effect of curvature of bl

7/26/2019 Effect of Curvature of BL

1/33

R . M , N o . 3 5 9 9

M I N I S T R Y

O F T E H N O L O G Y

d 3

i .

A E R O N A U T I C A L R E S E A R C H C O U N C I L

R E PO R T S A N D M E M O R A N D A

7 he Ef fec ts o f Curvature on the Tu rbu l en t B oun da ry

L a y e r

By V. C. PATEL

Cambridge Univers i ty Engineer ing Department

L O N D O N : H E R M A J E ST Y S S T A T IO N E R ~ ( O F F I C E

969

PRICE 17S . 6d. NET


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The Effects

o f C u r v a tu r e o n th e T u r b u l e n t o u n d a r y

Layer

B y V . C . P A TE L

C a m b r i d g e U n i v e r s i t y E n g i ne e ri n g D e p a r t m e n t

Repor t s and M emoran da N o. 3599

Augus t 1968

S u m m a r y

P re l imina ry mea s u reme n t s in the tu rbu len t bo und ary l aye r s on a c i rcu la r cy l inde r and the convex w a l l

o f a 90 deg ree cu rved d uc t a r e des c r ibed . The r e s u lt s ind ica te qu i t e s ign i fi can t e f fec ts o f long i tud ina l

s u r f ace cu rva tu re o n the deve lopm en t o f the boun dar y l aye r. I n pa r t i cu la r i t i s s how n tha t a ) cu rva tu re

has a m arked in f luence on the ve loc i ty p ro fi l es , b ) the u s ua l mo m en tu m in teg ra l equa t ion fo r fi a t s u r f ace

f low b reak s dow n in the p r es ence o f l a rge s t a ti c -p res su re va r i a t ions ac ros s the bou nda ry l aye r , and c )

n o n e o f th e e x i s ti n g b o u n d a r y - l a y e r c a l c u l a t i o n m e t h o d s a d e q u a t e l y p r e d i c ts t h e d e v e l o p m e n t o f th e

s hape f ac to r . A more genera l momen tum in teg ra l equa t ion has been de r ived f rom f i r s t p r inc ip le s . Th i s

g ives r eas ona b le ag reemen t w i th the exper imen ta l dev e lopm en t o f the mom en tu m th icknes s .

L IS T O F C O N T E N T S

S ec t i o n

1 . In t roduc t ion

2 . R e v i e w o f P r e v i o u s W o r k

2.1

2.2

2.3

2.4

F u l l y d e v e l o p e d f l o w s

C u r v e d j e t s a n d e n t r a i n m e n t

T u r b u l e n t b o u n d a r y l a y e r o n c u r v e d s u r fa c e s

G e n e r a l c o m m e n t s o n t h e r e v i ew o f li t er a t u re

.

Exper imen t s on a C i r cu la r Cy l inde r

3 .1 M eas u remen t s

3.2 R o d e v e l o p m e n t

3 .3 H deve lopm en t , ve loc i ty p ro f i le s and en t r a inm en t cons ide ra t ions

3 .4 Conc lus ions f rom cy l inde r mea s u rem en t s

*Re places A .R.C. 30 427.


3/33

.

6.

7.

Experiments on the Convex Surface of a Curved Channel

4.1 Introductory

4.2 Description of the apparatus

4.3 Measurements

4.4 Evaluation of fi, ~* and 0

4.5 Comparison of measured

R o

and H with flat surface predictions

The Momentum Integral Equation for Boundary Layers on a Curved Surface

Discussion and Conclusions

Acknowledgement

References

Appendix

Tables

Illustrations--Figs. 1 to 16

Detachable Abstract Cards

1. Introduction.

In the development of two- and three-dimensional boundary-layer calculation methods, whether

based on integral or differential equations, it has been customary in the past to assume that the curvature

of the streamlines in planes normal to the surface, and the associated variation of the static pressure

through the boundary layer, do not influence the flow significantly. The results of such calculation

methods, however, have often been compared with experimental measurements in which such an assump-

tion does not strictly remain valid. Large variations in static pressure through the boundary layer can

arise due to the curvature of the streamlines not only in a boundary layer growing on a curved surface

but also in the neighbourhood of separation on flat surfaces. In fact, in experiments such as those of

Schmidbauer 22, Schubauer and Klebanof f23, and others, which are still being used to guide the construct-

tion of present-day boundary-layer calculation methods, it is found that the curvature of the streamlines

is not small enough to be ignored.

If the usual two-dimensional curvilinear co-ordinate system formed by the distances along and normal

to a curved surface is used, and first-order curvature terms are retained, it is found that the Navier-Stokes

equations and the simplified boundary-layer equations contain terms involving the curvature of the

surface

see,

for example, GoldsteinT . These equations therefore suggest the use of a characteristic

surface-curvature parameter such as

6 / R ,

where 6 is the boundary-layer thickness and R the radius of

curvature of the surface, to correlate curved and flat surface boundary layers. For laminar flow several

analyses have in fact been carried out in the past using ~ / R as an additional parameter. It is, however,

worth pointing out that, while this ratio gives some indication of the effects of curvature, it is unlikely,

especially for turbulent flows, to correlate all the influences of curvature since, in general, the curvature of

the streamlines within the boundary layer is not simply related to and uniquely fixed by the curvature

of the surface and since it is the curvature of the streamlines which primarily influences the detail flow

mechanisms within the layer. This distinction between streamline curvature and surface curvature is


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essential for proper assessment of curvature effects, since in many flow situations of practical importance

substantial curvature of the streamlines can arise not as a result of surface curvature but due to other

influences such as a rapid growth in boundary-layer thickness due to, say, an adverse pressure gradient

or fluid injection from the surface.

The work to be described in this Report was carried out some three years ago and at the time the

results were not thought to be conclusive enough to merit a publication. In recent years, however, there

has been a resurgence of interest in the effects of curvature on turbulent boundary layers and since the

early results have not been superseded in any way it has been thought proper to make them more widely

available.

When the present work was started the various methods of calculating the development of the turbulent

boundary layer were based on integral equations and from the more successful of these methods, namely

those of Head 9 and Thompson 27, it appeared that surface curvature may have been one of the causes of

disagreement between calculations and some of the many experimental results with which they were

compared. In view of this, the experiments reported here were designed primarily to obtain some empirical

informat ion concerning the effects of curvature so that existing calculation methods could be improved

to take account of these effects.

Previous work on curved shear flows has been reviewed in the next Section in an effort to point out the

need for further work in this field and to indicate the most profitable lines of attack. Two types of prelimin-

ary experiment are then described. The first set of measurements were made in the turbulent boundary

layer developing on a circular cylinder. This case is an ideal example of flows in which the curvature of the

streamlines within the boundary layer is influenced both by the convex curvature of the surface and by

the rather severe pressure gradients. The second set of measurements were made on the convex wall of a

parallel curved duct o f constan t radius with a total turning angle of 90 degrees. In this case the curvature

of the streamlines within the boundary layer could easily be related to the curvature of the surface. Quite

substantial effects of curvature have been observed in both cases. The results showed that a) curvature

influences the mean velocity distribution in both the inner and the outer regions of the boundary layer,

b) the usual momentum integral equation for flat surface flows, namely

~ x 0 d U 1 C y

+ H+2) U1

d x - 2

breaks down in the presence of large static-pressure variations acro.ss, he boundary layer and c) none

of the existing boundary-layer calculation methods adequately p~ . c t s the development of the shape

factor of the velocity profiles. A more general momentum integral equation derived from first principles

including the first order curvature terms has been found to give reasonable agreement with the experi-

mental development of the momentum-thickness Reynolds number.

For reasons which will become apparent later, the restilts of the preliminary experiments referred to

above were not considered conclusive enough and therefore a more elaborate set of experiments was

undertaken in a parallel curved duct of constant radius with a total turning angle of 180 degrees. Although

this investigation has not been completed, the results which have already been obtained are described

in a separate paper 18. These results have some bearing on the measurements described here and also have

several features of interest to workers contemplating further experiments in tu rbulent boundary layers

on curved surfaces.

2 . R e v i e w o f P r e v i o u s W o r k .

2.1. Ful ly Deve loped F lows .

Fully developed turbulen t flow in parallel channels of constant curvature has been investigated quite

thoroughly by Wattendorf28, Eskinazi4, Eskinazi and Yeh 5 and Yeh, Rose and Lien a. There are several

features of this flow which may be expected to apply in turbulen t boundary layers with curved stream-

lines. Watt endorf made extensive measurements of mean velocity profiles and static-pressure variations

across two curved channels, with half-width to mean radius of curvature ratios of 1/19 Channel I) and


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1 / 9 ( C h a n n e l I I ) , a n d c o m p a r e d t h e m w i t h m e a s u r e m e n t s i n a s t r a i g h t c h a n n e l . H i s r e s u l t s s h o w e d ,

a m o n g s t o t h e r t h i ng s , th a t c u r v a t u r e h a s a m a r k e d i n f l u e n ce o n t h e m e a n v e l o c i ty d i s t r i b u t i o n a n d t h a t

t h e u s u a l l a w o f t h e w a l l o b t a i n e d f r o m s t r a i g h t c h a n n e l s d o e s n o t a p p l y . W a t t e n d o r f s i n n e r - la w r e s u l ts

o f C h a n n e l I I a r e r e p r o d u c e d , a l o n g w i t h s im i l a r m e a s u r e m e n t s m a d e l a t e r b y E s k in a z i , i n F i g u r e 1.

C o m p a r i s o n o f t h e s e r e s ul ts w i t h t h e v e l o c i t y d i s t r i b u t io n s m e a s u r e d i n s e v er e a d v e r s e a n d f a v o u r a b l e

p r e s s u re g r a d i e n t b o u n d a r y l a y e r s b y P a t e P 6 a n d P a t e l a n d H e a d ~8 s u g g e s ts a s t a r t li n g r e s e m b l a n c e

be t we en t he e f f ec t s o f s t r eam l i ne cu r va t u re a nd t hos e o f p res s u re g rad i en t s : t he i nne r - l aw ve l oc i t y p ro f il e s

o n t h e i n n e r c o n v e x s u r fa c e a p p e a r t o b e s i m i l a r to t h o s e m e a s u r e d i n s e ve r e a d v e r s e p r e s su r e g r a d i e n t s

wh i l e the p ro f i l e s on t he ou t e r conc ave s u r f ace a re s i m i l a r t o t hos e ob t a i n i ng i n s eve re f avou rab l e p res s u re

g r a d i e n ts . N o t i c e t h a t a s m a l l f a v o u r a b l e p r e s s u r e g r a d i e n t e x is ts o n b o t h s u rf a c e s o f t h e c u r v e d d u c t a n d

t h a t t h i s is v e r y m u c h s m a l l e r t h a n t h e p r e s s u re g r a d i e n t s w h i c h w e r e r e q u i r e d t o p r o d u c e d e p a r t u r e s o f

s i m i la r m a g n i t u d e f r o m t h e u s u a l , s e m i - l o g a r i th m i c i n n e r - l a w i n t h e e x p e r im e n t s o f P a t e l , a n d P a t e l a n d

H e a d . S i nc e t h e o b s e r v e d d e p a r t u r e s f r o m t h e u s u a l i n n e r - l a w a r e p r o d u c e d b y t w o e n t i r e ly d i f fe r e n t, a n d

a p p a r e n t l y u n r e l a t e d , p h e n o m e n a w e c a n n o t , i n t h e a b s e n c e o f f u r t h e r e v id e n c e , d r a w a n y f i r m c o n c l u s i o n s

r e g a r d i n g t h e r a t h e r r e m a r k a b l e s i m i l a r it y b e t w e e n t h e e f f ec ts o f c u r v a t u r e a n d t h o s e o f p r e s s u re g r a d i e n t s .

