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(max) eta approximate to 6, and additionally at R-lambda = 1300 with k(max) eta approximate to 3, where k(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 less than or similar to k eta <= 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 <= k eta <= 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 >> 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.