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-B´enard convection rolls in Rayleigh-B´enard 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 Moore- Greitzer 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.

In this paper, we study how to image both the location and the shape of extended targets using the response matrix obtained from inter-element response of an active array of transducers. In particular, the time reversal technique is used for efficient initial localization of the target and the level set method is used for shape reconstruction. We then show how to couple the location estimation and shape reconstruction in a complementary way to improve accuracy for range estimation. Resolution analysis for active arrays in remote sensing regime is also presented. We illustrate with numerical experiments which show that the method is capable of imaging objects with complicated shapes and that the method is robust with respect to noisy data. (C) 2004 Elsevier Inc. All rights reserved.