Local and nonlocal strain rate fields and vorticity alignment in turbulent flows

Local and nonlocal contributions to the total strain rate tensor $S_{ij}$ at any point $\mathbf{x}$ in a flow are formulated from an expansion of the vorticity field in a local spherical neighborhood of radius $R$ centered on $\mathbf{x}$. The resulting exact expression allows the nonlocal (background) strain rate tensor $S_{ij}^B\left(\mathbf{x}\right)$ to be obtained from $S_{ij}\left(\mathbf{x}\right)$. In turbulent flows, where the vorticity naturally concentrates into relatively compact structures, this allows the local alignment of vorticity with the most extensional principal axis of the background strain rate tensor to be evaluated. In the vicinity of any vortical structure, the required radius $R$ and corresponding order $n$ to which the expansion must be carried are determined by the viscous length scale ${\lambda }_{\nu }$. We demonstrate the convergence to the background strain rate field with increasing $R$ and $n$ for an equilibrium Burgers vortex, and show that this resolves the anomalous alignment of vorticity with the intermediate eigenvector of the total strain rate tensor. We then evaluate the background strain field $S_{ij}^B\left(\mathbf{x}\right)$ in direct numerical simulations of homogeneous isotropic turbulence where, even for the limited $R$ and $n$ corresponding to the truncated series expansion, the results show an increase in the expected equilibrium alignment of vorticity with the most extensional principal axis of the background strain rate tensor.

Figure 1

(Color online) Decomposition of the vorticity field in the vicinity of any point $\mathbf{x}$ into local and nonlocal parts; the Biot-Savart integral in Eq. (8) over each part gives the local and nonlocal (background) contributions to the total strain rate tensor $S_{ij}$ at $\mathbf{x}$.

Figure 2

(Color online) Equilibrium Burgers vortex with circulation $\Gamma$ and strain-limited viscous diffusion length scale ${\lambda }_{\nu }$ in a uniform, irrotational, axisymmetric background strain rate field $S_{ij}^B\left(\mathbf{x}\right)$.

Figure 3

(Color online) Similarity profiles of ${\omega }_z\left(\eta \right)$ and $S_{r\theta }\left(\eta \right)$ for any equilibrium Burgers vortex; wherever $-S_{r\theta }$ exceeds the horizontal line determined by the relative vortex strength parameter $\Omega$ in Eq. (29) the most extensional principal axis of the total strain rate $S_{ij}\left(\mathbf{x}\right)$ switches from the $\widehat{\mathbf{z}}$ axis to lie in the $r\text{−}\theta$ plane.

Figure 4

(Color online) Accuracy of the Taylor expansion for the local vorticity in Eq. (10) for a Burgers vortex, showing results for sixth-order approximation. In each panel, the solid curve shows actual vorticity profile, and the dashed curve gives approximated vorticity from derivatives at the location marked by the square.

Figure 5

(Color online) Convergence of background strain rate field $S_{ij}^B\left(\mathbf{x}\right)$ for a Burgers vortex, obtained from total strain rate field $S_{ij}\left(\mathbf{x}\right)$ using Eq. (20) for various $(n,\tilde{R})$ combinations, where $\tilde{R}\equiv R∕{\lambda }_{\nu }$. Shown are effects of increasing $\tilde{R}$ for fixed $n=6$ (top), increasing $n$ for fixed $\tilde{R}=0.65$ (middle), and increasing $n$ and $\tilde{R}$ simultaneously (bottom). The dashed horizontal lines follow from Eqs. (27, 29).

Figure 6

(Color online) Instantaneous snapshot of total strain rate component field $S_{12}\left(\mathbf{x}\right)$ in a two-dimensional slice through a highly resolved three-dimensional $\left({2048}^3\right)$ DNS of homogeneous, isotropic turbulence [17, 18]. Axes are given both in grid coordinates $(i=1,\dots ,2048)$ and normalized by the Kolmogorov length ${\eta }_K$. The box indicates the region in which background strain rate field $S_{ij}^B\left(\mathbf{x}\right)$ is computed in Fig. 7.

Figure 7

(Color online) Total strain rate component field $S_{12}\left(\mathbf{x}\right)$ (a), with corresponding results from Eq. (20) for nonlocal (background) field $S_{12}^B\left(\mathbf{x}\right)$ (left) and local field $S_{12}^R\left(\mathbf{x}\right)$ (right) for $(R∕{\eta }_K)=2.5$ [(b) and (c)], $(R∕{\eta }_K)=3.5$ [(d) and (e)], and $(R∕{\eta }_K)=4.5$ [(f) and (g)], all with $n=3$.

Figure 8

(Color online) Probability densities of alignment cosines for the vorticity with the eigenvectors of the strain rate tensor, showing results for $S_{ij}$ (a) and for $S_{ij}^B$ using $(R∕{\eta }_K)=2.5$ with $n=3$ (b) and $(R∕{\eta }_K)=3.5$ with $n=3$ (c).