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AD-A038 614 GENERAL ELECTRIC CO CINCINNATI OHIO AIRCRAFT ENGINE GROUP F/6 20/1 
SUPERSONIC JET EXHAUST NOISE INVESTIGATION. VOLUME III. COMPUTE=<ETC(U) 
JUL 76 D R FERGUSONe M A SMITH, P R KNOTT F33615-73-C-2031 
UNCLASSIFIED R74AEG452-VOL=-3 AF APL=TR-76-68-VOL-3 








WES WES ee 





AD No.—— 


ADAOQ38614 


NDE FILE COPY. 











AFAPL-TR-76-68 





SUPERSONIC JET EXHAUST NOISE INVESTIGATION 


Volume Ill 
COMPUTER USERS MANUAL FOR AEROACOUSTIC 
PREDICTIONS 


GENERAL ELECTRIC COMPANY 

AIRCRAFT ENGINE GROUP 

ADVANCED ENGRG. AND TECH. PROGRAMS DEPT. 
CINCINNATI, OHIO 45215 


JULY 1976 





TECHNICAL REPORT AFAPL-TR-76-68 
FINAL REPORT FOR THE PERIOD 1 DECEMBER 1972 - 23 SEPTEMBER 1975 


[_ Approved for public release; distribution unlimited 











AIR FORCE AERO PROPULSION LABORATORY 

AIR FORCE SYSTEMS COMMAND 
WRIGHT-PATTERSON AIR FORCE BASE, OHIO 45433 
and : ; 
DEPARTMENT OF TRANSPORTATION 

OFFICE OF NOISE ABATEMENT 

WASHINGTON, D.C. 





NOTICE 





When Government drawings, specifications, or other data are used for any 
purpose other than in connection with a definitely related Government pro- 
curement operation, the United States Government thereby incurs no responsibility 
nor any obligation whatsoever; and the fact that the Government may have 
formulated, furnished, or in any way supplied the said drawings, specifications, 
or other data, is not to be regarded by implication or otherwise as in any 
manner licensing the holder or any other person or corporation, or conveying 
any rights or permission to manufacture, use, or sell any patented invention 
that may in any way be related thereto. 


This report has been reviewed by the Information Office, ASD/OIP, and is 
releasable to the National Technical Information Service (NTIS). At NTIS, 
it will be available to the general public, including foreign nations. 


This technical report has been reviewed and is approved for publication. 


eget Pahl dd banman 
PAUL A. SHAHADY, DR. GORDON BANERIAN 


Project Engineer Project Engineer 
USAF Department of Transportation 


FOR THE COMMANDER 


ROBERT E. HENDERSON 
Manager, Combustion Technical Area 





Copies of this report should not be returned unless return is required by 
pedal considerations, contractual obligations, or notice on a specific 
ocument. : 








UNCLASSIFIED 
SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered) 


| __(/ 7) REPORT DOCUMENTATION PAGE READ | ee ete | 


BEFORE ea FORM 


GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 
pa wamlpereconvonm=sP P| 


Os 4 TITLE (and | Subtitle) 3. TYPE OF REPORT & PERIOD COVERED 


=f Supersonic Jet Exhaust Noise Ttaskioation., Technical Report (Final) 
Volume TIITe Computer User's Manual for Aero- 1 Dec 1972 - 23 a 1975 
= ) Acoustic Predictions» 1 : A 





AU THOR(=) * ° : 
PP) LA AS \ stad 
V/ ) David R, /Ferguson, Wachee A. /smi tn} and Os FSEGIS-73-6-aph | 
“/ Paul R./Knott 
7, - 


9 PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK 
General Electric Company Pp JE, 62203F, Project. 3066 

Aircraft Engine Group (AE&TPD) Task 306614, W. ; 
Cincinnati, Ohio 45215 U. 30661407 


" CONTROLLING OFFICE NAME AND ADORESS 12 


T DATE B 
Air Force Aero Propulsion Lab (TBC) VD, ULG_ 1976 
Wright-Patterson Air Force Base, Ohio 45433 


MONITORING AGENCY NAME & ADDRESS(s4-dislerent from Controlling Office) | 18. SECURITY CLASS. (of thie report) 


Y ’ ; , | ¢. echnical r« er Unclassified 
f f 


oF A 7 Sep i (Sa DECL ASSIFICATION/ DOWNGRADING 
t D = ae ~) SCHEOULE 


1QPERS 


“ a pad 
Approved tor Public Release; _ UY y, nlimited. 


1/6) SWEGE 


t7, OLSTRIBUTION STATEMENT (of the abstract entered in Block 20, If different trom Report) 


is OISTRIGUTION STATEMENT (of thie Report) 





On eee ak se report is Volume III of a four-volume Final] Technical 


Keport prepared by the Advanced Engineering and Technology Programs Department 
Aircraft Engine Group of the General Electric Company, under the joint sponsor 


ship of the Air Force Aero Propulsion Laboratory, pe Sg Air Force 
Base, Ohio, and the Department of Transportation, Washington, D.C. 


19 KEY WORDS (Continue on reverae aide if necessary and identify by block number) 


Supersonic Jet Noise, Far-Field Jet Noise, Near-Field Jet Noise, Aeroacoustics 
Acrodynamics of Subsonic/Supersonic Jets, Shocks. 


ABSTRACT (Continue on raverae aide il neceseaty and identity by block number) 

This report gives a detailed description of aerodynamic (Shock-free/Shocked 
flow) and acoustic turbulent mixing computer prediction programs developed by 
the General Electric Company for subsonic and supersonic simple exhaust jets. 
In addition to giving detailed descriptions of the aeroacoustic formulations 
and disesssions of computer manual instructions for operating the program, 
extensive theory/data comparisons are given, as well as computer program 


listings and sample test cases, 
\ 


DD 700%, 1473 Eortion oF | Nov 651s OBsoLETE UNCLASSIFIED 


ae en 
SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered) 


Kos YC S 











FOREWORD 


This is Volume III, The Computer Users Manual, of the final Technical 
Report prepared by the Advanced Engineering and Technology Programs Department, 
Aircraft Engine Group of the General Electric Company, Evendale, Ohio under 
the joint sponsorship of the Air Force Aero-Propulsion Laboratory, Wright- 
Patterson Air Force Base, Ohio and the Department of Transportation, Washington, 
D.C. under Contract F33615-73-C-2031. The inclusive dates for this work 
were December 1972 through August 1975. The work was accomplished under 
Project 3066, Task 14, Work Unit 07, with Mr. Paul A. Shahady (AFAPL/TBC) 
as Project Engineer. Dr. Paul R. Knott of the General Electric Company was 
technically responsible for the work. Other General Electric personnel were: 
Mr. David R. Ferguson, Applied Computer Methods, and Mr. Michael A. Smith, 
Acoustic Engineer. Additionally, Drs. C. L. Merkle and C. Y. Chen contributed 
to the work but, are no longer at the General Electric Company. 


iii PRECEDING my 


Section 


1.0 


2.0 


TABLE OF CONTENTS 


VOLUME III 


Page 

INTRODUCTION 473 
PROGRAM DESCRIPTION 474 
1.1 General Description 474 
1.2 JETMIX 474 
Ee. SSFD 476 
1.4 MERGE 478 
1.5 NOISE 478 
1.6 SSNOISE Calculation Steps and Flow Chart 479 
1.7 SSNOISE overlay Description 482 
Peter Overlay (0,0) - Entry MAIN 482 
Levad Overlay (1,0) - Entry JETMIX 482 
1.7.3 Overlay (1,1) - Entry JETMIX 482 
1.7.4 Overlay (1,2) - Entry JETMIX 484 
LeteS Overlay (2,0) - Entry SSFD 484 
1.7.6 Overlay (2,1) - Entry SSFDIN 484 
Boker Overlay (2,2) - Entry SSFDCA 484 
1.7.8 Overlay (3,0) - Entry MERGE 484 
1.7.9 Overlay (4,0) - Entry MAINNS 485 
PROGRAM INPUT 486 
2.1 General Description 486 
2.2 JETMIX Input 486 
2.2.1 Problem Specification 487 
2.2.2 Description of Primary and Secondary Jets 487 
2.2.3 External Boundary Conditions 488 
2.2.4 Fluid Properties 489 
2.2.5 Step-Size Controls/Restart 489 
2.2.6 Turbulence Scale Calculations 490 
2.2.7  Coannular/Coplanar Jet 491 
2.2.8 Transverse Mesh Control 495 
2.2.9 Species Diffusion Input 495 
2.2.10 Station Input 496 

2.3  SSFD Input 498 
2.3.1 Problem Description 498 
2.3.2 Program Controls 499 
2.3.3  Total-Pressure Input Stations 499 

2.4 MERGE 501 


~ 


PRE G 
CEDING PAGE fPELANK NOT FILMED 








TABLE OF CONTENTS (Concluded) 


Section Page 
2.5 NOISE Input 501 

2.5.1 General Input 502 

2.5.2 Acoustic Model Selection 503 

2.5.3 Microphone Selection 504 

2.5.4 Output Control Cards 504 

3.0 PROGRAM OUTPUT 505 
3.1 General Description 505 

3.2 JETMIX Program Output 505 

3.3 SSFD Program Output 508 

3.4 MERGE Program Output 509 

3.5 NOISE Program Output 509 

3.5.1 One-Third Octave Band Analysis 509 

3.5.2 Summary Acoustic Analysis 510 

3.5.3 Summary Acoustic Power Analysis eit 

3.6 Sample Case EL 

4.0 OPERATING PROCEDURES 515 
4,1 General Description 515 

4.2 Maintenance and Modification SLS 

4.3 Input and Output Files B15 

APPENDIX 6 - Analysis Incorporated in JETMIX 519 

APPENDIX 7 - Analysis Incorporated in SSFD/MERGE 549 

APPENDIX 8 - Analysis Incorporated in Noise 582 

APPENDIX 9 - Input Sheets 657 

APPENDIX 10 - Sample Cases 669 

APPENDIX 11 - Program Listings 902 
REFERENCES 1149 

vi 





LIST OF ILLUSTRATIONS 


VOLUME III 
Figure Page 

246, Schematic of General Electric Jet Aeroacoustic Computational 

Procedure. 480 
247. SSNOISE Calculation Flow Chart. 481 
248, Overlay Structure, Including All Subroutines. 483 
249, Subsonic Turbulent Jet. 520 
250. Supersonic Turbulent Jet. 521 
25k 6 Jet Plume Predictions, Free Turbulent Mixing Analysis, 

X/D = 1.145. 532 
252. Jet Plume Predictions, Free Turbulent Mixing Analysis, 

X/D = 3.8. 533 
ao Jet Plume Predictions, Free Turbulent Mixing Analysis, 

X/D = 6, 534 
254. Jet Plume Predictions, Free Turbulent Mixing Analysis, 

X/D = 3.8. 535 
255. Free Turbulent Mixing Data Comparisons, Total Temperature 

Decay Along Jet Axis. 536 
256. Free Turbulent Mixing Data Comparisons, Total Temperature 

Decay Along Jet Axis. 537 
Loh Free Turbulent Mixing Data Comparisons, Velocity Decay Along 

Jet Axis. 538 
258. Free Turbulent Mixing Data Comparisons, Mach Number Decay 

Along Jet Axis. 539 
259. 4.3" Conical Nozzle Exhaust Plume Total Temperature Versus 

Radius, JENOTS Wake Rake Data, L/Dg = 2.79. 540 
260. 4.3" Conical Nozzle Exhaust Plume Total Temperature Versus 

Radius, JENOTS Wake Rake Data, L/Dg = 19.53. 541 
261. 4.3" Conical Nozzle Exhaust Plume Velocity Versus Radius, 

JENOTS Wake Rake Data, L/Dg = 2.79. 542 


vii 


Figure 
262. 


263. 


264. 


265. 


266. 


267. 
268. 
269). 


270. 


271. 


py a 
273. 


274, 


275. 
276. 
277. 
278. 


279. 


LIST OF ILLUSTRATIONS (Continued) 


4.3" Conical Nozzle Exhaust Plume Velocity Versus Radius, 
JENOTS Wake Rake Data, L/Dg = 19.53. 


4.3" Conical Nozzle Exhaust Plume Mach Number Versus Radius, 
JENOTS Wake Rake Data, L/Dg = 2.79. 


4.3" Conical Nozzle Exhaust Plume Mach Number Versus Radius, 
JENOTS Wake Rake Data, L/Dg = 19.53. 


4.3" Conical Nozzle Exhaust Plume Total Pressure Versus 
Radius, JENOTS Wake Rake Data, L/Dg = 2.79. 


4.3" Conical Nozzle Exhaust Plume Total Pressure Versus 
Radius, JENOTS Wake Rake Data, L/Dg = 19.53. 


The Flow Field of an Ideally Expanded Viscous Set. 


The Flow Field of an Inviscid Two-Dimensional Supersonic Jet. 


Subdivision of the Jet into Inner and Outer Regions. 


Calculation of Boundary Conditions and Shock Strength and 
Location by Method of Characteristics. 


Method for Obtaining Fictitious Shock-Corrected Properties 
for Use in Calculating Field Points Adjacent to Shocks. 


Mach Disk Model. 
Passage of Turbulence Through a Shock. 


Amplification of Turbulence by a Shock Wave as Predicted by 
Ribner's Shock-Turbulence Theory. 


Outline of Complete Computational Procedure. 

Predicted Shock Shape and Outer Boundary Shape --- Inviscid. 
Intersection of Shock Shape with Axis ot Symmetry. 

Height of Mach Disk. 


Shock Shape Prediction on Schlieren Photograph. 


viii 


uw 
uo 


546 











Figure 


280. 


281. 


283. 


284. 


285. 
286. 
287 


288. 


289, 


290. 


291. 


292. 
293. 


294, 


LIST OF ILLUSTRATIONS (Continued) 


Cross-Stream Variation of Predicted Total Pressure and Static 
[Py = 
amb ref 


Pressure for Ideally Expanded Jet, M_,. = 1.60, P 
0.242. ws 


Cross-Stream Variation of Predicted Total Pressure and Static 
Pressure for Slightly Underexpanded Jet, Showing Total-Pres- 


sure Loss Due to Both Shocks and Mixing, M, = 1,25, 


= xit 
batt fece = 0.242, 


Cross-Stream Variation of Predicted Total Pressure and Static 
Pressure for Jet from Convergent Nozzle, Showing Total-Pres- 


sure Loss Due to Both Shocks and Mixing, M_,. = 1.0, 
PP /Pr__. = 0.242, ean 

amb ref 
Comparison Between Predicted Shock Shapes Using Inviscid 
Prediction Technique and Coupled (Inner-Outer Analysis) 
Viscous Technique, Shaded Region Represents Flow which is 


Significantly Affected by Mixing, M_., = 1.25, 
P /Pr_ . = 0.242, pe 
amb ref 
Shock Structure Theory Data Comparison, P5/Pamb = 2,1; 
M = 1.0, P flo = Onehs 
exit amb T 


Mean Velocity and Turbulent Velocity on Centerline. 
Prediction of Overall Power Level. 
Cold Jet Power Spectrum Prediction Measurement Comparisons. 


Hot Jet Power Spectrum Theory Data Comparisons for a 
Shock-Free Supersonic Jet. 


Cold Jet Power Spectrum Prediction Measurement Comparisons, 
2.17-Inch Cold Jet. 


Cold Jet Power Spectrum Prediction Measurement Comparisons, 
2-Inch Cold Jet. 


1/3-Octave-Band Directivity Patterns for a Cold Supersonic 
Jet. 


Predicted Power Distribution for a Cold Subsonic Jet. 


Predicted Powgr Distribution for a Cold Supersonic Jet. 


Directivity Patterns of Overall Sound Pressures for Hot 
Jets at a Distance of 320 Feet from the Jet Axis. 


ix 


Page 


576 


uo 
~I 
“I 


wo 
P<) 
“I 


629 


638 











Figure 
295. 


296. 


ye Te 


298. 


299. 


LIST OF ILLUSTRATIONS (Concluded) 


Hot Jet Sound Pressure Spectra at Various Angular Positions 
at a Distance of 320 Feet from the Jet Axis (D=45.6 inches, 
M, = 0.83). 


Near~Field Microphone Locations for a Full-Scale Jet Engine 
(D = 43 inches). 


Near~Field Sound Pressure Spectra at Various Microphone 
Locations (An Internal Turbulence Level, u'/Ug = 0.15, Was 
Assumed for the Calculations), Microphones 23 and 24. 


Near~Field Sound Pressure Spectra at Various Microphone 
Locations (An Initial Turbulence Level, u'/Ug = 0.15, Was 
Assumed for the Calculations), Microphones 1-4 and 27, 


Near~Field Sound Pressure Spectra at Various Microphone 
Locations (An Initial Turbulence Level, u'/Ug = 0.15, Was 
Assumed for the Calculations), Microphones 7 and 8. 


Near-Field Microphone Locations for a Scale Model Jet 
(D = 4.55 inches). 


Near-Field Sound Pressure Spectra at Various Microphone 
Locations. 


Comparison of Measurements and Predictions by Composite 
Acoustic Models. 


639 


641 


642 


643 


646 


647 


648 


651 





10. 





LIST OF TABLES 
VOLUMF. LIT 
Shear- and Self-—Noise Table of Experimentally 
Determined Constants, 
Fluid Shrouding for Unheated Jets. 


Fluid Shrouding for Heated Jets — Influence of 
Velocity. 


Fluid Shrouding for Heated Jets - Influence of 
Temperature. 


613 


617 





INTRODUCTION 


During the course of this Supersonic Jet Exhaust Noise Investigation, 
the General Electric Company has developed a number of computational aero- 
dynamic and acoustic predictive schemes. These computational methods were 
developed primarily to characterize key aerodynamic and acoustic features of 
unheated and heated subsonic and supersonic simple circular jets. Extensive 
accounts of these methods have been documented earlier. Some of the key 
documents are AFAPL-TR-72-52, AFAPL-TR-74-25, J. of S&V 25 (1972), ALAA 
73-188, Proceedings of HT&FMI (1974), as well as the results presented in 
Volume II of this final report. This report is a computer user's manual for 
the developed programs. 


The overall computer system is called the "Supersonic Jet Noise Pre- 
diction System" (SSNOISE). It is composed of four programs: JETMIX - 
an aerodynamic program for planar and axisymmetric compressible turbulent 
mixing jets. It also contains options for confined jet mixing, a coannular/ 
coplanar free jet, the influences of external flow on jet mixing, and 
composition of chemically inert species throughout the flowfield. SSFD - 
an inviscid analysis of supersonic turbulent jets including shock waves. 
MERGE - a collation program of the aerodynamic output of JETMIX and SSFD. 
NOISE - predictive schemes for turbulent mixing aerodynamic noise. 


Section 1.0 contains the general program concepts and descriptions of 
JETMIX, SSFD, MERGE, and NOISE. Section 2.0 discusses the detailed input 
requirements of the Supersonic Jet Noise Prediction Programs (SSNOISE), 
while Section 3.0 describes the output and a sample case. Section 4.0 
discusses the general operating procedures. Following Section 4.0, there 
is a series of appendices which are dedicated toward the mathematical 
analysis incorporated in SSNOISE, as well as a series of input sheets and 
program listings. The last enclosure is the Reference listing. 


173 








SECTION 1.0 


PROGRAM DESCRIPTION 


1.1 GENERAL DESCRIPTION 


The Supersonic Jet Noise Prediction System (SSNOISE) provides an analy- 
tical tool for determination of the aerodynamic flow field and the associated 
aerodynamic noise of supersonic turbulent jets. The SSNOISE system is com- 
posed of four programs which may be either run alone or in sequence. They 
are: 


12 JETMIX ~ Analysis of Free and Confined Jet Mixing 

Pag SSFD -—~ Inviscid Analysis of Supersonic Turbulent Jet Including 
Shock Waves 

Die MERGE ~ Collation of the Aerodynamic Output of JETMIX and SSFD 

4, NOISE ~ Prediction of Aerodynamic Jet Noise 


A functional description of each of these programs is included in the 
following sections. Also included is a detailed flow chart of the data flow 
in the system and a description of the overlay structure on the CDC 6400/6600 
computers. 


1.2 JETMIX 


The JETMIX computer program is a general aerodynamic analysis tool for 
the prediction of both planar and axisymmetric compressible turbulent mixing 
regions. The flow field may consist of either a single free jet, a coannular/ 
coplanar free jet, or a confined jet which is constrained to mix with a co- 
annular flow inside a duct of varying cross-sectional area. In addition to 
determining the aerodynamic properties of the mixing region, the JETMIX pro- 
gram also traces the rate of diffusion of chemically inert species throughout 
the flow field. 


The transport of momentum, energy, and individual species by turbulence 
effects is included in the analysis by means of a semiempirical turbulence 
model which relates the turbulent transport terms in the time-averaged equa- 
tions to the amount of kinetic energy which is contained in the turbulent 
eddies. This turbulence kinetic energy (TKE) is calculated, in turn, from a 
conservation equation which keeps track of the net effect of the production 
of turbulence energy by the local shearing forces in the flow, as compared 
with the decrease of turbulence energy by dissipation into heat, by diffusion, 
and by the streamwise convection of the turbulence energy with the time- 


474 





ee 





averaged flow field. It is important to note that the use of the turbulence 
kinetic energy in the turbulence model causes the turbulent transport terms 
to depend not only on the local flow properties (and their gradients) but 
also upon the history of the flow. 


This turbulence kinetic energy approach for modeling the turbulent 
effects is to be contrasted with the pure mixing length theories. Like the 
TKE approach, mixing length theories (such as G.I. Taylor's vorticity trans- 
fer theory and Van Karman's similarity hypothesis) have also yielded viable 
results when properly correlated with experimental data. However, the exten- 
sion of these theories over a wide range of flow conditions using a single 
empirically determined mixing length has not proven feasible. This lack of 
generality is partially due to using local flow properties to determine the 
local viscous transport effects in a turbulent flow where advection and dif- 
fusion are important. There is no flow history incorporated in the conven- 
tional mixing length models; the prior development of the flow is ignored. 


Recently, statistical approaches for handling turbulence have been con- 
sidered by Townsend (Reference 148), Hinze (Reference 149), Rotta (Reference 
3), and several others. The analysis of homogeneous isotropic turbulence can 
be handled quite readily by the statistical techniques, since the turbulent 
flow properties have a random distribution that tends to be Gaussian. The 
statistical methods have not proved tractable for nonisotropic turbulence in 
flows with large velocity gradients. Unfortunately, free turbulent mixing in 
jet plumes is characterized by large velocity and temperature gradients with 
nonisotropic turbulence. Consequently, the present analytical method was 
developed to incorporate a phenomenological approach in which the results 
were stated in measurable engineering quantities. At the same time, concepts 
from the statistical methods were utilized to develop universal models for 
the viscous transport properties. This led to adopting the turbulent kinetic 
energy (e) as another flow variable in the analysis. The inclusion of the 
conservation of turbulence energy permits the turbulence history to be moni- 
tored along with the local generation and decay. 


The turbulent kinetic energy can be related to the local turbulence in- 
tensity by assuming local isotropy (Reference 150). The analysis thus relates 
the turbulent mixing to the mean velocity and the turbulence intensity. The 
turbulence intensity represents the standard deviation of the random turbu- 
lent velocity distribution. Thus, the statistical nature of the flow is 
retained by utilizing representative population parameters, but the analysis 
is tractable to numerical solution. 


General Electric analysis is based on the work of Spalding and Patankar 
(Reference 5) as well as Bradshaw (Reference 151). Prior to the development 
of this plume-mixing model, the turbulent kinetic energy model had already 
been proven to be a valid approach for viscous flow in boundary layers 
(Reference 152). <A similar model for free jets is also being used by Harsha 
(Reference 153). 


475 





= 





The JETMIX analysis uses a numerical, finite difference technique to 
solve the partial differential form of the conventional boundary layer 
equations of motion. The use of the partial differential equations them- 
selves, plus the use of local turbulence properties for the determination 
of the Reynold's stresses, allows the JETMIX analysis to apply equally well 
to the case of a free jet mixing with a static ambient environment or with a 
moving external stream. For example, for the case of laminar flow (which 
can be calculated by "turning off" the turbulence calculations), the JETMIX 
program yields the "exact" solution to the constant pressure mixing problem 
regardless of the velocity of the external stream. Similarly, the only 
approximation in the turbulent mixing problem lies in the semiempirical han- 
dling of the turbulence, not in the mathematical analysis. Nevertheless, 
since the JETMIX program monitors the local turbulence properties in the 
mixing region, this turbulence model can be expected to apply over a rela- 
tively wide range of conditions. Specifically, one would expect that the 


local mechanisms for the production and dissipation of turbulence would be 
unchanged by the velocity, temperature, pressure, or composition of the 
external flow in which the jet is placed (although the actual production and 
dissipation would, in general, be quite sensitive to the jet environment). 
Comparison of JETMIX predictions with experimental data over a wide range of 
mixing conditions including changes in all of these parameters has verified 


this speculation. 


Finally, it should be noted that the JETMIX solution yiel¢* the local 
velocities, enthalpies, concentrations, and densities at ea: it » the 
flow field. Thus, once the JETMIX solution has been obtained, both the 
stream-wise variations and the cross-stream variations of all the sigmificant 


parameters are available. 


The JETMIX program is included in its full generality im the SSNOISL 
system, even though the acoustic model requires only a simple aerodynami 
flow field for a single free turbulent jet. The use of the turbulence inten- 
sity and the mean flow profiles for noise prediction is described in 
Section 2.4. 


1.3 SSFED 


The aerodynamic characteristics of a supersonic exhaust jet may resemble 
either an inviscid two-dimensional flow field or a viscous mixing region, 
depending on the back pressure to which the jet exhausts. In the case where 
the back pressure is substantially different from the static pressure at the 
nozzle exit, the characteristic features of the jet are similar to an invis- 
cid, two-dimensional flow field. This pressure mismatch generates an expan- 
sion fan or an oblique shock wave at the nozzle lip which adjusts the flow to 
the ambient pressure. However, as this initial disturbance propagates through 
the jet and reflects from the centerline, it causes the pressure to be 
overcorrected. This overcorrection represents the beginning of the familiar 
(nearly) periodic shock-cell structure. Quite naturally, the effects of 
friction are not altogether absent, but rather are restricted to a narrow 


476 








region on the outer edge of the jet. Because the viscous effects are confined 
to such a narrow region, the dominant characteristics of the flow field can 
be predicted by means of an inviscid two-dimensional analysis. 


By way of contrast, if the nozzle is just precisely matched to the ambi- 
ent pressure, no expansions or compressions take place at the nozzle exit, 
and the inviscid theory predicts a trivial, uniform, parallel flow field. 
However, experimental observations indicate that the flow field of this 
ideally expanded jet is dominated by viscous mixing in much the same way as 
are subsonic jets. Such a flow field can be predicted by means of a viscous, 
boundary layer analysis which assumes that the pressure throughout the entire 
plume is constant and is impressed by the external environment. 


