An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the...

8/19/2019 An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the …

1/18

REPORT 1111

AN ANALYSIS OF LAMINAR FREE-CONVECTION FLOW AND HEAT TRANSFER

ABOUT A FLAT PLATE PARALLEL TO

OF THE GENERATING BODY

B y SIMON @rBACH

SUMMARY

The

jree-conuedion

j low and hed

tianajer

g9n.eraiedby a

dyf or ce) abou t a j kt pki% pard?el t o t hed i r ect i on of t hebody

r ce ar ejor dy anal yzed and the type of @w h found to be

pend en t on w Gnw hof nu mbw al on e. F or l ur ge @z& f

um ber s (wh i ch ar e of i nter est i n aem nut i ), thej fow b of

boundary-luyer t ype and theproblem ix reduced in a jorma l

n ner , w hi ch b an ul ogow t o Pr an dt l’s j or ced -- bou nd .ar y-

yer t heor y, to i l w t im ui kzn eow sol ut i on of t wo or +~~

ren i iu l equa.t t i 8ubjeei t o theproper boundarycondt iaons.

v el oci t y an d t em p er a tu r e d i& i .bu i ti j or P r an d .t l n u m ber s

j 0 .01, 0 .72, 0 .7%?, 1 , $ ’, 10, 100 , and 1000 are computed ,and

i s shown t hai vel oci .t i a and Nussei ?tmu .i ber s oj t he order of

oj those encountered in forced-convect t i $ows may

e obt ai n ed i n j r e++ eon veet ion @w e. T h e t heor et ical an d

xperi mental velocity and temperatw xedM ri ln& m

a r e i n good

A$ow and a hea i -t r ana j erparamet er ,j r om wh i ch t he impor -

physica l guunt i t i .a such as shear st re+18nd hea$t rana jw

t e can be cm npu kxi ,ar e o%i oed m j un ct i om oj Pr a& n um -

r aki n . Compar i son oj t heorei%xd l yeompded vul ueaoj t he

abt ran-sjerparanwter w& $ values ob i%ind j rom an approx i -

e calculat ion and exper im ents yiefded good agreem-en$over a

zr ge r an ge oj Pr an dt i n um ber . Agr een wn i btd wetm th e

eor et i ad va i ?u a a nd i %m e obt i i m zi + om a j r egu en i l y u ed

em iem p& k al hea& t r an & r bw w as good on ly i n r at r i .eted

Prandti numbmr angm (depend ing on an arb i t ra rycandmv t).

.

INTRODUCTION

Two impor t an t t ypes of f lu id f low prob lems involving heq t

t ra nsfer a re t hose of forced a nd t hose of free convect ion.

f or ced


2/18

64

RJ 3PORT1111—NATIONALADVISORYCOMMJ 2TEEFOR m0NAuTIc6

Va rious t erm s a re om it ted from t he eq ua tions a t t he st ar t

on t he ba sis of eit her int uit ive ~ gument s or no a rgument s

a t a ll. Alt hough a t heor et ica l d evelopm en t m ad e in such a

ma nner led t o good fina l result s, t he sigdca ace of a ll t he

impor t a n t f a ct or s a s socia t ed w i th t h e f kee-con vect ion f low

phenomenon i s n ot ob ta i ned fr om such an an a ly sis .

The p rob lem of f ree-convect ion f low as produced by a body

for ce a bout a fla t pla t e in t he d ir ect ion of t he body for ce w a s

st udied a t t he NACA Lew is la bora tory during 1951 a nd is

t rea ted in a for ma l a nd more genera l ma nner herein . The

met hod u sed is somewh a t s im ila r t o t ha t u sed in r efer en ce 4

wherein cons ider a t i on was g iv en t o t he f ree-convect ion f low

a t ,h ig h G r a s hof number s in a h or iz on t a l cy lin der wh ich h ad a

v a r iab le sur fa c e-t empera t u re d is t ri bu t ion . The appl ica t i on

of t h is met h od t o t h e pr es en t pr ob lem lea d s t o a d ev elopmen t

t vh ich i s ~ n a l og ou s t o P r a n dt l’s t r ea tmen t of h ig h Reynold s

n umber for ced -con vect ion flo~ s . lt hoiq gh t he fin a l eq ua -

t ions obt a ined by t his method a re the sa me a s t hose of

S&mid t a n d Beckmanu, t h is mor e g en er a l a ppr oa ch not on ly

clea r ly dem on st ra t es t he sign iika nce of a ll t he im por ta nt

pa ra metem a nd a ssumpt ions a nd hence lea ds to a bet ter

underst a nding of t his t ype of flow but a lso iq dica tes the

qua nt it a tive limit at ions of the theory. In a ddit ion , t he

numer ica l s olut i on s of r ef erences 2 and 3 a re herein er t ended

tw cover a more complet i ra nge of pa ra meters. The new

ca lcu la t ions f i-e ld in format ion on the f ree-convect ion f low for

P r a ndt l n umbem cor respon din g t o t hose of liq uid m et a ls,

ga ses, liquids, a nd very viscous fluids. .,

ANALYSIS

STATEMENT OF PROBLEM AND BASIC EQUATIONS

The s t eady -s t a t e equa t ion s expres sing t he conse rv a t ion of

ma ss , momen t um , a n d en er gg for a compr es si ble,- vis cou s,

rmd h ea t -con duct in g flu id s ubject t o a bod y for ce t og et her

w ith a n equa tion of st a te govern t he flow a nd a ssocia ted

temper@w;

in C a r t es ia n

d is tr ibu tion a b;~ t t he pla t e. Th es e eq ua t ion s

tensor not a t ion are (see ref . 5), respect ive ly ,

&(Puj)=o

(1)

-+( + )1-~+(’ )-

=

Pf f+ & &

p ‘j

bxj

(2)

a p a

~ aT

X.= U j q+axl

Pcn Uj ax,

–( )

—+

axj

( + )-:( )1

‘3)

P= PV’,TI

(4)

(A complete lis t of the symbols used herein is given in

appendix A.) For t h e t “wo-d imen sion a l ca s e, eq ua t ion s (1)

t o (4) r ep resen t a s y st em of f iv e equa t ion s in t he f iv e depend-

ent var iab les CL,Zh, P, P, a nd T. F or kt or use, eq u~ t ion (4)

ca n be w r it t en

d p=p(K d P–@ d T)

(4a)

where K a n d ~ a r e t h e coef ficien t s of i sot h erma l compr

b il it y and volumet r ic expans ion , r espect iv el y (s ee re f. 6).

a ddit ion t o a genera l st a te equa tion, such a s is give

eq ua t ions (4) or (4a ), it w ill be convenient a t t im es in

d is cumion t o r efer t o s ome specific st a t e eq ua t ion . To

en d, t he-eq ua t ion of s ta t e for a n id ea l g a s

.P=PRT

w i ll b e u sed .

P a r t icu la r con sid er a tion is h er e given t o t he t w o-d im

s ion a l fr ee-con vect ion flow a bou t a s em i-t it e ver tica l

pla @ The X,-a xis of t he coordina te syst em is t aken a

t he, p la te a nd t he X2-a xis, norm al t o it . No dist inct io

m a de a s t o t he s pec c t ype of bod y for ce a ct in g, for exam

gr avit a tion al or cen tr ifuga l, but t he for ce is w sum ed t

a ct in g in t he ver tica l d ir ect ion only (t ha t is, pa r allel t o

pla t e). C en t rifu ga l a n d Cor ioli s for ces wh ich a r e conne

w ith flow s on curved pa ths a nd w ith rot a ting syst

genera lly va ry w it h posit ion a nd velocit y . H ow ever

or der n ot t o make t h e an a ly sis u nduly compl ica t ed , t h e b

for ce is t a ken t o be con st a nt .

I n or der t o d efin e t he pr oblem clea r ly , a ch oice must

be m a de of t he posit ion of t he or igin of t he coor din at e

t em . B efor e m a kin g a defin it e d ecision on t his poin t,

t ha t for con st a nt pla t e t emper a tu res t her e m -e fou r pcm

t a t ion s of t h e body-for ce d ir ect ion (ei th er upwa r d or clo

w ard) a nd the pla te t herma l condit ion (either hea ted

cooled ) wh ich w ill lea d t o fr ee-con vect ion flow s. On ce

posit ion of the edge of the pla te, w hich is CLlsoo be

origin of t he coor dina te syst em , is decided, t here m e

combin a t ion s of t h e body-for ce d ir ect ion and pla t e t h eu

con dit ion t ha t w ill y ield flow s wh ich pr oceed aw a y fr om

edge. I t is t his t ype of flow tha t is a ma na ble t o the typ

a n a ly sis t o be ma de h er e. Th is poin t w ill be mor e fully

cussed subsequent ly .

