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NASA Technical Memorandum 102293

ICOMP-89-

18

Calculation of Reattaching Shear Layers

in Divergent Channel With a Multiple-

Time-Scale Turbulence Model

S.-W. Kim

Institute for Computational Mechanics in Propulsion

Lewis

Research Center

Cleveland Ohio

August 1989

( N A S A - T M - 1 6 2 2 9 3 ) CALCULATIQN O F 6 A T T A C H I k G

S H E A R

L A Y E R S

I N U I V € R G € r J T

CHANIvEL k T 1 H A

HULTIPLE-TIpk-SCALE

T U H H U l t Y C E M 3 P E l ( N A S A ,

L e w i s Research

t e n t e r

7 p

c s c c

200

N93 2

ICOMP

CASE

WfSTfRN

REYRVf

WNERSlTY

18749

U r l c l as

0225964

3 / 3 4


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CALCULATI ON OF REATTACHI NG SHEAR LAYERS I N DI VERGENT CHANNEL W TH

A MULTI PLE- TI ME- SCALE TURBULENCE MODEL

S.-W Ki m*

I nst i t ut e f or Comput at i onal Mechani cs i n Pr opul si on

Lewi s Research Cent er

Cl evel and, Ohi o

44 35

SUMMARY

Numer i cal cal cul at i ons of t ur bul ent r eat t achi ng shear l ayer s i n a

di ver gent channel ar e pr esent ed. The t ur bul ence i s descr i bed by a

mul t i pl e- t i me- scal e t ur bul ence model . The t ur bul ent f l ow equat i ons ar e sol ved

by a cont r ol - vol ume based f i ni t e di f f er ence met hod. The comput at i onal r esul t s

ar e compar ed wi t h t hose obt ai ned usi ng k-E cur bul ence model s and al gebrai c

Reynol ds st r ess t ur bul ence model s. I t i s s own t hat t he mul t i pl e- t i me- scal e

t ur bul ence model yi el ds si gni f i cant l y i mpr oved comput ati onal r esul t s t han the

ot her t ur bul ence model s i n t he r egi on wher e t he t ur bul ence i s i n a st r ongl y

i nequi l i br i ums tat e.

*Work f unded under Space Act Agr eement C99066G.


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A,

A,

Cf

cP

CPf

CPJ

Ct R

CPf

f

H

k

kP

kt

P

Pr

Rt

u

a

NOMENCLATURE

coef f i c i ent f or t angent i al vel oci t y cor r ect i on

coef f i c i ent f or t ransverse vel oci t y cor r ect i on

f r i ct i on coef f i ci ent ( - rw(0. 5pU,

2

) )

pr essur e coef f i ci ent (- p/ (0. 5pU, 2 ) )

const ant coef f i ci ent f or eddy vi scosi t y equat i on ( - 0. 09)

t ur bul ence model const ant s f or equat i on

(R-1,3)

t ur bul ence model const ant s f or Et equat i on

(R-1,3)

const ant coef f i ci ent -0 .09 )

wal l dampi ng f unct i on f or eddy vi scosi t y equat i on

wal l dampi ng f unct i on f or

e w

equat i on

hei ght of backward- f aci ng s t e p

t ur bul ent ki net i c ener gy (k-$

+

kt)

P

t ur bul ent ki net i c ener gy of eddi es i n pr oduct i on range

t ur bul ent ki net i c ener gy of eddi es i n di ssi pat i on r ange

pr es sur e

pr oduct i on rate of t ur bul ent ki net i c ener gy

t urbul ent Reynol ds number (-k2 ( v e l ) )

i nl et f l ow vel oc i t y

t i me aver aged vel oci t y ( ={u, v) )

f r i c t i on vel oc i t y (==(rwp))

Reynol ds st r ess ( i =l , 2, 3and j =l , 2, 3)

vel oci t y vect or ( =( u, v) )

spat i al coor di nat es ( =( x, y, z) )

r eat t achment l ocat i on

wal l coor di nat e (=u, y/ v)

def l ect i on angl e of t he top wal l

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‘ P

Et

‘ 1

n

ener gy t r ansf er r at e f r omproduct i on r ange to

di ssi pat i on range

di ssi pat i on r at e of t ur bul ent ki net i c ener gy

di ss i pat i on rate i n near - Val 1 equi l i br i umr egi on

von Kar man const ant ( - 0. 41)

mol ecul ar vi scosi t y

ef f ect i ve vi scosi t y ( =p+pt )

t ur bul ent vi scosi t y

ki nemat i c vi scos i t y of f l ui d

t ur bul ent eddy vi scosi t y

densi t y

t ur bul ent Prandt l number f o r equat i on

t ur bul ent Prandt l number f o r kt equat i ons

t ur bul ent Prandt l number f or eP equat i on

t ur bul ent Prandt l number f or e t equat i on

wal l shear i ng st r ess

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INTRODUCTION

The exper i ment al st udy of a r eat t achi ng shear l ayer i n a di ver gent

channel

[ l ] was desi gned to t est t he pr edi ct i ve capabi l i t y of var i ous

t ur bul ence model s, t o i dent i f y any def i ci ency i n t ur bul ence cl osur e model s,

and thus t o i mpr ove pr edi ct i ve capabi l i t y of t ur bul ence model s.

The f l ow

geomet r y i s s hown i n Fi gur e 1.

The hei ght of t he backwar d- f aci ng st ep

i s

smal l er t han t he boundary l ayer t hi ckness of t he i ncom ng f l ow. Abr upt

br eakdown of t he boundar y l ayer gener at ed a str ongl y i nequi l i br i um

t ur bul ent f l ow. Fur t her mor e, a st r ong pr essur e gr adi ent was generat ed by

var yi ng the di ver gence angl e of t he t op wal l t o st udy i t s ef f ect on t he

devel opment of t he t ur bul ence f i el d, especi al l y t he Reynol ds st r ess, and

t he r eat t achment process.

