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TWiki> Saturne Web>SatTurbulenceModels>PhiModel (2011-07-07, FlavienBillard)

TWiki> Saturne Web>SatTurbulenceModels>PhiModel (2011-07-07, FlavienBillard)

The model is designed to obtain a measure of non-local effects via elliptic relaxation. In order to account for non-local effects in the framework of an eddy viscosity model, the elliptic relaxation approach of the full RSM is simplified in the case of a plane channel flow. Taking into account that the appropriate velocity scale near the wall (Durbin, 1991) is , the turbulent viscosity can be expressed as:

With the time and length scales defined by
In order to make the model more adaptable to an
industrial code but without sacrificing its performance, another
approach has been followed at the University of Manchester during the course of this work (See: Laurence *et al.* 2004).

By introducing a new variable, , defined as:

a transport equation can be solved for this new variable, which introduces some advantages. The resulting equations for and are:

In isotropic flow . A clear benefit is in the boundary condition for :

Although the resulting model has a boundary condition less stiff than the original one, it is non-zero at the wall but resulting from the ratio of three unknown variables, therefore still diminishing the robustness of the model. An interesting benefit is that the term is no longer in the transport equation, which can be difficult to reproduce correctly in the near wall region since becomes large and the ratio tends to zero.

By using the limiting value of near the wall and applying the theorem, the value of can be written in terms of :

The singularity near the wall is now second order only (ratio of two(1) |

Considering the limit , it is possible to show that . There are different possible substitutions to reach the homogeneous boundary condition for . Some of them, e.g. , can lead to an ill-posed problem for since in the right-hand side we obtain a negative diffusion term: .

Introducing the definition for , the equations of the model can be found to be:

The term is changed to:

Due to the introduction of the approximation in equation (1), the coefficients used in this formulation have been optimised using DNS data for a channel flow at the Reynolds as a target. The proposed coefficients are shown in the following table.
D. Laurence, J.C. Uribe, and S. Utyuzhnikov. A robust formulation of the v2-f model. *Flow, Turbulence and Combustion*, 73, 2004.

These are the cases that have results using this model in the CfdTm database:

Test Case | Case Results | Author | Model |
---|---|---|---|

Channel Flow | TestCase001Res001 | Juan Uribe | Phi-f, SST, EBM (Elliptic Blending Model) |

Channel Flow | TestCase001Res004 | J. Uribe | |

Vertical Heated Pipe | TestCase005Res004 | You et al [2003] | DNS |

Mixed convection in a vertical channel | TestCase012Res000 | Flavien Billard | |

Asymmetric plane diffuser | TestCase013Res000 | Juan Uribe, Flavien Billard | SST, V2f models |

Asymmetric plane diffuser | TestCase013Res002 | J .Uribe | |

Flow over 2D periodic hills | TestCase014Res000 | Juan Uribe | |

TestCase015Res000 | J. Uribe | ||

TestCase015Res001 | E. Moreau | ||

Parietal two-phase jet | TestCase038Res000 | M. Guillaud | |

Diurnal evolution of an atmospheric boundary layer | TestCase055Res000 | M. Milliez |

Number of topics: 11

Topic revision: r3 - 2011-07-07 - 10:32:20 - FlavienBillard

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