Stream Function _ Velocity Potential

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					                   Stream Function & Velocity Potential
              Stream lines/ Stream Function (Y)
                 Concept
                 Relevant Formulas
                 Examples
                 Rotation, vorticity

              Velocity Potential(f)
                 Concept
                 Relevant  Formulas
                 Examples
                 Relationship between stream function and velocity
                  potential
                 Complex velocity potential
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Stream Lines
               Consider 2D incompressible flow
               Continuity Eqn

                                
                  Vx   Vy   Vz   0
               t x       y       z

                                                          Vx   
                         Vx   Vy   0          Vy         dy
                      x        y                          x    
               Vx and Vy are related
               Can you write a common function for both?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Stream Function
           Assume                   
                                Vx 
                                     y
           Then
                                 Vx              2 
                         Vy           dy     xy dy
                                 x                     
                                      2         
                                        dy       
                                     yx        x 
               Instead of two functions, Vx and Vy, we need to solve
                for only one function   Stream Function
               Order of differential eqn increased by one

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Stream Function
       What does Stream Function  mean?

       Equation for streamlines in 2D are given by
            = constant
       Streamlines may exist in 3D also, but stream function
        does not
           Why?    (When we work with velocity potential, we may
            get a perspective)
           In 3D, streamlines follow the equation

                             dx dy dz
                                  
                             Vx V y Vz

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Rotation
         Definition of rotation
                                                                     


                Vx    y Dy

  y                        Dy            Time=t
                                                                                   

                Vx     y                   Dx
                                Vy                      Vy
                                     x                       x Dx

                                                    x
                                                                               d    
                                                              ROTATION   z 
   Assume Vy|x < Vy|x+Dx
                                                                                       
      and Vx|y > Vx|y+Dy                                                       dt  2 
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Rotation
       To Calculate Rotation

                          Dy1
                  tan  
                          Dx

              Dy1  Vy          x Dx
                                               
                                         Dt  Vy
                                                          x
                                                                  Dt       
                                                                                Dy1


                                                                           Dx

               arctan
                        V               y x Dx    Vy
                                                          x
                                                               Dt
                                           Dx
   Similarly
              arctan
                                     
                                   Vx y Dy Vx              y    Dt
                                                    Dy
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Rotation
         To Calculate Rotation


                          d      1                       t 
         ROTATION   z              lim                                 
                                                         t Dt
                                    2 Dt 0              Dt              
                          dt  2                                          


                               V
                        arctan 
                                   
                                y x Dx Vy        x
                                                         Dt 
                                                             
                                                             
                                                                 
                                                                                 
                                                                                V
                                                                                 x y Dy Vx
                                                                         arctan 
                                                                                                y    Dt 
                                                                                                         
                                                                                                         
                                                                                                             
                                       Dx                   1                       Dy             
            1                                                                                     
            lim                                               Dt 0
                                                                   lim
            2  Dxt 0 0
                 D                 Dt                          2                  Dt
                     
                                                                     Dx  0
                 Dy  0
                                                                     Dy  0
         For very small time and very small element, Dx, Dy
          and Dt are close to zero

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Rotation
        To Calculate Rotation
        For very small , i.e. ~ 0
           sin                      cos  1                       tan   

              arctan   
                  V
                   y
          arctan 
                                   x Dx
                                            Vy
                                                    x
                                                         Dt   V
                                                                        y x Dx   Vy
                                                                                            x
                                                                                                 Dt
                                                               
                                        Dx                                      Dx
                                                              

                              V
                              y
                      arctan 
                                        x Dx
                                                 Vy
                                                        x
                                                             Dt 
                                                                 
                                                                 
                                                                         V
                                                                         y
                                                                        
                                                                                  x Dx
                                                                                           Vy
                                                                                                  x
                                                                                                       Dt 
                                                                                                           
                                                                                                           
                                             Dx                                      Dx
                  lim                                            lim 
                                                                                                         
                                                                                                          
                 Dt 0                   Dt                           Dt 0                Dt
                Dx  0                                               Dx  0
                Dy  0                                               Dy  0
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Rotation
        To Calculate Rotation

                   lim
                       V   y x Dx   Vy
                                            x
                                                  V   y
                  Dx 0          Dx                 x

                             V       
                              y x Dx Vy
                      arctan                        x
                                                              Dt 
                                                                  
                                                                  
                                                                                      
                                                                                      V
                                                                                       x y Dy Vx
                                                                               arctan 
                                                                                                      y    Dt 
                                                                                                               
                                                                                                               
                                     Dx                          1                        Dy             
         1                                                                                              
   z    lim                                                       Dt 0
                                                                          lim
         2 DxDt 00
                                Dt                                  2 Dx  0           Dt
                Dy  0                                                  Dy  0


            Simplifies to
                       1   Vy Vx 
                 z              
                       2   x   y 
                                1
                          z     V 
                               2
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
            Rotation in terms of Stream Function
        To write rotation in terms of stream functions
                                             
             Vx                        Vy       
                     y                         x 

               1   Vy Vx   1       
                                         2   2
         z                   2  2 
               2   x   y   2   x  y 
                             1
                              2
                             2
        That is
                     2  2 z  0


        For irrotational flow (z=0)

                          2  0

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Rotation and Potential

          For irrotational flow (z=0)
                      1
                      2
                             
                  z    V  0      
                     V  0

                   Vy       Vx
                                0
                    x        y

         This equation is “similar” to continuity equation
         Vx and Vy are related
         Can we find a common function to relate both Vx
          and Vy ?

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                                                                           Vy        Vx
                                                                                         0
                  Velocity Potential                                        x         y


          Assume
                f                 Then             Vy V       2f       2f
           Vx                
                                                        x             
                x                                  x   y     y x     x y

                    f
            Vy 
                    y


         In 3D, similarly it can be shown that
                          f
                     Vz 
                          z
         f is the velocity potential

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Velocity Potential vs Stream
                  Function

                    Stream Function () Velocity Potential (f)
                    only 2D flow                all flows
                                                Irrotational (i.e. Inviscid or
                 viscous or non-viscous flows zero viscosity) flow
          Exists
                 Incompressible flow (steady or Incompressible flow (steady
          for
                 unsteady)                      or unsteady state)
                 compressible flow (steady      compressible flow (steady or
                 state only)                    unsteady state)

          In 2D inviscid flow (incompressible flow OR steady
           state compressible flow), both functions exist
          What is the relationship between them?

IIT-Madras, Momentum Transfer: July 2005-Dec 2005
                  Stream Function- Physical
                  meaning
                  2D (viscous or inviscid) flow
         Statement: In
         (incompressible flow OR steady state compressible
         flow),  = constant represents the streamline.
        Proof

        If  = constant, then d0
                                 
                 d       dx          dy
                       x         y 
                       Vy  dx  Vx  dy
                         0                                  Vy
        If  = constant, then
                            dy Vy                   Vx
                              
                            dx Vx
IIT-Madras, Momentum Transfer: July 2005-Dec 2005

				
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