A two-dimensional, incompressible, irrotational, linearized flow model is employed in this analysis of two supercavitating, flat-plate hydrofoils in the presence of a free surface. The cavities are taken to have finite lengths, and gravity is neglected. The ensuing boundary-value problem is converted, by means of conformal mapping, to a mixed-boundary-value problem for the complex velocity in the upper half-plane. This altered problem is solved by use of the methods of thin-aerofoil theory and the solution involves digital-computer evaluation of a large number of incomplete elliptic integrals of the first and third kinds. Typical results are presented in graphs, and the results of the present work are compared with Yim's (1964) theory for a single supercavitating body near a free surface.

The effect of thermal radiation is investigated for the axisymmetric flow over the blunt body associated with a given paraboloidal shock wave. Radiative transfer is treated by means of the differential approximation, which applies to multidimensional flow and is valid throughout the entire range of temperature and optical thickness. The gas is assumed to be perfect and optically grey, and molecular-transport processes are neglected. A semi-analytical solution for the flow and radiation fields is obtained by the method of series truncation.

Results are presented, in the strong-shock approximation, for various values of the appropriate dimensionless variables. In general, radiation is found to have a significant influence on temperature and density, moderate effect on velocity, and little effect on pressure. The stand-off distance between the shock wave and the body is found to decrease significantly with increasing radiation; the body shape is less affected. The anomalous behaviour of the gas temperature on the body streamline as obtained by earlier investigators in the optically thin case does not appear in the present work. The results thus show correct physical behaviour throughout the flow field for all values of optical thickness. The detailed flow quantities exhibit a number of features of multidimensional radiating flow. They also provide a check on the special assumptions made in other, more approximate treatments. Similarities between radiating flow and non-equilibrium reactive flow over blunt bodies are apparent.

A verification of some of the predictions of the theory of Phillips (1955) is presented, and the relation between one-dimensional and two-dimensional wavenumber spectra is discussed. The convection velocity of the irrotational field deduced from measurements of the frequency spectrum alone appears to be about 0.9U1 in the frequency range carrying most of the energy. It follows that the pressure-fluctuation spectrum is proportional to the velocity-fluctuation spectrum and varies as ω2 at low frequency. The discrepancy between this result and measurements of wall-pressure spectra is plausibly attributed to extraneous sound.

The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.

In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Bénard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.

The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).

In the flow-establishment region of an air jet issuing with an efflux velocity of about 35 ft./sec from a 1.0 ft. diameter nozzle into still air, measurements were made of mean axial and radial velocities, mean static pressure, turbulence intensities, turbulent shear, and pressure fluctuation. For the measurement of the latter a pressure probe using a ceramic piezo-electric tube was developed. Also included in the measurements were the temporal mean gradient and autocorrelation of the axial-velocity fluctuation and the intermittency factor. The fluctuating-pressure and turbulence-intensity fields were observed to be closely similar in form. Through use of the measured distributions of mean-flow and turbulence characteristics, all terms of the integral and differential forms of the momentum and mean-energy equations were evaluated throughout the region. The results are presented herein by curves of variation of each of the terms as they appear in the corresponding equations.

Solutions to the problem of a two-dimensional, laminar jet of incompressible fluid issuing into a uniform stream in the direction of the main flow are considered. Two co-ordinate-type expansions are developed. A direct expansion, when suitably transformed, predicts approximately the velocity along the plane of symmetry of the jet for all values of the abscissa, with a maximum error of 7·6% far downstream from the origin. This error is established by comparison with a second, asymptotic expansion valid only at large values of the abscissa. The two expansions are subsequently joined, permitting an approximate determination of a constant which multiplies a third-order term in the asymptotic series and which initially remained unknown even after satisfying all boundary conditions imposed on these series.

The decay of velocity excess along the plane of symmetry of the jet is accelerated by the presence of the external stream.

The slow motion of fluid exterior to a cylinder lying on a wall is considered for a variety of boundary conditions. In particular, the solution is obtained for the case when the motion far from the cylinder is one of uniform shear. Calculations are made for the force and the moment exerted by the fluid on the cylinder. The asymptotic form of the flow both far from the cylinder and near the cusped corners is presented. The flow sufficiently near a cusp consists of a sequence of eddies of rapidly diminishing strength, and the solution of another boundary-value problem supports the view that the nature of this eddy system is independent of conditions far from the cusp. The nature of inviscid flow with uniform vorticity in cusped corners is also considered.

The breakdown of the characteristics solution in the neighbourhood of the leading frozen characteristic is investigated for the flow induced by a piston advancing with finite acceleration into a relaxing gas and for the steady supersonic flow of a relaxing gas into a smooth compressive corner. It is found that the point of breakdown moves outwards along the leading characteristic as the relaxation time decreases and that there is no breakdown of the solution on the leading characteristic if the gas has a sufficiently small, but non-zero, relaxation time. A precise measure of this relaxation time is derived. The paper deals only with points of breakdown determined by initial derivatives of the piston path or wall shape. In the steady-flow case, the Mach number based on the frozen speed of sound must be greater than unity.

