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Comoving frames and symmetry-related motions in parallel shear flows

Published online by Cambridge University Press:  25 June 2014

Tobias Kreilos ,
Stefan Zammert  and
Bruno Eckhardt
Tobias Kreilos*
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
Stefan Zammert
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany
Bruno Eckhardt*
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany J. M. Burgerscentrum, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

Parallel shear flows have continuous symmetries of translation in the downstream and spanwise directions. As a consequence, flow states that differ in their spanwise or downstream location but are otherwise identical are dynamically equivalent. In the case of travelling waves, this trivial degree of freedom can be removed by going to a frame of reference that moves with the state, thereby turning the travelling wave in the laboratory frame into a fixed point in the comoving frame of reference. We here discuss a general approach, the method of comoving frames, by which the symmetry related motions can also be removed for more complicated and dynamically active states and demonstrate its application for several examples. For flow states in the asymptotic suction boundary layer (ASBL) we show that in the case of the long-period oscillatory edge state we can find local phase speeds which remove the fast oscillations and reveal the slow vortex dynamics underlying the burst phenomenon. For spanwise translating states we show that the method removes the drift but not the dynamical events that cause the big spanwise displacement. For a turbulent case we apply the method to the spanwise shifts and find slow components that are correlated over very long times. Calculations for plane Poiseuille flow show that the long correlations in the transverse motions are not specific to the ASBL.

JFM classification

Mathematical Foundations: General fluid mechanics Turbulent Flows: Turbulent boundary layers Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , pp. 685 - 697
Copyright
© 2014 Cambridge University Press 

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Kreilos et al. supplementary movie

The edge state in the ASBL at Re=400 in a box of size Lx=4pi, Lz=2pi. The left box shows the dynamics in the laboratory frame of reference, the right box in the comoving one. Color-coding is the downstream velocity component, which is also the blue isocontour. Black/white isocontours are vortices visualized by lambda2 and colored according to their downstream vorticity.

Download Kreilos et al. supplementary movie(Video)
Video 8.5 MB