Skip to main content Accessibility help
×
×
Home

Reappraisal of the velocity derivative flatness factor in various turbulent flows

  • S. L. Tang (a1) (a2), R. A. Antonia (a3), L. Djenidi (a3), L. Danaila (a4) and Y. Zhou (a1) (a2)...

Abstract

We first analytically show, starting with the Navier–Stokes equations, that the value of the derivative flatness is controlled by pressure diffusion of energy, viscous destructive effects and large-scale effects (decay and/or production). The latter two terms tend to zero when the Taylor-microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. We argue that the pressure-diffusion term should also tend to a constant at large $Re_{\unicode[STIX]{x1D706}}$ . Available data for the velocity derivative flatness, $F$ , in different turbulent flows are re-examined and interpreted in the light of the finite-Reynolds-number effect. It is found that $F$ can differ from flow to flow at moderate $Re_{\unicode[STIX]{x1D706}}$ ; for a given flow, $F$ may also depend on the initial conditions. The data for $F$ in various flows, e.g. along the axis in the far field of plane and circular jets, and grid turbulence, show that it approaches a constant, with a value slightly larger than 10, when $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. This behaviour for $F$ is supported, at least qualitatively, by our analytical considerations. The constancy of $F$ at large $Re_{\unicode[STIX]{x1D706}}$ violates the refined similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol.Β 13, 1962, pp.Β 82–85) to account for the intermittency of the energy dissipation rate. It is not, however, inconsistent with Kolmogorov’s original similarity hypothesis (Dokl. Akad. Nauk SSSR, vol.Β 30, 1941, pp.Β 299–303), although we contend that the power-law relation $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ (Kolmogorov 1962), which is widely accepted in the literature, has in reality been almost invariably used to β€˜model’ the finite-Reynolds-number effect for the laboratory data and has been strongly influenced by the weighting given to the atmospheric surface layer data. The inclusion of the latter data has misled previous investigations of how $F$ varies withΒ  $Re_{\unicode[STIX]{x1D706}}$ .

