二次元澱み点流れ
Navier-Stokes !"#$%&'(
)*+,-../012345678/ ,
:9;<=>?
@ABCDEF8
G"H
I
$J*<
+,
u = ax , v = −ay
(a > 0 )
(1)
&'(K( +#$%LM87NOP Navier-Stokes QR
u
 ∂ 2u ∂ 2u 
∂u
∂u
1 ∂p
+v
=−
+ν  2 + 2 
ρ ∂x
∂x
∂y
∂y 
 ∂x
(2)
u
 ∂ 2v ∂ 2v 
∂v
∂v
1 ∂p
+v
=−
+ν  2 + 2 
ρ ∂y
∂x
∂y
∂y 
 ∂x
(3)
J*<
+,STU+UVWXYZ@A[\
∂v
=0
∂x
(4)
∂2p
∂  ∂p  ∂  ∂p 
=
=
 =0
∂x ∂y ∂x  ∂y  ∂y  ∂x 
(5)
^]_`a,(3) x b>(4)cd +
feg +fhx ijkl^ y imn"\\!fo+,a
q_
∂p ( x , y )
∂x
= f (x )
(6)
"+,$(2)p$%n +o_r!p$%mstu,
X=
x
y
u
v
, Y= , U= , V=
xa
ya
ua
va
(7)
wvxp$%msyz{|*x a ^{u}^~€ ,(7)
(2)‚ƒ +
2
 u ∂ 2U ua ∂ 2U 
ua
∂U va ua ∂U
1 ∂p
+
=−
+ ν  a2
+ 2
U
V
2
2 
ρ ∂x
∂Y
∂
xa ∂X
ya
X
x
ya ∂Y 
 a
U
∂U va xa ∂U
x ∂p
ν ∂ 2U ν xa ∂ 2U
+
V
=− a2
+
+
∂X ya ua ∂Y
∂x xa ua ∂X 2 ya 2ua ∂Y 2
ρ
u
a
{
{
{
VW„…
[1]
[ 2]
(8)
[3]
y = 0 : u = v = 0

y → ∞ : u = ax , v = −ay
(9)
Y = 0 : U =V = 0

ax
ay
Y → ∞ : U = a X , V = − a Y
ua
va

{
{

[ 4]
[5]
(10)



vx
ν ν xa axa aya 

,
,
,
(U ,V ) = f  X ,Y , a a ,
ya ua xa ua ya 2ua ua va 

{ { { { {

[1]
[ 2]
[ 4]
[5] 
[3]

(11)
a\}p$%n +,
p$%34r†p$%‡ˆ +,
f
_[1];<[5]†p$%a\~€‰\u,
[4] = 1 tuf_ u = ax
a
a
[3] = 1
tuf_(12)‚ƒ y
[1] = 1
tuf_(12)(13)‚ƒ v
[2]
[5]
a
ν xa
=
ua
a
ν
xa ua
=
ν
ax a
aya
a
=
va
aν
2
≡
ν
a
1
Re
=
=
ν
a
ν a ⋅ax a
ya ua
=
= aν
xa
xa
(12)
(13)
(14)
‹ŠŒ;+^ffpŽ
=1
H<X = 1 tUST +,aq_(7)$!tU*<
+,
X=
u (x , y )
v (y )
x
y
a
= 1, Y = = y
≡ η , U (η ) =
, V (η ) =
xa
ya
ax
ν
aν
(15)
(15)‘d(2)†’“”+,
∂u ∂
∂U
dU ∂η
=
= aU + ax
= aU
(axU ) = aU + ax
∂x ∂x
∂x
dη ∂x
∂u ∂
dU ∂η
dU a
=
= ax
(axU ) = ax
∂y ∂y
dη ∂y
dη ν
∂ 2u
∂  ∂u  ∂
∂U
dU ∂η
=
=
=0
(aU ) = a = a


2
∂x
∂x  ∂x  ∂x
∂x
dη ∂x
‹ff^•–Re sŽ"u"+
∂ 2u ∂  ∂u  ∂  dU a 
a d 2U a a 2x d 2U
=
=
ax
=
ax
=


 
ν dη 2 ν
ν dη 2
∂y 2 ∂y  ∂y  ∂y  dη ν 
f
<(2)‚ƒ
axU ⋅aU + aνV ⋅ax
˜—$™+,
dU a
1 ∂p
a 2x d 2U
=−
+ν
dη ν
ρ ∂x
ν dη 2
f (x )
}
dU d U
1 ∂p
U 2 +V
− 2 =− 2
≡ −PX
dη dη
ρ
a x ∂x
144
42444
3 14243
2
function of y
(16)
function of x
f
;<Œ;+!šz y so_yz x so+;<›œI

