Fractal-like actuator disc theory for optimal energy extraction

The limit of power extraction by a device which makes use constructive interference, i.e. local blockage, is investigated theoretically. modelled using actuator disc theory in we allow the to be split into arrays and these then sub-arrays an arbitrary number times so as construct $n$ -level multi-scale original undergoes $n-1$ sub-divisions. alternative physical interpretation problem planar system arrayed turbines groups are homogeneously at smallest $n\mathrm {th}$ scale, spaced relative each other next $n-1\mathrm with this pattern repeating all subsequent larger scales. scale-separation idea Nishino & Willden ( J. Fluid. Mech. , vol. 708, 2012 b pp. 596–606) employed, assumes mixing within sub-array occurs faster than by-pass flow around that sub-array, -scale from inner scale outermost order. We investigate behaviour level device, determine arrangement discs devices) maximises coefficient (ratio extracted undisturbed kinetic energy flux through net frontal area). find optimal close fractal, fractal arrangements give similar results. With placed infinitely wide channel, zero global optimum tends unity scales infinity, 27/16 increase over Lanchester–Betz $0.593$ . For devices finite width channels, non-zero observations can made further uplift maximum coefficient. discuss fluid mechanics process examine distribution thrust wake velocity coefficients. Numerical demonstration performance due dynamics also provided. demonstrate bypass remixing ensuing losses above limits for single devices, although increase, expense overall efficiency decreases wake-scale necessarily rise. blockage two effects act extractable power; streamwise pressure gradient associated recoveries bypass-scale remixing.