The Flow Past a Pitot Tube at Low Reynolds Numbers

8/9/2019 The Flow Past a Pitot Tube at Low Reynolds Numbers

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M I N I S T R Y O F V I T I O N

A E R O N A U T I C A L . R ES E AR C H C O U N C I L

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

R . M . N o . 3 2 4 0

T h e F l o w p ast a P i t o t T u b e a t L o w R e y n o l d s

N u m b e r s

Par t I . T he N um er ica l So lu t ion o f the Nav ie r -S tokes Equa t ions fo r

S tead y V i sco u s A x i - sy mmet r i c F lo w

Par t I I . T he Ef fec ts o f Viscos ity and Or i f ice S ize on a P i to t Tub e a t

L o w R e y n o l d s N u m b e r s

By W. G. S. LESTER

DEPARTMENT OF ENGINEERING SCIENCE, UNIVERSITY OF OXFORD

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

I96I

PRICE

9s. 6d.

NET


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3/25

d i f f e r e n c e e q u a t io n s w h o se n o n - l i n e a r i t y h a s ma d e a t t e mp t s a t so lu t i o n e x t r e me ly l o n g a n d t e d io u s .

E v e n i n t h e s o l u t i o n f o r s l o w f lo w s , u s in g t h e S t o k e s ' a p p r o x i m a t i o n , b y n u m e r i c al m e t h o d s t h e

l a b o u r r e q u i r e d i s a lmo s t p r o h ib i t i v e a n d t h e i n c lu s io n o f t h e n o n - l i n e a r t e r ms so c o mp l i c a t e s t h e

p r o b l e m a s t o m a k e i t im p r a c t i c a b le f o r s o lu t i o n b y h a n d . H o w e v e r , t h e g r o w i n g u s e o f e le c tr o n ic

c o m p u t o r s r e m o v e s m u c h o f t h e d i ff i cu l ty a n d a f fo r d s e n c o u r a g e m e n t f o r t h e d e v e l o p m e n t o f a

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

s t r a i g h t f o r w a r d p r o b l e m . . T h e s u c c e ss o f a s o l u t io n m a y w e l l d e p e n d t o a la r g e e x te n t o n t h e c h o i c e

o f t h e i n i t i a l s t a r ti n g v a lu e s ; t o b u i l d u p a so lu t i o n w i th a l l f u n c t i o n v a lu e s i n it i al l y z e r o ma y s e e m

t h e b e s t p r o c e d u r e b u t h a s t h e d i s a d v a n ta g e o f t a k in g s o m u c h m a c h i n e t i m e . O n t h e o t h e r h a n d ,

a n u n f o r tu n a t e c h o i c e ca n e a s il y l e a d to d iv e r g e n c e i n t h e n u me r i c a l p r o c e s s e mp lo y e d . I t sh o u ld

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

h y d r o d y n a m i c s t a b i li t y s in c e s te a d y f l o w is p o s tu l a t ed . D i v e r g e n c e c a n u s u a l ly b e c o m b a t e d w h e n

w o r k i n g b y h a n d b u t c a n b e d i ff ic u l t t o d e a l w i t h o n a c o m p u t e r , t h e u s u a l t e c h n i q u e b e i n g e i th e r t o

r e d u c e t h e me sh s i z e o r e mp lo y o n ly a p a r t i a l mo v e me n t i n f u n c t i o n v a lu e s a t me sh p o in t s .

T h e f i rs t a t t e m p t t o o b t a i n a n u m e r i c a l s o l u t i o n w a s m a d e b y T h o m 3 w h o e x a m i n e d t h e s l o w

mo t io n o f a v i s c o u s f l u id t h r o u g h a n a b r u p t e x p a n s io n i n a c ir c u l a r p ip e, w o r k in g i n c y l i n d r i c a l

p o l a r c o - o r d in a t e s a n d so lv in g t h e S to k e s ' e q u a t i o n s . T h e o n ly w o r k v a l i d a t a f i n i te R e y n o ld s

n u m b e r a p p e a rs t o b e t h a t o f J e n s o n 4 w h o h a s p r o d u c e d s o l u ti o n s f o r t h e f l o w p a s t a s p h e r e p l a c e d

o n th e a x i s o f a c ir c u l a r p ip e a t R e y n o ld s n u m b e r s u p t o f o r ty ( b a se d o n t h e sp h e r e d i a me te r ) .

T h e c o - o r d in a t e sy s t e m u se d w a s sp h e r i c a l p o l a r s .

S o m e o f t h e m e t h o d s a p p l ic a b l e in t w o d i m e n s i o n s w i l l n o w b e e x t e n d e d t o t h e c a se o f a x ia ll y

sy m me t r i c f l o w s u s in g c y l i n d r ic a l c o - o r d in a te s . T h e w o r k in g v a r i a b l e s a r e t h e S to k e s ' s t r e a m

f u n c t io n ~b a n d t h e v o r t i c i t y ~ . I n t w o d ime n s io n s , t h i s tw o - f i e ld r e l a t e d sy s t e m o f v a r i a b le s , f i r s t

i n ' t r o d u c e d b y T h o m 1, h a s b e e n s h o w n b y F o x 5 t o h a v e a f a s t e r ra t e o f c o n v e r g e n c e t h a n i f t h e s i n g l e

v a r i a b l e ~b h a d b e e n u se d . I t i s to b e e x p e c t e d t h a t a s imi la r r e su l t w i l l h o ld f o r t h e a x i a l ly sy m me t r i c

c a se. A n a c c u r a t e f o r m u la t o r e l a t e ~ a n d ~b a t a so l i d b o u n d a r y , a n a lo g o u s t o W o o d s ' f o r m u la i n

t w o

d ime n s io n s ( R e f . 6 ) , i s d e r iv e d . T h i s f o r mu la h a s b e e n f o u n d to b e q u i t e s a t i s f a c to r y i n t h e

p r o b l e ms so f a r c o n s id e r e d a n d g iv e s a r e a so n a b le r a t e o f c o n v e r g e n c e . I t h a s u n f o r tu n a t e ly

n o t

y e t p r o v e d p o s s ib l e t o d e r iv e a n e a s i l y a p p l i e d c o n v e r g e n c e c r i t e r i o n a s f o r t h e tw o - d ime n s io n a l

f l o w ( R e f . 7 ) , b u t i n p r a c t i c e i t s e e ms th a t t h e tw o - d ime n s io n a l t e s t d o e s g iv e so me in d i c a t i o n a s

t o

w h e n d iv e r g e n c e c a n b e e x p e c t e d . ' '

A s an e x a m p le o f t h e m e th o d a n d t h e t e c h n iq u e u se d , t h e e f f ec t s o f v i s c o s i t y o n a p i t o t t u b e a t a

l o w R e y n o l d s n u m b e r h a v e b e e n i n v e s t i g at e d i n s o m e d e t a il i n P a r t I I .

