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Maximum palinstrophy growth in 2D incompressible flows

Published online by Cambridge University Press:  21 February 2014

Diego Ayala  and
Bartosz Protas
Diego Ayala
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1
Bartosz Protas*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1
*
†Email address for correspondence: bprotas@mcmaster.ca
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Abstract

In this study we investigate vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain. The issue of palinstrophy growth is related to a broader research program focusing on extreme amplification of vorticity-related quantities which may signal singularity formation in different flow models. Such extreme vortex flows are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing solutions parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigourous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex structures are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using the instantaneously optimal states with fixed energy and palinstrophy as the initial data saturates the upper bound for the maximum growth of palinstrophy in finite time. Possible implications of this finding for the questions concerning extreme behaviour of flows are discussed.

JFM classification

Mathematical Foundations: Variational methods Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 340 - 367
Copyright
© 2014 Cambridge University Press 

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References

⇒eferences

Adams, R. A. & Fournier, J. F. 2005 Sobolev Spaces. Elsevier.Google Scholar
Ayala, D. 2014 Extreme vortex states and singularity formation in incompressible flows. PhD thesis, McMaster University (in preparation).Google Scholar
Ayala, D. & Protas, B. 2011 On maximum enstrophy growth in a hydrodynamic system. Physica D 240, 1553–1563.Google Scholar
Ayala, D. & Protas, B. 2013 Vortices, maximum growth and the problem of finite-time singularity formation. Fluid Dyn. Res. (in press).Google Scholar
Brachet, M. E. 1991 Direct simulation of three-dimensional turbulence in the Taylor–Green vortex. Fluid Dyn. Res. 8, 1–8.Google Scholar
Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. & Frisch, U. 1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.Google Scholar
Bustamante, M. D. & Brachet, M. 2012 Interplay between the Beale–Kato–Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem. Phys. Rev. E 86, 066302.Google ScholarPubMed
Dascaliuc, R., Foias, C. & Jolly, M. S. 2010 Estimates on enstrophy, palinstrophy, and invariant measures for 2D turbulence. J. Differ. Equ. 248, 792–819.Google Scholar
Doering, C. R. 2009 The 3D Navier–Stokes problem. Annu. Rev. Fluid Mech. 109–128.CrossRefGoogle Scholar
Doering, C. R. & Gibbon, J. D. 1995 Applied Analysis of the Navier–Stokes Equations. Cambridge University Press.CrossRefGoogle Scholar
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. C. 1994 Krylov method for the incompressible Navier–Stokes equation. J. Comput. Phys. 110, 82–102.Google Scholar
Farazmand, M., Kevlahan, N. K. R. & Protas, B. 2011 Controlling the dual cascade of two-dimensional turbulence. J. Fluid Mech. 668, 202–222.Google Scholar
Fefferman, C. L. 2000 Existence and smoothness of the Navier–Stokes equation. Clay Millennium Prize Problem Description. Available at http://www.claymath.org/millennium/Navier-Stokes_Equations/navierstokes.pdf.Google Scholar
Foias, C. & Temam, R. 1989 Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369.Google Scholar
Gibbon, J. D., Bustamante, M. & Kerr, R. M. 2008 The three-dimensional Euler equations: singular or non-singular?. Nonlinearity 21, 123–129.CrossRefGoogle Scholar
Gibbon, J. D. & Titi, E. S. 2013 The 3D incompressible Euler equations with a passive scalar: a road to blow-up? Preprint, ArXiv:1211.3811.CrossRefGoogle Scholar
