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Mathematical and Numerical Analysis of an Alternative Well-Posed Two-Layer Turbulence Model

Published online by Cambridge University Press:  15 April 2002

Bijan Mohammadi  and
Guillaume Puigt
Bijan Mohammadi
Affiliation:
University of Montpellier II, Mathematics Department, ACSIOM Laboratory, France. ([email protected])
Guillaume Puigt
Affiliation:
University of Montpellier II, Mathematics Department, ACSIOM Laboratory , France. ([email protected])
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Abstract

In this article, we wish to investigate the behavior of a two-layer k - εturbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations.First, we explain the difficulties inherent in themodel. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulentviscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspectsof the model are preserved by the new formulation, and in particular, we show how the physicists can help us to provethe existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulencemodel and we show that the alternative model may be preferred to the original one, because of its good properties(existence of a solution of the coupled problems).

Keywords

Incompressible turbulent flowstwo-layer k - ε modelnear-wall treatment.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 6 , November 2001 , pp. 1111 - 1136
Copyright
© EDP Sciences, SMAI, 2001

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References

S. Clain, Analyse mathématique et numérique d'un modèle de chauffage par induction. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne (1994).
Clain, S. and Touzami, R., Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients. RAIRO Modél. Math. Anal. Numér. 31 (1997) 845-870. CrossRef
J. Cousteix, Turbulence et couche limite. Cepadues, Ed., Toulouse (1990).
R. Dautrey and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8. Masson, Ed., Paris (1988).
G. de Rham, Variétés différientiables. Hermann, Paris (1960).
Gallouët, T. and Herbin, R., Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 2 (1994) 49-55. CrossRef
T. Gallouët, J. Lederer, R. Lewandowski, F. Murat and L. Tartar, On a turbulent system with unbounded eddy viscosities. To appear in J. Non-Linear Anal. TMA.
M. Gómez Mármol and F. Ortegón Gallego, Existence of Solution to Non-Linear Elliptic Systems Arising in Turbulence Modelling. M 3 AS (Math. Models Methods Appl. Sci.) 10 (2000) 247-260.
M. Gómez Mármol and F. Ortegón Gallego, Coupling the Stokes and Navier-Stokes Equations with Two Scalar Nonlinear Parabolic Equations. ESAIM: M2AN 33 (1999) 157-167
R. Lewandowski and B. Mohammadi, Existence and Positivity Results for the Φ - θ and a Modified k - ε Turbulence Models. M 3 AS (Math. Models Methods Appl. Sci.) 3 (1993) 195-215.
R. Lewandowski, Analyse mathématique et océanographie. Masson, Ed., Paris (1997).
Lewandowski, R., The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity. J. Non-Linear Anal. TMA 28 (1997) 393-417. CrossRef
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villard, Eds., Dunod, Paris (1969).
B. Mohammadi and G. Puigt, Generalized Wall Functions for High-Speed Separated Flows over Adiabatic and Isothermal Walls. To appear in Internat. J. Comput. Fluid Dyn.
B. Mohammadi, A Stable Algorithm for the k - ε Model for Compressible Flows. INRIA, Report No. 1335 (1990).
B. Mohammadi and O. Pironneau, Analysis of the k - ε turbulence model. Wiley-Masson, Eds., Paris (1994).
Patel, V.C., Rhodi, W. and Scheuerer, G., Turbulence models for near-wall and low-Reynolds number flows: a review. AIAA J. 23 (1984) 1308-1319. CrossRef
R. Temam, Infinite Dimensional Systems in Mechanics and Physics. 2nd edn., Springer-Verlag, Eds., Berlin, Heidelberg, New York (1997).