Self-organization in purely viscous non-Newtonian turbulence

We investigate through direct numerical simulations (DNSs) the statistical properties of turbulent flows in the inertial subrange for non-Newtonian power-law fluids. The structural invariance found for the vortex size distribution is achieved through a self-organized mechanism at the microscopic scale of the turbulent motion that adjusts, according to the rheological properties of the fluid, the ratio between the viscous dissipations inside and outside the vortices. Moreover, the deviations from the K41 theory of the structure functions' exponents reveal that the anomalous scaling exhibits a systematic nonuniversal behavior with respect to the rheological properties of the fluids.

Figure 1

Normalized dissipated energy as a function of time for different values of $n$. In all cases, after shooting up, the dissipated energies for distinct values of $n$ drop quickly to eventually reach approximately the same stationary state. The normalization factor, $\langle \overline{{\epsilon }_1}\rangle$, corresponds to the average in space and time of the local dissipated energy for the Newtonian fluid ($n=1$). The average in time is computed at the stationary state, precisely, for $t/\tau >1000$ (region colored in cream).

Figure 2

Energy spectra, multiplied by $k^{5/3}$, each obtained for two values of ${\text{Re}}_{\lambda }$, for (a) $n=0.5$, (b) $n=1.0$, and (c) $n=1.5$. Each spectra is computed by averaging over 20 different snapshots, each separated by 150 turnover times.

Figure 3

Vortex identification using the ${\lambda }_2$-vortex-criterion [25]. A typical snapshot of the vortex structure at the stationary state of the turbulent flow of a shear-thickening fluid with $n=1.5$ is shown in panel (a). The isosurfaces are calculated for a threshold value ${\lambda }_2^{*}=-{10}^{-5}$ and the colors correspond to the vorticity amplitude. The highlighted plane in panel (a) indicates the cross section for the color maps in panels (b) and (c) for the vorticity amplitude and stress intensity, respectively. The white lines in panels (b) and (c) are the contours ${\lambda }_2={\lambda }_2^{*}$.

Figure 4

Vortex size distributions computed for lower ${\text{Re}}_{\lambda }$, for different values of the rheological exponent $n$, ranging from 0.25 to 1.75, and for two values of the threshold, namely, ${\lambda }_2^{*}=-{10}^{-4}$ (a), and $-5\times {10}^{-5}$ (b). For a given pair of distributions, we applied the Kolmogorov–Smirnov test [26] to verify if they can be considered as two particular statistical realizations of the same random variable (the null hypothesis). For all pairs, the obtained significance levels are smaller than 0.05. Consequently, the null hypothesis cannot be rejected, and we conclude that the vortex sizes are likely to be drawn from the same distribution. The insets show that this behavior is also observed at higher values of ${\text{Re}}_{\lambda }$.

Figure 5

(a) Energy dissipation rate per unit of mass for the same snapshot and plane highlighted in Fig. 3. The white lines correspond to isosurfaces at the threshold value ${\lambda }_2^{*}=-{10}^{-5}$. On average there is more dissipation outside the connected regions with ${\lambda }_2<{\lambda }_2^{*}$. (b) Spatial and temporal average of the energy dissipation ratio $\epsilon /{\epsilon }_0$ as a function of ${\lambda }_2$ for different values of $n$, averaged over several eddy-turnover times. (c) The change of the ratio ${\varphi }_n/{\varphi }_1$ as a function of the rheology exponent $n$ for different values of the threshold ${\lambda }_2^{*}$.

Figure 6

The scaling relation between $S_m^n$ and $S_3$ is approximately linear independent of the rheological exponent $n$, thus rendering the application of ESS for non-Newtonian turbulent flows suitable. The black dashed line represents the $1:1$ line.

Figure 7

Extended-self-similarity (ESS) plots for three values of the rheological exponent, $n=0.5$, 1.0, and 1.5, each showing the respective dependence of the structure functions $S_m^n$ on $S_3^n$ for three different values of the order $m$ and two different values of ${\text{Re}}_{\lambda }$. The black solid lines are the least-squares fits to the simulation data of the power law, $S_m^n\sim S_3^{{\xi }_m^n}$, where ${\xi }_m^n$ is the scaling exponent.

Figure 8

(a) Dependence of the scaling exponents ${\xi }_m^n$ of the structure functions on their corresponding order $m$ for different values of the rheological exponent $n$. The black solid line is the prediction of the K41 theory, $m/3$. (b) Relative deviations of the exponents ${\xi }_m^n$ from the K41 theory, ${\delta }_m^n=({\xi }_m^n-m/3)/(m/3)$, as a function of the order $m$, calculated for different rheological exponents $n$. The solid lines are the linear fits to the data sets, ${\delta }_m^n=a_{n}m+b_n$. (c) Dependence of the estimated values of the parameters $a_n$ and $b_n$ on the exponent $n$. The least-squares fits to these data sets of the functions $a_n={\alpha }_{1}lnn+{\alpha }_2$ and $b_n={\beta }_{1}lnn+{\beta }_2$, dashed lines, gives ${\alpha }_1=0.018\pm 0.001$, ${\alpha }_2=-0.018\pm 0.001$, ${\beta }_1=-0.054\pm 0.004$, and ${\beta }_2=0.055\pm 0.003$.