A solution is obtained for the motion of a thin oil sheet, of non-uniform thickness, under a boundary layer. The following points are deduced: (a) The oil flows in the direction of the boundary-layer skin-friction, except near separation, where the oil tends to indicate separation too early. These conclusions are independent of oil viscosity. (b) The effect of the oil flow on the boundary-layer motion is very small.

The application of the results to the interpretation of oil-flow patterns is briefly considered.

The method of characteristics is used to calculate numerical solutions for the one-dimensional motion of a plane shock-wave through a stationary gas which contains a region of non-uniform density. These solutions are compared with those given by the Chisnell-Whitham approach which ignores the effects on the shock-wave of the disturbances which are generated in the flow behind it, and also with the asymptotic solution given by the simple theory which regards the nonuniform region as a contact-surface discontinuity. It is concluded that the results of the simplified theories must be applied with caution.

A slender-body theory for the flow past a slender, pointed hydrofoil held at a small angle of attack to the flow, with a cavity on the upper surface, has been worked out. The approximate solution valid near the body is seen to be the sum of two components. The first consists of a distribution of two-dimensional sources located along the centroid line of the cavity to represent the variation of the cross-sectional area of the cavity. The second component represents the cross-flow perpendicular to the centroid line. It is found that over the cavity boundary which envelops a constant pressure region, the magnitude of the cross-flow velocity is not constant, but varies to a moderate extent. With this variation neglected only in the neighbourhood of the hydrofoil, the cross-flow is solved by adopting the Riabouchinsky model for the two-dimensional flow. The lift is then calculated by intergrating the pressure along the chord; the dependence of the lift on cavitation number and angle of attack is shown for a specific case of the triangular plan form.

In the flow past a slender delta wing at incidence one can observe a roughly axially symmetric core of spiralling fluid, formed by the rolling-up of the shear layer that separates from a leading edge. The aim in this paper is to predict the flow field within this vortex core, given appropriate conditions at its outside edge.

The basic assumptions are (i) that the flow is continuous and rotational, and (ii) that viscous diffusion is confined to a relatively slender subcore. In addition it is assumed that the flow is axially symmetric and incompressible. Together, these admit outer and inner solutions for the core from the equations of motion. For the outer solution the subcore is ignored, and the flow is taken to be inviscid (but rotational) and conical. The resulting solution consists of simple expressions for the velocity components and pressure. For the inner solution, which applies to the diffusive subcore, the flow is taken to be laminar, and certain approximations are made, some based on the boundary conditions and some analogous to those of boundary-layer theory. The solution obtained in this case is a first approximation, and has been computed.

A sample calculation yields results which are in good qualitative and fair quantitative agreement with experimental measurements.

The velocity potential for a simple source moving in a straight line at constant depth in a two-layer ocean is obtained by the Fourier transform method. It is used to develop a formula for the wave-making resistance of a ‘thin’ ship for both surface and internal waves. An asymptotic expansion is used to delineate quantitatively the internal wave system. It is shown that at speeds less than the critical speed transverse and divergent wave systems are excited, while at speeds greater than the critical internal wave speed only the divergent wave system is excited. Examples of the shape of wave crests and of wave heights are given.

The oscillating lift and drag on circular cylinders are determined from measurements of the fluctuating pressure on the cylinder surface in the range of Reynolds number from 4 × 103 to just above 105.

The magnitude of the r.m.s. lift coefficient has a maximum of about 0.8 at a Reynolds number of 7 × 104 and falls to about 0.01 at a Reynolds number of 4 × 103. The fluctuating component of the drag was determined for Reynolds numbers greater than 2 × 104 and was found to be an order of magnitude smaller than the lift.

Recently Kraichnan (1959) has propounded a theory of homogeneous turbulence, based on a novel perturbation method, that leads to closed equations for the velocity covariance. In the present paper, this method is applied to the theory of turbulent diffusion and closed equations are derived for the probability distributions of the positions of marked fluid elements released in a turbulent flow.

Two topics are discussed in detail. The first is the probability distribution, at time t, of the displacement of an element from its initial position. In homogeneous flows, this distribution is found to resemble that for classical diffusion but with a variable coefficient of diffusion which is proportional to $v^2_0 t$ for $t \ll l|v_0$ and which approaches a constant value [eDot ] lv0 for tt [Gt ] l/v0 (l = macroscale, v0 = r.m.s. turbulent velocity).

The second topic treated is the joint probability distribution of the displacements of two fluid elements. Particular attention is focused upon the probability distribution of relative displacement, i.e. Richardson's distance-neighbour function. This is found to be Gaussian for separations r which are large ([Gt ] l). For smaller separations (r [Lt ] l), its behaviour at high Reynolds numbers is found to be quite well expressed in terms of a variable diffusion coefficient K(r,t), as suggested by Richardson (1926). For all but extremely short times, K(r,t) is found to depend only on r and on the form of the inertial range spectrum E(k). On assuming $E(r) \propto v^2_0 l(kl)^{- \frac {3}{2}}$ as results from Kraichnan's approximation (1959), one finds $E(r) \propto v_0 l(r|l)^{ \frac {3}{2}}$. On the basis of similarity arguments of the Kolmogorov type, which give $E(r) \propto v^2_0 l(kl)^{- \frac {5}{2}}$, one finds $E(r) \propto v_0 l(r|l)^{ \frac {4}{3}}$ as, in fact, Richardson originally proposed. The dispersion r2 is proportional to $l^2(v_0 t|l)^4$ on Kraichnan's theory; while $\langle r^2 \rangle \propto l^2 (v_0 t|l)^3$ on the similarity theory. This illustrates that the behaviour of $\langle r^2 \rangle$ is very sensitive to the spectrum.

The effects of density variation and body force on the stability of a heterogeneous horizontal shear layer are investigated. The density is assumed to decrease exponentially with height, and the body force is assumed to be derivable from a potential; the velocity distribution in the shear layer is taken to be U(y) = tanh y. The method of small disturbances is employed to obtain a family of neutral stability curves depending on the choice of the Richardson number. It is demonstrated, furthermore, that the value of the critical Richardson number depends on the magnitude of the non-dimensional density gradient.

In this paper a study of the energy-transfer processes associated with the motion of a viscous, heat-conducting fluid is begun. The class of motions considered are unsteady, two-dimensional, vortical flows. After developing simplified equations of motion and energy appropriate to this type of flow in the low Mach-number limit, general solutions of the momentum equations are presented.

The concept of a line impulse of angular momentum is introduced as an example of this class of motions for which a solution of the energy field is obtainable in closed form. The solution for the line impulse can be viewed as a combination of velocity, pressure, and temperature waves concurrently radiating from the origin of the impulse and decaying with time. Particular examples of the development of the energy field of the impulse in both liquids and gases are presented for selected values of Prandtl number. The energy-transfer processes are discussed in some detail, and the resulting differences in the energy fields for liquid gases are emphasized.

The incompressible laminar boundary layer on a semi-infinite flat plate is considered, when the main stream has uniform shear. A solution is obtained for the first two terms of an asymptotic solution for small viscosity. It is shown that one of the principal effects of free-stream vorticity is to introduce a modified pressure field outside the boundary-layer region.