THE COLLEGE OF AERONAUTICS
CRANFIELD
A SIMPLIFIED THEORY OF SKIN FRICTION AND
HEAT TRANSFER FOR A COMPRESSIBLE
LAMINAR BOUNDARY LAYER
by
v:, .^ '^^ Kanaals»f»at 10 - D£,LFT M)TB WO. 9 3 . Jcmijanr. .1959. T H E C O L L E G E O P A E R O W A U T I O S C R A N F I E L D
A simplified theory of skin friction and heat transfer for a compressible
laminar boundary layer
b y
-G. M. Lilley, 11.Sc, D.I.C. of the
D e p a r t a c n t of Aerodynamics
SU1.MAEY
The oonrpressible LaiT±nar boundary l a y e r e q i a i i o n s f o r a p e r f e c t gas i n s t e a d y flow a t a r b i t r a r y ext^^rnal I'ach number and xvall t e m p e r a t u r e
d i s t r i b u t i o n are solved approximately by the ccmbined use of t h e Stewartson-IllingvTOrth transformation and a p p l i c a t i o n of L i g h t h i l l ' s method to y i e l d the skin f r i c t i o n and r a t o of heat t r a n s f e r .
Appendices are added \7hich give the necessary modifications to the method f o r t h e s e p a r a t e cases of very lovf P r a n d t l nvunber and f o r t h e flow-near a separation p o i n t . A f u r t h e r appendix describes Spr'.lding's method for improving the accuracy of t h e xvall value of shear s t r e s s and r a t e of heat t r a n s f e r d i s t r i b u t i o n s along a M/DII having a non-uniform tempoacature d i s t r i b u t i o n .
This paper was f i r s t -written i n January, 1959. A fev; minor a l t e r a t i o n s have been done during proof reading January, 19éO,
CCJOTEOTS ' Page Summary 1 L i s t of Si^mbols 4 1 . Introd\Jotion 7 2. Basic equations 9 3 . Ste-wai'tson-Illijngvvarth transformation 10
4 . Von Mises' transformation 13 5. Approximate solution of t h e transformed equation
of motion ^6 6. P l a t p l a t e a t zero pressxire gradient and constant
wall temperature 21 7. In5)roved r e l a t i o n be^v/een -wall shear s t r e s s and
' Mach number for v a r i a b l e -wall tenperatiore 22 8. Approximate i n v e r s i o n of t h e vrall shear s t r e s s
i n t e g r a l equation 22f 9. Approximate r e l a t i o n bet-ween -vTall shear s t r e s s
and Mach nimiber -v;-hen the -wall temperature
i s constant 27 10. Approximate s o l u t i o n of t h e transformed s t a g n a t i o n enthalpy eq-uation 32 1 1 , . Acknowledgements 37 ^ 12, Conclu'sions 37 References 3C Appendices
1. Hea.t transfer for fluids of very low Prandtl number 2^2
2. Heat transfer near a separation point 45 3. Approximate inversion of the v/all shear stress
3
-CO.MENTS (Continued) Page Appendices
if. Improved r e l a t i o n s f o r skin f r i c t i o n and heat t r a n s f e r 52 5 , Skin f r d c t i o n i n a l i n e a r adverse pressure gradient 58
6, Evaluation of a c e r t a i n i n t e g r a l 6l Pigures 1 t o 9
Tables 1 , 2 , 3 , 4 , 5
List of Symbols a speed of sovmd
a constant (see equ. k2) b constant (see equ.40) o constant (see equ. 37)
C^,C^_. Specific heats at constant pressure and constant volume
P V J.. T
•^ respectively C
ju T o
f(x), g(x) functions of x (see equ. 4'f")
P(x), G ( X ) , H ( X ) , J ( X ) functions of x (see equ, 34 snd 35)
f x
G(X, M^) = Z - I S(z, f) dü^ (z)
o 1
h stagnp.tion entlialpy i entlialpy
J(x) function of x (see equ. D.5) k thermal conductivity
Ic. Stanton heat transfer coefficient
K(X), K(X) functions of x (see equ. D.11 rndD.12) 1 body reference length
m external velocity gradient index M Mach number
n v/all temperature gradient indeot p pressure; Heaviside operator
ft rate of heat trj-insf er per unit area
Q^ overall rate of heat transfer (— / ^ (x) dx) •'o
- 5 L i s t of S.-^'mbols (Continued) S ( x ) = ^ 7 ^ ^ ) \7 X t = r (x) I X ^a ' ^ a ^ a
t i n d e p e n d e n t -variable ( s e e equ.,22 and 51) T Temperature
( u , v ) v e l o c i t y components i n c o m p r e s s i b l e flow
(U,V) v e l o c i t y components i n t r a n s f o r a i e d ( i n c o m p r e s s i b l e ) f levy ( x , y ) c o o r d i n a t e s i n c o m p r e s s i b l e flow ( Z , Y ) c o o r d i n a t e s i n t r a n s f o r m e d ( i n c o i i i p r e s s i b l e ) flow z durrmy v a r i a b l e of i n t e g r a t i o n z = u^ - U^ 1 y r a t i o of s p e c i f i c h e a t s 3. 7? = X - * /i - v i s c o s i t y V k i n e m a t i c v i s c o s i t y P d e n s i t y cr P r a n d t l number ^ stream function
T -wall shear stress w
r (x) X
^' 1 1 1
List of Symbols (Continued) Subscripts
0 stagnation -vc-JLue
1 val-ue outside the boundary layer \7 value at the wall
7
-1 • Introd.uction
Por bodies t r a v e l i n g a t very higji speeds through the atmosphere t h e r e e x i s t s neijr the b o ^ nose extensive regions of _aminar boundojry
l a y e r flo<,7. There i s , t h e r e f o r e , considerable i n t e r e s t i n finding rapid • accurate methods f o r estimating -the sldji f r i c t i o n ond heat t r a n s f e r i n a
Irminar boundary l a y e r of a p e r f e c t gas a t high speeds under conditions of a r b i t r a r y v e l o c i t y and -vi-a-ll ternperature d i s t r i b u t i o n s . Although i n p r a c t i c a l problems r e a l gas e f f e c t s are l i k e l y t o be of inrportance in t h e higher speed ranges, yet under c e r t a i n conditions, s o l u t i o n s obtained assuming t h e f l u i d in t h e boundary Layer i s a perfect gas, mny be used i n preliminary c a l c u l a t i o n s , provided they are i n t e r p r e t e d c o r r e c t l y .
I t i s g e n e r a l l y accepted t h a t t h e r a p i d accurate estimation aP t h e o v e r a l l c h a r a c t a r i s t i c s of a coiiTpressible laminar boundrr^'- l a y e r , f o r a r b i t r a r y d i s t r i b u t i o n s of e x t e r n a l veloci-fcy and -wall tenperature (or heat t r a n s f e r ) , i s best perfornEd by t h e use of the momenttun and energy
i n t e g r a l equations. For the case when the P r a n d t l nuiiiber (o) equcJLs
u n i t y ond the v i s c o s i t y - temperature index ( w) equals uni-ty Curie (l958b) describes a modified Pohlhausen method, analogous t o Thvmite's method i n incaiipressible flow, by wlxLoh the sl<±n f,riction can be evaluated for the case of heat t r a n s f e r with uniform wall tcanperature, and i n Curie (1958^) f o r non-miiform v.-all t e n p e r a t u r e . An e a r l i e r paper by Curie (1957a-) describes a s i m i l a r method f o r t h e case cf zero heat t r a n s f e r . For the l a t t e r case other methods e x i s t including those of Young (l949) and Tani (1954). Young's method has been extended by Luxton and Ycung (l958) t o deal m t h the e f f e c t of heat t r a n s f e r . For the s p e c i a l cases w = 1 and C = 1 , 0,7 Levy ( l 9 5 4 ) , and when w = o" = 1 Cohen and Reshotko (l95éa, 1956b), gi-ve r e s u l t s for e r b i t r a r y pressvire d i s t r i b u t i o n and cons-tant wall temperatirre. The v;ork of Le-»y stems from the Illing^.7orth (1949) T?:^ilst t h a t of Cohen and Reshotko stems from the Stevrartson (1949) transformation of t h e compressible boundary l a y e r equations whereby, for cr = u = 1 and zero heat t r a n s f e r , the oonrpressible flow equations are transformed
e x a c t l y i n t o t h e incompressible flo\7 equations. Thus knovm s o l u t i o n s of t h e l a t t e r equations can be used t o solve corresponding conipressible flow-problems. A modified form of t h i s transformation for a r b i t r a r y o* has been
described by Rott (1953) ''^-nd vaxs tised by Tani ( l 9 5 4 ) .
