The paper describes calculations of the statistical properties of turbulence-induced plate vibration, the computational scheme making no reference to Fourier synthesis. The problem considered is that of a local turbulent field acting on an infinite thin elastic plate, and the statistics of the response field are described at large distances from the forcing region. Effects of mechanical dissipation in the plate are examined, along with a discussion of the relevance of the results to problems involving plates of finite extent.

The effects of viscous dissipation are considered for external natural convection flow over a surface. A class of similar boundary-layer solutions is given and numerical results are presented for a wide range of the dissipation and Prandtl numbers. Several general aspects of similarity conditions for flow over surfaces and in convection plumes are discussed and their special characteristics considered. The general equations including the dissipation effect are given for the non-similar power law surface condition.

An inviscid flow with infinite gradients at the wall poses a problem for the accompanying boundary layer that is fundamentally different from the conventional one of bounded gradients. It turns out that in both cases the external gradients do not affect the first-order boundary layer, but the unbounded gradients generate second-order corrections that are of a lower order in Reynolds number than the conventional ones. The present singularity in stagnation-point reacting flow is of an algebraic type, and the boundary-layer corrections that it generates are proportional to non-integral powers of the Reynolds number, with exponents that vanish with the rate of reaction. The present example clarifies such matters as the matching of boundary layer and singular inviscid flow, the structure and decay of the new corrections, and their ranking in comparison with the conventional second-order effects. Numerical computations illustrate the problem and give quantitative results in a few selected cases.

In the design of dielectrophoretic liquid orientation and expulsion systems for zero-gravity environments, maximum electromechanical effect of an imposed electric field is obtained by concentrating the field gradients in the neighbourhood of liquid interfaces. In typical configurations, the electric field gradient plays the role of an electromechanical wall, with a stiffness and inertia represented dynamically by electrohydrodynamic surface waves. As an orientation system rotates, the liquid motions are characterized by these waves as they couple to inertial bulk oscillations and centrifugal surface waves resulting from the rotation. A study is made of configurations typified by an equilibrium in which a circular cylindrical column of inviscid liquid undergoes rigid body rotation. The equilibrium is made possible, even though the cylindrical interface is bounded from outside only by its vapour, because the interface is stressed by an essentially tangential axial electric field intensity, with a strong gradient in the radial direction. Dispersion equations are developed for the electrohydrodynamic centrifugal waves of small amplitude. Conditions for incipience of instability and the frequencies of normal modes of oscillation are given. Experimental observations, which demonstrate the destabilizing influence of the rotation and the effect of rotation and electric field intensity on the normal mode frequencies, are in satisfactory agreement with the theory.

Direct measurements of the energy transfer spectrum in locally isotropic grid turbulence have been used to determine the extent of validity for grid turbulence of the dynamical equation for the three-dimensional energy spectrum in isotropic turbulence. The extent of applicability of the isotropic energy balance is consistent with the usual local isotropy criterion based on energy spectra alone.

The present results are in general agreement with some previous measurements by Uberoi, who determined the transfer spectrum assuming the strict validity of the isotropic dynamical equation. The measured energy transfer spectra are quantitatively similar to those calculated by Kraichnan using the direct-interaction approximation.

Velocity and temperature boundary layers developed on a plane wall by ideal shock-tube flow are considered for weak shock and expansion waves. Analytically, the boundary layer consists of three regions, bounded by (1) expansion-wave head, (2) diaphragm location, (3) contact discontinuity, (4) shock. The flow fields (1, 2) and (3, 4) are, essentially, known. In the interaction region (2, 3), these flow fields merge, the governing equations are ‘singular parabolic’ and admit boundary conditions usually associated with elliptic equations. It is convenient to replace the weak expansion wave in the main flow by a line discontinuity. A consistent linearization scheme can now be devised to obtain the solution in the three regions. In (2, 3), the resulting linear singular parabolic equations for the first-order solutions are solved successfully by an iterative finite difference method, normally applied to elliptic equations.

A formula for the drag exerted on a cloud of spherical particles of a given particle size distribution in low Reynolds number flow is derived. It is found that the drag experienced by a particle depends only on the first three moments of the distribution function. A treatment of viscous interaction between N particles to the lowest order is carried out systematically. By appealing to the concept of ‘randomness’ of the particle cloud, equations describing the averaged properties of the fluid motion are derived. The averages are formed over a statistical ensemble of particle configurations. These mean flow equations so obtained are in a form resembling a generalized version of Darcy's empirical equation for the motion of fluid in a porous medium. The physical meaning of these equations is discussed.