1.2. T he Ax i a l l y Sym me t r i c For m o f the N av i er -S t okes Equa t ions. A b r i e f d e r iv a t i o n o f t h e

a x i al l y sy m m e t r i c f o r m o f t h e e q u a t io n s i n c y l i n d r i c a l c o - o r d in a t e s i s n o w in c lu d e d .

I n t e r m s o f S t o k e s ' s t r e a m f u n c t i o n t h e v e l o c i ty c o m p o n e n t s m a y b e w r i t t e n

a a a

r 8 ~; q r ' = r o z ' ' - 0

a n d t h e v o r t i c i t y c o m p o n e n t s

(cu rl q)~ = 0;

3qr 3q~

( c u r lq ) ~ - - O ; ( c u r lq ) ~ = ~ - 8 - 7 - - ~

I f

i~ i ~ a n d i r a r e u n i t v e c t o r s i n t h e m e r i d i a n

p l a n e a n d p e r p e n d i c u l a r

t o t h a t

plane ,

q - qzi~ + q~ir, an d ~ = ~i~.

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aT

a t V A q a g ) = - ~ V A V A g ) .

N o w

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{ a q o g ) a ~ )

~ +

~ +

a

a ( ,

a

v ~ v ~ ) = - t ~ ~ a ~ l + ~ ~ ) ) I i+

w h e r e

3qr Oq 1 t 3~ 32~b 1 3 I 1E ~ b

~ = ~ W - T / a z = + g ~ r ~ 7 g = 7

3s a 2

I 0

E 2 = - -

O z ~ O r r &

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

~ a ,q,,-, a q ; , ) _ o .

0 z 3 r

F r o m 2 ) a n d 5 ),

3

- - V A ( q A ~ ) = r l q z ~ - - z ( ~ ) -[ - q r ~ r ( ) I i "

F r o m 4 ) a n d 3 ),

V A ( V ~ ) = - - E ~ 4* i ~ .

S u b s t i t u t i n g i n 1 )

a t a , g/O .E~ = O.

r ~ + r 3 z,r)

A s s u m i n g s t e a d y m o t i o n

R 4 r a , g / r )

,, a z, r)

- 0 .

E q u a t i o n s 4 ) a n d 6 ) w i l l b e c o n s i d e r e d i n t h e f o r m

1 a

a ~ = r + - -

a r

w h e r e

A 2 ~ = 3 r + - -

7 - v r 3 z 3 r a r 3 z v r~ 3 z

A ~ - ~ r2 + az--~.

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4 )

5 )

6 )

7)

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T h i s e q u a t i o n e n a b le s th e p r e s s u r e d i f f e r e n c e b e t w e e n a n y t w o p o i n ts A a n d B i n t h e f l u i d t o b e

d e t e r m i n e d b y i n t e g r a t io n a l o n g a p a t h c o n n e c t i n g t h e m .

1 .5 . T h e N o n - D i m e n s i o n a l F o r m o f t h e E q u a t io n s . I t i s f r e q u e n t l y c o n v e n i e n t t o w o r k w i t h t h e

e q u a t i o n s in a n o n - d i m e n s i o n a l f o r m . I f a d a s h is u s e d t o d e n o t e a n o n - d i m e n s i o n a l q u a n t i t y ,

r L r ' , z L z ' , q U q ', F = U F , ,

U

= = = p = 1 y7 2~ ~ ,

~ . P ~ 2 , ~b = U L ' an d

~ = - ~ ,

where U i s a r ep res en t a t i ve ve l oc i t y and L a r ep res en t a t i ve l eng t h .

S ubs t i t u t i ng t hes e exp res s i ons i n Equa t i ons (4 ) and (6 ) o f S ec t i on 1 .2 :

E ' ~ ' = r ' ~ ' ( 11 )

r E 2 ( r ~ ) = R e 1 8 O ~ 8 ~ b a ~ ~ a - ~ t ( 1 2 )

8z ' 8r ' 8 r ' 8 z ' r ' Oz ' 1

w h e r e R e i s a R e y n o l d s n u m b e r U L / v , a n d

8 2 8 ~ 1 8

E ~ _ -_ + - -

8 Z 2

O r 2 r O r

T h e e q u a t i o n to d e t e r m i n e t h e p r e s s u r e d i ff e r e n c e b e t w e e n t w o p o i n t s b e c o m e s

2 ( F / d r ' + F : ' d z ' ) - ( P B ' - P f f ) = q B ~ - q ~ '2 _ 2 ~ t' t 7 ~ az. + ~ / , d r ' -

f I

e t \ O z ' + - - d r ' - + d z '

~ r S a z \ S t 7

w h e r e A a n d B a r e t h e p o in t s c o r r e s p o n d i n g t o A a n d B i n t h e n o n - d i m e n s i o n a l f o r m .

(13)

1.6.

T h e N u m e r i c a l S o l u t i o n o f t h e E q u a t io n s .

T h e t e c h n i q u e u s e d i n t h e n u m e r i c a l s o l u t i o n o f

t h e a x i a ll y s y m m e t r i c e q u a ti o n s i s s i m i la r t o t h a t u s e d b y , T h o r n i n t w o - d i m e n s i o n a l p r o b l e m s .

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

va l ues o f ~b and ~ a re p l aced . Th es e va l ues o f ~b and ~ a re t hen r eca l cu l a t ed a l t e rna t e l y by m eans o f

f i n i te d i f f e rence app rox i m at i ons t o t he Equ a t i ons (7 ) and (8) , w i t h a s s um ed va l ues o f ~ on t he s o l i d

b o u n d a r i e s . F r o m t h e n e w c o n f i g u r a ti o n s o f a n d ~ i n t h e n e i g h b o u r h o o d o f t h e b o u n d a r y n e w

bo und ary va l ues o f ~ have t o be ca l cu l a t ed . T he wh o l e p roces s i s t he n r e pea t ed un t i l t he va l ues o f ~b

a n d ~ a r e r e p e a t e d e v e r y w h e r e t o t h e r e q u i r e d d e g r e e o f a c c u r a c y .

1 .7 . F i n i t e D i f f er e n c e A p p r o x i m a t i o n s t o t h e N a v i e r - S t o k e s E q u a t io n s .

