Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-10T06:15:26.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Law of bounded dissipation and its consequences in turbulent wall flows

Published online by Cambridge University Press:  23 December 2021

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

Abstract

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number $Re_\tau$. This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of $Re_\tau$, owing to the natural constraint that the dissipation rate is bounded. Specifically, $\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$ where $\varPhi$ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript $\infty$ denotes the bounded asymptotic value of $\varPhi$, and the coefficient $C_\varPhi$ depends on $\varPhi$ but not on $Re_\tau$. Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form $\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$, where $\varphi$ represents the streamwise or spanwise velocity fluctuation, and $\alpha _q$ and $\beta _q$ are independent of $Re_\tau$. Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear $q$-norm Gaussian’, is proposed to explain the observed linear dependence of $\alpha _q$ on $q$.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers Boundary Layers: Pipe flow boundary layer
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A20
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.)

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
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity statistics in turbulent channel flow up to $Re_\tau =4000$. J. Fluid Mech. 742, 171–191.CrossRefGoogle Scholar
Bradshaw, P. 1967 ‘Inactive’ motion and pressure fluctuations in turbulent boundary layers. J. Fluid Mech. 30, 241–258.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.CrossRefGoogle Scholar
Chen, X. & Sreenivasan, K.R. 2021 Reynolds number scaling of the peak turbulence intensity in wall flows. J. Fluid Mech. 908, R3.CrossRefGoogle 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
Diaz-Daniel, C., Laizet, S. & Vassilicos, J.C. 2017 Wall shear stress fluctuations: mixed scaling and their effects on velocity fluctuations in a turbulent boundary layer. Phys. Fluids 29, 055102.CrossRefGoogle Scholar
Fiorini, T. 2017 Turbulent pipe flow – high resolution measurements in CICLoPE. PhD thesis, University of Bologna.Google Scholar
Hoyas, S. & Jimenez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to $Re_\tau =2003$. Phy. Fluids 18, 011702.CrossRefGoogle Scholar
Hultmark, M. & Smits, A.J. 2021 Scaling turbulence in the near-wall region. arXiv:2103.01765v1.Google 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
Hutchins, N., Nickels, T.B., Marusic, I. & Chong, M.S. 2009 Hot-wire spatial resolution issues in wall-bounded turbulence. J. Fluid Mech. 635, 103–136.CrossRefGoogle Scholar
Iwamoto, K., Suzuki, Y. & Kasagi, N. 2002 Database of fully developed channel flow. Tech. Rep. ILR-0201, see http://www.thtlab.t.utokyo.ac.jp.Google Scholar
Klewicki, J.C., Priyadarshana, P.J.A. & Metzger, M.M. 2008 Statistical structure of the fluctuating wall pressure and its in-plane gradients at high Reynolds number. J. Fluid Mech. 609, 195–220.CrossRefGoogle 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
Lozano-Durán, A. & Jimenez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to $Re_\tau =4200$. Phys. Fluids 26 (1), 011702.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. Phy. Fluids 22, 065103.CrossRefGoogle Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.CrossRefGoogle Scholar
Meneveau, C. & Marusic, I. 2013 Generalized logarithmic law for high-order moments in turbulent boundary layers. J. Fluid Mech. 719, R1.CrossRefGoogle 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
Monkewitz, P. 2021 Asymptotics of streamwise Reynolds stress in wall turbulence. J. Fluid Mech. 931, A18.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
Moser, R.D., Kim, J. & Mansour, N.N. 1999 DNS of turbulent channel flow up to $Re_\tau =590$. Phys. Fluids 11, 943–945.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. A 365, 755–770.CrossRefGoogle Scholar
Orlandi, P. & Leonardi, S. 2007 Technical Report for WALLTURB project. Tech. Rep. WT-071016-URS-1.Google Scholar
Örlü, R. 2009 Experimental studies in jet flows and zero pressure-gradient turbulent boundary layers. PhD thesis, KTH, Stockholm.Google Scholar
Panton, R.L., Lee, M. & Moser, R.D. 2017 Correlation of pressure fluctuations in turbulent wall layers. Phys. Rev. Fluids 2, 094604.CrossRefGoogle Scholar
Pirozzoli, S., Romero, J., Fatica, M., Verzicco, R. & Orlandi, P. 2021 One-point statistics for turbulent pipe flow up to $Re_\tau \approx 6000$. J. Fluid Mech. 926, A28.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 $Re_\tau = 20000$. J. Fluid Mech. 851, 391–415.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.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 11, 051702.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
Skote, M. 2001 Studies of turbulent boundary layer flow through direct numerical simulation. PhD thesis, KTH, Stockholm.Google Scholar
Smits, A.J., Hultmark, M., Lee, M., Pirozzoli, S. & Wu, X.H. 2021 Reynolds stress scaling in the near-wall region of wall-bounded turbulence. J. Fluid Mech. 926, A31.CrossRefGoogle 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.CrossRefGoogle Scholar
Sreenivasan, K.R. 1989 The turbulent boundary layer. In Frontiers in Experimental Fluid Mechanics (ed. M. Gad-el-Hak), pp. 159–209. Springer.CrossRefGoogle Scholar
Tardu, S. 2017 Near wall dissipation revisited. Intl J. Heat Fluid Flow 67, 104–115.CrossRefGoogle Scholar
Townsend, A.A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.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
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.CrossRefGoogle Scholar
Wilcox, D.C. 2006 Turbulence Modeling for CFD. DCW Industries.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.CrossRefGoogle Scholar
Yang, X.I.A., Hong, J., Lee, M. & Huang, X.L.D. 2021 Grid resolution requirement for resolving rare and high intensity wall-shear stress events in direct numerical simulations. Phys. Rev. Fluids 6, 054603.CrossRefGoogle Scholar
Yang, X.I.A. & Lozano-Durán, A. 2017 A multifractal model for the momentum transfer process in wall-bounded flows. J. Fluid Mech. 824, R2.CrossRefGoogle ScholarPubMed
Yao, J., Chen, X. & Hussain, F. 2019 Reynolds number effect on drag control via spanwise wall oscillation in turbulent channel flows. Phys. Fluids 31, 085108.CrossRefGoogle Scholar
Yeung, P.K., Sreenivasan, K.R. & Pope, S.B. 2018 Effects of finite spatial and temporal resolution in direct numerical simulations of incompressible isotropic turbulence. Phys. Rev. Fluids 3, 064603.CrossRefGoogle Scholar