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.

scholarly journals Evolution of the velocity gradient tensor invariant dynamics in a turbulent boundary layer

2017 ◽  
Vol 815 ◽  
pp. 223-242 ◽  
Author(s):  
P. Bechlars ◽  
R. D. Sandberg
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Velocity Gradient ◽  
Evolution Equations ◽  
Turbulence Models ◽  
Turbulent Structures ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
The Mean ◽  
The Individual

In order to improve the physical understanding of the development of turbulent structures, the compressible evolution equations for the first three invariants $P$, $Q$ and $R$ of the velocity gradient tensor have been derived. The mean evolution of characteristic turbulent structure types in the $QR$-space were studied and compared at different wall-normal locations of a compressible turbulent boundary layer. The evolution of these structure types is fundamental to the physics that needs to be captured by turbulence models. Significant variations of the mean evolution are found across the boundary layer. The key features of the changes of the mean trajectories in the invariant phase space are highlighted and the consequences of the changes are discussed. Further, the individual elements of the overall evolution are studied separately to identify the causes that lead to the evolution varying with the distance to the wall. Significant impact of the wall-normal location on the coupling between the pressure-Hessian tensor and the velocity gradient tensor was found. The highlighted features are crucial for the development of more universal future turbulence models.

scholarly journals Invariants of the velocity-gradient tensor in a spatially developing inhomogeneous turbulent flow

2017 ◽  
Vol 817 ◽  
pp. 1-20 ◽  
Author(s):  
O. R. H. Buxton ◽  
M. Breda ◽  
X. Chen
Keyword(s):  
Turbulent Flows ◽  
Velocity Gradient ◽  
Joint Probability ◽  
Shear Layers ◽  
Velocity Fields ◽  
Gradient Tensor ◽  
Drop Shape ◽  
Velocity Gradient Tensor ◽  
Flow Physics ◽  
Joint Probability Density

Tomographic particle image velocimetry experiments were performed in the near field of the turbulent flow past a square cylinder. A classical Reynolds decomposition was performed on the resulting velocity fields into a time invariant mean flow and a fluctuating velocity field. This fluctuating velocity field was then further decomposed into coherent and residual/stochastic fluctuations. The statistical distributions of the second and third invariants of the velocity-gradient tensor were then computed at various streamwise locations, along the centreline of the flow and within the shear layers. These invariants were calculated from both the Reynolds-decomposed fluctuating velocity fields and the coherent and stochastic fluctuating velocity fields. The range of spatial locations probed incorporates regions of contrasting flow physics, including a mean recirculation region and separated shear layers, both upstream and downstream of the location of peak turbulence intensity along the centreline. These different flow physics are also reflected in the velocity gradients themselves with different topologies, as characterised by the statistical distributions of the constituent enstrophy and strain-rate invariants, for the three different fluctuating velocity fields. Despite these differing flow physics the ubiquitous self-similar ‘tear drop’-shaped joint probability density function between the second and third invariants of the velocity-gradient tensor is observed along the centreline and shear layer when calculated from both the Reynolds decomposed and the stochastic velocity fluctuations. These ‘tear drop’-shaped joint probability density functions are not, however, observed when calculated from the coherent velocity fluctuations. This ‘tear drop’ shape is classically associated with the statistical distribution of the velocity-gradient tensor invariants in fully developed turbulent flows in which there is no coherent dynamics present, and hence spectral peaks at low wavenumbers. The results presented in this manuscript, however, show that such ‘tear drops’ also exist in spatially developing inhomogeneous turbulent flows. This suggests that the ‘tear drop’ shape may not just be a universal feature of fully developed turbulence but of turbulent flows in general.

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

Effect of the eigenvalues of the velocity gradient tensor on particle collisions

2016 ◽  
Vol 792 ◽  
pp. 36-49 ◽  
Author(s):  
Vincent E. Perrin ◽  
Harmen J. J. Jonker
Keyword(s):  
Turbulent Flows ◽  
Velocity Gradient ◽  
Stokes Number ◽  
Negative Eigenvalue ◽  
Collision Kernel ◽  
Flow Structures ◽  
Particle Collisions ◽  
Gradient Tensor ◽  
Local Flow ◽  
Velocity Gradient Tensor

This study uses the eigenvalues of the local velocity gradient tensor to categorize the local flow structures in incompressible turbulent flows into different types of saddle nodes and vortices and investigates their effect on the local collision kernel of heavy particles. Direct numerical simulation (DNS) results show that most of the collisions occur in converging regions with real and negative eigenvalues. Those regions are associated not only with a stronger preferential clustering of particles, but also with a relatively higher collision kernel. To better understand the DNS results, a conceptual framework is developed to compute the collision kernel of individual flow structures. Converging regions, where two out of three eigenvalues are negative, posses a very high collision kernel, as long as a critical amount of rotation is not exceeded. Diverging regions, where two out of three eigenvalues are positive, have a very low collision kernel, which is governed by the third and negative eigenvalue. This model is not suited for particles with Stokes number $St\gg 1$, where the contribution of particle collisions from caustics is dominant.

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