Non-uniform resistance in optimising energy extraction from a flow

Extracting energy from a flow is a fundamental problem in fluid mechanics of significant practical engineering importance. To generate power from a flow, a resistance must be applied. Open questions remain on how to optimise this resistance, particularly for non-uniform flows. In this paper, we extend the multi-streamtube theory to address this gap. The extended theory allows for an arbitrary resistance distribution across an actuator strip (representing either a single turbine or an array of turbines) and is formulated as a power-coefficient maximisation problem to determine the optimal resistance distribution for both uniform and non-uniform flows. When the undisturbed kinetic energy flux projected onto the strip’s frontal area is used to normalise the extracted power, a uniform resistance maximises the resulting power coefficient for both uniform and non-uniform incoming flows. When the upstream kinetic energy flux of the flow through the strip is used for normalisation, the same optimisation result is obtained for uniform incoming flow, regardless of the assumed resistance distribution. However, for a non-uniform incoming flow, the optimal resistance distribution is non-uniform, with greater resistance applied in regions of higher velocity within the shear flow. This different optimisation result for non-uniform flow arises physically because the kinetic energy flux used in the second power-coefficient definition depends on the resistance applied across the strip, whereas the first does not. Two-dimensional direct numerical simulations are employed to examine the applicability and limitations of the multi-streamtube theory. The numerical and optimisation results together demonstrate that optimising the resistance distribution requires accounting not only for the non-uniformity of the incoming flow but also for the local flow variability around the strip.