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Reynolds number scaling of the peak turbulence intensity in wall flows

Published online by Cambridge University Press:  15 December 2020

Xi Chen Open the ORCID record for Xi Chen [Opens in a new window]  and
Katepalli R. Sreenivasan
Xi Chen*
Affiliation:
Key Laboratory of Fluid Mechanics of Ministry of Education, Beihang University (Beijing University of Aeronautics and Astronautics), 100191Beijing, PR China
Katepalli R. Sreenivasan
Affiliation:
Tandon School of Engineering, Courant Institute of Mathematical Sciences, Department of Physics, New York University, New York, NY 10012, USA
*
†Email address for correspondence: [email protected]
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Abstract

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, $u^{\prime }$, is thought to be among the few exceptions. In particular, the near-wall mean-square peak, $\overline {u'u'}^+_p$ – where the superscript $+$ indicates normalization by the friction velocity $u_\tau$, the subscript $p$ indicates the peak value and the overbar indicates time averaging – is known to increase with increasing Reynolds number. The existing explanations suggest a logarithmic growth with respect to $Re$, where $Re$ is the Reynolds number based on $u_\tau$ and the thickness of the wall flow. We show that this boundless growth calls into question the veracity of wall-scaling and so cannot be sustained, and we establish an alternative formula for the peak magnitude that approaches a finite limit $\overline {u'u'}^+_\infty$ owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers Boundary Layers: Pipe flow boundary layer
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , R3
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Ahn, J.S., Lee, J.H., Lee, J., Kang, J.-H & Sung, H.J. 2015 Direct numerical simulation of a 30R long turbulent pipe flow at $Re_\tau = 3008$. Phys. Fluids 27, 065110.CrossRefGoogle Scholar
Cantwell, B.J. 2019 A universal velocity profile for smooth wall pipe flow. J. Fluid Mech. 878, 834–874.Google Scholar
Chen, X., Hussain, F. & She, Z.S. 2018 Quantifying wall turbulence via a symmetry approach. Part II. Reynolds stresses. J. Fluid Mech. 850, 401–438.CrossRefGoogle Scholar
Chen, X., Hussain, F. & She, Z.S. 2019 Non-universal scaling transition of momentum cascade in wall turbulence. J. Fluid Mech. 871, R2.Google Scholar
DeGraaff, D.B. & Eaton, J.K. 2000 Reynolds-number scaling of the flat plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.CrossRefGoogle Scholar
Hoyas, S. & Jimenez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to $Re_\tau =2003$. Phys. Fluids 18, 011702.CrossRefGoogle Scholar
Hultmark, M., Vallikivi, M., Bailey, S.C.C. & Smits, A.J. 2012 Turbulent pipe flow at extreme Reynolds numbers. Phys. Rev. Lett. 108, 094501.CrossRefGoogle ScholarPubMed
Iwamoto, K., Suzuki, Y. & Kasagi, N. 2002 Database of fully developed channel flow. Tech. Rep. ILR-0201. Available at: http://www.thtlab.t.utokyo.ac.jp.Google Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_\tau = 5200$. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Marusic, I., Baars, W.J. & Hutchins, N. 2017 Scaling of the streamwise turbulence intensity in the context of inner-outer interactions in wall turbulence. Phys. Rev. Fluids 2, 100502.CrossRefGoogle Scholar
Marusic, I., Chauhan, K.A., Kulandaivelu, V. & Hutchins, N. 2015 Evolution of zero-pressure-gradient boundary layers from different tripping conditions. J. Fluid Mech. 783, 379–411.CrossRefGoogle Scholar
Marusic, I., McKeon, B.J., Monkewitz, P.A., Nagib, H.M., Smits, A.J. & Sreenivasan, K.R. 2010 Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22, 065103.Google Scholar
