H 1 H 1 x x 1 1 x x a a x x u u - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline

• Expansion of a function
• Orthogonal set
• Wiener-Hermite orthogonal set
• Expansion of a random function
• Burger Equation
• W-H expansion for the Burger Eq.
• G. Ahmadi

ME 639-Turbulence

Let u(x) be an arbitrary function Let n(x) be an orthogonal set

n nm m n

dx     



n nm m n

dx     





  dx

2 n n



  dx

2 n n

   



 

n n n

x c x u 

 



 

n n n

x c x u

n n n

dx u c    

n n n

dx u c    

• G. Ahmadi

ME 639-Turbulence

Wiener -Hermite base Let a(x) be a white noise process

 

x a     x a  

     

2 1 2 1

x x x a x a       

  

2 1 2 1

x x x a x a    

  

1 x H 0 

  

1 x H 0 

  

 

x a x H 1 

  

 

x a x H 1 

 

      

2 1 2 1 2 1 2

x x x a x a x , x H    

 

      

2 1 2 1 2 1 2

x x x a x a x , x H    

SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

Wiener -Hermite base

 

                  

2 1 3 1 3 2 3 2 1 3 2 1 3 2 1 3

x x x a x x x a x x x a x a x a x a x , x , x H          

 

                  

2 1 3 1 3 2 3 2 1 3 2 1 3 2 1 3

x x x a x x x a x x x a x a x a x a x , x , x H          

   

H H

j i

 

   

H H

j i

 

j i  j i  Wiener -Hermite set is complete

• G. Ahmadi

ME 639-Turbulence

Wiener -Hermite base

     

1 x H x H  

     

1 x H x H  

 



 

  

2 1 2 1 1 1

x x x H x H    

 



 

  

2 1 2 1 1 1

x x x H x H    

 



 

       

3 2 4 1 4 2 3 1 3 3 2 2 1 2

x x x x x x x x x , x H x , x H          

 



 

       

3 2 4 1 4 2 3 1 3 3 2 2 1 2

x x x x x x x x x , x H x , x H          

• G. Ahmadi

ME 639-Turbulence

Wiener -Hermite Series

 

 



 



 



 



 

  

 



                                                          

Gaussian Non 3 2 1 3 2 1 3 3 2 1 3 Gaussian Non 2 1 2 1 2 2 1 2 2 1 2 Gaussian 1 1 1 1 1

... dx dx dx x , x , x H x x , x x , x x K dx dx x , x H x , x H x x , x x K dx x H x x K x u

 

  

         

 

 



 



 



 



 

  

 



                                                          

Gaussian Non 3 2 1 3 2 1 3 3 2 1 3 Gaussian Non 2 1 2 1 2 2 1 2 2 1 2 Gaussian 1 1 1 1 1

... dx dx dx x , x , x H x x , x x , x x K dx dx x , x H x , x H x x , x x K dx x H x x K x u

 

  

         

• G. Ahmadi

ME 639-Turbulence

Burger Equation

 

2 2

x u x u u t t , x u          



2 2

x u x u u t t , x u         

SLIDE 3

3

• G. Ahmadi

ME 639-Turbulence

 



 



 



 



 



 



 

                                                                             

 

x x K x x K x 2 1 x x , x x K x t x x , x x K x x K dx x 2 x x K x t

1 1 2 2 2 1 2 1 1 1 1 2 2

 



 



 



 



 



 



 

                                                                             

 

x x K x x K x 2 1 x x , x x K x t x x , x x K x x K dx x 2 x x K x t

1 1 2 2 2 1 2 1 1 1 1 2 2

• G. Ahmadi

ME 639-Turbulence

Concluding Remarks

• Expansion of a function
• Orthogonal set
• Wiener-Hermite orthogonal set
• Expansion of a random function
• Burger Equation
• W-H expansion for the Burger Eq.
• G. Ahmadi

ME 639-Turbulence