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3653 results in ebooks in fluid mechanics

Page 178 of 183

Vortex Dynamics
  • P. G. Saffman
  • Published online:
    05 August 2012
    Print publication:
    29 January 1993
  • The discovery of coherent structures in turbulence has fostered the hope that the study of vortices will lead to models and an understanding of turbulent flow, thereby solving or at least making less mysterious one of the great unsolved problems of classical physics. Vortex dynamics is a natural paradigm for the field of chaotic motion and modern dynamical system theory. The emphasis in this monograph is on the classical theory of inviscid incompressible fluids containing finite regions of vorticity. The effects of viscosity, compressibility, inhomogeneity and stratification are enormously important in many fields of application, from hypersonic flight to global environmental fluid mechanics. However, this volume focuses on those aspects of fluid motion which are primarily controlled by the vorticity and are such that the effects of the other fluid properties are secondary.

  • Epilogue

  • P. G. Saffman, California Institute of Technology
  • Book:
    Vortex Dynamics
    Published online:
    05 August 2012
    Print publication:
    29 January 1993, pp 295-295

  • Summary

    For God's Sake

    Let us sit upon the ground and tell sad stories

    Of vortex filaments.

    How some have been ill-posed, some singular,

    Some poisoned by their self induction, some core size killed,

    Some haunted by the mathematics they have involved.

    All murderous.

    For within the swirling motion that rounds the mortal circulation

    Of a vortex

    Keeps futility his court,

    And there the non-linearity sits

    Scoffing at his state and grinning at his theories

    Allowing him a breath, a little scene to linearize, compute and fill with approximations

    And then at last he comes and with a little inconsistency bores through the costly hopes and

    Farewell …

    Shakespeare, Richard II, Act 3, Scene 2 ‘Translated’ by H. C. Yuen

  • 2 - Singular distributions of vorticity

  • P. G. Saffman, California Institute of Technology
  • Book:
    Vortex Dynamics
    Published online:
    05 August 2012
    Print publication:
    29 January 1993, pp 20-45

  • Summary

    Vortex jumps

    It has been supposed so far that the velocity and vorticity fields are continuous and have continuous derivatives. In a real viscous fluid, these assumptions are generally regarded as appropriate, and the velocity field is assumed analytic everywhere, except possibly at an initial instant o, when the speed of boundaries changes in a non-analytic manner. However, if we go to the the limit of vanishingly small viscosity and consider ideal fluids, to which the Helmholtz laws apply, the mathematics allows non-analytic behaviour and we cannot assert on physical grounds that only continuous fields should be considered. Velocity or vorticity fields with singularities or discontinuities are indeed of considerable importance. The singularities are, of course, not arbitrary and must be consistent with integral forms of the Euler equations or equivalently with the conservation of mass, momentum and energy. In particular, the dynamical constraint that pressure is continuous across a surface of discontinuity must be satisfied unless there are also singularities in the external force fields. We suppose in this chapter that the density is uniform and put it equal to unity unless explicitly stated otherwise. Also, external forces are supposed conservative unless non-conservative forces are explicitly introduced.

  • Preface

  • P. G. Saffman, California Institute of Technology
  • Book:
    Vortex Dynamics
    Published online:
    05 August 2012
    Print publication:
    29 January 1993, pp xi-xiv

  • Summary

    In the past three decades, the study of vortices and vortex motions – which originated in Helmholtz's great paper of 1858, ‘Uber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen’ (translated by Tait [1867]), and continued in the brilliant work of Lord Kelvin and others in the nineteenth century, and Prandtl and his Göttingen school in the first half of this century – has received continuing impetus from problems arising in physics, engineering and mathematics. As aptly remarked by Küchemann [1965], vortices are the ‘sinews and muscles of fluid motions’. Aerodynamic problems of stability, control, delta wing aerodynamics, high lift devices, the jumbo jet wake hazard phenomenon, among other concerns, have led to a myriad of studies. Smith [1986] reviews some of this work. The realisation that many problems involving interfacial motion can be cast in the form of vortex sheet dynamics has stimulated much interest. The discovery (rediscovery?) of coherent structures in turbulence has fostered the hope that the study of vortices will lead to models and an understanding of turbulent flow, thereby solving or at least making less mysterious one of the great unsolved problems of classical physics. Vortex dynamics is a natural paradigm for the field of chaotic motion and modern dynamical system theory. It is perhaps not well known that the father of modern dynamics and chaos wrote a monograph on vorticity (Poincaré [1893]).

Perturbation Methods
  • E. J. Hinch
  • Published online:
    05 June 2012
    Print publication:
    25 October 1991
  • Perturbation methods are one of the fundamental tools used by all applied mathematicians and theoretical physicists. In this book, the author has managed to present the theory and techniques underlying such methods in a manner which will give the text wide appeal to students from a broad range of disciplines. Asymptotic expansions, strained coordinates and multiple scales are illustrated by copious use of examples drawn from all areas of applied mathematics and theoretical physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather to give a physical understanding of the subtle balances in a complex problem.

Page 178 of 183