Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T00:55:28.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe

Published online by Cambridge University Press:  12 January 2021

Josin Tom Open the ORCID record for Josin Tom [Opens in a new window] ,
Maurizio Carbone Open the ORCID record for Maurizio Carbone [Opens in a new window]  and
Andrew D. Bragg Open the ORCID record for Andrew D. Bragg [Opens in a new window]
Josin Tom
Affiliation:
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA
Maurizio Carbone
Affiliation:
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, Corso Duca degli Abruzzi, 24 10129Torino, Italy
Andrew D. Bragg*
Affiliation:
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Expressing the evolution equations for the filtered velocity gradient tensor (FVGT) in the strain-rate eigenframe provides an insightful way to disentangle and understand various processes such as strain self-amplification, vortex stretching and tilting, and to consider their properties at different scales in the flow. Using data from direct numerical simulation of the forced Navier–Stokes equation, we consider the relative importance of local and non-local terms in the FVGT eigenframe equations across the scales using statistical analysis. The eigenframe rotation rate (that drives vorticity tilting) is shown to exhibit highly non-Gaussian fluctuations even at large scales due to kinematic effects, but dynamically, the anisotropic pressure Hessian plays a key role. The anisotropic pressure Hessian conditioned on the eigenvalues and principal vorticity components exhibits highly nonlinear behaviour at low values of normalized local gradients, with important modelling implications. We derive a generalization of the Lumley triangle that allows us to show that the pressure Hessian has a preference for two-component axisymmetric configurations at small scales, with a transition to a more isotropic state at larger scales. Correlations between the sub-grid stress and other terms in the eigenframe equations are considered, highlighting the coupling between the sub-grid and nonlinear amplification terms, with the sub-grid term playing an important role in regularizing the system. These results provide useful guidelines for improving Lagrangian models of the FVGT, since current models fail to capture a number of subtle features observed in our results.

JFM classification

Turbulent Flows: Isotropic turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A24
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Present address: Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany.

