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

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MICROCOPY RESOLUTION TISI CHART

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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

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___________________________ 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

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00 FORM 1473, 84 MAR 83 APR etiton may be used until exlhaulted.All other etitons are obsolete.

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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

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- . ;- I I . I*. !*" ,. . . . .,.. . . . . . . . . . .- : -, l i Jh , ,,1 .- ,_ -- S ,-

-

ft.q/f-

r---

A'

'f,

aft

,-q

ft,%

-ft

40

;:1 - . -. - ~-- 77. -. ~ -

.Ak4: