to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

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The phenomenology of small-scale turbulence

The intermittency of the rate of turbulent energy dissipation e is investigated experimentally, with special emphasis on its scale-similar facets. This is done using a general formulation in terms of multifractals, and by interpreting measurements in that light. The concept of multiplicative processes in turbulence is (heuristically) shown to lead to multifractal distributions, whose formalism is described in some detail. To prepare proper ground for the interpretation of experimental results, a variety of cascade models is reviewed and their physical contents are analysed qualitatively. Point-probe measurements of e are made in several laboratory flows and in the atmospheric surface layer, using Taylor's frozen-flow hypothesis. The multifractal spectrum f(α) of e is measured using different averaging techniques, and the results are shown to be in essential agreement among themselves and with our earlier ones. Also, long data sets obtained in two laboratory flows are used to obtain the latent part of the f(α) curve, confirming Mandelbrot's idea that it can in principle be obtained from linear cuts through a three-dimensional distribution. The tails of distributions of box-averaged dissipation are found to be of the square-root exponential type, and the implications of this finding for the f(α) distribution are discussed. A comparison of the results to a variety of cascade models shows that binomial models give the simplest possible mechanism that reproduces most of the observations. Generalizations to multinomial models are discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091001830

The multifractal nature of turbulent energy dissipation

Velocity probability density functions of high Reynolds number turbulence

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

Axial flow in the core of a laminar steady trailing vortex from the tip of a semi‐infinite wing is analyzed assuming small departure of the axial velocity from the free‐stream velocity. It is further assumed that the axial pressure gradient is determined by the swirl velocities of an ideal infinite line vortex in which the radial and the associated axial velocity variations are neglected in the equation for the angular momentum. The axial and lateral variations of the axial velocity depend on the strength of the vortex and initial axial velocity distribution which must be specified at some station behind the wing except at the virtual origin of the vortex where a nonintegrable singularity exists. Numerical solutions for the axial velocity are obtained using the axial pressure gradient given by the line vortex and analytical solutions are obtained using an equivalent axial pressure gradient with good agreement between the two sets of axial velocity distributions. Resolution of the previous uncertainties in this field is given which were due to the unrecognized singularity at the virtual origin of the vortex. Using the calculated axial velocity the neglected radial and the associated axial fluxes of angular momentum are determined and the limits of validity of the theory presented here in terms of a suitably defined vortex Reynolds number and a nondimensional distance measured from the virtual origin of the vortex are given.

Axial flow in trailing line vortices

Preface Asymptotic Series and Expansions Regular Perturbation Methods The Method of Strained Coordinates/Parameters Method of Averaging The Method of Matched Asymptotic Expansions Method of Multiple Scales Miscellaneous Perturbation Methods References Answers to Selected Problems Index Permissions

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Perturbation Methods for Differential Equations

Special Functions Fourier Integrals and Fourier Transforms Applications Involving Fourier Transforms The Laplace Transformation Applications Involving Laplace Transforms The Melling Transform The Hankel Transform Finite Transforms Discrete Transforms Appendix A - Review of Complex Variables Appendix B - Table of Fourier Transforms Appendix C - Table of Laplace Transforms.

Integral transforms for engineers

This introduction to a wide range of theoretical studies in fluid dynamics, covers a great deal of material and offers updated information on topics such as stability and turbulence. It surveys nearly the entire field of classical fluid dynamics and discusses the various conceptual and analytical models of fluid flow.

Theoretical fluid dynamics

A new statistical theory involving the gamma distribution is presented for the local average rate of dissipation er. The gamma distribution for er leads to results for the high‐order moments of the velocity structure function that lie between those predicted by the lognormal model and β model and closely follow those of the multifractal model. Comparisons of results predicted by the gamma distribution are made with previously published experimental data, showing excellent agreement. Based on the gamma distribution for er, the one‐dimensional K distribution is developed as a plausible model for the distribution of energy dissipation e. Some laboratory measurements of velocity fluctuations and certain derived quantities from a modified airjet are also discussed. A good agreement is found by making comparisons of the measured cumulative probability distribution of the squared time derivative of the streamwise velocity component with that predicted by this new model.

Bhimsen K. Shivamoggi

Papers

Axial flow in trailing line vortices

Perturbation Methods for Differential Equations

Integral transforms for engineers

Theoretical fluid dynamics

A statistical theory for the distribution of energy dissipation in intermittent turbulence