Transcript Chap.2 - 永達技術學院

Chap.5
Incompressible Flow over
Finite Wings
OUTLINE
 Downwash and induced drag
 The Biot-Savart law and Helmholtz’s
theorems
 Prandtl’s classical Lifting-line theory
 Elliptical lift distribution
 General lift distribution
Downwash and induced drag
Aerodynamic difference
between finite wing and
airfoil

For finite wing, the flow
near wing tips tends to
curl around the tip, being
forced from the highpressure just underneath
the tips to the lowpressure region on top.


Due to the spanwsie component of flow from tip
toward to root, the streamlines over the top
surface are bent toward root. In contrast, the
streamlines over bottom surface toward tip.
A trailing vortex is created at each win tip.
Effect of downwash


Wing-tip vortices
downstream of the wing
induce a small component
of air velocity, called
downwash which is
denoted by the symbol w.
Downwash causes
inclining the local relative
wind in the downward
direction.



Effective angle of attack
 eff     i
The tilting backward of the lift vector induce a
drag, called induced drag Di which is a type of
pressure drag.
Total drag = Profile drag + Induce drag, therefore
Di
C D  c d  C D , i where C D , i 
q S
The Biot-Savart law and Helmholtz’s
theorems
The Biot-Savart law

The velocity at point P, dV, induced by a small
directed segment dl of a curved filament with
strength  is
 dl  r
dV 

4
r
3
The velocity at P by a
straight vortex filament
of infinite length is
V 

 dl  r

4

r
3

The magnitude of V
V 


4

r 
where


sin 
r
2
dl 

2 h
h
sin 
The velocity at P by a semiinfinite vortex filament
V 

4 h
Holmholtz’s vortex theorem


The strength of a vortex filament is constant along
its length.
A vortex filament cannot end in a fluid; it must
extend to the boundaries of the fluid (which can
be ) or form a closed path.
Lift distribution

Different airfoil sections may have geometric and
aerodynamic twist, that results in a lift distribution
along the span.
Prandtl’s classical Lifting-line theory
Horseshoe vortex


Horseshoe vortex
consists of a bound
vortex and two free
vortex.
The bound vortex
induces no velocity
along itself, however,
the two free vortices
contribute to the
downward velocity
along the bound
vortex.
Downwash

Downwash at point y along the bound vortex is
w( y)  

4  (b / 2  y )


4  (b / 2  y )
Lifting-line theory

Instead of a single
horseshoe vortex,
infinite number of
horseshoe vortices
with a vanishing
small strength d are
superimposed to form
the bound vortices a
single line which is
called lifting line.


The trailing vortices become a continuous vortex
sheet trailing downstream of the lifting line.
The velocity w induced at y0 by the entire trailing
vortex sheet is
1 b / 2 ( d  / dy ) dy
w( y0 )  

4

b / 2
y0  y
The induce angle of attack is
 i ( y 0 )  tan
  i ( y0 ) 
1
  w( y0 ) 


 V




1
4 V
for small angle

V
b/2
( d  / dy ) dy
b / 2
y0  y


 w( y0 )



The lift coefficient at y=y0 is
c l  a 0  eff ( y 0 )   L  0   2   eff ( y 0 )   L  0 
From the Kutta-Joukowski theorem, lift for the local
airfoil section located at y0 is
1
2
L    V  c ( y 0 ) c l   V   ( y 0 )
2
2 ( y0 )
 cl 
V c ( y0 )
Expression of effective angle of attack
 ( y0 )
 eff 
  L0
V c ( y0 )

Fundamental equation of Prandtl’s lifting-line
theory (integro-differential equation of )
 eff     i
  i ( y0 ) 

 ( y0 )
V c ( y0 )
  L0 ( y0 ) 
1
4 V
b/2
( d  / dy ) dy
b / 2
y0  y

The solution  gives the three main aerodynamic
characteristics of a finite wing
1. The lift distribution
L ( y 0 )   V   ( y 0 )
2. The lift coefficient
L   V  
b/2
b / 2
CL 
L
2

q S
 ( y ) dy
V S

b/2
b / 2
 ( y ) dy
3. The induced drag coefficient
D i  L i sin  i  L i i
C D ,i 
2
V S

b/2
b / 2
 ( y )  i ( y ) dy
Elliptical lift distribution
Charateristic

Elliptical circulation distribution
 ( y)  0

 2y 
1 

b


2
where 0 is the circulation at the origin.
Elliptical lift distribution
L ( y )    V   ( y )    V   0

Zero lift at the wing tips
b
b
2
2
 ( )   (
)0
 2y 
1 

 b 
2
Resulting aerodynamic properties

By using the transformation y=b/2 cos, we obtain
0
w ()  
2b

which states that downwash is constant over the
span for an elliptical lift distribution.
Induced angle of attack
i  
CL
 AR
, Aspect ratio  AR 
b
2
S

Induced drag coefficient
2
C D ,i 

CL
 AR
which states that CD,I is proportional to the square
of CL and inversely proportional to AR.
For an elliptical lift distribution, the chord must
vary elliptically along the span; that is, the wing
planform is elliptical.
General lift distribution
Characteristic

Consider the transformation
y
b
cos 
2
and assume
N
 (  )  2 bV   A n sin n 
1

Fundamental equation at a given location
 ( 0 ) 
2b
c ( 0
N

)
1
sin n  0
N
A n sin n  0   L  0 (  0 ) 
 nA
1
n
sin  0

We may choose N different spanwise stations,
then we can obtain N independent algebraic
equations with N unknowns, namely, A1, A2, AN.
Resulting aerodynamic properties

Lifting coefficient
CL 

2
V S

b/2
b / 2
 ( y ) dy  A1  AR
Induced drag coefficient
C D ,i 
C
2
L
 AR
N
(1   ) , where


2
 An 

n 

A
 1
2

Define span efficiency factor e
e  (1   )
1
2
C D ,i 

CL
 e AR
Note that =0 and e=1 for the elliptical lift
distribution. Hence, the lift distribution which yields
minimum induced drag is the elliptical lift
distribution.