An a l t e r n a t i v e approach has been used by L i g i i t h i l l (l950) fcr finding the skin f r i c t i o n and heat t r a n s f e r from a v/all of non-imiform temperature, The method, wMch nnkes use of Von Mises' form of the boundary'- l a y e r
equations and uses a l i n e a r approximation t o t h e v e l o c i t y d i s t r i b u t i o n near
t h e w a l l , i s applicable t o low Ma.ch n-umber flov/s of -variable e x t e r n a l » v e l o c i t y and t o high Mach nimiber flov/s of imif orm e x t e r n a l v e l o c i t y ,
L i g h t h i l l has applied t h i s method t o the problon of the vrall t a n p e r a t u r e d i s t r i b u t i o n on a f l a t p l a t e a t high Mach numbers v*iich i s l o s i n g heat by r a d i a t i o n only. Tifford ( l 9 5 l ) , pJid Tribxis and Klein ( l 9 5 5 ) , hive modified L i g j i t h i l l ' s method t o incliJde a b e t t e r approximation t o t h e v e l o c i t y d i s t r i b u t i o n near t h e v/all and so provide a more accurate method
f o r r e t a r d e d flov/s, i n v/idch t h e l i n e a r approxim-.tion t o t h e v e l o c i t y d i s t r i b u t i o n i s inadequate, A modified and improved frxjm of Tii'ford' s c o r r e c t i o n has been given by Spalding (1958), who shows t h a t with t h i s new c o r r e c t i o n the e r r o r i n t h e L i g h t h i l l method for I.eat t r a n s f e r can be reduced to l e s s tha.n 2,5% r e g a r d l e s s of pressure gradient, Liepmann (1958) has rederived L i g h t h i l l * s formula f o r t h e r a t e of heat t r a n s f e r using rn energy i n t e g r a l approach. Other methods, such as Chapman and Rubesin (1949), Schuh (1953^ Mid Imai (l958), for solving t h e heat t r a n s f e r
i n compressible flow i n Vidiich t h e v/all tonperature d i s t r i b u t i o n i s expressed as a polynomial i n x (the d i s t a n c e along the surface) do not have t h e
rrnge of a p p l i c a t i o n of L i g h t h i l l ' s method,
Illingv/orth (1954) has extended L i g h t h i l l ' s method t o deal vdth v a r i a b l e freestream and Vv'all t e n p e r a t u r e d i s t r i b u t i o n i n a compressible flow v/hen o" = a> = 1, However i n a p p l i c a t i o n s he considered only the case of constant v/all teirperature and mainstream v e l o c i t y d i s t r i b u t i o n s expressed as polynonials i n x.
The aim of the p r e s e n t paper i s t o produce an approximate r a p i d method for solving t h e compressible flav/ boundary l a y e r equations f o r a r b i t r a r y external Mach ntmiber aiad vi/all temperature d i s t r i b u t i o n . The P r a n d t l number (o) v d l l be taken as a r b i t r a r y , though not small compored vdth u n i t y , and although i t i s asstjned M a T across t h e boundary l a y e r
a more accurate -viscosity-tenrperature dependence for the vrall -viscosity v d l l be taken as suggested by ChapmD.n and Rubesin (1949), The proposed method makes use cf t h e Stev/artson-Illingworth transformation, but not r e s t r i c t i n g i t s use to zero heat t r a n s f e r , v/hereby the transfarmed equations are solved by the method of L i g h t h i l l (1950). A fev/ ex!?Jiiples of i t s a p p l i c a t i o n a r e given and the r e s u l t s are conpared vdth other known s o l u t i o n s . I t i s found t h a t even i n t h e severe t e s t of the a p p l i c a t i o n of t h e method t o t h e case of a small adverse pressure gradient t h e accuracy i s probably adequate for engineering purposes. The accuracy of t h e method can be improved by t h e a d d i t i o n of S p a l d i n g ' s c o r r e c t i o n and a b r i e f
account of t h i ö has been included in an appendix t o t h e p r e s e n t paper, I n other appendi.cos t h e r a t e of heat t r a n s f e r i s evaluated \asing d i f f e r e n t approximatioiis t o the v e l o c i t y d i s t r i b u t i o n close t o t h e v/all for the tv/o c a s e s ,
(a) -very a n a l l P r a n d t l nxMaer and ( b ; a t a s e r r a t i o n p o i n t .
9
-2, Basic equations
The steady two-dimensionr'.l bound.ar3- l a y e r equations of c o n t i n u i t y , motion and energy for a perfect gas t\re r e s p e c t i v e l y
ÈPÜ . 9pv _ 0 (1) a X + 6y - ^ -'
9u 9u 1 9 / 9 u \ /^\
''^ -SE
-^ ^^
^y+
^ ^ ^ to = -^ V# W +
'^W/ ^^^
v.iiere i i s t h e enthalpy and C i s the P r a n d t l number. Suffix (l ) denotes the 1OCP2 conditionr outside t h e boundoj:y l a y e r .
I f t h e flow e x t e r n a l t o the boundary l a y e r i s i s e n t r o p i c (and only t h i s case v / i l l be considered i n t h i s paper)
\ * ^ / 2 = const. (4)
or i f a i s the l o c a l speed cf sound and y i s the r a t i o of the s p e c i f i c heats
a'' + ^ ^ u^ = const, ( 4 ' ) The stagnation enthalpy (or as i t i s sometimes c a l l e d the s p e c i f i c t o t a l
energy) equation i s found by multiplying (2) by u and adding i t t o equc^-tion ( 3 ) . I f the stagnation enthalpy
h = i + u^2 the r e s u l t i s •
' " i * - f = f, [^ Ij(h-Cw) f)] (5)
The boundary conditions t o be applied are a t
and at
y = ~
9u OT
y = ~ u = u (x) T = T (x)
"Sy ~ 3y 0 ,
v/here suffix (x) denotes the V7all value ( y = 0),T is the v/all
shear stress and ö i s the r^te of heat transfer per unit area f ran the wall to the fluid in the boundary layer.
The -viscosity-temperature relation is assumoito be gi-ven by a la\7 of the form
/i ~ T where (w) is a constant, chosen so that in the range of temperatures considered the -viscosity agrees vdth that obtained from the more a.ccurate Sutherland relation . The Prandtl number (ö") is assu-iied to oe constant.
3. Stewartson - Illingv/orth transformation
In the compressible flay a streajn function (^) caji be introduced T/hich satisfies -the equation of continuity ( l ) . Thus
pu _ _9!^ . £v _ ^ (Q o " o
v/hore p is the density at sane constant reference condition.
No-// Stewartson (1949) oJ^d IllingV7orth (l949) hr<.ve shov-m tl-iat by means of the transformation from (x,y) coordinates in the compressible flor/ to (X,Y) coordinates, v/here
y-i (x)\ dx X = X ' o
o / V
(7)
ai^ Y = - i ~ - / ^ ^ ^ ^ dy a p oIf 77 " ^ ^^ equated bctv/een the Sutherland and the approximate viscosity relation i t i s found tliat w = v3 •>• /*^uf,h^ so that u varies
2(1 + T/_ ) betv/een the values / ^ u t h , 1.5 to 0.5
- 11
(Note 0
3y-i •"^—r G- P y -1 _ 1 ^1 a p o ^othe equations cf motion and energy become respectively
I K + aY - h ""i dX ^ Vx ^ 9A V''o'^o ^ ^ /
3h 3h ^o '^o 9 r _pjf 3 / , . l i s i ^ 1
v/here U ( X , Y ) = | | = u ( x ^
a / x ) u^(x) a
and U.(X) = f—rS. , I7e f u r t h e r note t h a t
1' V x )
a2 + y^ M? - af + - ^ U^ = c o n s t , ,
^ const / . ^ \
o
and i f (a ) i s chosen a s t h e stagnation speed of sound i n t h e extcarnal flow ( i . e . sirffix (o) r e f e r s t o sta.gna.tion conditions i n the i s e n t r o p i c e x t e r n a l flov/)
( 1 0 ' )
Thus i n tlie transformed flovy the constant f l u i d p r o p e r t i e s outside the boiondcry l a y e r are talcen a t the sta.gnation -values of t h e given
compressible flov/. a^ 1 a^ o = 1
2 al
0"?
o o
e q u a t i o n s ( 8 ) and ( 9 ) a r e e x a c t l y t h e e q u a t i o n s f o r an i n o c m p r e s s i b l e flow h a v i n g v e l o c i t y coniponents (U,V) i n c o o r d i n a t e s ( X , Y ) , p r o v i d e d t h e h e a t t r a n s f e r t o t h e Vi/all i s zc2?o. P o r i n t h a t c a s e o n l y h = c o n s t = h^ , I t follov7s t h a t known boundary l a y e r s o l u t i o n s i n i n c o m p r e s s i b l e flow ( X , Y ) c a n be u s e d t o f i n d t h e s o l u t i o n of c o r r e s p o n d i n g c o m p r e s s i b l e flows ( x , y ) . Ho7/ever t h i s method of a t t a c k o n l y a p p l i e s t o t h e c a s e of z e r o h e a t t r a n s f e r and -when o" = w = 1 . I t cannot be vised when h e a t t r a n s f e r i s p r e s e n t and vihen cr and w a r e not e q u a l t o u n i t y .
