The second-order problem of Helmholtz flow past lifting hydrofoils and symmetric struts has been formulated and solved. The solution involves elementary operations on the known solutions of the first-order problem. The second-order lift and drag coefficients are given in integral form. Results obtained for a flat plate at incidence and a symmetric wedge agree with the exact solutions up to the second order. In terms of quantitative improvements, the present second-order theory predicts a lift coefficient for a flat plate at 45° incidence with an error of 8%, and a drag coefficient for a symmetric wedge of 50° included angle with an error of 5%; the corresponding angles at which the linear theory would predict force coefficients incurring the same errors are 5° and 15° respectively.

The question considered here is whether turbulent wakes can undergo a simple self-preserving development when convected through a duct of varying crosssection. A flow is described as self-preserving if the distributions of its mean-value properties at successive sections have the same basic shape, differing only in magnitude and lateral extent. In such flows the scales of the mean velocity variation and turbulence have been found to be proportional.

The wakes studied are concentrated near a plane of symmetry through a duct of changing rectangular section but constant area; the portion of the wake near the other plane of symmetry is considered. Measurements in the flows behind three circular cylinders reveal a lack of universality of the scales associated with self-preserving solutions of the momentum equation. The wakes of the smaller cylinders adopt the predicted form, but that of the largest does not. As it moves through the channel, this wake is compressed less rapidly by the imposed lateral strain than is predicted.

Interest is then concentrated on the equation governing the kinetic energy of the turbulence and in particular on the relative size of its two production terms, that representing the action of the mean wake shear and that representing the effect of the distortion applied by the duct. It is found that self-preserving wakes can form only in a few kinds of distorting convecting flow. In the experimental duct, which has exponentially varying walls, one of these is set up. Attention is concentrated on this case. A stability analysis (based on the criterion that a stable wake is one in which the total turbulence production decreases as a result of more rapid wake expansion) suggests that, when production by wake shear is greater than that by distortion, a stable self-preserving development is possible in which the turbulence scale and the velocity defect scale remain proportional. But in the alternative, distortion-dominated, case the velocity defect decreases continually relatively to the turbulence.

The experimental results are in accord with these predictions. The transition from shear-dominated to distortion-dominated wakes appears to take place when the production terms are about equal. The failure of the one experimental wake to adopt a simple self-preserving form can be attributed to the relatively slight organization of its turbulence when distortion is begun.

The advancing front of a buoyant plume which is being established in uniform surroundings has some properties in common with the plume, while in other respects it behaves more like a ‘thermal’ released from rest. The solutions for these two cases cannot be matched directly, since the dependence of velocity on height is different. It is shown here that a similarity solution, which is consistent with the equations describing both parts of the flow, can be obtained once it is recognized that the velocity of the front may be less than that of the steady plume. The cap moves with a constant fraction of the plume velocity at the same level, and the total buoyancy in the cap is increasing, so a modification of the simple relations for thermals is required.

This prediction is verified experimentally, and numerical values for the ratio of the velocities and the rate of increase of the cap radius with height determined. The extreme front of plume cap advances at about 0·6 times the mean velocity on the axis of a steady plume, and it spreads at just over half the angle of a thermal. This implies a smaller rate of entrainment and therefore a smaller rate of dilution per unit height compared with a thermal, especially since about half of the fluid mixed into the cap comes from the plume below. The model allows one to estimate the time necessary for convection from a known steady source on the ground to lead to the formation of a cloud.

Recent investigations by Kraichnan (1961) and Ogura (1961) have raised doubts concerning the usefulness of the zero fourth cumulant approximation in turbulence dynamics. It appears extremely tedious to examine, by numerical computation, the consequences of this approximation on the turbulent energy spectrum although the appropriate equations have been established by Proudman & Reid (1954) and Tatsumi (1957). It has proved possible, however, to compute numerically the sequences of an analogous assumption when applied to an isotropic passive scalar in isotropic turbulence.

The result of such computation, for specific initial conditions described herein, and for stationary turbulence, is that the scalar spectrum does develop negative values after a time approximately $2 \Lambda | {\overline {(u^2)}} ^{\frac {1}{2}}$, Where Λ is a length scale typical of the energy-containing components of both the turbulent and scalar spectra and $\overline {(u^2)}^{\frac {1}{2}}$ is the root mean square turbulent velocity.

