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Reynolds-number dependence of line and surface stretching in turbulence: folding effects

Published online by Cambridge University Press:  14 August 2007

SUSUMU GOTO  and
SHIGEO KIDA
SUSUMU GOTO
Affiliation:
Department of Mechanical Engineering and Science, Kyoto University, Yoshida-Honmachi, Sakyo, Kyoto, 606-8501, Japan
SHIGEO KIDA
Affiliation:
Department of Mechanical Engineering and Science, Kyoto University, Yoshida-Honmachi, Sakyo, Kyoto, 606-8501, Japan
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Abstract

The stretching rate, normalized by the reciprocal of the Kolmogorov time, of sufficiently extended material lines and surfaces in statistically stationary homogeneous isotropic turbulence depends on the Reynolds number, in contrast to the conventional picture that the statistics of material object deformation are determined solely by the Kolmogorov-scale eddies. This Reynolds-number dependence of the stretching rate of sufficiently extended material objects is numerically verified both in two- and three-dimensional turbulence, although the normalized stretching rate of infinitesimal material objects is confirmed to be independent of the Reynolds number. These numerical results can be understood from the following three facts. First, the exponentially rapid stretching brings about rapid multiple folding of finite-sized material objects, but no folding takes place for infinitesimal objects. Secondly, since the local degree of folding is positively correlated with the local stretching rate and it is non-uniformly distributed over finite-sized objects, the folding enhances the stretching rate of the finite-sized objects. Thirdly, the stretching of infinitesimal fractions of material objects is governed by the Kolmogorov-scale eddies, whereas the folding of a finite-sized material object is governed by all eddies smaller than the spatial extent of the objects. In other words, the time scale of stretching of infinitesimal fractions of material objects is proportional to the Kolmogorov time, whereas that of folding of sufficiently extended material objects can be as long as the turnover time of the largest eddies. The combination of the short time scale of stretching of infinitesimal fractions and the long time scale of folding of the whole object yields the Reynolds-number dependence. Movies are available with the online version of the paper.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 586 , 10 September 2007 , pp. 59 - 81
Copyright
Copyright © Cambridge University Press 2007

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Goto and Kida supplementary movie

Movie 1. The movie version of figure 3(a). Temporal evolution of a S0022112007007240material line in three-dimensional homogeneous turbulence. Evolution until 10 tau, where tau is the Kolmogorov time, is shown. The cube shown is the simulation box (the side is about twice the integral length). The bottom chessboard pattern indicates 50 times the Kolmogorov length.

Download Goto and Kida supplementary movie(Video)
Video 3.1 MB

Goto and Kida supplementary movie

Movie 1. The movie version of figure 3(a). Temporal evolution of a S0022112007007240material line in three-dimensional homogeneous turbulence. Evolution until 10 tau, where tau is the Kolmogorov time, is shown. The cube shown is the simulation box (the side is about twice the integral length). The bottom chessboard pattern indicates 50 times the Kolmogorov length.

Download Goto and Kida supplementary movie(Video)
Video 1.4 MB

Goto and Kida supplementary movie

Movie 2. The movie version of figure 7(a). Temporal evolution of a material line in two-dimensional turbulence. Run IIA. The side length of the square shown is ten times the Kolmogorov length. Evolution until 12.5 tau, where tau is the Kolmogorov time, is shown together with the contour of vorticity magnitude. Red and blue regions correspond to positive and negative vorticity, respectively. It is observed that the local deformation of the line is governed by the smallest-scale eddies.

Download Goto and Kida supplementary movie(Video)
Video 3.5 MB

Goto and Kida supplementary movie

Movie 2. The movie version of figure 7(a). Temporal evolution of a material line in two-dimensional turbulence. Run IIA. The side length of the square shown is ten times the Kolmogorov length. Evolution until 12.5 tau, where tau is the Kolmogorov time, is shown together with the contour of vorticity magnitude. Red and blue regions correspond to positive and negative vorticity, respectively. It is observed that the local deformation of the line is governed by the smallest-scale eddies.

Download Goto and Kida supplementary movie(Video)
Video 3.6 MB

Goto and Kida supplementary movie

Movie 3. Same as movie 2 but also showing the contour of coarse-grained vorticity magnitude. The coarse-grained vorticity is obtained by the sharp low-pass filtering of the Fourier components of vorticity, where the cut-off wavelength is chosen to be four times the Kolmogorov length. It is observed that larger eddies than the Kolmogorov length contribute to the large-scale folding of the line.

Download Goto and Kida supplementary movie(Video)
Video 3.1 MB

Goto and Kida supplementary movie

Movie 3. Same as movie 2 but also showing the contour of coarse-grained vorticity magnitude. The coarse-grained vorticity is obtained by the sharp low-pass filtering of the Fourier components of vorticity, where the cut-off wavelength is chosen to be four times the Kolmogorov length. It is observed that larger eddies than the Kolmogorov length contribute to the large-scale folding of the line.

Download Goto and Kida supplementary movie(Video)
Video 2.5 MB