Experiments on injection into an incompressible turbulent boundary layer

THE COLLEGE OF A E R O N A U T I C S

C R A N F I E L D

EXPERIMENTS ON INJECTION INTO AN INCOMPRESSIBLE

TURBULENT BOUNDARY LAYER

by

CoA R e p o r t A e r o No. 177 O c t o b e r 1964 THE C O L L E G E OF AERONAUTICS CRANFIELD E x p e r i m e n t s on injection i n t o a n i n c o m p r e s s i b l e t u r b u l e n t b o u n d a r y l a y e r by T . N . Stevenson. B . S c . P h . D . , D . C . A e . SUMMARY An 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 into the c h a r a c t e r i s t i c s of an a x i s y m m e t r i c i n c o m p r e s s i b l e t u r b u l e n t b o u n d a r y l a y e r o v e r a p o r o u s cylinder with a i r injection i s d e s c r i b e d . T h e r e s u l t s show a g r e e m e n t with the t h e o r y of Black and S a r n e c k i and the e x p e r i m e n t a l r e s u l t s of Mickley and D a v i s but d i s a g r e e with the t h e o r y of R u b e s i n , T h e e x p e r i m e n t a l r e s u l t s for the m e a n v e l o c i t y d i s t r i b u t i o n r e v e a l two l a w s , one for the i n n e r r e g i o n and one for the o u t e r r e g i o n which a r e valid for both s u c t i o n and injection p r o v i d i n g t h e r e i s no p r e s s u r e g r a d i e n t . T h e l a w s r e d u c e to the 'law of t h e w a l l ' and t h e ' v e l o c i t y defect law' when the t r a n s p i r a t i o n v e l o c i t y i s z e r o .

Summary

List of Symbols Introduction Apparatus

The momentum equation for axisymmetric flow Experimental results

A comparison with previous theories and experiments, The law of the wall with suction or injection

The law for the outer region of turbulent boundary layers with suction or injection

The experimental r e s u l t s of Tewfik

The variation of skin friction with Reynolds number Axisymmetric flow

Conclusions References Table Figures

LIST OF SYMBOLS

Constant of Integration with respect to y. (equation 6.2). Skin friction coefficient, / i

-Constant of integration with respect to y. (equation 5.2). Universal function of /g only.

von Karman's constant (a value of 0.418 i s used in the calculations). defined by equation 5.6.

defined by equation 5.7.

p r e s s u r e recorded by Preston tube relative to the static p r e s s u r e radius of the cylindrical model (2r is the outside diameter.) Reynolds number, -^—^ .

Reynolds nunaber, —-•- . velocity in x-direction. friction v e l o c i t y ^ .

velocity in the y-direction at the wall (+ve.blowing and -ve. suction) coordinates in the free s t r e a m direction and normal to the wall r e s

-pectively.

defined by equation 5.4. boundary layer thickness. value of y at which F(^/g) = K. displacement thi<::kness. momentum thickness. density

defined by equation 5 . 8 .

shear s t r e s s at the wall (ix •— ) oy w kinematic viscosity.

free s t r e a m conditions

conditions at the edge of the sublayer region conditions when V^ = 0 at the same

R^-1. Introduction

The structure of the mean flow in turbulent boundary layers on i m p e r m -eable walls is now fairly well known. However information on the related structure with suction and blowing is s p a r s e . Hartnett et al.^ discuss the experimental results for incompressible flow (ref. 2) and compressible flow (refs.3 & 4) and com-pare these with the theoretical predictions of Dorrance and Dore (ref. 5) and Rubesin (ref. 6). Hartnett suggests that the skin friction results of Mickley and Davis a r e possibly low. (This is discussed in section 9)

This paper describes experiments which were intended to check those of Mickley and Davis. The measurements were made in an axisymmetric incompress-ible turbulent boundary layer over a porous cylinder through which there was a small injection velocity. (In section 10 there is a discussion of the likely differences between axisymmetric and flat plate flow.)

These experimental results have been used a s a critical check on the existing enapirical theories and as a result, a generalised law of the wall and a generalised velocity defect law to include suction or blowing have been derived.

