# Baldwin-Lomax model

## Introduction

The Baldwin-Lomax model is a two-layer algebraic 0-equation model which gives the eddy-viscosity, $\mu_t$, 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. The model is also commonly used in this type of application, especially for quick design iterations where robustness is more important than capturing all details of the flow physics.

## Equations $\mu_t = \begin{cases} {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ {\mu_t}_{outer} & \mbox{if} y > y_{crossover} \end{cases}$ (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

## Performance, applicability and limitations

The Baldwin-Lomax model is suitable for high-speed flows with thin attached boundary layers. Typical applications are aerospace and turbomachinery applications. It is a low-Re model and as such it requires a fairly well-resolved grid near the walls, with the first cell located at $y+ \approx 1$.

The model is popular in quick design-iterations due to its robustness and reliability. It seldom leads to any convergence problems and it seldom gives completely unphysical results.

The Baldwin-Lomax model should be used with great care in cases with large separations. It has been shown by several researcher that the Baldwin-Lomax model tends to overpredict separated regions (see for example the comments made by David Wilcox in his book Turbulence Modeling for CFD). However, there are ad-hoc modifications which reduce this problem. For instance, prediction of separation is sensitive to the value of the $C_{WK}$ coefficient and higher values than the original value have been shown to reduce the problems with too early separation. Also, the Granville corrections take partly into account adverse pressure gradient effects, which attenuate the original weaknesses.