2017 Spring Inviscid Flow 2017 Spring
INVISCID FLOW Week 8
Prof. Hyungmin Park
Multiphase Flow and Flow Visualization Lab.
Department of Mechanical and Aerospace Engineering Seoul National University
Joukowski Transformation
o Solutions for the flow around ellipses and a family of airfoils
– For zà±¥, z= z: the complex velocity in the two planes is the same far from the origin (identity mapping)
• if a uniform flow of a certain magnitude is approaching a body in the z-plane at some angle of attack, a uniform flow of the same magnitude and angle of attack will approach the corresponding body in the z-plane.
•
– Singularity point of transformation @ z= 0 (normally, inside the body and of no consequence)
– Critical points of transformation:
• Smooth curves passing through critical points may become corners in the transformed plane.
c2
z
z
= +
z
(c2: real number)/ 1 ( ) ( )
dz d
z
= ®W z =Wz
/ 0; = c dz d
z
=z
±2017 Spring Inviscid Flow
Joukowski Transformation
3 c
-c 2c
-2c
1 1
1 2 1 2
2 2
2 2 2
2 2
( ) 2( )
1 1 1 1
2 2 2 2
2
1 1
1 2 1 2
2 2
( ) ( ) 2
2 , 2
2 ,
, 2( )
i iv
i i v v
i iv
c c z c c
z c z c
z c c
R e e R
e e
R e e R
R v v
R
q q q
q
z z z
z z z
r r
r r
r q q
r
- -
æ ö
+ - - -
+ = - = ® + = çè + ÷ø
æ ö æ ö
= ç ÷ = ç ÷
è ø è ø
æ ö
® = ç ÷ - = -
è ø
Joukowski Transformation
o Therefore, a smooth curve passes through the point z= c, the corresponding curve in the zplane will form a knife-edge or cusp.
o Example: a circle centered at the origin of the z-plane (radius is c)
, 2 cos
iv iv iv
ce z ce ce c v x iy
z
= = + - = = +2017 Spring Inviscid Flow
Application of Joukowski Transformation
o Flow around Ellipses
– simple geometry of the circle, which we know details, will be placed in the z-plane, and the corresponding body in the z-plane will be
investigated
• c: real and positive
• Radius: a(> c), center at the origin
5
2
2 2 2
2 2
cos sin
cos , sin
iv
iv iv
ae z c
c c c
z ae ae a v i a v
a a a
c c
x a v y a v
a a
z z
z
-
= ® = +
æ ö æ ö
= + = ç + ÷ + ç - ÷
è ø è ø
æ ö æ ö
= ç + ÷ =ç - ÷
è ø è ø
Application of Joukowski Transformation
– Equation of ellipses in z-plane is obtained.
1) Complex potential for a uniform flow (U) approaching this ellipse at an angle of attack aßthe same flow to approach the circular cylinder in the z-plane
2 2
2 2 1
/ /
x y
a c a a c a
æ ö æ ö
\çè + ÷ø +çè - ÷ø =
x h z
-plane2
( ) a
F z U z x
æ ö
= ç + ÷
è ø
a
a
x
¢h
¢plane
z
¢ -2
( ) a , ' i
F z U z z ze a z
æ ö -
¢ = çè ¢+ ¢÷ø =
U
U
x
h
2017 Spring Inviscid Flow
Application of Joukowski Transformation
7
2
2 2
2 2 2
( )
0 :
2 2
i a i
F U e e
c z z
z z c c
a a
z z
z
z z z z
z
æ - ö
\ = ç + ÷
è ø
= + ® - + = = ± æ öç ÷è ø -
As zà¥, zàz:
2 2
2 2
z z
z
= + æ öç ÷è ø -c2 2
2
( ) 2
2 2
i a i i z z
F z U ze e e c
c
a a a
- -
é æ öæ æ ö öù
ê ç ÷ú
= êë +çè - ÷çøè - ç ÷è ø - ÷øúû
Application of Joukowski Transformation
2) Position of stagnation point
• In z-plane:
• Corresponding points in z-plane aeia
z
= ±2 2 2
cos sin
i c i c c
z ae e a i a
a a a
a -a æ ö
a
æ öa
= ± ± = ±ç + ÷ ± ç - ÷
è ø è ø
2
a
=p
a
=0 ???2017 Spring Inviscid Flow
Application of Joukowski Transformation
– Modified Joukouski Transformation
9
2 2
c , b
z
z
c ib zz
z z
= + = ® = -
2 2
2 2 1
/ /
x y
a b a a b a
æ ö +æ ö =
ç - ÷ ç + ÷
è ø è ø Equation of ellipses in z-plane
2 2
2
( ) 1 2
2 2
a z z
F z U z b
b
é æ öæ æ ö öù
ê ç ÷ú
= êë -çè + ÷çøè - ç ÷è ø + ÷øúû
Kutta Condition and Flat-plate Airfoil
o As a reminder,
– potential flow solution for flow around a sharp edge has a singularity (infinite velocity) at the edge itself àphysically impossible
o Two ways to correct this
– Place separation point at the edge, i.e., V= finite value
– Place stagnation point at the edge, i.e., V= 0 (Kutta Condition) o From the flow around an ellipse, if càa, then resulting ellipse in the z-
plane degenerates to a flat plate defined by the strip (-2a£x £2a).