Copyright

Corresponding author

†Email address for correspondence: shunlin.tang88@gmail.com

References

Hide All
Antonia, R. A., Anselmet, F. & Chambers, A. J. 1986 Assessment of local isotropy using measurements in a turbulent plane jet. J. Fluid Mech. 163, 365–391.
Antonia, R. A., Chambers, A. J. & Satyaprakash, B. R. 1981 Reynolds number dependence of high-order moments of the streamwise turbulent velocity derivative. Boundary-Layer Meteorol. 21, 159–171.
Antonia, R. A., Djenidi, L. & Danaila, L. 2014 Collapse of the turbulent dissipation range on Kolmogorov scales. Phys. Fluids 26, 045105.
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.
Antonia, R. A., Satyaprakash, B. R. & Hussain, A. K. M. F. 1982 Statistics of fine-scale velocity in turbulent plane and circular jets. J. Fluid Mech. 119, 55–89.
Antonia, R. A., Tang, S. L., Djenidi, L. & Danaila, L. 2015 Boundedness of the velocity derivative skewness in various turbulent flows. J. Fluid Mech. 781, 727–744.
Antonia, R. A., Zhou, T. & Romano, G. P. 2002 Small-scale turbulence characteristics of two-dimensional bluff body wakes. J. Fluid Mech. 459, 67–92.
Batchelor, G. K. & Townsend, A. A. 1947 Decay of vorticity in isotropic turbulence. Proc. R. Soc. Lond. A 190, 534–550.
Batchelor, G. K. & Townsend, A. A. 1949 The nature of turbulent motion at large wave-numbers. Proc. R. Soc. Lond. A 199, 238–255.
Belin, F., Maurer, J., Tabeling, P. & Willaime, H. 1997 Velocity gradient distributions in fully developed turbulence: experimental study. Phys. Fluids 9, 3843–3850.
Burattini, P., Lavoie, P. & Antonia, R. A. 2008 Velocity derivative skewness in isotropic turbulence and its measurement with hot wires. Exp. Fluids 45, 523–535.
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.
Dimotakis, P. E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 69–98.
Djenidi, L., Antonia, R. A. & Danaila, L. 2017a Self-preservation relation to the Kolmogorov similarity hypotheses. Phys. Rev. Fluids 2, 054606.
Djenidi, L., Antonia, R. A., Danaila, L. & Tang, S. L. 2017b A note on the velocity derivative flatness factor in decaying HIT. Phys. Fluids 29, 051702.
Djenidi, L., Antonia, R. A., Talluru, M. K. & Abe, H. 2017c Skewness and flatness factors of the longitudinal velocity derivative in wall-bounded flows. Phys. Rev. Fluids 2, 064608.
Friehe, C. A., Van Atta, C. W. & Gibson, C. H. 1971 Jet turbulence: dissipation rate measurements and correlations. AGARD Turbul. Shear Flows 18, 1–7.
Gauding, M.2014 Statistics and scaling laws of turbulent scalar mixing at high Reynolds numbers. PhD thesis, RWTH Aachen University.
Gibson, C. H., Stegen, G. R. & Williams, R. B. 1970 Statistics of the fine structure of turbulent velocity and temperature fields at high Reynolds number. J. Fluid Mech. 41, 153–167.
Gotoh, T., Fukayama, D. & Nakano, T. 2002 Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation. Phys. Fluids 14, 1065–1081.
Gotoh, T. & Nakano, T. 2003 Role of pressure in turbulence. J. Stat. Phys. 113, 855–874.
Gylfason, A., Ayyalasomayajula, S. & Warhaft, Z. 2004 Intermittency, pressure and acceleration statistics from hot-wire measurements in wind-tunnel turbulence. J. Fluid Mech. 501, 213–229.
Hill, R. J. 2001 Equations relating structure functions of all orders. J. Fluid Mech. 434, 379–388.
Hill, R. J. 2002 Scaling of acceleration in locally isotropic turbulence. J. Fluid Mech. 452, 361–370.
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high-Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2003 Spectra of energy dissipation, enstrophy and pressure by high-resolution direct numerical simulations of turbulence in a periodic box. J. Phys. Soc. Japan 72, 983–986.
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2007 Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics. J. Fluid Mech. 592, 335–366.
Jimenez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.
Kahalerras, H., Malecot, Y. & Gagne, Y. 1998 Intermittency and Reynolds number. Phys. Fluids 10, 910–921.
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–58.
Kim, J. & Antonia, R. A. 1993 Isotropy of the small-scales of turbulence at small Reynolds numbers. J. Fluid Mech. 251, 219–238.
Kolmogorov, A. N. 1941a Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 19–21.
Kolmogorov, A. N. 1941b Local structure of turbulence in an incompressible fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.
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, 82–85.
Kraichnan, R. H. 1991 Turbulent cascade and intermittency growth. Proc. R. Soc. Lond. A 434, 65–78.
