Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T23:20:13.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Velocity gradient statistics in turbulent shear flow: an extension of Kolmogorov's local equilibrium theory

Published online by Cambridge University Press:  21 October 2021

Yukio Kaneda Open the ORCID record for Yukio Kaneda [Opens in a new window]  and
Yoshinobu Yamamoto Open the ORCID record for Yoshinobu Yamamoto [Opens in a new window]
Yukio Kaneda*
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Yoshinobu Yamamoto
Affiliation:
Department of Mechanical Engineering, University of Yamanashi, Kofu 400-8511, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents an extension of Kolmogorov's local similarity hypotheses of turbulence to include the influence of mean shear on the statistics of the fluctuating velocity in the dissipation range of turbulent shear flow. According to the extension, the moments of the fluctuating velocity gradients are determined by the local mean rate of the turbulent energy dissipation $\left \langle \epsilon \right \rangle$ per unit mass, kinematic viscosity $\nu$ and parameter $\gamma \equiv S (\nu /\left \langle \epsilon \right \rangle )^{1/2}$, provided that $\gamma$ is small in an appropriate sense, where $S$ is an appropriate norm of the local gradients of the mean flow. The statistics of the moments are nearly isotropic for sufficiently small $\gamma$, and the anisotropy of moments decreases approximately in proportion to $\gamma$. This paper also presents a report on the second-order moments of the fluctuating velocity gradients in direct numerical simulations (DNSs) of turbulent channel flow (TCF) with the friction Reynolds number $Re_\tau$ up to $\approx 8000$. In the TCF, there is a range $y$ where $\gamma$ scales approximately $\propto y^ {-1/2}$, and the anisotropy of the moments of the gradients decreases with $y$ nearly in proportion to $y^ {-1/2}$, where $y$ is the distance from the wall. The theoretical conjectures proposed in the first part are in good agreement with the DNS results.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 929 , 25 December 2021 , A13
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

