Analysis of the dissipative range of the energy spectrum in grid turbulence and in direct numerical simulations

We present a statistical analysis of the behavior of the kinetic-energy spectrum in the dissipative range of fully developed three-dimensional turbulence, with the aim of testing a recent prediction obtained from the nonperturbative renormalization group. Analyzing spectra recorded in experiments of grid turbulence, generated in the Modane wind tunnel, and spectra obtained from high-resolution direct numerical simulations (DNSs) of the forced Navier-Stokes equation, we observe that the spectra decay as a stretched exponential in the dissipative range. The theory predicts a stretching exponent $\alpha =2/3$, and the data analyses of the numerical and experimental spectra are in close agreement with this value. This result also corroborates previous DNS studies which found that the spectrum in the near-dissipative range is best modeled by a stretched exponential with $\alpha <1$.

Figure 1

Spectra obtained from DNS, for the set of ${\mathrm{Re}}_{\lambda }$ and resolutions $N$ given in Table 1. In all the figures, INR, NDR, and FDR stand for inertial range, near-dissipative range, and far-dissipative range, respectively.

Figure 2

First logarithmic derivative $D_1$ of the spectra of Fig. 1, with the inset showing a zoom in the near-dissipative range. These results are very similar to those obtained in previous works [23, 24, 30]. In particular, $D_1$ is not linear in the NDR, hence it is not compatible in this range with a pure exponential behavior ($\alpha =1$).

Figure 3

Third logarithmic derivative, defined by (9), computed for the DNS spectra of Fig. 1, with the inset showing a zoom in the near-dissipative range. Only the curves for the highest resolution for each ${\mathrm{Re}}_{\lambda }$ are shown for clarity. They all show a plateau in the NDR, the value of which is $\alpha$, which approaches 2/3 as the ${\mathrm{Re}}_{\lambda }$ increases.

Figure 4

Exponent $K$ of the power law in the inertial range, from DNS and from experiments for both sample durations $\mathrm{\Delta }t=20$ and 60 s, as a function of the Taylor microscale Reynolds number. For the experimental points, the symbols represent the mean values of $K$ on each ${\mathrm{Re}}_{\lambda }$ bin, and the error bars represent the standard deviation within each bin. The numbers of spectra in each bin are not equal; they are given by the distribution shown in the inset.

Figure 5

Stretch exponent $\alpha$ as a function of the Taylor microscale Reynolds number from DNS and from experiments: ${\alpha }_2$ (black lines with diamonds) computed from $D_2$ and ${\alpha }_3$ (red lines with triangles) computed from $D_3$, for experimental spectra corresponding to both durations $\mathrm{\Delta }t=20$ (plain lines) and 60 s (dashed lines), and ${\alpha }_3$ (blue squares) from DNS. For the experimental points, the symbols indicate the mean values of $\alpha$ for each ${\mathrm{Re}}_{\lambda }$ bin, and the error bars represent the standard deviation within each bin. The numbers of spectra in each bin are not equal; the distribution of ${\mathrm{Re}}_{\lambda }$ for each set of spectra is represented in Fig. 6.

Figure 6

Distribution of the ${\mathrm{Re}}_{\lambda }$ corresponding to each set of data displayed in Fig. 5, for the two sample durations $\mathrm{\Delta }t=20$ and 60 s, and for the two methods $D_2$ and $D_3$.

Figure 7

Distribution of the stretch exponent $\alpha$: ${\alpha }_2$ (black lines with diamonds) computed from $D_2$ and ${\alpha }_3$ (red lines with triangles) computed from $D_3$, for spectra corresponding to both durations $\mathrm{\Delta }t=20$ (plain lines) and 60 s (dashed lines). The mean value and standard deviation of these distributions are reported in Table 2.

Figure 8

Distribution of the stretch exponent ${\alpha }_3$ for $\mathrm{\Delta }=60\mathrm{s}$ and for different precision criteria $Q$ from ${10}^{-1}$ to ${10}^{-4}$.

Figure 9

Parameters $\beta$ (main graph) and $\mu$ (inset) obtained for the DNS spectra of Fig. 1.

Figure 10

Result of the optimized fitting algorithm for the determination of $K$ from the log-log spectrum (red line with circles) and from the first logarithmic derivative $D_1$ (orange line with diamonds). The result from $D_1$ is reported on the spectrum $E$ for comparison.

Figure 11

Result of the optimized fitting algorithm for the determination of $\alpha$ from the second (red line with circles) and from the third (orange line with diamonds) logarithmic derivatives of the spectrum. The result from $D_3$ is reported on the $D_2$ curve for comparison.