Fredholm--residue selection of the unsteady Kutta amplitude

Abstract

We give an operator-theoretic interpretation of unsteady Kutta selection in trailing-edge acoustic receptivity. The inviscid acoustic--wake problem leaves one outgoing wake amplitude undetermined. We show that, under explicit structural hypotheses, this amplitude is the same scalar obtained from three representations: cancellation of the inverse-square-root edge singularity, Fredholm compatibility of the viscous lower-deck problem, and the residue of the Kutta-normalized transform solution at the downstream wake pole: $\displaystyle A = -\frac{C_-^{(0)}}{C_-^{(KH)}} = -\frac{\langle \mathbf F_{\rm inc},\Psi^\ast\rangle}{\langle \mathbf F_{KH},\Psi^\ast\rangle} = i\operatorname*{Res}_{\alpha=\alpha_{KH}}\mathcal M(\alpha)$. The inner Fredholm--edge mechanism is verified exactly in a linear-shear lower-deck model, where the primal shear and adjoint velocity are Airy fields and the edge concomitant is nonzero outside a discrete resonance set.