ROTARY-WING AERODYNAMICS Two Volumes Bound as One VOLUME I: BASIC THEORIES OF ROTOR AERODYNAMICS (With Application to Helicopters) by W. Z. STEPNIEWSKI VOLUME II: PERFORMANCE PREDICTION OF HELICOPTERS by C. N. KEYS Edited by W. Z. Stepniewski Dover Publications, Inc., New York This Dover edition, first published in 1984, is an unabridged, slightly corrected republication in one volume of the work originally published in two volumes by the Science and Technical Information Office of the National Aeronautics and Space Administration for the U.S. Army Air Mobility Research & Development Laboratory of the Aviation Systems Command. Volume I, “Basic Theories of Rotor Aerodynamics (With Application to Helicopters),” by W. Z. Stepniewski, was originally published in 1979. Volume II, “Performance Prediction of Helicopters,” by C. N. Keys, originally published in 1979, is being reprinted here from the 1981 edition revised and edited by W. Z. Stepniewski. The Volume II revision was prepared by Boeing Vertol Company. Indexes have been added to the Dover edition at the end of each volume. Wanda L. Metz was Associate Editor for this work. Library of Congress Cataloging in Publication Data Stepniewski, W. Z. (Wieslaw Zenon), 1909- Rotary-wing aerodynamics. “An unabridged, slightly corrected republication in one volume of the work originally published in two volumes by the Science and Technical Information Office of the National Aeronautics and Space Administration… in 1979. Volume II… is being reprinted here from the 1981 edition revised and edited by W. Z. Stepniewski [for the U.S. Army Air Mobility Research and Development Laboratory]”–T. p. verso. Includes bibliographical references and indexes. Contents: Basic theories of rotor aerodynamics (with application to helicopters)–Performance prediction of helicopters. 1. Helicopters–Aerodynamics. I. Keys, C. N. II. Title. TL716.S76 1984 629.132'3 83-20528 ISBN-13: 978-0-48664647-3 ISBN-10: 0-486-64647-5 Manufactured in the United States by Courier Corporation 64647508 www.doverpublications.com FOREWORD In recent years, there has been an increasing volume of reports, articles, papers, and lectures dealing with various aspects of rotary-wing aircraft aerodynamics. To those who enter this domain, either as graduate students with some background in general aerodynamics, or those transferring from other fields of aeronautical or nonaeronautical engineering activities, this vast amount of literature becomes a proverbial haystack of information; often with the result of looking for a needle that isn’t there. But even those who are professionally engaged in some aspects of rotary-wing technology may experience a need for a reference text on basic rotor aerodynamics. Through my experience both as an educator and practicing engineer directly involved in various aspects of industrial aeronautics, it became doubly apparent that there was a need for a textbook that would fulfill, if not all, at least some of the above requirements. With this goal in mind, the text entitled Rotary-Wing Aerodynamics was written under contract from USAAMRDL/NASA Ames. On one hand, the objective is to provide an understanding of the aerodynamic phenomena of the rotor and on the other, to furnish tools for a quantitative evaluation of both rotor performance and the helicopter as a whole. Although the material deals primarily with the conventional helicopter and its typical regimes of flight, it should also provide a comprehensive insight into other fields of rotary-wing aircraft analysis as well. In order to achieve this dual aim of understanding and quantitative evaluation, various conceptual models will be developed. The models will reflect physical aspects of the considered phenomena and, at the same time, permit establishment of mathematical treatment. To more strongly emphasize this duality of purpose, the adjective phystco-mathematical will often be used in referring to these models. It should be realized at this point that similar to other fields of engineering analysis, conceptual models—no matter how complicated in detail—still represent a simplified picture of physical reality. It is obvious, hence, that the degree of sophistication of the physicomathematical models should be geared to the purpose for which they are intended. When faced with the task of developing such a model, one may be advised to first ask the following two questions: (1) whether the introduction of new complexities truly contributes to a better understanding of the physics of the considered phenomena and their qualitative and quantitative evaluation, and (2) whether the possible accuracy of the data inputs is sufficiently high to justify these additional complexities. In this respect, one should determine whether a more complex model would truly lead to a more accurate analysis of the investigated phenomena or just, perhaps, that the procedure only looks more impressive while mathematical manipulation would consume more time and money. Furthermore, it should be realized that often in the more complex approach, neither intermediate steps nor final results can be easily scrutinized. With respect to rotary-wing aerodynamics in general, and performance predictions in particular, one should realize that aerodynamic phenomena associated with the various regimes of flight of an even idealized, completely rigid rotor are very complicated. Furthermore, the level of complexity increases due to the fact that in reality, every rotor is non-rigid because of the elasticity of its components and/or built-in articulations. As a result, a continuous interaction exists between aerodynamics and dynamics, thus introducing new potential complexities to the task of predicting aerodynamic characteristics of the rotor. Fortunately, even conceptually simple models often enable one to get either accurate trends or acceptable approximate answers to many rotary-wing performance problems. By following the development of basic rotor theories-from the simple momentum approach through the combined momentum and blade-element theory, vortex theory, and finally, potential theory-the reader will be able to observe the evolution of the physicomathematical model of the rotor from its simplest form to more complex ones. It will also be shown that an understanding or explanantion of the newly encountered phenomena may require modifications and additions and sometimes, a completely new approach to the representation of the actual rotor by its conceptual model. By the same token, a better feel is developed with respect to the circumstances under which a simple approach may still suffice. Finally, these simpler and more easily scrutinized methods may serve as a means of checking the validity of the results obtained by potentially accurate, but also more complicated ways which may be prone to computational errors. Presentation of the above-outlined theories, plus considerations of airfoils suitable for rotary-wing aircraft constitutes the contents of Vol I. “Reduction to practice” of the material presented in Vol I is demonstrated in Vol II, where complete performance predictions are carried out for classical, winged, and tandem configurations including such aspects as performance guarantees and aircraft growth. The existing need for a text conforming to the above outlined philosophy was recognized by representatives of USAAMRDL and in particular, by Mr. Paul Yaggy, then Director of USAAMRDL, and Dr. I. Statler, Director of Ames Directorate whose support made possible the contract for the preparation of the first two volumes. To perform this task, a team was formed at Boeing Vertol consisting of the undersigned as Editor-in-Chief and author of Vol I; Mr. C.N. Keys as the principal author of Vol II; and Mrs. W. L. Metz as Associate Editor. The course of the work was monitored by Mr. A. Morse and Dr. F.H. Schmitz of Ames Directorate; while Mr. Tex Jones from USAAMRDL-Langley Directorate provided his technical assistance and expertise by reviewing the material contained in both volumes. Thanks are extended to the abovementioned as well as the other representatives of USAAMRDL. Finally, my associates and I wish to thank the management of Boeing Vertol; especially, Messrs. K. Grina, J. Mallen, W. Walls, and E. Ratz for their support, understanding and patience. There are, of course many more people from this country and abroad who significantly contributed to the technical contents. Their individual contributions are more specifically acknowledged in the prefaces of the individual volumes. W. Z. Stepniewski Ridley Park, Pa. March 31, 1978 VOLUME I: BASIC THEORIES OF ROTOR AERODYNAMICS (With Application to Helicopters) PREFACE Volume I of the text entitled Rotary-Wing Aerodynamics is devoted in principle to Basic Theories of Rotor Aerodynamics. However, the exposition of the material is preceded by an introductory chapter wherein the concept of rotary-wing aircraft in general is defined. This is followed by comparisons of the energy effectiveness of helicopters with that of other static-thrust generators in hover; as well as with various air and ground vehicles in forward translation. While the most important aspects of rotor-blade dynamics and rotor control are only briefly reviewed, they should still provide a sufficient understanding and appreciation of the rotor dynamic phenomena related to aerodynamic considerations. The reader is introduced to the subject of rotary-wing aerodynamics in Ch II by first examining the very simple physicomathematical model of the rotor offered by the momentum theory. Here, it is shown that even this simple conceptual model may prove quite useful in charting basic approaches to helicopter performance predictions; thus providing some guidance to the designer. However, the limitations of the momentum theory; i. e., its inability to account for such phenomena as profile drag and lift characteristics of blade profiles and geometry, necessitated the development of a more sophisticated conceptual rotor model. The combined