- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Scaling of Lyapunov exponents in homogeneous isotropic turbulence

Prakash Mohan* and Nicholas Fitzsimmons†

Robert D. Moser‡

  • Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA

  • Institute for Computational Engineering and Sciences and the Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA

  • *prakash@ices.utexas.edu
  • †Present address: Applied Research Laboratories, the University of Texas at Austin, Austin, TX 78758; nfitz@arlut.utexas.edu
  • ‡rmoser@ices.utexas.edu
PDF

Phys. Rev. Fluids 2, 114606 – Published 16 November, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.114606

Abstract

Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysis of the linear disturbance used to compute the Lyapunov exponent, and its instantaneous growth, show that the instabilities do, as expected, act on the smallest eddies and that at any time, there are many sites of local instabilities.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (29)

  1. L. Keefe, P. Moin, and J. Kim, The dimension of attractors underlying periodic turbulent poiseuille flow, J. Fluid Mech. 242, 1 (1992).
  2. J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985).
  3. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D (Amsterdam, Neth.) 16, 285 (1985).
  4. J. A. Vastano and R. D. Moser, Short-time Lyapunov exponent analysis and the transition to chaos in Taylor-Couette flow, J. Fluid Mech. 233, 83 (1991).
  5. G. Comte-Bellot and S. Corrsin, The use of a contraction to improve the isotropy of grid-generated turbulence, J. Fluid Mech. 25, 657 (1966).
  6. G. Comte-Bellot and S. Corrsin, Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence, J. Fluid Mech. 48, 273 (1971).
  7. L. Mydlarski and Z. Warhaft, On the onset of high-Reynolds-number grid-generated wind tunnel turbulence, J. Fluid Mech. 320, 331 (1996).
  8. S. Chen and X. Shan, High-resolution turbulent simulations using the connection machine-2, Comput. Phys. 6, 643 (1992).
  9. J. Jiménez, A. A. Wray, P. G. Saffman, and R. S. Rogallo, The structure of intense vorticity in isotropic turbulence, J. Fluid Mech. 255, 65 (1993).
  10. R. S. Rogallo, Numerical Experiments in Homogeneous Turbulence, NASA STI Technical Report (1981).
  11. A. Vincent and M. Meneguzzi, The spatial structure and statistical properties of homogeneous turbulence, J. Fluid Mech. 225, 1 (1991).
  12. P. A. Durbin and B. P. Reif, Statistical Theory and Modeling for Turbulent Flows (Wiley & Sons, New York, 2011).
  13. A. Crisanti, M. H. Jensen, A. Vulpiani, and G. Paladin, Intermittency and Predictability in Turbulence, Phys. Rev. Lett. 70, 166 (1993).
  14. E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani, Growth of Noninfinitesimal Perturbations in Turbulence, Phys. Rev. Lett. 77, 1262 (1996).
  15. V. I. Oseledec, Multiple ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 197 (1968).
  16. J. Kim, P. Moin, and R. Moser, Turbulence statistics in fully developed channel flow at low reynolds number, J. Fluid Mech. 177, 133 (1987).
  17. P. R. Spalart, R. D. Moser, and M. M. Rogers, Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions, J. Comput. Phys. 96, 297 (1991).
  18. M. Lee, N. Malaya, and R. D. Moser, Petascale direct numerical simulation of turbulent channel flow on up to 786k cores, in Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis. (ACM, New York, 2013), p. 61.
  19. M. Lee, R. Ulerich, N. Malaya, and R. D. Moser, Experiences from leadership computing in simulations of turbulent fluid flows, Comput. Sci. Eng. 16, 24 (2014).
  20. S. B. Pope, Turbulent Flows (IOP Publishing, Bristol, UK, 2001).
  21. T. A. Oliver, N. Malaya, R. Ulerich, and R. D. Moser, Estimating uncertainties in statistics computed from direct numerical simulation, Phys. Fluids 26, 035101 (2014).
  22. E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, UK, 2003).
  23. H. Haario, M. Laine, A. Mira, and E. Saksman, Dram: Efficient adaptive mcmc, Stat. Comput. 16, 339 (2006).
  24. D. McDougall, N. Malaya, and R. D. Moser, The parallel c++ statistical library for Bayesian inference: Queso, in Handbook of Uncertainty Quantification, edited by R. Ghanem D. Higdon, and H. Owhadi (Springer International Publishing, Cham, 2017), pp. 1829–1865.
  25. E. E. Prudencio and K. W. Schulz, The parallel C++ statistical library QUESO: Quantification of Uncertainty for Estimation, Simulation and Optimization, in Euro-Par 2011: Parallel Processing Workshops (Springer, New York, 2012), pp. 398–407.
  26. S. C. C. Bailey, M. Hultmark, J. Schumacher, V. Yakhot, and A. J. Smits, Measurement of Local Dissipation Scales in Turbulent Pipe Flow, Phys. Rev. Lett. 103, 014502 (2009).
  27. J. Schumacher, Sub-Kolmogorov-scale fluctuations in fluid turbulence, Europhys. Lett. 80, 54001 (2007).
  28. J. Schumacher, K. R. Sreenivasan, and V. Yakhot, Asymptotic exponents from low-Reynolds-number flows, New J. Phys. 9, 89 (2007).
  29. V. Yakhot and K. R. Sreenivasan, Anomalous scaling of structure functions and dynamic constraints on turbulence simulations, J. Stat. Phys. 121, 823 (2005).

Outline

Information

Phys. Rev. Fluids 2, 114606– Published 16 November, 2017
  • Received 20 July 2017
Article part of CHORUS

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation