Finite Element 2D Wing [PDF]

Zitiervorschau

Mul�disciplinary  design  op�miza�on     in  computa�onal  mechanics     Applica�on  case  –  2D  wing   Piotr  Breitkopf   26.11.2014    

2  

Li�  

Angle  of  a�ack   Air  velocity  

Drag   Aerodynamic  center    

Chord   Reference  line  

3  

Hypothesis  


 

Simplifying  assump�ons   –  incompressible,  non-­‐viscous  (non-­‐rota�onal)  flow  

@v @u @v + = 0, @x @y @x

@u =0 @y

–  velocity  field  determined  by  stream  func�on  

Wortmain airfoil FX60.126 - computing domain

2

1.5

1

0.5

0

@ ,v = u= @y

@ @x

–  pressure  given  by  Bernoulli's  principle  

-0.5

-1

-1.5

-2

⇢ 2 P (x, y) = P1 + (Vwind V 2 (x, y)) 2
  far  field  pressure    P        1              ,  flow  velocity   Vwind

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

4  

3

Streamlines  
 

Curves  that  are  instantaneously  tangent  to  the  velocity  vector  of  the  flow  

0 1

n  

2

5  

Integral  (weak  formula�on)  
 

differen�al  formula�on  

∆ (x, y) = 0, (x, y) 2 ⌦
 

residual  weighted  by  test  func�on  

W =    
 

Z

δ

∆ (x, y)δ d⌦ = 0 ⌦

a�er  integra�ng  by  parts  and  using  Green’s  theorem  

W =

Z

⌦

(rδ , r )d⌦ −

I

@⌦

δ (r , n)d(@⌦) = W⌦ − W@⌦ = 0 6  

Finite  element  discre�za�on  
 

Integral  terms  

W⌦ = W@⌦ =
 

I

Z

⌦

ne Z X

(rδ , r )d⌦ =

δ (r , n)d(@⌦) = @⌦

e=1 ⌦e ne I X e=1

(rδ , r )d⌦e δ (r , n)d(@⌦e )

@⌦e

are  computed  over  surface  and  boundary  mesh   2 1.5 1 0.5 0 -0.5 -1 -1.5 -2

7   -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Finite  element  approxima�on   (x1 , y1 )

l1 l2

(x2 , y2 )

A3 (x, y)

(x, y)

(x1 , y1 )

A1

(x2 , y2 ) Ni (l) = li /L 2 X (x, y) = Ni (l) (xi , yi ) r (x, y) =

i=1 2 X i=1

rNi (l) (xi , yi )

A2

(x3 , y3 )

Ni (x, y) = Ai /A, i = 1...3 3 X (x, y) = Ni (x, y) (xi , yi ) r (x, y) =

i=1 3 X i=1

rNi (x, y) (xi , yi ) 8  

Explicit  form  of  shape  func�ons  and  deriva�ves  for  a  triangle  

1 A = ((x2 x1 )(y3 y1 ) (x3 x1 )(y2 y1 )) 2 1 ((x2 x)(y3 y) (x3 x)(y2 y)) N1 = 2A 1 ((x x1 )(y3 y1 ) (x3 x1 )(y y1 )) N2 = 2A 1 ((x2 x1 )(y y1 ) (x x1 )(y2 y1 )) N3 = 2A

1 @N1 = (y3 @x 2A 1 @N2 = (y3 @x 2A 1 @N3 = (y1 @x 2A

y2 )

1 @N1 = (x3 @y 2A

x2 )

y1 )

1 @N2 = (x1 @y 2A

x3 )

y2 )

1 @N3 = (x2 @y 2A

x1 ) 9  

Surface  term  

δ

r (x, y) =

3 X i=1

e

⇥

= N1 (x, y)

rNi (x, y)

i



N2 (x, y)



L y2 − y3 = 6 x3 − x2

1 y2 Be = 2A x3

W⌦e = δ

y3 x2

y3 x1

2

⇤ δ N3 (x, y) 4δ δ

y3 − y1 x1 − x3

y1 x3

T e Ke e , Ke

y1 x2

1 2 3

3 5 �

2

y1 − y2 4 x2 − x1

y2 x1

�

1 2 3

3

5 = Be

= AB T B

10  

e

Boundary  term  

⇥

⇤

(r , n) = N1 (s) N2 (s) (ny u¯n − nx v¯n ) �  ⇥ ⇤ δ 1 δ e = N1 (s) N2 (s) δ 2 W@⌦e = δ

T ¯n e Me (ny u

nx v¯), u ¯n = 

�

✓

u ¯1 u ¯2

◆

✓ ◆ v¯ , v¯n = 1 v¯2

L 2 1 Me = 6 1 2

11  

Finite  element  linear  system   W =

e X

We =

T

e=1


 

(K − F ) = 0 )K

=F

boundary  condi�ons   –  Neumann  at  the  external  boundary  –  intergrated  in  the  RHS   –  Dirichlet    



K11 T K12 1

K12 K22

�

= K111 (F

1

¯2

�

=



F1 0

�

K12 ¯2 )

12  

@ , ( @y

@ ) = (¯ u, v¯) @x

non-­‐rota�onal  flow  

=0

-2

-1

0

1

@ , ( @y 2

@ ) = (¯ u, v¯) 13   @x 3

=1

Circular  flow  

=0 =1

=1

=1

14  

Ku�a  (Joukowski)  condi�on  
 
 

Ku�a  condi�on:  no  circula�on  around  the  trailing  edge     resultant  velocity  follows  reference  line   y

0.25

y

0.25

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0

0

-0.05

-0.05

-0.05

-0.1

-0.1

-0.1

-0.15

-0.15

-0.15

-0.2

-0.2

-0.2

-0.25 0.75

0.8

0.85

 
 
 

0.9

0.95

1

1.05

1.1

1.15

1.2

1.2

-0.25 0.75

0.8

0.85

0.9

V1

0.95

1

1.05

p

0.25

1.1

1.15

1.2

1.25

-0.25

0.8

V2

0.9

1

g

1.1

1.2

V = V1 + ↵V2

weighted  sum  of  uniform  and  circular  flows   Li�  and  li�  coefficient  

L = ⇢V ,

I

L = (rφ, n), cL = 1 2 @⌦ 2 ⇢V l 15  

Summed  flows   p

2

g

1.5

1

0.5

0

-0.5

-1

-1.5

-2

16   -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Finite  element  2D  wing  example  summary  
 


 


 


 


 

generate  domain  mesh   –  surface  elements     –  linear  elements  on  the  boundary   solve  two  problems   –  uniform  flow   –  circular  flow   assemble  flows   –  compute  Ku�a  coefficient   –  sum  up  stream  func�ons   post-­‐process  quan��es  of  interest   –  veloci�es   –  pressure   –  li�,  drag,  pitching  moment   –  li�/drag  coefficients   –  ...   detailed  course  materials  at  :  

pressure

#10 4

0.8 10.5

0.6

10

0.4

0.2 9.5

0

-0.2

9

-0.4

8.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

–  h�p://www.utc.fr/~mecagom4/MECAWEB/EXEMPLE/EX07/SAAA1.htm  

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