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NJ\SI\ National Aeronautics and Space Administration
NASA CR·t14103 CHAM H3605/16
NASA-CR-174703 19840024359
RC)CKET INJECTOR ANOMALIES STUDY
Volume II: Results of Parametric Studies
by A.J. Przekwas, A. K. Singhal and L.T. Tam
CHAM of North America, Incorporated
August 1984
Prepared for
LIBRARY COpy
LANGLEY RESEARCH CENTER LIBRARY, NASA
HAMPTON, VIRGINIA
NJ~TIONAL AERONAUTICS AND SPACE ADMINISTRATION NASA - Lewis Research Center
Contract NAS3-23352
1111111111111 1111 11111 11111 11111 11111 11111111 : NF00432
https://ntrs.nasa.gov/search.jsp?R=19840024359 2018-05-16T20:22:20+00:00Z
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_CATEGORY 20 RPT*= NASA-CR-174703 P.I"",.....
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Cf\jT#;' l\lAS3-23352 84~./07/IG0 2 'vIOLS -,. II
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PAGES t~r~JCLASS I F I ED DOC1JMErJT UTTL: Rocket injector anomalies study. Results of parametric studies
. "r-\~. _ WnW· I fL.-',I-
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TLSP: Final Report A/PRZEKWAS1 A. J.; 8/SINGHAL~ A. K.; C/TAM1 L. CHAM of North Amerlca1 Inc.1 Huntsville, Ala. CJf.iLl ... · .. !:.~~ C:.i-i 1 ! 1'..1' f .... J II J ''''- 1..
/*COMBUSTION CHAMBERS/*LIQUID PROPELLANT r""nl..lr- T r"-I II"\~."" T /""\'.1'-'
l,{)PJr i t~ljKH: 1 L.!f'JO
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AvtAIL.f~TIS SAP! He
~/ COplPUTER PROGRArlS/ COf:,lPUTERIZED SIt~ULATIGr·J/ EULER-LAGRAf:·JGE EQljATIOp.!l EV'APORAT I Ofi,L·/ FLOtt! D I STR I BL~T IOf:·J ... / ;=·1ATHEf·1AT I CAL f~10DELS/1 SPRA\1 I rJG ,- A. It' .
i:.. H. t\. The employment of a existing computer program to simulate three d 1 rner}s 1 {Jfl.31 ttU() f=ifl,::;t;:;e 9aE; :=;t='t ,3~~: f i OUJS i fl 1 j c~u j d tJt'~Of:'e 1 1 Efnt r{)Ct:~e t eng: [-let:;. This was accomplished by modification of an existing three dimensIonal computer program (REFLAN3D) with Euler/Lagrange approach for simulating two phase spray flow, evaporation and combustion. The modified code is referred to as REFLAN3D-SPRAV. Computational studies of the model rocKet engine combustion chamber are presented. The parametric studies 0 the two f:'rldse lOl.lJ .3r}cl" c:()Ini)ust j or} E;fiOtJJE; cn .. Ja 1 j tE~t 1 \/e I}": <:c)t (Ere;! reSf:tonse fc) variat ons in geometrical and physical parameters. The injection
~---------'---------------r--------------------'-----'------------------'------~ 1, Report No. 12. Government Accession No. 3. Recipient's Catalog No.
CR-174iOa _ i--4.-T-ltI-e -an-d'Subtltle , ____________ ...1-__ _
Rocket Injector Anomalies Study, Results of Parametric Studies
7. Author(sl
A.J. Przekwas, A.K. Singhal and L.T. Tam
--9. Performing Organization Name and Address
CHAM of North America Incorporated 1525-A Sparkman Drive Huntsville, Alabama 35805
12. Sponsoring Agency Name and Address
NASA lewis Research Center
15. Supplementary Note:;
Project Manager: Dr. Larry P. Cooper NASA Lewis Research Center 21000 Brookpark Road Cleveland, Ohio 44135
5. Report Date
July 1984 6. Performing Organization Code
8. Performing Organization Report No.
H360f/16
10. Work Unit No.
11. Contract or Grant No.
NAS3-23352 13. Type of Report and Period Covered
Final
14. Sponsoring Agency Code
,---------------------,-----------------------------------------~ 16. Abstract
The objectlve of the presented research project was to employ existing computer pY'ogram for simulating three dimensional two-phase gas-spray flows in liquid propellant rocket engines. This has been accomplished by modifyin~J an existing three-dimensional computer program (REFLAN3D) with Eulerian Lagrangian approach for simulating two-phase spray flow, evaporation and combustion. The modified code is referred to as REFLAN3D-SPRAY. This report presents results of computational studies of the model rocket engine combustion chamber. The parametric studies of the two-phase flow and combust'ion showed qualitatively correct response for variations in geometrical and physical parameters. The injection nonuniformity test with blocked central fuel injector holes showed significant changes in the central flame core and minor influence on the wall heat transfer fluxes.
Mathematical background of the model is described in an accompanying report NASA CR 174702.
17. Key Words (Suggested by Author(s))
Rocket Eng'ines, Propulsion, Co.mbustor Models, Spray Combustion
18. Distribution Statement
1~SGu~CIMslt~~f~~~ls~re=p~~~)------~2~0~.S~e=~~rl~ty~C~~=~~If~.(~cl~th~ls~p=a=ge~)---------~12~1~N~~f-----~I--·--------~ . 0.o68Pages , 22. Price·
UNCLASSIFIED UNCLASSIFIED ____________ ~ ___________________________ _L _______ ~~ _______ ~
• For sale by the National Technical Information Service, Springfield, Virginia 22161
Ngcf~ 3;;L4~q1t;
FOREWORD
CHAM of North America Incorporated has performed a Rocket Injector Anomaly Study under tile NASA Contract NAS3-2~~352. The purpose of the study was to modify, test and demonstrate a computer code for predicting three-dimensional two-phase spray now and combustion in rocket engines. The modified computer code REFLAN:3D-SPRAY (REactive FLow ANalyzer 3-Dimensional , with two phase spray) and results of parametric studies have been described in the folloiwng two volumes:
Vo'lume 1: Description of the Mathematical Model and Solution Procedure; Volume 2: Results of Parametric Studies.
Transfer of the code to NASA LeRC computer center, and preparation of a user's manual are recommended as next steps of the study.
The authors wish to thank all those who have contributed to this work. In particular, thanks are due to Larry P. Cooper and Ken Davidian of the Communication and Propulsion Section of NASA LeRC; and to Laurence Keeton, Jack Keck, Kelli King, Janet Siersma, and Ronni Rossic of CHAM NA.
1
2
3
3.1 3.2
4. 4.1 4.2
4.3 4.4 4.5 4.6
5.
5.1 5.2 5.3
6
SUMMARY
INTRODUCTION
TABLE OF CONTENTS
CODE CHECK-OUT CALCULATIONS Symmetry Tests Uniformity Tests
PARAMETRIC STUDIES OF INJECTOR ANOMALIES Objective The Combustion Chamber and Injector Geometry Basic Test Case Specification Test Cases for the Parametric Evaluation Results of Base Test Case Results of Parametric Studies
MODEL IMPROVEMENTS AND RESEARCH NEEDS Improvements in Mathematical Models of Physical Processes Improvements in Numerical Method and Solution Algorithm Validation and Verification Study
REFERENCES
APPENDIX A INTERPRETATION OF RESULTS AND THE COMPUTER OUTPUT A.1 Geometry and Computational Grid A.2 Interpretation of the Printout A.3 Interpretation of the Graphical Plots
1-1
2-1
3-1 3-1
3-11 I
4-1 4-1 4-1 4-5 4-5 4-12 4-22
5-1 5-1 5-2 5-5
6-1
A-I A-I A-4 A-4
SECTION 1
SUMMARY
The liquid fuelled rocket engine combustors consist of an injector plate and a thrust chamber. The injector plate consists of a number of propellant injectors which are designed to atomize the liquid jets of reactants and to promote intensive mixing between the vaporized components.
Figure 1.1 shows the schematic of the rocket engine and injector' plate, with LOX-RPI-LOX unlike triplet injectors, considered. The purpose of the stud¥ was to demonstrate an analyti cal capabil ity to predi ct the effects of reactant injection non-uniformities (injection anomalies) on heat transfer to the walls. For this purpose an existing three-dimensional single-phase flow and combust"jon computer code (REFLAN3D: - REactive FLow ANalyzer, 3-Dimensional) has been modified for simul at"j ng two-phase flows in l"i qui d propell ent rocket engi nes. The modifi ed code is refelrred to as REFLAN3D-SPRJW.
Mathematica"1 basis and solution pY'ocedure of REFLAN3D-SPRAY are described in VolumE~ 1. This volume presents computational results to demonstrate the capability of the code. Reported results include numerical tests as well as several parametric test cases of the model rocket combustor (Figure 1.1).
The results of the effort undertaken in this program can be summarized as follows:
The REFLAN3D computer code with Eulerian-Lagrangean technique has b,een adapted for three-·d"imensional, elliptic, two-phase flow with evaporation, heat transfer and combustion in liquid fuel rocket engines.
1-1
....... I
N
('-I \--1 j ~--~/ .
R
-Flow
Injector Detail
~><)~'
2.60" Dia
Figure 1.1 Rocket Engine Geometry and Injector Plate Configuration
3.00"
The code check-out calculations satisfied uniformity and symmetry requirements for both nonreactive and reactive turbulent flows.
The parametric studies of the two-phase flow and combustion in rocket engi ne combustor showed qual itat'j vely correct response for variations of geometrical and phYsical parameters.
The injection nonuniformity test case, 'j .e. with blocked 25% fue'l injector holes near the chamber, showed significant changes in the central flame core and minor influence on the wall heat transfer ( Fig u re 1. 2) .
The areas for model improvements, in both physical and computational aspects, have been identified and recommended for further studies.
The recommendations for further stud'jes include: a) preparation of a user's manual and code transfer to NASA LeRC;
b) improvements in the physical models of evaporation, turbulent diffusion of droplets, and chemical reaction;
c) numerical improvements to increase the accuracy of solution method; and
d) Verification studies with and without comparisons with experimental data.
RO INJ ANAL, BASIC 21-XGRID XY PLANE 4 TEI'1P CONTOURS
PHIMIN 1.660E+02 PHII'1AX 2.S25E+03 CONTOUR LEVELS
1 o4.2S0E+02 2 6.901E+02 3 9.522E+02 4 1.2104E+03 5 1.o476E+03 6 1.73SE+03 7 2. 000E+03 8 2.263E:+03
ROINJAN,BLoce:D 25" FUEL !'tOLES :<Y PLANE 4 TEMF' CONTOUF!S
PHIMIN 1.61:19E+02 PHIMPIX 2.S1.7E+03 CONTOUR LEVELS
a) full injection
. ____ :=J
b) blocked 25% of the fuel injector holes 1 o4.298E,,·02
2 6. 907E,,·02 ; 3 9.516E+I2'2 .. 1.212E+I2'3
5 1.473E-.·03 l~'_~ ~~~~~~~~~~~~~~~§§~~~§~§~;;:::==:J 6 1. 7304E+03 ~ 7 1.995E+1213 i;:: 8 2. 256E+1213 ::
Figure 1.2 Temperature contours within the rocket eng inp
1-3
SECTION 2
INTRODUCTION
The objective of the present study was to demonstrate an analytical capabil ity to pr(~dict the effects of reactant injection-nonuniformity (injection anomal ies)
upon 'local and overall heat transfer' in a liquid propelled rocket engine combustion chamber. For this purpose the REFLAN3D' (REactive FLow ANalyzer
3-Dimensionall) computer code has been modified for the two-phase spray
calculations and applied in this study. The modified code is referred to as REFLAN3D-SPRAY.
Detailed description of the mathematical model is provided in Volume 1,
entitled "Description of the Mathematical Model and Solution Procedure".
This report constitutes the second volume of the documentation and describes the computational results and analysis of:
a) code check-out test cases (section 3), and b) five parametric test cases (section 4).
Based on the analYSis of these results further model refinements are identified and described in section 5. The need for model varification against experimental data is also discussed in section 5.
2-1
SECTION 3
CODE CHECK-OUT CALCULATIONS
In order to predict the effects of injection nonuniformity on the -fluid flow
and heat transfer pattern the numerical model has to be capable of responding
to sman changes in injection patt(~rn. Therefore, in addition to common checks
of computational stabil ity, converqence rate etc, a model accuracy for
handl ing cycl ic boundary conditions must be checked before it can be us'ed
for injection anomaly studies. For this purpose two test cases were selected
to check solution symmetry and uniformity characteristics.
3.1 SYMMETRY TESTS
Figure 3.1 presents the geometry configuration and computational grid used for
the symmetry test calculation. The fluid enters the cylindrical duct through a
partially blocked front plane. Two jets:
the inner jet with smaller velocity, and
the outer jet with larger velocity,
enter through a V-shaped opening in the front plane. The fluid leaves the
duct through fully open exit plane.
A 1800 sector of the duct has been used as the solution domain for the calculations
with uniformly distributed grid (NX: NY: NZ .. 21: 10:,4). Due to the symmetry of
the combustor' geometry and symmetrical boundary conditions, the results of the
computations should be symmetric about the 8=900 plane.
Results of the following two test cases are presented in this section:
A) A constant temperature (300o K), constant viscosity (.001 kg/ms)
flow with fixed inlet velocities (10 and 20~) and fixed exit pressure
(relative pressure, Pexit :: 0), specified as a boundary condition.
B) All conditions same as is case A, except that the larger inlet
velocity is changed from 20 to 50 m/s.
The k,··€ turbulence model has been used for both test cases o Figure 3.2 presents
calcu'lated v(~locity vectors in all four axial planes for test case Ao Note
that the min'imum (VELMIN) and maximum (VELMAX) velocity vectors at symmetric
planes (1 and 4) and (2 and 3) are identical. The shape and size of the
recirculation zones on plane 1 and 4 are also the same.
3-1
a: 2 , --
d co
l!X3 ~t I I
,.--r--- -r ~ , , C::) ,-
~f-+--~ , , , , , en , ,i-- --, , ' , , , , ,'- --
co
~ , , , , , J' , Z
~i-- --, , , , . , , , , 0 I-
<0 U W
! 'f-, I~~
. , -~ ,
~ l;r~ , , , ,
'f- -'l
rC!J ~ , , . , 'f- -,
C/l C/l 0
I!l a: I u > I I X
I '<1"
I M , , , ~ , , M I 'f- -, , . , . , . , , , , , ,
I N I
'f- -~ , ,
~ v////// '//-:'< I!l '<t M N ~I OalCOr--tO
~" " ~l I '\.. -.;
" I
" 0 I I!l a;
" OJ
U.
I I I I I I I I I
3-2
Figure 3,1 Chamber Geometry
/ /
/ /
/
/ /
/ /
x
/ /
/
/ /
/
':)J >, 41
w Z-~ B _.J (J
Cl. <1> C/l
>-cc ~ ~- OJ -,. ~ LU -
XY PLANE 1 l')ELOC I "Y PLOITS VELMIN 3. 160E-02 I..JELMAX 9. 825E+00
- -", -,,- - -I " \ -"- - --- "- -- • III'
\.." .-.. , ~ PLANE 2 I..,JELOCITV PLOTS I..)ELMIN 8.2~4SE-03 I,)ELMAX 1.908E+01
---. _ .. - --- - -'. .,
III' .. • • .. "-- --- - •
;-'~\' PLANE 3 UELOCITY PLOTS 1,}ELMIN 8.c~45E-03 VELMAX 1.908E+01
- -- po - ----.: .. ' ---- -, --.. .... --.. -_ .. po - • -'."' .. ' .···.1 PLANE 4
UELOCITY PLOTS I....JELl'lIN 3.1.61E-02 I...)ELl"lAX 9.825E+00
- - -", - -;' -- .... , " \ - -
" --- .. -- ... -
-- -.. •
- .. .. -... Po
- • .. .. .. ..
- --..
--.. .. -.. -- - - - -.-- -- - - --- -- - -. .-----.. r - - - ----- -- - - .. --- - .. .. -
-.
.---- ---- -- .. --- - po - ---- .. .. - ---- -- - - -
._-. _- --- ----- .. - - ----- • .. - ---- -- .. - -._- - - - -
--.. -- ..
... - - -~.- - - - --- -- - - ---- -- - - ---- - - - -.. --- - - - .... - - - - ...
Figure 3.2 Velocity vectors for the symmetry test case. 3-3
------ -• -III'
----
---
---- ----
To enable a thorough examination, results of test case B are presented in
the form of the foll owing tabl es:
Table 3.1 - u-axial velocity (m/s) for each ek plane Table 3.2 - v-radial velocity (m/s) Table 3.3 - rw-angular momentum (m 2/s)
Table 3.4 - k-kinetic energy of turbulence tm21s2)
Table 3.5 - global convective fluxes through the chamber cross section (kg/s)
Tables 3.1 to 3.4 indicate perfect symmetry about 6=900• Results are
symmetric at 81 and 84, and 82 and 830 Another symmetry verification is almost zero (10- 9) angular velocity at k=3 plane (8=900
). Note that "backward boomerang"
for velocity staggering is employed. Further guidance on interpretation of the computer output is provided in Appendix A of the present report.
Calculations were performed on a CRAVl computer and total execution time foy'
100 iterations was 5.6 seconds. Figure 3.3 pr~sent:s the convergence rate for test case B 'in the form of residual E!rrOr variation with number of iterations. The residual error E is calculated as the total mass inbalance for the entire calculation domain i.e.
E :: L: ( L:~d)
i j k d
where ijk denote qrid indices, and d = N, S, E, W, H, and L signifies grid cells boundary at which the convective flux Cd is calculated.
The numerical convergence can also be verified from table 2.5 where global or
total convective fluxes across various sections are presented. The notations used are:
Global net convective flux
FX 1 :: L: L: (pu A ) k j
Global positive flux
FXPi :: L: Emax (O.,puA) k j
Global negative (recirculating) flux
FXM. :: E E min (0. ,puA) 1 k j
k=l •••• N=4; j=l •••• M=lO
where u - axial velocity, p - density and, A - grid cell face area, and
i-axial, j - radial, k - c-i rcumferential coordinate indices.
