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Dissipation element analysis in turbulent channel flow

Published online by Cambridge University Press:  02 February 2012

Fettah Aldudak  and
Martin Oberlack
Fettah Aldudak*
Affiliation:
Department of Mechanical Engineering, TU Darmstadt, Petersenstrasse 30, 64287 Darmstadt, Germany
Martin Oberlack
Affiliation:
Department of Mechanical Engineering, TU Darmstadt, Petersenstrasse 30, 64287 Darmstadt, Germany Center of Smart Interfaces, TU Darmstadt, Petersenstrasse 32, 64287 Darmstadt, Germany GS Computational Engineering, TU Darmstadt, Dolivostrasse 15, 64293 Darmstadt, Germany
*
†Email address for correspondence: [email protected]
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Abstract

In order to analyse the geometric structure of turbulent flow patterns and their statistics for various scalar fields we adopt the dissipation element (DE) approach and apply it to turbulent channel flow by employing direct numerical simulations (DNS) of the Navier–Stokes equations. Gradient trajectories starting from any point in a scalar field in the directions of ascending and descending scalar gradients will always reach an extremum, i.e. a minimum or a maximum point, where . The set of all points and trajectories belonging to the same pair of extremal points defines a dissipation element. Extending previous DE approaches, which were only applied to homogeneous turbulence, we here focus on exploring the influence of solid walls on the dissipation element distribution. Employing group-theoretical methods and known symmetries of Navier–Stokes equations, we observe for the core region of the flow, i.e. the region beyond the buffer layer, that the probability distribution function (p.d.f.) of the DE length exhibits an invariant functional form, in other words, self-similar behaviour with respect to the wall distance. This is further augmented by the scaling behaviour of the mean DE length scale which shows a linear scaling with the wall distance. The known proportionality of the mean DE length and the Taylor length scale is also revisited. Utilizing a geometric analogy we give the number of DE elements as a function of the wall distance. Further, it is observed that the DE p.d.f. is rather insensitive, i.e. invariant with respect both to the Reynolds number and the actual scalar which has been employed for the analysis. In fact, a very remarkable degree of isotropy is observed for the DE p.d.f. in regions of high shear. This is in stark contrast to classical Kolmogorov scaling laws which usually exhibit a strong dependence on quantities such as shear, anisotropy and Reynolds number. In addition, Kolmogorov’s scaling behaviour is in many cases only visible for very large Reynolds numbers. This is rather different in the present DE approach which applies also for low Reynolds numbers. Moreover, we show that the DE p.d.f. agrees very well with the log-normal distribution and derive a log-normal p.d.f. model taking into account the wall-normal dependence. Finally, the conditional mean scalar differences of the turbulent kinetic energy at the extremal points of DE are examined. We present a power law with scaling exponent of known from Kolmogorov’s hypothesis for the centre of the channel and a logarithmic law near the wall.

JFM classification

Waves/Free-surface Flows: Channel flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 694 , 10 March 2012 , pp. 332 - 351
Copyright
Copyright © Cambridge University Press 2012

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