The propagation of unsteady disturbances in ducts of slowly-varying geometry,udsuch as those typical of an aero-engine, can be successfully modelled usinguda multiple scales approach. The multiple-scales approach has a number of distinctudadvantages over full numerical methods. Previous authors have validatedudthe accuracy and usefulness of the multiple scales approach by comparing withudresults obtained using the finite element method, using realistic aero-engineudconfigurations.udCut-on cut-off transition of acoustic modes in hard-walled ducts with irrotationaludmean flow is well understood. However, previous finite-element simulationsudof this phenomenon appear to indicate the possibility of energy scatteringudinto neighbouring modes at large Helmholtz numbers. In this thesis,udan attempt is made to explain such scattering phenomena in slowly varyingudaero-engine ducts using multiple-scales techniques.udIn order to model modal scattering a good understanding of cut-on cut-offudtransition is necessary. Here, the well known single turning point is revisited,udand our understanding of cut-on cut-off transition is extended to include anudanalysis of a double turning point. Then using a similar apparatus, modaludscattering in the case where a mode undergoes cut-on cut-off transition is investigated.udIt is found that, for sufficiently high frequencies, a mechanismudexists whereby a propagating incident mode can be scattered into neighbouringudmodes provided that a mean flow exists within the duct. An asymptoticudanalysis of this mechanism is presented and, by solving numerically a compositeudsolution, results in a duct of rectangular cross section are obtained. The energy distribution of the incident and neighbouring scattered modes revealsudan interaction and exchange of energy with the mean flow. This workudnow allows greater insight as well as more accurate and fast computations ofudhigh frequency mode propagation in slowly-varying hard walled ducts usingudmultiple-scales approaches.
展开▼
