(D) Conservation of Circulations in Turbulent Flow

We have emphasized the importance of developing a better understanding of the dynamical & statistical origin of the positivity of vortex-stretching rate 〈ω̄! S!ω̄!〉 > 0. For the fine-grained vorticity (! → 0), this is important in Taylor’s proposed mechanism for production of energy dissipation 〈ν|ω|2〉. For ! in the inertial-range, we have seen that the flux of energy to small-scales is proportional to the stretching rate: 〈Π!〉 ∝ !〈ω̄! S!ω̄!〉, to a first approximation, local in space and in scale. A deeper understanding of the vortexstretching process might help us to better predict and control turbulent flow.