Physically impossible to realize
2017 Spring Inviscid Flow
Kutta Condition and Flat-plate Airfoil
o Leading-edge: real airfoils have a finite thickness and so have a finite radius of curvature at the leading edge àwill not be a matter
o Trailing-edge: usually quite sharp
– if a circulation exists around the flat-plate and the magnitude of this circulation is enough to rotate the rear stagnation point to the trailing edge, this problem will be solved.
o Kutta condition
– For bodies with sharp trailing edges that are at small angles of attack to the free-stream, the flow will adjust itself in such a way that the rear stagnation point coincides with the trailing edge
11
Kutta Condition and Flat-plate Airfoil
o How to determine the amount of circulation?
4
p
Uasin (clockwise direction)a
G=sinqs 4 Ua p
=- G Recall
2
2 2
2
2 2 2
2 2
2 2
( ) 2 sin ln
, 2 2
( ) 2 sin ln 1
2 2 / 2 ( / 2) 2 2
i i
i i
F U e a e i Ua
a
a z z
z a
z z a e z z
F z U a e i a a a
z z a a
a a
a a
z z a z
z
z z
z
a
-
-
æ ö
= ç + ÷+
è ø
= + = + æ öç ÷ - è ø
ìé ù é æ öùü
ïê æ ö ú ê ç æ ö ÷úï
= íïîêë + ç ÷è ø - úû + + - + êë çè + ç ÷è ø - ÷øúûýïþ
2017 Spring Inviscid Flow
Kutta Condition and Flat-plate Airfoil
o Kutta-Joukowski Law (calculation of the lift force)
– Lift coefficient, CL
13
4 2 sin
L=
pr
U aa
2 2 2 sin ~ 2
0.5 0.5 (4 )
L
L L
C U l U a
p a pa
r r
= = =
(for small a)
Symmetrical Joukowski Airfoil
o A family of airfoils (in z-plane) obtained by Joukowski transformation from a series of circles in z-plane whose centers are slightly displaced from the origin àJoukowski Airfoils
– Variable: center and radius of the circle
o Let’s consider the displacement of the circle in real axis.
o As a reminder,
– if the circle passes through the two critical points of the Joukowski transformation, z= ±c, then a sharp edge is obtained in z-plane
o For the leading edge of the airfoil to have a finite radius of curvature and if there should be no singularities in the flow field, the point z= -cshould be inside the circle in z-plane.
o But, the circle should pass z= c for the trailing edge to be sharp.
Center:
Radius: (1 ), / ( 1)
m
a c m c m c z
e e
=-
= + = + = !
