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.

A wake model for the free-streamline theory is proposed to treat the two-dimensional flow past an obstacle with a wake or cavity formation. In this model the wake flow is approximately described in the large by an equivalent potential flow such that along the wake boundary the pressure first assumes a prescribed constant under-pressure in a region downstream of the separation points (called the near-wake) and then increases continuously from this under-pressure to the given free-stream value in an infinite wake strip of finite width (the far-wake). Application of this wake model provides a rather smooth continuous transition of the hydrodynamic forces from the fully developed wake flow to the fully wetted flow as the wake disappears. When applied to the wake flow past an inclined flat plate, this model yields the exact solution in a closed form for the whole range of the wake under-pressure coefficient.

The formation of vortex streets in the wake of two-dimensional bluff bodies can be explained by considering the non-linear interaction of two infinite vortex sheets, initially a fixed distance, h, apart, in an inviscid incompressible fluid. The interaction of such sheets (represented in the calculation by rows of point-vortices) is examined in detail for various ratios of h to the wavelength, a, of the initial disturbance. The number and strength of the concentrated regions of vorticity formed in the interaction depend very strongly on h/a. The non-linear interaction of the two vortex sheets explains both the cancellation of vorticity and vortex-street broadening observed in the wakes of bluff bodies.

This paper studies the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude. The effects are interpreted in terms of the radiation stresses in the waves.

The first example is of wave groups propagated in water of uniform mean depth. The problem is solved first by a perturbation analysis. In two special cases the second-order currents are found to be proportional simply to the square of the local wave amplitude: (a) when the lengths of the groups are large compared to the mean depth, and (b) when the groups are all of equal length. Then the surface is found to be depressed under a high group of waves and the mass-transport is relatively negative there. In case (a) the result can be simply related to the radiation stresses, which tend to expel fluid from beneath the higher waves.

The second example considered is the propagation of waves of steady amplitude in water of gradually varying depth. Assuming no loss of energy by friction or reflexion, the wave amplitude must vary horizontally, to maintain the flux of energy constant; it is shown that this produces a negative tilt in the mean surface level as the depth diminishes. However, if the wave height is limited by breaking, the tilt is positive. The results are in agreement with some observations by Fairchild.

Lastly, the propagation of groups of waves from deep to shallow water is studied. As the mean depth decreases, so the response of the fluid to the radiation stresses tends to increase. The long waves thus generated may be reflected as free waves, and so account for the 'surf beats’ observed by Munk and Tucker.

Generalle speaking, the changes in mean sea level produced by ocean waves are comparable with those due to horizontal wind stress. It may be necessary to allow for the wave stresses in calculating wind stress coefficients.

The solution is obtained to the problem of the steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. A no-slip condition on the velocity is assumed at the non-conducting wall. The solution is exact and thus valid for all values of the Hartmann number. Excellent agreement exists between the present theoretical results and the experimental values obtained by Hartmann & Lazarus (1937) in the low to medium Hartmann number range. The high Hartmann number case is treated by Shercliff (1962) in the following paper.

In a recent work Longuet-Higgins & Stewart (1961) have studied the changes in wavelength and amplitude of progressive waves of constant frequency as they are propagated into regions of surface divergence or convergence. In the work here described the complementary conditions are assumed. Standing waves of uniform wavelength, λ, exist in an area of uniform surface divergence. Changes in amplitude and wavelength are studied. These changes depend on the existence of the radiation stress which was discovered by Longuet-Higgins & Stewart but the physical interpretation of this stress is simpler for standing than for progressive waves. Three different ways of obtaining the same rate of strain in the direction of the current caused the amplitude to vary as $\lambda|^{-{\frac {1}{4}}}, \lambda|^{-{\frac {3}{4}}}$ and $\lambda|^{-{\frac {5}{4}}}$, respectively.

Experiments in which free-standing waves were generated in a tank one wave-length wide which was then made narrower verified the conclusion that contraction does not alter the periodic character of the waves, even though the ratio of amplitude to wavelength becomes so great that they can no longer be treated mathematically by the usual linearized approximation. The shape of the profile then appears to agree well with calculations of Penney & Price (1952).

The motion of an electrically conducting, incompressible, viscous fluid in the presence of a magnetic field is analyzed for flow between two parallel disks, one of which rotates at a constant angular velocity. The specific application to liquid metal lubrication in thrust bearings is considered. The two field configurations discussed are: an axial magnetic field with a radial current and a radial magnetic field with an axial current. It is shown that the load capacity of the bearing is dependent on the MHD interactions in the fluid and that the frictional torque on the rotor can be made zero for both field configurations by supplying electrical energy through the electrodes to the fluid.

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.

