{
  "id": "2012.14182",
  "version": "v1",
  "published": "2020-12-28T10:51:58.000Z",
  "updated": "2020-12-28T10:51:58.000Z",
  "title": "A pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation: tygers and thermalization",
  "authors": [
    "Sai Swetha Venkata Kolluru",
    "Puneet Sharma",
    "Rahul Pandit"
  ],
  "comment": "26 pages. 14 figures",
  "categories": [
    "physics.flu-dyn",
    "cond-mat.stat-mech"
  ],
  "abstract": "G.Luo and T.Y.Hou (Proceedings of the National Academy of Sciences 111.36 (2014): 12968-12973) have investigated a potentially singular solution of the radially bounded, 3D, axisymmetric Euler equations via a hybrid, 6th-order Galerkin and 6th-order finite-difference method on an adaptive (moving) mesh. Given the importance of this problem, we have developed a pseudospectral, Fourier-Chebyshev scheme to study this solution by an independent numerical technique, which leads to new insights into this (potentially) singular solution: (a) the time of singularity is preceded, in any, practical, spectrally truncated direct numerical simulation (DNS), by the formation of structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) these tygers herald the thermalization of our spectrally truncated system of equations. We demonstrate how we can analyse this (potential) singularity by using singularity-detection criteria based on (a) the Beale-Kato-Majda theorem and (b) a generalization of the analyticity-strip method. We find that the width of the analyticity strip, in the axial direction, decays faster than an exponential and, in the radial direction, it decays linearly; this linear decay can be extrapolated to obtain an estimate for the time at which the (potential) singularity occurs; our estimate is consistent with that of G.Luo and T.Y.Hou. We examine the DNS-resolution dependence of (a) the tyger-formation time, (b) thermalization, (c) energy and helicity conservation, and (d) our estimate for the singularity time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-28T10:51:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "three-dimensional axisymmetric incompressible euler equation",
      "potentially singular solution",
      "pseudospectral study",
      "thermalization",
      "truncated direct numerical simulation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}