Application of Model Order Reduction to Compressor Aeroelastic Models

A model order reduction technique that yields low-order models of blade row unsteady aerodyamics is introduced. The technique is applied to linearized unsteady Euler CFD solutions in such a way that the resulting blade row models can be linked to their surroundings through their boundary conditions. The technique is applied to a transonic compressor aeroelastic analysis, in which the high-fidelity CFD forced-response results are better captured than with models that use single-frequency influence coefficients. A low-speed compressor stage is also modeled to demonstrate the multistage capability of the method. These examples demonstrate how model order reduction can be used to systematically improve the versatility, fidelity, and range of applicability of the low-order aerodynamic models typically used for incorporation of CFD results into aeroelastic analyses. INTRODUCTION One of the foremost challenges in the study of free and forced blade vibrations in turbomachinery is encapsulation of aerodynamic effects in a form suitable for coupled aerostructural analysis. It is generally agreed that in modern aeroengines the unsteady aerodynamic effects governing the severity of vibrations are complex; at least some of the flow details (shock motion, blade loading, viscosity, boundary conditions, etc.) must be modeled to obtain realistic analyses. Because very little data exists to isolate the most important of these details, the current state of the art utilizes computational fluid dynamic (CFD) analysis to capture as much of the physics as possible. The CFD results are encapsulated as unsteady aerodynamic forces resulting from (usually prescribed sinusoidal) blade vibrations, which can then be coupled to the structural dynamics. Forced vibration analyses are often done in a similar two-step fashion, in which the aerodynamic effects are computed first and then incorporated into a coupled analysis (Chiang and Kielb, 1993). Non-reflecting boundary conditions are commonly used in these analyses for simplicity, although the boundary conditions can have a significant effect on the results (Hall and Silkowski, 1995). Pre-computation of aerodynamic effects, using canonical motions and forcing functions, invokes various assumptions based on the problem at hand. For stability analysis of a rotor, the fundamental assumption is that the blade vibration frequencies change by only a small amount due to aerodynamic coupling. Thus analysis of the aerodynamic loads using the most important structural dynamic eigenfunctions is justified. Forced vibration analyses often compute gust loads assuming sinusoidal gusts and fixed blades, assuming that the incoming gusts are not modified by unsteady blade motion (Manwaring and Wisler, 1993). Mistuning analyses commonly use simplified aerodynamic models (as in Crawley and Hall (1985)), although incorporation of CFD-based influence coefficients (based on sinusoidal, singlefrequency runs) has been reported (Dugundji and Bundas, 1984). Experience and order-of-magnitude analyses (mass ratio arguments, etc.) suggest that the engineering approximations discussed above are often valid, allowing trends and insight to be more easily obtained. However, the framework for encapsulation of aerodynamic effects is limited, and in some cases inconsistent with the goals of the analysis. Non-sinusoidal forcing, forced response with mistuning, and other analyses that involve combinations of effects stretch the assumptions of standard methods. In 1 Copyright  2000 by ASME addition, the effects of upstream and downstream boundary conditions (nearby blade rows, inlet duct acoustics) are increasingly coming under scrutiny. It is in these cases that a reduced-order model with generalized boundary conditions can play an important role. The CFD code is viewed as an input-output model: blade motions, incoming flow perturbations, and mean flow properties are viewed as inputs; outputs include outgoing flow perturbations and the blade forces and moments. A reduced-order model is a low-order model that duplicates the behavior of the CFD analysis over a limited range of conditions. The range of validity of the reducedorder model is determined by the set of CFD experiments performed during the model order reduction procedure. By making the model building procedure systematic, a single tool can yield reduced-order models for a wide range of applications. In the sense defined above, models based on spatially and temporally sinusoidal motion at a single frequency (we will call these “assumed-frequency” methods) are simple reduced-order models. They are limited to the boundary conditions chosen in the CFD analysis and are only valid at the precise mean flow conditions analyzed. They are limited to a small range of frequencies about the assumed frequency, and this range is unknown. Owing to their limited range of validity, assumed-frequency models are the most compact: they are coupled to the structural model as constant coefficients that are independent of changes in the mean flow, boundary conditions, frequency, etc. More detailed aerodynamic models, motivated by external aeroelastic (wing flutter) methods, have been suggested (Crawley, 1988), but do not appear to be in broad use. In this paper we present a systematic method for model order reduction from CFD, which can be applied to a broad range of problems. The method allows the analyst to specify the range of interblade phase angles, reduced frequencies, forcing functions, and boundary conditions that are appropriate to encapsulate into the problem at hand. This range is spanned by a series of “canonical” runs of the CFD code, which are then used to derive a low-order model. The model is in ordinary differential equation form, requiring approximately 100 to 200 states per blade row (in our examples), suitable for incorporation in a fully coupled aeroelastic analysis. Examples of application of this model are given for illustrative purposes. COMPUTATIONAL MODEL The starting point for the procedure is a computational model. In this paper, the nonlinear two-dimensional Euler equations, expressed on an unstructured triangular grid, are used: dU dt R U Ub x 0 (1) where R represents the nonlinear flux contributions at each node, which are functions of the problem geometry x, the unknown flow quantities on the grid points U and the boundary values U b. For a two-dimensional bladed disk with r blades with rigid cross sections, x can be written in terms of plunge displacement h i and pitch displacement αi: x x q q q1 q T 2 q T r T (2) where qi hi αi T . The vector Ub allows external flow disturbances and impedance properties to be modeled. These could be, for example, time-varying pressure or velocity distortions due to a neighboring blade row, inhomogeneity in the incoming flow, or acoustic boundary conditions. Given blade motion q and boundary disturbance d, we write Ub as: