Invariants Of Velocity Gradient Tensor Research Articles

A physics-informed deep learning closure for Lagrangian velocity gradient evolution

The pressure Hessian tensor is entangled with the inherent nonlinearity and nonlocality of turbulence; thus, it is of crucial importance in modeling the Lagrangian evolution of the velocity gradient tensor (VGT). In the present study, we introduce the functional modeling strategy into the classic structural modeling strategy to model the pressure Hessian tensor based on deep neural networks (DNNs). The pressure Hessian tensor and its contributions to the VGT evolution are set as, respectively, the structural and functional learning targets. An a priori test shows that the present DNN-based model accurately establishes the mapping from the VGT to the pressure Hessian tensor and adequately models the physical effect of the pressure Hessian tensor on VGT invariants. An a posteriori test verifies that the present model reproduces well the principal features of turbulence-like skewness and vorticity strain-rate alignments obtained via direct numerical simulations. Importantly, the flow topology is accurately predicted, particularly for the strain-production-dominant regions in the invariant space. Moreover, an extrapolation test shows the generalization ability of the present model to higher Reynolds number flows that have not been trained.

Prediction of polymer extension, drag reduction, and vortex interaction in direct numerical simulation of turbulent channel flows

Hydrodynamic and viscoelastic interactions between the turbulent fluid within a channel at Reτ=180 and a polymeric phase are investigated numerically using a multiscale hybrid approach. Direct numerical simulations are performed to predict the continuous phase and Brownian dynamics simulations using the finitely extensible nonlinear elastic dumbbell approach are carried out to model the trajectories of polymer extension vectors within the flow, using parallel computations to achieve reasonable computation timeframes on large-scale flows. Upon validating the polymeric configuration solver against theoretical predictions in equilibrium conditions, with excellent agreement observed, the distributions of velocity gradient tensor components are analyzed throughout the channel flow wall-normal regions. Impact on polymer stretching is discussed, with streamwise extension dominant close to the wall, and wall-normal extension driven by high streamwise gradients of wall-normal velocity. In this case, it is shown that chains already possessing high wall-normal extensions may attempt to orientate more in the streamwise direction, causing a curling effect. These effects are observed in instantaneous snapshots of polymer extension, and the effects of the bulk Weissenberg number show that increased WeB leads to more stretched configurations and more streamwise orientated conformities close to the wall, whereas, in the bulk flow and log-law regions, the polymers tend to trace fluid turbulence structures. Chain orientation angles are also considered, with WeB demonstrating little influence on the isotropic distributions in the log-law and bulk flow regions. Polymer–fluid coupling is implemented through a polymer contribution to the viscoelastic stress tensor. The effect of the polymer relaxation time on the turbulent drag reduction is discussed, with greater Weissenberg numbers leading to more impactful reduction. Finally, the velocity gradient tensor invariants are calculated for the drag-reduced flows, with polymers having a significant impact on the Q–R phase diagrams, with the presence of polymers narrowing the range of R values in the wall regions and causing flow structures to become more two-dimensional.

Physics-informed machine learning of the Lagrangian dynamics of velocity gradient tensor

Reduced models describing the Lagrangian dynamics of the Velocity Gradient Tensor (VGT) in Homogeneous Isotropic Turbulence (HIT) are developed under the Physics-Informed Machine Learning (PIML) framework. We construct the pressure Hessian and sub-filter contributions using the integrity bases and invariants of VGT and express them with extended Tensor Basis Neural Network (TBNN). We observe that the PIML model provides an improved representation for the magnitude and orientation of the small-scale pressure Hessian contributions. Statistics of the flow, as indicated by the joint PDF of second and third invariants of the VGT, show good agreement with the ``ground-truth'' DNS data.

Dynamics of Turbulent and Nonturbulent Interfaces in Cylinder and Airfoil near Wakes

Velocity gradient tensor invariants are used to extract flow physics near the turbulent and nonturbulent interfaces (TNTIs) of cylinder and airfoil wakes with vortex shedding in conjunction with spatially developing direct numerical simulation. Conditional sampling is performed on fuzzy-cluster-method-resolved TNTIs using a novel subzone approach in which each instantaneous TNTI is subdivided into four categories: trough, bulge, leading edge, and trailing edge. Results of the conditionally sampled statistics, topology, and orientation of TNTI local structures suggest that wake TNTI properties depend more heavily on the degree of vortex shedding and relatively less on the degree of wake symmetry. The present subzone-sampled joint probability density functions of the second and the third invariants of the velocity gradient tensor are compared with existing jet and mixing layer observations, and new insights are extracted. Random relative orientation between the vorticity vector and the TNTI normal is observed in the trough subzone of the present wake TNTIs, which casts doubts on the notion of full vortex structure confinement. The turbulent flow near the trailing edge subzone of wake TNTI is found to be the most effective in enstrophy production, whereas the turbulent flow in the leading-edge portion is the least effective.

