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Volume 74 - Issue 2 - 23 March 1976

Contents

Research Article

  • The far field of high frequency convected singularities in sheared flows, with an application to jet-noise prediction

  • Thomas F. Balsa
  • Published online by Cambridge University Press:
    29 March 2006, pp. 193-208

  • The purpose of the present paper is to derive expressions for the pressure fields of various high frequency convected singularities immersed in a unidirectional sheared flow. These expressions include the simultaneous effects of fluid and source convection and refraction. These results are then combined to predict the far-field directivity of cold round jets. It is found that the agreement between experiment and the present theory is quite good at a source Strouhal number of unity but that this agreement deteriorates as the source frequency is increased. Our theoretical results show the explicit form of the ‘refraction integral’ and that convective amplification for the pressure of a quadrupole is increased by a factor of (1 — MJ cos Θ)−1 over the classical results, where MJ is the jet Mach number and Θ is the angle from the jet axis. Thus acoustic/mean-flow interaction not only implies refraction but also additional convective amplification due not to source convection but to fluid motion.

  • On the two-dimensional mixing region

  • F. H. Champagne, Y. H. Pao, I. J. Wygnanski
  • Published online by Cambridge University Press:
    29 March 2006, pp. 209-250

  • An experimental investigation of the two-dimensional incompressible mixing layer was carried out. The measurements provide new information on the development of the mean and turbulent fields towards a self-preserving state and on the higher-order statistical characteristics of the turbulent field. The relevance of initial conditions to the development of the flow is discussed in the light of both present and previous data. Measurements of spectra, probability densities and moments to eighth order of all three velocity-component fluctuations at various transverse positions across the flow were carried out using an on-line digital data acquisition system. The probability density distributions of the derivative and the squared derivative of the longitudinal and lateral velocity fluctuations were also determined. Direct measurements of moments to eighth order of the velocity derivatives were attempted and are discussed in the light of the simultaneously measured histograms. The problems in obtaining higher-order statistical data are considered in some detail. Estimates of the integral time scale of many of the higher-order statistics are presented. The high wave-number structure was found to be locally anisotropic according to both spectral and turbulent velocity-gradient moment requirements. Higher-order spectra to fourth order of the longitudinal velocity fluctuations were measured and are discussed. Finally the lognormality of the squared longitudinal and lateral velocity-derivative fluctuations was investigated and the universal lognormal constant μ was evaluated.

  • Turbulence correlations in a compressible wake

  • A. Demetriades
  • Published online by Cambridge University Press:
    29 March 2006, pp. 251-267

  • Space-time correlations have been measured in the turbulent supersonic wake behind a slender axisymmetric body. The data were obtained with two independently-positioned hot-wire anemometers, and interpreted with the aid of an extension of compressible anemometry theory. Comparison with earlier single-point Eulerian spectra showed a breakdown of Taylor's hypothesis near the wake edge, and thus also the expected differences in the scale lengths. The eddy convection speed appears equal to that of the mean flow, and the eddy shape is inclined to the wake axis, although it becomes nearly parallel to it at the wake edge. As a consequence, radial and longitudinal space correlation functions appear nearly equal. An upstream-downstream asymmetry was noted in the latter.

  • The maintenance of turbulent shear stress in a mixing layer

  • Ian S. F. Jones
  • Published online by Cambridge University Press:
    29 March 2006, pp. 269-295

  • Wavenumber frequency spectra have been measured in a two-dimensional incompressible mixing layer, using linearized hot-wire anemometers. Spectra of two dimensions (frequency and wavenumber) have been measured for lateral and longitudinal turbulent velocities, and used to construct three-dimensional spectra. The validity of the separation assumption used to construct these spectra was tested. Spectra of the velocity product responsible for the mean shear stress and the lateral gradient of this spectrum have been determined, as has a structure constant for the lateral velocity fluctuations that Phillips (1967) suggested is relevant to the maintenance of the shear stress gradient. Phillips’ (1967) stress model fails the tests proposed in this study.