I n t h e g e n e r a l c a s e equ, ( 8 ) and (9) c a n be v / r i t t e n TT ^ do and U
^ = (1 - s) u,
3S dU^ dX •I. V _9S di V _ o _ 9 o- 3Y V. o "5Y S(0 f )
9 (^q (1-0-) U l a ? ^SÏ\^
+ 2 " ^ " T T " /
a
n.
(11) (12) VifhereJ5o
^ ^ T = C (X, Y ) , and1 -V,
S i n c e /i ~ T C 0)-ikT
b u t i n f i n d i n g s o l u t i o n s t o ( l l ) ajid ( 1 2 ) we w i l l assume C i s i n d e p e n d e n t of Y, a l t h o u g h n o t n e c e s s a r i l y of X, I n t h e finaJL answer we w i l l choose a -value of C v/hich g i v e s a b e s t f i t \ d t h knovm e x a c t s o l u t i o n s , E q u a t i o n s ( l l ) a n d (12) c a n nov/be m o d i f i e d , so t h a t C(x) i s e l i m i n a t e d , b y changing e q u a t i o n (7) t o r e a d X /•A a P X = / C(a) -i—- d-z. / ^ ^ a p J n o -^OHov/ever no advantages are gained as v/ill be seen in the next section.
The equations in this form were obtained by Cohen and Reshotko (1956b), For the case O" = 1, constant vrall temperature, and U '" A the equations were solved numerically.
Similar equations have baen obtained by Hayes (1956) for the case of imperfect gases, where -the transformation formulae between ( X , Y ) and (x,y) include a general term a(x) in place of a, / and a method for finding a(x)
is given. For a perfect gas a(x) becomesjf h, A ) , which is equal to a^ /
13
-4 , Von I vises' trajisforma.tion
Equations ( l l ) and ( l 2 ) i n terms of tlo independent -variables (X,Y) can be transformed i n t o equa.tions i n (X, \^ ) by means of Von Mises' transformation. This g i v e s , r e s p e c t i v e l y ,
i = s ^ , . „ - 0 (13)
o ^
since 0 i s a function of X only, and v/here Z = U* (X) - U*(X, f),
Equations (l3) and (14) are t h e transformed boundary la.yer equations for a pseudo-incompressible flo\7 having a density ajid kinematic -viscosity
of P and V r e s p e c t i v e l y . Along and a.cross the boundajy l a y e r a p r o p e r t y S, analogous t o tenporature i s con-vected and diffused,
Illingvyorth (1954) gives similr.ir equations t o (13) and (l4) above, e::cept t h a t i n Ms case he -uses (x,f ) as independent v a r i a b l e s and
puts 0"= G = 1, Illingv/orth uses g i n place of S and h f i n place of Z. Now ^ ~' • - <" -^
S (X,f ) ^
dX " "^ •' o so tliat equation (l3) can b e w r i t t e n
'X ^, fX
r X
cj / Y ,v > d U ^ ( X ) 9 / ,\ • ^^^ (z)
3X I ( 2 - £ S(z, ^) dU^ (z) ) = i^^ UO g ^ (Z - £ S ( z , 0 ) d U ; (z))
v/here f o r compactness t h e i n t e g r a l s a r e v / r i t t e n i n S t i e l t j e ' s form. But a t If- = 0
f^ fX
/ s(z, ff) auj (z) = / s(z,o) du; (z)
•' o .'o and a t \^ = 00 , since S(X,co ) = o
/ :
X
S(z,tf' ) d U^ (z) = 0. I t can a l s o be shown t h a t
—^ \ S(z,^f ) d u j (z) = O I t f o l l o w s t h a t n e a r ^P- = O v/e c a n v ; r i t e eq-usction ( l 5 ) a p p r o x i m a t e l y a s ax = "o ^ ° - ^ ^(^'^) (^6> 3 G(X.!^) ^ 9 ' 9iS' rX v/here G ( X , I ^ ) = Z - / S ( z , 0 ) d U * ( z ) v / h i l s t f o r l a r , e -va4ues of ^ i t h a s a. s i m i l a r form v d t h rX G(X,^) = Z - j S ( z , ^ ) d U j ( z ) .
The boundary c o n d i t i o r i s f o r G(X,\f'), f o r v/hich ( l 6 ) i s t o be s o l v e d , a r e t h e r e f o r e
^X
If- = 0
G(X,^) . U^ (X) - ƒ S(z,o) dU^ (z)
\f' = 00 G(X,\^) = 0 X H 0 G ( X , 0 ^ 0 v / h i l s t n e a r ï' = 0 'XG(X,,^) = Uj (X) - U'' (X,^) - ƒ (1 -h^(z)/h^) d u j (z)
o rX h ( z )= u; ( + 0 ) - u^ (x,^) ^ ƒ - ^ d u ; (z).
o •• E q u a t i o n ( l 6 ) , Vvdth t h e above b o u n d a r y c o n d i t i o n s , i s o n l y a.pproximately e q u a l t o (13) f o r a l l v a l u e s of f , a l t h o u ^ i t i s ex:ict a t if- = 0 aaid^ =r CO , S i n c e t h e s o l u t i o n of t h e e q u a t i o n of motion n e a r y = 0 (^f- = O) i s
o n l y r e q i d r e d ( s e e p a r a g r a p h 5 belo\7) i t v.dll be assumed t h a t , f o r t h i s
p u r p o s e , e q u a t i o n (16) v d l l be found a d e q u a t e . ( S e e f i r s t footno-fce on page 1 5 , A s i r r i p l i f i e d form fqp e o v n t i o n (14) v d l l nov/ b e obta^ined. S i n c e
z _ i ^ f, }M_ „^ N a^ a^ / 2 = -— h , = (1 + -—- -, U. ) . y-1 TT2 > • ' • 2 a ' ^ ' / v 'JA \ o 1 + ~2~j^2 Ui o \y -^) o i t c a n b e v , r i t t e n y-1 U 3X ~ cr 3
'' '° c.i (uf) = - ( ^ . c f - i 4 - - , ) - H ( u ^ ) 07)
^•^' 9^ '^ ^ o-U^ o ^ vM U f ; 3 ^ ^" 3 ^ °^ ^1 ^ * 2 a215
-"When O" = 1 -the r i g h t hand side of equation (17) v a n i s h e s . Por other values of cr , since U,Z are kncn-m functions of (X,i^), having been foT:ind from ( l 6 ) , equatioii ( l 7 ) oan be solved "oj tlie 'luethod of v a r i a t i o n of parameters' . I n •the case of an inconpressible flow, IT i s independent of ,1 and -the r i g h t hand side of (17) then gives the h e a t tr^uisfer c o r r e c t i o n -term t o allov/ for
the reco"very enthalpy. I t v d l l be assumed tliroughout -tliis paper t h a t the recovery enthalpy i s independent of the v/all tenperature d i s t r i b u t i o n and •that the r a t e of h e a t t r a n s f e r a t the w a l l can be obtained fi*om the s o l u t i o n t o (17) vd-th the r i g h t hand side put equal to z e r o , or*
Tlie above discussion hovye-ver onl;^ a p p l i e s to the case v/hen heat t r a n s f e r t o or from the v/all i s p r e s e n t . Ylhen the v/all heat t r a n s f e r i s zero the term on the r i g l i t hand side of equation (17) c o n t r i b u t e s s i g n i f i c a n t l y to -the value of S near -the v/all. Prom the v/orks of Pohlhausen (1921) for incompressible flcn7, and Brainerd and Emnions (1941) ^or coinpressible flor/ both f o r the f l a t pla-fce i n zero pressure gradien-t, and from the v/ork of Tifford and Chu (1952) i n conipressible flow with a pressure g r a d i e n t , we find -that the value of S
w i s approximately given by S„ = (1 -o-i) w o 1 + ^ if 2 1
a t l e a s t f o r values of the P r a n d t l n-umber near u n i t y . Thus v/hen cr ;^ 1 the v/all temperature v a r i e s according t o -tlie e x t e r n a l v e l o c i t y even v/hen the heat t r a n s f e r i s z e r o .
USSfe^^-J-s. I ^ i"-^6qu.(ll) S{X)f) and C(X,if') are replaced by constant mean values S (X) and C ( x ) , evaluated a t some ' intemiediate' entlialpy, an equation
s i m i l a r t o (16) can be derived. lica/ever such an equation cannot have -the same boimdary conditions l i s t e d above. In the l a t e r sections h (x) v,dll be
-v/
replaced by an intermediate en-fclialpy c o n s i s t e n t -vd-th, but not equal t o , the intermediate enthalpy a t -lyhich C i s eval-ua-ted, b u t -tlie boundary condition
G(X,CO)=0 v d l l always be used. In t h i s v/ay the value of G ( X , 0 ) can be
eniployed f o r a l l values of h.(X) including h (x)=0,
Footnot^e ^2.^ I n s e c t i o n 11 belov/ i t i s argued t h a t S i n equ,(18) should be replaced by tlie difference between the ac-fcual S and -tliat for zero heat t r a n s f e r . This v d l l be an adequate apj)roximation i n many problems. Hov/e-ver i t must not be overlooked t h a t (I7) can be solved exactly i f U(X,i/^) i s obtained from ( 1 6 ) ,
5» Approximate s o l u t i o n of the transformed eqijr.ticai of moticgi
The approximate fonn of t h e equr^tion cf motion in terms of Von Mises' v a r i a b l e s vra.s found above (equiticjn 16) to be
wliere G(X,OO) = 0 and near ^ = 0
rX h (z)
G(X,^) = u ; (0) -U^(X,^) + j ' ^ -f— dU^ (z) .
