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Symmetry transformation and dimensionality reduction of the anisotropic pressure Hessian

Published online by Cambridge University Press:  13 August 2020

M. Carbone Open the ORCID record for M. Carbone [Opens in a new window] ,
M. Iovieno Open the ORCID record for M. Iovieno [Opens in a new window]  and
A. D. Bragg Open the ORCID record for A. D. Bragg [Opens in a new window]
M. Carbone
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129Torino, Italy Department of Civil and Environmental Engineering, Duke University, Durham, NC27708, USA
M. Iovieno
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129Torino, Italy
A. D. Bragg*
Affiliation:
Department of Civil and Environmental Engineering, Duke University, Durham, NC27708, USA
*
†Email address for correspondence: andrew.bragg@duke.edu
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Abstract

The dynamics of the velocity gradient tensor in turbulence is governed in part by the anisotropic pressure Hessian, which is a non-local functional of the velocity gradient field. This anisotropic pressure Hessian plays a key dynamical role, for example in preventing finite-time singularities, but it is difficult to understand and model due to its non-locality and complexity. In this work a symmetry transformation for the pressure Hessian is introduced, such that when the transformation is applied to the original pressure Hessian, the dynamics of the invariants of the velocity gradients remains unchanged. We then exploit this symmetry transformation to perform a dimensional reduction on the three-dimensional anisotropic pressure Hessian, which, remarkably, is possible everywhere in the flow except on zero-measure sets. The dynamical activity of the newly introduced dimensionally reduced anisotropic pressure Hessian is confined to two-dimensional manifolds in the three-dimensional flow, and exhibits striking alignment properties with respect to the strain-rate eigenframe and the vorticity vector. The dimensionality reduction, together with the strong preferential alignment properties, leads to new dynamical insights for understanding and modelling the role of the anisotropic pressure Hessian in three-dimensional turbulent flows.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Intermittency

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A38
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., et al. 1999 LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
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
Ballouz, J. G. & Ouellette, N. T. 2018 Tensor geometry in the turbulent cascade. J. Fluid Mech. 835, 1048–1064.CrossRefGoogle Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1 (5), 497–504.10.1017/S0022112056000317CrossRefGoogle Scholar
Biferale, L., Meneveau, C. & Verzicco, R. 2014 Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence. J. Fluid Mech. 754, 184–207.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
Chertkov, M., Pumir, A. & Shraiman, B. I. 1999 Lagrangian tetrad dynamics and the phenomenology of turbulence. Phys. Fluids 11 (8), 2394–2410.10.1063/1.870101CrossRefGoogle 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. 2013 Orientation dynamics of small, triaxial-ellipsoidal particles in isotropic turbulence. J. Fluid Mech. 737, 571–596.CrossRefGoogle Scholar
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
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
Falkovich, G. & Gawȩdzki, K. 2014 Turbulence on hyperbolic plane: the fate of inverse cascade. J.Stat. Phys. 156 (1), 10–54.CrossRefGoogle Scholar
Girimaji, S. S. & Pope, S. B. 1990 A diffusion model for velocity gradients in turbulence. Phys. FluidsA 2 (2), 242–256.CrossRefGoogle Scholar
Ibbeken, G., Green, G. & Wilczek, M. 2019 Large-scale pattern formation in the presence of small-scale random advection. Phys. Rev. Lett. 123, 114501.10.1103/PhysRevLett.123.114501CrossRefGoogle ScholarPubMed
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, 617–658.10.1017/jfm.2016.238CrossRefGoogle 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, 659–711.10.1017/jfm.2016.227CrossRefGoogle 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, 170–177.10.1016/j.compfluid.2013.01.020CrossRefGoogle 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
Majda, A. J. & Bertozzi, A. L. 2001 Vorticity and Incompressible Flow. Cambridge University Press.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
Ohkitani, K. 1993 Eigenvalue problems in three-dimensional Euler flows. Phys. Fluids A 5 (10), 2570–2572.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle 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.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
Vlaykov, D. G. & Wilczek, M. 2019 On the small-scale structure of turbulence and its impact on the pressure field. J. Fluid Mech. 861, 422–446.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, 191–225.10.1017/jfm.2014.367CrossRefGoogle Scholar
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
Yu, H. & Meneveau, C. 2010 Lagrangian refined kolmogorov similarity hypothesis for gradient time evolution and correlation in turbulent flows. Phys. Rev. Lett. 104, 084502.CrossRefGoogle ScholarPubMed