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Baldwin-Lomax model

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== Introduction ==
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{{Turbulence modeling}}
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The Baldwin-Lomax model [[#References|[Baldwin and Lomax (1978)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, 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 (see below).
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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.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 the robustness is more important than getting capturing all details of the flow physics.
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== Equations ==
== Equations ==
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<table width="70%"><tr><td>
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<table width="100%"><tr><td>
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:<math>
:<math>
\mu_t =
\mu_t =
\begin{cases}
\begin{cases}
{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\  
{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\  
-
{\mu_t}_{outer} & \mbox{if} y > y_{crossover}
+
{\mu_t}_{outer} & \mbox{if } y > y_{crossover}
\end{cases}
\end{cases}
</math></td><td width="5%">(1)</td></tr></table>
</math></td><td width="5%">(1)</td></tr></table>
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Where <math>y_{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_{inner}</math> is equal to <math>{\mu_t}_{outer}</math>:
Where <math>y_{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_{inner}</math> is equal to <math>{\mu_t}_{outer}</math>:
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<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
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The inner region is given by the Prandtl - Van Driest formula:
The inner region is given by the Prandtl - Van Driest formula:
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<table width="100%"><tr><td>
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<table width="70"><tr><td>
:<math>
:<math>
{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|
{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|
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Where
Where
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
l = k y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
l = k y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
</math></td><td width="5%">(4)</td></tr></table>
</math></td><td width="5%">(4)</td></tr></table>
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<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}
\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}
</math></td><td width="5%">(5)</td></tr></table>
</math></td><td width="5%">(5)</td></tr></table>
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
\Omega_{ij} = \frac{1}{2}
\Omega_{ij} = \frac{1}{2}
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The outer region is given by:
The outer region is given by:
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
{\mu_t}_{outer} =  \rho \, K \, C_{CP} \, F_{WAKE} \, F_{KLEB}(y)
{\mu_t}_{outer} =  \rho \, K \, C_{CP} \, F_{WAKE} \, F_{KLEB}(y)
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Where
Where
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
F_{WAKE} = MIN \left( y_{MAX} \, F_{MAX} \,\,;\,\,
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<math>y_{MAX}</math> and <math>F_{MAX}</math> are determined from the maximum of the function:
<math>y_{MAX}</math> and <math>F_{MAX}</math> are determined from the maximum of the function:
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
F(y) = y \left| \Omega \right| \left(1-e^{\frac{-y^+}{A^+}} \right)
F(y) = y \left| \Omega \right| \left(1-e^{\frac{-y^+}{A^+}} \right)
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<math>F_{KLEB}</math> is the intermittency factor given by:
<math>F_{KLEB}</math> is the intermittency factor given by:
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
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
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<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.
<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.
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<table width="100%"><tr><td>
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<table width="70"><tr><td>
:<math>
:<math>
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})
</math></td><td width="5%">(11)</td></tr></table>
</math></td><td width="5%">(11)</td></tr></table>
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== Model constants ==
== Model constants ==
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</table>
</table>
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== Performance and applicability of the model ==
 