Abe, H. & Antonia, R.A. 2016 Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow. J. Fluid Mech. 798, 140–164.CrossRefGoogle Scholar
Antonia, R.A., Djenidi, L. & Spalart, P.R. 1994 Anisotropy of the dissipation tensor in a turbulent boundary layer. Phys. Fluids 6 (7), 2475–2479.CrossRefGoogle Scholar
Antonia, R.A., Kim, J. & Browne, L.W.B. 1991 Some characteristics of small-scale turbulence in a turbulent duct flow. J. Fluid Mech. 233, 369–388.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
Bolotnov, I.A., Lahey, R.T., Jr., Drew, D.A., Jansen, K.E. & Oberai, A.A. 2010 Spectral analysis of turbulence based on the DNS of a channel flow. Comput. Fluids 39, 640–655.CrossRefGoogle Scholar
Bradshaw, P. & Perot, J.B. 1993 A note on turbulent energy dissipation in the viscous wall region. Phys. Fluids 5 (12), 3305–3306.CrossRefGoogle Scholar
Browne, L.W.B., Antonia, R.A. & Shah, D.A. 1987 Turbulent energy dissipation in a wake. J. Fluid Mech. 179, 307–326.CrossRefGoogle Scholar
Cambon, C. & Rubinstein, R. 2006 Anisotropic developments for homogeneous shear flows. Phys. Fluids 18 (8), 085106.CrossRefGoogle Scholar
Chin, C., Philip, J., Klewicki, J., Ooi, A. & Marusic, I. 2014 Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows. J. Fluid Mech. 757, 747–388.CrossRefGoogle Scholar
Corrsin, S. 1958 Local isotropy in turbulent shear flow. N.A.C.A. Res. Memo. RM 58B11.Google Scholar
Folz, A. & Wallace, J.M. 2010 Near-surface turbulence in the atmospheric boundary layer. Physica D 239 (14), 1305–1317.CrossRefGoogle Scholar
George, W. & Hussein, H.J. 1991 Locally axisymmetric turbulence. J. Fluid Mech. 233, 1–23.CrossRefGoogle Scholar
Hinze, J.O. 1975 Turbulence, 2nd edn. McGraw-Hill.Google Scholar
Honkan, A. & Andreopoulos, Y. 1997 Vorticity, strain-rate and dissipation characteristics in the near-wall region of turbulent boundary layers. J. Fluid Mech. 350, 29–96.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
Ishihara, T., Yoshida, K. & Kaneda, Y. 2002 Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow. Phys. Rev. Lett. 88, 154501.CrossRefGoogle Scholar
Jiménez, J. & Moser, R.D. 2007 What are we learning from simulating wall turbulence? Phil. Trans. R. Soc. Lond. A 365, 715–732.Google ScholarPubMed
Kaneda, Y. 2020 Linear response theory of turbulence. J. Stat. Mech. 2020, 034006.CrossRefGoogle Scholar
Kaneda, Y., Morishita, K. & Ishihara, T. 2013 Small scale universality and spectral characteristics in turbulent flow. In Proc. Eighth International Symposium on Turbulence and Shear Flow Phenomena, Begel House.Google Scholar
Kaneda, Y. & Yoshida, K. 2004 Small-scale anisotropy in stably stratified turbulence. New J. Phys. 6, 34.CrossRefGoogle Scholar
Kida, S. & Orszag, S.A. 1990 Energy and spectral dynamics in forced compressible turbulence. J. Sci. Comput. 5, 85–125.CrossRefGoogle Scholar
Klewicki, J.C. 2010 Reynolds number dependence, scaling, and dynamics of turbulent boundary layers. Trans. ASME J. Fluids Engng 132, 094001.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. USSR 30, 301–305.Google Scholar
Livescu, D. & Madnia, C.K. 2004 Small scale structure of homogeneous turbulent shear flow. Phys. Fluids 16 (8), 2864–2876.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_{\tau } \approx 5200$. 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
Leslie, D.C. 1973 Developments in the Theory of Turbulence. Clarendon Press.Google Scholar
Loucks, R.B. & Wallace, J.M. 2012 Velocity and velocity gradient based properties of a turbulent plane mixing layer. J. Fluid Mech. 699, 280–319.CrossRefGoogle Scholar
Lumley, J.L. 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10 (4), 855–858.CrossRefGoogle 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
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219–245.CrossRefGoogle Scholar
Morishita, K., Ishihara, T. & Kaneda, Y. 2019 Length scales in turbulent channel flow. J. Phys. Soc. Japan 88, 064401.CrossRefGoogle Scholar
Pumir, A., Xu, H. & Siggia, E.D. 2016 Small-scale anisotropy in turbulent boundary layers. J. Fluid Mech. 804, 5–23.CrossRefGoogle Scholar
Pumir, A. 2017 Structure of the velocity gradient tensor in turbulent shear flows. Phys. Rev. Fluids 2, 074602.CrossRefGoogle Scholar
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
Sagaut, P. & Cambon, C. 2018 Homogeneous Turbulence Dynamics. 2nd edn. Springer.CrossRefGoogle Scholar
Schumacher, J., Sreenivasan, K.R. & Yeung, P.K. 2003 Derivative moments in turbulent shear flows. Phys. Fluids 15 (1), 84–90.CrossRefGoogle Scholar
Tardu, S. 2017 Near wall dissipation revisited. Intl J. Heat Fluid Flow 67, 104–115.CrossRefGoogle Scholar
Tsinober, A., Kit, E. & Dracos, T. 1992 Experimental investigation of the field of velocity gradients in turbulent flows. J. Fluid Mech. 242, 169–192.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. The MIT Press.CrossRefGoogle Scholar
Tsuji, Y. & Kaneda, Y. 2012 Anisotropic pressure correlation spectra in turbulent shear flow. J. Fluid Mech. 694, 50–77.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
Yamamoto, Y. & Kunugi, T. 2011 Direct numerical simulation of a high-Froude-number turbulent open-channel flow. Phys. Fluids 23 (12), 125108.CrossRefGoogle Scholar
Yamamoto, Y. & Kunugi, T. 2016 MHD effects on turbulent dissipation process in channel flows with an imposed wall-normal magnetic field. Fusion Engng Des. 109, 1137–1142.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
Yoshida, K., Ishihara, T. & Kaneda, Y. 2003 Anisotropic spectrum of homogeneous turbulent shear flow in a Lagrangian renormalized approximation. Phys. Fluids 15 (8), 2385–2397.CrossRefGoogle Scholar
Yoshizawa, A. 1998 Hydrodynamic and Magnetohydrodynamic Turbulent Flows. Kluwer Academic.CrossRefGoogle Scholar