Perfect energy transfer and mean-eddy interaction in incompressible fluid flows

Abstract

The mean-eddy interaction in a flow is investigated in terms of the energy transfer between its mean and eddy processes. In a Reynolds decomposition framework, energy transfer can be defined from the decomposed energy equations, but the definition is subjective with the classical energetics formalism. This paper shows that a faithful formalism can be rigorously established within the same framework. The resulting transfer sums to zero everywhere over the decomposed subspaces, representing a mere redistribution of energy between the mean and eddy processes, without generating or destroying energy as a whole. For this reason, it has been referred to as perfect transfer in distinction from other energy transfers one may have encountered. The perfect transfer has a simple form in expression, with the introduction of an eddy flow coupled with the mean and eddy processes of the field under concern. The eddy or turbulence production is then totally determined by the divergence of this flow. This formalism has been validated with a well-known barotropic instability model, and a wake control problem. It is expected to be useful for the harnessing of turbulence, in that it may help to identify the best locations to place passive controls, or design the performance functional for active controls.