An iterative algorithm between stream function and density for transonic cascade flow

Abstract

A set of Wu's equations governing the fluid flow along a given Si stream surface is solved by the use of artificial compressibility and a new iterative algorithm between the two variables stream function ^ and density p. The set of equations consists of a stream function equation and a new first-order partial differential equation for density derived from the continuity equation, momentum equation and Bernoulli's equation. The stream function equation proposed by Wu more than 30 years ago is written in general curvilinear coordinate system to allow for arbitrary boundary shapes. A strongly implicit approximate factorization algorithm (SI) of the multidimensional partial differential equations is used in numerical solutions of this set of equations. By the method proposed in this paper, the problem of nonuniqueness of density in the conventional stream function method is avoided. A number of transonic cascades are calculated and given herein to demonstrate the capability of the present method. N order to discretize the mixed-type full potential equation or stream function equation in a transonic flowfield, the artificial compressibility method1 could be considered. This method consists of modifying the density in such a way as to introduce the numerical dissipation necessary in the supersonic region. With a slight modification of the density inlhe supersonic region, the transonic flow problem becomes amenable to treatment by standard discretization techniques (central differencing) and some standard iterative procedures (SOR, ADI, explicit methods, and implicit algorithms212). The approximate factorization schemes, for example AF1, AF2, AF3, and SI, can be used to solve the conservative or nonconservative transonic full-potential equation3'5'79'12 and transonic stream function equation.2'6'10'11 For transonic flows, this scheme has proved to be fast and reliable and, therefore, is the only scheme considered in this study. For transonic cascade flow, the irrotational assumption is not valid when the entropy is nonuniform downstream of a strong shock. Without the assumption of uniform entropy or irrotational flow, the stream function equation is more suitable. However, the main difficulty with the solution of stream function in transonic flow lies in that the density is not uniquely determined in terms of mass flux. The purpose of this study is to develop efficient algorithm to overcome this difficulty. In the present paper, a set of Wu's equations1316 governing the fluid flow along Sl surface is solved by the combination of the artificial compressibility and a new iterative algorithm between the two variables \l/ and p. The set of equations consists of a stream function equation and a new first-order partial differential equation for density derived from continuity equation, momentum equation, and Bernoulli's equation. The SI algorithm is used in the numerical solutions of this set of equations. The problem of nonuniqueness of density in traditional stream function method is avoided in this study.