436

IN ZHENERNO-FIZICHESKII ZHURNAL

SIMPLE METHOD OF CALCULATING HEAT TRANSFER AND FRICTION FORCES IN A TURBULENT BOUNDARY LAYER FOR VARIABLE CONDITIONS A T THE W A L L A. D. Rekin I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 13, No. 6, pp. 831-836, 1967 UDC 536.24.532.517.4 The concept of wall influence zones in a turbulent boundary layer on a plate is introduced. Using these zones, the formulas obtained for caletflating heat transfer and friction forces for steady conditions at the wall are extended to include variable conditions. A l a r g e n u m b e r of t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n s have been p e r f o r m e d to study t u r b u l e n t b o u n d a r y l a y e r s on p l a t e s f o r s t e a d y conditions o v e r the e n t i r e l e n g t h of the p l a t e . S i m p l e e x p r e s s i o n s ,

vo, ro vs, rs

3r

F i g . i . S c h e m a t i c d r a w i n g of the p o s i t i o n of the influence zone at the w a l l . convenient f o r u s e in e n g i n e e r i n g c o m p u t a t i o n s , have b e e n obtained f o r the h e a t flow, s h e a r s t r e s s , and b o u n d a r y - l a y e r t h i c k n e s s f o r a g a s with constant p h y s ical p a r a m e t e r s in the b o u n d a r y l a y e r . F o r Reynolds n u m b e r s r a n g i n g f r o m l 0 b to 10 7, t h e s e e x p r e s s i o n s have, r e s p e c t i v e l y , the f o r m [1,2]:

qo(x) = O.02967,Y~176 x

pV~ ]~ Pr~

(1)

\T/

9o(X) = o.o296 y~ (P Voqo. x \T/ 60 (x) = 0,37x \---~--/

(2)

(3)

In m o s t c a s e s of p r a c t i c a l i n t e r e s t , h o w e v e r , the s u r f a c e t e m p e r a t u r e is not a constant. Then, the h e a t flow c a l c u l a t e d f r o m f o r m u l a (1), d e r i v e d u n d e r the a s s u m p t i o n of a constant s u r f a c e t e m p e r a t u r e , can d i f f e r a p p r e c i a b l y f r o m the actual flow. Allowance f o r the v a r i a b i l i t y of the s u r f a c e t e m p e r a t u r e can b e m a d e , in a r e l a t i v e l y s i m p l e way, with the aid of S e b a n ' s and R u b e s i n ' s f o r m u l a s , obtained by applying the i n t e g r a l method to the a n a l y s i s of the t h e r m a l b o u n d a r y l a y e r that d e v e l o p s within a d y n a m i c l a y e r [3,4].

The l i t e r a t u r e l a c k s f o r m u l a s f o r c a l c u l a t i n g s h e a r s t r e s s f o r flows p a s t a s u r f a c e , a p o r t i o n of which is in m o t i o n (for e x a m p l e , d u r i n g the f o r m a t i o n of a fluid f i l m on the s u r f a c e ) , while s h e a r s t r e s s c o m p u t a t i o n s f r o m f o r m u l a (2) can lead in this c a s e to r e s u l t s that a r e e r r o n e o u s even with r e s p e c t to the d i r e c t i o n in which t h e f r i c t i o n f o r c e s a r e a p p l i e d . It i s , t h e r e f o r e , d e s i r a b l e to obtain an a p p r o p r i a t e and, a t t h e s a m e t i m e , s i m p l e method with which h e a t flows and s h e a r s t r e s s e s can be c a l c u l a t e d f o r a v a r i e t y of boundary conditions. Definition of influence zones. It is p r o p o s e d to e x tend f o r m u l a s (1) and (2) to include s t e p w i s e - d i s c o n tinuous conditions at the wall by r e p l a c i n g the c h a r a c t e r i s t i c v a l u e s of the v e l o c i t y V0 and t e m p e r a t u r e T O at the b o u n d a r y - l a y e r b o u n d a r y 50 by the c o r r e s p o n d hag v e l o c i t y V s a n d t e m p e r a t u r e T s v a l u e s at the bounda r y of the i n t e r n a l b o u n d a r y l a y e r that d e v e l o p s f r o m the a r e a s w h e r e the conditions at t h e wall begin to v a r y ( s e e F i g u r e 1). In t h i s c a s e , the l i n e a r d i m e n s i o n x should be r e p l a c e d by x - s . S i m i l a r to the a p p r o a c h e s u s e d by R u b e s i n and Seban, t h i s a p p r o a c h is based 9 on the p h y s i c a l p r e r e q u i s i t e of the e x i s t e n c e of such an i n t e r n a l l a y e r t h a t would c o n c e n t r a t e in i t s e l f a l l the effects a s s o c i a t e d with the v a r i a b i l i t y of the b o u n d a r y c o n d i t i o n s . In the p r e s e n t p a p e r , the b o u n d a r y of this l a y e r i s d e t e r m i n e d f r o m f o r m u l a (3) with a l l o w a n c e for the a f o r e s a i d changes in the c h a r a c t e r i s t i c v a l u e s , i . e . , 6s(x)

