Effect of helicity on the correlation time of large scales in turbulent flows

Solutions of the forced Navier-Stokes equation have been conjectured to thermalize at scales larger than the forcing scale, similar to an absolute equilibrium obtained for the spectrally truncated Euler equation. Using direct numeric simulations of Taylor-Green flows and general-periodic helical flows, we present results on the probability density function, energy spectrum, autocorrelation function, and correlation time that compare the two systems. In the case of highly helical flows, we derive an analytic expression describing the correlation time for the absolute equilibrium of helical flows that is different from the $E^{-1/2}{k}^{-1}$ scaling law of weakly helical flows. This model predicts a new helicity-based scaling law for the correlation time as $\tau \left(k\right)\sim H^{-1/2}{k}^{-1/2}$. This scaling law is verified in simulations of the truncated Euler equation. In simulations of the Navier-Stokes equations the large-scale modes of forced Taylor-Green symmetric flows (with zero total helicity and large separation of scales) follow the same properties as absolute equilibrium including a $\tau \left(k\right)\sim E^{-1/2}{k}^{-1}$ scaling for the correlation time. General-periodic helical flows also show similarities between the two systems; however, the largest scales of the forced flows deviate from the absolute equilibrium solutions.

Figure 1

(Left) Correlation time as a function of the wave number for $1-\text{Kr}={10}^{-6}$. The results are represented with the full line with dark dots for the positive helical components and in bright dots for the negative helical components. The $k^{-1/2}$ scaling law is represented with the dashed line. The $k^{-1}$ scaling law is represented with the dotted line. (Center) Dependence of the correlation time for the positive helical component. The Kraichnan number is increased at fixed energy: $\text{Kr}\in \{1-{10}^{-1};1-{10}^{-2};1-{10}^{-4}\}$, represented by blue, red, and green lines, respectively. The dotted line represents the $k^{-\frac{1}{2}}$ scaling law, and the full line represents the $k^{-1}$ scaling law. (Right) Transition wave number as a function of $1-\text{Kr}$, the semianalytic prediction is represented with dots, and the $\frac{15}{8}(1-\text{Kr})ln(1-\text{Kr})$ scaling law is represented with the full line.

Figure 2

(Left) PDF of the amplitude of the velocity of modes with one degree of freedom of Taylor-Green symmetric DNSs in semilogarithmic scale. The color of the thin lines indicates the wave number of the mode as described in the legend box of the plot. (Right) PDF of energy of the modes of Taylor-Green symmetric DNSs of the truncated Euler equation in logarithmic scale: (thick) theoretical predictions, (thin dashed) one degree of freedom, (thin full) two degrees of freedom. The ${\chi }_1^2$ and ${\chi }_2^2$ plots are drawn using the formula presented in Eq. (21).

Figure 3

(Left) Average standard deviation of the velocity, $\sigma$, of Taylor-Green symmetric DNSs of the truncated Euler equation for different resolutions. (Right) Energy spectrum, $E\left(k\right)$, of Taylor-Green symmetric DNSs of the truncated Euler equation at different resolutions $N=\{64,128,256,512\}$ for a fixed energy. The results are consistent with the theoretical prediction that $E\left(k\right)\sim 4\pi k^2{\sigma }^2$. In both plots, the numeric data are represented with dots and the theoretical prediction with full lines.

Figure 4

Spatio-temporal spectrum of Taylor-Green symmetric DNSs of the truncated Euler equation. (Left) Power spectrum $S(k,\omega )$. (Right) Correlation function $\mathrm{\Gamma }(k,t)$.

Figure 5

Temporal correlation properties of Taylor-Green symmetric DNSs of the truncated Euler equation. (Left) Correlation function as a function of time. (Right) Correlation time as a function of wave number. The $k^{-1}$ power law is plotted with the normalization deriving from Eq. (20).

Figure 6

Energy spectrum for different Kraichnan numbers $\text{Kr}\in \{1-{10}^{-2};1-{10}^{-3};1-{10}^{-4}\}$ for general-periodic solutions of the truncated Euler equation. The numeric data are represented with dots and the theoretical prediction of Eq. (11) with full lines. (Left) Energy spectrum associated to the total velocity. (Center) Energy spectrum associated to the positive helical component of the velocity. (Right) Energy spectrum associated to the negative helical component of the velocity. In both panels, the full lines are associated with the energy of the total velocity; the dotted lines are associated with the energy of the positive helical component of the velocity; and the dashed lines are associated with the energy of the negative helical component of velocity.

Figure 7

Separation of the PDFs due to helicity in solutions of the truncated Euler equation. The PDFs have been rescaled to plateau at the same level near zero in order to compare the width of the tail of the distribution at large values. At $k=k_M$, the distributions of probability have an offset for $\text{Kr}=0.9$ because helicity greatly affects the distribution of energy in the small scales.