Despite the fact that each of the above classical approaches has applica- 
tion by itself, there are many problems which require the simultaneous inclu- 
sion of both two-dimensional and viscous mixing effects. For example, super- 
sonic jets are seldom uniform, parallel, ideally expanded jets. Further, 
even though the dominant characteristics of the flow field of the nonideally 
expanded jet may be calculable by means of an inviscid analysis, there may be 
an interest in the mixing region itself. This latter situation is particu- 
larly true for such contemporary uses of jet flow field calculation techniques 
as prediction of the acoustic radiation, the infrared signature, or the 
combustion emissions of the jet. 


A technique has been developed for calculating the flow field in a super- 
sonic jet which includes both the two-dimensional effects which are character- 
istic of supersonic flow fields as well as the viscous mixing losses which are 
caused by turbulent stresses in the highly sheared regions of the jet. This 
technique divides the flow into an "inner" region in which the flow is super- 
sonic and nearly inviscid, and an "outer" region where the viscous forces 
predominate. Coupling between the two regions is provided by including the 
viscous losses as known "right-hand-side" terms in the solution of the inner 
equations. Provision for handling discrete shock waves is included in the 
equations for the inner region, and an approximate technique which allows the 
shock to reflect at the centerline of an axisymmetric jet is also included. 
The viscous analysis of the “outer"™ region is referred to as the JETMIX com- 
puter program (Section 1.2). The analysis of the supersonic, nearly inviscid 
"inner" region is referred to as the Supersonic Finite Difference (SSFD) com- 
puter program. Other approaches to this problem have been considered in 
References 154 and 155, although their analyses were restricted to totally 
supersonic flow fields. 


Since the flow in the inner region is supersonic, the (hyperbolic) 
governing equations could be solved by the classical method of characteristics 
procedure. However, an explicit finite difference algorithm is used since 
it is somewhat simpler and is more easily adaptable to matching between the 
"inner" and "outer" solutions. Section 2.1 points out that the (parabolic) 
boundary layer equations characterizing the outer region are solved by a 
similar finite difference technique. Although the field points are calcu- 
lated by a finite difference technique, the enforcement of boundary conditions 
and calculation of shock propagation are accomplished with method of charac- 
teristics procedures. 


477 





* 


The equations for the inner region require that the "“right-hand-side" 
terms, reflecting variations in entropy and stagnation enthalpy due to the 
action of turbulent stresses, must be specified before a solution is calcu- 
lated. Thus, the JETMIX analysis must first be applied to the entire jet in 
order to allow approximate calculation of these terms. Then, the SSFD 
analysis if used to recalculate the flow field in the inner region. 


1.4 MERGE 


The MERGE computer program is used to combine the output of JETMIX and 
SSFD prior to its use in the NOISE program. Specifically, the velocity and 
turbulence energy profiles computed in the SSFD program for the supersonic 
portion of the jet plume override those computed by JETMIX. The JETMIX 
profiles in the subsonic portion of the plume are retained. 


1.5 NOISE 


In the previous sections, program descriptions were given for the aero- 
dynamic flow field predictive schemes for subsonic and supersonic (shock-free 
and shocked) exhaust jets. One of the key aspects of these aerodynamic 
programs, from a noise point of view, is that they are turbulent mixing 
analyses of which a primary output is the turbulent kinetic energy. With 
the output from such programs, it was quite feasible to utilize acoustic 
models whose source terms were functions of the aerodynamic input, thus 
the name aeroacoustic turbulent mixing. General Electric has devoted time 
to examining the advantages and disadvantages of such models. References 
1, 10, 77, and 156 discuss many of the details of this work. Appendix 6 
gives a detailed account of the models available in this document. 


The basic turbulent mixing (or velocity fluctuation) noise theory 
which can best serve as an example to illustrate the relationships between 
the acoustic far-field and the detailed aerodynamic properties of an exhaust 
jet is the Lighthill/Ffowcs-Williams convected quadrupole theory. This 
theory is founded on the hypothesis that individual and uncorrelated volumes 
emit acoustic energy given by the relationship: 








pre Ve ry 7? a4 4 wk \2 ~5/2 
ae * a Free (1 = M, cos 0)” + Pn, 


where R is the distance between the source term of turbulence and the point 

of observation; Ve, the uncorrelated eddy volume; w, the radiation frequency 
of fluctuation in a reference system moving at an eddy convection speed 

Ve; TZ, the mean square value of the quadrupole strength; 6, the angle 

between the direction of sound emission and the jet axis; Mc the ratio of 

eddy correction speed to the sound speed of the ambient gas; and, &%, the 

scale of turbulence. With the assumptions Ve = 23, T2 = o2u4 and a frequency- 
eddy-shear assumption due to P.O.A.L. Davies that w%~l.lu', our acoustic 


478 


equation transforms into a quite tractable form directly linking the local 
aerodynamic properties with the acoustic far-field (or near-field as will 
be shown in Appendix 6): 





2 4-4 . -5/2 
—2 : eu u = 9)2 + l.lu 2 
P Trr2 past [ « M. cos 8) ( a0 ) 


To further translate this simple expression to perform detailed acoustic 
predictions, the flowfield is divided into small-volume elements in the form 
of circular ring elements of incremental volume V = 2nrArAdx (see Figure 246), 
based on the compactness assumption that each volume element is considered 
to be a sound generator which emits acoustic energy at a specified frequency 
determined by the P.O.A.L. Davies frequency-eddy-shear assumption (or some 
similar assumptions). For the whole jet then, the mean square sound pres- 
sures from all circular ring regions can be combined computationally to give 
overall mean square sound pressure, sound pressure spectra, overall sound 
power spectra, or overall power level. 


The NOISE program contained in this document includes, in addition to 
the model described above, several other philosophically similar turbulent 
mixing aeroacoustic models. They are Ribners, a classical Lilley model, 
and a model by Pao. 


An item which is important to the predictive aspects of aerodynamic 
noise, as well as for understanding jet noise emission, is how to account 
for the influence of the physical flowfield on the noise source itself. The 
work of Mani, described in detail in Volume II of this final report, shows 
that the fluid shrouding of the noise source goes a long way toward accounting 
for the convective/refractive coupling of the acoustic radiation. Appendix 
6 will discuss a method by which this technique can be linked with the above- 
cited turbulent mixing models. 


1.6 SSNOISE CALCULATION STEPS AND FLOW CHART 


The processing flow chart depicting the flow of data between the various 
programs in the SSNOISE system is shown in Figure 247. The core program in 
the system is JETMIX, which provides aerodynamic data to be used in SSFD, 
MERGE, or NOISE. The files shown in Figure 247 may be either disc or tape 
and contain the following data: 


File 1 - JETMIX Restart file (normally tape) 
Contains aerodynamic data from a partial execution of JETMIX 
(File 2). 


File 2 - JETMIX Output file 
Contains aerodynamic data from a JETMIX execution. Used as an 
input file to SSFD, MERGE, or NOISE. 


¢ 


479 

















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480 


File 1 - JETMIX Restart 

anieee 2 - JETMIX Output 
3 - SSFD Output 
4 - MERGE Output 


- ieee 


oe 


NOISE 


Figure 247. SSNOISE Calculation Flow Chart. 


481 





File 3 - SSFD Output file 
Contains aerodynamic data from a SSFD execution. Used as an 
input file to MERGE. 


File 4 - MERGE Output file 
Contains collated data from JETMIX and SSFD. Used as an input 
file to NOISE. 


Each program in the system may be either run alone or in sequence with 
other programs; viz, a JETMIX file that has been saved on tape and may be 
used to execute either SSFD or NOISE. 


1.7 SSNOISE OVERLAY DESCRIPTION 


The SSNOISE system has been structured for execution on the CDC €6400/ 
6600 machines under the SCOPE 3.3 operating system. The basic OVERLAY 
features of SCOPE 3.3 have been utilized to reduce the memory requirements 
to a tractable level (< 110K octal locations). Shown in Figure 248 is the 
overlay structure, including all subroutines. As indicated in this figure, 
the program consists of a main overlay, four primary overlays, and four 
subordinate secondary overlays. A brief description of the processing in 
each overlay is given in the following section. 


1.7.1 Overlay (0,0) - Entry MAIN 


The main overlay (0,0) contains the main program MAIN, as well as 
general purpose routines which are called by subprograms in the subsequent 
primary and secondary overlays. The program MAIN provides control for execu- 
tion of the four programs (JETMIX, SSFD, MERGE, NOISE) by loading the appro- 
priate overlays in sequence. 


1.7.2 Overlay (1,0) - Entry JETMIX 


Overlay (1,0) is the primary overlay for loading of the JETMIX input and 
calculation overlays (Program JETMIX). Subroutines concurrently used by 
secondary overlays also reside in (1,0). 


1.7.3 Overla 1,1) - Entry JETMIX 1 


Overlay (1,1) consists of the subroutines to read card and/or tape input 
into the JETMIX program. The latter situation may occur if a tape has been 
saved from a previous JETMIX execution and the user desires to restart the 
problem and run further downstream. The main input routine is JETINP which 
calls JTINIT for initialization, JIFILE for tape input, and JTOUT1 for print- 
ing of the input and initial profiles generated in JETINP. 


482 





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1.7.4 Overlay (1,2) - Entry JETMIX 2 


The bulk of the numerical calculations of the mixing flow field is 
carried out in Overlay (1,2). The primary entry, JETMIX 2, initially calls 
the control routine JTCTRL which, in turn, calls the main calculation 
routine JTSTEP. At each output station, subroutine JETPRP and JTOUTS are 
called to calculate jet centerline parameters and print (if specified) cal- 
culated profile output. At this point, profiles are saved temporarily on 
File 3 (JTFILE). At the completion of the numerical calculations, JETMIX 2 
calls JTOUTS to print a summary output page. If an output file is requested, 
the JETMIX input and centerline properties are written on File 2. The data 
saved on File 3 are then appended to File 2 before returning to the main 
control routine in Overlay (0,0). 


1.7.5 Overlay (2,0) - Entry SSFD 


Overlay (2,0) is the primary overlay for loading of the SSFD input and 
calculation overlays. Subroutines concurrently used by secondary overlays 
also reside in (2,0). 


1.7.6 Overlay (2,1) - Entry SSFDIN 


Overlay (2,1) consists of the routines to initialize SSFD data regions, 
read card input, and read JETMIX flow field data from the disc (or tape) 
input File 2. Initially, INIT is called to read the card input and the 
centerline output record from JETMIX. INIT also initializes the principal 
SSFD data regions. Subroutine READT entry READT1 is then called to initialize 
the data regions which contain the JETMIX profile data. 


1.7.7 Overlay (2,2) - Entry SSFDCA 


The supersonic flow field calculation is carried out in Overlay (2,2). 
The main calculation routine SSFDM is called to carry out the stepwise 
solution of the supersonic inviscid flow equations. During the process of 
solution, imbedded shock waves are detected. Shocks emanating from a com- 
pression corner, as well as imbedded and reflected shocks, are carried as the 
solution proceeds downstream. At specified output stations, subroutine 
PRINT is entered to tabulate calculated output and store requisite output 
data on File 3 (if specified). 


1.7.8 Overlay (3,0) - Entry MERGE 


The MERGE overlay reads data from both the JETMIX output File and the 
SSFD output File 3. The profile data are then collated and written as File 
4 for subsequent input to the NOISE program. 


484 























1.7.9 Overlay (4,0) - Entry MAINNS 





fhe final overlay in the SSNOISE system contains the routines for calcu- 
lation of the aerodynamically generated jet noise. 


485 











2.0 PROGRAM INPUT 


2.1 GENERAL DESCRIPTION 


The following sections are concerned with the input to the four programs 
comprising the SSNOISE system. Each program may be either run alone or in 
sequence. The standard SCOPE file INPUT is used for card input. For each 
program, selected input will reside on a data file (TAPE1, TAPE2, TAPE3, 
TAPE4) which may be either disc or magnetic tape. The input sheets for the 
programs JETMIX, SSFD, MERGE, and NOISE are given in Appendix 9. An initial 
prefix input set, consisting of identification information, is read once in a 
given run using fixed-field format (6Al10). The first three cards of the 


input deck consist of the name and address of the user and the problem identi- 
fication. 


Card No. Cols. 
1 2-61 User name (1-60 alphanumeric characters) 
2 2-61 User address or location (1-60 alphanumeric 
characters) 
3 2-61 Problem identification (1-60 alphanumeric 
characters) 


Blank cards may be substituted for input quantities not required. 
The input parameters for each program are given in the following sections. 


Included in these descriptions are the input items appearing on the input 
sheets as well as input items not normally required. 


2.2 JETMIX INPUT 


The first card of the input set is a header card consisting of the program 
name JETMIX starting in Column 2, and aT or an F in both Columns 12 and 14, 


Card Column Description 
2 JETMIX - Denotes program name 
12 Input file (tape) ? (T or F) 
14 Output file (dise or tape) ? (T or F) 


The header card is followed by the NAMELIST SA and the associated JETMIX 
input data. The NAMELIST input is terminated with a $ in Column 2. 


486 











2.2.1 Problem Specification 


The input quantities MIX, AXI, and TWO define the type of problem. The 
allowable combinations are: 


MIX = F AXI = ad 
F 
sg 

Ji = 
TWO F 

MIX = T AXI = Ed 

F 
TWO = F 


If these quantities are not input, the case will be taken as a free, 
axisymmetric, single jet. This is normally the predominant type of JETMIX 
run within the SSNOISE system. 


Preset 
Parameter Description Value 
MIX Problem type F - Free jet mixing 14 
T - Confined jet mixing 
AXI Flow field type T - Axisymmetric T 
F - Plane (2D) 
TWO Jet configuration F - Single mixing region F 


tae 


- Coannular/coplanar 
mixing region 


Beene Description of Primary and Secondary Jets 


The jet streams must be defined in terms of either Mach number or stag- 
nation pressure. 


Optional inputs are turbulence intensity, static temperature, and veloc- 
ity. If turbulence intensities are not specified, the jet(s) will be assumed 
laminar. Either static temperature or velocity may be specified. If these 
quantities are not input, the velocity corresponding to SLS ambient conditions 
will be computed. 


487 





Parameter 


DIAJ 


MJET 
PTJET 
TIJET 
TJET 
VJET 


DIAO 


MJETO 
PTJETO 
TIJETO 
TJETO 


VJETO 


Description 


Diameter (AXI=T) or 2x half-height (AXI=F) 
of primary jet, in. 


Primary jet Mach number 

Primary jet stagnation pressure, psia 
Primary jet turbulence intensity 
Primary jet static temperature, ° R 
Primary jet velocity, fps 


Diameter (AXI=T) or 2x half-height 
(AXI=F) of secondary jet, in. 


Secondary jet 
Secondary jet 
Secondary jet 
Secondary jet 


Secondary jet 


Mach number 

stagnation pressure, psia 
turbulence intensity 
static temperature, ° R 


velocity, fps 


2.2.3 External Boundary Conditions 


For free-mixing problems, the external boundary conditions are constant. 
For confined~mixing problems, the external boundary condition input denotes 
quantities at the discharge plane of the jet. 


temperature, and either Mach number or velocity. 


Parameter 


PE 


TE 


TIE 


ME 


VE 





Specify static pressure, static 


Preset 


Value 


bits 


bits 


bits 
bits 
0 


518.688 . 


Preset 
Description Value 
Static pressure, psia 14.69594 
External stream static temperature, ° R 518.688 
External stream turbulence intensity 0 


External stream Mach number --- 


External stream velocity, fps --- 


488 








2.2.4 Fluid Properties 


The fluid viscosity is computed as a function of temperature using the 
Sutherland relationship: 


a, 


u = MUREF x oe 


* CIREF+SC) 
(T +SC) 


If no input is given, the fluid is assumed to be air. 


Preset 
Parameter Description Value 
RG Gas constant, ft lbf/lbm ° R 53.34 (air) 
PR Prandtl number 0.72 (air) 
PRT Turbulent Prandtl number 1.0 


Sc Sutherland constant for viscosity <a 
calculation, ° R 


MUREF Reference viscosity @ TREF, lbm/ft sec --- 


TREF Reference temperature, ° R --- 


2.2.5 Step-Size Controls/Restart 


The input parameters CXPC and CXTP control the streamwise step size in 
the potential core and transition/similar regions of the jet. In the potential 
core zone, the step size is taken as a fraction of the width of the primary 
mixing zone: 


AX = CXPC x b 
pe 


Following disappearance of the potential core, the step size is taken as 
a fraction of the jet radius or half-height: 


= CXTP x Y 
AX,, 7 CXIP x Yj 


As the stepwise solution proceeds downstream, streamlines are added at the 
outer edge of the jet. Specification of CXPC and CXTP is equivalent to artifi- 
cally specifying the rate of entrainment of ambient fluid into the jet. In 
certain cases, the step size may be too large (entrainment rate too small), 
and the solution for the effective edge of the jet may experience problems. 


489 








These problems are usually manifested by a square root cf a negative 
number in subroutine JTEDGE, and may be rectified by decreasing the value of 
CXPC and/or CXTP. The preset values CXPC = CXTP = 0.02 have been adequate 
for most cases run to date. 


A problem may be restarted from tape at a given X or XD station using 
the input RESTRI. The value of RESTRT must appear in the X or XD table, and 
the profiles must be stored on tape at the restart station. For continuation 
of confined mixing problems using the free-mixing option (MIX=F), input the 
normalized restart station. 


Preset 
Parameter Description Value 
CXPC Step size control - potential core region. 0.02 
Fractional % of mixing zone width 
CXTP Step size control - transition/similar 0.02 


region. Fractional % of jet radius or 
half-height 


RESTRT X or XD station for restart of problem --- 


2.2.6 Turbulence Scale Calculations 


The characteristic length scale of the jet turbulence is assumed indepen- 
dent of the transverse coordinate and is expressed in terms of the geometric 
parameters of the jet and the jet Mach number using the following semi- 
empirical relations: 


Single Jet 
1) Region 1 (Potential Core Present) 
. -1 
Mea it eg eee (i + Cre My) 


2) Region 3 (Similar Profiles) 


he oe * 


3) Region 2 (Transition Region) 


The turbulence length scale in the transition region is less well defined 
than those in the potential core and similar regions, in that experimental 
data are sparse or nonexistent. The input parameter LTERP may be used to 


select either a linear or an exponential variation of scale in the transition 
region: 


490 








—_—__ 





(Linear): 


-1 x x 
. aah e = .3 
Lig = (¢,, Cte, My) (2 x) eg (x. Ys 
(Exponential): 
-lyx 
= —_ y 
Lee Chg (C,, M)) () =p & (Chey "D] 


Pape ae 4 Coannular/Coplanar Jet 


The turbulence length scale for a coannular/coplanar jet is based on an 
empirical geometric model. A schematic diagram of the dual jet flow field, 
with pertinent symbols, is shown below: 


Giex A _—— 

Ms >y — 4 3 SECONDARY ae 
% aN MIXING ZONE man 

s Me Ys 


_— -”_ 
. * si¥m, = 
—— ees COMBINED 
=" x MIXING MIXIN NE 
Mp tS a —e 


= 
% & 
P SS so primary x ™ 
7" > CORE Se 
P > B 
Qip “N 
_ om xy - -_——> _ —_——_—_—______§ 





491 








The model is described as follows: 
Ky &x Cn 


The region Xj ¢ X _ upstream of the point where the secondary core dis- 
appears connect of ‘No independent jets which mix in somewhat the same manner 
as the previously described single jet. The turbulence length scale is de- 
fined in terms of the scale in the two mixing regions as follows: 


aid 
ul 


il 


bg 


-1 
Cg. ™ Ce.) 


«y 60 € Y K oY, 


L 


L 
P 


be) «oe ¥ WY, - 9b.) 


tho = hg 


— Ss iit 
Oh. t SS oa te 
s p s 


eu = 265 K Y 4 2 


L#=L 
s 


- 9Y 
> 


The factor 0.9 allows marginal interaction of the two mixing zones, in that 
the scale in the secondary core region varies linearly between the two 
extreme values. 


Xo 4 X a X 


When the outer edge of the primary jet has intersected the inner edge of the 

secondary jet, the secondary potential core disappears. The co-ordinates of 

the boundaries of the combined mixing zone are determined by fitting a linear 
curve between the jet corner co-ordinates and the merge point; viz, 


492 








(outer boundary) 


os) 
AB Y=Y,+(=>-—- 
2 Xo xy (Xx Xy) 


(inner boundary) 


Yad 
tn! = - 
A'B t= 4, + xX, (X - X)) 


For the situation shown in the sketch, the scale may be defined as: 


ae) ¥ 4 -9*(Y -b.) 


L=L 
P 


be) .9*( -b) K€ YK ¥,-.9(¥,-¥,) 
baa Se aii, lb "tee Bitekal ed 
es ae 


L=L+ 
9) i a 


[vy - .9*(Y,-b.)] 


ce) .. - HY, - Y 4 YY 
ee 2 


L=L 
s 


special situations occur as follows: 


(L, - Ly) 
a eS o * = 
ea Sis 90> +b) - BY, (y ~ tt, = 80) 


Ly Der rae 


L#=L 
s 


¢ 493 








If ¥ DY,» set Y_ = z. 


a.) o€y €.9* = d:) 
p= E 
Pp 
Day sae i bg ¥ 


(L. + i} 


he ee eae ot. 
p s s 


In the region downstream of the point where the primary core has disappeared, 
the combined mixing cone scale is redefined to: 


ms 
we +c 
Ly tn°p“ oP? 


The numerical values assigned to the constants for the single- and dual-jet 
geometries are; 


Preset 
Parameter Description Value 
LTERP Transition region scale selection T 
T - Linear variation 
F - Exponential variation 
cTl 0.23 
CT2 0.38 
cT3 0.23 
CT4 Constants for definition of turbulence 0.05 
CTS scale of single jet 0.38 
CT6 1.4 
CT7 0.43 
CT8 0.1875 
CTP Constants for definition of turbulence 0.175 
CTs scale of coannular/coplanar jet 0.23 
CTM 0.23 


494 











2.2.8 Transverse Mesh Control 


A semiuniform mesh is utilized in the potential core region. The number 
of streamlines in the initial discharge plane of the jet may be altered using 
the inputs NJ and NJO (NJ is the number of streamlines in the primary jet), 
whereas the difference between NJ and NJO is the number of streamlines in the 
secondary jet (if present). In the potential core zone, streamlines are added 
at a constant Ay until a maximum (NM) is reached. At this point, the number 
of streamlines is reduced and the mesh refined. The semiuniform streamline 
distribution is maintained until the potential core disappears. Downstream of 
this point, the jet grows rapidly, and it becomes advantageous to redefine the 
mesh. The streamlines are redistributed such that the ratio of Ay's for any 
two adjacent intervals is a constant. The distance of the i-th streamline is: 


where, K = mesh constant 


al = initial Ay 


¥;/¥, = 1@ 4 = NMSH 


Preset 

Parameter Description Value 
NJ No. of streamlines in primary jet 30 
NJO Streamline number of secondary jet edge 50 
NM Maximum number of streamlines before mesh 80 

redistribution occurs 
NMSH 71 
DY1 Mesh parameters (see above) O01 
CK 1.06064475 


fees? Species Diffusion Input 


The JETMIX program contains the option to consider the diffusion of up 
to six chemically inert constituents. The input for a case of ghis type 
given on JETMIX input sheet la, and includes variables to define the mole 
concentration of the jet streams, boundary conditions, effective Schmidt 
numbers, and coefficients for the molar heat capacity of each constituent. 
The NC, CNAME input variables are provided to designate names of the species. 
Note that this option may not be used if NAMELIST does not permit hollerith 
input. The molar heat capacities of the individual constituents are computed 
as a quadratic function of local temperature T; viz, Cy =a +bT + cT2. 


495 





Preset 
Parameter Description Value 
DIFF Species Diffusion F - No F 
T - Yes 
NC Number of constituents 3 
CNAME Name of constituents AIR, C02, H20 
ALJ Primary jet stream species mole 92, .04, .04 
fractions, moles i/mole mixture 
ALJO Secondary jet stream species mole 96, «O02, «02 
fractions, moles i/mole mixture 
ALE External (boundary) species mole -99934, .00033, 
fractions, moles i/mole mixture .00033 
SCM Effective Schmidt number for individual ah 
species 
CPC Coefficients in polynomial representation --- 
of molar heat capacities (see input sheet 
JETMIX/1a) 


2.2.10 Station Input 


The following parameters are termed "station input" and are used to de- 
fine the print/data file stations in the free-jet case and boundary coordinates 
in the confined-mixing case. In the latter situation, the streamwise coordi- 
nate also serves as identification for printing. Station data may be input 
in column format using the B array or in "free form" using the symbolic names 
associated with the columns of the B block. When no station input is supplied, 
the JETMIX program uses a set of input coordinates which has been optimized 
for use in acoustic calculations. 


Parameter Description 
B Input block for station data. See JETMIX input 


sheet 2 and associated notes on station data. 

The preset symbolic names assigned to each column 
of the B block are: 

Free mixing X, XPRN 

Confined mixing XD, RD, YCB, XPRN 

Data set to bits* will be interpolated relative to 
X, or the last value in the table will be auto- 
matically extended down the column. 

Up to 100 stations may be input. 

* bits = 1 x 10° (junk word) 


496 





xX Dimensionless streamwise coordinate (free mixing), x/dj 


XD Streamwise coordinate, in. (Confined mixing) 
RD Outer boundary coordinate, in. (Confined mixing) 
YCB Inner boundary coordinate, in. (Confined mixing) 
XPRN Profile print/file indicator: 
0 - Profile not printed or saved on file 2 
1 - Profile printed and saved on file 2 
-1 - Profile saved on file 2 but not printed 
XPRN(1)= 2 - All profiles printed and saved on file 2 
XPRN(1)=-2 - All profiles saved on file 2 but not printed. 


The various types of input are best illustrated by use of an example: 





Confined mixing in an annular duct - 5 stations (0, 2, 4, 6, 8 inches) 


with radii (1, 2, 3, 4, 5 inches). Centerbody radius - 0.1 inch. 


Print all stations. 


Sheet JETMIX-2 


Ww, XD RD YCB XPRN 

B(1)= 0., ie is Bis 
Pe. 2.5 i‘? Ns 
4., Dex i; Les 
®. or dy hag 
8 Ses 1 1 


- 


Free Form 


YW 


ot) Aine: Svar Sey Sees ee 

SU A13-@ Bis Zea Sek ee 
YCB (1) = 0.1, 

XPRN (1) = 2, 


This form of input is useful for modifying input data reclaimed from an 


input tape 1. Suppose it is desired to add 5 or more stations with duct and 
plug radii equal to those at the 5th station: 


7 


XD (6) = 10, 12, 14, 16, 18, 


497 











The balance of the RD and YCB arrays will be filled with the values of 5. 
and 0.1, respectively. As in the case of tabular input, missing input will be 
linearly interpolated or appended as required. 


Station data input via the B block will override corresponding free-form 
input. Inadvertent destruction of the free-form input may be avoided by 
selectively designating positions in the B array. 


2.3  SSFD INPUT 


The first card of the input set is a header card consisting of the pro- 
gram name SSFD starting in Column 2, and a T or an F in both columns 12 and 14. 