P e r h a p s a l l t h a t c a n b e s a i d a t t h is s ta g e i s t h a t u n d e r z e r o s t r e a m w i s e p r e s s u r e g r a d i e n t t h e b o u n d a r y

l aye r on a cu rv ed s u r f ace m a y s t i ll b ehave , i n ce r t a i n r e s pec t s, a s i f i t were deve l op i n g on a f l a t s u r f ace

u n d e r t h e i n f lu e n c e o f a s t r e a m w i s e p r e s s u r e g r a d i e n t . T h i s i s o n e p o i n t w h i c h n e e d s t o b e e x a m i n e d b y

f u r t h e r e x p e r i m e n t s .

E s k i n a z i, Y e h a n d c o - w o r k e r s h a v e m a d e v e r y d e t a i le d m e a s u r e m e n t s o f m e a n v e l o c i t y a n d t u r b u l e n c e

i n a pa ra l l e l cu rved duc t o f ha l f -w i d t h t o m ean - rad i u s r a t i o o f 1/ 19 . The i r m ean - f l ow res u l t s a r e qu a l i t a t i ve -

l y s i m i l ar t o t h o s e o f W a t t e n d o r f . I t is , h o w e v e r , t h e i r t u r b u l e n c e m e a s u r e m e n t s w h i c h a r e o f c o n s i d e r a b l e

i n t e r e st si n ce a c o m p a r i s o n w i t h t h e s t r a ig h t - c h a n n e l d a t a s h o w s t h a t t h e p r o d u c t i o n o f t u r b u l e n c e a n d t h e

t u r b u l e n c e i n t e n s it ie s a r e l a r g e r n e a r t h e o u t e r c o n c a v e w a l l a n d s m a l l e r n e a r t h e i n n e r c o n v e x w a l l t h a n

t h o s e o c c u r r i n g a t c o r r e s p o n d i n g p o i n t s i n a s t r a i g h t c h a n n e l . T h i s o b s e r v a t i o n is in a g r e e m e n t w i t h t h e

s t ab i l i ty c r i t e r i on fo r cu rved f l ows p ropos e d by R ay l e i gh~ 9 . Ac co rd i n g t o t h i s c r i t e r i on , wh i ch i s exp l a i ne d

i n s o m e d e t a i l b y W a t t e n d o r f , t h e m o t i o n o f a f l u i d w i t h m e a n t a n g e n t i a l v e l o c it y U a t a d i s t a n c e r f r o m

t h e c e n t r e o f c u r v a t u r e o f th e s t r e am l i n e , i s s t a b le i f t h e r a d i a l g r a d i e n t o f t h e p r o d u c t Ur) i s pos i t i ve an d

u n s t a b l e i f i t is n e g a ti v e . T h u s , i n t h e c a s e o f t h e c u r v e d c h a n n e l t h e f l o w in t h e n e i g h b o u r h o o d o f th e

conc ave s u r f ace is expec ted t o be un s t ab l e an d s how a m p l i f i ca t i on in t he l eve l o f t u rbu l ence . C onver s e l y ,

n e a r t h e c o n v e x s u r fa c e t h e f l ow i s e x p e c t e d t o b e s t a b l e a n d t u r b u l e n c e w i l l b e d a m p e d . F u r t h e r e x p e ri -

m e n t s a r e n e e d e d t o c o n f i r m t h a t t h e s e o b s e r v a t i o n s a p p l y a l s o to b o u n d a r y l a y e r s d e v e l o p in g o n c u r v e d

surfaces .

2.2. Curved Jets and Entrainment.

T u r b u l e n c e m e a s u r e m e n t s i n c u r v e d m i x i n g l ay e r s a n d je t s b l o w i n g t a n g e n t i a l l y o n c u r v e d s u r fa c e s

h a v e b e e n m a d e b y M a r g o l i s 1 . H i s r e s ul t s a re i n a g r e e m e n t w i t h t h e o b s e r v a t i o n s o f E s k i n a z i a n d Y e h .

H e a l s o f o u n d t h a t t h e t u r b u l e n c e i s a m p l if i e d in r e g i o n s w h e r e t h e g r a d i e n t o f Ur) i nd i ca t e s i n s t ab i l i t y

a n d d a m p e d i n r e g i o n s w h e r e t h e g r a d i e n t o f ( U r ) i n d i c a t e s s t a b il it y . P e r h a p s t h e m o s t i m p o r t a n t o b s e r v -

a t i o n w h i c h i s o f i m m e d i a t e i n t e r e s t in t h e s t u d y o f c u r v e d t u r b u l e n t b o u n d a r y l a y e r s is t h a t t h e i n v e s t i ga -

t i o n s i n c u r v e d j e t s m a d e b y N e w m a n 14, F e k e t e 6 , S a w y e r 21, S t r a t f o r d , J a w o r a n d G o l e s w o r t h y 26, M ar-

go l i s ~ , G u i t t on 8 and o t he r s , a l l s ugges t t ha t a je t on a convex wa l l g rows m ore r ap i d l y ( and t h a t on a

conca ve s u r f ace g rows l e s s r ap i d l y ) t han a co r r es po nd i n g j e t on a f l a t s u r f ace . Th i s r e s u l t was t o be expec t ed

f r o m t h e t u r b u l e n c e m e a s u r e m e n t s o f M a r g o l i s a n d a l s o f r o m R a y l e i g h s st a b i li t y c r i te r i o n . I n t h e c o n v e x

w a l l j e t t h e r e i s a l a r g e o u t e r r e g i o n w h i c h i s u n s t a b le a n d t h e r e f o re t h e t u r b u l e n t m o v e m e n t s a r e a m p l i fi e d ,

t h e e n t r a i n m e n t i s in c r e a s e d a n d t h e j e t s p r e a d s m o r e r a p i d ly . T h e c o n v e r s e i s t r u e f o r a c o n c a v e w a l l j e t.

I f t h e se i d e a s a r e e x t e n d e d t o b o u n d a r y - l a y e r f l o w t h e y l e a d u s t o e x p e c t i n s t a b i li t y a n d i n c r e a s e d e n t r a i n -

m e n t i n a b o u n d a r y l a y e r w i t h c o n c a v e s t r e a m l i n e s , a n d s t a b i l i t y a n d r e d u c t i o n i n e n t r a i n m e n t i n a

b o u n d a r y l a y e r w i t h c o n v e x s t r e a m l i n e s . T h i s o b s e r v a t i o n a l s o n e e d s f u r t h e r e x a m i n a t i o n i n f u t u r e

exper i m en t s .

T h e t h e o r e t ic a l s t u d i e s i n w a l l j e t s a r e r e s t r i c te d o n l y t o t h e p r e d i c t i o n o f m e a n f l o w p r o p e r ti e s a n d


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again the simplifying assumption of negligible static-pressure variation across the jet is made. This,

however, is one case of curve~t flow in which such an assumption may be justified since the centrifugal

force on the fluid elements within the jet decreases due to the decrease in velocity with distance from the

surface.

So far we have considered only curved wall jets developing in the absence of an external stream. It will

be clear that quite different considerations of stability and amplification of turbulence will apply when

an external st ream is present since markedly different distributions of Ur) may occur depending on the

magnitude of the stream velocity.

2.3. Turbulent Boundary Layer on Curved Surfaces.

The well known experiments of von Doenhoff and Tetervin 3, Schubauer and Klebanoff23 and Strat-

ford 25, which were primar ily devised to investigate the effects of pressure gradient on the development of a

turbulen t boundary layer, employed curved surfaces to produce the desired pressure distributions. Prior

to these experiments, Wilcken 29 and Schmidbauer22 had made measurements specifically to study the

effects of surface curvature. The ratio of boundary-layer thickness ~ (defined as the distance from the

surface at which U/U1 = 0 995) to the radius of curvature of the surface R did not exceed 0 025 in any of

these experiments. With the exception of Wilcken's measurements it is found that the effects of curvature

in these measurements are overshadowed by the rather large pressure gradients Which were present.

Wilcken, however, did find that the boundary layer on a convex surface grew at a slower rate, and that on

a concave surface at a faster rate, than on a corresponding flat surface. This result is in accordance with

the observations in curved channels and wall jets on curved surfaces noted previously.

Thompson 27 was perhaps the first author to include the influence of surface curvature in a calculat ion

method for turbulent boundary layers. F rom an examination of the various experimental investigations

he concluded that even though the curvature in these experiments was not large enough to cause sub-

stantial changes in the velocity profiles calculated by assuming negligible static-pressure variation (and

hence in the evaluation of the usual integral parameters such as the displacement and momentum thick-

nesses) the streamwise growth of the shape factor H and the momentum-thickness Reynolds number

Rodid show a consistent and fairly marked dependence on surface curvature. Thompson a ttribu ted this

dependence to the effects of streamline curvature on entrainment and proceeded to refine his entrainment

equation by including an empirical factor which was assumed to be a simple function of the ratio

6/R.

The

inclusion of this curvature term improved the agreement between experiments and the predictions of his

calculation method. While the actual procedure followed by Thompson was neither elegant nor wholly

staisfactory, his results suggested the important conclusion that the primary influence of streamline

curvature was not so much on the shape of the velocity profiles, skin-friction relation, the momentum

integral equation and the usual assumption of negligible variation in static pressure across the boundary

layer, as on the detailed turbulent motions and the entrainment of freestream fluid into the layer. It was

for this reason that the preliminary experiments reported here were undertaken. The main aim of these

experiments was to establish a suitable empirical correlation between the entrainment and a curvature

parameter such as

6/R

so tha t the then existing entrainment methods of Head 9 and Thompson 27 could be

improved to take account of surface curvature. It will be seen later that this aim was not realised due

partly to the fact that the effects of curvature were grossly underestimated and partly to the difficulty of

satisfactorily isolating the influence of curvature from that of pressure gradients.

In a recent paper Bradshaw 1 has drawn at tention to the analogy between meteorological parameters,

such as the Richardson number, and the parameters describing the effect of streamline curvature on

turbulen t flow. Using this analogy, in conjunction with existing meteorological data obtained from the

earth's boundary layer, Bradshaw has shown that the effects of curvature on the turbulence, and in

particular on the apparent mixing length, are appreciable if the boundary-layer thickness is greater than

roughly 1/300 of the radius of curvature. This limit on the ratio 6/R suggests that we may expect quite

substantial curvature effects in the experiments of the various authors mentioned above. Furthermore, it

leads to the conclusion that the influence of curvature can be large even when the static-pressure variation

across the bounda ry layer is negligibly small. Bradshaw has incorporated the modified distribution of

5


7/33

a p p a r e n t m i x i n g l e n g t h , o r m o r e p r e c is e ly , th e m o d i f i e d d i s t r i b u t i o n o f h i s d i s s i p a t io n l e n g t h p a r a m e t e r

L ~, i n t h e B r a d s h a w , F e r r i s a n d A t w e l l 2 c a l c u l a t i o n m e t h o d a n d r e c a l c u l a t e d t h e b o u n d a r y - l a y e r d e v e l o p -

m e n t s m e a s u r e d b y S c h m i d b a u e r a n d S c h u b a u e r a n d K l e b a n o ff . T h e i n c lu s io n o f th e c u r v a t u r e

m o d i f i c a t i o n l e a d s t o a m u c h b e t t e r a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l d e v e l o p m e n t s o f 6 a n d C ~ a n d

t h e c a l c u l a ti o n s . T h e d e v e l o p m e n t o f Ro r e m a i n s m o r e o r l e ss u n a ff e c t e d . B r a d s h a w ' s c a l c u l a t i o n s h a v e

t h e r e f o r e s h o w n q u i t e c o n c l u s i v e l y t h a t t h e i n f l u e n c e o f s u r fa c e c u r v a t u r e c a n b e q u i t e s i g n if i c a n t e v e n i n

e x p e r i m e n t s w h e r e c u r v e d s u r fa c e s a r e e m p l o y e d p r i m a r i l y t o p r o d u c e t h e d e s i r e d p r e s su r e d i s t r i b u t i o n s

a n d i n c a s e s w h e r e t h e s t a t i c - p r e s s u r e v a r i a t i o n t h r o u g h t h e b o u n d a r y l a y e r c a n s t i l l b e r e g a r d e d a s

negl ig ib le .

2.4.

General Comments on the Review of Literature

F r o m t h e s u r v e y o f p r e v i o u s w o r k o u t l i n e d i n t h e l a s t th r e e S e c t io n s i t w i ll b e c le a r t h a t p r i o r t o t h e

w o r k o f B r a d s h a w , w h i c h w a s p u b l i s h e d a f t e r t h e p r e s e n t w o r k h a d b e e n c o n c l u d e d , t h e i n f o r m a t i o n

a v a i la b l e o n t h e e f fe c ts o f c u r v a t u r e o n t u r b u l e n t b o u n d a r y l a y e r s w a s v e ry s k e t c h y . I n p a r t i c u l a r i t w a s

n o t k n o w n w h a t c o n s t i t u t e d la r g e c u r v a t u r e a n d h o w t h e s ta t i c - p re s s u r e v a r i a t i o n t h r o u g h t h e b o u n d a r y

l a y e r a f fe c t e d t h e v a r i o u s i n t e g r a l a n d d i f fe r e n t ia l e q u a t i o n s w h i c h a r e g e n e r a l l y u s e d t o p r e d i c t t h e

s tr e a m w is e d e v e l o p m e n t o f th e b o u n d a r y l ay e r. F r o m t h e t h e o r e ti c a l w o r k o f M u r p h y 11, S c h u l t z - G r u n o w

a n d B r e u e r 2 4, N a r a s i m h a a n d O j h a 12, a n d o t h e r s o n l a m i n a r b o u n d a r y l a y e r s o n c u r v e d s u r f a c e s it

a p p e a r e d t h a t q u i t e s i g n i f ic a n t c u r v a t u r e e f fe c ts c a n b e e x p e c t e d w h e n t h e r a t i o o f b o u n d a r y - l a y e r t h i c k -

n e s s t o r a d i u s o f c u r v a t u r e t a k e s v a l u e s o f t h e o r d e r o f 0 .0 5. T h e r e c e n t w o r k o f B r a d s h a w h a s , t o s o m e

e x t e n t , n a r r o w e d t h e g a p b u t a c c o r d i n g t o h i m c u r v a t u r e e f f e c ts b e c o m e s i g n i f ic a n t e v e n w h e n t h e r a t i o

b/R i s a s s m a l l a s 1 / 300 and t he s t a t i c -p res s u re va r i a t i on i s s t i l l n eg l i g i b l e . Th i s t he re fo re l eads t o t he

c o n c l u si o n t h a t w h e n 6/R i s m u ch l a rge r t h an 1 / 300 we m a y expec t qu i t e s eve re and f i rs t o rde r e f f ec ts o f

c u r v a t u r e a n d t h a t w e m a y h a v e t o r e v i se o u r e q u a t i o n s t o i n c l u d e t h e s t a t ic - p r e s su r e v a r i a t i o n a c r o s s t h e

b o u n d a r y l a y e r . I n t h e e x p e r i m e n t s t o b e d e s c r i b e d

6/R

t a k e s v a l u e s a s h i g h a s 0.1 a n d i t i s d e m o n s t r a t e d

t h a t f o r s u c h s ev e r e c u r v a t u r e s , w h i c h a r e n o t a l l t h a t u n c o m m o n i n p r a c t i c a l a e r o d y n a m i c s it u a t io n s , w e

n e e d t o r e c o n s i d e r a l m o s t a l l th e e q u a t i o n s w h i c h a r e g e n e r a l l y u s e d i n t h e a n a l y s i s o f tw o - d i m e n s i o n a l

t u r b u l e n t b o u n d a r y l a y e r s .

3 Experiments on a Circular Cylinder

T h e f ir s t s e t o f e x p e r i m e n t s w a s p e r f o r m e d o n t h e b o u n d a r y l a y e r d e v e l o p i n g o n a 6 i n. d i a m e t e r

c i r cu l a r cy l i nde r . Th i s s e t -up w as chos en m a i n l y fo r it s s im p l i c i t y . A 4 f t. l ong cy l i nde r was m ou n t e d

ver t i ca l l y i n t he 4 f t. x 5 f t. wo rk i ng s ec t i on o f t he C am br i d ge U n i ve r s i t y Eng i n ee r i ng D ep ar t m en t ' s

l o w - s p e e d w i n d t u n n e l . T h e m a x i m u m t u n n e l s p e e d w a s i n t h e r e g i o n o f 1 8 0 f t. /s e c. T h e s t a t ic p r e s s u r e

o n t h e c y l i n d e r su r f a ce w a s m e a s u r e d b y m e a n s o f a r o w o f s t a ti c t a p p i n g s s p a c e d a t 5 de g . in t e r v a ls a t m i d -

s pan . T ra ns i t i on o f t he bo un da ry l aye r was f i xed by m ean s o f two 0 .040 i n . d i am et e r w i res a t ~b = + 45

f r o m t h e f o r w a r d s t a g n a t i o n l i n e . A t y p i c a l p r e s su r e d i s t r i b u t i o n w i t h t h e t r i p w i re s i n p o s i t io n i s s h o w n

i n F i gu re 2 .

T h e t o p 1 f o o t l e n g t h o f t h e c y l i n d e r w a s s p r a y e d w i t h c h i n a - c la y t o i n v e s ti g a t e t r a n s i t i o n a n d s e p a r a t i o n

p o s i ti o n s . T r a n s i t i o n w a s f o u n d t o b e c o m p l e t e i m m e d i a t e l y b e h i n d t h e t r i p w i re s a n d s e p a r a t i o n , a s

de t ec t ed by t he ev apo ra t i on o f pa ra f f i n un i fo rm l y s p read on t he ch i na -c l ay , occu r re d a t ~b = 110 . The

p o s i t i o n o f t h e s e p a r a t i o n p o i n t w a s c o n s t a n t a l o n g t h e 1 f o o t l e n g th e x c e p t v e r y c lo s e t o t h e t u n n e l w a l l,

a n d n o n o t i c e a b l e c h a n g e i n t h i s p o s i t i o n o c c u r r e d o v e r t h e R e y n o l d s - n u m b e r r a n g e in w h i c h t h e

b o u n d a r y - l a y e r d e v e l o p m e n t s r e p o r t e d b e l ow w e r e m e a s u r e d .

T h e t r a v e rs e m e c h a n i s m u s e d t o m e a s u r e t h e t o t a l- p r e s s u r e p r o f il e s c o n s i s te d o f a s te m w i t h a s t r e a m -

l i ned l eg s p r i ng - l oaded on t o t he cy l i nde r s u r f ace . In s i de t h i s s t em was f i t t ed a m i c rom et e r s c rew wh i ch

c a r r i e d a f l a t te n e d p i t o t w i t h a m o u t h 0 .0 07 6 i n. d ee p . T h e s t e m w a s s u p p o r t e d f r o m a v e r t ic a l s t re a m l i n e d

s t r u t s p a n n i n g t h e t u n n e l a p p r o x i m a t e l y 1 5 i n. a w a y f r o m t h e c y l in d e r . T h e s p r i n g - l o a d e d l eg o n t h e

c y l i n d e r w a s k e p t 2 in . a w a y f r o m t h e c i r c u m f e r e n t ia l li n e a l o n g w h i c h t h e d e v e l o p m e n t o f t h e b o u n d a r y

l a y e r w a s m e a s u r e d . T h u s t h e i n t e r f e r e n c e f r o m t h e s t r u t a n d t h e s t e m w a s n e g l i g i b l e . T h e m i c r o m e t e r

6


8/33

screw and the pi tot were actuated in 0.001 in. steps from outside the tunnel by means of a flexible cable

drive.

3.1. Measurements.

Total pressure traverses were made a t six circumferential stations (q~ = 60, 70, 80, 90, 95 and 100)

at mid-span for two cylinder Reynolds numbers Re = Uo~ D/v = 3.53 x 105 and 5 01 x 10 s, where

Uoo is the undisturbed free-stream velocity, D is the cylinder diameter and v the kinematic viscosity).

