turbulence unclassified smmmmmmmmsm … · 13a. type of report 13b. time covered t4. date of report...
TRANSCRIPT
A-A17 628 A TUO-EGURTION TURBULENCE MODEL FOR A DISPERSED v iTWO-PHRSED FLOW NITH VAR.. CU) R AND D ASSOCIATESMARINA DEL REV CA G C YEN ET RL. 31 DEC *5
UNCLASSIFIED RDR-TR-135694-002 DNR-TR-86-42 F/O 20/4 MLsmmmmmmmmsmlmmmmnmmmmmmIollnlslollngIhmhlllllhh
r
I.
2..0
I111 l iiIII'
1.8.
MICROCOPY RESOLUTION TISI CHART
a.,,
AD-A 170 628DNA-TR-86-42 .-
A TWO-EQUATION TURBULENCE MODEL FOR A DISPERSEDTWO-PHASED FLOW WITH VARIABLE DENSITY FLUID ANDCONSTANT DENSITY PARTICLES
Gordon C. K. YehSaid E. ElghobashiR & D AssociatesP. O. Box 9695 1Marina del Rey, CA 90295-2095
31 December 1985
Technical Report
CONTRACT No. DNA 001-85-C-0022
Approved for public release;distribution is unlimited.
THIS WORK WAS SPONSORED BY THE DEFENSE NUCLEAR AGENCYUNDER RDT&E RMSS CODE B310085466 P99QMXDBOO099 H2590D
c.... Prepared forDirector
L'_DEFENSE NUCLEAR AGENCY -
•, Washington, DC 20305-1000 /
Lb
-- --... , - ,,
UNCLASSIFIED . -
SECURITY CLASSIFICATION OF THIS PAGE -
REPORT DOCUMENTATION PAGEI&. REPORT SECURITY CLASSIFICATION 1b. RESTRICTIVE MARKINGS
U14CLASSIFIED ___________________
2L. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABILITY OF REPORTN/A since Unclassified Approved for public release;
2. DECLASSIFICATION/DOWNGRAOING SCHEDULE distribution is unlimited.N/A since Unclassified
4. PERFORMING ORGANIZATION REPORT NUMBERIS) 5. MONITORING ORGANIZATION REPORT NUMBER(S)
RDA-TR-135604-002 DNA-TR-86-42
Ga. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION(if applicable) Director
R & D Associates Defense Nuclear Ag~ency6c. ADDRESS (C0tv. State and ZIP Code) 7b. ADDRESS (City. State and ZIP Code)
P.O. Box 9695 Tahntn C 23510,Ilarina del Rey, CA 90295-2095 ahntn DC 00-10
8a. NAME OF FUNDINGISPONSORING 8 b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION j(If api~tcable) DA01-5C02
Bc. ADDRESS Ii~ty, State and ZIP Code) 10. SOURCE OF FUNDING NUMBERSPROGRAM PROJECT TASK IWORK UNITELEMENT NO. NO. NO. ACCESSION NO.
___________________________ 1_6271511 9 9QrX DHjOOC67011. TITLE (Include Securitv Classification)
A TWO-EQUATION TURBULENCE MODEL FOR A DISPERSED TWO-PHASED FLOW WITHVARIABLE DENSITY FLUID A14D CONSTANT DENSITY PARTICLES12. PERSONAL AUTHOR(S)Yeh, Gordon C.11.; Elghobashi, Said E.
13a. TYPE OF REPORT 13b. TIME COVERED t4. DATE OF REPORT (Year, Month, Dayl 15. PAGE COUNT
TecnialFROM 850101 TO85J 3 851231 48
16S. SUPPLEMENTARY NOTATIONThis work was sponsored by the Defense Nuclear Agency underRDT&E RMSS Code B310065466 P99QMXDE00099 112590D.
17. COSATI CODES 18. SUBJECT TERMS (C.antinue on reverse if necessary and dentifytv bv oCk n~unicerlFIELD GROUP SUB-GROUP Dispersed Two-Phase Flow
20 4 Turbulence Closure Model10 3 IC-E Model
19. ABSTRACT (Continue on reverse it necessery and ,oentitv by block numnber)
A two-iiequation turbulence model has been developed for predicting a dis-persed twol-phase flow with variable density fluid and constant densityparticles. The two equations describe the conservation of turbulencekinetic energy and its dissipation rate for the fluid. They have beenderived rigorously from the momentum~ equations of the carrier fluid inNthe two-phase flow. Closure of the time4-mean equations is achieved bymodeling the turbulent correlations up to third order. The new modeleliminates the need to simulate in an ad hoc manner the effects of thedispersed phase on turbulent structure in situations where the comn-pressibility of the fluid must be taken into account.
20. DISTRIBUTION/AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATIONOUNCLASSIFIED/UNLIMITED [SAME AS RIFT. CDTIC USERS UNCLASSIFIED
22a. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (Include Area Coo@) ?2c. OFFICE SYMBOLBettv L. Fox (2 02 ) 3 25- 7042 0A, STTI
00 FORM 1473, 84 MAR 83 APR etiton may be used until exlhaulted.All other etitons are obsolete.
UjNCLASS :p :SECURITY CLASSIFICATION OF 7HIS PAGE
.............-. .. . . . . . . . . . . . . . . . . . . . . .
V
UNCLASS IF IEDSECURITY CLASSIFICATION OF THIS PAGE %
18. SUBJECT TERMS (continued)
Nonsteady FlowCompressible Viscous GasDilute Suspensions -,
Continuity EquationMomentum EquationReynolds Time-averagingKinetic EnergyDissipation Rate
*.*
-f
SECURITY CLASSIFICATION OF THIS PAGE
S S * -* .5.
*S ~5. .* ****.* . *~ .Z-~Q&&&.."- *i*. -
Lpig
PREFACE
This work was performed for the Defense Nuclear Agency under
contract DNA 001-85-C-0022. Dr. George W. Ullrich was the
DNA Contract Technical Manager. Dr. Allen L. Kuhl was the
RDA program manager. The authors wish to thank Dr. Ullrich
for his encouragement, Dr. Kuhl for suggestion of the
research subject and helpful discussions, and Mrs. Vivian
Fox for expert typing of the manuscript.