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(1). Yes, I think, the state of the art of the turbulence modeling is still not capable of predicting the separated flow accurately. (2). In David Wilcox's book, Section 3.6 Separated Flows, he stated "Menter(1992b) applied the Baldwin-Lomax model to an axisymmetric flow with a strong adverse pressure gradient. The experiment was conducted by Driver(1991). Inspection of the skin friction shows that the Baldwin-Lomax model yields a separation bubble nearly twice as long as the experimentally observed bubble. The corresponding rise in pressure over the separation region is 15% to 20% higher than measured. As pointed out by Menter, the Cebeci-Smith model yields similar results." (3). The question I have is: if an application engineer run the code with Baldwin-Lomax model and obtain the reault with flow separation, then based on the Menter's study, it is not possible for the application engineer to know whether the real flow is separated or not. Menter was very lucky because he is using the experimental data performed by Driver, so he was able to spot the poor performance of the Baldwin-Lomax model. But for the application engineer using the model, I don't think he will ever conduct the experiment to see whether his case is actually separated or not. The experiment to verify the flow separation in turbomachinery is very difficult to perform. (4). At the research level, this is just another opportunity to do research. But for the application engineer, what is going to happen to the results with flow separation? and also the subsequent designs in the real product? (I guess, it's his problem, if he is not reading this forum.)
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== Model variants ==
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You are quite right with your comments, as the original Baldwin-Lomax model indeed tends to overpredict separation bubbles. However, there are ad-hoc modifications which reduce this effect. For instance, the predictions of separation are quite sensitive to the Cwk coefficient and higher values than the original ones are known to restrict the early separation. Also, the Granville corrections take partly into account adverse pressure gradient effects, which attenuate the original weaknesses. This to mention that the model can be made to work in an efficient and adequate way for practical applications, without forgetting that, like all algebraic models, it contains less physics than two equation models, which in turn contain less physics than full Reynolds stress models. But particularly for separation predictions, much research is still needed to obtain reliable predictions with any turbulence model.  
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In order to improve the Baldwin-Lomax model modifications of the model-constants can be made in order to account for the effect of adverse pressure gradients. This has been done by Granville and Turner and Jennions. For further information see the references below.
 +
 +
== 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 <math>y+ < 1</math>.
 +
 +
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 researchers that the Baldwin-Lomax model tends to overpredict separated regions (see for example the comments made by David Wilcox [[#References|[Wilcox (1998)]]]). However, there are ad-hoc modifications which reduce this problem. For instance, prediction of separation is sensitive to the value of the <math>C_{WK}</math> coefficient and higher values than the original value tend to reduce the problems with too early separation. Also note that the Granville correction mentioned above, which attempts to account for adverse pressure gradient effects, increases the problem with too large separations.
 +
 +
The Baldwin-Lomax model does not account for the effect of a high free-stream turbulence level. Hence, it can not be used reliably when the free-stream turbulence has a signigicant effect on the boundary layer development.
 +
 +
 +
== Implementation issues ==
 +
 +
The computation of most of the model looks to relatively straightforward, but upon further examination, a few issues crop up.  First, the model is nonlocal in nature due to the presence of the damping function.  This means that for any location in the flow interior, we need a wall (or other suitable location) to compute a <math>y^+</math> from.  Further, the calculation of <math>y_{MAX}</math> and <math>F_{MAX}</math> is best suited to a structured grid in which grid lines emanate outward from a wall (or wakeline, etc.).  The model is thus best used in a structured grid setting, but has been used with unstructured grids via background grids [[#References|[Mavriplis (1991)]]].  Second, the determination of <math>y_{MAX}</math> and <math>F_{MAX}</math> is sensitive to gridpoint location, as the vorticity magnitude is typically only available pointwise.  One solution (perhaps with limited justification) is to do a fit of <math>F</math> to reduce any problems.  Finally, it is tempting to use the minimum of the two (inner and outer) eddy viscosity results instead of the correct crossover formula.  This simplifies the programming, but is not justifiable on any other grounds (and can lead to the use of the wrong eddy viscosity).  The (minimal) additional programming is required for correct model implementation.
 +
 +
''We need some further information here about what to think about when implementing this model in a CFD code. For example, there are some issues when computing the max and min values in the formulas - in complex 3D cases you can sometimes find several local mins/maxs. Can anyone add something about this?''
== References ==
== References ==
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* ''"Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows"'' by B. S. Baldwin and H. Lomax, AIAA Paper 78-257, 1978
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* {{reference-paper|author=Baldwin, B. S. and Lomax, H.|year=1978|title=Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows|rest=AIAA Paper 78-257}}
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* {{reference-paper|author=Granville, P. S.|year=1987|title=Baldwin-Lomax Factors for Turbulent Boundary Layers in Pressure Gradients|rest=AIAA Journal, Vol. 25, No. 12, pp. 1624-1627}}
 +
* {{reference-paper|author=Mavriplis, D. J.|year=1991|title=Algebraic turbulence modeling for unstructured and adaptive meshes|rest=AIAA Journal, Vol. 29, pp. 2086-2093}}
 +
* {{reference-paper|author=Turner, M. G. and Jennions, I. K.|year=1993|title=An Investigation of Turbulence Modeling in Transonic Fans Including a Novel Implementation of an Implicit <math>k-\epsilon</math> Turbulence Model|rest=Journal of Turbomachinery, Vol. 115, April, pp. 249-260}}
 +
* {{reference-book|author=Wilcox, D.C. |year=1998|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}
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----
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<i> Return to [[Turbulence modeling]] </i>
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 +
 
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[[Category:Turbulence models]]

Latest revision as of 09:15, 3 January 2012

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

The Baldwin-Lomax model [Baldwin and Lomax (1978)] 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. 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 (see below).

Contents

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


Model variants

In order to improve the Baldwin-Lomax model modifications of the model-constants can be made in order to account for the effect of adverse pressure gradients. This has been done by Granville and Turner and Jennions. For further information see the references below.


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+ < 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 researchers that the Baldwin-Lomax model tends to overpredict separated regions (see for example the comments made by David Wilcox [Wilcox (1998)]). 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 tend to reduce the problems with too early separation. Also note that the Granville correction mentioned above, which attempts to account for adverse pressure gradient effects, increases the problem with too large separations.

The Baldwin-Lomax model does not account for the effect of a high free-stream turbulence level. Hence, it can not be used reliably when the free-stream turbulence has a signigicant effect on the boundary layer development.


Implementation issues

The computation of most of the model looks to relatively straightforward, but upon further examination, a few issues crop up. First, the model is nonlocal in nature due to the presence of the damping function. This means that for any location in the flow interior, we need a wall (or other suitable location) to compute a y^+ from. Further, the calculation of y_{MAX} and F_{MAX} is best suited to a structured grid in which grid lines emanate outward from a wall (or wakeline, etc.). The model is thus best used in a structured grid setting, but has been used with unstructured grids via background grids [Mavriplis (1991)]. Second, the determination of y_{MAX} and F_{MAX} is sensitive to gridpoint location, as the vorticity magnitude is typically only available pointwise. One solution (perhaps with limited justification) is to do a fit of F to reduce any problems. Finally, it is tempting to use the minimum of the two (inner and outer) eddy viscosity results instead of the correct crossover formula. This simplifies the programming, but is not justifiable on any other grounds (and can lead to the use of the wrong eddy viscosity). The (minimal) additional programming is required for correct model implementation.

We need some further information here about what to think about when implementing this model in a CFD code. For example, there are some issues when computing the max and min values in the formulas - in complex 3D cases you can sometimes find several local mins/maxs. Can anyone add something about this?

References

  • Baldwin, B. S. and Lomax, H. (1978), "Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows", AIAA Paper 78-257.
  • Granville, P. S. (1987), "Baldwin-Lomax Factors for Turbulent Boundary Layers in Pressure Gradients", AIAA Journal, Vol. 25, No. 12, pp. 1624-1627.
  • Mavriplis, D. J. (1991), "Algebraic turbulence modeling for unstructured and adaptive meshes", AIAA Journal, Vol. 29, pp. 2086-2093.
  • Turner, M. G. and Jennions, I. K. (1993), "An Investigation of Turbulence Modeling in Transonic Fans Including a Novel Implementation of an Implicit k-\epsilon Turbulence Model", Journal of Turbomachinery, Vol. 115, April, pp. 249-260.
  • Wilcox, D.C. (1998), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..


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