T h e s i m p l e s t f i n i t e

d i f f e rence app rox i m at i ons t o t he Eq ua t i ons (7) and (8 ) o f S ec t i on 1 .2 a re bas ed on t w o p o i n t

d i f fe r e n ti a ti o n . W i t h a d i a m o n d

(i.e. ,

t he m es h i s o r i en t a t ed a s s ho w n i n F i g . 1 ), t he app rox i m at i ons

a re g i ven by

rn 2

~bo = C m - ( D - B ) - - ~ - ~o (14 )

I n ( d _ b ) _ l ~ . { ( A _ C ) ( d _ b ) _ ( D _ B ) ( a _ c ) } ,

(15)

~ n A - C ) = ~ m + ~ ;

~o 1 + 4r 2 8r2v

5

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I n o r d e r t o d e r i v e a f i n it e d i f fe r e n c e e q u a t i o n f o r o b t a i n i n g ~ o n a s o l i d b o u n d a r y s u b j e c t t o t h e

c o n d i t i o n s g i v e n i n E q u a t i o n ( 2 0 ), t h e c o n f i g u r a t i o n s h o w n i n F i g . 3 i s c o n s i d e r e d . S u p p o s e t h e

r a n d z a x e s a re r o t a t e d t h r o u g h a n a n g l e 0 so t h a t t h e n e w Z a x i s i s p a r a ll e l to t h e t a n g e n t t o

4` = c o n s t a n t a t 0, a n d t h e n e w R a x i s is p a r a l l e l to t h e n o r m a l a t 0, t h e n

Z = z c o s O + r s i n O

a n d

R

e

0

O r

= r c o s 0 - z s i n 0 ( 2 1 )

= Z s i n 0 + R c o s 0

0 O

= co s 0 ~ + s i n 0 0 -- 2 ( 22 )

0 0 O

- s i n + c o s 0 - - ( 2 3 )

O z 0 o R O Z

E q u a t i o n s ( 18 ) a n d ( 1 9 ) t r a n s f o r m t o b e c o m e

O34 034` 1 l O4` 04`l = C . (2 4)

O R~ + O Z3 ( Z s i n 0 + R c o s 0 ) c o s 0 ~ + s i n 0 ~ -~

O R+ O Z ~ ( Z s i n O + R c o s O ) c o s 0 ~ + s i n V f f Z t ' ( 2 5)

w h e r e f ' i s u s e d t o d e n o t e f ( r , z ) i n t h e t ra n s f o r m e d p l an e . O n t h e b o u n d a r y :

0 4 a 4

0- R - 0 ; f f Z = O ; 4 ` = 4 ` 0 . ( 26 )

n s i d e r i n g a p o i n t 2 o n t h e n o r m a l t o ~b = c o n s t a n t a t a d i s t a n c e h f r o m 0 i n t h e d i r e c t io n o f R

i n c r e a si n g , T a y l o r ' s t h e o r e m g i v e s

04`0 h a 02 h ~ h ~

' 0 0 3 4 `0 0

4`2 = 4`0 h -g ~ + 2 ~ + 6 OR + 24 OR + O (M ) . ( 2 7 )

D i f f e r e n t i a t in g E q u a t i o n ( 2 4 ) a n d e v a l u a t i n g a t t h e p o i n t 0 m a k i n g u s e o f t h e b o u n d a r y c o n d i t i o n s

( 2 6 ) , t h e f a c t t h a t a l l d e r i v a t i v e s w i t h r e s p e c t t o Z a r e z e r o a n d a ~4 ` - 0 , t h e f o l l o w i n g re l a t i o n s

OROZ 3

c a n b e o b t a i n e d :

0 3 4 ` 0

- ~ o * ( 2 8 )

OR 2

0 ~ 4 ` o 0 ~ . o c o s 0

+ ~0~ ( 2 9 )

OR 3 OR ( Z 0 s i n 0 + R 0 c o s 0 )

04~o 044 0 cosO 0. ~o* co s 0 0o*

OR+ O R 2 0 Z+ ( Z o s i n O + R

0 c o s 0 ) - ( Z o s i n 0 + R o c o s 0 )

O R

si n 0 085b0 0e~o*

( Z 0 s i n

O + R o

c o s 0 )

O R 2 0 Z = O R s

( 3 0 )

0 3 4 ` o 0 ~ o *

O R 2 0 Z O Z

( 3 1 )

a ~ o a~ o

R 2 a Z 2 a Z 2

( 3 2 )


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F r o m ( 3 0) , ( 3 1 ) a n d ( 3 2 ) , m a k i n g u s e o f ( 2 5) ,

0460

2

02~*

O R ~ - f o ' -

cs20 . ~o*

( Z o s in 0 + R o cos O)z

S u b s t i t u t i n g f r o m ( 2 8 ) , ( 2 9 ) a n d ( 3 3 ) i n ( 2 7 ) ,

6 9 . = 6 o + ~ ~0~ + ~ + Z 0 s i n 0 + R 0 c o s0

h ' I 32~o cos20 . ~0* _ f o , I

+ ~ 2 3 R 2 ( Z o s i n 0 + R o c o s0 ) 2

A l s o , b y T a y l o r ' s t h e o r e m ,

~2 = ~o + h ~ + - - - -

h 2 8 2 ~ o *

2 O R ~ + O (h 3 )

+ o h ~ ) .

(33)

(34)

3 5 )

E l i m i n a t i n g

O ~ o e /O R

f r o m ( 3 4 ) a n d ( 3 5 ) a n d r e a r r a n g i n g , t h e t e r m s i n

O 2 ~ o * / O R~

c a n c e l o u t a n d

l ; ( G - G ) - ~ * + - ~ fo ' l

~0 = h cos 0

2 + ( Z s i n 0 + R o c o s 0 ) - 4 ( Z o s i n 0 + R o c o s 0 )2

T r a n s f e r r i n g t o t h e o r i g i n a l ( r , z ) c o - o r d i n a t e s t h i s b e c o m e s

t 6 h I

~ 0 = t h2 ( ~ - 6 o ) - ~ 2 + y L

h cos 0

2 4ro 2 t

+

I 0

I n t h e a x i al ly s y m m e t r i c f o r m o f t h e N a v i e r - S t o k e s e q u a t i o n s

E ~ r ~ ) - r 0 6 , ~ / ~ )

v 3 ( z , r) - f ( r ' z )

A t th e b o u n d a r y

(36)

Or - 0; = 0

a n d s o f o i s z e r o .

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

~ e i n E q u a t i o n ( 36 ) b y

r~ ,

a n d f 0 b y z e r o . H e n c e

12+--hcS0r0

h _ 4 r ocs~Ol

(37)

T h i s f o r m u l a i s c o r r e c t to

O ( h ~ )

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

c a n b e u s e d .


10/25

A s a n e x a m p l e o f t h e u s e o f th i s f o r m u l a l e t u s c o n s i d e r t h e P o i s e u i l l e fl o w i n a c i r c u l a r p i p e .

W i t h a p ip e r a d i u s o f t w e n t y u n i t s a n d a c e n t r e - l i n e v e l o c i ty o f t w o u n i ts , t h e s t r e a m f u n c t i o n a n d

v o r t i c it y a r e g i v e n r e s p e c t i v e l y b y

= 0 . 0 0 1 2 5 r ~ - r 2

a n d

= 0 . 0 1 r .

S u p p o s e t h a t t h e v a l u e o f ~ o n t h e b o u n d a r y ( r = 2 0 ) h a s to b e f o u n d f r o m t h e v a lu e s o f ~ a n d ~ o n

t h e l i n e r = 1 9 . I n t h e b o u n d a r y f o r m u l a ( 3 7) t h e d i r e c t e d m e s h l e n g t h h i s - 1 a n d 0 = 0 .