Because of the rather small thickness encountered it was not possible to measure the static-pressure

variation through the boundary layer directly. Therefore the velocity profiles and corresponding integral

parameters 6* and 0 were all evaluated assuming that the static pressure remained constant through the

boundary layer and equal to its value measured at the surface of the cylinder. The justifications for making

this assumption are discussed later on. Velocity profiles at the higher Reynolds number are shown in

Figure 3 and the corresponding H and R odevelopments and U1 (the velocity at the edge of the boundary

layer) variation are shown in Figure 4. Figure 5 shows the H, R o and U1 values at the lower Reynolds

number.

3.2. Ro Development.

Using the measured value of Ro at q~ = 60, the measured H distribution, Thompson's 2v skin-friction

charts and the usual flat-surface momentum-integral equation,

dRo b H+l) RdU _ U1 C I

dx U1 d~ v 2 ' (1)

the subsequent development of R o has been predicted by quadrature for both Reynolds numbers. These

predictions are compared with the experimental values in Figures 4 and 5. The agreement between the

two can be considered satisfactory even though the predicted values tend to be slightly higher than the

observed ones. This agreement can be interpreted in at least two ways: (a) if the flow is assumed to be

two-dimensional then equat ion (1) can be considered satisfactory, and (b) if the validity of equation (1) is

accepted then the flow can be considered two-dimensional. At this stage, therefore, the experimental

evidence for or against the validi ty of the flat-surface momentum- integral equation cannot be considered

conclusive. This is particular ly so since we have taken no account of the static pressure variation through

the boundary layer even though the surface curvature is quite large (0-030 < 6/R < 0.051) compared

with the previous measurements of Schmidbauer and others. The procedure followed here does, however,

illustrate what conclusions may be drawn from using the standard approach of neglecting curvature

effects. The s tatic-pressure variation has no t been considered in this first experiment for several reasons:

(1) Such variations could not be measured directly owing to the very thin boundary layer.

(2) No reliable estimate for the curvature of the streamlines could be obtained. It will be clear that the

curvature of the streamlines is not simply related to that of the surface owing to the rather severe pressure

gradient effects. In fact, for ~b > 90 the rapid growth of the layer towards separation cancels, to some

extent, the curvature imposed by the surface. This may indeed lead to negligible pressure varia tion across

the boundary layer.

(3) Thompson's reanalysis of Stratford'szs measurements had shown that the inclusion of pressure

var iat ion led to differences in the values of 6* and 0 only of the order of 3 per cent. Thus, it was thought

that the direct effect on H and R o due to the evaluation of velocities using the wall static pressure will

be negligible.

3.3. H Development, Velocity Profiles and Entrainment Considerations.

Head 9 suggested a simple method of calculating H development based on considerations of the entrain-

ment of freestream fluid into the layer. IfQ is the quantity of fluid within a unit width of the layer, the rate

of entrainment is


9/33

u d y = u ~ 6 - 6 . ) } = v H * R o ) ,

2 )

dx 'dx o

where

H* = (6-6 ) /0.

Head proposed that the rate of entrainment is a function of the local velocity

profile shape, the external velocity U1 and some measure of the boundary-layer thickness. Assumption

of a one-parameter family of profiles allowed the choice of either H* or H as the parameter specifying the

profile shape, and the quantity (3 -6 ) was conveniently chosen as the measure of the boundary-layer

thickness. Thus,

d { U ~ ( 6 - 6 ) } =

f ( H *

o r

H , 6 - 6 , U ~ )

dx

1 d { U I ( 6- 6 * ) } , = F ( H * )

U1 dx

3 )

and H* =

G(H).

(4)

Functions F and G were obtained by Head from the experimental da ta of Newman 13 and Schubauer and

Klebanoff23. Thompson has applied Head's method as outlined here to boundary layer developments

measured by several authors and found it to be generally satisfactory provided three-dimensional ity and

curvature effect are absent. He also put forward a more refined calculation method, also based on the

entrainment process, and made empirical allowances for curvature and three-dimensionali ty. To obtain

agreement between the measurements of Schubauer and Klebanoff, Schmidbauer and von Doenhoff and

Tetervin and his new auxiliary equation he found the need to decrease the entrainment for convex surfaces

and to increase it for concave surfaces. The ratio

6/R

was used as the curvature parameter in these con-

siderations.

For the present experiments, the measured

R o

values and the value of H at ~ = 60 have been used in

Head's method to predict the subsequent development of H on the cylinder. Comparison of these calcula-

tions with the measured developments, as shown in Figures 4 and 5, indicates that the measured values

are consistently higher than the predicted ones. From Head's curves of F and G, reproduced in Figure 6,

it is seen that higher H values indicate a reduction in the rate of entrainment. Thus, the present results

can be interpreted as confirmation of the trends observed by Thompson and also, indirectly, the expect-

ations from the experiments on wall jets and fully-developed flows. As mentioned earlier, convex curvature

is expected to suppress the turbulence and therefore reduce the amount of fluid entrained by the large

eddies in the outer region of the boundary layer.

The cylinder results are also in agreement with the conclusions of Bradshaw 1 who showed that his flat

surface calculation method consistently underestimates the development of H in the convex surface

measurements of Schmidbauer22 and Schubauer and Klebanoff23.

The dependence of entrainment and the development of H on curvature had in fact been suspected at

the start of the present series of experiments. Examination of Head's F versus H* relationship (Figure 6a)

shows tha t the points corresponding to the convex surface in Schubauer and Klebanoff's measurements

lie below the mean line drawn by Head, indicating a decrease in entrainment, while Newman's results

lie above the mean line, indicating an increase in entrainment due to the concave curvature of the stream-

lines associated with the rapid growth of the boundary layer towards separation on a flat surface. Initially,

therefore, it was thought that Head's calculation method could be extended to take account of curvature

effects if the dependence of the entrainment function F on a suitable curvature parameter could be

established by experiment. The use of 6/R as this parameter cannot really be justified simply because the

increase or decrease of entrainment (which results from an increase or decrease in mixing due to the

centrifugal forces) depends on the curvature of the mean flow streamlines and not that of the surface.

As was mentioned previously the assumption of nearly-constant streamline curvature through the

boundary layer could lead to serious errors especially in regions where the rate of growth of the layer is

large.


10/33


11/33

1 f t. w id e a n d 5 f t. h i g h . A 9 0 d e g r e e c u r v e d d u c t o f t h e s a m e 5 ft . x 1 f t. c r o s s - s e c t i o n a n d i n n e r - w a l l

r a d i u s o f 2 ft . w a s f i tt e d o n t h e e n d o f t h e s t r a i g h t d u c t . C a r e w a s t a k e n t o e n s u r e t h a t t h e j o i n b e t w e e n t h e

t u n n e l a n d t h e c u r v e d s e c t i o n w a s s m o o t h a n d f r e e f r o m l e a k s. S a t i c - p r e s s u r e t a p p i n g s w e r e p r o v i d e d

a l o n g t h e m i d - s p a n o f t h e c o n v e x s u r f a c e a n d t h e s t r a i g h t w a l l o f t h e t u n n e l c o n t i n u o u s w i t h it . T h e

b o u n d a r y - l a y e r m e a s u r e m e n t s d e s c r ib e d b e l o w w e r e m a d e o n t h es e s u rf ac e s.

T r a n s i t i o n w a s p r o m o t e d a t t h e e n d o f t h e t u n n e l c o n t r a c t i o n b y m e a n s o f a 3 i n. w i d e g l a s s - p a p e r s t r ip

a n d a i n. d i a m e t e r t r i p w i r e ( th e l a t t e r b e i n g i n t r o d u c e d p r i m a r i l y t o t h i c k e n t h e b o u n d a r y l a y e r ) o n t h e

t e s t s i d e a n d o n ly t h e g l a s s - p a p e r s t r i p o n t h e o p p o s i t e s i d e .

B o t h t h e s t a t i c p r o b e a n d t h e t o t a l - h e a d t u b e w e r e m a d e o f 0 . 04 0 i n. d i a m e t e r h y p o d e r m i c t u b i n g a n d

w e r e m o u n t e d o n n . d i a m e t e r s t e m s a t t a c h e d t o t h e t r a v e r s e g e a r s it u a t e d o u t s i d e t h e t u n n e l T h e m o u t h

o f th e t o t a l - h e a d t u b e w a s f l a t t e n e d t o a n o v e r a l l h e i g h t o f 0 0 0 8 i n . T h e t r a v e r s e g e a r c o n s i s t e d o f a d i a l-

g a u g e c a p a b l e o f re g i s t e r in g d i s t a n c e s u p t o 2 2 i n . w i t h i n 0 .0 01 i n . O n e e n d o f t h e g a u g e s p i n d l e c a r r i e d

t h e p i t o t ( o r s ta t ic ) s t e m a n d t h e o t h e r w a s m o v e d b y m e a n s o f a s c r e w - t h r e a d .

4.3.

Measurements.

T o t a l a n d s t a t i c - p r e s s u r e t r a v e r s e s w e r e m a d e o n t h e t e s t w a l l b o u n d a r y l a y e r a t 1 6 s t a t i o n s s i t u a t e d

o n t h e t u n n e l c e n t r e li n e . T h e R e y n o l d s n u m b e r p e r f o o t a t S t a t i o n 1 ( 42 in . d o w n s t r e a m o f t r a n s i t io n ) w a s

k e p t c o n s t a n t a n d e q u a l t o 2 . 4 2 x l 0 s . T h e p o s i t i o n s o f t h e m e a s u r i n g s t a t i o n s a n d t h e s t a t i c - p r e s s u r e

v a r i a t i o n a l o n g t h e t e s t w a l l a r e s h o w n i n F i g u r e 8 , a n d f u r t h e r d e t a i ls a r e g i v e n i n T a b l e s 1 a n d 2 . T h e

f a v o u r a b l e p r e s s u r e g r a d i e n t a t t h e j o i n b e t w e e n t h e f l at a n d c o n v e x s u r f a c es is a ss o c i a t e d w i t h t h e c h a n g e

i n c u r v a t u r e o f t h e s t r e a m l i n e s a n d i s t h e r e f o r e u n a v o i d a b l e . A n a d v e r s e g r a d i e n t o f s i m i l a r m a g n i t u d e

e x i st s o n t h e o u t e r s u r fa c e . C o n v e r s e l y , a n a d v e r s e g r a d i e n t o n t h e c o n v e x w a l l a n d a f a v o u r a b l e g r a d i e n t

o n t h e c o n c a v e w a l l e x is t a t t h e e x i t f r o m t h e c u r v e d c h a n n e l . T h e m a g n i t u d e s a n d t h e e f fe c ts o f t h e s e

p r e s s u r e g r a d i e n t s a r e d i s c u s s e d a t a l a t e r s t a g e .

