Table Of Content-
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
TWO DIMENSIONAL BOUNDARY SURFACES
-
FOR
PLANAR EXTERNAL TRANSONIC FLOWS
by
Aharon Salama
MARCH, 1992
Thesis Advisor: 0. Biblarz
Approved for public release; distribution is unlimited
v_/111.Kinn11n_u
SECURITYCLASSIFICATIONOF THIS PAGE
REPORT DOCUMENTATION PAGE
Id REPORTSECURITYCLASSIFICATION lb RESTRICTIVE MARKINGS
Unclassified
2d SECURITYCLASSIFICATIONAUTHORITY 3 DISTRIBUTION/AVAILABILITYOF REPORT
Approvedfor publicrelease;distributionisunlimited
'b DECLASSIFICATION/DOWNGRADINGSCHEDULE
4 PERFORMINGORGANIZATION REPORTNUMBER(S) 5 MONITORINGORGANIZATION REPORT NUM6ER(S,
6a NAME OF PERFORMINGORGANIZATION 6b OFFICE SYMBOL 7a NAME OF MONITORINGORGANIZATION
NavalPostgraduateSchool (Ifapplicable) Naval PostgraduateSchool
AA
6c ADDRESS(City,State,andZIPCode) 7b ADDRESS(Oty,State,andZIPCode)
Monterey,('A 93943-5000 Monterey,CA 93943 5000
8a NAMEOFFUNDlNo/SPONSORlNG 8b OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER
ORGANIZATION (Ifapplicable)
8c ADDRESS(City. Slate andZIPCode) 10 SOURCE OFFUNDING NUMBERS
^grdmtlemenlNo KluH".!t«. »Voi
11 TITLE (IncludeSecurity Classification)
TWO DIMENSIONAL IiOUNlMKY SURFACES FOR PLANAR EXTERNAL TRANSONIC FLOWS
12 PERSONA! AUTHOR(S) AHARON SAI.AMA
13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (year, month, day) 15 PAGE COUNT
Master'sThesis From li MARCH 1992 67
16 SUPPLEMENTARY NOTATION
Theviewsexpressed inthisthesisarethoseol theauthoranddonoi reflecttheofficial policyorpositionot the Department ol Defenseorthe U.S.
Government.
17 COSATICODES 18 SUBJECTTERMS(continueonreverseifnecessaryandidentifybyblocknumber)
FIELD GROUP SUBGROUP Small perturbation transonic equation, boundary surfaces.
19 ABSTRACT(continueonreverseifnecessaryandidentifybyblocknumber)
flu- small perturbation, two-dimensional transonic equation is manipulated with a separation of variables approach to obtain two
ordinary, nonlinear, differential equations Numerical integration of these differential equations results in new transonic
boundary surfaces for planar external (lows. A key ingredient in these solutions is the identification ofdependence of two integration
constants, alpha and beta, on the parameter ( 1 M-i. The anticipated behavior lor both Mach number and the pressure coefficient is
used as a guide in the actual selection ofthe adjustable constants in the problem. The physical reality of the boundary surfaces is
examined by displaying the boundary conditions they satisfy. The strictly sonic flow ( Mach= 1.0 ) has an analytic representation
corresponding to a divergent surface which goes supersonic. The sonic solution is compared with an Euler-CFI) approach
confirming the validity ofour results over the region where small perturbations apply. Solutions are also shown i'oi Mach= 0.8, l) \) 1.1
and 1 2. These results are consistent with known behavior for both subsonic and supersonii external flows. Since the results of
this work yield actual transonic contour.-,, we can examine shockless surfaces for design applications The possibility ol starting with
transonic surface is of interest to present day CF1) approach. Finally an entire transonic upper surface is presented for Mach=0.8,
by patching a subsonic Mach number, which reaches a plateau at M=1.0, with sonic flow. This patching requires the careful
interpretation ofa nondimensiunal reference length, called Y„, which is a function of Mach number
20 DISTRIBUTION/AVAILABILITYOF ABSTRACT 21.ABSTRACTSECURITYCLASSIFICATION
Q
UNClASSIIItL)UNllMilli.' J >AMtASKH'Om J LHICUSLK5 unclassilieu
22d NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (IncludeAreacode) 22c OFFICE SYMBOL
() Ihblarz (408) 646 3096 AA/Bi
DD FORM 1473. 84 MAR 83 APReditionmaybeused untilexhausted SECURITYCLASSIFICATIONOF THISPAGE
Allother editionsareobsolete Unclassified
T259149
Approved for public release; distribution is unlimited.
Two Dimensional Boundary Surfaces
for
Planar External Transonic Flows
by
Aharon §alama
Major, Israeli Air Force
Submitted in partial fulfillment
of the requirements for the degree of
MASTER OF SCIENCE
IN AERONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
MARCH
1992
Department of Aeronautical Engineering
ABSTRACT
The small perturbation, two-dimensional transonic equation
is manipulated with a separation- of -variables approach to
obtain two ordinary, nonlinear, differential equations.
Numerical integration of these differential equations results
in new transonic boundary surfaces for planar external flows.
A key ingredient in these solutions is the identification of
dependence of two integrations constants, a and 0, on the
parameter (1-M^2 The anticipated behavior for both the Mach
) .
number and the pressure coefficient is used as a guide in the
actual selection of the adjustable constants in the problem.
The physical reality of our boundary surfaces is examined by
displaying the boundary conditions they satisfy. The strictly
sonic flow (M =1.0) has an analytic representation
oo
corresponding to a divergent surface which goes supersonic.
This sonic solution is compared with an Euler-CFD approach
confirming the validity of our results over the region where
small perturbations apply. Solutions are also shown for
Moo= . 8 , . 9 , 1 . 1 , and 1.2 . These results are consistent with
known behavior for both subsonic and supersonic external flow.
Since the results of this work yield actual transonic
contours, we can examine shockless surfaces for design
applications. The possibility of starting with transonic
surface is of interest to present day CFD approach. Finally an
entire transonic upper surface is presented for M =0.8, by
oc
patching a subsonic Mach number, which reaches a plateau at
M=1.0, with a sonic flow. This patching requires the careful
interpretation of a nondimensional reference length, called
Y which is a function of M„.
,
in
//
C.I
TABLE OF CONTENTS
I. INTRODUCTION 1
II. THE TRANSONIC EQUATION 3
III. THE GOVERNING DIFFERENTIAL EQUATIONS 5
IV. NUMERICAL INTEGRATION 11
V. BOUNDARY SURFACES 15
VI. PRESSURE COEFFICIENT 21
VII. THE LOCAL MACH NUMBER 23
VIII. BOUNDARY CONDITIONS 25
IX. CONCLUSIONS 31
LIST OF REFERENCES 32
APPENDIX A: COMPUTER PROGRAMS 33
APPENDIX B: FORM OF AND 37
OL /3
APPENDIX C: FIGURES 39
INITIAL DISTRIBUTION LIST 60
IV
