On the Sensitivity to the Filtering Radius in Leray Models of Incompressible Flow
One critical aspect of Leray models for the Large Eddy Simulation (LES) of incompressible flows at moderately large Reynolds number (in the range of few thousands) is the selection of the filter radius. This drives the effective regularization of the filtering procedure, and its selection is a trade-off between stability (the larger, the better) and accuracy (the smaller, the better). In this paper, we consider the classical Leray-\(\alpha \) and a recently introduced (by one of the authors) Leray model with a deconvolution-based indicator function, called Leray-\(\alpha \)-NL. We investigate the sensitivity of the solutions to the filter radius by introducing the sensitivity systems, analyzing them at the continuous and discrete levels, and numerically testing them on two benchmark problems.
The research presented in this work was carried out during LR’s visit at the Department of Mathematics and Computer Science at Emory University in the fall semester 2014. This support is gratefully acknowledged. This research has been supported in part by the NSF under grants DMS-1620384/DMS-1620406 (Quaini and Veneziani), DMS-1262385 (Quaini), Emory URC Grant 2015 Numerical Methods for Flows at Moderate Reynolds Numbers in LVAD (Veneziani), and DMS-1522191 (Rebholz).
- 3.Borggaard J, Burns J (1995) A sensitivity equation approach to shape optimization in fluid flows. In: Flow control (Minneapolis, MN, 1992), vol 68 of IMA Vol Math Appl, Springer, New York, pp 49–78Google Scholar
- 13.Geurts BJ, Holm DD (2002) Leray simulation of turbulent shear layers. In: Castro IP, Hancock PE, Thomas TG (eds), Advances in Turbulence IX: Proceedings of the Ninth European Turbulence Conference (Southampton, 2002), CIMNE, pp 337–340Google Scholar
- 17.Godfrey AG, Cliff EM (1998) Direct calculation of aerodynamic force derivatives: a sensitivity equation approach. In: 36th AIAA Aerospace Sciences Meeting and Exhibit. AIAA, Paper 98-0393, 12pGoogle Scholar
- 20.Gunzburger MD (1989) Finite element methods for viscous incompressible flows: a guide to theory, practice, and algorithms. Academic Press, BostonGoogle Scholar
- 26.Layton W (2008) Introduction to the numerical analysis of incompressible viscous flows. SIAMGoogle Scholar
- 36.Sagaut P, Lê T (1997) Some investigations of the sensitivity of large eddy simulation. Technical report, ONERAGoogle Scholar
- 37.Sagaut P, Lê TH (1997) Some investigations on the sensitivity of large eddy simulation. In: Chollet J-P, Voke PR, Kleiser L (eds) Direct and Large-Eddy Simulation II: Proceedings of the ERCOFTAC Workshop (Grenoble, 1996). Springer, Dordrecht, pp 81–92Google Scholar
- 38.Stanley L, Stewart D (2002) Design sensitivity analysis: computational issues of sensitivity equation methods. Number 25 in Frontiers in Applied Mathematics. SIAM, Philadelphia, PAGoogle Scholar