Download Two-Dimensional Potential Flows

VII. Analysis of Potential Flows
Contents
1.
Preservation of Irrotationality
2.
Description of 2D Potential Flows
3.
Fundamental Solutions
4.
Superposition
1. Preservation of Irrotationality
r
n
Stokes Theorem
ò
S
r
r
n ×(Ñ ´ u )dS =
Vorticity
r
W
S
r r
òÑu ×dl
C
Circulation
G
C
Kelvin’s Theorem
In the flow of an ideal fluid with constant
density, circulation along a fluid line is
invariant if body force is conservative
t +t
t
C
D G(C )
= 0
Dt
DG D
=
Dt
Dt
r r
òÑu ×dl =
C
D r r
òÑC Dt u ×dl
(
r
r
D r r
Du r r D dl
u ×dl =
×dl + u ×
Dt
Dt
Dt
(
)
)
( )
r
æ
Du
1
pö
÷
ç
= ÑY - Ñ p = Ñ çY - ÷
÷
çè
÷
Dt
r
rø
r
dl
r
r1
r
r2
r
D dl
( )=
Dt
r
r
r
r
D r
(r2 - r1 ) = u 2 - u1 = du
Dt
DG
=
Dt
é æ
ö r r rù
p
÷
êÑ ççY - ÷×dl + u ×du ú
òÑC ê çè r ÷÷ø
ú
ë
û
æ
ö
r
r
p
1
÷
ççY = ò
d
+
u
×
u
÷
ÑC çè r 2 ø÷÷
= 0
A piece of fluid is always irrotational if it is
initially irrotational
2. Description of 2D Potential Flows
2D Flow in x-y plane
w= 0
¶
= 0
¶z
Basic Equations for 2D Potential Flows
¶u ¶v
+
= 0
¶x ¶y
¶v ¶u
= 0
¶x ¶y
ìï ¶ u
¶u
¶u
1 ¶p
ïï
+u
+v
= fx ïï ¶ t
¶x
¶y
r ¶x
í
ïï ¶ v
¶v
¶v
1 ¶p
+
u
+
v
=
f
ïï
y
¶
t
¶
x
¶
y
r ¶y
ïî
¶f
1 2
p
2
+ (u + v ) + - Y = C
¶t
2
r
Velocity Potential
Irrotational flow
¶v ¶u
= 0
¶x ¶y
Definition of Velocity Potential
¶f
u=
;
¶x
¶f
v=
¶y
Continuity Equation
¶u ¶v
+
= 0
¶x ¶y
¶ 2f
¶ 2f
+
= 0
2
2
¶x
¶y
Stream Function
Incompressible fluid
¶u ¶v
+
= 0
¶x ¶y
Definition of Stream Function
¶y
u=
;
¶y
¶y
v= ¶x
Irrotational condition
¶v ¶u
= 0
¶x ¶y
¶ 2y
¶ 2y
+
= 0
2
2
¶x
¶y
Properties of Stream Function
y = constant represents a streamline
y = constant
along y = y (x )
¶y
¶y
dy =
dx +
dy
¶x
¶y
= - vdx + udy = 0
dx
dy
=
u
v
Properties of Stream Function
y2
q
y1
q = y2 - y1
y2
B
r
u
r
n
A
y1
q=
ò
B
ò
B
ò
B
ò
B
A
=
A
=
A
=
A
r r
u ×n dl =
B
ò (un
A
x
+ vn y )dl
(udy - vdx )
æ¶ y
ö
çç dy + ¶ y dx ÷
÷
çè ¶ y
¶x ÷
ø
dy = y B - y A = y 2 - y 1
Properties of Stream Function
Streamlines and equipotential
lines are always perpendicular
to each other
Along a streamline
¶y
¶y
dy =
dx +
dy
¶x
¶y
= - vdx + udy = 0
ædy ö÷
v
çç ÷ =
èdx ø÷y
u
Along an equipotential line
¶f
¶f
df =
dx +
dy
¶x
¶y
= udx + vdy = 0
ædy ö÷
u
çç ÷ = èdx ø÷f
v
ædy ö÷
çç ÷
èdx ø÷f
ædy ö÷
çç ÷ = - 1
èdx ø÷y
Complex Potential
Cauchy-Riemann Condition
ìï
¶f
¶y
ïï u =
=
ïï
¶x
¶y
í
ïï
¶f
¶y
= ïï v =
¶y
¶x
ïî
Analytic Function
F (z ) = f (x , y ) + i y (x , y )
[z = x + iy ]
3. Fundamental Solutions
a.
Uniform flow
b.
Source and sink
c.
Vortex
d.
Doublet
a. Uniform Flow
u=U
y
v= 0
x
¶u ¶v
¶U
+
=
+ 0= 0
¶x ¶y
¶x
¶v ¶u
¶U
= 0= 0
¶x ¶y
¶y
¶f
¶y
=
= U
¶x
¶y
¶f
¶y
= = 0
¶y
¶x
f = Ux
y = Uy
¶f
1
p
+ (u 2 + v 2 ) + - Y = C
¶t
2
r
U2 p
+ - Y= C
2
r
p
U2
=C+ Y= C + Y
r
2
u = U cos a
v = U sin a
y
U