S u b s t i t u t i n g i n t h e f o r m u l a ,

2 0~ 0 = 6 ( 2 0 0 - 1 9 8 . 0 9 8 7 5 ) - 1 9 x 0 . 1 9 = 4 . 0

2-

w h i c h g i v e s ~ = 0 2 0 a s r e q u i r e d b y t h e th e o r y . T h i s c o m p l e t e a g r e e m e n t is t o b e e x p e c t e d i n a

P o i s e u il l e fl o w , b e c a u s e t h e n e g l e c t e d t e r m s i n th e d e r i v a t io n o f t h e b o u n d a r y f o r m u l a a r e t h e n z e r o .

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P a r t I I T h e E f fe c ts o f V i s c o s i t y a n d O r i fi c e S i z e o n a P i t o t T u b e a t

L o w R e y n o l d s N u m b e r s

Summary T he e f fec ts o f v i s cos i ty and or i f i ce s i ze on a b lun t -nose d p i to t tube h ave been the ore t i ca l ly

i n v e s ti g a t ed u p t o a R e y n o l d s n u m b e r o f t e n , w h e r e t h e R e y n o l d s n u m b e r h a s b e e n b a s e d o n t h e r a d i u s o f

the tub e . R esu l t s a re expres sed in t e rm s o f a p res sure coe f f i c ien t

C I~ _ Po - P.~

pU~

whe re P 0 i s the p res su re m ea sured in the tu be , p the de ns i ty o f the f lu id , and p~ and U the s t a t i c p res sure and

ve loc i ty in an und is tu rbed f low a t the pos i t ion o f the tube .

T h e v a l u es o f f f: ~ f o r a b l u n t - n o s e d t u b e a r e f o u n d t o b e l e ss t h a n t h o s e f o r t u b e s w i t h h e m i s p h e r o i d a l

heads , b~ i t a lways g rea te r than un i ty in the range cons ide red . T he e f fec t o f the o r i f ice s ize i s to dec rease C2~

as the o r i f ice si ze inc reases, th i s dec rease i s ve ry sm a l l bu t inc reases wi th the Reyn olds num ber . At a Reyn olds

nu m ber o f t en the dec rease is a t m os t f ive pe r cen t o f the va lue o f Cp w hen th e re i s no o r if ice .

I t i s sugges ted tha t the d ec rease o f C~) be low un i ty foun d in som e exper im enta l inves t iga t ions a t a h ighe r

Reyn olds num ber cou ld be due to the e f fec t s o f o r i fi ce s ize .

2 . 1 . Introduction W h e n d e t e r m i n i n g v e l o c it i es i n a f l u i d f r o m i m p a c t p r e s s u r e m e a s u r e m e n t s

w i t h a p i t o t t u b e a s p e c ia l i n t e r p r e t a t i o n i s n e c e s s a r y w h e n t h e t u b e R e y n o l d s n u m b e r i s s m a ll .

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

a c t i o n o f v i s c o u s fo r c e s , a n d t h a t t h e p r e s s u r e m e a s u r e d a t t h e t u b e o r i f i c e i s t h e s t a g n a t i o n p o i n t

p r e ss u r e. T h e v e l o c it y c a n t h e n b e d e t e r m i n e d f r o m B e r n o u i ll i 's e q u a t i o n . W h e n t h e v i sc o u s a n d

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

i n t e r p r e t a t i o n i s r e q u i r e d t o d e t e r m i n e t h e v e l o c i t y c o r r e c t ly .

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

n u m b e r s . T h e t u b e s m o s t f r e q u e n t l y u s e d a r e t h e s t r ai g h t c y l in d r i c a l b l u n t - n o s e d t y p e a n d t h o s e

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

a t lo w R e y n o l d s n u m b e r s h a v e b e e n f o u n d e x p e r i m e n t a l ly to b e s m a l l e r t h a n t h o s e r e q u i r e d f o r

t u b e s w i t h h e m i s p h e r o i d a l h e a d s . I f th e t u b e o r i f i c e i s l a r g e a c o m p l e x f l o w p a t t e r n i s s e t u p i n s id e

a n d a r o u n d t h e t u b e w h i c h c a n s o d i s t u r b t h e f l o w t h a t th e p r e s s u r e r e c o r d e d i s n o t t h e t r u e

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

W h e n t h e t u b e o r i fi c e i s s m a l l c o m p a r e d w i t h t h e t u b e d i a m e t e r t h i s f l o w d o e s n o t o c c u r t o t h e

s a m e e x t e n t a n d t h e p r e s s u r e m e a s u r e d d i f fe r s l i tt le f r o m t h e t r u e s t a g n a t i o n p o i n t p r e s s u r e .

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

g e o m e t r y o n p i t o t tu b e s , t h e m a j o r i t y o f w h i c h h a s b e e n e x p e r i m e n t a l . T h e o n l y t h e o r e t i c a l b as e s

a v a il a b le f o r c o m p a r i s o n h a v e b e e n e i t h e r t h e f l o w n e a r t h e s t a g n a t io n p o i n t o f a s p h e r e o r a p r o l a t e

s p h e r o i d . R e a s o n a b l e a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t h as b e e n o b t a i n e d i n t h e c a se s o f

h e m i s p h e r i c a l h e a d e d a n d s o u r c e s h a p e d tu b e s . W i { h t h e b l u n t - n o s e d t u b e w e c a n e x p e c t t h e f l o w

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

s o m e a u t h o r s f o r c o m p a r i s o n t h e r . es ul ts h a v e n o t b e e n i n a g r e e m e n t a n d t h e r e i s l it tl e j u s t i f i c a ti o n

f o r m a k n g t h e c o m p a r i s o n .

T h e e a r li e st w o r k o n t h e e f fe c t s o f v i s c o s it y o n p i t o t t u b e s w a s t h a t o f M i s s B a r k e r n , w h o

e x p e r i m e n t e d w i t h s t r a ig h t c y l i n d r i ca l t u b e s i n a P o is e u i ll e fl o w . T h e v i s c o u s e ff e c ts w e r e f o u n d

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The origin of co-ordinates was taken at the stagnation point 0 on the cylinder

see

Fig. 4) With

the z-axis along the axis of the pipe and the r-axis perpendicular to it. We specified = 0 on the

cylinder surface and t~ = - 200 on the surface of the containing pipe. The upstream Poiseuille

flow was given by

= O 00125r 4 - r 2 (1)

= O.Olr where 0 ~< r ~< 20

and the limiting annular flow downstream by

= 0.0018716r ~ - 1. 14798r 2 log r + 0.245538r 2 - 0-247409 (2)

= 0.014973r - O 997125/r where 1 < r < 20.

2.3.

T h e N u m e r i c a l S o l u t i o n .

With an initial mesh length of unity, assumed values of and

were placed at mesh points throughout the field. Upstream, boundary values given by the

Poiseuille flow of Equations (1) were placed about two pipe radii from the cylinder. Downstream,

similarly about two pipe radii away, boundary values given by Equations (2) were used. These

artificial boundaries were completely arbitrary for it was obvious that the transition from Poiseuille

to annular flow could not take place over such a short distance. Placing the boundaries any further

away initially would have so increased the size of the field to be worked as to make an attempt at

solution impracticable.

