A Preconditioned Arbitrary Mach Number Scheme Applied to Rotating Machinery

Abstract

It is well known that the compressible flow equations face difficulties at low Mach numbers due to the large ratio of the acoustic and convective time scales, which leads to an illconditioned system when solving low-speed or incompressible flows. The time-dependent system of the Euler and Navier-Stokes equations exhibits stiffness that is strongly dependent on the Mach number and the Reynolds number. In this regard, Briley et al. (Briley et al., 1983) first introduced the preconditioning method using a simple constant preconditioning matrix added to a non-dimensional form of the isoenergetic equations. This generally improved convergence for a test case with reference Mach number Mr = 0.05 using the ADI factorization scheme in primitive variables. However, when applying this preconditioning to rotating flows in either fixed or rotating frame formulation, Sheng, et al. (Sheng & Wang, 2006; Wang & Sheng, 2005) observed the instability of the scheme due to the large variation of rotating speeds across the computational domain. Furthermore, it was found that the convergence of the preconditioned equations is sensitive to the selection of the reference Mach number, especially in rotating flows with a wide range of radial speeds and physical time scales. It was later proved using the Fourier footprint analysis (Wang & Sheng, 2005) that the eigensystem of the compressible governing equations can be significantly affected by both free stream and rotating speeds in rotating flows. A modified preconditioning scheme was thus proposed (Sheng & Wang, 2006; Wang & Sheng, 2005), in which both the global reference Mach number and the rotating Mach number are considered in the formulation of the preconditioning matrix. In general, this modified preconditioning scheme has improved the convergence and accuracy of compressible flows in subsonic, transonic and supersonic Mach number regimes. In this study, the modified preconditioning is further investigated and validated for predicting incompressible viscous flows in rotating machinery, such as a marine propeller P5168. One of the most important characters for marine propellers is the cavitation observed in high speed flows, which is of vital importance because of the damage of metal surfaces and degradation of performance of lifting surfaces. It is also a source of high-frequency noise in connection with acoustic detection of ships and submarines. Cavitation would take place when the local pressure drops to the vapor pressure. Therefore, accurate prediction of the velocity and pressure field is essential for understanding the process of cavitation inception and improving the hydrodynamic performance of the marine propeller. Since the condition for cavitation inception is related to the tip-vortex location, strength, convection, cavitation