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Baldwin-Lomax model

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

\mu_t = \left\{
{\mu_t}_{inner} & y \leq y_{crossover} \\[1.5ex]
{\mu_t}_{outer} & y > y_{crossover}

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}

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

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


Failed to parse (unknown function\renewcommand): \renewcommand{\exp}[1]{e^{#1}} l = k y \left( 1 - \exp{\frac{-y^+}{A^+}} \right)

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

\Omega_{ij} = \frac{1}{2}
 \frac{\partial u_i}{\partial x_j} -
 \frac{\partial u_j}{\partial x_i}

The outer region is given by:

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


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

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

:Failed to parse (unknown function\renewcommand): \renewcommand{\exp}[1]{e^{#1}} F(y) = y \left| \Omega \right| \left(1-\exp{\frac{-y^+}{A^+}} \right) (32)

F_{KLEB} is the intermittency factor given by:

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

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})

\begin{table}[ht] \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline Failed to parse (syntax error): A^+<math> & <math>C_{CP}<math> & <math>C_{KLEB}<math> & <math>C_{WK}<math> & <math>k<math> & <math>K<math> \\ \hline 26 & 1.6 & 0.3 & 0.25 & 0.4 & 0.0168 \\ \hline \end{tabular} \caption{Model Constants, Baldwin-Lomax Model} \end{center} \end{table} 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 == ''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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