I f t he edge of t he pla te (reca ll t h

s em i-in fin it e pla t e h a s but on e edge) is t a ken a t t h e bot t om

t he pla te (t ha t . is , t he pla te ext ends t o + ~ in t he Xl-dir

t ion ), t he t w o combin at ion s lea din g t o flow s in t he pr o

d ir ect ion (upw a r d in t his ca s e) a r e, r es pect ively , t he b

force a ct ing dow pw ard w it h a hea ted pla te a nd the b

force a ct ing upw ard w ith a cooled pla te. The equa t

d evel oped for on e of t h e ca a e s reduce d ir ect ly t o t h os e f or

ot her . Th e r ema in in g t w o permu ta t ion s, n amely , t he b

f or ce a ct in g downwa r d w i th a cool ed pla t e a n d t h e body f

a ct in g u pw a r d w it h a h ea t ed pla t e, w ou ld y ield flow s wh

proceed dow nw ard or t ow ard the edge of the pla te if

edge w ere t a ken a t the bot t om of the pla te. This t ype

now wou ld v iola t e a ph ys iw l condit ion of t h e pr ob lem wh

3tates

t ha t the flow st a rt s a t the pla te edge. The la

combina t i on s hence w i ll not be cons idered fur t her .

Becau se t he two accep tab le con fi gu r a t ion s can be redu

w en tia lly t o one, for t he developm en t t o be given h er e,

X@ of t he coordina te syst em w ill be t aken a t the bot t

>f a hea ted pla te, w ith t he body force a ct ing dow nw a

l?h e a ss umpt ion is n ow ma de t ha t t he vis cosi~ a n d t herm

:on duct ivi@ coeflicien ta a r e fu nct ion s of t he t emper a t

d on e a n d obey t he follow in g la w s:


3/18

FREE-CONVECTION F LOW

AND

HEAT

TRANSFER ABOUT

A FLAT

P LATE

6

(5)

ch oice of t he b od y-for ce d ir ect ion t og et her w it h eq ua -

on s (5) a l ter s eq ua t ion s (2) a n d (3) s o t ha t t hey become

(6)

ot e t ha t t he only nonzero com ponent of t he body force is

Xl-component.

BOUNDARY CONDITIONS

The bound a ry condit ion s a s soci a t ed w i th t h e g iv en pr ob -

m a r e t ha t :

(a ) Th e fluid must a dher e t o t he pla t e (t he n o4ip con di-

on of viscous flow s) a nd the pla te must be a st rea mline,

mathemat ica l ly ,

U,(x,,O)=,(x,,o)=o

(8)

(b) Th e t emper a tu re of t he flu id a t t he pla t e mus t be eq u~

t he pla t e t emper a tu re, t ha t is ,

T(X1,O)=TO

(9)

(c) Th e velocit y U , a t la r ge d is ta n ces fr om t he pla t e mus t

undi st u rbed , or

u,(x,,~)=o (lo)

(d) The tempera ture a t la rge dist a ncea from the pla te

s t be eq ua l t o t he u nd is tu rbed flu id t emper a t ur e, or

T (XI ,= )= T .

(11)

SIMPLIFICATION OF EQUATIONS

Let a sma ll qua nt it y e now be defied a s

c=/ ?(TO-Tm)

(12)

ch is a mea sur e of t h e magn it u de of t emper a t ur e v a ria t ion

t he flow field . Th e coefficien t of volumet ric expa n sion B

gen er a lly of t h e or d er of magn it ud e between 10-Z and 10-

ee t a ble 15 of r ef. 7, for emun ple) a nd for ga ses, p= l /T.

hus , for g a ses , i f P i s t a k en t o be con st a n t ,

C =(T o– T m )/ T .;

is , c is t he rela tive t empera ture difference.) The

efficient /? w ill be a ssum ecl const ant . B eca use in t he

a dy st a te flow ensues only w hen t here is a t em pw a ture

,va ria t ion in t he fluid, t he freea nvect ion velocit y shou

t hen d epen d d ir ect ly on C ,a n d t he

var ia t ions in

pressure an

d en sit y (fr om t he s ta t ic, 6= 0, ca s e) d ue. t o t he t emper a t u

d iffer en ces s hou ld a ls o d epen d on ~ . Th us

(1

P= P* +Pm W

(1

P= Pa+P. ~Q

(1

z’=T=(l+d)

(1

where

—fx

denot es t h e Xl-componen t of t h e body f or ce p

unit m aw , ut , u, p, a nd o denot e dim ensionless funct ion

(wh ich , in gener a l , can be funct ion s of c), 1 is some cha r a ct e

ist ic len gt h (for example, t he disia nce fr om t he ed ge of t

pla t e t o t he po” tit of in ter es t), P , a n d p, a r e t h e pr es su re a n

the densi~ , respect ively , for t he st a t ic ca se (Z7i= O

c= O), a nd

Pm

a nd

pm

denot e con st a n t v a lu es of t h e pr es su

a nd the densit y (t ha t is, t he va lue5 if no force field w e

pr esen t) defin ed by t he st a te eq ua t ion (in t he ca se of

a ga

in

part icular ,

P- =p~RT~ ):

B eca um t her e is n o ch a ra ct e

ist ic velocit y a ssocia ted w it h t he type of flow under co

sidera t ion , t he velocit y is dimensiona lized by t he fa ct

g iv en in pa r en t hes es on t h e r ig ht s id e of equa t ion (13).

I n or der t o det ermin e t he st a t ic q ua n tit ies, it w ill a t fir

be con ven ien t t o con sid er t he pa r ticula r ca se of a ga s. Th

pr oblem is t hen con sid er ed w it h t he t emper a tu re un ifor

t hroughout t he flow field a t t he va lue T - (therefore the

w ill be no flow a nd U ~ = O). For th is sit ua tion, equa tio

(4b) and (6) become

P,= P,R T.

(1

a nd

(1

(I t sh ou ld be n ot ed t ha t eq . (18) expr es ses t he ph ys ica l fa

previously st a ted tha t t he body force a nd hydrost a t

pr es su re a r e in equ ili br ium for t h e s t a t ic cn se.) Subs t it u ti

of eq ua t ion (17) in t o equa t ion (18) lea d s t o

‘~=p++7&x)

a nd equa tion (19) t ogether w ith equa tion

equa t ion deb .ing

P.

a n d p- y ield s

(1

(17) a nd t

P.=Pm em

(

-Ikix’)=’’-e+ )’) ’20

If the

exponen t ia l i n equa t ion (20) i s expr es sed in t erms

i t s s er ie s expans ion , t h a t equa t ion becomes

P*=P.

(

1—*X,+. ..

m )

(2

321a9D-G~


4/18

66

RE PORT1111—NATIONALADVISORY

,

cO~ FOR MJ RONAUMCS

A computa t ion of t he second t erm in the pa rentheses of

equa tion (21) for t he ca se of a ir under norm al condit ions

w ith jx= g a nd the fa ct tha t Xl is of the order of ma gnit ude

1 show t ha t

p .gl / P~ -

10-’l/oot . l’or t he t ype of problem

under considerat ion ,

1 wi ll shays be

of u nit or der of m a gn i-

tude so tha t even if the body force ~ x represent s a cen-

t rifuga l force ma ny t imes t ha t of gra vi~ , the inequa lib

PJxX/~.


5/18

l?RIZIE-CONVECTIONLOWAND 1311AT

.

TUNSEDR ABOUTA EMT PLATEl

6

r th er simplifim tion of t he eq ua t ion s w ou ld be d esir a ble.

st a s in the ca se of forced-convect ion flow s w here the

y nold s n umber d et ermin es t he t ype of flow or , in m a th e-

ica l t erms, t he t ype of solut ion , t he G ra shof n umber is

e pr ime fa ct or for fr ee-con vect ion flow s. F or t he ca se of

m a ll G ra shof number, it ca n be seen h or n eq ua tion s (32)

(34) t ha t a pert urba tion in t he sm all pa ra met er @ w ill

ield a syst em of linea r eq ua tions. F or G ra shof numbers

f u nit or der of magn it ud e, n o fu rt her impor t a nt s implif ica -

on can be made and t h e s olu tion swould h a ve t o be ob ta in ed

umerica lly . For the other limit ing ca se, tha t of la rge

a sh of number s (w hich is t he ca se un der consider at ion

r ein ), it w ould , a t fir st t hough t, a ppea r t ha t s ome simpli-

ca t ion cou ld be obt a in ed by per formin g a per tu rba t ion in

he sma ~ pa ra meter 1/(3.

How ever , t his w ould then

ply t h a t t he t erm con t a in ing t he h ighes t -order der iv a t iv es

t he l ef t t erm in equa t ion s (32) t o (34)) cou ld , among ot her s,

e neglect ed. (This a rgument w ould a lso imply t ha t t he

ody-force t er m q in equa tion (32), w hich is essent ia lly.

pusin g t he flow , could a lso be n eglect ed.) The om ission

f the h ighes t .ader der iva t ives f rom cons idera t ion , however,

uld lea d t o solut ions w hich w ould not sa tisfy a ll t he

undary condit ions .

P roblems of t his t ype a re referred

a s singula r perturba tion problems. For further dis-

u ss ion s of s in gu la r per t ur ba t ion pr ob lems, s ee r ef er en ces

a nd 9.