2. Apparatus

The model porous cylinder i s shown in figs. 1, 2 and 3. The model consisted of a 24" long, 4" diameter porous tube which was mounted as part of a long cylinder with an elliptical nose, in the College of Aeronautics 3 feet x 3 feet open circuit wind-tunnel. Compressed air was used a s the injection fluid. A

one inch thick felt filter was positioned upstream of the porous tube. The supporting wires allowed the model to be aligned with the free stream.

A material was required which was smooth to the boundary layer flow, and which was porous so that a uniform flow of air, rather than a s e r i e s of j e t s , could be forced through it. A material, which suits this specification is Porosint Grade A made by Sintered Products Ltd. Porosint is a sintered bronze material with a smooth surface in which the holes are about 20 microns in diameter. The material consists of spherical granules which are welded together at their points of contact, and through a microscope the holes in the surface have a bell-mouthed appearance. The p r e s s u r e drop a c r o s s the porous tube was far higher than the kinetic energy of the air passing through the tube, so that turning vanes inside the tube were not required.

A naicrometer screw traversing gear (fig.l), which was calibrated to 0.001 inch, enabled boundary layer profiles to be taken on the top and bottom s u r -faces of the cylinder at any longitudinal position. Mean velocity profiles through the boundary layer were measured with a pitot tube which had a rectangular c r o s s section 0.014 inch x 0.1 inch and the readings were corrected for the pitot tube displacement effect by the method of Young and M a a s ' . T r a v e r s e s using a static tube 0.064 inch diameter did not detect anychange in the static p r e s s u r e through the boundary layer.

?

-The air flow to the porous tube was measured with an orifice plate in the supply pipe. The orifice plate calibration was checked by taking momentum t r a v e r -ses a c r o s s the supply pipe. The approximate velocity distribution through the

porous tube was obtained with a hot wire anemometer when there was no flow through the wind-tunnel. The distribution of velocity (V^) with streamwise distance is shown in fig. 5.

A Betz manometer was used to record the p r e s s u r e difference a c r o s s the orifice plate and two Chattock gauges were used to measure the free stream velocity and the pitot tube p r e s s u r e .

Pitot and static t r a v e r s e s in the working section showed that the velocity outside the boundary layer was constant to within 0. 35%.

A s e r i e s of Preston tubes around the cylinder gave variations of only + 1% in (P - pto)(see fig. 4). (P - po) is the p r e s s u r e recorded by the Preston tube relative to the static p r e s s u r e . The Preston tubes consisted of tubes 3 inches in length and 0.064" outside diameter.

I 3. The momentum equation for axisymmetric flow.

The momentum integral equation for a steady incompressible turbulent boundary layer along a cylinder with zero p r e s s u r e gradient (ref. 8) is

^ . Z w = I w = f l 3 1

dx u, p u^' 2 '

where r ^ is the wall shear s t r e s s , u, is the free stream velocity and the momentum thickness, 8^ is defined

where r is the radius of the cylinder and y is measured from the surface of the cylinder.

The displacement thickness, &, , is defined (see ref. 8)

6, = | ( 1 + J ) ( l - M d y 3.3

4. Experimental r e s u l t s .

The mean velocity profiles a r e measured at different positions along the cylinder and at different values of the blowing velocity, V^, with a constant free-stream velocity of 50 feet per second. Curves of — ( 1 — ) (1 + ^ ) against y are

plotted (see figs. 7 & 8) and integrated graphically to find the momentum thickness, 6a . The variations in 6, along the cylinder a r e shown in fig. 6. These curves can be used to <ierive -r-^ and hence the skin friction can be evaluated using the integral

dx

momentum equation 3 . 1 . The skin friction variations are shown in fig. 10. The estimated accuracy of the skin friction measurements i s ±10% when Vy, = 0, and c.+ O. 0003 when there is a blowing velocity.

Some velocity profiles at a particular position on the cylinder for different blowing velocities a r e shown in fig. 11.