Using the methods of Part I, an approximate solution to the Stokes' equations i .e. , the case of

zero Reynolds number) was obtained working entirely by hand. The next step ~in the process was

to move the assumed boundaries further away and investigate the effect which this had upon the

function values in the neighbourhood in whic h we were particularly interested. After moving the

boundaries further away and settling the field again it sdon became apparent th at whilst the ups tream

flow was settling fairly rapidly and the disturbance caused by the insertion of the cylinder was not

being propagated very far upstream, the flow downstream could not settle to the annular flow for

many pipe diameters. Fort unat ely this was no serious disadvantage, for changes of quite large magni-

tude in the function values a short distance downstream were found to have almost no effect

on the upstream values. Since our main interest lay in the region around the end of the cylinder

and in the upstream flow it was found possible to fix the downstream function values when they

were only changing slightly after each recalculation from the finite difference equations, without

causing too serious a disturbance of the upstream flow. This procedure was adopted and the field

settled. The mesh length was then reduced in the region extending about one pipe radius up and

downstream of the stagnation point 0 and further reduced in the region near the cylinder and

stagnation point.

At the sharp corner of the cylinder the empirical method int roduced by Th om 17 for use in two

dimensions was used. In Fig. 7 two values of ~ were used at the poin t A, one of these was calculated

from the values of $ and ~ at B and used in the recalculation of the function values at B using a

diamond, and the other was calculated from the values at C and used in the diamond centred on C

for recalculating values there. The two values of ~ at A are artificial, effectively one is on the inner

wall of the annulus and the other on the blunt end of the cylinder and both are slightly removed

from the actual corner. Repeated subdivision of the field in this neighbourhood has been found to

raise these values considerably while hardly affecting the surrounding points.

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A sufficiently accurate approximat ion to the Stokes flow having been obtained this solution was

used to provide a series of starting values for the problem at a finite Reynolds number. The Reynolds

numbe r was based on the radius of the cylinder and the upstream centre-line velocity, v the kinematic

viscosity being the parameter used for varying the Reynolds number.

It was found that the flow away from the cylinder at a low Reynolds number deviated only

slightly from the solution given by the Stokes approximation and that changes in the function

values at some distance from the cylinder did not affect the solution closer in. This enabled a much

smaller field to be used for the finite Reynolds number solutions and the assumption was made that

the flow four radii upstream and downstream of the cylinder, and three radii out into the stream,

was given by the solution of the Stokes equations. Ini tially the field used was even smaller than

this and the boundary taken only two radii up and downstream, the extension to four radii still

caused only slight changes and it; seems that the assumpt ion is reasonably valid provided that it

is not used at too high a Reynolds number.

Solutions have been obtained under the assumptions already mentioned at Reynolds numbers of

0, 1.0, 2.5, 5.0 and 10. It was felt that the assumptions made should certainly be valid as far as

R e = 5 and whilst solutions above this could be obtained care should be taken in interpreting

them. The calculations at R e = 0 and R e = 1 were both initially worked by hand and later checked

on an electronic computer; those at R e = 2.5, 5.0 and 10 were entirely worked on a computer.

When the problem had been solved for a blunt-ended cylinder without an orifice, the geometry

was changed to that shown in Fig. 6 where the cylinder is assumed to have infinitesimally thin

walls, and the problem reworked. In this case three artificial values of ~ were required at the sharp

lip of the cylinder.

2.4. Th e C alculation of the Pressure C oeff icient It was convenient to express the pressure

measured at the stagnation point 0 on the cylinder in terms of the coefficient

Cp Po P~

p U

where Po is the pressure measured at O, p~ the static pressure at 0 in a Poiseuille flow undist urbe d by

the int roduction of the cylinder, p the density of the fluid, and U the centre-line velocity in the

undisturbed upstream flow.

A point A was taken on the pipe axis at such a distance from 0 that the flow there was practically

undisturbed. The pressure coefficient was then given by

= p U 2 - = ~ - 1 2

\ N pU ] Disturbed \ ~ p S ]Undislburbed

The pressure differences were then determined using Equat ion (10) of Part I. Between two points

and a on the pipe axis we have

P l - P - P P ) - q f2

+ f i ( )

On the pipe axis both ~ and r are zero, making ~/r an indeterminate form; L Hopital s rule gives

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The Reynolds number was based on the radius of the cylinder and the undisturbed centre-line

velocity (U = 2 cm/sec) whence

2

P Re

On substitution the following formula was arrived at for the calculation of C~):

0 0 2 l 2 f o

O~

C D = 1 R e R e ~ T r d Z

where l is the distance between 0 and A.

2.5.

Resul ts .

Numerical values for ~b and ~ m a portion of the field near to the cylinder at a

Reynolds number of five are recorded in Figs. 8 and 9; in Fig. 8 we have the values for the

blunt-ended cylinder and in Fig. 9 those for the cylinder with an orifice.

A careful numerical integration led to t he values of C~) set out in the Tables below being obtained.

TABLE 1

Bhmt-ended Cyl inder

Re Cp

0 -> 2.37/R e

1-0 3.21

2-5 1.818

5.0 1. 384

10.0 1.185

TABLE 2

Cylinder wi th Ori f ice

Re C~

1.0 3.17

2.5 1.765

5.0 1-326

10.0 1-128

The value of

C p

in the limiting case

R e ~ 0

is artificial. It should be realised that when we solve

the Stokes approximations by a numerical method the solution is still being obtained for a finite

Reynolds number. This Reynolds num ber varies with position in the field and is related to the order

of magnitude of the neglected terms in the finite difference approximations and to the residual at

each point in the field.

The variations of pressure and velocity on the stagnation streamline for the blunt-nosed cylinder

are shown in Figs. 10 and 11 respectively.

The change of position of the zero streamline with Reynolds number near the cylinder with

infinitely thin walls is depicted in Fig. 12. It is seen from Fig. 13 that the streamline enters the

tube and that a very weak eddy is set up inside the orifice.

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2.6. Comparison with Expe rim enta l Results . T h e v a r i a ti o n o f C ~ w i t h R e y n o l d s n u m b e r , b a s e d

o n th e i n t e r n a l r a d iu s o f t h e t u b e , i s p lo t t e d i n F ig . 1 4 . A l so sh o w n a r e t h e e x p e r ime n ta l r e su l t s o f

M i s s B a r k e r , S h e r m a n , H u r d

el al.

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

o f t h e m o d e l u s e d i n t h e p r e s e n t i n v e s ti g a ti o n . T h e e x p e r i m e n t a l p o i n t s a r e ra t h e r s c a t t e r e d b u t t h e

g e n e r a l a g r e e me n t w i th t h e t h e o r e t i c a l r e su l t i s q u i t e g o o d ; a l so sh o w n i s t h e t h e o r e t i c a l r e su l t o f

H o m a n n f o r t h e s p h e re , w h i c h l ie s c o n s id e r a b l y a b o v e th e p r e s e n t r e su l t.

