Table Of ContentBergman's Linear Integral Operator Method
in the
Theory of Compressible Fluid Flow
By
M. Z. v. Krzywoblocki
Se. D. (Lille), Ph. D. (Brooklyn), M. A. (Math., Stanford), M. S. (Appl. Math., Brown)
M. Aer. En. (Brooklyn), Dip!. lng. (Lemberg), Professor, University of Illinois
With an Appendix
by
Dr. Ph. Davis and Dr. Ph. Rabinowitz
u. S. Department of Commerce, National Bureau of Standards
With 3 Figures
Springer-Verlag Wien GmbH 1960
This book is based on the Author's papers
"Bergman's Linear Integral Operator Method in the Theory of Compressible Fluid Flow"
having been published in "Österreichisches Ingenieur - Archiv" Vol. VI (1952), No. 5,
Vol. VII (1953), No. 4, Vol. VIII (1954), No. 4, and Vol. X (1956), No. 1
ISBN 978-3-7091-3995-0 ISBN 978-3-7091-3994-3 (eBook)
DOI 10.1007/978-3-7091-3994-3
All rights reserved
This book, or parts thereof, may not be reproduced
in any form (including photostatic or microfilm form)
without permission of the publishers
To my beloved Wife
Preface
The reader who is somewhat acquainted with the field of compressible fluid
flow hears much about Stefan Bergman's method of integral operators. It took many
years for him to develop this method which is based primarily on the theory of
analytic functions and particularly on the theory of)functions of two complex variables.
The method, as a whole, is scattered throughout many papers in mathematical
journals, and as a matter of fact, in its present state, is accessible only to those who
are fully acquainted with mathematical literature. In one of their papers, Professors
R. von Mises and M. Schiffer greatly simplified the method in the subsonic casco
The purpose of the present work is to represent the method in all its variations in such
a way that a theoretical engineer or an applied aerodynamicist can use it in practical
applications. A professional mathematician will find the discussion too elementary
for him. The parts of Bergman's presentation which are most interesting mathe
matically-the proofs-are mostly omitted in the present work. The emphasis was
put upon the simplified representation of the final results and formulas, rather than
upon the derivation of those formulas.
In the preliminary remarks the author discusses various types of singularities
in a very elementary way. The first two parts of the work deal with the subsonic
case. In these sections the author followed mostly the paper of von Mises and Schiffer.
Part III contains the supersonic flow, Part IV the transonic flow and Part V the
axially symmetric flow. These sections are based on Bergman's original papcrs and
form a condensation of several of his papers on the subject. Part VI describes
singularities which play so important a role in all hodograph transformations. Part VII
contains a review of other methods available in the theory of compressible fluid flow.
The subsequent parts contain: Part VIII, a review of tables in Bergman's method;
Part IX, general remarks; Part X, a list of tables; Part XI, examples; and Part XII,
errata in previous papers. Part XIII contains generalization to diabatic flow,
Part XIV three-dimensional flow and Part XV by P. Davis and P. H,abinowitz from
U. S. National Bureau contains some computations of subsonic fluid flows by
Bergman's method of integral operators carried out on the U. S. National Bureau
of Standards Eastern Automatic Computer (SEAC), Washington, D. C.
The present work is the reproduction of the research reporV prepared by
the author in the years 1949-1951 for Harvard University and the late Professor
von Mises, done under Navy Contract Nord 10-449, Task 3, Bureau of Ordnance,
Operator Methods in the Theory of Compressible Fluids (H,eport 18). The work was
done under the supervision of the late Professor R. V. Mises and Professor S. Berg
man. The author is greatly indebted to the late Professor V. Mises for encouraging
him to undertake this task and for the permission of using freely all the works of
1 M. Z. v. Krzywoblocki: Bergman's Linear Integral Operator Method in the Theory of
Compressible Fluid Flow. Bureau of Ordnance, Operator Method in the Theory of Compressible
Fluids (Report 18), Contract Nord 10-449, Task 3, Harvard University, Cambridge, Massachusetts,
March, 1951.
VI Preface
Professor v. Mises as the source of information. He expresses his deep thanks to
Professor S. Bergman for numerous discussions on the subject of the work and for
permission to use all the papers and works pertinent to the integral method published
by Professor S. Bergman in numerous journals. His special thanks are due to the
Headquarters, Wright Air Development Center, for permission to publish the results
of the calculation applying Bergman's method, performed by Dr. P. Davis and
Dr. P. Rabinowitz, National Bureau of Standards, sponsored by the Wright Air
Development Center; to the National Bureau of Standards, U. S. Department of
Commerce for the permission to incorporate the calculations in the work; to
Dr. P. Davis and Dr. P. Rabinowitz for their kind agreement to publish their
calculations in the present work; to Professor Dr. Hilda von Mises for her kind
permission, as the successor in the rights of Professor von Mises, to use in Part I
and II the research report by R. von Mises and M. M. Schiffer, done under Navy
Contract Nord 8555-Task at Harvard University, published subsequently in
"Advances in Applied Mechanics" in 1948; to Professor Dr. M. M. Schiffer for his
kind permission to make any use of the above mentioned report and paper.
