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Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions

Published online by Cambridge University Press:  20 April 2015

Rachel M. Mulungye ,
Dan Lucas  and
Miguel D. Bustamante
Rachel M. Mulungye
Affiliation:
Complex and Adaptive Systems Laboratory, School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Dan Lucas
Affiliation:
Complex and Adaptive Systems Laboratory, School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Miguel D. Bustamante*
Affiliation:
Complex and Adaptive Systems Laboratory, School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
*
†Email address for correspondence: [email protected]
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Abstract

Motivated by the work on stagnation-point-type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (Physica D, vol. 132 (4), 1999, pp. 497–510) and the subsequent demonstration of finite-time blowup by Constantin (Int. Math. Res. Not. IMRN, vol. 9, 2000, pp. 455–465) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields: vorticity and stretching rate of vorticity. In particular, the value of the model’s parameter determines whether or not there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model’s parameter. We use a representative of this family of models, whose solution blows up at a finite time, as a benchmark for the systematic study of errors in numerical simulations. Using a high-order pseudospectral method, we compare the numerical integration of our ‘original’ model equations against a ‘mapped’ version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations. The mapping can be constructed explicitly whenever a Beale–Kato–Majda type of theorem is available therefore it is applicable to the 3D Euler equations (Bustamante, Physica D, vol. 240 (13), 2011, pp. 1092–1099). We show that the mapped system’s numerical solution leads to more accurate (by three orders of magnitude) estimates of supremum norms and singularity time compared with the original system. The numerical integration of the mapped equations is demonstrated to entail only a small extra computational cost. We study the Fourier spectrum of the model’s numerical solution and find that the analyticity strip width (a measure of the solution’s analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the fields’ supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet, Phys. Rev. E, vol. 86 (6), 2012, 066302) between the analyticity-strip method and the Beale–Kato–Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions to the 3D Euler fluid equations.

JFM classification

Mathematical Foundations: Computational methods Mathematical Foundations: Mathematical Foundations Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 771 , 25 May 2015 , pp. 468 - 502
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Copyright
© 2015 Cambridge University Press 

