Maximum propulsive efficiency of two pitching and plunging plates in tandem at low Reynolds number A numerical analysis

Purpose This paper aims to consider the thrust force generated by two plunging and pitching plates in a tandem configuration in forward flight to find out the configuration that maximizes the propulsive efficiency with high-enough time-averaged lift force. Design/methodology/approach To that end, the Navier-Stokes equations for the incompressible and two-dimensional flow at Reynolds number $500 are solved. As the number of parameters is quite large, the case of constant separation between the plates (half their chord length), varying seven non-dimensional parameters related to the phase shift between the heaving motion of the foils, the phase lag between pitch and heave of each plate independently and the frequency and amplitude of the heaving and pitching motions are considered. This analysis complements some other recent studies where the separation between the foils has been used as one of the main control parameters. Findings It is found that the propulsive efficiency is maximized for a phase shift of 180 degrees (counterstroking), when the reduced frequency is 2.2 and the Strouhal number based on half the plunging amplitude is 0.17, the pitching amplitude is 25 degrees and when pitch leads heave by 135 degrees in both the fore -plate and the hind plate. The propulsive efficiency is about 20 per cent, just a bit larger than that of an isolate plate with the same motion as the fore-plate, but the corresponding lift force is negligible for a single plate. The paper discusses this vortical flow structure in relation to other less efficient ones. Finally, the effect of the separation between the plates and the Reynolds number is also briefly discussed. Originality/value The kinematics of two flapping plates in tandem configuration that maximizes the propulsive efficiency are characterized discussing physically the associated vortical flow structures in comparison with less efficient kinematic configurations. A much larger number of parameters in the optimization procedure than in previous related works is considered.