Card Column Description 
2 SSFD - Denotes program name 
14 Input file (tape) ? (T or F) 
16 Output file (disc or tape) ? (T or F) 


The header card is followed by the NAMELIST, $INPUT, and the associated 
SSFD input data. The NAMELIST input is terminated with a $ in column 2, 


2.3.1 Problem Description 


The input quantities AXISYM, XMACH, and GAMMA define the type of problem. 
Note that XMACH is the actual initial Mach number, not the ideally expanded 
Mach number, and generally will be different from the input quantity MJET in 
JETMIX. Also note that the specific heat ratio must be input twice (i.e., 
GAMMA (1) = 1.4, 1.4), and that the two values must be identical for normal 
program operation. 


Preset 
Parameter Description Value 
AXISYM T - Axisymmetric flow field <= 
F - Plane (2-D) flow field 
XMACH Initial Mach number 1.05 
GAMMA (1) Specific heat ratio bees. Lat 


498 





Redee Program Controls 


The input quantities XL, STABIL, and IPRINT control the streamwise dis- 
tance over which the calculation proceeds, the streamwise step size used in 
the finite difference solution, and the program output, respectively. XL is 
preset so that SSFD will terminate upon reaching the end of the potential core 
region (as determined by JETMIX). The maximum allowable streamwise step size 
at any axial station is set by the requirement that characteristics from 
adjoining (cross-stream) points in the finite difference mesh must not inter- 
sect between stations (CFL condition). The step size which is used in SSFD is 
the product of this maximum step size multiplied by the stability parameter 
STABIL. If difficulties are encountered in obtaining an SSFD solution, it 
May be necessary to decrease the value of STABIL. 


Preset 
Parameter Description Value 
XL Final value of x/d4 (see above) 
STABLL Stability parameter for step-size control <5 2 
IPRINT 1 - Print profiles and shock pattern 1 


0 - No print 


2.3.3 Total-Pressure Input Stations 


SSFD uses total-pressure data from the JETMIX solution at the stations 
XPT. These are preset to: 


SWE sth ice Oo da aks woe eearais i ty Ry Be Sade Oa Be. Sy BR. 1.35 Gy 32, 
13, 15, 20. 


Preset 
Parameter Description Value 
NPT Number of stations for total-pressure 20 
input 
XPT Stations at which total-pressure data are (see above) 


taken from JETMIX solution, x/dj ~ 


Optional Input Parameters 
The following quantities were originally included as input variables 
either to allow SSFD to be run without direct coupling to JETMIX or for con- 


trol of diagnostic printout which was used in debugging SSFD. They have been 
retained to preserve full program capabilities, but are no longer required. 


499 





Parameter 


NJ 


X 


SSTRM 


NJJ 


RSTART 


TANTH 


2M 


PRESS 


THETA 


YIN 


PE 


TT 


XLOW 


YLOW 


XUP 


PUP 


PLOW 


PTOT 


PSLPT 


YBAR 


LPUNCH 


XMDISC 


XSAVE 





Definition 
No longer used 
Initial value of x/¥j 


Is there a slipstream at interface of inner/outer 
streams? (Logical) 


Number of initial grid points 


Is this a restart of a previous run? (Logical) 


Tangent (v/u) of flow angle at initial grid points 
Mach number at initial grid points 

Static pressure (p/ptj) at initial grid points 
Flow angle (tan v/u) at initial grid points 
Radial location (y/rj) of initial grid rene 
Total pressure (pt/Pt4) at initial grid points 


Total temperature (Ty/Ty;) of streams at initial 
station 


x/r4 coordinates of stream lower boundaries 
i 

y/rj coordinates of stream lower boundaries 

x/rj coordinates of stream upper boundaries 

y/rj coordinates of stream upper boundaries 

Static pressure (p/prj) on stream upper boundaries 
Static pressure (p/Pe4) on stream lower boundaries 
Total pressure (pr/prj) in JETMIX flow field 


Streamline coordinates (\)) for total-pressure input 


y/rj coordinates at initial station (in place of 
equal spacing). 


Are cards to be punched for restart? (Logical) 
x/rj coordinate of Mach disc 


x/Ty coordinate of profile to be saved for restart 


500 








Parameter Definition —s 
IDISC Control variable for Mach disc/restart 
MDISCC Is there to be a Mach disc? (Logical) 
BARPRT Print control (Logical) 
IPRTC Print control 
XBDY Print control (x/r4) 
XPR Print control (x/r4) 
SHKPRT Print control (Logical) 
XTKESH Print control (x/rj) 
XTKESF Print control (x/rj) 
XRESET Print control (x/rj) 
XENTRO Print control (x/rj) 
; XTKEPR Print control (x/rj) 


The subscript 'j'' has been used to denote nozzle exit-plane properties. 


2.4 MERGE 


No card input is required for the MERGE program with the exception of 
the following header card: 


Card Column * Description 
2 MERGE - Denotes program name 
12 T - Denotes input files (2 and 3) 
14 T - Denotes output file (4) 


2.5 NOISE INPUT 
The first card of the input set is a header card consisting of the 


program name NOISE starting in Column 2, and aT or an F in both columns 12 
and 14. 


501 





Card Column Description 
2 NOISE - Denotes program name 
14 Input file (tape)? (T or F) 
16 Output file (disc or tape)? (T or F) 


The header card is followed by the NAMELIST $A on the associated NOISE 
input data. The NAMELIST input is terminated with a $ in Column 2. 


2.5.1 General Input 


The input quantity MFILE denotes the input file code for the aerodynamic 
data: 


Nominal 
Parameter Description Value 
MFILE An input file code for either Z 
JETMIX (set to 2) or MERGE 
(set to 4) 


In addition to the aerodynamic data file selection, a series of 
quantities is necessary as input before selection of the acoustic models. 
These input parameters are defined as follows: 





Nominal 
Parameter Description Value 
SCALT Average Source-Receiver Doppler oS 
Shift 
SCALJ Jet Scale Factor (Model size to i. | 
full size) 
PREFN Reference Acoustic Pressure fo -0002 
Sound Pressure Level (dynes/cm’) 
BAND3 Selector for 1/3-Octave Band calcu- F 
lation (T selects 1/3-Octave Band; 
F selects Octave Band) 
MC Selection of Fixed or Variable Con- 0 
vection Mach Number (0 selects a | 
fixed convection Mach Number; l | 
selects a variable convection Mach 
Number) 
CVMACH The convection Mach No, Constant «Oo 
BETAIN Will BETA be input? (T or F) F 


502 











BETA An Acoustic Intensity Proportionality 00425 
Constant - .00425 for a cold jet, 
.002125 for a hot jet 


JETTEM Indicator for a Hot or Cold Jet 0 
(O - cold jet, 1 - hot jet) 


QCONV T or F Selector for the Convection F 
Singularity Function (F, q = 
La fag: Ty ¢ = @ w/a) 


2.5.2 Acoustic Model Selection 


The NOISE program allows the operator to select a number of far-field or 
near-field acoustic models. The far-field acoustic models are of the Lighthill 
(LIGHTH), Ribner (RIBNER), or Pao (PAO) forms. Appendix 8 discusses the 
analysis incorporated in each model. 


Nominal 
Parameter Description Value 
LIGHTH Lighthill selection (0, self-noise 0 
alone; 1, self-noise + shear-noise) 
' LILLEY T or F Selection of Classical Lilley F 
Model 
RIBNER T or F Selection of Classical Ribner F 
Model 
CRIB Acoustic Intensity Proportionality Pees Vi 
Constant in Ribner's Model (0, self- 
noise only) 
SE Selection for Shear Noise in Ribner E 
Model (0, shear noise only) 
PAG T or F Selection of Classical Pao F 
Model 
PSPEC Selection of Pao Pressure Spectrum F 
Calculation (T or F) 
MU Selection of Type of Pao Model 0) 


(0, self- and shear noise, l, 
self-noise alone, 2, shear 
noise alone) 


As for the far-field acoustic models, there are a number of near-field 
acoustic models the user may select. Their descriptions are contained in 
Appendix 8. 


503 





NEARFD = 


22d 


Isotropic turbulence model 


Lateral quadrupole model 


Longitudinal quadrupole model 


Microphone Selection 


Combination model using lateral quadrupole in transition 


Combination model using longitudial quadrupole model 
in transition region 


In order for the user to have acoustic prediction printed, the number 
of angle locations (NA) and the actual polar angles (referenced to the jet 
Additionally, an arc or sideline prediction can 


axis) must be specified. 


be made, 


Parameter 


NA 


ANGJ 


ARC 


SLINE 


ARCL 


2.5.4 


Description 


Number of input angles (1 through 
20) 


Angle (in degrees) input (a maxi- 
mum of 20 angles may be specified) 


F - Sideline configuration 
T - Arc configuration 


Sideline distance in feet 


Arc distance in feet 


Output Control Cards 


In addition to the normal program output of OASPL, SPL, 
detailed acoustic profiles can be computed and pointed out. 
acoustic profiles are controlled by the parameter ACSPAN and ACOUSP. 


Parameter 


ACS PAN 


ACOUSP 


Description 


(T cr F) Acoustic radial profile 
computation at a radial location 
for a given angle (specify T for 
each angle) 


(T or F) Acoustic profile indicator 
for axial locations specified from 
JETMIX (Specify T for each X 
station) 


504 


Nominal 
Value 


Must input 


Must input 


PWL, etc., 
These detailed 


Nominal 
Value _ 


F 


3.0 PROGRAM OUTPUT 


3.1 GENERAL DESCRIPTION 


The output from the SSNOISE system may be logically divided into the 
following two sections: 


1) Card input and preliminary printout 


2) Specific output generated by each program in the system 


The initial section of output consists of a card image print of the problem 
input and a designation of the tape (disc) input/output file selections; 
viz, TAPIN = } and TAPOT = 1. Upon completion of the card image print, 
the file TAPE5 is rewound to its original status. The output from the 
programs follows the section of preliminary printout and is described in 
the following sections: 


3.2 JETMIX PROGRAM OUTPUT 


The output from the JETMIX program consists principally of the dimen- 
sional and dimensionless profiles at designated calculation stations and the 
properties along the jet axis of symmetry. Profile printout will always 
occur at the stations inserted by the program. These inserted stations are: 


1) Potential core disappearance 

2) Supersonic core disappearance 

3) Merge station of coannular/coplanar jet 
The above profile printout is preceded by the defined and calculated reference 
conditions and the initial profiles (U, E, 6, a4) at the discharge plane 


of the jet. The calculated output variables in their literal order of 
appearance in the printout are as follows: 


505 


Print Heading 
y 
PSI 


UD 


LE 


TTD 


PTD 





MACH 
V 
¥ 3 
TOT 
PTOT 


(constituent) 
(DIFF=T) 





Dimensionless and Dimensional Profiles 


Description 


Dimensionless normal coordinate Y = 2y/dj 
Stream function ¥, tha/te* 

Dimensionless velocity, U = u/u; 
Dimensionless temperature, 6 = T/Tj 
Turbulence intensity, u'/u; 


Dimensionless temperature coefficient, 
(Ty pe Tex) / (Ty; i Tex) 


Dimensionless pressure coefficient 
(Py i Pex) / (Pr, - Pont 


Mach number 

Velocity (u), fps 

Static temperature (T), ° R 
Total temperature (Ty), ° R 
Total pressure (Py), psia 


Species mole fractions; the headings of the 
columns refer to the CNAME input parameters. 


506 





Jet Dimensionless Station Data (Summary) 


Point Heading Description 
X Dimensionless axial coordinate, X = x/dj 
B Dimensionless width of mixing zone 


Potential core - (Ye - Y, Pe 
Transition/similar - (2v3 j 


YJ Dimensionless edge of jet Y, = 2ye/d, 
UC Dimensionless velocity along y = 0 
TC Dimensionless temperature along y = 0 
FLC Turbulence intensity along y = 0 
PTC Dimensionless pressure coefficient along ~ = 0 
TIC Dimensionless temperature coefficient along y = 0 
YSONIC Dimensionless location of sonic line (if present) 
WJ Entrained flow ratio, (mass flow/mass flow at jet 


discharge plane) 


Continued Mixer Station Data - Summary (MIX = T) 


Print Heading Description 
XD Axial coordinate, in. 
RD Outer boundary normal coordinate, in. 

YCB Inner boundary normal coordinate, in. 

YD Dimensionless equivalent of RD 

YCD Iterated dimensionless location of outer boundary 
PD Iterated static pressure, psia. 

FLOW Flow conditions at a given station 


SUBSON - Subsonic 
SUPSON - Supersonic 
CHOKED - Choked 


507 





Continued Mixer Station Data - Summary (MIX = T) (Concluded) 


Print Heading 


THRUST 
MA2 
VE2 


TE2 


3.3 SSFD PROGRAM OUTPUT 


The output from the SSFD program consists of dimensionless profiles 
of aerodynamic properties at the stations designated in JETMIX, as well as 
information on the location and strength of shocks at each calculation 
Station. The calculated output variables are summarized as follows: 


Print Heading 
X/R 
Y 
Mach Number 
Flow Angle 
Total Pressure 
Static Pressure 
Density 
U - Velocity 
Shock Angle 
Turbulence Intensity 
XSHOCK 
YSHOCK 
P2 
Pl 
MACH2 
MACH1 





Description 


Integrated momentum, lbf. 
Mach number external to the jet. 
Velocity external to the jet. 


Static temperature external to jet. 


Description 





x/Tj 

y/rj 

M 

v/u (actually the tangent of flow angle) 
Pe/Pey 

Ps/Pt, 

plots Pty is density based on Pt and Trt 

Ww y/ar, (at is sonic velocity based on Trt) 
Ay/Ax (actually the tangent of shock angle) 
u'/uj 

x/T4 

y/rj at shock 

Ps/Pey downstream of shock 

Pg/Pej upstream of shock 

M downstream of shock 


M upstream of shock 


The subscript "j'' has been used to denote exit-plane properties. 


508 


3.4 MERGE PROGRAM OUTPUT 


The MERGE program is utilized to collate the aerodynamic output pro- 
duced by the JETMIX and SSFD programs. The output produced by the MERGE 
program is similar to that of JETMIX and consists of the merged dimension- 
less profiles at the JETMIX output axial stations. The SSFD output is only 


available for those stations where the flow is nominally supersonic. Downstream 


of this point, the printed profiles consist of only the JETMIX data. The 
collated variables in their literal order of appearance in the printout 
are as follows: 


Print Heading Description 

Y Dimensionless normal coordinate, Y = 2y/dj 
PSI Stream function jy, lbm/ft? 

UD Dimensionless velocity, u = u/u; 
THD Dimensionless temperature, 6 = T/Tj 

ED Dimensionless turbulence energy, E = e/ej 
RHO Density, lbm/ft? 
XLN Turbulence length scale, ft. 


3.5 NOISE PROGRAM OUTPUT 


The main output for the NOISE Program consists of the following cate- 
gories: One-Third Octave Band Analysis, Acoustic Summary of the Aero- 
dynamic and Acoustic Parameters, and the acoustic Power Summary. 


3.5.1 One-Third Octave Band Analysis 


The one-third octave band analysis consists of two parts; a tabulated 
list of 1/3 OBSPL's and a computer plot of the same data. The calculated 
output variables are summarized as follows: 


Print Heading Description 
x/D Axial location of microphone in rectangular 


coordinates (in diameters) 


Y/D Cross-stream location of microphone in rectangular 
Coordinate system (in diameters) 


R/D Radial location of microphone location in R,6 
polar coordinate system (in diameters) 








Print Heading 


ANGLE 


CENTER FREQ, Hz 
LOWER FREQ, Hz 
UPPER FREQ, Hz 


NPTS 


SPL 


Description 


Polar angle (8) from jet axis to microphone 
location (degrees) 

The center one-third octave band frequency (Hz) 
The lower limit for the one-third octave band 


The upper limit for the one-third octave band 


The number of volume elements used in performing 
the prediction of a 1/3 octave band 


The one-third octave band sound pressure level 
prediction 


3.5.2 Summary Acoustic Analysis 


The summary acoustic analysis for the NOISE Program consists of a summary 
ot the aerodynamic parameters and the far-field noise parameters. The out- 
put variables are summarized as follows: 


Print Heading 


TE 

PE 

VE 

ME 
TIE 
DIAJ 
MJET 
TJET 
PTJET 
VJET 
TIJET 
GAM 
RG 
PR 
PRT 
sc 
TREF 


Description 


Ambient static temperature, ° R 
Ambient static pressure, psia 
External velocity, fps 

External Mach number 

External turbulence intensity 
Diameter of jet, in. 

Primary jet Mach number 

Primary jet static temperature, ° R 
Primary jet stagnation pressure, psia 
Primary jet velocity, fps 

Primary jet turbulence intensity 
Ratio of specific heats 

Universal gas constant, ft lbf/lbm ° R 
Prandtl number 

Turbulent Prandtl number 

Sutherland constant, ° R 


Reference temperature, ° R 








Print Heading 


MUREF 

N 

ANGLE 

OASPL 
SOUND PRES. 

PNdB 

OBSPL 


Description 


Reference viscosity, lbm/ft sec 
Microphone number 

Microphone angle location, degrees 
Overall sound pressure level, dB 
Acoustic sound pressure, dynes/cn* 
Perceived noise level, dB 


Included in the acoustic summary are octave 
band sound pressure level predictions 


3.5.3 Summary Acoustic Power Analysis 


The acoustic power analysis consists of an overall power level pre- 
diction, power per unit length of jet slice, and a 1/3 OBPWL prediction. 
The output variables are summarized as follows: 


Parameter 


PWL 


X 

CENTER FREQ, Hz 
LOWER FREQ, Hz 
UPPER FREQ, Hz 


NPTS 


POWER 


3.6 SAMPLE CASE 


Description 


Overall power level {veto watts) per unit 
length of jet, radiating from station X 


Axial station in jet diameters 

The center one-third octave band frequency (Hz) 
The lower limit for the one-third octave band (Hz) 
The upper limit for the one-third octave band (Hz) 


The number of volume elements used in constructing 
the prediction 


The one~third octave band power spectra 
re 10713 watts 


The input/output for the Supersonic Jet Noise Prediction System 
(SSNOISE) is best illustrated with a sample case. The sample case is 
included in Appendix 10. The first page of Appendix 10 shows typical 
card input of JETMIX, SSFD, MERGE, and NOISE. This particular case was 


run in series as one job. 





511 











Following the requisite header cards and the program card, the 
essential input to JETMIX consists of the following items, describing the 
flow conditions at the discharge plane of the jet and the external 
boundary conditions: 


Axisymmetric - Isothermal - Single Jet 


Jet diameter 


Jet Mach no. 


Jet temperature 


Jet turbulence intensity 


Ambient pressure 


Ambient static temperature 


Ambient Mach no, 


Ambient turbulence intensity 


4.3 in. 

1.559 (supersonic) 
518.7° R 

0.1 


14.696 psia 
SL8.7" & 

0 

0 


The definition of jet temperature equal to ambient temperature causes th« 
JETMIX program to bypass the solution of the energy equation and to cal- 


culate an isothermal flow field. 
supplied for this case. 


Note that no station input (X, XPRN) was 
As discussed in the Input section, a set of 


optimized acoustic stations are built in and used if station input is not 
These 72 stations are as follows: 


given. 





0001 
-0002 
.0003 
.0005 
-0008 
001 
-002 
-004 
- 006 
.008 
-010 
015 
-017 


»45 
-50 
°55 
-60 
-65 
-70 
- 80 


2.0 
2.5 
3.0 
3.4 
4.0 
5.0 
6.0 
6.2 
6.5 
6.8 
7.0 
7.5 
8.0 
9.0 


XPRN X APRN 
1 14.0 ~1 
1 15.0 1 

~1 16.0 -1 
1 17.0 -1 
~1 18.0 -1 
1 19.0 ~1 
-1 20.0 1 
1 21.0 1 
-1 22.0 ~1 
-1 23.0 ~1 
at 24.0 ~-1 
1 25.0 ~1 
-1 26.0 ~1 
1 28.0 ~1 





X XPRN xX XPRN xX XPRN Xx XPRN 





-02 1 1.0 1 10.0 -1 30.0 =] 
-04 1 baz =-1 11.0 4. 32.0 : 
-06 Y.5 =. 12.0 -1 34.0 2 
-08 se RY: = 13.0 1 36.0 a 


The input of CXTP = 0.04 was utilized to speed the calculation in the 
transition/similar region downstream of the point where the potential core 
disappears. 


The program output of JETMIX is given on pages 672 to 719 and consists 
of the initial conditions at the discharge plane of the jet, the calculated 
aerodynamic profiles at the stations where XPRN = 1, and the final summary 
page with calculated station data and the overall jet properties. Referring 
to the summary page, it will be noted that stations have been inserted where 
the potential core disappears (X = 4.12861) and where the supersonic core 
disappears (X = 12.12730). The corresponding profile output at these stations 
is also printed. The output parameter headings are described in Section 3.2. 


The input to SSFD consists of the following items: 


Initial jet Mach number (XMACH) 1.05 
Stability parameter (STABIL) 0.6 
Final x/dj for calculation (XL) tay 


This case corresponds to an underexpanded jet (recall that the fully expanded 
"ideal" Mach number input to JETMIX was 1.559). The inputs of STABIL = 0.6 
and XL = 1.7 were utilized to speed the calculation by increasing the 

axial step size and decreasing the distance over which the calculation is 
performed, respectively. 


The program output for SSFD is shown on page 721. The output parameters 
and headings are described in Section 3.3. Radial profiles are given at each 
of the axial stations for which XPRN was specified as +l or -1 in JETMIX, and 
for which x/dj < XL. In addition, the progression of shocks through the flow- 
field is monitored at intermediate axial stations. 


The input to the MERGE program consists of a single card designating 
the MERGE program, with both a file input (file 2) and a file output (file 3). 
The output from the MERGE program consists of composite (JETMIX and SSFD) 
profiles at each of the JETMIX stations where XPRN # 0. The MERGE output is 


shown on page 766. The output parameter headings are described in Section 3.4. 


For the NOISE predictions, two cases were computed using a Lighthill 
model modified for a combination of self-noise and shear-noise (see pages 813 
to8s47). The first case is for a prediction using JETMIX results as input 





data alone, the second case is for a prediction using MERGE results as 
input (no-shock case versus a shock-flow case). The predictions are for a 
model scale jet on a 40-foot arc. The program output is seen to consist 
of 1/3 OBSPL summaries at jet angles of 10, 20, 30, 150, and 160°. This 
is followed by an overall summary page of the basic aerodynamic parameters 
and the far-field noise OASPL, PNdB, and OBSPL; and a summary page for the 
predicted jet power per unit slice, OAPWL, and power spectra. Section 2.5 
describes the output parameters. 


As an example test case for near-field noise predictions, see the last 
test case run in Appendix 10 (pages 848-901). There the lateral quadrapole 
near-field acoustic model is shown, This test case, with the instructions 
given in Appendix 9 - Input Sheets - should suffice as an example for running 
any of the other near-field models. 





4.0 OPERATING PROCEDURES 


4.1 GENERAL DESCRIPTION 


The SSNOISE system described herein may be run on any Control Data 
6400/6600 machine operating under SCOPE 3.0 or a higher level operating 
system. In general, operating procedures and control card setups will 
differ from site to site. The following comments on program modifications, 
deck setups, and operating instructions are restricted to the program as 
run through the CDC Cybernet System. Minimal changes should be necessary 
for successful installation at other CDC sites. 


4.2 MAINTENANCE AND MODIFICATION 


The SSNOISE system source deck consists of approximately 14,000 FORTRAN 
cards. The source copy contains *DECK cards as the first card of each sub- 
routine. These decks (tape) may be used directly to initialize an UPDATE 
file (tape or permanent disc) which may subsequently be used for maintenance 
and modification (standard SCOPE - UPDATE program). It is recommended that 
relocatable and absolute binary files also be maintained either on tape or 
permanent disc. Execution from the absolute binary file requires a central 
memory field length of 103000g locations. A typical deck setup for execution 
from an absolute binary file through CDC Cybernet System is as follows: 


7 


RFL (40000) 

LABEL (SSNOV, R, VSN = S2222) 
COPYBF (SSNOV, SSNOISE) 
REWIND (SSNOISE) 

COPYBF (INPUT, TAPES) 
RFL (103000) 

SET (0) 

SSNOISE 

67, (EOR) 

{Input Data] 

67g, (EOF) 


4.3 INPUT AND OUTPUT FILES 


As mentioned previous!y, each program, with the exception of NOISE 
program produces an output file which may be used as input to the other 
programs in the SSNOISE system. The files and their functional use are 
repeated here for convenience, 


u 
_ 
uy 





File Device Functional Usage 


TAPEL tape JETMIX restart file 
TAPE2 tape/disc JETMIX output file 
Input file to SSFD, MERGE, NOISE 
TAPE3 tape/disc SSFD output file 
Input file to MERGE 
TAPE4 tape/disc MERGE output file 


Input file to NOISE 


In a given run, if the file is not assigned via an appropriate tape card 
(REQUEST, LABEL), the file will be automatically assigned to disc when it 

is opened. The output file from JETMIX, SSFD, or MERGE may be saved on 
tape and used as input in a subsequent job. If the aerodynamic configura- 
tion does not change, it is economically feasible to run the aero program(s) 
once and save an output tape. The resulting file may then be used as 

input for parametric running of the NOISE program. In this case, there 

is no need to rerun the aero program(s) with each NOISE case. 


The following schematic input deck setups illustrate the flexibility 
available to the user: 


1) 





2) 


Execution of all programs in SSNOISE system. 


using both the JETMIX file and the MERGE file. 


[Control card stack] 
(EOR) 
7 
NAME = 
ADDRES = 
IDENT = 
JETMIX F E 
SA 
(JETMIX input) 
$ 
SSFD fi ‘EB 
SINPUT 
(SSFD input) 


MERGE 
NOISE 
SA 
(NOISE input - JETMIX output file) 


NOISE T F 

SA 

MFILE = 4, (NOISE input - MERGE output file) 
$ 

(EOF) 


Execution of JETMIX - Save output file on tape 


Control card stack 
LABEL (TAPE2, W, X, = SV, LABEL = JETMIX) 
(EOR) 
yee 
ADDRES = 
IDENT = 


NOISE executed 





JETMIX F T 
SA 
(JETMIX input) 
$ 
(EOF) 


Execution of SSFD and MERGE using previously saved JETMIX output 
tape. Save MERGE output file on tape. 


Control card stack 
LABEL (TAPE2, R, VSN = S1111) 
LABEL (TAPE4, X=SV, LABEL = MERGE) 
(EOR) 
7) 
NAME = 
ADDRES = 
IDENT = 
SSFD d T 
SINPUT 
(SSFD input) 


MERGE T T 
(EOF) 


518 








APPENDIX 6 


ANALYSIS INCORPORATED IN JETMIX 


Exhaust nozzles of contemporary gas turbine engines generally operate 
near their ideal expansion ratios. In view of this, General Electric's 
initial efforts at the prediction of the sound field of a supersonic jet 
relied on an aerodynamic analysis which includes only the effects of turbu- 
lent mixing. Specifically, the aerodynamic model is of the viscous, boundary- 
layer type. In this analysis, which is incorporated in the JETMIX computer 
program, the time-averaged turbulent boundary layer equations are solved 
using boundary conditions which are appropriate for free jets. The turbulent 
Reynold's stresses are included by means of a turbulence model which is based 
on a turbulent kinetic energy concept. 


