Simulating Periodic Unsteady Flows Using a Cubic-Spline-Based Time Collocation Method

Abstract

Time-periodical unsteady flows are typical in turbomachinery. Simulating such flows using a conventional time marching approach is the most accurate but is extremely time consuming. In order to achieve a better balance between accuracy and computational expenses, a cubic-spline-based time collocation method is proposed. In this method, the time derivatives in the Navier–Stokes equations are obtained by using the differential quadrature method, in which the periodical flow variables are approximated by cubic splines. Thus, the computation of a time-periodical flow is substituted by several coupled quasi-steady flow computations at sampled instants. The proposed method is then validated against several typical turbomachinery periodical unsteady flows, i.e., transonic compressor rotor flows under circumferential inlet distortions, single stage rotor–stator interactions, and IGV–rotor interactions. The results show that the proposed cubic-spline-based time collocation method with appropriate time sampling can well resolve the dominant unsteady effects, while the computational expenses are kept much less than the traditional time-marching simulation. More importantly, this paper provides a framework on the basis of a time collocation method in which one may choose more compatible test functions for the concerned specific unsteady flows so that better modeling of the flows can be expected.