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

One of the most frequently encountered numerical problems in scientific analyses is the solution of non-linear equations. Often the analysis of complex phenomena falls beyond the range of applicability of the numerical methods available in the public domain, and demands the design of dedicated algorithms that will approximate, to a specified precision, the mathematical solution of specific problems. These algorithms can be developed from scratch or through the amalgamation of existing techniques. The accurate solution of the full radial equilibrium equation (REE) in streamline curvature (SLC) through-flow analyses presents such a case. This article discusses the development, validation, and application of an 'intelligent' dynamic convergence control (DCC) algorithm for the fast, accurate, and robust numerical solution of the non-linear equations of motion for two-dimensional flow fields. The algorithm was developed to eliminate the large extent of user intervention, usually required by standard numerical methods. The DCC algorithm was integrated into a turbomachinery design and performance simulation software tool and was tested rigorously, particularly at compressor operating regimes traditionally exhibiting convergence difficulties (i.e. far off-design conditions). Typical error histories and comparisons of simulated results against experimental are presented in this article for a particular case study. For all case studies examined, it was found that the algorithm could successfully 'guide' the solution down to the specified error tolerance, at the expense of a slightly slower iteration process (compared to a conventional Newton-Raphson scheme). This hybrid DCC algorithm can also find use in many other engineering and scientific applications that require the robust solution of mathematical problems by numerical instead of analytical means.