E qua tions in w hich a sma ll pa ra meter mult iplies the

gh est -or der t erms a re sa id t o be of t he bounda ry-la yer

pe, beca use in order for solut ions w hich sa t isfy a ll t he

unda ry cond it ion s t o be ob ta ined , t he h ighes t -order t erms

st be con sider ed nea r t he bouhda ry. Th is fa ct im plies

e exist ence of a t hin region, ca lled t he bounda ry la yer ,

vherein the funct ions va ry ra pidly t im the va lue a t the

ounda ry t o t ha t in the flow out side this la yer . The con-

lusion t o be dr aw n fr om t he pr eceding discussion is t ha t

or la rge G ra shof numbers t he flow is of t he bounda ry-

yer t ype. Schmidt a nd B eckmann (ref. 2) a lso ma de

boundary-layer assumpt ions in thei r theoret ica l develop-

n t, a nd t hes e a s sumpt ion s w er e ju st ified on t he ba sis of

ir exper imen t a l ob ser va t ion s. The G r a sh of number s f or

i r exper imen t s we re of t he order of 8X106.

I n view of t he fa ct , pr eviously discussed, t ha t @gh&-

r der der iva t ives of ea ch depen den t va ria ble a s w ell a s of

ose t erm s of physica l import ance (a s, for exa m le, t he

F

od y-for ce t erm) mus t be r et a in ed in t he boun da r y a y er , it

s convenient t o ma ke bot h sides of ea ch of t he eq ua tiods

f the sa me order in G r. In t his w ay, a s w ill be show n, the

q ua tion s w ill be fur ther sim p%ed. I t is t hus convenient

o ma ke the follow ing t ra nsforma tions in the syst em of

q ua tions (32) t o (34) a nd (26) a nd t hen b ret ain only t he

omina nt pa rt s (t ha t is, t hose mult iplied by & to the

ghes t power) of ea ch -ind iv idua l t erm .

Let V= f3r ’y, +=GW$, u=WZ, P=?, a nd 8= e. Then

(38)

(39)

d~=K P=& ’dZ–~T.d~

(41

I t now ca n be seen tha t by proper choice of r , s, a nd t n

t ra nsforma tion of the t ype given providw a mea ns for

makin g t h e impor t a nt t erms in t h e d if fer en tia l eq ua t ion s o

the same order in t% Thus if r =%, s= —%, a nd t =—1,

equa t ion s (38) t o (41) become

(42

(44

dG+@Tm d~=O

(45

Mor e gen er a lly , if N is ver y much d iffer en t fr om un it or de

of m a gn it ud e, a va lue of tcan a lway s be chosen (depending

on iV) such t ha t eq ua t ion s (42) t o (45) a re obt a in ed. @’or

a ny n ega t ive

t less

t ha n —1, the la st t erm of eq . (42

w ill a lso d isa ppea r .) -

Th er e a re now sever al im por ta nt poin ts t o be discusse

con cer nin g t he t ra n sforma t ion ju st m a de a n d t he rw dt in g

simplified equa tions. F irst , it should be not ed t ha t th

t r a ns forma t ion i s mer ely a f orma l expr es sion of t he bound

a ry-I ayer wmunpt ion s fir st m ade by P ra ndt l a nd h eme t h

solut ions w ill be a symptot ic for la rge G% Second, th

secon d eq ua t ion of m ot ion her e a lso r ed uces t o st a te t ha

t he pr wure a cr oss t hq bounda ry la yer is con st a nt . Thir d

t he pressure t er ms in t he energy a nd st a t e eq ua tions a r

here found to be negligible. This fa ct veriiies a prior

a s sumpt ion s made by ot hem f rom the physi cs of t he prob lem

F in a lly , n ot e t ha t in teg ra t ion of t he gen er a l st a t e eq ua t io

(independen t of pr essur e) a a now given by eq ua tion (45

lea d s t o

,

g+ fl Tm i i=O (46

w here t he const ant of int egra tion ha s been t aken a s zer

w it hout a ny loss of genera lit y. F or t he pa rt icula r ca se o

a ga s, /3= 1/T. s o t ha t eq ua t ion (46) becomes

~ + ~ = o

Th e boun da ry con dit ion s (eq s. (35) t o (37)

wri t ten

~(z,o)=~=(z,o)=o

i(z,o)=&-

m

z(z,0)=7( m)=o

now ca n b

(47

(48

(49

If now it is a ssumed t ha t ~ s= O in equa tion (42) since con

sidera tion is here being given t o a fla t pla te, a nd if ~ i

elimina ted from equa tion (42) by use of equa tion (46)

t h er e r es ult s t he s ys tem of equa t ion s


6/18


7/18

FREE-CONVE~ IONFLOWAND HEAT

ysis on t h is t y pe of f low , t h is limit in g v a lu e i s t a ken t o be

0° , a s in di cm t ed in r ef er en ce 10. Con sid er a t ion of t h is lim-

t ion t hen .im pliw (see eq . (60)) t ha t for la rge la mina r

l ocit ies eit h er v. m ,y st b e la r ge or X mus t b e sma l l.

COMPARISON WITH

EXYERIMENT9

Care fu l exper iment s of f ree-convect ion f lows (as genera ted

y g rw it a t igma l for ces ) a bou t ver tica l fla t pla t es w er ema d e

Schm id t a n d Be ckmwm (r ef. 2) in wh ich v el ocit y mea sur e-

n ts a t va r iou s poin ts a lon g t he pla t e w er e ma de by mea n s

f r Lq um tz -illamen t a n emomet er a n d t he t emper a te mea s -

men t s we re ob t a ined by means of manganese-const an t an

E cker t (r ef. 1) per formed sim ila r exper i-

n ts in w hich t he m en sur sm en ts w er e m ad e by m ea ns of a

n

\,

3

I

4

~

“2

\

N

L

‘

“’

.

~-

.

.1

h

_e

0.01

-k

06 8

IO121416182022ZW

“

I

r)’

=7$

.7

.6

R- -

0 .01

.5

/ \

/ \

i

\

.3

/

.2 –

.1

0 I

2

3

4 5 6

7

~,

(z&++

FmuRE 1.—Dlmenskmlemvelwlty dlstxJ bntlonaor various

Pi-mill

numbers.

TRANSFERABOUTA FLATPMTE

6

Zeh nd er -Ma ch in ter fer omet er . Th e r es ult s of bot h s et s

exper im en ts a re in good a ~ eem en t, but sin ce t he da t a pr

sent ed in reference 2 by S chmidt a nd B ecknwm a ppea r

mor e d et a il, t hes e d a ta w ill be u sed for compa r is on w it h t

t heor y. ,

Th e exper im en ts of r efer en ce 2 w er e per formed on t w

d .i il er en t (in t h a t t h e edges wer e smoot h ed ei th er s ymmet

ca lly or n ot ) 12- by 25-cen timet er pla t es a n d on on e 50-b

50-cen timet er pla t e. I t s hou ld h er e be poin ted ou t t ha t t

r es ult s for t he t w o sma ller pla t es w er e a h nost id en tica l a

t ha t the flow w as ent irely la mina r

except nea r t he out

edge of t he bounda ry la yer w her e t he sligh t t ur bulen ce

t h e r oom a i r d is t ur bed t h e mea sur emen t s s omewh a t .

m

eilect w a s a ls o obs er ved by E cker t.) L a rg e per iod ic os ciU

t iom of t he flow n ea r t he d own st ream ed ge of t he la r ger pla

w er e obser ved in a dd it ion t o t he sigh t t ur bulen ce n ea r t

1.0

.8

I .p~e

:6

k

:0

.

~F

2

.4

.2

0 I

2 3

rx ~ 4 5

f)

4~

—

l?mwrm2.—DimmrsbJ rd=tempafhrm dlstrkmtlonsforvarlonsPrandtl nmnbma.

,


8/18

7

RDPORT 1 11 l—NATIONAL ADVTSORY

COMMT iT E1 3 FOR AJ 3RONAUT ICS

outer edge of t he bounda ry la yer . H ence t he da ta from t he

la r ger pla t e s hou ld n ot be expeobd t o y ield complet ely s a t is -

fa ct ory a greement w it h t he la mina r

t heor y a s pr esen ted

here.

S it e t he ph ya ioa l q ua n tit ies ca n be expr ess ed in t erms of

a sin gle va ria ble a s in eq ua t ions (60) a nd (61), it is t o be ex-

pect ed t ha t t he da ta t aken a t t he va rious point s a long t he

pla t es sh ou ld a ll lie on a sin gle lin e if t he d a ta a r e cor rela t ed

a ccor din g t o eq ua t ion s (60) a nd (61). Th us for t he sma ller

pla t es wher e (To-T_ )= 95.22° R a nd Tm= 518.68° R , eq ua -

t ions (60) to (62) become

u

=F’(q)

4.862-@

(63)

‘;:~268=H (T)

(64)

.

q=88.26+

(65)

The veloci ii y and t empera t u re d is t ri bu t ion s t i e so plot t ed in

figur es 3 a nd. 4, r espect ively , a s a re t he cur ves comput ed

t h eor et ica l ly for .P r =O .72. I t ca n be s een t h a t t h e a g reemen t

is in genera l very good for .m@l va lues of q a nd somew ha t

l es s s a t is fa ct or y t hou gh s tiI I r a t her g ood f or t h e ]@er va l ues

of q . The-sca tt er in the ra nge of t he la rger va lues of q is

believed t o be ca used by t he previously discussed room

3 .2

n

u

I

o= o

I I I

I

O.:

2 .8

(i’L

v

o

0 . 394

8

n

. 787

0

I . 575

/

1

Experimental

2.756

(ret 2)

a

2.4

D

4.33 I

4

— Theoretical

or R=O.72

z ‘

o

2 .0

4

\ .~

u

o

1.6

r

J

3

e

>

I .2 —G

\ b

n

6

.8

y

o 0

>

.4- 0“

I

1

0 Lo

2.0

50 4.0 5.0

6.0

/

%

q=(8 8 .2 6 ) Y X

~awws 3.-Oomparfwn ofsmall plataeqwrfmenkl end theorotkal

velwity dbtribrrtlmM

for

Pmndtl nnmberofoil.