A number of t ur bul ence model s , such as

k- e

t ur bul ence model s and al gebr ai c Reynol ds st r ess t ur bul ence model s

( ARSM) ,

wer e shown to yi el d poor comput at i onal r esul t s f or t he f l ow [ 1, 2] . I t i s

al so s hown i n Ref er ences 1 and 2 t hat a modi f i ed ARSM, wi t h modi f i cat i ons

i n t he di ssi pat i on r at e equat i on, yi el ded comput at i onal r esul t s whi ch ar e

i n good agreement wi t h measur ed dat a. However , gener al i t y of t he i mpr oved

pr edi ct i ve capabi l i t y f or ot her compl ex t ur bul ent f l ows has not been shown

yet .

I t has been shown pr evi ousl y t hat t he hi gh Reynol ds number

mul t i pl e- t i me- scal e t ur bul ence model yi el ds accur at e comput at i onal r esul t s

f or a number of compl ex t ur bul ent f l ows such as a wal l j et f l ow, a

wake- boundary l ayer i nt er act i on f l ow, a conf i ned coaxi al j et wi t hout swi r l

and a conf i ned coaxi al swi r l i ng j et to name a f ew

[ 3, 4] . I n t he

si ngl e- t i me- scal e ur bul ence model s such as k-c t ur bul ence model s,

al gebr ai c st r ess t ur bul ence model s, and Reynol ds s t r ess t ur bul ence model s,

a si ngl e t i me scal e i s used t o expr ess bot h the t ur bul ent t r anspor t and the

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di ssi pat i on of t he t ur bul ent ki net i c ener gy. However , t hi s pr act i ce i s

i nconsi st ent wi t h physi cal l y obser ved t ur bul ence i n t he sense t hat t he

t ur bul ent t r anspor t i s r el at ed t o the t i me scal e of ener gy cont ai ni ng l ar ge

eddi es and t he di ssi pat i on of t ur bul ent ki net i c ener gy

i s

r el at ed t o the

t i me scal e of f i ne scal e eddi es i n t he di ssi pat i on r ange. The

s i ngl e- t i me- scal et ur bul ence model s yi el d r easonabl y accur at e comput at i onal

r esul t s f or s i mpl e tur bul ent f l ows; however, t he pr edi ct i ve capabi l i t y

degenerat es r api dl y as

t ur bul ent f l ows t o be sol ved become more compl ex. I n

t he mul t i pl e- t i me- scal e ur bul ence model s

[3-71,

he t ur bul ent t r anspor t of

mass and moment um i s descr i bed usi ng t he t i me scal e of t he l arge eddi es and

t he di ssi pat i on r at e i s descri bed usi ng the t i me sca le of t he f i ne- s cal e

eddi es. The i mpr oved comput at i onal r esul t s obt ai ned usi ng t he

mul t i pl e- t i me- scal e t ur bul ence model f or compl ex t ur bul ent f l ows can

be

att r i but ed t o t he physi cal l y consi st ent nat ur e of t he t ur bul ence model s

di scussed above.

I n numer i cal cal cul at i ons of t ur bul ent f l ows, wal l f unct i on met hods

are most f r equent l y used t o model t he near - wal l r egi on. These met hods have

been der i ved f r om t he l ogar i t hm c vel oci t y pr of i l e based on exper i ment al

observat i ons t hat t he t ur bul ence i n t he near - wal l r egi on can be descr i bed

i n t er ms of t he wal l shear i ng st r ess.

Therefore, t hese met hods ar e not

val i d i f t he l ogar i t hm c vel oc i t y prof i l e no l onger prevai l s i n t he

near - wal l r egi on. For exampl e, t he l ogar i t hm c vel oci t y pr of i l e no l onger

pr evai l s i n t he near - wal l or i n t he wake r egi ons

of

unst eady t ur bul ent

f l ows [ 8 ] , t her ef or e wal l f unct i on met hods c an not be appl i ed.

Many

ot her

cases f or whi ch t he wal l f unct i on methods ar e i nval i d can be f ound i n

Ref er ences

9

and 10. Due to t hi s l i m t ed appl i cabi l i t y of t he wal l f unct i on

met hods, numerous al t ernat i ve appr oaches have been pr oposed. I n t he

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al t ernat i ve appr oaches, t he near - wal l l ow t ur bul ent Reynol ds number r egi on

i s i ncl uded i nt o numer i cal anal yses t o overcome t he shor t com ngs of t he

wal l f unct i ons methods. Vari ous t ur bul ence model s whi ch i ncl ude t he

near - wal l l ow t ur bul ence r egi on can be cl assi f i ed as t wo- l ayer (or

mul t i - l ayer) t ur bul ence model s [ l l ] and l ow Reynol ds number t urbul ence

model s [lo] based on t he way t he near - wal l r egi on i s t r eat ed. Mor e det ai l ed

di scussi on on t he advant ages and di sadvant ages of var i ous near - wal l

t urbul ence model s can be f ound i n Ref erences 9 and 10.

I n

t he pr esent st udy, t he near - wal l t ur bul ence i s descr i bed by a

par t i al l y l ow Reynol ds number appr oach. I n

t he

model

[ 9 ]

nl y t he

t ur bul ent ki net i c ener gy equati ons ar e extended t o i ncl ude t he near - wal l

l ow t ur bul ence r egi on and t he ener gy t r ansf er r at e and t he di ssi pat i on rat e

i nsi de the near - wal l l ayer ar e obt ai ned f r om al gebr ai c equat i ons. The

al gebr ai c equat i ons were obt ai ned f r om a k- equat i on t ur bul ence model

[ 12] .