3-4
W I
(J1
Table 3.1 Axial Velocity
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It:! J' c;!:· .~JI! I!J !lll/l III 4.ll! 1-'11...L.!1_'~_' lUI I'!/I.,,"-'I, C /,/ I" f! :,.'.,! 1 j III I 1IIIJ}II ... 8 111,',·' '.11'1" ,q 'I.' ".JII~"·"''' t 0/, .... ,.. '"
Q.O.lthUIE+OQ -1,02£+01, -0,89£'1'00. -2 ,5n~,'-oO ~T.43!+OQ".l,48E+OO "'4~47E.O-O '~4~~lE.OO; 5.iSE:.n O(t-:I~t~ln+oe.f,9lJE+OO _4,57E+00 -5.88E-01"'2.43£+00:- 4.1&E'.00 4:CJlhOO:'f:·S;UE+OO ';'5£+Otr OO,·.6~3ee+oo .S~HE+QO .2.lbEHO 1,1&E+OO ,'l,47E+00 -~,82E+OO 5,42E+OO 5,&2E'+001.7CE+OO 00 ·4.~4hOO .3~OZE+00 3.08E-01 3.14[+004.10[+00 5,&4E+00 5,UE+00 .. 5,95E+00 ;,88[+00; 00 ,·l~UhOG 3.29E .. Ol 3,35E+00 S.37E+OO o,2bE+00 o,S7!+OO -6~50[+00-&.-a8E+OO ~ o.03!+OO~ 00 1,lJE+OO 4.31E+00 &.11E+OO 7,81[+00 7,88£+00 7,55EtOO 7~OoE+00 o.oOE+OOo~loE+OO 00 ,,49E+00 0,84E+00 1.03E+01 1,03!+01 9,49E+00 8,48E+00 1,51E+00 o,86E+OO 0,25[+00 00 1,811+01 1.ibE+Ol 1,38[+01 1.2&E+Ol 1,09E+Ol 9.27EtOO_ 7,97E+00 .7,05E+00 ·o,30E+OO~
20,OOE~OO ·f.5~!+01 1,,8E+Ol 1,&0[+01 1,44[+01 1,19EtOI 9,79E+00 8,2iE+00 7.15E+OOo.3iE+OO~ '.J. _~.0~!!~!!5hOl.,_i.Q
W I
0'>
!l'll(i
INLfr
Table 3.2 Radial Velocity
·*-"'I!It •• * ••• *.!.'tJ*~*''''.CI''''*''*.*tpr*.'.*.*. V·VELOCITY - .*.*.* ••••••• * ••••••• *.* ••• * ••••••• *. .** Kill ***
'" <: " I: .... 7 S ., Q 10/ ,rF'! if'~ :"';b; Ii il{~(t~ //'1 ;o'bE':'o'(tu.,: ~~r':61 G' 'A ~ 19E;';'6 i G{ '6 ;~s~'..'o it. '4: 8'2E~~'i'{':( TsE .. O'( "3 ~5:ii--;',(o1 ~ (~~0~'1
-l,02E+00 o.22E-Ol 1.40E+00 1,59£+00 l,23E+00 9.31E-01 7.38E.01 o.84E-Ol 7.87E·01 -1.20[+00 Q.89E-01 2.01E+00 2.20E+00 1.77E+00 1.35E+00 1.08E+00 9.98£-01 1.00E+OO ~1.15E+OO 1.~lE+OO 2.55E+OO 2.70E+00 2.22E+00 1.72E+00 1.39E+00 1.29E+00 1.30E+OO -8.~9E-Ol 1.8bE+Oo 2.99[+Ou 3.10E+00 2.59E+00 2.03E+00 1,00[+00 1.54[+00 1.S2E+OC -2,58E-01 2.31E+00 3.31E+00 3,37E+00 2.80E+00 2.27E+00 1.88E+00 1,70E+00 l,71E+00
5,28E-01 2,08E+O~ 3,~QE+OO 3.~OE+00 ~.9'E+00 2.41E+00 2.03E+00 1,91E+OO 1,88E+00 1.20E+OO 2,84E+OO 3.~3E+00 3,40E+00 2,93E+00 2,39E+00 2.05E+00 1,90E+00 1.90E+00
Ek"
1.35E+00 2,40E+OO 2.85E+oO 2.80E+00 2,41£+00 1,99£+00 l,74E+00 1.08E+00 1.73£+00 ~ ~TOOE+ge g.~O ~E+o.Q-.-4.,OO~ 0.~0 ~+OO -G~E+OO ~E+OO -&-oOOE+09---' ~
"'''''' K = 2 "''''''' ~::t((.( ~fL::.rw .. (;r;~: H /( 'i ~ ~'h-ol 2. ~2E-0 1 1 • 8~[':O 2 -7, 4!E'~02 . ~ i ',2~E'~~ ~ .' ~ ~ ~ b! E'~O~' .~ i:;';~ ~O'; .~~ J~~~';2
1.20E+00 4.21E-01 -1.25E-02 -1.90E-01 -2,70£-01 -3.34E-Ol -3.57E-01 -2.40E-01 1.60E+oO 4,90E-Ol -1.08E-Ol -~.52E-Ol -4.54E-Ol -5,23E-Ol -5.55E.Ol -4.3bE-01 1.0~E+OC 3.70E-01 -3.08E-Ol -5.79E-01 -0.70E-01 -7,35E-Ol -7.72E-Ol -b.03E.01 1,28E~OO 1.99E-02 -0.49E-01 -8.84E-Ol -9.30£-01 .9,70E-Ol -1.01E+00 .9.15£-01
UJr 4.30E~Ol ~b.13E-Ol -1.14E+00 -1.20£+00 -1.22£+00 -1,22E+00 -1.25E+00 -I,19E+00
4 3.4eE~Ol 2.q5E~Ql -b.9QE-Ol .1.42£+00 -1.71E+00 -1.c7E+00 -1.53E+00 -1.47£+00 -1.48£+00 -1,40£+00 3 -2.71£+00 -1.11E+oO -l.'bE~OQ -2.13E+00 -2.19E+00 ~2,OOE+00 -1.70E+00 -1.05E+00 -1.05E+00 -1.01E+OO 2 -1,19£+00 -1,SlE+OO -1.Q9E+OO -2.2SE+00 -2.20E+00 -1.94E+00 -1.00E+00 -1.52£+00 -1.52£+00 .1.59£+00 1 ~ E~ .0 .... 0 (t£ +·0 0 ~ 0 E +-04-- 0 ~ 0 0 0 • 00 E + O.Q..----O • o.~+40 ".~.o..o.i + 0 0-- 0.0 0 E-"'~ 0 • o..o.uco 0 . -!l-rD 0 E.+.D..O- . _ .. ~
"'*'" I\. = 3 "''''''' ~ J 1- .1.,., n'£! '" 5 10 1.00£+0.0 1.03 +00 ,~2E-OI \.84£';-02 q~ l,48E+o.O 2,13E+00 ~.21E-Ol -1.25E-02 8~ 9.24E.Ol 2.96E+Oo. 4.9o.E-01 -l,08E-Ol 7j 1.17£+00 3.31E+Oo. 3.70E-01 -3.08E-o.l ~ 5.4'3E+00 3.06E+00 1.99E-02 -0.49E-01
5 2,05£+00 1.B1E+OO 4.3bE~Ol ~c.13E=Ol =l.l~E+OO 4 3.40E-Ol 2.9~E.Ol -0.99E-Ol -1.U2E+00 -1,71E+00 3 -2.711+00 -1.31E+00 .1.1&£+00 -2,13E+00 -2,19£+00 2 -1,19£+00 -1,S1E+OO -1. Q9E+CO -2,25E+00 -2.20E+00 1 ~6+~lOOE~0~Oo.~u~00E+~00~
-o,41E-01 -1,02E+OO -l,ilOE+OO "I,15E+00 -8.49E .. Ol -2,58£ .. 01
5,28E-01 1.20E+00
~~t1."E+OO 1,35E+OO 1; e,eeE490' ~(}O
5 J,
.0-0£"01 7,29E-01 8.19E-01 0.22E-01 l,4o.E+00 1.59E+00 9.89E-01 2.01E+00 2.20E+00 1.41E+00 2.55£+00 2.70E+00 l,80r-+Oe 2.99E+00 3.10E+OO 2,31£+00 3.31E+00 3.37£+00 2.&8£+00. 3.49E+o.O 3.50E+00 2,8uE+OO 3.u3E+00 3,40E+00 2,40E+00 2.85E+00 2,80£+00 ~+OO~OE+-*-O,O~O
;'!I·OJil .,' '~IJ./'7 I,· .,,,, • .'8 /r '~".,'" 9 "/'~"'/10f/'
-7.44E-02 -1.2&E.01 -1.blE-01 -1.73£-01 -8.23E-02 -1,90£-01 -2,70£-01 -3.34E-01 -3.57E .. 01 -2.40E-01 -3.52£-01 -4.S4~-01 -5.23£-01 -5.55E-01 -4.30E .. 01 -5,79E-01 -0.70E-01 .7.35E-01 -7.72£-01 -b.b3E-01 -8.84E-01 -9.30E.01 -9.70E-Ol -1,01E+00 .9.15E-01 -1.2eE+OO -1.22£+00 -1.22E+00 -1.25£+00 .1.19E+00 -1,07E+00 -1,53E+00 -1.47£+00 -1.45£+00 -1.40E+OO ~Jr--2.00.£+00 -1.70£+00 -1.05E+00 -1.05£+00 -1.&7E+00 -1,94E+00 -l.o&E+OO -1.52E+00 -1.52E+00 -1.59E+00 -t O~O ~OE+O~OOE~ O.~ O~OO-· .-
*** K = 4 ",,,,,,,' . bll//,I!ll!I,7}L/--"';IL..',.I// 8 f . .'~ J 1.1' / ..... 9/.':,/1.0" ·.10.'11
o.LrlIT~Of- 4.82E-01 3 ,78E';'-0 1 3.51E-01 4.00£-01 1,23E+OO 9.31£-01 7,38E-01 0,84E-01 7,87E-01 1,77E+00 1.35E+00 1.08E+00 9,98£-01 1.00E+OO 2.22E+00 1,72E+00 1,39E+00 1.29E+00 1,30E+00 2.59E+00 2,03£+00 1.boE+00 1.54E+00 1.52£+00 2.80E+00 2,27E+00 1,88E+00 1.70E+OU 1.71E+00 ~r-2. 99E+00 2.41E+00 2.03E+00 1,91E+00 1.88£+00 2.93£+00 2.39£+00 2,05£+00 1.90E+00 1.9&£+00 2.41£+00 1.99£+00 1,74E+00 1.&8E+00 1,73E+00 O~O ~E+OO~OOE~O,~ ~£+~- ---4:--
~~** CHICK2 'R!NT OF -VNS- 11. i IL.~1g:Jil:· i JLa OJ ,-.-J III 1· I J
4 O,OOi.O~' ,.0.1+0. _ b,OOE+OO '-'-'7'-'0.081.0.-'· I,OU+O'--"l, OOE+OO
2 O.OGE+oe 'o,ooe+ol a.OOE+OO O.OO'~OO C.Ot~~OC O.OO[~OO
" 5 0.00£+00 O,OOE+OO
··O;OOE~OO-·O,OOE.OO·
0.00£+00 O,OOE+OO O,OOE+OO O,OOhOQ
~ 7 ., 10 O,OOhOO O,OOE+OO 0,00£+00- 0.00£+00 : O.OOhOO· O~OOE+OO'- O~OOE+OO·"O,oO!ua·-oiOOE.OO~-- O,oo£.oa'~O,OOE.OO O,OOE+OO O,OOE+OO 0,00£.00 O,OOE+OO
. O. ~O~.~_~.~ __ .~ __ ~OE+.O.~ __ ~!~!~!~.! _~~!!.~ _~'-~~!~!!_.
W f
'-'
. r Table 3.3 Angular Momentum •• ~*-* •••• ~*.* ••• * •• ~*.*.* ••••••••• *.*. SWIRl- VEI-DCITY
**. K ill
.... -......... _._.-.-.@ ... -._ ... _.-1 ***
2 7 I" 9/h/I"",i10di& , • 0 ,~+OO ,+ ,+ ,+,., + 0, 0 +00 O,OOE+ 0
O,OOEtOo O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O,OOE+OO O,OOitOo O,OOE+OO O,OOE+Oa O.OOE+OO o.OOE+OO O.OOE+OO O,OOE+OO O,OOE+OO O.OOE+OO
JIO,OOf+oa C.Gaf+oa O.OOE+OO O.OOf+OO O.DDE+DD O.DOE+DO O,OOE+OO O.OOE+OO O,OOE~OO D,DDE.OO b O,OOE+OO O,OOE+OO O,OOE+OO O,oOE+OO O,OOE+OO O,OOE+OO O,OOE+OO O,OOE+OO O,OOE+OO O,OOE+OO ~ O,Oai+OO O,OO!+OO O,OOE+OO O,OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO ~ O.OOE+oO O,OOE+OO O,oOE+Oa o,OOE+OO O.OOf+OO O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO ~.' .O.OOE+O() O.O{lETOti o.OOE+OO 0.00£+00 C.OOHOO C.OCE700 O.OOEHIO O.OOEHIO O.OOEtOQ O.OOEtOO /~. ""\ ~O.OOi+OO O.OOE+OO O.OOE+OO O,OOE+OO O,OOE+OO O.OOE+OO O,OOE+OO O.OOf+OO 0,00£+00 O.OOE+OO :' \ ~IO,OO.E+.!L .O,OOE+OO _ O.OOE+.OO~O?E~.'OOE+OO . O,OOE+O,O ~+.O~O~+~,.OOE+OO . O.OOE+O~ __ J. f 4.+~:~ ... 1 _*'" K;: c; .*_ l.£
.~~~+OO -Z.86E+OO 2.1QE+OO -l,'~E+OO
~2'8bE+OO ·8.3~E-Ol
73,28E+OO -l,IIE-OI /3.11E+OO 1.b'E-Ol 2~01E+OO 2,'3E-Ol l,OiE+OO I.Z0E-Ot
-' I 1/-' J " II
+00 -1.29t+ -"S7E+OO -l,SSE+OO -Z,07E+OO -1,SSE+OO .l,~2E+OO -l.~~t+oo
-I,24E+OO -1.l2E+OO -9.ZBE-Ol -1 •. 13E+OD
, " ' ,S 'I
./loE-O! -7.2IE-02 -4.93E-Ol -7,S3E-Ol .B.77E-01 -B,92E-01 -B.24E-Ol
////1//.,,8 '/~'/"/.,, ':9 ",/,.'/",10/' , -, ... + OOC-~3 ff+OO -2 ;3b'H 0 0 -2~-3-2E+ 0 cf-Z. 07 Eta 0
1.00E+00 1.4SE+OO 1,SBE+OO 1,S8E+OO 1,37E+OO 3.B2E-Ol 8,l3E-01 9,b9E-Ol 9.91E.Ol 8.31E.Ol
-b.9~E-02 3.19E-Ol 4.80E-Ol S,IZE-Ol 4,02E-Ol -3.7IE-OI -U.UbE .. 02 1.0~E.Ol 1,39E.Ol b.82E-02 -S,UZE-Ol -Z.87E-Ol -1,b2E-01 -1.3IE-01 .1.73E-Ol -~,OOE-Ol -4,lbE-01 -3,ZOE-Ol -2,95E-Ol .. 3.20E-Ol
· 3,lJE-Ol -1.37E-Ol INLEt: 11.10£ .. 01 -2.3SE-Ol
~"?3 ~ .. ~.9,!,~ .. .Q.A!
.7.2lE-u1 -9.3Bt-Ol -S.8BE-Ol -7.32E-Ol -U.ISE-Ol .4.85E-Ol
-b,87E-Ol -4,72E-OI
-S.57E-Ol -4.3SE-Ol -3,b8E-01 -3.S1E-Ot .. 3,bbE.Ol /' ! '],:.' -4.07E-Ol -3.37E-Ol -Z,'HE-Ol -Z.87E-Ol .Z,9BE-Ol i "~. \~ •. ' \
e1.9BE~~1.23E-Ol -1.2~E~O 1 -1 .• 04E.Ol_-8.7I.IE~0~~~lE.0i! .-7.b7E-.0i! .• 8.0!~-OZkL-._. j..-' • "'** K = 3 *** ---t::
• :;-1. _ ~f'S.b1E .. 08 -S,2bE-01i' ";l;-3',r(-oa
.8.llE-OS .4.01E-Oa .2.SBE-oe b,SeE-Oe -J,Q3E .. OI -l,1ZE-oe 4.88E.08 -2,96£-08 -l,8bE-08
,3.04E-08 "Z.51E-08 -1.b7E·08 • -1,7JE.Og -i.1SE-08 .1.S2E.08 ~1.92E.OS -2,17E-01 -1.l7t-OS
INL€''1' .1.b86-08 .I,bQE-OB .CJ,S9E-OCJ ~ 1.81EcOq -l.oIE.OQ .1.7if-OCJ
5.UE~0L.J.~'T~ }I.ZSE-O?
q~l f .... ~ I' }
; '~.19E+OO ~, !.86E+Oo
:2>,,·",111/.1,3,-'/1 ,wU+oO- -l;-(j~
1.70E+00 2,S7E+00 8,34E-Ol 2.07E+OO 1,81E-01 l,b2E+OO
-1.b7E-01 1.24E+OO
" .', b ,If / / / ,. / / 7 /i /I' " ' " 8 "" "/ '.' q fl"'" ",' 1 0 /I!
.. .:.()~8 -1.7aE-OB ;;-Z.70E .. OB -4.32E.Oe -S.blE .. OS- -5,841-08 "4.~SE-OB ~1.S8E.OB -1.27E-OB -l.BIE-Oe -2.88E-Oe ·-3.78E-Oe .3.93E-08 .Z.9~E.OS -1.27E.Oe .9.74E-09 -l,29E.08 -Z.ObE-OB -2.74E-08 -Z.84E-08 -2,08E.Oe .1.09E-OB -7,79E-09 -9.77E-09 -1.59E-08 -Z.15E-08 -Z I ZIE-08 -l.5BE-oe .. 9.5BE.09 -b.37E-09 .7,BlE-09 -I,3lE-08 -1.7SE-OS -1,SlE-08 .1,27E.08 -8.3~E-09 .S.13E-09 -~.24E.09 -i.09E-08 -l.SOE-08 -l,S2E-OS -1.03E-08 -b,78E-09 -3.83E-09 -4.S5E-09 -7.S8E-09 -1.09E-OB -l.IZE-OS .7,03E.09 -a.3~E-oq -Z.3~E .. 09 -Z,38E-09 -Z,33E-09 .. Z.9SE-09 -3.92E.09 -8.83E-l0 ·l.2ZE-09 -9.43E-l0 3,8bE-I0 ~.80E-09 l10bE-08 8.20E.09 9.~OE_09 7.l2E-l0 -1.47E-l0 2.Z7E-09 1.34E.08 2.01E-Oe 1.~8E.08 1,71E.08 --. ---. ---.- • "",***K' = 4 *",.' -----, ---. --- , ---
.~T~+
, ~ .0 " .. ' . I ' I 7; J) ~. j /- /. 8 J /../ L./ / / .' /~il.L.I ." 1 ! ! tit_./' I I -b. 'l bt -01 -l,'f1 t + 01) -2,3 OE+O () .c-;-lb-~+O-o -2, 32E +0 0 .2. 07E + 00 7,21E-OZ -l,OOE+OO -l.~SE+OO -l.SSE+OO -1.S8E+00 -1.37E+OO 4,93E-Ol -3.82E-Ol -8,13E-01 .9,~qE-Ol -9.91E-Ol.8.31E .. Ol 7,53E-Ol b.9bE-02 -3,19E-Ol .4,80E-Ol -S,lZE-01 -4.0ZE-Ol b.77E-01 3,71E-Ol q.4bE-OZ -l,04E.Ol -1.39E-Ol -b,82E-OZ
SYl'1MtTeY 9 LAI'-JE
s= 90'0
(--~':.
, .IJ> / . . \. J ., , ~_._._.i \ ---.--4
~l."lllhDO
/ S.2ShOo ,I I.IIE+Oo · . !,01E+Oo - ~~,on+oo
-a,T3E-01 9.28E-Ol
1.SSE+OO l,SSE+OO 1.4bE+OO 1.3ZE+00 1.13E+00 9.38E-Ol 7.3ZE-Ol 1.l,8SE-01
8,9ZE-Ol S,42E-Ol 2.87E-Ol i,b2E-Ol 1.31E-01 1.73f-01 ~~ 8.2QE-Ol b,OOE-01 4,lbE-Ol 3,20E .. Ol Z,9SE-01 3,20E-Ol // ~. -.
I N("'fr ' ~i'17E.Ol · .•• 10E-01
: ,~t:"" _ ;1--.
-1.20E-Ot ~.21~-Ol 1,17E-01 S.8SE-01 2,lSE.Ol /.l,ISE-01 ~ r~ .,. t . ~ ':' 7: _. 1
b,8n-01 5,S7E-Ol 4.3SE-Ol 3,b8E-01 3,SlE-Ol 3,~~E.Ol L<t~: a,72f-Ol 4,07E-01 3,37E-Ol 2,97E.01 2.87E-01 Z,98E-Ol ~. ' . ~---'-'--'-.:~~-- ,.$.. ~ .. ~ ~'tt.._,,:'1 ~ ~."'~-""" ~~--{-'-'--<l --~.~. 1
~";, .1t: I?RINT OF .. HB 11= 1 IL=10 J1= l' JL.=10 t/C" II I / , '/ ,3 " U,. / " 4), J /I '-I I J 5 I.1li/
Q,OOE+OO O,OOE+OO--O.OOE:+oo O.OOE+OQ O.OOE+OO O,OOE+OO O.OUE+OO O,OOE+OO
~"OOEtOO O.OUE~oe 0,00£+00 O.OOE+OO O.OOE+OO
I ,OOE+OQ O.OQE+OO O.OOE+OO 0,00£+00 O,OOE+OO
',OO!+OO 0,00£+00 O.OOE+OO O.OOE+OO O,OOE+OO w x',OOE+OO O,OOE+OO O,OOE+OO O.OOE+OO O,OOE+OO
,00£+00 0,00E+06 O,OOE+OO O,OOE+OO O.OOE+OO I ,00E+Oo O,OOE+OO O,OOE+OO O.OOE+OO O,OOE+OO
',OOE*OO O,OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO ~ _, 0 glh 9 0 ~ 9 E + ~LO,O E ~o ,.0 0 E"~ . 0 , 0 0 H ~ 0,
o 1/",,.?/71!>-'··"; S J o-;OOE+oO O,OOE+OO O,OOE+OO O.OOE+OO O,OOE+OO O,OOE+OO a.OOE+OO O.OOE+OO O,OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO 0.00£+00 O,OOE+OO O.OOE+OO 0.00£+00 O.OOE+OO O,OOE+OO O,OOE+OO 0,00£+00 O,OOE+OO O,OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O,OOE~OO o,OO~tOO O,~OE+OO
/ / .-, q I"; .'." .·1 O'~ 0,00£+00 O,OOE+OO O,OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO O,OOE+OO 0,00£+00 O,OO~+OO O.OOE+OO 0,00£+00 O,OOE+OO O,OOE+OO 0.00£+00 O,OOE+OO O,OOE+OO ~,OOE+OO .. O.OOE+O~ -ill ... I .
