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The purpose of this book is to survey computational flow research on the design and analysis of supercritical wing sections supported by the National Aeronautics and Space Administration at the Energy Research and Development Administration Mathematics and Computing Laboratory of New York University. The work was performed under NASA Grants NGR 33-016-167 and NGR 33-016-201 and ERDA Contract EY-76-C-02-3077. Computer programs to be listed and described have applications in the study of flight of modern aircraft at high sub sonic speeds. One of the codes generates cascades of shockless tran sonic airfoi~s that are expected to increase significantly the effici ency of compressors and turbines. Good simulation of physically observed flows has been achieved. This work is a sequel to two earlier books [1,2] published by Springer-Verlag under similar titles that we shall refer to as Volumes I and II. New York November 1977 TABLE OF CONTENTS I. INTRODUCTION 1 1. Shockless Airfoils and Supercritical Wing Sections 1 2. Differential Equations of Gas Dynamics 2 II. THE METHOD OF COMPLEX CHARACTERISTICS 5 1. A New Boundary Value Problem 5 2. Topology of the Paths of Integration 8 3. Iterative Scheme for the Map Function 9 III. TRANSONIC AIRFOIL DESIGN CODE 10 1. Isolated Airfoils 10 2. Compressor Cascades 12 3. Turbine Cascades 13 4. Comparison with Experiment 14 IV. TWO-DU1ENSIONAL ANALYSIS CODE 16 1. Wave Drag 16 2. A Fast Solver 19 3. Remarks about Three-Dimensional Flow 24 V. REFERENCES 26 VI.
This handbook is a sequel to an earlier volume entitled "A Theory of Supercritical Wing Sections, with Computer Programs and Examples." Since the completion of the first volume, which we shall refer to as Volume I (cf. [ll), some effort has been made to improve our airfoil design program. A number of more desirable air foils have been designed. In addition several of our wing sections have been tested in wind tunnels. We should like to make this material available here, since it is more convenient to use the design program in conjunction with data for a fairly broad range of examples. Moreover, we have developed new analysis programs that supersede our previous work. Chapter I is devoted to a brief discussion of the mathematics involved in our additions and modifications. There is only a mini mum emphasis on theory, since the representation of important physical phenomena such as boundary layer shock wave interaction and separation is partly empirical. It is our contention, however, that the computer programs provide a better simulation than might have been expected. Chapter II presents numerical results found by our new methods, as well as comparisons with experimental data. Chapter III contains a discussion of the use of the program together with Fortran listings.
At present, there is considerable interest in supercritical wing technology for the development of aircraft designed to fly near the speed of sound. The basic principle is the suppression of boundary layer separation by shifting the shock waves that occur on the wing toward the trailing edge and making them as weak as possible. The purpose of this report is to make available to the engineering public mathematical methods for the design of supercritical wings. These methods depend on the numerical solution of the partial differential equations of two-dimensional gas dynamics. The main contribution is a computer program for the design of shockless transonic airfoils using the hodograph transformation and analytic continuation into the complex domain. Another contribution is a program for the analysis of transonic flow with shocks past an airfoil at off-design conditions. In our design work we include a turbulent boundary layer correction. Part I of the paper is devoted to a description of the mathemati cal theory and need not be studied by those primarily concerned with running the programs. Part II is a manual for users of our programs which is independent of the theoretical part. In Part III and in Appendices II and III we give numerical examples and discuss computa tional results. The main substance of the report, however, is contained in the listing of the computer programs themselves in Appendix IV. We have used the Fortran language throughout and we have included numerous comment cards in the listing.
In this monograph, noncooperative games are studied. Since in a noncooperative game binding agreements are not possible, the solution of such a game has to be self enforcing, i. e. a Nash equilibrium (NASH [1950,1951J). In general, however, a game may possess many equilibria and so the problem arises which one of these should be chosen as the solution. It was first pointed out explicitly in SELTEN [1965J that I not all Nash equilibria of an extensive form game are qualified to be selected as the solution, since an equilibrium may prescribe irrational behavior at unreached parts of the game tree. Moreover, also for normal form games not all Nash equilibria are eligible, since an equilibrium need not be robust with respect to slight perturba tions in the data of the game. These observations lead to the conclusion that the Nash equilibrium concept has to be refined in order to obtain sensible solutions for every game. In the monograph, various refinements of the Nash equilibrium concept are studied. Some of these have been proposed in the literature, but others are presented here for the first time. The objective is to study the relations between these refine ments;to derive characterizations and to discuss the underlying assumptions. The greater part of the monograph (the chapters 2-5) is devoted to the study of normal form games. Extensive form games are considered in chapter 6.
the economics of exhaustible " assets presents a whole forest of intriguing problems." 1 Harald Hotelling ) The two energy price shocks in 1973/74 and 1979/80 have arosed interest in the new area of resource economics. The affluent societies of Europe, North America and Japan were confronted with the new scarcity paradigm of the "space ship earth" with only a limited supply of natural resources aboard whereas population is growing and the environment can not accomodate the increasing volume of pollutants. The problem of natural resource scarcity gives rise to the question how resource-dependent economies like European coun tries and Japan are affected by an increase in resource prices and how they can adjust to rising energy prices. The new para digm also has focused new interest on the problem of the re source-extracting firm and of the resource-exporting country. The Hotelling revival of resource economics has given new im portance to the behavior and to the policy issues of resource exporting countries.
This book is an outgrowth of years of teaching and doing re search at the University of California, San Diego (UCSD), in the area of economic growth. Although there have been several books on this topic published in the last eight years, I have been dis satisfied with them for several reasons. First, books such as those by Wan, Burmeister and Dobell are uneven in their technical difficulty and, while they are excellent, are apparently difficult for first year graduate students and advanced undergraduates. Solow's expository book, on the other hand, is at the other ex treme. Furthermore, many of the books seem to be aimed at the authors' peers rather than the students. My primary objective in writing this book is to bridge this gap and to pitch, very appro priately I hope, at the level of a typical student enrolled in a beginning course in growth theory. Secondly, almost all the growth models in the literature can be recast in a single analyti cal framework. Although the various authors have not written so as to conform to any particular pattern, it -is the function of a textbook writer to identify such a pattern, if it exists, and pre sent the theory in that framework. Many authors make implicit as sumptions about their models which are either never specified or sometimes specified in footnotes.
The theory of Lie groups has proven to be a most powerful analytical tool in many areas of modern scientific endeavors. It was only a few years ago that economists discovered the usefulness of this approach in their study of the frontiers of modern economic theory. These frontiers include the areas of technical change and productivity, technology and preference, economic conservation laws, comparative statics and integrability conditions, index number problems, and the general theory of ~ observable market behavior (Sato [1980, 1981], Nono [1971], Sato and N~no [1983], Russell [1983]). 1 In Nono [1971] and Sa to [1981, Chapter 4] the concept of "G-neutral" (group neutral) technical change was first introduced as a natural extension of the well-known concepts of Hicks, Harrod, Solow and Sato-Beckmann-Rose neutrality. The present monograph contains a further extension of the G-neutral technical change to the case of non-constant-returns-to-scale technology and to the case of multiple factor inputs. The methodology of total productivity estimation by means of Lie group transformations is also developed in this monograph. We would like to express our sincere thanks to many individuals notably to Professor M. J. Beckmann, Professor F. Mimura, Professor G. Suzawa, T. Mitchell, K. Mino and P. Calem, for their numerous contributions at various stages of this work. We are also grateful to Marion Wathey for her usual superb typing of this difficult manuscript. Providence, R. I. , U. S. A.