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Lagrangian investigations of velocity gradients in compressible turbulence: lifetime of flow-field topologies

  • Nishant Parashar (a1), Sawan Suman Sinha (a1) and Balaji Srinivasan (a2)


We perform Lagrangian investigations of the dynamics of velocity gradients in compressible decaying turbulence. Specifically, we examine the evolution of the invariants of the velocity-gradient tensor. We employ well-resolved direct numerical simulations over a range of Mach number along with a Lagrangian particle tracker to examine trajectories of fluid particles in the space of the invariants of the velocity gradient tensor. This allows us to accurately measure the lifetimes of major topologies of compressible turbulence and provide an explanation of why some selective topologies tend to exist longer than the others. Further, the influence of dilatation on the lifetime of various topologies is examined. Finally, we explain why the so-called conditional mean trajectories (CMT) used previously by several researchers fail to predict the lifetime of topologies accurately.


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Armitage, P. J. 2010 Astrophysics of Planet Formation. Cambridge University Press.
Ashurst, W. T., Chen, J. Y. & Rogers, M. M. 1987a Pressure gradient alignment with strain rate and scalar gradient in simulated Navier–Stokes turbulence. Phys. Fluids 30 (10), 32933294.
Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987b Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30 (8), 23432353.
Atkinson, C., Chumakov, S., Bermejo, M. I. & Soria, J. 2012 Lagrangian evolution of the invariants of the velocity gradient tensor in a turbulent boundary layer. Phys. Fluids 24 (10), 105104.
Bechlars, P. & Sandberg, R. D. 2017a Evolution of the velocity gradient tensor invariant dynamics in a turbulent boundary layer. J. Fluid Mech. 815, 223242.
Bechlars, P. & Sandberg, R. D. 2017b Variation of enstrophy production and strain rotation relation in a turbulent boundary layer. J. Fluid Mech. 812, 321348.
Bhatnagar, A., Gupta, A., Mitra, D., Pandit, R. & Perlekar, P. 2016 How long do particles spend in vortical regions in turbulent flows? Phys. Rev. E 94 (5), 18.
Biferale, L. & Toschi, F. 2005 Joint statistics of acceleration and vorticity in fully developed turbulence. J. Turbul. 6, N40.
Buxton, O. R. H. & Ganapathisubramani, B. 2010 Amplification of enstrophy in the far field of an axisymmetric turbulent jet. J. Fluid Mech. 651, 483502.
Cantwell, B. J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4 (4), 782793.
Cantwell, B. J. 1993 On the behavior of velocity gradient tensor invariants in direct numerical simulations of turbulence. Phys. Fluids A 5 (8), 20082013.
Cantwell, B. J. & Coles, D. 1983 An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136, 321374.
Chevillard, L. & Meneveau, C. 2006 Lagrangian dynamics and statistical geometric structure of turbulence. Phys. Rev. Lett. 97 (17), 174501.
Chevillard, L. & Meneveau, C. 2011 Lagrangian time correlations of vorticity alignments in isotropic turbulence: observations and model predictions. Phys. Fluids 23 (10), 101704.
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.
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2 (5), 765777.
Chu, Y. B. & Lu, X. Y. 2013 Topological evolution in compressible turbulent boundary layers. J. Fluid Mech. 733, 414438.
Danish, M., Sinha, S. S. & Srinivasan, B. 2016a Influence of compressibility on the lagrangian statistics of vorticity–strain-rate interactions. Phys. Rev. E 94 (1), 013101.
Danish, M., Suman, S. & Girimaji, S. S. 2016b Influence of flow topology and dilatation on scalar mixing in compressible turbulence. J. Fluid Mech. 793, 633655.
Elghobashi, S. & Truesdell, G. C. 1992 Direct simulation of particle dispersion in a decaying isotropic turbulence. J. Fluid Mech. 242, 655700.
Elsinga, G. E. & Marusic, I. 2010 Evolution and lifetimes of flow topology in a turbulent boundary layer. Phys. Fluids 22 (1), 015102.
Falkovich, G., Fouxon, A. & Stepanov, M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419 (6903), 151.
Girimaji, S. S. & Pope, S. B. 1990a A diffusion model for velocity gradients in turbulence. Phys. Fluids A 2 (2), 242256.
Girimaji, S. S. & Pope, S. B. 1990b Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427458.
Girimaji, S. S. & Speziale, C. G. 1995 A modified restricted Euler equation for turbulent flows with mean velocity gradients. Phys. Fluids 7 (6), 14381446.
Kerimo, J. & Girimaji, S. S. 2007 Boltzmann–BGK approach to simulating weakly compressible 3D turbulence: comparison between lattice Boltzmann and gas kinetic methods. J. Turbul. 8 (46), 116.