Metzger, M.M. & Klewicki, J. 2001 A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys. Fluids 13, 692–701.CrossRefGoogle Scholar
Metzger, M.M., Klewicki, J.C., Bradshaw, K.L. & Sadr, R. 2001 Scaling the near-wall axial turbulent stress in the zero pressure gradient boundary layer. Phys. Fluids 13, 1819–1821.Google Scholar
Monkewitz, P.A. & Nagib, H.M. 2015 Large-Reynolds-number asymptotics of the streamwise normal stress in zero-pressure-gradient turbulent boundary layers. J. Fluid Mech. 783, 474–503.CrossRefGoogle Scholar
Monkewitz, P.A., Chauhan, K.A. & Nagib, H.M. 2007 Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19, 115101.CrossRefGoogle Scholar
Nagib, H.M., Chauhan, K.A. & Monkewitz, P.A. 2007 Approach to an asymptotic state for zero pressure gradient turbulent boundary layers. Phil. Trans. R. Soc. Lond. A 365, 755–770.Google Scholar
Örlü, R. 2009 Experimental studies in jet flows and zero pressure-gradient turbulent boundary layers. PhD thesis, KTH Royal Institute of Technology.Google Scholar
Örlü, R. & Alfredsson, P.H. 2013 Comment on the scaling of the near-wall streamwise variance peak in turbulent pipe flows. Exp. Fluids 54, 1431–1435.CrossRefGoogle Scholar
Rao, K.N., Narasimha, R. & Narayanan, M.A. 1971 The ‘bursting’ phenomenon in a turbulent boundary layer. J. Fluid Mech. 48, 339–352.CrossRefGoogle Scholar
Samie, M., Marusic, I., Hutchins, N., Fu, M.K., Fan, Y., Hultmark, M. & Smits, A.J. 2018 Fully resolved measurements of turbulent boundary layer flows up to $Re_\tau = 20\,000$. J. Fluid Mech. 851, 391–415.CrossRefGoogle Scholar
Schlatter, P., Örlü, R., Li, Q., Brethouwer, G., Fransson, J.H.M., Johansson, A.V., Alfredsson, P.H. & Henningson, D.S. 2009 Turbulent boundary layers up to $Re_\theta =2500$ studied through simulation and experiment. Phys. Fluids 21, 051702.Google Scholar
She, Z.S., Chen, X. & Hussain, F. 2017 Quantifying wall turbulence via a symmetry approach: a Lie group theory. J. Fluid Mech. 827, 322–356.CrossRefGoogle Scholar
Sillero, J.A., Jimenez, J. & Moser, R. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to $\delta ^+ = 2000$. Phys. Fluids 25, 105102.CrossRefGoogle Scholar
Smits, A.J. 2019 Experiments in high Reynolds number flows. Bull. Am. Phys. Soc. 64, 13.Google Scholar
Smits, A.J., McKeon, B.J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.CrossRefGoogle Scholar
Spalart, P.R. 1988 Direct simulation of a turbulent boundary layer up to $Re_\theta = 1410$. J. Fluid Mech. 187, 61–98.Google Scholar
Sreenivasan, K.R. 1989 The turbulent boundary layer. In Frontiers in Experimental Fluid Mechanics (ed. M. Gad-el-Hak), pp. 159–209. Springer.Google Scholar
Tardu, S. 2017 Near wall dissipation revisited. Intl J. Heat Fluid Flow 67, 104–115.CrossRefGoogle Scholar
Vallikivi, M., Ganapathisubramani, B. & Smits, A.J. 2015 Spectral scaling in boundary layers and pipes at very high Reynolds numbers. J. Fluid Mech. 771, 303–326.CrossRefGoogle Scholar
Vincenti, P., Klewicki, J., Morrill-Winter, C., White, C.M. & Wosnik, M. 2013 Streamwise velocity statistics in turbulent boundary layers that spatially develop to high Reynolds number. Exp. Fluids 54, 1629.Google Scholar
Wilcox, D.C. 2006 Turbulence Modeling for CFD. DCW Industries La Canada.Google Scholar
Willert, C., Soria, J., Stanislas, M., Klinner, J., Amili, O., Eisfelder, M., Cuvier, C., Bellani, G., Fiorini, T. & Talamelli, A. 2017 Near-wall statistics of a turbulent pipe flow at shear Reynolds numbers up to 40 000. J. Fluid Mech. 826, R5.CrossRefGoogle Scholar
Wu, X.H. & Moin, P. 2008 Direct numerical simulation on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 5–41.CrossRefGoogle Scholar
Yamamoto, Y. & Tsuji, Y. 2018 Numerical evidence of logarithmic regions in channel flow at $Re_{{\tau }}=8000$. Phys. Rev. Fluids 3, 012602(R).CrossRefGoogle Scholar