Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-20T11:44:04.783Z Has data issue: false hasContentIssue false

The Schur decomposition of the velocity gradient tensor for turbulent flows

Published online by Cambridge University Press:  13 June 2018

Abstract

The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the $\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$ diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in $\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$ space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions.

Type
JFM Papers
Copyright
© 2018 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.)

References

Ashurst, W. T., Kerstein, A. R., Kerr, R. A. & Gibson, C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30, 23432353.CrossRefGoogle Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497504.CrossRefGoogle Scholar
Biferale, L., Chevillard, L., Meneveau, C. & Toschi, F. 2007 Multiscale model of gradient evolution in turbulent flows. Phys. Rev. Lett. 98, 214501.Google Scholar
Cantwell, B. J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4 (4), 782793.Google Scholar
Cantwell, B. J. 1993 On the behavior of velocity gradient tensor invariants in direct numerical simulations of turbulence. Phys. Fluids A 5, 20082013.Google Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765777.Google Scholar
Dubief, Y. & Delcayre, F. 2000 On coherent-vortex identification in turbulence. J. Turbul. 1, N11.Google Scholar
Eberlein, P. J. 1965 On measures of non-normality for matrices. Am. Math. Mon. 72, 995996.Google Scholar
Elsinga, G. E. & Marusic, I. 2010 Evolution and lifetimes of flow topology in a turbulent boundary layer. Phys. Fluids 22, 015102.Google Scholar
Girimaji, S. S. & Pope, S. B. 1990 A diffusion model for velocity gradients in turbulence. Phys. Fluids 2 (2), 242256.Google Scholar
Golub, G. H. & van Loan, C. F. 2013 Matrix Computations, 4th edn. Johns Hopkins University Press.Google Scholar
Goto, S. 2008 A physical mechanism of the energy cascade in homogeneous isotropic turbulence. J. Fluid Mech. 605, 355366.Google Scholar
Henrici, P. 1962 Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices. Numer. Math. 4, 2440.Google Scholar
Higham, J. E., Brevis, W. & Keylock, C. J. 2016 A rapid non-iterative proper orthogonal decomposition based outlier detection and correction for PIV data. Meas. Sci. Technol. 27, 125303.Google Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Tech. Rep. CTR-S88. Center for Turbulence Research, Stanford University.Google Scholar
Jeong, E. & Girimaji, S. S. 2003 Velocity-gradient dynamics in turbulence: effect of viscosity and forcing. Theor. Comput. Fluid Dyn. 16, 421432.Google Scholar
Jimenez, J. 1992 Kinematic alignmernt effects in turbulent flows. Phys. Fluids A 4, 652654.Google 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, 387419.Google Scholar
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic, numerical turbulence. J. Fluid Mech. 153, 3158.Google Scholar
Keylock, C. J. 2017 Synthetic velocity gradient tensors and the identification of statistically significant aspects of the structure of turbulence. Phys. Rev. Fluids 2, 084607.Google Scholar
Kuo, A. Y.-S. & Corrsin, S. 1972 Experiment on the geometry of the fine-structure regions in fully turbulent fluid. J. Fluid Mech. 56, 447479.Google Scholar
Lawson, J. M. & Dawson, J. R. 2015 On velocity gradient dynamics and turbulent structure. J. Fluid Mech. 780, 6098.Google Scholar
Lee, S. L. 1995 A practical upper bound for departure from normality. SIAM J. Matrix Anal. Applics. 16, 462468.Google Scholar
Li, Y. & Meneveau, C. 2007 Material deformation in a restricted Euler model for turbulent flows: analytic solution and numerical tests. Phys. Fluids 19, 015104.Google Scholar
Li, Y., Perlman, E., Wan, M., Yang, Y., Burns, R., Meneveau, C., 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. 9, N31.Google Scholar
Lund, T. S. & Rogers, M. M. 1994 An improved measure of strain state probability in turbulent flows. Phys. Fluids 6 (5), 18381847.Google Scholar
Lundgren, T. S. 1982 Strained spiral vortex model for turbulent structures. Phys. Fluids 25, 21932203.Google Scholar
Lüthi, B., Holzner, M. & Tsinober, A. 2009 Expanding the [[()[]mml:mi[]()]]𝙌[[()[]/mml:mi[]()]]–[[()[]mml:mi[]()]]𝙍[[()[]/mml:mi[]()]] space to three dimensions. J. Fluid Mech. 641, 497507.Google Scholar
Martin, J., Dopazo, C. & Valiño, L. 1998 Dynamics of velocity gradient invariants in turbulence: restricted Euler and linear diffusion models. Phys. Fluids 10, 20122025.Google Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219245.Google Scholar
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, 6597.Google Scholar
Ohkitani, K. & Kishiba, S. 1995 Nonlocal nature of vortex stretching in an inviscid fluid. Phys. Fluids 7 (2), 411421.Google Scholar
Paul, I., Papadakis, G. & Vassilicos, J. C. 2017 Genesis and evolution of velocity gradients in a spatially developing turbulence. J. Fluid Mech. 815, 295332.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1987 Description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19, 125155.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr-Sommerfeld operator. SIAM J. Appl. Maths 53, 1547.Google Scholar
Schur, I. 1909 Über die charakteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen. Math. Ann. 66, 488510.Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476490.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578584.Google Scholar
Tsinober, A. 2001 Vortex stretching versus production of strain/dissipation. In Turbulence Structure and Vortex Dynamics (ed. Hunt, J. C. R. & Vassilicos, J. C.), pp. 164191. Cambridge University Press.Google Scholar
Tsinober, A., Shtilman, L. & Vaisburd, H. 1997 A study of properties of vortex stretching and enstrophy generation in numerical and laboratory turbulence. Fluid Dyn. Res. 21, 477494.Google Scholar
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125, 150162.Google Scholar
Wan, M., Chen, S., Eyink, G., Meneveau, C., Perlman, E., Burns, R., Li, Y., Szalay, A. & Hamilton, S.2016 Johns Hopkins Turbulence Database (JHTDB). http://turbulence.pha.jhu.edu/datasets.aspx.Google Scholar
Wan, M., Xiao, Z., Meneveau, C., Eyink, G. L. & Chen, S. 2010 Dissipation-energy flux correlations as evidence for the Lagrangian energy cascade in turbulence. Phys. Fluids 22 (6), 14.Google Scholar
Wilczek, M. & Meneveau, C. 2014 Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech. 756, 191225.Google Scholar