The flow field under consideration consists of a plane or axisymmetric 
turbulent jet exhausting into a region of constant (ambient) static pressure. 
This is illustrated schematically in Figures 249 and 250 for subsonic and 
supersonic jets, respectively. The flow of diameter or slot height (d,) dis- 
charges at an initial velocity (up) into a free stream of velocity (uex). 

The flow field is characterized by three distinct regions. Region 1 consists 
of a turbulent mixing layer which penetrates into the uniform parallel flow 
originating at the jet discharge. Upon disappearance of the potential core, 
the mixing characteristics undergo transition (Region 2), until, at some 
distance downstream of the discharge plane, the velocity profiles normal to 
the jet axis become similar (Region 3). 


The dependent variables of interest in the mixing problem are the stream- 
wise velocity (u), the transverse velocity (v), the static temperature (T), 
the turbulence energy (e), and the constituent mole fractions (aj). If trans- 
verse gradients are large with respect to streamwise gradients, the equations 
of motion describing the flow field may be reduced to boundary layer form. 
For plane or axisymmetric flow, the governing boundary layer equations appli- 
cable to the free-mixing problem are: 


CONTINUITY EQUATION 


a(puy) | ‘toys avy] «9 (232) 


ax 


: Plane (2-0) flow 
€ = 


1 axisymmetric flow 


519 





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521 





where u and v are the streamwise and transverse mean flow velocities, respec- 
tively. The term <p“v~> represents the induced transverse mass flux of the 
fluctuating portion of the flow. Using order-of-magnitude arguments, Bradshaw 
(Reference 157) and Mellor (Reference 158) have shown that this term should 

be properly retained in the boundary layer form of the compressible equations 
of motion. 


SPECIES CONTINUITY EQUATION 








& 
ele} Cle} u Ry do 
i ete ft. 0 i 
pu a5 +(ov +<p*v*> = (Fe ay ) (233) 
where: Sje = effective Schmidt number 


He effective viscosity 


X-MOMENTUM EQUATION 


The static pressure may be assumed constant for the free-mixing analysis. 


In this case, the streamwise momentum equation becomes: 


du RAs du aa € du 
=n + — = — 
pus + (ov +<p*v7> By ge ( y *) (234) | 


ENERGY EQUATION 


The energy equation in terms of enthalpy is: 


, ) a ' 
oe oe ob low + xp"e"> ea, y — 





ax oy y dy ay e ay 
h da ). u 2 
pi ni e i) de de a 
eee —-({=—) - ou —- (pv + <p*v’> 9 (235) 
i Sie °Y ay 8.J \ dy ox y 
where: hy = constituent molar enthalpy 
M = mixture molecular weight 


Te = exchange coefficient 


522 








The static enthalpy of the mixture may be eliminated as an independent 
variable by using the following definitions: 


h =+ Day hy 58) 
M i 
1 
Cp = =)oo4 Cpy (237) 
Mi 
Differentiating, 
sh dh, 3a 


LS i if nl 
=o go th 
a HT = RG ox 





l dh, da, 
ket HE yh Dihy iy 
M i i 


Ztile 


Using equation 236 and the species continuity equation, equation 233 
yields the energy equation in terms of the mixture Cp and the static 
temperature (T). 


on ee ee ee 
po Cp + (ov + <pov'>) Cy ay 5 (te » = 
1 3 9 Yow Sos 2% for) Ye foul? 
+e eT Yao tees ae bag) * a Lay oe) 
race aie fi 7 F VF Fe 
de ide BE 
=== (pv + <p*v~> ay 


The pressure, density, and temperature which appear in the above equa- 
tions are related to each other by means of the perfect gas law: 


P = oRT (239) 
where: R = mean gas constant for mixture. 
The mean mixture heat capacity is considered to be a function of the 


static temperature T. For constant R then, Cp may be related to the isen- 
tropic exponent y as: 








a 
perry (240) 
Specification of y as a function of temperature then enables calculation of 
Cp. 
The Prandtl/Glushko/Spalding model which is discussed in the following 
section has been incorporated into equations 233, 234, and 238. 


VISCOSITY MODEL 
Introduction of an eddy viscosity (ge) permits expression of the local 


shear stress in terms of an effective viscosity and the mean flow velocity 
gradient: 


du ee 
= + = —II}- < 
eS OP, (3) (pv) “u~> (241) 
t= (utu,) Ca du 


Phenomenological models, such as the Prandtl mixing length (Reference 
114) have been used to relate the "eddy" or "turbulent" viscosity (ut) to the 
local mean velocity field. These models imply that the turbulence adjusts 
immediately to changes in mean flow conditions and that a universal relation- 
ship exists between the turbulent stresses and the mean strain rates. Experi- 
mental data have indicated that there is a delayed response of the turbulence 
structure to sudden changes in mean conditions. The turbulent kinetic energy 
equation proposed by Prandtl (Reference 114), and utilized by Glushko (Reference 
4) and Spalding (Reference 118), provides a more fundamental modeling of the 
"eddy" viscosity. In the present work, the Prandtl-Kolmogorov relations, as 
given by Glushko and Spalding, are used to relate the “eddy” viscosity to the 
mean flow quantities. After Kolmogorov (Reference 115), the turbulent shear 
stress is taken as a universal function of the Reynolds number of turbulence: 


Te = ua R. (=) = Me (~) (243) 


where a = Constant = 0.2 ale L. 
a = Reynolds number of turbulence = ———— 
e B Turbulent kinetic energy 
Le = Length scale characterizing turbulence 


The "effective" viscosity is defined as the sum of the laminar and 
turbulent parts: 


= +u.2 ad +s R) (244) 


524 











Cu 

t W ' 

Defining a turbulent Prandtl number as Prt = -—, the "effective' 
thermal conductivity is given as: £ 


ke = Lo(*) 2 
e7 “pe Pipe “\u / Pr, (245) 


F 


The above relations introduce the turbulent kinetic energy as an additional 
dependent variable. The boundary layer equations cited previously may be aug- 
mented by an additional partial differential equation describing the conser- 
vation of turbulent kinetic energy. Specification or calculation of the char- 
acteristic length scale (Lt) then provides closure of the system of equations. 


TURBULENT KINETIC ENERGY 


The turbulent kinetic energy equation, discussed in Hinze (Reference 149), 
represents the balance between the advection, diffusion, production, and dis- 
sipation of turbulent kinetic energy. In the Prandtl/Glushko model (Refer- 
ences 114 and 4), the pressure-velocity correlation term and triple velocity 
correlation term arising in the development of the turbulence energy equation 
are combined as a "gradient diffusion" term. The resulting turbulent kinetic 
energy equation is: 


Je de t 2 e Je 3 2 
pu —— + (pv + <p*v*>) — = — —I Pr as oes ou} - D (246) 
ax dy y©& dy ey dy La dy - 


In modeling the dissipation term (De), it is assumed that the small-scale 
eddies responsible for the dissipation of mechanical energy are capable of 
handling all the energy transferred to them by the larger scale motion. The 
process is then assumed to be diffusion controlled, and both the exchange 
coefficient (Il.) and the dissipation term (De) are expressed in terms of an 
"intravortex" turbulent viscosity (D): 


Te = uD (247) 
= (CuD)e 
De x. (248) 
t 


The coetficient, D, is given by Glushko and Spalding as: 


D= 1+ an Rt (249) 


uw 
4 
a 








where: 
n = Constant = 0.586 (after Spalding) 


The constant (C) in the dissipation term of the turbulence energy equation 
is assigned the value 2.59 (Spalding) for application to the turbulent mixing 
problem. 


The principal uncertainty in the turbulence model resides in the charac- 
teristic length scale assigned to the turbulence (Lt). A partial differential 
equation for Lt, similar to the turbulence energy equation 246, has been 
derived by Rotta (Reference 3). In the present analysis, however, the char- 
acteristic scale of the jet turbulence is assumed independent of the trans- 
verse coordinate (y), and is expressed in terms of the geometric parameters 
of the jet. Experimental data are used to define the constants in the model. 


Referring to Figure 249, the mixing region of the single jet may be 
divided into three distinct zones. In zone 1, the flow consists of a mixing 
layer which penetrates into the uniform parallel flow emanating from the jet 
discharge. The turbulence scale in this region is assumed proportional to the 
width of the mixing layer: 


beqyy 7 — ie (250) 


L + Cho My 


i 


where: b 
My 
Ctr1, Ce2 = Empirical constants 


Width of mixing layer 


Jet discharge Mach number 


The above relationship, along with empirical values for Ct] and Ct2, is 
developed in References 49 and 159 where comparisons with experimental data 
are given. 


In Zone 3, the velocity profiles are known to be similar. The turbulent 
scale for this fully developed region is defined, after Spalding (Reference 
118), to be proportional to local radius or half-height of the jet (Yj). 


Lea) * “eg Y3 (251) 


The turbulence length scale in the transition zone (Region 2) is less 
well defined than those in Regions 1 and 3, in that experimental data are 
sparse or nonexistent. Two models are available for this region. [In the 
first, an exponential increase in length scale is assumed to occur upon 
disappearance of the potential core: 


526 








¢c 
L = £3 Yy >. “Ey (Cg z CE7 M,) 
t(2) M, (252) 


A ea Sr 
5 Ii 


The end of the transition zone is calculated as the axial station at which 
Lt(2) first becomes equal to or greater than Lt(3). The constants in this 
model must be determined from experimental data. 


The second model assumes that the length of the transition zone is equal 
to the length of the potential core (L<), and that the length scale varies 
linearly from the end of the core to the beginning of the fully developed 
region: 


Y 

= fh ae x 

Le) L+¢.. w (: L }* ee: * (; F : (253) 
2 3 € c 


Calculations based on this model generally have been found to show closer 
agreement with experimental data than those based on the model of equation 
(252). 


The numerical values assigned to the constants Cr] thru Crg are: 


Cea = 0.23 Ce5 = 0.38 
Cea = 0.38 Crg = (1.4 
Cez = (0.23 Cry = «0.43 
Ceg = «(0.05 Ceg = 0.1875 


Using equations 247 and 248, the turbulent energy equation 246 may 
be rewritten as: 


3 dy dy 


9 p) a € de 
pu = + (pv + <p*v>) <= a (0 ze 
x ye 








BOUNDARY CONDITIONS 


Considering the jet to be symmetric about the line y = 0, the applicable 
boundary conditions are: 


du de aT if 
@ = ——_> = — = — = — = c 
y = 0 By By Oy” By 0 (255) 
(a = = 
°F Vex * Yex 
fa Tf 
ex 
@e=e 
ex 
eg 1. 
i 1ex 


TRANSFORMATION OF DIFFERENTIAL EQUATIONS 


The preceding equations may be cast in a more convenient form by non- 
dimensionalizing with respect to the primary jet diameter and the discharge 
flow field variables up, ep, Tp. The requisite relations are: 


eo Bae y= 2 
P Pp 
u Vv Ay ward 

32 Vee , Vie (256) 
P p P 


Substitution in equations 232 through 235 and equation 254 yields: 


Continuity 


€ 
a a safe 
2G) + 2—> [(oV + <p*V*>) ¥°] = 0 (257) 








X-Momentum 











aU Heh eae & \9 e aU 
oy — + 2 (py + <p >) — = | ——  ) — Eel 
PU 3x fey + <p°W">) oy | oY (H z a4 (258) 
Wwod. ¥ 
P Pp 
Turbulent Kinetic Energy 
JE mes dE 4 3 € dE 
<= or Bs oh aero | eee) — 
eU xy 2 (pV + <p~V~>) aY 2 ar ay (uby a) 
\P P (259) 
4u ae uCDd_E 
+(——P- J u (2) - ——2- 
gJdoe, c oY ¢ L2 
pt 
Species Continuity 
da, da, » % u_Y° 30 
ay? Bean prem: same Ie ey ae ae 
OU 3X ad CON Fe <p o>) i =) 3rl3 7 (260) 
ud ¥ ie 
P Pp 


Energy 





06 gay ls i e 06 
ba ag SPU Sev OP ae on te (k, ve ) 
C_u 
P Pp 





The boundary conditions become: 


Ble! 


aU _ 9E _ 26 i 





a — => = = = 
Shed wy” oy” or” 0 
‘a > y J = ] 
@¥Y tan U vee (262) 
E = E 
ex 
= ex 
a ae “tex 


The continuity equation 257 may be identically satisfied by introduction 
of the stream function coordinate ()) as one of the independent variables 
(Von Mises transformation). Define a modified stream function, satisfying 
continuity as follows: 


iy J 3 Se 
ae ee =" - (pV + <p“V>) ¥© nc 


Using these relationships in equations 258 through 261 removes the 
transverse velocity components (V,V~). 


The system of nonlinear parabolic differential equations (257 through 
261), along with boundary conditions and suitably prescribed initial condi- 
tions, represents a properly posed initial value problem. The solution may 
be stepwise advanced in the positive X direction using a finite difference 
approach, The implicit numerical technique utilized in the present investi- 
gation closely parallels that of Patankar and Spalding (Reference 160). The 
general programs of these latter investigators were not available at the 
inception of this project. The differences between the two approaches arise 
primarily in the type of finite difference mesh and the methods of controlling 
the streamwise step size. 


The Patankar method uses a fixed dimensionless streamline grid. The 
stream-wise step size is controlled by explicitly monitoring the flow environ- 
ment over a given step. In the present work, streamlines are added in a 
systematic fashion to cover the entire flow field. The streamwise step size 
is related to the local geometric parameters of the jet, such as the width of 
the mixing zone of the distance to the effective edge of the jet. 


The partial differential equations are all of the general "diffusion 
equation" form: 


4 ¢ aF aF 
SF = qQ i (2 a) + v + 6F + Y) — (264) 
aX ay y 


530 











where F may represent any of the independent variables (U,E,4,aj). The 
equations are coupled principally through the normal derivative terms. Hence, 
a sequential solution technique may be utilized, as opposed to direct simul- 
taneous integration. The specific procedure consists of initially formulating 
the differential equations in terms of linear difference equations. The 
linearization is effected by evaluating the coefficients of the differential 
equations at the known upstream mesh voints. The X-momentum equation is 

first solved for values of U and 3U/3¥ at the downstream station. Using these 
quantities, the turbulent kinetic energy equation may be integrated to 
determine E and 3E/3¥. Finally, the species continuity and energy equations 
may be solved using the results of the preceding two solutions. 


VALIDATION OF MODEL BY COMPARISON WITH EXPERIMENTAL DATA 


The JETMIX analysis has been compared with a wide spectrum of experi- 
mental data. The comparisons (some of which are shown here) have, in general, 
been very good. In particular, the JETMIX analysis has proven capable of 
accurately predicting the effects of changes in the external environment of 
the jet, including changes in the velocity, temperature, composition, and/or 
pressure of the external stream. 


In Figures 251, 252, and 253, the measured cross-stream mean velocity 
profiles of Laurence (Reference 161) are compared with JETMIX predictions at 
each of three axial locations. A corresponding comparison for the turbulence 
intensity is shown in Figure 254. The excellent agreement between the JETMIX 
predictions and the experimental data for both the velocity profile comparisons 
indicates the potential of the JETMIX analysis as an exhaust-plume, flow-field 
prediction tool. 


Comparisons between hot-jet test results and the free turbulent mixing 
analysis are given in Figure 255 through 258. These comparisons, which 
show the axial variation of the total temperature, the total pressure, the 
velocity, and the Mach number on the jet centerline, are again very good. 
These data were obtained at General Electric using a traversing water-cooled 
rake. The experimental setup consisted of an exhaust jet issuing from a 
conical nozzle at a pressure ratio of 3.2. The facility temperature was 
2400° R. The actual rake data indicated that the nozzle was operating at a 
fully expanded Mach number, M = 1.45, and at a total temperature of 2550° R. 
These measured conditions in the supersonic core were used as the initial 
conditions for the mixing analysis predictions. 


Additional comparisons between these same hot-jet test results and the 
JETMIX analysis are given in Figures 259 through 266. Figures 259 and 260 
show typical radial profiles of total temperature at each of two axial locations. 
Figures 261 and 262 show the corresponding cross-stream velocity profiles, 
while Figures 263 and 264, and Figures 265 and 266, show the corresponding 
Mach number profiles and total-pressure profiles, respectively. It is 
emphasized that the empirical constants which were used in the turbulence 


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Total Temperature - Th 





3000 






































af, = 2, 48 
I Dy 2.79 
[L)O Test Data 
2500 pas - T si 2400 
2; T,, 8 
— — Predicted Data 
2000 
1500 
1000 
500 
O 





-3D =-2D =i O 1D 2D 3D 


ius/D 
Radius/I 8 


Figure 259. 4.3" Conical Nozzle Exhaust Plume Total Temperature Versus 
Radius, JENOTS Wake Rake Data. 


e 


540 


2500 


& 2000 
& 
1 
v 
I 
b= | 
~ 
a 

o 1500 
a 
E 
v 
= 
Cod 
& 

1000 

500 

0) 

Figure 





L/Dg = 19.53 


OOO Test Data 


vest ° 
Th = 2400” R 


Predicted Data 





























-3D -2D -1D 0 1D 2D 3D 
Radius/D 
adius/ 8 


260, 4.3" Conical Nozzle Exhaust Plume Total Temperature Versus 
Radius, JENOTS Wake Rake Data. 


541 


ft/sec 


Velocity, 


3500 


L/Dg = 2. 79 


OO Test Data 


“ x : = 2400 
3000 : T a 


Predicted Data 


R 


=e 











2500 





2000 











1500 





1000 











500 





- 3D -2D =i 





O 


Radius/D 
adius/ 8 


Figure 261. 4,3" Conical Nozzle Exhaust 


JENOTS Wake Rake 


Data, 


542 


Plume 


1D 


Velocity 














2D 3D 


Versus Radius, 


ft 


Velocity, 


2500 


2000 


L500 


L000 


500 


Figure 


2400 


ed Da 


L/D 19.53 
Ss 
7 mn 
©) Test Data 
P Pe” aes | 
R : 
Predict 
+ 
a= = 
| 


- 3D -2D 


262, 4,3" Conical 
JENOTS Wake 





= LI 


Nozz 


Rake 


R 


ta 


) a) 


Radius/D, 
bad 


le Exhaust 
Data. 


543 


Plume 


LD 


Velocity 


Versus 





Radius, 


M 


Mach Number, 


= 2,7 
L/D, 9 


TAS O Test Data 


= 2400° 
T= 2 R 


Predicted Data 





























-3D =-2D ~1D 0 1D 2D 3D 


Radius/D, 


Figure 263, 4,3" Conical Nozzle Exhaust Plume Mach Number Versus 
Radius, JENOTS Wake Rake Data. 








M 


Mach Number, 


Figure 
ania 





ES = 
De 19.53 








rAS ie Test Data 
= 2400° R 


— Predicted Data 












































-3D -2D -1D 0 1D 


adius/D 
Rac 8 


264. 4,3" Nozzle Exhaust Plume Mach Number Versus Radius, 
Wake Rake Data, 


uo 
a 
uw 





3D 


JENOTS 


Total Pressure - Pos psia 


50 


40 


35 


30 


20 


15 


10 


Figure 265. 





/ as 
L/D. a. 


<> (\ Test Data 


Pp = 3.2; Ty = 2400° R 


—— Predicted Data 










































































=-3D -2D =1D 0 1D 2D 3D 


Radius, JENOTS 


Radius/D, 


4.3" Conical Nozzle Exhaust Plume Total Pressure Versus 


Wake Rake Data, 


546 


P.. sia 
T p 


Pressure 


Total 





L/D 


Of Test Data 


= 3.2; T, = 2400° R —-L. 








R 


— Predicted Data 





















































-3D -2D -1D 0 1D 2D 3D 


Radius/D 
8 


Figure 266. 4,3" Conical Nozzle Exhaust Plume Total Pressure Versus 
Radius, JENOTS Wake Rake Data. 





model were not changed between the cold-jet predictions of Laurence's data 
and these hot-jet predictions. Consequently, the excellent agreement between 
the hot-jet data and the JETMIX predictions verifies that the analysis is 
capable of taking into account the effects of density changes in the flow. 





548 





APPENDIX 7 


ANALYSIS INCORPORATED IN SSFD/MERGE 


There are two distinct techniques which have been used to analyze the 
aerodynamic flow field in a supersonic jet. In the first approach, the jet is 
treated as a viscous, boundary layer flow. The resulting flow field is of the 
type depicted in Figure 267. According to the usual boundary layer approxima- 
tions, the radial velocity components are assumed small in comparison to their 
axial counterparts, and, in addition, the pressure is taken to be constant 
throughout the whole flow field. These approximations implicity assume that 
the static pressure at the jet exit plane is identical to the ambient pressure 
and that Prandtl-Meyer expansions and/or shock waves are not present in the 
flow field. Consequently, this viscous boundary layer analysis can only be 
applied to subsonic jets, or to supersonic jets which are ideally (or nearly 
ideally) expanded. 


In contrast to this viscous analysis, the second traditional technique 
for analyzing supersonic jets completely ignores the effects of turbulent 
mixing. In this second (inviscid) analysis, the full two-dimensional 
equations of motion are used, and strong radial and axial pressure gradients 
can occur. These pressure gradients have their origin at the nozzle exit 
plane where the static pressure generally is significantly different from the 
ambient pressure. In adjusting to the ambient pressure, the flow field 
generally develops a series of shock waves and Prandtl-Meyer expansions in a 
nearly periodic cell-like fashion. A schematic description of the quali- 
tative features of a jet described by this two-dimensional analysis is shown 
on Figure 268. 


As indicated above, both of these approximate models are applicable to the 
analysis of a certain class of supersonic jet. However, as might be expected, 
neither model applies to all supersonic jets. Thus, for example, the effects 
of friction can never be entirely removed from the jet. Further, supersonic 
jets are seldom uniform, parallel ideally expanded jets. Consequently, in 
order to obtain an acoustic prediction technique which is applicable to both 
ideally expanded and nonideally expanded jets, the aerodynamic model must 
include both two-dimensional effects and viscous mixing effects. This is 
accomplished by dividing the jet into an inner region and an outer region as 
shown in Figure 269. The outer region of the jet contains that part of the 
jet in which the effects of turbulent mixing are significant. Near the nozzle 
exit, the outer region is composed of a narrow annular portion of the flow 
field on the outer edge of the jet; downstream of the exit plane, the thick- 
ness of the outer region increases until eventually it includes the entire 
jet. In our analysis, this outer region is computed by our original viscous, 
boundary layer (JETMIX) computer program. Now, whereas the outer region of 
the jet is dominated by the effects of viscous mixing, the inner region of 
the jet is dominated by the familiar Prandtl-Meyer expansions and shock waves 
which characterize two-dimensional supersonic flow fields. In order to include 


549 














Fully 
Developed 
Region 


Transition | 
Region Region 


Figure 267. The Flowfield of an Ideally Expanded Viscous Jet. 


Figure 268. The Flowfield of an Inviscid Two-dimensional Supersonic Jet. 





"Boundaries" of 
Sonic Line VERes, SOE4On 





Viscous Mixing 

Region eT Se 
Nozzle oe i 7 
Plane ar i : 


Ae 









Shock Formed 


B ‘i 

M, y Coalescing ae ie 
J Characteristics oc 

ooo 


Inner Region 


Figure 269, Subdivision of the Jet into Inner and Outer Regions (Outer 
Region Shown Shaded). 


ao 
a 
_ 





these effects in our aerodynamic model, a new computer program has been written 
to handle this inner region. This new program is called the Supersonic Finite 
Difference (SSFD) computer program. Thus, in the refined aerodynamic model, a 
supersonic jet is analyzed in two parts by two separate computer programs. 

The outer portion of the jet is analyzed by the viscous JETMIX analysis. The 
inner portion of the jet is computed by the two-dimensional SSFD analysis. 


As indicated in Figure 269, these two separate parts of the flow field 
are matched along the sonic line. Thus, the inner portion of the flow field is 
supersonic while the outer flow is subsonic. (In actuality some constant Mach 
number line, which is slightly supersonic, is chosen as the matching line 
rather than precisely the sonic line.) However, it must be noted that the 
sonic line appears in the jet because viscous effects have reduced the Mach 
number of the formerly supersonic flow. This indicates that the outer edge of 
the supersonic region has experienced considerable viscous effects. Thus, in 
order to include the two-dimensional effects in as large a region as possible, 
and in order to enforce as smooth a match as possible between the inner and 
outer solutions, the effects of the viscous mixing are included in the inner 
(SSFD) analysis as known "right-hand-side" terms. The magnitude of the 
"right-hand~side" terms is estimated from the viscous JETMIX computer program 
as indicated later. This matching technique allows the total pressure to 
vary continuously from the outer edge of the jet (where the flow is essentially 
stagnated) through the sonic line and all the way to the jet centerline (where 
the flow is supersonic). Then, by matching the static pressure at the sonic 
line, we can be sure that all flow properties are continuous at the matching { 
line. 


DERIVATION OF THE EQUATIONS USED IN THE INNER REGION 


As indicated above, the equations used in the inner region include the 
viscous effects as "right-hand-side”" terms. In order to obtain the form of 
these "right-hand-side" terms, the equations for the inner region are obtained 
from the complete Navier-Stokes equations. An outline of the derivation 
follows. 


