A dynamic convergence control scheme for the solution of the radial equilibriumequation in through-flow analyses

摘要

One of the most frequently encountered numerical problems in scientific analysesis the solution of non-linear equations. Often the analysis of complex phenomenafalls beyond the range of applicability of the numerical methods available inthe public domain, and demands the design of dedicated algorithms that willapproximate, to a specified precision, the mathematical solution of specificproblems. These algorithms can be developed from scratch or through theamalgamation of existing techniques. The accurate solution of the full radialequilibrium equation (REE) in streamline curvature (SLC) through-flow analysespresents such a case. This article discusses the development, validation, andapplication of an 'intelligent' dynamic convergence control (DCC) algorithm forthe fast, accurate, and robust numerical solution of the non-linear equations ofmotion for two-dimensional flow fields. The algorithm was developed to eliminatethe large extent of user intervention, usually required by standard numericalmethods. The DCC algorithm was integrated into a turbomachinery design andperformance simulation software tool and was tested rigorously, particularly atcompressor operating regimes traditionally exhibiting convergence difficulties(i.e. far off-design conditions). Typical error histories and comparisons ofsimulated results against experimental are presented in this article for aparticular case study. For all case studies examined, it was found that thealgorithm could successfully 'guide' the solution down to the specified errortolerance, at the expense of a slightly slower iteration process (compared to aconventional Newton-Raphson scheme). This hybrid DCC algorithm can also find usein many other engineering and scientific applications that require the robustsolution of mathematical problems by numerical instead of analytical means.