2017 Spring Inviscid Flow
Symmetrical Joukowski Airfoil
15
Symmetrical Joukowski Airfoil
o
o The chord length, (chord length is not affected by the shift of circle) 2
c
c m
z z ì =
í =- - î
Trailing edge Leading edge
2
( 2 )
1 2
z c
z c e c
e ì =
ïí =- + -
ï +
î c2
z z
= + z
2 2
( 2 ) 1 2 ( ) 2 ( ) 2
z =- c+ e -céë - e +O e ùû =- c O+ e !- c 4
l = c
m a R
v
[ ]
2 2 2 2 2
2 2 2
2
1/ 2
2
2 cos( ) 2 cos
( ) 1 2 cos
( ) 1 2 cos 1 cos ( )
1 (1 cos )
a R m Rm v R m Rm v
m m
c m R v
R R
m m
c m R v R v O
R R
R c v
p
e e
= + - - = + +
æ ö
+ = ç + + ÷
è ø
æ ö æ ö
+ = ç + ÷ = ç + + ÷
è ø è ø
\ = + -
2017 Spring Inviscid Flow
Symmetrical Joukowski Airfoil
o Then, from the Joukowski transformation
– To find the maximum thickness,
17
[ ]
[ ]
2 2
2 2
1 (1 cos )
1 (1 cos )
1 (1 cos ) 1 (1 cos ) ( ) 2cos 2 (1 cos )sin ( )
2 cos
2 (1 cos )sin
iv
iv
iv iv
iv iv
c c
z Re
Re
c v e ce
v
c v e c v O e
c v i v v O
x c v
y c v v
z z
e e
e e e
e e
e
-
-
= + = +
= + - +
+ -
é ù
= + - + ë - - + û
é ù
= ë + - + û
ì =
í = -
î
2
2 1 1
2 2
x x
y c
c c
eæ ö æ ö
= ± ç - ÷ -ç ÷
è ø è ø
max
2 4 3 3
0 ,
3 3 2
3 3
dy v y c
dv
t c
p p e
e
= ® = ® = ±
=
Symmetrical Joukowski Airfoil
o Ratio of the thickness to chord:
o Equation for airfoil
o Circulation for Kutta condition
o Lift force and lift coefficient
max 3 3 max
or 0.77 4
t t
l l
e e
= =
2
max
0.385 1 2 1 2
y x x
t l l
æ ö æ ö
= ± çè - ÷ø -çè ÷ø
4 sin 1 0.77tmax sin
Ua Ul
p a p æ l ö a
G = = çè + ÷ø
2 max
max
1 0.77 sin 2 1 0.77 sin
L
L U l t
l C t
l
pr a
p a
æ ö
= ç + ÷
è ø
æ ö
= çè + ÷ø
2017 Spring Inviscid Flow
Circular-Arc Airfoil
o We will show that a circle whose radius is slightly larger than cand whose center is displaced along the imaginary axis of the z-plane produces an airfoil that has no thickness but has a camber.
19
-c c
-2c 2c
Circular-Arc Airfoil
o
2 2 2 2
2 2
2 2
2 2 2 2 2 2 2 2
2 2 2
cos sin
sin cos sin cos sin cos
4 sin cos
iv iv
c c c c
z Re e R v i R v
R R R
c c
x v y v R v v R v v
R R
c v v
z z
- æ ö æ ö
= + = + =ç + ÷ + ç - ÷
è ø è ø
æ ö æ ö
- =ç + ÷ -ç - ÷
è ø è ø
= equation of the airfoil surface in the z-plane
m a R
v
2 2 2 2 2
2 2
2
2 2 2 2
2 2
2 2
2 cos( ) 2 sin
2
sin , cos 1
2 2 sin 2
2 4
4
a R m Rm v R m Rm v
R c y y
v v
Rm m v m
x y c m y c
m
c m c m
x y c c
m c m c
= + - p - = + -
= - = = -
ö + + æç - ÷ =
è ø
é ù
é æ öù æ ö
+êë + çè - ÷øúû = êêë +çè - ÷ø úúû
-c c
2017 Spring Inviscid Flow
Circular-Arc Airfoil
o Using and linearize the equation, we get
o Chord length:
o Camber height:
o Circulation to satisfy Kutta condition
o Lift force and lift coefficient
21
2 2 2
2 2
2
1
4 m
c
c c
x y c
m m
e =
æ ö æ ö
+ç + ÷ = ç + ÷
è ø è ø
!
