Venkata Kolluru, SS and Pandit, R (2024) Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations. In: Physics of Fluids, 36 (9).
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Abstract
We investigate the complex-time analytic structure of solutions of the three-dimensional (3D)-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou �Potentially singular solutions of the 3D axisymmetric Euler equations,� Proc. Natl. Acad. Sci. U. S. A. 111(36), 12968-12973 (2014), to study a possible finite-time singularity. We use our pseudospectral Fourier-Chebyshev method Kolluru et al., �Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,� Phys. Rev. E 105(6), 065107 (2022), with quadruple-precision arithmetic, to compute the time-Taylor-series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. �Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,� Phys. Rev. E 105(6), 065107 (2022). We then perform asymptotic analysis of the Taylor-series coefficients, by using generalized ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex- t 2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the one-dimensional (1D) wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D Hou-Luo model, for which we use Fourier-pseudospectral methods to compute the time-Taylor-series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of partial differential equations, such as those we consider. © 2024 Author(s).
| Item Type: | Journal Article |
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| Publication: | Physics of Fluids |
| Publisher: | American Institute of Physics |
| Additional Information: | The copyright for this article belongs to the authors. |
| Keywords: | Asymptotic analysis; Chebyshev polynomials; Convergence of numerical methods; Fourier series; Galerkin methods; Holmium alloys; Integral equations; Iterative methods; Korteweg-de Vries equation; Polynomials; Solitons; Taylor series, Analytic structures; Axisymmetric; Euler's equation; Finite time singularity; Incompressible Euler equations; One-dimensional; Pseudospectral; Pseudospectral fourier; Singular solutions; Taylor-series, Euler equations |
| Department/Centre: | Division of Physical & Mathematical Sciences > Physics |
| Date Deposited: | 24 Oct 2024 08:15 |
| Last Modified: | 24 Oct 2024 08:15 |
| URI: | http://eprints.iisc.ac.in/id/eprint/86602 |
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