Nonlinear Analysis of Unsteady Flows in Multistage Turbomachines Using Harmonic Balance

Abstract

A harmonic balance technique for the analysis of two-dimensional linear (small-disturbance) and nonlinear unsteady flows in multistage turbomachines is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of multistage machines to both blade vibration (the flutter problem) and wake interaction (the forced response problem). In general, the flowfield may have multiple excitation frequencies that are not integer multiples of each other, so that the unsteady flow is (sometimes) aperiodic in time. Using our approach, we model each blade row using a computational grid spanning a single blade passage. In each blade row, we store several subtime level solutions. For flows that are periodic in time, these subtime levels span a single time period. For aperiodic flows, the temporal period spanned by these subtime level solutions is sufficiently long to sample the relevant discrete frequencies contained in the aperiodic flow. In both cases, these subtime level solutions are related to each other through the time-derivative terms in the Euler or Navier-Stokes equations and boundary conditions; complex periodicity conditions connect the subtime levels within a blade passage, and interrow boundary conditions connect the solutions among blade rows. The resulting discretized equations, which are mathematically steady because time derivatives have been replaced by a pseudospectral operator in which the excitation frequencies appear as parameters, can be solved very efficiently using multigrid acceleration techniques. In this paper, we apply the technique to both flutter and wake-interaction problems and illustrate the influence of neighboring blade rows on the unsteady aerodynamic response of a blade row.