Invariants Of Velocity Gradient Research Articles

Comparing local energy cascade rates in isotropic turbulence using structure-function and filtering formulations

Two common definitions of the spatially local rate of kinetic energy cascade at some scale $\ell$ in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’ $R>0$ , $Q>0$ quadrant, where $R$ and $Q$ are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when $R$ is large.

Evolution of the velocity gradient invariants in homogeneous isotropic turbulence with an inverse energy cascade

An investigation of topological features of homogeneous isotropic turbulence with an inverse energy cascade is performed by using a direct numerical simulation. The inverse energy cascade is induced by reversing the velocity field (i.e., under the transformation from ui to −ui) of a freely decaying isotropic turbulence. In the backward energy transfer process, the joint probability density function (PDF) of Q and R exhibits a novel shape, which seems to be symmetric with the well-known teardrop shape about the Q-axis (Q and R are the second and third invariants of the velocity gradients tensor, respectively). The predominance of the top-right (R > 0 and Q > 0) and the bottom-left (R < 0 and Q < 0) quadrants is observed in the backward energy transfer process. This observation is different from that of the forward energy cascade, in which the top-left (R < 0 and Q > 0) and the bottom-right (R > 0 and Q < 0) quadrants are dominant. The unexpected shape of the joint PDF of Q and R indicates that the turbulence with an inverse energy cascade is dominated by vortex compression and tube-like structures. The PDF of the intermediate eigenvalue of the strain-rate tensor in the inverse energy cascade is negatively skewed, which is opposite to the universal feature of the forward energy cascade. Nevertheless, the preferential alignment of vorticity with the intermediate eigenvector is rather robust, no matter whether the direction of the energy transfer is forward or backward. In addition, it is universal that the vorticity is mostly perpendicular to the eigenvector, which corresponds to a strain-rate eigenvalue with the maximum absolute value among the three eigenvalues. Since the velocity gradient invariants are closely related to the local flow topology, the numerical results reported in this work are expected to shed light on the intrinsic dynamics and mechanisms of inverse energy cascade.

The effect of large-scale forcing on small-scale dynamics of incompressible turbulence

The subtle but crucial effects of large-scale forcing on the small-scale velocity-gradient (VG) dynamics is examined using direct numerical simulation (DNS) data of incompressible turbulence. The interplay among large-scale forcing, inertia, pressure and viscous effects is characterised as a function of local streamline geometry and VG magnitude (Frobenius norm). When conditioned on local topology, forcing: (i) counteracts inertial and viscous action in the strain-dominated nodal topologies; and (ii) balances pressure action in the rotation-dominated unstable focal topologies. Unexpectedly, forcing acts to reduce VG magnitudes in regions of strong dissipation. In these regions, forcing balances the non-local pressure effects whereas viscous action offsets the nonlinear inertial effects. In regions of very low dissipation, forcing combines with inertia and pressure effects to offset viscous action. With regard to the probability distribution of the normalised VG invariants, the primary role of forcing is to nullify certain features (dilatational probability currents) of inertia, pressure and viscous action. This results in the emergence of universal statistical features (solenoidal probability currents) that are determined largely by inertia–pressure–viscous interactions. These findings serve to enhance our understanding of small-scale processes in turbulence and guide VG model development.

Revisiting turbulence small-scale behavior using velocity gradient triple decomposition

Turbulence small-scale behavior has been commonly investigated in literature by decomposing the velocity-gradient tensor (Aij) into the symmetric strain-rate (Sij) and anti-symmetric rotation-rate (Wij) tensors. To develop further insight, we revisit some of the key studies using a triple decomposition of the velocity-gradient tensor. The additive triple decomposition formally segregates the contributions of normal-strain-rate (Nij), pure-shear (Hij) and rigid-body-rotation-rate (Rij). The decomposition not only highlights the key role of shear, but it also provides a more accurate account of the influence of normal-strain and pure rotation on important small-scale features. First, the local streamline topology and geometry are described in terms of the three constituent tensors in velocity-gradient invariants’ space. Using direct numerical simulation (DNS) data sets of forced isotropic turbulence, the velocity-gradient and pressure field fluctuations are examined at different Reynolds numbers. At all Reynolds numbers, shear contributes the most and rigid-body-rotation the least toward the velocity-gradient magnitude (A2 ≡ AijAij). Especially, shear contribution is dominant in regions of high values of A2 (intermittency). It is shown that the high-degree of enstrophy intermittency reported in literature is due to the shear contribution toward vorticity rather than that of rigid-body-rotation. The study also provides an explanation for the non-intermittent nature of pressure-Laplacian, despite the strong intermittency of enstrophy and dissipation fields. The study further investigates the alignment of the rotation axis with normal strain-rate and pressure Hessian eigenvectors. Overall, it is demonstrated that triple decomposition offers unique and deeper understanding of velocity-gradient behavior in turbulence.

Joint Probability Distribution Functions for the Filtered Velocity Gradient Invariants in Homogeneous Isotropic Turbulence

Background:Turbulent flow is characterized by vortices with different scales. Extraction of various scales and filtering the turbulent field into coherent and incoherent parts are important processes that improve our understanding of turbulent characteristics.Objective:Joint probability distribution functions (JPDFs) for the filtered velocity gradient invariants are extensively studied for different scales as well as for the coherent and incoherent parts of each scale.Methods:The Fourier decomposition and the anisotropic diffusion model are used in the investigation. The extraction process is performed by employing the Fourier decomposition at different cutoff wavenumbers for the velocity field and three distinct scales (large, medium and fine scale) are identified. The velocity gradient invariants such as the second invariantQand the third invariantRfor the different scales are extracted. Then other important invariants such as the rate of rotation tensorQWand the rate of deformationQSare also identified for each scale. The anisotropic diffusion model is used to extract the coherent and incoherent parts of each invariant at each scale. Then the JPDFs of the coherent and incoherent invariants are compared. The scale decomposition and the filtering process are applied for turbulent flow fields that are simulated using the lattice Boltzmann method with resolution of 1283.Results:Results show that the (R-Q) space has a universal topological pear-like shape for the different scales as well as their coherent field. However, the (R-Q)-space for the incoherent fields are found different and no general shape can be observed. The (Qw-QS)-space results show self-similar shapes for coherent fields and for the incoherent fields no specific shape can be observed since the noise distributed as separated points everywhere.Conclusion:Two different methods for extraction and filtering of forced isotropic turbulence and the JPDFs of the velocity gradient invariants are studied. Some universal characteristics for the coherent parts were found. However, for the incoherent parts, no universal JPDFs were found.

Geometry and scaling laws of excursion and iso-sets of enstrophy and dissipation in isotropic turbulence

ABSTRACTMotivated by interest in the geometry of high intensity events of turbulent flows, we examine the spatial correlation functions of sets where turbulent events are particularly intense. These sets are defined using indicator functions on excursion and iso-value sets. Their geometric scaling properties are analysed by examining possible power-law decay of their radial correlation function. We apply the analysis to enstrophy, dissipation and velocity gradient invariants Q and R and their joint spatial distributions, using data from a direct numerical simulation of isotropic turbulence at Reλ ≈ 430. While no fractal scaling is found in the inertial range using box-counting in the finite Reynolds number flow considered here, power-law scaling in the inertial range is found in the radial correlation functions. Thus, a geometric characterisation in terms of these sets’ correlation dimension is possible. Strong dependence on the enstrophy and dissipation threshold is found, consistent with multifractal behaviour. Nevertheless, the lack of scaling of the box-counting analysis precludes direct quantitative comparisons with earlier work based on multifractal formalism. Surprising trends, such as a lower correlation dimension for strong dissipation events compared to strong enstrophy events, are observed and interpreted in terms of spatial coherence of vortices in the flow.

Joint statistics of the scalar gradient and the velocity gradient in turbulence using linear diffusion models

A model for the evolution of scalar gradients in turbulence is proposed. The molecular diffusion is approximated by a linear estimation, while the straining term is closed taking advantage of a previous model for velocity gradients [J. Martín, C. Dopazo, and L. Valiño, “Dynamics of velocity gradient invariants in turbulence: Restricted Euler and linear diffusion models,” Phys. Fluids 10, 2012 (1998)]. The joint statistics of the scalar gradients and velocity gradients obtained from this model are studied. In particular, the alignment of the scalar gradient with the principal strain axis, and the correlations between the strain rate and the scalar dissipation rate are investigated. The results show reasonably good agreement with data from direct numerical simulation, which indicates that this simple model captures the essential geometrical properties of the scalar-gradient/velocity-gradient coupling in turbulent flows.

