# Baldwin-Lomax model

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