A fundamental limit on the balance of power in a transpiration-controlled channel flow

This paper is a direct sequel to Bewley & Aamo (J. Fluid Mech., vol. 499, 2004, pp. 183-196). It was conjectured in that paper, based on the numerical evidence available at that time, that the minimum drag of a constant mass flux channel flow might in fact be that of the laminar flow. This conjecture turned out to be false; Mill et al. (J. Fluid Mech., vol. 558, 2006, 309318) discovered a curious control strategy which in fact reduces the time-averaged drag to sub-laminar levels. The present paper establishes rigorously that the power of the control input applied at the walls is always larger than the power saved (due to drag reduction below the laminar level) for any possible control distribution, including that proposed by Min et al. (2006), thus establishing that, energetically (that is accounting for the power saved due to drag reduction and the power exerted by application of the control), the optimal control solution is necessarily to relaminarize the flow.