Dynamics of the velocity gradient tensor invariants in isotropic turbulence

1998 ◽  
Vol 10 (9) ◽  
pp. 2336-2346 ◽  
Author(s):  
Jesús Martı́n ◽  
Andrew Ooi ◽  
M. S. Chong ◽  
Julio Soria
Keyword(s):  
Velocity Gradient ◽  
Isotropic Turbulence ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Tensor Invariants

scholarly journals Triple decomposition of velocity gradient tensor in homogeneous isotropic turbulence

2020 ◽  
Vol 198 ◽  
pp. 104389 ◽  
Author(s):  
Ryosuke Nagata ◽  
Tomoaki Watanabe ◽  
Koji Nagata ◽  
Carlos B. da Silva
Keyword(s):  
Velocity Gradient ◽  
Isotropic Turbulence ◽  
Homogeneous Isotropic Turbulence ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor

scholarly journals The Schur decomposition of the velocity gradient tensor for turbulent flows

2018 ◽  
Vol 848 ◽  
pp. 876-905 ◽  
Author(s):  
Christopher J. Keylock
Keyword(s):  
Turbulent Flows ◽  
Velocity Gradient ◽  
Evolution Equations ◽  
Isotropic Turbulence ◽  
Schur Decomposition ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Normal Part ◽  
Normal Term ◽  
Vorticity Vector

The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions.

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