I t woul d be appr opr i ate to cl assi f y the met hod as a part i al l y l ow Reynol ds

number approach to di st i ngui sh i t f r om ot her cl asses of met hods. Thi s

appr oach was f i r st used i n Chen and Pat e1 t o sol ve t urbul ent f l ows over

a i r f o i l s [ 1 3 ] . Advant ages of t he par t i al l y l ow Reynol ds number approach

over t he other met hods can be summar i zed as f ol l ows. The t urbul ence l engt h

scal e of t he ext er nal f l ows i s r el at ed to the f l ow f i el d char acter i st i cs

[14]. n t he ot her hand, t he t ur bul ence l engt h scal e

of

boundar y l ayer

f l ows i s st r ongl y r el at ed to t he nor mal di st ance f r om the wal l . Thi s

charact er i st i c of t he wal l bounded t ur bul ent f l ows can be descr i bed qui t e

natur al l y by t he pr esent cl ass of t ur bul ence model s. The l ow Reynol ds

number t urbul ence model s can al so be used t o descr i be t he wal l bounded

t urbul ent f l ows; however , mor e gr i d poi nt s have t o be used to r esol ve t he

st eep di ssi pat i on r at e

i n the near - wal l regi on. I t i s al so i nt erest i ng t o

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not e t hat var i ous si m l ar k- equat i on t ur bul ence model s, whi ch f or m t he

basi s of t he pr esent near - wal l t ur bul ence model , yi el d accur at e

comput at i onal r esul t s f or a cl ass of si mpl e t ur bul ent boundar y l ayer f l ows

[ 15] , t ur bul ent f l ows wi t h dr ag reduct i on

[16]

and

f ul l y

devel oped

unst eady t ur bul ent pi pe f l ows

[ 8 ] .

However , t he k - equat i on t ur bul ence model

i t sel f i s l ess usef ul f or separ at ed and/ or swi r l i ng t ur bul ent f l ows wi t h

compl ex geomet r y due t o l ack of a syst emat i c met hod t o eval uat e t he

t ur bul ence l engt h scal e. Devel opment

of

t he near - wal l t ur bul ence model and

i t s appl i cat i on t o f ul l y devel oped t ur bul ent channel and pi pe f l ows can be

f ound i n Ref erence

9.

I t has been shown i n t he r ef erence t hat t he pr esent

near - wal l t ur bul ence model can r esol ve t he over - $hoot phenomena of t he

t ur bul ent ki net i c ener gy and t he di ssi pat i on r at e i n t he r egi on ver y cl ose

t o t he wal l and t hat si gni f i cant l y i mpr oved comput at i onal r esul t s f or t he

t ur bul ence st r uct ur e i n t he near - wal l r egi on ar e obt ai ned. I ncor por at i on of

t he same near - wal l t ur bul ence model i nt o a

k- e

t ur bul ence model and i t s

appl i cat i on to compl ex turbul ent f l ows such as a super soni c t ur bul ent f l ow

over a compressi on r amp and a t r ansoni c f l ow over an axi symmet r i c cur ved

hi l l can be f ound i n Ref erences

17

and 18, r espect i vel y.

The numer i cal method used her ei n i s based on t he pr essure

cor r ect i on

met hod [ 19] whi ch has been used most ext ensi vel y

t o

sol ve i ncompr essi bl e

f l ows t he domai n of whi ch can be di scr et i zed

by

an or t hogonal mesh.

However , t he pr esent numeri cal met hod i s appl i cabl e f or bot h i ncompr essi bl e

and compr essi bl e f l ows wi t h ar bi t r ar y, compl ex geomet r i es. The capabi l i t y

t o sol ve compr essi bl e f l ows i s achi eved by i ncl udi ng a convect i ve

i ncr ement al pr essur e term i nt o t he pr essur e cor r ect i on equat i on [ 17, 18] .

n

t he met hod, t he vel oci t i es ar e l ocat ed at t he same gr i d poi nt s and t he

pr essure i s l ocat ed at t he cent r oi d of t he cel l f or med by t he f our adj acent

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vel oc i t y gr i d poi nt s .

Thi s gr i d l ayout was f ound t o be qui t e sui t abl e t o

sol ve f l ows wi t h compl ex geomet r i es

[ 17] . The accur acy and t he conver gence

nat ure of t he numer i cal met hod have been demonst r ated by sol vi ng

a

number

of f l ow cases. The exampl e probl ems consi der ed i n Ref er ences 17 and

18

i ncl ude: a devel opi ng channel f l ow, a devel opi ng pi pe f l ow,

a

t wo- di mensi onal l am nar f l ow i n

a 90

degr ee bent channel , pol ar cavi t y

f l ows, a t ur bul ent super soni c

f l ow over

a

compressi on r amp, and a shock

wave - t ur bul ent boundar y l ayer i nt er act i on

in

t r ansoni c f l ow over a cur ved

hi l l .

I t was f ound t hat t he numer i cal method used her ei n yi el ded accur ate

comput at i onal r esul t s even when hi ghl y s kewed, unequal l y spaced, cur ved

gr i ds were used.

TURBULENT

PLOW

EQUATIONS

The i ncompr essi bl e t ur bul ent f l ow equat i ons are gi ven as ;

a a

-((pu)

+

-((pv)

=

0.

ax aY

wher e eqs. ( 1- 3) f ol l ow f r om t he conser vat i on of mass, u- moment um and

v- moment um r espect i vel y.

