Tailwind turbulence: a bound on the energy available from turbulence for transit, tested in Kraichnan's model

Abstract

We investigate the unconstrained minimum energy required for vehicles to move through turbulence. We restrict our study to vehicles that interact with their environment through thrust, weight and drag forces, such as rotorcraft or submersibles. For such vehicles, theory predicts an optimum ratio between vehicle velocity and a characteristic velocity of the turbulence. The energy required for transit can be substantially smaller than what is required to move through quiescent fluid. We describe a simple picture for how a flight trajectory could preferentially put vehicles in tailwinds rather than headwinds, predicated on the organization of turbulence around vortices. This leads to an analytical parameter-free lower bound on the energy required to traverse a turbulent flow. We test this bound by computationally optimizing trajectories in Kraichnan's model of turbulence, and find that the energy required by point-models of vehicles is slightly larger than but close to our bound. Finally, we predict the existence of an optimum level of turbulence for which power is minimized, so that turbulence can be both too strong or too weak to be useful. This work strengthens previous findings that environmental turbulence can always reduce energy use. Thus, favorable trajectories are available to maneuverable vehicles if they have sufficient knowledge of the flow and computational resources for path planning.