1(:5 . /' ••••• ; 1
Table 3.4 Kinetic Energy of Turbulence "r~* ••• *_*.*_*_*.*_*.*.*.*_*.*_*.*~ •• K.E, OF TUR8U~ENCE .*.*.*.*.*.*~*-*-*~*.*.*.* ••• *.*.*.* •••
_** 1\ = 1 *** 1 a 3 4 5 0 7 8 9 10
~.7te:+oo 1.17E+Ol 1.22E.tOl 1.07E+Ol 0.55E+00 4,90£+00 3.89E+OO 3,04E+00 2,47E+00 2,23£+00 ,85E+Ol ',8'PE+Ol 1.1.17E ... 01 5,70E+Ul 4.0bE+Ol 2.90E+Ol 2,20E.+Ol 1.05Et01 1,28£+01 l,05E+01
'.ilE+Ol i.1SE+02 1.04E+02 7.79E+01 5.58£+01 4,05E.;.ul 2,'H1E+Ol 2.23E+01 1.1lf>HH 1;39E'tOl ';.75i+ 01 1,'15£+02 1,20Et02 9,32EH1 b.74E+01 1.I,881:t01 3.57E+Ol 2,1:I5E+01 2.02E+Ol l,02E+01 - .29E+02 1.1:11.1£+02 1,31E.t02 1.03E+02 7.59E+01 5,53E+01 4,04£+01 2,98£+01 2.25Et01 1.79E+01
,13E+ 02 1,58£+02 1.35E+02 1.07E+02 8,12El'Ol o.OlE+Ol 4. IJ OE+01 3.24E+01 2.42E+Ol 1,91E.01 7.956:+01 1.3aE+0i! 1.20E+02 l,OoEt02 8.35E+01 1I.33E+Ol 4,b7E+01 3,42E+01 2.55E+01 2.00E+Ol
,81E+01 1,25£+02 1,18E+02 1.03E+02 8,4IJE+01 0,54E+Ol 4.8bE+01 3,51:1E+Ol 2,04E+01 2,07E+01 .99£+01 1,1I!lE+oa l,12E+02 1.00E+02 S,48E.+Ol 0.70El'Ol 5,_00E+Ol 3,bbE+Ol 2,72E+01 2,13E+Ol
b,42E+01 1,09E+02 l,08E+02 9,80E+01 8,1J9E+01 b.70£+01 5.00E+01 3,70E+Ol 2,74E+Ol 2,15£+01 1t*1t ~ = 2 1t*1t
a 3 4 S b 7 8 9 10 -:.27E+OO 1.19E+Ol 7.t.l3E+OO 5.09E+OO 4. 'HE+OO 3,30E+00 2.52E+00 1.97E+00 1,bOE+00 1.43E+00
,48E+Oo 7,l1H01 'T,3I1E+Ot S.USE+Ol 3.7I.1E+Ol 2,501:+01 1.72£+01 l,24E+Ol 9,33E+00 7.08E+00 .75£+01 1.01E+02 l,OIE+02 7.77E+Ol 5,38E+Ol 3.S9E+Ol 2.1.10£+01 1.7bE+01 1,31E+Ol 1,ObE+Ol
,-,.HE+Ot 1,19E+02 1,201'::+02 9.lI9E+OI b.b9E+Ol 4,50£+01 3.07E+Ol 2017E+Ol 1,1:I1E+01 1,28E+Ol .11i:+OZ 1,33E+02 1.28El'02 1.05E+02 7,bbE+Ol S.25E+Ol 3.bOE'I'Ol 2.5"E+01 1,8bhOl 1.'I8E+Ol
(J.14E+Ol i.i4E+02 t.2 iE+02 1.Q7E+02 8,21E+Ol S.82E+Ol 4,CSE*Ol 2.8bE~01 :l t n".ni 1.#:5E+Ol ... ,."" ........ J·.32E+Ol 8.31E+vl 1.0SE+c2 1.o3E+02 8.l.!oE+Ol b.20E.Ol '1.'1lE+O! 3.1uE+Ol 2.30E+Ol 1.80E+Ol
w ,.15E+01 8.eaE+Ol 1,01E+Oa 9.93E+Ol 8.44E+Ol o,4bE+Ol 'I.70E+01 3.38E+Ol 2,48E+Ol 1,94E+Ol I -'.bU+Ot l,OSE-toc 1.0SE+oa 9.8I.1E+Ol 8.'I7E+Ol b.I:ISE+ol 4.92E+Ol 3,58E+Ol 2,b4E+01 2.07E+Ol co ',2bE ... Ol 1.0~E+02 1.0?E+u2 9,84E+OI 8.49E+Ol 0~7SE+Ol 5.0SE+01 3,b9E+Ol 2,7~E+Ol 2.1I.1E+Ol
It** K ::I 3 *It* ... • a 3 4 S 0 7 8 9 10
:.27E+Oo i.19E+Ol 7,"3E+OO 5.09E+OO 4,37E+OO 3,30E+OO 2~52E+OO l,97E+OO 1.bOE+OO 1.1.I1E+00 , .1I8E+Oo 7,71!:+Ol 7,3IJEtOl 5. lJ 8E .. 01 3.74E+Ol «.50E+Ol 1,72E+Ol 1.2I1E+01 9.33E+OO 7.08E+00 ,.75E",01 1.01E+02 1.0IE+02 7.77E+01 5.38E+Ol 3.59E+01 2,lJoE+01 1.7bE+Ol - 1,31E.01 1.0bE+Ol 1.?1E+Ol 1.19E+02 1,20E+02 9."QEl'01 b,1!l9E+Ol '1.50E+Ol 3.07E+Ol 2,17E+01 1,&1E+01 1.28E+Ol •• 17E+02 1.BE+oa 1,28E+02 1.05E+02 7,ooE+01 5,25E+Ol l.00E+01 2,54E+Ql 1.8U+iil 1.116E+01 ~.1I1hOl 1.11JEtOe 1,ilE+02 1.07E+02 8.21E+Ol 5,82E+01 1I,05E+01 2,80E+01 2,10E+Ol 1,05E+01 1,32E+01 8,31E+01 1.0!5E+02 l,03E+02 8,'10E+01 l!l,iOE+Ol 4.'I1E+01 3,14E+Ol 2,30E+Ol 1,80E+Ol ),151+ 01 8,8iiE+Ol 1.01E ... 02 9,93E+01 8,II'1E+01 o,lIoE+Ol 1I,70E+Ol 3~38E+Ol i!,lI8E+01 1~911E+Ol J.O;5E+Ol l,OSE+OI 1.05£+02 9.8'1E+Ol 8,1I7E+01 1:1,05£+01 4.92E+01 3.S8E+Ol i!.&lIhOl :i!.07EtOl
__ ~, ,_~!.!,!O 1 1,Oa15+01l _ ~,~~_E.Oi ~ •. 84E+Ol 8!~~E.!01 0,'5£+01 5-.05[+01 ·3,&9E+01 2,73£+01 '2.14E+01 - - - -0- 0--". •• K --,--4- ••• -- - --
;; 1 . a -J II 5 & 7 - 8 -'9 .- ·10 5.71hOe 1.171+01 I,UE.;o1 1.07E+Ol &,55E+OO ".90EtOO 3,89E+00 ·3,04E+00 :2.41£+00 'c.23E+OO 1.8!1&+01 7,e7E+Ol ',"7E+Ol 5.70E+01 4,00E+Ol 2,90E+01 2.20E+Ol 1,&SE+01 1~28E+Ol 1.05E+Ol 5,11£+01 1.11£+01 1,0IlE+02 7,79E+Ol 5,58E+01 II.05E+01 2,98E+Ol 2,23E+Ol 1.71EtOl 1.39£+01 S, ?!litO: t,Uhoa 1,20E+02 9.32E+01 b,74E+Ol 4,88E+Ol 3,57E+Ol 2.o5E+Ol i.oaE+Ol 1.&2E+Ol
-T~2qi+02 -1.616£+01- 1.i7E+02--- -1,03£+oa "~59E+Ol "5;53E+01 .. "~04E+01-' --2;98E+Or-:2-;c5E+Or-o~I-;19E.Ol . 1.1U+0!! 1.UE+0I l,35E+0« 1.0'E+02 8,12E+Ol 0.01E+Ol 41,40£+01 -3.24E+01 2,4iE+Ol 1,91[+01 7.95&+01 1,Uhoa 1,2oE+02 1,ObE+02 8.35E+01 0.33E+01 4,&7E+01 3,42£+01 2,55E+Ol 2.00E+01 7.81h01 l,iSE+oa 1,18E ... 02· 1,03E+02 8,1I4£t01 o~SllE+Ol 4,8&E+01 -3,5b£+01 ·-2,bl4[+01 i,OntOl 7 .9~l+0, 1, UE+oI 1.12£+02 1.00E+Oi 8,UE+01 8.70E+Ol - 1.00E+Ol -3.&&E+01 -2. 72E+O 1 :2~i3E+Ol ~.4Ih01 1.nE+Oi 1.08£,,02 9.80£+01 6.U~E"'01 8,7&E+01. 1.,Ofl~~O~ !.7~E~~1 2 •. 1~E~O~ 2.15E+~1
50
w 1--- 1 I 1= 2 '" 1: 3
i= 4 1= 5 1= b :~ 7 r~ 8 1~ 9 l~ 10 13: 11
g ci • x
1~1
-x
2 3 4
E ;h N I X
5 6 7 8 9 10
CONVECT IV! FLUXES THWOUGH THE cHAM~ER CROSSEC1IONS
)(1'1= O.OOOE+OO Flt: 1.(173138E+O i FXP= 1,473138E+Ol )(1'1= /). 3Soe:-0 1 Fl(:s t ,'I'73135E+01 FxP: 2.224907E+01 XM= 1.210E+OO Fl{= 1.'173133£+01 FxP= 2,112332E+01 :(M= 1.905E+OO Fl= 1.(173132£+01 FxP= 1,803385E+01 )(1'1= 2,540E+00 1')(= 1.1J73132E+Ol FXP= 1.5513b3£+01 J(M: 3.175£+00 FX: 1,473134E+01 FXP= 1,473134£+01 XM= 1.810E+OO "':= 1 ,(173139E+O 1 FXP= 1,473139£+01 XM= !I.4l15E+OO 1')(= 1 ~4731l1oE+Ol FXP: l,473111oE+Ol XM= 5,080E+OO FX= 1,475153£+01 FxP= 1,II73153E+01 )iM: S.715E+OO F:(~ 1.(113157£+01 FxP= 1,473157E+01 XM= 0.350(+00 FJ{: 1.(l73159E:+Ol F)(P= 1,II731SQE+Ol
E \l) M cD • x
FlCN= O.OOOOOOE+OO fXN= -7,517717E+00 FXN= -0,39t988£+00 F)(N: -3,302532£+00 FlN= .7.823103£-01 FXN= 0.000000£+00 F)(N: 0,000000£+00 FXN= 0.000000£+00 F)(N= 0,000000£+00 FlIN: O,OOOOOOE+OO FXN: O.OOOOOOE+OO
Table 3.5 Convective Fluxes Through the Chamber Crossections
FUL.FI.X= 0,000000£+0 FUI.FLX= O.OOOOOOE+l FULfLX= O,OOOOOOE+c FUl.Fl.X= O.,OOOOOOE+O FULFl.X= O,OOOOOOE+O FULFLX= O,OOOOOOE+O FULFl.X:: 0.000000£+0 FULFLX= O,OOOOOOE+C FUl.FLX= O,OOOOOOE+( FULFIoX= O,OOOOOOEH FULFLX= O.OOOOOOEH
Maximum Residual
Error ...
10-4~----------~------~----~----_--1~--_-~1 _____ �~ ____ ~ ____ ~I _____ J o 10 20 30 4050 60 70 80 90 100
Iteration Number
Figure 3.3 Convergence rate for the symmetry and uniformity test.
3-10
The global mass flux FX = 14.73138 should be, and is preserved, at each
axial location I = 1 . . • • • 10.
3.2 UNIFOBMITY TESTS
The unHormity tests have been performed on similar chamber geometry with axi
symmetric fluid entry (figure 3.4). A turbulent flow with propane-air combustion
in a chamber with sudden en1argment has been considered. Figure 3.4 presents the geometry and inlet conditions. Fuel enters the chamber through the axial
slot at x=O with u=I~, v=O, w=O, and air through the annular inlet with m s
u=5~, v=O, w=:O (see table 3.6). Results are presented by the following tables:
3.6 - u -axial velocity (m/s) 3.7 - v -radial velocity (m/s) 3.8 - rw-angular velocity (m2/s)
3.9 - k -kinetic enerty of turbulence (m2/s 2) 3.10- T-abso1ute temperature (OK)
3.11- mfu-fuel mass fraction (-) 3.12- FX, FXP, FXN and FULFLX fluxes (kg/s).
Calculated results are uniform on all ek (k = 1, 2, 3, 4,) planes. Note that the recirculation zone extends up to the middle of the chamber. The angular
-8 1 -12 momentum rw shaul d be zero throughout the chamber. It was predi cted as 10 .. 0 o -4 -6 0 0 at 8=90 and as 10 - 10 at e = 45 and 8 = 135 respectively. These values are
within acceptable ~rror limits.
The values k, T and mfu also indicate good uniformity. Table 3.12, in addition to global mass fluxes (FX, FXP, FXN), also provides total crosssection unburned fuel flow rate, which is defined as:
The convergence and conservativity of the numerical algorithm can be verified
by FX distribution. Note - the FX = 17.79037 kg/s value is exactly preserved at each axial i-plane (table 3.12).
The convergence rate for the uniformity test case is presented in figure 3.3. The residual error for test case B 'indicates better convergence for the flow
with uniform inflow. Intensi ve rec 'ircul at ion zones in case A (symmetry test) are responsib"'e for relatively slower convergence rate.
3-11
W I I-' N
..............
Y
Ie /x ~
..............
..............
tVNB
, /' / / /,,,,,,.,,,,, /"" / /""" ~I,,,,,,,,,,,,,,,,,,,,,,, ,-,,,,."""""""
/ //
10~ /9~
8~ 7~
//j,~ 5
//
// / Air 50 4
3
2 .... 1=1 --
..............
· · · I · I 0 0
0 0
0 0
0 0
0 · · 0
0 0
0 0
2
· · ! · · · · I · ! 0 I I 0 ! · I · I
0 · 0 · 0 0
0 0 U 0 0 0 0
0 °E ~ 0 0 0
0 · llJv 0 0 0
· 0 0 0 0 0
0 0 · 0 · 0 · 0 0 0 · · · 0 0 · 0
3 4 5 6 7 s x-v CROSSECTION
--- -- -- --..:-::.-r---_
" I
" I «- w- I • ............... .
........ L'--I. __
· · I · ·
0 0
0 0
0 0-
0 0
0 · 0 · 0 0
· · 9 10
I
1 2 3 4 5 6 7 8 9 10
ENTRY PLANE
Figure 3.4 Combustion Chamber Geometry
B
-<t
w I
t-> w
Table 3.6 Axial Velocity .*.*.*.*.*.*.*.*.* •• ~.-*.*.*.*.*.*.*.*. U-VEL.OCITY .*.*.*-•• *.* ••• *.* ••• *.*.*.*.*-*~*-*.*.* J 1= a 3 4 5
."'. K ;; o 1 "'*'" 7 8 9 10 O.OO~OO -8.49£+00 -8,59£+00 84.91£+00 -1.89£+00 3.90E+00 2.83£+01 3,20£+01 3.34£+01 ; o.oo-~Oc .S,3!f+OO -~.B~E*OO a.51E e o2 2.7SE+OO 7.45E+OQ 2.92E+Ql 3.37E+01 3.49E+01
0.00 00 -3.13E-01 3,13E+OO 5,01E+00 6,91E+00 1,05E+01 3,09E+Ol 3,46E+01 3,58E+01 O,OOE 00 6.16E+00 1,08E+01 1,12E+Ol 1.19E+01 1,38E+01 3,37E+01 3,59E+Ol -3,09£+01 o 00 00 l.13!+Ot 2.08£+01 1.88E+01 1,76£+01 1,74E+01 3,74E+01 3,77£+01 3,81£+01 5.00£+01 3.91E+01 3.29E+Ol 2.75E+Ol 2,39£+01 2,09£+01 4,16E+01 3,97E+01 3.95E+01
~iR4 5~OijE+01 4,62£+01 4.12E~01 3.50[+01 2.95E+01 2,38E+01 4.55E+01 4 i 15EtOi ".09£+01 3 5,00(+01 4,61E+Ol ".27E+01 3.78£+01 3,22E+01 2,51E+Ol 4,78£+01 4,29£+01 4.20E+01
r 2 I,oOt+Ol 3.18E+01 1,28E+01 3,20E+01 2,96E+01 2.34£+01 4,81E+01 4.35E+Ol 4.26E+Ol rU 1 -1,00E+01 2.17E+Ol 2,57£+01 2,69E+01 2.65E+01 2,13E+01 4,70E+01 4.34E+01 4.27E+01
-- - -- --- --_. -_. -- - -_._- .• "'. K;I' 2 *.'" - ---. ---J II: 1 3 4 5 0 8 9
10 l,3tE+01 3,57E+01 3,07E+01 1~7bE+01 3,80E+01 3,97E+Ol 1I.08E+Ol 4,17E+01 4,23E+01 4,~
10 0,00 ~ ,U9 +00 - ,60E+00 -4.97E+00 -1~90E+00 3,96E+00 i.83E+01 3,20E+Ol 3,34E+01 3,37E+01 9 0,00 -S,31E+OO .2,80E+00 U,21E-02 2,75E+00 7,45E+00 2,92E+Ol 3.37£+01 3,49E+01 3.57E+01 8 0.00 -3,75E-Ol 3,13E+00 5,00E+OO 0,91E+00 1,05E+01 3.09£+01 3.40E+013,58E+013.07E+Ol 7 0,00 &,lbE+OO !.08E+0! 1,12E+01 l,19E+01 1,38E+01 3~37E+01 ·3,59£+01 ·~3.09E+013;10E+01 o 0,00 +00 2,13E+Ol i.08E+01 1.86E+01 1,7oE+Ol 1,7~E+Ol 3,14E+013.11E+01 3,81£+01 .3~8oE+01 5 5.00E+Ol 3.97E+Ot 3.2QE+Ol 2,75E+01 2,39E+Ol i.D'E+01 4.10£+01 3,~U+Ol'3,95EtOl -.3~97E+01
AI~IJ, 5.00E+01 u.b2!+Ot 1I,1ZE+0! 3,50E.01 2,95£+01 i,l8£+01- '4,S5E+Ol- 4,11[+01 - 4,nU01:··-4.05hOl' 3 5.00E+Ot 4 1 61EtO! 4.41E+Ol 3.78E+01 3.22E+01 2.51£+01 4,78E+Ol ~,2~E+Ol ~,20E+Ol 4.17E+01
~~.OOi+Ol 3.16£+01 3.28E+Ol 3,20E+01 2,9bE+01 2,34E+01 4,81E+Ol 4.3!E+Ol 4,2&E+Ot 4,23E+01 1 1,00[+01 2,17E+0[ i.5'EtOl 2.69E+Ol 2.bSE+01 2,13£+01 4.76E+01 4.34E+Oi 4.27E+Ol 1.26EtOl --' -_. --_. • -----. " --".-... • ..... 1'\:: :\ 'If~1i " ---. ~. --
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00 "3,75E-01 3,13E+00 S,OOE+OO 0,91£+00 1.05E+01 3.09[+01 3.l.IoE+01 3,58E1'01 3.07£+01 00 a.lbE+OO 1.08£+01 1.12E+01 1.19[+01 1,38£+01 3,37E+01 3.59E+01 3,b9E+01 3.16E+01 00 2.13E+Ol 2,08E+0! 1,88E+01 1,76E+01 1,74£+01 3.74£+01 3,71E+Ol 3,SlE+Cl 3.80E+01
."lfOE"+ 01 3.9'E+01 1.29E+01 2.75E+01 2.39E+01 2.09E+01 1I,10E+01 3,97E+Ol 3,95E-+01 3.97£+01 AIfl.4 5,00i+01 4.62E+01 1I.12E+Ol 3.50E+Ol 2,9S£+01 2.38E+01 4,55E+01 4.15£+01 4.09E+01 4.08E+01
3 5 i °!lE+Ol II;OlE+Ot ".i7E+Ol 3.78E+01 3.22£+01 2,51E+Ol 4.78E+Ol tI,29E+Ol 4,20E+Ol 4.17E+01 2 l,OOh01 3.16£+01 3,28£+01 3.20E+01 2.9bE+Ol 2,14£+01 4,81E+01 4,35E+01 4.2oE+Ol 4.2SE+Ol
PJl 1. OOE'tD..L-2 t11E+O t 2,57E+01: 2,oqE+0~ 2.65E+~1 2,13E+01 4,76E+Ol 1.1,34£+01 4,27E+01 4,20E+Ol *"'* K :; zr *** --- . .-
.4:-
'"'k-
4-J 1:; 1_~~~~~~ __ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ __ 10 0,00 ~8,UqE+OO .8.SQE+00 e4.91E+00 -1,89E+00 3,Q6E+00 2,83E+01 3.2&E+01 3.34£+01 3,37E+01
q 0,00 RS,31E+OO -Z,86E+00 4,57E .. 02 2,75E+00 7,45E+00 2,92E+01 3,37E+01 3,49E+01 3.57E+01 e 0,00 -3.73E-01 3.13E+00 5.01E+00 0.91E+00 1.05E+01 3,09E+01 3."oE+01 3.58E+01 3,07E+Ol 7 0.00 8,lbE+OO l,08E+Ol 1.12E+01 1,19E+01 1,38E+Ol 3.37E+01 3,59E+01 3.09E+01 '3,70E+01 o 0 00 00 2.11E+01 2,08E+Ol 1.88E+01 1,76£+01 1,74E+Ol 3,74£+01 3,77£+01 3,81E+01 3,88E+01 5 5,00£+01 1,;?E+Ot 1,29E+01 2,75E+Ol 2.39E+01 Z,09E+Ol 4,loE+Ol 3,97E+01 3,95E+01 3.97£+01
AIR45,OOE+Ol 4.62£+Ot 4,liE+013,50E+Ol' 2,95£+01 2.38E+01 4,55E+01 4.15£+014.09£+01 4,08E+Ol 3 5,001+01 4,01(+01 4,a7E+013,78E+013.22E+Ol 2.51E+01 4,18E+01 4,zeE+Ol 4,20E+Ol 4,17E+01
