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Baldwin - Lomax algebraic model

The Baldwin-Lomax model (Baldwin and Lomax (1978), [10]) is a two-layer algebraic which gives the eddy viscosity $ \nu_T$ as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary-layers, typically present in aerospace and turbomachinery applications. It is commonly used in quick design iterations where robustness is more important than capturing all details of the flow physics. The Baldwin-Lomax model is not suitable for cases with large separated regions and significant curvature/rotation effects.
$\displaystyle \nu_T = \left\{ \begin{array}{cl} {\nu_T}_{inner} \mbox{ ... if } y \le y_{crossover} & \\ {\nu_T}_{outer} \mbox{ ... if } y > y_{crossover} & \\ \end{array} \right.$ (27.17)
where $ y_{crossover}$ is the smallest distance from the surface where $ {\nu_T}_{inner}$ is equal to $ {\nu_T}_{outer}$:
$\displaystyle y_{crossover} = MIN(y) : \ {\nu_T}_{inner} = {\nu_T}_{outer}$ (27.18)
The inner region is given by Prandtl – Van Driest formula:
$\displaystyle \fbox{\( {\nu_T}_{inner} = l_{mix}^2 \left\vert \Omega \right\vert\)}$ (27.19)
$\displaystyle l_{mix}= \kappa y \left(1 - e^{\frac{-y^+}{A^+}} \right)$ (27.20)
$\displaystyle y^+ = \frac{\rho_w v_\tau d_w}{\mu_w} = \frac{\sqrt{\rho_w \tau_w} d_w}{\mu_w}$ (27.21)
$\displaystyle \Omega_{ij} = \frac{1}{2} \left( \frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}\right) , \left\vert \Omega \right\vert = \sqrt{2 \Omega_{ij} \Omega_{ij}}$ (27.22)
The outer region is given by:
$\displaystyle \fbox{$ {\nu_T}_{outer}= \rho \, K \, C_{CP} \, F_{WAKE} \, F_{KLEB}(y) $}$ (27.23)
$\displaystyle F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\, C_{WK} \,y_{MAX} \,\frac{u^2_{DIF}}{F_{MAX}} \right)$ (27.24)
where $ y_{MAX}$ and $ F_{MAX}$ are determined from the maximum of the function:
$\displaystyle F(y) = y \left\vert\Omega \right\vert \left(1-e^{\frac{-y^+}{A^+}} \right)$ (27.25)
where $ F_{KLEB}$ is the intermittency factor given by:
$\displaystyle F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6 \right]^{-1}$ (27.26)
$ u_{DIF}$ is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero.
$\displaystyle u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i})$ (27.27)
The table below gives the model constants present in the formulas above.
$ A^+$ $ C_{CP}$ $ C_{KLEB}$ $ C_{WK}$ $ \kappa$ K
26.0 1.6 0.3 0.25 0.41 0.0168