From CFD-Wiki
The Baldwin-Lomax model is a two-layer algebraic model which gives
as a function of the local boundary layer velocity profile. The eddy-viscosity,
, 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.](/W/images/math/6/2/a/62aea5700d426d32d39f008c74eb985f.png)
| (1) |
Where
is the smallest distance from the surface where
is equal to
:
![y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}](/W/images/math/0/a/e/0aea083a70c0228df853cc09fa4d6aa1.png)
| (2) |
The inner region is given by the Prandtl - Van Driest formula:
![{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|](/W/images/math/5/3/b/53b46ed6398ab5bc23e63f84f566b189.png)
| (3) |
Where
- Failed to parse (unknown function\renewcommand): \renewcommand{\exp}[1]{e^{#1}} l = k y \left( 1 - \exp{\frac{-y^+}{A^+}} \right)
| (4) |
![\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}](/W/images/math/9/9/a/99ae36ac6bcb506dd9de94f3ad68fa52.png)
| (5) |
![\Omega_{ij} = \frac{1}{2}
\left(
\frac{\partial u_i}{\partial x_j} -
\frac{\partial u_j}{\partial x_i}
\right)](/W/images/math/f/2/a/f2a00d9b3ab8d744eee92f2309a91247.png)
| (6) |
The outer region is given by:
![{\mu_t}_{outer} = \rho \, K \, C_{CP} \, F_{WAKE} \, F_{KLEB}(y)](/W/images/math/5/e/8/5e85aaddbe6befa09de8997ec13ad708.png)
| (7) |
Where
![F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right)](/W/images/math/f/d/7/fd7818aa20d2df741dfe0b4b4cc22d86.png)
| (7) |
and
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) |
is the intermittency factor given by:
:![F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6
\right]^{-1}](/W/images/math/9/b/d/9bde77641b7e232fa3e083d3e59795c4.png) | (32) |
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})](/W/images/math/7/0/a/70a22d0d0496ef78a10ca0f129624163.png) | (32) |
\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