An analytical treatment is given of the problem of the establishment of the flow through a dam or levee with a horizontal underdrain, when the head behind it is raised and then kept at a constant value. The essential idea employed in the analysis is to consider the unsteady flow as a time-dependent perturbation of the final steady flow. The unsteady potential ϕ(x, y, t) is expanded in a power series of e−λt, of the form $\phi (x,y,t) = \phi_0(x,y) + \phi_1(x,y)e^ {\lambda t} + O(e^{-2\lambda t})$ where ϕ0(x, y) is the known steady-state potential, ϕ1(x, y) is a perturbation potential and O(e−2λt) = ϕ2(x, y) e−2δt + ϕ3(x, y) e−3λt + …. Each of the terms ϕn(x, y)e−nλt can be thought of as being a perturbation term of its precursor in the series, and the present approach is limited to the computation of the first perturbation term ϕ1(x, y)e−λt.

It is shown that ϕ1 satisfies Laplace's equation ∇2ϕ1 = 0 in a dimensionless hodograph plane. The free-boundary condition is linear but complicated, containing the eigenvalue λ, which is fixed by a determinantal equation. The amplitude of the displacement of the free surface is left undetermined; only the mode of the motion and the eigenvalue are computed. The results of a numerical example are summarized.

The effect of increasing the rate of mixing in turbulent boundary layers in a region of adverse pressure gradient has been investigated experimentally. Only the two-dimensional case was considered. The boundary layer was formed on a flat wall in a special wind tunnel in which a variety of adverse pressure gradients could be obtained. Speeds were low enough to justify the neglect of compressibility. The main objective was to compare the effect of increasing the rate of mixing with the effect of reducing the pressure gradient on boundary-layer development and separation. A Variety of mixing schemes was tried, all of them involving fixed devices arranged in a row on the surface in the region of rising pressure. While these differed considerably in effectiveness, they had a generally similar effect on the flow; and, except for effects arising from changes in displacement and momentum thickness introduced at the devices, their effect on the layer was basically equivalent to that of a decrease in pressure gradient. Apart from forced mixing, the shape of the pressure distribution was found to have a significant effect on displacement and momentum thickness, these being minimized and the wall distance decreased for a given pressure rise by a distribution with an initially steep and progressively decreasing gradient.

The mean velocity profile near the surface of turbulent flow in a broad open channel is discussed with dimensional arguments, and a new empirical constant m is introduced which is analogous to von Kármán's constant for flow near a rigid boundary. It is shown that, while the velocity profile depends only rather weakly on m, the dependence of the coefficient of apparent longitudinal diffusion is stronger, and measurements of diffusion could, in principle, provide an accurate determination of its value. The new profiles for various values of m are compared with those in current use, and finally the correction for finite Reynolds number is discussed.

The assumption of dynamic similarity is used to determine the velocity potential of an unsteady line source which lies on a plane separating two media of different density and sound velocity. The solution first derived is for a source whose strength varies with time as a step function; the pressure in this case may be identified with Hadamard's elementary solution which in a homogeneous fluid is (c/2π) (c2t2-r2)−½ if r < ct, and 0 if r > ct. We next derive the solution for a source whose strength has a delta-function time dependence; we then describe the results for a supersonic point source moving on the interface, and finally we transfer the results to solve the corresponding electromagnetic problem.

Measurements of spectra of sound emission to the far field from jets at high subsonic velocities are presented. The similarity relations found in the experiments suggest a mechanism of sound generation and scattering where the latter is of dominant importance. A possible mechanism is described.

The steady motion of an infinitely long solid cylinder parallel to its length in a conducting fluid in the presence of a uniform magnetic field is discussed. Due to Alfvén waves originating at the cylinder we find two opposite ‘wakes’ parallel to the applied magnetic field.

A formula which relates the total drag on the cylinder to the electric potential difference δΦ between the two undisturbed regions outside these two wakes is derived $D|\;|\delta \Phi| = 2\surd {\rho}v \sigma$ where ρν is the viscosity and σ is the conductivity of the fluid.

The reduction to a classical boundary-value problem is made for the case of an insulating cylinder.

Exact solutions are obtained for the case of a perfectly conducting or an insulating flat strip of semi-infinite width. These give a clear picture of the field, especially in the transition region near the edge of the strip.

