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The mean conformation tensor in viscoelastic turbulence

Published online by Cambridge University Press:  19 February 2019

Ismail Hameduddin Open the ORCID record for Ismail Hameduddin [Opens in a new window]  and
Tamer A. Zaki Open the ORCID record for Tamer A. Zaki [Opens in a new window]
Ismail Hameduddin
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T1Z2, Canada
Tamer A. Zaki*
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

This work demonstrates that the popular arithmetic mean conformation tensor frequently used in the analysis of turbulent viscoelastic flows is not a good representative of the ensemble. Alternative means based on recent developments in the literature are proposed, namely, the geometric and log-Euclidean means. These means are mathematically consistent with the Riemannian structure of the manifold of positive-definite tensors, on which the conformation tensor lives, and have useful properties that make them attractive alternatives to the arithmetic mean. Using a turbulent FENE-P channel flow dataset, it is shown that these two alternatives are physically representative of the ensemble. By definition, these means minimize the geodesic distance to realizations and exactly preserve the scalar geometric mean of the volume and of the principal stretches. The proposed geometric and log-Euclidean means have clear physical interpretations and are attractive quantities for turbulence modelling.

JFM classification

Waves/Free-surface Flows: Channel flow Turbulent Flows: Turbulent Flows Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 363 - 380
Copyright
© 2019 Cambridge University Press 

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References

Ando, T., Li, C. & Mathias, R. 2004 Geometric means. Linear Algebr. Applics. 385, 305–334.10.1016/j.laa.2003.11.019Google Scholar
Arsigny, V., Fillard, P., Pennec, X. & Ayache, N.2005 Fast and simple computations on tensors with log-Euclidean metrics. PhD thesis, INRIA.Google Scholar
Arsigny, V., Fillard, P., Pennec, X. & Ayache, N. 2006 Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56 (2), 411–421.10.1002/mrm.20965Google Scholar
Arsigny, V., Fillard, P., Pennec, X. & Ayache, N. 2007 Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Applics. 29 (1), 328–347.10.1137/050637996Google Scholar
Bhatia, R. & Karandikar, R. L. 2012 Monotonicity of the matrix geometric mean. Math. Ann. 353 (4), 1453–1467.10.1007/s00208-011-0721-9Google Scholar
Hameduddin, I., Gayme, D. F. & Zaki, T. A. 2019 Perturbative expansions of the conformation tensor in viscoelastic flows. J. Fluid Mech. 858, 377–406.10.1017/jfm.2018.777Google Scholar
Hameduddin, I., Meneveau, C., Zaki, T. A. & Gayme, D. F. 2018 Geometric decomposition of the conformation tensor in viscoelastic turbulence. J. Fluid Mech. 842, 395–427.10.1017/jfm.2018.118Google Scholar
Hiai, F. & Petz, D. 2009 Riemannian metrics on positive definite matrices related to means. Linear Algebr. Applics. 430 (11–12), 3105–3130.10.1016/j.laa.2009.01.025Google Scholar
Housiadas, K. D. & Beris, A. 2003 Polymer-induced drag reduction: effects of the variations in elasticity and inertia in turbulent viscoelastic channel flow. Phys. Fluids 15 (8), 2369–2384.10.1063/1.1589484Google Scholar
Knechtges, P. 2015 The fully-implicit log-conformation formulation and its application to three-dimensional flows. J. Non-Newtonian Fluid Mech. 223, 209–220.10.1016/j.jnnfm.2015.07.004Google Scholar
Lang, S. 2001 Fundamentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191. Springer.Google Scholar
Lee, S. J. & Zaki, T. A. 2017 Simulations of natural transition in viscoelastic channel flow. J. Fluid Mech. 820, 232–262.10.1017/jfm.2017.198Google Scholar
Masoudian, M., Kim, K., Pinho, F. T. & Sureshkumar, R. 2013 A viscoelastic k–𝜀–v 2 –f turbulent flow model valid up to the maximum drag reduction limit. J. Non-Newtonian Fluid Mech. 202, 99–111.10.1016/j.jnnfm.2013.09.007Google Scholar
Moakher, M. 2005 A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Applics. 26 (3), 735–747.10.1137/S0895479803436937Google Scholar
Nguyen, M. Q., Delache, A., Simoëns, S., Bos, W. J. T. & El Hajem, M. 2016 Small scale dynamics of isotropic viscoelastic turbulence. Phys. Rev. Fluids 1 (8), 083301.10.1103/PhysRevFluids.1.083301Google Scholar
Seo, Y. 2013 Generalized Pólya–Szegö type inequalities for some non-commutative geometric means. Linear Algebr. Applics. 438 (4), 1711–1726.10.1016/j.laa.2011.10.036Google Scholar
Wang, S., Graham, M. D., Hahn, F. J. & Xi, L. 2014 Time-series and extended Karhunen–Loève analysis of turbulent drag reduction in polymer solutions. AIChE 60 (4), 1460–1475.10.1002/aic.14328Google Scholar