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Baldwin-Barth model

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{{Turbulence modeling}}
==Kinematic Eddy Viscosity==
==Kinematic Eddy Viscosity==
:<math> \nu _t  = C_\mu  \nu \tilde R_T D_1 D_2 </math>
:<math> \nu _t  = C_\mu  \nu \tilde R_T D_1 D_2 </math>
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==Turbulence  Reynolds Number ==
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:<math>
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{\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 }}
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</math>
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== Closure Coefficients and Auxilary Relations ==
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:<math>
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  C_{\varepsilon 1}  = 1.2 
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</math> <br>
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:<math>
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    C_{\varepsilon 2}  = 2.0 
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</math> <br>
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:<math>
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  C_\mu  = 0.09 
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</math> <br>
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:<math>
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  A_o^ +  = 26 
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</math> <br>
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:<math>
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  A_2^ +  = 10 
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</math> <br>
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:<math>
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{1 \over {\sigma _\varepsilon  }} = \left( {C_{\varepsilon 2}  - C_{\varepsilon 1} } \right){{\sqrt {C_\mu  } } \over {\kappa ^2 }} 
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</math> <br>
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:<math>
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    \kappa  = 0.41
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</math> <br>
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:<math>
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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]
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</math>
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:<math>
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  D_1  = 1 - e^{{{ - y^ +  } \over {A_o^ +  }}}
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</math> <br>
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:<math>
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  D_2  = 1 - e^{{{ - y^ +  } \over {A_2^ +  }}}
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</math> <br>
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:<math>
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  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]
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</math> <br>
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== References ==
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#{{reference-book|author=Wilcox, D.C. |year=2004|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc}}
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#{{reference-book|author=Baldwin, B.S. and Barth, T.J. |year=1990|title=A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows|rest=NASA TM 102847}}
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[[Category:Turbulence models]]

Latest revision as of 09:34, 12 June 2007

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

Contents

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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