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Download fileThe Schur decomposition of the velocity gradient tensor for turbulent flows
The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions.
Funding
This research was supported by a Royal Academy of Engineering/Leverhulme Trust Senior Research Fellowship LTSRF1516-12-89 awarded to the author.
History
School
- Architecture, Building and Civil Engineering
Published in
Journal of Fluid MechanicsVolume
848Pages
876 - 905Citation
KEYLOCK, C.J., 2018. The Schur decomposition of the velocity gradient tensor for turbulent flows. Journal of Fluid Mechanics, 848, pp.876-905.Publisher
© Cambridge University PressVersion
- AM (Accepted Manuscript)
Publication date
2018-06-13Notes
This article has been published in a revised form in Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2018.344. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Cambridge University PressISSN
0022-1120eISSN
1469-7645Publisher version
Language
- en