Now showing 1 - 2 of 2
  • Publication
    Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions
    (Cambridge University Press, 2015) ; ;
    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.
      314Scopus© Citations 3
  • Publication
    Assessing late-time singular behaviour in models of three dimensional Euler fluid flow
    (University College Dublin. School of Mathematics & Statistics, 2016)
    The open question of regularity of the fluid dynamical equations is considered one of the most fundamental challenges of mathematics and physics [C. L. Fefferman. Existence and smoothness of the Navier-Stokes equation. The millennium prize problems, pages 57-67 (2000)]. While the viscous Navier-Stokes equations have more physical relevance, the inviscid Euler equations present the greatest challenge and exhibit the most extreme behaviours. For this reason, the numerical study of possible finite-time blowup is typically concerned with these inviscid equations. Extensive numerical assessment of finite-time blow up of 3D Euler has been carried out, albeit with conflicting yes and no conclusions with regard to the existence of finite time singularity. The fundamental difficulty of this important problem is the lack of analytic solutions or any a priori knowledge of asymptotic behaviour. A secondary obstacle is that the spatial collapse associated with intense vortex stretching results in numerical solutions becoming unresolved beyond a certain time. It is therefore imperative to devise a framework with nontrivial blowup dynamics and where analytic solutions are known in order to validate and compare various numerical methods, for the purposes of accurately solving the system and diagnosing blowup. In this regard, I have proposed investigating the issue of Euler finite-time blowup using a novel approach where the original system of equations is bijectively transformed to a new mapped system which is globally regular in time [M. D. Bustamante. 3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis. Physica D: Nonlinear Phenomena, 240(13):1092-1099 (2011)]. Since no known analytical solution for the full 3D Euler equations exist, I have studied the robustness of the proposed novel approach using the one-dimensional Burgers equation and a proposed new one-parameter family of models of the 3D Euler equations on a 2D symmetry plane. The proposed 2D symmetry plane model equations were motivated by the work on stagnation-point-type exact solution of 3D Euler equations by Gibbon et al. [J. Gibbon, A. Fokas, and C. Doering. Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations. Physica D: Nonlinear Phenomena, 132(4):497-510 (1999)]. I have shown that the mapped system’s numerical solution leads to more accurate estimates of the blowup quantities compared with the original system. I also established that only by using the mapped system can certain late-time behaviours be observed and asymptotic trends be established.