This note presents a discussion of the roles of axial momentum flux, flow force, angular momentum flux and circulation in determining the strength and hence characterizing the structure of such narrow rotating axisymmetric core flows as swirling jets, vortex jets, sink vortices and vortex wakes. The salient (though sometimes neglected) features of these core flows are that perturbation pressure plays an essential role both in the coupling of axial and azimuthal velocity fields and in the transmission of force along the core, and that flux of angular momentum is invariant only along cores with zero gross circulation. A number of existing solutions are brought into relationship by the discussion, including Long's similarity solution for draining vortices and Reynolds’ dimensional treatment of swirling wakes.

This is a study of the magnetohydrodynamic flow of an incompressible viscous fluid between coaxial disks, with a uniform axial magnetic field B. The fluid has density ρ, viseosity η and electrical conductivity σ. The flow is assumed to be steady, and to be similar in the sense that the radial and tangential components of velocity increase linearly with radial distance from the axis of rotation. Most of the work is concerned with disks which are electrical insulators, one of which rotates while the other remains stationary. The imposed conditions can then be represented by the Reynolds number R = ρΩad2/η and the Hartmann number M2 = σB2d2/η, where Ωa is the angular velocity of the rotating disk and d is the gap between the disks. Asymptotic solutions are given for R [Lt ] M2, and numerical solutions are obtained for values of R and M2 up to 512. Experimental measurements are presented which are in general agreement with the theoretical flows, and the results for small values of the Hartmann number provide the first known experimental support for the purely hydrodynamic solutions in the range 100 < R < 800.

It does not seem to be possible to prove analytically that an incompressible, inviscid free shear layer is less unstable with respect to spatially growing three-dimensional disturbances than to two-dimensional ones. For this reason a numerical calculation for the special case of the hyperbolic tangent velocity profile was performed. It was found that even for spatially growing disturbances the amplification of three-dimensional disturbances is smaller than for two-dimensional ones.

The Navier–Stokes hypersonic weak-interaction theory is presented for the flow of a viscous, heat-conducting, compressible fluid past a very slender axisymmetric body, when the ratio of the radius of the body to the radial thickness of the viscous region, produced and supported by the body, is much less than unity. The fluid is assumed to be a perfect gas having constant specific heats, a constant Prandtl number of order unity, and viscosity coefficients varying as a power of the absolute temperature. Solutions are studied for the free-stream Mach number, the free-stream Reynolds number based on the axial length of the body, and the reciprocal of the weak-interaction parameter much greater than unity.

It is shown that, for the viscosity-temperature exponent ω less than 1, seven distinct layers span the region between the shock wave and the body, which is of arbitrary shape. The leading approximations for the behaviour of the flow in these seven layers are analyzed, and the restrictions imposed on the theory are obtained.

A study is made of the transient flow which results from the impingement of a plane shock on a blunt body. The analysis is based on Taylor-series expansions in the space and time variables of the flow properties and of the shape of the reflected shock. Coefficients of the series are determined numerically so as to satisfy the exact equations of motion and shock jump conditions. Convergence problems are ameliorated by recasting the series into continued fractions. While the analysis does not treat the transition from regular to Mach reflexion, it remains valid for all time in the subsonic region of the flow if the incident shock is sufficiently strong. In any case, it is accurate enough where it is valid to be useful for evaluation of more conventional numerical methods.

Collapsing cylindrical metallic shells have been used to compress magnetic fluxes and generate megagauss magnetic fields. Such shells experience large, rapidly growing accelerations and their symmetry can be completely destroyed by Rayleigh-Taylor instabilities. This paper presents a theoretical study of the Rayleigh-Taylor instability for radially accelerated incompressible cylindrical shells submitted to the pressure of much lighter media. Low-amplitude flute perturbations are considered and Fourier-analyzed in the azimuthal angle. A fourth-order linear differential system with time-dependent coefficients is derived, which determines the two interface-displacements. Stability criteria are discussed. When the perturbation wavelength is much greater or much smaller than the shell thickness, the differential system splits into two independent differential equations and results are greatly simplified; analytical solutions are available for some cases.

The case of axial field compression (A.F.C.) is discussed as an application. Numerical solutions give the time behaviour of all possible initially given disturbances. The initial perturbations, which are able to reach the axis during their development and which are consequently dangerous in magnetic field compression experiments, have been calculated. Results are consistent with the few experimental data available. They show that the degree of symmetry of cylindrical devices has to be extremely good in order to get successful compressions.

Finally, non-linear and compressibility effects have been taken into account for some A.F.C. cases, solving the full non-linear fluid equations numerically.

Consideration is given to the effects of gravity which arise when a heated sphere, maintained at a steady uniform temperature, is located in a vertical uniform stream. Restricting analysis to a medium of unit Prandtl number (σ), the method of matched asymptotic expansions is employed in obtaining solutions for the velocity, temperature and pressure fields in the limit: G = o(R2), R ↓ 0 (G and R being, respectively, the Grashof and Reynolds numbers). Based on these results, conjectures are formed about the corresponding pure natural convection problem.