= 0.37(x --s) [P V~(x-- s)]-~

(4)

This i n t e r n a l l a y e r will be t e r m e d the "influence zone" of c r o s s s e c t i o n s. In a c c o r d a n c e with definition (4), the b o u n d a r y l a y e r with the b o u n d a r y 60(x) is a l s o the influence zone f o r s = 0. We a s s u m e t h a t the p a r a m e t e r s of the g a s beyond the influence zone (including t h e b o u n d a r y of the zone) r e m a i n constant r e g a r d l e s s of the n a t u r e of the changes in the boundary conditions at the w a l l . This condition m a k e s it p o s s i b l e to d e t e r m i n e the b o u n d a r y of the influence zone and, c o r r e s p o n d i n g l y , the v e l o c i t y and t e m p e r a t u r e v a l u e s at this b o u n d a r y for s t e a d y c o n d i t i o n s at the w a l l . If the v e l o c i t y and t e m p e r a t u r e d i s t r i b u t i o n s a r e a s s u m e d to obey the p o w e r taw

V

T--T.o(

vo--ro--T=o

8~ '/7

\~o'I

'

(5)

by analyzing relations (4) and (5) simultaneously, we get 6s

6o--(1--~)

7/9'

JOURNAL OF ENGINEERING PHYSICS Vs Vo

T~-- T~o To - - T~o

(

1--

+)"

437 .

(6)

The retations obtained for V s and T s will be used in the solution of problems with variable conditions at the wall to be examined below as applications of the method proposed. Heat t r a n s f e r in the c a s e of a variable wall t e m p e r ature. Let us e x a m i n e the flow past a plate of t u r b u l e n t gas having a t e m p e r a t u r e To and a velocity V0. The t e m p e r a t u r e of the plate f r o m its leading edge to c r o s s s e c t i o n s is kept c o n s t a n t and equal to Two. F u r t h e r d o w n s t r e a m , the s u r f a c e t e m p e r a t u r e of the plate changes a b r u p t l y to Tws. F o r s i m p l i c i t y , and for c l e a r e r r e p r e s e n t a t i o n of the r e s u l t s obtained, we shall a s s u m e in the following that the changes in the c h a r a c t e r i s t i c physical p a r a m e t e r s , which r e s u l t f r o m changes in the wall t e m p e r a t u r e , a r e negligible as c o m p a r e d with the absolute values of t h e s e p a r a m e t e r s (] Tws - Tw01 << Two). According to the method proposed, e x p r e s s i o n (1) for the heat flow at x > s m u s t be converted to the form

G(x) . 0.0296 . . ~. T ~ - - T ~ ' I P V ' ( ; - - s ) I ~ pro.43. X--S

(7)

