Baldwin-Lomax model

From CFD-Wiki

(Difference between revisions)

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

@@ Line 1: / Line 1: @@
-The Baldwin-Lomax model is a two-layer algebraic model which gives <math>\mu_t</math> as a function of the local boundary layer velocity profile. The eddy-viscosity, <math>\mu_t</math>, is given by:
+The Baldwin-Lomax model is a two-layer algebraic model which gives the eddy-viscosity <math>\mu_t</math> as a function of the local boundary layer velocity profile:
 <table width="100%"><tr><td>
 :<math>
-\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.
 </math></td><td width="5%">(1)</td></tr></table>
@@ Line 29: / Line 28: @@
 <table width="100%"><tr><td>
 :<math>
-\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)
 </math></td><td width="5%">(4)</td></tr></table>
@@ Line 60: / Line 58: @@
 F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
                 C_{WK} \, y_{MAX} \, \frac{u^2_{DIF}}{F_{MAX}} \right)
-</math></td><td width="5%">(7)</td></tr></table>
+</math></td><td width="5%">(8)</td></tr></table>
 <math>y_{MAX}</math> and <math>F_{MAX}</math> are determined from the maximum of the function:
-<table width="100%"><tr><td>:<math>
+<table width="100%"><tr><td>
-\renewcommand{\exp}[1]{e^{#1}}
+:<math>
-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)
-</math></td><td width="5%">(32)</td></tr></table>
+</math></td><td width="5%">(9)</td></tr></table>
 <math>F_{KLEB}</math> is the intermittency factor given by:
-<table width="100%"><tr><td>:<math>
+<table width="100%"><tr><td>
+:<math>
 F_{KLEB}(y) = \left[1 + 5.5 \left( \frac{y \, C_{KLEB}}{y_{MAX}} \right)^6
    \right]^{-1}
-</math></td><td width="5%">(32)</td></tr></table>
+</math></td><td width="5%">(10)</td></tr></table>
 <math>u_{DIF}</math> is the difference between maximum and minimum speed in the profile. For boundary layers the minimum is always set to zero.
-<table width="100%"><tr><td>:<math>
+<table width="100%"><tr><td>
+:<math>
 u_{DIF} = MAX(\sqrt{u_i u_i}) - MIN(\sqrt{u_i u_i})
-</math></td><td width="5%">(32)</td></tr></table>
+</math></td><td width="5%">(11)</td></tr></table>
+== Model constants ==
+The table below 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.
-\begin{table}[ht]
+<table cellpadding="5" cellspacing="1" border="1">
-\begin{center}
+<tr>
-\begin{tabular}{|c|c|c|c|c|c|}
+  <td><math>A^+</math></td>
-\hline
+  <td><math>C_{CP}</math></td>
-<math>A^+<math> & <math>C_{CP}<math> & <math>C_{KLEB}<math> & <math>C_{WK}<math> & <math>k<math>   & <math>K<math> \\
+  <td><math>C_{KLEB}</math></td>
-\hline
+  <td><math>C_{WK}</math></td>
-   & 1.6      & 0.3        & 0.25     & 0.4   & 0.0168 \\
+  <td><math>k</math></td>
-\hline
+   <td><math>K</math></td>
-\end{tabular}
+</tr>
-\caption{Model Constants, Baldwin-Lomax Model}
+<tr>
-\end{center}
+  <td>26</td>
-\end{table}
+  <td>1.6</td>
+  <td>0.3</td>
+  <td>0.25</td>
+  <td>0.4</td>
+   <td>0.0168</td>
+</tr>
+</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</math>
+* ''Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows'' by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978

Baldwin-Lomax model

From CFD-Wiki

Revision as of 12:28, 8 September 2005

Model constants

References

Views

My wiki

wiki navigation

wiki search

wiki toolbox