MATHEMATICAL MODELS AND METHODS OF LOCALIZED INTERACTION THEORY
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MATHEMATICAL MODELS AND METHODS OF LOCALIZED INTERACTION THEORY
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Series on Advances in Mathematics for Applied Sciences - Vol. 25
MATHEMATICAL MODELS AND METHODS OF LOCALIZED INTERACTION THEORY
Abram I. Bunimovich Moscow State University, Russia
Anatolii V. Dubinskii Ben-Gurion University of the Negev, Israel
WortGb80i@ntift€a/ Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office; Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Bunimovich, Abram I. Mathematical models and methods of localized interaction theory / by Abram 1. Bunimovich and Anatolii V. Dubinskii. p. cm. - (Series on advances in mathematics for applied sciences ; vol. 25) Includes bibliographical references and index. ISBN 9810217439 1. Aerodynamics-Mathematical models. I. Dubinskii, Anatolii V. II. Title. III. Title: Localized interaction theory. IV. Series. QA930.B787 1995 620.1'04-dc20 94-30339 CIP
Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA.
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Material
Preface The localized interaction theory (LIT) covers theoretical and applied problems dealing with determination of force and thermal effect of a medium on bodies moving in it or flowed around by it under physically different conditions. The basis of the investigations are so-called "localized interaction models" (LIM) describing the effect of a medium in each point of a body surface as independent one which is defined by the "local" geometric and kinematic characteristics of that point as well as by "global" parameters identical for all points of contact and representing the common characteristics of the medium, the body and the interaction process. In the most developed and practically used version of LIT which is oriented at translational body motion the angle between the body velocity vector and the normal to the body surface is considered as a local parameter. The models of considered type are widely used in aerodynamics. They cover all supersonic and hypersonic regimes from "continual" (continuum medium regime) to free-molecular flow of rarefied gas. They describe the effect of light flux on the body and that of plasma flow under some conditions. They are used to modelling pene tration of bodies into soils and metals, etc. The most traditional fields for numerous applications of the methods based on the local models is aerospace and aircraft engi neering. The subject of study in the framework of LIT is validation and analysis of different specific models and investigations of physical phenomena of their basis, defining the common properties of local interaction models, development of mathematical methods for solving the applied problems dealing with analysis, calculation and optimization of integral characteristics (IC) of effect of the medium on the body and featuring sufficient versatility in the framework of LIT. Of course, in these fields, LIT intersects the relevant sections if mechanics, physics and engineering sciences. Theoretical and applied research based on a general description of localized interaction rather than specific models holds a special place which enables one to obtain results of a fairly general nature which remain valid for various media and under various conditions of interaction of the medium and the body. This involves a significant abstraction from the specific physical phenomena since the model class becomes the basis for investigations. Therefore, on the one hand, LIT has been formed as a separate section of physics and mechanics. On the other hand, the LIM class is an important object for use of v
V]
Preface
m a t h e m a t i c a l methods which is orient at applied sciences. W h e n discussing the structure and contents of the book, the authors encountered t h e usual problem of selecting the material which was in this case intensified by t h e non-traditional interdisciplinary nature of the subject and consequently, by abun dance of publications dealing with it to a certain extent. T h u s , t h e authors worked out a n u m b e r of basic principles which they tried to use as guides. 1. T h e authors would like anyone who opens t h e book could rather easily and quickly get the notion of what LIT is about, its evolution, state-of-the-art and its potential. T h e authors realize that the enthusiasts which would want to struggle through awkward formulae for t h a t purpose in search for understand able text are hard to find. Therefore, starting from Chapter 1 all t h e necessary information needed for primary acquaintance with t h e subject is provided. 2. T h e authors paid their attention primarily on the investigations based on func tions of fairly general form which define LIM's. Therefore, numerous results based on specific models remain beyond t h e scope of consideration either for t h e principal volume of the book or the review or t h e bibliography. Exceptions are m a d e for the cases of non-traditional application of t h e specific LIM's as well as for modelling the rarefied gas flow in transition regime of flight altitudes. T h e latter is explained by t h e fact that LIT was formed, to a great extent as a result of d e m a n d s of aerodynamic calculation as applied to such conditions. Moreover, application of alternative approaches for transition region of flight altitudes still involves significant difficulties. 3. W h e n describing the investigations based on general LIM's, the authors encoun tered problems of terminological nature. Therefore, to avoid awkward word combinations for describing some characteristics, t h e highly developed termi nological vocabulary of aeromechanics is often used which n a t u r a l l y does not limit the general nature of the results obtained. 4. Although the present book is primarily devoted to m a t h e m a t i c a l aspects of LIT, t h e existence of numerous applications for the results of theoretical inves tigations enables the use of the d a t a of experiments and "precise" calculations carried out by other methods to illustrate and substantial those aspects. Such opportunities have not been missed in cases which the authors regarded as im portant. 5. Chapters 2-5 include only the investigations carried out by the authors either separately or together or, in exceptional cases, jointly with colleagues. In case of the latter references are given concerning t h e publications. 6. T h e authors have to reject the idea of complete unification of notation through the entire book since it would have caused an unjustified complication of for mulae due to the introduction of additional subscripts and superscripts. Nev ertheless, such unification is present inside the subsections. In t h e rest of cases t h e necessary explanations are provided.
