New fluid kinematics

Abstract

Fluid kinematics describes the fluid motion without consideration of any force. Classical fluid kinematics adopts Helmholtz velocity decomposition, which is equivalent to Cauchy-Stokes (CS) velocity gradient tensor decomposition. CS decomposes the velocity gradient tensor into a strain-rate (symmetric) tensor and a vorticity (anti-symmetric) tensor. However, several questions arise: (1) since vorticity cannot represent fluid rotation, the vorticity tensor is a mixture of vorticity shear and rigid rotation, (2) since the strain-rate tensor cannot represent fluid shear, the strain-rate tensor is a mixture of stretching and shear, (3) the stretching and shear in the CS decomposition are dependent on the selection of coordinate system and are therefore not Galilean invariant. On the other hand, Liutex is a new physical quantity to represent the rigid fluid rotation and a principal coordinate system can be set up based on Liutex. A principal decomposition of the velocity gradient tensor, or the rotation-stretching-shear decomposition, can be easily carried out in the principal coordinate system with a clear physical meaning, which represents the rigid rotation, stretching (compression) and shear (symmetric and anti-symmetric shear). In the principal decomposition, all elements in three sub-tensors are Galilean invariant and, therefore, the principal decomposition is unique, Galilean invariant and independent of coordinate system. The principal decomposition is then transformed back to the original xyz coordinate system. The Liutex-based principal decomposition creates the new fluid kinematics which is ready for building up new fluid dynamics. Since fluid kinematics is the foundation of the fluid dynamics, the new fluid kinematics could replace the classical fluid kinematics, Helmholtz or CS decomposition, and open a new gate to develop new fluid dynamics especially for vortex science and turbulence research.