Energy dissipation and flux laws for unsteady turbulence

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Direct Numerical Simulations of spatially periodic unsteady turbulence show that the high Reynolds number scalings of the instantaneous energy dissipation rate and interscale energy flux at intermediate wavenumbers are qualitatively different from the well-known $u'(t)^{3}/L(t)$ cornerstone scalings of equilibrium turbulence where $u'(t)$ and $L(t)$ are time-dependent rms velocity and integral length-scales. Instead, they both scale as $U_{0}L_{0}\:u'(t)^2/L(t)^2$ where $L_0$ and $U_0$ are length and velocity scales characterizing initial/overall unsteady turbulence conditions.