Unsteady Supersonic Cascade in Subsonic Axial Flow

This paper presents an analysis for determining the unsteady flowfield produced by an oscillating cascade placed in a supersonic stream which has a subsonic velocity component normal to the cascade. The analysis is based on the assumptions of an inviscid, two dimensional, linearized flowfield. Solutions for the velocity potential and the blade pressure distributions which satisfy the blade-to-blade periodicity condition are developed explicitly in terms of disturbance functions distributed on blade and wake surfaces. The boundary conditions of flow tangency at blade surfaces and continuity of pressure across wake surfaces provide integral relations which can be solved numerically to evaluate the disturbance functions. Predicted blade pressure distributions are in good agreement with results determined from a previous finite cascade solution. Further, in the two limiting cases of sonic axial velocity and zero frequency, the present solution approaches the lower limit of Lane's solution for supersonic axial flow, and it reduces to an Ackeret type of steady-state solution, respectively. The numerical examples indicate that a single~degree- of-freedom torsional instability will exist over a broad range of cascade parameter values.