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t a r c i a s m a Be w p o r ó w n a n i u z s i Ba m i b e z w Ba d n o [c i . W o d a i p o w i e t r z e - p By n y , z k t ó r y m i
n a j c z [c i e j s p o t y k a m y s i w r ó |n y c h z a g a d n i e n i a c h t e c h n i c z n y c h - o d z n a c z a j s i m a B
l e p k o [c i , a z a t e m j u | p r z y s t o s u n k o w o n i e w i e l k i c h p r d k o [c i a c h o t r z y m u j e m y p o d c z a s i c h
p r z e p By w ó w d u |e w a r t o [c i l i c z b y R e y n o l d s a . W r ó w n a n i a c h N a v i e r a - S t o k e s a , w m i a r
z w i k s z a n i a s i l i c z b y R e y n o l d s a , w y r a z y z a l e |n e o d l e p k o [c i s t a j s i c o r a z m n i e j s z e w
p o r ó w n a n i u z w y r a z a m i b e z w Ba d n o [c i o w y m i , a d l a l i c z b R e y n o l d s a R e ®ð ¥ð , w y r a z y t e
d | d o z e r a , a r ó w n a n i a r u c h u n i e r ó |n i s i j u | o d r ó w n a D d o t y c z c y c h p By n u d o s k o n a Be g o .
W o d r ó |n i e n i u o d p By n u d o s k o n a Be g o ( a w i c n i e l e p k i e g o ) , w k t ó r y m t y l k o s k Ba d o w a
n o r m a l n a p r d k o [c i m u s i z n i k a n a n i e p r z e p u s z c z a l n e j [c i a n i e , w p By n i e l e p k i m r ó w n i e |
s k Ba d o w e s t y c z n e p r d k o [c i z n i k a j n a t e g o r o d z a j u [c i a n i e . W a r u n k i t e n i e z a l e | o d
p r d k o [c i p By n u w p u n k t a c h o d l e g By c h o d g r a n i c y . O z n a c z a t o , |e p o d c z a s p r z e p By w u p By n u
p s e u d o d o s k o n a Be g o ( t z n . z d u | p r d k o [c i ) s i By l e p k o [c i s d o m i n u j c e w p o b l i |u [c i a n y
s z t y w n e j ( l u b g r a n i c y r ó |n y c h p By n ó w ) , c h o c i a | w g Bó w n e j m a s i e p By n u d o m i n u j s i By
b e z w Ba d n o [c i . R o z w i z a n i e t e g o p r o b l e m u p o l e g a n a p o d z i e l e n i u c a Be g o o b s z a r u
p o r u s z a j c e g o s i p By n u n a d w a n i e r ó w n e p o d o b s z a r y ( r y s . 1 . ) i p r o w a d z e n i u r o z w a |a D
o s o b n o d l a k a |d e g o z n i c h . P o d z i a B t e n - z a p r o p o n o w a n y p r z e z P r a n d t l a - p o l e g a n a
w p r o w a d z e n i u p o d o b s z a r u , w k t ó r y m s i By l e p k o [c i s c a Bk o w i c i e p o m i j a l n e , o r a z d r u g i e g o , w
k t ó r y m i c h w p By w j e s t d e c y d u j c y .
R y s . 1 . O b r a z w a r s t w y p r z y [c i e n n e j .
W a r s t w p By n u p o r u s z a j c s i b l i s k o g r a n i c y o [r o d k ó w ( n p . [c i a n y ) n a z y w a s i
w a r s t w p r z y [c i e n n . P r z e z t o p o j c i e r o z u m i e m y z a r ó w n o w a r s t w p r z y [c i e n n t w o r z c s i
n a o p By w a n e j n i e p r z e n i k l i w e j p o w i e r z c h n i , j a k i w a r s t w g r a n i c z n t w o r z c s i w
p r z e p By w i e s w o b o d n y m o t o c z o n y m p r z e p By w e m p o t e n c j a l n y m ( s t r u g a z a t o p i o n a , [l a d z a
o p By w a n y m c i a Be m ) .
