Long-term stable reduced models for hydraulic systems governed by Reynolds averaged Navier-Stokes equations

Realtime reconstruction of flow fields is a prerequisite to precise condition monitoring of hydraulic plant components. Because models based on Navier-Stokes equations are usually too complex for a realtime use, reduced order models (ROMs) are a viable alternative. Specifically, ROMs based on proper orthogonal decomposition and Galerkin projection are attractive, because they result in small models comprising a few (in the order of ten) ordinary differential equations. However, ROMs of this type often turn out not to be stable even for stable points of operation of the real system and the original partial differential equation model. Stability properties can sometimes be improved by fitting the ROM to the original data obtained from finite-volume or similar simulations for the partial differential equation model. This is unsatisfactory because a ROM is first derived in a systematic manner and then altered a posteriori in an ad hoc fashion. We propose to use a Petrov-Galerkin projection instead, i.e., we retain the optimal truncated basis to span the simulation data space but optimize the basis used in the projection separately. We demonstrate this improves the stability properties of the ROM considerably for a centrifugal pump, which is modeled with the Reynolds averaged Navier-Stokes equations.