The steady, two dimensional, incompressible flow of a viscoelastic fluid past a wedge of 90 degrees coated with the viscoelastic material is studied theoretically using constitutive equations proposed by Oldroyd in 1958. The effect of diffusion of the coating as well as its material properties (viscosity, relaxation time, retardation time, etc.) on the frictional force is investigated.

A boundary-layer analysis is performed on the constitutive equations as well as on the momentum equations. A similarity transformation is found for the set of boundary-layer equations. Series expansion and Laplace's method are employed to obtain the solution in the asymptotic series of gamma functions.

The results obtained show that:

1. The thinner displacement thickness does not necessarily imply a large frictional force for the viscoelastic flow.

2. For a homogeneous viscoelastic flow, the frictional force increases as the degree of dilatancy of the material increases, and decreases with increasing degree of pseudo-plasticity of the material.

3. For a non-homogeneous viscoelastic flow with given material constants, depending on whether the material is pseudoplastic or dilatant and on the ratio of the material concentration of outer flow and the concentration at the body, the frictional coefficient will decrease or increase from that of the homogeneous flow with the concentration at the body as the Schmidt number increases, and will approach a limit when the Schmidt number becomes very large. This limit is the frictional coefficient of the homogeneous flow with the concentration of the outer flow.

The Orr-Sommerfeld stability problem has been studied for velocity profiles appropriate to turbulent channel flow. The intent was to provide an evaluation of Malkus's theory that the flow assumes a state of maximum dissipation, subject to certain constraints, one of which is that the mean velocity profile is marginally stable. Dissipation rates and neutral stability curves were obtained for a representative two-parameter family of velocity profiles. Those in agreement with experimental profiles were found to be stable; the marginally stable profile of greatest dissipation was not in good agreement with experiments. An explanation for the apparent success of Malkus's theory is offered.

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.

At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.

Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

A formal expansion method for analysis of the non-linear development of an oblique wave in a parallel flow is presented. The present approach constitutes an extension and modification of the method of Stuart and Watson. Results are obtained for plane Poiseuille flow, and for a combination of plane Poiseuille and plane Couette flow. The Poiseuille flow exhibits finite-amplitude subcritical instability, and relatively weak but finite disturbances markedly reduce the critical Reynolds number. The combined flow, which becomes stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently great, remains unstable to finite disturbances.

We discuss a one-parameter family of general solutions of the hydrostatic equations for liquid crystals. For certain values of the parameter, the energy may take on extremal values. We discuss special features of such solutions. Configurations similar to those considered naturally occur in liquid crystals of cholesteric type, suggesting that some such state has a relatively low energy.

The wake behind a cone in incompressible flow has the form of a closed bubble. Measurements of velocity, turbulence and static pressure for various cone angles show that the wakes are all essentially similar. A wake Strouhal number may be defined, which is the same for all the models. A disk may be treated as a cone of 180° vertex angle.

Laminar incompressible flow past a semi-infinite flat plate is examined by using the method of series truncation (or local similarity) on the full Navier-Stokes equations. The first and second truncations are calculated at points on the plate away from the leading edge, while only the first truncation is calculated at the leading edge. The solutions are compared with the results from other approximate methods.

Temperature measurements in the test section of a hypersonic gun tunnel using a modified line-reversal two-beam optical pyrometer have been performed. Measurements were made in the shock layer at the front of an ‘end-on’ flat-faced cylinder using argon or air as the driven gas. The measured temperatures were greater than the calculated isentropic temperatures but much less than the originally predicted shock-heated values. The experimental values agreed reasonably well with the theoretical temperatures for shock-heated air corrected for boundary-layer and heat-loss effects in the gun barrel and also with previously reported experimental values found using heat-transfer methods. Preliminary studies of the vibrational relaxation of nitrogen and air in the test section have been carried out using a flat-sided ‘relaxation’ tube (a Pitot tube with glass windows). They demonstrated the feasibility of this approach.

The theory of stability of the flow of a viscous, electrically conducting fluid between rotating cylinders in the presence of an axial magnetic field is extended to the case where the cylinders are permeable and the primary flow includes a radial component. Numerical results pertaining to the stationary axially symmetric modes are presented, and the asymptotic stability behaviour for large values of the radial Reynolds number is derived.

A linear theory is developed for the motion of a viscous, incompressible fluid in a rotating cylinder with a sloping bottom.

An analysis of the normal modes of oscillation reveals that the presence of the bottom slope introduces a new set of low frequency inertial oscillations to replace the purely geostrophic modes which are not allowed in this geometry. The new waves possess mean circulation and are the mechanism by which the fluid adjusts to changes in the rotation rate of the container, a process discussed in detail.

The steady motion produced in the cylinder when the cylinder's upper surface rotates at a different rate than the bottom surface is studied. It is shown that the presence of the bottom slope inhibits the steady fluid motion in the body of the cylinder and introduces a non-symmetric, high velocity side wall boundary layer.

Experimental evidence, presented to validate the theory, reproduces certain important features of the oceanic circulation.

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a. An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that ε = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α0+α1ε) log ε + β0 + O(ε) where α0, α1 and β0 are constants which have been determined explicitly.