluctuation.

The region of greatest interaction is characterized by strong diffusional transport of turbulent shear stress and kinetic energy from the rough towards the smooth wall region, giving rise, inter alia, to an appreciable separation between the planes of zero shear stress and maximum mean velocity. The profiles of length scales of the larger-scale motion are, in contrast to the turbulent velocity field, nearly symmetric. Moreover, it appears that at high Reynolds numbers the small-scale motion may in many respects be treated as isotropic.