Minimum-domain impulse theory for unsteady aerodynamic force

TL;DR

This paper extends impulse theory for unsteady aerodynamics to finite and minimum domains, for both incompressible and compressible flows, showing that only certain vortical structures determine the force.

Contribution

It introduces a minimum-domain impulse theory for incompressible flows and a vorticity-moment theory for compressible flows, simplifying force calculations by focusing on key vortical structures.

Findings

Force determined by vortical structures connected to the body

Minimum-domain impulse theory proven as a theorem for incompressible flow

Numerical experiments confirm the theories' effectiveness

Abstract

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 (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 flow, we find that the global form in terms of the curl of momentum, obtained by Huang (Unsteady Vortical Aerodynamics. Shanghai Jiaotong Univ. Press, 1994), can be generalized to having arbitrary finite domain, but the formula is cumbersome and in general the curl of momentum no longer has discrete structure and hence no minimum-domain theory exists. Nevertheless, as the measure of transverse process only, the unsteady field of vorticity may still have discrete wake. This leads to a minimum-domain compressible vorticity-moment theory in terms of density-weighted vorticity (but it is beyond the classic concept of impulse). These new findings and applications have been confirmed by our numerical experiments. The results not only open an avenue to combine the theory with computation-experiment in wide applications, but also reveals a physical truth that it is no longer necessary to account for all wake vortical structures in computing the force and moment.