Derivation of the Viscous Moore-Greitzer Equation for Aeroengine Flow

The viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, is derived using a scaled Navier-Stokes equation. The method utilizes a separation of scales argument, based on the different spatial scales in the engine and the different time scales in the flow. The pitch and size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The motion of the stator and rotor blades in the compressor produces a very turbulent flow on a fast time scale. The leading order equation, for the fast-time and local scale, describes this turbulent flow. The next order equations, produce an axi-symmetric swirl and a flow-pattern analogous to Rayleigh-Bénard convection rolls in Rayleigh-Bénard convection. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. A higher order equation, in the small parameter, describes these global flow modulations, when averaged over the small (local) spatial scales, the fast time scale and the time scale of the vortex rotations. Thus a more general system of spatially global, slow-time equations is obtained. This system can be solved numerically without any approximations. The viscous MooreGreitzer equation is obtained when small inertial terms are dropped from these slow-time, spatially global equations, averaged once more in the axial direction. The new equations are simulated with two different simplifying assumptions and the results compared with simulations of the viscous Moore-Greitzer equations. Dept of Math, UCSB, Santa Barbara, CA 93106 [email protected], partially supported by the National Science Foundation grants number DMS 0072191 and DMS 0352563, and the University of Iceland, Reykjav́ık, 107 Iceland Dept of Math, Michigan State University, East Lansing, MI 48824 [email protected], research partially supported by NSF grant number DMS 0072191. Swedish Defence Research Agency, FOI, P.O.Box 1165, SE-581 11 Linköping, Sweden [email protected] 1