The author is particularly grateful to the Academic Press Inc., Publishers, New
York, N. Y., for their kind permission for quoting verbatim some sentences and
paragraphs from the paper by R. v. Mises and M. Schiffer "On Bergman's Integration
Method in Two-Dimensional Compressible Fluid Flow", published in "Advances in
Applied Mechanics", Volume I, 1948, pp. 249-285, Academic Press Inc., New York.
His thanks are also due to Dr. G. S. S. Ludford, previously of Harvard Uni
versity, at present with the University of Maryland for reading the manuscript
and making valuable comments concerning the material.
Fluid Dynamics Panel University of Illinois,
Urbana, Illinois, U. S. A.
April, 1960
M. Z. v. Krzywoblocki
Table of Contents
Page
Preface........ . ............................. . ............ ........................ V
Li"j of "YInbo]:.; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X
Part O. Preliminary remarks. Singularities ....................................... .
1. General properties and remarks on analytic functions ................. .
Part I. General theory of subsonic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1. Physical and hodograph planes, system of equations, Chaplygin's equation 6
2. Transformation of Chaplygin's equation, pseudologarithmic plane. . . . . . . . 7
3. Isentropic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4. The duality between the flows of an incompressible and a compressible
fluid. Logarithmic plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5. Method of generating stream functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
6. The domain of convergence of the development for p* . . . . . . . . . . . . . . .. 13
7. Discussion of the domain of convergence of the series representing p* .. 15
8. Transformation to the physical plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
9. Alternative formulas, additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
Part II. Simplified pressUI'e-density relation ....................................... 23
1. Definition of a series representing a stream function p* regular in the
whole half-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2. The physical relations corresponding to the assumption f C~ C }. -2. . . . . .. 25
3. Discussion of the obtained results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
Part III. Supersonic flow ........................................................ 30
1. Some remarks on different types of equations. . . . . . . . . . . . . . . . . . . . . . . .. 30
2. Differential equation in the supersonic flow. . . . . . . . . . . . . . . . . . . . . . . . . .. 31
3. Solution of the differential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
4. Operator obtained by the use of Riemann's function. . . . . . . . . . . . . . . . . .. 32
5. Evaluation of the funetion R(l) (&, 1); JO' 1)0) . . • • . . . • . . . . . . . . . . • . . . • . .. 34
G. Seeond type of integral operator ................................... " 34
7. The proof of the uniform eonvergenee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
8. The domain of convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
9. Solution regular at A = 7: = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
10. Transformation to thc physical plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
11. Alternative formulas ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
Part IV, Transonic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
1. General remarks . . . .. .............................................. 42
2. Behavior of the Gn at m= 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
3. Recursion formula for the functions r n' . . . • . . . • . . . . . • . • • . . . . . • . . . . . . .. 44
4. Methods of analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
5. Application of integral operators to transonic flow. Basic eqnations . . . .. 49
n, Integral operator of the second kind in the case of the simplified
compressibility equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
7. Integral operator of the second kind in the case of the "exact"
compressibility equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
8. Combination of the integral operator and Chaplygin's solution. . . . . . . . .. 58
9. Tables for the determination of transonic flow patterns. Alternative
formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. no
VIII Table of Contents
Page
Part V. Axially symmetric flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64
1. General equations, explanation of the method of infinite approximations
by a sequence of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64
2. Determination of (l)(0) ••••••••••••••.••.••••••••••••••••••••••••••••• 66
3. Determination of the solution of the equation in the transonic case..... 68
4. Determination of the solution in the sub- and supersonic regions .. . . . .. 71
5. Determination of (l)(1) in N p belonging entirely to the sub- or supersonic
region.............. . . ..... ....... ...... ..... . ..... . . . . ... .... . .. .. 73
6. General remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
Part VI. Singularities............................................................ 76
1. General properties and remarks on analytic functions. . . . . . . . . . . . . . . . .. 76
2. Properties of Bergman's operator in the subsonic range ............... , 76
3. The behavior of certain types of flows in the case of an incompressible fluid 79
4. The behavior of a subsonic compressible flow at infinity . . . . . . . . . . . . . .. 80
5. Fundamental solutions in the subsonic range. . . . . . . . . . . . . . . . . . . . . . . . .. 81
6. Modified integral operator in the subsonic range. . . . . . . . . . . . . . . . . . . . . . . 83
7. Subsonic compressible fluid flows with singularities of type S .......... 83
8. The transition from the pseudo-logarithmic to the physical plane in the
subsonic range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
9. Generalization of Blasius' formulae in the subsonic range. . . . . . . . . . . . .. 86
10. An integral formula representing a subsonic flow inside a domain in terms
of its values on the boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
11. A proof of an auxiliary lemma in the subsonic range. . . . . . . . . . . . . . . . .. 87
12. Singularities in transonic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Part VII. Review of other methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