W w a r s t w i e p r z y [c i e n n e j w y s t p u j i n t e n s y w n e z m i a n y p r d k o [c i o d z e r a n a [c i a n i e d o
w a r t o [c i r ó w n e j p r d k o [c i p By n u p o z a n i . M i d z y w a r s t w p r z y [c i e n n a g Bó w n m a s p By n u
n i e m a w y r a zn e g o r o z g r a n i c z e n i a , t o t e | n i e m o |n a [c i [l e z d e f i n i o w a z a s i g u w a r s t w y .
Z w y k l e p r z y j m u j e s i , |e w a r s t w a p r z y [c i e n n a s i g a d o m i e j s c a , w k t ó r y m p r d k o [ j e s t o 1 %
m n i e j s z a o d p r d k o [c i p r z e p By w u p o t e n c j a l n e g o , t j . p r d k o [c i , j a k a u s t a l i Ba b y s i w t y m
p u n k c i e p o d c z a s p r z e p By w u p By n u d o s k o n a Be g o .
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k t ó r e j z m i a n a p r d k o [c i p r z e p By w u w k i e r u n k u p r o s t o p a d By m d o p o w i e r z c h n i [c i a n y j e s t w
p r z y b l i |e n i u r ó w n a z e r u . G r u b o [ t e j w a r s t w y n a r a s t a s t o p n i o w o w m i a r o d d a l a n i a s i ( w
k i e r u n k u p r z e p By w u ) o d k r a w d z i n a t a r c i a ( m i e j s c a p o d z i a Bu s t r u g o p By w a j c y c h c i a Bo ) - r y s .
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d o m i n u j n a d s i Ba m i l e p k o [c i i w z w i z k u z t y m p By n u w a |a s i z a d o s k o n a By . P o d c z a s
r o z w a |a n i a p r z e p By w u w o k ó B [c i a n o k s z t a Bt a c h r e g u l a r n y c h p r z y j m u j e s i , |e p o z a w a r s t w
p r z y [c i e n n p r z e p By w k s z t a Bt u j e s i t a k j a k w r u c h u p o t e n c j a l n y m .
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p By n u i r o z w i z a r ó w n a n i e r u c h u w p o s t a c i N a v i e r a - S t o k e s a . W o b e c m a Be j g r u b o [c i w a r s t w y
p r z y [c i e n n e j , w p o r ó w n a n i u z d Bu g o [c i o p By w a n e j [c i a n y ( dð / l <ð<ð1
) , r ó w n a n i a N a v i e r a -
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r ó w n a n i a m i P r a n d t l a .
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w a r s t w t r a k t u j e j a k o p o t e n c j a l n y .
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( r y s . 1 ) , b d c y m k i e r u n k i e m p r z e p By w u s w o b o d n e g o - i n i e s k o Dc z o n e j , w p o p r z e c z n y m
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s k Ba d n i k ó w r ó w n a D i o d r z u c a j c s k Ba d n i k i n i |s z e g o r z d u , u k Ba d r ó w n a D p r z y j m u j e p o s t a
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p r z y [c i e n n e j j e s t s t a Be w z d Bu | n o r m a l n e j d o o p By w a n e j p o w i e r z c h n i , m a w i c t s a m w a r t o [
n a p o w i e r z c h n i , c o i n a g r a n i c y w a r s t w y . W y n i k a s t d d a l s z y w n i o s e k , |e w p o d o b s z a r z e
w a r s t w y p r z y [c i e n n e j c i [n i e n i e j e s t w y Bc z n i e f u n k c j z m i e n n e j x , i d e n t y c z n z f u n k c j
o k r e [l a j c c i [n i e n i e n a p o w i e r z c h n i o p By w a n e j p By n e m d o s k o n a By m . W n i o s k i t e ,
s f o r m u Bo w a n e n a p o d s t a w i e r o z w a |a D t e o r e t y c z n y c h , z o s t a By p o t w i e r d z o n e l i c z n y m i
d o [w i a d c z e n i a m i .
