Velocity Gradient Tensor Research Articles

Three-dimensional analysis of precursors to non-viscous dissipation in an experimental turbulent flow

Abstract

Turbulence topology evolution in weakly turbulent premixed flames

In turbulent premixed flames, not only the isotropy of velocity fluctuations is altered by the thermal expansion effect but also the dissipative structure of the turbulent flow field and the flow topology are also deeply influenced by the flame. Considering the joint probability density function of the second and third invariants of the velocity gradient tensor (VGT)—or its traceless counterpart—is a classical way to deduce the topology of turbulent flows at these smallest scales. These quantities are analyzed by considering direct numerical simulation databases of premixed flame kernel growth in homogeneous isotropic turbulence. Two conditions of turbulence–combustion interaction are considered, which correspond to two distinct values of the Bray number. The analysis of the VGT shows that the propagating premixed flame and its associated density variations significantly modify the turbulence structure and flow topology. To understand this behavior as the flow interacts with the flame front, Lagrangian dynamics of the VGT and its invariants are studied by considering the conditional mean rate of change vectors. Special emphasis is thus placed on the Lagrangian evolution equations of these invariants. To the best of authors' knowledge, this is first time that such budgets are scrutinized under premixed combustion conditions. The pressure Hessian contribution to the VGT invariant transport equations is shown to be one of the leading-order terms in this evolution, making it critically important to the flow dynamics and turbulence structure.

Relative motion of two neighboring points on inert and reactive scalar iso-surfaces in homogeneous turbulence

The study of the deformation and rotation of line and surface and volume elements, embedded in a mixing-reaction zone, is supposedly of a paramount importance to comprehend the evolution of inert and chemically reacting species in turbulent flows. This analysis examines the relative motion of two points on adjacent, nonmaterial, propagating, isoscalar surfaces, subject to the flow velocity, v, and the normal displacement speed vector, Sdn (n is the unit vector perpendicular to the isosurface). The methodology of Perry and Chong [“Topology of flow patterns in vortex motions and turbulence,” Appl. Sci. Res. 53, 357–374 (1994)], applied to the velocity gradient tensors of v, A, and Sdn, Aa, serves to characterize this motion. Small-scale flow and displacement topologies clearly emerge from this description. The invariants of these two velocity gradient tensors yield the total rate of change of the normal distance between two isosurfaces, the area stretch factor, the total volumetric dilatation rate, and the nodal or focal features of the motions. The influence of the first invariant on the displacement speed and the chemical depletion is also established. A simple mathematical formalism portrays the time rate of change of an infinitesimal nonmaterial vector, r, joining two points on adjacent isoscalar surfaces within a zone where turbulent mixing of inert and reactive species occurs. Databases of a 3D 5123 DNS of a statistically homogeneous and stationary turbulence with inert and reactive scalars undergoing random mixing in a constant density fluid are examined to illustrate the application of the previous conceptual framework. Small scales are well resolved. Various flow and displacement contributions to the two-point relative velocity components, normal and tangential to the isosurface, are compared to conclude that motions of isoscalar surfaces due to Sdn are, at least, as important as those caused by v. In particular, the additional vorticity, ωa, is one order of magnitude greater than the flow vorticity, ω, and tangential to the isoscalar surfaces, contributing significantly to their folding. While the dynamically passive scalars do not modify the conventional local flow motions, new additional topologies, induced by Aa and typified by its invariants, Pa, Qa, and Ra, appear and affect the displacement speed and the reaction rate. The mass entrainment rate per unit mass into a volume element between two adjacent isosurfaces is given by the first invariant, −Pa, and its influence on the displacement speed and the chemical reaction is explored.

On the Semiannual Formation of Large Scale Three‐Dimensional Vortices at the Stratopause

AbstractAn examination of the dynamics of the middle atmosphere as reconstructed in the NASA Modern‐Era Retrospective analysis for Research and Applications, Version 2 (MERRA‐2) reveals the formation of large scale three‐dimensional (3D) vortices in the tropical stratosphere following both the vernal and autumnal equinoxes at times when the jet associated with the westerly phase of the semiannual oscillation (SAO) is maximal and extratropical influences from planetary waves are weakest. An empirical orthogonal function (EOF) analysis of the second matrix invariant of the velocity gradient tensor applied to the shear zones about the SAO reveals statistically stationary wave‐5 vortex structures that span more than 3,200 km in length and up to 40 km in the vertical. Eliassen‐Palm fluxes suggests the vortices are maintained by a combination of local (shear zones) and remote (vertically propagating tropical) sources of momentum. These large‐scale coherent features appear to be unique to the stratopause.