^s"ž
Ÿ"<"\,yzH
+p$% X ijkl$ +,
PX ≡
k
1 ∂p f ( x )
= 2 = p 2 = const.
2
ρa x ∂x ρa x ρa
(17)
ff k ¡o_VWXZBCDEF8¢£;<‰”+f^U+,
(17) x ¤>$!j>¥^ x s y s¦/§H
+
f^H
+,
p
∂p
= kpx
∂x
1
⇒ p ( x , y ) = k p x 2 + p2 (y )
2
(18)
k “”+,BCDEF8@A[ž+ x iQR$J*<
+,
p
u
∂u
∂u
1 ∂p
+v
=−
∂x
∂y
ρ ∂x
f
(1)[¨(18)‚ƒ
ax ⋅a = −
1
ρ
⇒
kpx
k p = − ρa 2
(19)
p$%jkl(17);<
PX =
kp
− ρa 2
=
= −1
ρa 2 ρa 2
(20)
“q+,›œx iQR
dU
d 2U
U +V
= 1+ 2
dη
dη
2
(21)
q©ª †’^I

∂u
∂v
dV
= aU ,
=a
‹r€
∂x
∂y
dη
o+;<r!™<
+,
U+
dV
=0
dη
(22)
ª\(3)†’“”+,y iQRj“”+”«ž¬Œ
+^
)q ,
∂v ∂
∂V
dV ∂η
=
= aν
aνV ) = aν
=0
(
∂x ∂x
∂x
dη ∂x
∂v ∂
=
∂y ∂y
(
)
aνV = aν
∂V
dV ∂η
dV a
dV
= aν
= aν
=a
∂y
dη ∂y
dη ν
dη
∂ 2v
∂  ∂v  ∂  dV
=
=
a
2
∂y
∂y  ∂y  ∂y  dη

d 2V ∂η
a d 2V
=
a
=
a

dη 2 ∂y
ν dη 2

f
<›­(3)‚ƒ
ax ⋅ 0 + aνV ⋅a
˜— +
dV
1 ∂p
a d 2V
=−
+ν a
dη
ρ ∂y
ν dη 2
dV
1
∂p d 2V
V
=−
+
dη
ρa aν ∂y dη 2
j’®(18)‚ƒ +
2
dV
1
∂ 1
 dV
2
=−
k
x
+
p
y
+
(
)
p
2
 dη 2
dη
ρa aν ∂y  2
dV
1 dp2 (y ) d 2V
V
=−
+ 2
dη
dη
ρa aν dy
V
ff p (y ) = ρaν P (η ) tžŸ
2
dV
dP d 2V
V
=−
+
dη
dη dη 2
(23)
™+,¯°±p$% r²_o+,
a
η =y
ν
, u ( x , y ) = ax ⋅U (η ) , v (y ) = aν ⋅V (η ) ,
(24)
1
p ( x , y ) = − ρa 2x 2 + ρaν ⋅ P (η )
2
sq”
r©`³b>µ´¶"·u,
U +V ' = 0
V “”+
−U −VU '+ 1 +U '' = 0
U “”+
1
P = − V +V '
P “”+
2
2
2
B. C.
η = 0 : U =V = 0

η → ∞ : U = 1
"[(10)VW„…[\η → ∞ : V = −η "+^¸¹b>"
f
ºa»VW„…J*+fU"\,f„…"¼½™<
V
¡η ^=>wUu"+V ≅ −η + 0.648 o+¾¿ÀH δ (22)
Âq*r!"+,
1
∞


δ1 = ∫  1 −
0
u
ax
ν ∞
ν
ν
∞

 dy = a ∫0 (1 −U ) dη = a [η +V ]0 = 0.648 a

(25)
(1)J*<
BCDEF8
34>¥(24))Á +r!ÃÄ
H
+f^Œ;+,
u = ax , v = −ay + 0.648 aν = −a (y − 0.648 ν a ) = −a (y − δ )
(26)
f
_y i δ «žBCDEF8@A^EÅ7\+f^Œ;+,
Æs¡Çț­ ,η = 2.38 99ÊVWXÀHÉ +,:934kl U’
1.2326 oË,Ì345678/+
ÍÎÏn ,34&
'(Ðt;<:9ÑÒ"i‹x iÓwUu"+f^Œ;+^VWX
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1
1
ԛ­
2
1.75
1.5
-V
1.25
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1
0.75
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0.5
0.25
0
0
1
2
η
3
4
5
Æ 2 $%&'(
Çț­‹V P ÕÖ{ž/
0.15
0.2
0.25
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0.000000
0.500000
1.000000
1.500000
2.000000
2.500000
3.000000
3.500000
4.000000
4.500000
5.000000
/Æ s¡
U
0.000000
0.494650
0.777866
0.916169
0.973218
0.992852
0.998424
0.999716
0.999958
0.999995
1.000000
V
0.000000
-0.133579
-0.459217
-0.887317
-1.361961
-1.854415
-2.352543
-2.852160
-3.352096
-3.852087
-4.352086
P
0.000000
-0.503539
-0.883284
-1.309823
-1.900681
-2.712278
-3.765654
-5.067125
-6.618233
-8.419284
-10.470328