The effect of chordwise flexibility on the thrust and efficiency of a flapping foil

We show experimentally that properly selected chord-wise flexibility can have a significant effect on the propulsive efficiency of two-dimensional flapping (heaving and pithing) foils, up to a 36% increase, compared to the efficiency of a rigid foil, with small loss in thrust. Two different foil kinematics are employed in the experiments: the first using simple harmonic heave and pitch motions; and the second using a multi-harmonic heave motion combined with a harmonic pitch motion, selected to produce a harmonic angle of attack variation. For both types of motion, chordwise flexibility improves efficiency, the Shore A60 producing the highest efficiency; the second mode of kinematics causes the thrust coefficient to increase significantly for high Strouhal numbers. A non-dimensional flexibility parameter is developed which provides a scaling law for the effect of flexibility. I. NOMENCLATURE • c,s = hydrofoil chord and span lengths [m] • f = heave frequency [Hz] • ω = oscillating frequency [rad/s] • U = horizontal towing velocity [m/s] • dh dt = heave velocity [m/s] • φ = the angle of attack due to heave motion [rad] • θ = the physical pitch angle [rad] • α = the total angle of attack [rad] • ψ = the phase angle by which the pitch motion leads the heave motion [rad] • ν = the kinematic viscosity of water [m/s] • ρ = the water density [kg/m] • h0 = heave amplitude, limited by a travel of the linear drive [m] • θ0 = pitch amplitude [rad] • h0/c = heave amplitude to chord ratio • Rec = Reynolds number ( Uc ν ) • St = Strouhal number (2h0f/U ) • X(t) = horizontal instantaneous force [N ] • Y(t) = vertical instantaneous force [N ] • τ(t) = instantaneous applied torque [N ·m] • F̄x, F̄y = average thrust and lift forces [N ] • P = power input to the foil [Watt] P = 1 T ( ∫ T 0 Y (t) dh dt dt + ∫ T