Preface
VII
The book comprises the Preface, five chapters, the Appendix, and the bibliog raphy. Each chapter is divided into sections which are divided into subsections when needed. Chapter 1 is the introduction to LIT. It comprises discussion of the basic prerequi sites of LIT, mathematical description of the general localized model of media—body interaction, presentation of classes of specific models describing the effect of different medium on a body under various conditions. In the final section of Chapter 1 the evolution of LIT and its mathematical apparatus is traced up to the present time and the process of integration of separate studies into a general theory is shown. Also, the basic directions of studies are classified and the most important results are out lined. This section also includes the survey of the principal publications which allows the reader to get more specifically acquainted with the applications to those fields of physics, mechanics and engineering sciences which are of interest to him. To avoid duplication, the following sections devoted to studies in specific branches of LIT, as a rule do not contain surveys and feature only those references which are needed to present the material. Described in Chapter 2 are methods for calculation of integral force and energy characteristics of the effect of a medium on a body which are specific for LIT. Devel oped in Section 1 are methods for calculation of integral characteristics of the effect of a medium depending on the parameters defining the body orientation. They provide for reduction of the problem to a study of a system of differential equations. Although the methods presented in Section 1 are fairly general in the framework of LIT, the potential for their use is stipulated by the knowledge of a specific model of interaction between a. medium and a. body surface. A different approach intended to be used when there is no information sufficient for direct calculation, is developed in Section 2. Sets of bodies having integral char acteristics related by linear relationships invariant with respect to selection of any specific model, are proven to exist. Original calculation methods are developed on this basis. Examined in Section 3 are some generalities of change of IC's for bodies with small deviation from symmetrical flow around. In Section 4 the effect of small variations in body geometry on IC's is studied for the LIM class and the the generalized area rules are validated. Presented in Chapter 3 is a comprehensive examination of the problem of design of bodies having invariable longitudinal static stability factor which does not depend on specific conditions of local interaction between a medium and a body as well as its orientation. Equations for surface elements featuring this property are obtained on the basis of mathematical analysis of LIM. The general method for designing the bodies from separate elements is substantiated and specific algorithms are developed for different configuration classes. Chapter 4 deals with the problems of body configurations optimization which may be solved on the basis of LIM. It is shown in Section 1 that in the framework of
Vlll
Preface
L I T , there are some common solutions for t h e problems concerning t h e bodies with o p t i m u m characteristics. T h e problem dealing with bodies of revolution of m i n i m u m drag when moving in gas of various rarefaction is studied in Section 2 by using b o t h analytical and computational methods. Suggested in Section 3 is a generalization of LIT oriented at modelling of processes of penetration (gradual submersion) of bodies into various media (soils, metals, etc.). Problems of o p t i m u m configuration of p e n e t r a t i n g bodies are also studied. C h a p t e r 5 deals with the development of LIT and its applications for t h e case of nontranslational motion of bodies when velocities of different points of body surface are different and t h e angle between t h e normal to its surface and velocity vector cannot serve as a single local parameter. T h e relevant generalization of classic LIT is given in Section 1. Section 2 features m e t h o d s for calculation of body IC's which are based on the use of rotary derivatives, i.e. coefficients of expansion into Maclourin series of the relevant force and m o m e n t characteristics of b o d y — m e d i u m interaction considered as functions of component of low angular velocity. T h e problem is studied on the basis of general LIM for bodies of revolution. Formulae for calculation of IC's in different coordinate systems which widely ap pear in t h e main text of the book are given in the Appendix. Two-index system of numbering of formulae and figures is used in each chapter of the book. T h e first number indicates the section n u m b e r while t h e second one, the number of formula (figure) in t h e given section. W h e n referring to formula (figure) from another chapter, additional number of t h e chapter is introduced on the left. W h e n referring to formula (figure) which appears in t h e Appendix, letter "A" appears on t h e left. T h e bibliography is arranged in alphabetical order. T h e titles of Russian papers published in journals and collections have been translated into English. W h e n de scribing Russian books, journals, collections of papers, conference proceedings, etc. their titles, the names of publishing houses and their locations are given in Latin transliteration. Sometimes, English version of those titles is given in square brackets. Investigations in t h e field of LIT on which the book is based carried out at Moscow State University. Guided by Prof. A. I. Bunimovich, directly associated with him and actively participating in the studies were V. G. Chistolinov, V. R. Dushin, 0 . A. Khots, N. I. Sazonova, M. A. Vorotyntsev, G. E. Yakunina, et al. T h e authors are grateful to t h e m and regret the fact t h a t the results of their work are not fully represented in t h e book but are referred to by brief abstracts and references. T h e authors are grateful to all who participated in discussions over their reports on the subject m a t t e r of the book at numerous conferences, congresses and seminars by paying attention both to theoretical and applied aspects of t h e study. T h e authors are especially grateful to Prof. N. Bellomo (Italy) and Prof. A. V. Bobylev (Russia) who supported the idea of writing and publishing t h e book.
Contents Preface 1
v
Introduction to localized interaction theory (LIT) 1 1 Mathematical model of localized interaction between a medium and a body surface . . . .... 1 2 Development and state-of-the-art of LIT . 6 2.1 Origin and general evolution tendencies of LIT 6 2.2 Development of "localized" modelling methods 12 2.3 Methods of calculation and optimization of integral characteristics on the basis of specific local interaction models (LIM) . . . 2 0 2.4 Research of general properties of LIM and their applications 23 2.5 Non-traditional fields of the LIT application . . 27
2 M e t h o d s of calculation of integral characteristics of environment ef fect on b o d y moving in it 1 Differential equations method 1.1 Basic equations. General solutions 1.2 The case of small angles a and ip 1.3 Analysis of integral characteristics behavior at small angles a and (p . . 2 Invariant relationships method .... ... 2.1 Invariant relationships for a pair of bodies 2.2 Generalization for several bodies . . 2.3 The inverse problem . .... .... 2.4 The case of three-dimensional bodies 3 On some properties of integral characteristics resulting from body sym metry . 4 Generalization of area rules . . . . . . 3
29 29 29 36 41 52 52 58 66 77 83 88
Design m e t h o d s for bodies with invariable longitudinal static stabil ity factor 95 1 Problem statement 95 2 Surface elements with the invariable center of pressure position . . 100 ix
X
Contents
3
4 5
2.1 General solution . . . ... • 2.2 Flat wings . . . . . . . . . . . . . 2.3 Cylindrical and conical surfaces . . . 2.4 Flat elements . P y r a m i d a l bodies . . 3.1 Typical geometric configurations . . . 3.2 General algorithm . . . . Bodies with surface comprising conical and fiat elements . . . Bodies of complex configuration with elliptical cross-section . . 5.1 Cylinder and cone with spherical bluntness . . . . . 5.2 Segmental-conical and some other bodies . . .
100 102 103 107 109 109 113 116 122 123 124
4
V a r i a t i o n a l p r o b l e m s of o p t i m u m b o d y c o n f i g u r a t i o n d e t e r m i n a t i o n 1 Some general solutions for local models . . . . . . . . 1.1 T h e class of the o p t i m u m three-dimensional bodies 1.2 T h e o p t i m u m wing class 2 Bodies of revolution of m i n i m u m drag in gas of different rarefaction 2.1 Problem definition . . . 2.2 Analytical study of variational problem . . . . . 2.3 T h e numerical study m e t h o d . 2.4 Calculation results . . 2.5 Study of powerlaw bodies . 3 On the o p t i m u m configuration of penetrating bodies 3.1 Specific character of penetration problems 3.2 Modelling of body penetration into media on the basis of LIM 3.3 Configuration optimization for conical three-dimensional bodies
129 129 129 133 138 138 140 142 146 153 159 159 160 165
5
G e n e r a l i z a t i o n of L I T 1 Generalized model . . 2 Calculation of integral characteristics for bodies in 2.1 Problem definition 2.2 T h e case of body rotating about its axis . 2.3 General case. Basic relationships 2.4 General case. "Main" components of rotary 2.5 Corrections resulting from the exposed area
169 169 172 172 174 175 177 179
A
. . combined motion . derivatives modification
.
