Some Applications of a Time-Linearized Euler Method to Flutter & Forced Response in Turbomachinery

Abstract

This paper presents the theory and a numerical algorithm for the prediction of flutter and forced response using the three-dimensional linearized Euler equations. The nonlinear steady-state flow field is first calculated by solving the steady Euler equations using a standard numerical method. The linearized unsteady Euler equations are derived by assuming that the unsteady flow solution is a small perturbation to the steady flow. Due to linear superposition, the unsteady part of the flow can then be split into a sum of components of differing frequencies, each of which can be calculated independently. The harmonic equations for a single frequency are solved by a pseudo-time-marching approach, which allows the use of standard numerical algorithms. Computational results are presented to validate the implementation of the numerical algorithm. Comparisons are made to two-dimensional, linear unloaded flat plate cascade theory for test cases involving forced response due to wakes and pressure waves, as well as bending and torsion. The method is also applied to two 3D test cases — a high pressure turbine stage, and a fan flutter problem.