This paper describes Fortran programs that give the solution to the two-dimensional, subsonic, nonviscous flow problem on a blade-to-blade surface of revolution of a turbomachine. Flow may be axial, radial, or mixed. There may be a change in stream channel thickness in the through-flow direction. Either single, tandem, or slotted blades may be handled as well as blade rows with splitter vanes. Also, small regions may be magnified to give more detail where desired, such as around a leading or trailing edge or through a slot. The method is based on a finite difference solution of the stream function equations. Numerical examples are shown to illustrate the type of blades which can be analyzed, and to show results which can be obtained. Results are compared with experimental data.

A method of analyzing flow through a turbomachine i s summarized which i s r l s u i t a b l e f o r computer programming. The method, which has been reported i n NASA t r a r y quasi-orthogonal r a t h e r than t h e normal t o t h e streamline as used i n previous methods. A quasi-orthogonal is defined t o be any curve that i n t e r s e c t s every streamline between t h e flow boundaries e x a c t l y once, as does an orthogonal t o t h e streamlines. With t h i s method a streamline a n a l y s i s can be made on any given stream surface. A quasi-three-dimensional s o l u t i o n can be obtained by using t h e method f o r a hub-to-shroud analysis, followed by blade-to-blade analyses at hub, mean, and shroud. As an example, t h e method was applied t o a radial inflow t u r b i n e w i t h s p l i t t e r blades. The complete quasi-three-dimensional SOYutioii was o'uiained i n a s i n g l e computer run.

A finite-difference method for the solution of symmetric positive linear differential equations is developed. The method is applicable to any region with piecevvise smooth boundaries. Methods for solution of the finite-difference equations are discussed. The finite-difference solutions are shown to converge at essentially the rate 0(h1'2) as h-> 0, h being the maximum distance between adjacent mesh-points. An alternate finite-difference method is given with the advantage that the finite-difference equations can be solved iteratively. However, there are strong limitations on the mesh arrangements which can be used with this method. Introduction. In the theory of partial differential equations there is a fundamental distinction between those of elliptic, hyperbolic and parabolic type. Generally each type of equation has different requirements as to the boundary or initial data which must be specified to assure existence and uniqueness of solutions, and to be well posed. These requirements are usually well known for an equation of any particular type. Further, many analytical and numerical techniques have been developed for solving the various types of partial differential equations, subject to the proper boundary conditions, including even many nonlinear cases. However, for equations of mixed type much less is known, and it is usually difficult to know even what the proper boundary conditions are. As a step toward overcoming this problem Friedrichs [1] has developed a theory of symmetric positive linear differential equations independent of type. Chu [2] has shown that this theory can be used to derive finite-difference solutions in two-dimensions for rectangular regions, or more generally, by means of a transformation , for regions with four corners joined by smooth curves. In this paper a more general finite-difference method for the solution of symmetric positive equations is presented (based on [3]). The only restriction on the shape of the region is that the boundary be piecewise smooth. It is proven that the finite-difference solution converges to the solution of the differential equation at essentially the rate 0(h112) as h-> 0, h being the maximum distance between adjacent mesh-points for a two-dimensional region. Also weak convergence to weak solutions is shown. An alternate finite-difference method is given for the two-dimensional case with the advantage that the finite-difference equation can be solved iteratively. However , there are strong limitations on the mesh arrangements which can be used

A method for calculating turbine stage performance is described. The usefulness of the method is demonstrated by comparing measured and predicted efficiencies for nine different stages. Comparisons are made over a range of turbine pressure ratios and rotor speeds. A quasi-3D flow analysis is used to account for complex passage geometries. Boundary layer analyses are done to account for losses due to friction. Empirical loss models are used to account for incidence, secondary flow, disc windage, and clearance losses. Nomenclature