Calculation of unsteady flows in turbomachinery using the linearizedEuler equations
A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small, the unsteady Euler equations are linearized about the mean flow to obtain a set of linear variable-coefficient equations that describe the small-amplitude harmonic motion of the fluid. These linear equations are discretized on a computational grid via a finite-volume operator and solved directly, subject to an appropriate set of linearized boundary conditions. An important feature of the present analysis is the use of shock fitting to determine steady and unsteady shock positions. Use of the Euler equations in conjunction with the Rankine-Hugoniot shock-jump conditions correctly models the generation of entropy and vorticity at shocks. Results of this method are presented for both channel and cascade flows. Unsteady flows produced by blade motion (the flutter problem) and incoming disturbances (the gust-response problem) are predicted. A comparison of the present unsteady flow predictions to those of semi-analytical and time-marching numerical methods shows good agreement. Furthermore, the linearized Euler method requires substantially less computational time than the time-marching procedures. © 1989 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
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