0 0 . . 09 09 T ~ T ~ Viscosity u u Plane - PowerPoint PPT Presentation

Outline  Introduction to Turbulence Modeling  Eddy Viscosity Models  Prandtl Mixing Length Model  One Equation Model  Two-Equation Model  Stress Transport Model ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Boussinesq             U U U U 2 2 U U U U     Eddy                 j j     2 2   u u u u i i k k u u ~ ~   m  m  Eddy           T T m m i i j j T T ij ij y y x x x x 3 3 y y     Viscosity j j i i Viscosity       1 1 U U U U U U                   2 2         k k u u m u m u u u v v Thin Shear Layer u u v v T  T      m m m m 2 2 y y y y y y Eddy Mixing Length m  m        0 0 . . 09 09   T ~ T ~ Viscosity u u Plane Jet m m m  m      0 0 . . 075 075 Circular Jet Mixing Length  m  Mixing Layer m  m  m     0 0 . . 07 07 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

      2 2 4 4 y y                 Mixing Length Nikuradse 0 0 . . 41 41   y y y y     0 0         y y           m m 0 0 . . 14 14 0 0 . . 08 08 1 1 0 0 . . 06 06 1 1                     r r   r r     r r   Boundary Layer Pipe Flows   0 0 0 0 0 0 m m y y       0  0        0 0 0 0 . . 09 09   0 0        m =0.4 y Nikuradse  m /r o U / U 1 o   / m y / r  y / o  /  1 1 o ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi   U  U    T  T    T  T  Mixing 0 0 0 0 0 0 When   Length y y     0  0  . . 8 8 | | Experiment T T T T max max  Cold U T     0 T T   T  T  Mixing Reattachment 0 0 Hot Point Length Maximum Heat Experiment Transfer ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

Turbulence Kinetic Energy 1 1 Kolmogorov-     k k   2 2 T T Prandtl               d d u u u u P P U U                 i i i i   i i k k u u ( ( ) ) u u u u   j j   i i j j   1 1 dt dt x x 2 2 x x     Turbulence       j j j j k k u u i u i u Kinetic Energy i i           2 2 2 2 u u u u k k         i i i i         x x x x x x x x j j j j j j j j ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Thin Shear Layer 3 3 Dissipation k k 2 2                   dk dk u u u u P P U U c c                   i i i i   v v ( ( ) )   u u v v D D         dt dt y y 2 2 y y     Closed k-Equation   U U           u u v v T  T  3 3 Turbulence Stress 2 2 y y                 dk dk k k U U k k 2 2                 T T c c               T T   D D   dt dt y y   y y     y y             1 1 P P k k k k             Diffusion     v v u u u u T T           i i i i   2 2   y y k k ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3

Near a Wall Distribution of Length Scale 3 3 3 3 2 2           2 2 2 2 U U k k 2 2 ( ( u u v v ) ) k k 2 2 1 1       1 1     1 1     1 1   U U           U U c c             c c         2 2     u u v v c c 2 2     k k 2 2 c c 2 2   u u v v c c 2 2 k k T T D D   D D   y y     D D       D D   y y   D D y y T T 1 1     2 2 ( ( u u v v ) )       c c   * *   k k 2 2 U U u u D D 2 2   2 2 1 1 1 1 k k         T T * * u u v v u u               c c 4 4 y y c c 4 4 y y y y D D D D m m     c D  c D  u u v v 0 0 . . 07 07 ~ ~ 0 0 . . 08 08     0 0 . . 25 25 ~ ~ 0 0 . . 3 3 k k c D  c D        0 0 . . 08 08   0 0 . . 4 4 0 0 . . 2 2 y y   k  k  1 1 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Achievements of the Model  Heat transfer in the heat exchanger  Separated flows Short Comings  Transport of the turbulent length scale  Little advantage over the mixing length ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 4