A study is made about the energy spectrum $E\left(k\right)$ of turbulence on the basis of high-resolution direct numerical simulations (DNSs) of forced incompressible turbulence in a periodic box using a Fourier spectral method with the number of grid points and the Taylor scale Reynolds number $R_{\lambda }$ up to $12{288}^3$ and approximately 2300, respectively. The DNS data show that there is a wave-number range (approximately $5\times {10}^{-3}<k\eta <2\times {10}^{-2}$) in which $E\left(k\right)$ fits approximately well to Kolmogorov's $k^{-5/3}$ scaling, where $\eta$ is the Kolmogorov length scale. However, a close inspection shows that the exponent is a little smaller than $-5/3$, and $E\left(k\right)$ in the range fits to $E\left(k\right)/\left[{\langle \epsilon \rangle }^{2/3}{k}^{-5/3}\right]=c{\left(kL\right)}^m$, where $\langle \epsilon \rangle$ is the mean energy dissipation rate per unit mass; $L$ is the integral length scale; and $m\approx -0.12$. The coefficient $c$ is independent of $k$, but has a $R_{\lambda }$ dependence, such as $c=C{R}_{\lambda }^{\zeta }$, where $C\approx 0.9$ and $\zeta \approx 0.14$.

• Received 4 September 2016

DOI:https://doi.org/10.1103/PhysRevFluids.1.082403