Self-similar turbulence evolution and the dissipation rate transport equation

The dissipation rate transport equation is analyzed in the setting of time-dependent isotropic turbulence driven by a statistically unsteady force. In the limit of slow spectral variation, the balance between vortex stretching and enstrophy destruction postulated by Tennekes and Lumley [A First Course in Turbulence (MIT Press, Cambridge, MA, 1972)] is verified and spectral closure is used to identify the O(Re0) difference between them; however, no definite formulation of an ϵ-equation results. The ϵ equation is usually calibrated to predict self-similar unit flows such as decaying turbulence and homogeneous shear flow. The limitations of this approach are shown by constructing classes of self-similar states of forced isotropic turbulence. Any choice of constants in the ϵ equation yields a model that is consistent with some self-similar states, but not with all possible states: two-equation models of the standard form select a class of admissible self-similar states and rule out the others.