Energy spectrum in the inertial and dissipation ranges of two-dimensional steady turbulence

Abstract

The energy spectrum in the inertial and dissipation ranges in two-dimensional steady turbulence is examined theoretically and by high resolution direct numerical simulations (DNS) up to ${N=4096}^{2}$. A theoretical spectrum smoothly joining the two ranges is derived using the K\'arm\'an-Howarth-type equation. In the inertial range we obtain an asymptotic form of the energy spectrum as $E(k)=C{\ensuremath{\eta}}^{2/3}{k}^{\ensuremath{-}3}{(k/k}_{d}{)}^{\ensuremath{-}\ensuremath{\delta}}[\mathrm{ln}{(k/k}_{I}){]}^{\ensuremath{-}(2\ensuremath{-}\ensuremath{\delta})/(6\ensuremath{-}\ensuremath{\delta})}$ with small $\ensuremath{\delta}$. It is found from the DNS that $\ensuremath{\delta}$ decreases slowly with the microscale Reynolds number and the constant $C$ is of the order of unity but increases with the microscale Reynolds number. In the far dissipation range, we derive $E(k)\ensuremath{\propto}{k}^{\ensuremath{-}(3+\ensuremath{\delta})/2}{e}^{\ensuremath{-}{\ensuremath{\alpha}}_{2}{(k/k}_{d})}$, which agrees with the DNS results. The slope ${\ensuremath{\alpha}}_{2}$ depends explicitly on the microscale Reynolds number and agrees with the DNS values. Universality of the spectrum in the two ranges is also discussed.