The focusing of very weak and slightly concave symmetrical shock waves is examined. The equation that describes this focusing is derived and the resulting similitude discussed. The initial conditions come from a formal matching of this nonlinear description with the linear solution. The maximum value of the pressure coefficient is shown to be proportional to the two-thirds power of both the initial strength of the wave front and a parameter characterizing its rate of convergence.

Some of the dense fluid at the front of an advancing gravity current is observed to be mixed with the ambient fluid. This process continues when the cross-stream non-uniformities at the head of the current are suppressed by advancing the floor beneath the head. In the resulting two-dimensional flow regular billows are visible. This paper considers experimentally and analytically the inviscid gravity current head and specifically includes the observed mixing at the head. Experimental results were obtained with an apparatus in which the head of the gravity current was brought to rest by an opposing uniform flow. The mixing appears to occur through Kelvin-Helmholtz billows generated on the front of the head and controls the dynamics of the head. A momentum balance is used to analyse the flow and the problem is closed by quantitatively introducing the billow structure.

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.

In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

A series of initially planar shock waves was allowed to expand from a shock tube into a near half-space. The strength of the wall shock was measured at two positions on the slightly concave front wall. These measurements are compared with shock strengths predicted by a shock-dynamic model based on the cylindrical expansion of a critical shock. Chisnell's (1957) theory is used to account for the effect of the increasing surface area of the expanding wall shock, and Whitham's (1957) treatment to correct for the curvature of the wall. The critical shock strength is obtained from Skews’ (1967) experimental measurements of the Mach number of the self-similar wall shock following two-dimensional diffraction at a 90° edge.

The model predicts the relatively small degree of attenuation observed between the measuring stations, but overestimates the absolute shock strength. The most likely cause is that, in the early stages of expansion, the wall shock experiences further attenuation owing to its interaction with the expanding flow. These effects are shown to be short range and of negligible importance at the first measuring station, 1·86 tube diameters from the axis. Thus, using the experimental results at this station as the starting point, the model predicts accurately the shock strength at 3·76 diameters. It is concluded that Chisnell's theory can be applied to the weakening of the wall shock in ducts with large abrupt changes in cross-section only when the wall shock is some distance from the entrance to the area change.

In this paper calculations are made of the two-dimensional flow field of an incompressible viscous fluid in a long parallel-sided channel whose walls pulsate in a prescribed way. The study covers all values of the unsteadiness parameter α and the steady-streaming Reynolds number. The wall motion is, in general, assumed to be of small amplitude and sinusoidal. Particular attention is given to the steady component of the flow at second order in the amplitude parameter ε. The results for the corresponding problem in axisymmetric geometry are given in an appendix.

Next the following problem is considered: the calculation of the wall motion which will result, in response to prescribed unsteady pressures imposed at the ends of the channel and outside its walls, if the walls are assumed to respond elastically to variations in transmural pressure. It is found that the system has a natural frequency of oscillation, and that resonance will occur if this frequency is close to a multiple of the frequency of the external pressure fluctuations. Finally the preceding work is applied in a discussion of blood flow in the coronary arteries of large mammals.

Turbulent motions within the wind-mixed layer, which is advected by near-surface inertial oscillations, excite internal gravity waves in the underlying ocean layers. Momentum transport in the radiated wave field results in a drag force on the inertial currents. Because the magnitude of the inertial currents is large compared with the turbulence intensity, the resultant rate of dissipation of inertial oscillation energy is approximately equal to the energy flux in the radiated wave field. Using linear internal wave theory, asymptotic results are derived for the energy flux in terms of the Brunt-Väisälä frequency N below the mixed layer, the magnitude U0 of the inertial current, the integral length scale l of the mixed-layer turbulence and the mean-square displacement 〈ζ20〉 of the base of the mixed layer. For representative conditions, we estimate an energy flux of 1-10 erg/cm2 s into relatively short (wavelength of order 2πU0/N) high frequency (of order, but less than, N) internal waves. The resultant decay times for inertial oscillation energy range from a day to a week or so, in agreement with reported observations on the decay of inertial oscillations in the upper ocean. The estimated energy flux is comparable in magnitude to estimates for other internal wave generation mechanisms, indicating that, in addition to being a significant sink of inertial energy, this process may locally represent a significant source of internal wave energy in the open ocean.

