# Baldwin-Barth model

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 Revision as of 09:02, 26 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Latest revision as of 09:34, 12 June 2007 (view source)Jola (Talk | contribs) (7 intermediate revisions not shown) Line 1: Line 1: + {{Turbulence modeling}} ==Kinematic Eddy Viscosity== ==Kinematic Eddy Viscosity== :$\nu _t = C_\mu \nu \tilde R_T D_1 D_2$ :$\nu _t = C_\mu \nu \tilde R_T D_1 D_2$ Line 30: Line 31: {1 \over {\sigma _\varepsilon  }} = \left( {C_{\varepsilon 2}  - C_{\varepsilon 1} } \right){{\sqrt {C_\mu  } } \over {\kappa ^2 }} {1 \over {\sigma _\varepsilon  }} = \left( {C_{\varepsilon 2}  - C_{\varepsilon 1} } \right){{\sqrt {C_\mu  } } \over {\kappa ^2 }} [/itex]
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+ + :$:[itex] \kappa = 0.41 \kappa = 0.41$
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+ + + :$+ P = \nu _T \left[ {\left( {{{\partial U_i } \over {\partial x_j }} + {{\partial U_j } \over {\partial x_i }}} \right){{\partial U_i } \over {\partial x_j }} - {2 \over 3}{{\partial U_k } \over {\partial x_k }}{{\partial U_k } \over {\partial x_k }}} \right] +$ + + + :$+ D_1 = 1 - e^{{{ - y^ + } \over {A_o^ + }}} +$
+ :$+ D_2 = 1 - e^{{{ - y^ + } \over {A_2^ + }}} +$
+ + + :$+ f_2 = {{C_{\varepsilon 1} } \over {C_{\varepsilon 2} }} + \left( {1 - {{C_{\varepsilon 1} } \over {C_{\varepsilon 2} }}} \right)\left( {{1 \over {\kappa y^ + }} + D_1 D_2 } \right)\left[ {\sqrt {D_1 D_2 } + {{y^ + } \over {\sqrt {D_1 D_2 } }}\left( {{{D_2 } \over {A_o^ + }}e^{{{ - y^ + } \over {A_o^ + }}} + {{D_1 } \over {A_2^ + }}e^{{{ - y^ + } \over {A_2^ + }}} } \right)} \right] + +$
+ + == References == + + #{{reference-book|author=Wilcox, D.C. |year=2004|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc}} + #{{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}} + + [[Category:Turbulence models]]

## 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.