Acoustic energy balances for sound radiated from duct exit with mean flow

An old model problem for the exchange of energy between sound field and mean flow by vortex shedding has been worked out in numerical detail. The analytically exact solution of the problem of reflection, diffraction and radiation of acoustic modes in a semi-infinite annular duct with uniform subsonic mean flow, including shedding of unsteady vorticity from the duct exit, allows a precise formulation of Myers’ energy for perturbations of an inviscid mean flow. The transmitted power P t in the duct and the radiated power P f in the far field differ by the amounts of hydrodynamic far field powers P H i inside and P H o outside the wake (vortex sheet) emanating from the duct edge, plus the power P w that disappears into the vortex sheet. This last component represents the source term in Myers’ energy equation. This is evidence of the non-conserved character of acoustic energy in mean flow, owing to the coupling of the acoustic field with the mean flow. P f , P H i and P H o are always positive. This is normally the case too for P w and P t . But for not too high frequencies or other circumstances where shed vorticity produces more sound than was necessary for its creation, P w and even P t may also be negative.