4 l = c
2 h= m
2 2 2 2
2
1 2
8 4 16
l l l
x y
h h
æ ö æ ö
+ç + ÷ = ç + ÷
è ø è ø
4 sin m 4 sin m
Ua Uc
c c
p æa ö p æa ö
G = çè + ÷ø = çè + ÷ø
4 2 sin
8 sin 2 sin 2
L
L U c m
c
c m h
C l c l
pr a
p a p a
æ ö
= çè + ÷ø
æ ö æ ö
= çè + ÷ø= çè + ÷ø
Joukowski Airfoil
o Cambered airfoil of finite thickness
– Joukowski transformation in conjunction with a circle in the z plane whose center is in the second quadrant.
2017 Spring Inviscid Flow
Joukowski Airfoil
o Equation for the upper and lower surfaces of the airfoil
o Lift coefficient
o Complex potential
o Circulation for Kutta condition
23
2 2 2 2
2 2 max
1 0.385 1 2 1 2
4 16 8
l l l x x
y x t
h h l l
æ ö æ ö æ ö
= çè + ÷ø- - ± çè - ÷ø -çè ÷ø
Thickness effect Circular-arc centerline
max 2
2 1 0.77 sin
L
t h
C = pæçè + l ö÷ø æçèa + l ö÷ø
Thickness effect Camber effect
2
( ) ( ) ln
2
i i
i i
i
a e i me
F U me e
me a
a d
d a
d
z z z
z p
é - ù G æ - ö
= êë - + - úû+ çè ÷ø
max 2
1 0.77t sin h
Ul l l
p æ ö æa ö
G= ç + ÷ ç + ÷
è ø è ø
* Limit in increase of thickness and camber àstall
Schwarz-Christoffel Transformation
o maps the interior of a closed polygon in the z plane onto the upper half of the z-plane, while the boundary of the polygon maps onto the real axis of the z plane.
/ 1 / 1 / 1
( ) ( ) ( )
( 2)
dz K a b c
d
n
a p b p g p
z z z
z
a b g p
- - -
= - - -
+ + + = -
!
!
2017 Spring Inviscid Flow
Schwarz-Christoffel Transformation
25 o Example flow field
– Flow around a flat plate of finite length oriented vertically (90 °angle of attack)
1/ 2 1 1/ 2
2 2
( 1) ( 0) ( 1)
1 1 dz K d
K
z K D
z z z
z z z
z
- -
= + - -
= -
= - +
(D: constant of integral, complex)
1 0
1 0
0 2
z z z i a z
z z
= « = ìï =- « = íï = « = î
2 2 1
z a z
\ = -
Schwarz-Christoffel Transformation
o To find a magnitude of uniform velocity,
– As zà¥, zà2az andW( )z ®2aW z( ) 2= aU ( ) 2
F z aUz
\ = Complex potential for horizontal uniform flow
2
2 2 2 2
2 1
1 4 , ( ) 4
2 z
a
z a F z U z a a
z z
ö
= ± æçè ÷ø +
\ = + = +
From mapping function zà¥, zà¥
This result can be confirmed by the flow around an vertical ellipse using a modified Joukowski transformation
2017 Spring Inviscid Flow
Source in a Channel
27 o Flow generated by a line source located in a two-dimensional channel
– Complex potential in z-plane
z-plane z-plane
first quadrant only, source is located at z= 0
1/ 2 1/ 2
2 1
( 1) ( 1)
1 cosh
dz K
d K
z K D
z z
z z
z
- -
-
= + - =
-
= +
1 0
1 z
z il z
z
= « = ìí =- « = î
cosh 1 , cosh
l z
z l
z z p
p -
\ = =
( ) ln( 1), ( ) ln cosh 1
2 2
m m z
F F z
l
z z p
p p
æ ö
= - = çè - ÷ø
Source in a Channel
o Using cosh(X Y+ ) cosh(- X Y- ) 2sinh sinh= X Y
2
2
cosh 1 2sinh 2
( ) ln 2sinh ln sinh ln 2
2 2 2 2
ln sinh 2
z z
l l
m z m z m
F z l l
m z
l
p p
p p
p p p
p p
- =
æ ö æ ö
= çè ÷ø= çè ÷ø+
æ ö
= çè ÷ø
this term does not affect the velocity
4 m= Ul