I .2

I

Jx

I

I

I

1.

(in.)

o 0.394

0

,

0

t P

1

~~~

x~;~tm;;fal

4

2:756

.8 D

4.33 I

-e

— Theoretlccrlfor ‘ * 0.72

;

.

a

.6

q ru

a) Cy

\: ~

,.

s In

, m

a

k

.4

.

a

2

,

b

~

I IY

).

I

o

LO

2 .0 3 0

. , 40

5 .0

/“

=(88.26) Y X7

FrGIJBE4.-Campd.mn of mnell plateeqxdraenbl andthwetlml tomporntumd

tfonsforPmndtl namborof0.71.

turbulence.

I t s hou ld a ko be n ot ed t ha t t he poin ts fa r

a wa y from the t heoret ica l a re those mea sured nea

l ead ing edge. These point s shou ld not , of cou r se, be oxQ

t o a gr ee t oo w ell w it h t he t heor y sin ce t he bou nd ar y-

a s sumpt ion s made in t he t heoret i ca l developmen t imply

t he dist ance a long t he pla te is la rge a s com pa red w it

bounda ry-l a yer t h iclmess . Hence, t h is a s sumpt ion i s in

nea r t he lea din g edge.

S chmid t a n d B ecknmnn obt a

closer ag reemen t between the t heory and t he mTer imen

t h e t emper a t ur e d a t a a n d poor er a g reemen t for t h e vel

d a t a by ba sing t he k inema t i c v is cos it y coef fi ci en t in oqu

(62) on the pl a t e t empera t u re r a t her t h an on ”t he undi st u

st ream t emper a tu re a s w a s don e h er e.

F or t h e la r ger pla t e, (TO– T .) =83 .7°R and T . = 627

R , s o t h a t eq ua t ion s (60) t o (62) become

u

=F’(q)

4.622@

‘–8~;.1 4=H (q)

q=83.93~

The v elocit y a n d t emper a t ur e d is tr ibut ion s for t h is ex

men t a r e plot t ed in figu res 6 a n d 6, r espect ively ,

a nd

t he t heor et ica l ct ies for Pr=O.72 a r e i nclu ded . I n fig ur

ca n be seen tha t for la rge q the a greement is ra ther

pa r t icu la r ly for t h e d a t a for bot h sma l l a n d la r ge v a lu es

The poor a greem ent for snd va lues of X is a ga in due t

t heory limit at ion nea r t he edge of t he pla te rm d for

va lu es of X, t o t he fa ct t ha t t he flow w a s becom in g t ur b

there.


9/18

F RJ 3 E-CONV3WI’ION F LOW AND E EAT

FLOW AND HEAT-TRANSFER PARAMETERS

In a d dit ion t o t h e v elocit y a nd t emper a t ur e d is t rib ut ion s,

t is oft en d esir a ble t o compu te ot h er ph ysica lly im por ta n t

t it ies (s uch M shea r s t res s, d ra g , h ea t -t r amfer r a t >, a nd

trrmsfer coeff ic ient) associa ted with the free-convect ion

low . To t his end, t wo pa ra met ers, a flow pa ra met er a nd a

a t -t r a ns fer pa r amet er , m e d er iv ed in a ppendixes C a nd D ,

The f low pa r ameter

i-

=F’’(o)

(4(3r=~$(vmw/ X)

s present ed a a a funct ion of P ra ndt l number in figure 7.

hus , t h e v a riou s flow q ua n t it ies for a g iv en s et of ect n dit ion s

a n ea sily be compu ted by a pplica t ion of fig ur e 7.

The loca l hea t-t r ans fer pa ramete r

Nu

=

–H’(o)

(Qr x/4)+

s d et ermin ed h er e is given a s a fu nct ion of P r a nd ti n umber

in figu re 8. A ca lcu la t ion of t he loca l N uss elt n umber h or n

t h is eq ua t ion for

Pr=O.72

a nd Qr= = 10g y ields a va lue of

63.6,

which indica te s tha t l a rge hea t-t r ama fe rcoe ff icien ts ean

a l so be ob t a ined w i th f ree-convect ion f low s .

3,2

2.8

2.4

2.0

~P

-:

r

1.6

a:

q

w 1.2

.8

.4

c

I

I

1

I I

o D

(i’L

o

1.969

0

3.939

@

4

1’

I~78xwgfm ~tal

D

8:817

V 15.756

VD

— Theoretkol for R“ 0.72

0

7

v.

D

b

,4

u

7

\

b

4

r

‘v,

v

\

>

D

?

\

v

c

-

c.

o

v

v

o

,,

n

\

v

Q

D 4

D 8

d

3

\

D

8

b

\

Lo

2 .0 3 .0

, 4 .0

5 .0

6 13

/’ .

=(83.93) Y X7 ‘

FIOVBES.-corn-n

Offwo P~@ W T@II I~ ~d

Owreifd VowaY dfstrfbltloru

forProndtlmunbmof0.72.

TRANSFER ABOUT A FLAT PLATE

7

On the ba sis of a simplified theory (tha t iej by use o

in tegr at ed m omen tum a nd en er gy eq ua t ion s a nd a ssum ed

veloci~ a n d t emper a tu re d ist r ibu tion s), E ek er t (see p. 162

of r ef. 7) obt a in ed t h e a ppr oxim a t e r ela t ion

N u ~ 0.718(P

(w x/ 4)i (0.962+P

I .2

x

4--

.8

Y.L I

I

I I

V 15.756 ]

.

F

-t-l

Theoretical for /?= 0.7?

I I

I

I

.2

\ -v

P

De

v

q

Q

v

~

o

0

1.0

.20 3.0

, 40 5.0. 60

,—

I

From 7.—Dtm8ml0IIlm f low fmrameti u fnmtkm of Prandtf number.


10/18

72

REPORT 1 11 l—NATIONAL ADVISORY COMMTMIEO

FOR AERONAUTICS

Th e cur ve r epr esen tin g t his eq ua t ion is a lso pr esen ted in

figure S , a nd it closely a pproxima tes (to w ith in a bout 10

per cen t ) t h e cu rv e det erm in ed by t h e mor e exa ct Con sid er x

t ion s of t his r epor t over t he en tir e P r a n dt l n umber r a ng e. A

s em iempi ri ca l eq ua t ion a s g iv en in r ef er en ce 11 r ela t i ng t h e

a ver age (over t he len gt h X) NuAelt n umber t o t he P r a ndt l

a nd G ra shof numbers w hich ha s been used in t he hea t-

t r a n sf er ca l cu la t ion s up t o t h e pr es en t i s

Num= o.54s [(B-)(%]*

The const ant 0.54S pert a ins spec.ilica lly t o a ir ; for oil it

s hou ld be 0.555 (see ref . 12) and for mercu ry, approt i a t el y

0.33 (see ref. 13). In order t o obt a in loca l va lues fim the

a , ver ageones g iv en by t he l a s t equa t ion , i t i s merel y necemry

t o mul tiply t h e a v er a ge va l ues b y 0.75. (The det erm in a t ion

of t h is r ed uct ion fa ct or of 0.75 is d is cumed in a ppen dix D .)

Thus in t erm s “of t he loca l q ua nt it ies t he sem iem pir ica l

rela t ion becomes

NU= O.411 [(Pr) t7rx]*

or

Nu

=0.5s1(R-)*

(Qrx / 4)+

The cu rve g iven by t his eq ua t ion is a ls o pr es en tid in fig ur e

S a n d t he a g reemen t w it h t he t heor et ica l cu rve is ver y good

for P r a n dt J n umber s n ea r unit y , n ot s o g ood for la r ge P r a n d tl

num bem, a nd very poor for t he sm all P ra ndt l num bers. Of

course, changes of the cons t an t s in the semiempir ica l re la t ion

a s p rev iou sl y d is cu ssed for t he l a r ge or sma ll P r and t l number

ca ses (oil a nd m er cu ry , r espect ively , for example) w ould

cause the semiempir ica l curve to apprtmimate the theore t ica l

curve more closely . The v a lu es of t h e h ea tAr an sf er pa r am-

et er obt a in ed exper im en ta lly for m er cur y (P r = O.03), a ir

(P r =O.72), w a t er (P r = 7), a n d oil (P r = 75.5, 115:190,224,

275, 31S , 368, a nd 442) a re h er e r educed by t he fa ct or 0.75

from t he a vera ge va lue report ed. The va lue for m er cury

is a n a ver age t a ken of four r ea din gs fr om a cur ve, sin ce t his

exper im en t w a s t he on ly on e n ot r epor ted in t a bula r form.

F rom iigure S it ca n be seen t ha t a ll of the exper iment a l

va lu es except t hos e for t he oil exper imen ts a r e in ver y g ood