B a s i c f o r m u l a e for c a l c u l a t i o n of i n t e g r a l c h a r a c t e r i s t i c s o n t h e b a s i s of L I M 185 A.l General relationships . . . 185 A.2 Cylindrical coordinate system—I . 189 A.3 Cylindrical coordinate system—II . . . . 192 A.4 Cartesian coordinate system—I . . . 194 A.5 Cartesian coordinate system—II 195
Contents
Bibliography Subject Index
xi
1" 225
Chapter 1 Introduction to localized interaction theory (LIT) 1
Mathematical model of localized interaction be tween a medium and a body surface
Assume a medium comes to a body at rest with constant velocity v,*, and the problem is to calculate the characteristics of the medium effect on a body surface. The desired characteristics in the first place include forces and heat flux. The body at rest and moving medium are naturally considered here for definiteness. It may be done vice versa that is to consider the body as moving in a still medium which is accomplished in Chapter 5 where it is more convenient. Mathematical models and methods for calculation of the effect of a medium essen tially depend on the kind of the medium (liquid, gas, soil, etc.) and also the properties and the state of a surface and other factors. The relevant fields of physics, mechanics and technology deal with the development of calculation methods as applied to spe cific classes of "medium-body" situations. As far as "classic" LIT is concerned, the problems are studied in its framework for conditions of interaction between a medium and a surface of the following form 1 AF c F = — lim — - = fipv(a,i)n° + n T ( a , 0 ' r o , qx AS-o A S * i -
v
oo ■
n
9t» =
.
—g—,
(1.1)
where A F is the force acting from the medium on a small surface element dS which is tangent to the body surface in the given point; cp is the local force coefficient; q^ is the dynamic head; p^ is the density; n°, r ° are the vectors of inner normal and the tangent line in the point of the surface, respectively; a is the vector of "global" parameters characterizing the interaction as a whole; Qp, fiT are the functions char1
2
Chapter 1. Introduction
to localized interaction
theory
(LIT)
acterizing t h e model of medium-body interaction; subscript oo refers to t h e m e d i u m ; subscript "zero" refers to a unit vector. Expression (1.1) yields formulae for pressure coefficient c p and tangent force coef ficient c T , Cp
= c F - n ° = ^ —^ ^ = n p ( a , it ) , 1), expressions (2.5) may be transformed in the following fashion [12, 258]: .p - 2 (2 - o„) cos2 9 + an
a,
*"(7-l)
-,1/2 COS0,
QT = 2 aT sin 9 cos 9, QQ = aell
—t w ) cos 9,
(2.6)
where tw = Tw/T0 while TQ is adiabatic stagnation temperature, whereas T0 = TX
! + •7
1
If it is assumed [219] that a fraction of particles (1 — a) is reflected mirror-like from the surface and the remaining part (CT) is ejected with Maxwell distribution which is characterized by temperature T r , then the so-called mirror-diffusion reflection pattern is realized for which p„ = (2- o-)pi +o-pr, TS = an. In such a case "local" expressions may be obtained to calculate integral characteristics similar to (2.5). However, instead of Tw there appears temperature TT. The effect of a light flux with parallel incidence rays is also described by the "local" model [139] ftp = 2 [1 + e (1 - &„)] cos2 9 + - [1 - e (1 - bn)} cos 9, J1T = 2[1
-e(l-bT)]sm9cos9,
(2.7)
where e is a reflectivity, bn and br are coefficients of normal and tangent momentum accommodation, respectively, whereas for completely diffusion reflection 6„ = br = 1 and for completely mirror reflection bn = bT = 0. In transition to nondimensional parameters, the analog for dynamic head qw = Eaol{2c) is used. In this expression, Ex, is energy flux which is incident on any perpendicular surface element of unit area and c is light velocity. The example of using model (2.7) for solving specific problems appears in [296]. As a rule, the models belonging to the LIM class emerged "by chance" as a result of different physical phenomena analysis (often in limiting cases) while the research based on them was limited by the framework of corresponding field of science. Up to the end of the 1960s "locality" was not considered a basis for integration of research
10
Chapter 1. Introduction to localized interaction theory (LIT)
for physically different conditions of interaction between a m e d i u m and a body surface and for development of sufficiently universal methods of calculation and analysis of q u a n t i t a t i v e characteristics of such interaction. This of course does not mean t h a t "locality'' as intrinsic property of t h e models was overlooked altogether earlier. It is appropriate to mention in this context analogy between effect of a free-molecular flow and a light flux. Also, [122] may be cited as an example; in t h e comparison of the Newton model and of t h e tangent-cone m e t h o d t h e "locality" n a t u r e of interaction between flow and body surface is stressed whereas the mentioned models are interpreted as possible ways to quantitatively describe such an interaction. A number of such examples is also given in [52, 61]. An i m p o r t a n t role in shaping the conception of LIM as a specific class of models for continuum mechanics was played by [64] published in 1969 and dealing with aerodynamic calculation for transition region in rarefied gas flow. In t h a t period there emerged practical problems related to aerospace technology which required development of efficient methods for calculation of aerodynamic characteristics (AC) of bodies which cover t h e whole range of flight altitude. Meanwhile t h e problem which used to be and still remains t h e most complex one consists in modelling of flow around bodies in intermediate range of altitudes located between t h e region where t h e Navier-Stokes equations are valid and the region of free-molecular flow. Considering t h e "local" n a t u r e of models for hypersonic "dense" flow around bod ies (Newton's and other models) and free-molecular flow, authors of [64] suggested to "interpolate t h e locality" also for intermediate range. Similar approach may also be found in [147] published openly in 1977, although, as t h e authors a d m i t , t h e research was carried out in 1968. T h e lack of rigorous theoretical pre-conditions for selection of one or the other specific kind of model in transition regime of flow to a great extent stimulated interest in "local" models and p r o m p t e d shaping of conception of LIM as a subject of study. Rich pre-history of using specific LIM in mechanics and physics notwithstanding, t h e conception of LIT results from deliberate consideration of physical process simulation through t h e "locality prism", i.e. by including the corresponding approach to the arsenal of previously known tools used to choose t h e proper method when conducting theoretical or applied research. Such an understanding began taking shape from the end of t h e 60s. From then on such terms as "locality", "local m e t h o d " , "theory (law, hypothesis) of locality (local interaction)" have been included to scientific lexicon and have become widespread as applied to t h e approach under consideration. These expressions become common in publication titles (see bibliography), they emerge in names of work groups in scientific conferences and results of research in the corresponding field become t h e subject of special analysis [20, 53-57, 82-84, 94, 96, 232]. T h u s , there are serious reasons to record the history of creating and developing the LIT theory from the end of the 1960s considering the fact of emergence of this field of mechanics as a manifestation of general trends in progress of science which is
Section 2. Development and state-of-the-art of LIT
11