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

Abstract

Parametric numerical study of passive scalar mixing in shock turbulence interaction

Turbulent mixing of passive scalars is studied in the canonical shock–turbulence interaction configuration via shock-capturing direct numerical simulations, varying the shock Mach number ( ), turbulence Mach number ( ), Taylor microscale Reynolds number ( ) and Schmidt number ( , 1, 2). The shock-normal evolution of scalar variance and dissipation transport equations, spectra and probability density functions (PDFs) are examined. Scalar dissipation, its production and destruction increase across the shock with higher , lower and lower . Mixing enhancement for different flow topologies across the shock is studied from changes in the PDFs of velocity gradient tensor invariants and conditional distributions of scalar dissipation. The proportion of the stable-focus-stretching flow topology is the highest among all the topologies in the flow both before and after the shock. Unstable-node/saddle/saddle topology is the most dissipative throughout the flow domain, despite variations across the shock. Preshock and postshock distributions of the alignment between the strain-rate tensor eigenvectors and the scalar gradient, vorticity and the mean streamwise vector conditioned on flow topology are studied. A novel barycentric map representation is introduced for a more direct visualization of the alignments and conditioned scalar dissipation distributions. Interaction with the shock increases alignment of the scalar gradient with the most extensive eigenvector, decreasing it with the most compressive, which is still dominant. The barycentric map of the passive scalar gradient also reveals that, across the shock, the most probable alignment between scalar gradient and strain eigendirections converges towards the alignment that provides the most dissipation. This also leads to an enhancement of scalar dissipation immediately downstream of the shock.

Effect of compressibility on the local flow topology in homogeneous shear turbulence

The local flow topology based on the invariants of the velocity gradient tensor in stationary compressible homogeneous shear turbulence (HST) is studied by numerical simulations. In the compressible homogeneous shear turbulence, local compressibility decreases the flow volume fraction occupied by the focal, eddy, and shear flow structures both in compression regions and in strong expansion regions. The joint probability density function (pdf) of the second and third invariants of the deviatoric velocity gradient tensor exhibits a similar teardrop shape as for the homogeneous isotropic turbulence (HIT), and the tail of the joint pdf alongside the right branch of the null-discriminant curve is elongated as the turbulent Mach number increases. When conditioned on dilatation, the statistical preference for points in the fourth quadrants of the joint pdf is enhanced significantly by the compression motion. It is found that the shape of the joint pdf shows a good similarity between HST and HIT in strong compression regions, which is dependent on the root mean square dilatation, rather than the turbulent Mach number. In strong expansion regions, the shape of the joint pdf in HST has a long tail in the third quadrant, which is related to sheetlike expansion structures and does not exist in HIT. After the Helmholtz decomposition, the properties of local flow topology associated with the solenoidal component of the velocity field are found to be very similar to those in incompressible turbulence and are insensitive to the change in local dilatation and turbulent Mach number.

Topological evolution near the turbulent/non-turbulent interface in turbulent mixing layer

The topological evolution near the turbulent/non-turbulent interface (TNTI) in turbulent mixing layer is studied by means of statistical analysis of the invariants of velocity gradient tensor (VGT) based on direct numerical simulation data. The dynamics of topological evolution is investigated in terms of the source terms of the evolution equations for the invariants, including the pressure effect term, viscous effect term and interaction term among the invariants. It is found that the local topology of fluid particles at the TNTI evolves from non-focal region to focal region in the plane of the second (Q) and the third (R) invariants of the VGT. The topological evolution is mainly associated with the pressure effect term in the TNTI region. According to the analysis of the evolution of enstrophy and dissipation, the enstrophy increase and the dissipation decrease are revealed in the TNTI region, which are caused by viscous vorticity diffusion near the TNTI. A weak correlation between the strain rate and the rotation rate is found in the TNTI region which is related to the reduction of enstrophy production. The alignments between vorticity and strain near the TNTI are investigated and a strong alignment of the vorticity with the extensive eigenvector direction is identified in the TNTI region.

