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Euler equation existence, non-uniqueness and mesh converged statistics.

James Glimm1, David H Sharp2, Hyunkyung Lim3

  • 1Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794-3600, USA Computational Science Center, Brookhaven National Laboratory, Upton, NY 11793-6000, USA.

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This study examines the Euler equation for fluid flow, highlighting non-uniqueness issues that challenge accurate simulations. A strategy to mitigate these problems is presented and illustrated with turbulent flow data.

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Area of Science:

  • Fluid dynamics
  • Mathematical physics

Background:

  • The Euler equation governs inviscid fluid flow.
  • Existence and uniqueness of solutions are fundamental for physical modeling.
  • Non-uniqueness poses challenges for computational fluid dynamics (CFD) simulations.

Purpose of the Study:

  • To review existence and non-uniqueness results for the Euler equation.
  • To contextualize these mathematical findings within physical fluid flow models.
  • To propose and illustrate a strategy for mitigating non-uniqueness in simulations.

Main Methods:

  • Literature review of theoretical results on the Euler equation.
  • Analysis of mathematical solutions in the context of physical models.
  • Examination of mesh-converged turbulent statistics.
  • Comparison of simulation results with laboratory experiments.

Main Results:

  • Non-uniqueness of solutions is a significant theoretical challenge.
  • This non-uniqueness directly conflicts with the requirements for reliable CFD simulations.
  • A strategy to address non-uniqueness is outlined and demonstrated.

Conclusions:

  • Understanding non-uniqueness is crucial for accurate fluid flow modeling.
  • Mitigation strategies are necessary for practical application of CFD.
  • Validation against experimental data is essential for confirming simulation reliability.