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w b a r d z i e j o g ó l n y c h p r z y p a d k a c h d w u w y m i a r o w y c h p r z e p By w ó w , z t y m j e d n a k , |e p r o m i e D
k r z y w i z n y [c i a n y w k a |d y m j e j p u n k c i e j e s t w i e l o k r o t n i e w i k s z y o d g r u b o [c i w a r s t w y
p r z y [c i e n n e j w t y m p u n k c i e .
Z d r u g i e g o r ó w n a n i a ( 2 ) w y n i k a , |e c i [n i e n i e n i e z m i e n i a s i w p o p r z e k w a r s t w y
p r z y [c i e n n e j , a z a t e m , m i m o j e j i s t n i e n i a , c i [n i e n i e s t a t y c z n e m o |n a m i e r z y b e z p o [r e d n i o n a
o p By w a n e j p o w i e r z c h n i .
M e t o d a r o z w i z y w a n i a z a g a d n i e D p r z e p By w ó w p By n ó w l e p k i c h , z w y k o r z y s t a n i e m
k o n c e p c j i w a r s t w y p r z y [c i e n n e j i r ó w n a n i a P r a n d t l a , p o l e g a n a :
·ð W y z n a c z e n i u p r z e p By w u p By n u n i e l e p k i e g o w o k ó B c i a Ba , a s z c z e g ó l n i e r o z k Ba d u
c i [n i e n i a p ( x ) n a [c i a n i e .
·ð R o z w i z a n i u r ó w n a D P r a n d t l a z u w z g l d n i e n i e m w y z n a c z o n e g o r o z k Ba d u c i [-
n i e n i a i o d p o w i e d n i c h w a r u n k ó w b r z e g o w y c h .
T a k a d r o g a p o s t p o w a n i a j e s t w p r a w d z i e d o k Ba d n a , a l e d o [ u c i |l i w a . N i e b d z i e
z a t e m p r z e d s t a w i o n e r o z w i z a n i e u k Ba d u r ó w n a D P r a n d t l a , a j e d y n i e z o s t a n n a s z k i c o w a n e
t y l k o g Bó w n e w y n i k i t e g o r o z w i z a n i a .
R o z w i z a n i e u k Ba d u r ó w n a D ( 3 ) u m o |l i w i a z n a l e z i e n i e r o z k Ba d u p r d k o [c i w w a r s t w i e
p r z y [c i e n n e j , a s t d n a p r |e n i a s t y c z n e g o . W p r z y p a d k u o p By w u [c i a n y p Ba s k i e j ( l u b o
n i e w i e l k i e j k r z y w i zn i e ) m o |n a p o k a z a , |e n a p r |e n i a s t y c z n e w w a r s t w i e p r z y [c i e n n e j
p r z y j m u j p o s t a ( s p o s ó b w y p r o w a d z e n i a t e j z a l e |n o [c i z n a j d u j e s i w d o d a t k u A d o
w i c z e n i a ) . J e s t t o t z w . z a l e |n o [ c a Bk o w a K a r m a n a
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C e l e m w i c z e n i a j e s t b a d a n i e w a r s t w y p r z y [c i e n n e j p r z y o p By w i e p Ba s k i e j p By t k i ,
w y z n a c z e n i e r o z k Ba d u p r d k o [c i w t e j w a r s t w i e o r a z w y z n a c z e n i e n a p r |e D s t y c z n y c h n a
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a ) W y z n a c z y p r o f i l e p r d k o [c i u ( y ) w f u n k c j i w y s o k o [c i y w w a r s t w i e
p r z y [c i e n n e j w z d Bu | p o w i e r z c h n i p Ba s k i e j p By t k i , g r u b o [ w a r s t w y p r z y [c i e n n e j
n a k a |d e j s t a c j i p o m i a r u o r a z l o k a l n l i c z b R e y n o l d s a ,