x
u = U cos a
v = U sin a
¶u ¶v
+
= 0
¶x ¶y
¶v ¶u
= 0
¶x ¶y
¶f
¶y
=
= U cos a
¶x
¶y
¶f
¶y
= = U sin a
¶y
¶x
f = U (x cos a + y sin a )
y = U (y cos a - x sin a )
b. Source and Sink
Source
In polar coordinates
u r = f (r )
uq = 0
y
u
r
v
ur
x = r cos q
y = r sin q
u = u r cos q - u q sin q
v = u r sin q + u q cos q
x
u
x2 + y2
x = r cos q
r =
y = r sin q
q = t an - 1 y x
¶r
x
= = cos q
¶x
r
¶r
y
= = sin q
¶y
r
( )
¶q
y
sin q
= - 2 =
¶x
r
r
¶q
x
cos q
= 2 =
¶y
r
r
¶u ¶v æ
¶r ¶
¶ q ¶ ö÷
+
= çç
+
(u r cos q - u q sin q)
÷
÷
è
ø
¶x
¶y
¶x ¶r
¶x ¶q
æ¶ r ¶
ö
¶q ¶ ÷
+ çç
+
÷
(u r sin q + u q cos q)
÷
çè¶ y ¶ r
¶ y ¶ qø
=
¶ ur
u
1 ¶ uq
+ r +
¶r
r
r ¶q
ö
¶v ¶u æ
¶r ¶
¶q ¶ ÷
= çç
+
(u r sin q + u q cos q)
÷
÷
è
ø
¶x ¶y
¶x ¶r ¶x ¶q
æ¶ r ¶
ö
¶q ¶ ÷
ç
- ç
+
÷
÷(u r cos q - u q sin q)
çè¶ y ¶ r
¶ y ¶ qø
=
¶ u q u q 1 ¶ ur
+
¶r
r
r ¶q
¶u ¶v
¶ ur
ur
1 ¶ uq
+
=
+
+
¶x
¶y
¶r
r
r ¶q
df (r ) f (r )
=
+
= 0
dr
r
f
(r ) =
c
r
¶v ¶u
¶ u q u q 1 ¶ ur
=
+
= 0
¶x ¶y
¶r
r
r ¶q
u r = f (r )
uq = 0
Discharge
Q=
ò
2p
ò
2p
0
=
0
= 2p c
u r rd q
c
rd q
r
Q
Qx
cos q =
2p r
2p (x 2 + y 2 )
Q
ur =
2p r
u=
uq = 0
Q
Qy
v=
sin q =
2p r
2p (x 2 + y 2 )
Q
f =
log r
2p
Q
y =
q
2p
Sink (Q < 0)
Source or Sink at (x0,y0)
Q
f =
log
2p
( (x - x ) + (y - y ) )
2
0
2
0
c. Vortex
Vortex
In polar coordinates
ur = 0
u q = f (r )
¶u ¶v
¶ ur ur
1 ¶ uq
+
=
+
+
= 0
¶x ¶y
¶r
r
r ¶q
¶v ¶u
¶ u q u q 1 ¶ ur
=
+
¶x ¶y
¶r
r
r ¶q
df
f
=
+ = 0
dr r
f
(r ) =
c
r
ur = 0
u q = f (r )
Circulation
G=
ò
2p
ò
2p
0
=
0
u qr d q
æ cö
çç- ÷
r dq
÷
è rø
= - 2pc
ur = 0
G
uq = 2p r
G
Gy
u=
sin q =
2p r
2p (x 2 + y 2 )
G
Gx
v= cos q = 2p r
2p (x 2 + y 2 )
G
f =
q
2p
G
y = log r
2p
Clockwise Vortex
(G < 0)
Vortex centered at (x0,y0)
G
y = log
2p
( (x - x ) + (y - y ) )
2
0
2
0
c. Doublet
y
Q
+Q

x

e® 0
Q® ¥
eQ ® pm
Velocity Potential
f =
Q
Q
log (x + e)2 + y 2 log
2p
2p
(x
- e )2 + y 2
(x + e )2 + y 2
Q
=
log
2
(x - e )2 + y
4p
é
æ e2 ÷
öù
Q
x
e
çç
ú
=
log ê1 + 4 2
+
O
÷
2
2
2
÷
ê
çèx + y øú
4p
x +y
ë
û
Q
=
4p
é xe
ù
æ e2 ö
mx
÷
ç
ê4
ú
+ Oç 2
® 2
÷
2
2
2
÷
ê x +y
çèx + y øú x + y 2
ë
û
y
r
Stream Function
A