1. Methods of expansion in powers of the Mach number. . . . . . . . . . . . . . . . .. 90
2. Methods of expansion in powers of a thickness parameter (small perturbations) 93
3. Variational methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4. Hodograph method....................................... ...... .... 97
5. Slender bodies and bodies of revolution .............................. 106
6. Southwell's method ................................................. 107
7. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107
Part VIII. Review of tables and particular formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 108
1. Subsonic flow ...................................................... 108
2. Supersonic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109
3. Transonic flow ...................................................... 110
4. General remarks .................................................... 110
Part IX. General remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. III
1. Application of the hodograph method in the three-dimensional case and
to flows with shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. III
2. The method of orthogonal functions and the theory of integral operators III
3. General remarks .................................................... 113
Part X. List of tables ......................................................... " 113
1. Subsonic flow ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113
2. Transonic flow .............................. , . . . . . . . . . . . . . . . . . . . . . .. 115
Part XI. Examples .............................................................. 116
1. Subsonic flow .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116
2. Use of punch-card machines ......................................... 119
3. Flow past an oval-shaped obstacle ................................... 122
4. Transonic flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 124
Part XII. Errata in previous papers ................................................ 125
Additional contributions ................................................. 127
Part XIII. Generalization of Bergman's method to diabatic flow ....................... 128
1. Basic equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 129
2. The stream function equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130
Table of Contents IX
Pag('
3. The "generalized" potential equation ................................. 131
4. Hodograph transformation ........................................... 132
5. General considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133
6. Generalized Chaplygin's equation and its transformation. . . . . . . . . . . . . . .. 134
t
7. Computation of for diabatie flow ................................... 135
8. Method of generating stream functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137
9. The domain of convergence of the development for 'P* ................. 138
10. Discussion of the domain of convergence of the series representing 11'* . .. 140
11. Behavior of an near m = m* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 140
12. Recurrence formula for the functions r" (Tl) ........................... 142
13. Discussion and conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 143
14. Discussion of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14:>
Part XIV. Integral operators in the case of cel·tain differential equations in thre(' "adabll's 14(;
1. Some examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14(i
Part XV. Some SEAC computations of subsonic fluid flows by D1'8. P. Davis and P. Rabinowitz 14H
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 148
2. Basic equations of flow ............................................. 14!:J
3. Generation of particular solutions of the differential equation. . . . . . . . . .. 150
4. Fluid flow of type S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5. Singular solutions of type S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15a
6. Numerical computation of particular solutions and singular solutions 155
7. Computation of F and P as functions of a complex variable ........... 156
8. Estimation of truncation errors in the computation of particular solutions 157
9. Computation of particular solutions. Method A . . . . . . . . . . . . . . . . . . . . . . .. 157
10. Local development of the solutions. Method B .... . . . . . . . . . . . . . . . . . . .. 159
11. Development of the singular solutions ................................ HiO
12. Computation of particular solutions. Method B . . . . . . . . . . . . . . . . . . . . . . .. Hi2
13. Computation of singular solutions. ::\iethod B. . . . . . . . . . . . . . . . . . . . . . . . .. lfi3
14. Method for the solution of the boundary value problem in the pseudo-
logarithmic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. I G3
15. Sources of error in Method B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 166
16. Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. lG6
17. Tables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 H7
Final J'(·mal'ks..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 173
Additional bibliography .......................................................... " 177
Index of authors ................................................................ 185
SUbject index ................................................................... 186
List of Symbols
F, compressibility function for stream eq. [see (15.2.16)].
Fl = ReF.
F2 = ImF.
F*, real approximation to F.
=(k-l)'/2
h
k+l . r
(1 + 1
H = (1 - M2)-'/. k 21M2 2(k-l).
k, gas constant.
1 = (1 - M2)/ri.
M, local Mach number.
p, pressure.
P, compressibility function for potential eq. [see (15.2.17)].
P(ZN), Integral operator [see (15.3.26)].
q, speed.
S*, fundamental singularity of eq. (15.2.14).
T = (1 - M2)'/2.
1
w = 2 (z - Ao)·
1
w* = 2 (z* - Ao)·
W 1> W~, singular points in the logarithmic plane.
+ e.
z = A i
e.
z* = A - i
f) , angle of flow.
fJ=-Imz*.
A, pseudologarithmic variable [see (15.2.5)].
A = Rez*.
(2, density.
(J = arg T.
ITI.
i =
([J, potential function.
([J* = ([J H, modified potential function.
X, singular stream function.
"P, stream function.
"P* = "PIH, modified stream function.
"PN*, particular solutions of (15.2.14).
"P(L, 2), stream function of source.
