scholarly journals Genesis and evolution of velocity gradients in near-field spatially developing turbulence

2017 ◽  
Vol 815 ◽  
pp. 295-332 ◽  
Author(s):  
I. Paul ◽  
G. Papadakis ◽  
J. C. Vassilicos
Keyword(s):  
Reynolds Number ◽  
Strain Rate ◽  
Velocity Gradient ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Grid Element ◽  
Velocity Gradients ◽  
Developing Flow ◽  
Vorticity Vector ◽  
Spatially Developing Flow

This paper investigates the dynamics of velocity gradients for a spatially developing flow generated by a single square element of a fractal square grid at low inlet Reynolds number through direct numerical simulation. This square grid-element is also the fundamental block of a classical grid. The flow along the grid-element centreline is initially irrotational and becomes turbulent further downstream due to the lateral excursions of vortical turbulent wakes from the grid-element bars. We study the generation and evolution of the symmetric and anti-symmetric parts of the velocity gradient tensor for this spatially developing flow using the transport equations of mean strain product and mean enstrophy respectively. The choice of low inlet Reynolds number allows for fine spatial resolution and long simulations, both of which are conducive in balancing the budget equations of the above quantities. The budget analysis is carried out along the grid-element centreline and the bar centreline. The former is observed to consist of two subregions: one in the immediate lee of the grid-element which is dominated by irrotational strain, and one further downstream where both strain and vorticity coexist. In the demarcation area between these two subregions, where the turbulence is inhomogeneous and developing, the energy spectrum exhibits the best$-5/3$power-law slope. This is the same location where the experiments at much higher inlet Reynolds number show a well-defined$-5/3$spectrum over more than a decade of frequencies. Yet, the$Q{-}R$diagram, where$Q$and$R$are the second and third invariants of the velocity gradient tensor, remains undeveloped in the near-grid-element region, and both the intermediate and extensive strain-rate eigenvectors align with the vorticity vector. Along the grid-element centreline, the strain is the first velocity gradient quantity generated by the action of pressure Hessian. This strain is then transported downstream by fluctuations and strain self-amplification is activated a little later. Further downstream, vorticity from the bar wakes is brought towards the grid-element centreline, and, through the interaction with strain, leads to the production of enstrophy. The strain-rate tensor has a statistically axial stretching form in the production region, but a statistically biaxial stretching form in the decay region. The usual signatures of velocity gradients such as the shape of$Q{-}R$diagrams and the alignment of vorticity vector with the intermediate eigenvector are detected only in the decay region even though the local Reynolds number (based on the Taylor length scale) is only between 30 and 40.

scholarly journals The kinematics of the reduced velocity gradient tensor in a fully developed turbulent free shear flow

2015 ◽  
Vol 767 ◽  
pp. 627-658 ◽  
Author(s):  
P. K. Rabey ◽  
A. Wynn ◽  
O. R. H. Buxton
Keyword(s):  
Shear Flow ◽  
Strain Rate ◽  
Velocity Gradient ◽  
Mixing Layer ◽  
Joint Probability ◽  
Strain Rate Tensor ◽  
Gradient Tensor ◽  
Local Isotropy ◽  
Velocity Gradient Tensor ◽  
Kinematic Behaviour