W i t h t h e t u b 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 e x t e rn a l r a d iu s t h e r e s u l ts a r e g iv e n in F i g . 1 5 .

M i s s B a r k e r s r e s u k s h a v e b e e n p l o t t e d o n t h e a s s u m p t i o n t h a t t h e t u b e w a s s h a r p l i p p e d a n d t h e

in t e r n a l a n d e x t e r n a l r a d i i w e r e t h e s a me .

M a c M i l l a n h a s d r a w n f a i r ed c u r v e s th r o u g h h i s o w n e x p e r i m e n t a l p o i n t s a n d a l s o c u r v e s th r o u g h

t h e p o i n t s o f S h e rm a n a n d H u r d et al. w i t h t h 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 e x t er n a l d i a m e t e r

o f t h e p i t o t t u b e . P l o t t e d o n t h i s b a s i s i t i s a p p a r e n t t h a t t h e v a l u e s o f C ~ f o r b l u n t - n o s e d t u b e s l ie

b e lo w th o se f o r h e misp h e r o id a l h e a d e d t u b e s , a n d t h a t a s t h e o r i f i c e s i z e i n c r e a se s so C ~ d e c r e a se s .

T h e r e s u l t s o b t a i n e d u s i n g t h e p r e s e n t a p p r o a c h c o n f i r m b o t h o f t h e s e e x p e r im e n t a l f a c ts . W h e n

t h e i n t e r n a l r a d i u s i s u s e d i n d e f i n i n g t h e R e y n o l d s n u m b e r a l l t h e e x p e r i m e n t a l r e s u l t s t e n d t o

l ie m u c h c l o se r t o g e t h e r a s h a s b e e n s h o w n b y M a c M i l l a n . C e r t a in l y i n t h e c a se o f t h e b l u n t - n o s e d

t u b e i t s e e m s t h a t t h e i n te r n a l r a d i u s s h o u l d b e u s e d i n t h e d e f i n it i o n o f t h e R e y n o l d s n u m b e r ;

w i t h o u t f u r t h e r i n v e s ti g a t io n o n h e m i s p h e r o i d a l h e a d e d t u b e s i t is d i ff ic u lt t o d e t e r m i n e w h i c h

d im e n s io n sh o u ld b e u se d i n t h e i r c a se . T h e f a c t t h a t a s t h e o r i f i c e s i ze i n c r e a se s so C 2~ d e c r e a se s

c o u l d p o s s i b l y e x p l a in t h e d e c r e a s e o f C ~ b e l o w u n i t y a s f o u n d b y H u r d et al. i t m a y b e t h a t t h e

o r i f ic e e ff e c t o u tw e ig h e d t h e v i s c o u s e f f e c t a r o u n d a R e y n o ld s n u m b e r o f 6 0 . S in c e t h e o r i f i c e a n d

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

t o e x t e r n a l r a d iu s o f th e t u b e , t h e v a l u e o f th i s r a ti o u s e d b y M a c M i l l a n m a y n o t h a v e b e e n h ig h

e n o u g h to i n f lu e n c e su f f i c ie n t l y t h e o r i f i c e ef f e c t i n h i s e x p e r im e n t s w i t h c i r c u l ar t u b e s , a n d t h e r e

w a s n o d e c r e a s e Of C ~ b e l o w u n i t y . T h e t u b e s w i t h a r e ct a n g u l a r c r o s s s e c t io n w i t h w h i c h M a c M i l l a n

f o u n d C ~ < 1 h a d b e e n f o r m e d b y f l a tt e n in g c i rc u l a r t u b e s w h o s e o r i g i na l r a ti o o f i n t e r n al t o

e x t e rn a l r a d i u s w a s b e t w e e n 0 . 8 a n d 0 . 9 . I f t h e f l a tt e n e d t u b e s c o u l d b e r e g a r d e d a s e q u i v a l e n t

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

f o r t h e o r i fi c e ef fe c t t o o u t w e i g h t h e v i s c o u s ef fe c t. T h e t u b e g e o m e t r y o f S h e r m a n w a s s l ig h t ly

d i f f e r e n t a n d f r o m h i s r e s u l t s w i t h a n o p e n - e n d e d t u b e t h e r e a p p e a r s t o b e s o m e M a t h n u m b e r

e f fe c t e v e n a t s u b s o n i c s p e e d s . S h e r m a n s t a te s t h a t a t s u b s o n i c s p e e d s t h e r e w a s n o t e n d e n c y f o r

C ~ to f a ll b e lo w u n i ty ; h o w e v e r , o n a clo se e x a min a t io n o f h i s r e a d in g s i n t h e r a n g e o f R e y n o ld s

n u m b e r s b e t w e e n 5 0 a n d 1 0 0 0 w i t h t h e o p e n - e n d e d t u b e , o u t o f s o m e s ix t y r e a d in g s a b o u t h a l f

w e r e e i t h er l e s s t h a n o r e q u a l t o u n i t y . A t s u p e r s o n i c s p e e d s C ~ w a s l e s s t h a n u n i t y o v e r a w i d e r a n g e .

T h i s su g g e s t i o n t h a t C ~ d e c r e a se s be lo w - u n i ty a t so m e v a lu e o f t h e r a t io o f i n t e r n a l t o e x t e r n a l

t u b e r a d i u s w h e n t h e R e y n o l d s n u m b e r i s h i g h e n o u g h f o r t h e v i s c o u s e f fe c ts t o b e o u t w e i g h e d

b y th e o r i f i c e e f f e c t i s v e r y t e n t a t i v e . T h e e x p e r im e n ta l e v id e n c e i s b y n o m e a n s c o n c lu s iv e a n d

mu c h s c o p e i s s t i l l a v a i l a b l e f o r f u r th e r e x p e r ime n t .

Acknowledgement . T h e a u t h o r w i s h e s t o t h a n k P r o f . A. T h o m f o r hi s v a lu a b l e h e lp a n d c r i ti c is m .

T h a n k s a re a ls o d u e to t h e O x f o r d U n i v e r s i t y C o m p u t i n g L a b o r a t o r y f o r p r o v i d i n g f a c il it ie s o n th e

M e r c u r y C o m p u t e r w h i c h e n a b l e d p a r t o f t h is w o r k t o b e d o n e.

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No. Au thor

1 A. Tho m ..

2 H. Schlichting .. . .

3 A. Thorn .. ..

4 V.G. Jenson . . . . .

5 L. Fox . . . . . .

6 L.C. Woods . . . .

7 A. Thorn and C. J. Apelt ..

8 H.B. Squire . . . . . .

9 H. L . Agrawal . . . . . .

10 A. Thorn . . . . . . .

11 M. Barker . . . . . .

12 F. Homann . . . . . . .

13 C. W. Hurd, K. P. Chesky and

A. H. Shapiro .. ..

14 F.A . MacMillan . .. .

15 F.A . MacMillan . .. .

16 F.S. Sherman . . . . . .

17 A. Thorn

REFERENCES

Title etc.

