Low Speed Aerodynamics Questions

SUPPLEMENTARY

LOW SPEED AERODYNAMICS (DEN233)

 Thin Boundary Lauer Equations

0 y

V

x

U









0

1

2

2





y

P

y

U

x

P

y

U V

x

U U

























 Momentum Integral Equation:

2 2

2

f

e

we

e

C

U )H(

xd

Ud

Uxd

d 





 Thwaites Method

xdu45.0 u x

0

5 e

6 e

2

 



xd

Ud m e



 2  09.0mseparation  0750.mstag 

2 e

w f

u

2 C



 



xu Re ex  )m(S

ue

w  



088.2 m14.0

0731.0 )m(H

m107.0

m018.0 m402.122.0)m(S

0.0m0.1For

 



 



2m24.5m75.361.2)m(H

2m8.1m57.122.0)m(S

1.0m0For







 Transition, Michel’s Method

4092 . trxtrx

Re.)(Re  

tre trx

xU Re 



 

eURe 

 Turbulent wall shear stress, you may assume this equation is valid for

transitionxx 

202028850 .xeW ReUρ.τ 

 Blasius Solution

)(fUx)y,x( e   x

U y e

  

y U



 

x V



 

  δη)η(fη e

δ

e

*

U

xν yd)

U

U (δ 0

0

1  

  δ

ee

yd) U

U (

U

U θ

0

1   δ η

)η(f)η(f)η(f)η(f U

xν

e 0

2 



* H  =

∂

∂ =

y

U μτ

x e

Re fUρ

12  ν

xU Re ex 

ρ

μ ν

 f f  f 

0 0 0 0.33206 0.2 0.00664 0.06641 0.3319 0.4 0.02656 0.13277 0.33147 0.6 0.05974 0.19894 0.33008 0.8 0.10611 0.26471 0.32739 1.0 0.16557 0.32979 0.32301 1.2 0.23795 0.39378 0.31659 1.4 0.32298 0.45627 0.30787 1.6 0.42032 0.51676 0.29667 1.8 0.52952 0.57477 0.28293 2.0 0.65003 0.62977 0.26675 2.2 0.78120 0.68132 0.24835 2.4 0.92230 0.72899 0.22809 2.6 1.07252 0.77246 0.20646 2.8 1.23099 0.81152 0.18401 3.0 1.39682 0.84605 0.16136 3.2 1.56911 0.87609 0.13913 3.4 1.74696 0.90177 0.11788 3.6 1.92954 0.92333 0.09809 3.8 2.11605 0.94112 0.08013 4.0 2.30576 0.95552 0.06424 4.2 2.49806 0.96696 0.05052 4.4 2.69238 0.97587 0.03897 4.6 2.88826 0.98269 0.02948 4.8 3.08534 0.98779 0.02187 5.0 3.28329 0.99155 0.01591 5.2 3.48189 0.99425 0.01134 5.4 3.68094 0.99616 0.00793 5.6 3.88031 0.99748 0.00543 5.8 4.07990 0.99838 0.00365 6.0 4.27964 0.99898 0.00240 6.2 4.47948 0.99937 0.00155 6.4 4.67938 0.99961 0.00098 6.6 4.87931 0.99977 0.00061

  



  





 



  









 



 



 

 

 







 



 



 



 

2

2

2

2 11

11

0

1

1



 





















rr r

rr

)V(

rr

)rV(

r

)V(

r

)rV(

r V,

r V

r V,

r V

r z

r

r

r

2

2

2

2 2

0

yx

y

U

x

V

y

V

x

U

y V,

x U

x V,

y U

z



 



 



 



 

 

 







 



 



 



 







Elementary Flows

Uniform Flow





cosrUxU

sinrUyU









Source/Sink

rln π2

λ )yx(ln

π4

λ φ

θ π2

λ

x

y arctan

π2

λ ψ

22 =+=

==

Vortex

θ π2

Γ

x

y arctan

π2

Γ φ

rln π2

Γ )yx(ln

π4

Γ ψ 22

-=-=

=+=

Doublet

r

cos

yx

x

r

sin

yx

y









 









 

22

22

22

22

 



 

