Periodic Navier Stokes Equations for a 3D Incompressible Fluid with Liutex Vortex Identification Method

Abstract

The Incompressible Navier-Stokes Equations (NSEs) are on the list of Millennium Problems, to prove their existence and uniqueness of solutions. The NSEs can be formulated in a periodic 3D domain, where they are termed the Periodic Navier Stokes (PNS) Equations, and can be treated on a subspace spanning a 3-dimensional torus, or T3. Treating the PNS Equations in T3-space, this article demonstrates that a decaying of turbulence occurs in the 3D case for the z component of velocity when non-smooth initial conditions are considered for x, y components of velocity and that ‘vorticity’ sheets in the small scales of 3D turbulence dominate the flow to the extent that non-smooth temporal solutions exist for the z velocity for smooth initial data for the x, y components of velocity. Unlike the Navier-Stokes equations, which have no anti-symmetric vorticity tensor, there are new governing equations which have vorticity tensor and can be decomposed into a rotational part(Liutex), antisymmetric shear and compression and stretching. It is shown that under these recent findings, that there is a strong correlation between vorticity and vorticies for (PNS).