[ Search | Site Map | Contact ]

Center for Scientific Computation and Mathematical Modeling

Research Activities > Programs > Incompressible Flows 2006> Robert Kerr

Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

CSIC Building (#406), Seminar Room 4122.
Directions: home.cscamm.umd.edu/directions


A New 2D Model For 3D Euler


Robert Kerr

                             Department of Mathematics, University of Warwick

Abstract: Several two-dimensional models have been proposed in recent years that contain aspects of the underlying dynamics of the three-dimensional incompressible Euler equations while being more tractable. This presentation will introduce a new model in this class that is more directly inspired by fully three-dimensional solutions as well as a new conditional restriction upon Euler [Gibbon et al. (2006)] that shows that only certain symmetrical alignments are allowed if there is to be a singularity. The model will be similar to [Gibbon et al. (1999)], but an equation for curvature of vortex lines in the plane instead of stretching out of the plane will be the equation in addition to vorticity. The importance of studying models such as this has been highlighted by a new pseudospectral calculation of collapsing Euler vortices [Hou and Li (2006)] that has called into question the long-term conclusions of singular behavior described earlier in [Kerr (1993), Kerr (2005)]. Because similar differences between well-resolved numerical solutions using similar initial conditions have appeared before, it seems that ultimately numerical solutions cannot resolve this issue and can only act as a guide for analytic work. Guidelines for comparing results from high resolution calculations using different numerical methods will be discussed.


[Gibbon et al. (1999)] J.D. Gibbon, A. Fokas, and C.R. Doering, “Dynamically stretched vortices as solutions of the Navier-Stokes equations”, Physical D 132, 497 (1999). 

[Gibbon et al. (2006)] Gibbon, J.D., D.D. Holm, R.M. Kerr, I Roulstone, “Quaternions and particle dynamics in the Euler fluid equations”, Nonlinearity 19, 1969 (2006).

[Hou and Li (2006)] T.Y. Hou and R. Li, “Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations”, Accepted J. Nonlin. Sci. (2006).

[Kerr (1993)] R.M. Kerr, “Evidence for a singularity of the threedimensional, incompressible Euler equations”, Phys. Fluids 5, 172 (1993).

[Kerr (2005)] R.M. Kerr, “Velocity and scaling of collapsing Euler vortices”, Phys. Fluids 17, 075103 (2005).