Revisiting turbulence small-scale behavior using velocity gradient triple decomposition

Abstract

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 rotationrate (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 DNS data sets of forced isotropic turbulence, the velocitygradient 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 (A 2 ). Especially, shear contribution is dominant in regions of intermittency (high values of A 2 ). It is also 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 absence of intermittency of the pressure-Laplacian, despite the strong intermittency of enstrophy and dissipation fields. Overall, it is demonstrated that triple decomposition offers unique and deeper understanding of velocity-gradient behavior in turbulence. *

Velocity gradient analysis of a head-on vortex ring collision

“…However, while the q A -r A plane efficiently characterizes local streamline geometries at critical points, additional parameters are required to fully describe all geometries (Das & Girimaji 2020a). Following Das & Girimaji (2020b), we consider the local streamline geometry in the context of the modes of deformation of a fluid parcel: extensional straining, (symmetric and antisymmetric) shearing and rigid rotation. The well-known Cauchy-Stokes decomposition of the VGT, Ã = S + W , disambiguates contributions from the symmetric strain rate tensor, S = ( Ã + ÃT )/2, and the antisymmetric vorticity tensor, W = ( Ã − ÃT )/2.…”

“…For example, the original triple decomposition (Kolář 2007) has been used to show that lifetimes of fundamental flow structures at macroscopic scales (where viscosity can be neglected) can be related to stability of rigid rotation, linear instability of pure shearing and exponential instability of irrotational straining (Hoffman 2021). At small scales, the more recent triple decomposition has been used to show that pure shearing is typically the dominant contributor to energy dissipation (Wu et al 2020) and intermittency (Das & Girimaji 2020b) in turbulent flows. Further, the symmetric and antisymmetric components of γ * are given by γ * S = ( γ * + γ * T )/2 and γ * W = ( γ * − γ * T )/2, respectively.…”

“…Local streamline topology underlies many common geometry-based vortex criteria, including the Δ (Chong et al 1990) and λ ci (Chakraborty et al 2005) criteria. Like the geometry-based methods, and unlike the symmetry-based methods, the rigid vorticity criterion ( φz * > 0) (Tian et al 2018) captures all rotational local streamline geometries under the assumption that rigid rotation is an essential ingredient of a vortex (Liu et al 2019a;Das & Girimaji 2020b). The philosophical distinction between symmetry-based and geometry-based criteria also underlies the so-called 'disappearing vortex problem' in which, fixing the VGT configuration and strain rate, increasing only the vorticity magnitude can remove a geometry-based vortex from the flow (Chakraborty et al 2005;Kolář & Šístek 2020Kolář & Šístek , 2022.…”

“…The transition to turbulence (0.90t * t t * ), which is associated with the generation Table 2. Comparison of the equilibrium partitioning of the velocity gradients for the present vortex ring collision with the partitioning computed for forced isotropic turbulence (Das & Girimaji 2020b). Here, the equilibrium partitioning is computed as the mean over the turbulent decay regime (t t * ) and it is insensitive to the length of the averaging interval.…”

Rigid vorticity transport equation and its application to vortical structure evolution analysis in hydro-energy machinery

“…To distinguish between rigid rotation and pure shearing in total vorticity, the residual vorticity was proposed by Kolář (2007), and the precise mathematical expression of the rigid vorticity ω R was given by Liu et al (2018Liu et al ( , 2019, which is referred to as Liutex (originally known as Rortex). Moreover, according to the studies of Das and Girimaji (2020b), it is found that low-pressure regions are strongly associated with rigid rotation motions, while pure shearing is associated with nearly zero pressure fluctuations. Vortex-induced pressure fluctuation and vibration are important topics in system stability studies of hydroenergy machinery.…”