5. A comparison with previous theories and experiments

Mickley and Davis^ have published a very comprehensive set of experi-mental r e s u l t s of the mean flow in a turbulent boundary layer in a wind-tunnel which had a porous wall 12 feet long and 1 foot wide. Blowing velocity r a t i o s , _2* , which were constant in a particular experiment, ranged between 0 and 0,01 and free s t r e a m velocities between 17 and 60 feet per second. A small p r e s s u r e gradient was

present in the experiments but does not seenn to have been included in the skin friction calculations. This is discussed in more detail in reference 9 where it i s

shown that the effect of the small p r e s s u r e gradient i n c r e a s e s the skin friction by Vw

a s much a s 80% when —!^ = 0.003, although its effect is negligible when there is no blowing velocity.

Rubesin used the mixing length hypothesis to obtain equations for the inner region of a compressible turbulent boundary layer over a porous wall. When the flow is incompressible Mickley and Davis show that the equations reduce to

log Z = 2Kur

(l.X-^^)^ - ( l . I ^ ) H ,

ur* Uf* J 5 . 1

where the subscript '« ' r e f e r s to conditions at the outer edge of the sublayer, Mickley and Davis compare their experimental results with equation 5.1 by plotting log (^—^ ) against ( 1 + ^ ) ^ and they show that a straight line of the predicted

V Uy2

2Kur /• •

slope, — , is obtained and also that von Karman's constant, K, is independent ^w

of Vw . However it was not possible to deduce satisfactory variations in the conditions at the edge of the sublayer where y = 3{« • Mickley and Davis showed that the

velocity defect t e r m , -i-^— , only correlated the velocity profiles in the outer region

" T

when Vw was z e r o .

A few of the present experimental r e s u l t s a r e plotted a s log *—^ against . Vwu 2

(1 + —— ) in fig. 12 and a r e in agreement with equation 5 . 1 . (The straight lines i^/ 2Ku

in fig. 12 have gradients of ^ •) Vw

. 4

-Black and S a r n e c k i ^ use the mixing length hypothesis to derive their bilogarithmic law,

U r ' + V w u = ( § loge l> ' • 5-2 where d is independent of y. This equation can be rewritten a s

" 1 Vw ,. „ Ü1 " 4K» 1Ï7 ^ ^ ^

H ! l . \ * - 1 Vw ,, „ Uid .a ur* 1

where the left hand side of the equation contains only the quantities which are easily measured, and the right hand side is linear in log ^ . When there is injection, Black and Sarnecki introduce the substitution

and hence equation 5.3 becomes

- - Y* = ( r^ - p,*) + 2nY , 5.5 , -1 I" Vw , M ^ _ where " = 2K NJ "^ log e - f r . 5-6 u^ ° - V and f^' = "' Vw", 5.7 u

If experimental results a r e plotted a s ( - - Y' ) against Y, a straight line is obtained in the inner region where the mixing length hypothesis applies and the skin friction is obtained from the gradient and the intercept. Some of the present

experimental results are plotted in this way in fig. 13 and the skin friction which is predicted by Black and Sarnecki's theory is in good agreement with that measured by momentum t r a v e r s e s (see table 1).

Black and Sarnecki introduce the parameter X in place of d, the unknown constant of integration with respect to y, or in place of the conditions at y = ya which Mickley and Davis considered. When there is injection

Several non-dimensional parameters which might provide a possible criterion for specifying conditions at y = y^ were considered by Black and Sarnecki.

Eight possible equations for X were obtained by considering the sublayer equation Vw

and the bilogarithmic law, equation 5.2. X was then plotted against —^ and compared

Uj-with experimental r e s u l t s . There i s a considerable scatter in the experimental r e s u l t s , but Black and Sarnecki chose the equation for X which predicted most accurately the actual variation for layers on sm.ooth and nearly homogeneous walls. The equation is not the same a s that predicted by the present experiments and the theory outlined in the next section.

Black and Sarnecki's theory enables the skin friction to be determined from a velocity profile without assuming a form for d, the unknown constant of integration with respect to y. This was not possible with Mickley and Davis' theory.

