Performance augmentation mechanism of in-line tandem flapping foils

The propulsive performance of a pair of tandem flapping foils is sensitively dependent on the spacing and phasing between them. Large increases in thrust and efficiency of the hind foil are possible, but the mechanisms governing these enhancements remain largely unresolved. Two-dimensional numerical simulations of tandem and single foils oscillating in heave and pitch at a Reynolds number of 7000 are performed over a broad and dense parameter space, allowing the effects of inter-foil spacing ($S$) and phasing ($\unicode[STIX]{x1D711}$) to be investigated over a range of non-dimensional frequencies (or Strouhal number, $St$). Results indicate that the hind foil can produce from no thrust to twice the thrust of a single foil depending on its spacing and phasing with respect to the fore foil, which is consistent with previous studies that were carried out over a limited parameter space. Examination of instantaneous flow fields indicates that high thrust occurs when the hind foil weaves between the vortices that have been shed by the fore foil, and low thrust occurs when the hind foil intercepts these vortices. Contours of high thrust and minimal thrust appear as inclined bands in the $S-\unicode[STIX]{x1D711}$ parameter space and this behaviour is apparent over the entire range of Strouhal numbers considered $(0.2\leqslant St\leqslant 0.5)$. A novel quasi-steady model that utilises kinematics of a virtual hind foil together with data obtained from simulations of a single flapping foil shows that performance augmentation is primarily determined through modification of the instantaneous angle of attack of the hind foil by the vortex street established by the fore foil. This simple model provides estimates of thrust and efficiency for the hind foil, which is consistent with data obtained through full simulations. The limitations of the virtual hind foil method and its physical significance is also discussed.