Dissipation, intermittency, and singularities in incompressible turbulent flows

We examine the connection between the singularities or quasisingularities in the solutions of the incompressible Navier-Stokes equation (INSE) and the local energy transfer and dissipation, in order to explore in detail how the former contributes to the phenomenon of intermittency. We do so by analyzing the velocity fields (a) measured in the experiments on the turbulent von Kármán swirling flow at high Reynolds numbers and (b) obtained from the direct numerical simulations of the INSE at a moderate resolution. To compute the local interscale energy transfer and viscous dissipation in experimental and supporting numerical data, we use the weak solution formulation generalization of the Kármán-Howarth-Monin equation. In the presence of a singularity in the velocity field, this formulation yields a nonzero dissipation (inertial dissipation) in the limit of an infinite resolution. Moreover, at finite resolutions, it provides an expression for local interscale energy transfers down to the scale where the energy is dissipated by viscosity. In the presence of a quasisingularity that is regularized by viscosity, the formulation provides the contribution to the viscous dissipation due to the presence of the quasisingularity. Therefore, our formulation provides a concrete support to the general multifractal description of the intermittency. We present the maps and statistics of the interscale energy transfer and show that the extreme events of this transfer govern the intermittency corrections and are compatible with a refined similarity hypothesis based on this transfer. We characterize the probability distribution functions of these extreme events via generalized Pareto distribution analysis and find that the widths of the tails are compatible with a similarity of the second kind. Finally, we make a connection between the topological and the statistical properties of the extreme events of the interscale energy transfer field and its multifractal properties.

Figure 1

Wavelet structure functions. (a) Plots of $S_p/{\epsilon }^{p/3}{\eta }^{\zeta \left(p\right)}$ vs $\ell /\eta$ for the first nine orders: $p=1$ (light blue), $p=2$ (orange), $p=3$ (yellow), $p=4$ (purple), $p=5$ (green), $p=6$ (pale blue), $p=7$ (red), $p=8$ (strong blue), and $p=9$ (deep orange). The black dashed lines indicate the power-law behavior ${(\ell /\eta )}^{\zeta \left(p\right)}$ [see Table 2 for the values of $\zeta \left(p\right)$]. (b) Plots of the rescaled structure functions $S_p/S_3^{p/3}$ vs $\ell /\eta$. The black dashed lines indicate the power law ${(\ell /\eta )}^{\zeta \left(p\right)-p/3}$. $\eta$ is the Kolmogorov dissipation length scale and $\epsilon$ is the dimensionless energy injection rate. The structure function of any order $p$ is obtained by combining the five different experimental data sets, shown here by using different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). A comparison with the DNS results is shown by plotting $S_p$ of different orders as solid curves with the same color scheme as used for the experimental data sets. The structure functions have been shifted by multiplying with an arbitrary factor for visual clarity.

Figure 2

Plots of the probability distribution function of the instantaneous values of the (a) local energy transfer ${\mathcal{D}}_{\ell }^{\mathrm{I}}/\epsilon$ and (b) viscous dissipation ${\mathcal{D}}_{\ell }^{\nu }$ at different scales. The PDFs at the different scales are shown by using different colors: $\ell /\eta =680$ blue curve (data set A), $\ell /\eta =240$ red curve (data set B), $\ell /\eta =90$ yellow curve (data set C), $\ell /\eta =30$ purple curve (data set D), and $\ell /\eta =3$ green curve (data set E). $\eta$ is the Kolmogorov dissipation length scale and $\epsilon$ is the dimensionless energy injection rate. Inset (a1): Same as in panel (a) but ${\mathcal{D}}_{\ell }^{\mathrm{I}}$ normalized with the standard deviation of the distribution. Inset (b1): A log-log plot of the PDF of viscous dissipation ${\mathcal{D}}_{\ell }^{\nu }$ at different scales for the positive values only.

Figure 3

Plot of the probability distribution function of the instantaneous values of $log({\epsilon }_v=\nu {\partial }_i{u}_j{\partial }_i{u}_j)$ for the experimental data set E. The black curve indicates a Gaussian with the mean and variance obtained from the experimental data set.

Figure 4

Pseudocolor plots of the instantaneous ${\mathcal{D}}_{\ell }^{\mathrm{I}}$ field in the plane of measurement from the experimental data sets E and C at the scales $\ell =\eta$ and $\ell =12\eta$, respectively. Top panels (a) data set E and (b) data set C: extreme events are absent and the location of the maximal local dissipation is shown by a white dot. Bottom panels (c) data set E and (d) data set C: in the presence of the extreme events, the white dot indicates the location of an extreme event, where the local energy transfer is 100 times larger than the mean. The measurement domain is shown in units of the radius of the cylindrical tank $R$.

