Comptes Rendus
no. 5
Partial differential equations/Numerical analysis
On non-linear Stokes problems with viscosity depending on the distance to the wall

Presented by: the Editorial Board

Jacques Rappaz1; Jonathan Rochat1
1 EPFL – SB – MATHICSE, CH-1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 499-502.

In fluid mechanics, the RANS modeling (Reynolds Averaged Navier–Stokes equation) assumes that the period of the averaged solutions to the Navier–Stokes equations is several orders of magnitude larger than the turbulent fluctuations. A type of simple model often used by engineers is a mixing-length model called “Smagorinsky modeling”. In this paper, we present some theoretical and numerical results on a mixing-length model in which the eddy viscosity is depending on the strain tensor and on the distance to the wall of the fluid flow domain. In particular, we show that the so-called von Karman model becomes an ill-posed problem when the laminar viscosity tends to zero.

En mécanique des fluides, la décomposition de Reynolds appliquée aux équations de Navier–Stokes suppose que la période des solutions moyennées est bien plus grande que les fluctuations turbulentes locales. Un modèle de longueur de mélange populaire chez les ingénieurs est appelé « modèle de Smagorinsky ». Dans cette courte note, nous présentons quelques résultats théoriques et numériques sur un modèle de longueur de mélange dont la viscosité turbulente dépend à la fois du tenseur des déformations et de la distance à la paroi la plus proche. En particulier, nous montrons que le modèle dit de von Karman devient mal posé lorsque la viscosité laminaire tend vers zéro.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.01.022

Jacques Rappaz 1; Jonathan Rochat 1

1 EPFL – SB – MATHICSE, CH-1015 Lausanne, Switzerland 
@article{CRMATH_2016__354_5_499_0,
     author = {Jacques Rappaz and Jonathan Rochat},
     title = {On non-linear {Stokes} problems with viscosity depending on the distance to the wall},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {499--502},
     publisher = {Elsevier},
     volume = {354},
     number = {5},
     year = {2016},
     doi = {10.1016/j.crma.2016.01.022},
     language = {en},
}
TY  - JOUR
AU  - Jacques Rappaz
AU  - Jonathan Rochat
TI  - On non-linear Stokes problems with viscosity depending on the distance to the wall
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 499
EP  - 502
VL  - 354
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2016.01.022
LA  - en
ID  - CRMATH_2016__354_5_499_0
ER  - 
%0 Journal Article
%A Jacques Rappaz
%A Jonathan Rochat
%T On non-linear Stokes problems with viscosity depending on the distance to the wall
%J Comptes Rendus. Mathématique
%D 2016
%P 499-502
%V 354
%N 5
%I Elsevier
%R 10.1016/j.crma.2016.01.022
%G en
%F CRMATH_2016__354_5_499_0
Jacques Rappaz; Jonathan Rochat. On non-linear Stokes problems with viscosity depending on the distance to the wall. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 499-502. doi : 10.1016/j.crma.2016.01.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.022/

[1] J. Baranger; K. Najib Analyse numérique des écoulements quasi-Newtoniens dont la viscosité obéit à la loi de puissance ou la loi de Carreau, Numer. Math., Volume 58 (1990), pp. 35-49

[2] B. Dacorogna Direct Methods in the Calculus of Variations, Springer, 2008

[3] P. Drabek; A. Kufner; F. Nicolosi Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyer, 1997

[4] P. Gurka Continuous and compact imbeddings of weighted Sobolev spaces, Czechoslov. Math. J., Volume 39 (1989) no. 1, pp. 78-94

[5] A. Kalamajska Coercive inequalities on weighted Sobolev spaces, Colloq. Math., Volume LXVI (1993) no. Fasc. 2

[6] A. Kufner Weighted Sobolev Spaces, John Wiley, 1985

[7] J. Rappaz; J. Rochat On some weighted Stokes problems. Applications on Smagorinsky models, March 2016 https://infoscience.epfl.ch/record/217038 (Preprint)

[8] P. Sagaut; C. Cambon Homogeneous Turbulence Dynamics, Cambridge University Press, 2008

Cited by Sources:

☆ Project supported by Rio-Tinto Alcan Company, with grant number CW2088943.

Comments - Policy