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b ) W y z n a c z y p r o f i l e p r d k o [c i w b e z w y m i a r o w e j f o r m i e v e r s u s
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w i c z e n i e p r z e p r o w a d z a n e j e s t z w y k o r z y s t a n i e m t u n e l u a e r o d y n a m i c z n e g o
e d u k a c y j n e g o A e r o l a b t y p u o t w a r t e g o . W k o m o r z e t e s t o w e j t u n e l u n a l e |y u m i e [c i p Ba s k
p By t k w y p o s a |o n w s p e c j a l n y p r ó b n i k ( 1 0 - k a n a Bo w y m a n o m e t r ) t z w . " m o u s e " . O t w a r t e
k o Dc ó w k i p r ó b n i k a " m o u s e " u m i e s z c z o n e s k o l e j n o n a 0 . 0 1 8 , 0 . 0 2 5 , 0 . 0 3 0 , 0 . 0 4 0 , 0 . 0 6 0 ,
0 . 1 0 0 , 0 . 1 2 0 , 0 . 1 6 0 , 0 . 2 0 0 c a l a o d p o w i e r z c h n i p By t y . D r u g i e k o Dc ó w k i p r ó b n i k a n a l e |y
p o d p i n a d o m u l t i - m a n o m e t r u t u n e l u . D r u g i e g a Bz k i m a n o m e t r u s o t w a r t e d o a t m o s f e r y w
z w i z k u z t y m m a n o m e t r m i e r z y r ó |n i c c i [n i e n i a c a Bk o w i t e g o n a p r z e k r o j u p o m i a r o w y m i
c i [n i e n i a a t m o s f e r y c z n e g o . N a l e |y p a m i t a o o d p o w i e d n i e j k o n w e n c j i z n a k o w e j
o d c z y t y w a n i a w y s o k o [c i s Bu p k ó w c i e c z y w m a n o m e t r a c h . W y s o k o [c i p o w y |e j p r z y j t e g o
p o z i o m u z e r o w e g o b i e r z e m y z e z n a k i e m ( - ) ( p o d c i [n i e n i e w s t o s u n k u d o c i [n i e n i a
a t m o s f e r y c z n e g o ) n a t o m i a s t w y s o k o [c i p o n i |e j p o z i o m u z e r o w e g o z e z n a k i e m ( + )
( n a d c i [n i e n i e ) . A b y o b l i c z y c i [n i e n i e d y n a m i c z n e w [l a d z i e a e r o d y n a m i c z n y m o t r z y m a n e
w y n i k i p o m i a r u c i [n i e n i a n a l e |y o d n i e [ d o c i [n i e n i a s t a t y c z n e g o s e k c j i t e s t o w e j ( c i [n i e n i e
d y n a m i c z n e j e s t r ó |n i c m i d z y c i [n i e n i e m c a Bk o w i t y m a c i [n i e n i e m s t a t y c z n y m ) . D o
o t r z y m a n y c h w y n i k ó w n a l e |y d o d a w i c r ó |n i c m i d z y c i [n i e n i e m o t o c z e n i a a c i [n i e n i e m
s t a t y c z n y m s e k c j i t e s t o w e j Dðp P 0 -ð1 =ð p 0 -ð p s t . R ó |n i c t y c h c i [n i e D o d c z y t u j e m y k o r z y s t a j c
z s o n d y P r a n d t l a u m i e s z c z o n e j w s e k c j i t e s t o w e j t u n e l u . O t r z y m a n e w y n i k i b d w ó w c z a s
s t a n o w i By c i [n i e n i e d y n a m i c z n e p d 3 = ”p + ”p P 0 - 1 = ( p - p 0 ) + ( p 0 - p s t ) = p - p s t . . Z a p o m o c s o n d y
P r a n d t l a w y z n a c z a m y d o d a t k o w o c i [n i e n i e d y n a m i c z n e p r z e p By w u n i e z a k Bó c o n e g o
p d ¥ð =ð 0 , 0 2 5 rðH O g Dðz P 1 -ð2 .