This equation i s simila.r to Von Mises' equation for the v e l o c i t y f i e l d i n cji incompressible lajaiin--^r boundary laj^er. I t i s i d e n t i c e l with i t i f G i s replaced by Z = U^ - I j s and C = "^1.
I f only the vroll shear s t r e s s C . ) i s required as a function af U2(x) and h ( x ) , and not the v e l o c i t y profile; o-ver t h e e n t i r e boundary' l a y e r , an ap5)roxima.te solution of ( l 6 ) can be obtained "by rex^lacing U by i t s approxima.te form near the v/all. Tnla method of approach %7as used, by Page and Falkner ( l 9 3 l ) i n obtainijig approximate s o l u t i o n s of t h e incompressible energj'- equation i n t h e case of varia.ble v/adl teniperature, and by L i g h t h i l l (195O) f o r aTjproximate s o l u t i o n s of both the equations of motion ajnd energy i n incoinprossiblo flov/. I f we then follow L i g h t h i l l ' s method of s o l u t i o n v/e find on using the approximate form for U(X,if') near the surfa.ce, namely,
(s^ ino^
u
u
5G•Sx
0 2 u 1 and If- = / U dY ) Jo (X) C (X)^ *^ ^ (19) t h a t ^ ^ ^ ^ ' W ( X ) C ( X ) ^ * - - ^ (20) vdth the boujidary conditions G ^ 0 a.s ^ ^ co , and a s X ^ 0, andrX h (z) 2T (X) _/ G = U^ (0) + / -f— d u ; (z) - - ^ f + 0 ( / ^ ) (21)
' 0 1 o
0.
K A s i m i l a r method vms used by Illingv/orth ( l 9 5 4 ) , T^*IO used t l i i s a.pproximation i n the ccmpressible flov/ eqijatp ons i n Von Mises farm.
17
-If
t = / J - ^ y . ) c (z) dz
o ^ o(22)
and p is the Heaviside op<:ra.tci''""' corresponding to /QL., then equation (20) becomes
i ^ (23)
pG = 1^ 3^2
This equation i s similar t o eq-uation (66) i n the pa.per by L i g i i t h i l l (1950) and s a t i s f i e s siirdlar boimdary condi-fcions. The s o l u t i o n of (23) s a t i s f y i n g (21 ) i s tlierefore (see L i g h t h i l l )
G = ( | p ^ ) ' fh'^y. 1 4 ( f P ^ ^ ' ) ut(o)
h (z) w h . dz .2 ± Y% + l 3 P '(f)!
I.?, é p^ / )
3 •> 2 T (X) W^ ' (24)Since G •* 0 as \ir •* <o t h e c o e f f i c i e n t s of I 2. and I2 must be equal and opposite and t h e r e f o r e f^ h ( z ) , ^^ /?N, 2
U(o)^ * - r ^ 'iUj(z) = V - ^ P"^" r(X) (25)
' i o h^ 1 2-3 ( 4 ) ! /^. ' ' X ^ fe) v/^ r(t - t, )•
dX d X 3^, 2(-f)!(Vo)^
0 (x^) V ^ i )
^ ^j r j z ) C(z)' dz
dX (26)H I:: and I g are Bessel functions, 3 " 3
3ü£ The opera.ti.onal form of a function f ( t ) v d l l be denoted by f ( p ) , viiere
This i n t e g r a l eoua-tion f o r t h e v.nll shear s t r e s s i s i d e n t i c a l vdth L i g h t h i l l ' s equation C69) if C = 1 and h = h , ( i n L i g h t h i l l ' s equation t h e § power of P U was omitt ed) .
I f v/e put U ~ X^ and h = const, in equation (26) v/e caji ocmpare the r e s u l t s -idth -those of Cohen end Reshotko (1956). I t can be shewn t h a t e r r o r s of l e s s than ^(M in the value of T are obtained for causes of
w
a c c e l e r a t e d flovy and v/all entlxilpies ( i ) of the same order, or g r e a t e r than t h e mainstream stagnation enthalpy. I n cases of r e t a r d e d flov/ or very cool surfaces t h e e r r o r s i n c r e a s e ajid t h e r e f o r e a c o r r e c t i o n term must be added t o iriprove t h e accuracy as outlined in s e c t i o n 9 and in appe:idices 4 and 5.
Jn the p r e s e n t section t h e a n a l y s i s -tvill be continued vdthout any attempt being made t o improve t h e accuracy,
The conipressible flow solution i s nov/ obtained by a p p l i c a t i o n of t h e Stewartson-Illingworth transfcrma.tion (7) t o equation ( 2 6 ) . I f v/e p u t , for convenience, y = 1,4
then t h e necessary transforma.tion r e l a t i o n s a r e ax ^ 1
^ ^ (1 4- M ; ( X ) / 5 ) *
(27)
U (X) = a^ M (x) (28)
v/here, consistent vdth t h e other approximations, C i s i d e n t i c a l to C,
9 2y — 1
K I n t h i s equation ~ ^ — and i n the equa.tions belov/ ^ y - 1
19
-Thus equation (26) becomes
/ , r^ h (z) \
-o ( ^^°) ^ 4 "V- ^ '^^'V
''(-l)!(Vo)' ''o °K)'^' ^4(i ^ <(^)J^
(30) I f suffix (a) denotes an a r b i t r a r y constant reference condition ajid
't (x) = T (x)
W^ V/^ A 3 P iU U
a '^a a then equation (28) ca.n bo v.ritten
M«(of . £ 4 1 aH:w =
M : ( ^ ) ,
> d 1£ 3/of M^(x)V^ / / V , X-» ^"^
f -ajlA ^^ 2 [
„ H L I L ,I I § / VVo ^ / ax
( r n r - j X , 1 z^ (1 + M Ü ) f^
V/ O / \ ' 5 (30') v/here / a \ 2 / " i^ \ V 3 ' / P ^ a ' \ / ^ / T A i \ ^ / 3 / 10/_ a y \ P /i y = V P i" a3 / = \ T /i / / ^' + ' V 5 ' 0 0 0 0 0 0 a o ^ I f f u r t h e r we now replace fp° by ( TTT ) e t c . and noting t h a tT h h T
r -
TT = ï" '-^ -r = ^ * "^5
•w w- v/ a ^ equation ( 3 0 ' ) becomesa
34.2 r *w(-,)'^' / \ ( ^ ) \ ^ /1*M^(x,)
W^ 1' 1 + Mf/c / (-1)! •'o x / \ a / \ 1 + M J / 5 ± 1-cj , . 7/, \ " ^ / r ^ t ( z ) 2 / i x-2" M + M%^ ^. / ^ 't' i - ^ - ( i T z ) ) ^ ^ 1 dz dx, (31)
•; •'x^ 2^'^ \ V ^ v \^ + M j ( z ) / 5 / / which i s a convenient non-dimensional form of t h e i n t e g r a l equation f o r t (x) i n terms of M (x) and h ( x ) .3^ . 2 1,^4^2. 2 , ^-,o
( - § ) ! ~~ ^-^^^ =
T.lien y = 1 equ-^.tion (31) red-uces t o a form s i m i l a r to t h a t in a heated or cooled incompressible floiy. I n tliis case since a = a = a and
J. 1 0 a 1^ = c o n s t . ,
^1 M\^ CbllL ^a^"^
-t ^ a ' , \ 13 1:± / 1 1-« ^^'*3
^^ ^ px t ( x ) /2 i ( x ) 2 / z»^ t ( z ) 2 / i rr \
.3i_.__2 / w^ 1 ^ w^ 1 ^ f / _ i / ^ / -^1 \ dz I dx (31')21
-Thus f i n a l l y , i n t h i s s e c t i o n , Xfhen y = 1 , a = ^ v/e find t h a t
u
(+o)
Y r^ i
(z) / u ( z )
(^j^)\ rbt
d i -u a 23^.2 / - V i i ' ^ ' / ƒ " v ^ ) ' ^•*
i-ihich gives t h e approximate extension t o L i g h t h i l l ' s equation (7)) t o allov/ for v a r i a b l e v/all t e n p e r a t u r e . Equation (31") shows, as no-ted by many workers, t h a t i n an incompressible tmiform flow (u. = const) t h e
skin fricjtion parameter, t , i s independent of the -wall temperature. 6, P.lat p l a t e a.t zero pressure gra.dient and constant v/all tempera.ture
I f M = M (+o) i s t h e constant e x t e r n a l Mach number t o the boundary
a 1
l a y e r on a f l a t p l a t e \iJiose constant v/all enthalpy i s i^^, then from equa.tion (31 ) vye find t h a t
K
w' 13^
( 4 ) !