The equations of motion for steady, compressible, viscous flow are: 





Vepve=QO (265) 

o(v.Vv+%pe2V.T (266) 

~pv.Ve=tT:VW-pyV-v 0-q (267) 
where: e = internal energy 


= pressure 


= heat flux vector | 


Qi vo 





v= velocity vector 


= density 


ne 


= viscous stress tensor 


an 


Two vector identities which are useful are: 


ve (vw. VY Vev.e VW. v)/2)] (268) 
ve Oh te we. Oe eo eS (269) 


If we dot equation (266) by the velocity vector, v, and use identity (268), 
we obtain: 


ov. * fo. v1 eo. tp ee. a td (270) 


Then, combining equations (265) and (267) and using identity (269), the 
energy equation becomes: 


aM. Tew Ve tp Ou «TRO Rs Ws = Oa (271) 


where we have also converted from internal energy to enthalpy. Then, adding 
equations (270) and (271) gives: 


° - os me 

5¥.vh @%. Ww. tT) -%. 4q (272) 

where: h = enthalpy 
h°= stagnation enthalpy 
e 

We now define the scalar function, Q, as: 

Q (x,y) =p Vv. Vb? (273) 
and, by equation (272), we also have: 

Qiy)ev. WwW. - 0. Gg (274) 


If we now combine equation (274) with equation (271) and use the thermo- 
dynamic equation of state, the energy equation becomes: 


oT (¥. Ve) @ev. (V. 7) +Q (275) 


553 





where: s 
E 


entropy 


temperature 


Finally, defining the scalar function, 4, as: 


¢ (x,y) =v. (V.T) (276) 


we can write the entropy variation along a streamline as: 


pF Vive =-— d+ 0 GATT) 


Thus, the equations of motion can now be rewritten as: 


V.pv=0 (278) 
0 (v. VW v+VpeR (279) 
APY. Genes. BSD (280) 


where the vector R is defined as: 


R=eV.t 


(281) 
so that: 
@ (x, ye © v.R (282) 
At present, the SSFD program limits the function, Q (X,Y), to the 
trivial function: 
Q (x,y) = 0 (283) 


This implies that only flow fields which have uniform total temperature 
throughout can be calculated. The extension of the computer program to 
include an arbitrary specification of the stagnation enthalpy is relatively 
simple. Note that the function, $, is not restricted; it can (in principle) 
be any function. When coupled to the JETMIX viscous analysis, the SSFD 
program automatically determines ¢ from the JETMIX-predicted entropy gain due 
to the turbulent stresses, 


w 
w 
Ps 

















The equations of motion (278), (279), and (280) can be expressed in x-y 
coordinates as: 


3 € a € 
ax (PUY dD +a levy ) = O (284) 
Cre eee ae 
ou a + OV ay + ee Ry (285) 
ov Bd 22 
a a By + ay R, (286) 
3s 3s 
oo ty Pee ee eee (287) 
where: e = 0 Plane (2-D) flow 
emi Axisymmetric flow 
Rj, R2 = Axial and transverse components of R 


Axial and transverse components of velocity, v 


u,v 


Since the inner region is restricted by definition to be a supersonic 
flow region, it follows that equations (284) through (287) form a system 
which is hyperbolic in character. Consequently, these equations, like the 
(parabolic) boundary layer equations which are used in the outer region, can 
be solved by a marching technique. Although the classical method-of- 
characteristics (MOC) procedure could be used to solve these equations, we 
have chosen to use an explicit finite-difference algorithm, since it is some- 
what simpler and is more easily adapted to the matching procedure. During 
the calculation, the inner solution is determined on a series of axial planes 
at successively greater distances from the nozzle exit. The use of axial 
computation stations in the inner solution is convenient, since the outer 
solution also is determined on axial planes. Similarly, both analyses use 
the stream function, , as the cross-stream variable. Streamline-following 
simplifies the enforcement of the desired entropy and stagnation enthalpy 
variations along the streamlines. 


TRANSFORMATION OF THE DIFFERENTIAL EQUATIONS 


Before being replaced by their finite-difference equivalents, equations 
(284) through (287) are transformed to the x- coordinate system. The 
stream function, , is defined by: 


é 


ay 4. puy' oy == pvy- (288) 


ay Ix 


uw 
ow 
wu 


iene ca 


The dependent variables also are replaced by the more convenient quantities: 


e logarithm of pressure, In p 

e tangent of the local flow angle, t = = 
° entropy, s 

® stagnation enthalpy. h° 


These four quantities represent the classical dependent variables for two- 
dimensional supersonic flow calculations, with two minor changes. First of 
all, the pressure has been expressed as the natural logarithm of the pressure, 
rather than as the pressure itself. This variation has been suggested in 
Reference 162 as a means of obtaining more accurate results in finite- 
difference calculations. Our experience has verified this finding, particu- 
larly in regions of high gradients such as those near the focal point of 
expansion fans. An unfavorable aspect of using ln p rather than p is that 

an increase in processing time is required, but the improved accuracy and 
reliability of the code seems to justify this added expense. The second 
change, the use of t rather than the flow angle, 6, is based on computational 
economy. There seems to be no accuracy advantage for using either t or § as 
the second dependent variable; consequently, t has been chosen because it 
eliminates extensive use of the trigonometric functions. The remaining two 
primary variables (entropy, s) and (stagnation enthalpy, h°) are chosen since 
the variation of these functions will be specified along the streamlines. 


Transforming to the x-) coordinate system and introducing the new depen- 
dent variables leads to the system of equations: 


oTu = = = ¢ (x,¥) +Q (x,¥) (289) 
ah° 

pera SN (x,) (290) 
2 3t 2 e 3 3 

ou oP eS hoa = ee & xk, (291) 

a « 2) 3p + {eis aty't) Pn a. ae (1 + uy R (292) 

cz 3x ceater ay Pvy On = h 1 


where we have used the assumptions of a perfect gas with constant specific 
heats. Note that, since partial differentiation with respect to x implies 
differentiation along a streamline, equations (289) and (290) represent 
the substantial derivatives of entropy and stagnation enthalpy. These 
equations explicitly relate the variation of s and h® along streamlines to 
the viscous effects, 


556 


Equations (289) through (292) are solved numerically by the MacCormack 
two-step finite-difference algorithm (Reference 163). This procedure is 
simpler and faster than the classical Lax-Wendroff technique (see Reference 
164), but still maintains second-order accuracy. Being explicit, the 
algorithm is only conditionally stable, and the maximum stream-wise step size 
is limited by the Courant-Friedrichs-Lewy (CFL) condition (see Reference 164). 


The marching calculation starts by determining the entropy and stagnation 
enthalpy at the new x-station for a particular streamline from equations (289) 
and (290). This is basically a table-lookup operation. Then, the pressure at 
the new x-station is determined from equation (292); and, finally using this 
value of 3p/3~, the flow angle at the new station is determined from equation 
(291). 


The SSFD computer program is limited to equal spacing in the stream 
function, ». Changing to variable cross-stream-mesh spacing is possible; but, 
with the MacCormack algorithm, a change to variuble-mesh spacing would render 
the calculation first-order accurate. (Note that each downstream step size 
is calculated as the computation proceeds so that, in general, no two-stream- 
wise step sizes are identical but rather only the cross~stream-mesh widths are 
the same.) Equal intervals in | implies equal amounts of mass flow between 
each of the mesh points in the field. In an axisymmetric flow calculation, 
this equal spacing in stream-function means that the grid points will become 
less dense in physical space as the axis is approached. This tendency is 
further amplified in an underexpanded exhaust jet because the flow responds 
to the lower ambient pressure by expanding and, by so doing, moves the points 
nearest the axis of symmetry further from the centerline. In order to 
counteract this grid-point spreading near the centerline, an existing SSFD 
capability for running two parallel flows was used. Thus, any jet calculation 
is run as a coannular pair of jets with a complete slip-line boundary between 
them. Since the coannular jets are really comprised from a single jet, no 
discontinuity in temperature or stagnation pressure is used across the slip 
line. This coannular jet artifice is used only to increase the density 
grid points near the axis of symmetry while, at the same time, maintai 
convenience and economy of equal cross-stream spacing. As the prograr 
currently set up, it will automatically divide any axisymmetric jet 
coannular ones. 


BOUNDARY CONDITIONS 


Although the field points are calculated by a fi 
the boundary conditions are enforced by means of a met 
approach similar to the one used by Moretti (Reter 
finite-difference calculations. The desirabil 
conditions in finite-difference calculation 
study by Abbett (Reference 166), in whi: 
approach (along with one other method) w 


an entire series of other numerical 











“ AD-A036 614 GENERAL ELECTRIC CO CINCINNATI OHIO AIRCRAFT ENGINE GROUP F/6 20/1 
re SUPERSONIC JET EXHAUST NOISE INVESTIGATION. VOLUME III. COMPUTE--ETC(U) 
JUL 76 D R FERGUSONs M A SMITHe P R KNOTT F33615-73-C-2031 
UNCLASSIFIED R74AE6452-VOL<-5 AFAPL-TR-76-66-VOL=3 


2 i 








ile 


mw 








In applying the boundary conditions, we first note that the characteristic 
directions are unaffected by the viscous "right~hand-side" terms and so remain 
the same as in inviscid flows. The viscous terms, however, do affect the 
compatibility conditions which must be enforced along the characteristics. 


By using standard techniques, the compatibility conditions can be shown to be: 


1 uw dt + MZ =] dp (293) 


T -1 Ig 
eae [ opuc - = (-¢+0) + say (t M4-1 + »| dg 


where: = speed of sound 


( 
M = Mach number 
Y 


= ratio of specific heats 


At the outer edge of the jet, the pressure is specified; and, the single 
characteristic from the new boundary point to the previous x-station is 
sufficient to determine the flow angle, v/u, as shown in Figure 270. Along 
the axis of symmetry, the flow angle must go to zero. In this case, it is the 
pressure which is determined from the compatibility relation. 


In either of these MOC calculations, it is necessary to know the value of 
the flow properties where the characteristic intersects the plane correspond- 
ing to the previous x-station. In all cases such as these, the values are 
found by means of linear interpolation. The use of higher order interpolation 
formulas is not recommended, since they frequently lead to increased errors 
arising from the nonanalytic nature of the supersonic flow variables. 


METHOD FOR INCLUDING DISCRETE SHOCKS 


The presence of discrete shock waves in a supersonic flow field, and the 
calculation of their location and strength, introduces two distinct problems. 
First, since the shock wave, in general, will lie between two field points 
(rather than being concident with one of them), additional shock-point 
storage locations must be introduced. Once these additional storage locations 
are provided, the propagation of the shock can be accomplished in a rela- 
tively straightforward manner. The particular method we have chosen is 
analogous to our technique for calculating boundary points, in that an MOC 
procedure is used, Thus, in computing the shock locus from the previous 
x-station to the new x-station, the new radial position of the shock is first 
approximated. The properties on the upstream side of the shock are then 
calculated by using the complete MOC equations along both the left-running 
and the right-running characteristics, as shown in Figure 270. Then, by 
geometrical relationships, the shock angle at the downstream station can be 








(a) Boundary Calculation (b) Shock Calculation 
—---Characteristic 


Figure 270. Calculation of Boundary Con- 
ditions and Shock Strength 
and Location by Method of 
Characteristics. 








determined. The known conditions in front of the shock, plus the shock 
angle, are then sufficient to define all properties downstream of the shock 
via the Rankine-Hugoniot jump conditions. Then, finally, the compatibility 
condition for the single characteristic which overtakes the shock from the 
downstream side can be used to improve the predicted location of the shock. 


A second problem which must be surmounted when there is a shock in the 
flow field is the calculation of the regular field points which are adjacent to 
the shock. Calculation of such points requires a finite-difference technique 
which "reaches across" the shock. In our analysis, we adjust the properties on 
the opposite side of the shock by the jump values for that particular quantity, 
as shown in Figure 271. Then, using these adjusted values, we proceed to 
calculate these near-shock points in the normal manner. This procedure 
implicitly assumes that the gradients of the flow variables change by only 
small amounts across the shock. Although this is not necessarily the case, 
we have obtained good results with this method even in rotational flows. We 
have also tried a variation of the method suggested in Reference 167, but have 
returned to the present technique. 


At present, the SSFD computer program is capable of including only a 
single shock in the flow field at any axial location. When there are no 
shocks in the field, the program monitors the developing solution to detect 
the generation of shock waves, either from a sharp corner in a solid boundary, 
or due to the coalescence of characteristics. Since the program operates on 
a finite-difference technique rather than along characteristic lines, the 
characteristics are not readily available; however, their slopes are calcu- 
lated at each axial station in order to determine the allowable step size to 
the next axial station. The shock search procedure observes the angle change 
between characteristics of the same family which emanate from adjacent grid 
points. When these characteristics converge toward each other at faster than 
a prespecified rate (currently set at 5 degrees), a shock is inserted and the 
coalescing waves are represented by a discontinuous Rankin-Hugoniot jump. 
This shock search procedure is quite simple and, to date, has proved to be 
reliable. 


SHOCK REFLECTION FROM AXIS OF SYMMETRY 


As a shock wave in an axisymmetric flow field approaches the centerline, 
the shock becomes increasingly steeper. Because of this steepening, the 
axisymmetric equations will not allow the shock to reflect from the symmetry 
axis in a regular fashion. Instead, some sort of "strong" reflection must 
occur. As a result, a local pocket of subsonic flow appears behind the shock, 
and any computational procedure which relies on the hyperbolic character of 
the equations becomes invalid and has to be terminated. However, experimental 
Schlieren photographs show that this subsonic region is frequently small or 
even nonexistent (Reference 168). Thus, although the reflection shows up as 
a nearly normal "Mach disc" or "Rieman wave" in some cases, an apparently 
regular reflection takes place in other cases when the shock is sufficiently 
weak (even though the inviscid equations will not allow this). In order to 


560 





@ jr+l @ Property Values 
at Regular Grid 
3 Points 
QO Fictitious Values 
ge jt+l Used for Differ- 
oj encing Across 


Shock 





Yj-1 Yj Yj+l1 Yj+2 
Stream Function - Y 


or rs 


= - 

Jj 5 rt 

Via ° Ga 
Shock Jump for 
Property 9% 


i 
Ss 
" 


Figure 271, Method for Obtaining Fictitious Shock~corrected 
Properties for Use in Calculating Field Points 
Adjacent to Shocks, 


561 











provide a means for continuing the flow field calculation beyond the location 
at which the shock first reflects from the axis of symmetry, we have incor- 
porated two approximate techniques for "calculating through" this presumably 
small, localized subsonic pocket. First, when the incoming shock is weak, a 
"regular" reflection procedure is used. However, for stronger incoming shocks, 
we switch over to a "Mach disc" reflection procedure. Which of the two 
techniques is to be used must be determined by the problem at hand. 


The "regular" reflection procedure utilizes a suggestion by Oswatitsch 
(Reference 169) that the axis of symmetry be "enlarged" near the shock in- 
pingement point so that the radial coordinate becomes small but still remains 
finite. The radial size of this "enlargement" is determined by the program, 
depending on the local strength of the shock. (Stronger shocks require more 
"fattening" of the axis of symmetry.) It is emphasized that these "enlarge- 
ments" generally encompass less than one half of one percent of the original 
mass flow so that they are scarcely detectable on a "blown up" plot of the 
shock locus. 


The "Mach disc" reflection technique (which is considerably more compli- 
cated than the "regular" reflection procedure) involves the insertion of a 
triple point and the use of an iterative technique to determine its location. 
In this analysis, a triple point is inserted at a chosen location on the 
shock, and the oblique shock which is moving radially inward is forced to 
branch into a second outward-running shock and a normal shock which extends 
to the axis as shown in Figure 272. The normal shock represents the Mach 
disc. A slip line is also generated at the "lambda" intersection. Downstream 
of the Mach disc, this slip line serves as a boundary between the supersonic 
flow and the subsonic flow. The supersonic flow is handled by the standard 
SSFD algorithm, while the subsonic flow is analyzed by a one-dimensional 
approximation. The height of this one-dimensional channel at succeeding 
axial locations is determined by requiring the pressure to be balanced across 
the slipstream, and by requiring the supersonic flow to be tangent to the slip 
line. This matching requirement causes the Mach number in the one-dimensional 
stream to vary as it flows downstream. The axial position of the Mach disc 
is then iteratively determined based on the behavior of the flow in this 
one-dimensional channel. The Mach disc is said to have been correctly 
positioned when the slip line forms a "throat" which reaccelerates the sub- 
sonic flow through sonic velocity in a smooth, continuous fashion. This 
Mach disc model is very similar to the ones used by Abbett (Reference 170), 
Averenkova, et al, (Reference 162),and Fox (Reference 171). Comparisons 
between this Mach disc model and experimental results, have shown reasonable 
agreement, but the iterative procedure is quite expensive (in terms of 
computer processing time) and tends to be unreliable. Most of the results 
presented in this report are based on the "regular" reflection. 


CALCULATION OF THE TURBULENCE FIELD 
The previous sections have described the model which is used to predict 


the velocity field in a nonideally expanded jet. However, before the acoustic 
characteristics of the jet can be determined, it is necessary to know some- 


562 








Oblique 






_— 


~ 
a 
— ee ee ee 





Slipline 
Mach Disk 


Figure 272, Mach Disk Model, 











thing of the turbulence field in the jet. As indicated in Appendix 6 (or, see 
Reference 6), the turbulence model which is used in the JETMIX computer 
program is based on a turbulent kinetic energy approach. For ideally 

expanded jets, the magnitude of this turbulent kinetic energy has been used to 
evaluate the source terms in the classical Lighthill acoustic equations 
(Reference 10). Once these source terms are evaluated, the acoustic sig- 
nature of the jet can be readily determined. Since this acoustic formulation 
is based on the local mean and fluctuating properties of the jet, it should 
also be directly applicable to nonideally expanded jets. (This, of course, 
does not imply that the model would predict the same acoustic radiation from 
ideally expanded jets, because both the mean velocity field and the turbu- 
lence field depend on the expansion ratio of the jet.) Thus, it remains to 
determine the turbulent kinetic energy in the nonideally expanded jet. 


The conversation of turbulent kinetic energy is governed by the balance 
between the production, dissipation, convection, and diffusion of turbulence 
energy throughout the flow field. The form of the turbulent kinetic energy 
equation which is used in the JETMIX analysis is: 


de Ue ee 30 € de 
ail ax se ay ry ye ay (C)u, y 3y> sal 
3/2 
iv, aabe eS 
Me oy Ly 


where Cy, Cy constants 


e = turbulent kinetic energy 


Ly = turbulent length scale 
1. * turbulent eddy viscosity 


The turbulent kinetic energy is related to the three components of fluctuating 
velocity by: 


2 


é@ 1/2 ( a" > + < v's + < w?*>) (295) 


As used in the JETMIX analysis, equation 294 applies to ideally expanded 
jets. However, there is one flow phenomenon which affects the turbulence 
levels, and which is unique to nonideally expanded jets, that is not included 
in the conservation equation (294). This phenomenon is the presence of shock 
waves in the flow field. These effects have been included in our analysis by 
means of Ribner'’s shock-turbulence interaction model (References 172-174). 





Ribner's analysis starts by decomposing the turbulence field into an 
infinite number of elementary vorticity waves of all wavelengths and orien- 
tations. Then for any one of these elementary waves, he calculates the 
manner in which the vorticity of the wave is altered as it is convected 
through a normal shock wave. The results of his calculation show that the 
Magnitude of the vorticity is increased as the wave goes through the shock. 
(Besides the increased vorticity, two new waves are generated, an entropy 
wave and an acoustic wave.) A summation over all wave numbers of the 
effects of the shock on each individual wave then yields an amplification 
factor for the turbulence as it is transmitted through the shock. Conver- 
sion from turbulence convected through a normal shock to turbulence convected 
through an oblique shock is made by a transformation of coordinates (Figure 
273). It should be noted that, although Ribner's analysis strictly applies 
only to straight shocks, it can also be applied to curved shocks (such as 
occur in supersonic jets), as long as the radius of curvature of the shock 
is significantly larger than the longest wavelength of the turbulence. 


In Ribner's analysis, the three components of turbulence amplification 
are defined by: 


T 





<2 
<u, > 2 2 2 
= 32 f \s| cos 68, cos 6, dé, (296) 
' 
<u > ° 
1 
2 «2 aul 
+< 
bs Mids Thee 1+ f* is] 7 sin 6, cos 6, de, (297) 
<ut?> ° 


Figure 274 shows the amplification of turbulence by a shock in terms of 
the ratio of turbulent kinetic energy in front of and behind the shock. The 
turbulence amplification is plotted as a function of the ratio of the normal 
components of velocity in front of and behind the shock. As can be seen, the 
amplification is unity at a velocity ratio of unity (shock of vanishing 
strength) but quickly increases to a maximum of some 20% amplification for 
moderate shock strengths (normal component of incoming Mach number about 1.5). 


In our computer model, the turbulent kinetic energy is monitored at each 
point in the flow by means of equation (294), suitably transformed to x-y 
coordinates. When a shock wave is encountered, the turbulence amplification 
is determined from Ribner's theory, and the turbulent kinetic energy is 
increased locally by the amount of turbulence which is generated at the 


565 












Refracted 
Shear-Entropy 
Wave 


Shock 


Sound Wave 


U 





Initial Shear 
Wave 


Figure 273. Passage of Turbulence Through a Shock, 


566 





1 


Turbulence Amplification Across Shock Wave, e,/e 


Figure 274. 


Outgoing Shear Wave 


"Rippled"” Shock Wave 


Incoming 
Shear 
Wave 





2 3 4 5 6 
Ratio of Normal Components Across Shock, u,/u, ‘ 


Amplification of Turbulence by a Shock Wave as Predicted by 
Ribner's Shock-Turbulence Theory, 


567 





shock. The resulting turbulence energy profiles have a discontinuous jump 
across the shock. Equation (294) is solved numerically using an implicit 
Crank-Nicholson procedure. Since this numerical algorithm is unconditionally 
stable, there is no restriction on the axial step size which must be used. 
Consequently, the step-size is determined by the supersonic flow equation, 
and the same (Ax) step size is used for both the hyperbolic supersonic flow 
equations and the parabolic turbulent kinetic energy equation. It should be 
noted that despite their differing mathematical character, both the super- 
sonic flow equations and the turbulent kinetic energy equation can be 

solved by a "marching" process, and that the simultaneous solution of the 
two equations therefore can be performed quite compatibly. 


In calculating the turbulent kinetic energy equation, the velocity pro- 
files which are used as coefficients are obtained from the SSFD solution, 
while the mixing length and the outer boundary condition on the turbulent 
kinetic energy are obtained from the original JETMIX solution. 


MATCHING BETWEEN THE INNER AND OUTER SOLUTIONS 


As described previously, the equations for the inner region require 
that the variations in entropy and stagnation enthalpy due to the turbulent 
shearing stresses be specified along each streamline before the solution is 
calculated. In the computational procedure, these variations are first 
estimated from a solution of the viscous boundary layer equations for the 
entire mass flow in the jet (i.e., both inner and outer flows). This viscous 
calculation also establishes the value of the stream function at which the 
flow becomes sonic and, hence, the location of the boundary between the inner 
and outer regions. Then, using the calculated entropy/enthalpy variations 
as a first approximation to their actual behavior, the velocity field in the 
inner region is recalculated by means of the two-dimensional (inner) equa- 
tions. (The velocity field in the outer region is left unchanged except for 
repositioning the streamlines so that they match with the streamlines in the 
inner region.) In principle, this sets up an iterative process which would, 
upon convergence, yield the "exact" solution (except that the viscous terms 
would be included only to the boundary layer approximation). Note that the 
iteration would proceed by assuming that a known static pressure field is 
impressed on the outer (boundary layer) equations. The boundary layer calcu- 
lation then defines an entropy/enthalpy field which is impressed on the inner 
(two-dimensional) "inviscid") equations which, in turn, redefine the static 
pressure field, and so forth. Nevertheless, for the problem at hand, it is 
assumed that the use of a constant pressure field in the boundary layer 
equations will give the entropy/enthalpy field to sufficient accuracy that an 
improved approximation need not be determined. The computational procedure 
is described schematically on Figure 275. 


The boundary conditions along the sonic line complete the matching of 
the inner and outer solutions. The viscous solution assumes that the static 
pressure is constant throughout the outer region and is equal to the ambient. 
At the matching (sonic) line, the static pressure in the inner region is 





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569 





required to approach the ambient pressure. Thus, by requiring the static 
pressure to be continuous across the sonic line, and by obtaining the entropy/ 
enthalpy field for the entire jet from the outer solution, we are assured 
that all other flow and thermodynamic properties are continuous at the inter- 
face also. 


RESULTS FOR SHOCK STRUCTURE COMPUTATIONS 
Inviscid Calculations 


Some typical predictions of the aerodynamic flow field are given in 
Figures 276 through 279. The results in these figures have been obtained 
from completely inviscid calculations. Figure 276 shows the predicted shock 
shape and outer boundary shape based on the inviscid calculation. The shock 
originates near the outer edge of the jet due to coalescing characteristics 
coming from the curved outer boundary. The shock moves radially inward and 
eventually reflects from the axis of symmetry and returns to the outer 
boundary. The "> ;ular" reflection technique has been used in this case. 
Figures 277 and 2/8 show the composite result of a number of computations 
similar to that of Figure 276. In Figure 277 the distance from the nozzle 
exit to the point at which the shock first crosses the axis of symmetry is 
plotted as a function of pressure ratio, Pjet/Pamb- These results are for 
both "regular" and Mach disc reflection. Also shown on Figure 277 is a line 
representing the experimental data of Love, et al (Reference 168). The 
inviscid predictions agree quite well with the experimental results; however, 
this is to be expected, since the viscous effects don't start to have signi- 
ficant effects on the shock shape until after it reflects from the axis and 
nears the outer boundary. Figure 278 is similar to Figure 277 except that 
it shows the height of the Mach disc as a function of pressure ratio. Again, 
Love's experimental data are shown for comparison. The predicted Mach disc 
heights are only in fair agreement with the experimentally observed values. 
Nevertheless, the qualitative agreement is sufficient to show that the Mach 
disc model can be used as an artifice to allow the two-dimensional supersonic 
flow calculation to proceed beyond the point where the shock hits the axis. 


An overlay of the inviscid shock shape prediction of Figure 276 with a 
Schlieren photograph taken under the experimental portion of this contract, 
is shown as Figure 279. The agreement between the computed shock and the 
experimental shock is excellent, except for two points. First of all, the 
computed shock starts considerably closer to the nozzle than does the experi- 
mental shock. However, the computed version of the shock represents a Mach 
number jump of only about 0.02 until very near the centerline. A shock this 
weak would not be expected to show up on a Schlieren photograph. Secondly, 
the predicted shock does not turn normal to the flow near the edge of the 
jet. This difference is due to the neglect of the viscous effects in the 
outer region of the jet. Some calculations which do include the effects of 
viscous mixing are shown in the next section. 


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O 
M = 1.5 
4 
3 
O 
= .O 
4 M 1 
d Oo 
_ fe) ——Data of Love, 
et al. (Ref. B-7) 
© SSFD 
Calculation 
1 
a 
0 
0 1 2 3 4 5 
P./P 
j am 


Figure 277. Intersection of Shock Shape with Axis of Symmetry. 


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Figure 279. Shock Shape Prediction on Schlieren Photograph, 


Turbulent Flow Field Calculations 


Figures 280 through 285 present the results of calculations based on 
the fully coupled viscous-inviscid analysis. Figures 280, 281, and 282 
show the effect of varying amounts of underexpansion on a jet plume. In all 
three figures, the total-to-ambient pressure ratio, PT/Pamb = 4.10. The 
static-to-ambient ratio, Pj/Pamb, is, however, different in each figure. 
Figure 280 shows the radial variation of both the total pressure and the 
static pressure for the ideally expanded jet, Py;/Pamb = 1.0. Here, the 
pressure is constant (and equal to the ambient) throughout the entire jet. 
Consequently, both the complete inner-outer analysis and the purely boundary 
layer analysis give identical results for this case. At the axial locations 
shown, x/R = 1.90 and 2.35, the total pressure near the centerline of the jet 
has remained at its original upstream value, indicating that the inviscid 
core is still present. Near the outer edge of the jet, the total pressure 
falls off quite rapidly due to mixing. This decrease continues until the 
total pressure approaches the static (ambient) pressure signifying that the 
velocity has dropped to zero. 


A slightly underexpanded jet (Py/Pamb = 1.6) is shown in Figure 281. 
This flow field contains a weak shock, which, at the axial location shown, 
x/R = 2.65, has just reflected from the axis of symmetry and is moving back 
toward the outer boundary. Because of the shock, there are now two sources 
of total-pressure loss. Since the shock has already reflected from the axis 
of symmetry, the flow in the center of the jet has experienced a finite, shcck- 
induced total-pressure loss as shown by the smaller shaded region. Between 
this region and the outer mixing-loss region (also shown shaded) lies a por- 
tion of the gas which is unaffected by mixing and has been traversed by only 
a very weak shock so that its total pressure remains equal to its upstream 
value. The radial variation of the static pressure is no longer trivial in 
this case as it was in Figure 280. The pressure near the center is relatively 
high, then drops across the shock to a below-ambient value and finally 
asymptotically approaches the ambient value at the interface between the inner 
and outer regions. The location of this interface, as well as the location 
of the sonic point, is also shown in Figure 281. 