a gr eem en t w it h t he t heor et ica lly comput ed va lues.. Th e

da ta from t he oil experiment s, though not so good,

r ea son able a gr eem en t (m tium er ror of a ppr oxim a t

per cen t) w it h t he t heor et ica l cu rve rmd g ood a g reeme

is t o be expect ed , w it h t he s em iempir ica l cu rve. Th e d

en ce bet w een t he t heor et ica l va lu es a n d t he oil exper

result s ca n possibly be due t o t he fa ct t ha t t he visc

changes in oi l a r e l a r ge even for sma l l t empera t u re d if fe r

or d ue t o t he en d effect s in t he mea s ur emen ts .

CON~USIONS

An analyaia was

made of t he fr ee-con vect ion flow a b

fla t pla te orient ed in a direct ion pa ra llel t o tha t o

gen er a tin g bod y for ce u nd er t he pr ime a s sumpt ion t ha

r ela t ive t emper a tu re d ifler on ce is sm a ll. I t w a s foun d

t he G r as&of n umber w a s t he pr in cipa l fa ct or d et wm

t he t ype of flow a nd t ha t for la r ge G ra sh of n umber s t he

w a s of t he bound ar y-la yer t ype. Th e t heor et ica l dev

m en t w a s t hen con tin ued t o con sid er on ly t he ca a ea of

G ra shof number beca use t hese a re of m ost im port an

amonaut ics .

Velocit y a n d t emper a t ur e pr dles for P r a n dt l n umbe

0.01, 0.72, 0.733, 1, 2, 10, 100, a n d 1000 wer e compu te

the ba sis of a const ant body force a nd pla te t emper

a nd a greement w ith experiment s w here the fluid w b

(P ra ndt l number of 0.72) -w a s g ood. I t vm .s a lso de

st ra t ed t ha t velocit ies a nd Nuss elt n umber s of t he or d

magnit u de of t h os e ob ta i ned in for ced -con vect ion cou

obt a ined in f ree-convect ion f lows .

A flow pa r amet er a n d a h ea t -t ra n sfer pa r amet er wh ic

fu nct ion s of t he P r a nd tl n umber a lon e w er e d er ived . C

la t ion s of t he im por ta n t ph ysica l q ua n tit ies such a s

s t res s, h ea t -t r a ns fer r a t e, a n d t h e l ik e ca n be compukd

t hese pa r am et er s. Va lues of t he h ea t -t ra n sfer pa r a

ob ta i ned f rom an appr oxima t e t h eor et ica l d ev elopmen

f rom exper imen t s compa r ed w i th va l uea compu t ed f rom

pr esen t d evelopm en t sh ow ed good a gr eem en t, over a

ra nge of P ra ndt l number (0.01 t o 1000). I t is show n

t he common ly u sed sem iempir ica l r ela t ion ’ for t he

t r a ns fer coeff icien t w i ll y iel d g ood r es ul ts on ly in r es t

P r a n d tl n umber r a n ges .

Lmvrs

FLIGRT

PROPULSION LABORATORY

NATIONAL ADVI SORY COZJM IT PE E FOR AERONAUT ICS

CLEVELAND, OHIO, October3, 1961

APPENDIX

A

SYMBOLS

The f ollow i ng not a t ion i s u sed in t h is r epor t :’

Atj(n), B j(a ),

coef fi ci en t s in numer ica l d ifb ren t ia t ion -a n d

@), D i(fll

in t egra t ion formulas

Cp

speci fi c hea t a t con s t an t p res su re

F

dimensionless velocity function

ff

componen t s of body form per unit ma ss,

i= l, 2, 3

.f x

n eg a tive of X-compon en t of bod y ~ f or ce per

u nit m a ss

G

G .

g

H

h

K

G ~ of number EMz~

G ra sh of n umber ba s~ ~ on X

g ra v it a t ion a l f or ce per unit ma s s (or a cce

t ion d ue t o gr i@y )

d imens ioi lew tempera ture funct ion

heat -tra nsfer coeflicie~t

isothermal compressibi li ty coeff icient ,

_p

a(l / p I

[1

ap =


11/18

FRJ J E -CONVECTION FLOW AND H33AT

TRANSFER ABOUT A FLAT PLATE

, n

u

, s, t

i

f

therma l-conductivity coeiiicient

character is t ic length

arb i t r a ry esponent s

a n umber , d efied follow in g eq ua t ion (26)

Nussel t number,

hX/ k

aver age Nussel t number

pressure

P r a n d tl n umber

ga a cons t rmt

arb i t r a ry esponent s

absolute tempera ture

velocit y compon en ts , i= 1, 2, 3

d imen sior d eiwv elocit y componen t s, ;= 1 , 2, 3

d imens ion les s veloci t y componen t in zd ir ec-

tion

d ime.n s ion lw veloci t y componen t in y -d ir eb

tion

Cm%e.s iancoord ina tes , ~ = 1, 2, 3

d imens ion lessC ar t e9ian coord ina tes , i = 1,2,3

dimensionless Ca rtes ian coordinate

Car t es ian coord ina te

dimensionless Ca rtes ian coordinate

B

‘r

A

E

v’

e

K

P

v

P

u

7-

;

Subscripts:

~ Jj

8

0

w

coefl?icientof volumetric expansion,

r a t io of s pecif ic h ea t s

La pla cia n opera t or ‘

rela t ive temperat ure difference, f?(Z’O—

T.)

s im i la r i t y variable

d im en sion lem t em per a t ur e fu n ct ion

s t ep s ize

used in nwmer ica l ca lcu la t ions

a b sohh v is cos it y

kinema t ic v iscosi ty

d~ i t y

dimemiordws pressure funct ion

shea r s t r es s

d imens ionkas dens it y funct ion

st ream funct ion

Carte . s iaq

t e n sor an d sum ma t ion su bs cr ip t s

d en ot e s eva lu a t ion a t s t a t ic con d it ion s (c=O

den ot es eva lu a t ion a t p la t e su r fa ce

d en o t e s eva lu a t ion a t u n d is t u rbed con d it ion

Subs cr ipt n ot a t ion i s u sed t o d enot e pa r t ia l d if fer en t ia t i

Superscripts:

P r imes denote ord inary d i fferent i a t ion .

B a r s (a s ; or ~ d enot e t r a ns formed d imen si on les s q u ant it

APPENDIX B

NUMERI CALSOLU TIONOF SIMP U l?IED BOUNDARY-VALURROBLEM

By

LYNN U. Armms

The met hod is pr es en ted h er ein by wh ich s olu tion s t o t he

undary-value problem

obt a ined

, 2, 10,100,

~ “’+ 3FF’’-22+ H= 0=0

(B1)

H “+3PTFH ’=0 (B2)

F (0) =F ’(0) =0 H (0) =1

F’(=)=H(0)=’o

for t he ca ses of

Pr

equa l t o 0.01, 0. 72, 0. 733,

a nd 1000. This discussion w ill ena ble t he

sult s t o be clea rly eva lua ted a nd w ill perha ps ser ve a a a

ide in t he numer ica l s olut i on of s im i la r p rob lems .

Ea ch of the ca ses of the problem ha s a solut ion for a

a rt icula r set of va lues for

F “ (0)

a nd

H’(0),

hereinafter

ed eigenvalues.

Th e ba s ic a ppr oa ch t o t he pr oblem wa s

est im a te t he eigen va lues a nd t o in tegr a te out fr om zer o,

bt a in in g fu nct ion s wh ich s a tis fied eq ua t ion s @l) a n d (B 2)

ea ch s tep. Th e in teg ra t ion w a s con tin ued u nt il t he fun c-

F ’

a nd

H

beh aved in a fa sh ion in con sist en t w it h t he

und a ry v a lu es a t in ii nit y ; f or exampl e, wh en t h ey became

ega t ive or d iverged t o in 6 n it y.

igen va lu es we re t h en m ade on

re c ed in g ru n s an d t h e p roce s s

n t il a s olu t ion was obt ain ed .

Improv ed es t ima t es of t h e

the ba sis of the result s of

wa s repea t ed succes si ve ly

Modh ica t i on s requ ir ed t o over come speci fi c obst a c le s w

be d is cu ss ed a ft er s ufficien t d et a ils of t he ba s ic pr oced

h ave been given. Th en a n eva lua t ion of t he a ccur acy of

numer ica l r es ult s w i ll b e made.

