Velocity gradient partitioning in turbulent flows

TL;DR

This work investigates how the velocity gradient tensor (VGT) in turbulent flows can be partitioned into normal straining, pure shear, and rigid rotation components using a normality-based triple decomposition. By applying the real Schur-based decomposition to DNS data from forced isotropic turbulence and wall-bounded flows, it computes averaged partitioning metrics $\langle A_\zeta^2 \rangle_{A^2}$ and their fluctuation counterparts, revealing that isotropic partitioning holds far from walls and in well-developed flows, while near-wall regions are dominated by mean shear. The study finds that increasing the friction Reynolds number $Re_\tau$ expands the isotropic-like region and that a practical collapse occurs at $Re_\tau \gtrsim 700$, with mean shear $\bar{A}_\gamma^2 / \bar{A}_{\gamma,w}^2$ providing a useful control parameter. The results underscore the expressivity and broad applicability of the normality-based partitioning for turbulence modelling, including RANS, LES, and Lagrangian or input-output analyses, and point to higher-$Re_\tau$ data and explicit mean-flow-based closures as fruitful directions.

Abstract

The velocity gradient tensor can be decomposed into normal straining, pure shearing and rigid rotation tensors, each with distinct symmetry and normality properties. We partition the strength of turbulent velocity gradients based on the relative contributions of these constituents in several canonical flows. These flows include forced isotropic turbulence, turbulent channels and turbulent boundary layers. For forced isotropic turbulence, the partitioning is in excellent agreement with previous results. For wall-bounded turbulence, the partitioning collapses onto the isotropic partitioning far from the wall, where the mean shearing is relatively weak. By contrast, the near-wall partitioning is dominated by shearing. Between these two regimes, the partitioning collapses well at sufficiently high friction Reynolds numbers and its variations in the buffer layer and the log-law region can be reasonably modelled as a function of the mean shearing strength. Altogether, our results highlight the expressivity and broad applicability of the velocity gradient partitioning as advantages for turbulence modelling.

Figures (3)

• Figure 1: Total (a,c,e) and fluctuation (b,d,f) partitioning profiles for the channels and boundary layers in terms of wall-normal location in inner units (a,b) and mean shearing strength (c--f). The vertical lines represent the isotropic values. The BL0729 and BL1024 profiles are shown for $Re_\tau \approx 729$ and $1000$, respectively, and the top boundary of the log-law region represents Ch1000. The dashed white lines in (a,b) represent the locations of the partitioning values reported in \ref{['tab:iso']} for BL0729 and BL1024. In (a,b), the markers are used to distinguish between the profiles and, in (c--f), they represent actual data points. In (c--f), the mean shearing axis is reversed and the dashed lines represent comparable linear-log trends for each dataset, with the partitioning as the dependent variable.
• Figure 2: Streamwise development of BL0729 (a,b) and BL1024 (c,d) in terms of $Re_\tau$, where the colour axis represents $\bar{A}_\gamma^2 / \bar{A}_{\gamma,w}^2$. The white, grey, and black contours represent $\Delta_{iso}^{} = 1\%$, $\Delta_{iso}^{} = 2\%$ and $\Delta_{iso}^{} = 5\%$, respectively, for the total partitioning (a,c) and $\Delta_{iso}' = 1\%$, $\Delta_{iso}' = 2\%$ and $\Delta_{iso}' = 5\%$, respectively, for the fluctuation partitioning (b,d). The dashed and dash-dotted black lines represent the top of the viscous sublayer $(y^+ = 5)$ and the top of the buffer layer $(y^+ = 30)$, respectively, and the dotted black lines represent $y^+ = 100$ and $y^+ = 150$. The black circles represent the locations of the partitioning values reported in \ref{['tab:iso']} and $\delta$ represents the boundary-layer thickness.
• Figure 3: Symmetry-based total (a) and fluctuation (b) partitioning profiles for the channels and boundary layers in terms of wall-normal location in inner units. The plots are in the same style as those in \ref{['fig:profiles']}.