The last figure of this series represents a still larger degree of under- 
expansion than did Figure 281. Figure 282 corresponds to flow from a con- 
vergent nozzle with sonic velocity at the exit. The pressure ratio is 
Pj/Pamb = 2.1. This figure again shows radial variations of both total and 
static pressures at each of two axial stations, x/R = 1.90 and x/R = 2.65. 

The rate at which the mixing region spreads with distance from the nozzle exit 
can again be seen, as can the increasing total-pressure loss due to shocks. 
Note the relatively large levels of static pressure variation, even though the 
underexpansion is still mild. Finally, note that the viscous boundary layer 
analysis by itself would predict the same flow field for all three jets in 
Figures 281 and 282, (assuming the impressed pressure were taken as the 
ambient pressure in all cases). Also note that the considerable effect of the 
mixing-induced total-pressure loss on the flow field would be ignored by pure 
inviscid analyses, 


575 





1.0 






8 
HH 
Fa 
Aa 
& 6 Total Pressure 
A, 
o 
he 
a 
a 4 
o 
& 
a, 






—<-- 


Static Pressure 


0 2 4 .6 .8 1.0 1.2 1.4 1.6 
Radial Distance, rr. 


Figure 280, Cross-Stream Variation of Predicted Total Pressure and 
Static Pressure for Ideally Expanded Jet. 
M 


= 1.60, P_|/P = 0.242 


exit amb  Tref 


Total Pressure Loss Due to Shock 







1.0 








Total 
Pressure 


ag Affected by 

0 Mixin 

hi 
Ay 
zz ue 

© Cuter 
5 se Edge 
. .4 Sonic Point of Jet 
3 | 
rv 





Inner Outer 
Region “ ++ Region 


0 2 4 6 .8 1.0 1.2 1.4 1.6 
Radial Distance, r/r, 










Static 
Pressure 





Figure 281. Cross-Stream Variation of Predicted Total Pressure 
for Slightly Underexpanded Jet. Showing Total Pre- 
ssure Loss Due to Both Shocks and Mixing. 


Moxit 7 2°25s Poan/Pores * 9-242 


576 





























8 
H 
© 
Mt 
& .6 
Oy 
- Total Pressure 
bs 
a 4 
e amb 
é = ci 
rs —-—- = ja — 
Static Pressure 
0.0 
10) 0.2 0.4 0.6 0.8 1.0 Daw 1.4 1.6 
Radial Distance, r/r, 
Figure 282 . Cross-Stream Variation of Predicted Total Pressure for Jet 
from Convergent Nozzle, Showing Total Pressure Loss Due to 
= 1.0, P = 0, 242. 
Both Shocks and Mixing, Mo xit r amb’ Tref 2 
‘" “a Inviscid Boundary 
1.5 Sonic Line 
a ; (M = 1.1) 
~ 
he 
§ 1,0 
& Viscous 
+ Calculation 
on 
 » 8 
3 Inviscid " 
3 Calculation Shock 
-  o0 , 
(8) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 


Axial Distance, x/T, 


Figure 283. Comparison Between Predicted Shock Shapes Using Inviscid Pre- 
diction Technique and Coupled (Inner-Outer Analysis) Viscous 
Technique. Shaded Region Represents Flow Which is Significantly 


= 1,25 = 0,242. 
Affected by Mixing, M,,, = 1-25, Plii/Prieg = 0-242 


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3000 


2500 


2000 





1500 


Le) 


1000 


500 


Figure 285. 


07 
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‘7 fe) 
Oo 1) 
ie) 
08 8 
Predicted Velocity 
LV Measurement 
P = 
P/ aks 2.1 
M cit aa 
P amb! rT = 0,24 


Axial Distance, X/R 


Mean Velocity and Turbulence Velocity on Centerline, 


579 


400 


300 


200 


100 


c 


sdj 





The next two figures show the predicted shock-wave shapes for the jets 
of Figures 281 and 282. Figure 283 shows a comparison between the shock 
shape which is predicted by the complete inner-outer analysis and the shape 
predicted by a completely inviscid analysis for the jet of Figure 281. The 
inviscid calculation was made by specifying the entropy to be constant along 
all streamlines (except for shock losses). As Figure 283 shows, the two cal- 
culations give nearly identical shock shapes before and immediately after the 
reflection of the shock from the centerline. Indeed, the minor differences 
between the two calculations in this region is more due to small errors in 
calculation (stemming mostly from undesired interactions between the initial 
part of the mixing layer and the expansion fan at the nozzle lip) than from 
the physics of the problem. However, the sharply curved portion of the shock 
near the outer boundary is due to real effects. This sharp curvature comes 
about as the shock enters the strongly rotational flow region which has been 
created by the viscous mixing. As the shock traverses this mixing layer, the 
Mach number in front of the shock approaches unity so that even as the shock 
turns normal to the flow, its strength decreases until it eventually fades 
out. 


Also shown in Figure 283 are the outer boundaries of the inviscid calcu- 
lation (which, of course, is a streamline) and the "sonic" matching line 
(M = 1.1) which was used in the coupled analysis. Finally, the mixing region 
is shown by the shaded area. 


Figure 284 shows the predictions of the coupled analysis for the jet of 
Figure 282. superimposed on a Schlieren photograph of the first two shock 
waves for a jet at the same conditions. As can be seen, the agreement is 
excellent. Note that the curved portion of the shock in the mixing region 
agrees quite well with the Schlieren result, and that the predicted size of 
the "Mach disc" agrees with the photograph (although without the theoretical 
prediction superimposed - the photograph appears to show a regular reflection). 
Again note that, although the predicted shock starts much too close to the 
nozzle exit, it remains extremely weak until it nears the axis and so would 
not be expected to be visible on the Schlieren photograph. Finally Figure 
284 shows that both the predicted and the experimental outer boundaries show 
a point of inflection at about the axial distance from the nozzle exit where 
the shock reflects from the centerline. This inflection in the outer boundary 
is caused by the displacement of the viscous mixing region by the inner 
inviscid core of the plume. At the exit plane, the inviscid flow turns out- 
ward through an expansion fan. Then the axisymmetric effects force this flow 
to again turn and approach the axis (see outer boundary shape of the inviscid 
calculations in Figure 276). The superposition of an everwidening mixing 
region on these curved inviscid streamlines generates the inflection of the 
boundary. 


Figure 285 shows the predicted axial variation of mean and turbulence 
velocity on the centerline for the jet of Figures 282 and 284 (Pj/Pamb = 2.1). 
Also shown are laser velocimeter measurements of the mean and turbulent 
velocity distributions. What is noticed in both the analytical predictions 
and the data is that the mean velocity has a wide excursion (over and below) 


580 








about the ideal isentropic velocity for this nozzle, while the turbulence 
velocity distribution remains relatively uniform. On the average, the mean 
velocity distribution may be equivalent (in an acoustical sense) to the ideal 
velocity distribution. If the shock waves have little influence in turbulence 
amplification as observed in the centerline case, these two factors may 
combine to give a reason why shock-free and shocked flow acoustic characters 
on an overall basis are essentially the same. 


581 








APPENDIX 8 


ANALYSIS INCORPORATED IN NOISE 


Over the last two decades, a considerable amount of theoretical and 
experimental research effort has been directed toward investigating sound 
generating mechanisms in subsonic and supersonic jets. As a result, several 
distinctly different theoretical schemes have been advanced to describe the 
jet noise generation problem. Most of the schemes are intended to deal with 
the same general concept (that of a finite region of turbulent flow sur- 
rounded by a uniform, homogeneous, stationary, acoustic medium), while 
other schemes deal with the jet stability and nonlinear properties. 


The first scheme was introduced by Lighthill. He described the sources 
of sound in turbulent flow by means of an acoustic quadrupole distribution in 
a uniform medium at rest. Undoubtedly, Lighthill's theory has exerted the 
most profound influence on the subject. Ribner, Ffowcs-Williams, and many 
others have followed this method of attack and have made significant contri- 
butions to the theory. Mathematically, Lighthill's acoustic analogy has re- 
duced the aerodynamic noise problem to essentially the solution of the clas- 
sical wave equation with a forcing term. The fundamental result of the theory 
can be expressed in the form of the retarded potential solution: 


298 
a(o-Po) = paces - T, ay (ve bez), y a7) 


where: 


2 


Ty = sss + T4 + (p - a, 0) ee 
This equation represents the instantaneous radiated sound pressure 

p' = az (op - Oo) in terms of turbulent fluctuations in the jet flow. Many 

conclusions of the theory have already been verified experimentally and some 

of the observed characteristics of jet noise are explained by Lighthill's 

theory. For example, References 1 and 10 have demonstrated how Lighthill's 

model (modified for supersonic flow) can be applied to spectral predictions 


of supersonic jets. 


However, Lighthill's theory is not without limitation. One of the 
difficulties is that it does not effectively account for the refractive or 
coupled convective/refractive properties of sound. In this respect, Ribner 
(Reference 175) and Schubert (Reference 176) have attempted to offer some 
improvements. Mani (References 20 and 177) has illustrated that more 
realistic moving sources exist which can explain more clearly the influences 
of the jets shrouding influences on aerodynamic noise. 


582 








A second approach has been advanced independently by several authors 
(References 178-180) based on the method of matched asymptotic expansions. 
In Lighthill's fundamental solution (see equation 298), the instantaneous 
sound pressure p' = a, (9 - p,) is given in terms of the quadrupole strength 
Tjj, which involves the unknown density fluctuation. However, in Lighthill's 
acoustic analogy, the quadrupole strength Ti4 is regarded as a known 
quantity. The essence of Lighthill's approach, therefore, is that the 
density fluctuation in Tj; can reasonably be ignored. This approximation 
has been considered as rather unsatisfactory in some quarters and has led 
several investigators to examine the problem on the basis of matched 
asymptotic expansions. Using inner and outer expansions to describe respec- 
tively the turbulent and sound fields, they concluded that Lighthill's neglect 
of the density fluctuation in T;; indeed gives the first term of an asymptotic 
expansion for the density field. While higher approximations have yet to be 
generated, these studies have clearly illustrated that there are limitations 
in the acoustic analogy scheme. 


A third approach applicable to the jet noise problem was given by 
Phillips (Reference 21) on noise generation from supersonic shear layers. 
He formulated the problem in such a way that the effects of convection and 
variation in the local speed of sound are displayed explicity; whereas, in 
Lighthill's acoustic analogy, they were described in the source term. 
Therefore, an important issue for supersonic flow ~ the refraction of sound 
in travelling from its point of generation through velocity and temperature 
gradients into the ambient air is taken into account in this formulation. 
Furthermore, the problem of neglecting the density fluctuations in the 
source term in Lighthill's approach does not arise in the formulation. Pao 
(Reference 181) has presented a generalization of the Phillip's theory in 
which the range of validity has been extended to include the low supersonic 
and transonic ranges. 


A fourth method of attack was proposed by Liepmann (Reference 182). 
This scheme is to extend the principle established in steady viscous flow 
analysis, i.e., that the flow external to a body is that established in an 
ideal potential flow around the hypothetical body formed when the boundary- 
layer displacement thickness supplements the real body dimension. Liepmann 
proposed that the radiation field could be driven by an ideal boundary 
faithfully following the profile of the instantaneous displacement thick- 
ness. The concept is sound; however, all the emphasis is placed on the 
computation of the instantaneous displacement thickness - a very difficult 
task. Laufer, Ffowcs-Williams, and Childress (Reference 183) have made 
probably the most comprehensive attempt at solving this scheme. 


A fifth new approach is based on the concepts of the existence of large- 
scale instabilities in jet flows. From the earlier work of Landau (Reference 
184) on the plane vortex sheet problem, Sedelnikov's (Reference 185) eddy 
Mach wave concept based on instability waves in oscillating jet boundaries, 
to the more recent instability theories of Tam (References 80 and 186), it 
is becoming more possible that instability concepts could be useful in the 
understanding of the heretofore unexplored physical aspects of supersonic 








and subsonic jet noise. From numerous aerodynamic experimental investigations 
(References 11, 34, 46, 187, and others), the existence of these low fre- 
quency instability waves cannot be denied. However, to date there has been 

no clear cut acoustic experimental evidence illustrating by how much, if any, 
these instability waves influence the acoustic radiation field. 


Of all the techniques described, the basic turbulent mixing concepts 
which rely on the jet being composed of compact quadrupole sources (in the 
Lighthill, Ribner, Flowcs-Williams sense) modified by inclusion of the jets 
fluid shrouding, as proposed by Mani, offered the most unified aeroacoustic 
predictive scheme. Described below is an account of the basic turbulent 
mixing acoustic analysis contained in NOISE and how it can be used for predictive 
purposes. 


GOVERNING EQUATIONS 


Without restrictions, we postulate a general fluid motion in a continuous 
medium being governed by the conservation equations of mass and momentum and 
by the equation of state: 


d 4 > 
at * divepu) = o (299) 
— + div (puu) = div 7 + pf, (300) 
3 dp 
p =p (p,S)3; do = 22 dp a) ds (301) 
s P 
where: 
a me 
ap 
s 


f,, = Body forces 


t = Viscous stress tensor 


584 


Lighthill's original idea was to combine two equations similar to equations 
299 and 300 in order to derive a wave equation for the fluid density op. 
Ribner formulated an equivalent equation with p replacing p as the dependent 
variable. to do this, we take the time derivative of (298) and the diver- 
gence of (299) and subtract: 


oa 
ba 3 


2 a 
2 - Vp = Div Div (uu - t') (302) 
t 


2 
Adding a, V'p to both sides of (302) yields: 


97 2 2 > = 2 2 
oer ay Vp = Div Div (puu - t') + Vo(p- ay p) (303) 


2 
or similarily adding 1 3 p to Equation 302 vields: 
a2 at 


° 
1 92 2 = = 2 32 2 
— +2 _y pP = Div Div (uw - ct") +> -— > @= @ p) (304) 
“Zue 2 2 fe) 
a, at a, at 


Equations 303 and 304 can be expressed in index notation as: 


m2 a ne 
=p. 22. 80 a0 ou,u,-t',.)+ 9 er: (305) 
at2 *0 ax" 4 9x ; OX; ; ij ij ox,* (P a9) 
and, 
wv? 2 2 2 
3 3 9 ' 1 3 2 (306) 
a cf, eee (OU. | A tp = A. 
2 
at ox, OX, 9X, 2 ij - 2 se fe) 


oO 


Far from the flow region of the jet itself, the right-hand~side of equations 
303 through 306 must vanish identically leaving the well-known homogeneous 
wave equation, which, under homogeneous isentropic conditions, governs 

linear acoustics in a uniform medium of rest - the implied "acoustic 
analogy."" One can now imagine the medium as being at rest at any point in 
Space and interpret all the additional effects caused by the flow as a result 
of inhomogeneities which are continuously distributed throughout a limited 
part of the medium. By neglecting the last right-hand term of equations 303 
through 306, the sound attenuation and variation in the speed of sound are 
neglected. 


585 


GENERAL INTEGRAL OF THE WAVE EQUATION 


The formal transformation of the differential equations such as presented 
by equations 303 through 306 into an integral equation may be performed using 


the well-known Kirchhoff integral. For the pressure perturbation field we 
may write: 







Observer 


Source 
Pe. 





\ 
a 
_ 
By 

No 

pos 
a. 
x“ 
i 


i 
a 
a 
+ 

< 
— 
i 
= 


Sketch of Coordinate System 


p' (r,t) = (p-p,) s aff, Div, (T] dV, 
| ——__ 


Noise resulting from fluctuating shearing stresses 
1 3 [p] 9R 1_ {3p} aR 7 
+ —— —P_} + —- += —  )ds 
at] R 8n, ap Lae} on, o (307) 


Noise resulting from the effect of solid boundaries 
on the flow 


where the bracket, [ ], means evaluation at the retarded time t - R/a (the 


finite time for the sound emitted to travel the distance R_from the source to 
the observer), and the tensor T is the Lighthill tensor [puu - 1t* + 
(p- ao) Tf). 





Neglecting the influences from solid bodies, (307) may be written in 
index notion as? 


* 1 i ere 
p'(r,t) = ie dV (308) 
4n R dyydyy ° 


noting that, 


3 1L 3 1 1 jo... 
3X1 (2 (.-.1)+ (3 [--I) LA 


and the integral divergence theorem: 


Is ee dV, “ff.-- ds | 


equation 308 becomes: 


2 
Sp 1) 3 ‘I 3 1 
p'(r,t) = mee ox, IE i) 3, * alle it, 1%, 
eee a 
“fesleg = 
i 


The first term on the right-hand side of (309) represents Lighthills 
integral for an unbounded flow, as an equivalent quadrupole distribution. 
The other surface integrals would contribute to back reaction of a solid 
body to continuous flow (the impact of sound waves from the quadrupole 
distribution on the solid surface to hydrodynamic flow). This development 
concerns the first term only: 


2 
eae oo Sees L (310) 
p’ (r,t) 4n Ni IT, 5] ars 


Carrying out the double differentiation under the integral sign, the 
instantaneous sound pressure due to the distribution of quadrupoles becomes: 


‘ (x,-y,) (x.-y,) T Tis Tj 
. ff rae rl i i 

t= a ——— ald tl, +3 —nk gg ok 
et 4q 2 Ra, R2a, R3 








R 


Ts 5 he 
roe pe Tn) a (311) 


S87 











where the dot denotes partial differentiation with respect to time, and the 
stress Tj; and its derivatives are evaluated at the retarded time t - R/ao. 
In this form, the far-field sound pressure term is Ty /R, the induction near- 
field sound pressure terms are represented by T44/R3, and Ty ,/R? terms are 
for the transition near~field. 


APPROXIMATION FOR THE FAR-FIELD TURBULENT MIXING NOISE 


To evaluate the acoustic intensity involves squaring (311) and time 
averaging the product. The evaluation of such a term is a formidable task. 
For the far-field terms, Lighthill pictured the turbulent flow divided into 
regions such that strengths of quadrupoles within any one region are cor- 
related perfectly, but strength at points in different regions are uncorre- 
lated. The extent, 2, of each independent quadrupole distribution is 
assumed to be eouskts the size of the energy-bearing Midy. The output of a 
single region of this kind is: 


xX.X 


a A 
4na2R? 
° 


ve [T,.] (312) 


p'(r,t) = ij 


Since the outputs of the volume elements Ve are perfectly uncorrelated, 
their acoustic energy intensities, <p'2>/p oa» must be added to give the 
radiation field. 


Therefore, the sound intensity for each uncorrelated volume is: 


2- 


or, 


where, 


The sound intensity including source convection, as discussed by Lighthill 
(References 44, 45 and 46), is written as: 


1 ~ ve wv (1-M cose)? +( 2 Feeds (313) 
Po a> R2 < ay 


588 





where M, is the convection Mach number, M, number, M, = U,/a 
° 


g\2 |-5/2 
The convection factor [a - i. cost)" + (=) | / 
o 


replaces the original Lighthill form of (1-M, Cos ey”. This convection 
term accounts for the neglect of variation in retarded time within an eddy, 
and allows Lighthills subsonic analysis to be extended for convective Mach 
numbers greater than one. 


FAR-FIELD ACOUSTIC MODELS AVAILABLE IN NOISE 


Lighthill Acoustic Computational Model 


In order to compute the quantities included in equation 313, approxima- 
tions are made (See References 1 and 187). The eddy volume was taken as 23, 
The quadrupole strength was assumed proportional to p“u"; the quantity», »%, 
was approximately by 1.lu' (p.0. A.L. Davies frequency - eddy - shear 
assumption). With these assumptions, the mean square pressure fluctuation 
due to an individual ring volume element (See Figure 46) is: 


8 Prac 
ee. g*a'a aM cose) 2 i 1l.lu' si 
P 4mR2 ° p9aa % Cc a 


oO 


where 8; is a proportionality constant determined experimentally: 


8 = ,A7B« 107° Hot Jets 


.956 x 10°72 Cold Jets 


Empirical Self-Noise, Shear-Noise Model 


The aeroacoustic relationship expressed by equation 313 describes what 
is commonly referred to as "self-noise" generation; i.e., noise generated 
directly by a turbulence-turbulence interaction. However, the directivity 
characteristics predicted by the self-noise model show some discrepancies 
with the experimental results. Reasons for this disagreement with experi- 
ment can be ascribed to refraction effects, quadrupole orientation, 
quadrupole source terms other than self-noise, or any combination thereof, 


589 














The term “shear noise“ was probably first used by Lilley (Reference 187), 
and it reflects the idea that the acoustic source term can also be described 
in terms of a turbulence and mean "shear" interaction field. Related work 
concerning the mathematical descriptions of there terms can be found in the 
work of Ribner (References 11 and 34), Maestrello and McDaid (Reference 188) 
and Csanady (Reference 26). Jones (Reference 30) showed that, for subsonic 
jets, the "shear-noise" acoustic intensity directivity character should be of 
the form I ~ (1 -M_. cos g)73, as opposed to the self-noise form of I ~ (1 - 
M_ cos 8)-5, . 


With concepts such as this, a model was formulated for NOISE. Refer- 
ence 1 and 10 document these concepts in more detail. The model was form- 
ulated as a composition of shear-noise and self-noise of the following form: 


rh 7 ata 
wir sean 9 C +2 4 
P 4m™R_ og, at Q (x Cyn y \xer . C,) (314) 
o “oO 
where: 
C = Cos* 9 E -M, Cos 6)? + ale 
xX a 
oO 
, a 2 2 2 for) 2 )-3/2 
C.- = Sin 6 Cos 6 CL = M. Cos 8) + ) | 
ys wk \ 2}-5/2 
= - t +(— 
Cy [a M, Cos 4) (3 ) | 


The factors X, Y and Z are empirically determined factors based on sub- 
sonic jet noise data. Tables 7 and 8 show these factors: they are functions 
of Strouhal number based on the ambient speed of sound (fD/a,), and of the 
jet angle 6; they are fixed separately for cold and hot jets and are 
independent of jet Mach number. 


Ribners Self- and Shear-Noise Model 


Ribner (Reference 3) also formulated a jet noise theory to calculate 
the relative contributions of self-noise and shear-noise cross coupling. 
He postulated isotropic turbulence superimposed on a mean shear flow and 
assumed a two-point velocity correlation separated in space and time. Ribner 
showed that the expression of sound intensity from a unit volume of jet is: 


590 


Table 7. Shear and Self Noise Table of 
Experimentally Determined Constants. 


a) Cold Jets 











£D/ag ae x Y} i, 24 Zo 
0.01672 0 I 3.18 3.18 7.94 7.94 
0.02105 0 1 2.95 2.95 7.38 7.38 
0.02650 0 E 1.45 1.45 3.63 3.63 
0.03368 0 1 1.49 1.49 Se FZ Sere 
0.04203 0 1 1.63 1.63 4.07 4.07 
0.05293 0 1 0.60 0.60 1.51 1.51 
0.06658 0 20 0.91 0.91 1.30 1.30 
0.08360 0 20 Osae 0.37 0.53 0.53 
0.10526 0 20 0.16 0.16 0.40 0.40 
0.13287 20 20 0 0.29 Pay 0.977 
0.16721 20 20 0 0.40 7.56 1.326 
0.21051 20 20 0 0.65 12.4 pa 
0. 26500 20 20 0 0.93 12.05 3.09 
0. 33368 30 20 0 1.03 13.4 3.43 
0.42027 30 20 0 1.42 18.4 4.73 
0.52926 40 40 0 1.44 15.8 4.79 
0.66587 40 40 0 1.44 15.9 4.81 
0.83607 40 40 0 Bras S502 10.7 
1.05255 50 200 0 2.93 30.3 9.77 
1.32875 50 200 0 2.63 2162 8.76 
1.67214 50 200 0 mee 26.1 8.41 
2.10511 50 200 0 2.92 30.1 9.72 
2.65004 50 200 0 2, OL 19.8 6.38 
3.3318 50 200 0 1.91 19.8 6. 38 
11.7527 50 200 0 aE 8 19.8 6.38 
60.606 50 200 0 1.91 19.8 6.38 
Sin2@ Cos26 a= 
Note: Cy, = [am cose)? ‘ i) where: 
“ ate ae 
, 
+ = 1 max. 
4 } Ys, 3 2 = 
Cos 6 Z — “max. 
Cy * 2793/2 
2 wh 
(1-Me Cos6)? + ( ) 
4o @< 6 
i ls max. 
2, swg\2779/2 aa: ee oa 
Cc, = [a-m. Cos6)” + Vea) | ~ — “max. 
o 


591 


b) Hot Jets 
£D/ao 


0.02000 
0.76268 
0.96016 
1. 20872 
1.52196 
1.91692 
1.91692 
3.03711 
3.81342 
4.80042 
6.06061 
7.62683 
9.60163 
12.08716 
15. 21961 
19. 16922 
24.14028 
30. 37113 
38.13415 
48.00817 
60. 60606 
76. 26830 
96.01634 
120. 87164 
152.19612 


Table 7. 





Shear and Self Noise Table of Experimentally 


Determined Constants. (Concluded) 





Yy Yo 
1.0 1.0 
1.0 1.0 
0.18 0.18 
0.53 0.53 

0 1.4 

0 1.65 

0 1.9 

0 LS 

0 3.3 

0 2.1 

0 Zen 

0 1.4 

0 1.4 

0 0.57 

0 0.54 

0 0.468 

0 0.456 

0 0.446 

0 0.144 

0 0.75 

0 0.96 

0 £.2 

0 1.5 

0 1.89 

0 2.37 

592 


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secaaiech de 4 2 
t« (P=Po) vi. (. +B contd ¢ e's) ly eos 9)? ma 
Pom R2 T ao 
Self Shear Convection factor 
Noise Noise 


2,2) -3/2 
2 (315) 


where: 


A= /72 pw ‘ Pl 
o f fe) 
= he ee 2 5 3 
B = Po SO 0/8 1 a, (1 + a) 


/2 


o = .45 


If the characteristic radian frequency (wf) is approximated by the radian 
frequency (w) of turbulence, and w is approximated by 1.1 u'/2, the sound 
intensity can be written as: 


8 Zz 
2 eee 4 
E * aeae au-|a toe ZU (cos 8 + cos” »| E ~ M, cos 9)? 


2\ 2] -5 
4 =) /2 (316) 
(0) 


where: 
h = o/(1 + 3) 3/2 = .258 
— j 0.4 Hot Jet 
11.6 Cold Jet 


2 
The additional directional effect Con” 68+ cos 6), due to shear noise, is 
displayed. 


Lilley Turbulent Mixing Model 


Lilley (Reference 187) formulated a turbulent mixing model. His model 
is derived from the Lighthill theory. Lilley's work was divided into a 
contribution from self-noise in the turbulence, and the interaction between 
the turbulence and mean shear. The first part of the calculation relies on 


601 





Proudman's results, while the second part is developed from an approximation 
for 3p/3t covariance in incompressible sheat flow turbulence. The form of the 
acoustic intensity used in NOISE is: 


au 


2 6 s 

Eats oF 3. a aa 2 2. kaxio? Gm 
oe eee Sin Oo Ces. 6 Fiche cos: 8)" te te 

4a? Py R é ag 


where 6; is as given earlier. 