Kuo, A. Y.-S. & Corrsin, S. 1971 Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. J. Fluid Mech. 50 (02), 285–319.
Metzger, M., McKeon, B. J. & Holmes, H. 2007 The near-neutral atmospheric surface layer: turbulence and non-stationarity. Phil. Trans. R. Soc. Lond. A 365, 859–876.
Mi, J., Xu, M. & Zhou, T. 2013 Reynolds number influence on statistical behaviors of turbulence in a circular free jet. Phys. Fluids 25, 075101.
Moisy, F., Tabeling, P. & Willaime, H. 1999 Kolmogorov equation in a fully developed turbulence experiment. Phys. Rev. Lett. 82 (20), 3994–3997.
Mydlarski, L. & Warhaft, Z. 1996 On the onset of high-Reynolds-number grid-generated wind tunnel turbulence. J. Fluid Mech. 320, 331–368.
Pearson, B. R. & Antonia, R. A. 2001 Reynolds-number dependence of turbulent velocity and pressure increments. J. Fluid Mech. 444, 343–382.
Pearson, B. R. & Krogstad, P. A. 2001 Further evidence for a transition in small-scale turbulence. In 14th Australasian Fluid Mechanics Conference, Adelaide.
Qian, J. 1983 Variational approach to the closure problem of turbulence theory. Phys. Fluids 26 (8), 2098–2104.
Qian, J. 1986 A closure theory of intermittency of turbulence. Phys. Fluids 29, 2165.
Saddoughi, S. G. & Veeravalli, S. V. 1994 Local isotropy of turbulent boundary layers at high Reynolds number. J. Fluid Mech. 268, 333–372.
Shen, X. & Warhaft, Z. 2000 The anisotropy of the small scale structure in high Reynolds number (R πœ† ∼ 1000) turbulent shear flow. Phys. Fluids 12, 2976–2989.
Sinhuber, M., Bodenschatz, E. & Bewley, G. P. 2015 Decay of turbulence at high Reynolds numbers. Phys. Rev. Lett. 114 (3), 034501.
Sreenivasan, K. & Antonia, R. A. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472.
Sreenivasan, K. R. 1995 Small-scale intermittency in turbulence. In Proc. Twelfth Australasian Fluid Mechanics Conference, University of Sydney, Australia, pp. 549–556.
Tabeling, P. & Willaime, H. 2002 Transition at dissipative scales in large Reynolds number turbulence. Phys. Rev. E 65, 066301.
Tabeling, P., Zocchi, G., Belin, F., Maurer, J. & Willaime, H. 1996 Probability density functions, skewness, and flatness in large Reynolds number turbulence. Phys. Rev. E 53, 1613–1621.
Tang, S. L., Antonia, R. A., Danaila, L., Djenidi, L., Zhou, T. & Zhou, Y. 2016 Towards local isotropy of higher-order statistics in the intermediate wake. Exp. Fluids 57, 111.
Tang, S. L., Antonia, R. A., Djenidi, L., Abe, H., Zhou, T., Danaila, L. & Zhou, Y. 2015a Transport equation for the mean turbulent energy dissipation rate on the centreline of a fully developed channel flow. J. Fluid Mech. 777, 151–177.
Tang, S. L., Antonia, R. A., Djenidi, L. & Zhou, Y. 2015b Transport equation for the isotropic turbulent energy dissipation rate in the far-wake of a circular cylinder. J. Fluid Mech. 784, 109–129.
Thiesset, F., Antonia, R. A. & Djenidi, L. 2014 Consequences of self-preservation on the axis of a turbulent round jet. J. Fluid Mech. 748, R2.
Thiesset, F., Danaila, L. & Antonia, R. A. 2013 Dynamical effect of the total strain induced by the coherent motion on local isotropy in a wake. J. Fluid Mech. 720, 393–423.
Tong, C. & Warhaft, Z. 1994 On passive scalar derivative statistics in grid turbulence. Phys. Fluids 6 (6), 2165–2176.
Van Atta, C. W. & Antonia, R. A. 1980 Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives. Phys. Fluids 23, 252–257.
Wang, L.-P., Chen, S., Brasseur, J. G. & Wyngaard, J. C. 1996 Examination of hypotheses in Kolmogorov refined turbulence theory through high-resolution simulations. Part 1. Velocity field. J. Fluid Mech. 309, 113–156.
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge University Press.
Wyngaard, J. C. & Tennekes, H. 1970 Measurements of the small-scale structure of turbulence at moderate Reynolds numbers. Phys. Fluids 13, 1962–1969.
Xu, G., Antonia, R. A. & Rajagopalan, S. 2001 Sweeping decorrelation hypothesis in a turbulent round jet. Fluid Dyn. Res. 28 (5), 311–321.
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2005 High-Reynolds-number simulation of turbulent mixing. Phys. Fluids 17, 081703.
Yeung, P. K. & Zhou, Y. 1997 Universality of the Kolmogorov constant in numerical simulations of turbulence. Phys. Rev. E 56, 1746–1752.
Zhou, T. & Antonia, R. A. 2000 Reynolds number dependence of the small-scale structure of grid turbulence. J. Fluid Mech. 406, 81–107.
Zhou, T., Antonia, R. A. & Chua, L. P. 2005 Flow and Reynolds number dependencies of one-dimensional vorticity fluctuations. J. Turbul. 6, N28.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
Γ—
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed