Abstract: In this paper, by applying the tensor analysis and semigeodesio coordinate system, we have established a partial differential equation and conditions satisfied by the stream function on any stream surface of three dimensional flow in turbomachines. We have also provided computational examples by use of finite element method, and the error estimate of Galerkin's finite element approximation solution which depends on Mach number.
In addition, we have drawn the conclusion that the transonic equation for three dimensional compressible flow in the turbomachines is equivalent to an optimal control problem of a linearly distributed parameter system. This problem can be disoretized by the finite element method and solved by a conjugate gradient algorithm. A new class of method for solving the transonic equation is particularly well adapted to problems with complicated three dimensional geometries and shocks, and at the same time it can be used to improve the precision of approximation solution.