Pao Turbulent Mixing Shear Layer Model 


The Pao model (References 2 and 189) is based on the convected wave equa- 
tion introduced by Phillips in 1960. The convected wave equation itself is 
derived through the basic principles of fluid mechanics, and it is a natural 
extension of the Lighthill equation of aerodynamic noise. The linearized 
version of the general equation has the form of a simple wave equation in 
Lagrangian coordinates. The right-hand side of this equation contains four 
terms: a turbulent quadrupole, shear flow and turbulence interaction, entropy 
fluctuations, and viscous effects. If the flow field is free of shocks, the 
acoustic pressure fluctuation can be assumed to be decoupled from the entropy 
fluctuations. It is tacitly assumed in the analysis that all terms on the 
right-hand side of the wave equation are known quantities and the contribu- 
tions of individual terms can be considered as independent of each other, 
as with all the models presented. 


The Pao acoustic intensity (per unit volume) expressions contained in 
NOISE are as follows: 





2 ate 
M4 an \<5/2 
Pe C 3v2 i Vo : we a! . 2s (Ye) Eu —— (x2? <2) » SO 
P94, R42 AZ 2 _ M (1-M. Ccs 6) 
Po? L60V2r> vena? (O)a 13/3 says 
= 9 as 4 r(3 ) (cos #4) Fu 
Pu%o gr? a2 ¢ 
I = M 2 = 
(Be)13/3 [ate Cos0)* + at Mo? Cos a] ie Sla (318) 








cos 4 ory 


2 4 
Po . 3v2 nl? v 4 v8 a 2x) (7) F(b) 
F(O 
(ome) R2 Azg 1g (0) 
x .. 
. fam, cose)” + s Me" Cos 25 | -5/ Slb 


602 





ee cold jet 
«dy Rot, jet 


. 
ms 


1, self noise 
22 (Coa* Gack tou 86) (2) 
3 
2V2 og? M2 é 


ld (M 
M dy \A | 


2v72 Mow (yo, 0) a Cos 6 
2 





2 Vu [(1-Mc Cos 6)2 + a” M2 cos? @]1/? 


a i Self noise 


2, shear noise 


603 














% | a2 (1-Cos* ¢ 


“1 (1-Me Cos 8)2 - A2 Cos? 6 
Ay (0) = 0.35503 


F(b) -f eb*-u2 (u-b) 4 du 
b 


c 5 - A Cos 6 8 = 1 h 6 < ) 
os O * 1-M. Cos 6° cos ‘ when S, 


y 
Wlyo, 9) = ° w(d-) dy = Q/k 
K wu 
‘ag § 
tp 


2 


1 2 
ra je .) ‘ “ =| oA 1 
ay A (y) M M2 


The choice of solutions is obtained through a selection process defined 
below: 





Cos, = ea 5 6, < 1/23 0, =01fM,-AK1 
io 





we 


1 ul é wis = 
3 MomA 5 < 84 Rr hE M. A> = 


604 








The quantities tp» Q, and W(yo, @) are determined as follows: 


tp Computation: 


For a given source volume and far-field angle 6, determine tp from the 


equation (y = tp): 


Meee eee 
Cos 8 ¥_@) + AG) ( ») 


This means that, at a given angle 6, (6 < 69), there is a unique y = tp at 


a given axial location X. See sketch below: 
y 


Source Volume at : 


t 
P 


Slice of Jet i 
=| fax 


Q Calculation: 


y 
) 
Q -f Mq dy, note W(yo,6) = Q/Ky 
t 
P 
where Mq is given by: 
a 
a, _|cos_ 2 + Mc on] | 
ky A(y) M m2 


ky = - Mw Cos 6 


Spectrum Calculations: 


There exist two options for computing the spectrum. One is the way 
described for all the previously discussed turbulent mixing models; the 
second method is via Pao's spectrum calculation. The latter is performed 
as follows: 


605 


(SO) 


(Sla) 


(S1b) 





| 


ht 


"ne Vo4 M8 Lt (=) (Wo L1)4 





poz (Ye) = 32 p2 a6 2 Fo! (1-M, Cos 8) 
re ere 

exp {-+ [ety + wtr|} (319) 
mar 1 Se Se 2 A 
p' Beate my" M lL A, (0) tan & (wo Ly Cos 9) 13/3 
a See a... eee 16/3 
=" er? a2 e@/3 ol/3 (1-M, Gos 6) 16/ 
Oo 

a“ © « 2 
* exp {2 fay? + (wl) | : (320) 


Pe a = 4 
pr? (y,w) _ 1 v4 M8 L ( = ~290 (Wo Li Cos 8) 
Sa f —) ON 


2 — 
hs 32x? a2 2 4, (1-Me Cos 8)5 
m ~ 9 
+ exp \- - (on + (wL,) |} (321) 
where: 
Lag 8 
. uj aB 
8 = width of mixing zone 
w = 2n St 


o = 2m St (1-M, Cos 8), Source Frequency 
Ly = 2/B 
k, = -Mw Cos 6 


= “a, M/A 


-b + 52 + 84%, Self Noise 
st == 


2 ss = 
one » +¥h2 + 4a2, Shear Noise 


606 














o> 
i] 


M W (y,» 6) Cosé 


i°) 
" 


‘ [a-m Cose)* + om - cose | 
c c 


eR 


METHODS OF ACOUSTIC CALCULATION IN NOISE 
Sound Pressure Calculation 


The mean square pressure fluctuation due to an annular ring volume 
element of jet AV = 27rAx, as shown in Figure 46 of the text, can be 
written as: 


2 By : as 2 we - ae 
pave —y FY (1-M_ Cose)~ + { — 2nrArdx (322) 
4nR- a 4 ° * 
oO 


for the basic Lighthill model (similar type expressions are used for the 
other acoustic models discussed earlier). 


The radian frequency at which each volume element emits energy is 
approximated by: 


w = 1.1 u'/2 


where the turbulence velocity u' and the turbulent length scale (%) are 
determined for each local volume from the JETMIX and SSFD programs described 
in Appendices 6 and 7, The frequency, w, is the frequency in the eddy frame 
of reference. The frequency at the observation point, w,, is related to it 
through a Doppler shift: 


2 2 . 
w = (1-M, Cos6)” + =) W, (323) 


607 











The best correlation with data was found, however, by assuming: 
w= 1/2 us 

This amounts to assuming that frequencies in the moving frame of reference 
at u, are half of those received by the stationary observer. 

For the whole jet then, the mean square sound pressures from all circular 
ring regions combine to give the overall mean square sound pressure. The 
1/3 octave (or 1 octave) band analysis sums the mean square pressures whose 
frequencies fall into the range of a frequency band. This gives tne sound 
pressure level for the frequency band. Thus, the sound pressure level 
frequency spectrum of the jet is constructed by calculating the sound pressure 
levels in the various frequency bands. 


Acoustic Power of the Jet 


To obtain the acoustic power output from the circular ring of jet we 
may write: 


20 1 
Il -f f T. R Sinedqrde 
re ro) (324) 


TT 
‘ ann? [ I, Sineds 
oO 


or, for an incremental volume: 
T 


a2 

' 
(x, t,0)* 2 PB sinode (325) 
A J oO o 6 


608 


The geometry is defined by the sketch below: 


- Source Flement 







Observer 


The sound power of the jet for one jet element is obtained by integr- 
ation of the sound intensity over a large spherical surface enclosing the jet. 


Integrating across the jet results in: 


= 
1, (x, 0) = J ™) (x,1 54) dr (326) 


This represents a power spectrum generated by a slice of jet at an axial 
Station X. 


Integrating with respect to frequency gives: 


7, (x) = 1, (x50) dw (327) 


oO 


which represents the total acoustic power output from a slice of jet at X. 


For the entire jet: 


eo 


7 a ™ (x) dx (328) 


oO 


All of these quantities are available in the NOISE program. To illustrate 
some of the explicit formulations, again consider the Lighthill acoustic 
model: 


609 





= ai? 
—,2 V wee 1 2 inf 2 
PP = + ae (1-M_ Cos6)° + i (329) 
Po% P63 4mR . \ oO 








Then, integration of a volume element at x, r yields the following acoustic 
power per unit volume of jet: 





Wea 45 2 T 
re a! Sing dé es 
1, (154) et a re | : a2 5/2 dé andr Axx, 
°o%o fo) lam, Cosé) + os 
% (330) 
Integration yields: 
2=- 2 
Vio. F M -1 
m(er,u) * SAS m) Pearce Coed 
- + 1-M_)* + 
20 585 3M. [a M.) q | (1-M,) q 


M +1 1 2 
+ ares st lat - oe aur Ar Ax, 
+ P 
fas? + | ae ee (331) 


where: 


Thus, through simple summation, it is possible to calculate the power 
spectrum (72) generated by a slice of jet at x, the acoustic power (13) 
radiated from the slice of jet at x, and the total power (mp). 


610 














Mani Slug Flow Analysis 


Mani's model (see Section 1 of Volume II of this final report) represents 
the sound field produced by a convected-point quadrupole embedded in, and 
moving along the axis of, a round plug jet. The essential feature of this 
model is the incorporation of mean flow shrouding effects on the radiation 
of the convective quadrupoles, 


The starting point of Mani's analysis is the Lilley equation: 





Der! ab 3 -2 dar' D 3 -2 odr' 
ae TAGE Poa) Rae A 
Se x, 9x, 3X5 Dt ax, ax, 
dv 2 3 (u" sla 
=-2y — 9 (u' “ut ) y ll ¢ Si is: Sais | 
dx dx ,OX 2 kK Dt dx, dx 
2 1 aa es 


Mani's work deals with the second term on the right-hand side of the 
above equation (the self-noise term). The details of the solution are not 
given here, since they are completly described in Section 1, Chapter I, 
Volume II of this final report. Suffice to say that there exists two 
fundamental solutions; one basic solution for unheated jets, and a second 
solution for heated jets. 


Each solution is composed of an acoustic intensity expression consisting 
of quadrupole solutions which include the influence of fluid shrouding. The 
composite quadrupole solutions are formulated from six basic quadrupoles 
(x-x, x-y, x-z, y-z, y-y, and z-z), weiphted according to Ribner's 1970 
Studies. The basic difference between this model and Ribner's work is that 
the weak cross quadrupole contributions (i.e., of type xx-yy, xx-2z, vy-zz, 
etc.) are neglected, 


For both the unheated jet and the heated jet, the acoustic intensity is 
evaluated from the following formula: 


I «(mean square pressure of x-x quadrupoles) + 4 (circumferential 
average of mean square pressure of x-y or x-z quadrupoles) + 2 (cir- 
cumferential average of mean square pressure of y~-y or z2-z quadrupoles) 


+ 2 (circumferential average of y-z quadrupole). 


The far-field acoustic pressure expressions are given by equations 43-51 
of Section 1.0 of Volume II, for the unheated jet. For the heated jet 
solution, representative average estimates of jp/ir, 2p /ar2, etc. are incor- 
porated whenever a quadrupole singularity for the heated jet generates 
additional dipole and source-like terms. For the heated jet, then, the 











interference between different order multipole singularities is neglected 
(see equations 55 - 59 of Section 1.0 of Volume II of this final report). 


A recommended procedure for predictive purposes would be to use the 
Ribner turbulent mixing model available in NOISE, modified for the inclusion 
of fluid shielding by Mani's model. To perform the corrections for fluid 
shrouding, use Tables 9 and 10 given here. 


NEAR-FIELD ACOUSTIC MODEL AVAILABLE IN NOISE 


The near sound field of a jet can be considered as a region within 20 
to 30 diameters from the jet exhaust nozzle. More precise definitions are 
the induction near field at distances less than a wavelength from the source 
and the geometric near field at distances from the sources that are less than 
the order of the geometric extent of the source. In many situations, both 
induction and geometric near fields overlap each other and, for practical 
purposes, a region within 20 to 30 diameters from the jet exhaust includes 
both types of near field. 


The near and far sound field of a jet display quite different properties. 
For example, the familiar far-field sound pressure dependence on the inverse 
of distance and the simple relationship between the sound pressure and intensity 
i = (p= Po) 2/008 are no longer valid for the near field. In the far field, 
the distances from the observation point to different parts of the source 
distribution in the jet flow field are approximately equal. In the near field, 
the spatial distribution of sources along the jet flow becomes very important 
because distances are different from different parts of the jet flow to the 
observer. 


612 








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620 


Experimental investigations (References 13, 190-193) have provided some 
useful information on the sound pressure spectra and constant sound pressure 
contours of the near noise field of jets under particular operating condi- 
tions. To estimate near noise field, there exist semiempirical methods based 
on near-field noise measurement from full-scale jet engines (Reference 194) 
and based on tests from scale-model hot jets (Reference 195). However, there 
is definitely the lack of a basic understanding of the near noise field and, 
therefore, the lack of an accurate prediction procedure, 


Here, the method of calculation as described for the far-field acoustic 
approximations is applied to the near sound field. The formulations are 
based on quadrupole distributions studied by Franz (Reference 196). Extensive 
calculations with the models to be discussed may be found in References 78 
and 197. 


Discussion of Models 
Equation 311 was a general solution to Lighthills inhomogeneous wave 


equation for far-field and near-field sound due to a distribution of 
quadrupoles: 


me (x,-y,) (x.-y,) T.. t. Tv. 
Keon [ff Se et A, gg cig y 
R 


Ra . “e R? 
fe) fe) 
t Ps 
-s,,{ Se +—t dv 
ij 2 3 ° 
R ay R 


Under the assumptions that the turbulence is essentially steady and the 
distance R is large compared to the eddy size, Franz (Reference 196) shows 
that the mean-square sound pressures due to the distribution of different 
types of quadrupoles can be written in terms of the radiated acoustic power 
W as: 


2 4 
Sisco W a a 
: es 2 
= PY? - — ; (15 sino swos"6 cos >) 1+ 3 ; 5 9 r Z (332) 


4nR R w Rw 











621 


for a lateral quadrupole: 

















a W a 2 
re A Ss 4 1 
@- pF =——-5 G cos*é) }1 + + eb eee Se eer 
47R Rw \ cos 6 cos @ 
a . 
der ee: (» e es it ee (333) 
R w cos 6 cos 8 
for a longitudinal quadrupole, and: 
2 4 
nS 2 Fev 36 45 : 
(p - PY? = 5} I + 2 72 roe =< % (334) 
4mR \ R w R w 


for quadrupole radiation from an isotropic turbulence. 


These expressions were obtained based on a small flow velocity, so that 
the effects of source convection do not appear. For non-negligible convec- 
tion speed, the expressions can be written as (Appendix C of Reference 196): 

















2 
—_—_ p aw 5 2 a D 
(p - p , = i (15 sin’6 cos 9) ee + 2 (3 Q° cos 6 
oO 2 2 2 
4mR R w 
4 
2 #6 4 2 3 
=- 6 MQ cos6 + 4M }) +—— (8 Q cos @- 18 M O «cosd 
c c Raw? c 
Z = 
+ 9 M_” QD) 16 ’ (335) 
for a lateral quadrupole: 
2 
—— p_aW a 4 ‘ 
(p - p a —< we cos 6 + —2- S Q? cos 6 - 12 M Q cos 
fo) 2 2 2 c 
47R R w 
6 4 
y y 4 
- (4 - 9M * cust +6M_cosé + 1 + —2 9 a’ cos 6 
c c 44 
R w 
622 





- 36 M. Q? kegs - 6(1 -~ 9 M_” )o” ewes 


2 
+ 12(% = am.” ) QM, cosé + Lis 3 u_’) | e° (336) 


for a longitudinal quadrupole: 


2 
Po ane ew a5 ae -5 
{p- ae i 2 = Wi OP fa precemeacli u lly  r c (337) 





4nRe Rew" 3° 


for quadrupole radiation from an isotropic turbulence, where: 


a 
i! 
f — 
— 
1 
iK< 
ie) 
re) 
° 
n 
@D 
are 
sts 
ia 
m IE 
Oo je 
~Seee 
N 
S| 


The factor C used here, which is obtained by Ffowcs-Williams for high- 
speed flow, replaces Franz' original form (1 - M, cos6é). The factor c-> in 
equations (335) to (337) also replaces c-® of Franz’ form to account for the 
increased number (C) of eddies whose sound arrives at the field point 
simultaneously. It is easy to see that, when M, is small compared to unity, 
both C and Q approach unity and equations (335) to (337) reduce to equations 
(332) to (334). Thus, equations (335) to (337) are valid for both low-speed 
and high-speed flows. 


Method of Calculation 


The mean square sound pressures given above can be considered as contri- 
butions due to a unit volume of jet flow. The idealized structure of a 
turbulent jet may be divided into several regions. The initial region consists 
of a potential core enclosed by a mixing region of strong shear. Downstream 
of it are the transition region and the fully developed turbulent region. 
Different parts of the jet may be considered to generate sound by different 
types of quadrupoles. In the present investigation, each type of quadrupole 
is considered as an acoustic model for prediction, assuming that the entire 
jet flow is represented by one type of quadrupole. This is followed by 
investigation of compositions of different quadrupoles for different parts 
of the jet. 


623 








To express the near-field sound pressure in terms of mean flow and 
turbulence parameters throughout the entire jet flow, the following approxima- 
tions are made: 


iy 


W 1 ies: u'/2 


where the terms are the same as those discussed earlier for the far-field 
noise approximations. The method of calculation for the far-field OASPL 
and SPL is also applied for the near-field calculations of NOISE. 


From extensive theory/data comparisons it was found that, at certain 
near-field locations, one particular model may compare better with measure- 
ments than another. References 78 and 197 describe these results. To 
improve predictions by use of individual models, composite acoustic models 
were also considered. To do this it was necessary to consider the structure 
of the turbulent jet as divided into several regions: the initial region 
(consisting of a potential core enclosed by a mixing region of strong shear), 
the transition region (downstream of the potential core), and the fully 
turbulent region. The different parts of the jet are considered to generate 
sound by different acoustic sources. 


Composite Model I 


The isotropic turbulence model is used in the potential core and the 
fully turbulent region (see sketch below). The lateral quadrupole, which is 
usually considered to be predominant in any region with large mean shear, is 
used in the mixing region enclosing the potential core and in the transition 
region which is approximately one core length downstream of the potential core. 


Lateral Quadrupole 







Mixing Region 
Potential 
Core 
Isotropic 
Turbulence 





| 

| 

| 

| 

| 

| 
bx 
ec 


Isotropic 


I 

\ 

| 

| 

Turbulence 





Initial Transit ion —geleg—Fully Turbulent 
Region | Region Region 


624 





Composite Model II 


The lateral quadrupole is used in the mixing region, and the isotropic 
turbulence model is used elsewhere for the jet flow. 


Predictions with these two composite acoustic models have been made for a 
Supersonic jet. Model I was found to be almost the same as the lateral quad- 
rupole model except at low frequencies (f < 1000 Hz). Model II, the isotropic 
turbulence model, was found to be dominant for f < 2500 Hz. 


Although these composite models were not found to yield much improvement 
on predictions, it was found that there exists the possibility of changing the 
predicted characteristics by changing the composition of the quadrupole models 
and, thus, through a better selection, possibly arriving at a better pre- 
diction. 


Using any of the models described above, it was found that, by replacing 
the term (wk/ag)2 (1.1 u'/ag)2 of the convective amplification terms by 
q2 = 0.5 Mc, improvements for the low frequency spectra were obtained. This 
option is available in NOISE. 


It was also found that a much simplified model for the near field could 
be obtained by using any of the above single or composite models with measured 
far-field spectra. 


Simplified Near-Field Prediction Procedure 


As can be observed from the earlier work, the near-field prediction models 
can be written in the following functional form: 


iz r 

aoe - 5 3 aay 
pt (r,u) =~ ( ta ata, Gy | FM) 

4mR R Wd R W 

where: 
P(w) = the far-field acoustic power spectrum 
F(6,M,) = the near-field directivity factors appropriate for any of the 
models described earlier 

Aj, Ay = the appropriate coefficients for the discussed near field 


Further, if the source frequency distribution is assumed as a unique 
function of axial location, then, by using a measured far-field power 


spectrum, the near field can be simply computed for any observation point. 
An appropriate source frequency distribution is (Reference 156): 


625 


where D is the jet diameter, uy is the isentropic exhaust speed, f is the 
observed frequency, and x is the axial location. The empirical constants 
are ao = 1.25 and @ = 1.22 for circular jets. 


EXAMPLES OF PREDICTIVE SCHEMES IN NOISE 


Extensive comparisons have been performed using the far-field and near- 
field acoustic models described above. References 1, 10, 78, and 197 contain 
many such examples. Included here are some of the results obtained. 

Far-Field Acoustic Predictions 

Figures 286 to 292 show the prediction of NOISE for the Lighthill and the 
Lighthill Self-Noise/Shear-Noise Models. Figures 293 to 295 show predictions 
for the basic Lighthill and Ribner models. 

Near-Field Acoustic Predictions 


Figures 296 to 302 show example predictions using the developed near- 
field prediction schemes. 


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My 


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Arc 





Figure 291. 1/3 





100 150 
Angle 


O Nagamatsu's Data 

—Lighthill's/FFOWCS Williams 
Self Noise Model 

-~--Self-Noise/Shear Noise Model 





from Jet Axis 


Octave Band Directivity Patterns for a Cold Supersonic 











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170 


Radius Length, 


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160 


Acoustic Power 


Figure 293, 


Cold Supersonic Jet 
My = 3.0 


Ribner's 
Asymptotic 
Form 

OW. Se 
a ta 























| 
¥ 
Poros 


G.E. 's Aero Acoustic Model 



































10 50 100 
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ao 


Predicted Power Distribution for a Cold Supersonic Jet. 


634 





eD 
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M 
110 


Lo. 


2 


90 





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et Diameter = 2 inches 
eS = «99 


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ao 


.6° from Jet Axis 





j++ 1 | | _ 











f 
| 











—-—-Prediction (Lighthill's Model) 
Prediction (Ribner's Model) 
O Measurement 








1/3 Octave Band Sound Pressure Level, 


Figure 293, 


























25 50 100 200 400 800 1600 3150 6300 12500 
Frequency, cps 


Predicted Power Distribution for a Cold Supersonic Jet 
(Cont inued). 


635 


2 


re 0.0002 dynes/cm 


1/3 Octave Band Sound Pressure Level, dB, 


110 


43.8° from Jet Axis 


<2) 
o 








~I 
° 











a 
o 


---Prediction (Lighthill's Model) 
Prediction (Ribner's Model) 
OQ Measurement 





110 


60° from Jet Axis 








© 
o 








J 
°o 





5 
92.5 25 50 100 200 400 800 1600 3150 6300 12500 
Frequency, cps 


Figure 293, Predicted Power Distribution for a Cold Supersonic Jet 


(Continued). 


636 


























110 
90° from Jet Axis 

a. - ae 

E 

3) 

SS 

9) 

v 

s 

is 

ae, 

N 

2 70 

ro 

oy 

uM 

as 

zs 

J 50 

2 —--Prediction (Lighthill's Model) 
is Prediction (Ribner's Model) 
® O Measurement 
& 220 

e 120° from Jet Axis 
i 

[oF 

ce. 

5 

fe) 

n 90 

c 

Fst 

33] 

m 

o 

> 

ts>] 

se 

v 

oe 7% 

” 

et 

50 





12.5 25 50 100 200 400 800 1600 3150 6300 12500 
Frequency, cps 


Figure 293, Predicted Power Distribution for a Cold Supersonic Jet 
(Concluded), 


637 














140 

D 456° in. 
Mo= 0.83 
130 : Up= 1337 fps 


T= 1271 °R 
u'/Us 0.15 (Assumed) 


120 [— a ~ : = ox = ee) ae aaa ta a 


110 











100 


90 














re 0.0002 dynes/cm 








80 


—--Prediction (Lighthill's Model) 
— Prediction (Ribner's Model) 
Oo Measurement 


140 





fps 
°R 


130 ).15_ (Assumed) 











120 


Overall Sound Pressure Level, dB, 





100 














0 40 80 120 160 200 


Angle from Jet Axis, Degrees 


Figure 294, Directivity Patterns of Overall Sound Pressures for Hot 
Jets at a Distance of 320 feet from Jet Exit. 


638 








N 


cm 


0,0002 dynes 


Level, dB, re 


Pressure 


Band Sound 


3 Octave 


1 


120 


(a) 30° from Jet Axis 
I 


eas 


110 -+K—— $= + =~ - - 





100 





90 





80 





70 








60 


——— Prediction (Lighthill's Model) 
Prediction (Ribner's Model) 
Oo Measurement 





110 

(b) 60° from Jet Axis 
100 
90 


80 


70 





60 
100 1000 
Frequency, cps 








10000 


Figure 295, Hot Jet Sound Pressure Spectra at Various Angular 


Positions at a Distance of 320 feet from 
(D = 45.6 in., M 0.83). 
oO 


639 





Jet Exit 


2 


dynes/cm 


0.0002 


re 


4B, 


1/3 Octave Band Sound Pressure Level, 


110 


(c) 90° from Jet Axis 











100 


90 


80 





70 





60 


—--Prediction (Lighthill's Model) 
—— Prediction (Ribner's Model) 
O Measurement 


(d) 120° from Jet Axis 

















100 1000 
Frequency 


» eps 





10000 


Figure 295. Hot Jet Sound Pressure Spectra at Various Angular 


Positions at a Distance of 320 feet 


from Jet Exit 


(D = 45.6 in., M 0.83) (Concluded). 
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Band 


1/3 Octave 


160 


150 


140 


Microphone No. 


OQ Measured Data 





— 


-~-==~{sotropic Turbulence 


Lateral Quadrupole 





—:— Longitudinal Quadrupole 





= eee 


























150 








(a) 














50 200 800 
Frequency, Hz 


3150 12500 


Microphone No. 23 and 24 


| 


43 inches 





140 


1.4612 
3300 fps 
3035° R 





130 


























50 200 800 3150 12500 
Frequency, Hz 


Near-Field Sound Pressure Spectra at Various 
Microphone Locations (An Initial Turbulence 
Level, u'/U 0.15, was Assumed for the Cal- 
culations) .° 


642 








ssurg 
s/em 


150 


e 


140 


0.0002 dyn 


Band Sound Pre 


re 


dB, 


130 


3 Octave 


1 
Level, 


120 


(ec) Microphone No. 27 


Microphone No. 