Substituting the v a l u e s for V s and T s f r o m the r e l a tions (6) into r e l a t i o n (7) for the condition Tws = Two , we get qs(x) - q0(x). Consequently, to calculate the heat ftow for steady conditions at the wall, one m a y s u b s t i t u t e into f o r m u l a (1) a n a r b i t r a r y l i n e a r d i m e n sion x - s and the c o r r e s p o n d i n g values for the v e l o c ity and t e m p e r a t u r e at the boundary of the influence zone of s e c t i o n s. Although, s t r i c t l y speaking, the "1/7 law" for the d i s t r i b u t i o n s of V and T is not a p p l i cable to a l a m i n a r s u b l a y e r , the fact that r e l a t i o n (7) holds for the c a s e Tws = Two j u s t i f i e s to a c e r t a i n extent the use of the heat t r a n s f e r model proposed for values of 1 - (s/x) < 0,001, which c o r r e s p o n d to the region of the l a m i n a r s u b l a y e r . F o r the condition Tws ~ Two in the p r o b l e m u n d e r c o n s i d e r a t i o n , e x p r e s s i o n (7) yields qs (x) = 1 - - T~, - - T,o qo(x) To - - T~,o

1 --

~.~ ) --1.,'9

(8)

In the c a s e w h e r e the wall t e m p e r a t u r e f r o m x = 0 x = s coincides with the gas t e m p e r a t u r e (Two = To), f o r m u l a (8) takes the f o r m

F i g u r e 2 gives the r e s u l t s of calculations f r o m f o r m u l a s (9) and (10), and for c o m p a r i s o n also e x p e r i m e n t a l data [3] obtained at Reynolds n u m b e r s ranging

It

-

~ ^0"

t.5

0

!

ot

%

[.2 1

f.!

O..a

-

2

3

1r 5 B 78..q/0 "t

f

2

3

4, .q 8

f-s/x

Fig. 2. C o m p a r i s o n of t h e o r e t i c a l and e x p e r i m e n t a l data for a stepwise v a r y i n g s u r f a c e t e m p e r a t u r e : 1) f r o m f o r m u l a (9); 2) f r o m Seban's f o r m u l a (10); 3) e x p e r i m e n t a l data [3]. f r o m 5 9 105 to 4 9 106. It can be s e e n that the r e l a t i o n s (9) and (10) c o r r e l a t e well with the e x p e r i m e n t a l data. C o m p a r i n g the r e s u l t s obtained f r o m f o r m u l a s (9) and (10), it b e c o m e s evident that the difference b e tween them is m a x i m u m , and equal to 1.2% for x ~ s, while for x > s, it d e c r e a s e s m o n o t o n i c a l l y . Since this d i f f e r e n c e l i e s within e x p e r i m e n t a l u n c e r t a i n t y , it should be noted that f o r m u l a (9) d e s e r v e s p r e f e r e n c e over f o r m u l a (10) i n a s m u c h as it involves l e s s c o m putational l a b o r . In the case of an a r b i t r a r y wall t e m p e r a t u r e d i s t r i bution for x _> s, the r e s u l t s obtained for a stepwise v a r y i n g t e m p e r a t u r e s can be g e n e r a l i z e d (as shown i n [4]) by s u m m i n g up the heat flux i n c r e m e n t s of all the a r e a e l e m e n t s . G e n e r a l i z a t i o n of f o r m u l a (8)to include the case of a continuously v a r y i n g wall t e m p e r a t u r e leads to the e x p r e s s i o n

q~(x) - v cp G Sto(x) x

to

St(x)= Sto (l(x)

s_)

-)/o

(9)

where Sto (x) = qo (x)/P cp Vo (To - - T~o);

St, (x) = q, (x)/9 c o Vo (To - - T ~ ) . F o r m u l a (9) does not differ s i g n i f i c a n t l y f r o m S e b a n ' s formula

st, (x) _ s_L_)9/~~ Sto(X) Ii (

(:to)

s Shear s t r e s s at a moving s u r f a c e . Let us examine a t u r b u l e n t gas flowing at a velocity V0 past a plate.. The f a i r i n g of the plate between the leading edge and x = s is at r e s t , while f u r t h e r d o w n s t r e a m it m o v e s at a constant velocity Vws in the s a m e d i r e c t i o n as the gas. In a c c o r d a n c e with the method proposed, e x p r e s sion (2) for the s h e a r s t r e s s at x > s should be t r a n s formed as ~(x) = 0.0296~

V~ - - V~,s [ P (V~ - - V ~ ) (x - - s) ]~ x--s ~

"

438

IN ZHENERNO-FIZICHESKII ZHURNAL

Substituting V s f r o m (6) into this f o r m u l a , we get -

1 . . . .

