Three applications of scaling to inhomogeneous, anisotropic turbulence

The energy spectrum in three examples of inhomogeneous, anisotropic turbulence, namely, purely mechanical wall turbulence, the Bolgiano-Obukhov cascade, and helical turbulence, is analyzed. As one could expect, simple dimensional reasoning leads to incorrect results and must be supplemented by information on the dynamics. In the case of wall turbulence, a hypothesis of Kolmogorov cascade, starting locally from the gradients in the mean flow, produces an energy spectrum that obeys the standard $k^{-5/3}$ law only for ${\mathrm{kx}}_3>1\mathrm{}$, with $x_3$ the distance from the wall, and an inverse power law for ${\mathrm{kx}}_3<1\mathrm{}$. An analysis of the energy budget for turbulence in stratified flows shows the unrealizability of an asymptotic Bolgiano scaling. Simulation with a Gledzer-Ohkitani-Yamada shell model leads instead to a $k^{-\alpha }$ spectrum for both temperature and velocity, with $\alpha \simeq 2$, and a cross correlation between the two vanishing at large scales. In the case of non-reflection-invariant turbulence, closure analysis suggests that a purely helical cascade, associated with a $k^{-7/3}$ energy spectrum cannot take place, unless external forcing terms are present at all scales in the Navier-Stokes equation.