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Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields

Published online by Cambridge University Press:  01 September 2014

Michael Wilczek  and
Charles Meneveau
Michael Wilczek*
Affiliation:
Department of Mechanical Engineering and Institute for Data Intensive Engineering and Science, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA
Charles Meneveau
Affiliation:
Department of Mechanical Engineering and Institute for Data Intensive Engineering and Science, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

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Understanding the non-local pressure contributions and viscous effects on the small-scale statistics remains one of the central challenges in the study of homogeneous isotropic turbulence. Here we address this issue by studying the impact of the pressure Hessian as well as viscous diffusion on the statistics of the velocity gradient tensor in the framework of an exact statistical evolution equation. This evolution equation shares similarities with earlier phenomenological models for the Lagrangian velocity gradient tensor evolution, yet constitutes the starting point for a systematic study of the unclosed pressure Hessian and viscous diffusion terms. Based on the assumption of incompressible Gaussian velocity fields, closed expressions are obtained as the results of an evaluation of the characteristic functionals. The benefits and shortcomings of this Gaussian closure are discussed, and a generalization is proposed based on results from direct numerical simulations. This enhanced Gaussian closure yields, for example, insights on how the pressure Hessian prevents the finite-time singularity induced by the local self-amplification and how its interaction with viscous effects leads to the characteristic strain skewness phenomenon.

JFM classification

Turbulent Flows: Intermittency Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 756 , 10 October 2014 , pp. 191 - 225
Copyright
© 2014 Cambridge University Press 

References

Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30 (8), 2343–2353.Google Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1 (05), 497–504.Google Scholar
Cantwell, B. J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4 (4), 782–793.CrossRefGoogle Scholar
Chertkov, M., Pumir, A. & Shraiman, B. I. 1999 Lagrangian tetrad dynamics and the phenomenology of turbulence. Phys. Fluids 11 (8), 2394–2410.CrossRefGoogle Scholar
Chevillard, L. & Meneveau, C. 2006 Lagrangian dynamics and statistical geometric structure of turbulence. Phys. Rev. Lett. 97, 174501.CrossRefGoogle ScholarPubMed
Chevillard, L., Meneveau, C., Biferale, L. & Toschi, F. 2008 Modeling the pressure Hessian and viscous Laplacian in turbulence: comparisons with direct numerical simulation and implications on velocity gradient dynamics. Phys. Fluids 20 (10), 101504.Google Scholar
Dopazo, C. 1994 Recent developments in PDF methods. Turbulent Reacting Flows II. Springer.Google Scholar
Friedrich, R., Daitche, A., Kamps, O., Luelff, J., Vosskuhle, M. & Wilczek, M. 2012 The Lundgren–Monin–Novikov hierarchy: kinetic equations for turbulence. C.R. Phys. 13 (9–10), 929–953.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press.Google Scholar
Girimaji, S. S. & Pope, S. B. 1990 A diffusion model for velocity gradients in turbulence. Phys. Fluids 2 (2), 242–256.CrossRefGoogle Scholar
Haken, H. 2004 Synergetics: Introduction and Advanced Topics. Springer.CrossRefGoogle Scholar
Holzer, M. & Siggia, E. 1993 Skewed, exponential pressure distributions from Gaussian velocities. Phys. Fluids A 5 (10), 2525–2532.Google Scholar
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2007 Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics. J. Fluid Mech. 592, 335–366.Google Scholar
Jeong, E. & Girimaji, S. S. 2003 Velocity-gradient dynamics in turbulence: effect of viscosity and forcing. J. Theor. Comput. Fluid Dyn. 16, 421–432.Google Scholar
Li, Y., Perlman, E., Wan, M., Yang, Y., Meneveau, C., Burns, R., Chen, S., Szalay, A. & Eyink, G. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. N31.CrossRefGoogle Scholar
Lund, T. S. & Rogers, M. M. 1994 An improved measure of strain state probability in turbulent flows. Phys. Fluids 6 (5), 1838–1847.CrossRefGoogle Scholar
Lundgren, T. S. 1967 Distribution functions in the statistical theory of turbulence. Phys. Fluids 10 (5), 969–975.CrossRefGoogle Scholar
Lüthi, B., Holzner, M. & Tsinober, A. 2009 Expanding the $Q\text {--}R$ space to three dimensions. J. Fluid Mech. 641, 497–507.Google Scholar
Martin, J., Dopazo, C. & Valino, L. 1998 Dynamics of velocity gradient invariants in turbulence: restricted Euler and linear diffusion models. Phys. Fluids 10 (8), 2012–2025.Google Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43 (1), 219–245.Google Scholar
Meyers, J. & Meneveau, C. 2008 A functional form for the energy spectrum parametrizing bottleneck and intermittency effects. Phys. Fluids 20 (6), 065109.Google Scholar
Monin, A. S., Yaglom, A. M. & Lumley, J. L. 2007 Statistical Fluid Mechanics: Mechanics of Turbulence, Statistical Fluid Mechanics, vol. 2. Dover Publications.Google Scholar
Naso, A. & Pumir, A. 2005 Scale dependence of the coarse-grained velocity derivative tensor structure in turbulence. Phys. Rev. E 72, 056318.CrossRefGoogle ScholarPubMed
Nomura, K. K. & Post, G. K. 1998 The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence. J. Fluid Mech. 377, 65–97.CrossRefGoogle Scholar
Ohkitani, K. & Kishiba, S. 1995 Nonlocal nature of vortex stretching in an inviscid fluid. Phys. Fluids 7 (2), 411–421.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Shtilman, L., Spector, M. & Tsinober, A. 1993 On some kinematic versus dynamic properties of homogeneous turbulence. J. Fluid Mech. 247, 65–77.Google Scholar
Tsinober, A. 1998 Is concentrated vorticity that important? Eur. J. Mech. B/Fluids 17, 421–449.Google Scholar
Tsinober, A. 2009 An Informal Conceptual Introduction to Turbulence. Springer.CrossRefGoogle Scholar
Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. France 43 (6), 837–842.Google Scholar
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125 (1), 150–162.Google Scholar
Wilczek, M., Daitche, A. & Friedrich, R. 2011 On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity. J. Fluid Mech. 676, 191–217.Google Scholar