Euler-based inverse method for turbomachine blades. Part I: Two-dimensional cascades

A three-dimensional inverse method for the aerodynamic design of turbomachine blades using robust timemarching algorithms for the numerical solutions of the Euler equations is proposed. In this inverse method, the circumferential mass-averaged tangential velocity (or the blade loading) is the primary specified flow quantity, and the corresponding blade geometry is sought after. The presence of the blades is represented by a periodic array of discrete body forces which is included in the equations of motion. A four-stage Runge-Kutta time-stepping scheme is used to march a finite volume formulation of the unsteady Euler equations to a steady-state solution. Modification of the blade geometry during this time-marching process is achieved using the flow-tangency conditions along the blade surfaces. In this paper, the method is demonstrated for the design of two-dimensional infinitely thin cascaded blades ranging from the subsonic to the supersonic flow regimes, including cases with rotational flows and complex shock structures in the flow passage.