Th e in teg ra t ion pr oce9s con sis ts of t w o pa r t s,

a

s t a r t

pha se a nd a n ext ension pha se. The st ar ting pha se beg

w it h a n eMr na t e of t he eigen va lues

F“(0)

a nd

H’(0)

a n

decision on t he st ep size K to be used. I t cont inues w it h

i ter a t iv e pr oces s of a l ter na t ely compu t in g F ’” a nd H “

t he t it four point s a nd integra ting t hem by five-po

formula s. This process a nd t ha t in the extension pha

a r e s o. a r ra n ged t ha t t he d iffer en tia l eq ua t ion s a r e s a tis

a t ea ch in teg ra l mult iple of t he s tep s iz e.

The ext en sion ph a se u sed pr eced in g d a t a t o in t eg ra t e s

by st ep beyond t he fourt h point . D ia gra ms of bot h pha

w i ll b e g iv en a f ter a few pr elimina r y expla n a t ion s.

All in teg ra t ion formu la s u sed a r e ba s ed on t he s ame id

I f a funct ion , f or ex ample, F“’”, iskn own

at five points, there

a

un iq ue four th degr ee poly nom ia l w hich a gr ees w it h it

t hese five poin ts. Mor eover , if t he su ccessive a n tid er i

t i ve s (i it eg r a ls ) F“, l “, a nd F of F ’t ’ ar e know n a t

poin t, t her e a r e un iq ue fift h-, sixt h-, a nd seven&d egr

poly nom ia ls w hich he successive, a nt id er iva t ives of t

f ou rt h deg ree pol yn om ia l a h d wh ich a g ree w i th

F“, F ’,

a

F,

r espect ively , a t t he on e poin t. I t is t hen a sim ple a lgb

pr oblem t o d ed uce fr om t he va lu es of

F ’” a t five po in t s a


12/18

74

REPORT 1 11 l—NATIONAL D~ORY

F , F ’, a nd F“ a t a single poin t the va lues of a ny of these

four poly nom ia ls a t a ny poin t. Th ese r esult s w ill a ppr oxi-

ma t e t he funct i on s

F , F ’, F“,

a nd

F ’” ta a

degree dependent

on s tep s iz e, t he r ela t ive pos it ion s of t he poin ts in q ues tion ,

a nd the ma gnitude of t he iift h der iva tive of

F ’” i n the

neighborhood of these poin t s .

The pr eceding a lgebra pr oblem ca n be pr esolved in a ll

s it ua t ion s t ha t a r is e in t he s ta r t in g a n d ext en sion ph a ses of

t he pr es en t pr oblem a n d s pecific in teg ra t ion formu la s m a y

be deduced. These formula s a re discussed in the next

paragraph.

Let

F ’”

be denoted a t five successive point s by

Fe’”,

F 1’”, F ~’”, F ~’”,

a nd

F.’”.

I n the

s ta r t in g ph a se, t hes e

poin ts a re O,

K, 2K, 3K,

a n d 4K , a nd

FO,FO’,

a nd

F O” a r e a l so

known.

Then t he five s et s of for .umk r eq uir ed in t he s ta r t -

in g ph a se a r e.

E i =Ho+ iKHo ’ +

&

A. i jmE; ’

;= 1, 2, 3, 4 (I36)

Fi =Fo+W+~ Fo”+& ~ A,jm F i ” “

i=l,,2, 3, 4 (B 7)

wh er e t he s uper scr ipt s on t he

A; ‘a)

a nd

D i @

refer t o t he

order of in tegra t ion .

The cons t an t s

A,j@)

a nd

Di @I )

ma y be rea d from t he fol-

10~ t a bles:

For A~ j (l J and Di(l) :

.

COMMFFPEE FOR AERONAUTK

For At i@)

a nd

D i @?

Au($)

1

W?)

,J ~ ~ .2 ~ 4

1

1017 1070 -618 263

-47 mm

-m

: lE 3% -480

–19

M -81

lE

4 744 n?o 93 334 -40

316

I t is n ow pos sib le t o d ia g ram t he s teps of t he s ta r t in g

of t he int egra t ion . I f ea ch ba r a bove a funct ion denot

im pr oved &im a te of it , a nd t he fir st estimates of F1 ’”,

F3’”, a nd F4’” a re W equa l t o Fe’”, a nd sim ila rly fo

H “,

t hen t he s ta r t in g ph a se d ia gmms a r e

(1) (F o, F e’, F O”, F e’”, F ,’”, F ,’”, F a’”, F 4’’’)+ F,, F 1

(T hi s d i agr am mea ns t ha t the va lues in pa rent hese

used w it h a ppropr ia te int egr at ion for mula s fr om @3

(B7) t o ob ta in F , F ’, a nd F “ at q=z.)

(2) (H o, H {, H o”, H ,”, H ,”, H ,”, H [’)dH ,, H I ’

(3) (F ,; F ,’, F ,”, H ,, H ,’) -~,’”, ~1”

(The preceding d iag r am means t h a t t he v a lues in pa ren t

a re subst it ut ed in t he d iffer en tia l eq ua t ion s @l) a nd

t o ob ta i n F ’” a nd H “ a t q= K .)

(4)

(F ,, F {, F ;’, F ,:, ~,’”, F ,’”, F s’”, F 4’’’)=3F ,, F ,

(5) (H O,H ;, H o”, H I”, H ,”, H i’, H ,’ ’]* H ,, H ,’

(6) (F ~, F~’, F s”, H ,, H ,’)~F ,’”, E ,”

(7) (Fo, Fe’, F J ’, Fe’”, ~1’”,

~ ,’”, F ,’”, F4 ’’’+F~j~ j Fa

(8) (H O,H o’, H o”, g,”, B ,”, H ,”, H 4’’)+7,, H 3’

(9) (F a, F s’, F s”, H ,, H 2’)+7Y”, Z “

(10) (FO,F,’, F, ’ ’, FO’’’,”, ’”,

l?,’”, F3’’ ’,F)+F4,1F4,,’4 ’,

(11) (H o, H o’, H o”, ~,”, E ,”, E ,”, H 4’’)~H 4, H 4’

(12) (F ,, F ,’, F ,”, H ,, F4’)+,’”, ~,”

I t ma y be not ed here tha t a ll four va lues of F ’” a n

have

been improv ed , a n d fu rt h er improv emen t w i ll r e

it er a tion of s teps 1 t o 12. Th e s ta r t of t he s econ d it er

isdi

amamrned s s f ol lows :

(13;

(14)

(15)

(16)

.

.

,.

.

. .

.

.

.

Su cces si ve s et s of 12 s t eps a r e per formed unt il t h e v rd

F,

‘“

a n d E t ~ ” n o lon ger ch a ng a .

On t he I IXM Ca r d-P r og rammed E lect ron ic C a lcu la

deck of punched cwds 2 inches t h ick su. fi icedt o pe rf orm

1 to 12. Three runs of t h is st a rt er deck a t 3 minut e

r un a ccomplis hed complet e con ver gen ce in mos t ca s es

t h e end of t h e s ta r t in g pr oces s t h er eh a v e been compu t e

stored

F Fd ’, F4 / ’,.Hh ,

a n d H4f, a n d &ml es t ima t es of

F S’”, F S’”, F d’”, H ,”, H ,”, H S”, a n d H4”.


13/18

FREWCONVDCTION FLOW AND HEAT

The ext ension pha se ha s now been rea ched. I t used a

ffer en t s et of in teg ra t ion formu la s ba s ed on t he same gen -

ra l idea a s equa tions (B 3) to (B 7). I f FO’”, F1’”, .F~ ’” ,

s ’” , a nd F ,’” now designa te F“r

a t a ny five succw sive

oin ts , a n d t he su bs cr ipt 5 d en ot es t he n ext poin t,

(B8)

(B1O)

(611)

F5= ~4+ KF4’+; F;’+& , B,WF’”

(3312)

t he .B j@J a n d C@) a r e given in t he follow in g t a ble:

l?+)

c(=)

j o 1

2

3 4

a

1 ml –la74

W6 –2U4 1931 Tm

-s92

1446 -16%3 1427

M –2116

4478 -Ke4 M74

4?%

I

1 1

1 1 1 1

(

The &xt en sion ph a a e ma y t hen be d ia g ra rmn ed s imply a s

(1)

(F ,, F i , F :’, F {”, F ,’”, Pi ”, F :”, FJ ’’)~F,, FL , F ,”

(2) (H i , H i , H i ’, H ~, H :;, H :;, H 4’’j~H ,, H ; “

(3) (F ~, F(, F (’) H ,, H :)+F :”, H :’

The v a lu es of t h e f un ct ion s a t t h e n ext poin t a r e compu t ed

n simila r ma nner , w here the la test set s of five va lues of

’”

a nd

H “

a re used. Th is pr ocess a dva nces st ep by st ep

t owa rd int i it y .

The ext ended deck of punched ca rds m a a bout 3 inches

t hick a n d t ook a lit t le over 3m i nu tes per r un . F or P r =O.72,

s t ep s iz e of 0,1 wa s u sed , t h e s t a r t in gph a se t ook 10minut es ,

d the extension pha se, a bout 30 minutes. When it is

a l iz ed t h a t a bou t 11,000 oper a t ion s wer e per formed in t h e