6. The law of the wall with suction or injection.

The equation for the inner turbulent region, equation 5.2, may be written

6.1 When Vw = 0, this equation reduces to the familiar law of the wall equation;

- = ^ l o g e ^ + B. . 6.2

u r K ^^ V

The experimental curves for flow over a permeable or impermeable wall may now be compared on one figure if log„ ' ^ ^ is plotted against rf- ] {—^— + 1 ) ^ - 1

V Vw L UT-^ J

and the inner turbulent region should plot as a s e r i e s of parallel lines if von Karman's constant K is independent of Vw. The present experimental r e s u l t s were plotted in this way and it was found that they fell close to the accepted impermeable wall

On

curve (see fig. 9). The experimental r e s u l t s show that the t e r m (d + -r— ) in ''w equation 6.1 changes very little with injection. Therefore

2ur Vw

In reference 9 this equation, 'the law of the wall with suction or injection', is discussed in detail and it is shown to be in reasonable agreement with the suction r e s u l t s of Black and Sarnecki and Dutton . It is also shown how the skin friction may be estimated from the measured mean velocity profile for a given suction or injection velocity.

6

-7. The law for the outer region of turbulent boundary layers with suction or injection. Equation 6.1 is used to derive the equation for the outer region (ref. 12) by methods analogous to those of von Karman and Millikan. The equation for the outer region is

2ur

V w L Ur' ' Ur* J K

where F i s a function of ^/6 . When Vw = 0 the equation reduces to the velocity defect law for zero p r e s s u r e gradient.

_u^ ^ F(y/6 )

u^ K 7.2

The present experimental r e s u l t s a r e plotted as :rp j ( 1 + ^ * )^ - ( 1 + —^—)^ j against (^/öp) in fig. 14, ( 6o is the value of y at which — = 1) and the r e s u l t s fall close to the zero p r e s s u r e gradient and zero blowing curve. It is thereby deduced that Fi^l^ ) is independent of Vw- The law of the outer region is discussed in detail in references 12 and 13 and it is shown to agree with the experimental r e s u l t s of Mickley and Davis.

The mean velocity distribution can easily be obtained from the following rearranged form of equation 7,1

F ( y / 6 ) ^ ( 1 + ^ )^ + 4 ^ [ F ( y / 6 ) ^ ]

^ = 1 - F ( J ' / R ) ^ ( 1 + ^ ^ a ^ )2 + . - ^ 1 F(J'/&)' 1 7.3 1

and may be evaluated for a s e r i e s of values of Cf for any transpiration velocity. 14

8. The experimental r e s u l t s of Tewfik.

The rig which Tewfik used i s very similar to that used in the present experiment except that the cylinder was 2"diameter. The velocity profiles which were measured by Tewfik for a particular blowing velocity, collapsed onto one curve which is shown in fig. 16. The curve is similar in shape to that given by equation 7. 3 and the skin friction appears to be in the range 0. 001 to 0. 0015. This range is again obtained if the profiles a r e compared with the law of the wall with suction or injection' (fig. 15). Tewfik measured the skin friction by momentum t r a v e r s e s and obtained values considerably higher in the range 0. 0015 to 0. 002. The discrepancy between these r e s u l t s and those of the present method needs further investigation.

9. The variation of skin friction with Reynolds number.

In reference 12 equations 6.1 and 7.1 a r e used to evaluate the variation of skin friction with Reynolds number, and curves of cf against Rs^and Rx a r e

p r e s e n t e d for different blowing v e l o c i t i e s . Some of t h e s e c u r v e s a r e shown in f i g u r e s 17 and 1 8 , and a r e shown to c o m p a r e r e a s o n a b l y w e l l with the p r e s e n t e x p e r i m e n t a l r e s u l t s , and with the r e s u l t s of Mickley and D a v i s .

