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

Abstract

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 [Formula: see text] in the duct and the radiated power [Formula: see text] in the far field differ by the amounts of hydrodynamic far field powers [Formula: see text] inside and [Formula: see text] outside the wake (vortex sheet) emanating from the duct edge, plus the power [Formula: see text] 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. [Formula: see text], [Formula: see text] and [Formula: see text] are always positive. This is normally the case too for [Formula: see text] and [Formula: see text]. But for not too high frequencies or other circumstances where shed vorticity produces more sound than was necessary for its creation, [Formula: see text] and even [Formula: see text] may also be negative.