Dissipation range of the energy spectrum in high Reynolds number turbulence

We seek to understand the kinetic energy spectrum in the dissipation range of fully developed turbulence. The data are obtained by direct numerical simulations (DNS) of forced Navier-Stokes equations in a periodic domain, for Taylor-scale Reynolds numbers up to $R_{\lambda }=650$, with excellent small-scale resolution of $k_{\text{max}}\eta \approx 6$, and additionally at $R_{\lambda }=1300$ with $k_{\text{max}}\eta \approx 3$, where $k_{\text{max}}$ is the maximum resolved wave number and $\eta$ is the Kolmogorov length scale. We find that for a limited range of wave numbers $k$ past the bottleneck, in the range $0.15\lesssim k\eta \lesssim 0.5$, the spectra for all $R_{\lambda }$ display a universal stretched exponential behavior of the form $exp(-k^{2/3})$, in rough accordance with recent theoretical predictions. In contrast, the stretched exponential fit does not possess a unique exponent in the near dissipation range $1\lesssim k\eta \lesssim 4$, but one that persistently decreases with increasing $R_{\lambda }$. This region serves as the intermediate dissipation range between the $exp(-k^{2/3})$ region and the far dissipation range $k\eta \gg 1$ where analytical arguments as well as DNS data with superfine resolution [S. Khurshid et al., Phys. Rev. Fluids 3, 082601 (2018)] suggest a simple $exp(-k\eta )$ dependence. We briefly discuss our results in connection to the multifractal model.

Figure 1

(a) Compensated kinetic energy spectra $E\left(k\right)$ as a function of $k\eta$ for various Taylor-scale Reynolds number $R_{\lambda }$. (b) The log-derivative of the energy spectra, i.e., $\varphi \left(k\right)=dlogE\left(k\right)/dlogk$.

Figure 2

(a) The negative of log-derivative of the energy spectra for various $R_{\lambda }$. (b) Zoomed in version of the same plot. The dashed blacks line in both panels represent a power law with exponent $2/3$. In the range $k\eta >1$, we observe only power laws with only a varying exponent from $0.70\pm 0.03$ at $R_{\lambda }=140$ to $0.50\pm 0.01$ at $R_{\lambda }=1300$.

Figure 3

The exponent $\gamma$ as a function of $R_{\lambda }$ on (a) linear scales, and (b) log scales. The (blue) triangles correspond to the data of Ref. [15]. The dashed lines correspond to fits shown in the legend.

Figure 4

Log-derivative of various stretched exponential functions $f\left(x\right)$, plotted vs $x^{2/3}$. All curves exhibit a visually perceptible range where they look like a straight line. Similar behavior is observed if some other power of $x$ is used instead of $2/3$.