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Technische Universität
Darmstadt (TU Darmstadt), |
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Abstract: In recent years it became clear that at least for "clean" canonical turbulent flows such as pipe, channel or boundary layer flow Lie group theory provides the axiomatic building block for deriving turbulent scaling laws. The basis of the analysis is the infinite set of multi-point correlation equations which solely originate from the Navier-Stokes equations. The set of mean velocity profiles obtained includes the classical logarithmic law of the wall, an algebraic law in the center of a channel flow, the linear mean velocity in the center of a Couette flow, the linear mean velocity in the center of a rotating channel flow, and an exponential mean velocity profile which is an explicit form of the law of the wake. Beside the original work in Oberlack (JFM, vol 427, 2001) the new scaling laws were verified by Lindgren, Osterlund & Johansson (JFM vol 502, 2004) using the high Reynolds number KTH data base and Khujadze & Oberlack (TCFD vol 18, 2004)) also further validated them employing high resolution DNS data. All the latter classical and new scaling laws were, however, derived under idealized and rather "clean'' conditions in particular an infinite Reynolds number was presumed. Further, scaling laws were limited to the mean velocity. The content of the talk is twofold. First, employing DNS and experimental data we show that the turbulent scaling laws are not limited to mean velocity but scaling may also be observed in correlation function data. Second, we introduce approximate group theory to derive the turbulent scaling laws which explicitly depend on large but finite Reynolds number. Approximate groups provide a new mathematical methodology which unifies group theory with asymptotics. Finally it is shown that in the limit of infinite Reynolds number all scaling laws converge to the classical laws. |
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