. _.2 .1..00i+01--1.1~£+Ol -.:1.28£+01· -,3,20hOl --2,90£+01 ... 14£+'01 ,4,81E+Ol '4,35£+01- 4,20E+Ol-.:4i23hOlF\}1---r;OOhO!-I~t"+OJ-·i~51E.~C·-'C;ou .• or-c.o5.£+01- a.1.lE~01--4.'0E~+Olu -4;34E.!Of-4'27~+Ol . 4-_~~~E_.Ol ~;
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.. J. l.c c'" "'<''''.:''' 'a-"~· -:~c3 ~'_." -'-5' .. 1 8 , to " 4 •• i68£.;1 ",1~8tE.Ol-·-4,1388E+Ol .. -4,0525£+01 -3,94CJ1E+Ol ,1.84 .. £+01 .,1,7594E+01 -,3.o9iilEtO, ,1,U09hCl - J,53040E+Ot::; 3 -4;Z26tE+81 -.; 198!E+U .. '~U89E+Ol -:4,0527[+01 - l,'493E+or- J;U70E+U -1;nnU01-=,;o9nnOl----:).641SE+CJr-:I;SUUlOr-Z 4.l"O!+91 4,l"l£tl1 4,138tE+014.05lT£+0l 3~~49JE+Ol -s. .. 10hO\ ·3, n"E+01·,3'.'S1E+0l ,3.UlSbOl ,Ji5SUC.U
- .. 1-~4.Z~.'£.U- -... ltatUn -·I t UI8E+0l· ... OSZlh01J.9491E+01--·J.U.U+Gl ~,3, 75~4hOl-l~ •• Uh01l ... 09E+01J.SSO.[." -------~.~-:;:;-:~:,,--:'".: .. -.-~ . -!it: . 'J-Lt!.. -'~'. ---•. --'----.. -- ..... -- .---- .. - .. ---~-!'..-:" -~-~'.--~-::-"-- .~--~:~., ... _:".' ~-' .... "~
W I
i--> +:>
Table 3.7 Radial Velocity
-*-*-.-*-*-*-.-*-*-*-*-.-*-*-.-•.. -.-•. V-VEL.OCITY -*-*-*-.-.-..•.. -*-.-•.• -.-•.•. *-*_ •••••
J 1= 10 .. l.''!i'!oOO 9 .. 3.0~i+OO 8 -3,51E+OO 7 .. 2,3ItE+00 o 1,01E+oO 5 =2.7jE~01 ~ -1,2'1[+00 3 .. 3.10E+Oo 2 -1,00E+00 1 O,OOE+OO
J 1= 1 10 -1.711[+00 9 -3,OUIi+OO 8 .. 3,53E+00 7 "2,30E+OO o 1,07E+OQ 5 -2,73E .. Ol 4 -1.211[+00 3 -3,10E+00 2 -1,00[+00 1 O,OOE+OO
J I;: 1 10 -1,74E+Oo 9 .. 3.04[+00 8 -3.51hOO 7 -2,3bE+OO o 1,07hOO 5 -2,7lE: .. Ot II -1;24i+OO 3 .. 3,106+00 2 -1.00hOO 1 O.OOE+OO
J I;:, 1
2 -2.1!l[-02 ",09E-01 1,25[+00 2,0IE+00 2,29E+00 1.2iE+OO II.!HE-Ot
-«,o2E-01 -3,10E-Ol
O.OOE+OO
2 .. 2.19E .. 02 II,~9E-Ot 1.25E+00 2.01E+oO 2,29E+oO 1.28E+OO ".BH:-Ol
-2.02E-OI -3.10E-01
O,OOE+OO
a -2,II1E .. 02 4.on-01 1,2SE+OO 2,01E+00 2,29[+00 1,2aE+OO 1I;1l7E-Ol
"2,oiE .. Ol -1,10E-Ol
G,OOhOG
3 7."'!E-Ol 1.UIE+00 1.98E+00 2.38E+00 2.u3E+00 t,81E+OO 9.a3E-01 9.3'1E .. 02
-~,2'1E-02 O.OOE+OO
3 7.uUE-01 I,UIE+OO 1.98E+00 2.38f::+00 2.1I3E+OO 1.81E+00 9.2'1E-Ol Q,31E-02
.1.22E-02 O.OOE+OO
3 7.uI.IE .. 01 I.U1E+oO 1.98E+OO 2.38E+00 2.u3E+OO 1.81E+00 ~.allE-Ol 9.17E .. 02
·;.a2E",02 O.OOE+OO
4 o.49E.Ol 1.20E+00 l,7Qe:+00 2.10E+00 2.20E+00 1.Q4E+OO 1.21E+00 3.80E-Ol 0.82E.02 O.OOE+OO
U -0. 49E-0 1
1.20E+00 1 I HE+OO 2.1&E+00 2.20E+00 1,9UEtOO 1.21E+00 3.81E-Ol °180E-02 O,OOE+OO
'I 0.U9E-Ol 1.20E+OO 1.79E+OO 2.10E+00 2.2&E+00 1.9I1E+00 1.21EtOO 3.8lE-01 &.80£ .. 02 O,OOE1000
5 1.18E+oO 2.1eEtOO 2.115E+00 3.45E+00 3.57E+00 3,23E+OO 2.41E1000 1.38E+00 o.ObE-OI O.OOE+OO
5 1.18EtOO 2.10E+OO 2.95E+00 3.'I5E1000 3.58E+00 3.23£+00 2.IIIE+00 1.38E+00 0.07E-01 O.OOE+OO
5 1.18E+00 2.1bE+00 2.95E1000 3.1.15£tOO 3,58E1000 3.23E+00 2.111E+00 1.38£+09 b.07E-01 0.00£+00
11 " . 5 -i,liE-Oc 1,414£-01 o,4'E.Ql 1,18E+OO
4.o,E-Ot 1.41£+00 l,2~E+OO 2,16£+00 t~2SE+0. --1.'Ie+OO -liHHOO "2.-95(+00 2.0tE+00 2,J8£.00 2.10E+00 ,3,415[+00 2.292+08 l,_JE+OO '2.iU.u0 ·3,57E+00 1,2tE+Oe 1.U£+00 1,94£+00 '-3~c3E+00 4.aTE.Ol ',llE~OI 1,21E+0$ 2,41£tOO
-a.biE-Oi 9.34£-02 3.80E-01 1,38E+00
*** K. 1 *** & 7
u.03E+OO 0.72E-01 7;32EtOO 1:33E+00 1,02E+Ol 1.90E+00 1,24E+Ol 2.32E+00 1,35E+Ol 2.50E+00 1.2~E+Ol 2.28E+OO 8.58E+00 1.80EtOO 1.I.12EtOO 1.28EtOO 1,32E+00 0.09E-01 O,OOEtOO O.OOE+OO
*** K iii 2 *** o 7
U.03EtOO b.73~.01 7.33E+00 1.33E+00 1.02EtOI 1.90E+00 1.2 4 E+Ol 2.32EtOO 1,35E+Ul 2.50E+00 1.2'1E+Ol 2.28£+00 8.58E+OO 1.80E+00 '1.12£+00 1.28E+00 1,33£+00 0.72E-01 0,00£+00 0.00£+00
*** ... ;: 1 *"'* 07
4,03E+00 0.73E-01 7.33E+ou 1.33E+00 1.02E+Ol 1.90EtOO 1.4I.1E+Ol 2.32E+00 1,35E+Ol 2.50E+00 1.2I1E+01 2.28E+OO 8;58EtOO 1.80E+OO 4,12E+OO 1.28E+00 1.33E+00 0.72e-01 O,OOE1000 0.00E1000
.*'" K. 4 '111** o ' 7 4,01[+00 o,7iE-01 7.~2E+OO 1.33E+00
8 5.11E-02 l;73E .. Ol 3.01E-Ol ".08E-01 ~,77E.01
4.89E·Ol ~, 77E-01 1.'HE-01 2,31E .. 01 O.OOE+OO
8 5.3I1E-02 1,7UE-01 3.02E.01 U.09E-01 U.79E .. Ol 4,91E-01 4.80E-Ol 1.I.00E-01 2.35E .. 01 O.OOE+OO
B S.3I1E-Oil 1.7I1E-01 3.02E-01 U.09E-01 4,79E-01 4,91E-Ol 4;80E .. 01 II.OOE .. Ol "2,35E .. Ol 0.00£+00
9 -3.00E-02
1J,0qe:.Oll 5.6&E·02 l,l11E .. 01 1.71E·01 2,21E"'Ol 2,eOE-01 2.30E-01 1,37E-01 O,OOE+OO
9 -3.b3E-02
1,11E-03 5,70E-02 1.10E-01 1,73E"01 2,30E-Ol 2.e2E-01 2,33E.01 1.1I0E-Ol O.OOE+OO
9 -:;.03E.Oi
1.11E·03 5.7&E·02 1,10E-01 1,73E.01 2,lOE-01 2.b2E-0! 2,33E-Ol 1.40i-Ol O,OOE~OO
10 2.50E-01 3.1q[eOl 3.14E-01 2.79E-Ol 2.50E.Ol 2.51[·01 2.1J3E-Ol 2,00E-Ol 1.17E.01 O,OOEtOO
10 2.50E-01 3.20E .. 01 3.15E-01 2.80E.Ol 2.S1E-01 2.53[·01 2.1J5E-Ol 2.03E-01 1.19[ .. 01 O.OO[tOO
10 2.S0E.01 3.20E-01 3.15E .. 01 2.80E-Ol 2.51E-Ol 2.53E-Ol 2.45E.Ol 2.03E-01 1.19£-01 O.OOE+OO
8 9 10 . 5.31£-02-] .bbf-Oa.2. soe-o 1 1, 73£.014,09E-04 ·l~ 1qe-Ot
-r,OIE-01 .. , S-ioU-OZ '3;14£;'01' 4,08!-011,14£-012.79E-Ol 4,77E-Ol 1~71£-Oi2i50E.Ol 4,8C1E-01 2,an-Ol -2~51E.Ol
10 .. 1.741&+00 9 .. 3.041+00 8 -3-;5ihOQ 7'-a.3U+00 o hUhOO 5-2. 7U.0 1 ' " -l,cl/E+OO 3 -3,101+00 2-1.00hOIl 1 O,OOhoo
-3,101-01 -.';IU60.- oi 82E-oa-6,Oo!-01'
-1;02E+01 1~90E+OO 1,24£+012,32E+OO 1 ~ 15E+0 1 a. 50£+00 1,24£+01 i.28E+00 8.58E+00 1.80E+00 4.12E+00 1.a8E~OO I'il'2hOO -, a,oUitOt O.OO£+O~ 0,00£+00
4,77E-Ol 2~00E.Ol 2.413£.01 ],97E-012,30£-01 2.00£-01 c,ll£.Or-·r;ln-01 ~ l,17E-01 0.00£+00 0,00[+00 0,00£+00 0,0'£+01 O,OOE+QO 0.00£+00 0,00[+00
... CHICKl '''INT 0, VNI, -11., n .• 1QJ1,· 1JLa "
II 3 2
'"
Ie, 1 '. 3 III ' 's 0.000 ohao - 0 ~ 000 DEtOO' 0,0000£1000--0. OOOOE+ 00 0.0000£+00 0,0000[1000 O,OOOOE+OO 0.0000[+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOEtOO 0.0 D.O 0·0 0'0
CI 0;0000£+00 O,OOOOE+OO O.OOOOE+OO 0,0
7 O.OOOOE+OO' O.OOOOE+OO O.OOOOE+OO 0.0
e ,9 .10 O,OOOOE+OO '0,0000£+00 0,0000[+00 0,0000£.00 O.OOOOE+OO O~OOOOE+OO [J,O 0,0
0,0000[+00 - O,OoOO~+OO --0,0000£+00 0,00001.00 O.OOOOE.OO 0.0000(+00 0.0 o. ()
W I ......,
<.n
,*_*_*_*_*_._*_._*_*_*_*_*_*_*_*_*_*_*e Si<lIRL Vr' ,)CITl' .*~*.*.*~#.*-*~ •• *.*.*.*.*.*.*.* ••••••••
J 1= 1 10 O.OOE+Oa
q O.OOE+OO 8 O,OOE+OO 7 0.00£+00 b O.OOE+OO :; O.OOf+OO /I O.OOE+OO 3 O.OOf+Oo 2 O.OOE+OO
2 O.OOE+OO 0,00£+00 O,OOE+OO O,OOE+OO 0.00£+00 O,OQE+OO O,OOE+OO O,OOE+OO o.oa£+oo o,OOf.00
3 O,OOE+OO O.OOE+OO 0,00£+00 O.OOE+OO O.OOE+OO 0.00£+00 O.OOE+OO O.OOE+OO O.OQ£+OO O.OOE.OO
" O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO 0,00£+00 O,OOE+OO O.OOE+OO O,OOE+OO O.OOE.OO
5 O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O,OOE+OO O.OOE+Oa O.OOE+OO O,OOE+OO O.OOE.OO
*** K:; 1 *** b 7
O,OOE+OO O.OOE+OO O.OOE+OO 0,00£+00 O,OOE+OO O.OOf+OO O,OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO 0.00£+00 0,00£+00 O,OOE+OO 0,00£+00 O.OOE+OO O.OO£~OO O.OOEtOO
6 O.OOE+OO O,OOE+OO O.OOE+OO
·O.OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO 0.00£+00 C.OOHOO O.OOf+OO
9 O.OOE+OO 0.00£+00 O.OOE+OO 0.00e:+00 O,OOE+OO 0.00£+00 O,OOE+OO O.OOE+OO O.OOE+OO 0.001::+00
10 0.00[+00 0,00£+00 O,OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO O,OOE+OO O.OOE+OO O.OOE+OO
Table 3.8 Angular Momentum
O.OOi+OO
, 1= 1 :.0 -8.71£-011
q -1.OBe.03 8 -1.21E-03 7 -1.2"E .. 03 b -S.3UE .. OIl 5 -1.BbE-OIl 4 -1l.b8E-OS 3 -l.qOE-Os 2 -8.3SE-Ob 1 -2.38E-Ob
I 1= If\ \ u S.23f .. li 9 £I.01E .. ll 8 1.8bE .. 11 7 -4.bl£-12 b -Z.lH:-lt 5 ... 2.28E-It 4 -1.b5E-l1 3 .. 7.3SE .. 12 2 ,,0.5IlE .. !} 1 -5.1QE"13
J 1= 1 10 B.17E .. OIl 9 1,08E .. 03 8 1,2n"01 7 1.24E .. 03
~o '8~34["0"
5 1.8bE-01l 4 4.b86-05 3 1.9Qi-05 2 8.3'hOb 1 a,3Si.Ob
2 -2."5E-oa -2,7I1E-OI.! -3,21E-01l .. 3,QQe:-O" "3,1I2E-OII -1.21E-OII -S,1l2E-01,
1,105:-u5 1,IH-OS 2.05E.Ob
a q,blE-il e,96E e l1 2,91E-l1
-1:1,43£-12 -2.8bE-l1 .. 3,65E-l1 -3,ObE-ll "l,bbE-lt -1,21E-l1 "1,HE .. 11
;I
2,"5£-0" 2.hE-01l 3,i!lE-0" 3,99e: .. 04
, 1;1I2E .. 04' ' , 1.23E .. O" . S.lIai-Oo "1.10E-OS -1.1t1E-OB -a.051i"'Oo
3 .. 1,18E-0'4 -S.9IlE-04 -1l.48E.04 -3.21E·O" -2,OSE-01l -1,7SE-OS
q.93E."Ob c.oSE-OS Z,2QE-OS b.OlE-Oo
3 :. 10"''''' •• ' .... /CO-ll
1,C4E e l0 7.87E-ll l.88E.-ll 2.77£-12
-1.97E-ll -2.53E-11 .. 1,b8E-11 .1,1l8E-ll -c,10E-11
3 'I',18E·04 5,94E.-04 4,48E.04 3,21E-04 a,OSE-C'iI 1,75E-05
.9.93E-06
.Z~6!iE.05 -Z.29E-05 -6.Q7E .. 06
" -0.b2E.0" -5.27E .. 04 -3.84E-04 -2.49E.OIl -1.3IE-0" -3.22E-05
3,19E.05 4.70E-OS 3.59E.05 I,03E,.05
Il
5 -1,77£-011 .. 1.02E-06
8.00E-05 1,20E"04 1.41E-U4 1,511;"04 1.1l8E .. 01l 1.22E-01l 7,91E-05 2,13E·05
5
, *** K = -2 *** b 7
-3.7bE-05 2.b8E-05 3.43E .. 04 3.94E~04
5.38E-04 5.80£-0" b.32E-04 0.4QE-04 0.55E-01l b.20E-04 b 1 3lE-04 5.27E.04 5.Q8E-OIl 4.3I1E-OII 4.92E-0" 3.38E-OIl 3.07E-01l 2.43E-04 8.04E-05 7.77E-05
*** K = 3 **" 6 7
8 .. 4.00E-04 ~5.55E=05
1.IlSE-OIl 2.b2E.04 3.03E-OII 2,9"e.-01l 2.75E-04 2.bO[ .. 04 2.32£ .. 04 7.7bE .. OS
8 .2.o1E.1Q -l.QbE-OY -l.QaE-OQ -S,ibE.OIf -i.20E-Oa =-1.40E=10 -5,85E .. l1 -1.32E-11
7,83E-13 -1.90E-12 -b.8bE-tO! .. 7,36E.12 -I,35E-l1 .3,52E-l1
=7.30E=lO -1l.59E-l0 -2.55E"10 -1,26E-l0 .. 5.blE-11 -2."8E c l1 -1.L/4E-l1 -2015E-ll -3,7IlE-l1
~1.49E=;:oq
-1,04E-09 ,,0.72E-l0 -1I,04E-l0 -2.27E-l0 -1.IlOE-l0 -1.01E-l0 -8.48E-l1 -8,02E-ll
*** K
."'.80E.OQ -Il,18E-09 .. 3.52E.OQ -2.B8E-09 .. 2,26E-09 .. 1.70E .. 09 -I,18E-09 .. 8.40E-l0 -7,33E .. 10 = "***
=-1,18E·08 -1,09E-08 .9,72E-O<l -8.43E-09 "o,99E-09 .. S.30E=09 ·3,41E.09 "l.boE .. OQ -5,47E .. l0
9 a .. 5. 7Sf.04 ~.S8S.'04 .2.53E-OII .2.b6E~04 -II,81E .. OS ~8.58E-05
1,QOE-0, 2,211E.05 1,50E-0~ 8,63E.05 1.85£ .. 01l 1.2QE.O/f 2.13E.04 i~b9E.~1I 2,45E.OII 2.10E-OII 2.33E"01l 2.04E-04 7.42E-05 5.95E-05
9 -;::,50E .. 08 .,2.l.£7E u 08 .. 2,3lE-08 .2,14E.08 .1,97£·08 .1,74E .. 08 01.3"E=08 .b,55£·09
1l,20E"09 l,42E.08
10 .li,OiE.08 .. 3.82E-Oa .. 3.55E·OS .. 3.35E-08 -3.22E.08 .3.03E-08 .. 2.117E .. 08 .. 1.07E.08
1.b2E .. 08 4,11E.U8
II 5 1:1 7 8 9 10 5.S8E-04 2,60E.O'l 8.58E-05
o.ocE-O" 1,77E-01l 3,7bE .. 05 S.27E .. 04 1,02E-Ob -3_113E-OII 3,84E-04 .. 8,OOE-05 -5.38E-04 2,1l9E.04 -1,20E-0" -b.32£·04
'{ , 3'1 E'O!T4--'-r, 4 fE - 0 a- "0;5'5£-011-3,22E.05 .1.S1E-04 -b,llE-04
-3.19E-05 "1,48E-OII -5,98E-04 .4,70E-05 -1,22£-04 -4,92E-04 ~3,59E-OS .7.91£-05 -3,07£-011 -~,03Ee05_·2,13E-05-8.0llE-05
.2.08E.05 3,99E.OII 5.75E-04 -3.94£.011 5.55Ee05 2,53E.OU -5.BOE-04 -1.48E .. 04 4,80E-OS .b.49E.04 -a.beE-04 .7,90E-05 -0-,-20E-04 oii3;o-Jr-04_r;SO-e:-.04 .5,27E-0" .2.94E-OII .l,85E.04 .II,311E.OII .2,75E.04 .2.13E.04 .3.38E.04 -2,bbE.OU -2,45e-04 .2,43E.OII -2.32E.04 .2,33E-04 .7,77E.05 .7,1~E.OS .7,42E.tS
.2,25E.O~;
.8~b4E.05
-1.29E.04 .1,b9E.OII .2,10E.04 -2,04£-04 .5,911E.05
**** CHECK2 ~NINT OF WH8 UIII 1 II.lil0 3 . 4
Jla 1 JL=10 , ",--' 7
O,OOOOE+OO O,OOOOE+OO O,OOOOE+OO O,OOOOE+OO O,OOOOE+OO O.OOOOE+OO O,OOOOE+OO O,OOOOE+OO
, J 1::1 1 i ' 10 O,OOOOE+OO O,OOOOE+OO '9' 0.0000E+80 -O~OOOOE+OQ 8 0.0000£+00 G.OOOaF.+oo 7 O,OOOOE+QO 0,0000£+00 6 O,OOOOE+OO O,OOOOE+OO 5 O,OOOOE+OO O.OOOOE+OO II O,OOOOE+OO O,OOOOE+OO 3 O,OOOOE+OO O.OOOOE+OO 2 o.OOnnE.oo ~,o~~~~.nn
O.OOOOE+OO O,OOOOE+OO O,OOOOE+OO O.OOOOE,+OO 0.0000£+00 O.OOOOE+OO O,OOOOE+OO O,OOOOE+OO ,.., ("Ir-,I,\t".J,.(""
, .. 5
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 0.0000£+00 O,OOOOE+OO O,OOOOE+oO O,OOOOE+OO O,OOOOE+OO ,", '''If'f''o''C'" .... ''···
6 .... ---,-O,OOOOE+OO O,OOOOE+OO O,OOOOE+OO 0,0000£+00 O,OOOOE+OO O,OOOOE+OO O,OOOoE+Ou O.OOOOE+OO
f." C. ,"",
: >,."' 8 ~- 9 10 o,ooooe+oo O,OOOOE+OO O.OOOOE+OO 0,0000£+00 O,OOOOE.OO' O,OOOOE+OO OiOOOOE+OO O,OOOOE+OO O,OOOOE+OO O,OOOOE+OO
~OiOOOOE+OO 'OiOOOOE+OO O,OOOOE+OO O,OOOOE+OO 0,0000£+00 O.OOOOE+OO
O,OOOOE+OO OpOOOOE+OO O,OOOOE+OO O,OOOOE+OO O,OOooHoo O,OOOOE+OO O,OOOOE+OO O.OOOOE+OO
0,0000£+00 0.0000[+00 O,OOOOhOO
.0,0000£+00 0,0000£+00 ~O,OOOOhOO
0,0000£+00 O.OOOOE+OO
Table 3.9 Kinetic Energy of Turbulence
-*-*.*-*-*-*.*.*.*-*-*.*~.-*~~=*=*-* ••• K.Ej o~ TUR8ULENCE .* ••••• *.*.*_*.*.* ••••• *w •• *.* ••• * ••••• **'" K :I 1 1111*
J I: 1 2 ! ij 5 0 7 6 9 10 10 5.92£:+00 1.32E+Ol 1.'iijE+Ol 1.c1E+01 1:>,99E+OO 2.58E+01 2.3lE+01 2.19E+Ol 2,ltE+Ol 2,09E+01
