Hostname: page-component-6b989bf9dc-md2j5 Total loading time: 0 Render date: 2024-04-15T02:40:46.242Z Has data issue: false hasContentIssue false
  • English
  • Français

On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity

Published online by Cambridge University Press:  12 April 2011

MICHAEL WILCZEK ,
ANTON DAITCHE  and
RUDOLF FRIEDRICH
MICHAEL WILCZEK*
Affiliation:
Institute for Theoretical Physics, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
ANTON DAITCHE
Affiliation:
Institute for Theoretical Physics, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
RUDOLF FRIEDRICH
Affiliation:
Institute for Theoretical Physics, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
*
†Email address for correspondence: mwilczek@uni-muenster.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the single-point probability density function of the velocity in three-dimensional stationary and decaying homogeneous isotropic turbulence. To this end, we apply the statistical framework of the Lundgren–Monin–Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework, the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier–Stokes equation are used to complement the theoretical results and assess deviations from Gaussianity.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 676 , 10 June 2011 , pp. 191 - 217
Copyright
Copyright © Cambridge University Press 2011

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

Batchelor, G. K. 1953 The theory of homogeneous turbulence. Cambridge University Press.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1948 Decay of isotropic turbulence in the initial period. Proc. R. Soc. Lond. A, Math. Phys. Sci. 193 (1035), 539–558.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Ching, E. S. C. 1996 General formula for stationary or statistically homogeneous probability density functions. Phys. Rev. E 53 (6), 5899–5903.Google Scholar
Falkovich, G. & Lebedev, V. 1997 Single-point velocity distribution in turbulence. Phys. Rev. Lett. 79 (21), 4159–4161.Google Scholar
George, W. K. 1992 The decay of homogeneous isotropic turbulence. Phys. Fluids A: Fluid Dyn. 4 (7), 1492–1509.CrossRefGoogle Scholar
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 (3), 1065–1081.Google Scholar
Hosokawa, I. 2008 One-point velocity statistics in decaying homogeneous isotropic turbulence. Phys. Rev. E 78 (6), 066312.CrossRefGoogle ScholarPubMed
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397.CrossRefGoogle Scholar
Ishida, T., Davidson, P. A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.Google Scholar
Jimenez, J. 1998 Turbulent velocity fluctuations need not be Gaussian. J. Fluid Mech. 376 (1), 139–147.CrossRefGoogle Scholar
Kármán, T. de & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. A, Math. Phys. Sci. 164 (917), 192–215.Google Scholar
Kolmogorov, A. N. 1941 On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk SSSR 31 (6), 538–541.Google Scholar
Lundgren, T. S. 1967 Distribution functions in the statistical theory of turbulence. Phys. Fluids 10 (5), 969–975.CrossRefGoogle Scholar
Monin, A. S. 1967 Equations of turbulent motion. Prikl. Mat. Mekh. 31 (6), 1057.Google Scholar
Noullez, A., Wallace, G., Lempert, W., Miles, R. B. & Frisch, U. 1997 Transverse velocity increments in turbulent flow using the relief technique. J. Fluid Mech. 339 (1), 287–307.Google Scholar
Novikov, E. A. 1968 Kinetic equations for a vortex field. Sov. Phys. Dokl. 12 (11), 1006–1008.Google Scholar
Novikov, E. A. 1993 A new approach to the problem of turbulence, based on the conditionally averaged Navier–Stokes equations. Fluid Dyn. Res. 12 (2), 107–126.Google Scholar
Novikov, E. A. & Dommermuth, D. G. 1994 Conditionally averaged dynamics of turbulence. Mod. Phys. Lett. B 8 (23).Google Scholar
Perot, J. B. 2010 Determination of the Decay Exponent in Mechanically Stirred Isotropic Turbulence. arXiv:1007.5043.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Robertson, H. P. 1940 The invariant theory of isotropic turbulence. Proc. Camb. Phil. Soc. 36, 209.Google Scholar
Saffman, P. G. 1967 Note on decay of homogeneous turbulence. Phys. Fluids 10 (6), 1349–1349.Google Scholar
Shu, C. & Osher, S. 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (12), 379–397.CrossRefGoogle Scholar
Tatsumi, T. & Yoshimura, T. 2004 Inertial similarity of velocity distributions in homogeneous isotropic turbulence. Fluid Dyn. Res. 35 (2), 123–158.Google Scholar
Tennekes, H. & Lumley, J. L. 1983 A First Course in Turbulence. MIT Press.Google Scholar
Ulinich, L. R. & Lyubimov, B. Y. 1969 The statistical theory of turbulence of an incompressible fluid at large Reynolds number. Sov. Phys. JETP 28 (3), 494–500.Google Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225 (1), 1–20.CrossRefGoogle Scholar
Wilczek, M. & Friedrich, R. 2009 Dynamical origins for non-Gaussian vorticity distributions in turbulent flows. Phys. Rev. E 80 (1), 016316.Google Scholar