N
I-I
ii i'"'
2I
CONVERSION TABLE
Conversion factors for U.S. Customary to metric (Sl) units of measurement
MULTIPLY B BY TO GET :%TO GET B DY 4 DI"IDE
angstrom 1. 000 000 X 9 -10 moters (mi)
atmosphere (normall 1 01325 X Z .2 kilo Pascal (kPa)
bar 1 000 000 X E .2 kilo pascal tkPa)
barn 1. 0 0 X E 28 meter2 im
2
British thermal unit (thaermochemicsl) 1.064 350 X E +3 joule 1J
calorie tthermochemicaul 4 164 000 )oule J)
Cal tthermochemicalt/cm2
4 154 000 X E -2 mega joule/m2 (M J/m
2
curse 3 700 00) X E * I 1tga becquerel lGBq)degree tangle) 1. 745 329 X E -2 redian (redl)
degree Fahreniheit . - 0'f * 459 67)/1. S degree kelvin (K)
electron volt 1.602 19 X i -19 jouls ()
erg 1. 000 XE-7 ouis IJ .
erg/second 1.000 000 X E -7 wtt (WI
oot 3. 048 000 X E -i moter (ml
foot-pound-force 1. 355 616 joule J)
gallon It S iquidl 3 765 412 X E -3 meter3 (im
3
ich 2 540 000 X E -2 metr (ml
jerk 1 000 000 X E .9 joule (W)
joule/kilogram J/19) radiation doseaboorbed) 1.000 000 Gray (Gy)
kilotons 4 183 terajoules
kip 0000 Ib) 4 448 222 XE3 newton N)
kap/iach (Ul) 6 94 757 X E . 3 kilo pascal (kPalkLa~p ft w - €ond/m
2 ,.
1.0tr 000X00 -2 X/ 9 2
macran 1 000 000 X E -6 meter (mi)
mit 2 540 000 X E -5 mtor iml
mile interuational) 1.609 344 X E -3 meter (in)
ounce 2. 634 952 X E -2 kilogram (ji)
pound-force ibs soirdupoisa 4. 448 222 newton (N)poWd-force inch 1. 129 S41 X E -1 newton-meter (N-m)
pound-force/ich 1 751 26£ X E -2 newton/meter tN/mipouatd-force/foot
2 4. 76 026 X E -2 kilo pascal (klPs)
pound-forcc/inch2
Ipsl 6 894 757 kilo pascal (kPal
pound-mass (Ibm solrdupoist 4 S35 924 X E -1 kilogram 1ki
pound-mass-(oo2
imomont of iniertiai kilogram-meter2
4.214 011 X E -2 (Jq.m2 )
pound-mass/oot3 kilogram/m ret r 3
1 601 $49 X E .1 (kg/m3
)
rad (radiation dos absorbed) . 000 000 X E -2 *Gray IGy)
r'oenitll ,coulomb/k logr am2 S79 740 X E -4 W/ As)
sham 1 000 004 X E -8 second (a)
slug 1.459 390 X E .1 kilogram (kg)
torr imm Ig. 0- Ct 1. 333 22 X E -1 kilo pascal (kPs)
*The becluerel 1BqJ is the SI unit of radioactiviry; 1 Bi t 1 vSent/s.-The Grav (Gy) is the SI unit of absorbed radiation.
iv
.........................................Ip ,r *" ',* .-.-".,",r'-"".'" "'" " "*'+ -" "" "," .* -" " "-" "--------,---"---"--"----"------""---"-"-----"---------------"---.---"----------------------"--"-
• + . + • l+ - |.
TABLE OF CONTENTS
Section Page
PREFACE ii
CONVERSION TABLE iv
INTRODUCTION 1
2 EQUATIONS OF MOTION 2
3 TURBULENCE-KINETIC-ENERGY ANDDISSIPATION-RATE EQUATIONS 8
4 CLOSURE OF THE MOMENTUM EQUATIONS 9
5 CLOSURE OF THE TURBULENCE KINETIC-
ENERGY EQUATION 12
6 CLOSURE OF THE TURBULENCE-ENERGYDISSIPATION-RATE EQUATION 17
7 A SAMPLE APPLICATION 20
8 CONCLUSION 22
9 LIST OF REFERENCES 23
Appendices
A EXACT EQUATIONS OF KINETIC ENERGYOF TURBULENCE AND DISSIPATION RATEOF THAT ENERGY 27
.~ .
. * * * v
~. ~ ~.
"" '" " '' "" " .. . " '" '''' ''' ° ° "'' ''' ''' "" "'" " "" " "' " "'" '"- "" ." ' ° .' . i' .""." "'.'"-°"" "'.p. -- .*.
...
TABLE OF CONTENTS (CONCLUDED) '
Appendi ces •
B THE MODELED FORM OF k and E ,EQUAT IONS 31
C NOMENCLATURE 33
,%4
.
.- 4
4°
5-
I%
.4 5.°.
-1 7- 9. W.~~** 7
SECTION 1
INTRODUCTION
There are many problems of practical interest which require a
mathematical model for a two-phase :urbulent flow of suspended
particles in a viscous fluid. Most one- and two-equation
* models for two-phase flows (e.g., Refs. 1 and 2) are based on
ad hoc modifications of the single-phase turbulence-kinetic-
energy and length-scale equations and fail to adequately
predict the physical behavior of two-phase flows. Recently,7-
Elghobashi et al. (Ref. 3) presented a rigorous derivation of
the turbulence-energy and dissipation-rate equations from the
momentum equations for an incompressible dispersed two-phase
flow and successfully predicted the main features of a round
gaseous jet laden with uniform-size solid particles (Refs. 4
and 5). In situations such as the dust transport by turbulence
in nuclear burst flow fields (Ref. 6) and supersonic nozzle
flows (Ref. 7) the compressibility of the fluid must be taken
into account. The objective of the present work is to extend
the two-equation turbulence model in Ref. i to become appli-
cable to a compressible dispersed two-phase flow in which the
fluid has variable density and the particles have constant
density. The extended model can also be used to study such
problems as compressible adiabatic mixing and low-speed iso-
thermal mixing of two dissimilar two-phase flows (Ref. 8).
SECTION 2
EQUATIONS OF MOTION
We begin the formulation of the problem by stating the
assumptions involved in deriving the equations. These are:
1. Both phases behave macroscopically as a continuum,
but only the carrier fluid behaves microscopically
as a continuum with variable density. This means
that the volume-averaged equations are based on a
control volume larger than the particle spacing but
much smaller than the characteristic volume of the
flow system. Mutuial exclusion of the phases is
* also ensured.
2. The dispersed phase consists of rigid particles
spherical in shape, uniform in size and constant in
density. The uniformity of size reduces the
magnitude of bookkeeping at this stage of the work,
and thus concentrates the effort on understanding
the mechanisms of interactions between the two
phases. Extension to nonuniform size distribution
is a straightforward matter (Ref 9).
3. The volume fraction of the dispersed phase is
such that no collisions occur between the
particles. This assumption renders the equations
valid only for dilute suspensions. 6
4. Neither the suspended matter nor the carrier fluid
undergoes any phase changes. Although this
assumption rules out some situations of practical
2
interest, it is necessary to investigate
complexities in a stepwise manner.
5. Additional assumptions on modeling of some of the
turbulent correlations are stated in Sections 4, 5
and 6. The sparseness of experimental data for
variable density flows allows us to assume that
the forms and values of the coefficients of the
turbulent correlations of constant-density flows
apply to variable-density flows as well.
The instantaneous, volume-averaged momentum equations, in
Cartesian tersor notations, of the carrier fluid are thus
(Refs. 3 and 10)
(QiUi),t + (QiUjUi),j = -(1-K+ 2 )P,i - KF+ 2 (Ui-V i)
2:;3(MllU,.Q) + Q1gi + Fi 1
The corresponding equations for the particle phase are
(Q2Vi),t + (Q2VjVi),j = - 2Pi + F+ 2 (Ui-V i )
+ [M 2 +2(Vi,j+Vj,i)],j - 2(M2 2V.A, ),i
+ Q 2g i + F 2 i (2)
3
. .. . . . . . . . . .. . . . . . . . . . . . -
- - -.. .-. V '
. V.
Th-e z-ontainuitv ecuatiocns are V
~(0
N4for the fluid and
1p ad2dnt th ' dan at- - hs
or theoatl c haatse. ThUae --hba v ocotity pieuaz: e-
p.u4d Vi ar the (3)i,, ocoet o
:ncaen Eqs.' . +1 's5)an thoughout.he frc ort thP usc't
is ant2dee throen fi and aticnle achase reoetie 7.
Partia derivative's re reprsente by a-o, subesritcsito
of- a omm n an ide F Fs-e gu ,?n'as f r t c~ nt C:e
satial coordinates-h fuid are the veloJctye~ coinentsa
flid t~r thet veloit copnsohe w cuatc I aS. n
coou ae te aterial djaesity andme visoit. 3=
duhae tos Irv'v an Fn, in %h, nrzasIae be:t o ooe r
-~~ Is tiend ona 'ves or moetm ras'!--
1%%
turbulence, the slip velocity, and the particulate size and
concentration.
The mean flow equations are now obtained from the
instantaneous ones, for variable pl, constant P2 and pl and
zero M2 (consistent with the dilute suspension approximation
stated in Ref. 10, p. 256), by performing the conventional
Reynolds time-averaging of Eqs. (1) to (5). (The density-
weighted averaging of Favre, Ref. 11, does not render
significant simplification. We use Reynolds averaging mainly
because most experimental data refer to time averaging
correlations). The mean momentum equations of the fluid are
(Q1Ui + qjuj) t + (QiUjUi),j = - (1-K+ 2 )Pi + K02P i
- KF[ 2 (Ui-Vi) + 0 2 (ui-vi)]
+ A 1 [+1U i j+u j i + €~i5u~ ) , -
-01 uI(UJj + #lUAA) ,i -
- (Qluiuj + Ui qjuj + Uj qjui + qluiuj),-
+ Qlg 1 + Fli (6) R
C.o
-C. o
° -'C
5
. . .....
. .. . . . . . ..: " " "'" . ". " .,' " " " ," " ' "' " " " : " " -" " " " - ' " " " " - .' - ' ' "
" " " "
L7
The mean momentum equations of the particle phase are
2 ( 2- ) ,t 2 ( i),j 2 1i +2P,i
+ F($ 2 (U ..V ) + 2( - )
(0 v v+ + Vqq-v. + q- 27ivj 1v J j 2 i+q 2viv.),j
+ Q 2g + 21 (7)
The mean continuity equation of the fluid is
Qit (Q1 Ui + qlui) , = 0 . (8)
The mean con',4nulty equation of the particle phase is
(9)"
Q2,t (Q2Vi q 2 vi) ,i = 0 (9)
which can be written as
+2,t +(4 2Vi + 2vi),i 0 (10)
since P2 is constant. The mean global continuity equation is
+1+ +2= 1 (l
which, when substrated from Eq. (5), gives
01 + 0b2 0 (12)
In Eqs. (6)-(12) capital letters (except K and F) denote
•. time-mean quantities, lower-case letters (except p, and g4)
designate fluctuating components and overbars indicate
..
-6
"'"-
Reynolds . ..- -. - av ra e cor el ti ns Fo co st n quanti ies (p
Reynods-aertaed corrluatin. Formonstt quanitie Q (q +u
pi + P1' ... etc.).
7-d
°°
SECTION 3
TURBULENCE-KINETIC-ENERGY AND DISSIPATION-RATE EQUATIONS
The first step in the derivation of the equations of the
fluid's turbulence kinetic energy (k Muui/2) and its
dissipation rate (E vlui,k uik where v,= p1/p ) is to
obtain a transport equation for ui by subtracting Eq. (6)
from Eq. (1). The k equation is produced by multiplying the
ui equation throughout by u i and then time-averaging. The e
equation is obtained by differentiating the ui equation with
respect to xk, multiplying throughout by 2viui,k and finally
time-averaging.
The resulting k and e equations are given in Appendix A.
The closure of these equations is discussed in Sections 5
and 6, respectively.
°.~
o.
8
' d-
.. ~.. . . . a ,. . .
a,. a '..,
SECTION 4
CLOSURE OF THE MOMENTUM EQUATIONS
The turbulent correlations appearing in Eqs. (6) and (7) are
of five types:
1. Correlation of velocity fluctuations with those of
the volume fraction or apparent density, e.g., Ojui
or qjui;
2. The pressure interaction correlation 02P,i;
3. Multiple correlations among various components of
velocity fluctuations with those of the apparent
density, e.g., qluiuj
4. Correlations of strain rate fluctuations with those
of the volume fraction, e.g., 0jui,j
5. Multiple correlations of various components of
velocity fluctuations, e.g., uiuj
The first four types occur only due to the presence of the
second phase; their modeling is discussed below.
According to Ref. 3 (Eq. (10)) and Ref. 12, we model the
turbulent flux 0jui by a gradient transport term and a
convective transport term such that
olu= -(Vt/a#)+,i i , (13)
4..*. . . . . . . . . . . . . ..
4 where v is the kinematic eddy viscosity (= c with c
0.09, Ref. 13, under the provisional assumption that it has the
same value as in the case of constant density) and a is the
turbulent Schmidt number of *. Similarly we model q.u 4 bvNit
qjuji t/qQ, - Q, (Vt/W ),1 (14)
where a. Is the turbulent Schmidt number of q. As long as
experimental data for aq are not available we make a
provisional assumption that aq is equal to (= 1.0 for the
sample calculation in Ref. 3).
According to Ref. 3 (Eqs. (11), (12), (14), (15) and (16)) and
Refs. 14, 15 and 16 we model *2P by
2P,i= - 01 kcpi uii - 2- i-
" p 1 (0.8 u U;, - 0.2 u 1 , ua..
--co 30 ik /2 U~mq emC 3/ /"
0C4Pi kI /E Unm uml enj ,i 5
where em and en are unit vectors and the approximate values of
the coefficients are
C = 4. 3, c = 3.2, c0 3 = 1.0, c04 = . 1 -
.3°o
-- -.- - .- -. . .-. . . . _ .-. • • , - . . -. .- - - . -.- - _.- . .-. - . . % .
According to Ref. 16 (Eq. (6.45)) we model qjujuj by
qluiuj = -5(k/e) uiuA (ujql),a + ujuA (uiql),j] , (17)
,- where the proportionality constant c05 is approximately
.0 equal to 0.1.
The strain-rate volume-fraction correlations of the type
OlUij only appear multiplied by the molecular viscosity of
the fluid and therefore will be neglected due to its relatively
small magnitude.
The last correlation to be modeled in the momentum equations
is that of the form uiuj. Again, to be consistent with the
present level of closure, this quantity will be calculated
from (Eq. (18) in Ref. 3)
uiuj = _ vt(Ui,j+uj,i) + 3 Sijumum - 2 Vt .ijUjj (18)
This completes the modeling of the momentum equations.
°.1°
SECTION 5
CLOSURE OF THE TURBULENCE KINETIC-ENERGY EQUATION
The exact equation of the turbulence energy k for the carrier
fluid appears in Appendix A and consists of 34 terms. They
are classified into groups enclosed by square or curly
brackets; each group is labeled according to its particular
contribution to the conservation of k.
The various correlations in these groups range from second tofourth order. We decide at the outset on neglecting all
fourth-order correlations such as qlui(uiuj),j and uiuiujql,4.
Also, the contribution to the diffusion of turbulence energy
due to the pressure interaction (uiP) i will be neglected as
it is of relatively small magnitude (Ref. 17). Now the
remaining terms will be modeled.
The five transient terms will be collectively approximated by
(Qlk),t. The convection terms require no approximation. The
production group contains the correlations qjujuj and uju j
which have been evaluated earlier by Eqs. (17) and (18). The
pressure velocity-divergence correlation Pui'i in the
turbulent-diffusion group cannot be neglected here since ui i
does not vanish in two-phase flows. PUi' i is evaluated
following the approach of Ref. 18, thus (Eas. (19) and (20) of
Ref. 3))
Pu. C (u.12k (c2+8)Q 1-~ ~ 2~ k 3 lT()(ii- 22 (Pi )
(15ci1)0 ( (4c2 -i) (i- (1+ 55(2k Ui i) - iDii _ (19),..55 3-
:2
. ,
where
P = -2(uIuk U i,k)
P = P i",
2 ii j
D ii -2(uiu k Uki) (20)
C1 1.5 2 = 0.4,- 2
The last two terms in the turbulent-diffusion group can be
modeled as (Eqs. (21) to (25) and Eq. (10) of Ref. 3)
-Uu( ,- UU = Q t/k,,
+ 2 k (vt ,'2 'tKJ k
where the turbulent Schmidt number of k is taken as ak 1.0
(Ref. 13, under the provisional assumption that it has the
same value as in the case of constant density).
There are eight terms in the extra production and transfer
group, the last two of which are neglected for being of fourth
order. The remaining six terms are modeled next.
The second term, -QIUi uiuj , is modeled following the
proposal of Ref. 19 as Q1Ui tUy ij. The correlation of the
form ui(qjuj) , which appears in the third and fourth terms is
expanded as
u (quj) = (uquj)j -qujij (22)
:3
73 .7 .-.
I '77*
where the first term on the right Is evaluated using Eq. (17),
and the second term is modeled as
qlujui, j - c Ql i + Q1 T (23) N%qq
where the provisional assumption on a has already been
mentioned in Section 4. The correlations 0lU, (in the first
term) and q ul (in the fifth term) have been discussed earlier
(Eqs. (13) and (14)). The sixth term is approximated by
Ui uj , Ui J[(uiq1 ) , . qju, ] (24)
where q1Ui j is neglected for being relatively smaller than
(uq), '
. The extra dissipation group contains three terms which exist
only due to the slip between the two phases.. According to
Ref. 3 (Eq. (39)) and Refs. 20 and 21 we model the correl'at-on
u (vi-u4 ) by
ui (vi-ui) = -1 . 1 fMd 25)
where w is the frequency of turbulence ana,
i "/'i
2 ( 3 .'2
:4:
i,;- - -:.':, - .- ; . '.'... ,-.
2= -2 ,/)2 + .p1 -(W/ 3/2 + 3(w/a)
+ 419 (w/' )1/2 + . (28)
R = (1lp)w/ C2 (29) S
f(w) 4(T)3/2 A 3) e (30)
in which
12,,1 P d 3-= 12'..l/p 1d
2 ( 3: )
P= 3p,/(2 2 + ,) (32)
and X is Taylor's microscale. The last term in the extra
dissipation group contains the triple correlations 2uiui and
*2 uivi which can be modeled by Eqs. (:7) and (42) of Ref. 3
asI -
2 2 u =-2c05 (k/e) ui (ui,02 ), 1331.
and
2~~~~ . •uI
where the double correlations on the right sides hav:e been
modeled earlier (Eqs. (13) and (18)).
The term I,+. u4(u 4 , 4+ u constitutes the d'ss-pat4on of :<
due to iiiscous action, and iff *. is set to unity 4t is reduced"
.5
to the single-phase dissipation terms. This term is modeled as
(Eq. (43) in Ref. 3)
11i 1 ui(ui,j+uj,i) ,j ( -P 135)
The six terms in the viscous diffusion and dissipation group
will be neglected due to their relatively small magnitudes as
compared to the terms in the turbulent diffusion group.
The first term in the field forces effects group is modeled by
Eq. (14). Similarly the second term ir, the group can be
modeled as
Ui-(vt/U)Fi,i - Fli(vt/af),i (36)
where we make a provisional assumption that af is equal to
16. .. *
- - - - - - - - - - - - - - - -o".1 . -
SECTION 6
CLOSURE OF THE TURBULENCE-ENERGY DISSIPATION-RATE EQUATION
The exact equation of the dissipation rate of turbulence
energy e for the carrier fluid appears in Appendix A and
consists of 52 terms. They are classified into groups similar
to those of the k equation.
Again we neglect all fourth-order correlations as mentioned
in the previous section.