An investigation of fluid flow in two dimensions.

A.R .C .R . M. 1194. 1929.

Boundary Layer Theory.

Pergamon Press. 1955.

An arithmetical solution of certain problems in steady viscous flow.

A.R .C. R. M. 1475. 1932.

Viscous flow round a sphere at low Reynolds numbers.

Proc. Roy. Soc. A. 249. 1959. 346.

The numerical solution of elliptic differential equations when the

boundary conditions involve a derivative.

Phil . Trans. Roy. Soc.

A. 242. 1950.

A note on the numerical solution of fourth-order differential

equations.

Aero. Quart .

V. Pt. 3. September, 1954.

Note on the Convergence of numerical solutions of the Navier-Stokes

equations.

A.R .C. R. M. 3061. June, 1956.

The round laminar jet.

Quar t . J . Mech . App . M ath .

IV. Pt. 3. 1951.

A new exact solution of the equations of viscous flow with axial

symmetry.

Quar t . J . Mech . App . Math .

X. Pt. 1. 1957.

The arithmetic o f field equations.

Aero. Quart .

IV. Pt. 3. August, 1953.

On the use of very small pitot tubes for measuring wind velocity.

Proc. Roy. Soc.

A. 101. 1922. 435.

The effect of high viscosity on the flow around a cylinder and

around a sphere.

N.A.C.A. Tech. Memo. 1334. 1952.

Influence of viscous effects on impact tubes.

J . App . Mech .

Vol. 20. No. 2. 1953.

Viscous effects on pitot tubes at low speeds.

A.R.C. 16,866. June, 1954.

Viscous effects on flattened pitot tubes at lowspeeds.

A.R.C. 17,106. October, 1954.

New experiments on impact pressure interpretation in supersonic

and subsonic rarefied air streams.

N.A.C.A. Tech. Note. 2995. 1953.

Arithmetical solution of equations of the type V4~ = constant.

A.R .C .R . M. 1604. 1934.

17


19/25

( 3 o

m

B

F I G . I

F I G . 2 -

2

F I G . 3 .

: ~ - Z

F I G . 4

o /

"~ = 2 0 0

l r

F I G , , 5 t

O

F I G . 6

F

O

F I G . ?

~ z

~ ' - = - - z

~ z

= ' ' l ~ g = ' : ' i " i : ' q : r ' " P ~ ' q i ' = ' ' = ' ' : = i l ~ l l i: " ' I l l ' I ; j ' l T i l r l ' I = ' ~ t l i " i ' ' I r ' " r s ' " = ~ " = " a p r , : : : , ; , , ; r l : i i i , ~ l , , " r T " d ' = = , ' ~ r ; i t ' ] ' 1 , = ' ] I : , i r i: i l r, , i ,] , ~ i , , , i , , = , , i i [ s ~ , i , , , , , , , i i = i , i i ; l l , i ~ , , i : , l , , i ; , l , i i ~ l i l l l l i , = i ;, l l :m i , = i , i = , i = , l i l l q ' l i l' ~ i ' i ' ' l i ; ' = " i * " l i l " 'l l r l l i l " ' ' ; M l ' ; i i : l i i i ' r ; l l I I ' ] i l ' l i r = ; l l T i ' i i


20/25

-2 , 23 3 -1 "95 1 - 1 "6 IS

- 0 '051 -0 " 102 -0 .2 57

- I , 6 6 0 - I - 5 4 3 - I - 4 1 0 - 1 " 2 6 0 - 1 " 0 9 6 -0 " 9 2 7

- 0 .. 0 4 7 - 0 " 0 7 0 - 0 " 1 1 6 - 0 " 2 0 2 - 0 . 3 4 5 - 0 - 5 5 6

1.187 -1-084 -0 969 -0 836 -0-6 86 -0-530

- 0 " 0 4 5 - 0 .0 7 4 - 0 ' 1 3 5 - 0 - 2 5 4 - 0 " 4 6 9 - 0 "8 1 2

- 0 . 7 9 4 - 0 ' 7 1 7 - 0 ' 6 2 4 - 0 '5 1 4 - 0 " 3 8 6 - 0 " 2 4 9

- 0 " 0 4 3 -0 - 0 7 7 - 0 " 1 5 2 - 0 - 3 0 7 - 0 " 6 1 3 - I . 1 8 0

- 0 : 4 9 2 - 0 " 4 3 7 - 0 ' 3 7 0 - 0 - 2 8 8 - 0 " 1 9 2 - 0 - 0 8 8

- 0 " 0 4 0 - 0 -0 7 6 - 0 " 1 5 7 - 0" 3 31 - 0 " 6 9 7 - 1 ' 4 9 4

- 0 " 2 6 9 - 0 " 2 3 5 - 0 " 1 9 4 - 0 " 1 4 3 - 0 " 0 8 6 - 0 -0 5 1

- 0 ' 0 3 4 - 0 " 0 6 7 - 0 " 1 4 0 - 0 . 2 9 0 - 0 " 5 7 0 - 1 " 0 1 8

- 0 " 1 1 7 -- 0" 10 1 - 0 - 0 8 1 - 0 " 0 5 8 - 0 " 0 3 2 - 0 " 0 1 0

- 0 ' 0 2 5 - 0 . 0 5 0 - 0 " 1 0 3 - 0 " 2 0 7 -0 " 3 7 8 - 0 " 5 8 5

- 0 " 0 2 9 - @ 0 2 4 - 0 " 0 1 9 - 0 - 0 1 5 - 0 - 0 0 7 - 0 " 0 0 2

, 0 " 0 1 3 - 0 " 0 2 6 - 0 ' 0 5 4 - 0 " 1 0 6 - 0 - 1 8 4 - 0 " 2 6 3

0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 ' 0 0 0 0 "0 0 0

0 . 0 0 0 - - 0 0 00 ~ 0 0 0 0 ~ 0 O 0 0 Q 0 - 0 0 0 ~ 0 . 0 0 0 - - 0 - 0 0 ( ]

- I 2 6 6 - 1 .0 21 - 0 8 8 0

- 0 . 5 6 9 - 0 . 7 0 9 - 0 . 7 1 5

- 0 " 7 7 2 - 0 ' 6 5 2 - 0 - 5 6 9 - 0: 5 0 1 - 0 " 4 7 2

- 0 " 7 9 2 - 0 " 9 1 4 - 0 " 9 2 7 - 0 " 9 0 6 - 0 " 8 6 0

- 0 - 3 91 - 0 " 5 0 1 - 0 , 2 4 7 - 0 . 2 1 6 - 0 , 1 9 7

- 1 " 2 1 9 - 1 " 2 7 7 - 1 " 1 9 4 - 1 " 0 9 2 - 1 " 0 0 4

- 0 " 1 2 9 -0 " 0 7 9 - 0 " 0 5 9 - 0 " 0 5 0 - 0 " 0 4 5

- 2 " 0 9 8 - 1 "8 0 7 - 1 " 4 8 6 - 1 "2 7 4 - ] ' 1 4 2

- 4 ' 5 7 0 T - 2 " 5 9 5 - 1 "6 9 7 - 1 "4 1 7 - 1 " 2 8 4

- 3 " 4 6 0

- 1 460

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V a lu e s a r e w r i t te n V s l ~ , e x c e p t o n t h e b o u n d a r y w h e n o n l y ~ is g i v e n

