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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

  • Rachel M. Mulungye (a1), Dan Lucas (a1) and Miguel D. Bustamante (a1)


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.


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Ayala, D. & Protas, B. 2014 Maximum palinstrophy growth in 2D incompressible flows. J. Fluid Mech. 742, 340367.
Bardos, C. & Titi, E. S. 2007 Euler equations of incompressible ideal fluids. Russ. Math. Surv. 62, 409451.
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, 6166.
de Boor, C. 1978 A Practical Guide to Splines. Springer.
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.
Bustamante, M. D. 2011 3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis. Physica D 240 (13), 10921099.
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.
Bustamante, M. D. & Kerr, R. M. 2008 3D Euler about a 2D symmetry plane. Physica D 237 (14–17), 19121920.
Cantwell, B. J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4 (4), 782793.
Constantin, P. 2000 The Euler equations and nonlocal conservative Riccati equations. Int. Math. Res. Not. IMRN 9, 455465.
Cottet, G.-H. & Koumoutsakos, P. D. 2000 Vortex Methods. Cambridge University Press.
Cottet, G.-H., Salihi, M. L. O. & El Hamraoui, M. 1999 Multi-purpose regridding in vortex methods. ESAIM Proc. 94103.
Deng, J. H., Thomas, Y. & Yu, X. 2005 Geometric properties and nonblowup of 3D incompressible Euler flow. Commun. Part. Diff. Equ. 30 (1–2), 225243.
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, 316331.
Euler, L. 1761 Principia motus fluidorum. Novi Commentarii Acad. Sci. Petropolitanae 6, 271311.
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), 325351.
Gibbon, J. D. 2008 The three-dimensional Euler equations: where do we stand? Physica D 237, 18941904.
Gibbon, J. D. 2013 Dynamics of scaled norms of vorticity for the three-dimensional Navier–Stokes and Euler equations. Procedia IUTAM 7, 3948.
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), 497510.
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), 18231831.
Gibbon, J. D. & Ohkitani, K. 2001 Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics. Nonlinearity 14 (5), 12391264.
Grafke, T. & Grauer, R. 2013 Finite-time Euler singularities: a Lagrangian perspective. Appl. Math. Lett. 26 (4), 500505.
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), 19321936.
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), 639664.
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226 (1), 379397.
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids 5, 17251746.
Kerr, R. M. 2013 Bounds for Euler from vorticity moments and line divergence. J. Fluid Mech. 729, R2.
Kiselev, A. & Zlatos, A.Blow up for the 2D Euler equation on some bounded domains. Preprint 2014, arXiv:1406.3648v1 math.AP.
Knott, G. D. 2000 Interpolating Cubic Splines. Birkhäuser.
Kolmogorov, A. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Dokl. Akad. Nauk SSSR 30, 301305.
Koumoutsakos, P. 2005 Multiscale flow simulations using particles. Ann. Rev. 37, 457487.
Kuznetsov, E. A. 2006 Vortex line representation for the hydrodynamic type equations. J. Nonlinear Math. Phys. 13 (1), 6480.
Kuznetsov, E. A. & Ruban, V. P. 2000 Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems. Phys. Rev. E 61 (1), 831841.
Li, D. & Rodrigo, J. 2009 Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation. Commun. Math. Phys. 286 (1), 111124.
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), 26932728.
Luo, G. & Hou, T. Y. 2014 Potentially singular solutions of the 3D axisymmetric Euler equations. Proc. Natl Acad. Sci. USA 111 (36), 1296812973.
Mailybaev, A. A. 2013 Blowup as a driving mechanism of turbulence in shell models. Phys. Rev. E 87, 053011.
Monaghan, J. J. 1985 Extrapolating B splines for interpolation. J. Comput. Phys. 60 (2), 253262.
Ohkitani, K. & Gibbon, J. D. 2000 Numerical study of singularity formation in a class of Euler and Navier–Stokes flows. Phys. Fluids 12 (12), 31813194.
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), 731751.
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.
Sulem, C., Sulem, P. L. & Frisch, H. 1983 Tracing complex singularities with spectral methods. J. Comput. Phys. 50 (1), 138161.
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125 (1), 150162.
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