Dynamics of the velocity gradient tensor invariants in isotropic turbulence

The evolution of the invariants (R and Q) of the velocity gradient tensor in homogeneous isotropic turbulence is investigated using data from direct numerical simulation (DNS). The concepts of conditional average time rate of change of the invariants and conditional mean trajectories (CMT) in invariant phase space are introduced to study the dynamics of this flow. The resulting dynamical system in the (R,Q) phase space is a clockwise spiral with a stable focus at the origin, illustrating that in the mean, the cyclic sequence of topological evolution following a fluid particle is unstable-node/saddle/saddle (UN/S/S)→stable-node/saddle/saddle (SN/S/S)→stable-focus/stretching (SF/S)→unstable-focus/contracting (UF/C). The mean rates of change of R and Q, i.e., Ṙ, Q̇, are found to be negligible near the right branch of the null discriminant (D=0) curve, indicating that this curve is an attractor in the (R,Q) space. The effects of both the diffusion term and the anisotropic part of the pressure Hessian term on the dynamics of the invariants have also been analyzed using the conditional averages. Both contributions are found to be important in the dynamics of the velocity gradient invariants. Based on these results the extent of the validity of the model equations governing the evolution of R and Q proposed by Cantwell [Phys. Fluids A 4, 782 (1992)] and Dopazo et al. [“Velocity gradients in turbulent flows. Stochastic models,” Ninth Symposium on “Turbulent Shear Flows,” Kyoto, Japan, 1993, pp. 26-2-1–26-2-5] are discussed.

Dynamics of velocity gradient invariants in turbulence: Restricted Euler and linear diffusion models

A complete system of dynamical equations for the invariants of the velocity gradient, the strain rate, and the rate-of-rotation tensors is deduced for an incompressible flow. The equations for the velocity gradient invariants R and Q were first deduced by Cantwell [Phys. Fluids A 4, 782 (1992)] in terms of Hij, the tensor containing the anisotropic part of the pressure Hessian and the viscous diffusion term in the velocity gradient equation. These equations are extended here for the strain rate tensor invariants, RS and QS, and for the rate-of-rotation tensor invariant, QW, using HijS and HijW, the symmetric and the skew-symmetric parts of Hij, respectively. In order to obtain a complete system, an equation for the square of the vortex stretching vector, Vi≡Sijωj, is required. The resulting dynamical system of invariants is closed using a simple model for the velocity gradient evolution: an isotropic approximation for the pressure term and a linear model for the viscous diffusion term. The local topology and the resulting statistics implied by this model reproduce a number of trends similar to known results from numerical experiments for the small scales of turbulence.

Lagrangian statistics from direct numerical simulations of isotropic turbulence

A comprehensive study is reported of the Lagrangian statistics of velocity, acceleration, dissipation and related quantities, in isotropic turbulence. High-resolution direct numerical simulations are performed on 643and 1283grids, resulting in Taylor-scale Reynolds numbers Rλin the range 38-93. The low-wavenumber modes of the velocity field are forced so that the turbulence is statistically stationary. Using an accurate numerical scheme, of order 4000 fluid particles are tracked through the computed flow field, and hence time series of Lagrangian velocity and velocity gradients are obtained.The results reported include: velocity and acceleration autocorrelations and spectra; probability density functions (p.d.f.'s) and moments of Lagrangian velocity increments; and p.d.f.'s, correlation functions and spectra of dissipation and other velocity-gradient invariants. It is found that the acceleration variance (normalized by the Kolmogorov scales) increases as R½λ- a much stronger dependence than predicted by the refined Kolmogorov hypotheses. At small time lags, the Lagrangian velocity increments are distinctly non-Gaussian with, for example, flatness factors in excess of 10. The enstrophy (vorticity squared) is found to be more intermittent than dissipation, having a standard-deviation-to-mean ratio of about 1.5 (compared to 1.0 for dissipation). The acceleration vector rotates on a timescale about twice the Kolmogorov scale, while the timescales of acceleration magnitude, dissipation and enstrophy appear to scale with the Lagrangian velocity timescale.