Non-normal effect of the velocity gradient tensor and the relevant subgrid-scale model in compressible turbulent boundary layer

The non-normal effects of the velocity gradient tensor (VGT) in a compressible turbulent boundary layer are studied by means of the Schur decomposition of the VGT into its normal and non-normal parts. Based on the analysis of the relative importance of them, it is found that the non-normal part significantly affects the dynamics of the VGT in the wall-bounded turbulent flow and the relevant non-normal effect has a dominant influence on the enstrophy and dissipation. It is revealed that the deviatoric part of the pressure Hessian is associated with the non-normal effect and the isotropic part is associated with the normal effect. The pressure Hessian significantly influences the vortex stretching. The non-normal effect reinforces the preferences for the vorticity vector to align with the intermediate strain-rate eigenvector and to be perpendicular to the extensive and compressive strain-rate eigenvector in the near-wall region. The non-normal effect also reduces the intermediate eigenvalue of the strain-rate tensor. Furthermore, a subgrid scale (SGS) model that separately considers the normal and non-normal effects is proposed based on the above characters and is verified to give a better prediction of the SGS dissipations in the wall-bounded turbulent flow.

Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe

Abstract

Consititutive modeling of time-dependent flows in frictional suspensions

This paper summarizes recent joint work towards a constitutive modelling framework for dense granular suspensions. The aim is to create a time-dependent, tensorial theory that can implement the physics described in steady state by the Wyart-Cates model. This model of shear thickening suspensions supposes that lubrication films break above a characteristic normal force so that frictional contact forces come into play: the resulting non-sliding constraints can be enough to rigidify a system that would flow freely at lower stresses [1]. Implementing this idea for time-dependent flows requires the introduction of new concepts including a configuration-dependent ‘jamming coordinate’, alongside a decomposition of the velocity gradient tensor into compressive and extensional components which then enter the evolution equation for particle contacts in distinct ways. The resulting approach [2, 3] is qualitatively successful in addressing (i) the collapse of stress during flow reversal in shear flow, and (ii) the ability of transverse oscillatory flows to unjam the system. However there is much work required to refine this approach towards quantitative accuracy, by incorporating more of the physics of contact evolution under flow as determined by close interrogation of particle-based simulations.

Reynolds number dependence of heavy particles clustering in homogeneous isotropic turbulence

In homogeneous isotropic turbulence, the degree of heavy particle clustering is reduced with increasing Taylor Reynolds number (${R}_{\ensuremath{\lambda}}$) for 52 \ensuremath{\le} ${R}_{\ensuremath{\lambda}}$ \ensuremath{\le} 139 when the Stokes number (St) is small. Contrarily, the clustering of high-St particles tends to stegthen with increasing ${R}_{\ensuremath{\lambda}}$. Quantities invoked for low-St particle clustering include both the strength of and characteristic time of particle trapping by ``shear structures'' quantified by the second invariant of the velocity gradient tensor. The enhancement of high-St particle clustering results from the increasing large-scale eddies at higher ${R}_{\ensuremath{\lambda}}$.

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, λci, λ2 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.

A FENE-P k–ε Viscoelastic Turbulence Model Valid up to High Drag Reduction without Friction Velocity Dependence

A viscoelastic turbulence model in a fully-developed drag reducing channel flow is improved, with turbulent eddies modelled under a k–ε representation, along with polymeric solutions described by the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive model. The model performance is evaluated against a wide variety of direct numerical simulation data, described by different combinations of rheological parameters, which is able to predict all drag reduction (low, intermediate and high) regimes with good accuracy. Three main contributions are proposed: one with a simplified viscoelastic closure for the NLTij term (which accounts for the interactions between the fluctuating components of the conformation tensor and the velocity gradient tensor), by removing additional damping functions and reducing complexity compared with previous models; second through a reformulation for the closure of the viscoelastic destruction term, Eτp, which removes all friction velocity dependence; lastly by an improved modified damping function capable of predicting the reduction in the eddy viscosity and thus accurately capturing the turbulent kinetic energy throughout the channel. The main advantage is the capacity to predict all flow fields for low, intermediate and high friction Reynolds numbers, up to high drag reduction without friction velocity dependence.