characterized by tightly interrelated processes of integration and differentiation of its separate divisions. In evolution of LIT two interrelated trends become boldly apparent. On the one hand, the results for the whole LIM class are generalized while on the other hand, fairly general new methods covering the comprehensive set of physically different conditions of interaction between a medium and a body surface are revealed and in vestigated. Tightly interrelated in their development with these processes are specific fields of LIT essentially based on features of simulated concrete physical processes. As far as the latter is concerned, research results for intermediate region of rarefied gas flow are of particular methodological interest since LIT was conceived and significantly evolved (especially in the first stage) in the framework of this way of thinking. Here the whole sequence of theoretical and applied research from postulation of locality to development of program systems for conducting systematic practical calculations is the most apparent. Therefore, special interest exhibited below when analyzing the LIT development in the problems of simulation in transition region of rarefied gas flow appears methodologically justified although, of course, presentation of LIT as a separate field of continuum mechanics is primarily associated with sufficiently common nature of approaches developed in its framework. The "latest history" of LIT spans less than 30-year period over which complex and sometimes contradictory process of integration of separate relatively independent investigations into a common theory was implemented. The choice of specific fields of work and research methods in the framework of LIT was influenced to a great extent by traditions of relevant scientific schools and orientation of institutions active in the research as well as scientific interests of particular scientists. Relative independence in choosing subjects and inner logic of investigations in different fields resulted in nonuniform development of separate fields of LIT. Such effects are essentially smoothed out by "averaging" when considering fairly long periods of time since strategic laws of science evolution come into effect. As applied to a short period of time, such nonuniformity hampers the chronological principle of the LIT development analysis as a whole and justifies its breaking down into separate fields for the same purpose. A brief analysis of development of the LIT main fields is given below. It cannot be claimed that the review fully covers the problem which is dealt with in many hundreds of publications. So far as the major attention is paid to the methodological side of the problem as well as the mathematical apparatus of LIT, authors were compelled to exclude from consideration or only slightly mention many rather interesting works directly or indirectly dealing with the problems under examination but based on traditionally used models as well as experimental investigations. Authors also rejected separate detailed analysis of the most part of works in preference to their classification on the basis of uniformity of approach or lead of research in the framework of LIT.
12
2.2
Chapter 1. Introduction to localized interaction theory (LIT)
Development of "localized" modelling methods
Although, as shown, a number of approaches under development to creating specific models feature a certain kind of common ground as far as LIT is concerned, they evolved mainly as applied to transition region of rarefied gas flow which naturally is displayed in the contents of the present section. The problems of creating specific LIM's became a study subject from the concep tion of LIT and were considered in two aspects: finding out the reasonably common expressions for functions fi p and 0 T defining the model as a basis for development of efficient calculation methods for aerodynamic characteristics and identification of parameters of specific models; and as a matter of fact, modelling of specific conditions of interaction between a medium and a body surface. In one of the first publications on LIT [64] selection of the form of relationships is conducted in the class of functions
np = £A'*' is proven for sharp cones and wedges. In limit case of free-molecular flow, the "local" nature of interaction between a flow and a body surface features a fairly high level of theoretical validation [50-51, 75, 188, 193] although uncertainty as far as the choice of some parameters of the models is concerned, leaves much to be improved. Among different approaches to this problem
19
Section 2. Development and state-of-the-art of LIT
(for example, [72, 283]) taking account of depending accommodation coefficients on angle 0 ([178, 189, 300, 235, 241]) should be noted as a method to increase the model accuracy. In this connection, arguments contained in [70] are demonstrative. Stating that integral characteristics of a sphere are more conservative relative to the choice of interaction model than the ones for a plate, the authors explain it, in particular, by the fact that in the first instance a wide range of values of angle 9 takes place and the errors cancel one another as a result of approximate allowance for dependence of model parameters on 9\ in case of plate, these errors are summed up. In essence, as applied also to free-molecular regime, the point is to choose (revise) a specific type of model in the framework of the certain LIM subclass. The above clearly illustrates the basic difference between the assumption of local nature of effect a medium has on a body surface and the proposal of validity of some specific model supposed to describe such an interaction. Model (2.5) which is usually used to calculate a heat flux to a body surface with free-molecular gas flow, belongs to the LIM class. The existence of "local" models as applied to a "dense" gas flow provided a basis for a number of investigations [3638] based on general presentation of heat transfer dependence of angle 9. In the mentioned works there are references to specific relationships for convective heat transfer [266, 292], radiative heat transfer [125, 274], as well as to the case of effect of intensive external radiation [27]. In [40-43] distribution of radiation heat transfer was studied over a surface of three-dimensional bodies and bodies of revolution for air flow around in flight speeds ranging between 8 and 18 km/s and altitudes of 40 to 80 km. Body dimensions were varied in a wide range. As a result, the following formula was suggested to calculate the local heat transfer at three-dimensional and axisymmetrical flow around: Q(6.) = Q_ ■ (cos6>,)",
n = 1.811 +(0.051 Woo-0.43)- 1 ,
where Q- is a heat transfer in a stagnation point which depends on the average value of curvature radius for this point at fixed flight conditions, 9, is a local angle between a normal to the leading shock wave and the forward flow velocity vector. For a general case of flow around three-dimensional bodies a marked deviation form "locality" was detected. However, for axisymmetrical flow an universal relation ship between 9, and 9 holds true which means that the model becomes "local". In such a case it is recommended to use formula [280] for function 6,(9): [ „ 0.164 sin 9 9, = 0 — arctan cos 9 -\ . „ [ y/l - 0 . 6 9 8 sin2 6, or the expression 9„ = 0.91861 -0.0075 93,
0 < TT/3.
The work [39] contains an overview on modeling the radiative heat transfer.
20
Chapter 1. Introduction to localized interaction theory (LIT)
It should be noted that for some classes of three-dimensional bodies "the locality" as far as heat flux are concerned, is observed with a fairly high accuracy. As an example, the following formula [301] may be quoted: Q(0) = Q- • [0.53 + 0.47cos(2.20)],
0 < 9 < 80°.