Figure 5

Plots of the logarithm of the joint probability distribution function of the local energy transfer ${\mathcal{D}}_{\ell }^{\mathrm{I}}$ and the viscous dissipation ${\mathcal{D}}_{\ell }^{\nu }$ at different scales: (a) $\ell /\eta =540$, (b) $\ell /\eta =250$, (c) $\ell /\eta =70$, and (d) $\ell /\eta =3$. $\eta$ is the Kolmogorov dissipation length scale and $\epsilon$ is the dimensionless energy injection rate.

Figure 6

Spatiotemporal averages of the local energy transfer and viscous dissipation: (a) plots of $\langle {\mathcal{D}}_{\ell }^{\mathrm{I}}\rangle /\epsilon$ (red symbols) and $\langle {\mathcal{D}}_{\ell }^{\nu }\rangle /\epsilon$ (green symbols) vs $\ell /\eta$. (b) Analogous plots for the absolute values of the local energy transfer and the viscous dissipation. The black horizontal dashed line indicates a constant value of $1/3$. The black dotted line indicates the power law ${\ell }^{-4/3}$. A comparison with DNS is shown by plotting in (a) $\frac{1}{3}\langle {\mathcal{D}}_{\ell }^{\mathrm{I}}\rangle /\epsilon$ (red curve) and $\langle {\mathcal{D}}_{\ell }^{\nu }\rangle /\epsilon$ (green curve) vs $\ell /\eta$; (b) $2\langle \left|{\mathcal{D}}_{\ell }^{\mathrm{I}}\right|\rangle /\epsilon$ (red curve) and $\langle \left|{\mathcal{D}}_{\ell }^{\nu }\right|\rangle /\epsilon$ (green curve) vs $\ell /\eta$. $\eta$ is the Kolmogorov dissipation length scale and $\epsilon$ is the dimensionless energy injection rate. The different experimental data sets are represented by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars).

Figure 7

Spatiotemporal averages of the local energy transfer and viscous dissipation from the DNS: plots of $\langle {\mathcal{D}}_{\ell }^{\mathrm{I}}\rangle /\epsilon$ (red curve) and $\langle {\mathcal{D}}_{\ell }^{\nu }\rangle /\epsilon$ (green curve) vs $\ell /\eta$. Analogous plots for the average of the absolute values of local energy transfer and the viscous dissipation are shown by dashed red and green curves, respectively. The black horizontal dashed line indicates a constant value of 1. The black dotted line indicates the power law ${\ell }^{-4/3}$. $\eta$ is the Kolmogorov dissipation length scale and $\epsilon$ is the dimensionless energy injection rate.

Figure 8

Plots of the structure function of the (a) local energy transfer ${\mathrm{\Sigma }}_p^{\mathrm{I}}$ vs $\ell /\eta$ and (b) viscous dissipation ${\mathrm{\Sigma }}_p^{\nu }$ vs $\ell /\eta$. Different orders of the structure functions are distinguished by using different colors: $p=1/3$ (light blue), $p=2/3$ (orange), $p=1$ (yellow), $p=4/3$ (purple), $p=5/3$ (green), $p=2$ (pale blue), $p=7/3$ (red), $p=8/3$ (strong blue), and $p=3$ (deep orange). For the structure function of a given order, the DNS result is represented by a continuous line, while different experimental data sets are represented by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). The dotted lines indicate the power law ${\ell }^{\tau (p/3)}$ (for ${\mathrm{\Sigma }}_p^{\mathrm{I}}$) and ${\ell }^{\gamma (p/3)}$ (for ${\mathrm{\Sigma }}_p^{\nu }$); see Table 2 for the values of $\tau (p/3)$ and $\gamma (p/3)$. Note: The structure functions shown in (a) and (b) have been rescaled by ${\left({\mathrm{\Sigma }}_1^{DR}\right)}^{p/3}{\eta }^{\tau (p/3)}$ and ${\left({\mathrm{\Sigma }}_1^{\nu }\right)}^{p/3}{\eta }^{\gamma (p/3)}$, respectively; $\eta$ is the Kolmogorov dissipation scale.

Figure 9

Role of the extreme events of inertial dissipation. Plots of the structure functions: $S_p^C$ vs $\ell /\eta$ conditioned (symbols with black filling) either on the set $\mathcal{A}$ (less intense events of ${\mathcal{D}}_{\mathrm{\Delta }}^{\mathrm{I}}x$; shown in the left panel) or $\mathcal{B}$ (extreme events of ${\mathcal{D}}_{\mathrm{\Delta }}^{\mathrm{I}}x$, shown in the right panel) and the unconditioned $S_p$ vs $\ell /\eta$. The black dashed lines indicate ${\ell }^{\zeta \left(n\right)}$. The colored lines show ${\ell }^{p/3}$: $p=1$ (light blue), $p=2$ (orange), $p=3$ (yellow), $p=4$ (purple), $p=5$ (green), $p=6$(pale blue), $p=7$ (red), $p=8$ (strong blue), and $p=9$ (deep orange). The structure functions have been shifted by multiplying with an arbitrary factor for visual clarity.