9 0(x)

~ V0 /

while e x p r e s s i o n (8) for the heat flow and e x p r e s s i o n (11) for the s h e a r s t r e s s will t r a n s f o r m to (11)

x

where r0(x) is the s h e a r s t r e s s at the c r o s s section u n d e r c o n s i d e r a t i o n for a s u r f a c e in the state of r e s t . F r o m an a n a l y s i s of e x p r e s s i o n (11), it follows that, in the case of V0 > Vws, the ratio T s / r 0 for x ~ s is negative, i . e . , that the f r i c t i o n f o r c e s applied at this point to the moving s u r f a c e a r e d i r e c t e d opposite to the gas flow. If f o r m u l a (11) is g e n e r a l i z e d to include a c o n t i n uously v a r y i n g velocity of the s u r f a c e , in the s a m e m a n n e r as f o r m u l a (8) was extended to include an a r b i t r a r i l y v a r y i n g wall t e m p e r a t u r e , one can obtain the following e x p r e s s i o n for the s h e a r s t r e s s at a moving surface: x

s

This e x p r e s s i o n and f o r m u l a (11) s t i l l r e q u i r e e x p e r i mental verification. All the r e l a t i o n s in this paper were obtained u n d e r the a s s u m p t i o n that the velocity and t e m p e r a t u r e d i s t r i b u t i o n s in the b o u n d a r y l a y e r a r e governed by law (5) with an exponent of 1/7. S i m i l a r r e l a t i o n s can be obtained also for an a r b i t r a r y exponent 1 / n (n _> 1). In this case, in a c c o r d a n c e with [2], the exponents of the Reynolds n u m b e r s in the e x p r e s s i o n s (1)-(3) m u s t be

(n + l)/(n + 3), (n + l)/(n + 3), and -(2/(n + 3)), respectively. With the aid of operations similar to those performed with formulas (I)-(5), the expressions (6) for the boundary of the influence zone and for the velocity and temperature at this boundary take the form

60 T~--r.o

Vo

To-- T.o

T~s - - T ~ o To ~ Two 2n+4

1 --

' --2

NOTATION V is the velocity; T is the t e m p e r a t u r e ; p is the d e n s i t y ; Cp is the specific heat; # and k a r e the v i s cosity and heat conduction coefficients, r e s p e c t i v e l y ; q is the heat flow; r i s t h e s h e a r s t r e s s ; 6 is t h e b o u n d a r y l a y e r t h i c k n e s s ; x is the d i s t a n c e f r o m the plate leading edge to the c r o s s section u n d e r c o n s i d e r a t i o n ; s is the length of the initial p o r t i o n of the plate with stable conditions at the wall; St is the d i m e n s i o n l e s s h e a t - t r a n s f e r coefficient (Stanton number) ; s u b s c r i p t 0 r e f e r s to p a r a m e t e r s in the case of stable conditions at the wall; s u b s c r i p t s s r e f e r to p a r a m e t e r s in the case where a boundary l a y e r f o r m s at c r o s s section s.

REFERENCES i. M. A. Mikheev, Fundamentals of Heat Transfer [in Russian], Energoizdat, 1956. 2. H. Schliehting, Boundary Layer Theory [Russian translation], 3. W. C. S. J. T r a n s . 4. M. W. 1951.

IL, 1956. Reynolds, W. M. K a y s , and S. J. Kline, ASME, s e r . C, 82, no. 4, 1960. R u b e s i n , NASA technical note, 2345,

x I

v~_

--I

q~ -- 1 qo

21 F e b r u a r y 1967

- { 1 -- ~+~

~

x }

'

Baranov C e n t r a l Institute of Aviation Engine C o n s t r u c t i o n , Moscow