Multifractality in a nested velocity gradient model for intermittent turbulence

Small-scale intermittency is an omnipresent feature of turbulent flows, and the velocity gradient tensor in three dimensions provides a rich characterization of its salient statistical and geometric features. A recent model for isotropic turbulence involving a set of stochastic differential equations was derived for the Lagrangian time evolution of the velocity gradient tensor at multiple scales. Dominant terms in the model could be derived directly from the Navier-Stokes equations while also requiring phenomenological models for unclosed pressure and viscous terms. In this work we show that instead of having to integrate this stochastic model in time at multiple levels and scales, the same predictions can be obtained based on a single level that recursively modulates the model terms interpreted at arbitrarily higher levels. The equivalence is explored based on actual realizations of the modeled stochastic process and then extended to probability densities and moments at all levels and scales. Statistical properties compare very well to direct numerical simulation and experimental data at different Reynolds numbers. The proposed formalism represents a new type of multifractal intermittency model that can be connected directly to terms in the Navier-Stokes equation and is capable of predicting many intricate statistical and geometric features of turbulent flows over a wide range of Reynolds numbers.

Figure 1

Multifractal model of temporal intermittency growth in turbulent flows. Panel (a) shows an illustration of such a hierarchical process based on an synthetic prescribed signal with three constant intervals [Eq. (11)]. Different color solid lines correspond to different steps (black: ${\tau }^{(n=0)}\left(t\right)$, red: ${\tau }^{(n=1)}\left(t\right)$, blue: ${\tau }^{(n=2)}\left(t\right)$, green ${\tau }^{(n=3)}\left(t\right)$. Panel (b) shows ${\left({\tau }^{(n=3)}\right)}^{-2}\left(t\right)$, proportional to the corresponding dissipation signal. Panels (c) and (d) show the same as panels (a) and (b) but obtained from a master time-series arising from the single-scale RDGF model, Eq. (3).

Figure 2

Panels (a) and (c): Sample $A_{11}\left(t\right)$ (a) and $A_{12}\left(t\right)$ (c) signals from five levels of the same trajectory in the five-level model obtained from a single (master) level using Eqs. (7, 8, 9, 10). From top to bottom: level 1 to level 5 are shown, from the coarsest level to the finest level. Panels (b) and (d): Comparisons of results from Ref. [23] to results obtained using a single (master) level and then constructing recursively gradient signals from which the probability density functions (PDFs) are measured. PDFs of the longitudinal component $A_{11}$ and transverse component $A_{12}$ elements are shown in panels (b) and (d), respectively, for $n=1$ (blue line), 2 (red line), 3 (yellow line), 4 (purple line), and 5 (green solid line). These are compared with the numerically obtained multilevel results [23] for $n=1$ (☆), 2($\circ$), 3($\square$), 4($▵$), and 5($▿$). We also show data from ${\mathrm{Re}}_{\lambda }=610$ DNS results (gray solid line), to be compared to the $n=2$ case since $n=1$ corresponds to ${\mathrm{Re}}_{\lambda ,1}\approx 60$ and ${\mathrm{Re}}_{\lambda ,n}\approx {\beta }^{n-1}{\mathrm{Re}}_{\lambda ,1}$ [23] with $\beta =10$.

Figure 3

Fourier transform of master level PDF of the variable $y^{*}=ln\left({\tau }^{*}\right)$ measured from the master level signal of RDGF model. Writing $\mathcal{F}\left[f_{*y}\right]\left(\omega \right)={\int }_{-\infty }^{\infty }{f}_{*y}\left(\mathcal{Y}\right)exp(-i\mathcal{Y}\omega )d\mathcal{Y}$ as $\mathcal{F}\left[f_{*y}\right]\left(\omega \right)=R\left(\omega \right)exp\left[i\varphi \left(\omega \right)\right]$, the amplitude $R\left(\omega \right)$ is shown in panel (a), while the angle $\varphi \left(\omega \right)$ is shown in panel (b) as a function of frequency $\omega$.

Figure 4

Panel (a): Probability density function of $y_n=ln\left({\tau }_n\right),n=2,3,...,8$. Panel (b): Probability density function of dissipation for $n=1,2,...,8$, where the corresponding Reynolds number are ${\mathrm{Re}}_{\lambda }=6\times {10}^n$. Panel (c): Premultiplied probability density functions of $y^{*}$, multiplied by ${\left({\tau }^{*}\right)}^{-m}=exp(-m{y}^{*})$ to confirm that moments up to $m=8$ computed from these PDFs are converged.

Figure 5

Panels (a) and (b): Probability density functions of $A_{11}$ (a) and $A_{12}$ (b) for $n=1$ (blue), 2 (red), 3 (yellow), 4 (purple), 5 (green solid line), 6 (green), 7 (blue), and 8 (red dash line) compared with Ref. [23] for $n=1$ (☆), 2 ($\circ$), 3 ($\square$), 4 ($▵$), and 5 ($▿$). The ${\mathrm{Re}}_{\lambda }=610$ DNS results are shown with a gray solid line. Panels (c) and (f): Probability density functions of dissipation and enstrophy, respectively, for $n=1$ (blue), 2 (red), 3 (yellow), 4 (purple), 5 (green solid line), 6 (green), 7 (blue), and 8 (red dash line). Again, the ${\mathrm{Re}}_{\lambda }=610$ DNS results are shown with gray solid line, while the Ref. [23] case corresponding to level n = 2 is shown as $\circ$. Panel (d): Probability density functions of $s^{*}$ parameter. Panel (e): Probability density functions of alignment of vorticity vector with the $j\mathrm{th}$ strain-rate eigenvector ordered by decreasing eigenvalues: ${\lambda }_1$ (green line), ${\lambda }_2$ (blue line), and ${\lambda }_3$ (red lines). In panels (d) and (e) the solid lines indicate model results (these are independent of level $n$) while the dashed line shows the DNS results at ${\mathrm{Re}}_{\lambda }=610$.

Figure 6

Logarithmic representation of the joint PDF, $f_{nQR}(Q,R)$, of R-Q from (a) proposed approach at level $n=2$, compared with R-Q joint PDF from Ref. [23] (b) and results from the DNS data at ${\mathrm{Re}}_{\lambda }=610$ (c). The logarithmically spaced isocontours shown are at $f_{2QR}(Q,R)={10}^1$, ${10}^0$, ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$. The gray line represents the Vieillefosse line: $(27/4)R^2+Q^3=0$.

Figure 7

Panels (a) and (b): Skewness (a) and flatness (b) factors of velocity gradient components as a function of ${\mathrm{Re}}_{\lambda }$ compared with experimental, DNS and explicitly evaluated multi-time RDGF data. Red filled circles ($A_{11}$ skewness and flatness) and squares ($A_{12}$ flatness) represent the results of the proposed approach based on a single level, blue filled circle ($A_{11}$ skewness and flatness), and green filled squares ($A_{12}$ flatness) represent MT-RDGF data from Ref. [23]. DNS data from Ref. [19] ($\square ,\circ$), [3] ($▵$), and a compilation of experimental data from Ref. [1] ($+$). Panel (c): Scaling exponents $\alpha \left(m\right)$ from the proposed model (red $+$) with ratio $\beta$ = 10, compared with MT-RDGF model [23] (green filled $▹$), Kolmogorov 1941 [31] (black solid line) log-normal $\mu =0.2$ (dashed magenta line) and $\mu =0.25$ (dot-dashed green line), She-Leveque [10] (continuous blue line), p-model [6] with $p_1=0.7$ (black dotted line), and DNS data from Ref. [3] ($▿$), [4] ($▵$), and experimental data from Ref. [32] ($\square$).

Figure 8

Probability density functions of $A_{11}$ (a), $A_{12}$ (b), and dissipation (c) of $n=1.85$ (red solid lines) and $n=2.33$ (purple solid lines) compared with the ${\mathrm{Re}}_{\lambda }=430$ (gray dot line) and ${\mathrm{Re}}_{\lambda }=1300$ (black dot line) DNS results.

Figure 9

Hyperskewness (a) and hyperflatness (b) and (c) factors of velocity gradient components as a function of ${\mathrm{Re}}_{\lambda }$. Colored $▵\mathrm{s}$ indicate the interpolation model for noninteger $n$ while colored solid lines are drawn connecting the hyperflatness values at six discrete Reynolds numbers (separated by powers of 10) generated by the integer levels of the model [Eqs. (33) and (35)].