# Baldwin-Lomax model

(Difference between revisions)
 Revision as of 11:10, 8 September 2005 (view source)Jola (Talk | contribs)← Older edit Revision as of 12:28, 8 September 2005 (view source)Jola (Talk | contribs) Newer edit → Line 1: Line 1: - 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: + 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:
:$:[itex] - \mu_t = \left\{ + \mu_t = - \begin{array}{ll} + \begin{cases} - {\mu_t}_{inner} & y \leq y_{crossover} \\[1.5ex] + {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ - {\mu_t}_{outer} & y > y_{crossover} + {\mu_t}_{outer} & \mbox{if} y > y_{crossover} - \end{array} + \end{cases} - \right. +$(1)
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:$:[itex] - \renewcommand{\exp}[1]{e^{#1}} + l = k y \left( 1 - e^{\frac{-y^+}{A^+}} \right) - l = k y \left( 1 - \exp{\frac{-y^+}{A^+}} \right) +$(4)
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Line 60: Line 58: F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\, F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\, C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right) C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right) - [/itex]
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+ [/itex]
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$y_{MAX}$ and $F_{MAX}$ are determined from the maximum of the function: $y_{MAX}$ and $F_{MAX}$ are determined from the maximum of the function: -
:$+ - \renewcommand{\exp}[1]{e^{#1}} + :[itex] - F(y) = y \left| \Omega \right| \left(1-\exp{\frac{-y^+}{A^+}} \right) + F(y) = y \left| \Omega \right| \left(1-e^{\frac{-y^+}{A^+}} \right) -$(32)
+ [/itex]
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$F_{KLEB}$ is the intermittency factor given by: $F_{KLEB}$ is the intermittency factor given by: -
:$+ + :[itex] F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6 F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6 \right]^{-1} \right]^{-1} -$(32)
+ [/itex]
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$u_{DIF}$ is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero. $u_{DIF}$ is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero. -
:$+ + :[itex] u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i}) u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i}) -$(32)
+ [/itex]
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+ + + == 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. - \begin{table}[ht] + - \begin{center} +
$A^+[itex]C_{CP}[itex]C_{KLEB}[itex]C_{WK}[itex]k[itex]K 261.60.30.250.40.0168 - \begin{tabular}{|c|c|c|c|c|c|} + - \hline + - [itex]A^+[itex] & [itex]C_{CP}[itex] & [itex]C_{KLEB}[itex] & [itex]C_{WK}[itex] & [itex]k[itex] & [itex]K[itex] \\ + - \hline + - 26 & 1.6 & 0.3 & 0.25 & 0.4 & 0.0168 \\ + - \hline + - \end{tabular} + - \caption{Model Constants, Baldwin-Lomax Model} + - \end{center} + - \end{table} + + + + + + + - Table 1 gives the model constants present in the formulas above. Note that [itex]k[itex] is a constant, and not the turbulence energy, as in other sections. It should also be pointed out that when using the Baldwin-Lomax model the turbulence energy, [itex]k[itex], present in the governing equations, is set to zero. == References == == References == - ''Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows'' by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978$ + * ''Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows'' by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978

## Revision as of 12:28, 8 September 2005

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