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Fundamental Performance Limitations

Some of our most interesting and unique foundational work in the field of flow control involves the derivation of fundamental performance limitations for controlled systems governed by the Navier-Stokes equation.  Such analyses provide rigorous bounds on the performance of systems for a given class of actuation scheme (e.g., zero-net blowing/suction from the wall), regardless of the (linear or nonlinear) feedback scheme devised to coordinate the actuators with the flowfield fluctuations. 

We have derived two such fundamental performance limits in canoncial flow systems (channel flows), one related to heat flux [BZ07], and one related to the power saved due to drag reduction [B09].  In both cases, we have proved that, effectively, you can not do better than relaminarizing the flow.  Stated mathematically, the second paper proves that:​

That is the power of the control input applied at the walls, 𝜙(x,t), is always larger than the power possibly saved due to drag reduction, for any possible control distribution 𝜙(x,t) - even in the peculiar case when the infinite-time-averaged drag is less than the laminar drag, ⟨D < DL.  Cool.
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