Supersonic-Transonic Flow Generated by a Thin Airfoil in a Stratified Atmosphere.

摘要

This paper deals with the steady, frictionless, nonheatconducting flow field of a thin airfoil moving supersonically in an atmosphere with a weak wind gradient. Because of this wind gradient, the flow gradually becomes transonic in the far-field below the airfoil. It is shown that a solution which describes both supersonic and transonic regimes can be derived by matching two expansions. One of these describes the flow in the 'outer' supersonic region, and is a multiple-scale expansion which can be calculated analytically. Since this expansion becomes singular as the local Mach number approaches unity, one then considers an 'inner' transonic expansion valid in a layer of the atmosphere which is thin compared to the distance between the airfoil and the sonic line. Although the solution of the transonic equation cannot be calculated analytically for the present case where there are shocks in the flow, an asymptotic representation of this solution is derived in the supersonic region of the far-field, and this is used to show that the supersonic and transonic solutions match. The flow field in the transonic region is then calculated numerically using the method of characteristics and incorporating the appropriate shock relations. (Author)