2
N a l e |y u s t a w i p r d k o [ n a w a r t o [ 1 5 - 2 0 m / s i p r z e p r o w a d z i p o m i a r y d l a 3 p o z y c j i
z d e f i n i o w a n e j n a p r o w a d n i c y p r ó b n i k a " m o u s e " . W y n i k i p o m i a r u w r a z z o d l e g Bo [c i
o t w o r ó w p o m i a r o w y c h s o n d y o d k r a w d z i p By t y n a l e |y z a p i s a w t a b e l c e w i c z e n i o w e j .
5 . O p r a c o w a n i e w y n i k ó w p o m i a r u .
W y n i k i p o m i a r ó w n a l e |y p r z e d s t a w i w f o r m i e w y k r e s u p r d k o [c i p r z e p By w u u ( y )
v e r s u s o d l e g Bo [ o d p o w i e r z c h n i p By t y y d l a k a |d e j s t a c j i p o m i a r o w e j ( z a d a n i e a ) ) .
P r z y k Ba d o w y w y k r e s t e g o r o d z a j u d l a j e d n e j s t a c j i p o m i a r o w e j p r z e d s t a w i a r y s . 2 .
O t r z y m a n e p r o f i l e p r d k o [c i o d p o w i a d a j c e p o s z c z e g ó l n y m s t a c j o m n a l e |y p r z e d s t a w i w
f o r m i e w y k r e s y z r y s . 1 .
y
v x
R y s . 2 . P r z y k Ba d r o z k Ba d u p r d k o [c i w w a r s t w i e p r z y [c i e n n e j .
D l a k a |d e j s t a c j i p o m i a r o w e j n a l e |y w y z n a c z y g r u b o [ w a r s t w y p r z y [c i e n n e j o r a z
l o k a l n l i c z b R e y n o l d s a . - w z o r y ( 7 ) i ( 6 ) .
W z a d a n i u b ) n a l e |y w y z n a c z y
p r o f i l e p r d k o [c i w b e z w y m i a r o w e j f o r m i e
y / dð u / V ¥ð
v e r s u s ( z a d a n i e b ) ) . P r z y k Ba d
t e g o t y p u w y k r e s u z o s t a B p r z e d s t a w i o n y
n a r y s . 3 .
R y s . 3 . P r z y k Ba d p r o f i l u p r d k o [c i w
w a r s t w i e p r z y [c i e n n e j p r z e d s t a w i o n y w
y / dð
f o r m i e b e z w y m i a r o w e j v e r s u s
u / V ¥ð
.
Z a d a n i u c ) p o l e g a n a w y z n a c z e n i u n a p r |e D s t y c z n y c h n a p o w i e r z c h n i p By t k i z
p r z y b l i |o n e g o c a Bk o w e g o w z o r u K a r m a n a . W a r t o [ p o c h o d n e j w y z n a c z a m y w s p o s ó b
p r z y b l i |o n y k o r z y s t a j c z i l o r a z u r ó |n i c o w e g o d l a d w ó c h k o l e j n y c h s t a c j i p o m i a r o w y c h .
Wðn -ð Wðn -ð1
tð =ð
( 9 )
0
Dðx
dð
g d z i e : Wðn =ð rðu ( V ¥ð -ð u ) d y - c a Bk a w y s t p u j c a w e w z o r z e K a r m a n a w y z n a c z o n a d l a
òð
0
n - t e j s t a c j i p o m i a r o w e j ,
Dðx - o d l e g Bo [ p o m i d z y d w i e m a k o l e j n y m i s t a c j a m i p o m i a r o w y m i .
W a r t o [ c a Bk i w y s t p u j c e j w e w z o r z e K a r m a n a m o |e m y w y z n a c z y s t o s u j c n p .
m e t o d t r a p e z ó w .
dð
n -ð1
yð -ðyði
i +ð1
rðu ( V ¥ð -ð u ) d y »ð rð +ð ) Dðy i ( 1 0 )
åð( yð i
òð
2
i =ð1
0
g d z i e : yði =ð u ( V ¥ð -ð u i ) - i l o c z y n w y s t p u j c y p o d c a Bk w e w z o r z e K a r m a n a ;
Dðy i =ð ( y i +ð1 -ð y i ) - r ó |n i c a o d l e g Bo [c i m i d z y d w o m a k o l e j n y m i p r ó b n i k a m i
s o n d y " m o u s e " .
