On the flow field of a rapidly oscillating airfoil in a supersonic flow

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Abstract
This paper examines the features of the flow field off the surface of an oscillating flat-plate airfoil immersed in a two-dimensional supersonic flow Although the exact linearized solution for a supersonic unsteady airfoil has been known for a long time, its expression in the form of an integral is not convenient for a physical interpretation. In the present paper, the quintessential features of the flow field are extracted from the exact solution by obtaining an asymptotic expansion in descending powers of a frequency parameter through the repeated use of the stationary-phase and steepest descent methods. It is found that the flow field consists of two dominant and competing signals: one is the acoustic ray or that component arising from Lighthill's ‘convecting slab’ and the other is the leading-edge disturbance propagating as a convecting wavelet. The flow field is found to be divided into several identifiable regions defined by the relative magnitude of the signals, and the asymptotic expansions appropriate for each flow region are derived along with their parametric restrictions. Such intimate knowledge of the flow field in unsteady, supersonic flow is of interest for interference aerodynamics and related acoustic problems.

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