Water-tunnel experiments on two cones of taper ratio 36:1 and 18:1 have shown that the frequency of vortex shedding is controlled by the local diameter and has a value which is slightly lower than that on a parallel cylinder of the same size. The shedding process is modulated by a low-frequency oscillation which has a frequency dependent only on U2/v and is independent of any physical dimension of the model. A mathematical analogue of the wake, which helps to explain these features of the flow, has been constructed. The pattern of vortices in the wake of a cone suggested by this model is in reasonable agreement with the photographic evidence.

A conducting liquid with a free surface is subjected to a vertical magnetic force due to imposed, horizontal, magnetic and current fields in the liquid. Because the current field is modified differently by differently oriented surface waves, the propagation of gravity waves becomes strongly anisotropic. The cases of shallow and deep fluid are explored. The group velocity shows features that are reminiscent of magnetoacoustic waves. The need for stability of the surface sets limits on the magnetic force which may be imposed. The feasibility of experiments is discussed and the effect of ohmic damping and surface tension is found to be relatively unimportant under suitably chosen conditions.

Existing analytical treatments of the hypersonic strong interaction problem adopt the two-region structure of the classical boundary-layer theory; however, the uneven heating and external vorticity created by the highly curved, leading-edge shock wave leads to the question of the uniform validity of the boundary-layer approximation near the boundary-layer edge. Recently Bush (1966), using a non-linear viscosity-temperature law (μ ∞ Tω, ω < 1) instead of the linear one (μ ∞ T) used by previous investigators, demonstrated the need to analyze separately a transitional layer intermediate between the inviscid region and the boundary layer. In this paper, an asymptotic analysis of the Navier-Stokes equations in von Mises's variables, allowing a three-region structure, is carried out for the case of μ ∞ T. Results, with the second-order effects associated with heating and vorticity accounted for, show that a separate analysis of the transitional region is not strictly necessary in this case, and hence the equivalence of the two-region approach is confirmed. On the other hand, it is shown that the second-order boundary-layer correction owing to the heating and vorticity effects, not considered by Bush, is necessary in order to determine a uniformly valid temperature distribution in the physical variables. Numerical results for an insulated and a cold flat plate, considerably different from those of others, are obtained.

The axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers. It was established that the jet was truly self-preserving some 70 diameters downstream of the nozzle and most of the measurements were made in excess of this distance. The quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity. The r.m.s. values of the various velocity fluctuations differ from those measured previously as a result of lack of self-preservation and insufficient frequency range in the instrumentation of the previous investigations. It appears that Taylor's hypothesis is not applicable to this flow, but the use of convection velocity of the appropriate scale for the transformation from temporal to spatial quantities appears appropriate. The energy balance was calculated from the various measured quantities and the result is quite different from the recent measurements of Sami (1967), which were obtained twenty diameters downstream from the nozzle. In light of these measurements some previous hypotheses about the turbulent structure and the transport phenomena are discussed. Some of the quantities were obtained by two or more different methods, and their relative merits and accuracy are assessed.

An asymptotic solution is obtained to the problem of internal wave propagation in a horizontally stratified inhomogeneous fluid of non-uniform depth. It also applies to fluids which are not stratified, but in which the constant density surfaces have small slopes. The solution is valid when the wavelength is small compared to all horizontal scale lengths, such as the radius of curvature of a wavefront, the scale length of the bottom surface variations and the scale length of the horizontal density variations. The theory underlying the solution involves rays, a phase function satisfying the eiconal equation, and amplitude functions satisfying transport equations. All these equations are solved in terms of the rays and of the solution of the internal wave problem for a horizontally stratified fluid of constant depth. The theory is thus very similar to geometrical optics and its extensions. It can be used to treat problems of propagation, reflexion from vertical cliffs or from shorelines, refraction, determination of the frequencies and wave patterns of trapped waves, etc. For fluid of constant density, it reduces to the theory of Keller (1958). The theory is applied to waves in a fluid with an exponential density distribution on a uniformly sloping beach. The predicted wavelength is shown to agree well with the experimental result of Wunsch (1969). It is also applied to determine edge waves near a shoreline and trapped waves in a channel.

The problem considered is that of two-dimensional viscous flow in a straight channel. The steady Navier-Stokes equations are linearized on the assumption of small disturbance from the fully developed flow, leading to an eigenvalue equation resembling the Orr-Sommerfeld equation. This is solved in the limiting cases of small and large Reynolds number R, and an approximate method is proposed for moderate R. The main results are (i) the dominant mode of the disturbance velocity (i.e. that which persists longest) is antisymmetrical; (ii) for large R there are two sequences of eigenvalues. Both sequences are asymptotically real as R → ∞. The members of the first sequence are O(1) as R → ∞ and are complex for all finite R. The members of the second sequence are O(R−1) and the imaginary part is O(R−N) for all N. It is the eigenvalues of the second sequence which will dominate the flow at large R.