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Baldwin-Lomax model

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The Baldwin-Lomax model is a two-layer algebraic model which gives <math>\mu_t</math> as a function of the local boundary layer velocity profile. The eddy-viscosity, <math>\mu_t</math>, is given by:  
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The Baldwin-Lomax model is a two-layer algebraic model which gives the eddy-viscosity <math>\mu_t</math> as a function of the local boundary layer velocity profile:  
<table width="100%"><tr><td>
<table width="100%"><tr><td>
:<math>
:<math>
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\mu_t = \left\{
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\mu_t =
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\begin{array}{ll}
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\begin{cases}
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{\mu_t}_{inner} & y \leq y_{crossover} \\[1.5ex]
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{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\  
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{\mu_t}_{outer} & y > y_{crossover}
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{\mu_t}_{outer} & \mbox{if} y > y_{crossover}
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\end{array}
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\end{cases}
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\right.
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</math></td><td width="5%">(1)</td></tr></table>
</math></td><td width="5%">(1)</td></tr></table>
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<table width="100%"><tr><td>
<table width="100%"><tr><td>
:<math>
:<math>
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\renewcommand{\exp}[1]{e^{#1}}
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l = k y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
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l = k y \left( 1 - \exp{\frac{-y^+}{A^+}} \right)
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</math></td><td width="5%">(4)</td></tr></table>
</math></td><td width="5%">(4)</td></tr></table>
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F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
               C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right)
               C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right)
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</math></td><td width="5%">(7)</td></tr></table>
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</math></td><td width="5%">(8)</td></tr></table>
<math>y_{MAX}</math> and <math>F_{MAX}</math> are determined from the maximum of the function:
<math>y_{MAX}</math> and <math>F_{MAX}</math> are determined from the maximum of the function:
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<table width="100%"><tr><td>:<math>
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<table width="100%"><tr><td>
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\renewcommand{\exp}[1]{e^{#1}}
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:<math>
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F(y) = y \left| \Omega \right| \left(1-\exp{\frac{-y^+}{A^+}} \right)
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F(y) = y \left| \Omega \right| \left(1-e^{\frac{-y^+}{A^+}} \right)
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</math></td><td width="5%">(32)</td></tr></table>
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</math></td><td width="5%">(9)</td></tr></table>
<math>F_{KLEB}</math> is the intermittency factor given by:
<math>F_{KLEB}</math> is the intermittency factor given by:
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<table width="100%"><tr><td>:<math>
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<table width="100%"><tr><td>
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:<math>
F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6
F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6
   \right]^{-1}
   \right]^{-1}
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</math></td><td width="5%">(32)</td></tr></table>
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</math></td><td width="5%">(10)</td></tr></table>
<math>u_{DIF}</math> is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero.
<math>u_{DIF}</math> is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero.
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<table width="100%"><tr><td>:<math>
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<table width="100%"><tr><td>
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:<math>
u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i})
u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i})
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</math></td><td width="5%">(32)</td></tr></table>
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</math></td><td width="5%">(11)</td></tr></table>
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== Model constants ==
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The table below gives the model constants present in the formulas above. Note that <math>k</math> is a constant, and not the turbulence energy, as in other sections. It should also be pointed out that when using the Baldwin-Lomax model the turbulence energy, <math>k</math>, present in the governing equations, is set to zero.
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\begin{table}[ht]
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<table cellpadding="5" cellspacing="1" border="1">
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\begin{center}
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<tr>
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\begin{tabular}{|c|c|c|c|c|c|}
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  <td><math>A^+</math></td>
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\hline
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  <td><math>C_{CP}</math></td>
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<math>A^+<math> & <math>C_{CP}<math> & <math>C_{KLEB}<math> & <math>C_{WK}<math> & <math>k<math>  & <math>K<math> \\
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  <td><math>C_{KLEB}</math></td>
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\hline
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  <td><math>C_{WK}</math></td>
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26   & 1.6     & 0.3       & 0.25     & 0.4  & 0.0168 \\
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  <td><math>k</math></td>
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\hline
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   <td><math>K</math></td>
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\end{tabular}
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</tr>
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\caption{Model Constants, Baldwin-Lomax Model}
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<tr>
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\end{center}
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  <td>26</td>
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\end{table}
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  <td>1.6</td>
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  <td>0.3</td>
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  <td>0.25</td>
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  <td>0.4</td>
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   <td>0.0168</td>
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</tr>
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</table>
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Table 1 gives the model constants present in the formulas above. Note that <math>k<math> is a constant, and not the turbulence energy, as in other sections. It should also be pointed out that when using the Baldwin-Lomax model the turbulence energy, <math>k<math>, present in the governing equations, is set to zero.
 
== References ==
== References ==
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''Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows'' by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978</math>
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* ''Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows'' by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978

Revision as of 12:28, 8 September 2005

The Baldwin-Lomax model is a two-layer algebraic model which gives the eddy-viscosity \mu_t as a function of the local boundary layer velocity profile:


\mu_t =
\begin{cases}
{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ 
{\mu_t}_{outer} & \mbox{if} y > y_{crossover}
\end{cases}
(1)

Where y_{crossover} is the smallest distance from the surface where {\mu_t}_{inner} is equal to {\mu_t}_{outer}:


y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
(2)

The inner region is given by the Prandtl - Van Driest formula:


{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|
(3)

Where


l = k y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
(4)

\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}
(5)

\Omega_{ij} = \frac{1}{2}
\left(
 \frac{\partial u_i}{\partial x_j} -
 \frac{\partial u_j}{\partial x_i}
\right)
(6)

The outer region is given by:


{\mu_t}_{outer} =  \rho \, K \, C_{CP} \, F_{WAKE} \, F_{KLEB}(y)
(7)

Where


F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
               C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right)
(8)

y_{MAX} and F_{MAX} are determined from the maximum of the function:


F(y) = y \left| \Omega \right| \left(1-e^{\frac{-y^+}{A^+}} \right)
(9)

F_{KLEB} is the intermittency factor given by:


F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6
  \right]^{-1}
(10)

u_{DIF} is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero.


u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i})
(11)


Model constants

The table below gives the model constants present in the formulas above. Note that k is a constant, and not the turbulence energy, as in other sections. It should also be pointed out that when using the Baldwin-Lomax model the turbulence energy, k, present in the governing equations, is set to zero.

A^+ C_{CP} C_{KLEB} C_{WK} k K
26 1.6 0.3 0.25 0.4 0.0168


References

  • Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978
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