In a recent paper Tadjbakhsh & Keller (1960) have predicted that two-dimensional finite standing gravity waves in a rectangular container will have lower frequency than infinitesimal standing waves in deep water but have higher frequency below a certain mean depth to wavelength ratio. This is in strong contrast to the frequency results for finite progressive waves obtained by many investigators. Experimental confirmation of this prediction is reported together with estimates of the magnitude of the frequency effects at several depths. The frequency effect reversal appears to occur at a depth ratio of 0·14, somewhat less than the predicted ratio of 0·17.

The paper presents an improved, second approximation for the laminar motion of a conducting liquid at high Hartmann number in non-conducting pipes of arbitrary cross-section under uniform transverse magnetic fields. A satisfactory comparison with the author's previously experimental pressure gradient/flow results is made for the case of a circular pipe.

A theory is proposed in which the stress tensor is a function of the components of the rate of deformation tensor and a symmetric tensor describing the microscopic structure of a fluid. The expression for the stress tensor can be written in closed form using results from the Hamilton-Cayley theorem. This theory is shown to contain Prager's theory of dumbbell suspensions as a special case. By limiting the type of terms in the constitutive equations, various stress components can be evaluated for simple shear. These exhibit non-Newtonian behaviour typical of certain higher polymer solutions.

Some of the results of the anisotropic fluid theory are compared with experimental measurements of normal stress and apparent viscosity. Certain high polymers in solution show good agreement between theory and experiment, at least for low enough values of the rate of shear.

The density distribution in the relaxation regions of shock waves in carbon dioxide were determined in the Mach number range 1·4 to 4·0 using an interferometer. The over-all density ratios were found to agree with the theoretical final equilibrium values. Detailed analysis of the relaxation regions showed that the simple relaxation equation is inadequate, the relaxation frequency depending on departures from equilibrium as well as on temperature.

It is the purpose of this paper to solve a boundary-value problem posed by induction electromagnetic pumps and generators. Solutions are obtained by an expansion technique and a momentum method for the laminar, incompressible flow problem. For large values of the interaction parameter (μ2σH20λ/ρμe viscous effects are shown to be restricted to periodic boundary layers. In regions of high-field strength a local Hartmann solution is valid. Where the applied field is weak an inertial boundary layer is present which thickens in the upstream direction.

A logical explanation of this phenomenon is given. The condition that a boundary-layer type flow exist is obtained and is shown to be in general satisfied. The results show that inviscid theory may be used to calculate the overall performance of electromagnetic pumps and generators while the boundary-layer theory developed here may be used to obtain the wall shear stress.

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 effects of compressibility, temperature-jump and gaseous adsorption are considered in an attempt to predict the temperature-time history at the interface between a gas and a solid when both the temperature and the pressure of the gas are suddenly changed. It appears that temperature-jump will be the dominant effect, with adsorption contributing significantly in some circumstances. Compressibility is of minor importance during practically-resolvable time intervals. Some experimental evidence which gives tentative support to the results of the analysis is commented upon.

Vibrational relaxation effects are examined for hypersonic nozzle air flows. A simple method of estimating the distribution of vibrational temperature along a nozzle is described and higher-order approximations discussed. Results are presented for hyperbolic axisymmetric nozzles with reservoir conditions 1000 ≤ p0 ≤ 4000 p.s.i., 1000 ≤ T0 ≤ 3000 °K. Vibrational freezing is shown to occur and to cause significant changes in the nozzle exit flow conditions.

The flow of air in the lowest few metres of the atmosphere is examined in the case when a neutrally stratified logarithmic profile encounters a sudden change in surface roughness. Examination of the available evidence suggests that the ratio of new to old friction velocity is given by the ratio of the roughness lengths raised to a power about equal to 0.09. Some field observations and wind-tunnel measurements over a change in roughness indicate that vertical displacement of the streamlines at all heights within the range considered commences very near the surface transition. Calculations are made of the fetch necessary for the new wind profile to be established up to a given height and indicate that its ratio to the height is not constant but varies with the height and amount of change in roughness. Over a fairly wide range this ratio is about 100-150.

Two simultaneous first-order non-linear equations are derived to give a ’high-flow-rate’ model for the displacement of oil by hot water in a porous medium. The solution of these equations is analysed by the method of characteristics and it is shown that in problems for which thermal capacity dependence on temperature is neglected, the solution will have the properties of a simple wave. The simple wave behaviour gives a rapid method for solving practical systems. When temperature dependence is included in the thermal capacities, it is found that a simultaneous shock in temperature and saturation develops, but the solution will usually approximate quantitatively the simple wave result. Decoupling the equations by using an average saturation in the heat transport equation gives results in reasonable agreement with the coupled case.

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.