Multiscale simulation of a polymer melt flow between two coaxial cylinders under nonisothermal conditions

We successfully extend a multiscale simulation (MSS) method to nonisothermal well-entangled polymer melt flows between two coaxial cylinders. In the multiscale simulation, the macroscopic flow system is connected to a number of microscopic systems through the velocity gradient tensor, stress tensor and temperature. At the macroscopic level, in addition to the momentum balance equation, we consider the energy balance equation, where heat generation plays an important role not only in the temperature distribution but also in the flow profile. At the microscopic level, a dual slip-link model is employed for well-entangled polymers. To incorporate the temperature effect into the microscopic systems, we used the time-temperature superposition rule for the slip-link model, in which the temperature dependence of the parameters is not known; on the other hand, the way to take into account the temperature effect in the macroscopic equations has been well established. We find that the extended multiscale simulation method is quite effective in revealing the relation between nonisothermal polymeric flows for both steady and transient cases and the microscopic states of polymer chains expressed by primitive paths and slip-links. It is also found that the temperature-dependent reptation-time-based Weissenberg number is a suitable measure for understanding the extent of the polymer chain deformation in the range of the shear rate used in this study.

Hairpin vortices in the largest scale of turbulent boundary layers

We have conducted direct numerical simulations of a turbulent boundary layer for the momentum-thickness-based Reynolds number Reθ = 180–4600. To extract the largest-scale vortices, we coarse-grain the fluctuating velocity fields by using a Gaussian filter with the filter width comparable to the boundary layer thickness. Most of the largest-scale vortices identified by isosurfaces of the second invariant of the coarse-grained velocity gradient tensor are similar to coherent vortices observed in low-Reynolds-number regions, that is, hairpin vortices or quasi-streamwise vortices inclined to the wall. We also develop a percolation analysis to investigate the threshold-dependence of the isosurfaces and objectively identify the largest-scale hairpin vortices in terms of the coarse-grained vorticity, which leads to the quantitative evidence that they never disappear even in fully developed turbulent regions. Hence, we conclude that hairpin vortices exist in the largest-scale structures irrespective of the Reynolds number.

Analyzing the effect of dilatation on the velocity gradient tensor using a model problem

The effect of variable mass density on the velocity gradient tensor is addressed by means of a model problem. An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. The model results show the evolution of the velocity gradient tensor as the density front is approached and are relevant to the physics of flame fronts.

The effect of initial conditions on mixing transition of the Richtmyer–Meshkov instability

Abstract

Particle flows around an intruder

Particle flows injected as beams and scattered by an intruder are numerically studied. We find a crossover of the drag force from Epstein's law to Newton's law, depending on the ratio of the speed to the thermal speed. These laws can be reproduced by a simple analysis of a collision model between the intruder and particle flows. The crossover from Epstein's law to Stokes' law is also found for the low-speed regime as the time evolution of the drag force caused by beam particles. We also show the existence of turbulent-like behavior of the particle flows behind the intruder with the aid of the second invariant of the velocity gradient tensor and the relative mean square displacement for the high-speed regime and a large intruder.

