Fast Estimation of Unsteady Flows in Turbomachinery at Multiple Interblade Phase Angles

T IME-LINEARIZED computational fluid dynamic (CFD) models for the computation of unsteady flows in turbomachinery are now used routinely in the design and analysis of turbomachinery blade rows, particularly to predict the onset of flutter. A typical flutter analysis of a rotor requires one to compute unsteady flow solutions over the full range of interblade phase angles, and hence requires significant computational time even for relatively efficient time-linearized flow solvers. Typically, onemight compute the unsteady aerodynamic damping at several dozen interblade phase angles to ensure that no single interblade phase angle will flutter. In this note, we describe a fast technique that significantly reduces the computational time required to perform such a flutter clearance analysis. In recent years, researchers have developed models to reduce the time required to compute multiple unsteady flow solutions when one or more parameters is varying. In many (but not all) cases, the varying parameter is the frequency of airfoil vibration [1,2]. For example, Florea and Hall [3] computed the dominant eigenfrequencies and mode shapes of the unsteady fluid motion within a twodimensional cascade of airfoils using a nonsymmetric Lanczos algorithm, and then used these eigenmodes to construct a lowdegreeof-freedom reduced-ordermodel of the unsteadyflowfield around an airfoil. Kim [4] developed a frequency-domain proper orthogonal decomposition (POD) technique and applied it to both mechanical and fluid dynamic models. Hall et al. [5] developed reduced-order models of small-disturbance unsteady flows around airfoils and turbomachinery cascades using a POD technique. Similarly,Willcox et al. [6] developed an efficient frequency-domain proper orthogonal decomposition method for low order aeroelastic control of turbomachines. Cizmas and Palacios [7] used a POD technique to construct reduced-order models to examine a turbine rotor-stator interaction problem. (For a complete review of ROM/PODmethods, see Lucia et al. [8]). In most of these investigations, the reduced-order model is constructed in such a way that estimates of the unsteady flow can be constructed rapidly over a wide range of (possibly undetermined) vibrational frequencies. In the turbomachinery flutter problem, however, because of the high mass ratio associated with turbomachinery blading, the frequency of flutter is known a priori. Instead, the interblade phase angle of flutter is not known, and one must examine all possible interblade phase angles to insure that a rotor is flutter free. This suggests that for flutter problems, reducedorder model should be constructed in which the interblade phase angle is the free parameter. One notable exception is the work of Epureanu et al. [9]. Here, a POD technique was used to form a reduced-order model of two-dimensional viscous flow based on an inviscid boundary-layer coupled flow solver. Both frequency and interblade phase angle were allowed to vary. In this note, we present a fast technique that allows us to estimate the unsteady flows for many interblade phase angles very efficiently. Note that the base CFD solver used here is an iterative, threedimensional time-linearized flow solver.