Introduction. Centrifugal compressors are widely used for distributing city gas, for transporting natural gas through long pipelines, for liquidizing natural gas, for pressurizing gas at various kinds of chemical plants. Also, they are indispensable as turbo-superchargers for reciprocating engines to increase output power dramatically. In many of these cases, demand for flow rate at high pressure varies during operation at a fixed shaft speed.As complex configurations of turbo-machines make it difficult to analyze the behavior of fluid flow through them theoretically, a lot of empirical descriptions have been used for design. If better understanding on flow in turbo-machines makes it possible to design machines with better efficiency, a considerable amount of energy resources will be saved throughout the world.Basic problems on turbo-machine. As opposed to reciprocating compressors, which are popular as high pressure-ratio and low capacity compressors, turbo-compressors are compact for the capacity and quiet without vibration, as they continuously pressurize fluid by means of high-speed rotating impeller. Consequently they are widely used and especially for large capacity applications in industry turbo-compressors are exclusively adopted.In steady flow, fluid flows toward a zone of low pressure, being accelerated by favorable pressure gradient, but in a diverging passage flow increases the pressure by reducing the velocity. In cases of centrifugal compressors in Fig. 1, this principle is directly applicable only to the diffuser. Flow through a rotating impeller is not steady unless it is observe in the rotating system. That is, if relative velocity in an impeller is reduced, static pressure increases. Using this principle the pressure of flow through a compressor increases pressure both in the impeller and in the diffuser by reducing the relative velocity successively.In cases of axial compressors where diameter of impeller and diffuser remains constant from the inlet to the exit, phenomena induced by rotating impeller and by stationary diffuser are identical to those in cases of centrifugal compressors except that the flow remains on a cylindrical surface.Inlet geometry of vanes. Specifically, in the case of a rotating impeller, straight axial or radial flow toward a rotating impeller is recognized as diagonally incoming flow relative to the impeller. Therefore, at the inlet all the vanes of the impeller should be oriented toward the direction of incoming flow relative to the impeller. Furthermore, the vanes must be curved smoothly toward axial or radial direction so that passages between vanes become wider downstream. Consequently fluid increases pressure while it flows Proc. Japan Acad., 81, Ser. B (2005) Abstract: Turbo-machines (e.g. compressors, blowers and pumps) obtain energy transfer from their rotating impellers to fluid flowing through them, and they are used for a wide range of purposes obtaining pressurized fluid flow. The author created theoretical models to clearly describe the complicated flow...