Invariants of velocity gradient tensor in supersonic turbulent pipe, nozzle, and diffuser flows

Velocity gradient tensor (VGT) analysis of high-order accurate direct numerical simulation data of supersonic pipe, nozzle, and diffuser flows at computationally moderate Reynolds numbers is performed. Joint probability density functions of second and third invariants of the VGT conditioned on positive and negative dilatation levels are presented in the near-wall viscous layer, buffer layer, log layer, and core region of these flows. For flow regions with positive dilatation, there is a preference for unstable flow topologies, while regions with negative dilatation show a preference for stable flow topologies.

Analysis of coherent structures in Rayleigh–Bénard convection

Using direct numerical simulation, we investigate characteristics of coherent structures in Rayleigh–Bénard convection in a soft turbulence regime. The role of thermal plumes, essential structures in Rayleigh–Bénard convection, is studied by splitting flow regimes into thermal plume and background by investigating joint probability density function (PDF) of invariants of velocity gradient tensor. The contribution to thermal dissipation rate by these two regions is analysed separately. Through the joint PDF of invariants, we also examine the thermal effect on velocity structures.

Characterisation of a horizontal axis wind turbine’s tip and root vortices

The vortical near wake of a model horizontal axis wind turbine has been investigated experimentally in a water channel. The objective of this work is to study vortex interaction and stability of the helical vortex filaments within a horizontal axis wind turbine wake. The experimental model is a geometrically scaled version of the Tjaereborg wind turbine, which existed in western Denmark in the late 1980s. Here, the turbine was tested in both the upwind and downwind configurations. Qualitative flow visualisations using hydrogen bubble, particle streakline and planar laser-induced fluorescence techniques were combined with quantitative data measurements taken using planar particle image velocimetry. Vortices were identified using velocity gradient tensor invariants. Parameters that describe the helical vortex wake, such as the helicoidal pitch, and vortex circulation, were determined for three tip speed ratios. Particular attention is given here to the root vortex, which has been investigated minimally to date. Signatures of the coherent tip vortices are seen throughout the measurement domain; however, the signature of the root vortex is only evident much closer to the rotor plane, irrespective of the turbine configuration. It is postulated that the root vortex diffuses rapidly due to the effects of the turbine support geometries.

Experimental analysis of a turbulent boundary layer at high Reynolds numbers

In this work, time-resolved two-dimensional particle image velocimetry measurements in a zero pressure gradient turbulent boundary layer at high Reynolds numbers (Re θ = 3940 and 7257, based on momentum thickness) are presented. The boundary layer is obtained on the flat wall of a very large recirculating water channel and is investigated on a streamwise–wall-normal plane. Statistical moments of velocity are determined with the aim to investigate high-Reynolds-number effects in profiles obtained in a direction orthogonal to the wall. A careful analysis of such a profile for the measured average velocity indicates departures from the classical logarithmic law towards the power law or parametric models. Such a departure seems to be independent on the specific parameters used in model laws. On the other hand, instantaneous velocity fields are considered to derive information on the dynamic processes involving near-wall vortical structures. Different vortex eduction methods are implemented including Reynolds decomposition, vorticity, invariants of velocity gradient tensor and wavelet tools and compared. The agreement between these methods is rather good, so all of them are used to derive information on wall dynamics. At both Reynolds numbers, packets of vortical structures are observed to form and to evolve aligned more or less at an angle of 30° to the wall. In order to quantify the importance of such ‘coherent’ phenomena over the entire range of turbulent structure, probability density functions of vortical structure size (obtained from wavelet analysis) are determined. The results show the existence of a hierarchical relation between coherent vortices, the average vortex size being almost equal to 1/10 of the boundary layer displacement thickness. The measured data are consistent with the formation of about seven large ‘children’ vortices from each ‘parent’ vortex.

Topological visualisation of focal structures in free shear flows

This paper describes a method for identifying and visualising the three-dimensional geometry of focal (vortex) structures in complex flows. The method is based primarily on the classification of the local topology as it is identified from the values of the velocity gradient tensor invariants. The identification of the local topology is reference frame invariant. Therefore, focal (vortex) structures can be unambiguously identified in these flows. A novel flow visualisation method is introduced whereby focal structures are rendered using a solid model view of the local topology. This new approach is applied to the identification of focal structures in three-dimensional plane mixing layer and plane wake flows.