Inverse Computation of Profile Shapes for Given Transonic Flow-Configurations With and Without Detached Bow-Shocks in Two-Dimensional Turbomachinery Cascades

A method is presented for solving the inverse problem to compute the profile shape of turbomachinery cascades for given transonic pressure distributions. Weak normal shocks on the profile surface and detached bow shocks may appear in the flow field. A finite difference method based on the small disturbance formulation of the nonlinear potential flow problem is used. Different finite difference algorithms are used appropriate to the behavior of the differential equation at each mesh point. The method involves iteration between the direct solution and the design solution. In the direct solution, the Neumann problem for the potential is solved. In the design step, the Neumann condition is replaced by a Dirichlet condition. That is, the pressure distribution is specified. According to Langley’s idea [18] and based on the irrototionality of the flow, the φ_xy values on the profile are calculated by extrapolation and differentiation of the pressure distribution. Integrating φ_xy in x-direction gives φ_y, which is used in the same way as in the direct problem. After convergence, an integral boundary layer calculation is used to determine the displacement thickness and to correct the profile coordinates. Due to this correction and the changes of the finite difference operators according to the flow type, there is a good agreement between the calculated and the real profile shapes of the test cases.