scholarly journals Scalar imaging velocimetry measurements of the velocity gradient tensor field in turbulent flows. I. Assessment of errors

1996 ◽  
Vol 8 (7) ◽  
pp. 1869-1882 ◽  
Author(s):  
Lester K. Su ◽  
Werner J. A. Dahm
Keyword(s):  
Turbulent Flows ◽  
Velocity Gradient ◽  
Tensor Field ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor

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.

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.

Invariants of the reduced velocity gradient tensor in turbulent flows

2013 ◽  
Vol 716 ◽  
pp. 597-615 ◽  
Author(s):  
J. I. Cardesa ◽  
D. Mistry ◽  
L. Gan ◽  
J. R. Dawson
Keyword(s):  
Strain Rate ◽  
Turbulent Flows ◽  
Velocity Gradient ◽  
Three Dimensional ◽  
Joint Probability ◽  
Strain Rate Tensor ◽  
Gradient Tensor ◽  
Local Isotropy ◽  
Velocity Gradient Tensor ◽  
Asymmetric Shape

AbstractIn this paper we examine the invariants $p$ and $q$ of the reduced $2\times 2$ velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of $p$ and $q$ exhibit a common characteristic asymmetric shape consistent with $\langle pq\rangle \lt 0$. An explanation for this inequality is proposed. Assuming local homogeneity we derive $\langle p\rangle = 0$ and $\langle q\rangle = 0$. With the addition of local isotropy the sign of $\langle pq\rangle $ is proved to be the same as that of the skewness of $\partial {u}_{1} / \partial {x}_{1} $, hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of $p{{\ndash}}q$ stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of $Q{{\ndash}}R$ obtained from the full $3\times 3$ VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full $3\times 3$ strain-rate tensor.

STATISTICS OF THE VELOCITY GRADIENT TENSOR IN SPACE PLASMA TURBULENT FLOWS

2015 ◽  
Vol 812 (1) ◽  
pp. 84 ◽  
Author(s):  
Giuseppe Consolini ◽  
Massimo Materassi ◽  
Maria Federica Marcucci ◽  
Giuseppe Pallocchia
Keyword(s):  
Turbulent Flows ◽  
Velocity Gradient ◽  
Space Plasma ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor

scholarly journals Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence

2014 ◽  
Vol 754 ◽  
pp. 184-207 ◽  
Author(s):  
L. Biferale ◽  
C. Meneveau ◽  
R. Verzicco
Keyword(s):  
Turbulent Flows ◽  
Velocity Gradient ◽  
Large Deviation ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Time Histories ◽  
Kolmogorov Scale ◽  
Critical Capillary Number ◽  
Neutrally Buoyant ◽  
Deviation Theory

AbstractSmall droplets in turbulent flows can undergo highly variable deformations and orientational dynamics. For neutrally buoyant droplets smaller than the Kolmogorov scale, the dominant effects from the surrounding turbulent flow arise through Lagrangian time histories of the velocity gradient tensor. Here we study the evolution of representative droplets using a model that includes rotation and stretching effects from the surrounding fluid, and restoration effects from surface tension including a constant droplet volume constraint, while assuming that the droplets maintain an ellipsoidal shape. The model is combined with Lagrangian time histories of the velocity gradient tensor extracted from direct numerical simulations (DNS) of turbulence to obtain simulated droplet evolutions. These are used to characterize the size, shape and orientation statistics of small droplets in turbulence. A critical capillary number is identified associated with unbounded growth of one or two of the droplet’s semi-axes. Exploiting analogies with dynamics of polymers in turbulence, the critical capillary number can be predicted based on the large deviation theory for the largest finite-time Lyapunov exponent quantifying the chaotic separation of particle trajectories. Also, for subcritical capillary numbers near the critical value, the theory enables predictions of the slope of the power-law tails of droplet size distributions in turbulence. For cases when the viscosities of droplet and outer fluid differ in a way that enables vorticity to decorrelate the shape from the straining directions, the large deviation formalism based on the stretching properties of the velocity gradient tensor loses validity and its predictions fail. Even considering the limitations of the assumed ellipsoidal droplet shape, the results highlight the complex coupling between droplet deformation, orientation and the local fluid velocity gradient tensor to be expected when small viscous drops interact with turbulent flows. The results also underscore the usefulness of large deviation theory to model these highly complex couplings and fluctuations in turbulence that result from time integrated effects of fluid deformations.

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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