References

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.CrossRefGoogle Scholar
Ayyalasomayajula, S., Warhaft, Z. & Collins, L. R. 2008 Modeling inertial particle acceleration statistics in isotropic turbulence. Phys. Fluids 20 (9), 095104.CrossRefGoogle Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1 (5), 497–504.CrossRefGoogle Scholar
Beylkin, G. 1995 On the Fast Fourier Transform of functions with singularities. Appl. Comput. Harmon. Anal. 2 (4), 363–381.CrossRefGoogle Scholar
Biferale, L., Chevillard, L., Meneveau, C. & Toschi, F. 2007 Multiscale model of gradient evolution in turbulent flows. Phys. Rev. Lett. 98, 214501.CrossRefGoogle ScholarPubMed
Borue, V. & Orszag, S. A. 1998 Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 1–31.CrossRefGoogle Scholar
Buaria, D., Pumir, A., Bodenschatz, E. & Yeung, P. K. 2019 Extreme velocity gradients in turbulent flows. New J. Phys. 21 (4), 043004.CrossRefGoogle 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
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Mechanics. Springer.Google Scholar
Carbone, M. & Bragg, A. D. 2020 Is vortex stretching the main cause of the turbulent energy cascade? J. Fluid Mech. 883, R2.CrossRefGoogle Scholar
Carbone, M., Iovieno, M. & Bragg, A. D. 2020 Symmetry transformation and dimensionality reduction of the anisotropic pressure Hessian. J. Fluid Mech. 900, A38.CrossRefGoogle Scholar
Cerutti, S., Meneveau, C. & Knio, O. M. 2000 Spectral and hyper eddy viscosity in high-Reynolds-number turbulence. J. Fluid Mech. 421, 307–338.CrossRefGoogle Scholar
Cheng, W.-P. & Cantwell, B. 1996 Study of the velocity gradient tensor in turbulent flow. NASA Tech. Rep. JIAA TR 114.Google 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.CrossRefGoogle Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2 (5), 765–777.CrossRefGoogle Scholar
Chong, M. S., Soria, J., Perry, A. E., Chacin, J., Cantwell, B. J. & Na, Y. 1998 Turbulence structures of wall-bounded shear flows found using DNS data. J. Fluid Mech. 357, 225–247.CrossRefGoogle Scholar
Danish, M. & Meneveau, C. 2018 Multiscale analysis of the invariants of the velocity gradient tensor in isotropic turbulence. Phys. Rev. Fluids 3, 044604.CrossRefGoogle Scholar
Dresselhaus, E. & Tabor, M. 1992 The kinematics of stretching and alignment of material elements in general flow fields. J. Fluid Mech. 236, 415–444.CrossRefGoogle Scholar
Elsinga, G. E. & Marusic, I. 2010 Evolution and lifetimes of flow topology in a turbulent boundary layer. Phys. Fluids 22 (1), 015102.CrossRefGoogle Scholar
Eyink, G. L. & Aluie, H. 2009 Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining. Phys. Fluids 21 (11), 115107.CrossRefGoogle Scholar
Girimaji, S. S. & Pope, S. B. 1990 a A diffusion model for velocity gradients in turbulence. Phys. Fluids A 2 (2), 242–256.CrossRefGoogle Scholar
Girimaji, S. S. & Pope, S. B. 1990 b Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427–458.CrossRefGoogle Scholar
Ireland, P. J., Bragg, A. D. & Collins, L. R. 2016 a The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects. J. Fluid Mech. 796, 617–658.CrossRefGoogle Scholar
Ireland, P. J., Bragg, A. D. & Collins, L. R. 2016 b The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects. J. Fluid Mech. 796, 659–711.CrossRefGoogle Scholar
Ireland, P. J., Vaithianathan, T., Sukheswalla, P. S., Ray, B. & Collins, L. R. 2013 Highly parallel particle-laden flow solver for turbulence research. Comput. Fluids 76, 170–177.CrossRefGoogle Scholar
Jeong, E. & Girimaji, S. S. 2003 Velocity-gradient dynamics in turbulence: effect of viscosity and forcing. Theor. Comput. Fluid Dyn. 16, 421–432.CrossRefGoogle Scholar
Johnson, P. L. & Meneveau, C. 2015 Large-deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence. Phys. Fluids 27 (8), 085110.CrossRefGoogle Scholar
Johnson, P. L. & Meneveau, C. 2016 A closure for lagrangian velocity gradient evolution in turbulence using recent-deformation mapping of initially Gaussian fields. J. Fluid Mech. 804, 387–419.CrossRefGoogle Scholar
Johnson, P. L. & Meneveau, C. 2017 Turbulence intermittency in a multiple-time-scale Navier–Stokes-based reduced model. Phys. Rev. Fluids 2, 072601.CrossRefGoogle Scholar
Lawson, J. M. & Dawson, J. R. 2015 On velocity gradient dynamics and turbulent structure. J. Fluid Mech. 780, 60–98.CrossRefGoogle Scholar
Leppin, L. A. & Wilczek, M. 2020 Capturing velocity gradients and particle rotation rates in turbulence. Phys. Rev. Lett. 125 (22), 224501.CrossRefGoogle Scholar
Lumley, J. L. 1979 Computational modeling of turbulent flows. In Advances in Applied Mechanics (ed. C.-S. Yih), vol. 18, pp. 123–176. Elsevier.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
Lüthi, B., Holzner, M. & Tsinober, A. 2009 Expanding the Q–R space to three dimensions. J. Fluid Mech. 641, 497–507.CrossRefGoogle Scholar
Martın, J., Dopazo, C. & Vali no, L. 1998 Dynamics of velocity gradient invariants in turbulence: restricted Euler and linear diffusion models. Phys. Fluids 10 (8), 2012–2025.CrossRefGoogle Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43 (1), 219–245.CrossRefGoogle 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
Orszag, S. A. 1971 On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. J. Atmos. Sci. 28 (6), 1074–1074.2.0.CO;2>CrossRefGoogle Scholar
Pennisi, S. & Trovato, M. 1987 On the irreducibility of Professor G. F. Smith's representations for isotropic functions. Intl J. Engng Sci. 25 (8), 1059–1065.CrossRefGoogle Scholar
Pereira, R. M., Moriconi, L. & Chevillard, L. 2018 A multifractal model for the velocity gradient dynamics in turbulent flows. J. Fluid Mech. 839, 430–467.CrossRefGoogle Scholar
Pope, S. B., Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
She, Z. S., Jackson, E., Orszag, S. A., Hunt, J. C. R., Phillips, O. M. & Williams, D. 1991 Structure and dynamics of homogeneous turbulence: models and simulations. Proc. R. Soc. Lond. A 434 (1890), 101–124.Google Scholar
Towns, J., Cockerill, T., Dahan, M., Foster, I., Gaither, K., Grimshaw, A., Hazlewood, V., Lathrop, S., Lifka, D., Peterson, G. D., et al. 2014 XSEDE: accelerating scientific discovery. Comput. Sci. Engng 16 (5), 62–74.CrossRefGoogle Scholar
Tsinober, A. 2001 An Informal Introduction to Turbulence. Kluwer Academic Publishers.Google Scholar
Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. France 43 (6), 837–842.CrossRefGoogle Scholar
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125 (1), 150–162.CrossRefGoogle Scholar
Wilczek, M. & Meneveau, C. 2014 Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech. 756, 191–225.CrossRefGoogle Scholar