The formula was obtained for elliptic paraboloids as applied to hypersonic flow around. However, from the viewpoint of general approach in the framework of LIT, numerous cases of "locality" for narrow classes of configurations are not so interesting. Thus, "the locality" of heat effect takes place for free-molecular flow and in a num ber of situations for "dense'' gas flow. Therefore, there are the same prerequisites for assumption of "local" nature of thermal effect in transition region of rarefied gas flow as in the case of force effect modelling. This allows [5] to extend the techniques devel oped earlier for momentum transfer, to the process of energy transfer for transition regime. Some approximate models are suggested in [207, 234] and multiparameter model of form (2.8) is used in [267]. As a whole, however, the problem of heat effect modelling in transition region of rarefied gas flow is in research stage.
2.3
Methods of calculation and optimization of integral cha racteristics on the basis of specific local interaction mo dels (LIM)
If the model of interaction between a medium and a body surface allowing to deter mine local characteristics of force or heat effect in any point is known, then calculation of integral characteristics may be carried out directly by integrating the correspond ing characteristic over the body surface. In case of specific models of the form (2.8) "shape functions" for typical bodies may be arranged in a table form to simplify the calculations. That is how it was done, for example, in [20] for the case of transi tion region of rarefied gas flow. To increase the speed of effect of the corresponding programs, other techniques may be employed [18, 107, 290]. Problems emerge as applied to bodies of complex configuration (in particular, orbiting spacecraft with open solar panels) due to the difficulty in determining the exposed windward surface. Even dealing with the simplest local model (without allowing for the effect of rebounded molecules), the necessity to cut the time required for calculation, especially in cases where it is important to take a timely account of changes in orientation of a body moving in a flow, results in need to develop special methods and algorithms [6, 67, 304]. This range of problems has a common character in terms of LIT and the measure of efficiency for developed techniques is invariant under the choice of any specific LIM. Differential equations method is intrinsic for LIT and is used for calculation of integral characteristics (IC) of bodies. Its principle lies in reducing the problem of integral characteristics calculation over an entire range of angles defining body
Section 2. Development and state-of-the-art of LIT
21
orientation relative to characteristic motion of a medium to solution of recurrent system of differential equations of Poisson type (for three-dimensional bodies) or Legendre type (for bodies of revolution). So far as the form of general solutions for such equations is known, the problem is reduced to determining particular solutions and constants. Advantages of the technique as compared to regular integration over a surface, are defined by the potential to decrease the time it takes for a calculation and to obtain approximate relationships by presenting solutions in the form of series. Two main stages in evolution of the method may be highlighted. The first stage represented by [121, 173-174, 251-253] covers a period from 1965 to 1973 when the method was developed as applied to the simplest Newton model. The beginning of the second stage involves transition to the LIM class of the form (2.10) proposed in [80] in connection with generalization of the method. Theoretical principles of such generalized approach were developed in [80-81, 114, 254]. Subse quent publications (68, 85-88, 112, 124, 212, 279] dealt with development of different aspects of the approach, development of methods, algorithms and computer programs on its basis, etc. Thus, in [68] numerical methods were suggested for solving differential equations to which the problem of integral characteristic calculation is reduced along with ana lytical methods, especially for bodies of complex configuration. The calculations con ducted for satellite "Cosmos-230" as applied to free-molecular regime of flow around showed that when using calculation techniques based on differential equations it is possible to essentially reduce the time required for calculation as compared with reg ular integration over a surface. Work [279] dealt with combined analytical-numerical method for solving the equations [80] and studies of possibility to adapt the model of interaction between a flow and a body surface to the change of its orientation in the flow. Therefore, by the end of 1970-s, the development of theoretical and practical aspects of the method for the class of models (2.10) was accomplished to the extent allowing to include the method into the arsenal of standard methods of IC calculation for LIT. Further development of the method [184, 225, 228, 228-230] involves using func tions of more general form in expression (2.8) than in (2.10). The corresponding approach was dealt with consistently in [232]. Practical requirements naturally stipulated the emergence of the problem of de termination of aerodynamic characteristics of bodies in nontranslational motion. Using "local" models for the purpose has a long history as applied to calculation of forces and moments with free-molecular flow [69, 146, 198, 291, 298] while in such cases a simplified model was usually used [265]. Because of the known analogy, many of the obtained formulae are also valid for the case of light flux effect [23, 285]. More detailed analysis of aerodynamic characteristics behavior was conducted on the basis of calculation and study of so-called rotary derivatives [73]. Among works based on local models, [169, 299] (Newton model), works [70, 170, 172-173, 295] (free-molecular
22
Chapter 1. Introduction to localized interaction theory (LIT)
flow), [3] which numerically confirmed analytical formulae [170, 291, 298] in compared conditions should be noted. One of the complex problems which make themselves evident when calculating non-steady AC on the basis of LIM, is the fact that boundary of exposed surface appears dependent on angular velocity of a body. Therefore, to simplify the calcula tions, this dependence is often neglected. Study of the problem is one of the essential features of [295] where it was shown that as applied to approximation [265], rotation does not affect the structure of exposed area on the body surface. From the middle of the 1970s, investigations in the discussed field begin to evolve as applied to transition region of the rarefied gas. The work [257] being among the first known publications on the problem, was dealt with the results of numerical calculations aimed at determination of rotary derivatives for some classes of bodies. Further studies [111, 113, 118, 149, 264] were carried out using analytical methods based on the model described in [147-148]. They resulted in appearance of expres sions for calculation of aerodynamic characteristics and rotary derivatives of first and second orders as well as determination of particulars of the effect body rotation has on the values of components of force and moments. It should be noted that up to now there are almost no results of experimental investigations and "accurate" calculations which could enable to analyze the adequacy of theoretical results for nonstationary motion of bodies in transition region of rarefied gas flow based on models evaluated as applied to stationary motion of bodies. Similar situation holds true for investigations [93, 104, 106, 109-110, 115, 134, 136, 223, 245] pertaining to optimizing the body configuration in transition region of rarefied gas flow which began to evolve from the end of 1970s. Nevertheless, the results obtained for the blunt-nosed bodies of revolution in all likelihood may be considered as relatively reliable due to comprehensive evaluation testing of the model [147] on which the investigations were based. Systematic analysis [93, 134] of variational problems for the body of revolution having minimum drag over the entire region of hypersonic flow (for "limit" regimes the solutions were obtained earlier [278]) using both analytical and numerical methods, pointed to optimality for flat-nosed bodies and good aerodynamic properties for powerlaw bodies. As far as variational problems of optimization of body configuration based on local models as applied to free-molecular and "dense" gas flow are concerned, the corresponding research evolved mainly in the framework of approaches [199, 278]. Authors of some publications turn to problems considered in [278] with the purpose to study them more thoroughly. Thus, in [76] the problem of three-dimensional body of minimum drag was studied on the basis of the Newton model in compliance with different requirements to its configuration. The primary objective of the investigation was analysis of correctness for different problem definitions. It was shown, in particu lar, that defining limitation of surface area gives correctness and strict solution to the problem. On the other hand, noncorrectness of the problem was proven for some other definitions. In [200] a new solution was given for the minimum drag body of revolution
