Principal coordinates and principal velocity gradient tensor decomposition

Helmholtz velocity decomposition and Cauchy-Stokes tensor decomposition have been widely accepted as the foundation of fluid kinematics for a long time.However,there are some problems with these decompositions which cannot be ignored.Firstly,Cauchy-Stokes decomposition itself is not Galilean invariant which means under different coordinates,the stretching(compression)and deformation are quite different.Another problem is that the anti-symmetric part of the velocity gradient tensor is not the proper quantity to represent fluid rotation.To show these two drawbacks,two counterexamples are given in this paper.Then“principal coordinate”and“principal decomposition”are introduced to solve the problems of Helmholtz decomposition.An easy way is given to find the Principal decomposition which has the property of Galilean invariance.

Stretching and shearing contamination analysis for Liutex and other vortex identification methods

The newly developed vortex-identification method,Liutex,has provided a new systematic description of the local fluid rotation,which includes scalar,vector,and tensor forms.However,the advantages of Liutex over the other widely used vortexidentification methods such as Q,Δ,λ2,andλci have not been realized.These traditional methods count on shearing and stretching as a part of vortex strength.But,in the real flow,shearing and stretching do not contribute to fluid rotation.In this paper,the decomposition of the velocity gradient tensor is conducted in the Principal Coordinate for uniqueness.Then the contamination effects of stretching and shearing of the traditional methods are investigated and compared with the Liutex method in terms of mathematical analysis and numerical calculations.The results show that the Liutex method is the only method that is not affected by either stretching or shear,as it represents only the local fluid rigid rotation.These results provide supporting evidence that Liutex is the superior method over others.

Correlation analysis among vorticity,Q method and Liutex

Influenced by the fact that vorticity represents rotation for rigid body,people believe this idea also works for fluid flow.However,the vortex predictions by vorticity do not match experimental results,which drove scientists to look for more appropriate methods to identify vortex.All vortex identification methods can be categorized into three generations.The vorticity-based method is classified as the first generation.Methods relying on eigenvalues of velocity gradient tensor are considered as the second generation.People still believe vorticity is vortex since vorticity theory looks correct in mathematics,but all other methods are only scalars and unable to indicate the swirl direction.Recently,a new vortex identification method called Liutex is innovated.It is regarded as the third-generation method,not only overcoming all previous methods’drawbacks but also having a clear physical meaning.The direction of Liutex represents the swirl axis of rotation,and its strength is equal to twice the angular speed.In this paper,we did a correlation analysis between vorticity,Q,λ_(ci),λ_(2)methods and Liutex based on a direct numerical simulation(DNS)case of boundary layer transition.The results show that the correlation between vorticity and Liutex is very small or even negative in strong shear regions,which demonstrates that using vorticity to detect vortex lacks scientific foundation and vorticity is not appropriate to represent vortex.The correlation analysis also shows that the second generation is contaminated too by shear and thus is not accurate to identify the vortex structure.

Correction to: Stretching and shearing contamination analysis for Liutex and other vortex identification methods

Correction to:Adv Aerodyn 3,8(2021)https://doi.org/10.1186/s42774-020-00060-9 After publication of this article[1],it is noticed letter‘a’is missing from the below sentence in section 6.2‘Adding stretching components’:The subscript“”means“adding”the stretching components.The correct sentence should be。