O b l i c z e n i a n a p r |e D s t y c z n y c h n a l e |y w y k o n a d l a d w ó c h p o c z t k o w y c h i d w ó c h
k o Dc o w y c h s t a c j i p o m i a r o w y c h
L i t e r a t u r a
1 . K a b s c h K , H . S z e w c z y k : M e c h a n i k a P By n ó w , O f i c y n a W y d a w n i c z a P o l i t e c h n i k i W r o c Ba w s k i e j ,
W r o c Ba w 2 0 0 1 .
2 . B u k o w s k i J , K i j k o w s k i P . : K u r s m e c h a n i k i p By n ó w , P W N , W a r s z a w a 1 9 8 0 .
P a Ds t w o w a W y |s z a S z k o Ba Z a w o d o w a w C h e Bm i e
L a b o r a t o r i u m M e c h a n i k i P By n ó w
L a b o r a t o r i u m m e c h a n i k i p By n ó w , w i c z e n i e n r 5
T e m a t w i c z e n i a : B a d a n i e p r z e p By w u w w a r s t w i e p r z y [c i e n n e j . "
I m i i n a z w i s k o G r u p a S e m e s t r / r o k a k a d e m i c k i
P r o w a d z c y D a t a w y k o n a n i a w i c z e n i a / g o d z . O c e n a
Z a d a n i e d o w y k o n a n i a :
1 . W y z n a c z y p r o f i l e p r d k o [c i u ( y ) w f u n k c j i w y s o k o [c i y w w a r s t w i e p r z y [c i e n n e j
w z d Bu | p o w i e r z c h n i p Ba s k i e j p By t k i , g r u b o [ w a r s t w y p r z y [c i e n n e j n a k a |d e j s t a c j i
p o m i a r u o r a z l o k a l n l i c z b R e y n o l d s a ,
y / dð u / V ¥ð
2 . W y z n a c z y p r o f i l e p r d k o [c i w b e z w y m i a r o w e j f o r m i e v e r s u s
3 . W y z n a c z y n a p r |e n i a s t y c z n e n a p o w i e r z c h n i p By t k i k o r z y s t a j c z c a Bk o w e g o
w z o r u K a r m a n a
V ¥ð
t o t = . . . . . . . [ ° C ] ; p o t = . . . . . . . . . . . [ P a ] ; q = . . . . . . [ P a ] ; = . . . . . . [ m / s ] ;
rð =ð . . . . . rðH O =ð1 0 0 0
[ k g / m 3 ] ; [ k g / m 3 ] ;
p o w
2
Dðp P 0 -ð1 =ð p 0 -ð p ¥ð =ð 0 , 0 2 5 rðg Dðz P 0 -ð1 =ð . . . . . . . . . . ;
Dðz P 0 -ð1 - w y s o k o [ s Bu p a c i e c z y m a n o m e t r u o d p o w i a d a j c a r ó |n i c y c i [n i e n i a
a t m o s f e r y c z n e g o i c i [n i e n i a s t a t y c z n e g o s e k c j i t e s t o w e j [ c a l ] .
T a b e l a w i c z e n i o w a d o z a d a n i e a ) i b )
S t a c j a p o m i a r o w a n r 1 . x 1 = . . . . . . . [ m m ] ;
p i D y n =ð
2 p i d y n
yð =ð Dðy i
u i =ð yð -ðyð
i
i +ð1 i
rð ( yð +ð ) Dðy i
y i Dðz 0 , 0 2 5 rðH g Dðz +ð Dðp P 0 -ð1 p o w
u ( V ¥ð -ð u i )
L p O =ð y i +ð1 -ð y i i 2
u i / V ¥ð
2
.