2 A
1 - w= 0.312
(^Y
1 - 0 ) (32)This r e l a t i o n i s s i m i l a r to t h a t given by Young (1948) except t h a t 0,312 i s repLTced by 0.332 (Blasius» value)
i j _
and \ i s replaced by ( 0 , 4 5 + 0 , 5 5 •— + O.O36 ll? cr ^),
K
a °'
a
t o give the b e s t f i t with Crooco's exact r e s u l t s .
This comparison suggests t h a t a more a c c u r a t e form of t h e i n t e g r a l equation (31) can be obtained i f
± A
•^—-— i s replaced by u n i t y , and -r- on the r i g h t hand side
(4)1 _ "-a —
i (x) J. i s replaced by (0,45 + 0.55 T^^— + O.O36 H? <T 2).
Por t h e vinheated inoannressible flov/ case (M •» O and h * h ) m w 1 ' L i g h t h i l l (1950) showed t h a t v/hen u (x) = ox e r r o r s af l e s s than about I5ê i n t h e value of t vAien m > 0 v/oild be obtained i f the constant i n equation (31) vra.s s u i t a b l y mod-ified,
7. Improved r e l a t i o n bctv/een wall shear s t r e s s and Mach number f o r Vfiriable -wall tanpera-ture
I n view of t h e coniparison between r e s u l t s obtained from equation (31) and t h e exact r e s u l t s f o r the fla.t p l a t e t h e following improved form f o r
(31) i s proposed ( y = 1,4) e /-x
•^K(-/^ / !...(-.) . . t X ^ . , . . . "A
fx t {^x ) / i ( x ) i \ -^—- '1 7T^ ( 0.45 4. 0.55 f^—^ + 0,036 M^ a-2 J ^ / I
+ M ; ( X ) / 5 \
'' J- /1-w\ 7/. ^' - ^ ^ ' " ' / i ( ^ ) i x " ^ — V i + M V 5 V^ >
( 0.45 + 0,55 rr— + ^'^36 M^ o- 2 ( 1— 1 dz J
\ \ ^ / v>, +M'(z)/y' /
. . ^(33)
R e s u l t s f o r values cf y other t h a n 1,4 can s i m i l a r l y be obtained. Por instajit Vi*ien y = 1 the terms i n (1 + M*/5) vanish a s v/ell a s t h e term invol-ving t h e P r a n d t l number,
H L i g h t h i l l quoted t h e modified value of the constant a s 1,157 "but a b e t t e r values vrould be 0,98.
23 -I f v/e p u t ' P(x) ' ' G ( X ) H(x)
/M(o)\'' r \ . ( z ) /M(z)\''
=
(0.45
+
0,55
- V - +
°-°5^
K '^^)
ai (x) 1 '^'t
= (0,45 + 0.55 -f— + 0,036 M* 0- 2)
a then (33) becomes P(x) / 1 .. I f (x)/3X A V i + M ^ ( x ) / 3 /p t,/x) / ' , ,xt (.)4 ,-i '
'L TT =(==.' (/ "i ''(^'^) *=. (*)
1 X,v/here, i n g e n e r a l , P ( x ) , G(X) and H(X) v/ill be known functions of x and eq-uation (34) i s "to be inverted t o find t ( x ) ,
8, Approximate inversiori of t h e v/o.31 shear s t r e s s integyal equation L i g h t h i l l (1950) has shoi-.n hov/ t h e incompressible form of (33) or (34) can be inverted i f as an approximation
ƒ.
H(Z) dz i s replajsed by=f -^
t ( x . ) 2 H(x.)
(
\ W 1 1X - X,; V ' . The r e s u l t i n g s o l u t i o n
f a r t^^(x) i n the case u , ( x ) = ox differed from the exact s o l u t i o n by "^^ 1 C^ when m > 0, I f then e r r o r s of t h a t order of magni-tude are acceptable T/e can r e p l a c e equation (33) or {3li) (but r e t a i n i n g the constant term
3 ^ 2 ")
VW
J
t y ( - ! ) !V3
3^ 2 r V ^ ) J ^ )
'T-I S05)
x , ^ ( x - X , ) \U) 2 ± K l - w ) / I + M^(x)/ \ T^ere j ( x ) = ( 0 . 4 5 + 0 . 5 5 - ^ + O.O36 M O" 2) ( 1 L2. ]a \ 1 + M /„ /
73
V 5
which i s i n v e r t i b l e a s \{x) j ( x ) ' ^ -V3 or t (x) T/^ ' J ( T ) " ^ —^—r-— 2 •"3 r / M , ( o )(2.3^(4)!) L ^
M J - ^xh^^(z) d M^(z) M^ Jo "1 ( x - z ) ^ ( 2 . 3 \ ( 4 ) ! ) ' / M , ( o ) Y 2 r ' ' \ ( z ) h /M/z) Y -1V M — > ^ Jo 7 — W A'M—y
a ° ( x - z ) * \ a / , (36)25 -NOVT 2.3^ , (-4)! = 0,360 s o tha.t e q u a t i o n (36) becomes t ( x ) 0,360
i-Xx)
v/ 1 1-u 0.45 + 0 , 5 5 - f + 0,036 M* o- 2 1 + MJ(X)^3 ( 3 6 ' ) M , ( o ) x * M Jo (x-z)^ ' M ( Z ) d » J -M a 4 On comparison of e q u a t i o n (3^^) v d t h e x a c t s o l u t i o n s f o r t h e f l a t p l a t e a t z e r o p r e s s u r e gr-adient we s e e t h a t t h e c o n s t a n t 0 , 3 6 0 s h o u l d b e r e p l a c e d b y 0 , 3 3 2 .Hov/ever L i g h t h i l l (1950) h a s shown t h a t i n i-ncompressible f l o v / w i t h u , ( x ) = e x t h e e r r o r introduced, b y eq-uation ( 3 6 ' ) -varies from + 8,k.fo vixen m = 0 t o - 10,6/? v/hen m = co , A l s o t h e a c c u r a c y i s p o o r v/hen m i s n e g a t i v e . Prom t h e s e r e s u l t s i t would a p p e a r t h a t l i t t l e a c c u r a c y v / i l l be g a i n e d b y a change i n t h e v a l u e of t h e oonstajit 0,360. F o r t h e s p e c i a l c a s e Tviien M = M (+0) and 1 * H ; ( ^ ) / 5 = 1 + cx m (37) , v/here m > 0, t h e n H, (x) ' M = 1 4 . ^ (1 + M S , ) X ' " a M
V5^
a and / M ' ' ( X ) \ ^_d {-V-)= °('' * K^
dx \ M^ ^ ^ a m X m-i (38) (39)I f i n a d d i t i o n
\Uo) - h./x)
= b X n (40), where n > 0,
then eqtif.tion (3^^) becomes
tXx)
v/ 1 1 + Ml^s^
V^ ' (0.45 * 0.55 ^ ^ * 0.056 M > 4) ^ ^ ^ * "^'^Vs ^ '
a )'72
MJ(X)ir
- 1 h (+o) T/^ -' h (+o) - h (x) W^ ' V/^ ''"^ (-4)!
( m - l ) !
m. ( n + m (n + m • 1 ) ^ (^-)! 1 - § ) ! -* 3. 4 (41) The values cf t h e constants f o r various values of m and n a r e given in t a b l e 1. (Similar r e s u l t s for other forms of external v e l o c i t yd i s t r i b u t i o n and v/all enthalpy d i s t r i b u t i o n caji e a s i l y be obtained), As aji exanrple we have taken m = 1 i n both an a c c e l e r a t e d and a r e t a r d e d f l a v and, n = 1 and. 10. These res-ults a r e p l o t t e d i n f i g u r e s 6 and 7 respecti-voly. Since eqijation (41 ) does not contain the c o r r e c t i o n terras, discussed i n the next paragraph, i t i s unlilcely t h a t tlie numerical accuracy of t h e s e r e s u l t s v/ill be good. Ila'/ever t h e res-ults do show some interesting^; t r e n d s . I n a c c e l e r a t e d flow, v,^en the V^TII t e n p e r a t u r e i s roughly constant except neaj? x = 1, (n = 1 0 ) , t h e skin f r i c t i o n i s greater a t a c e r t a i n d i s t a n c e f ran t h e o r i g i n than for t h e case vdiere t h e v/all tejnperature f a l l s lineai:ly frcm t h e o r i g i n , n = 1, On the other hand
i n r e t a r d e d flov/ v/e find tha.t sepa.ra-tion i s earlie.r when t h e v/all temperature i s roughly constant. These ca.lculaticns do not i n fact p r e d i c t sei)a.ration f a r the ca.se n = 1 al-bhough beyond "/l = 0.4 t h e value cf t i s very small, Hov/ever t h e s e p a r a t i o n po-.'.nt i s not v/ell p r e d i c t e d i n t h e a^ove a n a l j ' s i s , as -vdll be shov/n i n tlie next pa.ra.gra.ph, and a small c o r r e c t i o n term must be introduced i n order t o inipro-/-e t h e accuracy. But t h e pred.iction tliat t h e v/all must be cooled s i g n i f i c a n t l y , immediately dov/nstream of the o r i g i n , i n order t o delay s e p a r a t i o n i s an important conclusion,
2 7
-9» Approximate r e l a t i o n betv/een tfte -v?a.ll shear s t r e s s and Macïh number •VThen t h e v;a.31 tenrpGr^atm-'e i s constajit
When t h e VTall tanpera-ture i s constant and _ d dx m, ra=0 m a X m (42)
equation (36^) becomes ( i f M (+o) = M )
.2*160.