I n

numer i cal cal cul at i on, the conser vat i on of

mass equat i on i s r epl aced by a pr essure cor r ect i on equat i on gi ven as:

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wher e t he l ast t er m r epr esent s the mass i mbal ance,

and

t he f i r s t t wo

convect i on t erms are unnecessar y f or i ncompr essi bl e f l ows. The pr essur e

cor r ect i on equat i on can be der i ved f ol l owi ng the st andar d SI MPLE procedure

[ 191. I n t he pr esent numer i cal met hod, al l f l ow var i abl es, except pr essur e,

are l ocat ed at t he same gr i d poi nt s and t he pr eoour e node has been l ocat ed

at t he cent r oi d of t he cel l .

The cont r ol vol ume f or

the

pr essur e cor r ect i on

equati on i s def i ned as t he cel l encl osed by t he f our nei ghbori ng gr i d

poi nt s. The vel oci t y- pr essur e decoupl i ng i s el i m nat ed by t r eat i ng the

pr essur e corr ect i on equati on as a cont i nuous f o rmpar t i al di f f er ent i al

equat i on r at her t han t r eat i ng

i t

as a const r ai nt condi t i on. I n t he f or mer

case, t he di scret e pr essur e cor r ect i on obt ai ned f r om eq.

4 )

becomes a

f i ve- di agonal syst em of equat i ons f or r ect angul ar gr i ds. On t he ot her hand,

t he di scret e pr essur e cor r ect i on equat i on obt ai ned by di r ect l y subst i t ut i ng

t he i ncr ement al pr essur e

-

i ncrement al vel oci t y rel at i ons i nt o t he

conser vat i on of mass equat i on yi el ds a ni ne- di agonal system of equat i ons.

The l at t er di scret e pr essur e cor r ect i on equat i on can yi el d a

vel oci t y- pr essur e decoupl ed sol ut i on, wher eas t he former equat i on does

no t

[ 1 7 1 .

I n cont r ol - vol ume based f i ni t e di f f er ence met hods, t he di scret e syst em

of equat i ons i s der i ved by i nt egr at i ng t he gover ni ng di f f er ent i al equat i ons

over t he cont r ol vol ume [19]. For cur vi l i near gr i ds, t he number of

i nt er pol at i ons r equi r ed to obt ai n f l ow var i abl es at t he cel l boundar i es i s

si gni f i cant l y r educed by usi ng the pr esent gr i d l ayout . Enhanced

conver gence r ate i s par t l y at t r i but ed t o t he gr i d l ayout whi ch r equi r ed

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f ewer i nt er pol at i ons [ 17]. I n sol vi ng the di screte syst em of equat i ons, t he

of f - di agonal t erms ar e moved t o the l oad vect or t er m and t he r esul t i ng

syst em of equat i ons can be sol ved usi ng a t r i - di agonal mat r i x al gor i t hm

( TDMA)

.

TURBULENCE EQUATIONS

For cl ar i t y, t he mul t i pl e- t i me- scal e ur bul ence model suppl ement ed

wi t h t he near - wal l t ur bul ence model i s summar i zed bel ow. The t ur bul ent

ki net i c energy and t he ener gy t r ansf er r at e equati ons f or t he ener gy

cont ai ni ng l ar ge eddi es are gi ven as;

ut 8

cP

( u + - )

- Pr

-

akp axj

j -

- -

axj

axj

(5)

wher e t he pr oduct i on r ate i s gi ven as;

The t ur bul ent ki net i c ener gy equat i on and the di ssi pati on r at e equat i ons

f or t he f i ne scal e eddi es are gi ven as:

akt

a

ut

akt

uj -

- ( u

+ - )

-

€p - €t

axj axj akt axj

( 7 )

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The t ur bul ent ki net i c ener gy equat i ons, eqs. (5) and ( 7 ) , ar e def i ned f or

t he ent i r e f l ow domai n whi l e t he ener gy t r ansf er r ate and the di ssi pat i on

r at e equat i ons are val i d f or t he f l ow domai n away f r om t he near - wal l

r egi on. The t ur bul ence model const ant s are gi ven

as;

Ukp- o. 75, Ukt- 0. 75,

~, ~- 1 . 15 , Ct - 1. 15, cpl - 0. 21, cp2- 1. 24, cp3- 1. 84, ct l - 0. 29, ct2- 1. 28, and

ct3- 1. 66.

These t ur bul ence model const ant s appr oxi mat el y sat i sf y t he

near - wal l equi l i br i um t urbul ence condi t i on,

t he decay r at e of t he gr i d

t ur bul ence

[ 20] , and the t ur bul ence i nt ensi t y gr owt h r at e i n a const ant

shear f l ow [ 21] . Fur t her di scussi on on t he est abl i shment of t hese

t urbul ence model const ant s can be f ound i n Ref erences 3.

The energy t r ansf er r ate and t he di ss i pat i on rate i ns i de the near - wal l

l ayer ar e gi ven as;

wher e

f -

1- exp( - A, Rt )

k2

Rt I

V e1

Not e t hat

€1

i n eq. (l O) r epr esent s t he st andar d di ssi pat i on r at e f or

near - wal l t ur bul ent f l ows i n equi l i br i um state. The di ss i pat i on rat e gi ven

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as eq.

(9)

i s f ormal l y i dent i cal t o t he one pr oposed by Wol f sht ei n [ 12] .

For y=O, eq.

(9)

t akes t he l i m t val ue gi ven as 2uk/ y2, whi ch i s

an

anal yt i cal sol ut i on of t he t urbul ent ki net i c ener gy equat i on f or a l i m t i ng

case as

y

appr oaches the wal l . Sl i ght l y away f r om t he wal l where t he

t ur bul ence i s i n the equi l i br i um s t at e, f, becomes uni t y. For near - wal l

equi l i br i um t ur bul ent f l ows, t he pr oducti on r at e (P) i s appr oxi matel y

equal

to

di ss i pat i on r at e et ) and hence t he energy t r ansf er r at e

( E

) f r om

t he l ow wave number product i on range t o t he hi gh wave number di ssi pat i on

r ange has t o be approxi mat el y equal to t he pr oduct i on and di ssi pat i on

P

r ates. Recal l t hat t he pr oduct i on rat e vani shes on t he wal l and gr ows

to

a

peak val ue at y+=15. Hence eq.