All the terms in the first group, labeled Transient, are
approximated collectively by (QlE),t
The convection group consists of eight terms of which only
the first and the second are of higher magnitude than the
other six at large Reynolds number. This is based on an order
of magnitude analysis (Ref. 22) which shows that the first and
second terms are greater than the others by at least a factor
of (A/k) which is of order (R )1/2 Here . is the length
scale of the energy containing eddies, k is a Taylor's
microscale, and Rj is the Reynolds number based on 1.
The third term in the production group is decompcsed as
-2 vlQl(uiuj,kui,k),j = -2 v1Ql(ui,jUj,kui,k uiu j, kU k
Suiuj,kUi,kj) 37'
The third term on the right side of Eq. (37) and the second
term in the production group differ only in their signs and
thus cancel each other. The first term on the right side of
* . . .• .. ,
. . . . .. . . . . ....... :: .... .... .. . . . . .
Eq. (37), which represents the production of e by self-
stretching of the vortex tubes, is the dominant one at large
Reynolds number. It is larger than the second term by a factor
of Rj and larger than the first term in the production group
by a factor of R1/2
We, therefore, retain only -2v 1 Q1 ui,jujkui,k as the main
generation of e. This term and the extra production terms are
modeled collectively as
-2v1 Q1 ui jukuik + extra production terms = c eGke/k , (38)
where Gk is the total production of k discussed in Section 5,
and cel is a constant of value 1.43 (Ref. 13, under the provi-
sional assumption that it has the same value as in the case of
constant density). "Total" here means the production terms which
are common to the single-phase and two-phase k equations in
addition to the extra production and transfer terms.
The turbulent diffusion group contains six terms. At high
Reynolds number only the last two terms will be retained; they
are larger by at least a factor of R.0 than the other terms.
These two terms will be modeled collectively as
-2 v1(Qlujui,kui,jk + QI,j ujui,kUi,k)
S[QVt/e *ej +t (39)
All the terms in the extra production group except the mean
pressure gradient term are smaller than the main production
term, modeled in Eq. (38), by at least a factor of R-1 /2 and
I::
18. . . .
thus can be neglected. The mean pressure gradient term is
included in the production term (Eq. (38)).
The first term in the viscous diffusion and dissipation group
represents the main dissipation of e; it reduces to the
single-phase form when *1 equals unity. This term is larger
than the other terms in the group by a factor of R3 / 2 and thus
it is the only one retained. Now the total dissipation of e
includes this term in addition to the extra dissipation terms.
They are modeled collectively as Ql(E/k)(c. 2e + ce3ee) where
ee represents the extra dissipation terms appearing in the k
equation, cE2 is a constant of a value about 1.92 (Ref. 13) and
c,3 is a constant of value 1.2 (Ref. 3) (both under the
provisional assumption that they have the same values as in
the case of constant density).
Both terms in the field forces effects group appear multi-
plied by the kinematic molecular viscosity of the fluid and
therefore will be neglected due to their relatively small
magnitudes.
19
I....
. . . . . . . . . . . . . . .. . . . . . . . . .
SECTION 7
A SAMPLE APPLICATION
4'
As an example of the application of the modeled k and e equa-
tions, let us consider the motion of the dusty air during a
nuclear explosion (Ref. 6). In order to understand the essen-
tial features of the complicated phenomena, some highly ideal-
ized and well-controlled laboratory tests have been performed
or planned such as high speed wind above a sand bed in a wind
tunnel (Ref. 23) and shock wave sweeping a sand bed in a shock
tube (Ref. 24). To interpret and correlate the results from
such tests, we may use the two-dimensional version of the
present model in Cartesian coordinates x and y. The final
form of the modeled k and e equations is given in Appendix B
(under the assumptions that the diffusional fluxes in y direc-
tion are much larger than those in x direction, that the
effective coefficient K = I and that gy = g and gx = F lx =Fly
= 0). The remaining modeled equations contain two mean con-
tinuity equations (from Eqs. (8) and (9)), one mean global
continuity equation (Eq. (11)), four mean momentum equations
(from Eqs. (6) and (7) and Section 4) and one mean equation
of state such as the perfect gas equation
P = RplT 1 (40)
where R is the specific gas constant and T1 is the absolute
temperature of the air. Thus, for isothermal problems (T1 =
constant) we have ten equations for ten unknowns Q,, Q2, P1,
Ux , Uy, Vx, Vy, P, k and e (from which + = Qi/pi and +2 =
Q2/P2 can be readily obtained). With proper initial and
boundary conditions, these equations can be solved numerically
by a marching finite-difference procedure described in Ref. 5.
20
The results of such calculation will be presented in a forth-
coming report.
If T1 is variable and the energy exchange between the air and
dust can be neglected we need to include a mean energy equa-
tion for the air in the numerical solution procedure.
If the energy exchange between the air and dust (at tempera-
ture T2 ) is not negligible, we need to include the mean energy
equations for both air and dust in the numerical solution.
21
- - . . . - -.
-i
SECTION 8
CONCLUSION
The k - e turbulence model for an incompressible dilute
suspension of Ref. 3 has been extended to a compressible
dispersed two-phase flow by introducing the apparent densi-
ties Q1 and Q2 and the material density pI as new variables.
The fluid has variable density and the particles have con-
stant density. This allows the application of the model to a
wider class of practical problems.
As in Ref. 3, the k and e equations are first rigorously
derived from the two-phase momentum equations and then their
closure is provided. This is in contrast to the usual
approach based on ad hoc modifications of the single-phase
turbulence-kinetic-energy and length-scale equations.
The proposed closure of the equations accounts for the inter-
action between the two phases and its influence on the turbu-
lence structure. Sparseness of experimental data for variable-
density flows necessitates some provisional assumptions that
forms and values of the coefficients in the turbulent corre-
lations of constant-density flows apply to those of variable-
density flows. Such assumptions indicate areas of needed
experimental investigations which, when completed, can in turn
modify the present work and enhance its validity.
22
........
............................
SECTION 9
LIST OF REFERENCES
1. H. Danon, M. Wolfshtein and G. Hetsroni, "NumericalCalculations of Two-Phase Turbulent Round Jet," Int. J.Multiphase Flow, 3, 223-234 (1977).
2. Zh. D. Genchev and D. S. Karpuzov, "Effects of theMotion of Dust Particles on Turbulent TransportEquations," J. Fluid Mech., 101, 833-842 (1980).
3. S. E. Elghobashi and T. W. Abou-Arab, "A Two-EquationTurbulence Model for Two-Phase Flows," Phys. Fluids,26, 931-938 (1983).
4. D. Modarress, J. Wuerer and S. Elghobashi, "An Experi-mental Study of a Turbulent Round Two-Phase Jet,"AIAA/ASME Third Joint Thermophysics, Fluids, Plasma andHeat Transfer Conference, St. Louis, Missouri (1982).