biade-eiement and momentum theory presented in Ch III represents a new approach which demonstrates that indeed, greater accuracy in performance predictions is achieved, and this would also become a source of more-detailed guidelines for helicopter design. Even with this improvement, many questions regarding flow fields (both instantaneous and time averaged) around the rotor still remain unanswered. In the vortex theory discussed in Ch IV, a rotor blade is modeled by means of a vortex filament(s) or vorticity surface; thus opening almost unlimited possibilities for studying the time-average and instantaneous flow fields generated by the rotor. Unfortunately, the price of this increased freedom was computational complexity usually requiring the use of high-capacity computers. It appears that some of the rotor aerodynamic problems amenable to the treatment of the vortex theory may be attacked with a somewhat reduced computational effort by using the approaches offered by the velocity and acceleration potential theory. This subject is presented in Ch V which also contains a brief outline of the application of potential methods to the determination of flow fields around three-dimensional, nonrotating bodies. Considerations of airfoil sections suitable for rotors, as presented in Ch VI, completes the sequence on fundamentals of rotary-wing aerodynamics. This material provides a basis for development of the methods for helicopter performance predictions used in Vol II. In order to create a complete series on Rotary-Wing Aerodynamics the author anticipates a third volume devoted to the application of the basic theories established in Vol I. This volume would include (1) selected problems of helicopter flight mechanics (e.g., ground effect, flight maneuvers, performance limitations, and autorotation); (2) establishment of a link between aerodynamics and design optimization; and (3) development of techniques leading to performance maximization of existing helicopters. In fairness to the aeronautical engineers and designers who have been anxiously awaiting for the publication of this series, the first two volumes are being released prior to the writing of the proposed third volume. Returning to the present volume, the reader’s attention is called to the fact that both SI metric and English unit systems are used in parallel; thus expediting an acquaintance with the metric approach for those who are not yet completely familiar with this subject. In conclusion, I wish to express my indebtedness to the following persons who generously contributed to this volume: Professor A. Azuma of the University of Tokyo, Japan for his review of the appendix to Ch IV; and to Drs. R. Dat and J.J. Costes of ONERA, France for their valuable inputs and review of Ch V. W. Z. Stepniewski NOTES ON METRIC SYSTEM In order to assist the reader in making the transition from English units to equivalent SI metric units of measure, some important aspects of the SI system encountered in applied subsonic aerodynamics and rotary-wing mechanics of flight are listed below and briefly reviewed. The reader’s attention is called to the fact that as long as metric units are input into relationships designated as SI, and no special conversion factors are incorporated into the formulae, the obtained forces will be in newtons, pressures in newtons per meter squared, work or energy in newton meters, and power in newton meters per second. It should be noted however, that in many countries, the kilogram of force (symbolized in this text as kG) is widely used for the determination of weight (including that of aircraft), while such quantities as disc and/or wing loadings are measured in kG/m2. The popularity of the kilogram as a unit of force stems from the fact that prior to the establishment of the newton as a unit of force, the kilogram was generally accepted in engineering practice as well as in everday life. It was defined as a force resulting from the acceleration of earth’s gravity acting on one kilogram of mass. However, the earth’s g value varies with altitude over sea level and geographic latitude. Consequently, the definition of the kilogram of force as the weight of a kilogram of mass on the earth surface required additional specifications of earth coordinates as to where the weight is measured. The newton (kg m/s2) as a unit of force is more universal in the context of its not being directly related to the gravity conditions encountered on this particular planet. It should also be mentioned that in addition to the direct use of the kilogram of force in engineering practice, its indirect influence can be consistently found when dealing with the metric horsepower defined as hp = 75 kG m/s. It should also be noted that the so-defined metric horsepower amounts to 0.9863 of its English counterpart defined as hp = 550 ft-lb/s. One final note – The most important characteristics of air according to the International Standard Atmosphere are given in the following table. THE INTERNATIONAL STANDARD ATMOSPHERE