AbstractThis paper examines the kinematic behaviour of the reduced velocity gradient tensor (VGT),$\tilde{\unicode[STIX]{x1D608}}_{ij}$, which is defined as a$2\times 2$block, from a single interrogation plane, of the full VGT$\unicode[STIX]{x1D608}_{ij}=\partial u_{i}/\partial x_{j}$. Direct numerical simulation data from the fully developed turbulent region of a nominally two-dimensional mixing layer are used in order to examine the extent to which information on the full VGT can be derived from the reduced VGT. It is shown that the reduced VGT is able to reveal significantly more information about regions of the flow in which strain rate is dominant over rotation. It is thus possible to use the assumptions of homogeneity and isotropy to place bounds on the first two statistical moments (and their covariance) of the eigenvalues of the reduced strain-rate tensor (the symmetric part of the reduced VGT) which in turn relate to the turbulent strain rates. These bounds are shown to be dependent upon the kurtosis of$\partial u_{1}/\partial x_{1}$and another variable defined from the constituents of the reduced VGT. The kurtosis is observed to be minimised on the centreline of the mixing layer and thus tighter bounds are possible at the centre of the mixing layer than at the periphery. Nevertheless, these bounds are observed to hold for the entirety of the mixing layer, despite departures from local isotropy. The interrogation plane from which the reduced VGT is formed is observed not to affect the joint probability density functions (p.d.f.s) between the strain-rate eigenvalues and the reduced strain-rate eigenvalues despite the fact that this shear flow has a significant mean shear in the cross-stream direction. Further, it is found that the projection of the eigenframe of the strain-rate tensor onto the interrogation plane of the reduced VGT is also independent of the plane that is chosen, validating the approach of bounding the full VGT using the assumption of local isotropy.

Lagrangian investigations of velocity gradients in compressible turbulence: lifetime of flow-field topologies

2019 ◽  
Vol 872 ◽  
pp. 492-514 ◽  
Author(s):  
Nishant Parashar ◽  
Sawan Suman Sinha ◽  
Balaji Srinivasan
Keyword(s):  
Mach Number ◽  
Velocity Gradient ◽  
Direct Numerical Simulations ◽  
Compressible Turbulence ◽  
Lagrangian Particle ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Conditional Mean ◽  
Velocity Gradients ◽  
Decaying Turbulence

We perform Lagrangian investigations of the dynamics of velocity gradients in compressible decaying turbulence. Specifically, we examine the evolution of the invariants of the velocity-gradient tensor. We employ well-resolved direct numerical simulations over a range of Mach number along with a Lagrangian particle tracker to examine trajectories of fluid particles in the space of the invariants of the velocity gradient tensor. This allows us to accurately measure the lifetimes of major topologies of compressible turbulence and provide an explanation of why some selective topologies tend to exist longer than the others. Further, the influence of dilatation on the lifetime of various topologies is examined. Finally, we explain why the so-called conditional mean trajectories (CMT) used previously by several researchers fail to predict the lifetime of topologies accurately.

scholarly journals Measurements of the coupling between the tumbling of rods and the velocity gradient tensor in turbulence

2015 ◽  
Vol 766 ◽  
pp. 202-225 ◽  
Author(s):  
Rui Ni ◽  
Stefan Kramel ◽  
Nicholas T. Ouellette ◽  
Greg A. Voth
Keyword(s):  
Strain Rate ◽  
Velocity Gradient ◽  
Strong Dependence ◽  
Joint Probability ◽  
Weighted Average ◽  
Seeding Density ◽  
Full Description ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
The Mean

AbstractWe present simultaneous experimental measurements of the dynamics of anisotropic particles transported by a turbulent flow and the velocity gradient tensor of the flow surrounding them. We track both rod-shaped particles and small spherical flow tracers using stereoscopic particle tracking. By using scanned illumination, we are able to obtain a high enough seeding density of tracers to measure the full velocity gradient tensor near the rod. The alignment of rods with the vorticity and the eigenvectors of the strain rate from experimental results agree well with numerical findings. A full description of the tumbling of rods in turbulence requires specifying a seven-dimensional joint probability density function (jPDF) of five scalars characterizing the velocity gradient tensor and two scalars describing the relative orientation of the rod. If these seven parameters are known, then Jeffery’s equation specifies the rod tumbling rate and any statistic of rod rotations can be obtained as a weighted average over the jPDF. To look for a lower-dimensional projection to simplify the problem, we explore conditional averages of the mean-squared tumbling rate. The conditional dependence of the mean-squared tumbling rate on the magnitude of both the vorticity and the strain rate is strong, as expected, and similar. There is also a strong dependence on the orientation between the rod and the vorticity, since a rod aligned with the vorticity vector tumbles due to strain but not vorticity. When conditioned on the alignment of the rod with the eigenvectors of the strain rate, the largest tumbling rate is obtained when the rod is oriented at a certain angle to the eigenvector that corresponds to the smallest eigenvalue, because this particular orientation maximizes the contribution from both the vorticity and strain.

