Supercritical Wing Sections III

187 Pages·1977·5.37 MB·English

by  Frances Bauer

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continuation on page 185
Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Kunzi
150
Frances Bauer
Paul Garabedian
David Korn
Supercritical Wing Sections III
Springer-Verlag
Berlin Heidelberg New York 1977
Editorial Board
H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor)
P. Dhrymes J. Green W. Hildenbrand W. Krelle
H. P. KUnzi (Managing Editor) K. Ritter R. Sato H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. KUnzi
Brown University Universitat ZUrich
Providence, RI 02912/USA 8090 ZUrich/Schweiz
Authors
Frances Bauer
Paul Garabedian
David Korn
New York University
Courant Institute of Mathematical Sciences
251 Mercer Street
New York, N.Y. 10012/USA
AMS Subject Classifications (1970): 76H05, 65P05, 35M05
ISBN-13: 978-3-540-08533-1 e-ISBN-13: 978-3-642-48852-8
DOl: 10.1007/978-3-642-48852-8
This work is subject to copyright. All rights are reserved, whether the whole or
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be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1977
PREFACE
The purpose of this book is to survey computational flow research
on the design and analysis of supercritical wing sections supported by
the National Aeronautics and Space Administration at the Energy
Research and Development Administration Mathematics and Computing
Laboratory of New York University. The work was performed under NASA
Grants NGR 33-016-167 and NGR 33-016-201 and ERDA Contract
EY-76-C-02-3077. Computer programs to be listed and described have
applications in the study of flight of modern aircraft at high sub
sonic speeds. One of the codes generates cascades of shockless tran
sonic airfoi~s that are expected to increase significantly the effici
ency of compressors and turbines. Good simulation of physically
observed flows has been achieved.
This work is a sequel to two earlier books [1,2] published by
Springer-Verlag under similar titles that we shall refer to as Volumes
I and II.
New York
November 1977
TABLE OF CONTENTS
I. INTRODUCTION 1
1. Shockless Airfoils and Supercritical Wing Sections 1
2. Differential Equations of Gas Dynamics 2
II. THE METHOD OF COMPLEX CHARACTERISTICS 5
1. A New Boundary Value Problem 5
2. Topology of the Paths of Integration 8
3. Iterative Scheme for the Map Function 9
III. TRANSONIC AIRFOIL DESIGN CODE 10
1. Isolated Airfoils 10
2. Compressor Cascades 12
3. Turbine Cascades 13
4. Comparison with Experiment 14
IV. TWO-DU1ENSIONAL ANALYSIS CODE 16
1. Wave Drag 16
2. A Fast Solver 19
3. Remarks about Three-Dimensional Flow 24
V. REFERENCES 26
VI. USERS MANUAL FOR THE DESIGN CODE 29
1. Introduction 29
2. The Input Deck 29
3. Closure 34
4. Achieving a Good Design 35
5. Boundary Layer Correction 37
6. Error Messages 38
7. Glossary of TAPE7 Parameters 39
8. Glossary of Output Parameters 43
VII. PLOTS AND TABLES OF RESULTS 45
1. Airfoils Designed Using the New Code 45
2. Data from Analysis and Experiment 72
VIII. FORTRAN LISTINGS OF THE CODES 90
1. The New Design Code K 90
2. Update of the Analysis Code H 167
I, INTRODUCTION
1. Shockless Airfoils and Supercritical Wing Sections
Supercritical wing technology is expected to have a significant
influence on the next generation of commercial aircraft. Computation
al fluid dynamics has played a central role in the development of new
supercritical wing sections. One of the principal tools is a fast and
reliable code that simulates two-dimensional wind tunnel data for
transonic flow at high Reynolds numbers (see Volume II). This is used
widely by industry to assess drag creep and drag rise.
Codes for the design of shockless airfoils by the hodograph method
have not been so well received because they usually require a lot of
trial and error (see Volume I). However, a more advanced mathematical
approach makes it possible to assign the pressure as a function of the
arc length and then obtain a shockless airfoil that nearly achieves
the desired distribution of pressure [11]. This tool should enable
engineers to design families of transonic airfoils more easily both
for airplane wings and for compressor blades in cascade.
There are plans to use the supercritical wing on commercial air
craft to economize on fuel consumption by reducing drag. Computer
codes have served well in meeting the demand for new wing sections.
This work is an example of the possibility of replacing routine wind
tunnel tests by computational fluid dynamics.
An effective approach to the supercritical wing is through shock
less airfoils. An advanced design code implementing the concept of
designing a shockless airfoil so that its pressure distribution very
