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The phenomenon of cut-on cut-off transition of acoustic modes in ducts with mean flow is examined using an analytical multiple-scales solution, and compared to solutions obtained from a numerical finite-element method. The analytical solution, derived for an arbitrary duct with irrotational mean flow, remains valid to leading-order throughout the duct. In other words, it is a composite solution, encompassing both the inner boundary-layer solution in the neighbourhood of the transition point and the outer slowly varying modal solution far upstream and downstream. Several test cases are defined and presented within a geometry representative of a high-bypass turbofan engine. The cases span a wide realistic range of frequencies and circumferential mode numbers both with and without mean flow, including one numerically-challenging investigation of cutoff cut-on transition of a mode. The agreement is in most cases remarkably good. Slight differences in position of the pressure pattern can be observed for cases with mean flow, which seem due to the slight variations in mean flow fields obtained from both methods. When cut-on cut-off transition occurs for high Helmholtz number and high radial mode number a certain amount of modal scattering is observed. An attempt is made to explain this by incorporating the presence of neighbouring modes in the asymptotic scaling arguments for the turning point region. The composite solution should enable designers to continue to use multiple-scales theory to examine flow pressure and noise transmission inside an engine duct, whilst now being able to include directly the contributions of modes undergoing transition without encountering singular behaviour.