<; 1,8ZE+01 i,90E+Ol Q,e.2E+Ol 8.39E+Ol 1:>,75E"'01 1.03E+02 8.09E+Ol c.50E+Ol 5,ijillETOt 1I,71E+01 8 2.07E+01 1,13E+oa 1.29E+02 1.12E+02 8.83E+ul 1.1C;E+02 Q.Q3EtOl A. 111.10./\ I .., .. ., ...... 0.81[+01 5.89E+Cl 7 3.21h01 l,311E+02 1.118E+02 1.30E+02 9,95E+01 l,21E+02 l,07E+02 8.95E+Ol 7.58E+Ol o.oOE+Ol I:> 1:>,19E+01 1.38E+02 1.ij9E+02 1.3I1E+02 1,02H02 1.15E+02 1.07E+02 9,19E+Ol 7.91E+Ol b,90E+o1 5 5,1181:+01 1I,20E+01 1.15E+02 1.15E+02 9.2I1E+Ol 1,05E+02 1,OlE+02 8,90E+Ol 7.83E+Ol 7.00E+01 II 1,4I1E+01 3.2ijE+01 s.qO~+Ol 7.I:>L1E+Ol 7.32E+01 9.22E+Ol 8.9I:>E+01 8,17E+Ol 7.41E+Ol b.77E+Ol 3 2.07E+01 2.Ll9E+Ol 3.3 QE+Ol 1I.1:>8E+Ol 5,43E+01 1,70E+Ol 7,I:>OE+Ol 7,23E+01 0,80E+Ol 1:>.381::+01 2 1I.01E+Ot 1I.29E+01 1I,15EH1 1I.20E+01 L1,I:>OE+Ol 1:>,53E+Ol b,S7E+01 o,IIIIE+Ol 0,211E+01 b,OOE+Ol 1 2,17E+Ol L1.08E+01 II.2bE+Ol ij,lOE+Ol 1I,30E+01 o,19E+01 o,12E+01 1:>,03E+Ol 5,92E+Ol 5,78E+01
*"'''' K : 2 "'** J 1= 1 2 3 II 5 ° 7 8 9 10 10 5,92E+Oo 1.3<1E+Ol t,94E+Ol 1.01E+Ol 6.98E+OO 2.58E+Ol 2,33E+Ol 2,19E+Ol 2,11E+Ol 2,10E+Ol 9 1.82E+Ol 7.90E+Ol 9,02E+Ol 8.3QE+Ol 1:>,75E+Ol I,03E+02 8.10E+Ol 1:>,50E+Ol 5,II2E+Ol 1.I,71E+Ol 8 2,07E+Ol 1,13E+02 1.29E+02 1.12Et02 8,83E+Ol 1,19E+02 9.93E+Ol 8.13E+Ol 1:>,81£+01 5,89E+Ol 7 3,23E+01 i.34E+02 1.Ll8E+o2 1.30E+02 9,95EtOl 1.21E+02 1.07E+Oc 8,95Et01 7,58E+01 b,bOE+O! 0 0,1I1E+01 1,38EtC2 l.4qE~U2 1.311£+02 1.02E+02 1.15E+02 1,01£+02 9,19E+Ol 7.91E+Ol b.9b£+01 5 s.~eE+01 9 .20EtO 1 1.15£+02 1.15E+02 9,21.1EtOI 1,05E:+02 1,01£'1'02 8,90£+01 1.83£+01 1.00£+01 1.1 l,44E+01 3.2L1E+01 5,90E+Ol 7.1:>11£+01 7.32E+Ol 9.22£+01 8,91:>£+01 a.i'i'EtOi 7.l.iiE+Ol b~77E"Ol
w :3 2.0H+01 2. 49E+Ol 3.39EtOI ".08EtOI 5."3E+Ol 7,70E+Ol 7.I:>OE+Ol 7,23E+Ol 0,80E+Ol 0.38£+01 I ........ 2 LI.01E+O! 1.1.29£+01 1.1. 151:.tOI ".20E+1l1 LI.oOEtOI 1:>,53E+01 1:>,57E+Ol 1:>,I.iIlE+01 o,cIlE+Ol 1:>.00E+Ol 01 1 a,l7i.+Ol ",08EtO\ ij,21:>Et01 1.1,10£+01 1I.30E+01 o,1 9EtOl 0.12£+01 0,03£+01 5,92£+01 5.78£+01
*"'''' II. : 3 *** J I= 1 2 :3 4 5 I:> 7 8 9 10 10 5,92E+Oo 1.32E+01 1.911£+01 1.b1EtOI o,98E+OO 2,58£+01 2.33E+Ol 2,19£+01 2,11E+Ol 2.10E+01
9 1,82E+Ol 1 8 90£+01 9,02E.tOI 8.39E+Ol b,75E+Ol 1.03E.02 8,10E+Ol 0,50E+Ol 5,42E+01 4.71E+01 8 2,07E+Ol 1.13f!+02 t.29E+02 1.12E+02 8.83E+01 1,19£+02 9,93E+Ol 8,13E+01 0,81£+01 5,89E+Ol 7 3.23i+01 i.34E+Ot! 1.w6E+02 1.301::+02 9,Q5E+01 1.21E+02 1.07E+02 8.95E+01 7.58E+01 1:>,1:>0E+Ol I:> col9e:tOI 1.38i!:+oa 1,/l9E+02 1,3'1E+02 1.02E+02 1,15E+02 1.07E+02 9.19£+01 7,91E+01 b,91:>£+01 5 5,"8hOt 9.20!+ol 1.15E+02 1.15E+0.2 9.2'1Et01 1.05Et02 1,01£+02 8,90E+Ol 7,83E+01 7,00£+01 II 1,1I1I!+01 3.2/1E+Ol 'ii.90E+01 7.oIlE+01 7,32E+Ol 9.22E+01 8,9OE+01 8,17£+01 7.~lE+01 0,77E+01 3 2,07£;.01 2."9e:+01 3,39E+01 4,08£+01 5,"3E+01 7.70E+01 7,I:>OE+01 7,23E+01 b,80E+01 0,38E+01 2 II,OlhOl ~.2ge+Ol ~,15E+Ol (i,cOE+Ol 4,I:>OE+01 o,5lE+Ol 0,57£+01 o,,,"e:+01 1:>,2~E+01 0.00£+01 1 2,17£+01 II,OaE+01 1I.20f+01 lI.l0E+Ol 4,30E+01 0,19E.Ol 0.12E+01 0,03E+Ol 5,92E+01 5,78£+01
*** K :If 4 *** J III 1 a 3 'I 5 0 7 8 9 10 10 ' 5,9<1(+00 l,32E+Ol 1,9ij£+01 1.01E+Ol 0,99E+00 2,58E+01 2~33E+01 2.19£+01 2.11£+01 2,09£+01
9 1,82E+Ol 1,90E+01 9,&2E+01 8,39E+Ol o,75E+Ol 1,03E+02 8,09E+01 0,50£+01 5.42E+01 ",71E+Ol 8 2,OTE+01 1,lU+OIl 1.29E+02 1,12E+Oi 8,83E+Ol 1.19E+02 9,93E+01 8,13E+01 0,81£+01 5,09£+01 7 3,23i+o 1 1.3~!+oa 1,Q8E+0i! 1.30E+O. 9.95£+01 1.21E+02 1,07£+02 8,95£+01 7,58E+01 &,oOE+Ol 0 o,1CJ!+OI loUhOI 1"jI~E+02 1.3Q£+0i! 1,02£+02 1,15E+02 1.07£+02 9,19E+Ol 7,91E+01 0,90£+01 5 5,4aE+Ol 9,20E+Ol 1,15E+02 1.15E+Oi 9.24E+Ol 1,05£+02 1,01E+02 8,90E+01 7,83E+Ol 7.00E+Ol 4 1~44E+Ol' !ii!4E+Ol 5 ,90E+0 1 . 7,04£+01 7.32E+or9~22E+Ol 8,90E'+Dr 8.17£+01 '1,41001 ~o'·17HOl' 3 2.0ntOl 2,Qge+ol l.39E+Ol ",b8E+Ol 5,43£+01 7.70£+01 7.00h01 7,23E+Ol '0,80£+01 ·& .. 38E+Ol 2 1I,01hOl 4.a9E+Ol lI,15Et 01 4, i Of. iil It.bOE+Ol to.51hOl =.57E+Ol ~i"4E+Q! o.24E+Ol &.00E+01 1 c,11E+Ol 4.06E+Ot 4,coEt01 11,10£+01 4.30E+Ol 0,19E+01 0.12£+01 0.03E+01 5,92£+01 5.78E+01
Table 3.10 Absolute Temperature
.*-*~*~*.*~*.*~*-.".-*.*.*~*-~ ••• *-* ••• TEMPERATURE .*W*Q*.*=*O*~*~*~*.*.*W*.*.*.*.*.*.* ••• * *** K :; ***
J 1= ~ :3 I.i 5 b '1 S q 10 " 10 5.50£+02 5.52£'1'02 5.52E+02 5.b8E ... Oi b.3~~"'02 1.39E'I'03 1,I.II.1E ... 03 1.l.IbE+03 1.1.18E+03 1.50E+03 9 5.51E ... Oa '3.1.I1E+02 5.1.I0E+002 5.5I.1E+02 b.BE.+02 1,1.I1E+O~ 1.50E+03 1.53E+03 1.5bE+03 1.58E+03 8 5.1.I1E+02 5.35E+02 5.34E'I'U2 5,lIoE+02 b,22E+02 1.l.IbE+O~ 1.5bE+03 1,bOE+03 1.b2E+03 1.05E+03 7 5."Oi+02 5.30E+02 5.119E+02 5.3<1E'I'02 b,09E+02 1.'31.1£+03 1.b5E+03 1,08E+03 1,71E+03 l,73E+03 b 5,2"E+02 5,23E+o2 5.2"E+02 5,33£+02 5,97E+02 1,boE+Oj 1.7bE+03 1.79E+03 1,80E+03 1,82E+03 5 5,HE+02 5.13E+02 5,18E+Q2 5.29E+02 5.87£+02 1.79£+03 1.88E+0~ 1. 9OE+03 1.91E+03 1.91E+03 1.1 5.02E+02 5.08E+02 5.18E+oa 5.30E+02 5.82E+02 1.90E+03 1,95E+03 1.95E+03 1.95E+03 1.9bE+03 3 5,15E+02 5.28E+02 5.31E.02 5,45E+02 5,84E+02 1.87E+03 1.89E+03 1.90E+03 1,92E+03 1.93E+03 2 b.07E+02 5.CldE+02 5.8'OE+02 5.79E+02 5,9bE"02 1.7bE+03 1.811::+03 1.85E+03 1,88£+03 1.90£+03 1 b.31E+02 1:I,IQE+02 b,08E+02 5.'ObE+02 b,05E+02 1.07E+03 1.75E+03 1,80E+03 1.85E+03 1,88E+03
*** K = 2 "'.* J 1= 1 2 3 4 5 0 7 8 9 10 10 5.5bh02 5.52E+02 5,52E+02 5.08E+02 b.39£..o2 1.39E ... 03 1.41.1E+03 1.'1bE+03 1.1.18E+03 1.50E+03
'0 5.51E+O! 5.41E+02 S,lI0E+o2 5.54Et02 b,33~+02 1.1.I1E+0~ 1.50E+03 1.51Et03 1,50E+03 1.58E+03 8 5."7E.02 5.356:+02 5.3"E+02 5."6E+02 0.22E ... 02 1.4bE+03 i.5bE+03 i,bilEtO} 1,b2E+03 1.115E';'03 7 5,40! ... 02 5.30E+02 S.2'Ot+02 :'.39£+02 e,09E"02 1.5QE>+03 1,b5E+03 1.08£+03 1,71E+03 1.73E+03
w 0 5.27E+02 5,23E+02 5.211~t02 S.33b02 5.97E+02 1,bbE+03 1.7eE+03 1, HE+03 1.80E+03 1.82E+03 I 5 5.07E+02 5,13E+02 5.18t"'02 5.29E+02 5.87£+02 1.79E+0~ 1.88E+03 1.'OOE+03 1,91E+03 1.91E+03 ~
" 5.02h02 5,08E+02 5,18E+02 5.30£+02 5.82E1'02 1,90E+03 1.'OSE+0.s 1.95£+03 1,95E+03 1,'0&£+03 --.l 3 5,1Sh02 5.2iE+02 5.37E.02 5,45E+02 5,8'1E+02 1.87£+03 1,89E+O~ 1,'00£+03 1,92E+03 1.'03E+03 2 b.01E+02 5,98&:+02 S.!lQE.o2 5.7QE+02 5,'001:.+02 1.701:":+03 1.81E+03 1,85E+03 1.88E+03 1,90E+03 1 b.31E+Oa 0,19E.02 o,08E+02 5.'OoE1'02 0.05E+02 1.07Et03 1.75E+03 1.80E+03 1.85£+03 1.88E'I'03
**'" i'. = 3 "''''* J 1= 1 2 3 II 5 b 7 8 9 10 10 5.50E+02 5.52E+02 '3.52E+Q2 5.08Et02 b.39E+02 1.39E+03 1.'14E+03 1.l.IoE+03 1.1.18£+03 1.S0E+03 9 5,51E+02 5,"IE+02 5,,,OE+02 5.54E+02 b.33E.+02 1."lE+03 1.50E+03 1,53E+03 1.50E+03 1.58E ... 03 8 5,'I1E+02 S.35EtOi/ 5.34E.+02 5."6E+02 b.22E+02 1.l.IoE+03 1.5bE+03 1.&OE+03 1.02E+03 1.05E+03 7 5.1.1\)£+02 '3,306+02 5.2~e:t02 5,39E+u2 0.09E+02 1,5aE+0.3 1.&5E+03 1.&8Ei'03 l,71E+03 1,73E.03 b 5.iHE+02 S,i!3E+02 S.2aE+02 5.33E+02 5.97E"'02 1.b&E+03 1.7bE+03 1.7QE+03 1.80E+03 1.82E+03 5 5,071:+02 ':5. 13E+02 5,18E+02 5.2'OE+02 5.87E+02 t,79E+0~ 1,88E+03 1,90E ... 03 1.91E+03 1.91E.03 Ii 5,02E+02 5.08E+0. S.t8E.+02 5.30Et02 5.82E.02 1.90E+03 1,95E+03 1.95£+03 1,95E+03 l,90E+03 3 5.15E+02 5.28E+02 5.37E+02 5.45E+02 5.8I.1E+02 1.87E+03 1.89E+03 1.90E+0~ 1.92E+03 1,93E+03 2 b.O'l£+02 5.9SEto,a 5,89E .. 02 5,79E+02 5.90E+02 1.7&Ei'Oj 1.81E+03 1.8SE+03 1,86E+03 1,90E+03 1 0.31£+02 0,19E+02 /),08~+02 S.9bE+02 &,05E+02 1,07E+03 1.75E+03 1~80E+03 1.85E+03 1,88E+03
*"'''' K ;;; 1.1 *"'* J 1= 1 C 1 'I 5 b 7 8 9 10 10 5.50E+02 5,52E+02 5.52E+02 5.08E+02 b~39E+02 1.39E+03 1.44E+03 1,4bE+03 1.48E+03 1.50E+03 9 5.51[+02 5."IE+Oil S.1.I0E+Oc 5,51.1E.02 &.33E+02 1.1.I1E.03 1.50E+03 1,51E.03 1.50E+03 1.58E+03 8 5."'1+02 5,3!5f+02 5,34E+O& 5.I.I&E+02 &.22E+02 1,I.I&E+03 1.5&E+03 1,00E+03 1.&2E+03 1.&SE+03 7 5.40E+0 4 5.30E+Oi S.Z9E+Oi 5,39E+02 - 0,09E+02 1.5I.1E+03 1.&5E+03 1.b8E+03 1.71E+03 1.73E+03 0 5.21Et02 5.i!3!:+02 5.2'1E+02 5.33E+02 5.nE+02 1,00E+03 1, '!bE+03 1,79E+03 1.80E+03 1.82E+03 5 5,(,.'£*02 5. !3h02 5.18E.,.Oi 5.a9E",Q2 5.87E+O!1 1.7QE+03 1.88E+03 1.'OOE+03 1.91E+01 1.91E+03 'I 5.021+02 5,08E+O! S,18E+OZ 5.30£+02 -S~82E+02 1~90E+03 -l,95E+03 1~9SE+03 - 1,95£+03 1~9b£+03 3 5.1Sh02 S,iU+O! ',HE+O! 5.45E+Oi 5.84E+02 1,87£+03 l,89E+03 1~90E+Ol 1.92[+03 1.93E+01 2 &.0'£+02 5.96E+oa S.e'iE+02 5.7~E+0! 5,9&E+02 1,70E+03 l,81e+03 1.85E+03 l,88E+03 1.90E+03 1 &.31E+OZ e.19E+oa b.Q8E+Oi! 5.9oe+02 o,05E+02 1-~07E+03 l,75E+Ol 1.80Et01 l,85E+03 1.88E+Ol
Table 3.11 Fuel Mass Fraction
~*.*.*.*.*.*-*-*.*.*.* ••• *.*-*.*-*-*~.- ~UEL CUNCENTxATTION e*~*~*e*_*e*e._.~*~a_~_~~~~~~~~~_~_~_~
*** K = 1 *** J 1= 1 2 b 7 8 9 10 10 .1911:-03 1.b6E-03 ,IIIE-03 • • • ·~-r;·2SE"04· T;~S t,4tE .. OC '-r~7
9 b.80E .. 03 b.SSE-O! 1.'13E-03 1.30e:=OZ 1.7BE=02 1.b2E .. 03 le40E·O~ t.t2E .. OS B.llE-07 5.blE-08 ~ b,31E .. 03 5,1QE .. 03 '1.'12E-03 1,28E .. 02 l,8bE-02 1.7I1E"03 1.18E-01I 7.25£.Ob 1I,33E"07 3.b4E.OS
7 5.35!-03 S.OIE-O] 7.SSE-Ol 1.32E.02 2.0bE-02 1.70E-03 8.92E-OS 1I,77E-Ob b.19E-07 3.37E-07 3.05E-03 1I.17E.03 7.7I1E-03 1."9E-02 2,IISE-02 1.7SE-03 9.22E-OS 1.Q3E-05 1.1tE.05 8,99E-Oo
A\K5 1.08E-0! 3.S0E-OJ 9.S3E-Ol 1.9SE-02 3.22E-02 2.bOE-Ol S.lOE-OIi 3.7bE .. 01I l.28E-OIl 2.07E-01l '\~II 1.SijE-Ol ? l6E-03 1.83E-02 3.2I1E-02 II.BOE-02 8.93E-0.3 8,SbE .. 0.3 8.2SE-0.3 7.05E-03 S,SOE-O.3
.......L~.18E .. 02 u.30E-02 o,02E-0~ 7,lI;JE-02 8.1SE-02 3. 11 8E-02 l.27E-02 2.80E-02 2,Z7E .. 02 1. HE-OZ we;:{...2 3.3I1E-OI 2.711£-01 i.2bE-01 1.78E-Ot 1.III1E-Ol 8.01E-02 0,lIbE-02 S.07E-02 3.92E"02 3,01E-02 . -1 S,71E .. Ol 1I.32E .. Ol 3,37E-Ol 2.S7E-O! 1,93E-01 1,lbE-01 8.93E-02 b.7:3E.02 S.0IlE-02 3,79E.02
*** K = 2 **'" J I:; ! 2 3 U 5 b Z §-.-".~. ~~- lQ 10 • ['-63 , • 6 7t .. iJ~-==W:urE -153 1.3I1E-02 1,77E-02 1.08E-03 1.2I1E-01i 1.38E-OS 1.1I0E-OQ 1.29E-07
9 b.80E-OJ 6,SbE.O] i.43E-03 l,JOE-02 1.78E-02 1.o2E-0.3 l.lIOE-OIl 1.11E-OS 8.1SE-07 5,SbE.08 8 b.JIE .. 03 5,7bE-03 1,93E-03 1.28E.02 1.8bE .. 02 1.7I1E-03 1.t7E.OII 7.22E .. Ob 4.31E .. 07 3.03E.08 7 5.He:.03 S.OlE-Oj 7.58E-03 1, BE-Oit 2.0bE-02 1.70E.03 8,90E .. 05 1I.7be:-Ob o,19E"07 3,38E"07 .. 3 b"E-"~ 4.17E=o3 ',75E~03 1.£.!qE~02 2.'ISE-02 1.75E-03 9.21E.OS 1.BE-OS 1.lZE-OS 9.03E-Oo --?- . v -v~
1.OBE.O:; 3.S0e-O] 9.53E-03 1,9SE .. 02 3.23E-02 2.o0E-03 S.11E .. Oll 3.77E-01I 3.30E-01I 2.b9E.OII A 4 l,seE-03 7,30E-03 1.83E-02 3,2UE .. 02 1I,8n:-02 8.94E .. 0.3 8.StlE-03 a,30E.03 7~ObE .. 03 S,52E-03
w .3 2.18E-02 1I.30E-02 b.02E-02 7.19E-02 8,lSE .. 02 3.48E-02 3,27E.02 2.BOE-02 2,2t1E-02 1,79E-02 I -Z 3.3I1E.Ot 2.7i1E .. Ol Z.ZbE-01 1.78E.Ol l.uIIE·Ol 8.01E .. 02 Q.UbE-02 S.07E-02 3.92E-02 3.01E-02 i-' ";:.
1 S.17E.Ot lI.32E·01 3 • .HE-O 1 2,57E-Ol 1.93E-Ol 1,1oE-Ol B.93E-02 b.73E.02 StOIlE-02 3.79E-02 :0 *"'* !< = 3 "'.'"
J 1= a 3 " S b 7 8 9 10 1O' .1 q ·.03 ,o7E-O! .41E·03 1. 4E';;'02 1. 77E .. 02 1.08E-03 1.211E.04 1,38E-OS 1.uOE-Ob 1.29E"07 9 b,801i-03 b,5bE-03 8,113£ .. 03 1,30E-02 1,78E-02 1,02E-03 1.40E-04 1.11E-OS 8.1SE-07 S,SbE.08 8 b.31E.03 S.7bE-OJ 7,93E-03 1.28E-02 1.8bE"02 1.74E-03 1.17E-01.I 7.22E-Oo 1I,31E-07 3.Q3E.08
~ S.30E-03 S,01E-O} 7,S8E-03 1,33E-02 2,ooE-02 1.70E-03 S,90E"05 4,70[-00 o,19E .. 07 3.38E-07 3.00E.03 1.1. 17E-03 7,7SE-03 1,IIQE-02 2,IISE-02 1.7SE.03 1;J,21E .. OS 1,Q3E.OS t,12E-OS 9,03E-Ob
5 1.OU,,03 3,SOE-03 9,S3E-03 1.95E-02 3,23E"02 2.bOE.03 S.lIE.OII 3.77E-01I 3,30E-01I 2,b9E.OII A 1.1 1.SU.03 1,31;1E-03 !.83E-02 3,2I1E-Oi! 1I.81E.02 8.9I1E-03 8,S8E.03 8.30E-03 7,ObE-03 5,S2E-03 --l. 2.1/;!-02 'I,30E-02 Q.02E-02 7.t9E-02 8.1SE-02 3,1I8E-02 3.27£-02 2.80E.02 2.2SE.02 1.79E.02 F2 3,3I1E.01 2,7I1E-Ot 2.2bE-01 1,78E-Ol 1.III.1E-Ol 8.01E.02 &.lIoE-02 S,07E·02 3,92E"02 3.01E-02
1 S.71~ .. Ol 1I,32e:-01 3.37E-01 2,Sin.ol 1,93E-Ol 1,lbE.01 8,93E-02 b, nE .. 02 5.04E-02 3,79E-02 "'*'" K = /I ***
1 , 3 II 5 c 7 8 <1 10 .19£-03 1,60E-03 9.II1E"03 1.3I1E,,02 1.77E-02 l,08E-03 l,2SE-01I 1.3QE"05 1. /HE-Ob 1,lOE.07
b.80[.03 b.SSE-OJ 8,1I3E-03 1,30E.02 1.78E-02 1,b2E-03 1,1I0E"01l 1.lZE.OS 8.21E .. 07 5.blE-08 0.3U-03 ',7bE-OJ ,.I;J2E-03 1,28E.Oc 1,8bE-02 1.74E .. 03 1. 18E.01.I 7,2liE-Ob 1I.3JE.07 3,OIIE.05 5.35£-03 5.01E-0] 7,S8E-03 1,32E-02 2.0bE-02 1.70E .. 03 8.92E-OS 1I.77E-Oo 0.19E-07 3.37£-07 3.05E-03 1I.1TE-0:) ',7I1E-03 1.1I9E.02 2,IISE-02 1.75E-03 9,22E-05 l.UE-OS 1.11E-05 8,99E-Oo 1.OiE.03 1.50E-0] 9.53E-03 l,95E.O' 3.22E·02 2,bOE-03 5,10E-OII 3.70E .. 01l 3.28E-01l 2,07E-04
All 1.5111-03 1,3&£.03 l.nE-oZ 3,ZIIE.02 1I,80E-OZ 8.93E-03 8.50£.03 8.28E-03 7,05E-03 5.50E-03 3 2.1U-02 4,36&:-01 0,02E-02 7.1QE .. oa 8.15E .. 02 3,118[-02 3,27E.02. l.80E-02 2,l7E-02 1.7U-Oc '-:z ,3.341-01 2,Ur-Ol 2.hE-01 l,78E.Ol--i~II4E.Ol 8.0it:·O;:-·o~ilbf.Oc -S~01E-Ol . 3,9cE.Oc3~OiE.OZ
f 1 5,11bO i ",UE-Ol l,37E-01 2,57E,.01 1,93£ .. 01 1.10E-01 1.93h02 0,7]£ .. 02 S!04E .. 02 l!79E=O~ .
W I
>-' \.0
o q 'i' x
\=1 2 3 4
E ~ ~ X
5 6 7 8 9 10
E ID M cO • x
'««,f! " I "~'''<'t''« 1 ' , 1"'''''<''<
ro ~ I I t I I r TUf"f"'1 'X'''''M
= 1 1 {:U 1 1 1 1 1 I· r 10 .' ~
-x
CONVECTIV~ FLUxES T~ROUGH T~E CHAMBER CRUSSECT1UNS
I~ 1 XM: O,OOOE"'OO Fh l,779037E+Ol FXP: 1,7H037E ... Ol FlCN: O,OOOOOOE+OO 1= 2 l(M= c>,350E-Ol F"ltlll 1,7H037E ... Ol FXP:; 2,lbl037HOl F~N= -3,819Q99E+OO 1:; 3 )(1'1:; 1.'170!::+OC FlClI 1. 7H037E+O 1 FXP: 2,09319aE+Ol F~N= -3.141554E+OO 1= II xf'l:; 1.90SE+OO FX: 1.7HO.57E+Ol FXP= 1,Cil14738E+Ol FXN: -l,3S7011E+OO 1= 5 XM= 2.540E+OO FX: 1.779037E+Oi FXPli 1.82S1OZE+Ol F~~= =4.bOb51aE e Ol I= b X 1'1= 3,17iE+OO F"~:: 1,779037E+01 FxP:: 1.779037E+Ol FXN:: O.OOOOOOE+OO 1= 7 XM:: 3,810E+OO F~= 1.77Ci1037E+Ol FXP:: l,779037E+Ol FXN:: O.OOOOOOE+OO III 8 XM= '.l.III1SE+OO Fl(:: 1, 779037E ... 0 1 FXP= 1, 77Ci1037E+O t FXN= O,OOOOOOE+OO I: q ltM= 5,OeO~+ol) FX:: 1, 7H0371:.+0 1 FXP= 1,779037E"'Ol FXN= O.OOOOOOE+OO I= 10 )(1'1= 5.715E+OO FX= 1.779037E+Ol FXP= 1. 779037E+0 1 FlCN: O,OOOOOOE+OO I= 11 XM:: tI,350EtOO F":: 1.1790.571:.+01 F)(P:: l,779037E"'Ul FXN:: O,OOOOOOE+OO
FULFLX= FULFLX= FULFLX= FUL.FLX= FULFl.X= F"UL.FL.X= FULFLX:: FUL.FLX= F"UL.FLX= FULFLX: FUI.FLX::
Table3.12Convective Fluxes Through the Combustion Chamber Crossections
7,b14280E w01 7,b87377E-Ol 7.72bS1bE-Ol 7.7782741::-01 7.790212E-Ol 7. 4bHbSE-0 1 1.S370bbE-01 9.8oS331E-02 7.b75099e:-02 S.9S0782E-02 4.519Q4bE"02
SECTION 4 PARAMETRIC STUDIES OF INJECTOR ANOMALIES
4.1 OBJECTIVE
The objective of the present study is to demonstrate a capability to simulate
two-phase spray flow, heat transfer, evaporation and combustion within the rocket engines for predicting the effects of reactant injections nonuniformity
upon local and overall heat transfer. For this purpose, six test cases have
been considered. The basic test case has been set up based on the data specified by NASA. First, four model sensitivity tests were performed to
examine the influence of physical and numerical parameters on the fluid flow
and combustion .. The final sixth test case considered an injection nonuniformity,
in which 25% of the central part of the fuel "injector had been blocked.
All results ay'e presented in graphical form.
4.2 THE COMBUSTION CHAMBER AND INJECTOR GEOMETRY
The configuration of the model combustion chamber employed in the present calculations is shown in Figure 4.1. The combustion chamber is O.33m long. Chamber diameter at the injection plane is d = 121.92mm and at the throat o dt = 66.04mm. Contraction ratio is approximately 4.
Figure 4.2 presents details of the fuel and oxidizer injector plane. Fuel is
supplied to the injector through the fuel connector (marked FUL on figure 4.2) and ax'ially enters the triplet injectors. Liquid oxygen enters radially (connector OX on Figure 4.2) and is distributed through the annular collector to all ;oxygentriplet inlets.
Detail C on Figure 4.2 presents the LOX-RPI-LOX triplet. The triplet
arrangement within the injector plane is also shown in Figure 4.2. Note that due to the specific hole arrangement the injection plane has two symmetry 1 ines. Therefore, a 900 sector has been used for the numerical simulation studies.
The grid arrangement for the 900 sector of the combustor is shown in Figure 4.3.
A uniform nonorthogonal grid in x-axial, y-radial and z(=8) circumferential
directions with NX:NY:NZ - (21:8:4) "is employed for all test cases. Figure 4.3 presents the injector hole distribution and the y-8 grid.
The calculation domain in the axial direction extends up to the throat section of the nozzle.
4-1
I
t~
g ~---:!LY .,
4-2
.~
_l
~ +-> OJ
5 OJ
C!)
OJ r-N N o
Z
. «:t
OJ s.... ::s 0'1
u..
C06<2"9"Y 7
;7;'/ O~LL X THRU C.ORILL .2..50 0'''' OtPT... '.9~ j' n HOL"'!>
(su .. ~~~~~
"'-:J.g!> -
ox
·--Hi !--.~2:2. / I /
.as -11
// .196
I ./
"r-Drn_l Y ~RU TO / 2SO O~ -:::"_E.
("74 HCU'.:' ~,:)T"''-)
~·RU \
r---E3:;'±;:!-:ff-----L-L..---_ -_J.~___,t_ q:;a liP'
.6450 REF
>--~\ __ 3 __ _
O&.TNL. C: ~\
o.ooc
.9S0
1.650
··-Z-.:'10
1.£oSO i
c..~O
Figure 4.2 LOX-RPI-LOX Injector geometry and Injection . triplet arrangement,(X = 1.70~mm,'Y = 1.85 mm)
4-3
+:> I
+:>
y
Nyt
8 7 6 5 4
3 2
I z 1
NX -x
K=4
2 3 4 5
Injector Plane 4
7
5
4
3
K=1 2
1-1
-r
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Figure 4.3 Grid Arrangement for the Rocket Injector Anomalies Study.
4.3 BASIC TEST CASE SPECIFICATION
A fully coupled two-phase spray flow, evaporation heat transfer and combustion with a uniform reactant injection distribution has been considered. Table
4.1 presents a summary of input data specifications for the basic test case. It includes:
geometrical data,
- physical parameters, and
injector data.
During the calculations it has been assumed that oxygen enters the combustion
chamber in a fully vaporized form, while kerosene (RPl) enters in the form of discrete jets of the liquid fuel sprays. To describe the gas phase motion
(oxygen + combustion products) Eulerian frame of coordinates has been used. The liquid phase (fuel) is described in Lagrangian coordinates. The spray is represented by 80 individual droplet parcels (16 slot and 5 droplet
diameters). The liquid spray jet, at each injection slot, is represented by 5-·initial
droplet sizes· Do.
Figure 4.4 presents assumed droplet distribution function and selected
droplet diameters (DI!D = 5, 10, 20, 30, 40 and 60]Jm).
40
35
30
25
dm - 20 m·
[%J 15
10
5
0 0 10 20 30 40 50 60 70 80
Do [11 m J
Figure 4.q Droplet Distribution Function
4-5
Table 4.1 Basic Test Case Data Specification for Combustion in Rocket Engine Calculations
SPECX FI CA TI ON
Chamber radius at injection plane Chambey' radius at the throat Chamber length (to the throat) Injector type UNLIKE TRIPLET Total number of triplets Number of fuel holes in 900 sector Number of oxygen holes in 900 sector Number of grids (NX:NY:NZ)
Total fuel flBw rate ,through 90 sector
Total oxygen flow rate through 90° sector
Fuel type (Cn.96H23.23) Fuel molecular weight Fuel boiling temperature (RP1) Heat of Vaporization (RP1)
Heat of Combustion (RP1)
Liquid Fuel Inlet Temperature
Density of Liquid Fuel at 300 oK Laminar Vapor Diffusion Coefficient Vapor Specific Heat
Oxygen Inlet Temperature (assumed vapor oxygen temperature)
Operating Pressure
Droplet Median Diameter (for distribution function see Figure 4.5)
4·-6
DATA
60.36 33.02
330 284
37 9.25 19.0
21:8:4
4.632 (0.525) 13.924 (1.578)
RPI 167 487 56.5 236700
(10457)
60 (288.6)
800 1.5 . 10-4
2000
150(83) 150
41. 37 . 105 (600)
27
UNIT
mm mm mm
----------
1 bm/sec kg/sec
1 bm/sec kg/sec
--g/mol oK
keal/kg J/kg
( kca 1/ kg)
of (OK)
kg/m3
kg/ms J/kg K
R (K) K
N/m2 (psia)
11m
The mass-median droplet diameter of the droplet distribution function
(Dpm ~ 20-3~m) for the triplet injector has been selected based on experimental
data of FANG (Reference 1.). F6r jet diameter 1.85 mm the approximate Dpm is 27um (See Figure 4.5). All dr6plets are assumed to be injected axailly from the
liquid fuel nozzl~.
4.4 TEST CASES FOR THE PARAMETRIC EVALUATION.
The intent of the model sensitivity study was to asses the behavior of the
model when flow conditions, empirical factors and numerical parameters were
altered about selected mean values: Under the current contract this task was
limited to five test cases. The selected five test cases included variation in:
- droplet evaporation rate formula,
- vapor fuel reaction rate formula,
- droplet diameter distribution function, and
- liquid fuel injection distribution.
Table 4.2 presents the basic test case (TO) default parameters and five
test cases (Tl to T5) parameter specification summary. Note that in test
cases 11 - T5 only one parameter at a time has been altered; the remaining
parameters were sp~!cified as in the basic t(~st TO. In all cases a 21: 8: 4
grid has been used.
,------,_._-_.-TEST TEST C
- -
NtfMBER
TO Base c
T1 Double T2 Halved T3 Increa T4 Al tere
but ion T5 Blocke
near a
/ Table 4.2 Test Cases for Parametric Evaluation
. ASE DESCRIPTION o IFF. COEF REACTION RATE
(
III ::s
.r-'0 ro
0::
0-0 S-o c ro ..... '0 OJ
::E I
III III ro
::E
10000
1000
100
I.~ 4
Jet Diameter, ern
Figure 415 DROP SIZE DETERMINED FROM EXPERIMENTAL .LOX/HEPTANE ENGINE PERFORMANCE (REFERENCE 1)
4-A
"t,
10
Influe!,!ceof Diffusion CoeffiCient in the Evaporation'Rate ForrtJula
First two test cases Tl and T2 were designed to investigate the influence of the vapor diffusion coefficient of 0 on the rate of evaporation of the liquid droplets. The evaporation rate formula is exressed as:
where ~'d (T.:. Bat) B = --~L---L
and in the basic test case, the diffusion coefficients 0 has been estimated -6 2 ( ) to be: 10 m /s see reference 2 page 204. The pO is obtained as:
(4.1)
pO :: P·M 0 ~ 41. x 105.196 10-6 ~ (1.+ 2.) x 10-4 l9. (4.2)
. 8314'Tsat ' = 8314·600 . ms
( ) -4 kg In the basic test case TO value of pO equal to 1.5'10 ms has been used.
For model sensitivity studi es, thi s value has been respectively doubl ed (Test Case Tl) and reduced to half (Test Case T2).
Influence of the Reaction Rate Constant
The two step combustion reaction rate scheme employed in the model assumes Arrhen'ius reaction rate expressions in the fo"llowing form:
where a fu' bfu ' aco ' bco ' ceo = 1 and
Efu = 18000 (cal/g)
E = 1860 co
(4.3)
(4.4)
The fuel reaction constant Pfu in the reaction mechanism is less reliable. At the same time this reaction provides most of the hear release. In the present study a test case with P fu increased to 1.1012 (two order increment) has been considered to study changes in the flame structur.
4--9
Influen~e of the Droplet Distribution Function
Liquid fuel injected into the combust-jon chamber through the axial hole of the "triplet LOX-RPI-LOX" injector is atomized and sprayed in the form of droplets
of di fferent d'iameters. Exact specif-ication of the spray characteristics for the chamber operating conditions is very difficult. Approximate
specification of the droplet diameter distribution function at the injection location requires knowledge of median droplet diameter (Dm), and minimum and
maximun diameters (Dmin , Dmax )' Experimental data by Fang [11 indicate that the median droplet diameter for the triplet injector and RPI fuel can be taken equal to 27 urn. The maximum droplet diameter is estimated as 80 -120 urn.
Figure 4.6 presents two assumed droplet distribution functions at the injection point for basic test case T9 (sol id 1 ine) and for test
case T4 (dotted line). In both cases median droplet diameter is similar. In the test case T4 maximum droplet diameter has been reduced and number
of smaller diameter droplets (15 to 30um) increased.
Table 4.3 presents discrete droplet distribution function (D - droplet o
diametev' and P··dropl et population number) for both test cases.
1\ 30 \
I \ -- Basic (TO) dm
/ - 20 -- - Alternative (T4) m
10 ~ 15
«} --..Ii.-..;.;.JI..-....;..J_....;..J_..;.. ! ID _ 10 20 30 ~--:":".-;.;...J::::....::t:: .. __ _
40 50 80 70 80 Do [pm 1
Figure 4.6 Droplet Distribution Functions at the Inlet (Injection) to the Chamber for Test Cases TO - T4.
4-10
Table 4.3 Droplet Distribution Function For Basic Test Case TO and Test Case T4.
TO - TEST CASE T4 - TEST CASE \
Do P 0 P 0
11m 11m -
5 20 5 10
10 35 10 30
20 25 20 40
40 15 35 15
60 5 50 5
Infl uence 0 f ~!ljec tion Nonu niform i ty"
The injE!ctor
following:
a)
b)
c)
nonuniformity, in general, would require specification of the
location of partially or' fully blocked fuel holes,
location of partially or fully blocked oxygen holes, and
percentages of blockages.
--
--
--
During the present study, only one test case (T5) of the injection nonuniformity has
been considered for the computations.
It has been assumed that 25% of the fuel holes have been blocked. The blocke!d
holes are located symmetrically near' the chamber axis so the calculation
domain stays unchanged as in the basic case TO.
Figure 4.7 presents the injector hole arrangement, fuel holes blockage
and (y-8) grid distribution at the injector plane. The blocked fuel entryaY'ea
comprises 24.3% of the nominal fuel entry area. The fuel mass flow rate through open
holes has increased so as the total flow rate is maintained to be the same
as in basic test case TO.
4.5 RESULTS OF BASE TEST CASE
Presentation of the Results
Figure 4.8 presents typical velocity and pressure contours on an axial (XV) .-
plane of the combustor. Figures 4.9 and 4.10 show contours of absolute temperature
(in OK) and contours of vaporized fue"j concentrations in all four (X-V) planes
4-11
Blocl(ed Fuel Holes .c-, __
INJECTOR PLANE
Figure 4.7 Injector Plane with Blocked Fuel Holes Near the Axis of the Chamber (Test Case T6). Total No. of Holes in 900 Sector: 91 Blocked No. of Holes 21 % Blockage 24.3%
4-12
.;::. I
....... W
RO INJ ANAL, BASIC 21-XGRID ~<'.. PLANE 1
t}ELOCITV PLOTS ,-lELMIN l·)ELMAX
1.856E+00 4. 436E+02
------. ---- --------------------.,..".,- .---. ,-... ~ .---~ ~
R0 INJ ANAL, BASIC 21-XGRID :"<V PLANE 3 PRES CONTOURS
PHIMIN -6.57~E+05 PHIMAX 9.601E+03 CONTOUR LEVELS
1 -5.964E+05 2 -S.358E+05 3 -4.752E+05 4 -4.146E+05 5 -3.540E+05 6 -2.934E+05 7 -2. 328E+05 8 -1. 722E+05 9 -1.116E+05
10 -5.099E+04
Figure 4.8 Velocity Vectors and Pressure Contour? on an Axial Plane (XV) of the Combustor. (Basic Test Case Tl).
I
:,\, PLANE 1 TEMP CONTOURS
PHIMIN 1.631E+02 PHIMAX 2.S23E+03 CONTOUR LEVELS
1 o4.253E+02 ;3 6. 8704E+02 3 9. 496E+02 4 1.212E+03 5 1.o474E+03 6 1.736E+03 7 1.998E+03 8 2. 260E+03
;" .... PLANE 2 TEMP CONTOURS
PHIMIN 1.738E+02 PHIMAX 2.513E+03 CONTOUR LEVELS
1 4. 337E+02 2 6. 937E+02 3 9.536E+02 4 1.213E+03 5 1. 473E+03 6 1.733E+03 7 1.993E+03 8 2. 253E+03
f' :<V PLANE 3 ~ TEMP CONTOURS ~ PHIMIN 1.728E+02
PHIMAX 2.527E+03 CONTOUR LEVELS
1 4.344E+02 2 6.960E+02 3 9.575E+02 4 1.219E+03 .5 1.481E+03 6 1.742E+03 7 2. 004E+03 8 2.265E+03
XV PLANE 4 TEMP CONTOURS
PHIMIN 1.660E+02 PHIMAX 2. 525E+03 CONTOUR LEVELS
RO INJ ANAL,B~SIC 21-XGRID
1 4.280E+02 )1 2 6.901E+02 3 9. 522E+02 4 1. 214E+03 5 1.476E+03 6 1. 738E+03 7 2. 000E+03 8 2.263E+03
Figure 4.9 Temperature Cantors in Axial (XV) Planes; Basic
Test Case (Tl).
~ I I-' (]1
:<'r' PLANE 1 FUEL CONTOURS
PHIMIN 3. 650E-07 PHIMAX 3.268E-01 CONTOUR LEVELS
1 3.632E-02 2 7. 263E-02 3 1.egeE~el
4 1.453E-01 5 1.816E-01 6 2. 179E-01 7 2.542E-01 8 2. 905E-01
:,s PLANE 2 FUEL CONTOURS
PHIMIl'l 7. 712E-08 PH If'lAX 4. 333E-01 CONTOUR LEVELS
1 4.815E-02 2 9.630E-02 3 1.444E-01 4 1.926E-01 5 2. 407E-01 6 2. 889E-01 7 3.370E-01 8 3.852E-01
;-<'\' PLANE 3 FUEL CONTOURS
PHIMIN 4.956E-08 PHIMAX 5. 496E-01 CONTOUR LEVELS
1 6. 106E-02 2 1.221E-01 3 1. 832E-01 4 2. 442E-01 5 3. 053E-01 6 3.664E-01 7 4.274E-01 8 4. 885E-01
:<\' PLANE 4 FUEL CONTOURS
PHIMIN 8.417E-08 PHIMAX 3. 121E-01 CONTOUR LEVELS
1 3. 468E-02 2 6.935E-02 3 1.040E-01 4 1.387E-01 5 1. 734E -211 6 2.081E-01 7 2. 427E-01 8 2. 774E-01
RO INJ ANAL, BASIC 21-XGRID
I ¥
- -----I m:
,. t
Figure 4.10 Contours of Gaseous Fuel Mass Fraction in XY Planes. (Basic Case Tl).
of the combustor. Figure 4.11 presents fuel concentrations at three select€!d cross-sections of the combustor. Figure 4.12 shows the distribution of the convective heat transfer coefficient on one (X -Z) plane along the
combustor wan (developed cylindrical surface). Figure 4.13 presents the
heat tr'ansfer coeffic ients variation along the cyl indrical wall for each e - plane. Some of the results of the basic test case TO, such as total
gaseous fuel flow rate, liquid spray rate, etc, are presented and discussed in
the section describing parametric evaluation.
Discussion of the Results --"--" "-
The flow field represented by the velocity vecotor and pressure contours
is typical of nozzle flows. Largest pressure gradient exists near the combustion throat, and the lowest pressure is also located in the same region.
In the front of the combustor, the gas velocities are relatively low. At a distance approximately equal to the chamber radius, a large increase in velocities takes place. At that distance the flame fronts of the individual triplet injectors are joined and large temperature gradients occur (see figure 4.9).
Figure 4.9 indicates that the flame front is nonuniform in both circumferential and axial directions. Temperature rise takes place in shorter axial distance at mid-,-radii than at small radii (i.e. near the axis). Note that at 2nd and 3rd XV - plane near the chamber wal'l there is a recirculation region carring hot combustion products towards the injection plane.
The lar'gest fuel concentration grad'ients exi st in the front part of the chambeY1 where droplets enter hot combustion zone and intensive evaporation takes place. The smallest droplets evaporate in a short distance after entering the reaction zone creating a region of significant interphase mass transfer. The largest
droplets penetrate further into the chamber and create a zone of reacting vapor fuel along the trajectory. F'igure 4.14- presents qual itative comparison between the experimental results (photograph of s i n91 e tripl et fl arne [1J and carcul ated gaseous fuel concentration profiles at K = 2 plane. In both diagrams (Figure 4.14), an enlongated flame shape can be seen. Note that the photograph represents on"ly a IIhot 1 uminous ll zone where unburned hydrocarbon creates a luminous zone. No conclusion can be drawn regarding high temperature zone from this photograph. Figures 4.10 and 4.11 present the gaseous fuel mass fraction
4-1n
.j:::> I ~
'-J
\'= PLANE 2 FUEL CONTOURS
PHIMIN 9. 489E-e3 PHIMAX 5. 496E-e1 CONTOUR LEVELS
1 6.95eE-132 2 1. 295E-01 3 1. 89SE-e1 4 2.49SE-e1 s 3.139SE-el 6 3.695E-el
4.29SE-e1 4.89SE-131
\'Z PLANE S FUEL CONTOURS
PHIMIN 4. 739E-e5 PHIMAX 2. 176E-131 CONTOUR LEVELS
1 2. 423E-e2 <= 4.840E-e2 3 7.2S8E-e2 4 9. 676E-e2 5 1.2e9E-e1
1~451E-el 1.693E-131 1. 935E-e1
., 2 .5 4 S £, 7 ag,o II I, 1.3 14 IS 16 '7 Ie. ,g ~o .zt RO INJ ANAL~ BASIC 21-XGRID
Figure 4.11 Contours of Gaseous Fuel Mass Fraction in YZ-Planes. (Basic Case Tl).
\'Z PLANE 113 FUEL CONTOURS
PHIMIN 1.3geE-e7 PHIMAX 1.seeE-el CONTOUR LEVELS
1 1.667E-e2 3. 333E-!.a2 s.ee13E-e2 6. 667E-e2 8.334E-e2 1.eeeE-e1 1.167E-e1 1.333E-01
.j:::. I
f-'
co
RO INJ ANAL, BASIC 21-XGRID XZ PLANE S GAMN CONTOURS
PHIMIN S.862E+01 PHIMAX 3. 762E+03 CONTOUR LEVELS
1 4.702E+02 2 8.817E+02 3 1.293E+03 4 1.70SE+03 5 2. 116E+03 6 2.S28E+03 7 2.939E+03 8 3.3S1E+03
3
K=4
I njector Plane
~
'" .... ~
.Zr '"- ~2 ~ 'F'-'-F . ---1-.I..--~---LJ1 I r~r p --- --------~ "'-~ It
2
X -t+>\IA-L
Figure 4.12 Heat Transfer Coefficient on the Cylin~rical Wall (Development of X-Z.Plan~). Case Tl
q" (BTU/ft2h)
-+::> x 106 I
....... \.0
8
7/ 6
51
4
I
3
I 2
4
~
~
K=2
K=3
\ K=l
LEGEND: /:'. K=l o K=2 X K=3 o K=4
o I t=: a0 u// I .. 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-1
Figure 4.13 Distribution of convedtive heat fluxes along the cylindrical wall.
X/L
ox
FUEL
ox
HIGH-SPEED PHOTOGRAPH OF OFO TRIPLET CARBON FORMA TI ON PHENOMENA, TEST 116
• t
------------------
-
4.14 Comparison between predicted fuel mass function contours and
experi ta 1 fl arne photograph.
