Hostname: page-component-788cddb947-wgjn4 Total loading time: 0 Render date: 2024-10-15T01:37:55.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of developing and decaying two-dimensional turbulence

Published online by Cambridge University Press:  29 March 2006

D. K. Lilly
D. K. Lilly
Affiliation:
National Center for Atmospheric Research, Boulder, Colorado
Get access Rights & Permissions [Opens in a new window]

Abstract

Two-dimensional isotropic turbulence is investigated in its development from an arbitrarily specified initial flow through its transformation into a statistically self-preserving decaying flow. Numerical simulation is the principal method of investigation. The early development is characterized by rapid growth of the mean squared vorticity gradient, and this growth is found to be predicted satisfactorily by the quasi-normal hypothesis. During the later states of decay the numerical results are found to be generally consistent with the predictions by Kraichnan, Leith and Batchelor of a k−3 inertial range spectrum. The dimensionless constant of the spectrum is found to be near 2, about half the value found earlier for turbulence maintained by a constant forcing amplitude. The results are also consistent with Batchelor's predictions of the time-dependent behaviour of certain quadratic moments: An inconsistency in those predictions is pointed out, however, which can be resolved by altering the inertial range spectrum by a logarithmic term, as suggested by Kraichnan. The most important two-point Eulerian correlation functions are exhibited. An investigation is made of the Gaussianity of the flow with results indicating a strong tendency toward intermittency in the enstrophy dissipation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 2 , 30 January 1971 , pp. 395 - 415
Copyright
© 1971 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

Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion. Two-dimensional incompressible flow. Part 1 J. Comp. Phys. 1, 119–143.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. High-speed computing in fluid dynamics. Phys. Fluids Suppl. II, 233–239.Google Scholar
Gibson, C. H., Stegen, G. R. & Williams, R. B. 1970 Statistics of the time structure of turbulent velocity and temperature fields measured at high Reynolds number J. Fluid Mech. 41, 153–167.Google Scholar
Kraichnan, R. 1962 The closure problem of turbulence theory Proc. Symposium Appl. Math. 13, 199–225.Google Scholar
Kraichnan, R. 1967 Inertial ranges in two-dimensional turbulence Phys. Fluids, 10, 1417–1423.Google Scholar
Kraichnan, R. 1970 Inertial-range transfer in two- and three-dimensional turbulence. To be published.
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence Phys. Fluids, 11, 671–674.Google Scholar
Lilly, D. K. 1969 Numerical simulation of two-dimensional turbulence. High-speed computing in fluid dynamics. Phys. Fluids Suppl. II, 240–249.Google Scholar
Ogura, Y. 1952 The structure of two-dimensionally isotropic turbulence J. Meteor. Soc. Japan, 30, 59–64.Google Scholar
Ogura, Y. 1958 On the isotropy of large scale disturbances in the upper troposphere J. Meteor. 15, 375–382.Google Scholar
Ogura, Y. 1962 Energy transfer in a normally distributed and isotropic turbulent velocity field in two dimensions Phys. Fluids, 5, 395–401.Google Scholar
Orszag, S. A. 1969 Numerical methods for the simulation of turbulence. Phys. Fluids Suppl. II, 250–257.Google Scholar
Proudman, I. & Reid, W. H. 1954 On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. Roy. Soc A 247, 163–189.Google Scholar
Reid, W. H. 1955 On the stretching of material lines and surfaces in isotropic turbulence with zero fourth cumulants Proc. Cam. Phil. Soc. 51, 350–361.Google Scholar
Stewart, R. W., Wilson, J. R. & Burling, R. W. 1970 Some statistical properties of small-scale turbulence in an atmospheric boundary layer J. Fluid Mech. 41, 141–152.Google Scholar