Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-01T00:22:55.586Z Has data issue: false hasContentIssue false
  • English
  • Français

Intermediate scaling and logarithmic invariance in turbulent pipe flow

Published online by Cambridge University Press:  23 February 2021

Sourabh S. Diwan Open the ORCID record for Sourabh S. Diwan [Opens in a new window]  and
Jonathan F. Morrison Open the ORCID record for Jonathan F. Morrison [Opens in a new window]
Sourabh S. Diwan*
Affiliation:
Department of Aeronautics, Imperial College London, SW7 2AZ, UK
Jonathan F. Morrison
Affiliation:
Department of Aeronautics, Imperial College London, SW7 2AZ, UK
*
Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A three-layer asymptotic structure for turbulent pipe flow is proposed revealing, in terms of intermediate variables, the existence of a Reynolds-number-invariant logarithmic region for the streamwise mean velocity and variance. The formulation proposes a local velocity scale (which is not the friction velocity) for the intermediate layer and results in two overlap layers. We find that the near-wall overlap layer is governed by a power law for the pipe for all Reynolds numbers, whereas the log law emerges in the second overlap layer (the inertial sublayer) for sufficiently high Reynolds numbers ($Re_{\tau }$). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low $Re_{\tau }$ and the coexistence of power and log laws at higher $Re_{\tau }$. The classical von Kármán ($\kappa$) and Townsend–Perry ($A_1$) constants are determined from the intermediate-scaled log-law constants; $\kappa$ shows a weak trend at sufficiently high $Re_{\tau }$ but falls within the commonly accepted uncertainty band, whereas $A_1$ exhibits a systematic Reynolds-number dependence until the largest available $Re_{\tau }$. The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is proportional to $\sqrt {Re_{\tau }}$ (rather than $Re_{\tau }$) and therefore the approach to an asymptotically invariant state can be expected to be slow.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent boundary layers Boundary Layers: Pipe flow boundary layer
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , R1
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Present address: Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012.

References

REFERENCES

Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Ing.-Arch. 52, 355–377.CrossRefGoogle Scholar
Bailey, S.C.C., Vallikivi, M., Hultmark, M. & Smits, A.J. 2014 Estimating the value of von Kármán's constant in turbulent pipe flow. J. Fluid Mech. 749, 79–98.CrossRefGoogle Scholar
Barenblatt, G.I. 1993 Scaling laws for fully developed shear flows. Part 1. Basic hypotheses and analysis. J. Fluid Mech. 248, 513–520.CrossRefGoogle Scholar
Diwan, S.S. & Morrison, J.F. 2019 Reynolds-number dependence of the Townsend-Perry ‘constant’ in wall turbulence. In 11th International Symposium on Turbulence and Shear Flow Phenomena. Begel House.Google Scholar
Hultmark, M. 2012 A theory for the streamwise turbulent fluctuations in high Reynolds number pipe flow. J. Fluid Mech. 707, 575–584.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
von Kármán, T. 1930 Mechanische Ahnlichkeit und Turbulenz. Gott. Nachr. 1, 58–76.Google Scholar
Klewicki, J.C. 2013 Self-similar mean dynamics in turbulent wall flows. J. Fluid Mech. 718, 596–621.CrossRefGoogle Scholar
Long, R.R. & Chen, T.-C. 1981 Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59.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. Phy. Fluids 22, 065103.CrossRefGoogle Scholar
Marusic, I., Monty, J.P., Hultmark, M. & Smits, A.J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
McKeon, B.J. 2004 High Reynolds number turbulent pipe flow. PhD thesis, Princeton University.Google Scholar
McKeon, B.J., Li, J., Jiang, W., Morrison, J.F. & Smits, A.J. 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147.CrossRefGoogle Scholar
Morrison, J.F. 2007 The interaction between inner and outer regions of wall-bounded flow. Phil. Trans. R Soc. Lond. A 365, 683–698.Google ScholarPubMed
Morrison, J.F. & Fernandez Vicente, J. 2019 The energy budget at the outer peak of $\overline {u^2}$ in turbulent pipe flow. In APS DFD G11.00006. American Physical Society.Google Scholar
Morrison, J.F., McKeon, B.J., Jiang, W. & Smits, A.J. 2004 Scaling of the streamwise velocity component in turbulent pipe flow. J. Fluid Mech. 508, 99–131.CrossRefGoogle Scholar
Nagib, H.M. & Chauhan, K.A. 2008 Variations of von Kármán coefficient in canonical flows. Phy. Fluids 20, 101518.CrossRefGoogle Scholar
Perry, A.E. & Chong, M.S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Perry, A.E., Henbest, S. & Chong, M.S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199.CrossRefGoogle Scholar
Sreenivasan, K.R. & Sahay, A. 1997 The persistence of viscous effects in the overlap region, and the mean velocity in turbulent pipe and channel flows. In Self-Sustaining Mechanisms of Wall Turbulence (ed. R. Panton), pp. 253–272. Computational Mechanics Publications.Google Scholar
Townsend, A.A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Vallikivi, M. 2014 Wall-bounded turbulence at high Reynolds numbers. PhD thesis, Princeton University.Google 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
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327.CrossRefGoogle Scholar
Wosnik, M., Castillo, L. & George, W.K. 2000 A theory for turbulent pipe and channel flows. J. Fluid Mech. 421, 115–145.CrossRefGoogle Scholar
Zagarola, M.V. & Smits, A.J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.CrossRefGoogle Scholar
Supplementary material: File

Diwan and Morrison supplementary material

Diwan and Morrison supplementary material

Download Diwan and Morrison supplementary material(File)
File 214.2 KB