Numerical treatment of unsteady subsonic flow past an oscillating cascade

Abstract

A numerical procedure for determining unsteady subsonic flow past a finite-deflection, oscillating cascade is described. Based on the assumption of small-amplitu de blade motions, the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady flow. These equations are solved on a nonorthogonal, body-fitted, and periodic grid which facilitates the implementation of blade, wake, and cascade-periodicity conditions but disallows the use of standard difference approximations. Instead, difference approximations are based on an implicit least-squares development applicable on arbitrary grids. This permits flexibility in the choice of difference neighbors and the simultaneous approximations of differential equation and boundary conditions at boundary points, strategies which can significantly improve numerical accuracy. Sample results illustrating the effects of blade thickness, motion frequency, and inlet Mach number on the unsteady response of staggered cascades are presented. N unsteady aerodynamic analysis which accounts for the effects of mean flow deflection due to blade geometry and flow turning and is capable of providing efficient predictions of unsteady response coefficients is an important requirement for turbomachinery applications. A general aerodynamic model which fully includes the effects of nonuniform mean or steady flow on a small-disturbance unsteady flow has been formulated and described in Refs. 1 and 2. The steady flow is governed by the full potential equations, and the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady flow. In contrast to classical linear theory, an unsteady aerodynamic formulation which includes the effects of nonuniform mean flow requires numerical solutions of the governing steady and unsteady boundary value problems. Numerical procedures for determining steady subsonic3 or transonic 4 potential flow through cascades are currently available. A solution procedure for the linear, variable-coefficient, unsteady problem is described in the present paper. Since the unsteady aerodynamic model treats realistic cascade configurations, the numerical approximation must take into account nonrectangular geometries. Generally, approximations to differential equations in nonrectangula r regions are developed in one of two ways. A mapping may be defined so that in mapped variables calculation points can be placed in a rectangular grid; the transformed equations are then approximated using rectangular difference expressions in the usual way. Alternatively, more general difference ap- proximations may be defined so that differential equations can be approximated on nonrectangula r meshes in physical variables. Thompson, Thames, and Mastin5 have defined global mappings for arbitrary two-dimensional bodies, but these mappings are inappropriate for cascades since the resulting meshes generally do not allow for cascade periodicity. Ives and Liutermoza4 have determined global mappings which transform infinite two-dimensional cascade regions into rectangular computational regions while allowing for cascade periodicity. However, the necessary mappings are not presently known for all cascades, particularly those with