12 


Figure 298, 


——+ ae 





Measured Data 

Isotropic Turbulence 
Lateral Quadrupole 
Longitudinal Quadrupole 








43 inches 
1.4612 
3300 fps 
3035" R 














50 200 800 


Frequency 


Near-Field Sound Pressure Sp 
Locations (An Initial Turbul 
Was Assumed for the Calculat 


643 











3150 12500 
» BZ 
ectra at Various Microphone 


ence Level, u'/U, = 0.15, 
ions), Microphenes 1-4 and 27, 





2 











re 0.0002 dynes/cm 


dB, 


. 

o 12 

? 

v 

a 

o 

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a 

an 

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w 

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3 

n 

z 

& 135 

eo 

> 

fas} 

~ 

7) 

° . 

q im 
115 

12 


O Measured Data 





Microphone No, 2 





50 200 800 3150 12500 
Frequency, Hz 


Microphone Nos. 1 and 2 














43 inches 
1.4612 
3300 fps 
3035 ° E 


90 200 800 3150 12500 


Frequency, Hz 





Lateral Quadrupole 


socees Isotropic Turbulence —--—Longitudinal Quadrupole 


Figure 298, 


Near-Field Sound Pressure Spectra at Various Microphone 
Locations (An Initial Turbulence Level, u'/U, = 0.15, 
Was Assumed for the Calculations), Microphones 1-4 and 
27 (Continued). 


644 





Microphone No, 3 O Measured Data 
saen=== Tsotropic Turbulence 
Lateral Quadrupole 
—-— Longitudinal Quarrupole 
! 
- ee ee ae 


T0%}°0Po00500 


140 
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o 

& 

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- 

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Zz 140 

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n 

TC 

S 

a 

© 

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- 

a 

~~ 

y 

fo) 

isp] 

= 320 
110 


Figure 298, 





Microphone Nos. 


800 3150 12500 
Frequency, Hz 


3 and 4 


43 inches 
1.4612 








= 3300 fps 
= 3035 ° R 





800 3150 12500 
Frequency, Hz 


Near-Field Sound Pressure Spectra at Various Microphone 
Locations (An Initial Turbulence Level, u'/U, = 0.15, 
Was Assumed for the Calculations), Microphones 1-4 and 
27 (Concluded). 


645 








150 

N 

r= 

yn 

@ 

~ 140 
N 

2 130 
ree) 

i 

) 3 
> 120 
vo 

4 

v 

a 

an 

o 

= 140 
a 

L 

Z 

a 

nde 
vo 

- 

fa] 

pe 

y 

ro) 

120 

110 


© Measured Data 
eeeee- Isotropic Turbulence 


Figure 299, 


12 50 200 800 


Microphone No, 7 

















Frequency, Hz 








12500 
































(g) Microphone Nos. 7 and 8 
Microphone No. 8 | 
+ t : aS | 
ce) roses 
OOQO0O 4 
v = ‘009 w\ 
a ‘ae Pot 
| x a ee & 0d 
= ae a 
OF, 
By D = 43 inches 
oO o\ M = 1, 4612 
U_ = 3300 fps 
Of \ t x, = 3035 ° R 
oO . 
: \ 
v6 
12 50 200 800 3150 12500 


Frequency, Hz 
—— Lateral Quadr 
—-— Longitudinal 


Near-Field Sound Pressure Spectra at 
Locations (An Initial Turbulence Level, u 


646 





upole 
Quadrupole 


Various Microphone 
'/U 
Was Assumed for the Calculations), Microphones 7 and 8. 


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647 


re 0.0002 dynes/cm 


dB, 


1/3 Octave Band Sound Pressure Level, 


170 


169 


150 


140 


130 


12 50 200 800 3150 12500 50000 
Frequency, Hz 
(a) Microphone Nos. 1 and 2 
160 
Microphone No, 1 
150 een ee 
O Measured Data 
---Isotropic Turbulence f NEN 
—tLateral Quadrupole As 
140 = 
—--Longitudinal Quadrupole i ee a al Weantog. 
oF), Pan ! ‘ 
oOo fF /=000002 peti 
130 a 7 C000 = 
Oo to 
O0° yf O90 
120 © : — 900 
/ O5 
/ 
110 
2 50 200 800 3150 12500 S0000 
Frequency, Hz 
Figure 301. Near-Field Sound Pressure Spectra at Various 


Microphone No. 2 
































Microphone Locations, 


648 











160 


Microphone No. 4 


150 
ow D = 4.55 inches 
—< M = 1.3559 
n a 
U_ = 2781 fps 

vo 
st 1407 7° _ 2407° R 
z u" /U = 0.05 (Assumed) 
nN oO 
S 130 
o 
io) 
2 120 





110 
12 50 200 800 3150 12500 50000 
Frequency, Hz 


160 


Microphone No. 3 
150 —l 
O Measured Data 
---Isotropic Turbulence 
——Lateral Quadrupole 
—--Longitudinal Quadrupole 





140 


130 


1/3 Octave Band Sound Pressure Level, dB, 











12 50 200 800 3150 12500 50000 
Frequency, Hz 


(b) Microphones No, 3 and 4 


Figure 301. Near-Field Sound Pressure Spectra at Various 
Microphone Locations (Continued), 


- 





i/3 Octave Band Sound Pressure Level, dB, re 0.0002 dynes /cm* 











130 





120 





110 








12 50 200 800 3150 
Frequency, Kz 


Microphone No, 7 





O Measured Data 
ee--[sotropic Turbulence 
Lateral Quadrupole 











12500 





12 50 200 800 3150 
Frequency, Hz 
(d) Microphones No. 7 and 8 


Figure 30). Near-Field Sound Pressure Spectra 
Microphone Locations (Continued), 


650 








12500 


50000 


50000 


at Various 





— a 


2 


re 0.0002 dynes/cm 


1/3 Octave Band Sound Pressure Level, dB, 


130 


Microphone No. 10 


120 








110 





100 





90 





12 50 200 800 
Frequency, 


licrophone No, 9 


130 Oweasured Data 
----[sotropic Turbulence 
— Lateral Quadrupole 

120 }—--Longitudinal Quadrupole oe 





110 














4,55 inch 
M 1. 3559 
Uo = 2781 fps 
Ty; = 2407 ° R 
u'/U, = 0.05 (Assumed) 


3150 12500 50000 
Hz 











100 








90 . 
12 50 200 800 


Frequency ,; 


(e) Microphones No. 9 and 











3150 12500 50000 
Hz 


10 


Figure 301. Near-Field Sound Pressure Spectra at Various 


Microphone Locations 


651 


(Cone luded), 











Microphone No, 1 





2 





150 
4.55 inch 

5 = 1.3559 
3 Ua = 2781 
Q 140 Tt = 2407 ° R 
= u'/Uo = 0.05 (Assumed) 
S 130 
So 
3S 
° 
2 120 





110 
12 50 200 800 3150 12500 50000 


Frequency, Hz 


170 


Microphone No. 





170 








160 
O Measured Data 
~—=--Composite Acoustic 
150 Model II 


Composite Acoustic 
Model I 


1/3 Octave Band Sound Pressure Level, CB, 








140 














130 
12 50 200 go 3150 12500 50000 
Frequency, Hz 


(a) Microphones No. 1 and 2 


Figure $02, Comparison of Measurements and Predictions by Composite 
Acoustic Models. 


Microphone No, 3 








2 


150 
© Measured Data 


---- Composite Acoustic 
Model II 





140 


Composite Acoustic 


Model] I OYWO 


130 





120 





12 50 200 800 3150 12500 50000 
Frequency, Hz 


Microphone No, 4 





150 + 
= 4,55 inches 
= 1.3559 
140 eS = 2781 a 
Tt = 2407 R 
u'/U, = 0.05 (Assumed) 








130 





1/3 Octave Band Sound Pressure Level, dB, re 0.0002 dynes/cm 





120 








12 50 200 800 3150 12500 50000 
Frequency, Hz 


(b) Microphones No. 3 and 4 
Figure 302. Comparison of Measurements and Predictions by Composite 


Acoustic Models (Continued). 


- 








—EE 8 = 


AD-A038 614 GENERAL ELECTRIC CO CINCINNATI OHIO AIRCRAFT ENGINE GROUP F/6 20/1 
SUPERSONIC JET EXHAUST NOISE INVESTIGATION. VOLUME III. COMPUTE=-ETC(U) 
JUL 76 DR FERGUSONs M A SMITH, P R KNOTT F33615-73-C-2031 
R74AEG6452-VOL=-3 AFAPL-TR-76-68-VOL-3 





pile 
Me 


P. L2S LA Wes 








2 


Microphone No. 5 


150 
© Measured Data 


--=--Composite Acoustic 
140 Model If 
Composite Acoustic 


Model I 
130 


120 





110 
12 50 200 800 3150 12500 50000 


Frequency, Hz 


Microphone No. 6 


1/3 Octave Band Sound Pressure Level, dB, re 0,0002 dynes/cm 





130 
120 
110 
D = 4.55 inches 
M = 1.3559 
Uo = 2781 fps 
100 fe) 
Tt = 2407 ° R 
u'/Ug = 0.05 (Assumed) 
90 
12 50 200 800 3150 12500 50000 


Frequency, Hz 


(c) Microphones No, 5 and 6 


Figure 302. Comparison of Measurements and Predictions by Composite 
Acoustic Models ( Continued ). 


654 


130 


2 


120 


110 


110 Ug = 2781 
Ty = 2407 ° R 
u'/U, = 0.05 (Assumed) 








100 
12 50 200 800 3150 12500 50000 


Frequency, Hz 


Microphone No. 8 


O Measured Data 
----Composite Acoustic 
Model II 
Composite Acoustic 


1/3 Octave Band Sound Pressure Level, dB, re 0.0002 dynes/cm 





12 50 200 800 3150 12500 50000 
Frequency, Hz 


(d) Microphones No, 7 and 8 


Figure 302. Comparison of Measurements and Predictions by Composite 
Acoustic Models ( Continued), 





130 


2 


120 


110 


O Measured Data 
----Composite Acoustic 





100 Model II — 
—— Composite Acoustic 
Model I 
90 
12 50 200 800 3150 12500 50000 


Frequency, Hz 








4.55 inches 
1,3559 





90 


1/3 Octave Band Sound Pressure Level, dB, re 0.0002 dynes/cm 


12 50 200 800 3150 12500 50000 
Frequency, Hz 


(e) Microphones No. 9 and 10 





Figure 302. Comparison of Measurements and Predictions by Composite 
Acoustic Models (Concluded). 


656 











APPENDIX 9 





INPUT SHEETS 


: 
{ 
i 


657 














JET MIXING ANALYSIS Page of ___ 
CDC JET MIX 
General Flow Properties Sheet 1 
74 
NAME = 
ADDRESS = 
IDENT = 
input tape? output tape? 
; T or F TorF 
74 12 14 
JETMIX 
#Aa Problem Type 
(F) free jet (T) axisymmetric (F) single mixing region 
T confined mixer F plane, 2-D T coannular or coplanar 
MIX= »AXI= » TWO= 
Primary Jet Description 
diameter, in. 
DIAJ= Pe 


-specify one line of reference quantities/specify either TJET or VJET- 


Mach number turb. intensity (0.) velocity, fps 
MJET= »TIJET= »TJET= » VJET= ; 
total pressure, psia ® 
PTJET= » LIJET= »TJET= »VJET= ‘ 
Secondary Jet Description | 


diameter, in. 
DIAO= ’ 


-specify one line of reference quantities/specify either TJETO or VJETO- 





Mach number turb. intensity (0.) velocity, fps 
MJETO= » LIJETO= » TJETO= » VJETO= ; 
total pressure, psia 
PTJETO= » LIJETO= »TJETO= » VJETO= 


External Boundary Conditions { 


-specify static pressure and temperature/specify either Mach number or velocity- 
static pressure, psia,static temp., ° R, turb. intensity, Mach number, velocity, 


fps. 
(14.69594) (518.688) (0.) 
PE= ’ TE= , TIE= »ME= » VE= > 
Fluid Properties and Program Controls 
gas constant ft lbf/lbm ° R Prandtl number turbulent Prandtl number 
(53.34) (.72) (1.0) 
RG= »PR= »PRT= ; 


-data for Sutherland viscosity formula~+(air) 


SC= » TREF= » MUREF= acetates 9 
step controls, restart station, and transition region scale interpolation 

(.02) (.02) (T) 
CXPC= »CXTP= »RESTRT= » LTERP= ’ 


658 








JETMIX/Sheet 1 (back) 
NOTES: 


1. Supply the 3 prefix cards for identification (user name, address, problem 
identification). 


2s For no input tape file (01) or output tape (disc) file (02), set columns 
12 and 14 of the JETMIX card to F. For either an input or an output file 
set the proper column to T. 


as Parenthesized quantities denote initialized values. 


4. The confined mixer option (MIX = T) is restricted to a single mixing 
region (TWO = F). 


Oe For imcompressible cases, input MJET = 0. 
6. For free-jet cases, the static pressure PE is constant. For confined- 
mixing cases, PE denotes the static pressure at the discharge plane of 


the jet. The preceding remarks also apply to TE, TIE, ME, and VE. 


qT. The step-size controls CXPC and CXTP apply to the potential core and 
transition regions of the jet, respectively. 


* Ye 

ransition Po 

em EX = D, 
% ap 





In region I, AX = CXPC*b, where b = width of mixing zone. 
In region II, AX = CXTP*y 


8. The isentropic exponent is computed as a function of temperature using 
the following empirical relation: 


(-.070271) 


y = 2.23708T 800 < T ¢ 3600 
y=1.4 T < 800 
y = 1.254 T > 3600 


9. RESTRT may be used to restart a mixing problem at a given X or XD 
station. The value of RESTRT must appear in the X or XD table, and pro- 
files must be stored on tape at the RESTRT station. For continuation of 
confined-mixing problems using the free-mixing option (MIX = F), input 
the normalized restart station (XD/DIAJ). 


10. LTERP determines the variation of turbulence scale in the transition 
region. 
LTERP = T -- linear variation to fully developed region. 
LTERP = F -- exponential variation to fully developed region. 


659 














JET MIXING ANALYSIS Page of 
CDC JETMIX 
Species Diffusion Input SHEET 1A 


T -- Species diffusion 


(F) - No diffusion no. of constituents 
(3) 
DIFF= ’ NC= > 
(ATR) (c02) (H20) 
CN AME= > ’ > > eee 


jet stream mole fractions/external boundary conditions 


primary jet 
ALJ= ’ ’ ’ ’ ’ 


ALJO= > 9. > 


OSS SSeS 


ALE= > ? > 


fluid properties 
effective Schmidt number 


(.7) (.7) (.7) 
SCM= ’ ’ ’ ’ Se ae 
coefficients in polynominal for species molar C, 

a b c a b c 
cPC(1)= > > > > eS, 
CPC(7)= ’ ’ ’ ’ ’ ao 
CPC(13)= > ’ > > > 


Oe 


L. The program is initialized to consider AIR, C02, and H20 as constituents 
1, 2, and 3, respectively. These settings may be overridden by using the 
CNAME input. In this instance, the coefficients for the molar C, of each 
constituent must be specified. Note that this option may not be used if 
the NAMELIST input routine does not read hollerith data. 


2. Heat capacities are specified as a quadratic function of absolute tempera~ 
ture: 


C. = a+t+bT + eT” Btu/lb mole ° R 


i cal./gm mole ° K 


660 








JET MIXING ANALYSIS Page of 
CDC JETMIX 
Station Input Data Form Sheet 2 





Free-Jet Mixing 


X XPRN 

B(1)= ’ ’ 

? = 

? ’ 

’ ’ 

’ ’ 

’ ’ 

’ > 

’ ’ 

’ ’ 

’ > 

> ’ 

9 

Confined-Jet Mixing 

7 XD RD YCB XPRN 
B(1)= ’ ’ So Se 


661 





JETMIX/Sheet 2 (back) 


NOTES : 


Es 


For free-jet cases, the required input consists of the dimensionless 
axial stations (X = X, in./DIAJ, in.) at which a print of the jet 
properties is desired. 


The coordinates for the confined-mixing case (XD, RD, YCB) are input in 
dimensionai form. Select suitable intervals in XD to adequately describe 
the intended curve. If bits = 1, E+15 (junk word) appears in the lists 
of RD and YCB, the missing values will be supplied by linear interpola- 
tion against the independent variable XD or (if no subsequent data are 
given) the last value in the list is extended down the colum. 


Specify XPRN(1)=1 at those stations for which tabular printout of 

the jet mixing profiles is desired. If profiles are to be printed at all 
stations, specify XPRN(1)=2 only at the first station. Also, if profiles 
at a given station are to be saved on file (2), but not printed, set 
XPRN(1)=1. To save all profiles, set XPRN(1)=-2. 


The centerbody coordinate is assumed 0 if the YCB colum is left blank. 
If desired, these data lists may be input in "Free form" under their 
symbolic names X, XD, RD, YCB, and XPRN using FORTRAN IV Namelist conven-~ 
tions. If data are specified in both the B~block and in the Free form, 
the B-block data will take precedence. 


The maximum number of X or XD stations is 100. 


If no station input are provided, the program will use a set of X's, 
XPRN's which have been optimized for acoustic cases. 


662 








FLOW FIELD ANALYSIS 
SUPERSONIC TURBULENT JETS 








FILE MERGE 
Page of 
(CDC) SSFD MERGE 
SSFD/MERGE Sheet 1 of 1 
———$—<—<———— | 
output tape? 
wy or F 
SSFD J 
SINPUT 
Problem Description 
(T) axisymmet ric initial Mach No. Specific heat ratios 
F 2-D (1.05) (1.4) (1.4) 
AXISYM= ,XMACH= »GAMMA(1) = , , 
Program Controls 
oy eee San a Cee ae eM an es ae Page ce 
final x/D, stability parameter (1)print profiles & 
shock pattem 
(.5) O-no print 
XL= »STABIL= , IPRINT= ‘ 
SSFD Total Pressure Input Stations 
SIC Se eee Sh ees ee eee 
no. of stations 
(20) 
NPT= > 
x/D from JETMIX Tables (see NOTES) 
XPT(1)= ’ ‘J > > ’ 
EE Ce ee | ’ 
: | ’ 
’ ’ ’ > — 
MERGE z 


663 





SSFD 

MERGE/Sheet 1 (back) 
NOTES: 
Je SSFD always requires a JETMIX input file for execution. If SSFD is 





being run alone, using an input tape saved from a previous JETMIX run, 
supply the 3 prefix identification cards (see JETMIX input sheets). If 
an output file is to be generated (file 3), set colum 14 to T. 


If GAMMA is input, two equal values must be supplied. 


Parenthesized quantities denote initialized values. The only item which 
may normally require revision is STABIL. This input controls the step 
size used in the finite difference solution. If XL is not input, the 
final station will be taken as the JETMIX X/D where the potential core 
disappears. 


SSFD uses total pressure data from the JETMIX solution at the stations 
XPT. These stations are initialized to: XPT(1)=0, .1, .2, .3, .4, .5, 
aby (Mary. Zag cate, Sag Gels Deis Cees, Pedis oes, has PS. BS. 2, 20), 


The MERGE program is used to collate data from JETMIX and SSFD for input 


to the NOISE program. Include the MERGE T T card if the collated data 
are to be saved for a NOISE run (file 4). No namelist input are required. 


664 








JET NOISE PREDICTION 
Page of 
NOISE/Sheet 1 








NOISE T F 
SA 


General Input 


(2) JETMIX input file 
4 MERGE input file 


MFILE= 3 
source-receiver jet diameter scaling 
frequency shift factor ref. pressure 
C25) Cz) (.0002) 
SCALT= »SCALJ= , PREFN= 
- 1/3-octave band analysis 
BAND3= ; 
(0) fixed convected Mach No. convected Mach No. constant 
1 variable convected Mach No. C263) 
MC= ; CVMACH= ; 
(F) tint 2 acoustic intensity proportionality constant 
T -00425 cold jet 
-002125 hot jet 
BETAIN= ; BETA= : 
(0) cold jet (Eq = GEL a" )faq 
1 hot jet tT q@= aM 
JETTEM= , op rr 
Acoustic Models-Far Field 
(0) Lighthill self-noise model (F) ‘ 
1 Empirical self-+shear-noise a 
LIGHTH= LILLEY= n 
(F) : 0 - self-noise only O - shear-noise only 
r Ribner's model (.2577) (1.0) 
RIBNER= ,CRIB= ,SE= ; 
(0) self- +shear-noise 
(F) ' (F) Pao Press. 1 self-noise only 
Hs —— T spectrum model 2  shear-noise only 
PAO= »PSPEC= »MU= Se es 


Acoustic Model-Near Field 


- Isotropic turbulence model 

- Lateral Quadrupole model 

- Longitudinal Quadrupole model 

- Combination model using Lateral Quadrupole model 
in transition region 

5 - Combination model using Longitudinal Quadrupole 

model in transition region 


1 
2 

3 
NEARFD= , 4 


X/D where potential core disappears 
XCORE= ’ 665 





NOISE/Sheet 1 (back) 
NOTES: 





Le [If NOISE is being run alone, using an input tape from JETMIX or MERGE, 
supply the 3 identification prefix cards (see JETMIX input sheets). 


N 


Parenthesized quantities denote initialized values. These items need 
not be input unless different from preset values. 


as MFILE denotes the input file code for the aerodynamic data. Set MFILE = 


4 if aero input is from the MERGE program. 


> 


The proportionality constant BETA may be determined from experimental 
data. If input, set BETAIN = T and supply the empirically determined 
constant. 


aie If no acoustic model is selected, the Lighthill far-field model will be 
used. Only one model may be selected per case. 


666 

















JET NOISE PREDICTION 

















Page of 
NOISE/Sheet 2 
Microphone Geometry Configuration 
\?/ no. of input angles 
NA= - 
angle from jet exhaust axis, degrees 
ANGJ(1) = > > > >. ’ > % > > ’ 


(F) sideline configuration 
T are configuration 


ARC= Fr 
specify either sideline distance or arc radius, ft 
SLINE= ' ARCL= Z 


Output Control 
specify T for profile printout at a given angle (ANGJ) 


NY OY YY. ? — ” 


ACQUSP(1)= ee ae ee RE ee ’ ’ ’ ’ ’ 


667 











NOISE/Sheet 2 (back) 


NOTES: 


ee eee 
OECeEECCECElElelee eel ll l__ 


1. 


Acoustic calculations may be performed at a maximum of 20 angles 
(ANGJ, degrees). 


Specify the microphone configuration as either a sideline or an arc. 
Also, specify the sideline distance or the arc radius in feet. 


The normal program output consists of a summary of the @ASPL, SPL, PWL, 
etc. at the axial stations of the jet for each angle. This output may 
be augmented by 1/3-octave-band PWL's. If detailed output is desired, 
ACSPAN and AC@USP may be used. Detailed profile printout will be 
obtained when ACSPAN(I)=T and any of the AC@<USP(J)'s are true. 


668 











APPENDIX 10 


SAMPLE CASES 


669 





670 


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x/O s 


2900 Y/0 = 


SIOEFLINE NISTANCE= 


LOWFR FREQ,H2Z 


11.2 
14,1 
17.8 
22.4 
28.2 
35.5 
4u,7 
56.3 
70,9 
89.2 
112.0 
141.0 
178.0 
224,0 
282.0 
355.0 
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563.0 
709.0 
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1120.0 
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4470.0 
5630.0 
7090.0 
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11200.9 
14100,0 
17A09,0 
22400.0 
28200.9 
35500.0 


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16.0 
20.0 
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31.5 
40.0 
50.0 
63.0 
80.0 
100.0 
125.0 
169.0 
200.0 
250.0 
315.0 
400.0 
500.0 
630.0 
800.0 
1000.0 
1250.0 
1600.0 
2009,0 
2500.0 
3150.0 
4000.0 
5000.0 
6300.0 
8000.0 
10000,0 
12500.0 
16000.0 
20000,0 
25000.0 
31500,0 
40000,0 


CENTER FREQ,HZ 


1/3 OCTAVE BAND ANALYSIS 


6.457 R/D = 
2.60 


14.1 
17.8 
22.4 
28.2 
35.5 
44,7 
56.3 
70.9 
89.2 
112.0 
141.0 
178.0 
224.0 
262.0 
355.0 
447.0 
563.0 
709.0 
892.0 
1120.0 
1410.0 
1780.0 
2240.0 
2820.0 
3550.0 
4470.0 
5030.0 
7090.0 
8920.0 
11200.0 
14100.0 
17800.0 
22400,0 
28200.0 
35500,0 
44700.0 


UPPER FREOQ,HZ 


6.857 


NPTS 


210 
264 
246 





ANGLE= 90.90 


SPL , OB 
0.0090 
12.3334 
26.0390 
14,9565 
29.6719 
40.4346 
47.6497 
58.2277 
71.6895 
75.5581 
81.3884 
65.7414 
90.4295 
94.9076 
99.6651 
104.3096 
110.1286 
113.8932 
119.642A 
124.1747 
126.8332 
130.1194 
131.2254 
131.1840 
131.6952 
129.9682 
129.8927 
127.4655 
123.0744 
122.uAuS 
118.7219 
116.4661 
112.2212 
101.9524 
74,4468 
9.0000 











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X/D = #13,693 


14/3 OCTAVE RAND 


Y/0 = 


SIDELINE DISTANCE= 


LOWER FREQ,HZ 


11.2 
14.1 
17.8 
22.4 
ek.e 
35.5 
44u.7 
56.3 
70.9 
89.2 
112.0 
141.0 
178.0 
224.0 
262.0 
355.0 
447,0 
563.0 
709.0 
892.0 
1120.0 
1410,0 
1780.0 
2240.0 
2820.0 
3550.0 
4470.0 
5630.0 
7090.0 
8920.0 
11200.9 
14100,0 
17600.0 
22400.0 
28200.0 
35500.0 


CENTER FREQ,H2Z 


12.5 
16.0 
20.0 
25.0 
31.5 
40.0 
50.0 
63.0 
80.0 
100.0 
125.0 
160.0 
200.0 
250,0 
315.0 
400.0 
500.0 
630.0 
800,0 
1000.0 
1250.0 
1600.0 
2000.0 
2500.0 
3150.0 
4000.0 
5000.0 
6300.0 
8000.0 
10000.0 
12500,0 
16000,0 
20000.0 
25000.0 
31500.0 
40000.0 


6.857 R/D = 
2,60 


14.1 
17.8 
22.4 
28.2 
35.5 
44,7 
56.3 
70.9 
R9.2 
112.0 
141.0 
178.0 
224.0 
282.0 
355.0 
447.0 
563.0 
709,0 
892.0 
1120.0 
1410.0 
1780.0 
2240.0 
2820.0 
3550.0 
4470,0 
5630.0 
7090.0 
8920.0 
11200.0 
14100,0 
17800,0 
22400.0 
28200,0 
35500,0 
44700,0 


898 


ANALYSIS 


15,314 


UPPER FREO,WZ NPTS 


210 
284 
246 
234 
teu 
172 
195 
130 
161 

Bo 
144 
131 
434 

36 


39 





ANGLE= 153,40 


SPL . DB 
0.0000 
1.5605 

14.4717 
0.0000 
18.4729 | 
29.7013 
39.2610 
50,8691 
65,1495 
68.5694 
74,9417 
77.8628 
82.0509 
85.9428 
90.1270 
94.0747 
99.2437 
102.2861 
107.4037 
111.1046 
112.9323 
115.4782 
115.9907 
115.7544 
116,504} 
115.5696 
116.5754 
115.9823 
113.5279 
115,0344 
113.7349 
114.1018 
113.5339 
108.6402 
94.,84R4 
0.0000 | 





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APPENDIX 11 


PROGRAM LISTINGS 


902 





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