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.

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 Orientation dynamics of small, triaxial–ellipsoidal particles in isotropic turbulence

2013 ◽  
Vol 737 ◽  
pp. 571-596 ◽  
Author(s):  
Laurent Chevillard ◽  
Charles Meneveau
Keyword(s):  
Stochastic Model ◽  
Strain Rate ◽  
Velocity Gradient ◽  
Isotropic Turbulence ◽  
Strain Rate Tensor ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Rotation Rates ◽  
Ellipsoidal Particles ◽  
Tracer Particles

AbstractThe orientation dynamics of small anisotropic tracer particles in turbulent flows is studied using direct numerical simulation (DNS) and results are compared with Lagrangian stochastic models. Generalizing earlier analysis for axisymmetric ellipsoidal particles (Parsa et al., Phys. Rev. Lett., vol. 109, 2012, 134501), we measure the orientation statistics and rotation rates of general, triaxial–ellipsoidal tracer particles using Lagrangian tracking in DNS of isotropic turbulence. Triaxial ellipsoids that are very long in one direction, very thin in another and of intermediate size in the third direction exhibit reduced rotation rates that are similar to those of rods in the ellipsoid’s longest direction, while exhibiting increased rotation rates that are similar to those of axisymmetric discs in the thinnest direction. DNS results differ significantly from the case when the particle orientations are assumed to be statistically independent from the velocity gradient tensor. They are also different from predictions of a Gaussian process for the velocity gradient tensor, which does not provide realistic preferred vorticity–strain-rate tensor alignments. DNS results are also compared with a stochastic model for the velocity gradient tensor based on the recent fluid deformation approximation (RFDA). Unlike the Gaussian model, the stochastic model accurately predicts the reduction in rotation rate in the longest direction of triaxial ellipsoids since this direction aligns with the flow’s vorticity, with its rotation perpendicular to the vorticity being reduced. For disc-like particles, or in directions perpendicular to the longest direction in triaxial particles, the model predicts noticeably smaller rotation rates than those observed in DNS, a behaviour that can be understood based on the probability of vorticity orientation with the most contracting strain-rate eigendirection in the model.

scholarly journals Structure and dynamics of small-scale turbulence in vaporizing two-phase flows

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Radouan Boukharfane ◽  
Aimad Er-raiy ◽  
Matteo Parsani ◽  
Nilanjan Chakraborty
Keyword(s):  
Strain Rate ◽  
Velocity Gradient ◽  
Liquid Interface ◽  
Small Scale ◽  
Two Phase Flows ◽  
Gradient Tensor ◽  
Two Phase ◽  
Velocity Gradient Tensor ◽  
Carrier Liquid ◽  
Wide Range

AbstractImproving our fundamental understanding of multiphase turbulent flows will be beneficial for analyses of a wide range of industrial and geophysical processes. Herein, we investigate the topology of the local flow in vaporizing forced homogeneous isotropic turbulent two-phase flows. The invariants of the velocity-gradient, rate-of-strain, rate-of-rotation tensors, and scalar gradient were computed and conditioned for different distances from the liquid–gas surface. A Schur decomposition of the velocity gradient tensor into a normal and non-normal parts was undertaken to supplement the classical double decomposition into rotation and strain tensors. Using direct numerical simulations results, we show that the joint probability density functions of the second and third invariants have classical shapes in all carrier-gas regions but gradually change as they approach the carrier-liquid interface. Near the carrier-liquid interface, the distributions of the invariants are remarkably similar to those found in the viscous sublayer of turbulent wall-bounded flows. Furthermore, the alignment of both vorticity and scalar gradient with the strain-rate field changes spatially such that its universal behaviour occurs far from the liquid–gas interface. We found also that the non-normal effects of the velocity gradient tensor play a crucial role in explaining the preferred alignment.

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