1 4 >| , (0.45 + 0.55 ^ + 0,036 M^ cr ^ ) 1 ^1-tJ \1 +
MJ/ 5 ^
'72 1 -»• TT (3 a„ X + 2.25 a x** + 1.9286 I x ' + 1.7357 \ x* + 1.6022 S ai?) n o 1 s 9 4 (43)I
V/here 0,360 must be replaced by 0,332 i f agreement vdth the exact solution i s desired when M = M = c o n s t .
1 a
A more accurate s o l u t i o n can be obtained following t h e method o u t l i n e d i n appendix 4 . I n t h e s p e c i a l case of zero heat t r a n s f e r vihen M ( + O ) Ï; M and cr = u = 1 we f i n d thxit 1 * 1 1 ^ , , N ' / 2
-! a ^ ^
2 t (x) = . ,^ + V 5
,,2 dx ^ a dx MV 2 ^ ( 4 ) t = ' 7 ^ L
d ' a (x^ - -.^) -a:\ 3 z . . . ( 4 3 ' )K This form of Mach nimiber d i s t r i b u t i o n i s chosen t o f a c i l i t a - t e t h e evaluation of t h e i n t e g r a l i n equ, ( 3 6 ' ) . However any otlier Mach number d i s t r i b u t i o n can be e q m l l y v/ell be used.
and v;hen M *, i s s u b s t i t u t e d from equaticn (k2) ' "a 2 t (x)
= (--^-^f [( )^
^ 1 + Mwc ^ L «» = o - Dl(.1a x
mf1i
•)(-„I,^^)
r
- mf1\ / a X ^ \ / mntrO
)(-i-S)..'(;is')(..
KfcrOm=o -n 1
]l|Mty;
(43") -where 16 1 2 8 . ( - | - ) ! ' = 0.1511 and values of m^.(^)!(4)!
(^)! (-§)!
P^!
0 3.533are given i n the folloiving t a b l e , 1 2 3 2.687 2,314
Our r e s u l t s above can now b e conipar-ed with those of Luxtcan and Yoxaig (1958) and others for -^-arious d i s t r i b u t i o n s of Mach nuiriber. Thus v/hen
or = td = 1 and M,
M
- = 1+71
X/_ , where 1 i s a reference l e n g t h , and the v/all teniperature i s constant \re obtain from equation (43)
5.
2 t (x) = 0.720 ( . ^ W ' 1 + M/5
h V^-
^
(6S
+4.5
5*) j
^ * t;
(44) where x = / I ,29
-2 t (x) = w^ '
1 + M 2 ^ / 2
S i m i l a r l y f a r zero heat t r a n s f e r (h = h ) , v/e obtain frcan equation (43")
x(l + x ) ' \|[S'(1 4.x)« +
l ' —(440
^ 1 + M 1/5 - 2 \ 3 0.1511 (1 + 7.066 5 + 5.374 x^ ) vdi,ich oan be v/ritten approxin-iately3 4
(w)
2 t (x) ;^ 0.624 (1 + f ( x ) ) (1 + 7,066 X + 5.374 x ) v/ vAiere f (S) = F 1 + ^ ^ ( l + ^ ) 1 .3 ^/'-•^ 1 x 2 ( i + x)^* , and g (x}= - 5 - ^ ^ , X ^ ; 77oè6 X T 3 ' . 1 7 4 l ? T ^ / 2From equations (Vh') and ( W ) i^e see t h a i an inrproved form of (Z^) i s , i f 2 t (o) = 0.664,
W"
2 t (x) ~ 0,664
1 + I (6 5 + 4.5 x"*) J
(W)
for acro^ heat t r a n s f e r , vAiile f o r the case of heat t r a n s f e r i t i s suggested'/5 tliat"§^/,-. i s replaced by .K W h. 6 Y/O
5 \\
5
7
1 - 1 wo v/ (see appendix 5),These modified r e l a t i o n s are p l a t t e d DJI f i g u r e 1 together v/ith Luxton and Young's resiiLts. The agreement i s vary good. The conclusion from both s e t s of r e s u l t s i s tlia.t marked red.uctions i n sldn f r i c t i o n a r e obtained by ccraling of tlie v/all.
I n a retajrded flow, H = M (I - ^ l ) , r e s u l t s can be obtained i n a * 1 a^ I ly
similar v/ay. Thus from eq-uation (43) v/ith tlie above c o r r e c t i o n term added, v/e obtain t h e r e s u l t s p l o t t e d i n f i g u r e 2. S e m r a t i o n i s delaj^ed by
cooling the v/all and i t i s a l s o noted t h a t t h e v/all shear s t r e s s m u l t i p l i e d by the square root of the Reynolds number, i n i t i a l l y increases s l i p h t l y i n the cooled v/csll case so t h a t for a cer-tain distance c_ f» V •'/x as i n t h e oase of tlie f l a t p l a t e i n zero pressure g r a d i e n t .
Luxton and Young (1958) ajid. others have considered the following retarded flov? case, u = u (^1 - / l ) , which is not so ameriable to treatment, "by the present method, as is the case of M = M (1 - / l ) . Tl-ie reason for this difference in appl.ication lies in the fact that for the linear velocity gLvadient equation (2f2) becomes an infinite series, v.h.ich is only slowly convergent even -when /l << 1.
?hen cr = CO - 1 (the case treated by Luxton and Young) the modified form of equation 36') becomes
2 t (x) w^ ' = 0,664
1 : 1 ^
1 + M' 1/5'72
1 - 2 X -Ji
r* (i - X g) dz(1 - ^ ) V 3 ( ^ ^ < - 5 ( , ^2|)''
(45)
For t h i s form of e x t e r n a l v e l o c i t y d i s t r i b u t i o n i t i s not convenient t o use equ. (43) since a very l a r g e number of terms fire r e q u i r e d for even si.iall values cïf x, T.1ien x « 1 and M = 4 the i n t e g r a l i n equ. (45)
a reduces t o 6 40,96 X
7.4
6 a
2 / 2 . 7 - 6,4 5^ 1 2 / 2 . 7 • • 6 , 4 : 9 \ \ + 6,4 X « 3 n dtp —T a^-l * 2 n (j[3 r."3 + 1/3 arc t a nA
1 + 2liJi
viicre a = — - — - - — , The exact value of the i n t e g r a l i s given i n 6,4 X
Appendix 6.
The r e s u l t s f o r t h e cases of zero heat t r a n s f e r and t h e cooled v/all •fdth i, = i , ai-e p l o t t e d i n figures3a and 3b resi)ectively and are compared v/ith the r e s u l t s of Luxton and Young, Ciui-le (l953b) and, Cohen and
Reshotko ( l 9 5 6 ) . The agreement v/ith Luxton and Young's r e s u l t s i s good for zero h e a t t r a n s f e r b u t not so good f o r the cooled wall c a s e s . The r e s u l t s a r e
lov/er, for small values of x, than those of the ' e x a c t ' s o l u t i o n obtained by F.P.L, for t h e cooled v/all c a s e , ^ t a.greement could be obtained i f a s l i g h t l y dtffei'ent value of [ i A i were used,
^ e n t h e pressi-ure d i s t r i b u t i o n , i n place of t h e v e l o c i t y d i s t r i b u t i o n , i s defined as a f\jnction of / I the above method needs only small modification,
31
-R e s u l t s a r e given i n Appendix 5 for the ca.se of a l i n e a r adverse p r e s s u r e gradient v/he:r"e i t i s shor/ti t h a t a rela!d-vely siirrple r e s u l t i s obtained, i n closed form, when y = 1,5 and t h e wall tempera-ture i s constant. Pigures 4 and 5 shov/ Jhe r e s u l t s obtained both from t le unmodified and t h e modified foriffilae"' for the oases of ze.ro heat tra.nsfer ond t h e cooled T/all r e s p e c t i v e l y , t o g e t h e r vdth the r e s u l t s obtained by o t t e r v/arkers f o r y = 1,4. In b o t h casos i t i s found t h a t only r e l a t i v e l y minor differences e x i s t between these r e s u l t s and those obtained using the
modified fcxrraula. ( i n making t h i s cctirparison i t i s aa^umed t h a t changing y from 1,4 t o 1,5 does not sericrosly modify the r e s u l t s ) ,
I n t h e l i g h t of t h e s e conparisons with other knovai accurB.te r e s u l t s f o r the case of consta.'-^t -vvaJ.1 t a n p e r a t u r e i t i s proposed t h a t in the genera.l case a mor.-, a-.-^rrate form of equation (36') i s v/hen y = 1,4, 2 t (x) = w^ •' (0.45 0,664 w / 1 + M k - ~ \ 1 + M
v5
M ( z ) ' - . ^ (46) where J ( z ) = 5 i (z) wo ^ ' 1: 5 7 i (z) wo^ ^i
(z)
(47)I t i s noted t h a t equation (46) can be a p p l i e d t o tlie cases of
a c c e l e r a t e d and retaxded flows as well as t o cases of constant and v a r i a b l e w a l l temperature. The effect of v a r i a t i o n s of cr and w fran u n i t y are a l s o approximately ?. .-•.v'.uded. I t might all.so be noted t h a t the r e s u l t above can be vised f o r a dlsücciated gas i n equilibrium, provided -the Leivis
number for the gas i s equal to u n i t y ,
Now from equa.tion (46) i t i s seen tliat separation occurs v/hen the terms inside the square bracket equal zero, and t h e r e f o r e f o r constant v/all teiTperatvL-e the dist;ance t o s e p a r a t i o n w i l l , in g e n e r a l , be a function botli of \/h. and M , af3 shovm by Gadd (l957b). I^Then M = M (I - / l ) , hov/ever, we see from P i g , 8 thg.t t h e d i s t a n c e t o separa-tion i s independent of M for a constant value of "V/h • The t r e n d s are simila.r t o t h o s e
X ^
shoT/n by Gadd for u = u (1 - / l ) apart from t h e l a t t e r r e s u l t .