Near wall Prandtl number effects on velocity gradient invariants and flow topologies in turbulent Rayleigh\u2013B\xe9nard convection

The statistical behaviours of the invariants of the velocity gradient tensor and flow topologies for Rayleigh–Bénard convection of Newtonian fluids in cubic enclosures have been analysed using Direct Numerical Simulations (DNS) for a range of different values of Rayleigh (i.e. Ra=10^7-10^9) and Prandtl (i.e. Pr=1 and 320) numbers. The behaviours of second and third invariants of the velocity gradient tensor suggest that the bulk region of the flow at the core of the domain is vorticity-dominated whereas the regions in the vicinity of cold and hot walls, in particular in the boundary layers, are found to be strain rate-dominated and this behaviour has been found to be independent of the choice of Ra and Pr values within the range considered here. Accordingly, it has been found that the focal topologies S1 and S4 remain predominant in the bulk region of the flow and the volume fraction of nodal topologies increases in the vicinity of the active hot and cold walls for all cases considered here. However, remarkable differences in the behaviours of the joint probability density functions (PDFs) between second and third invariants of the velocity gradient tensor (i.e. Q and R) have been found in response to the variations of Pr. The classical teardrop shape of the joint PDF between Q and R has been observed away from active walls for all values of Pr, but this behavior changes close to the heated and cooled walls for high values of Pr (e.g. Pr=320) where the joint PDF exhibits a shape mirrored at the vertical Q-axis. It has been demonstrated that the junctions at the edges of convection cells are responsible for this behaviour for Pr=320, which also increases the probability of finding S3 topologies with large negative magnitudes of Q and R. By contrast, this behaviour is not observed in the Pr=1 case and these differences between flow topology distributions in Rayleigh–Bénard convection in response to Pr suggest that the modelling strategy for turbulent natural convection of gaseous fluids may not be equally well suited for simulations of turbulent natural convection of liquids with high values of Pr.

Symmetry transformation and dimensionality reduction of the anisotropic pressure Hessian

Abstract

Characteristics of shearing motions in incompressible isotropic turbulence

Shearing motions in isotropic turbulence are studied with a triple decomposition of velocity gradient tensor. A mean flow around the shearing motions exhibits a thin shear-layer pattern sustained by a biaxial strain. The thickness of each shear layer is well predicted by Burgers' vortex layer. Interplay between the shear and biaxial strain causes enstrophy production and strain self-amplification.

Dynamics and invariants of the perceived velocity gradient tensor in homogeneous and isotropic turbulence

The perceived velocity gradient tensor (PVGT), constructed from four fluid tracers forming a tetrahedron, provides a natural way to study the structure of velocity fluctuations and its dependence on spatial scales. It generalizes and shares qualitatively many properties with the true velocity gradient tensor. Here, we establish the evolution equation for the PVGT, and, for homogeneous and isotropic incompressible turbulent flows, we analyse the dynamics of the PVGT in particular using its second- and third-order invariants. We show that, for PVGT based on regular tetrads with lateral size , the second-order invariants can be expressed solely in terms of the usual second-order velocity structure functions, while the third-order invariants involve the usual third-order longitudinal velocity structure function and a less well known three-point velocity correlation function. For homogeneous and isotropic turbulence, exact relations between the second moments of strain and vorticity, as well as enstrophy production and the third moments of the strain, are derived. These generalized relations are valid for all ranges of , and reduce to classical results for the velocity gradient tensor when is in the dissipative range. With the help of these relations, we quantify the importance of the various terms, such as vortex stretching, as a function of the scale . Our analysis, which is supported by the results of direct numerical simulations of turbulent flows in the Reynolds-number range , allows us to demonstrate that strain prevails over vorticity when is in the inertial range.

Flow structure segmentation for vortex identification using butterfly convolutional neural networks

Vortex identification is important for understanding the physical mechanism of turbulent flow. The common vortex identification techniques based on velocity gradient tensor such as [Formula: see text] criterion will consume a lot of computing resources for processing great quantity of experimental data. To improve the vortex identification efficiency and achieve real-time recognition, we present a novel vortex identification method using segmentation with convolutional neural network (CNN) based on flow field image data, which is named “Butterfly-CNN”. Considering that the view of flow field is small, it is necessary to integrate both the local and global feature maps to achieve higher precision. The architecture consists of an encoded–decoded path, which is similar to [Formula: see text]-net but with different superimposed network part. In the Butterfly-CNN, the cross-expanding paths are designed with the global information to enable precise localization, and the feature maps after each convolution are regarded as the original pictures, then convolute to the size of the last feature map and upsample to the original size again. Finally, the decoded and cross-expanding networks are added up. The Butterfly-CNN can be trained end-to-end from a few images, and it is useful and efficient for vortex identification.