Kumar, G., Girimaji, S. S. & Kerimo, J. 2013 WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence. J. Comput. Phys. 234, 499523.
Liao, W., Peng, Y. & Luo, L. S. 2009 Gas-kinetic schemes for direct numerical simulations of compressible homogeneous turbulence. Phys. Rev. E 80 (4), 046702.
Lüthi, B., Tsinober, A. & Kinzelbach, W. 2005 Lagrangian measurement of vorticity dynamics in turbulent flow. J. Fluid Mech. 528, 87118.
Martín, J., Ooi, A., Chong, M. S. & Soria, J. 1998 Dynamics of the velocity gradient tensor invariants in isotropic turbulence. Phys. Fluids 10, 23362346.
Martín, M. P., Taylor, E. M., Wu, M. & Weirs, V. G. 2006 A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220 (1), 270289.
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219245.
Ohkitani, K. 1993 Eigenvalue problems in three-dimensional Euler flows. Phys. Fluids A 5 (10), 25702572.
O’Neill, P. & Soria, J. 2005 The relationship between the topological structures in turbulent flow and the distribution of a passive scalar with an imposed mean gradient. Fluid Dyn. Res. 36 (3), 107120.
Ooi, A., Martin, J., Soria, J. & Chong, M. S. 1999 A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J. Fluid Mech. 381, 141174.
Parashar, N., Sinha, S. S., Danish, M. & Srinivasan, B. 2017a Lagrangian investigations of vorticity dynamics in compressible turbulence. Phys. Fluids 29 (10), 105110.
Parashar, N., Sinha, S. S., Srinivasan, B. & Manish, A. 2017b GPU-accelerated direct numerical simulations of decaying compressible turbulence employing a GKM–based solver. Intl J. Numer. Meth. Fluids 83 (10), 737754.
Pater, I. D. & Lissauer, J. J. 2015 Planetary Sciences. Cambridge University Press.
Pinsky, M. B. & Khain, A. P. 1997 Turbulence effects on droplet growth and size distribution in clouds-A review. J. Aerosol Sci. 28 (7), 11771214.
Pirozzoli, S. & Grasso, F. 2004 Direct numerical simulations of isotropic compressible turbulence: influence of compressibility on dynamics and structures. Phys. Fluids 16 (12), 43864407.
Pope, S. B. 2000 Turbulent Flows, pp. 483489. Cambridge University Press.
Pope, S. B. 2002 Stochastic Lagrangian models of velocity in homogeneous turbulent shear flow. Phys. Fluids 14 (5), 16961702.
Pumir, A. 1994 A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient. Phys. Fluids 6 (6), 21182132.
Samtaney, R., Pullin, D. I. & Kosovic, B. 2001 Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids 13 (5), 14151430.
Sarkar, S., Erlebacher, G. & Hussaini, M. Y. 1991 Direct simulation of compressible turbulence in a shear flow. Theor. Comp. Fluid Dyn. 2 (5-6), 291305.
da Silva, C. B. & Pereira, J. C. F. 2008 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids 20 (5), 55101.
Soria, J., Sondergaard, R., Cantwell, B. J., Chong, M. S. & Perry, A. E. 1994 A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids 6 (2), 871884.
Suman, S. & Girimaji, S. S. 2009 Homogenized Euler equation: a model for compressible velocity gradient dynamics. J. Fluid Mech. 620, 177194.
Suman, S. & Girimaji, S. S. 2010 Velocity gradient invariants and local flow-field topology in compressible turbulence. J. Turbul. 11 (2), 124.
Suman, S. & Girimaji, S. S. 2012 Velocity-gradient dynamics in compressible turbulence: influence of Mach number and dilatation rate. J. Turbul. 13 (8), 123.
Toschi, F., Biferale, L., Boffetta, G., Celani, A., Devenish, B. J. & Lanotte, A. 2005 Acceleration and vortex filaments in turbulence. J. Turbul. 6, N15.
Vaghefi, N. S. & Madnia, C. K. 2015 Local flow topology and velocity gradient invariants in compressible turbulent mixing layer. J. Fluid Mech. 774, 6794.
Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. 43 (6), 837842.
Wang, L. & Lu, X. Y. 2012 Flow topology in compressible turbulent boundary layer. J. Fluid Mech. 703, 255278.
Wilczek, M. & Meneveau, C. 2014 Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech. 756, 191225.
Xu, H., Pumir, A. & Bodenschatz, E. 2011 The pirouette effect in turbulent flows. Nature Phys. 7 (9), 709712.
Xu, K. 2001 A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171 (1), 289335.
Yeung, P. K. & Pope, S. B. 1988 An algorithm for tracking fluid particles in numerical simulations of homogeneous turbulence. J. Comput. Phys. 79 (2), 373416.
Yeung, P. K. & Pope, S. B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531586.
Zhou, Y., Nagata, K., Sakai, Y., Ito, Y. & Hayase, T. 2015 On the evolution of the invariants of the velocity gradient tensor in single-square-grid-generated turbulence. Phys. Fluids 27 (7), 075107.
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Lagrangian investigations of velocity gradients in compressible turbulence: lifetime of flow-field topologies

  • Nishant Parashar (a1), Sawan Suman Sinha (a1) and Balaji Srinivasan (a2)


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