[ c a l ]
[ m ] [ m 2 / s 2 ] [ m 3 / s 2 ]
[ m ]
[ m / s ]
[ p a ]
1
2
3
4
5
6
7
8
9
1 0 - - - - - - - - - - - - - - - -
i =ð9
åð
i =ð1
S t a c j a p o m i a r o w a n r 2 . x 2 = . . . . . . . [ m m ] ;
p i D y n =ð
2 p i d y n
yð =ð Dðy i
u i =ð yð -ðyð
i
i +ð1 i
rð ( yð +ð ) Dðy i
y i Dðz 0 , 0 2 5 rðH g Dðz +ð Dðp P 0 -ð1 p o w
u ( V ¥ð -ð u i )
L p O =ð y i +ð1 -ð y i i 2
2 u i / V ¥ð
.
[ c a l ]
[ m ] [ m 2 / s 2 ] [ m 3 / s 2 ]
[ m ]
[ m / s ]
[ p a ]
1
2
3
4
5
6
7
8
9
1 0 - - - - - - - - - - - - - - - -
i =ð9
åð
i =ð1
S t a c j a p o m i a r o w a n r 3 . x 3 = . . . . . . . [ m m ] ;
p i D y n =ð
2 p i d y n
yð =ð Dðy i
u i =ð yð -ðyð
i
i +ð1 i
rð ( yð +ð ) Dðy i
y i Dðz 0 , 0 2 5 rðH g Dðz +ð Dðp P 0 -ð1 p o w
u ( V ¥ð -ð u i )
L p O =ð y i +ð1 -ð y i i 2
2 u i / V ¥ð
.
[ c a l ]
[ m ] [ m 2 / s 2 ] [ m 3 / s 2 ]
[ m ]
[ m / s ]
[ p a ]
1
2
3
4
5
6
7
8
9
1 0 - - - - - - - - - - - - - - - -
i =ð9
åð
i =ð1
D o d a t e k d o w i c z e n i a A .
R o z w a |m y u k Ba d r ó w n a D P r a n d t l a o p i s u j c y p a r a m e t r y p r z e p By w u w w a r s t w i e
p r z y [c i e n n e j
¶ðv x ¶ðv x 1 ¶ðp ¶ð2 v x
v x +ð v y =ð -ð +ðnðl e p k o s
A 1
¶ðx ¶ðy rð ¶ðx ¶ðy 2
¶ðv x ¶ðv y
+ð =ð 0
A 2
¶ðx ¶ðy
Z r ó w n a n i a B e r n o u l l i e g o d l a p r z e p By w u n i e z a k Bó c o n e g o ( p o z a w a r s t w p r z y [c i e n n )
m a m y
1
2
p +ð rðV ¥ð =ð c o n s t
A 3
2
r ó |n i c z k u j c t o w y r a |e n i e w z g l d e m w s p ó Br z d n e j x
¶ðV ¥ð 1 ¶ðp
V ¥ð =ð -ð
A 4
¶ðx rð ¶ðx
Z z a l e |n o [c i N e w t o n a w i e m y , |e n a p r |e n i a s t y c z n e s p r o p o r c j o n a l n e d o g r a d i e n t u
p r d k o [c i w k i e r u n k u p r o s t o p a d By m d o p r z e p By w u
æð d v x öð
tð =ð -ðmðçð ÷ð
A 5
çð ÷ð
d y
èð øð
mð =ð rðnðl e p k o s c
r ó |n i c z k u j c t o w y r a |e n i e w z g l d e m x o r a z p a m i t a j c , |e o t r z y m u j e m y
1 ¶ðtð ¶ð2 v x
=ðnðl e p k o s c
A 6
rð ¶ðy ¶ðy 2
p o d s t a w i a j c o t r z y m a n e w y r a |e n i a d o z a l e |n o [c i A 1 o r a z c a Bk u j c o b i e s t r o n y p o
z m i e n n e j y o t r z y m u j e m y
h
éð ¶ðv x ¶ðv x ¶ðV ¥ð ùð 1
A 7
x
êðv ¶ðx +ð v y ¶ðy -ðV ¥ð ¶ðx úðd y =ð -ð tð0
òð
0
rð
ëð ûð
