The Theories of Turbulence

The theory of turbulence reached its full growth at the end of the 19th century as a result of the work by Boussinesq and Reynolds. It then underwent a long period of stagnation which ended under the impulse given to it by the development of wind tunnels caused by the needs of aviation. Numerous researchers, attempted to put Reynolds' elementary statistical theory into a more precise form. During the war, some isolated scientists - von Weizsacker and Heisenberg in Germany, Kolmogoroff in Russia, Onsager in the U.S.A. - started a program of research. By a system of assumptions which make it possible to approach the structure of turbulence in well-defined limiting conditions quantitatively, they obtained a certain number of laws on the correlations and the spectrum. Since the late reports have improved the mathematical language of turbulence, it was deemed advisable to start with a detailed account of the mathematical methods applicable to turbulence, inspired at first by the work of the French school, above all for the basic principles, then the work of the foreigners, above all for the theory of the spectrum

The Hopf Bifurcation and Its Applications

The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations. Historically, the subject had its origins in the works of Poincaré [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term "Poincaré-Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle-Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper. These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and Kirchgässner and Kielhoffer [1] (contributed by G. Childs and O. Ruiz) are given. The contributions of S. Smale, J. Guckenheimer and G. Oster indicate applications to the biological sciences and that of D. Schmidt to Hamiltonian systems. For other applications and related topics, we refer to the monographs of Andronov and Chaiken [1], Minorsky [1] and Thom [1]. The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value. In Hopf's original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation. We have given explicit algorithms for this calculation which are easy to apply in examples. (See Section 4, and Section 5A for comparison with Hopf's formulae). The method of averaging, exposed here by S. Chow and J. Mallet-Paret in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly. In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for t > O. This enables the invariant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]). To aid in determining smoothness in examples we have presented parts of the results of Dorroh-Marsden. [1]. Similar ideas for utilizing smoothness have been introduced independently by other authors, such as D. Henry [1]. Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here. We have selected results of Chafee [1] and Ruelle [3] (the latter is exposed here by S. Schecter) to indicate some generalizations that are possible. The subject is by no means closed. Applications to instabilities in biology (see, e.g. Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf. Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field (cf. Durand [1]) are appearing at a rapid rate. Also, the qualitative theory proposed by Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9). In this direction, the papers of Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important. Stable oscillations in nonlinear waves may be another fruitful area for application; cf. Whitham [1]. We hope these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods. After we completed our stability calculations we were happy to learn that others had found similar difficulty in applying Hopf's result as it had existed in the literature to concrete examples in dimension ≥ 3. They have developed similar formulae to deal with the problem; cf. Hsü and Kazarinoff [1, 2] and Poore [1]. The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type. The new proof, relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations. These notes originated in a seminar given at Berkeley in 1973-4. We wish to thank those who contributed to this volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive. Many other references are contained in the lengthy bibliography in Cesari [1]. We also thank those who have taken an interest in the notes and have contributed valuable comments. These include R. Abraham, D. Aronson, A. Chorin, M. Crandall., R. Cushman, C. Desoer, A. Fischer, L. Glass, J. M. Greenberg, O. Gurel, J. Hale, B. Hassard, S. Hastings, M. Hirsch, E. Hopf, N. D. Kazarinoff, J. P. LaSalle, A. Mees, C. Pugh, D. Ruelle, F. Takens, Y. Wan and A. Weinstein. Special thanks go to J. A. Yorke for informing us of the material in Section 3C and to both he and D. Ruelle for pointing out the example of the Lorentz equations (See Example 4B.8). Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript. Jerrold Marsden Marjorie McCracke

Prediction of the position and velocity of a satellite after many revolutions

Position and velocity prediction method for satellite after many revolution

All done by mirrors: reflectivity in the novels of Elizabeth Taylor (1912-75)

Elizabeth Taylor's texts are elusive and allusive. Part 1 offers a biographical sketch of the writer, then applies theory, drawing on and adapting the work of Harold Bloom, Roman Ingarden, Michael Riffaterre, Linda Hutcheon and Patricia Waugh, among others, to describe and account for the bare style and the high degree of referentiality. Seven kinds of "reflectivity" are proposed, with special attention to self-reflectivity - "reflexivity." The presence of "structural reflexivity" is shown in the first novel.Part 2 discusses the other 11 novels and one novella in pairs linked by the texts which they reflect (upon), as follows: Palladian (1946) and "Hester Lilly" are located within the Gothic tradition in women's writing, but their application of a Bloomian kenosis to its major text, Jane Eyre (1847), is noted. A View of The Harbour (1947) and A Wreath Of Roses (1949) are discussed in terms of their refusal of Woolfian "vision. " A Game Of Hide And Seek (1951) and The Sleeping Beauty are interrelated with the Grimm fairy tale. Two major characters in Angel (1957) and The Wedding Group (1968) are shown to be based on real people and their psychology to define one theme. Two Henry James novels are detected as the parallel texts for In A Summer Season (1961) and The Soul Of Kindness (1964), with two allegories also drawn in by allusion. Lastly, Mrs Palfrey At The Claremont (1971) and Blaming (1976) are contrasted in their reflectivity and the latter's application of kenosis to itself and the whole oeuvre

