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Approach to the 4/3 law for turbulent pipe and channel flows examined through a reformulated scale-by-scale energy budget

Published online by Cambridge University Press:  26 November 2021

Spencer J. Zimmerman Open the ORCID record for Spencer J. Zimmerman [Opens in a new window] ,
R.A. Antonia ,
L. Djenidi Open the ORCID record for L. Djenidi [Opens in a new window] ,
J. Philip Open the ORCID record for J. Philip [Opens in a new window]  and
J.C. Klewicki Open the ORCID record for J.C. Klewicki [Opens in a new window]
Spencer J. Zimmerman*
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
R.A. Antonia
Affiliation:
School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
L. Djenidi
Affiliation:
School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
J. Philip
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
J.C. Klewicki
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia
*
†Email address for correspondence: zimmerman@jhu.edu
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Abstract

In this study, we propose a scale-by-scale (SBS) energy budget equation for flows with homogeneity in at least one direction. This SBS budget represents a modified form of the equation first proposed by Danaila et al. (J. Fluid Mech., vol. 430, 2001, pp. 87–109) for the channel centreline – the primary difference is that, here, we consider the role of pressure along with the errors associated with the isotropic approximations of the interscale divergence and Laplacian of the squared velocity increment. The term encompassing the effects of mean shear is also characterised such that the present analysis can be extended straightforwardly to locations away from the centreline. We show, based on a detailed analysis of previously published channel flow direct numerical simulations and pipe flow experiments near the centreline, how several terms in the present SBS budget equation (including the third-order velocity structure function) behave with increasing Reynolds number. The behaviour of these terms is shown to imply a rate of emergence and subsequent growth of the 4/3 law scale subrange at the channel centreline and pipe axis. The analysis also suggests that the peak magnitude of the third-order velocity structure function occurs at a scale that is fixed in proportion to the Taylor microscale at sufficiently high Reynolds number.

JFM classification

Turbulent Flows: Pipe flow Turbulent Flows: Isotropic turbulence Turbulent Flows: Homogeneous turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 931 , 25 January 2022 , A28
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