C a Bk u j c r ó w n a n i e c i g Bo [c i A 2 m a m y
h
¶ðv x
æð öðd y
v y =ð -ð
çð ÷ð
A 8
òð
0
¶ðx
èð øð
R o z w a |m y t e r a z d r u g i s k Ba d n i k z a l e |n o [c i A 7 . C a Bk u j c g o p r z e z c z [c i o t r z y m u j e m y
h h ¶ðv y
¶ðv x
h
v y d y =ð [ðv x v y ]ð -ð v x d y
A 9
òð òð
0
0 0
¶ðy ¶ðy
W y k o r z y s t u j c w a r u n k i b r z e g o w e :
v x =ð v y =ð 0 y =ð 0
d l a
o r a z
h
¶ðv x
æð öðd y
çð ÷ð
v x =ð V ¥ð v y =ð -ð 0 y =ð h
i d l a
òð
¶ðx
èð øð
o t r z y m u j e m y
h h h ¶ðv y
¶ðv x ¶ðv x
v y d y =ð -ðV ¥ð 0 æð öðd y -ð v x d y
çð ÷ð
A 1 0
òð òð òð
0 0
¶ðy ¶ðx ¶ðy
èð øð
a w y k o r z y s t u j c r ó w n a n i e c i g Bo [c i A 2 m a m y
h h h
¶ðv x ¶ðv x ¶ðv x
v y d y =ð -ðV ¥ð 0 æð öðd y +ð v x d y
çð ÷ð
A 1 1
òð òð òð
0 0
¶ðy ¶ðx ¶ðx
èð øð
P o d s t a w i a j c t o w y r a |e n i e d o A 7 o t r z y m u j e m y
h
¶ðv x ¶ðv x ¶ðV ¥ð 1
éð2 v +ðV ¥ð -ðV ¥ð ùðd y =ð -ð tð
A 1 2
x 0
òð êð úð
0
¶ðx ¶ðx ¶ðx rð
ëð ûð
m o |n a z a u w a |y , |e
¶ðv x ¶ðv x 2
2 v x =ð A 1 3
¶ðx ¶ðx
o r a z
¶ðv x V ¥ð ¶ðV ¥ð ¶ðv x ¶ðv x ¶ðv x V ¥ð ¶ðV ¥ð
=ð v x +ðV ¥ð s t d V ¥ð =ð -ð v x A 1 3
¶ðx ¶ðx ¶ðx ¶ðx ¶ðx ¶ðx
p o d s t a w i a j c t o d o A 1 2 i o d p o w i e d n i o g r u p u j c o t r z y m u j e m y
h h
¶ð d V ¥ð 1
[ðv x ( V ¥ð -ð v x ) ]ðd y +ð ( V ¥ð -ð v x ) d y =ð tð A 1 4
0
òð òð
0 0
¶ðx d x rð
J e s t o g ó l n a p o s t a z a l e |n o [c i K a r m a n a d l a p r z e p By w u w w a r s t w i e p r z y [c i e n n e j .
D l a o p By w u [c i a n y p Ba s k i e j z a k Ba d a m y , |e p r d k o [ s t r u m i e n i a n i e z a k Bó c o n e g o n i e
V ¥ð =ð c o n s t
z m i e n i a s i . S t d d l a t a k i e g o p r z y p a d k u w z ó r K a r m a n a u p r a s z c z a s i
h
d
tð =ð rðu ( V ¥ð -ð u ) d y A 1 5
0
òð
0
d x
u =ð v x
g d z i e :
y >ð dð
P o n i e w a | d l a ( d l a o b s z a r u p o z a w a r s t w p r z y [c i e n n a ) w a r t o [ c a Bk i j e s t s t a Ba
m o |e m y o s t a t e c z n i e z m i e n i g r a n i c c a Bk o w a n i a
dð
d
tð =ð rðu ( V ¥ð -ð u ) d y A 1 5
0
òð
0
d x
P o w y |s z z a l e |n o [ w y p r o w a d z i m o |n a r ó w n i e | s t o s u j c z a s a d z a c h o w a n i a
s t r u m i e n i a p d u .
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