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Baldwin-Barth model

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Kinematic Eddy Viscosity

 \nu _t  = C_\mu  \nu \tilde R_T D_1 D_2

Turbulence Reynolds Number


{\partial  \over {\partial t}}\left( {\nu \tilde R_T } \right) = U_j {\partial  \over {\partial x_j }}\left( {\nu \tilde R_T } \right) = \left( {C_{\varepsilon 2} f_2  - C_{\varepsilon 1} } \right)\sqrt {\nu \tilde R_T P}  + \left( {\nu  + {{\nu _T } \over {\sigma _\varepsilon  }}} \right){{\partial ^2 } \over {\partial x_k \partial x_k }} - {1 \over {\sigma _\varepsilon  }}{{\partial \nu _T } \over {\partial x_k }}{{\partial \left( {\nu \tilde R_T } \right)} \over {\partial x_T }}


Closure Coefficients and Auxilary Relations


   C_{\varepsilon 1}  = 1.2

    C_{\varepsilon 2}  = 2.0

   C_\mu   = 0.09

   A_o^ +   = 26

   A_2^ +   = 10



 {1 \over {\sigma _\varepsilon  }} = \left( {C_{\varepsilon 2}  - C_{\varepsilon 1} } \right){{\sqrt {C_\mu  } } \over {\kappa ^2 }}



    \kappa  = 0.41



P = \nu _T \left[ {\left( {{{\partial U_i } \over {\partial x_j }} + {{\partial U_j } \over {\partial x_i }}} \right){{\partial U_i } \over {\partial x_j }} - {2 \over 3}{{\partial U_k } \over {\partial x_k }}{{\partial U_k } \over {\partial x_k }}} \right]



   D_1  = 1 - e^{{{ - y^ +  } \over {A_o^ +  }}}

  D_2  = 1 - e^{{{ - y^ +  } \over {A_2^ +  }}}



   f_2  = {{C_{\varepsilon 1} } \over {C_{\varepsilon 2} }} + \left( {1 - {{C_{\varepsilon 1} } \over {C_{\varepsilon 2} }}} \right)\left( {{1 \over {\kappa y^ +  }} + D_1 D_2 } \right)\left[ {\sqrt {D_1 D_2 }  + {{y^ +  } \over {\sqrt {D_1 D_2 } }}\left( {{{D_2 } \over {A_o^ +  }}e^{{{ - y^ +  } \over {A_o^ +  }}}  + {{D_1 } \over {A_2^ +  }}e^{{{ - y^ +  } \over {A_2^ +  }}} } \right)} \right]

References

  1. Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.
  2. Baldwin, B.S. and Barth, T.J. (1990), A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows, NASA TM 102847.
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