Q
Q
Q
y = q1 +
q2 = [q1 - q2 ]
2p
2p
2p
Q AB
Q 2e sin q
; ; 2p r
2p
r
Q 2ey
my
= ® - 2
2
2
2p x + y
x + y2


B
x
Streamlines
my
y = - 2
=C
2
x +y
m
x +y + y= 0
C
2
2
2
2
m
m
æ
ö
æ
ö
÷
x 2 + ççy +
= çç ÷
÷
÷
è
2C ø è2C ø
4. Superposition
a. Circular Cylinder without Circulation
Uniform Flow
f = Ux
y = Uy
mx
x2 + y2
mö
æ
ç
= çUr + ÷
÷cos q
è
ø
r
f = Ux +
Doublet
mx
f = 2
x + y2
my
y = - 2
x + y2
my
x2 + y2
mö
æ
ç
= çUr - ÷
÷sin q
è
ø
r
y = Uy -
mö
æ
ç
y = çUr - ÷
sin q
÷
è
ø
r
y = 0
m
Ur = 0
r
Þ
r =
sin q = 0
Þ
q = 0, p
mU
æ a2 ÷
ö
m÷
æ
ö
ç
f = ççUr + ÷cos q = U çr + ÷
cos q
÷
ç
è
ø
r
r ø
è
æ a 2 ö÷
mö÷
æ
y = ççUr - ÷sin q = U ççr sin q
÷
÷
ç
è
rø
r ø
è
é a 2 (x 2 - y 2 )ù
2
æ
ö
¶f
a
ê
ú
÷
çç1 u=
= U ê1 =
U
cos 2q÷
ú
2
2
÷
2
2
¶x
ø
êë
(x + y ) úû çè r
¶f
2a 2xy
Ua 2
v=
= - 2
U = - 2 sin 2q
2 2
¶y
r
(x + y )
æu ö÷
çç ÷=
çèv ø÷
÷
écos q - sin qùæu r ö
÷
ê
úçç ÷
êsin q cos q úèçu q ø÷
÷
êë
úû
æu r ö÷
çç ÷=
÷
çèu q ø÷
é cos q sin q ùæu ö
÷
ê
úçç ÷
ê- sin q cos qúèçv ø÷
÷
êë
úû
æ a2 ÷
ö
ç
u r = U ç1 - 2 ÷
cos q
÷
çè
r ø
æ a 2 ö÷
u q = - U çç1 + 2 ÷
sin q
÷
çè
r ø
On surface of cylinder
Velocity
ur = 0
u q = - 2U sin q
u q = - 2U sin q
2U
Stagnation Point
2U
r
p + u ×u = C
2
Pressure
2
rU
2
2
p¥ +
= p + 2rU sin q
2
cp =
p - p¥
( )
1 2 rU 2
2
= 1 - 4 sin q
D’Alembert Paradox
r
F= 0
Drag due to viscosity
►
Skin friction
►
Form drag
b. Circular Cylinder with Circulation
Uniform Flow
y = Ur sin q
Doublet
m sin q
y = r
æ a2 ÷
ö
G
r
ç
y = U çr sin q log
÷
÷
çè
r ø
2p
a
Vortex
G
r
y = log
2p
a
æ a2 ÷
ö
u r = U çç1 - 2 ÷
cos q
÷
çè
r ø
æ a 2 ö÷
G
u q = - U çç1 + 2 ÷
sin
q
+
çè
r ø÷
2p r
On surface of the cylinder
ur = 0
G
u q = - 2U sin q +
2pa
Stagnation point on cylinder
ìï
ïï
G ïï
sin q =
í
4pUa ïï
ïï
ïî
< 1 2 solut ions
= 1
1 solut ion
> 1 no solut ion
r
p + u ×u = C
2
Pressure
2
ö
rU
ræG
÷
p¥ +
= p + çç
- 2U sin q÷
÷
ø
2
2 è2pa
2
cp =
2
æ G
ö
= 1 - çç
- 2 sin q÷
÷
÷
2
è
ø
2
p
Ua
1 2 rU
p - p¥
( )
Lift
r
F = -
r
p
p
n
(
¥ ) dS
ò
S
= -
ò
2p
0
r
r
(p - p¥ ) i cos q + j sin q a d q
(
rU 2a 2 p éê
= 1ò
ê
0
2
ë
r
= - rU G j
)
2ù
r
r
æ G
ö
ú i cos q + j sin q d q
çç
- 2 sin q÷
÷
÷
è2pUa
øú
û
(
)