Mathematical and numerical analysis of an alternative well-posed two-layer turbulence model

Bijan Mohammadi; Guillaume Puigt

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 6, page 1111-1136
  • ISSN: 0764-583X

Abstract

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In this article, we wish to investigate the behavior of a two-layer k - ε turbulence 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 the model. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspects of the model are preserved by the new formulation, and in particular, we show how the physicists can help us to prove the existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulence model 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).

How to cite

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Mohammadi, Bijan, and Puigt, Guillaume. "Mathematical and numerical analysis of an alternative well-posed two-layer turbulence model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.6 (2001): 1111-1136. <http://eudml.org/doc/194088>.

@article{Mohammadi2001,
abstract = {In this article, we wish to investigate the behavior of a two-layer $k-\varepsilon $ turbulence 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 the model. Then, we present a new variable $\theta $ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspects of the model are preserved by the new formulation, and in particular, we show how the physicists can help us to prove the existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulence model 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).},
author = {Mohammadi, Bijan, Puigt, Guillaume},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {incompressible turbulent flows; two-layer $k-\varepsilon $ model; near-wall treatment; two-layer -epsilon model},
language = {eng},
number = {6},
pages = {1111-1136},
publisher = {EDP-Sciences},
title = {Mathematical and numerical analysis of an alternative well-posed two-layer turbulence model},
url = {http://eudml.org/doc/194088},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Mohammadi, Bijan
AU - Puigt, Guillaume
TI - Mathematical and numerical analysis of an alternative well-posed two-layer turbulence model
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 6
SP - 1111
EP - 1136
AB - In this article, we wish to investigate the behavior of a two-layer $k-\varepsilon $ turbulence 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 the model. Then, we present a new variable $\theta $ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspects of the model are preserved by the new formulation, and in particular, we show how the physicists can help us to prove the existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulence model 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).
LA - eng
KW - incompressible turbulent flows; two-layer $k-\varepsilon $ model; near-wall treatment; two-layer -epsilon model
UR - http://eudml.org/doc/194088
ER -

References

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  1. [1] S. Clain, Analyse mathématique et numérique d’un modèle de chauffage par induction. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne (1994). 
  2. [2] S. Clain and R. Touzami, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients. RAIRO Modél. Math. Anal. Numér. 31 (1997) 845–870. Zbl0894.35035
  3. [3] J. Cousteix, Turbulence et couche limite. Cepadues, Ed., Toulouse (1990). 
  4. [4] R. Dautrey and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8. Masson, Ed., Paris (1988). Zbl0749.35004MR1016605
  5. [5] G. de Rham, Variétés différientiables. Hermann, Paris (1960). Zbl0089.08105
  6. [6] T. Gallouët and R. Herbin, Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 2 (1994) 49–55. Zbl0791.35043
  7. [7] 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. Zbl1013.35068MR1941245
  8. [8] 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. Zbl1023.76017
  9. [9] 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 Zbl0921.76039
  10. [10] 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. Zbl0773.76036
  11. [11] R. Lewandowski, Analyse mathématique et océanographie. Masson, Ed., Paris (1997). 
  12. [12] R. Lewandowski, 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. Zbl0863.35077
  13. [13] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villard, Eds., Dunod, Paris (1969). Zbl0189.40603MR259693
  14. [14] 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. Zbl1079.76557MR1895352
  15. [15] B. Mohammadi, A Stable Algorithm for the k - ε Model for Compressible Flows. INRIA, Report No. 1335 (1990). 
  16. [16] B. Mohammadi and O. Pironneau, Analysis of the k - ε turbulence model. Wiley-Masson, Eds., Paris (1994). MR1296252
  17. [17] V.C. Patel, W. Rhodi and G. Scheuerer, Turbulence models for near-wall and low-Reynolds number flows: a review. AIAA J. 23 (1984) 1308–1319. 
  18. [18] R. Temam, Infinite Dimensional Systems in Mechanics and Physics. 2nd edn., Springer-Verlag, Eds., Berlin, Heidelberg, New York (1997). Zbl0871.35001MR1441312

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