(9)

may not be a good approxi mat i on f or

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t he cubi c power of t he di st ance f r om t he wal l . I t can be f ound i n Ref erence

10 t hat t he near - wal l anal ysi s yi el ds t he same gr owt h rat e of t he eddy

vi scosi t y i n t he r egi on ver y cl ose to the wal l . However , t her e al so exi st a

f ew l ow Reynol ds number t ur bul ence model s i n whi ch t he eddy vi scosi t y

var i es i n pr opor t i on to t he f our t h power of t he di st ance f r om t he wal l , see

Ref erences 9 and 10 f or mor e di scussi on.

COMPUTATIONAL RESULTS

The exper i ment al data f or t he r eatt achi ng shear l ayer c an be f ound i n

Ref er ence

1.

The i nl et f r ee st r eamvel oci t y was

40

m sec, t he boundary

l ayer t hi ckness was 0.019 met er s, and the hei ght of t he backwar d- f aci ng

st ep was

0. 0127

met ers . The top wal l was def l ect ed f r om - 2 degrees t o 10

degr ees t o generat e a st r ong adver se pr essure gradi ent .

I n numer i cal cal cul at i ons, the i nl et boundar y was l ocat ed at f our

st ep- hei ght s upst r eam of t he expansi on cor ner and t he exi t boundar y was

l ocat ed at appr oxi mat el y

35

st ep- hei ght s downst r eam of t he expansi on

corner . The f l ow domai n was di scr et i zed by

a

105 by 85 mesh wi t h

concent r at i on of gr i d poi nt s near t he expansi on corner and i n t he bott om

wal l r egi on, see Fi gur e 2-(a).

The gr i d i n t he vi ci ni t y of t he expansi on

corner i s f i ne enough t o r esol ve det ai l s of t he l ar ge eddi es s ubj ect ed to

st r ong shear and sudden expansi on, see Fi gur e 2- (b). The i nl et boundary

condi t i ons f or t he tangent i al vel oci t y, t he turbul ent ki net i c ener gi es, and

the di ss i pat i on r at es

ep

and

e t )

wer e obt ai ned f r om experi ment al dat a f or

a f ul l y devel oped boundar y l ayer f l ow over a f l at pl at e [ 3, 22] . The

non- di mensi onal vel oci t y and t he t ur bul ent ki net i c ener gy pr of i l es wer e

scal ed to

boundar y.

yi el d a boundar y l ayer t hi ckness of

0. 019

met er s at t he i nl et

The no- sl i p boundar y condi t i on f or vel oci t i es and vani shi ng

13


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t ur bul ent ki net i c ener gi es wer e pr escri bed at t he sol i d wal l boundar y. At

t he exi t boundar y, a vani shi ng gr adi ent boundar y condi t i on was used f or al l

f l ow var i abl es except t he pr essur e. A uni f or m pr essur e was pr escr i bed at

t he exi t boundar y. The part i t i on bet ween t he near - wal l

l ayer and t he

ext er nal r egi on was l ocat ed at appr oxi matel y y+- l OO, 12 gr i d poi nt s were

al l ocat ed i nsi de t he near- wal l l ayer . The mesh si ze of t he f i r st gr i d poi nt

on t he bot t om wal l was Ay+- 2 and t he gr i d si ze i n t he normal t o t he wal l

di r ect i on was

i ncr eased by a f act or of appr oxi mat el y 1. 15. Fur t her det ai l s

on t he comput at i onal pr ocedur e can be f ound i n Ref erence 17.

The cal cul at ed st r eam i ne cont our s ar e shown i n Fi gur e 3 . The f l ow

f i el d consi st ed of t wo r eci r cul at i on zones. The pr i mar y r eci r cul at i on zone

ext ended f r om t he separat i on corner t owar d the downst r eam di r ect i on; and

t he secondary reci r cul at i on zone was ver y smal l and conf i ned i n t he cor ner

r egi on. The r eatt achment l ocati on ver sus t he top wal l def l ect i on angl e i s

shown i n Fi gur e 4. I t can be seen i n t he f i gur e t hat t he k- c and

ARSM

t ur bul ence model s l argel y under - pr edi ct t he r eat t achment l ocati on. The

modi f i ed

ARSM

yi el ded a si gni f i cant l y i mpr oved comput at i onal r esul t ,

however , t he pr esent computat i onal r esul t compar ed more f avorabl y wi t h t he

measur ed dat a t han di d t he modi f i ed ARSM.

The st at i c pr essur e cont our l i nes are shown i n Fi gur e 5, wher e t he

pr essur e has been normal i zed by t he i nl et t otal pr essur e and t he

i ncr ement al pr essure between the cont our l i nes i s

0. 005.

I t can be s een i n

t he f i gur e t hat a f ew cont our l i nes pass t hr ough the expansi on cor ner , and

t hus

t her e exi st s a m l d base pr essur e i n t he backward- f aci ng st ep r egi on.

The cal cul at ed st at i c pr essur e on t he wal l i s compared wi t h exper i ment al

dat a as wel l as t he numer i cal r esul t s of Ref erence

1

i n Fi gur e

6 .

The m l d

pr essure dr op at x/ H- 0 r epr esent t he base pr essur e. For a=Oo t he pr esent

14


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comput at i onal r esul t compared sl i ght l y more f avorabl y wi t h t he measured

wal l pr essur e t han t he ot her comput at i onal r esul t s. For a-6 , al l the

comput at i onal r esul t s compared decent l y wi t h t he measured dat a; however ,

t he sl ope of t he wal l pr essur e i n t he cont i nuousl y di ver gi ng downst r eam

r egi on obt ai ned i n t he pr esent st udy compar ed more f avor abl y wi t h t he

measur ed dat a than t he ot her r esul t s. Thi s di f f erence may due to the

di f f erent numer i cal met hods used.

The cal cul at ed wal l shear i ng st r esses ar e shown i n Fi gur e 7. I t can be

seen i n t he

f i gur e t hat t he l ocat i on of t he peak wal l shear i ng st r ess

obt ai ned usi ng t he

k- e

t ur bul ence model i s grossl y i n er r or . I t i s

i nt er est i ng t o note t hat t he modi f i ed ARSH under - pr edi ct s t he peak val ue

and t he pr esent t ur bul ence model over - pr edi ct s t he peak val ue even t hough

t he rel at i ve di f f er ences ar e al most t he same f or bot h def l ect i on angl es.

The mean vel oci t y, t he t ur bul ent ki net i c ener gy, and t he Reynol ds

st r ess pr of i l es at f our downst r eam l ocat i ons are compar ed wi t h experi ment al

dat a and wi t h t he cal cul at ed r esul t s usi ng t he modi f i ed ARSM

[ 1, 2] i n

Fi gur es

8- 10,

espect i vel y. The experi ment al t ur bul ent ki net i c ener gy shown

i n Fi gure

9

was est i mat ed usi ng t he measured val ue of uf 2+v f 2 nd an

assumpt i on t hat w 2=(u82+v' 2) / 2. A s shown i n Fi gur es

8- 10,

bot h

comput at i onal r esul t s exhi bi t f ai r compari son wi t h t he exper i ment al dat a.

I t can be s een i n Fi gur e 9 t hat t he peak val ue of t he t ur bul ent ki net i c

ener gy and t he shape of t he t ur bul ent ki net i c energy pr of i l e obt ai ned usi ng

t he mul t i pl e- t i me- scal e ur bul ence model compar e sl i ght l y bet t er wi t h t he

measured dat a t han t hose obt ai ned usi ng t he modi f i ed ARSM at x/H-1.0 wher e

t he t ur bul ence i s i n a st r ongl y i nequi l i br i um st at e. I t has been shown

pr evi ousl y t hat t he i mpr oved comput at i onal r esul t s f or compl ex t ur bul ent

f l ows ar e at t r i but ed to t he capabi l i t y of t he mul t i pl e- t i me- scal e

15


17/27

t ur bul ence model t o r esol ve t he i nequi l i br i um t ur bul ence

[ 3 ] .

The s ame

argument can be appl i ed f or t he pr esent f l ow case. At f ur t her downst r eam

l oc at i ons ,

t he pr esent comput at i onal r esul t s compar ed sl i ght l y l ess

f avor abl y wi t h t he exper i ment al dat a, a r esul t per haps due

t o t he near - wal l

t ur bul ence model whi ch can not t ake i nt o account of t he i nequi l i br i um

t ur bul ence.

The r at i o of t ur bul ent vi scosi t y t o mol ecul ar vi scosi t y at t hr ee

downst r eam l ocat i ons ar e shown i n Fi gur e 11.

I t

can be seen

i n

t he f i gur e

t hat t he J ones- Launder k-6 t ur bul ence model over - est i mat es t he r at i o so

t hat t he r eat t achment l ocat i on i s l ar gel y under - pr edi ct ed. On the ot her

hand, t he pr esent comput at i onal r esul t s compar e qui t e f avorabl y wi t h t he

measur ed dat a

so

t hat t he r eat t achment l ocat i on i s cor r ect l y pr edi ct ed. The

cal cul at ed pr oduct i on and di ssi pat i on r at es of t he t ur bul ent ki net i c ener gy

at t he same downst r eam l ocat i ons wer e qual i t at i vel y and quant i t at i vel y

al most t he same as t hose of r ef erence

1.

CONCLUSI ONS

Numer i cal cal cul at i ons

of

r eat t achi ng shear l ayer s i n a di ver gi ng

channel usi ng a mul t i pl e- t i me- sc al e ur bul ence model suppl ement ed wi t h a

near - wal l t ur bul ence model have been pr esent ed. The cal cul ated reat t achment

l ocat i on ver sus t he top wal l def l ect i on angl e obt ai ned usi ng t he pr esent

t urbul ence model was i n excel l ent agr eement wi t h measured dat a. The

cal cul at ed wal l pr essure and t he wal l shear i ng st r ess wer e al so

i n good

agr eement wi t h t he exper i ment al dat a. The r est of t he pr esent comput at i onal

r esul t s such as t he nor mal i zed vel oci t y pr of i l es and the Reynol ds s t r ess

pr of i l es compared f avorabl y wi t h exper i ment al data.

The comput at i onal

r esul t s obt ai ned usi ng t he mul t i pl e- t i me- scal e ur bul ence model compar ed

16


18/27

sl i ght l y more f avorabl y wi t h t he experi ment al dat a t han t hose obt ai ned

usi ng t he modi f i ed al gebr ai c Reynol ds st r ess t ur bul ence model . I t has been

shown t hat pr edi ct i on of t he cor r ect r eat t achment l ocat i on depends on t he

pr edi ct i on of t he cor r ect l evel of t he t ur bul ent vi scosi t y, whi ch depends,

i n t ur n, on t he capabi l i t y of a t ur bul ence model t o resol ve t he ent i r e

t ur bul ence st r uct ur e of t he f l ow f i el d cor r ect l y. Thus t he i mpr oved

comput at i onal r esul t s ar e at t r i but ed t o the capabi l i t y of t he

mul t i pl e- t i me- scal et ur bul ence model t o r esol ve t he st r ong i nequi l i br i um

t ur bul ence i n t he vi ci ni t y of t he expansi on cor ner and i n t he f ol l owi ng

shear - l ayer r egi on.

