Energy fluxes and spectra for turbulent and laminar flows

Two well-known turbulence models to describe the inertial and dissipative ranges simultaneously are by Pao~[Phys. Fluids {\bf 8}, 1063 (1965)] and Pope~[{\em Turbulent Flows.} Cambridge University Press, 2000]. In this paper, we compute energy spectrum $E(k)$ and energy flux $\Pi(k)$ using spectral simulations on grids up to $4096^3$, and show consistency between the numerical results and predictions by the aforementioned models. We also construct a model for laminar flows that predicts $E(k)$ and $\Pi(k)$ to be of the form $\exp(-k)$, and verify the model predictions using numerical simulations. The shell-to-shell energy transfers for the turbulent flows are {\em forward and local} for both inertial and dissipative range, but those for the laminar flows are {\em forward and nonlocal}.