4-20
contours at different cross-sections of the combustor. Long wakes of the
unburned vapor fuel are generated along the 1 iquid jet trajectories (see figure
4.10). In Figure 4.11, the streams of unburned vapor fuel are represented
by conc~!ntric semicircles.
Distribution of heat transfer coefficients r at the cyl indrical wall is
shown in FigurE~ 4.12 (contour map) and in Figure 4. 13 (distribution curves).
From Fi!Jure 4.13, it is seen that the heat transfer coefficient is increasing
with the distance, with the maximum being near the throat region. Similar
observation has been reported in experiments. Note that r is quite nonuniform
in the circumferential direction, especially in the upstream part of the
chamber. The heat transfer curves at k = 2, 3, and 4 are quite similar. At
k = 1, howev~r, heat transfer coefficients are significantly smaller (see also
Figure 4.12).
The outline of the grid and injection hole arrangement, shown in Figure 4J2"
reveals that as compared with k = 2, 3, and 4 in the k = 1 plane, oxygen
holes are at the shortest distance from the wall. This results in lower
velocity and temperature (cool ing effect) and lower heat transfer coefficient r: The cooling effect may have been overpredicted due to grid coarseness at
the large radii. Additionally, the influence of the oxygen jet is "averaged"
(or smothered) over the entire grid volume. For better resolution, calculat"ions
with finer grid or alternative injection specificat"ion would be required.
4.6 RESUU:~:; OF PARAMETRIC STUDlli
TEST nand T2: Influence of Diffusion Coefficient 0 in Evaporation Formula
The evaporation rate dD/dt is proportional to the diffusion coefficient 0
(see Equation 4.1). Assumed variants in the diffusion coefficient, viz: -4 n pO = 3.0x 10 kg/ms -4 TO pO = 1.5x 10 kg/ms -4 T2 p:O=0.75xl0 kg.ms
is approximately equal to the limits :of uncertainty for this factor.
Figure 4.15 presents the axial variations of liquid fuel flow rate calculated
for test cases: TO, n, and T2. The results show that:
a) larger 0 creates higher evaporation rates and shorter penetration
of liquid fuel jets;
4-21
b) in case T1, liquid is totally evaporated within 3/4 of the chamber
ll~ngth;
c) in the other two cases, only droplets with the largest initial
diameter Do can be found at the throat. The liquid fuel flow
rates at the exit are:
for T1 rfILiq = O. (i.e. 0% of rfIL' lq. inj) ;
for TO -4 (i.e. 0.01% of rfIL. i nj) ; rfIL' = 3.10 lq 1 q.
for T2 -3 (i. e. 0.7% of rfIL' i nj ) • IflLiq = ,4.10 1 q.
Detailed flow patterns indicate no major differences between the flow field
and heat transfer in all three cases.
Test T3: Infl uence of the Fuel React; on Rate Constant PFU
Figure 4.16 presents the axial variations of gaseous fuel flow rate for two
different fuel reaction rate constants; viz:
Case TO with PFU = 1010 kg/m3s ; and
Case T3 with PFU = 1012 kg/m3s.
In the case of larger PFU (Case T3) vapor fuel flow rates in the front part
of the chamber are singificantly smaller that those in the basic case TO. From
temperature contours (Figures 4.9 and 4.17) it can be seen that in the case
of larger PFU ' the flame front is located closer to the injector plane. This
seems physically plausible.
Test T4: Infulence of Alternative Droplet Distribution Function
By changing the droplet distribution, there are no significant changes in the
calculated flow field and heat transfer characteristics. In Test T4, flame
front ·is slightly closer to the injector plane. Inspection of the droplet
distribution functions (Figure 4.6) reveals that in Test Case T4 there is a
smallel' amount of the finest drops (O-101lm) and therefore less vapor fuel
will b(~ evaporated in the front part of the combustor. This probably creates
mixture conditions in Case T4 closer to the stoichiometric than that in base
case TO.
Test T5: Influence of Injector Nonuniformity {Blocked Fuel Holes Near Axist
Figure 4.7 prE~sents injector plane with four blocked fuel holes in the 900
sector. Blocked fuel entry area is 24.3% of the nominal fuel entry area. As
4-22
0.5 ... -------- ° u 3.0 x 10-4 [ kg J (T1) o ms
0.4 - 00
= 1.5x 10~4 (TO)
--,--- Do = 0.75 x 10-4 IT2) mL1Cl 0.3
['!9. 1 s
0.2
0.1
0.0 '-_.L.._..r:....:::::~. ±= -b 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X/L
4.15 Axial Variation of Liquid Fuel Flow Rate (for 450 Sector of the Chamber); for (Tests TO, T1, and T2).
0.2
MFU
[ _~!l) !;
0.1
0.0 [ -
0.0 0.1 0.2 0.3 0.4
---- PFU = 1010 [~J (TO) ms
IT3)
0.5 0.6 0.7 0.8 0.9 X/L
..J 1.0
4.16 Axial variation of Gaseous Fuel Flow Rate (for 450 Sector of the Chamber); for (Tests TO and T3).
4-23
+=> I N +=>
TEMP CONTOURS RLIINJAN PREXP < FU) INCRESAED :';v PLANE 1 -
PHIMIN 1.873E+02 r-::::J~~r===============~---------------------------------~~~~~~R t.~~~~E+03 JUJJlr 1 4.476E+02 2 7.079E+02 3 9.682E+02 4 1.228E+03 5 1.4S9E+03 6 1.749E+03 7 2. 009E+03 8 2. 270E+03
XV PLANE 2 TEMP CONTOURS r-.;;:;------------------------------------------------------------
PHIMIN 2.02SE+02 I Ri\ ~--~ PHIMAX 2.514E+03 CONTOUR LEVELS
1 4.596E+02 2 7. 164E+02 3 9.732E+02 4 1. 230E+03 5 1.o487E+03 6 1.704o4E+03 7 2.001E+03 8 2.2S7E+03
X'{ PLANE 3 TEMP CONTOURS
PHIMIN 1.987E+02 PHIMAX 2. 525E+03 CONTOUR LEVELS
1 4.572E+02 2 7. 156E+02 3 9.741E+02 4 1.233E+03 5 1.o491E+03 6 1.7049E+03 7 2. 008E+03 8 2.266E+03
~y PLANE 4 TEMP CONTOURS
PHIMIN 1.810E+02 PH I MAX 2. 529E+03 CONTOUR LEVELS
1 4. 420E+02 2 7. 029E+02 3 9.639E+02 4 1.225E+03 5 1.o486E+03 6 1.7047E+03 7 2. 008E+03 8 2.269E+03
Figure 4~17 Fuel Mass Fraction Contours Test Case T4-Higher Fuel Reaction Rate Constant.
-t=:> I N (J"l
XV PLANE 1 TEMP CONTOURS
PHIM!N 1.71SE+0C! PHIMAX 2. 523E+03 CONTOUR LEVELS
1 4.328E+02 2 6. 940E+02 3 9. 552E+02 4 1.216E+03 5 1.478E+03 6 1.739E+03 7 2. 000E+03 8 2.26lE+03
x:Y PLANE 2 TEMP CONTOURS
PHIMIN 1. 783E+02 PHIMAX 2.513E+03 CONTOUR LEVELS
1 4. 377E+02 2 6.971E+02 3 9. 565E..,02 -4 1. 216E+03 5 1.4?SE..,03 6 1.735E..,03 7 1. 994E+03 8 2. 254E..,03
XV PLANE 3 TEMP CONTOURS
PHIMIN 1. 765E+02 PHIMAX 2.S28E+03 CONTOUR tEJELS
1 4. 378E+02 2 6.991E+02 3 9. 603E+02 4 1. 222E+03 5 1. 483E+03 6 i.744E+03 7 2. 005E+03 8 2. 267E+03
XV PLANE 4 TEMP CONTOURS
PHIMIN 1.689E+02 PHIMAX 2.517E+03 CONTOUR LEVELS
1 4. 298E+02
ROINJAN#BLOCKED 25% FUEL HOLES
2 6.907<+02 I ~ -----3 9.516£+02 • I -4 1. 212E+03 5 1. 473E+03 6 1.734E+03 7 1.995E+03 - - _ : 8 2.256E+03
Figure 4.18 Temperature Contours in XV-Planes. Test Case T5- Near Axis Blockage of 25% of the Fuel Entry.
~ I
N C)
RINJAN BLOCED 25% FUEL HOLES XZ PLANE 8 GAMN CONTOURS
PHIMIN 5.S15E+01 PHIMAX 3. 78SE+03 CONTOUR LEVELS
1 4.697E+02 2 8.842E+02 3 1.299E+03 4 1.713E+03 5 2. 128E+03 6 2.542E+03 7 2.957E+03 8 3.371E+03
z (e)
K=4
I njector Plane
I -1
~ x _ axial - - - I""
Figure 4e19 Heat Transfer Coefficient on the cylindical Wall Test Case T6-
Blocked 25% of the Fuel Entry.
compared to Cases T1 to T4, this test case (T5) shows much more singificant
differences in the heat transfer and combustion within the chamber. Figure
4.18 presents the temperature contours (oK) in four axial plans. The central,
oxygen rich core is significantly cooler than the annular, fuel rich, zone.
Steep gradients in the radial direction in the middle of the combustor
represent the enlongated flame front.
Comparison of Figure 4.12 and 4.19 indicates that convective heat transfer
coeffic'ients at the combustor wall are nearly the same for the basic case TO
and blocked fUE!l entry case T5. This is due to the "blanketing" of the cylindrical
wall by the hot reactive streams of operating fuel spray jets. An equivalent
blockagE! near the combustor wall, rather than near the axis, is likely
to have a much larger influence on the "wall" heat transfer coefficients.
Further parametric studies, were not included in the current work scope, and are
recommended fOl~ future.
4-27
SECTION 5
MODEL IMPROVEMENTS AND RESEARCH NEEDS
The applications reported in Sections 3 and 4 of this report have demonstrated the bask capabil ity of simulating three-dimensional, two;;.;phase flows with
evaporation and combustion. The use of Langrangian technique for liquid fuel drops has proviided necessary fl exi!)il ity of predicting effects of small
changes in injector plate ,geometry and/or flow conditions. Results of parametric studies have shown that the numerical model responds to changes in phys'ical pat'ameters in a plausible manner, and thus it can be util ized for the identification of sensitive parameters.
Several improvE!ments and studies can and should be made for the code to be
a valuable tool! in design and performance analysis of rocket engines. The improvements are desirable for both mathematical models of physical processes
built into the code as well as for numerical methods.and solution algorithm employed. The code also requires verification and sensitivity study for both
single and two-phase flows with and without mass transfer and combustion.
The fol'Iowing three subsections describe recommendations for (a) physical model improvememts; (b) numerical model improvements and (c) code val idation and verification studies.
5.1 IMPROVEMENTS IN MATHEMATICAL MODELS OF PHYSICAL PROCESSES
There are several possible improvements in simulating the spray flow, evaporation and combustion processes implemented in current version of the code. Further refinements can be made in simulating the injector and spray atomization. This section briefly describes basic ideas of selected possible improvements.
Spray A~omizat_ion in Triplet Injector~
The unlike impinging elements accomplish mixing and atomization by direct
impingement of fuel and oxidizer jets. Atomization takes place in the immediate vicinity of the impingement point. The effect of oxygen jet impingement on the central fw:!l stream scatters the fuel stream in the direction normal to the injection plane (Figure 5.1).
5-1
(a) Inner and outer jets momentum ba lanced
o OUTER ORifiCE --OUTER ORifiCE flUX
• INNER ORifiCE _ INNER ORifiCE flUX
(b) Outer st r(~am momentum much greater than inner stream momentum
Figure 5.1 Triplet Injector Spray Mass Distribution
Experimental data exists (see Review in [3]') for specifying a range of droplet
beams at the atomization point. Such information should be utilized in future
studies and incorporated in if necessary in the code for regular use in analysis
of injector anomalies.
Turbul e!lt Drop'! et DiffuSion
In the present version of the code, turbulent droplet diffusion has been
neglected. This must be improved for future studies. The task of including
turbulent particle diffusion into the model require~ investigations of various
possible approaches. One diffulty is that the fluid properties are constant
within a cell, while the partic,les are tracked through a cell in a series of
time steps srnalller than the c-elldimensions. -Thus the particles at several time
step positions within a cell see only one set of mean flow properties.
There are two basic approximate techniques for simulating the turbulent
droplet diffusion.
a) Y'andom walk method (Dukowicz [4J ) ; and
b) diffusive drift method of Jurewicz[5J and Stock [6J.
In both techniiques, an isotropic turbulence is assumed. The second technique
seems to be more adequate for "fully coupled" calculations of spray combustion
modeling, and is recommended for inclusion in REFLAN3D-Spray code.
Accurate Eva'!lUati on of Thermod.vnam"J c and Transport Properti es
In the present version of the code, most of the liquid fuel properties
(C l' ,T t' Pl' ,etc) have been assumed to be constant. More accurate p lq sa lq representation would require sepcification of their functional dependence
on local pressure and temperature.
5-2
The gaSE!OUS species specific heat formula coefficients (5-order polynomial)
are valid for the "high temperature range" between 500-3000ok. For the lowtemperature liquid propellants, two sets of specific heat constants are dpsired.
5.2 IMPROVEMENTS IN NUMERICAL METHOD AND SOLUTION ALGORITHM
The computational process of the coupled Eulerian-Lagrangian analysis is highly
nonlinear. In the present calculations, no special relaxation or linerization of interphase mass, momentum and energy transfer source terms has been undertaken.
Calculations with (11 x 8 x 4) and (21 x 8 x 4) grid caused no numerical
instabilities. However, trial runs with finer grids in the circumferential
direction (21 )( 8 x 8) showed slow convergence and oscillatory behavior. These difficulties must be investigated and remedied before the code can be regarded suitable for studies of injector anomalies.
Relaxation Practices
Investigations of the relaxation practices and line~rization methods are desired.
For the two-phase flow with mass transfer, possible relaxation practices include: a) rE!l axation of the interphase mass transfer rate ;
b) rE!laxation of the average liquid property ¢liq in the transfer rate expression;
c) relaxation of velocities and enthalpies; and d) combination of a, b, and c.
Accuracy' Improvements
The accuracy of the three dimensional calculations in the Eulerian frame can be improved by refining the grid in the region of interest and/o·r by using higher order differencing methods.
The usefulness of the first practice is often limited by the computer storage,
especia"'ly for the reactive flow calculations where twelve or more differential equations must be solved.
The second approach is higher order f1nite differencing and has attracted more attention in the recent publ ications :[7,8,9]. One of the most promising
methods of the accuracy improvement has been developed by the authors [10,11}. The new method called "Multiflux Conservative Differencing" (MCD?, has been successfully applied in a 2-d~mensional calculation. Results obtained with the coarse grids 10 x 10 and 20 x 20 compare very well with resul ts on 50 x 50 or finE!r grids obtained with the other (e.g. upwind differencing) methods. It is recommended that: (a) MCD be incorporated in REFLAN3D-Spray Code, and (b) comparative studies be performed to evaluate the improvements dlle to MCD.
5-3
The second part of the computational algorithm (Lagrangian part) is relatively
new in comparison with the Eulerian part and the computational experience
is still very limited. Some basic studies in the area of Eulerian-Lagrangian
approach would greatly enhance the computational fluid dynamics. New solution
schemes for the homogenous and heterogenous combustion and fluid flow
calculations could be explored.
For th(~ rocket engine calculations, the first improvement should probably D'e miilde
in th~ calculations of the interphase mass source. In the present version of
the code, the source terms are calculated at the grid cells through which
the particles are passing (see Figure 5.2a).·
eN eN
a) CURRENT b) PROPOSED
Figure 5.2 Lagrangian Interphase Source Term Calculations for
Eulerian Transport Equations.
Therefore, the mass release is "lumped" in the souy'ce at point "P" with no
source for point N. In real ity, howE~ver, the droplet trajectory represents
an average dropl et path and one cou·1 d expect to have the mass transfer to
the N cell as well. The practice in Figure 5.2b presents an alternate
approach for the source term calculations around the average particle path.
The mass transfer from the 1 iquid to the gaseous ptlase will be distributed to
both Nand P points based on
a) distance weighting factors or,
b) volume weighting factors.
With this practice, implementation of the droplet turbulent diffusion models
would also be more realistic and economical. Simple test cases on a 2-dimensional
grid are recommended as the first step. 5-4
Improveri)ents' iii' the' Notiorthogotia 1 . Gri d . sys tern
The nonorthogonal grid system and associated velocity components in the present version of the code are shown in figure 5.3a. The orthogonal velocity components, u and v are used on the nonorthogonal grid. Within this practice it has been assumed that the up velocity is driven by the (pp - pw) pressurr:! gradiE~nt. This assumption is valid only if the velocity components are aligned with the grid lines, or the degree of departure from this condition is small (say ~: 300
). A new practice is proposed in which the grid lines and velocity resolutes (nonorthogonal) are used figure (5.3b). Most of the derivat"j ons for prel imi nary testi ng have be~n done under CHAW s in-hou~e development project. Its implementation and numerical test in REFLAN30-SPRAY are recommended to enhance the code capability and accuracy .
. -t--
.-t--•
CURRENT PROPOSED
Figure 5.3 A Nonorthogonal Grid Systems
Modification of the Code Structure
In the present version of the REFLAN3D Code, COMMON statements are used to transfer the vari abl es between subrouti nes themsel VE~S.
A more economical method of passing data to a subprogram is through the argument list in the CALL statement. This practice is more economical (smal"l storage allocation) and offers greater flexibility by IIdynamic storage reservation ll option. This practice is also preferred in recently specified IINASA Standard Requirements for Oig'ital Computer Programs ll
The modification does not require any changes in the bulk of the coding, and is recommended for REFLAN3D-SPRAY code.
s- 5
Improv!~ments __ in Solution Al gorithm
The current solution algorithm emp'!oys a modified version of the SIMPLE
procedure [12J for calculating the hydrodynamic field (u-v-w-p). Fiqure 5-4 presents the flow chart of the iter'ative scheme employed, Note that after
solving the pressure correction equation-, velocities are corrected by using approximate velocity-pressure coefficients based on truncated momentum equations.
Therefore, the u, v, and w do not satisfy the full momentum equations with bod.v forces, large shear stress and interphase momentum transfer.. An alternative
scheme which can be useful for flow with high shear stresses and/or with
intensive interphase momentum transfer is shown in Figure 5.5. The scheme differs from the SIMPLE algorithm in that after the correction of p, u, v, and w, a second correction is performed based on the PoissoY-l equation for pressure, obtained from the full (not truncated) momentum equations. Velocities are corrected using these secondary pressure corrections and the subiterative loop returns to the Ilcontinuity conserving" pressure correction
equation. It is recommended that in the next stage of study, a few variations of the proposed scheme be tested and the most ecol1lomical one employed for
retention as a permanent feature of REFLAN3D··SPRAY Code.
5.3 VALIDATION AND VERIFICATION STUDY
The calculations of the three-dimensional two-phase flow in rocket engine
combustion chambers undertaken in the present project represent the first attempt to model 3-D combustion by the Eulerian-Lagrangian algorithm.
Additional validation and verification tests of the present model are required and rE~commencled. The computational studies should be performed for:
tests for improvements in physical model and numerical algorithm (as described in Sections 5.1 and 5.2); wider range of physical parameters; prediction of single triplet injector flow and combustion (Experimental results of the relevant case are available in NASA CR 1169006. The geometry of the combustion chamber and the
photograph of the triplet jet flame are shown in Figure 5.6); Injector nonuniformity study for several hole arrangements and injection hole blockages.
CALCULATE THE I;:;:;'ERPHASE MASS MOMENTUM I AND ENERGY TRANSFER SOURCE TERMS
_----- t CALCULA~ DENSITY AND VISCOSI:=J
I SOLVE FOR ".' AND w VELOCITIE~
__ ----- I CALCULATE CONVECTIVE FLUXE.~
I SOLVE PRESSURE GORRECTION (p') EQUATION I
I . CO:::T p, u, v AND w :=J
'------, I CALCUL~:C-O-N~V~EC-T-I-V-E'-F-LU-X-E-S:=J
I ' SOLVE REMAINING 1/>- TRANSPORT EQUATIONS I
NO
Fig~re 5.4 Flow Chart of Current Solution Algorithm
5-7
• CALCUL~TE THE INTERPHASE MASS MOMENTUM I AND ENERGY TRANSFER SOURCE TERMS
['---CA-~CULATE DEN":Y AND VISCOSITY =:J I
[CALCULA:E CONVECTIVE AND DIFFUSIVE FLUXES I
[.' _ I
SOLVE FOR u, v, AND w VELOCITIES ~ ,-_.- * [ ~~LCULATE CONVECTIVE FLUXES ~
['. ~ . } .
SOLVE PRESSURE CORRECTION (p') EQUA~
._-- * [ .- CORRECT P AND U,V AND w =:J 1.
"- t =~ ___ ~LCULATE CONVECTIVE FLUXE~------1
YES
NO
NO
YES
I SOLVE FOR THE p' EQUATION. ---, ~ORRECT PRESSURE AND VELOCITIES--.l
t
CALCULATE THE COEFFICIENTS AND J SOURCE FOR POISSON EQUATION
OF SECONDARY PRESSURE CORRECTION
r
Figure 5.5 Flow Chart of Proposed Double pressure Correction Algorithm.
5-8
ox
FUEL
ox
OFO TRIPLET INJECTOR COLD-FLOW SPRAY PATTERN
TEST CHAMBER ASSEMBLY
WIN DOW ----o/~ PURGE INL.ET INJECTOR
''JOUNTI NG RING
30'
-k'I---""E----+--30·
r-\ SINGLE ELEMENT INJECTOR'
\ I
\ \
2" DIA QUARTZ VIEWING PORT' 4 PLACES
-2.75" DIA
"- /NOZZLE _ ,~ ~i~~INING
). -r~-{!J I .,'
WINDOW RETAINING RING
COPPER NOZZLE
FiqurE~ 5.6 Experimental Test Chamber Geometry and OFO Triplet
Injection Spray Pattern (Reference 1).
5-9
SECTION 6
REFERENCES
1. J. Fang "LOX/Hydrocarbon Fuel Carbon Formation and Mixing Data
analys'is", NASA CR 169006, by Aerojet Liquid Rocket Co, November 1983.
2. G. M. Faeth "Current Status of Droplet and L.iquid Combustion"
Progress in Energy and Combustion Science, Vol 3, 1977.
3. G.S. Gill and W. H. Nurick illiquid Rocket Engine Injectors"
NASA SP-8089, March 1976.
4. J. K. IDukowic:z "A Particle-· Fluid Numerical Model For Liquid Sprays"
Journal of Compo Phys. v35, (1980), pp 229-253
5. J.T. Jurewicz and D. E. Stock, "A Numerical Model for Turbulent Diffusion
in Gas Particle Flows", Paper 76-S/FE-33, Winter Annual Meeting, NY 1976
6. G. Arnason and D. E. Stock "Dispersion of Particles in Turbulent Pipe
Flow" in "Gas Solid Flow" ASME FED-vol 10, ed by J. T. Jurewicz 1983.
