Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition [15] [Phys. Fluids 5 (1993), 1725]. We use a pseudo-spectral method with resolution up to 1536 × 1024 × 3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations [15], [22]. The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.

[1]  Steven A. Orszag,et al.  Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes , 1993 .

[2]  Robert McDougall Kerr.  The outer regions in singular Euler , 1999 .

[3]  Russel E. Caflisch,et al.  Singularity formation for complex solutions of the 3D incompressible Euler equations , 1993 .

[4]  T. Hou,et al.  Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation , 2006 .

[5]  Richard B. Pelz,et al.  DIRECT NUMERICAL SIMULATION OF TRANSITION TO TURBULENCE FROM A HIGH-SYMMETRY INITIAL CONDITION , 1994 .

[6]  Robert McDougall Kerr.  Evidence for a Singularity of the Three Dimensional, Incompressible Euler Equations , 1993 .

[7]  Richard B. Pelz,et al.  Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications , 1992 .

[8]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[9]  Peter Constantin,et al.  Geometric Statistics in Turbulence , 1994, SIAM Rev..

[10]  F. Hussain,et al.  Simulation of vortex reconnection , 1989 .

[11]  Norman J. Zabusky,et al.  Vortex intensification and collapse of the Lissajous-elliptic ring: single- and multi-filament Biot-Savart simulations and visiometrics , 1995, Journal of Fluid Mechanics.

[12]  F. Hussain,et al.  Cross‐linking of two antiparallel vortex tubes , 1989 .

[13]  Jerrold E. Marsden,et al.  Diffeomorphism groups, hydrodynamics and relativity , 1972 .

[14]  Richard B. Pelz,et al.  Locally self-similar finite time collapse in a high-symmetry vortex filament model , 1997 .

[15]  E. Siggia,et al.  Collapsing solutions to the 3‐D Euler equations , 1990 .

[16]  Robert McDougall Kerr.  Velocity and scaling of collapsing Euler vortices , 2005 .

[17]  Grauer,et al.  Numerical computation of 3D incompressible ideal fluids with swirl. , 1991, Physical review letters.

[18]  The evolution of a turbulent vortex , 1982 .

[19]  C. Fefferman,et al.  Geometric constraints on potentially singular solutions for the 3-D Euler equations , 1996 .

[20]  Kai Germaschewski,et al.  ADAPTIVE MESH REFINEMENT FOR SINGULAR SOLUTIONS OF THE INCOMPRESSIBLE EULER EQUATIONS , 1998 .

[21]  C. Greengard,et al.  The vortex ring merger problem at infinite reynolds number , 1989 .

[22]  T. Hou,et al.  Geometric Properties and Nonblowup of 3D Incompressible Euler Flow , 2004, math-ph/0402032.

[23]  Numerical scaling analysis of the small-scale structure in turbulence , 2005, math/0509027.

[24]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .