Unsteady transonic flow past a quarter-plane

Abstract

One of the most significant mechanisms of noise generation in prototype counter-rotation propeller systems is the unsteady interaction between the rear row and the wake and trailing tip vortices shed by the front row. A crucial part of predicting this noise is the determination of the resulting unsteady lift distribution on the rear row; since much of the interaction occurs in the vicinity of the rear-row blade tips, however, two-dimensional airfoil response theory cannot be applied exclusively, and some account of the presence of the blade tip must be taken. With this in mind, we solve the model problem of the unsteady interaction between a convected harmonic velocity gust and a quarter-plane, for the case of mean flow Mach number in the transonic range. The detailed lift distribution near the leading edge and corner is analysed, revealing the complicated nature of the lift singularity at the corner, and allowing the lift distribution throughout a narrow region along the leading edge to be determined. A closed-form expression for the practically important acoustically weighted lift is derived, which could easily be incorporated into existing noise prediction schemes in order to correct the rear-row blade response calculations for the presence of the blade tips. The radiation in this quarter-plane problem is also considered, and the field is seen to possess two components, one arising from the interaction between the gust and the semi-infinite leading edge, and the other from the interaction between the gust and the blade corner. The acoustic energy associated with this second term, corresponding to the conversion of vortical energy into sound by the corner, is considered in detail, and the directivity and parametric scaling determined. Our exact solution to this model problem is also used to assess the accuracy of a strip-theory approximation, which is seen to be accurate in this case over only a restricted range of observer positions.