K The c o r r e c t i o n t-;i;!i i s modified s l i g h t l y t o allow far the difference i n y between these r e s u l t a and the value of y = 1,4 used pre-viously,
A d i r e c t comparison of G a d d ' s r e s u l t s v d t h t h o s e o b t a i n e d from e q u , ( 4 5 ) h a s n o t b e e n made, a l t h o u g h c l e n r l y i n -this c a s e t h e d i s t a n c e t o s e p a r a t i o n w i l l be a f u n c t i o n of b o t h M and \/ii • The i n c r e a s e i n t l i e d i s t a n c e t o s e p a r a t i o n a s t h e v / a l l t e i a p e r a t u r e i s iov/ered i s i n ö u a l i t a t i v a a g r e s n e n t -with tiie r e s u l t s of I l l i n g i v o r t h ( l 9 5 4 ) a n d Gadd. ( 1 9 5 7 b ) . I n i n c o n i p r e s s i b l e flov/ t h e d i s t a n c e t o s e p a r a t i o n , a s a f u n c t i o n of v / a l l t e n p e r a . t u r e , f o r t h e e x t e r n a l v e l o c i t y d i s t r i b u t i o n u = u (1 - ^/l) i s p l o t t e d i n P i g . 9 t o g e t h e r w i t h t h e r e s u l t s of I l l i n g w o r t h ( l 9 5 4 ) and
t h e t e n t a . t i v e r e s u l t s of Gadd ( l 9 5 7 b ) . I t i s s e e n t h a t t h e p r e s e n t metliod p r e d i c t s s e p a r a t i o n d i s t a n c e s g r e a t l y i n e x c e s s of t h e l a t t e r r e s u l t s f o r t h e c o o l e d v / a l l whereas f a r t h e h e a t e d -wall t h e agreement i s b e t t e r ,
1 0 , A-pprcximate s o l u t i o n of t h e t r a n s f o r m e d s t a g n a t i c n e n t h a l p y equatican The approximate form f o r t h e t r a n s f o r m e d s t a g n a t i o n e n t h a l p y
equa.tian i n terms cf Von M i s e s ' v a r i a b l e s v/as found above ( e q u a t i o n I 8 ) t o b e
9X (x ölf" \ ^^
where S(X,co ) _ 0 , and S •» 0 a s ][ •• 0 , and
\(x) _ vx) rj:;:
s .
,
-
^
. . . ^ ^ . ^ ^ . (^)
a s ^P' •• 0, The r a t e of h e a t t r a n s f e r from t h e w a l l t o t h e f l u i d , i s
•W-' - "0 V - . . Y ^ 0
U x ) = 'K \dY
and t h e P r a n d t l n-umber ( cr) i s given b y ^^ 0
o- = k . o
I n tlie cja.se of z e r o h e a t t r a n s f e r we must u s e t h e f u l l e q u a t i o n (17) and v/e -write t h e s o l u t i o n of t h i s e q u a t i o n a s S (X, f),
I f i n a f i r s t a p p r o x i m a t i o n , t h e c h a n g e s i n t h e v e l o c i t y d i s - t r i b u t i c n U(X, i^) and. C(x) a r e n e g l e c t e d be-tx?een tlie c a s e s of h e a t t r a n s f e r a n d
z e r o h e a t t r a n s f e r , ^ v/e s e e t h a t a s o l u t i o n of t h e complete e q u a t i o n (17) i s S = S «* S , v-iiere S (X, if') s a t i s f i e s t h e f o l l o v d n g e q u a t i o n , (v/hich i s a n improved form of ( 1 8 ) ,
H T h i s i s ta nt am ou n t t o s a j d n g t h r . t t h e v / a l l s h e a r s t r e s s i s approxima^tely independent of h e a t t r a n s f e r . T h i s i s trvie i n a n i n c a n p r e s s i b l e f l a / / , s i n c e t h e v e l o c i t y d i s t r i b u t i o n i s t h e n i n d e p e n d e n t of t h e t e m p e r a t u r e d i s t r i b u t i o n , I t i s however n o t t r u e i n t h e c a s e cf t h e p s e u d o - - ' n c o m p r e s s i b l e flov/, vAiose e q u a t i o n of m o t i o n i s ( 1 3 ) , on a c c o u n t cjf t h e t e r m i n S, The e r r o r \ d . l l b e g r e a t e s t when t h e v / a l l i s c o o l e d . But b e c a u s e we a r e going t o assume t h a t a good a p p r o x i m a t i c n t o t h e r a t e of h e a t t r a n s f e r c a n be o b t a i n e d from a c r u d e a p p r o x i m a t i o n t o t h e v e l o c i t y d i s t r i b u t i o n , v/e -vdll c o n c l u d e , v d t h o u t p r o o f , t l i a t t h e e r r o r s due t o t h e one a p p r o x i m a t i o n a r e no g r e a t e r t h a n t h e e r r o r s due t o t h e o t h e r .
• 33
-8B 1
Gv-)
(49)
with t h e boundary conditions S • • O as f ^ co ^ and as X -•• O, and
(50) h _ (X) - h (X) -WO ^ ' w -^
n
~ E —
2 if-1 if-1 ' O Was ^ H o, h i s t h e v/all enthalpy a t zero heat t r a n s f e r . v/O
I f , follov/ing L i g h t h i l l , we assume t h a t an approximate s o l u t i o n of
(49) i s found by using an approximate form for U(X, \^), such as equation ( l 9 ) , and put
t =
Xcij)
'o equation 49 becomes n - / 2 /^ r (z) dz P,.. \ o w^ ' o c. (51)P s.
_9i
3 S
3 f (52) v/here p i s t h e Heaviside operator co]n.-esponding t o -rr , This equation f o r S and i t s boundary conditions a r e s i m i l a r t o L i g h t h i l l ' s equation (21) f o r t h e tenperat-ure d i s t r i b u t i o n i n an inccmpressible flow. The s o l u t i o n of (52) s a t i s f y i n g (50) Ic-a.ds t o^w
^l"7^7^) " '"' (4)! ^ V h ;
oryx) = 1 ^ 4 3t_i
( 3 0 - ) ^ ( ^ ) . C(z) / r ^ ( z ) dz Xƒ C(z)/yz) dz
o o -Videre t h e l a t t e r integ^ral i s a S t i e l j e s i n t e g r a l .Y d(h (X ) - h (X ) V; {53)
If v/e na-/ transfonü equation (53) back into the compressible flow coordinates (x,y) then for y = 1,4 v/e find tliat
^s^)
V (hUo)^hUo)''^'a^a^'a w^ wo' vr ' (^)! 33 0-3 X 1 — T X 4 1—r
X 4 X / i aTTz)
W^ '' -^ w^ ' X, ^ 7 2 ^ V-O)"T"
v/ V ï j z ) / I +Mf/^ \ V 4 v/^ 4 1 aTTz)
v/^ ' 1-co 2 Ë/L 1 + M^(2)/5 1 + Mt(x)/34
dz\V4
1 + M*' ^"^^ J dz 1 + M , ( z ) ^ 5 ^ 1 "5 d(h (x ) - h (x ) ) ^ w'- 1 "^ wo'' 1 ^ ' ( h (+0) - h ( + 0 ) ) V w^ ' v/o^ ' ' • • w t » » c I (54)and can be e\'aluated vdien t , M , i a,re given as functicais of x, v/' 1 ' v/ ^
YHien M = const. = M v/e found, from equa.tion ( 3 6 ' ) tliat 0,360
[0.i
1-0) 2VT ^ 1 (x) , ±v,
.45 + 0.55 T^— + 0.036 i r a« I - a a J
liiieixs the term i n the denominator replaced \yV.^^ U+ a/5^ "y/''"^^ , h = i
1 a
I f v/e then assume t h a t t and h / /^ .,,2 \
•w vli (1+ M V.) '^ w^ a y
can '-«e tal'.en as constants during i n t e g r a t i o n we find fran (54) t h a t
% / - ) "X"' (0,360)^ (j)- " l - C O (h^(+o)-h^^(+o)) xl a a ""a Q" 3 5S ( l ) ! ( o , 4 5 + 0,55 t ^ + 0,036 l / cr^ ( x ) 1 +
a[Vx,) -h^^(x,)j ^
o (1 - ( " V , ) ^ ) ^ ( h , ( . o ) - h^^(.o; ) J 0,348 , a (55) 4, ( 0.360 , 3 r 1^33
-The constant 0.348 d i f f e r s from t h e value 0.339 given by L i g h t h i l l (1950) for he used t h e more accurate value 0,332 i n the expression f o r t
w i n place of 0,360 used hjere.
As previously s t a t e d the value of h for zero heat t r a n s f e r ( h ) w ^ wo i s given approximately by the Pohlhausen r e l a t i o n
( 5 6 ' )
I f f u r t h e r the v/all tempera-ture i s constant then the Stanton heat t r a n s f e r coefficient (k. ) i s given by iVD
i
1 o r , " l l 1 f o r cr new uni-ty, 1 4. ^ M^ 0 - ^ 2 1 ( I - O ^ ) ^ M ; (x) W ^ M^ (x) where p /i w w a a ^ x = ) * ^a ^ a ^ ^ais the unmodified value and
w
I n t h e case of a flow conmencing frcm a stagnaticn point t t e use of the reference Mach number (M ) i s not convenient unless i t i s , say, the
freestream Mach number,
An alter'native form of (54) v ^ c h i s s u i t a b l e i n t h i s case i s
1 -0) i -JL
y^o r x r _j
(t.Y l^^A^
(h,^(+o) - h , > o ) ) ^i ^ ^ ^ , • (±)! (30-)^ V^w / (1 ^ yiA^f/^9^
i ^ / \ w ^ ^ dz 1 ^ J o 3(34.ÓJ) - J ! ^ (1 . Mj(z)/3) ' - t ^ ^ - V
i ^ / ^ ) x ( z ) M ^ ( z ) / 2
I 3i3r5r
5C, (1 .^ Mj(z)^,_) 4 dz (h^(4.0) - h^^(+0) ) (58) -ïshere ' ' . ( x ) = r (x) v r ' \, X P /i u7 1 1 1But i n p a r a g r a p h 8 v/e have found t h a t viiien t h e i v a l l t o n p e r a t u r e i s c o n s t a n t and M (+c) = O, >:(x) = 0,360 _ i 1 v/ X o ( x - z ) ^ d M ^ ( z )
Mt(x)
3. 4 p r o v i d e d -that i does n o t a p p r o a c h z e r o . w i \ ^ _ w 1(55)
Thus v/hen M (x) = cx' (m > O) 3 1 - c o x(x) = 0.360 [^^ i \ 2 i -»7 (2m)j_H(2n--i)!
:M
(60) and when m = 1 5f(x) = 1.112 ( w 3. •4 i i w 1 - C J 2 (61)I f i n e q u a t i o n (58) v/e omit t h e t e r m s i n ^ + llf y^ and r e p l a c e them b y u n i t y , t h e n f o r const.ant v / a l l t e m p e r a t u r e ( n o t i n g t h a t ^ i s a p p r o x i m a t e l y
independent of x ) ajid assijming t h a t h , i s a c o n s t a n t ,
h - h V/ 'v/O 1 1 1 v/o 1 / 0 , 3 6 0 ( m + 1) 2 1 ^ 2
(£•! <^
1 - ( j 1.' ^ ' • ' ( M L . H I L ^ '
(2n - f ) ! w (62)But t h e l e f t liand s i d e e q u a l s k.,(x) ^ , v/here R = 1 ^ ^ ,
h X X ^ so t h a t ^-vhen m = 1 ^ (^) ^ 0.6.37^ K^a 0 - 3 0.637 ""zn T" ^^x ^ v/ p ^ w v/ p iti 1 1 i \ 2 1 w i ^4 w h 1 (63)
3 7
-IIcw for the case u ~ x Cohen and Reshotko (l956) find f o r
deduce t h a t
1
cr = CO = 1 (tvro-dimensional body)
k^(x) = -^f^^—Z vdillc Pay anu Riddoll (1958) o- • -'^^
/ P U \ ° ' ^
V
\ 0 ^U I V "W \ • • x) = —**:--—— I -~ for a i r under equilibrium0,6^ \ p H J
o" V R 1 1 X
d i s s o c i a t i o n conditions vdth the Lewis number equal t o u n i t y ,
The difference between the values of the constants 0,637 and 0.54 i s partHy the r e s u l t of u?ing t h e -value of 1.112 as the constant i n the sliear s t r e s s parameter term. Of g r e a t e r importance i s the difference i n
/ P /^ \
the powers of f -^—" J, The r e l a t i o n s are only similar when u has a
^ 1 1 ''^
value near 0 , 4 . However i f the -viscosity i s e-valuated frcm the Sutherland formula i t i s found, on comparison vdth t h e r e a t i o n u ~ T t h a t w v a r i e s frcm 1.5 a t very low temperatures to 0*.5 a t very h i ^ teniperatures.
/
P
Ut \^ ?Kr\^
Thus t h e r e l a t i o n I • "^ ) U ^ J may not i n f a c t d i f f e r markedly
. , 0.1 ' "' '
from 1 —^'—1 a t high temperatures. 1 1 ^
I n appendix 4 Spald:l-^g's method i s described whereby tb^ accuracy of t h e above c a l c u l a t i o n s can be im.proved,
11<> Aclmov/ledr^ement
The c a l c u l a t i o n s v/ere performed by Miss R, P u l l e r and Mrs, Lennard, 12, Ccnolusiona
The compressible flov/ laminar boundary l a y e r equations have been solved approximately f a r a r b i t r a r y p r e s s u r e gradient and w a l l temperature d i s t r i b u t i o n
hy
(a) reducing t h e compressible flov/ equations t o eqintions s i m i l a r t o t h e incanpressible flow equations using the Sv/etartson-Illingivorth transformation, and
(b) sol-vdng t h e r e s u l t i n g equations by L i g h t h i l l ' s method t o obtain t h e slcin f r i c t i o n and r a t e of heat t r a n s f e r ,
The method i s p r o b a b l y s u f f i c i e n t l y a c c u r a t e f o r e n g i n e e r i n g p u r p o s e s i n r e g i o n s of nega.ti-ve presi^ure g r a d i e n t and f o r s m a l l a d v e r s e p r e s s u r e g r a d d e n t s , p r o v i d e d s e x n r a t i a n i s not approached. The method
i s hov/e-ver improved i n a.ccuracy i n t h i s r e g i o n by u s i i g t h e m o d i f i c a t i o n t o L i g h t h i l l ' s method i n t r o d u c e d b y S p a l d i n g ( 1 9 5 8 ) . RIirriREMOSG 1 . B r a i n e r d and 1941 T a n p e r a t u r e e f f e c t s i n a l a m i n a r EmmoriS 1942 c o m p r e s s i b l e - f l u i d boundary l a y e r a l o n g a f l a . t p l a t e . J.Appl.Mech. V o l . 8 (1941) p p . 1 0 5 - 1 1 0 . E f f e c t of v a r i a b l e v i s c o s i t y on Boundary l a y e r s , v/ith a d i s c u s s i o n of d r a g measurements. J,Appl.Mech. Vol. 9 ( l 9 4 2 ) p p . 1 - 6 , 2 . Chapnan D . R . , and 1949 Rvibesin, H.ïï,
3 . Cohen C . B , , and 1956a R e s h o t k o , E. 4 . Cohen, C B . , and 1956b R e s h o t k o , E. 5 , C u r i e , N, 1957a 6 . C u r i e , N. 1958a 7 . C u r i e , N. 1958b Tenipera-ture and v e l o c i t y p r o f i l e s i n t h e c o m p r e s s i b l e l a m i n a r boundary l a y e r -vdth a r b i t r a r y d i s t r i b u t i o n of s u r f a c e t e n p e r a - t u r e . J n , A e r o , S c . V o l . 1 6 , p p . 5 4 7 - 5 6 5 . S i m i l a r s o l u t i o n s f o r t h e c o m p r e s s i b l e l a m i n a r boundary l a y e r w i t h h e a t t r a n s f e r and p r e s s u r e g r a d i e n t . N.A.C.A. r e p . 1 2 9 3 ( f o r m e r l y N.A.C.A. T.N. 3325 ( 1 9 5 5 ) ) . The c o m p r e s s i b l e l a m i n a r b o u n d a r y l a y e r v d t h h e a t t r a n s f e r a n d a r b i t r a r y p r e s s u r e g r a d i e n t .
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