Minimum-domain impulse theory for unsteady aerodynamic force

We extend the impulse theory for unsteady aerodynamics from its classic global form to finite-domain formulation then to minimum-domain form and from incompressible to compressible flows. For incompressible flow, the minimum-domain impulse theory raises the finding of Li and Lu [“Force and power of flapping plates in a fluid,” J. Fluid Mech. 712, 598–613 (2012)] to a theorem: The entire force with discrete wake is completely determined by only the time rate of impulse of those vortical structures still connecting to the body, along with the Lamb-vector integral thereof that captures the contribution of all the rest disconnected vortical structures. For compressible flows, we find that the global form in terms of the curl of momentum ∇ × (ρu), obtained by Huang [Unsteady Vortical Aerodynamics (Shanghai Jiaotong University Press, 1994)], can be generalized to having an arbitrary finite domain, but the formula is cumbersome and in general ∇ × (ρu) no longer has discrete structures and hence no minimum-domain...