On the added-mass force and moment and the vortex projection method. Application to thin airfoils

Abstract

The force and moment on a solid body in non-uniform motion in an incompressible fluid are derived following the projection technique of Quartapelle and Napolitano (1983 AIAA J. 21, 911–913) and Chang (1992 Proc. R. Soc. A-Math. Phys. Engng Sci. 437, 517–525) in a compact vector form, identifying added-mass terms equivalent to those resulting from potential-flow theory. The expressions are formally much simpler in a non-inertial frame fixed relative to the body, though general relations for an inertial frame are also given. Approximate analytical expressions are obtained for the force and moment on two-dimensional elliptic bodies at high Reynolds numbers, each one consisting of just two terms: an added-mass component and a vortex term (integration of a projection of Lamb’s vector). The formulation is applied to a flat plate in steady motion and in non-uniform oscillatory motion, recovering classical results in both cases, but adding new physical insight about the origin of the different elements of the force and moment. It is shown that, in the non-steady motion, the added-mass components of the force and moment cancel out with part of their respective vortex terms, so that all the force and moment come from appropriate projections of Lamb’s vector, including the supposedly added-mass force and moment. This result indicates that one has to be cautious about simplified aerodynamic models for flapping foils based on a simple separation of the added-mass and vortical components of the force and moment. The present results can be used to obtain analytically the force and moment on an ellipse with arbitrary motion through an incompressible fluid at high Reynolds numbers.