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Dissipation and enstrophy statistics in turbulence: are the simulations and mathematics converging?

Published online by Cambridge University Press:  18 May 2012

R. M. Kerr
R. M. Kerr*
Affiliation:
School of Engineering, University of Warwick, Coventry CV4 7AL, UK
*
†Email address for correspondence: R.M.Kerr@warwick.ac.uk
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Abstract

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Since the advent of cluster computing over 10 years ago there has been a steady output of new and better direct numerical simulation of homogeneous, isotropic turbulence with spectra and lower-order statistics converging to experiments and many phenomenological models. The next step is to directly compare these simulations to new models and new mathematics, employing the simulated data sets in novel ways, especially when experimental results do not exist or are poorly converged. For example, many of the higher-order moments predicted by the models converge slowly in experiments. The solution with a simulation is to do what an experiment cannot. The calculation and analysis of Yeung, Donzis & Sreenivasan (J. Fluid Mech., this issue, vol. 700, 2012, pp. 5–15) represents the vanguard of new simulations and new numerical analysis that will fill this gap. Where individual higher-order moments of the vorticity squared (the enstrophy) and kinetic energy dissipation might be converging slowly, they have focused upon ratios between different moments that have better convergence properties. This allows them to more fully explore the statistical distributions that eventually must be modelled. This approach is consistent with recent mathematics that focuses upon temporal intermittency rather than spatial intermittency. The principle is that when the flow is nearly singular, during ‘bad’ phases, when global properties can go up and down by many orders of magnitude, if appropriate ratios are taken, convergence rates should improve. Furthermore, in future analysis it might be possible to use these ratios to gain new insights into the intermittency and regularity properties of the underlying equations.

Type
Focus on Fluids
Information
Journal of Fluid Mechanics , Volume 700 , 10 June 2012 , pp. 1 - 4
Copyright
Copyright © Cambridge University Press 2012

References

1. Chen, S., Sreenivasan, K. R., Nelkin, M. & Cao, N. 1997 Refined similarity hypothesis for transverse structure functions in fluid turbulence. Phys. Rev. Lett. 79, 2253–2256.CrossRefGoogle Scholar
2. Clay Mathematical Institute, AMS-Clay Prize: www.claymath.org/millennium/.Google Scholar
3. Gibbon, J. D. 2009 Estimating intermittency in three-dimensional Navier–Stokes turbulence. J. Fluid Mech. 625, 125–133.CrossRefGoogle Scholar
4. Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high-Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.CrossRefGoogle Scholar
5. Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–58.CrossRefGoogle Scholar
6. Kerr, R. M., Meneguzzi, M. & Gotoh, T. 2001 An inertial range crossover in structure functions. Phys. Fluids 13, 1985–1994.CrossRefGoogle Scholar
7. Nelkin, M. 1999 Enstrophy and dissipation must have the same scaling exponent in the high Reynolds number limit of fluid turbulence. Phys. Fluids 11, 2202–2204.CrossRefGoogle Scholar
8. Orszag, S. A. & Patterson, G. S. 1972 Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76–79.CrossRefGoogle Scholar
9. Schumacher, J., Eckhardt, B. & Doering, C. R. 2010 Extreme vorticity growth in Navier–Stokes turbulence. Phys. Lett. A 374, 861–865.CrossRefGoogle Scholar
10. Siggia, E. D. 1981 Numerical study of small-scale intermittency in three-dimensional turbulence. J. Fluid Mech. 107, 375–406.CrossRefGoogle Scholar
11. Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–25.CrossRefGoogle Scholar
12. Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2012 Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers. J. Fluid Mech. 700, 5–15.CrossRefGoogle Scholar