This paper considers the acoustic field inside an annular duct carrying mean axial and swirling flow and with either acoustically hard or lined walls. The particular aim is to compute the Green's function, which is required for predicting the noise generated by known acoustic sources, both approximately and numerically. Asymptotic approximations for first the eigenmodes and then the Green's function are derived in the realistic limit of high reduced frequency, which are found to agree very favourably with results determined numerically, even for relatively modest frequencies. Using a blend of uniform WKB asymptotics and numerics, singular cases in which multiple turning points are present are treated, allowing computation of accurate results across a very wide range of parameter values and flows.