Is the Smagorinsky coefficient sensitive to uncertainty in the form of the energy spectrum?

• Published in: Physics of fluids (1994) 2011-12, Vol.23 (12), p.125109-125109-14
• Main Authors: Meldi, M., Lucor, D., Sagaut, P.
• Format: Article
• Language: English
• Subjects: Engineering Sciences; Exact sciences and technology; Fluid dynamics; Fluids mechanics; Fundamental areas of phenomenology (including applications); Mechanics; Physics; Turbulence simulation and modeling; Turbulent flows, convection, and heat transfer
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/1.3663305

We investigate the influence of uncertainties in the shape of the energy spectrum over the Smagorinsky ["General circulation experiments with the primitive equations. I: The basic experiment," Mon. Weather Rev. 91 (3), 99 (1963)] subgrid scale model constant C S : the analysis is carried out by a stochastic approach based on generalized polynomial chaos. The free parameters in the considered energy spectrum functional forms are modeled as random variables over bounded supports: two models of the energy spectrum are investigated, namely, the functional form proposed by Pope [ Turbulent Flows (Cambridge University Press, Cambridge, 2000)] and by Meyers and Meneveau ["A functional form for the energy spectrum parametrizing bottleneck and intermittency effects," Phys. Fluids 20 (6), 065109 (2008)]. The Smagorinsky model coefficient, computed from the algebraic relation presented in a recent work by Meyers and Sagaut ["On the model coefficients for the standard and the variational multi-scale Smagorinsky model," J. Fluid Mech. 569 , 287 (2006)], is considered as a stochastic process and is described by numerical tools streaming from the probability theory. The uncertainties are introduced in the free parameters shaping the energy spectrum in correspondence to the large and the small scales, respectively. The predicted model constant is weakly sensitive to the shape of the energy spectrum when large scales uncertainty is considered: if the large-eddy simulation (LES) filter cut is performed in the inertial range, a significant probability to recover values lower in magnitude than the asymptotic Lilly-Smagorinsky model constant is recovered. Furthermore, the predicted model constant occurrences cluster in a compact range of values: the correspondent probability density function rapidly drops to zero approaching the extremes values of the range, which show a significant sensitivity to the LES filter width. The sensitivity of the model constant to uncertainties propagated in the small scales of the energy spectrum is noticeable and a wide range of possible Smagorinsky model constant values is observed, if the LES filter cut is performed close to the dissipation region.