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Title: Scaling of Lyapunov exponents in homogeneous isotropic turbulence

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Publication Date:
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
Grant/Contract Number:
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Physical Review Fluids
Additional Journal Information:
Journal Volume: 2; Journal Issue: 11; Related Information: CHORUS Timestamp: 2017-11-16 10:14:47; Journal ID: ISSN 2469-990X
American Physical Society
Country of Publication:
United States

Citation Formats

Mohan, Prakash, Fitzsimmons, Nicholas, and Moser, Robert D. Scaling of Lyapunov exponents in homogeneous isotropic turbulence. United States: N. p., 2017. Web. doi:10.1103/PhysRevFluids.2.114606.
Mohan, Prakash, Fitzsimmons, Nicholas, & Moser, Robert D. Scaling of Lyapunov exponents in homogeneous isotropic turbulence. United States. doi:10.1103/PhysRevFluids.2.114606.
Mohan, Prakash, Fitzsimmons, Nicholas, and Moser, Robert D. 2017. "Scaling of Lyapunov exponents in homogeneous isotropic turbulence". United States. doi:10.1103/PhysRevFluids.2.114606.
title = {Scaling of Lyapunov exponents in homogeneous isotropic turbulence},
author = {Mohan, Prakash and Fitzsimmons, Nicholas and Moser, Robert D.},
abstractNote = {},
doi = {10.1103/PhysRevFluids.2.114606},
journal = {Physical Review Fluids},
number = 11,
volume = 2,
place = {United States},
year = 2017,
month =

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on November 16, 2018
Publisher's Accepted Manuscript

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  • The Hasegawa-Wakatani model equations for resistive drift waves are solved numerically for a range of values of the coupling due to the parallel electron motion. The largest Lyapunov exponent, {lambda}{sub 1}, is calculated to quantify the unpredictability of the turbulent flow and compared to other characteristic inverse time scales of the turbulence such as the linear growth rate and Lagrangian inverse time scales obtained by tracking virtual fluid particles. The results show a correlation between {lambda}{sub 1} and the relative dispersion exponent, {lambda}{sub {ital p}}, as well as to the inverse Lagrangian integral time scale {tau}{sub {ital i}}{sup {minus}1}. Amore » decomposition of the flow into two distinct regions with different relative dispersion is recognized as the Weiss decomposition [J. Weiss, Physica D {bold 48}, 273 (1991)]. The regions in the turbulent flow which contribute to {lambda}{sub 1} are found not to coincide with the regions which contribute most to the relative dispersion of particles. {copyright} {ital 1996 American Institute of Physics.}« less
  • One of the hallmarks of turbulent flows is the chaotic behavior of fluid particle paths with exponentially growing separation among them while their distance does not exceed the viscous range. The maximal (positive) Lyapunov exponent represents the average strength of the exponential growth rate, while fluctuations in the rate of growth are characterized by the finite-time Lyapunov exponents (FTLEs). In the last decade or so, the notion of Lagrangian coherent structures (which are often computed using FTLEs) has gained attention as a tool for visualizing coherent trajectory patterns in a flow and distinguishing regions of the flow with different mixingmore » properties. A quantitative statistical characterization of FTLEs can be accomplished using the statistical theory of large deviations, based on the so-called Cramér function. To obtain the Cramér function from data, we use both the method based on measuring moments and measuring histograms and introduce a finite-size correction to the histogram-based method. We generalize the existing univariate formalism to the joint distributions of the two FTLEs needed to fully specify the Lyapunov spectrum in 3D flows. The joint Cramér function of turbulence is measured from two direct numerical simulation datasets of isotropic turbulence. Results are compared with joint statistics of FTLEs computed using only the symmetric part of the velocity gradient tensor, as well as with joint statistics of instantaneous strain-rate eigenvalues. When using only the strain contribution of the velocity gradient, the maximal FTLE nearly doubles in magnitude, highlighting the role of rotation in de-correlating the fluid deformations along particle paths. We also extend the large-deviation theory to study the statistics of the ratio of FTLEs. The most likely ratio of the FTLEs λ{sub 1} : λ{sub 2} : λ{sub 3} is shown to be about 4:1:−5, compared to about 8:3:−11 when using only the strain-rate tensor for calculating fluid volume deformations. The results serve to characterize the fundamental statistical and geometric structure of turbulence at small scales including cumulative, time integrated effects. These are important for deformable particles such as droplets and polymers advected by turbulence.« less
  • We perform a direct numerical simulation (DNS) of forced homogeneous isotropic turbulence with a passive scalar that is forced by mean gradient. The DNS data are used to study the properties of subgrid-scale flux of a passive scalar in the framework of large eddy simulation (LES), such as alignment trends between the flux, resolved, and subgrid-scale flow structures. It is shown that the direction of the flux is strongly coupled with the subgrid-scale stress axes rather than the resolved flow quantities such as strain, vorticity, or scalar gradient. We derive an approximate transport equation for the subgrid-scale flux of amore » scalar and look at the relative importance of the terms in the transport equation. A particular form of LES tensor-viscosity model for the scalar flux is investigated, which includes the subgrid-scale stress. Effect of different models for the subgrid-scale stress on the model for the subgrid-scale flux is studied.« less
  • The Lagrangian velocity autocorrelation and the time correlations for individual wave-number bands are computed by direct numerical simulation (DNS) using the passive vector method (PVM), and the accuracy of the method is studied. It is found that the PVM is accurate when [ital K][sub max]/[ital k][sub [ital d]][ge]2 where [ital K][sub max] is the maximum wave number carried in the simulation and [ital k][sub [ital d]] is the Kolmogorov wave number. The Eulerian and Lagrangian time correlations for various wave-number bands are compared. At moderate to high wave number the Eulerian time correlation decays faster than the Lagrangian, and themore » effect of sweep on the former is observed. The time scale of the Eulerian correlation is found to be ([ital kU][sub 0])[sup [minus]1] while that of the Lagrangian is [[integral][sub 0][sup [ital k]] [ital p][sup 2][ital E]([ital p])[ital dp]][sup [minus]1/2]. The Lagrangian velocity autocorrelation in a frozen turbulent field is computed using the DIA, ALHDIA, and LRA theories and is compared with DNS measurements. The Markovianized Lagrangian renormalized approximation (MLRA) is compared with the DNS, and good agreement is found for one-time quantities in decaying turbulence at low Reynolds numbers and for the Lagrangian velocity autocorrelation in stationary turbulence at moderate Reynolds number. The effect of non-Gaussianity on the Lagrangian correlation predicted by the theories is also discussed.« less