5. S. Elghobashi, T. Abou-Arab, M. Rizk and A. Mostafa,"Prediction of the Particle-Laden Jet with a Two-Equation Turbulence Model," Int. J. Multiphase Flow, 10,697-710 (1984).
6. G. Ullrich, "Simulation of Dust-Laden Flow Fields,"Seventh International Symposium, Military Applicationsof Blast Simulation, Medicine Hat, Alberta, Canada,June 13-27, 1981.
7. K. Fujii and P. Kutler, "Computations of Two-PhaseSupersonic Nozzle Flows by a Space Marching Method,".AIAA Paper, Orlando, Florida, January, 1982.
8. P. A. Libby, "A Provisional Analysis of Two-DimensionalTurbulent Mixing with Variable Density," Proceedings ofNASA Conference on Free Shear Flows, Langley Field,Hampton, VA, July 20-21, 1972, pp. 427-462.
9. A. A. Mostafa and S. E. Elghobashi, "A Two-Equation . -
Turbulence Model for Jet Flows Laden with VaporizingDroplets," Int. J. Multiphase Flow, 11, 515-533 (1985).
10. S. L. Soo, Fluid Dynamics of Multiphase Systems,(Blaisdell, Waltham, Ma, 1967).
23- . .- - - - - - -- - - - - - - - - - - - - - - - - . . . . . . .
--
11. A. Favre, "Equations des gaz turbulents compressibles,"Journal de M~canique, 4, 363 (1965). 04
12. J. L. Lumley, "Modeling Turbulent Flux of Passive ScalarQuantities in Inhomogeneous Flows," Phys. Fluids, 18,619-621 (1975).
13. B. E. Launder, A. Morse, W. Rodi and D. B. Spalding,"The Prediction of Free Shear Flows-A Comparison ofthe Performance of Six Turbulence Models," Proceedingsof NASA Conference on Free Shear Flows, Langley Field,Hampton, VA, July 20-21, 1972, pp. 361-426.
14. J. L. Lumley, "Computational Modeling of TurbulentFlows," Adv. Appl. Mech. 18, 123-177 (1978).
15. J. L. Lumley, "Pressure-Strain Correlation," Phys.Fluids, 18, 750 (1975).
16. B. E. Launder, "Heat and Mass Transport," in Turbulence,edited by P. Bradshaw (Springer-Verlag, Berlin, 1976).
17. K. Hanjalic and B. E. Launder, "A Reynolds Stress Modelof Turbulence and Its Application to Thin Shear Flows,"J. Fluid Mech. 52, 609-638 (1972).
18. B. E. Launder, G. J. Reece and W. Rodi, "Progress in theDevelopment of a Reynolds-Stress Turbulence Closure,"J. Fluid Mech. 68, 537-566 (1975).
19. F. H. Harlow and P. I. Nakayama, "Transport of Turbu-lence Energy Decay Rate," Report LA-3854, Los AlamosScientific Laboratory of the University of California,Los Alamos, NM, 1968.
20. B. T. Chao, "Turbulent Transport Behavior of SmallParticles in Dilute Suspension," Osterr. Ing. Arch. 18.7-21 (1964).
21. C. M. Tchen, "Mean Value and Correlation ProblemsConnected with the Motion of Small Particles Suspendedin a Turbulent Fluid," Martinus Nijhoff, The Hague(1947), Ch. 4, p. 72.
22. H. Tennekes and J. L. Lumley, A First Course inTurbulence, (MIT, Cambridge, MA, 1972).
23. B. Hartenbaum, "Lofting of Particulates by a High-SpeedWind," DNA Report No. 2737, Defense Nuclear Agency,Washington, D.C. 20305, September 1971.
24
J.
24. D. R. Ausherman, "Initial Dust Lofting: Shock TubeExperiments," DNA Report No. 3162F, Defense Nuclear
Agency, Washington, D.C. 20305, September 1973.
25
APPENDIX A
EXACT EQUATIONS OF KINETIC ENERGY OF TURBULENCEAND DISSIPATION RATE OF THAT ENERGY
The exact equation of the turbulence kinetic energy (k uiui/2)
of the carrier fluid is
(Q, uiui/2) ,t + Qi,t uiui/2 + Uit ujql + Ui uiql1 tTransient
+ ui(qlui) t + [(QUjk) ,j + k(QUj) , -"
Convection
[(Qiui), uiuj + Ui jqjuiuj + Ujjqiujui]
Production
- (1-K+2)(uip),i-pui,i+Koluipi + uiui(Qluj),j
Turbulent Diffusion
+ Qluiujui,J [KP,ji 1 ui + QIUi Uiuj,j + Ui ui(qluj),j
+ Uj ui(qlui),j + (UiUj),jqlui + UiUjuiql, j
Extra Production and Transfer
+ qlui(uiuj),j + uiuiujql, j + KF[ 2 ui(vi-ui)
4-(i-Ui)0 2ui +- 02ui(vi-ui)] + ~i~ i(ui j+uj~i) '9Extra Dissipation Dissipation
27 PREVIOUSPAGE
............. G.
+ +,Jui~ui,j4Uj,i) + (U~+ji)1Ij
+ ~(uij~uj~i)~1J~ui L(+1uA,A),iui+(UA),u
Viscous Diffusion and Dissipation
+ (Oiuj~i),iui~j [i uiq, " if ii
Field Forces Effects
The exact equation of the dissipation rate of k
(E v 1 ui,kui,k) is
2ai (Qlui) ,ktui,k + (qt 1 ) ,ktui,k + (qlui) ,ktui,k
Transient
4-IQ+e, + e(Q1 Uj),j + 2v, [(Q1 U±) ,j uiku~
+(QlUi),kuj,jui,k + (Q1Ui),jkui,kuj +(QlUj),kui,kui,jConvection
+(Qlt~j),Jk uiui,k + QltJ1 ui,kuj,jk] = 2 1 Q1 ui,kui,kui,.4
Q1 uiui,jkuj,k + Q(uiuj,kui,k),j]
Production
+ 211+[ 2 (vj-uj)] lkuik + [02(Vi-Ui)],kUi,k
Extra Dissipation
" [02(vi-ui)],kui,k - 2v, [Ql,jk ujui~k
28
. . . . n
+Ql,k(uiuj j+ujui,j)Ui.,k+ Q1 ,j Uiuj kujkl O,k';i,k
Turbulent Diffusion
+ Qlujuikui,jk + Q1,jujui,kui,kl + 2vaj (K+2P, i;) ~
-KP,ki Olui,k P,i ui,k(KOl),k -(K~jp'i),k'al,k
- ~jLj~1.q~j),kuj,jj ku4
- ujtuik(qUi) ,jk - t qlui kui jk -ui(q 1 J~j) ,ki
Extra-
- [(qUj),jtikuik + CqU),kui~iuik]
- Production
- [UiUq 1 jk4qlk(Uitj~j+u *V*) i~
- [q1 (Ui,tj~j),+UiUji kqlj+q1 (Uj,jUi) ,k]ui,k
- U2 Ui kqljjik - uiujql1jk+qkuiJJujui~j)j
- [ql(uiuj k) j+uiujkqlj4ql(ujukjuukl i
+ [01(ui,j+uj,i)],jkui,kc U ~L1~ ikui,k
+ (01 UP,A) ikui, k + (0 lul, ),ikui, k1
+ 2v, giql1kuik+fli:kuik]
Field Forces Effects
29
. . .. . . . . . . ... ... ... ..