17


19/27

REFERENCES

1.

D. M. Dr i ver and H. L. Seegm l l er , Feat ur es of a Reat t achi ng Tur bul ent

Shear Layer i n Di ver gent Channel Fl ow,

J .

AI AA, vol . 2 3 , pp. 163- 171,

1985.

2 . M

Si ndi r , Numer i cal St udy

of

Separat i ng and Reat t achi ng Fl ows i n a

Backward- Faci ng St ep Geometr y, Ph. D. Thesi s, Uni ver si t y of Cal i f or ni a

at Davi s, CA, 1982.

3.

S.-W.

Ki m and C. - P. Chen, A Mul t i pl e- Ti me- Scal eTurbul ence model

Based on Vari abl e

Par t i t i oni ng of t he Tur bul ent Ki net i c Ener gy

Spect r um , To appear i n the Numer i cal Heat Tr ansf er , 1989, Al so

avai l abl e as NASA CR- 179222, 1987 and AI AA Paper 88- 0221, 1988.

4. S.-W. Ki m and C.-P. Chen, A Revi ew on Var i ous Fl ow- Sol i d

I nt eract i on Anal ysi s Methods wi t h Emphasi s on Recent Advances i n

Tur bul ence Model s and Fl ow Anal ysi s Met hods, AI AA Paper 88- 3685, 1988.

5 . K.

Hanj el i c, B. E. Launder , and

R.

Schi est el , Mul t i pl e- Ti me- Scal e

Concept s i n Tur bul ent Shear Fl ows i n

L.

J . S . Br adbur y, F . Dur st ,

B. E. Launder , F .

W

Schm dt , and

J . H.

Whi t el aw, ( eds. ) , Tur bul ent

Shear F l ows, vol . 2 , pp. 36- 49, Spr i nger - Ver l ag, New Yor k, 1980.

6. R.

Schi est el , Mul t i pl e- Ti me- Scal eModel i ng of Tur bul ent Fl ows i n One

poi nt Cl osure , Phvsi cs

of

Fl ui ds , vol .

30,

pp. 722- 731, 1987.

7.

R.

Schi est el , Mul t i pl e- Scal e Concept i n Tur bul ence Model l i ng,

11. Reynol ds St r esses and Tur bul ent Fl uxes of a Passi ve Scal ar ,

Al gebr ai c Model l i ng and Si mpl i f i ed Model usi ng Boussi nesq Hypot hesi s ,

J our nal de Mechani aue t heor i aue et appl i auee, vol . 2, pp. 601- 628,

1983.

8. S . W

Tu and B. R. Ramapr i an, Ful l y Devel oped Per i odi c Tur bul ent Pi pe

Fl ow, Par t

I .

Mai n Exper i ment al Resul t s and Compar i son wi t h

18


20/27

Predi c t i ons ,

J.

F1ui d Mech,

, V O~.

37, pp. 31- 58, 1983.

9 . S. - W Ki m A Near - Wal l Tur bul ence Model and I t s Appl i cat i on to Ful

Devel oped Tur bul ent Channel and Pi pe Fl ows, To appear i n Numer i cal

Beat

Transf er ,

1989. Al so avai l abl e

as

NASA TM- 101399, 1988.

10.

V.

C. Pat el , W Rodi and G. Scheuarer , Tur bul ence Model s f or Near

Wal l and Low Reynol ds Number Fl ows: A Revi ew , J. A I M, 23,

pp. 1308- 1319 (1985) .

11.

R.

S . Amano, Devel opment of

a

Tur bul ent Near - Wal l Model and I t s

Appl i cat i on t o Separated and Reat t ached

Fl ows,

Pumer i cal

He

at

Tr ansf er , vol . 7 , pp. 59- 75, 1984.

12.

M

Wol f sht ei n, T h e Vel oci t y and Temper at ur e Di st r i but i on i n

One- Di mensi onal Fl ow wi t h Tur bul ence Augment at i on and Pr essure

Y

Gr adi ent , I J nt .

J .

Heat and Mass Tr ansf er , vol . 12, pp. 301- 318, 1969.

13. H. C. Chen and V. C. Pat el , Pr act i cal Near - Wal l Tur bul ence Model s f or

Compl ex Fl ows I ncl udi ng Separ at i on, AI M- 87- 1300, 1987.

14.

A . Roshko, St r uct ure of Tur bul ent Shear Fl ows: A New Look,

J .

AI AA,

V O~.

4,

NO. 10, pp. 1349- 1357, 1976.

15.

Gi bson, M M. , Spal di ng, D.

B. ,

and Zi nser , W , Boundary Layer

Cal cul at i ons usi ng t he Hassi d- Por eh One- Equat i on Ener gy Model ,

Let t ers i n Heat and

Mass

Tr ansf er

,

V O~. , pp. 73- 80, 1978.

16.

Hassi d, S . , and Por eh, M. , A Tur bul ent Ener gy Model f or Fl ows wi t h

Dr ag Reduct i on,

J .

Fl ui d Enei

neer ng

,

Transact i ons of ASME,

pp. 234- 241, 1975.

17.

S. - W

Ki m A Cont r ol - Vol ume Based Reynol ds Aver aged Navi er - St okes

Equat i on Sol ver Val i d at Al l Fl ow Vel oci t i es, NASA TM- 101488, 1989.

