General Transformation on the
Equations
• Consider two-dimensional unsteady flow
• Transform the independent variables in physical
•
space ( x, y, t ) to a new set of independent variables in
transformed space ( , , )
The transformation must be given as some type of
specific analytical function or specific numerical
relation for an actual application
   ( x, y , t )
   ( x, y , t )
   t   t
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General Transformation on the
Equations (1st order derivative)
• To transform PDEs from physical space to
computational space requires transformation of the
derivatives
            
     


x     x      x 
            
     


y     y      y 
     
  
t     t
•
7
            

    

     t      t 
Metrics: the coefficients of the derivatives with
respect to  , , and 
General Transformation on the
Equations (2nd order derivative)
            
A
     


x     x      x 
 2 A   

 
2
x
x x  
     2
  2
    x
          

   

x
x





  

      2       2       2 
   



 2   
x
x
x
x
x











 

   
 



 
B
2
  
 
B
x x  
   2       2
   2    
     x   
C
   


  x 
2
      2       2    
 
C
    2 


x x        x      x 
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General Transformation on the
Equations (2nd order derivative)
 2 A      2

   2
2
x
x     x
      2    2    

 2    2  
     x      x 
  2    
 2
  2 
  2
    x 
 
2
    


 
y 2     y 2
2
2
      

 
  x   x 
             

 2    2  
     y      y 
2
2
2
2
2
     
  2       
  2 

  2
 


y



y




  y 




2
     2       2    2       
2
  

   2   

xy     xy      xy      x   y 
  2          2              
  2 
 



    
y
x
y
x
y



x










   
 



 
9