A s a t y p i ca l e x a m p l e o f t h e m e a s u r e m e n t s o n t h e c o n v e x s u r f ac e , F i g u r e 9 s h o w s t h e t o t a l a n d s t a t ic -

p r e s s u r e v a r i a t i o n s t h r o u g h t h e b o u n d a r y l a y e r a t S t a t i o n 1 4 ( w h e r e

x - x o

= 92.25 in. and q5 = 4 4.3) ,

a n d t h e c o r r e s p o n d i n g v e l o c i t y p r o f i l e i s s h o w n i n F i g u r e 1 0 .

4.4.

Evaluation o f b, 6 and O.

F r o m t h e t y p i c a l v e l o c i t y p r o f i l e s h o w n i n F i g u r e 1 0, i t w il l b e c le a r t h a t t h e v e l o c i t y i n t h e p o t e n t i a l

f l o w o u t s i d e t h e b o u n d a r y l a y e r i s n o t c o n s t a n t w i t h d i s t a n c e f r o m t h e s u r f ac e . F o r t h i s r e a s o n w e c a n n o t

o b t a i n fi, ~ * a n d 0 i n t h e u s u a l w a y . T o c a l c u l a t e m e a n i n g f u l v a l u e s o f t h e s e q u a n t i t i e s w e n e e d t o c o n s i d e r

t h e d i ff e r e n c e b e t w e e n t h e m e a s u r e d v e l o c i t y d i s t r i b u t i o n a n d t h e v e l o c i ty d i s t r i b u t i o n o b t a i n i n g u n d e r

i d e n t i c a l c o n d i t i o n s i f t h e f l o w w e r e p o t e n t i a l .

U s i n g a c u r v i l i n e a r c o - o r d i n a t e s y s t e m f o r m e d b y t h e d i s t a n c e s a l o n g ( x) a n d n o r m a l t o ( y) a w a l l o f

r a d i u s o f c u r v a t u r e R ( R b e i n g p o s i t i v e fo r c o n v e x c u r v a t u r e ) i t is e as i ly s h o w n t h a t , f o r t w o - d i m e n s i o n a l

f lo w , th e c o n d i t i o n f o r z e r o v o r t i c i t y in t h e p o t e n t i a l f l o w f o r

(y /R) 2

< < 1 c a n b e w r i t t e n

Uv 4 oUr R ~Vv

R + y Oy R + y ~ x - 0

(5)

S u f fi x p d e n o t e s p o t e n t i a l f l o w a n d U a n d V a r e th e v e l o c i t y c o m p o n e n t s i n t h e x a n d y d i r e c t i o n s

r e s p e c ti v e l y . B y t a k i n g e x a m p l e s s u c h a s t h e f l o w p a s t a c i r c u l a r c y l i n d e r it c a n b e s h o w n t h a t t h e t h i r d

t e r m i n t h i s e q u a t i o n i s sm a l l p r o v i d e d t e r m s o f t h e o r d e r (y /R) 2 c a n b e n e g l e c te d i n c o m p a r i s o n w i t h

u n i t y . T h u s , t h e p o t e n t i a l - f l o w v e l o c i t y f ie l d c a n b e a p p r o x i m a t e d b y

OUp Up

- O, (6)

O y ~ R + y

i.e. Uv(1 + y/R ) = Uvw ,

(7)

10


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where w denotes the value at the wall (i.e. at y = 0). The velocity distribution in a boundary layer on a

curved surface must therefore approach Up at its outer edge in the manner shown in Figure 10.

We now adopt the conventional definitions of 6, 6* and 0. The boundary-layer thickness 6 is defined

as the value of y at which

U/Up

= 0.995.6* and 0 represent the mass and momentum flux defects within

the boundary layer, and are therefore defined by the relations.

f 6 f , J

Updy = (U p- U)dy

(8)

0 0

and

f o U ~dy f '~

U(U p- U)dy . (9)

Substitution of Up from equation (7) then leads to

f ( u p - v )

In (1 + 6*/R) = dy (8.1)

0 U p w

0 f

nd

I + O / R - o-U--~w,, -O~p,~ dy.

(9.1)

So far in the analysis we have retained terms of the order 6/R. But since from flat surface boundary layers

we expect 6*/6 and 0/6 to be of the order of 1/5 to 1/10, a* /R and O/R will be small in comparison with

unity provided

6/R

does not exceed 1/10. We can therefore approximate equations (8.1) and (9.1) by

Ups, ) dy

8 . 2 )

and

o = f '~ u@~ U p - U ) dyp,

(9.2)

In the present experiments, the distribut ion of UJUpwwas first found using equation (7) in conjunction

with the observed velocity at some y greater than 6. The values of 6* and 0 were then computed from the

measured profiles with the use of equations (8.2) and (9.2). For the measurements on the flat surface

6 and 0 were obta ined in the usual way by assuming Up =

Upw

= U1, the free-stream velocity. The

entire development of momentum-thickness Reynolds number R o (defined as U~O/v)and the shape

parameter H (= 6 /0) is shown in Figure 11.

It will be recalled that the values of 6* and 0 quoted earlier for the circular cylinder measurements were

not evaluated in the manner described above. To assess the effects of assuming constant static pressure

through the boundary layer in those results, the measurements on the present convex surface have been

reanalysed using only the wall static pressure and the total-pressure profiles. This procedure was found

to give values of 6* and 0 which are approximately 10 per cent and 12 per cent, respectively, LOWER

than the values computed using the correct definitions of equations (8.2) and (9.2). The corresponding

values of Uo/vwere about 4 per cent HI GH ER than the correct ones and therefore the net result was an

overestimation in

Ro

of about 8 per cent and an underestimat ion in H of about 3 per cent.

These calculations suggest that the H development in the cylinder experiments would remain sub-

stantially unaltered but the measurements of Ro, if corrected for static-pressure variation, would lie below

the values shown in Figures 4 and 5. The correction will of course be smaller than the 8 per cent in the

channel results due to the reduction in streamline curvature caused by the rather strong adverse pressure

gradients on the rear of the cylinder.

11


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4.5. Comparison of Meas ured R o and H with Flat-Surface Predictions.

Following the procedure outlined in Sections 3.2 and 3.3 for the cylinder experiments, two calculations

have been performed: (a) the flat-surface momentum-integral equat ion (1), the Ludwieg-Tillmann

skit friction formula and the measured H distribution have been used to predict the growth of Ro down-

stream of the first measuring station, and (b) the measured Ro has been used along with Head's F and G

curves to predict the development of H. The results of these calculations are compared with the measure-

ments in Figure 11. It will be seen that the flat surface predictions are in good agreement with exper iment

only up to ( x- x0 ) = 50 in., and tha t thereafter the momentum integral equation gives

R o

values which

are too high while Head's method considerably underestimates H. We also notice that the favourable

pressure gradient which, as seen from Figure 8, exists from (X-Xo) = 50 in. to 85 in. is not large enough

to account for the sudden drop of R o in this region.

Thompson 27 and others have suggested that any disagreement between R o predicted by using the flat

surface momentum integral equation and the measured R o can be explained by the existence of three-

dimensionality. Following Thompson, then, the rather low values of

R o

measured on the convex surface

lead to the conclusion that the flow is divergent. This, however, is contrary to what will be expected from

considerations of secondary flows within the curved duct. The radial centrifugal pressure gradient will

give rise to inward (i.e. from the concave to the convex surface) cross-flows within the boundary layers

on the roof and the floor of the channel and these in turn lead to cross-flows towards the outer corners

on the concave surface and away from the inner corners on the convex surface. Thus, the flow on the

centreline of the convex surface, if at all influenced by the inner corners, is expected to be

convergent

leading to measured values of Ro higher than those predicted by the momentum integral equation. In the

present experiments no check on the two-dimensionali ty of the flow has been made but later experiments

with a 180 degree channel (of the same cross-section and mean radius) which are described in a separate

paper by the au tho r I v have substantiated the cross-flow picture just outlined and in particular shown

that the flow in the region of the centreline of the convex wall is fairly two-dimensional for at least the

first 80 degrees. The disagreement in R o development of the present experiments cannot therefore be

attributed to three-dimensionality.

The above considerations are of course based on the assumption that the usual momentum integral

equation, equation (1), is valid for two-dimensional flow in the present situation. The rather gross dis-

agreement in the Ro development therefore leads one to question the validity of this equation. Newman 13

has in fact observed that the static-pressure variation through a rapidly growing boundary layer can

make a significant contribution to the growth of R o. By integrating the equations of motion on a flat

surface and assuming p = p(x,y) within the boundary layer, he has shown that the integrated momentum

equation can be written

dO 0 dU1 Zw 1 d ( ~

- - - J p ~ - p ) d y , 1 0 )

x~' (H+ Z) u a dx pU 2 pU~ dx o

where po is the static pressure at y = 6. This equation is applicable only if the pressure remains constan t

with y outside the boundary layer. While this condition is not strictly satisfied in the present experiments

on the convex surface, it is possible to est imate the influence of the additional pressure variation term for

the flat surface boundary layer between (x-Xo) = 55 in. and 73.75 in. Since the radius of curvature of the

streamlines (and hence the centrifugal pressure field) changes very abruptly in this region, the correction

term, which is a rate of change of the integrated pressure force across the boundary layer, is large in

comparison with both the pressure gradient and the skin-friction terms in the momentum integral

equation. Figure 12 shows the measured static-pressure distributions at Stations 10 to 16 and Figure 13

shows the variation of the integral in the last term of equation (10). This latter figure has been used to

obtain plausible estimates of the last term in equation (10) and starting from Station 7 the subsequent

development of R o has been calculated using the same procedure as before. The result of this calculation

is shown in Figure 11. It will be clear that owing to the rather large streamwise distances between the

measuring stations and graphical differentiation of the faired curve in Figure 12, the accuracy of such a

calculation is very small. Nevertheless, the agreement between the calculated and measured development

12


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o f Ro in the region 55 in. < ( X - X o ) < 73 .75 in . i s qu i t e encourag ing and l eads unmis takab ly to the

conc lus ion tha t the u s ua l m om en tum in teg ra l equa t ion , eq ua t ion (1 ), canno t be u s ed no t on ly on f la t

s u r f aces w here s t a t i c p r es s u re does no t r emain cons tan t th rough the boundary l aye r bu t a l s o on heav i ly

curved surfaces.

The deve lop men t o f the s hape f ac to r , H , me as u red on the f l a t and convex s u r f ace i s com pared in

F igu re 11 w i th tha t p r ed ic ted b y us ing H ea d s me thod . A s in the cy l inde r exper imen ts i t is obs e rved tha t

the mea s u red va lues a r e cons ide rab ly h ighe r than the p red ic ted ones on the convex s u rf ace. The ve loc i ty

p r o fi le s m e a s u r e d o n t h e c o n v e x s u r f a ce w h e n c o m p a r e d w i t h T h o m p s o n s t w o - p a r a m e t e r f a m i l y a l s o

ind ica te the s ame t r ends a s thos e obs e rved fo r the cy l inde r (F igu re 3 ) . Th i s s ugges t s the re fo re tha t a

cu rva tu re pa ramete r i s r equ i r ed , in add i t ion to H and

Ro,

to des c r ibe thes e p ro f i l e s comple te ly . F o r

r eas ons men t ioned p rev ious ly the f ac t tha t the meas u red H va lues a r e low er than thos e p red ic ted by

H e ad s m e tho d can no t be r ega rded as s u f f ic i en t ev idence fo r the expec ted r educ t ion in en t ra inmen t .

5. Th e Mo me ntum Integ ral Equation fo r Boundary La yers on a Curved Surface.

The exper imen ta l ev idence o f the l a s t tw o s ec t ions s how s qu i t e conc lus ive ly tha t the u s ua l m om en tu m

in teg ra l equa t ion ,

d ~ + ( H + l ) 0

dU1 zw

u 1 d x p u ~ - O ( 1)

break s dow n w henever the re i s a l a rge va r i a t ion o f s t a t ic p r es s u re ac ros s the bo und ary l aye r . I t has

a l s o been s ho w n tha t N e w m an s m od i f i ed r e la t ion , equa t io n (10 ), can be u s ed w here s t a t ic -p res s u re

va r ia t ion a r i s es on f la t s u rf aces due to s t r eaml ine cu rva tu re caus ed by a r ap id g row th o f the laye r . There

remains , how ever , a need to r econs ide r the bo und ary l aye r on a cu rved s u r f ace w here s ubs tan t i a l s t a t i c -

p res s u re va r i a t ions can occu r w hen the g row th o f the l aye r i s neg l ig ib ly sma ll .

T w o s e p a r a te a t t e m p t s w e r e m a d e t o o b t a i n a m o m e n t u m i n te g r al e q u a t i o n f o r cu r v e d -s u r fa c e b o u n d a r y

laye r s . I n the f i r s t a t t emp t , boundary - laye r equa t ions in cu rv i l inea r co -o rd ina tes ( r e t a in ing f i r s t o rde r

cu rva tu re t e rms ) w ere in teg rated . Th e r e s u l t, how ever , w as a r a the r e l ab o ra te in teg ra l equa t ion in w h ich

t h e v a r i o u s c o r re c t i o n t e r m s c o u l d n o t b e r e d u c e d t o c l o s e d f o r m s a n d h a d t o b e e s t i m a t e d b y m a k i n g

arb i t r a ry a s s um pt ions a bou t the bou ndary - laye r ve loc i ty p ro fi l es . The de ta i ls o f th is equa t ion have been

d e s c r ib e d b y P a t e l 5. I n t h e s e c o n d a n d m o r e f r u it fu l a t t e m p t b o u n d a r y - l a y e r e q u a t i o n s i n p o l a r c o -

o rd ina tes h ave been cons ide red . The de ta i l s o f the de r iva t ion o f the co r r es pond ing in teg ra l equa t io n a r e

s e t ou t in the A ppend ix . There i t i s s how n tha t fo r a s u rf ace o f CO N S T A N T rad ius o f cu rva tu re , R ,

t h e m o m e n t u m ( o r a n g u l a r - m o m e n t u m ) i n te g r al e q u a t i o n c a n b e w r it t e n

d19 l d~ ~R 2

d { ~ 2 r~ ff r

dq~ ~-(A*+219) ddp p - ~ - - - p - ~ ~ 2 P I n - ~ + p rd ,

11)

w h e r e

~ 0 dr

=U lr l , o ) = Ur , 19 = 1 -

A * = 1 -

r

and the s u ff ix 1 r e fe r s to cond i t ions on a l ine o f C O N S T A N T rad ius r 1 imm ed ia te ly ou t s ide the bou nd ary

laye r. I n the o u te r i r ro ta t iona l f low f~ i s cons tan t w i th r an d the re fo re the va lues o f 19 and A * a re inde -

pen den t o f the ou ter l imit of in tegrat ion . 19 and A* are re la ted to the u sual defect th icknesses 0 and (5*

of equat io ns (9) and (8) by

o - ] + ~ / R ) O / R

a n d zX * = , W R . 1 2 )

13


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Usin g these re la t ions i t is easy to show th at th e lef t -hand s ide of equ at ion (11) , whe n pu t equa l to zero ,

reduces ident ical ly to equat ion (1) for suf f ic ient ly smal l values of ~ /R . The cu rva tu re co r r ec t ion t e rm

on the r igh t -hand s ide o f equa t ion (11 ) i s s imi la r to the p res s u re t e rm in t roduced by N ew m an in equa t ion

(10 ) and can r ead i ly be ca lcu la ted e i the r f rom the kno w n s ta t ic -p res s u re p ro f i l es o r f rom the meas u red

veloci ty prof i les us ing the radia l eq ui l ibr ium equ at ion ,

U z 1 ap

r p d r

13)

to re la te the ve loci ty and s ta t ic-pressure f ie lds .

Prof i les of 09/f2 me asured a t the s ix s t reamw ise s ta t ions on the 9 0 degree convex w al l are sho wn in

F igu re 14. Thes e w ere in teg ra ted to ob ta in the exper imen ta l deve lop me n t o f and A * s how n in F igu re

15. To calcula te the de velo pm ent o f a ccord ing to e qua t ion (11) , use has been m ade of (a) r~ = 26 inches ,

(b) the m easured d is t r ibut io n of fR~b) , (c) the m easure d dev elop me nt of A*, (d) the me asured s ta t ic-

pressure prof i les show n in F igure 12 , and (e) the Lud wieg -Ti l lm ann skin-f r ic t ion form ula to re la te Zw/~plU2

to H ( _= A*/) and R o ( =- U 1 R/v) . Equa t ion (11 ) w as in teg ra ted in the u s ua l w ay by a s t ep -by - s tep

p roced ure s t a r ting w i th the exper imen ta l va lue o f a t S ta t ion 11 . The deve lopme n t o f p red ic ted in th i s

man ner i s com pared w i th the exper imen ta l r e s u l t s in F igu re 15. A s imi lar ca lcu la t ion w as pe r fo rmed

us ing

d O 1 d O R 2

d--~ + ( A * + 2 o ) f ~ dq~ p f ~ = 0 (1 4)

in p lace o f equ at ion (11) to s how clear ly the ef fect of the las t term in equ at ion (11) . The resul t of th is

ca lcu la t ion i s a l s o s how n in F igu re 15. Even though the g raph ica l d i f f e ren t i a tion o f exper imen ta l d a ta

s omew ha t r educes the accu racy o f these ca lcu la tions , i t w i ll be s een tha t equa t ion (11 ) g ives qu i t e good

agreemen t w i th the meas u red deve lop men t o f w h i le equa t ion (14 ) g ives va lues o f w h ich a r e ve ry

much h igher than the meas u red ones . The d i f f e r ence be tw een the tw o ca lcu la ted deve lopmen ts s how s

qu i t e conc lus ive ly tha t the s t a t i c -p ress u re va r i a t ion th roug h the boun dary l aye r has a m arked in f luence

on the ove ra l l g row th o f the l aye r . F o r s u r faces o f l a rge cu rva tu re i t is the re fo re neces s a ry to inc lude the

curv ature (and s ta t ic-pressure v ar ia t ion) term s in both th e in tegral as wel l as the d if ferentia l forms of the

boundary - laye r equa t ions .

6. Discuss ion and Conc lus ions .

The p re l imina ry exper imen ts on the c i r cu la r cy l inde r and the 90 deg ree convex w a l l r epo r ted he re

s uggest tha t w e may expec t qu i t e s ubs tan t i a l e f f ec ts o f s tr eaml ine cu rva tu re on the ove ra l l deve lopm en t

o f the bo und ary l aye r w hen the r a t io o f the bound ary - laye r th ickness , 6 , to the r ad ius o f cu rva tu re o f the

surface , R, is in the region of 1 /30 to 1 /10 . The values of 6 / R occu r r ing in thes e exper imen ts a r e s om ew h a t

h igher than thos e encoun te red in p rev ious meas u remen ts and ve ry muc h h igher than the 1 /300 quo ted b y

Brads ha w 1 as be ing the h ighest va lue be low w h ich cu rva tu re e f fec ts ma y be neg lec ted .

F o r s u ff i ci en tly l a rge cu rva tu re i t becom es neces s a ry to r e -de fine the conven t iona l in teg ra l pa ra mete r s

6* and 0 ow ing to the f ac t tha t the ve loc i ty in the i r ro ta t iona l f low imm ed ia te ly ou t s ide the bou nda ry

layer no longer remains cons tant wi th d is tance f rom the surface . In the present work logical def in i t ions

a re s ugges ted and thes e have been us ed w herever pos s ib le .

I t has been s how n tha t the u s ua l f ia t - su r f ace mom en tu m in teg ra l equ a t ion b rea ks dow n in cas es w here

the s ta t i c p r es s u re does no t r em ain s ubs tan t i a l ly cons tan t ac ros s the bou nda ry l ayer . N e w m an s m od i f i ed

re la t ion has been s how n to app ly in boundary l aye r s w here l a rge s t a t i c -p res s u re va r i a t ions occu r a s a

r es u lt o f s tr eaml ine cu rva tu re on f la t s u rf aces . A new in teg ral eq ua t ion has been ob ta ined fo r the bou nda ry

laye r on a s u r f ace o f cons tan t cu rva tu re . Th i s eq ua t ion g ives goo d ag reem en t w i th exper imen ts .

V a lues o f the s hape f ac to r H m eas u red on convex s u r f aces a r e found to b e ve ry muc h h igher than thos e

14


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predicted by the method of Head 9. Since this method has been shown to be fairly successful in predicting

the development of H on flat surfaces, it is concluded that the observed differences between calculation

and measurements are a direct result of curvature effects. Such differences have also been observed in the

convex surface boundary-layer measurements of Schmidbauer22 and Schubauer and Klebanoff23 by

Bradshaw. The observation that the values of H measured on convex surfaces are consistently higher than

flat-surface predictions superficially suggests tha t convex curvature reduces the en trainment of free-stream

fluid into the boundary layer. This result was to be expected from considerations of Rayleigh s stability

criterion for curved flows and also from previous measurements in curved wall jets and in fully developed

flow in curved channels, but it has no t been possible to obtain any direct evidence of this here.

The measured velocity profiles show a marked dependence on curvature in both the inner and the outer

regions. In particular, it is found that they are no t adequately described by the two-parameter family of

profiles constructed for flat surface boundary layers by Thompson 27. The influence of curvature on the

entire profile suggests that the detail turbulent motions within the boundary layer are affected by stream-

line curvature. Some evidence in support of this observation has al ready been put forward by Bradshaw.

While the present investigation has led to many important conclusions it will be obvious that much

further work is required to clarify several points. The experiments reported here, being preliminary in

nature, have suffered from several drawbacks: (a) We had grossly underest imated the influence of curvature

and therefore employed rather large values of 6/R In future experiments it will be more profi table to work

with smaller curvature, say 1/50 > 6 / R > 1/300. (b) It has not been possible to isolate satisfactori ly the

effects of streamwise pressure gradients f rom those of curvature. In the cylinder experiments the curvature

of the streamlines could not be estimated accurately enough, while the pressure gradients associated with

the abrupt changes in surface curvature in the duct experiments made the interpretation of curvature

effects rather difficult. In future work, therefore, it is suggested tha t an a ttempt should be made to produce

a predominantly zero streamwise pressure gradient boundary layer on a curved surface. Comparison of

measurements in such a layer with the well-established results of flat plane boundary layers will shed

much light on the influence of curvature. (c) To obtain the conditions just mentioned it is necessary to

use curved ducts. Other experiments reported by the au tho r 17 suggest that the absence of three-

dimensional ity must be rigorously checked and if necessary the end-wall boundary layers must be con-

trolled. (d) Experiments have not so far been carried out to assess the influence of concave curvature.

In future attempts to study boundary layers on concave surfaces the G6rtler-Taylor type instability

vortices must be taken into account. A study of such vortices in turbulent boundary layers may in itself

be profitable. (e) More refined and detailed measurements are obviously needed to check (i) the new

momentum integral equation, (ii) the expected dependence of entrainment on streamline curvature, and

(iii) the influence of curvature on the turbulence properties and mean velocity distributions in order to

provide more direct evidence for Bradshaw s mixing-length modifications.

7 Acknowledgement

The author would like to express his sincere gratitude to Dr. M. R. Head for his many invaluable

suggestions and discussions during the course of this work.

15


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N o . A u t h o r s )

1 P . B r a d s h a w

P . B r a d s h a w , . .

D . H . F e r r i s s a n d

N . P . A tw e l l

3 A . E . v o n D o e n h o f f a n d

N . T e t e r v i n

4 S . E s k i n a z i . . . .

5 S . E s k i n a z i a n d H . Y e h

6 G . I . F e k e t e . . . .

7 S . G o l d s t e i n . . . .

8 D . E . G u i t t o n . .

9 M . R . H e a d . . . .

1 0 M a r g o l i s, D . P . . .

11 J . M u r p h y . .

1 2 R . N a r a s i m h a a n d . .

S . K . O j h a

13 B . G . N e w m a n . .

R E F E R E N C E S

Ti t le , e tc .

T h e a n a l o g y b e t w e e n s t r e a m l i n e c u r v a t u r e a n d

t u r b u l e n t s h e a r f lo w .

A.R .C . 29 , 048 . 1968 (unpubl i shed) .

b u o y a n c y i n

C a l c u l a t i o n o f b o u n d a r y - l a y e r d e v e l o p m e n t u s in g t h e t u r b u l e n t

e n e r g y e q u a t i o n .

J . F l u i d M e c h . Vol. 28, p. 593. 1967.

D e t e r m i n a t i o n o f g e n e r al r e l a ti o n s f o r t h e b e h a v i o u r o f t u r b u l e n t

b o u n d a r y l a ye rs .

N . A . C . A . R e p o r t 7 7 2 . 1 9 43 .

A n i n v e s t i g a t i o n o f f u l ly d e v e l o p e d t u r b u l e n t f l o w i n a c u r v e d

c h a n n e l .

J o h n s H o p k i n s U n i v . R e p o r t 1 - 2 0 . 1 9 5 4 .

A n i n v e s t i g a t i o n o f fu l ly d e v e l o p e d t u r b u l e n t f l o w i n a c u r v e d

c h a n n e l .

J . Aero . Sc i . Vo l. 23, p. 23. 1956.

C o a n d a f l ow o f a t w o - d i m e n s i o n a l w a ll j e t o n t h e o u t s i d e o f a

c i r c u l a r c y l i n d e r .

M c G i l l U n i v . R e p o r t 6 3 - 1 1 . 1 9 63 .

M o d e r n D e v e l o p m e n t s in F l u i d D y n a m i c s .

V o l . 1 , p . 1 1 9, O x f o r d U n iv . P r e s s ( a l s o D o v e r P u b l i c a t i o n s ) . 1 9 38 .

T w o - d i m e n s i o n a l t u r b u l e n t w a l l j e t s o v e r c u r v e d s u r fa c e s.

M c G i l l U n i v . R e p o r t 6 4 - 7 . 1 96 4.

E n t r a i n m e n t i n t h e t u r b u l e n t b o u n d a r y l a y e r.

A . R . C . R . M. 3 1 5 2 . 1 9 58 .

A s t u d y o f t h e e f fe c t o f c u r v a t u r e o n t u r b u l e n t f lo w .

P e n n . S t a t e U n i v . P r o j e c t N o . 3 2 2 7 - - E R e p o r t . 19 63 .

S o m e e ff e ct s o f s u r f a ce c u r v a t u r e o n l a m i n a r b o u n d a r y - l a y e r fl o w .

J . Aero . Sc i . Vol. 20, p. 338. 1953.

E f f e ct o f l o n g i t u d i n a l s u r f a c e c u r v a t u r e o n b o u n d a r y l a y e rs .

I n d i a n I n s t . Sc i. B a n g a lo r e , R e p o r t A E 1 6 4 A . 1 96 6 .

S o m e c o n t r i b u t i o n s t o t h e s t u d y o f t h e t u r b u l e n t b o u n d a r y l a y e r

n e a r s e p a r a t i o n .

A u s t . D e p t . S u p p l y , R e p o r t A C A - 5 3 . 1 95 1.

16


18/33

No. Au thor s )

1 4 B . G . N e w m a n . .

1 5 V . C . P a t e l . . . .

1 6 V . C . P a t e l . . . .

1 7 V . C . P a t e l . . . .

18 V . C . P a t e l a n d . .

M . R . H e a d

1 9 J . W . S . R a y l e i g h . .

2 0 A . J . S a r n e c k i . .

2 1 R . A . S a w y e r . .

2 2 H . S c h m i d b a u e r . .

2 3 G . B . S c h u b a u e r a n d

P . S . K l e b a n o f f

2 4 F . S c h u l t z - G r u n o w a n d

W . B r e u e r

2 5 B . S . S t r a t f o r d . .

26

B . S . S t r a t fo rd , . .

Z . M . J a w o r a n d

G . I . G o l e s w o r t h y . .

2 7 B . G . J . T h o m p s o n . .

R E F E R E N C E S - - c o n t i n u e d

Title, etc.

T h e d e fl e ct io n o f p l a ne je t s b y a d j a c e n t b o u n d a r i e s ~ 2 o a n d a

effect.

Boundary Layer Control , E d . G . V . L a c h m a n n , P e r g a m o n P r e s s ,

p. 232. 1961.

P h . D . T h e s i s , C a m b r i d g e U n i v e r s i t y . 1 96 5.

C a l i b r a t i o n o f th e P r e s t o n t u b e a n d l i m i t a t i o n s o n i t s u s e in

p r e s s u r e g r a d i e n t s .

J . F lu id Mech .

Vol. 23, p. 185, 1965.

M e a s u r e m e n t s o f s e c o n d a r y f lo w i n th e b o u n d a r y l ay e r s o f a 1 80

c u r v e d c h a n n e l .

A .R .C .C .P . 1043 . 1968 .

R e v e r s i o n o f t u r b u l e n t t o l a m i n a r f l ow .

A.R.C. 29 ,859 (unpubl i shed) .

J . F lu id Mech . Vol. 34, p. 371. 1968.

T h e d y n a m i c s o f r e v o l v in g f l u id s .

Proc. Roy. Soc.

Vo l. 6A, 19. 14 8. 1916.

P h . D . T h e s i s , C a m b r i d g e U n i v e r s i t y . 1 95 9.

T w o - d i m e n s i o n a l r e a t t a c h i n g j e t f l o w s i n c l u d i n g t h e e f fe c ts o f

c u r v a t u r e o n e n t r a i n m e n t .

J . F lu id Mech .

Vol. 17, p. 481. 1963.

T u r b u l e n t f r i c t i o n l a y e rs a l o n g c o n v e x s u rf a c e s.

N . A . C . A . T . M . 7 91 ( t r a n s l a t i o n o f L u f t f a h r t f o r s c h u n g , V o l . 1 3,

p. 160). 1966.

I n v e s t ig a t i o n o f s e p a r a ti o n o f th e t u r b u l e n t b o u n d a r y l ay e r .

N.A.C.A. Repor t 1030. 1951.

L a m i n a r b o u n d a r y l a y er s a t c u r v e d s u rf a ce s .

I n s t i t u t f ii r M e c h a n i k , T e c h . U n i v . A a c h e n , R e p o r t 3 . 19 63 .

A n e x p e r i m e n t a l f l o w w i t h z e r o s k i n - f r i c t i o n t h r o u g h o u t i t s

r e g i o n o f d e v e l o p m e n t .

J. Flu id Mech.

Vol. 5, p. 17. 1959.

T h e m i x i n g w i t h a m b i e n t a i r o f a c o l d a i r s t r e a m i n a c e n t r i f u g a l

field.

A .R .C .C .P . 687 . 1962 .

P h . D . T h e s i s , C a m b r i d g e U n i v e r s i t y . 1 9 63 .

17


19/33

No. Au thor s )

2 8 F . L . W a t t e n d o r f . .

2 9 H . W i l c k e n . . . .

3 0 H . Y e h , W . G . R o s e a n d . .

H . L i e n . . . . . .

R E F E R E N C E S - - c o n t i n u e d

Title, etc.

A s t u d y o f t h e e f f ec t o f c u r v a t u r e o n f u l l y d e v e l o p e d t u r b u l e n t f lo w .

Proc. Roy. Soc.

Vol. 148A, p. 565. 1935.

T u r b u l e n t e G r e n z s c h i c h t e n a n g e w 6 1 b t e n F l g c h e n .

Ing. Archiv.

Vo l. 1, p. 357. 1930.

F u r t h e r i n v e s t i g a t i o n s o n f u l ly d e v e l o p e d t u r b u l e n t f l o w s in a

c u r v e d c h a n n e l.

J o h n s H o p k i n s U n iv . , C o n t r a c t N o n r - - 2 4 8 3 3) R e p o r t . 1 95 6.

18


20/33

A P P E N D I X

Mo m en tum In tegr a l Equa t ion fo r a Boundar y L ayer on a Sur face o f Cons tan t Curva ture .

U s i n g t h e n o t a t i o n a n d t h e c o - o r d i n a t e s y s te m s h o w n i n F i g u r e 1 6, t h e t w o - d i m e n s i o n a l b o u n d a r y -

l a y e r e q u a t i o n s c a n b e w r i t t e n

u OU + v OU.+ U V 1 Op f o2 u l OU u ~

~o~ a~

- 7 - =

p r ~ ~ ~ ~ - J 7

r

rTJ

A.1)

U 2 1 Op

an d - A .2)

r ' p Or

T h e c o n t i n u i t y e q u a t i o n i s

0 U ~r

r a ~ F ~ V r ) = O .

A.3)

W e n o w c o n s i d e r b o u n d a r y - l a y e r f lo w o n a s u r f ac e o f c o n s t a n t r a d i u s o f c u r v a tu r e , R , a n d a s s u m e t h a t

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, 6 , i s sm a l l e n o u g h f o r t e r m s o f t h e o r d e r 6 / R ) 2 t o b e n e g l e c t e d in c o m -

p a r i s o n w i t h u n i t y . T h e n i t is r e a d i ly s h o w n t h a t i n t h e i r r o t a t i o n a l f l o w o u t s id e t h e b o u n d a r y l a y e r th e

p r o d u c t

U r

b e c o m e s i n d e p e n d e n t o f r, i.e .

U l r ~ = f~ q~), A.4)

w h e r e t h e s u f fi x 1 r e p r e s e n t s c o n d i t i o n s o u t s i d e t h e b o u n d a r y l a y e r . N o t i c e t h a t e q u a t i o n A .4 ) i s e x a c t

w h e n t h e s t r e a m l i n e s i n t h e o u t e r f l o w a r e c i rc u l a r.

W i t h t h e u s e o f t h e s u b s t i t u t io n

U r

= co, A.5)

t h e b o u n d a r y c o n d i t i o n s

O U _

z,,,at r =__R

U = ,0, Or

~

a n d

U --~ U1 Ulr I

= ~~ ~ ) fo r

r = r l ,

a n d t h e e q u a t i o n o f c o n t i n u i t y , e q u a t i o n A .1 ) c a n b e i n t e g r a t e d w i t h re s p e c t to r f r o m t h e w a l l r

t o s o m e r r. = r l ) o u t s i d e th e b o u n d a r y l a y e r to o b t a i n ,

c3q5 R - ~ - - d r - f~ --9dr

R r ' P O ~ R

A.6)

R

zwR

plr l dr1

pr dr - - 2vf~-t . A.7)

p p d 4

T h i s e q u a t i o n c a n a ls o b e d e r i v e d b y c o n s i d e ra t i o n s o f t h e fl u x o f a n g u l a r m o m e n t u m t h r o u g h a s m a l l

c o n t r o l v o l u m e in t h e u s u a l w a y . T h e t h i r d , v i s c o u s t e r m o n t h e r i g h t - h a n d s i de o f t h i s e q u a t i o n a r i s e s

f r o m t h e f a c t t h a t t h e f l o w o u t s id e t h e b o u n d a r y l a y e r is n o t f r e e f r o m s h e a r . T h i s t e r m c a n , h o w e v e r , b e

n e g l e c t e d si n c e i t is re c i p r o c a l o f th e R e y n o l d s n u m b e r b a s e d o n t h e v e l o c i t y a n d r a d i u s o f c u r v a t u r e o f

t h e s t r e a m l i n e s o u t s i d e t h e b o u n d a r y l a y e r .

I f w e n o w d e f i n e tw o i n t e g r a l l e n g t h s c a le s b y t h e r e l a t i o n s

19


21/33

; , ) f o

* = co d r a n d O = g l _ O ~ d r

R

1 ~ r R

f l J r

(A.8)

e q u a t i o n ( A .7 ) c a n b e r e d u c e d t o t h e f o r m

d i d f ~ zw R 2 1 d f ~ f ~ }

dq~ ~-(A* + 2 0 ) de pf~2 pf~2 de pf~2 n + prd

I 2 I dr1

p~ z (~P ~ + Pl r2) r~ d--C (A.9)

T h i s i s t h e b a s ic f o r m o f t h e n e w m o m e n t u m i n t e g r a l e q u a t i o n . S o m e o f th e n o t a b l e f e a t u r e s o f t h is

equ a t i on a re l i st ed be l ow.

( 1) T h e i n t e g r a l p a r a m e t e r s A * a n d a r e b a s e d o n t h e d e f ec t s o f U r a n d a r e d e f i n e d t o e n s u r e t h a t

t h e i r v a l u e s a r e i n d e p e n d e n t o f t h e c h o i c e o f r~ . I t i s, h o w e v e r , e a s y t o s h o w t h a t t h e y a r e r e l a t e d t o t h e

u s u a l m a s s a n d m o m e n t u m - f l u x d e f e c t th i c k n e s s e s ( d ef in e d b y e q u a t i o n s (8 ) a n d (9 ) i n t h e t e x t , S e c t io n

4 .4 ) by t h e r e l a t i on s

A* =

6* /R

a n d

(1 + ~5/R)O/R + t e r m s o f o r d e r

~ / R ) 2 .

(2) Since dx = Rd, 6 = RA* , 0 - (1 + 6 /R) - 1 R @, f~ = Ulrx an d ( r 1 - R ) i s o f t he o rde r o f 6, i t can

r e a d i l ly b e s h o w n t h a t t h e l e f t - h a n d s id e o f e q u a t i o n ( A.9 ) ( w h e n p u t e q u a l t o z e r o ) r e d u c e s i d e n t i c a ll y t o

t h e u s u a l f l a t s u rf a c e m o m e n t u m i n t e g r a l e q u a t i o n , e q u a t i o n ( 1) , f o r s u f f ic i e n tl y s m a l l v a l u e s o f 6/R.

(3) Th e f i r s t t e rm on t he r i gh t -h and s i de o f equ a t i on (A:9 ) i nc l udes t he s t a t i c -p res s u re va r i a t i on ac ro s s

t h e b o u n d a r y l a y e r a n d i s in m a n y w a y s s i m i la r t o t h e p r e s s u re t e r m o c c u r r i n g i n t h e m o d i f i e d i n t e g r a l

e q u a t i o n ( 10 ) s u g g e s t e d b y N e w m a n . T h e 9 0 c o n v e x w a l l b o u n d a r y - l a y e r m e a s u r e m e n t s d e s c r i b e d i n t h e

t e x t s u g g e s t t h a t t h i s t e r m c a n b e q u i t e l a r g e in c o m p a r i s o n w i t h t h e o t h e r t e r m s o n t h e l e f t -h a n d s i de .

T h e m a g n i t u d e o f t h is t e r m d e p e n d s o n r l .

(4) Th e l a s t t e rm i n equ a t i on (A .9 ) a r i s es due t o t he f ac t t ha t t he cho i ce o f the ou t e r l i m i t o f i n t eg ra t i on

( r = r l ) h a s b e e n l e ft a r b i t r a r y w i t h t h e o n l y r e st r i c ti o n t h a t ( r l - R ) i s o f t h e o r d e r o f th e b o u n d a r y - l a y e r

t h ic k n e s s . W e c a n o f c o u r s e c h o o s e r 1 = 6 , a s i s c u s t o m a r y , b u t t h e n t h i s t e r m c a n n o t b e e v a l u a t e d w i t h

a n y g r e a t a c c u r a c y d u e t o t h e u n c e r t a i n t y o f d e f i n i n g 6 . I t is t h e re f o r e b e t t e r t o c h o o s e r~ = c o n s t a n t

w i t h s o a s t o m a k e t h i s t e r m z e r o . N o t i c e a l s o t h a t t h e q u a n t i t y w i t h i n th e b r a c k e t s i n t h e l aS t t e r m i s

n o m o r e t h a n 2

ours, Po~be i ng the t o t a l p res s u re a t r = r l .

(5) E qua t i on (A .9 ) can be r e -cas t i n t o s eve ra l d i f f e ren t fo rm s . In pa r t i cu l a r , w e m a y s ubs t i t u t e t he

s t a t i c -p res s u re va r i a t i on g i ven by t he eq ua t i o n o f r ad i a l eq u i l i b r i um , equ a t i on (A .2) , i n t he i n t eg ra l

o c c u r r i n g i n t h e s e c o n d t e r m o n t h e r i g h t - h a n d s id e t o o b t a i n t e r m s i n v o l v i n g A * , O a n d o t h e r q u a n t i t i e s.

W e have , however , choos en t he above fo rm s i nce ( a ) i t s hows how t he u s ua l f l a t s u r face equa t i on i s r e -

p r o d u c e d f o r s m a l l 6/R, (b) a s i ng le ' cu rva t u re co r re c t i on t e r m ' i s ob t a i n ed w hen r~ = con s t an t , and ( c)

t h i s c o r r e c t i o n t e r m c a n e a s i l y b e e v a l u a t e d f r o m t h e m e a s u r e m e n t s w h i c h h a v e b e e n m a d e .

(6) Eq ua t i o n (A .9 ) w i t h con s t an t r~ has been u s ed i n the t ex t t o ca l cu l a t e t he dev e l op m e n t o f o n t he

90 deg ree conve x s u r face .

20


22/33

TABLE 1

Details of Measuring Stations in

the

90

de ree Curved Duct

Station

number

4

5

6

7

8

9

10

11

12

13

14

15

16

Distance

from

transition

x inches

42-0

53.0

62-0

71.0

77-0

83 0

~89.0

95.5

102.0

108.0

116.25

122.25

128.25

134.25

140.25

146.25

Distance

from

Station 1

( x - xo) inches

0

11-0

20.0

29 0

35.0

41.0

47-0

53-5

60 0

66 0

74.25

80 25

86.25

92.25

98.25

104.25

21


23/33

TABLE 2

Static Pressure istribution on the Test Wall

of the 90 degree uct

inches

42

45

48

51

54

P s t - P r e f

mm water

3'37

3.34

3.33

3 29

3.28

P s t - P

Cp -T277~

~pU1

0

- 0033

- .0045

- .0089

-

.0100

57

60

63

66

69

72

75

78

81

84

87

90

93

96

99

102

105

108

111

117.75

120 75

123.75

126.75

129.75

132.75

135.75

138.75

141.75

144.75

147.75

150.75

3.27

3.27

3.23

3.25

3-27

3-26

3.21

3.20

3.26

3.25

3.21

3.16

3.07

3.01

2-93

2.78

2.60

2.42

2 09

0

-0.84

-

1.51

- 1 9 3

- 1 - 9 9

effect of curvature of bl

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