# 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: $\mu_t = \left\{ \begin{array}{ll} {\mu_t}_{inner} & y \leq y_{crossover} \\[1.5ex] {\mu_t}_{outer} & y > y_{crossover} \end{array} \right.$ (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

 Failed to parse (unknown function\renewcommand): \renewcommand{\exp}{e^{#1}} l = k y \left( 1 - \exp{\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 $F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\, C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right)$ (7) $y_{MAX}$ and $F_{MAX}$ are determined from the maximum of the function:

 :Failed to parse (unknown function\renewcommand): \renewcommand{\exp}{e^{#1}} F(y) = y \left| \Omega \right| \left(1-\exp{\frac{-y^+}{A^+}} \right) (32) $F_{KLEB}$ is the intermittency factor given by:

 : $F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6 \right]^{-1}$ (32) $u_{DIF}$ is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero.

 : $u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i})$ (32)

\begin{table}[ht] \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline Failed to parse (syntax error): A^+[itex] & [itex]C_{CP}[itex] & [itex]C_{KLEB}[itex] & [itex]C_{WK}[itex] & [itex]k[itex] & [itex]K[itex] \\ \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 [itex]k[itex] 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, [itex]k[itex], 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