7. B.P. Leonard, "A Stable and Accurate Convective Modeling Procedure
Based on Quadratic Upstream Interpolation," Compo Meth. App.
Mech. Eng., Vol. 12 (1979), 59.
8. C. J. Chen, J. Naseri-Nashat, and P. Li, "Finate Analytic Method,"
Iowa Inst. of Hydraulic Res., University of Iowa, Rep. E-CJC-1-80, 1980.
9. R. W. Claus, G. M. Neely, S. A. Syed, and A. D. Gosman, "Reducing Numerical
Diffusion for Incompressible Flow Calculations", preprint copy to be
published as a NASA TM, 1984.
10. A. J. Przekwas, A. K. Singhal, and L. T. Tam, "A Multi-Flux Conservative
Differencing Scheme to Reduce Numerical Diffusion in Fluid Flow
Calculations," submitted for publication in Computer Methods in Applied
Mechanics and Engineering.
11. A. J. Przekwas and L. T. Tam IIA Multiflux Conservative Differencing
Method for Fluid Flow and Heat Transfer Calculations" to be
presented at ASME Winter Annual Meetihg, New Orleans, December 1984.
12. S. V. Patankar "Numerical Heat Transfer and Fluid Flow" Hemisphere
Publ. Corp., Washington, 1980.
6-1
APPENDIX A
INTERPRETATION OF RESULTS AND THE COMPUTER OUTPUT
The computational results discussed in this report are presented in the form
of tables of computer output· and graphical plots of velocity vectors and contour maps of the dependent Variables. This appendix describes geometry,
grid distribution notation and interpretation of graphical and tabulated results
obtained from the REFLAN3D code.
A.l GEOMETRY AND COMPUTATIONAL GRID
A sample geometry for elliptic flow calculations is presented in figure A.l.
In this particular case a symmetry plane is assumed and only the 1800 sector
of the flow region is considered as a calculation domain.
The fluid enters the cylindrical channel through the front plane which is partially blocked (Figure A.l), and leaves the channel through the fully opened exit plane.
The selected grid distribution for this geometry is shown in Figures A.2a and
A.2b respectively for two cross-sections viz: a) XY - axial plane - aligned with the main stream direction, and b) VZ - cross-section plane - perpendicular to the main stream
direction.
The coordinate directions are selected as: X - axial direction,
Y - radial direction, and
Z - C"ircumferential direction, (in radians for polar coordinates).
The boundries of the calculation domain are identified as: West Boundary - located at X = 0,
East Boundary - located at X = Xmax South Boundary- located at Y = 0
North Boundary- located at Y = Y max Low Boundary - located at Z = 0
High Boundary - located at Z = Z max
A-l
Exit Plane
cylindrical Wall
__ symmetry Plane
Inlet
Front Plane
Figure A.l Geometry for test Flow Calculations.
A-2
J:> I
W
................
' ........ v ................
I z/X ~
//
................
t VNB North Bcundai"'J ,uuru r U/ //ur ,1// / a.:rU'r";rur"ul ;o~ . . . I' . I' . . . . / 9'1
/ /
/ / West
/ Boundary-
Inle
-
. ~ ·
.~ · i.~ 0
5 · 4 · 3 0
2 · 1=1
I I
0 0
· · 0 · 0 0
0 0
· · · · 0 · 2 3
I I I
0 · 0
0 Uij • 0
'E E"ilij · · .l~Vii · · · · · · · · · · · · · 4 5 6
South Boundart
x-v CROSSECTION
North Boundary -----------
3
· · · 0
· · 0 · 0 · · 0-
· · 0 · · · 0 0
· · · · · · · · · 0 · 7 8 9 10
........ Ll' -L--1-::-,-;;--;-o ............... I '"","I I ................ I I I I ........... '--
WHB High Boundary Low Boundary
x-z PLANE
Figure A.2 Coordinate Direction and Grid Distribution in Two Planes: XY- axial plane; and YX cross-sectional plane.
B
East oundary
-G..
FigurE! A.2 al so presents staggered grid arrangement and the storage location
of velocity components. Note that "backward boomerang" is used for storing
all three velocity components viz: U-axial, V-radial and W-circumferential.
Seperate arrays are allocated for velocities at the last X-face (UEB), last
y-fact~ (VNB) and last z-face (WHB). All scalar variables (p,T,k,E,mfu ••• )
are stored in the middle of the grid cell.
A.2 INTERPRETATION OF THE PRINTOUT
FigurE! A.3 pY'esents the typical printout of the axial velocity components,
and appropriate diagrams for the output interpretation.
All variables (including velocity) are printed in the form of tables; each
table for a particular axial plane (Z = Const) numbered K=l, K=2, etc. For
veloc"ity components additional arrays for:
east boundary axial velocity UEB,
north boundary radial velocity VNB, and
high boundary circumferential velocity WHB are printed too.
Velocities ay'e calculated and printed at appropriate cell faces of the control
volume. Scalar quantities are printed at the grid centers. All variables are
pri ntE~d in SI-units i.e.
k - kinetic energy of turbulence m2/s
2
E - dissipation rate m2/s
3
h •. tota 1 enthalpy J/kg
T - absolute temperature OK
f - mixture fractions
mfu - fuel mass fraction
p - pressure (N/m2) relative to the reference point"
J1 .. effective viscosity kg/ms
A.3 INTERPRETATION OF THE GRAPHICAL PLOTS
Graphical representations of the calculation results include:
velocity vectors ~n XV, VZ or XZ plane
lsoline contours of scalar variables in XV, VZ or XZ plane.
The velocity vectors IT, are calculated and plotted at the grid centers based
on the linear interpolation from the velocity components located at the grid
cell faces (see Figure A.4).
A-4
or-S
o...
or-u o r-QJ
>
or-
>< c:x:: QJ
...c: +J
SO 4-
s::: o 'r-+> ttl +J QJ So. SQJ +J s:::
1-1
QJ ...c: +J
4-o QJ r-0. E ttl ><
W
(Y) o
« QJ S:::5 en
0,-lL.
,I I
<.
\:,
.... ',
& • • • · • • • .' , ,. , .. • .' • · I • • · • • • .' • • ! • ·
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O,...,....~.rt_'O.Il~.n_
_"'::1'0.--.0"10 ..... ::1'.0 .................... """-'\1::1'.110""0.0
nJ_oo=>:»o,:::.?? 00'00'::>0000,:> It+ .. +-+-++- ..... ..... "'.&J .u ~'u J..I ..... ..tJ,iJ,
,. ::t·O ~:1'."\I_::t,..1'1 .... .0 '0 '::).-.0 '\J .... ,.."O '\J .. .. .. .. .. .. .. .. .. . "",'\I _"\I "I.rt ~ .... :o:7' • ............. 0.:;10':> .:> ::'I 0'0 ':) 0 ::> :) ? =- :" ::) '::).:;. 'II"'+-+-""'''+......... .1J,u . .u.u..tJ.u,tJ..tJ
X)-o,...:::r"10o-0 ......... ...... 1"0',110"',. ........ ::>.1'1 .. .. .. .. .. .. .. .. .. .. .... -" - "l.~ 0 .0 C1' 7-• •
'" _'0 -;::r..:lo:> ::;) ::::too ":>010:>':>000'00 • I ........ +- of- ....... .AJ.1JJ.J..tJ £'!'JJ..tJ ..tJ . .aJ.LI :7'or...-nO::tV_,._ 0-;1::;'::1...,.0''''','''' .................. .t'I"' .... '"\,I~01)::r-:7'" · _"l."":>:>:;):;) .... _0::- • 0":>-:>::>::>0::>'00::> • •• +- + ... +- ........ U.JJ.LJ...o.J..u..uU..tJ.I..I..u
~::)'--O.(l""'-ru-"'l -IlOOI'l.o.l'l.l'l::;):::::or-................... -00'\1":#-"' __ ,.
• I
'\1_.:.:» "., '0 :)'0:) ;) -=-:. Co:;:l ::>'0";:' Co , ................ . .... ..., .. , ..... .., ...... .a.I.aJ.
ot-..~J'I7'r--._,....O_:::r
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"' .... 00:).:;30000 ::>;)0::>::-0:):)0:> • ............ +++. .aJ.1J,.l.I' ..... ,u,d.&l,u.l..l,.,U • a- 0 ........... 11 0'0,'" _ I)' ....
'! ~. ~ ::t.,';'.o.. ~ ""..~ "!' :.I1N_.·'\I:;t DI'O 0' D'I)', · -"t·-"lO':::J';:,·O· .... '_·O, o·::t:>:> :>'0'0'00 0., ••• + ... + ........ ........ ..tJ ..tJ • ..a.J .u .d LU ,.u .... 'Oo- __ o.tlr--. ... ", .... ..,
~.~ ~ ~ ~ ~ ~ ~ ~ to: -O'O"'\I~t-,., .... _",·
• •
( \ \ , , , /
'/
'''" ''---... --.., ~
IV"':> :t :t :t.C)·0" 0' ~·o:> ~ 0'00'0 :t'00'01 • t ............... . 1IJ.u d."'.d .... .&J '" ,AJ,&I
::tt-.. ..... :JI/t ... ."III'INII'I_ -~ ~ ~ "'; ~ ;~~.': ~ ~ .,.. ""-'\I":t",.., t-..IO'U'
r\,I-:tooooooo ':> -;:, 0 0 ::;, 0 0'0 0'0' I , ................. . .IJ'''' .u,.u ,AJ ..... '.u'M,.lJ1tlJ1
0- :3".o",~ """":1" "'''' .... ~ ~ ~"'! ~.'! ': ~ ~~~ .., "\I-I'\J~"".o .... «J •• • -.- -·~''!:J·:>'QIOI~IOI :0 0 0'0_0:;) 0'0 0'0.' • • , .... + .......... . .LI,JJ .aJ ..... .u.u u .&J'lIJ ..... ,
02)'O""'"~.."OO'OMt-.. .....
"; ~~ ~ ,,!~,'"; ~ ~<w:.' - - ::t'_ "'1::1' . .., "OIIJ""" • • · ----~~~~o~ ---~~~---- ___ ~oo____ ----000000
41 'iI' :::> 0 :) 0 0 0> 0 ':) :::> ::0 ~, :> J ::> ::> =>.J ':) =>::;1 ::::0 _, :) ':l>::> .:> :;) ::>:1 :;) 0'0'.' ':):;).:> 000'00'0'0' ..... ,..................... II ...... ................. _. .,.................... • It. t ..... + 'f. + ... . .' .... .&JIAJ·..tJ.1J..u-AJ..tJ.1..I.1JtI IJ.1JJ..I.u.aJ·.&J.aJ.&.I.IJ ..... ' o.IJ .... .1J .... ..tJ\lJ<Al ............... ,aJ.uwlaJ.&Jl&I.aJW;aJW, ~-~O~OO~O~~ -~~--o~n~~O ~~~--~~D~~O' ~_~O~O~~~~~
~~~~~~~~~~~ ~~~~~~~~~~~ ~~~~~~~~~~~ ~~~~~~~~~~ ~~~~~O-___ ~~~~~~____ ~~~~N.~~.~.
• I • • It ••• C -O::>_.::JI"::>~O'O'::>tt:
· ... --:>o .... ____ ~ ::>::>:;J J '0::> =' :;)'0::;) ':> :;. :::::0:::::0 :>::> 0'0 '.:>:;:,
ft ..... , ............. fI • •• + ................ -II ..... .LIlLJ.LILI.LI..u.u..tJ.u.ufl - iII·O ::J ..... O;t- .., ~.11 .0 '\f "l iI 'O=1_"10 __ 'O .... :X)
U ................. .. :::J .t1---"\J:;fO ..... "..:7-J .,. I
~ _0 O~-"-;I"" • 'O::;):;JOO-:>':>':)O'::I :l '''''' .. f- .. +- ...
..tJ.J..I41.1..l..tJ-U.LI...L.I.JJ.LI 111.0--,.....1>.00-0:;:> ....
----- ~-:-:I~~~~;,.·~ .A- ';" i' ~u..:.:.: .:::r" ,.
t · ! I · • · I · I · • · : ! ! • · I • • • I · • · • · : · • I · !
-00::> .... 0,:>000 ::> 0,:;::»:;::» J":> :> '0::>':;" , ................ + .a.I.LI~..u .... ..u.lJ.IJ.1I..u
-:tl"'lt'\l-=10_:10:Q0_ :OO(()O'.", ..... O::r:tr--.
0n.Jt'\I_"' .... .:20COO' , •• I I
-0'000'\1::>:::::0:;'::> OOO.::;oO~:::>O=>O • .. .. .. .. I+- ..... +
.AJ.&J.lJ,jJ,jJ.LI..tJ .... .u,jJ ""~O.n,.... ..... ""'_O~~
:7'lI'IOOOCt'\J::>.o:1 .. .. .. .. .. .. .. .. .. .. ~"''''':'\J-:J'I'\I.n!'-:7' I' • I I •
.... OOO::> __ .J;:,IC OO:::::>CoO'?C"I-:J':>C , + .......... . uJl&.ILiJ .... WoJJ.vuJ,.1J~
f\J"1.oC1''''l"\J''OJ1 :1 ...... '\1 Jl ..... ,.., :t 0" .f) -:t _ 0" ..... .. .. .. .. .. - ........ :t-ru~-O"""'"I,-"..c • • • • • • O"::>::>:>::>:>jo:;tO COOOOccacc> .. +- ............ +- .. .1..I ..... WuJ....,.....,.L!..M~~
-0.000000000'0:> O::;)O~::>"':JOO:>"':J .. .. .. .. .. .. .. .. .. . 000-=>0:>0000
"
.U.4J.&l <LI .&J 1J·..tJ ~..tJ..tJ -II 0'" J' J' .0 3' ,'\I "\J '2 -Il :'\J t( "l.:2-_.n::>'\1"'_~ .. .. .. .. .. .. .. .. .. .. "''1-",0 .... - .... -_
• I ............ .:>-:::.----::>-':>:::':>::>0'.:>0:::>:::> t •• +-+-++- ......... JJ..u.AJA.J..L.:A.J~.J.J.u.1J
"'.:t'""\JJlO-ll:or--iIl""" ':7' -::r.:o:o .... .",."' ..... ::> .. .. .. .. .. .. .. . . . ....,.(13''''',, ____ ..... , I
~ ...... o::>::>_ ..... __ _
:):>::>::>"::> =..::> ':):;,0.:> o I ............ + .........
.JJ ..u ..&.I 01.1 ..., .AJ .&1 .IJ UJ .tJ ::r.Q.n:1'\lf1'l''\fO''Jl.:2
~r 0 <01\ ...... _ ..... .f\ ::r '\I ::;.
::Ji;i'::-::t";::::":'=; <t-'_'~J _____ _ -...,; .:) :> ~.:.> ::- :J;' ::> 0 0 > •• +-+++- .........
,u..jJ.J.!.JJ.u..oJ..o.u..o ..u ~.." ....... '\I .... (:.....,::>.:2 Xl ::.t -- ::> Jl 0 . .0 "'l .::,. · ............ ..
\..n -Jl -' .... - __ .... .... I, •• , ......... "2> ___ .......... :>
-:; =: ':"J -=- ::>:";':;;: =-::::>::> • • , ........... +- + ...
.a.J ~~ ...... /J"..JJ...., .J.!-4./W
""~::,i,~~~;;~;,~ .. .. .. .. . .. .. .. .. .. :1' .... ..,.- __ ...... _:r-• I
:;:>:::."".:>:.-,:::.- ..... _ ..... -c:oooo::;-.oc>oo ... of- .... +- ............. ..&JU.1.-.aJ.4J<JJ..u..a.J.J.JW
.... o::>~:>:;;:..;.::>::>.::;,::> oo'~OO?o:::>oo , ................. .. :;:I:;:JOoc-=-"\J'\tf\J ..... .... 11---
..... .::0:7' J() ...... 01", :1....,"\J ....
-, - 'I \-
~i
A-5
- __ 00 _____ • -oo-ooo~oa,
~ ':).':;, ::;J -:> :J ':>·0 00'00 ':;:I':> :;) 0'0:) ::>.~.::Jo 0' .••• +.f- ....... +tI: ....... +++++. .JJ.lJ W·.aJ.&.I.u..,.aJ W,.., tI .... tLJ.aJ..t.I "'·.LIW >&J .... 1.&1.
o .,. ,.. :7' 0 ':1' .. '\1 N ::t "".'\1 ... '0 =t r--. 0 ~ • .., ::t·1n .0 ~ "" I ", . .;:t .... _.n :::. r'IA 1'\1 .... 0 IJ ~ .... "I :::) _ ..... :0 _:01 .. .. .. .. .. .. .. .. .. .. .. .............. .. ~~~~ONN___ ~ ___ ~.~~~~.
• • • • I - __ .:>0 ____ _
0::;)0':;:10000':::>::) · , ................ ... .aJ.aJ .... ..tJ.AJ,ulA.lw..u.u
""O'I'\I...n::O""(OI'--:.oO""" :7'_::7:00""""""_0 .. .. .. . .. .. . .. .. .. -Ilo.ll:1""" ..... ____ _
• • --,:,.:;,0 ____ _
-:;'-:;'0'0'="0'.:>;:'0::-'0 · .................. .. ILl .... .u ...., .JJ .v laJ .aJ'M .AI
~"()1Il::t!'V1Il.;:t1'\l0".S'I" ::t 0 . ...,,..., _..,,/\ ~ "':0 . .. .. .. .. . .. .. .. .. ::1:1-::10' ____ _
• I --:»0--- __ _ co-:..:ooo.oOOO!;) • • +- ................ ... ..J.u..a.JJ.J&J:.uUJ.u.u~
"" ..... ::t..".""1 N -:0"'" !;)::t ",:0::1":"0111.0.0""'0 .. ................ .. .t"I:.n .... ...n- .... ___ .... I. .... --0- ......... __ 0
O:;-::'?OOOOOo.o , I • + ............. .. . .u·..uwUJ'..., .... uJw..u.&J
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~~~~"!~~~":~ 11"1:)-_..., _____ 0-
I • ::;,00"00- ___ _
cae·acc-oooo ++-+-"+""+-++flJ.lJ.LI.LIo.lJ.LI.&J.LI..u~
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.... 00_000000 ":;) 0 0 ::;) 0 0 :;:, 0 0> ,0 · ................. .. .... 11J.LI..tJ.aJ..tJoIJlaJ ..... .&.I
II1D""'-"","O:O.o",.oo_ .0 __ 0 0"'I"'l""0" 0:0' .. . .. .. . .. .. .. .. .. -" '\I a- ""I"'ltn ""'" ~ 3" • ••• -00-::>0-000000 ~ 0 :> ::> 0::;):;:' ::;:, 00 00' , .......... + ...... ltJ..AIIAJ\lJW...,III~1JoI1&J
:::J,..,r"\I:;fO_~OIO'O_ '2)'OoX):J"oIll ..... O>~:f",..,. · ................ . iXl~'U_'""'.-4~·O:l)O' · , ... -OOO~NOOOO ';;) 00 !:;:I 0 ":)00000
• .. + ... I ........ ..aJ14J..u .... wlLlw.u~IIJ .
""o.o",,,,,,,,,,",,_COQoCJo (J-V'lo>O.o"''''O-D:t ................... 1-N"'1I'\1_!1'f\ltn ..... CJo • ••• I , .... 0.:::000 __ 000
-::> 0 0 0 0 0::> GOO O' I ......... , , ...... ..tJw.aJWuJWwWWW
N,.,,'OO' ..... nII~\Il:3',...rv U"\ ... ....,":!DtG.::t_o-,... ................. :f-ruru-:C)",H'\&f'I1O • ••••• ~~O~O:>~oOO OCOOOOOQOD .......... ++ .... ++ lLJUJ,&Jw;d!lJWWlaJw
-0.000000000 00::>0000"::30000 .. .. .. .. . . .. . .. . o::'o'!)'Oooooo
............... 0"0"',",' , . , . ., ,;,a ,\aI '\d, 0'11' __ ",
-'~~":.":. ct'ne ",'~I
0"''''0 o o·~ 0' ." a •. .. ,"'-', ..... c",_"'''''' .... ~'~.~'~' ............ -
0000' ~ 0' 0'0' .' .... 1&1,111 1M", .... .,.,"" ..... tJ"'o':tO' ...... "~NI",
0'0'00 00'0'0'1 ........ id IaI YJ oW·
elO'IJ"'eJo'O'
""!.~ ~~--: ct'~,",,'~1
0000 0'0' 0'0 .. .... lIJ·ld £I !.r •
"_10 __ 0
J ""~",,,,,, .., ........ 1ft WI '" tn·
•. 0000 ... 00'00'01 .., ......
I» ""1M W o~ ... ,.,.,.,..., - 04"0 II ....... ;:: •• .0 .0 .0
0000 00'00'
II ........ - ~caI\d .. - .... :t"."'~
~~~~ :tI .... ,.. .... r-. .. ::J
00>00 0000
Ia. ++++ o klWW ...
nII_Ion",,_ t- .... _ .... _
z ...... .. ,... IIO.HOIlO
'" <I.' OClfOO
N 0000 x ........ U lLIWWW· IW_ ............. ..... X ~,.,.. ... .. U .... ..
."""110 OH .. «. :::t..,.. "' .... « • .."
Figure A.4: Calculation un Velocity Vectors.
'* . The vector U length is taken as:
'--------.....
Figure A.5 presents a typical distr-ibution of velocity vectors for the first XV plane with an overlayed grid line system and contours of the scalar quantity (temperature) on the second XV pl am~.
Additional information for velocity vectors includes printout of minimum (VELMIN) and maximum (VELMAX) velocity vectors at that plane.
The legend for the contours of scalar property ¢ includes:
plane orientation eg: XV Plane 2;
name of the contoured quanity: TEMP;
minimum and maximum value of ¢ within that plane (PHIMAX and PHIMIN); and
contour numbers and appropriate contour levels.
11-6
)::> I
-....J
Figure A.5 Example of the Graphical Results Interpretation
."<'l PLANE 1 I.. IELOC I T\~ PLOTS UEL!'"lIN 4. 056E-01 l)ELMAX 2.260E+01
- -" -'" -, . , -..... -- -
><'/ PLANE 2 TEf'1P CONTOURS
----, ---PH If'lAX PHIMIN'
---, .. -- •
4.9519E+02 3.0457E+02 CONTOUR LEVELS ARE
1 3.2190E+02 2 3.3923E+02 3 3.5656E+02 4 3.7389E+02 5 3.9122E+e2 6 4.085SE+02 7 4.2S87E+02 8 4.4320E+02 9 4.6053E+02
1121 4.7786E+02
I
- - - . . T #
# - - - - .1 -- - - - -- - - - - T -- - - - - l -- - - l --- .. - -
I I 1 I I J 2 E
End of Document