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References

Ayala, D. & Protas, B. 2014 Maximum palinstrophy growth in 2D incompressible flows. J. Fluid Mech. 742, 340–367.CrossRefGoogle Scholar
Bardos, C. & Titi, E. S. 2007 Euler equations of incompressible ideal fluids. Russ. Math. Surv. 62, 409–451.Google Scholar
Beale, J. T., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3-d Euler equations. Commun. Math. Phys. 94, 61–66.Google Scholar
de Boor, C. 1978 A Practical Guide to Splines. Springer.Google Scholar
Brachet, M. E., Bustamante, M. D., Krstulovic, G., Mininni, P. D., Pouquet, A. & Rosenberg, D. 2013 Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor–Green symmetries. Phys. Rev. E 87 (1), 013110.Google Scholar
Bustamante, M. D. 2011 3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis. Physica D 240 (13), 1092–1099.CrossRefGoogle 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 (6), 066302.Google Scholar
Bustamante, M. D. & Kerr, R. M. 2008 3D Euler about a 2D symmetry plane. Physica D 237 (14–17), 1912–1920.Google Scholar
Cantwell, B. J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4 (4), 782–793.Google Scholar
Constantin, P. 2000 The Euler equations and nonlocal conservative Riccati equations. Int. Math. Res. Not. IMRN 9, 455–465.Google Scholar
Cottet, G.-H. & Koumoutsakos, P. D. 2000 Vortex Methods. Cambridge University Press.Google Scholar
Cottet, G.-H., Salihi, M. L. O. & El Hamraoui, M. 1999 Multi-purpose regridding in vortex methods. ESAIM Proc. 94–103.Google Scholar
Deng, J. H., Thomas, Y. & Yu, X. 2005 Geometric properties and nonblowup of 3D incompressible Euler flow. Commun. Part. Diff. Equ. 30 (1–2), 225–243.CrossRefGoogle Scholar
Donzis, D. A., Gibbon, J. D., Gupta, A., Kerr, R. M., Pandit, R. & Vincenzi, D. 2013 Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations. J. Fluid Mech. 732, 316–331.Google Scholar
Euler, L. 1761 Principia motus fluidorum. Novi Commentarii Acad. Sci. Petropolitanae 6, 271–311.Google Scholar
Frisch, U. & Villone, B. 2014 Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow. Eur. Phys. J. H 39 (3), 325–351.Google Scholar
Gibbon, J. D. 2008 The three-dimensional Euler equations: where do we stand? Physica D 237, 1894–1904.CrossRefGoogle Scholar
Gibbon, J. D. 2013 Dynamics of scaled norms of vorticity for the three-dimensional Navier–Stokes and Euler equations. Procedia IUTAM 7, 39–48.CrossRefGoogle Scholar
Gibbon, J. D., Fokas, A. S. & Doering, C. R. 1999 Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations. Physica D 132 (4), 497–510.Google Scholar
Gibbon, J. D., Moore, D. R. & Stuart, J. T. 2003 Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations. Nonlinearity 16 (5), 1823–1831.Google Scholar
Gibbon, J. D. & Ohkitani, K. 2001 Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics. Nonlinearity 14 (5), 1239–1264.CrossRefGoogle Scholar
Grafke, T. & Grauer, R. 2013 Finite-time Euler singularities: a Lagrangian perspective. Appl. Math. Lett. 26 (4), 500–505.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 (14), 1932–1936.Google Scholar
Hou, T. Y. & Li, R. 2006 Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16 (6), 639–664.CrossRefGoogle Scholar
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226 (1), 379–397.Google Scholar
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids 5, 1725–1746.CrossRefGoogle Scholar
Kerr, R. M. 2013 Bounds for Euler from vorticity moments and line divergence. J. Fluid Mech. 729, R2.Google Scholar
Kiselev, A. & Zlatos, A.Blow up for the 2D Euler equation on some bounded domains. Preprint 2014, arXiv:1406.3648v1 math.AP.Google Scholar
Knott, G. D. 2000 Interpolating Cubic Splines. Birkhäuser.Google Scholar
Kolmogorov, A. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Koumoutsakos, P. 2005 Multiscale flow simulations using particles. Ann. Rev. 37, 457–487.Google Scholar
Kuznetsov, E. A. 2006 Vortex line representation for the hydrodynamic type equations. J. Nonlinear Math. Phys. 13 (1), 64–80.CrossRefGoogle Scholar
Kuznetsov, E. A. & Ruban, V. P. 2000 Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems. Phys. Rev. E 61 (1), 831–841.CrossRefGoogle ScholarPubMed
Li, D. & Rodrigo, J. 2009 Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation. Commun. Math. Phys. 286 (1), 111–124.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 (6), 2693–2728.CrossRefGoogle Scholar
Luo, G. & Hou, T. Y. 2014 Potentially singular solutions of the 3D axisymmetric Euler equations. Proc. Natl Acad. Sci. USA 111 (36), 12968–12973.Google Scholar
Mailybaev, A. A. 2013 Blowup as a driving mechanism of turbulence in shell models. Phys. Rev. E 87, 053011.Google Scholar
Monaghan, J. J. 1985 Extrapolating B splines for interpolation. J. Comput. Phys. 60 (2), 253–262.CrossRefGoogle Scholar
Ohkitani, K. & Gibbon, J. D. 2000 Numerical study of singularity formation in a class of Euler and Navier–Stokes flows. Phys. Fluids 12 (12), 3181–3194.Google Scholar
Orlandi, P., Pirozzoli, S., Bernardini, M. & Carnevale, G. F. 2014 A minimal flow unit for the study of turbulence with passive scalars. J. Turbulence 15 (11), 731–751.CrossRefGoogle Scholar
Perlin, M. & Bustamante, M. D.A robust quantitative comparison criterion of two signals based on the Sobolev norm of their difference. Preprint 2014, arXiv:1412.6977.Google Scholar
Sulem, C., Sulem, P. L. & Frisch, H. 1983 Tracing complex singularities with spectral methods. J. Comput. Phys. 50 (1), 138–161.Google Scholar
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125 (1), 150–162.CrossRefGoogle Scholar