R u b e s i n ' s t h e o r y , which i s a l s o for c o m p r e s s i b l e flow, g i v e s h i g h e r v a l u e s of skin friction t h a n the author's t h e o r y , the p r e s e n t e x p e r i m e n t s and the e x p e r i m e n t s of M i c k l e y and D a v i s . R u b e s i n ' s t h e o r y c o m p a r e s v e r y well with the e x p e r i m e n t a l r e s u l t s in c o m p r e s s i b l e flow (ref. 3) and it w a s for t h i s r e a s o n t h a t H a r t n e t t et a l s u g g e s t e d that M i c k l e y and D a v i s ' r e s u l t s w e r e p o s s i b l y low ( s e e fig. 20). R u b e s i n ' s t h e o r y i s b a s e d , not on a t r u e v e l o c i t y p r o f i l e , but on a n e x t e n d e d law of the w a l l r e g i o n with the c o n s t a n t s in the law of the wall a d j u s t e d t o give the c o r r e c t skin f r i c t i o n - R e y n o l d s n u m b e r v a r i a t i o n when Vw = 0.

One s o u r c e of e r r o r in e x p e r i m e n t s of t h i s kind i s t h a t the d i s t a n c e f r o m t h e effective o r i g i n of the b o u n d a r y l a y e r i s insufficient for a fully developed

t u r b u l e n t b o u n d a r y l a y e r to e x i s t . C o l e s 1^ h a s shown t h a t an i n c o m p r e s s i b l e t u r b u l e n t b o u n d a r y l a y e r i s not in an e q u i l i b r i u m (fully d e v e l o p e d , n o r m a l , i d e a l , a s y m p t o t i c ) s t a t e until t h e R e y n o l d s n u m b e r . R e , i s g r e a t e r t h a n about 3 , 0 0 0 . T h e i n c o m p r e s s i b l e t u r b u l e n t b o u n d a r y l a y e r in Mickley and D a v i s ' e x p e r i m e n t s would p r o b a b l y be in e q u i l i b r i u m b e c a u s e t h e i r p o r o u s s e c t i o n w a s 12 feet in l e n g t h . On the m o d e l u s e d in the p r e s e n t e x p e r i m e n t s the b o u n d a r y l a y e r w a s c e r t a i n l y not in an e q u i l i b r i u m s t a t e for the f i r s t 12 i n c h e s of the p o r o u s c y l i n d e r but w a s p r o b a b l y c l o s e t o e q u i l i b r i u m when x w a s g r e a t e r t h a n 12 i n c h e s . ( T h i s w a s i n d i c a t e d by the r a p i d change in P r e s t o n tube r e a d i n g s o v e r the f i r s t 1 2 " (ref. 24)). T h e skin f r i c t i o n m e a s u r e m e n t s in c o m p r e s s i b l e flow ( r e f s . 3 and 4) w e r e o v e r a l l v a l u e s and t h e r e f o r e t h e y i n c l u d e d t h e b o u n d a r y l a y e r which w a s not fully d e v e l o p e d .

1 R

S q u i r e d i s c u s s e s a v a i l a b l e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s and c o m p a r e s t h e m in t h r e e f i g u r e s which a r e r e p r o d u c e d in t h i s r e p o r t a s f i g u r e s 19, 20 and 2 1 . T h e f i g u r e s h a v e a x e s - — a g a i n s t ^ ^ ^ ^ , ^ a g a i n s t ^P ^ w

^ T-wo ^ Pi ujo cfo ^ P,u,cf„

Cf V

and — a g a i n s t ^^ ^ w h e r e the s u b s c r i p t s r e f e r t o :

0 conditions when the blowing v e l o c i t y i s z e r o 1 f r e e s t r e a m conditions

w w a l l c o n d i t i o n s .

T h e a u t h o r ' s t h e o r y f r o m r e f e r e n c e 12 i s a l s o shown in t h e f i g u r e s and it i s s e e n t h a t the plot of — a g a i n s t ~——— i s the only one which c o l l a p s e s the t h e o r e t i c a l c u r v e s . ' o ' ^"

10. A x i s y m m e t r i c flow.

In t h e p r e v i o u s s e c t i o n s , a p a r t f r o m u s i n g the modified i n t e g r a l m o m e n t u m e q u a t i o n , no account h a s b e e n t a k e n of the .difference b e t w e e n a x i s y m -mietric and flat plate flow. A few r e p o r t s on the subject h a v e b e e n published and t h e y i n d i c a t e that the b o u n d a r y l a y e r in the p r e s e n t e x p e r i m e n t i s a l m o s t the s a m e a s that on a flat p l a t e . L a n d w e b e r ^ ' and E c k e r t ^° a s s u m e d that the v e l o c i t y p r o f i l e s on a c y l i n d e r could be r e p r e s e n t e d by a V ? p o w e r law and t h a t the

relationship between the wall shear s t r e s s and the boundary layer thickness for the cylinder i s identical to that on a flat plate, from which they calculated the boundary

6

layer growth using the i n t e g r a l momentum equation. If — , the ratio of the boundary layer thickness to the radius of the cylinder, is 0.5 then Eckert'.s theory suggests that the skin friction on the cylinder is 5% greater than that on a flat plate.

19 20 Ginevskii and Solodkin and Sparrow et al consider the boundary layer to be composed of a laminar region near the wall and a turbulent outer region. Ginevskii and Solodkin follow the analysis of Prandtl and assume that the mixing length is proportional to the distance from the wall and Sparrow follows the analysis of Deissler ^^ and a s s u m e s that the logarithmic region extends to the outer edge of the boundary layer. The theories suggest that the skin friction in the present experiments would be 5 to 10% greater than on a flat plate.

When y, the distance from the wall, is small compared with the radius of the cylinder very little difference is to be expected in the turbulent boundary layer characteristics in flat plate and axisymmetric flow and the velocity profiles a r e close to those predicted by the law of the wall. Richmond ^^ and Yasuhara measured velocity profiles on cylinders and estimated the skin friction by comparing the profiles with the law of the wall equation. Their results a r e roughly in agreement with the theories.

The theories and experiments for axisymmetric flow indicate that the skin friction in the present experiments may be slightly higher, perhaps 5% higher, than that on a flat plate a s previously mentioned. The curves of cf against Reynolds number in figures 17 and 18 confirnas that the present skin friction r e s u l t s a r e slightly higher than the theoretical curves, whereas Mickley and Davis' flat plate r e s u l t s ,

corrected for p r e s s u r e gradient as described in reference 9, a r e very close to the theoretical curves.

Conclusions.

The experimental investigation into the characteristics of an axisym-m e t r i c turbulent boundary layer with air injection has shown:

(1) The mean velocity distribution a c r o s s the layer agrees with the measured r e s u l t s of Mickley and Davis for the flat plate case.

(2) In confirmation with theory the measured values of Cf a r e slightly higher than those on the flat plate(when there is no injection).

(3) The experimental r e s u l t s confirm the assumptions made by Black and Sarnecki in deriving an empirical theory for the mean velocity distribution in the inner region but disagree with the theory of Rubesin.

(4) The available experimental r e s u l t s show good agreement with the mean velocity distribution as predicted from an extended law of the wall and velocity defect law to include suction and blowing. In particular these laws correlate the experimental r e s u l t s of the author with those of Mickley and Davis, Black and Sarnecki and Dutton.

1. Hartnett, J . P . . Masson, D . J . G r o s s , J . F . and Gazley, C . J r .

Mass - transfer cooling in a turbulent boundary layer.

J . Aero. Space Sci. Vol.27, p.623. 1960.

2. Mickley, H . S . and Davis, R . S .

Momentum, transfer for. flow over a flat plate with blowing, NACA TN4017 1957. 3. Tendeland, T. and

Okuno, A . F .

The effect of fluid injection on the compres-sible turbulent boundary layer - the effect on skin-friction of air injected into the boundary layer cf a cone at M=2. 7.

NACA RM A56D05. NACA/TIL 5126. 1956. 4. Pappas, C . C . and

Okuno, A . F .

Measurements of the skin-friction of the compressible turbulent boundary layer on a cone with foreign gas injection. J n l . Aerospace Sci. Vol.27 p . 3 2 1 . 1960. Dorrance, W. H. and

Dore, F . J .

The effects of m a s s transfer on compressible turbulent boundary-layer skin friction and heat transfer. J n l . of Aero. Sci. Vol. 21 pp. 404-410 June 1954.

6. Rubesin, M.W, An analytical estimation of the effect of transpiration cooling on the heat transfer and skin friction characteristics of a compressible turbulent boundary layer. NACA TN 3341 1954.

7. Young, A.D. and Maas, J . N .

The behaviour of a pitot tube in a t r a n s v e r s e total p r e s s u r e gradient.

R & M. No. 1770. 1936.

8. Young, A . D . The calculation of the total and skin friction drags of bodies of revolution at zero incidence. R & M. No. 1874. 1939.

9. Stevenson, T . N , A law of the wall for turbulent boundary layers with suction or injection. College of Aeronautics Report Aero. No.166, 1963. 10. Black T . J . and

Sarnecki, A . J .

The turbulent boundary layer with suction and injection. ARC 20,501. 1958.

- 1 11. Dutton, R . A . 12. Stevenson, T . N . 13. Stevenson, T . N . 14. Tewfik, O . E . 15. Coles, D. 16. Squire, L . C . 17. Landweber, L. 18. Eckert, H.U. 19. Ginevskii, A . S . and Solodkin, E. E. 20. Sparrow, E . M . , Eckert, E . R . G . and Minkowycz, W . J . 21. D e i s s l e r , R . G . and Loef fier, A. L. J r . 22. Richmond, R . L .

The effects of distributed suction on the development of turbulent boundary l a y e r s . R & M No. 3155, 1960.

A modified velocity defect law for turbulent boundary layers with injection. College of Aeronautics, Rpt. Aero. No. 170. 1963. A note on the outer region of turbulent boundary l a y e r s . (To be published). Some characteristics of the turbulent boundary layer with air injection. AIAA. J.1.pp.1306-1312, 1963.

The turbulent boundary layer in a c o m p r es-sible fluid. The Rand Corporation, Santa Monica, p. 2417. 1961.

Some notes on turbulent boundary layers with fluid injection at high supersonic speeds. R . A . E . Tech. Note Aero. 2904, ARC 24211, ' F M 3 3 7 1 . 1963.

Effect of transverse curvature on frictional r e s i s t a n c e . David W. Taylor Model Basin, United States Navy, Rpt. 689. 1949.

Simplified treatment of the turbulent boun-dary layer along a cylinder in compressible flow. Jnl. Aero. Sci. Vol. 19, pp. 23-29. 1952.

Effect of transverse curvature of the surface on the characteristics of an axi-symmetrical turbulent boundary layer. Prikladnaya Matematika i Mekhanika Vol.22, No. 6. pp. 819 - 825, 1958.

Heat transfer and skin friction for turbulent boundarylayer flow longitudinal to a c i r -cular cylinder. Jnl. Applied Mech, Vol. 30 pp. 37-43. 1963.

Analysis of turbulent flow and heat transfer on a flat plate at high Mach numbers with variable fluid properties. NASA Rpt.R-17 1959.

Experimental investigation of thick, axlally symmetric boundary layers on cylinders at subsonic and hypersonic speeds. Guggenheim Aero. Lab. , California Instit. of Tech. Memo No. 39, 1957.

23. Yasuhara, M.

24. Stevenson, T . N .

25. Townsend, A.A.

26. Nash. J . F .

Experiments on axisymmetric boundary l a y e r s along a long cylinder in incompres-sible flow. T r a n s , of Japan Soc. for Aero. and Space Sci. Volume 2, pp. 72-76. 1959. The use of Preston tubes to measure the skin friction in turbulent boundary layers with suction and injection. College of Aeronautics Rpt. Aero. No. 173. 1964. The Structure of Turbulent Shear Flow. Cambridge University P r e s s 1956.

A correlation of skin friction measurements in compressible turbulent boundary layers with injection. ARC 22,386. 1960.

Acknowledgements.

The author wishes to thank Professor G. M. Lilley of the College of Aeronautics and Professor A.D.Young of Queen Mary College, the University of London for many helpful discussions during the course of this work.

12

-TABLE

Comparison of Skin Friction Results

,Vw Ul 0 . 0 0 1 3 0 . 0 0 2 0 . 0 0 2 9 0 . 0 0 4 0 . 0 0 5 7 cf F R O M MOMENTUM TRAVERSES 0 . 0 0 2 6 ± 0 . 0 0 0 3 0 . 0 0 2 0 " 0 . 0 0 1 4 " 0 . 0 0 1 0 " Cf FROM THE LAW O F T H E W A L L EQUATION 6 . 1 0 . 0 0 2 4 5 0 . 0 0 1 9 5 0 . 0 0 1 4 5 0 . 0 0 0 8 5 0 . 0 0 0 3 3 Cf USING BLACK AND SARNECKI'S METHOD 0 . 0 0 2 3 ± 1 5 % 0.0016 " 0 . 0 0 1 5 " 0.0009 " 0.0006 "

R O O F WIND TU

FIG.I. MODEL

FIG.2. THE MODEL-VIEW OF THE WORKING SECTION.

(:->") < f , — — - — ^ . o° so MODEL ISO 2 7 0 _ , • 3 6 0

FIG.4. PRESTON TUBE READINGS ROUND THE M O D E L .

' / > ^ ^ \ JOINTS POROUS

1

i

FIG 5, THE APPROXIMATE INJECTION V E L O C I T Y D I S T R I B U T I O N .

m O , u X I-2 3 t-Z b j 2

i

V

c

sv^

c

t

c^

V. N

^,0-it;)0-^/r)

INCHES

0 - 4

0 - 2

0 ' I 6

7,0-OCi-y.)

FIG 8. MOMENTUM CURVES. Vw

2 0 - K

ii:^

l

' 1 ^

f,

A

Y \

FIG.9. THE LAW OF THE WALL REGION.

X in

FIG.IO. VARIATION OF SKIN F R I C T I O N COEFFICIENT WITH STREAMWISE DISTANCE.

0 - 3 INCHES 0 - 0 6 O O I "A/ o - 4 INNER REGION 0 - 6 _{0 - 4} OUTER REGION

L^i

•V

'k^

X,

it

r<

FIG.I4. THE OUTER REGION.

FIG.15. THE INNER REGION- CTEWFIK REF 14}

OUTER REGION INNER REGION

P R E S T O N TUBES ( R E F 2 4 ) • EXP RESULTS O F MICKLEY t, ^

DAVIS (SEE BEF i) V « = : ! 0 0 0 3

FIG.17 VARIATION O F SKIN FRICTION WITH R E Y N O L D S

0 - 0 0 4 0 0 0 3 O O O I 0 - ^ 1 • -~—_^ " ~ " ^ = ^ - - ^ " ^ ~ " " — — -—.^ ^^•-^ ^

5^

-ï : ^

—^:iJ

FIG.18. VARIATION O F SKIN FRICTION WITH REYNOLDS N U M B E R , R^2

P, U T

3-0

FIG.20. COMPARISON OF THEORETICAL AND EXPERIMENTAL EFFECTS OF AIR INJECTION ( F R O M REFERENCE 16) PAPPAS TENDELAND MICKLEY & i OKUNO R 11 II I OKUNO u DAVIS II Mc 0 - 3 O- 3 3 - 21 4 - 35 2 • 55 2- 55 M - O M = O «, R Rt « lO 0 - 8 7 4 0 3 5 - 8 6 , 4 - 4 2 4 - 6 8 , 3 - 2 5 6 - 2 5 8 - 5 6 - 0 - 5 X l o ' , - 3 X l o '

(g t h NOT CORRECTED FOR PRESSURE GRADIENT)

CONES ( O V E R A L L SKIN F R I C T I O N ) F L A T P L A T E ( L O C A L S K I N F R I C T I O N )

FIG.21.CORRELATION OF SKIN FRICTION MEASUREMENTS WITH INJECTION (FROM REFERENCE 16)

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