nearly takes on prescribed data has been written recently. It has
turned out to be so successful that we hope it may ultimately gain the
same acceptance as the better established analysis code.
In this book we shall describe the new design code in detail and
we shall give an update of the analysis code, which incorporates new
2
features decreasing the execution time. Fortran listings are includ-
ed, together with directions for running the codes. A selection of
examples and comparisons with experiment complete the work. They
include shockless airfoils in cascade suggesting a new technology for
turbomachinery that may contribute significantly to energy conserva-
tion.
2. Differential Equations of Gas Dynamics
The partial differential equation for the velocity potential ~
describing isentropic flow of a compressible fluid can be derived from
a variational principle asserting that the integral over the flow
field of the pressure p, considered to be a function of the speed
q = Iv~l, is stationary in its dependence on ¢. The variational prin-
ciple is useful in the formulation of finite element and finite
difference methods. Here it will suffice to state the equation for
~ in the quasilinear form
2 2 2 2
(c -u l¢xx - 2uv ~xy + (c -v l~yy 0
for plane flow, where u = ~ ,v = ~ and c is the speed of sound
x y
2
related to q by Bernoulli's law
const.
The normal derivative of ~ is set equal to zero at the boundary of
the flow.
For many transonic flows of practical interest, including flows
around airfoils, we can assume entropy to be conserved across shock
waves, so the velocity potential can be retained in the formulation
of the equations of motion. Moreover, when shockless flows appear
they can often be viewed as physically relevant weak solutions of the
equations for which shock losses have been successfully eliminated.
We proceed to bring the equation for ~ into a canonical form
3
suitable for computation. The physical coordinates x and yare con-
nected with the velocity potential <P and the st.ream function 1jJ by the
relation
x + iy (d<p + i d1jJ )
p ,
where e is the flow angle and p is the density given by the equation
of state p = pY, or c2 ypY-l. We find it convenient to solve first
for <p and 1jJ and obtain x and y afterwards from this integral.
The ordinary differential equation for the characteristics, or
Mach lines, of the flow is known to be
In terms of <p and 1jJ it becomes
where M = q/c is the local Mach number. We introduce characteristic
coordinates sand n associated with the integrals of this equation
(see Volume I). In terms of sand n we obtain for <p and 1jJ the
canonical system
- i/l-M2 1jJ /p .
n
These partial differential equations will be integrated numerically by
a finite difference scheme of the form
4
Here '+ and, stand for suitable mean values of the coefficients
:f:. i/l_M2 /p, which become known functions of ~ and Tj ,,,,hen the corres-
ponding system of partial differential equations for the hodograph
variables u and v is solved in closed form. In Volume I it has been
shown how the finite difference scheme can be implemented for sub-
sonic as well as supersonic flow through analytic extension into the
complex domain.
II. THE METHOD OF CHAIU\CTERISTICS
CO~1PLEX
1. A New Boundary Value Problem
We have seen that physically realistic transonic flow computations
can be based on partial differential equations for the velocity poten-
tial and stream function that presuppose conservation of entropy.
In terms of characteristic coordinates ~ and n we have
=
cP n T - 1jJ n •
The coordinates ~ and n can be specified in terms of the speed q and
the flow angle S by the formulas
J J
log f (~) Il-r.12 d~ - is log f (ii) = Il-M2 d~ + is ,
where f is any complex analytic function. Prescription of a second
arbitrary function g serves to determine cp and 1jJ as solutions of the
characteristic initial value problem
where ~O = nO is a fixed subsonic point in the complex plane. With
these conventions it turns out that 1jJ(~,n) = 1jJ(~,ii) , as can be seen
from the uniqueness of the solution. Hence for subsonic flow the real
hodograph plane corresponds to points in the complex domain where
~ = ii. To calculate cp and 1jJ paths of integration are laid down in the
complex plane, and then a stable finite difference scheme is applied
to solve the characteristic initial value problem (see Volume I).
Consider the nonlinear boundary value problem of designing an air-
foil on which the speed q has been assigned as a function of the arc
length s. To construct such an airfoil it is helpful to view f as a
function mapping the unit circle I~I < 1 onto the region of flow.
Then both log f and g have natural expansions as power series