Vorticity statistics and the time scales of turbulent strain

Time scales of turbulent strain activity, denoted as the strain persistence times of first and second order, are obtained from time-dependent expectation values and correlation functions of Lagrangian rate-of-strain eigenvalues taken in particularly defined statistical ensembles. Taking into account direct numerical simulation data, our approach relies on heuristic closure hypotheses which allow us to establish a connection between the statistics of vorticity and strain. It turns out that softly divergent prefactors correct the usual “$1/s$” strain time-scale estimate of standard turbulence phenomenology, in a way which is consistent with the phenomenon of vorticity intermittency.

Figure 1

Panels (a) and (b) are produced from direct numerical simulation (DNS) data addressed in Sec. 3. (a) A typical profile of the piecewise continuous function $\overline{s}\left(t\right)$. (b) The compactly supported configurations $s_1\left(t\right)$ (red), $s_2\left(t\right)$ (green), and $s_3\left(t\right)$ (blue) belong all to the ensemble ${\Lambda }_s$, with $s=-13.6\pm 0.1$, since $s_1\left(0\right)=s_2\left(0\right)=s_3\left(0\right)=s$. The total time span of the DNS is $\Delta t=2048\times {10}^{-3}$ in arbitrary time units.

Figure 2

${\sigma }_{\omega }\left(s\right)=\sqrt{{\langle {\omega }^2\left(0\right)\rangle }_{{\Lambda }_s}}$, i.e., the standard deviation of the projected Lagrangian vorticity $\omega \left(0\right)$, is plotted as a function of standardized rate-of-strain eigenvalues $s/{\sigma }_{\pm }$. The standard deviations of the positive and negative rate-of-strain eigenvalues $s$ are, respectively, ${\sigma }_{+}=4.63$ and ${\sigma }_{-}=6.52$.

Figure 3

The horizontal dotted lines indicate the mean values $-3.84$ ($\equiv -g$) and $15.81$ ($\equiv g^{\prime }$) of the ratios $I_1(-s)/I_1\left(s\right)$ and $I_2(-s)/I_2\left(s\right)$, which are, in fact, approximately constant in the range $3.4\le s\le 21.0$, the region between vertical dashed lines, where the ensembles ${\Lambda }_s$ are large enough for the evaluation of reasonable statistical averages.

Figure 4

Plots which provide support for the validity of Eqs. (2.3) and (2.4) with the use of the strain persistence times $T_1^{\pm }\left(s\right)$ and $T_2^{\pm }\left(s\right)$, as defined from Eqs. (2.28, 2.29, 2.30, 2.31). The dotted lines in the plots for $I_1\left(s\right)$ have unit slope, while the ones in the plots for $I_2\left(s\right)$ have slope 2. The upper plots in each pair of plots correspond to negative $s$. The vertical dashed lines indicate the range $3.4\le s\le 21.0$ (see Fig. 3).

Figure 5

A test of the relations (2.6) and (2.7), where $T_1^{\pm }\left(s\right)$ and $T_2^{\pm }\left(s\right)$ are given by (2.28) and (2.29) with $\alpha =1.64$, $\beta =0.73$, $g=3.84$, and $g^{\prime }=15.81$.

Figure 6

The numerical evaluation of the square root of the right-hand side of (2.18), which is assumed to yield ${\sigma }_{\omega }\left(s\right)$, is compared to the result reported in Fig. 2 (straight lines).