Figure 8

Correlation time of DNSs of general-periodic solutions of the truncated Euler equation for three Kraichnan numbers. The red dots represent the correlation time computed using the total velocity field, the yellow X crosses and the blue + crosses represent the correlation time computed using the positive and negative components of the helical decomposition, respectively. The black dashed line and full black line represent the $k^{-\frac{1}{2}}$ and $k^{-1}$ power laws, respectively. (Left) Nonhelical flow $\text{Kr}=0$. (Center) Slightly helical flow $\text{Kr}=0.85$. (Right) Highly helical flow $\text{Kr}=0.9$,

Figure 9

PDF of modes of Taylor-Green symmetric DNSs of the forced Navier-Stokes equation with a forcing wave number at $k_f=11\sqrt{3}$. (Left) PDF of even modes with one degree of freedom at scales larger than the forcing scale. (Right) Standard deviation of the velocity modes. As the total energy is constant in the different DNSs, a $k_f^{3/2}$ normalization factor has been introduced in the plot to take into account the increase of the number of modes in the scales larger than the forcing scale.

Figure 10

Energy spectrum of Taylor-Green symmetric flows solutions of the forced Navier-Stokes equation. (Left) Fixed forcing mode $k_f=11\sqrt{3}$ at different Reynolds numbers $\text{Re}\in \{11\sqrt{3};35\sqrt{3};59\sqrt{3}\}$. (Right) Fixed Reynolds number $\text{Re}=6.56$ at different forcing modes $k_f\in \{11\sqrt{3};35\sqrt{3};59\sqrt{3}\}$.

Figure 11

Correlation time of Taylor-Green symmetric DNS of the forced Navier-Stokes equation. (Left) Different Reynolds numbers at fixed forcing wave number $k_f=11\sqrt{3}$. (Right) Different forcing wave numbers $k_f\in \{11\sqrt{3};35\sqrt{3};59\sqrt{3}\}$ at fixed Reynolds number.

Figure 12

Statistical characteristics of general-periodic DNSs of the Navier-Stokes equation. (Left) $ABC$ forcing, PDF of modes before the forcing scale. The full lines represent the positive helical components of the velocity, and the dashed lines represent the negative helical components of the velocity. (Center) $ABC$ forcing, energy spectrum of positive and negative helical components of the velocity. (Right) $CBA$ forcing, energy spectrum of positive, and negative helical components of the velocity.

Figure 13

Spectrum of general-periodic DNSs of the Navier-Stokes equation with an $ABC$ forcing. (Left) Helicity spectrum in the dotted line and the absolute equilibrium power law in the full line. (Right) Energy spectrum in the dotted line with absolute equilibrium power law in the full line and absolute equilibrium fit in the dashed line. The insert represents the local determination of the Kraichnan number, $k_f\langle h_{\mathit{k}}\rangle /\left(k^2\langle e_{\mathit{k}}\rangle \right)$, resulting from Eq. (10). The asymptotic value is used to make the absolute equilibrium fit.

Figure 14

Correlation time of general-periodic DNSs of the Navier-Stokes equation. (Left) $ABC$ forcing. (Right) $CBA$ forcing.

Figure 15

Comparison of the correlation time of ${[0;2\pi ]}^3$-periodic DNSs solutions of the Navier-Stokes equation forced with $ABC$ and $CBA$ forcings. (Left) Correlation time. (Right) Ratio of the correlation time.

Figure 16

Diagram of a cut of the integration domain. The dark surface corresponds to the integration domain. The dotted lines correspond to the limit of the circles of radius $k_M$ and of center 0 or $k$. The thick full line corresponds to radii of the previously described circles. The dark dashed line corresponds to the distance between the center of the two circles. The bright dashed line corresponds to the maximal length possible for $q_{\perp }$.

Figure 17

Triadic interaction as a function of the wave number and associated correlation time. (Left) At $\text{Kr}=1–{10}^{-6},{\mathcal{S}}^{--}$ plotted with squares, ${\mathcal{S}}^{+-}$ plotted with triangle, ${\mathcal{S}}^{++}$ plotted with disks, $k^1$ plotted with a dashed line, $k^2$ plotted with a full line. (Center) ${\mathcal{S}}^{++}$ plotted at $\text{Kr}=1–{10}^{-8}$ with disks, at $\text{Kr}=1–{10}^{-6}$ with triangles, at $\text{Kr}=1–{10}^{-2}$ with squares, at $\text{Kr}=1–{10}^{-2}$ with stars, $k^1$ with a dashed line, $k^2$ with a full line. (Right) ${\mathcal{T}}^{++}$ plotted at $\text{Kr}=1–{10}^{-8}$ with disks, at $\text{Kr}=1–{10}^{-6}$ with triangles, at $\text{Kr}=1–{10}^{-2}$ with squares, at $\text{Kr}=1–{10}^{-2}$ with stars, $k^{-\frac{1}{2}}$ with a dashed line, $k^{-1}$ with a full line.