Unsteady flow past a cascade of staggered thin profiles

Abstract

Many studies have been made of plane flow of an incompressible inviscid fluid past a cascade of profiles with arbitrary stagger angle β≠0. For example, in the particular case of the motion of a cascade with the stagger angle β at zero-oscillation phase-shift angle α=0 Khaskind [1] determined the unsteady lift force theoretically by isolating the singularities with the Sedov method [2], applying a conformal mapping to the cascade of unstaggered flat plates. Belotserkovskii et al. [3] calculated the over-all unsteady aerodynamic characteristics of a cascade in the particular case α=0 and for any β on a computer by the method of discrete vortices, and for the more general case (α≠0) Whitehead [4] has done the same using a vortex method. Gorelov and Dominas [5] calculated the over-all unsteady force and moment coefficients of a profile in a cascade with stagger angle β≠0 and phase shift α≠0. The calculation method was based on unsteady theory for a slender isolated profile whose flow pattern is known, with subsequent account for the interference of the profiles and the vortex wakes behind them. In the present study the singularity isolation method [2] is extended to slender profiles with arbitrary stagger angle β≠0 and arbitrary phase shift α≠0 of the oscillations between neighboring profiles. It is shown that the solution reduces to the solution of a Fredholm integral equation of the first kind in terms of the sum of the tangential velocity components along the profile. It is found that the relative effect of the unsteady flow due to the system of vortex trails behind the cascade with staggerβ may be determined without solving this integral equation. However, this solution must be found to calculate the added masses of the cascade and the total magnitudes of the unsteady forces. It is found that regularization transforms the integral equation of the first kind to an integral equation of the second kind, for which solution methods are known. Thus the expressions for the unsteady forces are determined in the form of separate terms, each of which has a physical significance: as a result we obtain finite formulas (improper integrals) for calculating the variable forces; from these formulas are derived the asymptotic expressions for the forces in the limiting cases of high and low solidities and Strouhal numbers, which as a rule are lost in numerical calculations. The proposed method may be considered as one of the techniques for improving the convergence of the numerical methods (elimination of singularities). Moreover, this method may be used to solve the problems of unsteady flow past cascades of arbitrary systems of slender profiles for various profile incidence angles relative to the x-axis and in the presence of a finite cavitation zone on the profiles. The limited practical application of this method is explained by the extreme theoretical difficulties in its applications to cascades with stagger angle. In the present studies these difficulties are examined using the example of a cascade with stagger angleβ.