Figure 10

Quantile threshold parameter from the GPD analysis: (a) ${\mu }_{-}/\epsilon$ vs $\ell /\eta$ and (b) ${\mu }_{+}/\epsilon$ vs $\ell /\eta$ from the negative and positive tails of the PDF of ${\mathcal{D}}_{\ell }^{\mathrm{I}}$, respectively. (c) Parametric plot of ${\mu }_{-}/\epsilon$ vs ${\mu }_{+}/\epsilon$. Different colors indicate different quantile values: 0.975 (cyan), 0.99 (orange), and 0.995 (green). The dotted line indicates $|{\mu }_{+}|=|{\mu }_{-}|$. Different experimental data sets are indicated by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). $\eta$ is the Kolmogorov length scale.

Figure 11

Intermittency parameter from the GPD analysis: (a) ${\sigma }_{-}/\epsilon$ vs $\ell /\eta$ and (b) ${\sigma }_{+}/\epsilon$ vs $\ell /\eta$ for the negative and positive tails of the PDF of ${\mathcal{D}}_{\ell }^{\mathrm{I}}$, respectively. Insets: (a.1) and (b.1) show the analogous plots of $\sigma$ obtained by using the self-similarity of the second kind, i.e., plots of $ln(\sigma /\epsilon )/ln\left(\text{Re}\right)$ vs $ln(\ell /\eta )/ln\left(\text{Re}\right)$, where Re is the Reynolds number. Different colors indicate different quantile values: 0.975 (cyan), 0.99 (orange), and 0.995 (green). Different experimental data sets are indicated by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). $\eta$ is the Kolmogorov length scale.

Figure 12

Intermittency parameter: Parameteric plot of ${\sigma }_{-}/\epsilon$ vs ${\sigma }_{+}/\epsilon$. Different colors indicate different quantile values: 0.975 (cyan), 0.99 (orange), and 0.995 (green). Different experimental data sets are indicated by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). ${\sigma }_{-}$ (${\sigma }_{+}$) denotes the intermittency parameter for the negative (positive) tail of the PDF of ${\mathcal{D}}_{\ell }^{\mathrm{I}}$.

Figure 13

Power-law parameter from the GPD analysis: (a) ${\xi }_{-}$ vs $\ell /\eta$ and (b) ${\xi }_{+}$ vs $\ell /\eta$ for the negative and positive tails of the PDF of ${\mathcal{D}}_{\ell }^{\mathrm{I}}$, respectively. Insets: (a.1) and (b.1) show the analogous plots of $\xi$ vs $\ell /R$, where $R$ is the radius of the cylindrical tank. Different colors indicate different quantile values: 0.975 (cyan), 0.99 (orange), and 0.995 (green). Different experimental data sets are indicated by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). $\eta$ is the Kolmogorov length scale.

Figure 14

Power-law parameter from the GPD analysis: Parametric plot of ${\xi }_{-}$ vs ${\xi }_{+}$. Different colors indicate different quantile values: 0.975 (cyan), 0.99 (orange), and 0.995 (green). Different experimental data sets are indicated by different symbols: A (circles), B (squares), C (diamonds), D (triangles), and E (stars). ${\xi }_{-}$ (${\xi }_{+}$) is the power-law parameter for the negative (positive) tail of the PDF of ${\mathcal{D}}_{\ell }^{\mathrm{I}}$.

Figure 15

Multifractal spectrum: Plot of $D\left(h\right)$ vs $h$ computed by using the values of the $\tau \left(p\right)$, scaling exponents of the local energy transfer structure functions (see Table 2), in Eq. (24). The red dash-dot line indicates the tangent $f\left(h\right)=4(h-h_{\min })$, where $h_{\min }=-0.88$.

Figure 16

Patchwork of mean velocity field in experiments A and C (in the black square). Color codes $u_y$, while $u_x$ and $u_z$ are in arrows. The coordinates have been nondimensionalized by $R$ the radius of the cylinder.

Figure 17

Radial (a) and vertical (b) velocity profiles at $z=0$ (resp. $x=0$). Red: case A; green: case B; blue: case C; gray: case E. Square: $u_r$; circles: $u_y$; stars: $u_z$. The coordinates have been nondimensionalized by $R$ the radius of the cylinder.

Figure 18

Wavelet spectrum for the 5 different cases of Table 1: blue circle: case A; red square: case B; yellow diamond: case C; purple triangle: case D; green stars: case E. The dotted line is ${(\eta /\ell )}^{-5/3}$. The spectrum and the the scale $\ell$ have been made nondimensional using $\eta$, the Kolmogorov scale, and $\epsilon$, the energy dissipation.

Figure 19

Convergence of ${\mathcal{D}}_{\ell }^{\mathrm{I}}$ for cases A (a), B (b), C (c), D (d), and E (e).

Figure 20

Convergence of ${\mathcal{D}}_{\ell }^{\nu }$ for cases A (a), B (b), C (c), D (d), and E (e).

Figure 21

Convergence of $|{\mathcal{D}}_{\ell }^{\mathrm{I}}|$ for cases A (a), B (b), C (c), D (d), and E (e).