del Álamo, J.C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41–L44.CrossRefGoogle Scholar
del Álamo, J.C., Jiménez, J., Zandonade, P. & Moser, R.D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
Antonia, R.A. & Burattini, P. 2006 Approach to the 4/5 law in homogeneous isotropic turbulence. J. Fluid Mech. 550, 175–184.CrossRefGoogle Scholar
Antonia, R.A., Djenidi, L. & Danaila, L. 2014 Collapse of the turbulent dissipative range on Kolmogorov scales. Phys. Fluids 26 (4), 045105.CrossRefGoogle Scholar
Antonia, R.A., Djenidi, L., Danaila, L. & Tang, S.L. 2017 Small scale turbulence and the finite Reynolds number effect. Phys. Fluids 29 (2), 020715.CrossRefGoogle Scholar
Antonia, R.A., Ould-Rouis, M., Anselmet, F. & Zhu, Y. 1997 Analogy between predictions of Kolmogorov and Yaglom. J. Fluid Mech. 332, 395–409.CrossRefGoogle Scholar
Antonia, R.A. & Pearson, B.R. 2000 Reynolds number dependence of velocity structure functions in a turbulent pipe flow. Flow Turbul. Combust. 64 (2), 95–117.CrossRefGoogle Scholar
Antonia, R.A., Tang, S.L., Djenidi, L. & Zhou, Y. 2019 Finite Reynolds number effect and the 4/5 law. Phys. Rev. Fluids 4 (8), 084602.CrossRefGoogle Scholar
Antonia, R.A., Zhou, T., Danaila, L. & Anselmet, F. 2000 Streamwise inhomogeneity of decaying grid turbulence. Phys. Fluids 12 (11), 3086–3089.CrossRefGoogle Scholar
Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Boschung, J., Gauding, M., Hennig, F., Denker, D. & Pitsch, H. 2016 Finite Reynolds number corrections of the 4/5 law for decaying turbulence. Phys. Rev. Fluids 1 (6), 064403.CrossRefGoogle Scholar
Champagne, F.H. 1978 The fine-scale structure of the turbulent velocity field. J. Fluid Mech. 86 (1), 67–108.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E. & Casciola, C.M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436–451.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E., Jiménez, J. & Casciola, C.M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417–436.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E., Schlatter, P., Brethouwer, G., Talamelli, A. & Casciola, C.M. 2015 Sources and fluxes of scale energy in the overlap layer of wall turbulence. J. Fluid Mech. 771, 407–423.CrossRefGoogle Scholar
Danaila, L., Anselmet, F. & Zhou, T. 2004 Turbulent energy scale-budget equations for nearly homogeneous sheared turbulence. Flow Turbul. Combust. 72 (2), 287–310.CrossRefGoogle Scholar
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R.A. 1999 A generalization of Yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence. J. Fluid Mech. 391, 359–372.CrossRefGoogle Scholar
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R.A. 2001 Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87–109.CrossRefGoogle Scholar
Davidson, P.A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Djenidi, L. & Antonia, R.A. 2012 A spectral chart method for estimating the mean turbulent kinetic energy dissipation rate. Exp. Fluids 53 (4), 1005–1013.CrossRefGoogle Scholar
Djenidi, L. & Antonia, R.A. 2020 Assessment of large-scale forcing in isotropic turbulence using a closed Kármán–Howarth equation. Phys. Fluids 32 (5), 055104.CrossRefGoogle Scholar
El Khoury, G.K., Schlatter, P., Noorani, A., Fischer, P.F., Brethouwer, G. & Johansson, A.V. 2013 Direct numerical simulation of turbulent pipe flow at moderately high Reynolds numbers. Flow Turbul. Combust. 91 (3), 475–495.CrossRefGoogle Scholar
Gatti, D., Chiarini, A., Cimarelli, A. & Quadrio, M. 2020 Structure function tensor equations in inhomogeneous turbulence. J. Fluid Mech. 898, A5.CrossRefGoogle Scholar
Graham, J., et al. 2016 A web services accessible database of turbulent channel flow and its use for testing a new integral wall model for LES. J. Turbul. 17 (2), 181–215.CrossRefGoogle Scholar
Hill, R.J. 2001 a Equations relating structure functions of all orders. J. Fluid Mech. 434 (1), 379–388.CrossRefGoogle Scholar
Hill, R.J. 2001 b Mathematics of structure-function equations of all orders. Tech. Rep. OAR ETL-303. NOAA.CrossRefGoogle Scholar
Hill, R.J. 2002 a Exact second-order structure-function relationships. J. Fluid Mech. 468, 317–326.CrossRefGoogle Scholar
Hill, R.J. 2002 b Structure-function equations for scalars. Phys. Fluids 14 (5), 1745–1756.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to $Re_\tau = 2003$. Phys. Fluids 18 (1), 011702.CrossRefGoogle Scholar