Section 2. Development and state-of-the-art of LIT
23
at predetermined volume. As before, the Newton model is widely used to solve the problems of body configuration optimization especially when studying new promising aerodynamic configurations (in this context one may refer to works [269-271] where the corresponding material is collected and extensive bibliography is included) and complicated flight conditions [246]. Lately, the ever-growing attention was given to studying optimization problems for configuration of bodies flying in "dense" gas when heat flux to a surface appears as an objective functional or a limitation [45-47, 190-191, 211]. To calculate pressure distribution on a body surface, the "local" Newton model is used while to calculate heat flux, both local and more complex models are used. The work [45] in which model (2.16) was used to calculate radiative heat flux to a three-dimensional body surface by means of "area rule" [36], may serve as an example of using local model. The problems of body configuration optimization are essentially of multipriority nature. The traditional method to take account of this factor is to take the main demand into the objective functional and to allow for the rest by limitations. The approach [151] was taking into account multipriority nature of the problem while formalizing it and was more adequate for the situation. Successes in the field of model development and appearance of efficient calculation methods formed a basis for development of program systems [1, 18, 28, 34, 196-197, 248, 260, 294], whereas the 1980-s were marked by transition to wide practical appli cation of "local" methods. Apart from the application side, this trend is important also for development of LIT proper since the existence of "feedback" appears to be the source for obtaining substantial estimates for analysis of "local" models and methods stimulating the development of LIT in the most vital fields.
2.4
Research of general properties of LIM and their appli cations
From the end of 1970s, development of LIT is characterized by naturally increasing interest in research of the most general properties of LIM resulting precisely from "locality" rather than specific model describing the effect of a medium on a body under certain conditions. In particular, the basic difference between the requirements for validity of local interaction hypothesis (LIH) and availability of the specific LIM lies in the following: in the first case, it is only assumed that surface elements of different bodies identically oriented in a flow at identical "global" exterior conditions of interaction between bodies and a medium, experience the same effect of the medium; in the second case, the model which enables one to calculate quantitative characteristics of the effect, is presumed to be known. To develop a specific model, a deeper insight into the phenomenon is needed, in particular, about the mechanism of interaction between a flow and a body surface. Finding out the generalities is important not only due to the possibility to directly
24
Chapter 1. Introduction to localized interaction theory (LIT)
extend t h e m to all known LIM's (and, accordingly, physical p h e n o m e n a ) . It is also i m p o r t a n t to note t h e fact t h a t LIT is often used in conditions where traditional theoretical validation of model form is lacking which makes t h e results following from "locality'' assumption appear as more reliable t h a n those based on specific model which lacks convincing validation. Let us consider now t h e investigations which led to t h e development of calculation m e t h o d s for integral characteristics of interaction between a m e d i u m and a surface of a body which is moving in it, t h a t do not require knowledge of t h e specific interaction model. Some analogy to similarity theory methods may be traced in such approach. In fact, similarity laws establish a certain system of relationship between param eters which describe a m e d i u m , a body configuration and a process of interaction between t h e body moving in the medium and t h e m e d i u m itself whereas such rela tionships are valid for a certain range of variation of t h e p a r a m e t e r s . T h e existence of similarity laws apart from scientifically cognitive interest, has a considerable practical value since results of experiments or "accurate" calculations based on the laws may be "transferred" to other conditions of interaction between a, m e d i u m and bodies (other media a n d / o r other bodies). Therefore, t h e calculation based on similarity laws is generally more simple t h a n direct calculation on the basis of initial model. More over, to enunciate similarity laws, a complete m a t h e m a t i c a l problem formalization is not necessarily needed [266] which is especially important when development of such models is difficult for t h e present level of knowledge and involves introduction of major errors. Using similar consideration in such cases is probably t h e only method which enables one to extend t h e results of some experiment outside t h e framework of conditions under which it was conducted. In fact, t h e local interaction hypothesis postulates local similarity of effect on identically oriented elements interacting with a medium in "identical conditions''. Prerequisites for development of calculation methods based on invariant relation ships between integral characteristics of bodies, were defined in [63, 89-92] in which the a p p a r a t u s for obtaining linear relationships based on t h e knowledge of t h e spe cific LIM is developed. T h e works mentioned are, in t u r n , an extension of investiga tions [175-176] which were carried out on the basis of t h e Newton model. Basically new results were obtained in investigations [33, 94, 96, 132-133] so far as they led to universal (invariant) relationships which remain valid for variation of t h e model of interaction between a m e d i u m and bodies (retaining the generality of locality) while for bodies of revolution and some bodies of more complex configuration, also for angle of attack (the same for all bodies). Existence of such relationships allows one to calculate integral characteristics for some body by "factorization" it into basic bodies for which they are known. Therefore, the method works when there is no spe cific model necessary for direct calculation. T h e corresponding theory is developed in the above mentioned works both in general case and as applied to specific classes of configurations. Its practical efficiency is confirmed by comparison with results of experiments and "accurate calculations". T h e ideas inherent to t h e m e t h o d were
Section 2. Development and state-of-the-art of LIT
25
further developed as applied to obtaining various approximate relationships [31-32]. The approach based on general properties of LIT appears to be a constructive one also as applied to the study of static stability of bodies [138, 203]. So far as the longitudinal static stability factor is defined by the position of center of pressure (CP) with respect to the center of mass, then along with the study of behavior of force and moment components effected on a body by a flow, an important subject of analysis generally also lies in finding out the regularities in variation of the CP position depending on the free-stream flow parameters, orientation of the body in the flow and design elements of the vehicle. Usually, such regularities are established on the basis of a series of calculations or experiments. For some simple configurations of bodies (cone of revolution, wedge, plate) independence of the CP position with respect to the angle of attack is determined on the basis of the Newton model or more accurate models [150, 194, 201-203, 213]. Works [8-9, 180, 272-273] should also be mentioned for confirming that the CP position of a cone of revolution is independent of the angle of attack and for studying the influence of viscosity on the CP position. In addition, some publications [143-144, 242, 244, 286, 302-303] should be noted which directly or indirectly dealt with the study of stability problems out of extensive series of works on the study of star-shaped bodies which enjoyed unabated interest since publication [153]. Although independence or slight dependence of the position of center of pressure was repeatedly noted as applied to the specific flight conditions for some classes of bodies, the work [242] pioneered the way for purposeful study the problem. Some classes of bodies of complex configuration were considered in [242] on the basis of the Newton model, however, in the discussed aspect it is interesting mainly in that part which is based not on a specific model but on the assumption of the conical nature of flow on the body surface. In the framework of such general model, independence of the position of center of pressure with regard to the angle of attack and conditions under which a flow acts on a body along the normal to its surface was established for sharp elliptical cones. Further investigations [94-97] indicated that, apart from cones, the following classes of bodies with elliptical cross-section exhibit the property of unchanged position of center of pressure: segments, cylinders as well as combined bodies including cylindrical and conical elements with flat bluntness or (with adequate choice of geometric dimensions) ellipsoidal bluntness (segmentalconical and segmental-cylindrical bodies, double cones etc.). It was established that this property exists also for a wide range of conical bodies having cross-section which is limited by straight-line segments and segments of elliptic curves, in particular for regular pyramid and rhombic wings. Mathematical apparatus was developed for geometric design of such bodies. It is shown that the revealed properties of CP are in a good agreement with the results of experiments and "accurate" calculations and shed light on numerous phenomena discovered empirically. The investigations based on general guidelines of LIT were developed also in a number of other fields. Some of them are described below. Study of variation of integral characteristics of the effect a medium has on a body
26
Chapter 1. Introduction to localized interaction theory (LIT)
when varying its configuration, poses a problem which is of significant theoretical and applied importance. Investigations which serve this objective are aimed, in particular, at finding the conditions which, provided they are observed, ensure that the IC values are maintained (with certain accuracy) when varying the body configuration. Such conditions involve using the area rule stating that IC's of bodies (more often the drag) are barely differ if the laws of cross-section area variation are identical and a number of additional conditions are observed. Area rules are known for a wide range of conditions of flow around, various configurations, integral characteristics and conditions of flow around [35-36, 51, 137, 169, 187, 202, 204, 209-210, 215-216, 246, 255]. In a number of works, local approaches or those close to them were used to that end. Area rules with respect to drag are obtained [213] on the base of the Newton model. The investigations founded on locality concept are represented in [36] as applied to heat transfer coefficient and in [255] for a three-dimensional body wake. Brief review on the discussed problem is given in [44]. Validity of area rules for bodies close to bodies of revolution by configuration as well as their analogy for a number of other three-dimensional configurations are established in [94, 104, 135] as applied to the general class of LIM. Consideration of problems dealing with optimization of body configuration on the basis of general LIM [100, 106, 136] resulted in determining aerodynamic configura tions featuring retention of optimum properties for different specific local models. An important stage of development of general LIT is its generalization for nonstationary motion of bodies [98]. If the body motion is translational when velocity of each point of a body surface is the same and its effect on integral characteristics may be taken account of either directly or indirectly through "global parameters" of interaction between a body and a medium. Accounting velocity distribution over the body surface is absolutely necessary in the case of rotating body. In general, allowing this factor results in the need to introduce a second local parameter into the model. However, it appears that a significant number of local models may be described in the framework of some subclass of LIM which enables one to leave the description intact using functions of only one variable when making generalization for the case of rotat ing bodies. Further investigations [99, 101-102, 105] were conducted for fairly general subclass of LIM. Expressions were obtained for calculation of rotary derivatives of the first and second order for bodies of revolution and low angular velocity. This helps one to generalize a number of regularities of behavior of integral characteristics which were discovered earlier for specific models. So far as the LIT general approaches are discussed in this section, it is necessary to mention [232]. The following LIM formalization is adapted as a basic one: +i
C{a,V) = j f0(t)Mt,a,/\vv\
= sin ip +cos ipz°
(1.2)
31
Section 1. Differential equations method while
dv | v a | = l, 6 = a, Sp and coefficients Bj with subscripts j < 0 or j > R? are assumed t o equal zero; p a r a m e t e r s Rn and RD are chosen on t h e condition t h a t begins with i = R„ + 1 and j = RD + 1 all a, and bj equal zero. Let us introduce functions which will be called "support":
Gk(a,V) = —JJMt)ds, (S) 0 M) O = M = t**k = = (v • nii0 ))*, *,
v v == v(a v(a))V >). V >).
(1.8) (1.8)
It then follows from (1.4) t h a t Cxa(ot, 2), the following expression [114] is obtained: v){m + v+ \)GV + v{v - 1)G„_».
Lm Gu = (m-
(1.13)
In the particular case when m = v = k, (1.13) is transformed into the system of recurrent relations for determining Gk'. (1.14)
LkGk = k(k-l)Gk-2
which was obtained in [114] independently of [254] and later reiterated in [222]. On the basis of (1.13) equations [80] for C°a may be derived, where RD
C£. = X >
G
"
C„ = C^a + b0G0 + blG1.
(1.15)
It is sufficient to write (1.13) for n = RD, " — 2,3, — , RD, to multiply i/-th relation ship by 6„ and to add the equations obtained. The result is: = *,
LRDC°S
(1.16)
where * = bRv ■ RD{RD - \)GRD
+ £
b„[(RD - v)(Ro + v + 1}GV + v{v - l)Gv-i\.
(1.17)
i/=2
For many models, R^ + 1 = RD and expressions (1.10) and (1.11) may be presented in the following form [80] allowing for (1.15):
M Cya =-5—, da.
i
a
*
G2a = — : •-=—, sin a Of
U-ioJ
where
SoPJL + £ h*0, 0 T/2
L
r r W3 I I —dxd.j
h=
/ l3 =
-TT/2
> 0,
raw J -ljj-dxd6>0, 0 -,r/2
*
jjH*trW(xt7 + *W osO)W + $Wcos 6)W = C ax atdxde
'--IS
2
0 -*/2 £
I4=ff
T/2
$(t7Cos8
/ / 0
+
x$x)dxd8,
-„/2
55$ $ #. = _ ,
W = **x,
w = ^x,
d$ a$ ** = -
». = —, *, = -
(1.35) (1.35)
whereas functions 0,
Bi>Q,
A2>BX.
So we get from (1.31), (1.33) and (1.35) Ho > 0,
fi2 > 0.
So far as t h e following estimate
w/ 0
-,11
WdxdO
. + (i/*,) 2 + (*«/w)2 -
/ / wdxde = i0 0 -»/2
Section 1. Differentia/ equations method
43
Figure 1.2: Characteristic variation of coefficients C*, C„, m, against angle of attack a at a < a, for bodies with horizontal symmetry plane. is valid for t h e integral h,
then
C „ ( 0 ) = Cx(0)
=110 + ^
= (A2 - 5 , ) / ! +
B1M)dxl"\ J o
$ = f * l dx^"'
( 2 .1)
where function $' t ''(:r'''') characterizes the configuration of a body of length L'"' in longitudinal direction, function H includes information on t h e LIM type; functions
Section 2. Invariant relationships method
53
If" a r e defined by t h e class of body configurations under consideration; subscript indicates t h e IC n u m b e r out of w-"> considered; t h e rest of t h e functions and parameters characterizing t h e body configuration and its orientation are not specified as arguments. As follows from A p p e n d i x , formula (2.1) may incorporate components of force of m e d i u m effect for a wide range of bodies including wings with curvilinear edges, bodies of revolution and others. Consider t h e problem dealing with t h e search for t h e transformation which can be used to o b t a i n t h e corresponding body (also in some predetermined class of configurations) for a r b i t r a r y original one of t h e given class in such a way t h a t IC's of bodies of form (2.1) satisfy t h e relationship: 1
n(">
E E « . M ^ = ». i/=o ; = i
2 2 ((2.2) -)
i/=o ; = i
where a) are certain arbitrary chosen coefficients. T h e requirement of invariance (2.2) with respect t o choice of model is a principal one, i.e. expression (2.2) should b e valid irrespective of t h e selection of t h a t or another function H. Let us prove t h e existence of invariant relationships (2.2) and validate the m e t h o d for d e t e r m i n a t i o n of functions which characterize the configuration of t h e corresponding body. Transition to p a r a m e t r i c definition of function $ W ( x M ) in t h e form:
$0) $(1) = = * $ (( x* )) ,,
x ( 1 ') = x(x), x(x), x'
x(0)= x(0) = 00 ,,
x(L) x(L) = L Lw(,1 \
x x < ° >== x
enables one to transform expression (2.1) to t h e form: p . ( i ) = / j i W Lx,$,— $ *1 J L x i x' 0 o
H\%]-x'(x)dx, H 7 • x'(x) dx, ix.I .J
ii = =
M ,, . , It l,2,-.\,2,...,n
where derivatives with respect to x of functions characterizing t h e corresponding body configuration are m a r k e d by prime. Requiring t h e observance of condition {x)
*' -= 6(x) ¥$ *(*) 9{X x'(x) x'[x)
(2.3) (2-3)
>
t h e following combination is obtained: l
1
»W Til"'
^E EEL«f!t M i/=0 i = l
L L
") p(")
r
„(D
) = / #/ /((*§)) x' x'J^a'i £ i)4TyTP(X, {x,^,i)
0
i=l
11
„«»
0) #, 4) ++ Y£iai°a!)T0)t{0)r/(x,^,i) (x, $, $) i=l
rfx dx
54
Chapter 2. Methods of calculation of IC of environment effect on body moving in it
whence it transpires that condition G = 0 is valid irrespective of the form of function H, should the following expression hold true: x> £ af^d,
*,*)
= -£
aflj^x,«, *).
(2.4)
The corresponding function can be determined from equations (2.3) and (2.4). The following conditions may be assumed as the initial ones: x(0) = 0,
1(0) = r « ,
(2.5)
where r' 1 ' is some parameter which, as a rule, characterizes dimension of correspond ing body nose section. Therefore, the problem of search for the corresponding body is reduced to solving the system of differential equations (2.3), (2.4) with initial conditions (2.5). For many classes it may be easily solved. We outline the modification of the problem dealing with determination of body configuration having n integral characteristics Pi related between themselves by in variant relationships of the form:
£>.Pj=o. 1=1
In this case, function $(x) which defines body configuration may be found from
£>7i[*,»(*),*(*)] = 0 i=i
with the desired initial conditions. Function H may be of an even more general form than that shown above. This modification of the method is further used when examining the problem of designing the bodies with invariable static stability factor. To explain the essence of the method validated above using formal transforma tions, some considerations are given below regarding the simplest situation when velocity vector of a free-stream medium flow with respect to a body is parallel to the axis of a body of revolution effected on by the medium. In this case, the angle of tangent inclination at each point of frontal part of the body surface conclusively defines the angle between vectors vj^ and n° Condition (2.3) provides mapping the points of initial curve described by function $(x) onto the points of the corresponding curve described by function $i l '(a^*') so that the angles of tangent inclination are equal in the corresponding points and consequently, for the considered classes of bod ies (the original and the corresponding ones) quantitative characteristics of medium effect are identical. As far as relationship (2.4) is concerned, it represents additional requirements for the corresponding points (they may be tentatively called geometric ones) which are intended to validate (2.2).
Section 2. Invariant
relationships
method
55
Figure 2.1: Coordinate system and notation for bodies with affine cross-sections.
Now let us turn to using the method for solving specific classes of problems. It appears that its applicability stretches far beyond the framework of the simplest situation just considered. The terminology used below is typical for aerodynamics and deals with the effect of a flow on a body. This does not limit the general nature of results but makes them more demonstrative. As an example, a situation is considered when both the original and the corre sponding bodies have affine cross-sections (Fig. 2.1) while the configuration of trans verse contour of the bodies is known and identical. Bodies of revolution, in particular, belong to this class. In cylindrical coordinate system x'1'' p'"' 9^ the surface for such a body may be given in the form: pM = n{0M) ■ $ ( ">(x ( "'), where function r) characterizes the transverse contour configuration and function $'")—the longitudinal contour configuration. Being divided by dynamic head, projection of aerodynamic force F'"' on some direction defined by vector 1° lying in plane i'">, j / ' " ' and forming angle x with axis a;'"', is considered as characteristic P ' a body-axis coordinate system, free-stream flow velocity vector v ^ lies in plane x'"', 2/(l/) at angle a to axis x(") (Fig. 2.1). The effect on the body side surface is considered.
56
Chapter 2. Methods of calculation of IC of environment effect on body moving in it The expression for P'"' using formulae (A, 2.6) may be written in the form: PM(a,x)
= (F^/qoo) C\"H] = (F ( ">/ 9oo ) ■ 1° = S* is of the form (A, 2.6) while functions U are calculated by formulae (A, 2.5) at / = $