New fluid kinematics

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.

Speculation of fluid dynamics equations based on Liutex theory and constitutive relation of symmetric shearing deformation

The fluid kinematics of Liutex decomposes a velocity gradient tensor(VGT)of∇v into four components,including rotation(R),stretching/compressing(SC),anti-symmetric shear(Santi-sym)and symmetric shear(Ssym),as oppose to the traditional Cauchy-Stokes decomposition where a VGT was decomposed into the strain rate and vorticity tensors.The current study limpidly clarified the physical meanings of these deformations in the newly-proposed decomposition from the perspectives of both fluid kinematics and dynamics.With in-depth understanding the physical connotations of these deformations,the present study further suggests that the Ssym be the only deformation appropriately correlated to the stress tensor,leading to the establishment of a new constitutive relation for Newtonian fluids with the modified model assumptions originated from Stokes in 1845.Moreover,the present research finds that the“principal decomposition”proposed by Liu is not mathematically unique when a VGT has three real eigenvalues(TR).Within the context,a new decomposition method is introduced to avoid the non-uniqueness issue arising from using the principal decomposition to establish fluid dynamics equations.Based on the modified Stokes assumptions and the novel VGT decomposition method,a set of new fluid dynamics momentum equations are obtained for Newtonian fluid.The added stress tensor of Fadd is identified as the key difference between the newly-derived governing equations and the conventional Navier-Stokes(N-S)equations,which is caused by excluding the SC correlation to the stress tensor in the new constitutive equation.Finally,a preliminary analysis of Fadd is conducted using the existing channel turbulence direct numerical simulations(DNS)data based on the traditional N-S equations.The Fadd is found widely existing in turbulence and is of the same order of magnitude with the other force terms.Therefore,the Fadd is expected to have some nonnegligible effects on altering the current DNS data based on the traditional N-S equations,which will be further verified by performing the“DNS”simulation using the newly-derived fluid dynamics equations in near future.

Uniform decomposition of velocity gradient tensor

In this paper,the principal decomposition of the velocity gradient tensor[∇v]is discussed in 3 cases based on the discriminant∆:∆<0 with 1 real eigen value and a pair of conjugate complex eigen values,∆>0 with 3 distinct real eigen values,and∆=0 with 1 or 2 distinct real eigen values.The velocity gradient tensor can also be classified as rotation point,which can be decomposed into three parts,i.e.,rotation[R],shear[S]and stretching/compression[SC],and non-rotation point,we defined a new resistance term[L],and the tensor can be decomposed into three parts,i.e.,resistance[L],shear[S]and stretching/compression[SC].Example matric are also displayed to demonstrate the new decomposition.Connections of principal decomposition between 3 different cases,and between Resistance and Liutex will also be discussed.

New ideas on governing equations of fluid dynamics

Classical fluid kinematics or Cauchy-Stokes decomposition mistreated vorticity as fluid rotation and mixed flow stretching with shearing.Classical fluid dynamics or Navier-Stokes(N-S)equations are based on the classical kinematics and treated vorticity as null in contribution of forces and mixed the stretching force with the shearing force,which is not consistent with the Galilean invariancy.N-S equations also neglect the flow rotation.It is believed that N-S equations may work for incompressible and laminar flow but are not satisfied for turbulent flow and compressible flow especially for high-speed flow.Based on Liutex,new fluid kinematics has been established by Liu in 2021,which gives a Liutex-based principal coordinate system and a new principal decomposition in that system,which has been transferred back to the original Cartesian coordinate system.The principal decomposition of velocity gradient tensor has four parts which are called rotation,stretching,anti-symmetric shearing and symmetric shearing.Four forces are derived according to the four parts of the velocity gradient tensor.According to the new fluid kinematics,it is reported in this letter that based on the principal decomposition,a new relation between the velocity gradient tensor and stress-rate tensor has been established to form a new fluid dynamics equation to govern fluid flow.The new governing equation may be applicable to both laminar flow and turbulent flow,and both incompressible flow and compressible flow including high-speed flow for reasonable results with reasonable grids.Further numerical experiment is needed to verify.