Finite Time Euler Singularities: a Lagrangian Perspective

• Applicant: Professor Dr. Rainer Grauer
• Subject Area: Fluid Mechanics ; Mathematics
• Term: from 2008 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 71493972

• Final Report Year: 2012

Final Report Abstract

The question of finite time singularities in the incompressible Euler equations was investigated. The main result obtained from a combination of geometric arguments and numerical simulations is that in flows with highly symmetric initial conditions no finite time singularities can occur. The analysis using a combined knowledge of geometry and vortex stretching is much more robust compared to other types of criteria. This is a major result and improvement in the search for finite time singularities. A second result of this project is the important observation that the assumption of isotropy for the deviations of a smoothed trajectory from the turbulent one is not isotropic. Based on this numerical observation, a new modified Lagrangian α-model was introduced which does not suffer from the insufficiency’s of the original one and produces an energy transfer that reduces the formation of “rigid bodies” in the turbulent flows.

Publications

• “Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods”, Physica D 237 (2008), 1932–1936
  Tobias Grafke, Holger Homann, Jürgen Dreher, Rainer Grauer
  
• “Global Solutions and Dynamics for a modified Navier- Stokes Equation”, DPG-Jahrestagung (2010), Regensburg
  Tobias Grafke, Rainer Grauer
  
• “High resolution simulations of the incompressible 3-D Euler equations: A Lagrangian perspective”, University of Minnesota (2010), IMA workshop “Analysis and Computation of Incompressible Fluid Flow”
  Tobias Grafke, Rainer Grauer
  
• “Turbulence properties and global regularity for a modified Navier-Stokes equation”, University of New Hampshire (2011)
  Tobias Grafke, Rainer Grauer
  
• “Finite time Euler singularities: A Lagrangian perspective”, Les Houches (2012), workshop “New Challenges in Turbulence Research II”
  Tobias Grafke, Rainer Grauer
  
• “Three short Stories in Turbulence and Singularities”, Les Houches (2012), workshop “New Challenges in Turbulence Research II”
  Rainer Grauer, Tobias Grafke