. . .-
a.
a.
30
S.
a ~ I
. . . . .r VT 7% . . .- , .7 7
APPENDIX B
THE MODELED FORM OF < AND E EQUATIONS
The modeled conservation equations of the kinetic energy and
the dissipation rate of that energy for the carrier fluid in
the sample application are listed here.
(i) The k equation:
Ql~tj+ Q, Uxkx+U ky
Transient Convection
i vtUxY(QIUx),y -2 k(QIUy) y l]
+ c 5(1)(vt Q), (v.,U2,y - 4 y
Production (P)
Q .4 U 2, -2 k Uyy- 43- k Uyy
+ [1 ()k, + Q. Qk(t),Y]~+ Ql a 'y 2 1l 0 'y'Y "Diffusion
' ,y tQUyUy,yy
IV
+i c UP-X ISV~xY- t
-2c;5*y[ (U)( Q + o.3f( Q1 ) .
Extra Production (Pe)
+ (U + U2 [(_ .+
31
.-..-..... i
•-p
Extra Dissipation (ee)
Fe+ c~5 ~)~(- T_; f(w)dw)t Q 4j
41
-Qo- +] ,.'r
Q E: t Q +1 Ql(t)Y
Dissipation Field Forces Effects
(ii) The e equation:
1itI + Q,{£y +
Transient Convection
E c / ( + [Ql(Vt/%)£,y + 21 QC (Vt/-.) j{Total Production Diffusion
S[ /k) (cE2 E + c 3 Eed
Total Dissipation
The notations used in the partial derivatives have been
explained in Section 2 of the text, thus ( means
9( )/at, ( ) means 3( )/Dx and ( ) means 9( )/ay, where
x and y are the distances along the horizontal and vertical
directions, respectively. The values of the constants
appearing in the two equations are:
c E= 1.43, c = 1.92, c 3 = 1.2, c = 0.1, a= 1.0,
" =1.3, T = 1.0
32
.".-.-. . .. .. ,. .. .- .'... .. . . .. ,-•g-. -. •. ."., - . - -- -
APPENDIX C
NOMENC LATURE V
.'. 4
, C c c c c c c2. 2 e l E e2 e3 01' '2 C03 C04 ' 5
constants in the turbulence model
d particle diameter
Dii an expression defined in Eqs. (20)
em, en unit vectors
f(w) an expression defined in Eq. (30)
fi fluctuating component of body forces other than
that due to gravity
F i instantaneous (in Eqs. (1) and (2)) or time-mean
component of body forces other than that due to
gravity
F interface friction coefficient
component of gravitational acceleration
Gkc total production of k
k turbulence kinetic energy of the fluid
K local effectiveness of momentum transfer from the - -
particle phase to the fluid
j length scale
p pressure fluctuation
P instantaneous (in Eqs. (1) and (2)) or time-mean
pressure
P an expression defined in Eqs. (20)
P production (in Appendix B)
e extra production (in AppEndix B)
q fluctuation of the apparent density
Q instantaneous (in Eqs. (l)-(5)) or time-mean
apparent density
R specific gas constant
Reynolds number based on I
33
- . '. --.-
...........-.-.-.-.:..- ....... -.. : -......... . ... ..... . .... . .. . .
t time
T absolute temperature
ui fluctuating velocity component of the fluidUi instantaneous (in Eqs. (1)-(3)) or time-mean
velocity component of the fluid
Vi fluctuating velocity component of the particle
phasei instantaneous (in Eqs. (1)-(4)) or time-mean
velocity component of the particle phase
Xi rectangular spatial coordinate
x horizontal coordinate
y vertical coordinate
Greek symbols
E an expression defined in Eq. (31)
an expression defined in Eq. (32)
* ij Kronecker symbole dissipation rate of k
ee extra dissipation terms in the k equation
X Taylor's microscale
viscosity
kinematic viscosity
Wt kinematic eddy viscosity
p material density
, aq, ark, f , ae
turbulent Schmidt numbers
fluctuation of the volume fraction
* instantaneous (in Eqs. (1)-(5)) or time-mean volume
fraction
frequency of turbulence
an expression defined in Eq. (27)
02 an expression defined in Eq. (28)
OR an expression defined in Eq. (29)
34
F.: Subscripts1 fluid phase
2 particle phase
It partial derivative with respect to t
partial derivative with respect to xj
Superscript
time-averaged value
fluctuating component
* a%
35
. . . .
.2
%1!
-.
"...
F-S
.4.
•".
3 6"'
°.4.
-- e
DISTRIBUTION LIST -P
DEPARTMENT OF DEFENSE HARRY DIAMOND LABORATORIESATTN: SLCHD-NW-RH
AIR FORCE SOUTHATTN: US DOC OFFICER U S ARMY MATERIAL COMMAND
ATTN: DRCDE-DASST TO THE SECY OF DEFENSE ATOMIC ENERGY
ATTN: EXECUTIVE ASSISTANT U S ARMY MATERIAL TECHNOLOGY LABORATORYATTN: DRXMR-HH
COMMANDER IN CHIEF, ATLANTICATTN: J36 U S ARMY STRATEGIC DEFENSE CMD
ATTN" OACS-BM/TECH DIVDEFENSE INTELLIGENCE AGENCY
ATTN: DT-2 U S ARMY STRATEGIC DEFENSE COMMANDATTN: RTS ATTN: ATC-D WATTSATTN: RTS-2B ATTN: ATC-R ANDREWS
DEFENSE NUCLEAR AGENCY US ARMY WHITE SANDS MISSILE RANGEATTN: SPAS ATTN: STEWS-NR-CF R VALENCIAATTN: STSP
4 CYS ATTN STI-CA DEPARTMENT OF THE NAVY
DEFENSE TECHNICAL INFORMATION CENTER NAVAL ELECTRONICS ENGRG ACTVY. PACIFIC12 CYS ATTN: DO ATTN: CODE 250 D OBRYHIM
FIELD COMMAND DNA DET 2 NAVAL ORDNANCE STATIONLAWRENCE LIVERMORE NATIONAL LAB ATTN: .J MILLER CODE 5422 1
ATTN: FC.1NAVAL RESEARCH LABORATORY
FIELD COMMAND DEFENSE NUCLEAR AGENCY ATTN: CODE 2627 TECH LIBATTN: FCPR ATTN: CODE 4040 D BOOKATTN: FCT COL J MITCHELLATTN: FCTT W SUMMA NAVAL SEA SYSTEMS COMMANDATTN: FCTXE ATTN: SEA-0351
JOINT CHIEFS OF STAFF NAVAL SURFACE WEAPONS CENTERATTN: J-5 NUC & CHEM DIV ATTN: CODE K82
ATTN: CODE R44 H GLAZJOINT STRAT TGT PLANNING STAFF
ATTN: JLK DNA REP OFC OF THE DEPUTY CHIEF OF NAVAL OPSATTN: JLKS ATTN: NOP 654 STRAT EVAL & ANAL BRATTN: JPPFM
STRATEGIC SYSTEMS PROGRAMS (PM-1)UNDER SECY OF DEF FOR RSCH & ENGRG ATTN: SP-272
ATTN: STRAT & SPACE SYS (OS)ATTN: STRAT & THR NUC FOR J THOMPSON DEPARTMENT OF THE AIR FORCEATTN: STRAT & THEATER NUC FORCES
AIR FORCE SYSTEMS COMMANDDEPARTMENT OF THE ARMY ATTN: DLW
DEP CH OF STAFF FOR OPS & PLANSATTN: DAMO-NCZ
37
-- . ~ .- - - - - - - - ---
-- .-m('., *...- " - , . '. ",,' . . '-., -- - - - - - - - - - - - - - - - - - - - --- -. .". .. -" ' - .' ." . . - - • .- - - .-- . . .
• " " " " " "" -" " -. " -2- -'Z'.,
DEPARTMENT OF THE AIR FORCE (CONTINUED) ATTN: L-122 S SACKETTATTN: L-203 T BUTKOVICH
AIR FORCE WEAPONS LABORATORY, AFSC ATTN: L-22 D CLARKATTN: NTED J RENICK ATTN: L-8 P CHRZANOWSKIATTN: NTED LT KITCHATTN: NTED R HENNY LOS ALAMOS NATIONAL LABORATORYATTN: NTEDA ATTN: Al12 MS R SELDEN .J6
ATTN: NTES ATTN: M T SANDFORDATTN: SUL ATTN: R WHITAKER
AIR FORCE WRIGHT AERONAUTICAL LAB SANDIA NATIONAL LABORATORIES .'ATTN: FIBC ATTN: D J RIGALIATTN: FIMG ATTN: ORG 7112 A CHABAI
ATTN: R G CLEMAIR FORCE WRIGHT AERONAUTICAL LAB
ATTN: AFWAL/MLP OTHER GOVERNMENTATTN: AFWAL/MLTM
CENTRAL INTELLIGENCE AGENCYAIR UNIVERSITY LIBRARY ATTN: OSWR/NED
ATTN: AUL-LSE -,
DEPARTMENT OF THE INTERIORBALLISTIC MISSILE OFFICE/DAA ATTN: D RODDY ..
ATTN: CAPT T KING MGENATTN: CC MAJ GEN CASEY DEPARTMENT OF DEFENSE CONTRACTORSATTN: ENSNATTN: ENSR ACUREX CORP
ATTN: C WOLFDEPUTY CHIEF OF STAFF
ATTN: AF/RDQI AEROSPACE CORPATTN: H MIRELS
DEPUTY CHIEF OF STAFFATTN: AFRDS SPACE SYS & C3 DIR APPLIED RESEARCH ASSOCIATES, INC
ATTN: N HIGGINS
FOREIGN TECHNOLOGY DIVISION, AFSC
ATTN: SDBG APPLIED RESEARCH ASSOCIATES, INCATTN: S BLOUIN
STRATEGIC AIR COMMAND 41ATTN: NRI/STINFO APPLIED RESEARCH ASSOCIATES, INC
ATTN: D PIEPENBURGSTRATEGIC AIR COMMAND
ATTN: XPFS BELL AEROSPACE TEXTRONATTN: C TILYOU
STRATEGIC AIR COMMANDATTN: XPQ CALIFORNIA RESEARCH & TECHNOLOGY, INC
ATTN: K KREYENHAGEN161 ARG ARIZONA ANG ATTN: M ROSENBLATT
ATTIN: LTCOL SHERER "-CALIFORNIA RESEARCH & TECHNOLOGY, INC
DEPARTMENT OF ENERGY ATTN: F SAUER
UNIVERSITY OF CALIFORNIA H-TECH LABS, INC-, LAWRENCE LIVERMORE NATIONAL LAB ATTN: B HARTENBAUM
ATTN: D BURTONATTN: L-10 H KRUGER INFORMATION SCIENCE. INCATTN: L-10 J CAROTHERS ATTN: W DUDZIAKATTN: L-122 G GOUDREAU
• .'
38
N .
DEPT OF DEFENSE CONTRACTORS (CONTINUED S-CUBEDATTN: A WILSON
KAMAN SCIENCES CORPATTN: L MENTE S-CUBEDATTN: R RUETENIK ATTN: C NEEDHAMATrN: W LEE
SCIENCE APPLICATIONS INTL CORPMAXWELL LABORATORIES. INC ATTN: H WILSON
ATTN: J MURPHYSCIENCE APPLICATIONS INTL CORP 4%
MERRITT CASES, INC ATTN: J COCKAYNEATTN: J MERRITT ATTN: W LAYSON
NEW MEXICO ENGRG RESEARCH INSTITUTE SCIENCE APPLICATIONS INTL CORPATTN: G LEIGH ATTN: A MARTELLUCCI
R & D ASSOCIATES SCIENCE APPLICATIONS INTL CORPATTN: A KUHL ATTN: G BINNINGER
ATTN: C K B LEE2 CYS ATTN: G YEH TRW ELECTRONICS & DEFENSE SECTOR
ATTN: J LEWIS ATTN: G HULCHERATTN: P RAUSCH ATTN: P DAI
2 CYS ATTN: S ELGHOBASHIWEIDLINGER ASSOC, CONSULTING ENGRG
R & D ASSOCIATES ATTN: P WEIDLINGERATTN: P MOSTELLER
39
...............................
- . ;- I I . I*. !*" ,. . . . .,.. . . . . . . . . . .- : -, l i Jh , ,,1 .- ,_ -- S ,-
-
ft.q/f-
r---
A'
'f,
aft
,-q
ft,%
-ft
40
;:1 - . -. - ~-- 77. -. ~ -
.Ak4: