On the aerodynamic forces of a plunging airfoil

The generation of aerodynamic forces by a plunging NACA0012 airfoil at a Reynolds number of 20,000 was studied for a range of non-dimensional plunge frequencies k and amplitudes h using a 2D unsteady compressible Navier-Stokes solver, an unsteady panel method (UPM) and Garrick's analysis. Calculations using these two methods indicate that the forces collapse reasonably well with kh (or equivalently the Strouhal number), but are only weak functions of k. In contrast results from the NS code indicate that the forces are dependent on both k and kh independently, with large variations at low frequencies. The frequency dependence was found to be a result of vortex shedding from the leading edge of the airfoil, and appears to result from two factors. Firstly at high plunge frequencies k, the leading edge vortex does not have sufficient time to grow, whereas at low k the vortex can become a sizeable fraction of the airfoil chord before separating. Secondly once the vortex separates, it is convected downstream over the Surface of the airfoil. Due to the low pressure in the vortex core, thrust is maintained while the vortex is upstream of the airfoil maximum thickness point (where the airfoil surface is tilted upstream and the vortex low pressure creates an upstream suction force). Once past this point, the airfoil surface is tilted downstream and the vortex contributes to drag rather than thrust. At high plunge frequencies the vortex cannot be convected far downstream before the motion cycle creates another leading edge vortex on the opposite side of the airfoil, so the impact is lessened. At low k however the vortex travels far downstream over the airfoil surface, causing drag for a larger portion of the flapping cycle, and therefore lower propulsive efficiency. These results have strong implications on how the thrust and the propulsive efficiency can be maximised by controlling the relative amplitudes and phases of combined pitching and plunging motions of an airfoil.