Grafke, T., Homann, H., Dreher, J. & Grauer, R. 2008 Numerical simulations of possible finite-time singularities in the incompressible Euler equations: comparison of numerical methods. Physica D 237, 1932–1936.Google Scholar
Gunzburger, M. D. 2003 Perspectives in Flow Control and Optimization. SIAM.Google Scholar
Hassanzadeh, P., Chini, G. P. & Doering, C. R. 2013 Wall to wall optimal transport. Preprint, ArXiv:1309.5542.Google Scholar
Hou, T. Y. 2009 Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations. Acta Numerica 18, 277–346.Google Scholar
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5, 1725–1746.Google Scholar
Kiselev, A. 2010 Regularity and blow up for active scalars. Math. Modelling Nat. Phenom. 5, 225–255.Google Scholar
Kreiss, H. & Lorenz, J. 2004 In Initial-Boundary Value Problems and the Navier–Stokes Equations Classics in Applied Mathematics, vol. 47, SIAM.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
Lu, L. 2006 Bounds on the enstrophy growth rate for solutions of the 3D Navier–Stokes equations. PhD thesis, University of Michigan.Google Scholar
Lu, L. & Doering, C. R. 2008 Limits on enstrophy growth for solutions of the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2693–2727.Google Scholar
Luenberger, D. 1969 Optimization by Vector Space Methods. John Wiley and Sons.Google Scholar
Majda, A. J. & Bertozzi, A. L. 2002 Vorticity and Incompressible Flow. Cambridge University Press.Google Scholar
Matsumoto, T., Bec, J. & Frisch, U. 2008 Complex-space singularities of 2D Euler flow in Lagrangian coordinates. Physica D 237, 1951–1955.Google Scholar
Ohkitani, K. 2008 A miscellany of basic issues on incompressible fluid equations. Nonlinearity 21, 255–271.Google Scholar
Ohkitani, K. & Constantin, P. 2008 Numerical study of the Eulerian–Lagrangian analysis of the Navier–Stokes turbulence. Phys. Fluids 20, 1–11.CrossRefGoogle Scholar
Orlandi, P., Pirozzoli, S. & Carnevale, G. F. 2012 Vortex events in Euler and Navier–Stokes simulations with smooth initial conditions. J. Fluid Mech. 690, 288–320.CrossRefGoogle Scholar
Pelinovsky, D. 2012a Enstrophy growth in the viscous Burgers equation. Dyn. Partial Diff. Equ. 9, 305–340.Google Scholar
Pelinovsky, D. 2012b Sharp bounds on enstrophy growth in the viscous Burgers equation. Proc. R. Soc. A 468, 3636–3648.Google Scholar
Pelz, R. B. 2001 Symmetry and the hydrodynamic blow-up problem. J. Fluid Mech. 444, 299–320.Google Scholar
Pouquet, A., Lesieur, M., André, J. C. & Basdevant, C. 1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72, 305–319.Google Scholar
Protas, B., Babiano, A. & Kevlahan, N. K. R. 1999 On geometrical alignment properties of two-dimensional forced turbulence. Physica D 128, 169–179.Google Scholar
Protas, B., Bewley, T. & Hagen, G. 2004 A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems. J. Comput. Phys. 195, 49–89.Google Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Variational identification of minimal seeds to trigger transition in plane Couette flow. J. Fluid Mech. 712, 244–272.Google Scholar
Ruszczyński, A. 2006 Nonlinear Optimization. Princeton University Press.Google Scholar
Scott, R. K. 2011 A scenario for finite-time singularity in the quasigeostrophic model. J. Fluid Mech. 687, 492–502.Google Scholar
Serrin, J. 1963 The initial value problem for the Navier–Stokes equations. In Nonlinear Problems (ed. Langer, E. R.), pp. 69–98. Mathematics Research Center, United States Army, The University of Wisconsin Press.Google Scholar
Siegel, M. & Caflisch, R. E. 2009 Calculation of complex singular solutions to the 3D incompressible Euler equations. Physica D 238, 2368–2379.Google Scholar
Sulem, C., Sulem, P. L. & Frisch, H. 1983 Tracing complex singularities with spectral methods. J. Comput. Phys. 50, 138–161.Google Scholar
Tran, Ch. V. & Dritschel, D. G. 2006 Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit. J. Fluid Mech. 559, 107–116.CrossRefGoogle Scholar