18. S. - W Ki m Numer i cal Comput at i on of Shock Wave - Tur bul ent Boundary

Layer I nt eract i on i n Tr ansoni c Fl ow over an Axi symmet r i c Cur ved Hi l l ,

19


21/27

NASA TM- 101473, 1989.

19.

S . V.

Pat ankar , Numer i cal Heat Tr ansf er and Fl ui d F l ow, McGr aw- Hi l l ,

New Yor k, 1980.

20.

F.

H. Har l ow and

P.

I .

Nakayama, Tr anspor t of Turbul ence Ener gy Decay

Rat e, os Al amos Sci . Lab. , LA- 3854, 1968.

21.

V.

G . Har r i s ,

J . A .

H. Gr aham and

S .

Cor r s i n, Fur t her

Exper i ment s i n Near l y Homogeneous Tur bul ent Shear Fl ow, J . Fl ui d

Nec h. , vol .

81,

pp. 657- 687, 1977.

22.

P. S .

Kl ebanof f , Char act er i st i cs of Tur bul ence i n a Boundar y Layer

wi t h Zer o Pr essur e Gr adi ent , NACA Report 1247, 1955.

20


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FIGURE

1.

- NWNCLATURE OF THE REATTACHING SHE R LAYER,

H: HEIGHT OF THE BACKWARD-FACING STEP. a: TOP WALL DEFLEC-

TION ANGLE.

FIGURE

3.

-

S T R W C I E

ColllouR.

12

10

8

x

6

4

b )

VICINITY

OF

EXPANSION CORNER.

FIGURE 2. - DISCRETIZATIM OF FLCU WMIH.

0

XPER PTNTAL DATA 1

PRESENT COMPUTATIONAL RESULT

MODIFIED ARSN

0

TANDARD k -E AND

RSN

-

I

I

I

I I

1 J

a

FIGURE 4 . - REATTACHKNT LOCATION VERSUS DEFLECTION

ANGLE.

21


23/27

FIGURE

5. -

PRESSURE CONTOUR.

-.15

0

EXPERIMENTAL DATA [ 1

I

RESENT COMPUTATIONAL

% ---- STANDARD

k-E

AND ARSM

RESULT

MODIFIED ARSM

0 -

\

a

I

I I

V

I

I I

5 10 15 20 25

30

X/H

b)

a

=

6 .

FIGURE 6 .

-

PRESSURE ON BOTTOM WALL.

22


24/27

2 -

0 EXPERIENTAL DATA 111

RODIFIED AR M

STANDARD k -E AND A R M

RESENT CWUTATIONAL RESULT

Do

0

0 EXPERIENTAL DATA 111

RODIFIED AR M

-1


STANDARD k -E AND A R M

o

-2

a )

a =

o0

'r

-2

0 5 10 15 20 25 30

X/H

b )

a = 6 .

FIGURE

7. -

SHEAR STRESS ON Bolloll UALL.

4

0 EXPERIEIITAL DATA


-

-

-

RODIFIED ARM

WH =

1.0

4.0 7.0

12.0

a ) a = oo

WH = 1.0

4.0

7.1 12.0

u/u

00

b )

a

= 6 .

FIGURE 8. - VELOCITY PROFILES.

23


25/27

0

EXPERIENTM DATA


--- NODIFIED A R M

a )

a

= 0

k/Um2

(b)

a

=

6 .

FIGURE 9. - TURBMNT KlWETlC ElERGY

PROFILES.

0

EXPERIENTAL DATA

RESENT COMPUTATIONAL RESULT

ODIFIED

A R M

'

c

WH =

1 . 0

4.0 7.0

1 2 . 0

a )

a

= 0

e

r

9 X/H = 1.0 4.0 7 .1 12 .0

i T / U m 2

b ) a =

6

F W 10. - REYNOLDS STRESS PRQILES.

24


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.

1

0

e

0 EXPERIENTAL DATA


----

CEBECI ALGEBRAIC W E L

[ 1 I

JONES AND LAUNDER

k-E

IK)DEL [ I

a ) a =

oo

0 0 0 lo

2000

b )

a

=

6 .

C I t 4

FIGURE

11.

- RATIOS OF TURBULENT VISCOSITY OVER MOLECULAR

VISCOSITY

25


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Natlonal Aeronautics and

Space

Administration

Report Documentation Page

2. Government Accession No.

NASA TM-102293

ICOMP-89- 18

1. Report No.

4.

Title and Subtitle

Calculation of Reattaching Shear Layers in Divergent Channel

With a Multiple-Time-Scale Turbulence Model

7. Author(s)

S.-W. Kim

9. Performing Organization Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C . 20546-0001

3. Recipient s Catalog No.

5. Report Date .

August 1989

6. Performing Organization Code

8.

Performing Organization Report No.

E-4965

10. Work Unit No.

505-62-2 1

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

5.

Supplementary Notes

S. W. Kim, Institute for Computational Mechanics in Propulsion, Lewis Research Center, Cleveland, Ohio 44135

work funded by Space Act Agreement C99066G). Space Act Monitor: Louis A. Povinelli

6. Abstract

Numerical calculations of turbulent reattaching shear layers in a divergent channel are presented. The turbulence is

described by a multiple-time-scale turbulence model. The turbulent flow equations are solved by a control-volume

based finite difference method. The computational results are compared with those obtained using k-c turbulence

models and algebraic Reynolds stress turbulence models. It is shown that the multiple-time-scale turbulence model

yields significantly improved computational results than the other turbulence models in the region where the

turbulence is in a strongly inequilibrium state.

7. Key Words (Suggested by Author@))

Multiple-time-scale turbulence model

Algebraic Reynolds stress model

Shear layer

Diverging channel flow

18. Distribution Statement

Unclassified Unlimited

Subject Category 34

backward facing shear layer

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