F I O 8 F l o w p a s t p i t o t t u b e a t R e = 5

-2 "2 37 - 1 -957 - - 1 -620

- 0 "0 5 0 - O ' I 0 0 - 0 "25 3

- 1 " 6 65 - 1 "5 4 9 - I - 4 1 6 - 1 " 2 6 6 - 1 " 1 0 2

- 0 ' 0 4 6 - 0 "0 6 8 - 0 - 1 1 3 - 0 "1 9 7 - 0 " 3 4 0

- 1 - 1 8 7 - I - 0 9 0 - 0 " 9 7 6 - 0- ~,4 3 - 0 - 6 9 3

- 0 " 0 4 3 - 0 "0 7 1 - 0 " 1 2 8 - 0 " 2 4 4 - 0 " 4 5 9

- 0 " 7 9 9 - 0 - 7 2 4 - 0 " 6 3 2 - 0 - 5 2 2 - 0 - 3 9 4

- 0 - 0 41 - 0 " 0 7 2 - 0 ' 1 4 1 - 0 ' 2 8 9 - 0 " 5 9 1

- 0 " 4 9 7 - 0 "4 4 3 - 0 " 3 7 7 - 0 ' 2 9 7 -0 " 2 0 2

- 0 - 03 7 - 0 -0 6 9 - 0 " 1 4 1 - 0 " 2 9 9 - 0 " 6 4 7

-0 272 -0 240 -0 200 -0 152 0 097

-0 030 -0-058 -0 I19 -0 237 -0 439

-0 119 -0-I05 -0 085 -0 063 -0'039

- 0 0 2 2

- 0 "0 4 2 - 0 - 0 8 3 - 0 " 1 5 7 - 0 " 2 6 0

- 0 " 0 2 9 : - 0 '0 2 5 - 0 " 0 2 0 - 0 " 0 15 - 0 " 0 0 9

- 0 -0 1 1 - 0 " 0 2 2 - O ' 0 4 Z - 0 " 0 7 7 - 0 " 1 2 1

- I "268 - 1 "021 -0 "8 79

- 0 " 5 7 0 -0 . 7 1 1 - 0 . 7 1 6

- 0 . 9 3 1 - 0 " 7 7 4 - 0 . 6 5 3 - 0 " 5 6 9 - 0 ' 5 1 0 - 0 " 47 2

- 0 - 5 5 3 - 0 . 7 9 4 - 0 , 9 1 7 - 0 . 9 3 0 - 0 . 9 0 8 - 0 " 8 6 2

- 0 " 5 3 5 - 0 . 3 9 3 - 0 . 3 0 1 - 0 "2 4 7 - 0 - 2 1 6 - 0 - 1 9 6

- 0 . 8 0 8 - 1 ' 2 2 5 -~ 1" 28 4 - 1" 1 9 8 - 1 " 0 9 5 - I . 0 0 5

- 0 - 2 5 5 - 0 . 1 3 1 - 0 " 0 8 0 - 0 " 0 5 9 - 0 . 0 5 0 - 0 "0 4 5

- I . 1 7 3 - 2 " 1 1 8 - I . 819 - I - 4 9 1 - 1 " 2 7 5 - I ' r 4 2

- 0 " 09 5 ~ . ~ 4R 4 ~ 3 ~ - 2 " 40 6 - I - 6 9 5 -

1 414

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- 0 - 6 6 2 - 0 " 5 3 1 - -0 " 1 1 12 - 0 " 0 2 9 2 - 0 " 0 0 5 3 0 " 00 0 0

- 0 " 0 1 8 - 0 " 0 0 5 - 0 " 0 0 0 6 0 "0 00 3 0 - 00 03 0 0002

- 0 " 3 3 Z - - 0 ' 2 1 9 - - 0 " 0 4 2 - - 0 " 0 4 0 9 - 0 - 0 1 3 8 - 0 . 0 0 3 6

- 0 " 0 0 0 1 0 " 0 0 0 1 0 ' 0 00 1 0 " 0 00 1

- O- 0577 - O- 0258 - O- O0 99 - O- 0030

- 0 " 0 0 4 - 0 " 001

- 0 . 1 4 5 - 0 " 1 0 6

O ' O 0 0 Q O . 0 0 0 0 . 0 0 0_ _ .. 0 .0 0 0_ __ _ 0 . 0 0 0 O 0 0 0 O . O 0 0 q O . 0 0 0 O . 0 0 00 _ _. 0 0 0 00 _ _. 0 -0 0 00 ._ _

0 0 0 0 0 0 0 0

0 "0 0 0 0 "0 0 0 0 - 0 0 0 0 "0 0 0 0 - 0 0 0 0 "0 0 0 0 0 . 0 0 0 0 0 "0 0 0 0 0 .0 0 0 0

V a l u e s a r e w r i t t e n ~ / ~ p e xce p t o n t h e b o u n d a r y w h e n o n l y ~' is 5 i ve n

F r o 9 F l o w p a s t o p e n p i t o t t u b e a t R e = 5

19


21/25

3 ,0

2 .0

2 0

3 5

. o

0

Fm. 10.

0 ' 5

2 3 4

z i n c m

Pressure variation on stagnation streamline

for blunt-ended tube.

1 5

1.0

u

E

~9

U

0 - 5

0

I

1,0

Fm. 11.

2 . 0 3 0

z i n c m

Velocity variation on stagnation streamline for

blunt-ended tube.

i " , " " " : ' " : i ' ~ " ' " i 1 ' 1 : i ~: ' ' ' ' : , , ' i ; ' l " ' i ' ~ " l " " i i " " i l ' "i T , I : " ~ [ i r ' l l li ' i l i l " I ' :' l [ ' ~ : , ; ," l ' u ~ [ ' , ~ l " l " i i ' ' It ' ' ' : '' I l l ' l ,, , , l i ' l ' : l i , ' :' l i i ' r i i l " '1 1 l ; , j l ' il ' l ' ll ' e i l r ' i l i ~ i ' :: , ' i l , l , l r , ~ ' , ~ : l i " l i F I : "I ~ " ' l [ ' l l ' i I l ' i i 'l ' l l l i ' i : i l ' r i a ~ i ] l ? i ' " l l l " " i l i ' i ~ l [ ' r " ' l i " l ~ ,, , , i , ' , , i i : r ' , : i I l l l [ I r l l ' : r ' , r I l , l i , " r ' i r i ' i I i i,


22/25

R

FIG 12 Position of zero streamlines

O

Fic 13 Flow at

R e = 1

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24/25


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Documents

The Flow Past a Pitot Tube at Low Reynolds Numbers