The boundary-layer equations are solved numerically for mainstreams \[ U(x) = x(1-x^2)^{-\alpha}\quad {\rm and}\quad U(x) = (1-x)^{-\alpha}, \] which are both O((1 − x)−α) near x = 1. Series expansions are derived near x = 1. For α > 1, where, for the similarity solution at x = 1, the outer boundary condition is approached through exponentially small terms, a straightforward expansion in powers of 1 − x is possible. For 0 < α < 1, where the decay is only algebraic (Brown & Stewartson 1965), the outer boundary condition cannot be satisfied even with algebraic decay by the higher-order terms in the series and this must be regarded as only an inner expansion. An outer expansion is required which matches with this inner expansion and which approaches the outer boundary condition with exponential decay. For α = 1, the decay is exponential, but not of the same form as for α > 1, and again the outer boundary condition cannot be attained by the higher-order terms in the series. An outer expansion for this case is also derived.

Erdogan & Chatwin (1967) derived a nonlinear diffusion equation \[ \partial_tc = \partial_z([D_0+(\partial_zc)^2D_2]\partial_zc) \] which models the effect of buoyancy upon the longitudinal dispersion of a solute in pipe flow. The same equation arises more widely as a limiting form in which only the first buoyancy correction is retained. In this paper long-term asymptotic solutions are obtained both for the smearing-out of a concentration jump and for the approach to normality of a finite discharge. A variant of the method provides an approximate solution to the initial-value problem, and a comparison is made with Prych's (1970) experimental results.

The fully developed laminar flow in a heated curved pipe under the influence of both centrifugal and buoyancy forces is studied analytically. The pipe is assumed to be heated so as to maintain a constant axial temperature gradient. Both horizontal and vertical pipes are considered. Solutions for these two cases are obtained by regular perturbations in the Dean number and the product of the Reynolds and Rayleigh numbers; the solutions are therefore limited to small values of these parameters. Predictions of the axial and secondary flow velocities, streamlines, shear stress, temperature distribution and heat transfer are given for a representative case.

The Lagrangian-history method in turbulence theory (Kraichnan 1977) is modified such that triple moments are expanded in functional powers of the Lagrangian covariance of the symmetric rate-of-strain field instead of the Lagrangian covariance of the velocity field. The simplest approximation which results corresponds to the abridged Lagrangian-history direct-interaction approximation. It is illustrated by application to the Lagrangian properties of a random velocity field whose Eulerian values are frozen in time. Then it is formulated for isotropic Navier-Stokes turbulence. The new approximation is expected to give reduced energy transfer in the dissipation range because the rate of strain along a fluid-element trajectory is statistically stationary in stationary homogeneous turbulence while the derivatives of the Lagrangian velocity with respect to initial position tend to grow and thereby have a longer correlation time. The correlation times of these two entities play corresponding roles in the new and old approximations for energy transfer, respectively.

We study the time-dependent solutions of a nonlinear cascade model for homogeneous isotropic turbulence first introduced by Novikov & Desnyansky. The dynamical variables of the model are the turbulent kinetic energies in discrete wave-number shells of thickness one octave. The model equations contain a parameter C whose size governs the amount of energy cascaded to small wavenumbers relative to the amount cascaded to large wavenumbers. We show that the equations permit scale-similar evolution of the energy spectrum. For 0 ≤ C ≤ 1 and no external force, the freely evolving energy spectrum displays the Kolmogorov k power law, and the total energy decreases in time as a power t−w, where the exponent w depends on the value of C. Grid-turbulence experiments seem to favour a value of C in the range 0·3-0·6. In the presence of an external stirring force acting near a wavenumber k0, the model predicts, in addition to the Kolmogorov k spectrum for k > k0, a scale-similar flow of energy to wavenumbers k < k0. This backward energy flow falls off as a power law in time, and establishes a stationary energy spectrum for k < k0 which is a power law in k less steep than k. We discuss the similarity of the behaviour of the model for C > 1 to the behaviour of turbulent fluid for a spatial dimensionality near 2. The model is shown to approach the Kovasznay and the Leith diffusion approximation equations in the limit in which the thickness of the wavenumber shells approaches zero. However, the cascade model with finite shell thicknesses appears to behave in a more physically reasonable way than the limiting differential equations.

The force on a small sphere translating relative to a slow viscous flow is found to O(Re½) for two different fluid flows far from the sphere, namely pure rotation and pure shear. For pure rotation, the O(Re½) correction to the Stokes drag consists of an increase in the drag. For pure shear, the O(Re½) force contains a component perpendicular to the Stokes drag force.