Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil flow with vortex production — A general model

By using a special momentum approach and with the help of interchange between singularity velocity and induced flow velocity, we derive in a physical way explicit force formulas for two-dimensional inviscid flow involving multiple bound and free vortices, multiple airfoils, and vortex production. These force formulas hold individually for each airfoil thus allowing for force decomposition, and the contributions to forces from singularities (such as bound and image vortices, sources, and doublets) and bodies out of an airfoil are related to their induced velocities at the locations of singularities inside this airfoil. The force contribution due to vortex production is related to the vortex production rate and the distance between each pair of vortices in production, thus frame-independent. The formulas are validated against a number of standard problems. These force formulas, which generalize the classic Kutta–Joukowski theorem (for a single bound vortex) and the recent generalized Lagally theorem (for problems without a bound vortex and vortex production) to more general cases, can be used to identify or understand the roles of outside vortices and bodies on the forces of the actual body, optimize arrangement of outside vortices and bodies for force enhancement or reduction, and derive analytical force formulas once the flow field is given or known.