Klewicki, J.C. & Falco, R.E. 1990 On accurately measuring statistics associated with small-scale structure in turbulent boundary layers using hot-wire probes. J. Fluid Mech. 219, 119–142.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 a Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18.Google Scholar
Kolmogorov, A.N. 1941 b The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Kolmogorov, A.N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13 (1), 82–85.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $\textit {Re}_\tau \approx 5200$. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886–938.CrossRefGoogle Scholar
Li, Y., Perlman, E., Wan, M., Yang, Y., Meneveau, C., Burns, R., Chen, S., Szalay, A. & Eyink, G. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9 (31), 1–29.CrossRefGoogle Scholar
Lindborg, E. 1999 Correction to the four-fifths law due to variations of the dissipation. Phys. Fluids 11 (3), 510–512.CrossRefGoogle Scholar
Lundgren, T.S. 2002 Kolmogorov two-thirds law by matched asymptotic expansion. Phys. Fluids 14 (2), 638–642.CrossRefGoogle Scholar
Lundgren, T.S. 2003 Linearly forced isotropic turbulence. CTR Annual Research Briefs, p. 461.Google Scholar
Mansour, N.N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.CrossRefGoogle Scholar
Marati, N., Casciola, C.M. & Piva, R. 2004 Energy cascade and spatial fluxes in wall turbulence. J. Fluid Mech. 521 (191–215), 290.CrossRefGoogle Scholar
Meldi, M. & Vassilicos, J.C. 2021 Analysis of Lundgren's matched asymptotic expansion approach to the Kármán–Howarth equation using the EDQNM turbulence closure. Phys. Rev. Fluids 6, 064602.CrossRefGoogle Scholar
Mizuno, Y. 2016 Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. J. Fluid Mech. 805, 171–187.CrossRefGoogle Scholar
Morrison, J.F., Vallikivi, M. & Smits, A.J. 2016 The inertial subrange in turbulent pipe flow: centreline. J. Fluid Mech. 788, 602–613.CrossRefGoogle Scholar
Perlman, E., Burns, R., Li, Y. & Meneveau, C. 2007 Data exploration of turbulence simulations using a database cluster. In Proc. Supercomp. SC07, ACM, IEEE, pp. 1–11.Google Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Qian, J. 1997 Inertial range and the finite Reynolds number effect of turbulence. Phys. Rev. E 55 (1), 337.CrossRefGoogle Scholar
Qian, J. 1999 Slow decay of the finite Reynolds number effect of turbulence. Phys. Rev. E 60 (3), 3409.CrossRefGoogle ScholarPubMed
Saddoughi, S.G. & Veeravalli, S.V. 1994 Local isotropy in turbulent boundary layers at high Reynolds number. J. Fluid Mech. 268, 333–372.CrossRefGoogle Scholar
Tang, S.L., Antonia, R.A., Djenidi, L., Danaila, L. & Zhou, Y. 2017 Finite Reynolds number effect on the scaling range behaviour of turbulent longitudinal velocity structure functions. J. Fluid Mech. 820, 341–369.CrossRefGoogle Scholar
Taylor, G.I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151 (873), 421–444.CrossRefGoogle Scholar
Tchoufag, J., Sagaut, P. & Cambon, C. 2012 Spectral approach to finite Reynolds number effects on Kolmogorov's 4/5 law in isotropic turbulence. Phys. Fluids 24 (1), 015107.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT.CrossRefGoogle Scholar
Vreman, A.W. & Kuerten, J.G.M. 2014 Statistics of spatial derivatives of velocity and pressure in turbulent channel flow. Phys. Fluids 26 (8), 085103.CrossRefGoogle Scholar
Vukoslavčević, P.V. & Wallace, J.M. 2013 The influence of the arrangements of multi-sensor probe arrays on the accuracy of simultaneously measured velocity and velocity gradient-based statistics in turbulent shear flows. Exp. Fluids 54 (6), 1537.CrossRefGoogle Scholar
Wyngaard, J.C. 1968 Measurement of small-scale turbulence structure with hot wires. J. Phys. E: Sci. Instrum. 1 (11), 1105.CrossRefGoogle Scholar
Yaglom, A.M. 1949 On the local structure of a temperature field in a turbulent flow. Dokl. Akad. Nauk SSSR 69, 743–746.Google Scholar
Zhou, T., Antonia, R.A., Danaila, L. & Anselmet, F. 2000 Approach to the four-fifths ‘law’ for grid turbulence. J. Turbul. 1, 005.CrossRefGoogle Scholar
Zimmerman, S., Morrill-Winter, C. & Klewicki, J. 2017 Design and implementation of a hot-wire probe for simultaneous velocity and vorticity vector measurements in boundary layers. Exp. Fluids 58 (10), 148.CrossRefGoogle Scholar
Zimmerman, S., et al. 2019 A comparative study of the velocity and vorticity structure in pipes and boundary layers at friction Reynolds numbers up to $10^4$. J. Fluid Mech. 869, 182–213.CrossRefGoogle Scholar