Abstract: The expressions of the governing equations of turbulent flow with contravariant physical velocity components in a general non orthogonal curvilinear coordinate system are derived by means of tensor theory.A numerical method and a computer program for turbulent recirculating flow in the general non orthogonal curvilinear coordinate system are developed.Since the velocity and the displacement vectors are decomposed into their contravariant components,the method has the main advantages of its obviousness in physical meanings,easiness in treatment of boundary conditions and the suitability to the numerical calculations of turbulent flow in various complex geometries.The flow fields in some kinds of complex geometries are calculated.