Multiscale analysis of the invariants of the velocity gradient tensor in isotropic turbulence

Knowledge of local flow-topology, the patterns of streamlines around a moving fluid element as described by the velocity-gradient tensor, is useful for developing insights into turbulence processes, such as energy cascade, material element deformation, or scalar mixing. Much has been learned in the recent past about flow topology at the smallest (viscous) scales of turbulence. However, less is known at larger scales, for instance, at the inertial scales of turbulence. In this work, we present a detailed study on the scale dependence of various quantities of interest, such as the population fraction of different types of flow-topologies, the joint probability distribution of the second and third invariants of the velocity gradient tensor, and the geometrical alignment of vorticity with strain-rate eigenvectors. We perform the analysis on a simulation dataset of isotropic turbulence at ${\text{Re}}_{\lambda }=433$. While quantities appear close to scale invariant in the inertial range, we observe a “bump” in several quantities at length scales between the inertial and viscous ranges. For instance, the population fraction of unstable node-saddle-saddle flow topology shows an increase when reducing the scale from the inertial entering the viscous range. A similar bump is observed for the vorticity−strain-rate alignment. In order to document possible dynamical causes for the different trends in the viscous and inertial ranges, we examine the probability fluxes appearing in the Fokker-Plank equation governing the velocity gradient invariants. Specifically, we aim to understand whether the differences observed between the viscous and inertial range statistics are due to effects caused by pressure, subgrid-scale, or viscous stresses or various combinations of these terms. To decompose the flow into small and large scales, we mainly use a spectrally compact non-negative filter with good spatial localization properties (Eyink-Aluie filter). The analysis shows that when going from the inertial range into the viscous range, the subgrid-stress effect decreases more rapidly as a function of scale than the viscous effects increase. To make up for the difference, the pressure Hessian also behaves somewhat differently in the viscous than in the inertial range. The results have implications for models for the velocity gradient tensor showing that the effects of subgrid scales may not be simply modeled via a constant eddy viscosity in the inertial range if one wishes to reproduce the observed trends.

Figure 1

Six zones and associated streamline patterns on the $Q\text{−}R$ plane. Figure adapted from Refs. [18, 19]. We distinguish also between positive and negative $Q$.

Figure 2

Negative of longitudinal derivative skewness coefficient as function of filtering scale $\mathrm{\Delta }/\eta$ in isotropic turbulence. The open symbols are from the filtered fields, and the solid circle is for the unfiltered DNS value (corresponding to a very small filtering scale so it is placed here arbitrarily somewhere at $\mathrm{\Delta }/\eta <1)$.

Figure 3

Isocontours of $\mathcal{P}\left(Q,R\right)$ at discrete levels (a) 0.1, (b) 0.01, (c) 0.0032, and (d) 0.001. Different line types and colors represent the contours at various scales $\mathrm{\Delta }/\eta$ as indicated in the label at right.

Figure 4

Population fraction of various local flow topologies as function of filter scale in isoptropic turbulence. The symbols associated with each zones are shown in the $Q\text{−}R$ plane sketch to the right.

Figure 5

Average cosine-square of the angle between vorticity vector and the three eigenvectors $\left({\widehat{e}}_i\right)$ of a strain-rate tensor. Here $i=1,2,\text{and}3$ represent eigenvectors corresponding to the largest, intermediate, and smallest eigenvalues of strain-rate tensor, respectively.

Figure 6

Streamline plot of $\vec{V}$ for RE effect, i.e., the vector field $(-3R,\frac{2}{3}{Q}^2)$, which is the same at all scales.

Figure 7

Streamline plot for $\vec{V}$ for the PR effect at scales $\mathrm{\Delta }/\eta =\left(\mathrm{a}\right)70$, (b) 35, (c) 18, (d) 9, (e) 4, and (f) 2.

Figure 8

Streamline plot $\vec{V}$ for the VIS effect at scales, $\mathrm{\Delta }/\eta =\left(\mathrm{a}\right)70$, (b) 35, (c) 18, (d) 9, (e) 4, and (f) 2.

Figure 9

Streamline plot for $\vec{V}$ for the SGS effect at scales, $\mathrm{\Delta }/\eta =\left(\mathrm{a}\right)70$, (b) 35, (c) 18, (d) 9, (e) 4, and (f) 2.

Figure 10

Representative circle with radius equal to 2.5 centered at the origin of the $Q\text{−}R$ plane, shown together with the streamlines for the (a) RE, (b) PR, (c) VIS, and (d) SGS terms. The contour plot indicates the joint PDF $\mathcal{P}(R,Q)$.

Figure 11

Normalized fluxes for the RE, PR, VIS, and SGS terms across circles as function of filter scale $\mathrm{\Delta }/\eta$, with radii (a) $r=2.5$ and (b) $r=3.25$ units. Note that the RE term (red) is positive, while PR, VIS, and SGS terms are negative and are shown with opposite sign. The sum of all contributions equals 0 by construction.

Figure 12

Power law for VIS effects.

Figure 13

Energy spectra obtained from fields which are filtered using the Eyink-Aluie filter at various filter wave numbers defined as ${\kappa }_c=\pi /\mathrm{\Delta }$.

Figure 14

Population fraction of the UNSS zone as a function of $\mathrm{\Delta }/\eta$ obtained from spectral cutoff (circles), top-hat (crosses), Gaussian (plus), and Eyink filters (red squares).