  • Laminar film flow on a cylindrical surface

  • Ernst Becker
  • Published online by Cambridge University Press:
    29 March 2006, pp. 297-315

  • The paper deals with steady laminar film flow which is set up at the cylindrical surface of an idealized horizontal ‘road’ when homogeneous ‘rain’ is falling onto the road in a vertical downward direction. It is shown that a particular solution of the Navier-Stokes equations is possible for which the depth of the liquid film is constant. In that case the Navier-Stokes equations reduce to the equations governing plane stagnation-point flow. However, the boundary conditions differ from those for the classical stagnation-point problem. Solutions for nearly inviscid flow and predominantly viscous flow are derived analytically. In particular, simple formulae for the depth of the film are found in both cases. Finally, the importance of the particular solution as a member of a whole class of solutions is discussed on the basis of a momentum integral approximation.

  • The distortion of a flexible inextensible thread in a shearing flow

  • E. J. Hinch
  • Published online by Cambridge University Press:
    29 March 2006, pp. 317-333

  • Using the slender-body theory for Stokes flow, the equations of motion are developed for a small flexible inextensible thread. The nearly straight thread is examined analytically, and is shown to straighten rapidly. In a simple shear flow the distortions decay less rapidly, but rapidly enough not to rotate the thread through the plane of flow. Numerical studies of simple shear with more substantially distorted threads show the same qualitative behaviour. Additionally some differences are revealed in the nonlinear regime between the buckling and stretching processes which occur in the compressive and tensile quadrants of the flow.

  • On a rotational flow disturbed by gravity

  • H. P. Greenspan
  • Published online by Cambridge University Press:
    29 March 2006, pp. 335-351

  • We examine a rapidly rotating flow that exhibits periodic vortex detachment. Specifically, the rotation/symmetry axis of a fluid-filled cylinder is set perpendicular to gravity. A free buoyant cylindrical float placed within the container is acted upon by both centrifugal and gravitational forces, the competition of which causes fluid motion and, in certain parameter ranges, flow instability. The motion is determined, a criterion for separation is advanced, preliminary experiments and data are described and the relationship of this phenomenon to other examples of vortex shedding in rotating fluids is discussed.

  • Nonlinear effects in edge waves

  • G. B. Whitham
  • Published online by Cambridge University Press:
    29 March 2006, pp. 353-368

  • Nonlinear corrections to Stokes's linear edge-wave solution are obtained by means of perturbation expansions in the amplitude. The shallow-water formulation is considered first, but even for small beach angles β the behaviour in the deep water offshore becomes important and this formulation is limited. In the full formulation, amplitude dependence is required in the dispersion relation and in the exponents for the exponential decay away from the shore. There is a non-uniformity in the results as β → ½π, which is corrected by a special perturbation expansion.

  • Nonlinear edge waves and shallow-water theory

  • A. A. Minzoni
  • Published online by Cambridge University Press:
    29 March 2006, pp. 369-374

  • Nonlinear effects are considered for shallow-water edge waves on beaches with a general depth distribution. The case of uniform depth away from the shoreline is considered in detail. It is shown that the results obtained are in qualitative agreement with those obtained by Whitham (1976) using the full nonlinear theory for a beach of constant slope.

  • Nonlinear energy transfer and the energy balance of the internal wave field in the deep ocean

  • Dirk J. Olbers
  • Published online by Cambridge University Press:
    29 March 2006, pp. 375-399

  • The source function describing the energy transfer between the components of the internal wave spectrum due to nonlinear interactions is derived from the Lagrangian of the fluid motion and evaluated numerically for the spectral models of Garrett & Munk (1972a, 1975). The characteristic time scales of the transfer are found to be typically of the order of some days, so that nonlinear interactions will play an important role in the energy balance of the wave field. Thus implications of the nonlinear transfer within the spectrum for generation and dissipation processes are considered.