Robust data analysis for factorial experimental designs: Improved methods and software

Factorial experimental designs are a large family of experimental designs. Robust statistics has been a subject of considerable research in recent decades. Therefore, robust analysis of factorial designs is applicable to many real problems. Seheult and Tukey (2001) suggested a method of robust analysis of variance for a full factorial design without replication. Their method is generalised for many other factorial designs without the restriction of one observation in each cell. Furthermore, a new algorithm to decompose data from a factorial design is introduced and programmed in the statistical computer package R. The whole procedure of robust data analysis is also programmed in R and it is intended to submit the library to the repository of R software, CRAN. In the procedure of robust data analysis, a cut-off value is needed to detect possible outliers. A set of optimum cut-off values for univariate data and some dimensions of two-way designs (complete and incomplete) has also been provided using an improved design of simulation study

Numerical methods for ordinary differential equations with applications to partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The thesis develops a number of algorithms for the numerical solution of ordinary differential equations with applications to partial differential equations. A general introduction is given; the existence of a unique solution for first order initial value problems and well known methods for analysing stability are described. A family of one-step methods is developed for first order ordinary differential equations. The methods are extrapolated and analysed for use in PECE mode and their theoretical properties, computer implementation and numerical behaviour, are discussed. Lo-stable methods are developed for second order parabolic partial differential equations 1n one space dimension; second and third order accuracy is achieved by a splitting technique in two space dimensions. A number of two-time level difference schemes are developed for first order hyperbolic partial differential equations and the schemes are analysed for Ao-stability and Lo-stability. The schemes are seen to have the advantage that the oscillations which are present with Crank-Nicolson type schemes, do not arise. A family of two-step methods 1S developed for second order periodic initial value problems. The methods are analysed, their error constants and periodicity intervals are calculated. A family of numerical methods is developed for the solution of fourth order parabolic partial differential equations with constant coefficients and variable coefficients and their stability analyses are discussed. The algorithms developed are tested on a variety of problems from the literature.British Governmen

On the numerical evaluation of finite-part integrals involving an algebraic singularity

Thesis (PhD)--Stellenbosch University, 1975.ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields of aerodynamics or electron optics, involve certain singular integrals which do not exist classically. The problems can, however, be solved pLovided that such integrals are interpreted as finite-part integrals. Although the concept of a finite-part integral has existed for about fifty years, it was possible to define it rigorously only by means of distribution theory, developed about twenty-five years ago. But, to the best of our knowledge, no quadrature formula for the numerical eva= luation of finite-part integrals ha~ been given in the literature. The main concern of this thesis is the study and discussion of.two kinds of quadrature formulae for evaluating finite-part integrals in= volving an algebraic singularity. Apart from a historical introduction, the first chapter contains some physical examples of finite-part integrals and their definition based on distribution theory. The second chapter treats the most im= portant properties of finite-part integrals; in particular we study their behaviour under the most common rules for ordinary integrals. In chapters three and four we derive a quadrature formula for equispaced stations and one which is optimal in the sense of the Gauss-type quadra= ture. In connection with the latter formula, we also study a new class of orthogonal polynomials. In the fifth and.last chapter we give a derivative-free error bound for the equispaced quadrature formula. The error quantities which are independent of the integrand were computed for the equispaced quadrature formula and are also given. In the case of some examples, we compare the computed error bounds with the actual errors. ~esides this theoretical investigation df finite-part integrals, we also computed - for several orders of the algebraic singularity the coefficients for both of the aforesaid quadrature formulae, in which the number of stations ranges from three up to twenty. In the case of the equispaced quadrature fortnu1a,we give the weights and - for int~ger order of the singularity - the coefficients for a numerical derivative of the integrand function. For the Gauss-type quadrature, we give the stations, the corresponding weights and the coefficients of the orthogonal polynomials. These data are being published in a separate report [18] which also contains detailed instructions on the use of the tables

Annual research briefs, 1993

The 1993 annual progress reports of the Research Fellow and students of the Center for Turbulence Research are included. The first group of reports are directed towards the theory and application of active control in turbulent flows including the development of a systematic mathematical procedure based on the Navier Stokes equations for flow control. The second group of reports are concerned with the prediction of turbulent flows. The remaining articles are devoted to turbulent reacting flows, turbulence physics, experiments, and simulations