0 minut es per run, it ma y be seen tha t solut ion of t he

r es en t pr ob lem wou ld h a ve been pr oh ib it iv ely d if fi cu lt on

sk-t ype cr i lcu la tom. S impl if ica t ions in method would have

r if iced a ccu ra cy or r equir ed sma l ler s t ep s iz e.

I n t wo-poin t bound a ry -va l ue pr ob lems wher e on e poi nt i s

int l ni ty , s ome prob lems of judgmen t a re i ri volved a s t o where

infin it y is, a nd a s t o w hen a sa t isfa ct or y a ppr oxim a tion t o a

solut ion ha s been obt ained. h m ost w cs t his q uest ion w a s

s et t led for t he pr es en t pr oblem by ca llin g a r un s a tisfa ct or y

w hen it fell bet ween t wo runs for w hich

F ’

a nd

H

d id n ot

d iffer a t im por ta nt poin ts in t he four th d ecim a l pla ce, a nd

TRANS FE R ABOUT

for w h ich

F ’

a nd

dechmd places.

A FLAT PLATE

H fla t t ened out a t zero, corr ect t o fo

Cert a in difE cuMes w ere met in the a tt empt t o use t

b a si c pr ocedur e pr ev iou sly d is cu ss ed . Thes e n eces sit a t

certa in modif icat ions .

For P r= 2, 10, 100, a nd 1000,

H

-w ou ld s et t le d own

zero a t a n ea rly st a ge; but w hile

F ’ w as st i l l com in g

dow

H “ wou ld beg in t o os ci ll a t e and t lmse osci ll a t ion s increa s

a nd fed ba ck int o a ll other funct ions. This t rouble w

a void ed by t he follow in g mod ifica t ion s: I t is a con seq uen

of eq ua t ion (B 2) t ha t

.

( IF@

’ (q)=H ’(O) exp –3PT

=H ’ (q –

K) exp

-3p’c=F@4 ’13)

The ex t en sion phase waa mod if ied t o requ ir e t he add it ion

integrat ion formuhw

5

H ,=H ,+ ~ A i H :

(B1

(B

where Al= –19, A~ = 106, A3= -264, A4= 646, a nd 4= 251

Th ese formula s w er e used a lon g w it h eq ua t ion s (B 8)

(B1O) a ccor d.ir g t o t h e fol low in g d ia g ram :

(1) (F4,

F i , F~’, F :”, F ,’”, F ,’”, FJ ”, FJ ’’)~F5, FJ , F

(2) (FI , F ?, F ,, F A, F ,, H4 ’ )+H ’ by mea ns of

(1314)

a

(B13)

(3) (H I ’, H9’, H ;, H ;, H :, H&H~ by mea ns of @15)

(4) (Fs , F s ’, F {’,

H,)+F:”

The va lue

F.

‘“ i s

discarded and

F ’”

a t t h e l a s t fiv e poin

is used to repea t the whole process a ga in a nd a ga in

infin it um . As long a s F s t ays pos it ive , H ’ i s gua r a n teed

appr oa ch z er o an d H wi l l flatten out to some value a nd n

oscillate.

F or P r =O.01, 0.72, 0.733, Wd 1, t he

F ’”

began t o osci ll

a t a n a d va n ced poin t a n d t hes e os cilla t ion s g r ew a n d fed in

t h e ot h er fu nct ion s. F or a l l ca s es but .F%=O .01, t h e os ci

,t ions a ppea red very la te, nea r the end of the run, a nd

su it a ble h a lvin g of s tep siz e when oscilla t ion w a s d et ec

in t he fou rt h d iffer en ces of

F ’” was

.m fE cien t t o a void

d iff icu lt y . Bu t for t h e 0.01 ca s e, os ci ll a t ion s o f

F ’”

appea

ea rly in t he run, na mely , soon a ft er t he pea k in

F.

Th

oscilla t ion s w er e found t o be st ep-size con nect ed , so t

reduct ion of the st ep t o 0.02 a voided t hem. Even th

osci l la t ions in

Fr t l

would begin t o a ppea r ever y 25 st eps

s o, a n d t hese w er e smoot hed ou t r eg ula r ly by r epea t ed r u

of a deck sim ila r t o t he st ar ter deck. E a ch run under t h

con dit ion s t ook a bout 16 h ou rs, m a kin g t his t he mos t d

cu lt ca s e t o s ol ve.

.-


14/18

76

REPORT

1 1 11—NATIONAIJ ADVISORY CO~El FOR AERONAUTICS

APPENDIX c

DERIVATIONOF FLOW PARAMETER

I

Subs tit u ti on of t h is expr es si on in t o equa t ion (C l) y ield

B y d efin it ion t he s hea r s tr es s i s given by

f low pa r amet er

.(Cl)

To exprq s @?7/bY), in t er ms of t he know n @ct ion F(7),

use ca n be ma de of eq ua tions (60) a nd (62). Then

T

=F’’(o)

(4Qrx~*(vmp0/ AV)

Not e t ha t fr om t he gen er al d er iva t ion, t he flow pm am

cont ains t he viscosit y eva lua ted a t t wo different po

R eca ll, h ow ever , t ha t t he a na ly sis h as sh ow n t ha t t o a

a ppr oxim a tion t he va r ia t ion of vis cos it y w it h t emper a

&n be neglect ed.

Thus the viscosit y cun be t a ke

con st a nt in t he en tir e flow field .

APPENDIX D

DERIVATION OF HEAT-TR ANS FE R

PARAMETER

The l oca l Nus sel t n umber i s d efin ed a s

~u=hX_ –x a

(+.(TO–T.)~,

(D1)

To es pr es s @T/3Y)0 in t erms of t he kn own fun ct ion

H q ) ,

use is ma de of eq ua tions (61) a nd (62). Thus

The hea t-t ra nsfer pa ra met er a a given by eq ua tion

is, a s w a s previously st a t ed, a loca l pa ra met er . I t is o

d esir ed t o comput e t he a ver age (over ,t he lengt h X) v

. of t his pa r am et er . To t his end, the Nusselt number

given in equa t ion @l)) must be defined in terms o

a ver a ge h ea t -t ra n sfer coefficien t a n d t he q ua n tit y t hu

t a ined must then be int egra t ed over the length X

d ivided by X. Th is p~ ocedur e y ield s t he r esult

S ubst it ut ion of t his expr ession in to eq ua t ion (I II ) y ield s

t he hea t -t ra nsfer pa rameter .

Nu= : (Nu)m

N I L

=–H ’ (0)

(D2)

It iS from this last equation that th e

(@=/ 4)4

prev ious ly d iscussed wa s obt a ined .

REFERENCESh

]. Eokert, E. R. G., and Soehngen, E. E .: S tudfea on H ea t Transfer

7.

in Laminar

Free Convection with the Zebnder-Mach Interferom-

eter. Tech. Rep. hTo. 5747, A. T. I. ATO. 44580, Air Materiel

&

03 rmn a n d

(D a y t on , Ohio), D ec. 27, 1948.

2. %hxrddt , E rnst , und B eckma q Wiielm: Da s Tempera tw und

Ges chw ind fgke it ef dd vor e iner WWrmeabgebenden s enk re ch t er

P la tt e bei n at tlr lich er K on vekt ion . Tech Mcch .

U- The~@

9.

dyna rn& B d. 1, Nr. 10, Okt . 1930, pp. 341-349; cont ., B d. 1,

N r. 11, Nov . 1930, pp. 391-406.

3. S chuh, H .: B ounda ry Layers of Tempera tum. Reps. & Tmm.

~ (j

1007,

AVA Mon og ra ph s, B r it ish M . A. P ., Apr il 15, 1948.

4. Ost ra ch S im on : A B ou nd ar y L ay er P roblem in t he Th eor y of F ree

C on vect ion . D oct or a l Th esis , B r ow n U n iv., Au g. 1950.

5. Lew is, J . A.: B ounda ry byer in Compressible Fluid. Mono-

ll.

gr aph V, Tech . R ep. N o. F –TR-1179-ND (G DAM A-9-h f V),

Afr M at er iel C Omma n

d (D FLyt mj Ohio), Feb. 1948. (Ana lysis 12.

D iv ., I n tel lig en ce D ep t. C km t ra ct W’33-033-a e150M (16%51)

t it h B row n U niv.)

& jkrnansky, Ma rk W.: H ed a nd Thprmodyna rnics. S econd d.,

la .

hlcGraw-Hilf

Book Co., hlC.,

1943, pp. 23-27.

0.75 reduct ion fa

E oker t , E . R. G .: I nt roduct ion t o t he Tra na fer of H ea t a nd M

McG r aw -H fll B ook (2a , I ns ., 1950.

L a ge rs t rom , P a ce, C ol e, J u lia n D ., a n d Tr illin g, L eon : P r ob lem