The differential equations representing the supersonic flow of a gas past a cone of any cross-section are integrated numerically, using a method similar to those used for bluff-body problems. A stream function is used as one of the independent variables and this is particularly suitable for determining the singular ‘vortical layer’. The method is here applied to the cases of elliptic cones at zero yaw and circular cones at incidence. The results are compared with experiment and with other numerical solutions.

The impulsive movement of a plate (Rayleigh problem) is considered for a compressible flow in which, prior to initiation of the motion, both velocity and enthalpy gradients exist normal to the plate. The solution is valid for large times, and the external gradients are chosen of such magnitude that their effects enter to the same order as displacement effects due to induced vertical velocity. This displacement effect is influenced by the external enthalpy gradient. Both insulated wall and a step change in wall enthalpy are considered. For the insulated wall it is found that the part of the displacement solution which is uninfluenced by the external gradient requires a term in the logarithm of Reynolds number (based on time). This differs in principle from the case for a constant wall enthalpy. Displacement with uniform outside flow affects heat transfer but not skin friction, just the opposite from the corresponding results for the steady two-dimensional semi-infinite flat plate. The influence of external gradients on skin friction, heat transfer and adiabatic wall enthalpy is given.

The equations of motion and continuity are solved together with the slow-flow stress equations for an incompressible Rivlin-Ericksen fluid. The boundary conditions for slow flow past a sphere are satisfied by matching inner (Stokes) and outer (Oseen) Reynolds-number expansions of the stream function. The terms in the inner expansion are the solutions of non-linear partial differential equations which are solved approximately by expanding in terms of a non-Newtonian parameter λ. The drag force on the sphere is obtained from the solution.

The inviscid Orr-Sommerfeld equation for ϕ(y) in y > 0 subject to a null condition as y → ∞ is attacked by considering separately the intervals (0, y1) and (y1, ∞), such that the solution in (0, y1) can be expanded in powers of the wave-number (following Heisenberg) and the solution of (y1, ∞) regarded as real and non-singular. Complementary variational principles for the latter solution are determined to bound an appropriate parameter from above and below. It also is shown how the original differential equation may be transformed to a Riccati equation in such a way as to facilitate both the Heisenberg expansion of the solution in (0, y1) and numerical integration in (y1, ∞). These methods are applied to a velocity profile that is linear in (0, y1) and asymptotically logarithmic as y → ∞, and it is found that the mean of the two variational approximations is in excellent agreement with the results of numerical integration of the Riccati equation.

The transfer of energy from wind to short surface waves through the viscous Reynolds stress in the immediate neighbourhood of the surface is explored. Resonance between the Tollmien-Schlichting waves for a given wind profile and the free-surface waves is shown to be an important possibility. Numerical results are given for water wave generation by a wind having a velocity profile that is linear in the neighbourhood of the surface and asymptotically logarithmic.

The flow field near a stagnation point in two-dimensional, incompressible, viscous flow is considered to change with time in such a way that the inviscid flow is steady after some given finite instant of time. The final approach to steady flow throughout the field is shown to be characterized by exponential decay with time of perturbations from the steady velocity field. The characteristic factors in the exponents arise from the solution of an eigenvalue problem in ordinary linear differential equations.

Similar behaviour exists for the axially symmetric case. A comparable analysis furnishes, however, a meaningless result in the case of a two-dimensional, semiinfinite flat plate which is moving in its own plane, normal to its leading edge.

Using an electrically driven shock tube with initial pressures of 0.1 to 1.0 mm Hg and shock speeds of about Mach 12 to 15, the resistance of an air plasma between two parallel probes has been measured by two different techniques and the results compared. In one, external voltages of from 0 to 100 V were applied to the probes and in the other, electromagnetically induced voltages of from 0 to 25 V were produced by the plasma's motion in a magnetic field of up to 3500 G. In either case the resistance was found to decrease as the current flow increased and was consistent with the equilibrium electronic conductivity of the air plasma at high current densities.