The case of a strip of finite width is also discussed with special reference to the viscous and the magnetic drags, Df and Dm. We find that Df + ½Dm, on a perfectly conducting strip, is equal to the viscous drag on an insulating strip for which Dm is zero. Precise values of these drags are given.

The motion of bodies in a direction parallel to an applied magnetic field and through a perfectly conducting fluid is considered. It is shown that the perturbation in the state of the fluid cannot remain small except in the particular case when the velocity U of the body is much smaller than that of the Alfvén waves in the fluid. In this case, however, the perturbation is not confined to the neighbourhood of the body, and extends to infinity inside planes which touch the body and are parallel to the undisturbed magnetic field. In addition the body experiences a drag.

The oscillating plate problem is investigated for the case of an incompressible, electrically conducting fluid in the presence of a magnetic field. The boundary conditions are examined in detail, and a solution is found with the aid of suitable approximations. The motion of the fluid is shown to consist mainly of magnetohydrodynamic waves, but there is also a viscous boundary layer in which the solution agrees with that given by other writers.

By a consideration of the relationships holding along the characteristics in an unsteady motion involving plane, axially or spherically symmetrical flow of compressible inviscid fluid, it is shown that the existence of a region of compression anywhere in the flow must lead eventually to the breakdown of continuity. The paper generalizes and unites previous work on this topic, and discusses some recent numerical calculations in which the expected discontinuity was not found.

An examination of the Newtonian inviscid theory of hypersonic flow past a sphere is made in the neighbourhood of the singular point which occurs at θ = 60°. A uniformly valid theory is developed in this neighbourhood which exhibits the limiting detached ‘free-layer’ behaviour postulated by Hayes and Lighthill in the limit as γ → 1. A complete solution is obtained which gives details of streamline and shock wave shape, pressure distribution, etc. for γ − 1 small.

Reasons are given why the empirical ‘modified’ Newtonian theory of Lees proves to be a good approximation to experimentally determined pressure distributions.

A novel check to the theory is provided by a simple power-law relation between the pressure on the sphere surface and the distance of the shock wave away from sphere at the same point.

By use of appropriate approximations the incompressible stagnation-point ablation rate for ice is determined theoretically. The theory includes both melting and vaporization or condensation. To verify the theory hemispheres of ice were melted in a subsonic wind tunnel with controlled humidity. It is found that the effects of heat transfer and condensation are of equal importance in determining the melt rate. The agreement between theory and experiment is adequate.

The hydromagnetic stability of a basic two-dimensional parallel flow of an incompressible conducting fluid in a uniform magnetic field parallel to the flow is considered. By use of the generalization of the Orr–Sommerfeld equation for an electrically conducting fluid, it is shown that any given small wave disturbance can be stabilized by a sufficiently strong magnetic field if the Reynolds number is finite and the magnetic Reynolds number small.

Stability of velocity profiles with a point of inflexion at small magnetic Reynolds number and infinite Reynolds number is considered in detail. Perturbation methods are developed to find stability characteristics in two cases, when the magnetic field is weak, and when the disturbance is a long wave. These methods are applied to the jet and the half-jet, which are both found to be unstable to long-wave disturbances, however strong the magnetic field. Nonetheless, these two flows can be stabilized for any given harmonic disturbance of finite wavelength. The analysis of the jet reveals the surprising result that the magnetic field makes inviscid long-wave disturbances more unstable.

In a recent paper, Stratford has described a turbulent boundary layer with continuously zero wall stress and has developed a theory to describe the flow based on two assumptions. The first is that the flow in the outer part of the layer is affected only by the original Reynolds stresses during the initial development, and the second is that flow in the equilibrium layer close to the wall is determined by the pressure gradient and is independent of upstream conditions. In this paper the same assumptions are used, but more careful consideration of their limitations has led to the elimination of some inconsistencies in the original work and to a theory that gives a better description of some of the experimental results. The principal results are: (i) a criterion for zero wall stress in an adverse pressure gradient of sufficient strength, (ii) the form of the pressure distribution for a self-preserving flow with zero stress, (iii) the mean velocity distribution in this flow, (iv) an estimate of the constant in the 'square-root’ velocity distribution for flow near a wall with zero stress.