On Bardina and Approximate Deconvolution Models

Roger Lewandowski[1]

  • [1] IRMAR — UMR CNRS 6625 Department of Mathematics University of Rennes 1 Campus Beaulieu 35042 Rennes cedex France

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-12
  • ISSN: 2266-0607

Abstract

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We first outline the procedure of averaging the incompressible Navier-Stokes equations when the flow is turbulent for various type of filters. We introduce the turbulence model called Bardina’s model, for which we are able to prove existence and uniqueness of a distributional solution. In order to reconstruct some of the flow frequencies that are underestimated by Bardina’s model, we next introduce the approximate deconvolution model (ADM). We prove existence and uniqueness of a “regular weak solution” to the ADM for each deconvolution order N , and then that the corresponding sequence of solutions converges to the mean Navier-Stokes Equations when N goes to infinity.

How to cite

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Lewandowski, Roger. "On Bardina and Approximate Deconvolution Models." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-12. <http://eudml.org/doc/251182>.

@article{Lewandowski2011-2012,
abstract = {We first outline the procedure of averaging the incompressible Navier-Stokes equations when the flow is turbulent for various type of filters. We introduce the turbulence model called Bardina’s model, for which we are able to prove existence and uniqueness of a distributional solution. In order to reconstruct some of the flow frequencies that are underestimated by Bardina’s model, we next introduce the approximate deconvolution model (ADM). We prove existence and uniqueness of a “regular weak solution” to the ADM for each deconvolution order $N$, and then that the corresponding sequence of solutions converges to the mean Navier-Stokes Equations when $N$ goes to infinity.},
affiliation = {IRMAR — UMR CNRS 6625 Department of Mathematics University of Rennes 1 Campus Beaulieu 35042 Rennes cedex France},
author = {Lewandowski, Roger},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {incompressible Navier-Stokes equations; turbulence model; Bardina's model; regular weak solution},
language = {eng},
pages = {1-12},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {On Bardina and Approximate Deconvolution Models},
url = {http://eudml.org/doc/251182},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Lewandowski, Roger
TI - On Bardina and Approximate Deconvolution Models
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 12
AB - We first outline the procedure of averaging the incompressible Navier-Stokes equations when the flow is turbulent for various type of filters. We introduce the turbulence model called Bardina’s model, for which we are able to prove existence and uniqueness of a distributional solution. In order to reconstruct some of the flow frequencies that are underestimated by Bardina’s model, we next introduce the approximate deconvolution model (ADM). We prove existence and uniqueness of a “regular weak solution” to the ADM for each deconvolution order $N$, and then that the corresponding sequence of solutions converges to the mean Navier-Stokes Equations when $N$ goes to infinity.
LA - eng
KW - incompressible Navier-Stokes equations; turbulence model; Bardina's model; regular weak solution
UR - http://eudml.org/doc/251182
ER -

References

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  1. H. Ali, Mathematical study of some turbulence models. PhD Thesis of the University of Rennes 1, 2011. 
  2. J. Bardina, J.H. Ferziger and W.C. Reynold. Improved turbulence models based on large eddy simulation of homogeneous, invompressible, turbulent ows. Technical Report No.TF-19, Department of mechanical Engineering, Stanford University, Stanford, 1983. 
  3. F.K. Chow, R.L. Street, M. Xue and J.H. Ferziger. Explicit filtering and reconstruction turbulence modeling for large-eddy simulation of neutral boundary layer ow, Journal of the Atmospheric Sciences, 62(7): 2058-2077, 2005. 
  4. L. Berselli and R. Lewandowski. Convergence of Approximate Deconvolution Models to the mean Navier-Stokes equations Annales de l’Institut Henri Poincare (C), Non Linear Analysis, 29:171-198, 2012. Zbl1302.76083MR2901193
  5. A. Dunca and Y. Epshteyn. On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows. SIAM J. Math. Anal., 37(6):1890–1902 (electronic), 2006. Zbl1128.76029MR2213398
  6. W. Layton and R. Lewandowski. A simple, accurate and stable scale similarity model for LES: Energy balance and existence of weak solutions. Appl. Math. Lett., 4:1205–1209, 2003. Zbl1039.76027MR2015713
  7. W. Layton and R. Lewandowski. On a well posed turbulence model. Discrete and Continuous Dynamical Systems series B, 6(1):111-128, 2006. Zbl1089.76028MR2172198
  8. W. J. Layton and R. Lewandowski. Residual stress of approximate deconvolution models of turbulence. J. Turbul., 7:Paper 46, 21 pp. (electronic), 2006. Zbl1273.76206MR2254595
  9. W. Layton and R. Lewandowski, A high accuracy Leray-deconvolution model of turbulence and its limiting behavior Analysis and Applications 6(1):23-49, 2008. Zbl1210.76084MR2380885
  10. S. Stoltz and N. A. Adams. An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids, 11:1699–1701, 1999. Zbl1147.76506
  11. S. Stolz, N. A. Adams, and L. Kleiser. An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows. Phys. Fluids, 13(4):997–1015, 2001. Zbl1184.76530

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