Efficient mode-matching based on closed form integrals of Pridmore-Brown modes

A new mode-matching method is developed for acoustic modes in axially sectioned lined ducts with parallel, but otherwise arbitrary mean flow. The method is evaluated in detail for a circularly symmetric configuration (where the mean flow depends only on the radial coordinate), with the so-called Pridmore-Brown modes, satisfying the radial Pridmore-Brown equation. Classically, the matching of the modal representations at the interfaces between the sections is based on continuity of pressure and velocity in combination with projection to a suitable but unrelated set of test functions by means of a standard integral inner product. The alternative proposed here uses essentially the same Pridmore-Brown modes, but instead of the standard inner product it applies a particular bilinear form that can be evaluated in closed form. Apart from numerical efficiency, this approach features also a higher accuracy because it avoids the inherently inaccurate numerical quadrature of oscillating functions. The results are compared with results obtained by an implementation of the classical approach, and the agreement is excellent, with higher accuracy and with greater computational efficiency.