t h e Theor y of Vis cou s C ompr em ib le F lu id s. G u gg enh eim

La b., C . I . T., Ma rch 1949, pp. 38-40. (OfE ce of Na vnl

Con t r a ot N60ru -Ta sk VI I I .)

F riedr fch s, K . O., a nd lVa sow , W. R.: Singular P e rt u rb a tion

N on -L in ea r Os cilla t ion s. D uk e Ma t h. J ou r., v ol. 13, 1946

367–381.

E ck er t, E . R . G ., a n d J a ck son , Th om a s W.: An a ly sis of Tu rbu

F ree-C onvect ion B ounda ry La yer on Fla t P la te. NAC A

1015, 1951. (S uper wdes NACA TN 2207.)

McAd+ s , Willia m H .: H ea t Tr an sm ission . S econ d cd ., MoG

~

B ook CO.,

hlC.,

1942, p. ~ .

Loren .z, H ans H .: D ie Wiirmet lber t rsgun g von ein er eb

sdzrech ten P la tt e a n, U 1 b ei n at tt rlioh er K on vekt ion . 2

t eoh . Phy s ., N r . 9, He ft 16, 1934, pp. 362-366.

S aundem , O. A.: Na tura l con vect ion in liquids. P roo. Roy

(London ), s er . A, v ol . 172, no. 948. J u l y 19, 1939, pp. 56-’71.


15/18

REE -CONVTlmON FLOW

AND HEAT TRANSFER ABOUT A FLAT PLATE

TABLE I .—FUNCTIONS F XND H AND DERIVATIVES FOR VARIOUS P RANDTL NUMB ERS

(a)

Przndtl nnmlxr, 0.01

F

F“ H F

L 9319

1.m

LW

L W40

L8769

1.ms7

2 OU4

2mm

Low

21037

z 14U

217$0

2 a43

2 ml

22853

232m

~g

2617u

2s787

26s2.4

21ma

z 7624

28037

28m3

2W4

&mm

a 1411

&m

&m

3.371b

:ij

Z6E3

3.6022

X7M6

3.7514

x mu

;~

$%

4.0M4

4.0591

4.aw3

4.Km

4.1226

4.13m

4.1343

F

o.4m7

.4024

.4012

.=

.m

.3939

.3916

. ml

.s$2 2

.3774

.3n6

.36M

.2804

.3348

.3494

:g

:%%

.3m5

; ~;

.m

.mn

.X%3

.2423

.2%?II

.X23

. 1s34

. 18M

.1724

. lm

.1601

. 13W

.I m7

.lxn

.1114

. Iml

.M52

.0877

.0773

.mm

.M42

.0513

.0441

.M36

.M46

.0171

.Olm

.M57

.C4116

1-’

-a MM

—.m14

—.0612

—.Oem

—.mm

—.m

—.0.w9

—.mm

—.065-7

—.mm

—.0572

—.C&4

~.

—.W-41

—.0s34

—.06m

—.m

—.04m

—.0479

;.

—,0i42

—.0434

—.6419

—.mm

—.mm

—.0366

—.0?37

—.0319

—.mm

—.0236

—.0270

—.0254

—.0241

—.02m

—.0216

—.am

—.0192

—.Olm

~ m)

—.0133

—.0126

—.0114

—.0m7

—.CO u

—.6m3

—.mm

—.0046

–. 0m7

H

a6741

%J

.M54

.& m

.6697

.m

.6497.6427

.m67

:=

.6149

.Msl

.m13

.M78

.5746

.M16

.mm

.S3m

.5236

.6113

.4m3

.4n4

.4643

.44XI

.4205

.=

.Wm

.3m7

.3423

.3246

.3WQ

. 2916

.27E3

.Mlo

.24m

.2232

.Lm2

.mls

.1847

. lm-i

. lm

.13M

. llwl

.mm

.mm

.0694

.Wa5

.0452

H ’

-0.0723

—.0722

-. Om

—.07m

—.0n 8

—.on6

—.n4

—.on3

—.Oms

—.0704

—.mm

—.0m5

—.Omil

—.06s6

—. m

—.0676

—.0667

—.m67

—.m48

—.Ms3

—.mm

—.mm

—.w

—.06m

—.ms3

—.w

—.W7

—.m27

—.Oms

—.64s9

:. 04~

~.

—.m

—.mso

—.M64

—.0w3

—.M92

—.0317

—.0m6

—.0276

—.m57

—.OmJ

—.0223

—.0197

—.0175

—.0161

—.0137

—.Olx

—.0107

O.oml

.W6

. in’4

. Zm

.3164

.37%

.41s3

.4M6

.4919

.Sl%

.5379

.6527

.bml

.M97

.5731

.6739

;=

.M50

.5693

:%

.m

.6W

.Ez53

.6178

.6164

.W?3

.4W

.4m2

.4810

.4736

.4e@

.46m

.4mo

.4496

.4462

:$H

.4?s2

: %

.4264

:%%

.41@a

.4X36

.4106

.4074

.4m4

.4049

amm

.M69

. ml

.m42

.M30

. 61m

.4349

;%

.1734

.Iz59

.Ilm9

.64s9

.01s3

—.0m9

—.02m

—.mm

—.04%3

—.m

—.0641

—.w

—.on4

—.m

—.0742

—.0746

—.0745

—.

0741

—.0733

—.0728

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16/18

78

REPORT 1 11 l —NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

TABLE I .—F~TCTIONS F AND H AND D ERIVATIVE S FOR VARIOU S P RANDTL NUMB EIW-C orit ixmed

V

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.5.W

.0184

.6916

.575

-.6478

.Olw

:% –. mm –. ml

.m .0w6 .6137 –. m :% –.4s37

.es

.02.27 .0433

–. 0111 .0417 -.3496

.70 .0240 .04m –. Olm :=

.75. :~

.0420

–. 1440

–. o135 –. 1657

.m

.65

–. 0140 .0111 –. lm

.0311

:26

.Q3

–. 0142 . mm

-. mm

.ml .03$9 –. 0142 .C039

.ml :%

-. 04m

i:

–. o141 ;~

–. 6256

.a3n –. 0140

L 10

.04u9 .03n

-,0149

–. 0136

:~

–. 0046

L20 .6445 :% –. 0132

.04wl

–. 0013

–. 0W3 .Oom

k%l

.0514 ;=

-. m

–. o124 :=

.W6

-. ml

–. o119

;: .0578 :~ –.0116 .0600

Lm

:M

L80

-.0112 .COoo .KQo

:s –. 0108

.0665

.0a30

:%% -.0104

;$’

%$’ .Ixm

.02?3

–. 0101 .Ooca

21 :%% .0255

.CKm

–. 0097

22

.0743

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-. m

23

:E .COou

.0767 :=6

24

–. owl .Woo

. Om ;g –. W@ .mo

26 .W4

:%%

–. 06s2 .Cmo .mo

.as74 .Olos –. Oom .Cwa

H

.0912

.mo

.Olm –. Oon .Wm .@wl


17/18

FR1313-CONVEUMON

FLOW AND HE&T TRANSFER ABOUT A FIAT PIMI? E

TABLE I .—FUNCTIONS F AND H AND DE RIVATIVE S FOR VARIOU S P RANDTL NU iU BE RS-Ca na luded

(g) Prandtl nnmk, lW-Ckndud6d

F

:&7

.11X6

. lm6

.lml

.1106

.1143

.1176

.Kx 3

. lm

. 126

;&fi

.1315

.1326

.1332

. Iw

.1336

.

F+

:

m3

.0142

. Olzl

. mm

. moz

.Cw@

[email protected]

.m63

.Cm3

.mu

.mm

.Cm2.2

. m14

.m

.W

.m

.m

~: g

–.

mm

–. ma

–. m49

–. mn

–. m

-,. ml

–.

W.M

–.

mm

–. mls

–. 0314

–. Cnlo

–. mo7

–. W

–. m

–. m

-. m

,.

E h“

am

.axll

.Cml

.mxl

.Cml

.a m

.Cm3

.a m

.m

.aml

.lwm

.m

.C o

.a m

.a m

.a m

awl

.m o

o.

.0?3

.m o

.075

. lm

.126

. 1E41

.176

.m

.226

.2W

.276

.W1

.X26

.W

.375

.4m

.426

.460

.476

.ml

.&%

.m

.676

.m)

.626

;: g

1:m

22.73

2@l

&o

2.64.2

&o

&8

7.0

$;

no

14.0

16.0

18.0

Zao

2z0

m.6

F

(h) Przndtl nwnkr, lCKkl

:Cs

.m’

.a m

.Cm+

.m

.

Cm2

.

am

.@18

.m

.Ur26

.ME9

.Ixt32

.K137

,:%6

.w3

.W

.mm

.wu

.wa

.m

.mn

.C075

.m?s

.m

.0107

.0135

. m87

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.m79

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.02m

. [email protected]

.&w

.0649

.Om

.-

. cknl

.0727

.07m

.0771

.on36

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: ~6

O.m

.Cm2

.mm

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.0102

. m16

.0127

.0136

.0142

.0146

.0149

. mm

.0162

. 01s3

.(OS

.m62

. m62

.0162

.mm

.mm

.mm

. m49

.m4s

. m47

. m47

. m46

.m41

.0136

. :E

. mm

.m

.Cw3

.m.Wo

.m

.mm

.aml

.m

.Can

.am

. mu

.m

.m)7

.aw

.C015

.a06

0.1460

.

lmz

.0s99

.mll

. m47

:$%

.am

.02m

. m40

.ma6

.mm

.m

.alla

–. m

–. m

–. 0019

–. m

–. m26

–. m

–. m

–. m

–. WL9

–. cum

–. m

–. mm

-. mm

–. m27

–. m

–. Ooia

–. cum

—. m

–. m19

–. m17–. m15

–. M12

–. 0011

–. m

~.

-. m

–. m

–. ml

.axa

.mxl

.m

.axo

H

E’

Lam

.Wc3

.Etrzl

.7’W6

.mm

.61$3

.433

.3649

.m7

.2?2.6

.1717

.lzm

.W

.0675

. w

.C@l

. 021s

.0134

.W

.mm

.m

.mra

. mu

:M

.m

.m

.mm

.COm

.m

.Cmxl

.m

.m

.CBwl.m

.CK03

.m

.mxl

.m

.m

.m

:R

.COm

.CIxm

.m

.m

,

0

An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the...

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